Handbook of Mathematical Relations in Particulate Materials Processing

MATHEMATICALRELATIONS INPARTICULATEMATERIALS PROCESSING

MATHEMATICALRELATIONS INPARTICULATEMATERIALS PROCESSING

Ceramics, Powder Metals, Cermets,Carbides, Hard Materials, and Minerals

RANDALL M. GERMANSEONG JIN PARK

Copyright # 2008 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc.

Published simultaneously in Canada

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Library of Congress Cataloging-in-Publication Data:

German, Randall M., 1946–Mathematical relations in particulate materials processing : ceramics, powder metals, cermets, carbides,

hard materials, and minerals / Randall M. German, Seong Jin Parkp. cm.

Includes bibliographical references and index.ISBN-13: 978-0-470-17364-0

1. Powder metallurgy—Handbooks, manuals, etc. 2. Powder metallurgy—Mathematicalmodels. I. Park, Seong Jin, 1968– . II. Title.TK695.G4694 2008671.307—dc22

2008000684

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

Dedicated toYoung Park and Jackson German

CONTENTS

Foreword xxxi

About the Authors xxxiii

A 1

Abnormal Grain Growth 1

Abrasive Wear—See Friction and Wear Testing 3

Acceleration of Free-settling Particles 3

Activated Sintering, Early-stage Shrinkage 4

Activation Energy—See Arrhenius Relation 5

Adsorption—See BET Specific Surface Area 5

Agglomerate Strength 5

Agglomeration Force 6

Agglomeration of Nanoscale Particles—See NanoparticleAgglomeration 6

Andreasen Size Distribution 6

Apparent Diffusivity 7

Archard Equation 7

Archimedes Density 8

Arrhenius Relation 9

Atmosphere Moisture Content—See Dew Point 10

Atmosphere-stabilized Porosity—See Gas-generated Final Pores 10

Atomic Flux in Vacuum Sintering 10

Atomic-size Ratio in Amorphous Metals 10

Atomization Spheroidization Time—See Spheroidization Time 11

Atomization Time—See Solidification Time 11

Average Compaction Pressure—See Mean Compaction Pressure 11

Average Particle Size—See Mean Particle Size 11

Avrami Equation 11

vii

B 13

Ball Milling—See Jar Milling 13

Bearing Strength 13

Bell Curve—See Gaussian Distribution 13

Bending-beam Viscosity 14

Bending Test 16

BET Equivalent Spherical-particle Diameter 18

BET Specific Surface Area 18

Bimodal Powder Packing 20

Bimodal Powder Sintering 21

Binder Burnout—See Polymer Pyrolysis 22

Binder (Mixed Polymer) Viscosity 23

Bingham Model—See Viscosity Model for Injection-molding Feedstock 23

Bingham Viscous-f low Model 23

Boltzmann Statistics—See Arrhenius Relation 24

Bond Number 24

Bragg’s Law 24

Brazilian Test 25

Breakage Model 26

Brinell Hardness 27

Brittle Material Strength Distribution—See Weibull Distribution 28

Broadening 28

Brownian Motion 29

Bubble Point—See Washburn Equation 30

Bulk Transport Sintering—See Sintering Shrinkageand Surface-area Reduction Kinetics 30

C 31

Cantilever-beam Test—See Bending-beam Viscosity 31

Capillarity 31

Capillarity-induced Sintering—See Surface Curvature–DrivenMass Flow in Sintering 32

Capillary Pressure during Liquid-phaseSintering—See Mean Capillary Pressure 32

Capillary Rise—See Washburn Equation 32

Capillary Stress—See Laplace Equation 32

Case Carburization 32

Casson Model 32

viii CONTENTS

Cemented-carbide Hardness 33

Centrifugal Atomization Droplet Size 34

Centrifugal Atomization Particle Size 34

Charles Equation for Milling 35

Chemically Activated Sintering—See Activated Sintering,Early-stage Shrinkage 36

Closed-pore Pressure—See Spherical-pore Pressure 36

Closed Porosity—See Open-pore Content 36

Coagulation Time 36

Coalescence—See Coagulation Time 37

Coalescence-induced Melting of Nanoscale Particles 37

Coalescence of Liquid Droplets—See Liquid-droplet Coalescence Time 38

Coalescence of Nanoscale Particles—See Nanoparticle Agglomeration 38

Coble Creep 38

Coefficient of Thermal Expansion—See Thermal Expansion Coefficient 39

Coefficient of Variation 39

Coercivity of Cemented Carbides—See MagneticCoercivity Correlation in Cemented Carbides 39

Cold-spray Process—See Spray Deposition 39

Colloidal Packing Particle-size Distribution—See AndreasenSize Distribution 40

Combined-stage Model of Sintering 40

Comminution—See Grinding Time 40

Comminution Law—See Charles Equation for Milling 40

Compaction-induced Bond Size—See Contact Sizeas a Function of Density 41

Compaction-induced Neck Size 41

Compaction Pressure Effect on Green Density—See Green-densityDependence on Compaction Pressure 41

Complexity 41

Complex Viscosity 42

Composite Density 43

Composite Elastic Modulus 44

Composite Thermal Conductivity 45

Composite Thermal Expansion Coefficient 46

Compression Ratio 47

Conductive Heat Flow 47

Conductivity 48

Connectivity 49

CONTENTS ix

Constitutive Equations for Sintering—See Macroscopic Sintering ModeConstitutive Equations 49

Constructive Reinforcement in X-ray Diffraction—See Bragg’s Law 49

Contact Angle 50

Contact Pressure—See Effective Pressure 50

Contact Size as a Function of Density 51

Contacts Per Particle—See Coordination Number and Density 51

Container-size Effect on Random-packing Density 52

Contiguity 52

Continuum Theory of Sintering 53

Continuum Theory for Field-activated Sintering 54

Convective Heat Transfer 55

Cooling Rate in Atomization—See Newtonian Cooling Approximation,Gas Atomization Cooling Rate, and Secondary Dendrite Arm Spacing 56

Cooling Rate in Molding 56

Cooling Time in Molding 56

Coordination Number and Density 57

Coordination Number and Grain Size—See Grain-size Affecton Coordination Number 58

Coordination Number for Ordered Packings 58

Coordination Number from Connectivity 59

Coordination Number in Liquid-phase Sintering 59

Costing and Price Estimation 60

Coulomb’s Law for Plastic Yielding 62

Courtney Model for Early-stage Neck Growth inLiquid-phase Sintering 63

Creep-controlled Densification 63

Critical Solids Loading—See Solids Loading 64

Cross Model 64

Curved-surface Stress—See Neck Curvature Stress 65

Cyclone Separation of Powder 65

Cylindrical Crush Strength—See Bearing Strength 66

D 67

Darcy’s Law 67

Debinding—See Polymer Pyrolysis, Solvent Debinding Time, ThermalDebinding Time, Vacuum Thermal Debinding Time, and Wicking 68

Debinding Master Curve—See Master Decomposition Curve 68

Debinding Temperature 68

x CONTENTS

Debinding Time—See Solvent Debinding Time, Thermal Debinding Time,Vacuum Thermal Debinding Time, and Wicking 70

Debinding by Solvent Immersion—See Solvent Debinding Time 70

Debinding Weight Loss 70

Delubrication—See Polymer Pyrolysis 70

Densification 71

Densification in Liquid-phase Sintering—See Dissolution-inducedDensification 71

Densification in Sintering—See Shrinkage-induced Densification 71

Densification Rate 71

Densification Ratio 73

Density Calculation from Dilatometry 74

Density Effect on Ductility—See Sintered Ductility 75

Density Effect on Sintered Neck Size—See Neck-size Ratio Dependenceon Sintered Density 75

Density Effect on Strength—See Sintered Strength 75

Dew Point 75

Die-wall Friction 76

Diffusion—See Vacancy Diffusion 78

Diffusion-controlled Grain Growth in Liquid-phase Sintering—SeeGrain Growth in Liquid-phase Sintering, Diffusion Controlat High Solid Contents 78

Diffusional Neck Growth—See Kuczynski Neck-growth Model 79

Diffusional Homogenization in Sintering—See Homogenizationin Sintering 79

Diffusional Translation—See Stokes–Einstein Equation 79

Dihedral Angle 79

Dihedral Angle–Limited Neck Growth—See Neck Growth Limitedby Grain Growth 80

Dilatant Flow Momentum Model 80

Dilatant Flow Viscosity Model 81

Dilute Suspension Viscosity 82

Dimensional Change—See Sintering Shrinkage 82

Dimensional Variation—See Gaussian Distribution 82

Dimensional Precision and Green Mass Variation 82

Direct Laser Sintering—See Laser Sintering 84

Disk Crush Test—See Brazilian Test 84

Dislocation Climb-controlled Pressure-assisted Sintering Densification 84

Dislocation Glide in Sintering—See Plastic Flow in Sintering 85

CONTENTS xi

Dispersion Force—See London Dispersion Force 85

Dissolution Induced Densification 85

Dorn Technique 86

Drainage—See Wicking 87

Droplet Cooling in Atomization—See Newtonian Cooling Approximation 87

Ductility Variation with Density—See Sintered Ductility 87

E

Effective Pressure 89

Ejection Stress—See Maximum Ejection Stress 89

Elastic Behavior—See Hooke’s Law 89

Elastic-deformation Neck-size Ratio 90

Elastic-modulus Variation with Density 91

Elastic-property Variation with Porosity 91

Electrical-conductivity Variation with Porosity 92

Electromigration Contributions to Spark Sintering 93

Elongation 94

Elongation Variation with Density—See Sintered Ductility 95

Energy-governing Equation for Powder Injection Molding 95

Energy in a Particle 95

Enhanced Sintering—See Activated Sintering,Early-stage Shrinkage 96

Equilibrium Constant 96

Equivalent Particle Size Based on Area—See BETEquivalent-spherical-particle Diameter 97

Equivalent Sintering—See Temperature Adjustmentsfor Equivalent Sintering 97

Equivalent Spherical Diameter 97

Error Function for Cumulative Log-normal Distribution 98

Euler Relation 99

Evaporation 99

Evaporation–Condensation—See Initial-stage Neck Growth 100

Exaggerated Grain Growth—See Abnormal Grain Growth 100

Exothermic Synthesis—See Self-propagatingHigh-temperature Synthesis 100

Expansion Factor for Tooling—See Tool Expansion Factor 100

Experimental Scatter—See Gaussian Distribution 100

Exponential Distribution Function 100

Extrusion Constant 101

xii CONTENTS

F 103

Feedstock Formulation 103

Feedstock Viscosity—See Suspension Viscosity and Viscosity Modelfor Injection-molding Feedstock 103

Feedstock Viscosity as a Function of Shear Rate—See Cross Model 103

Feedstock Yield Strength—See Yield Strength of Particle–PolymerFeedstock 104

Fiber-fracture from Buckling 104

Fiber-fracture Probability 104

Fiber Packing Density 105

Fick’s First Law 106

Fick’s Second Law 106

Field-activated Sintering 107

Filtration Rating 109

Final-stage Densification 109

Final-stage Liquid-phase Sintering Densification 110

First-stage Neck Growth in Sintering—See Initial-stage Neck Growth 112

Final-stage Pore Size 112

Final-stage Pressure-assisted Densification 112

Final-stage Pressure-assisted Viscous Flow 113

Final-stage Sintering by Viscous Flow 114

Final-stage Sintering Grain Growth and Pore Drag 114

Final-stage Sintering Limited Density 115

Final-stage Sintering Pinned Grains—See Zener Relation 117

Final-stage Sintering Stress 117

First-stage Shrinkage in Sintering—See Initial-stageShrinkage in Sintering 117

First-stage Sintering Surface-area Reduction—See Surface-areaReduction Kinetics 118

Fisher Subsieve Particle Size 118

Flatness—See Particle-shape Index 119

Flaw Effect on Green Strength—See Green StrengthVariation with Flaws 119

Flow Governing Equation during Powder Injection Molding 119

Fluidized-bed Processing 121

Force Distribution in Randomly Packed Powder 122

Four-point Bending Strength—See Transverse-rupture Strength 122

Fractional Coverage of Grain Boundaries in Supersolidus Sintering 122

Fractional Density 123

CONTENTS xiii

Fragmentation by Liquid 123

Fragmentation Model—See Breakage Model 124

Freeform Spraying—See Spray Deposition 124

Frenkel Model—See Two-particle Viscous Flow Sintering 124

Friction and Wear Testing 124

Funicular-state Tensile Strength 125

G 127

Gas-absorption Surface Area—See BET Specific Surface Area 127

Gas-atomization Cooling Rate 127

Gas-atomization Melt Flow Rate 128

Gas-atomization Particle Size 129

Gas-generated Final Pores 130

Gas Permeability—See Kozeny–Carman Equation 131

Gate Strain Rate in Injection Molding 131

Gaudin–Schuhmann distribution 132

Gaussian Distribution 133

Gel-densification Model 134

Gessinger Model for Intermediate-stage Liquid-phase Sintering 136

Glass Viscosity Test—See Bending-beam Viscosity 137

Grain Accommodation—See Grain-shape Accommodation 137

Grain Bonding—See Contiguity 137

Grain Boundary–controlled Creep—See Coble Creep 137

Grain-boundary Energy and Misorientation Angle 137

Grain-boundary Fraction 137

Grain-boundary Groove—See Dihedral Angle 138

Grain-boundary Misorientation—See Grain-boundary Energyand Misorientation Angle 138

Grain-boundary Penetration—See Fragmentation by Liquid 138

Grain-boundary Pinning—See Zener Relation 138

Grain-boundary Wetting 138

Grain Coordination Number in Liquid-phase Sintering—See CoordinationNumber in Liquid-phase Sintering 140

Grain Diameter Based on an Equivalent Circle 140

Grain Growth 140

Grain Growth in Liquid-phase Sintering, Diffusion Controlat High Solid Contents 141

Grain Growth in Liquid-phase Sintering, Dilute Solids Contents 143

Grain Growth in Liquid-phase Sintering, Interfacial Reaction Control 144

xiv CONTENTS

Grain-growth Master Curve 145

Grain-growth Master Curve, Interfacial Reaction Control 147

Grain Number Changes in Liquid-phase Sintering 149

Grain Pinning by Pores in Final-stage Sintering 149

Grain Separation Distance in Liquid-phase Sintering 150

Grain Separation in Cemented Carbides—See Mean Free Path,Carbide Microstructure 151

Grain-shape Accommodation 151

Grain Size 152

Grain-size Affect on Coordination Number 152

Grain-size Distribution for Liquid-phase Sintered Materials 153

Grain-size Distribution for Solid-state Sintered Materials 154

Grain-size Effect on Strength—See Hall–Petch Relation 155

Grain Size to Pore Size in Final-stage Liquid-phase Sintering 155

Granulation Force—See Agglomerate Force 155

Granule Strength—See Agglomerate Strength 156

Green Density Dependence on Compaction Pressure 156

Green Density Dependence on Punch Travel 157

Green Density From Repeated Pressing 157

Green Strength 158

Green Strength Distribution—See Weibull Distribution 158

Green Strength Variation with Flaws 158

Grinding Time 159

Growth—See Sintering Shrinkage 159

H 161

Hall–Petch Relation 161

Hardenability Factor 161

Hardness 162

Hardness Variation with Grain Size in Cemented Carbides 163

Heating-rate Effect in Transient Liquid-phase Sintering 164

Heat Transfer in Sintered Materials 164

Heat-transfer Rate in Molding—See Cooling Rate in Molding 165

Herring Scaling Law 165

Hertzian stress—See Elastic Deformation Neck-size Ratio 166

Heterodiffusion—See Mixed-powder Sintering Shrinkage 166

Heterogeneous Nucleation 167

CONTENTS xv

High Solid-content Grain Growth—See Grain Growth in Liquid-phaseSintering, Diffusion Control at High Solid Contents 167

Homogeneity—See Segregation Coefficient 167

Homogeneity of a Microstructure 168

Homogeneity of Mixed Powders—See Mixture Homogenization Rate 168

Homogeneous Nucleation 168

Homogenization in Sintering 169

Homogenization Rate in Powder Mixing—See MixtureHomogenization Rate 170

Hooke’s Law 170

Hot Pressing in the Presence of a Liquid Phase—See Pressure-assistedLiquid-phase Sintering 170

I 171

Impregnation—See Infiltration Pressure 171

Inertial-flow Equation 171

Infiltration Depth 172

Infiltration Pressure 172

Infiltration Rate 173

Inhibited Grain Growth—See Zener Relation 173

Initial-stage Liquid-phase Sintering Stress—See Sintering Stress inInitial-stage Liquid-phase Sintering. 173

Initial-stage Neck Growth 174

Initial-stage Sintering—See Surface Diffusion–ControlledNeck Growth 176

Initial-stage Sintering Model—See Kuczynski Neck-growth Model 176

Initial-stage Sintering Stress—See Sintering Stress in Initial-stageSolid-state Sintering 176

Initial-stage Sintering Surface-area Reduction—See Surface-areaReduction Kinetics 176

Initial-stage Shrinkage in Sintering 176

Injection-molding Viscosity—See Viscosity Modelfor Injection-molding Feedstock 177

In Situ Sintering Strength 177

Integral Work of Sintering—See Master Sintering Curve 178

Interdiffusion—See Mixed-powder Sintering Shrinkage 178

Interface-controlled Grain Growth 178

Intermediate-stage Liquid-phase Sintering Model 179

xvi CONTENTS

Intermediate-stage Liquid-phase Sintering Shrinkage—SeeSolution-reprecipitation-induced Shrinkage in Liquid-phase Sintering 180

Intermediate-stage Pore Elimination 180

Intermediate-stage Sintering-density Model 182

Intermediate-stage Surface-area Reduction 183

Interrupted Heating-rate Technique—See Dorn Technique 183

Inverse Rule of Mixtures—See Composite Density and MixtureTheoretical Density 183

J 185

Jar Milling 185

Jet Mixing Time 186

K 187

Kawakita Equation 187

Kelvin Equation 188

Kelvin Model—See Viscoelastic Model for Powder–Polymer Mixtures 189

K-Factor 189

Kingery Intermediate-stage Liquid-phase Sintering Model—SeeIntermediate-stage Liquid-phase Sintering Model 189

Kingery Model for Pressure-assisted Liquid-phase Sintering—SeePressure-assisted Liquid-phase Sintering 190

Kingery Rearrangement Shrinkage Kinetics—See RearrangementKinetics in Initial-stage Liquid-phase Sintering 190

Kissinger Method 190

Knoop Hardness 191

Knudsen Diffusion—See Vapor Mean Free Path 192

Kozeny–Carman Equation 192

Kuczynski Neck-growth Model 192

L 195

Laminar Flow Settling—See Stokes’ Law 195

Laplace Equation 195

Laplace Number—See Suratman Number 196

Laser Sintering 196

Lattice Diffusion—See Vacancy Diffusion 196

Lifschwiz, Slyozov, Wagner Model 196

Ligament Pinching—See Raleigh Instability 197

Limiting Neck Size 197

CONTENTS xvii

Limiting Size for Sedimentation Analysis 198

Liquid and Solid Compositions in Prealloy Particle Melting 199

Liquid Distribution in Supersolidus Liquid-phase Sintering 200

Liquid-droplet Coalescence Time 203

Liquid-droplet Viscous Flow—See Viscous Flow of a Liquid Droplet 203

Liquid Embrittlement—See Fragmentation by Liquid 203

Liquid Penetration of Grain Boundaries—See Melt Penetrationof Grain Boundaries on Liquid Formation 204

Liquid-phase Sintering Grain Growth—See Grain Growthin Liquid-phase Sintering, Interfacial Reaction Control,and Grain Growth in Liquid-phase Sintering, Diffusion Controlat High Solid Contents 204

Liquid-phase Sintering Grain-size Distribution—See Grain-sizeDistribution for Liquid-phase Sintered Materials 204

Liquid-phase Sintering Intermediate-stage Shrinkage—See GessingerModel for Intermediate-stage Shrinkage in Liquid-phase Sintering 204

Liquid-phase Sintering Neck Growth—See Neck Growth Early inLiquid-phase Sintering and Courtney Model for Early-stageNeck Growth in Liquid-phase Sintering 204

Liquid-phase Sintering Rheological Model—See Rheological Modelfor Liquid-phase Sintering 204

Liquid-phase Sintering Surface-area Reduction—See Surface-areaReduction during liquid-phase Sintering 204

Liquid Velocity in Atomization 204

Logarithmic Viscosity Rule—See Binder (Mixed Polymer) Viscosity 205

Log-normal Distribution 205

Log-normal Slope Parameter 206

London Dispersion Force 207

Low-solid-content Grain Growth—See Grain Growthin Liquid-phase Sintering, Dilute-solids Content 207

Lubricant Burnout—See Polymer Pyrolysis 208

Lubricant Content—See Maximum Lubricant Content 208

M 209

Macroscopic Sintering Model Constitutive Equations 209

Magnetic Coercivity Correlation in Cemented Carbides 211

Mass Flow Rate in Atomization—See Gas-atomization Melt Flow Rate 212

Master Decomposition Curve 212

Master Sintering Curve 213

xviii CONTENTS

Master Sintering Curve for Grain Growth—See Grain-growthMaster Curve 214

Maximum Density in Pressure-assisted Sintering 214

Maximum Ejection Stress 215

Maximum Grain Size in Sintering 215

Maximum Lubricant Content 216

Maxwell Model—See Viscoelastic Model forPowder–Polymer Mixtures 216

Mean Capillary Pressure 216

Mean Compaction Pressure 217

Mean Free Distance—See Pore-separation Distance 218

Mean Free Path, Carbide Microstructure 218

Mean Free Path in Liquid-phase Sintering—See Grain SeparationDistance in Liquid-phase Sintering 218

Mean Free Path, Sintering Atmosphere 218

Mean Particle Size 219

Mean Time Between Particle Contacts—See Brownian Motion 220

Measure of Sintering—See Sintering Metrics 220

Melting-temperature Depression with Particle Size—See NanoparticleMelting-point Depression 220

Melt Penetration of Grain Boundaries 220

Mercury Porosimetry—See Washburn Equation 222

Mesh Sizes—See Sieve Progression 222

Microhardness—See Vickers Hardness Number 222

Micromechanical Model for Powder Compact 222

Microstructure Homogeneity—See Homogeneity of a Microstructure 223

Microwave Heating 224

Migration of Particles 224

Milling Energy—See Charles Equation for Milling 225

Milling Time—See Grinding Time 225

Mixed Grain Boundary and Lattice Diffusion—See Apparent Diffusivity 225

Mixed Lattice and Grain-boundary Diffusion—See Apparent Diffusivity 225

Mixed-powder Segregation 225

Mixed-powder Sintering Shrinkage 227

Mixed-powder Swelling during Sintering—See Swelling Reactionsduring Mixed-powder Sintering 227

Mixing Optimal Rotational Rate—See Optimal MixerRotational Speed 228

Mixture Homogenization Rate 228

CONTENTS xix

Mixture Theoretical Density 228

Modulus of Rupture—See Bending Test and TransverseRupture Strength 229

Moisture Content—See Dew Point 229

Molecular Mean Free Path—See Mean Free Path andSintering Atmosphere 229

Multiple-mechanism Sintering 229

Multiple-stage Model of Sintering—See Combined-stageModel of Sintering 230

N 231

Nabarro–Herring Creep-controlled Pressure-assisted Densification 231

Nanoparticle Agglomeration 232

Nanoparticle Melting-point Depression 233

Nanoscale Particle-agglomerate Spheroidization 234

Nanoscale Particle-size Effect on Surface Energy—See Surface-energyVariation with Droplet Size 235

Neck-curvature Stress 235

Neck Growth Early in Liquid-phase Sintering 236

Neck Growth–induced Shrinkage—See Shrinkage Relationto Neck Size 237

Neck Growth Limited by Grain Growth 237

Neck-growth Model—See Kuczynski Neck-growth Model 238

Necklace Microstructure—See Fragmentation by Liquid Penetration 238

Neck-size Ratio Dependence on Sintered Density 238

Newtonian Cooling Approximation 239

Newtonian Flow 239

Normal Curve—See Gaussian Distribution 240

Nucleation Frequency in Small Particles 240

Nucleation Rate 240

Number of Features—See Complexity 241

Number of Particles in Agglomerates 241

Number of Particles per Unit Weight 242

O 243

Open-pore Content 243

Optimal Packing Particle-size Distribution—See AndreasenSize Distribution 243

Optimal Mixer Rotational Speed 243

Ordered Packing 244

xx CONTENTS

Osprey Process—See Spray Deposition 244

Ostwald Ripening 245

Oxide Reduction 245

P 247

Packing Density for Log-normal Particles 247

Particle Cooling in Atomization—See Newtonian CoolingApproximation 248

Particle Coordination Number—See CoordinationNumber and Density 248

Particle Diffusion in Mixing 248

Particle Fracture in Milling 249

Particle Packing 249

Particle-shape Index 250

Particle Size—See Equivalent Spherical Diameter andMean Particle Size 251

Particle-size Analysis—See Sieve Progression 252

Particle size and Apparent Density 252

Particle size by Viscous Settling—See Stokes’ Particle Diameter 252

Particle-size Control in Centrifugal Atomization—See CentrifugalAtomization Particle Size 252

Particle-size Distribution—See Andreasen Size Distribution,Gaudin–Schuhman Distribution, Log-normal Distribution,and Rosin–Rammler Distribution 252

Particle-size Effect on Initial-stage Sintering—See HerringScaling Law 252

Particle-size Effect on Packing Density—See Particle Packing 253

Particle-size Effect on Solubility—See Solubility Dependenceon Particle Size 253

Particle-size in Atomization—See Water Atomization Particle Size 253

Particle-size Measurement Error 253

Peak Broadening—See Broadening 254

Peak Stress for Ejection—See Maximum Ejection Stress 254

Pendular-bond Capillary Force 254

Percolation in Semisolid Particles 255

Percolation Limits 256

Perimeter-based Particle Size—See Grain Diameter Basedon Equivalent Circle 256

Permeability-based Particle Size—See FisherSubsieve Particle Size 257

Permeability Coeff icient 257

CONTENTS xxi

Phase Transformation—See Nucleation Rate 257

Plasma-spraying Particle Size 258

Plastic Flow in Hot Compaction 258

Plastic Flow in Sintering 259

Plastic Working—See Strain Hardening 259

Poiseuille’s Equation 260

Poisson’s Ratio 260

Polymer-blend Viscosity—See Binder (Mixed-polymer) Viscosity 261

Polymer Pyrolysis 261

Pore Attachment to Grain Boundaries 261

Pore Closure 262

Pore Drag—See Migration of Particles 262

Pore Filling in Liquid-phase Sintering 262

Pore-free Composite or Mixed-phase Density 264

Pore Mobility during Sintering 264

Pore Pinning of Grain Boundaries—See Grain Pinning by Poresin Final-stage Sintering and Zener Relation 266

Pore-separation Distance 266

Pore Separation from Grain Boundaries 266

Pore Size and Grain Size in Final-stage Sintering—See Grain Sizeto Pore Size in Final-stage Liquid-phase Sintering 267

Pore Size and Grain Size in Intermediate-stage Sintering 267

Pore Size in Final-stage Sintering 267

Pore Size in Viscous-flow Final-stage Sintering 268

Porosimetry—See Washburn Equation 269

Porosity—See Fractional Density 269

Porosity Effect on Ductility—See Sintered Ductility 269

Porosity Effect on Elastic Behavior 269

Porosity Effect on Sonic Velocity—See Ultrasonic Velocity 270

Porosity Effect on Strength—See Sintered Strength 270

Porosity Effect on Thermal Conductivity—See Thermal-conductivityDependence on Porosity 270

Porosity in Swelling Systems with Limited Solubility 270

Powder-forging Height Strain and Densification 270

Powder Injection-molding Feedstock Viscosity—See ViscosityModel for Injection-molding Feedstock 271

Power-law Creep 271

Prealloyed-particle Sintering—See Liquid and Solid Compositionsin Prealloy-particle Melting 272

xxii CONTENTS

Prealloyed-powder Liquid-phase Sintering—See SupersolidusLiquid-phase Sintering Shrinkage Rate 272

Precision 272

Pressure-assisted Liquid-phase Sintering 272

Pressure-assisted Sintering Maximum Density—See MaximumDensity in Pressure-assisted Sintering 273

Pressure-assisted Sintering Semisolid System—See Viscous Flowin Pressure-assisted Sintering 273

Pressure Effect on Feedstock Viscosity 274

Pressure Effect on Final-stage Sintering by Viscous Flow 274

Pressure-governing Equation in Powder-injection Molding 275

Pressure-governing Equation for Powder Injection Molding with Slip Layer 277

Pressure-governing Equation in 2.5 Dimensions for Powder InjectionMolding with Slip Velocity 278

Pressure Gradients in Compaction—See Die-wall Friction 280

Pressure-induced Neck Flattening—See Compaction-inducedNeck Size 280

Price Estimation—See Costing and Price Estimation 280

Process Capability 280

Projected Area–based Particle Size—See EquivalentSpherical Diameter 280

Proof Testing—See Weibull Distribution 280

Pycnometer Density 281

Q 283

Quantitative-microscopy Determination of Surface Area—See SurfaceArea by Quantitative Microscopy 283

Quasi-3-dimensional Energy-governing Equation for PowderInjection Molding—See Energy-governing Equation for PowderInjection Molding 283

Quasi-3-dimensional Pressure-governing Equation for PowderInjection Molding—See Pressure-governing Equationin Powder Injection Molding 283

Quasi-3-dimensional Pressure-governing Equation for PowderInjection Molding with Slip-layer Model—See Pressure-governingEquation in Powder Injection Molding with Slip-layer Model 283

Quasi-3-dimensional Pressure-governing Equation for PowderInjection Molding with Slip-velocity Model—See Pressure-governingEquation in 2.5 Dimensions for Powder Injection Molding withSlip-velocity Model 283

CONTENTS xxiii

R 285

Radial Crush Strength—See Bearing Strength 285

Radiant Heating 285

Raleigh Instability 286

Random Packing Density 287

Random Packing Radial-distribution Function 288

Reaction-controlled Grain Growth—See Grain-growth MasterCurve, Interfacial Reaction Control, and Interface-controlledGrain Growth 289

Reaction-rate Equation—See Avrami Equation 289

Reactive Synthesis 289

Rearrangement Kinetics in Liquid-phase Sintering 290

Recalescence Temperature 291

Reduction in Area 291

Reduction of Oxides—See Oxide Reduction 292

Reduction Ratio in Extrusion—See Extrusion Constant 292

Reynolds Number in Porous Flow 292

Rheological Model for Liquid-phase Sintering 293

Rheological Response—See Complex Viscosity 294

Rockwell Hardness 294

Rosin–Rammler Distribution 296

S 297

Saddle-surface Stress—See Neck-curvature Stress 297

Scherrer Formula 297

Screen Sizes—See Sieve Progression 298

Secondary Dendrite–Arm Spacing 298

Secondary Recrystallization—See Abnormal Grain Growth 298

Second-stage Liquid-phase Sintering Model—See Intermediate-stageLiquid-phase Sintering Model 298

Second-stage Sintering Densification—See Intermediate-stageSintering-density Model 298

Second-stage Sintering Pore Elimination—See Intermediate-stagePore Elimination 298

Second-stage Sintering Surface-area Reduction—See Intermediate-stageSurface-area Reduction 299

Sedimentation Particle-size Analysis—See Stokes’ LawParticle-size Analysis 299

Segregation Coefficient 299

xxiv CONTENTS

Segregation of Mixed Powders—See Mixed-powder Segregation 299

Segregation to Grain Boundaries during Sintering 299

Self-propagating High-temperature Synthesis 300

Semisolid-system Viscosity—See Viscosity of Semisolid Systems 301

Shapiro Equation 301

Shear Modulus 301

Shear-rate Effect on Viscosity—See Cross Model 302

Shrinkage 302

Shrinkage Factor in Injection-molding Tool Design 302

Shrinkage-induced Densification 303

Shrinkage in Intermediate-stage Liquid-phase Sintering—SeeSolution-reprecipitation-induced Shrinkage in Liquid-phase Sintering 303

Shrinkage in Sintering—See Sintering Shrinkage (Generic Form) 304

Shrinkage Rate for Supersolidus Liquid-phase Sintering—SeeSupersolidus Liquid-phase Sintering Shrinkage Rate 304

Shrinkage Relation to Neck Size 304

Sieve Progression 304

Sintered Ductility 305

Sintered Strength 305

Sintering Atmosphere-flux Equation—See Vacuum Flux in Sintering 306

Sintering Grain Size—See Maximum Grain Size in Sintering 306

Sintering Metrics 306

Sintering Shrinkage (Generic Form) 307

Sintering Shrinkage for Mixed Powders—See Mixed-powderSintering Shrinkage 308

Sintering Shrinkage in Supersolidus Liquid-phase Sintering—SeeSupersolidus Liquid-phase Sintering Shrinkage Rate 308

Sintering Stress, Bulk 308

Sintering Stress in Final-stage Sintering for Small Grains and Faceted Pores 309

Sintering Stress in Final-stage Sintering for Small Grainsand Rounded Pores 309

Sintering Stress in Final-stage Sintering for Spherical Pores Inside Grains 310

Sintering Stress in Initial-stage Liquid-phase Sintering 311

Sintering Stress in Initial-stage Solid-state Sintering 311

Sintering Swelling with Mixed Powders—See Swelling Reactionsduring Mixed-powder Sintering. 311

Sintering Viscous Flow—See Viscosity during Sintering 312

CONTENTS xxv

Size Distribution—See Andreasen Size Distribution,Gaudin–Schuhman Distribution, Gaussian Distribution,Log-normal Distribution, and Rosin–Rammler Distribution 312

Slenderness 312

Slip Characterization of Powder–Binder Mixtures 312

Slip Flow in Pores 315

Slope of the Log-normal Distribution—See Log-normalSlope Parameter 315

Small Particle–Induced X-ray Line Broadening—SeeScherrer Formula 316

Solidification Time 316

Solids Loading 316

Solubility Dependence on Particle Size 317

Solubility Ratio 318

Solution-reprecipitation-controlled Liquid-phase Sintering—SeeDissolution-induced Densification 318

Solution-reprecipitation-induced Shrinkage in Liquid-phase Sintering 318

Solvent Debinding Time 320

Sound Velocity—See Ultrasonic Velocity 320

Spark Sintering—See Field-activated Sintering 320

Specific Surface Area 321

Spherical Pore Pressure 321

Sphericity 322

Spheroidization of Nanoscale Particles—See NanoscaleParticle-agglomerate Spheroidization 322

Spheroidization Time 322

Spouting Velocity 323

Spray Deposition 324

Spray Forming—See Spray Deposition 325

Spreading 325

Standard Deviation 326

Stiffness—See Elastic-modulus Variation with Density 326

Stokes–Einstein Equation 326

Stokes’ Law 327

Stokes’ Law Particle-size Analysis 328

Stokes’ Particle Diameter 329

Strain Hardening 329

Strain Rate in Injection Molding—See Gate Strain Ratein Injection Molding 331

xxvi CONTENTS

Strength—See Sintered Strength 331

Strength Distribution—See Weibull Distribution 331

Strength Evolution in Sintering—See In Situ Sintering Strength 331

Strength-evolution Model 331

Strength of Pressed Powder—See Green Strength 333

Stress Concentration at a Pore 333

Stress in Liquid-phase Sintering—See Sintering Stress in Initial-stageLiquid-phase Sintering 334

Stripping Stress—See Maximum Ejection Stress 334

Subsieve Particle Size—See Kozeny–Carman Equation 334

Superplastic Forming 334

Supersolidus Liquid-phase Sintering Liquid Distribution—See LiquidDistribution in Supersolidus Liquid-phase Sintering 335

Supersolidus Liquid-phase Sintering Shrinkage Rate 335

Surface Area–Based Particle Size—See EquivalentSpherical Diameter 337

Surface Area by Gas Absorption—See Specific Surface Area 337

Surface Area by Quantitative Microscopy 337

Surface-area Reduction during Liquid-phase Sintering 337

Surface-area for Broad Particle-size Distributions 338

Surface-area Reduction Kinetics 338

Surface Carburize—See Case Carburize 339

Surface Curvature–Driven Mass Flow in Sintering 339

Surface Diffusion–Controlled Neck Growth 341

Surface-energy Variation with Droplet Size 342

Surface-transport Sintering—See Surface Area–Reduction Kinetics 343

Suratman Number 343

Suspension Viscosity 343

Swelling—See Shrinkage and Shrinkage-induced Densification 344

Swelling Reactions during Mixed-powder Sintering 344

T 347

Tap Density—See Vibration-induced Particle Packing 347

Temperature Adjustments for Equivalent Sintering 347

Temperature Dependence—See Arrhenius Relation 348

Terminal Density—See Final-stage Sintering Limited Density 348

Terminal Neck Size—See Neck Growth Limited by Grain Growth 348

Terminal Neck Size in Sintering—See Limiting Neck Size 348

Terminal Pore Size—See Final-stage Pore Size 348

CONTENTS xxvii

Terminal Settling Velocity—See Stokes’ Law 348

Terminal Sintering—See Trapped-gas Pore Stabilization 348

Terminal Velocity—See Acceleration of Free-settling Particles 348

Tetrakaidecahedron 348

Theoretical Density for Mixed Powders—See MixtureTheoretical Density 350

Thermal Conduction—See Conductive Heat Flow 350

Thermal Conductivity 350

Thermal Conductivity Dependence on Porosity 350

Thermal Conductivity from Electrical Conductivity 351

Thermal Convection—See Convective Heat Transfer 351

Thermal Debinding—See Polymer Pyrolysis and VacuumThermal Debinding 351

Thermal Debinding Master Curve—See Master Decomposition Curve 352

Thermal Debinding Time 352

Thermal Diffusivity—See Thermal Conductivity 352

Thermal Expansion Coefficient 352

Thermally Activated—See Arrhenius Relation 353

Thermal Shock Resistance 353

Theta Test 353

Third-stage Sintering Densification—See Final-stage Densification 354

Third-stage Sintering Stress—See Final-stage Sintering Stress, SinteringStress in Final-stage Sintering for Small Grains and Faceted Pores,Sintering Stress in Final-stage Sintering for Small Grains and RoundedPores, and Sintering Stress in Final-stage Sintering for SphericalPores Inside Grains 355

Three-point Bending Strength—See Transverse-rupture Strength 355

Three-point Bending Test—See Bending Test 355

Time for Thermal Debinding—See ThermalDebinding Time 355

Time to Solidify in Atomization—See Solidification Time 355

Time to Spheroidize in Atomization—See Spheroidization Time 355

Tool Expansion Factor 355

Tortuosity—See Darcy’s Law 356

Transformation Kinetics—See Avrami Equation 356

Transient Liquid-phase Sintering 356

Transverse-rupture Strength 357

Trapped-gas Pore Stabilization 359

Truncated Octahedron—See Tetrakaidecahedron 360

xxviii CONTENTS

Two-dimensional Grain Contacts—See Connectivity 360

Two-particle Sintering Model—See KuczynskiNeck-growth Model 360

Two-particle Viscous-flow Sintering 360

U 363

Ultrasonic Velocity 363

V 365

Vacancy Concentration Dependence on Surface Curvature 365

Vacancy Diffusion 366

Vacuum Debinding—See Vacuum Thermal Debinding 367

Vacuum Distillation Rate 367

Vacuum Flux in Sintering 368

Vacuum Thermal Debinding 368

Vapor Mean Free Path 369

Vapor Pressure 371

Vibration-induced Particle Packing 372

Vickers Hardness Number 372

Viscoelastic Model for Powder–Polymer Mixtures 373

Viscoelastic Response 374

Viscosity 376

Viscosity Dependence on Shear Rate—See Cross Model 377

Viscosity during Sintering 377

Viscosity Model for Injection-molding Feedstock 379

Viscosity of Semisolid Systems 380

Viscosity of Suspension—See Suspension Viscosity 381

Viscosity Variation with Hydrostatic Pressure—See PressureEffect on Feedstock Viscosity 381

Viscous Flow in Pressure-assisted Sintering 382

Viscous Flow of a Liquid Droplet 382

Viscous Flow Sintering 384

Viscous Flow Sintering of Glass 385

Viscous-phase Sintering—See Viscosity ofSemisolid Systems 386

Viscous Settling—See Stokes’ Law 386

Viscous Sintering, Viscous-phase Sintering—See Two-particleViscous-flow Sintering 386

Voigt Model—See Viscoelastic Model for Powder–Polymer Mixtures 386

CONTENTS xxix

Volume-based Particle Size—See Equivalent Spherical Diameter 387

Volume Diffusion—See Vacancy Diffusion 387

Volume Diffusion-controlled Creep Densification—SeeNabarro–Herring Creep-controlled Pressure-assisted Densification 387

W 389

Washburn Equation 389

Water-atomization Particle Size 389

Water Immersion Density—See Archimedes Density 390

Weber Number 390

Weibull Distribution 390

Wetting Angle 392

Wicking 393

Work Hardening—See Strain Hardening 393

Work of Sintering—See Master Sintering Curve 393

X 395

X-ray Line Broadening—See Scherrer Formula 395

Y 397

Yield Strength in Viscous Flow—See Bingham Viscous-flow Model 397

Yield Strength of Particle–Polymer Feedstock 397

Young’s Equation—See Contact Angle and Wetting Angle 398

Young’s Modulus—See Elastic Modulus 398

Z 399

Zener Relation 399

Zeta Potential 401

Appendix 403

References 409

xxx CONTENTS

FOREWORD

Computer simulations and mathematical models are important aspects of modernengineering. The technical journals abound with examples of how mankind isgaining predictability, even moving to virtual design of aircraft, spacecraft, auto-mobiles, and other engineered systems. Accordingly, software sales to the engineer-ing community have accelerated to shorten design times while accurately predictingwhat will happen. It is clearly desirable to move from observation to prediction, if wecan establish that the underlying principles are known. In particulate materials proces-sing, many mathematical relations have been identified by research over the pastcentury. These relations describe the systems and the interactions during processing.This book collects that knowledge into a compilation geared to many users. Theobvious application is in support of computer simulations, where constitutiverelations are required to feed discrete and finite element analysis. A related area isin the analysis of experimental data, where underlying patterns are extracted fromdesigned experiments. Another important area is in sensitivity analysis, and theunderstanding of uncertainty. The relations presented here provide a context for sup-porting all of these activities and to help students find the needed relations withoutdelving into many different papers, books, and handbooks dating from the 1800s.

On the one hand, this book reflects where our knowledge is firm enough to providea mathematical description. On the other hand, areas were our knowledge is shallowwill obviously be targets for future studies. Indeed, this organization provides a fertiledelineation of areas needing attention. Here we give attention to techniques widelyemployed in ceramics, powder metallurgy, cemented carbides, and related particulatematerials. Entries are included that deal with many aspects of powder technology,such as the following:

Powder production and powder characterization

Powder shaping via compaction, injection molding, and extrusion

Powder consolidation via sintering, hot pressing, and hot isostatic pressing

Finishing operations, microstructure analysis, material testing

Performance linkages to structure–property relations.

Although modeling is fundamental to materials processing, little organizationoccurs in the field. This book is written for those already exposed to the conceptsassociated with particulate materials processing. It will be most useful for researchers,production engineers, students, faculty, and quality personnel.

xxxi

The book is an addition to the Wiley Series on Materials Processing. After muchdiscussion, we elected to organize the sections alphabetically. This was in part due toearly concerns by external reviewers about trying to organize the information aroundprocessing sequences, especially in light of the uneven developments up to now.Consequently, an alphabetical organization is employed, keeping in mind thatmany topics arise at several points; for example, grain growth concepts arise inpowder formation, sintering, hot isostatic pressing, and heat treatment. The rigorand depth of knowledge was nonuniform by topic, leaving some areas with poorcoverage, so this organization seemed to provide rapid access to key points withless repetition.

Our efforts were assisted by several people. Jennifer Brou provided the line draw-ings and Jay Chae was most helpful with drafting the text. We are very thankful toSukyoung Ahn, Paul Allison, Arockiasamy Antonyraj, Pavan Suri, and LauraTucker for providing helpful reviews on the manuscript.

RANDALL M. GERMAN

SEONG JIN PARK

xxxii FOREWORD

ABOUT THE AUTHORS

RANDALL M. GERMAN ([email protected]) is the Center for AdvancedVehicular Systems (CAVS) Chair Professor in Mechanical Engineering andDirector of the Center for Advanced Vehicular Systems at Mississippi StateUniversity, Mississippi State. Rand was previously the Brush Chair Professor inMaterials at Pennsylvania State University, University Park, and Robert HuntProfessor in Materials Engineering at Rensselaer Polytechnic Institute, Troy, NY.He is the author of over 900 articles, 15 books, 23 patents, and editor of 19 books.He received his Ph.D. degree from the University of California at Davis, and hehas an Honorary Doctorate from University Carlos III de Madrid, Spain. Otherdegrees include an M.S. from The Ohio State University, Columbus, and a B.S.from San Jose State University, San Jose, CA. His teaching includes courses suchas sintering theory, rheology of particulate solids, and powder metallurgy and particu-late materials processing.

SEONG JIN PARK ([email protected]) is Associate Research Professor in theCenter for Advanced Vehicular Systems at Mississippi State University, MississippiState. Prior to that he held research positions at Pennsylvania State University,CetaTech, Fine Optics, and LG Electronics. His research focus is on modelingmaterials processing and performance problems relevant to industry. He received aPh.D. in Mechanical Engineering at POSTECH, Pohang, Korea, and dealt withdesign optimization for injection-molding systems using boundary-elementmethods. Seong Jin is the author of over 155 articles in engineering journalsand has worked extensively in developing electronic learning technologies. AtMississippi State he teaches Advanced Strength of Materials and specialty courseson modeling materials forming processes.

xxxiii

A

ABNORMAL GRAIN GROWTH (Worner et al. 1991; Kang 2005)

Abnormal grain growth involves the excessively rapid growth of a few grains in anotherwise uniform microstructure. It is a particular problem in the later stages ofsintering. It is characterized by certain grains or crystallographic planes exhibitingfaster growth than average. Figure A1 is a sketch of a microstructure formed as aconsequence of abnormal grain growth where one large grain at the top is growingat the expense of the surrounding smaller grains. Abnormal grain growth is favoredwhen segregation changes the grain-boundary mobility or grain-boundary energy.When grain growth occurs, there is an interfacial velocity Vij for the grain boundarybetween the i– j grain pair given by the product of the mobility Mij and the force perunit area on the grain boundary Fij,

Vij ¼ MijFij

where the grain-boundary velocity varies between individual grain boundaries, asindicated by the subscript. The force Fij is given by the product of the interfacialenergy and the curvature,

Fij ¼ �gij1Gi� 1

Gj

� �

where Gi and Gj are the grain size for contacting grains, and gij is the correspondinginterfacial energy for the i– j interface. Although not routinely recorded, the inter-facial energy depends on the misorientation between grains. Effectively, the energyper unit volume scales with the inverse grain size, so if Gi . Gj, then the force ispushing the grain boundary toward the smaller grain center. A critical conditionoccurs when the mobility of an individual grain boundary, Mij, greatly exceeds the

Mathematical Relations in Particulate Materials: Ceramics, Powder Metals, Cermets, Carbides,Hard Materials, and Minerals. Edited by Randall M. German and Seong Jin ParkCopyright # 2008 John Wiley & Sons, Inc.

1

average or when the individual grain-boundary energy is excessively low. This criticalcondition is expressed by the following inequality:

Mij

Mm.

169

gij

gm

� �

where Mm is the mean grain-boundary mobility, gij is the individual grain-boundaryenergy, and gm is the mean grain-boundary energy. With respect to abnormal graingrowth, the two situations of concern are a twofold higher individual grain-boundarymobility, for example, because of a segregated liquid, or a twofold lower individualgrain-boundary energy, for example, due to segregation or near coincidence in grainorientation. In sintering practice, most examples of abnormal grain growth are causedby impurities that segregate on the grain boundaries even at the sintering temperature.For example, in sintering alumina (Al2O3), abnormal grain growth is favored by ahigh combined calcia (CaO) and silica (SiO2) impurity level.

Fij ¼ grain-boundary force per unit area between the i– j grain pair, N/m2

Gi, Gj ¼ grain size for corresponding grain, m (convenient units: mm)

Mij ¼ grain-boundary mobility between the i– j grain pair, m3/(s . N)

Mm ¼ mean grain-boundary mobility averaged over the body, m3/(s . N)

Vij ¼ interfacial velocity for the grain boundary between the i– j grain pair, m/s

gij ¼ individual grain-boundary energy between the i– j grain pair, J/m2

gm ¼ mean grain-boundary energy averaged over the body, J/m2.

Figure A1. Abnormal grain growth during sintering is evident in sintering by the formation ofa very large grain growing into a matrix of much smaller grains. The resulting nonuniformmicrostructure is evident in this reproduction from a sintered (Sr, Ba)Nb2O6 ceramic afterheating at 12608C for 4 h, where the grain at the top of this image is much larger than thesurrounding small grains.

2 CHAPTER A

ABRASIVE WEAR

See Friction and Wear Testing.

ACCELERATION OF FREE-SETTLING PARTICLES (Han 2003)

An assumption in Stokes’ law, as applied to both particle-size classification andparticle-size distribution analysis, is that the particles instantaneously reach terminalvelocity. However, this is not the case in practice, and the acceleration of the particleto the free-settling terminal velocity adds an error in a particle-size analysis. Theapproach to the Stokes’ law terminal velocity vT is described by the followingequation for spherical particles initially at rest:

v ¼ vT 1� exp � 18th

rD2

� ��

where v is the velocity after time t when the particle starts from rest, h is thefluid viscosity, r is the theoretical density of the particle, and D is the particlediameter. A plot of this equation is given in Figure A2, where the actual velocity isnormalized to the terminal velocity for the case of a 1-mm stainless steel particlesettling in water.

Figure A2. A plot of the relative particle velocity (when starting from rest) versus time to showthe acceleration of a particle settling by Stokes’ law. The particle velocity is relative to the term-inal velocity. This calculation is for a 1-mm stainless steel ball settling in water.

ACCELERATION OF FREE-SETTLING PARTICLES 3

D ¼ particle diameter, m (convenient units: mm)

t ¼ time, s

v ¼ velocity (starting with v ¼ 0 at t ¼ 0), m/s

vT ¼ Stokes’ law terminal velocity, m/s

h ¼ fluid viscosity, Pa . s

r ¼ theoretical density of the particle, kg/m3 (convenient units: g/cm3).

ACTIVATED SINTERING, EARLY-STAGE SHRINKAGE(German and Munir 1977)

Activated sintering is associated with a treatment, usually by an additive, that greatlyincreases sintering densification at lower temperatures than typically required. In acti-vated sintering the initial sintering shrinkage depends on the rate of diffusion in theactivator, which is segregated to the interparticle grain boundary. Figure A3 providesa schematic of the sintering geometry used to model first-stage activated sintering.The growth of the interparticle bond results in attraction of the particle centers,which gives compact shrinkage DL/L0 as follows:

DL

L0¼ L� L0

L0¼ gVdCgSVDAt

D4RT

where DL is the change in length, L0 is the initial length, L is the instantaneous lengthduring sintering, g is a collection of geometric terms, V is the atomic volume, d is thewidth of the second-phase activator layer coating the grain boundary, C is the solubi-lity of the materials being sintered in the second-phase activator, gSV is the solid–vapor surface energy, DA is the diffusivity of the material being sintered in theactivator (note this changes dramatically with temperature), t is the sintering time,D is the particle size, R is the gas constant, and T is the absolute temperature.

Figure A3. Simple two-particle geometry for activated sintering, where the activator is segre-gated to the interparticle grain boundary to form a layer of width d for a neck of diameter X anda grain or particle of diameter D.

4 CHAPTER A

Faster diffusion in the activator induces early sintering gains, but this mandates thatthe solid be soluble in the activator. The controlling step is the diffusivity in the acti-vator layer. The difference in effectiveness between various activators is explained bytheir differing diffusivities and solubilities.

C ¼ volumetric solubility in the activator, m3/m3 (dimensionless)

D ¼ median particle size, m (convenient units: mm)

DA ¼ diffusivity of the base material in the activator layer, m2/s

L ¼ instantaneous length, m (convenient units: mm)

L0 ¼ initial length, m (convenient units: mm)

R ¼ universal gas constant, 8.31 J/(mol . K)

T ¼ absolute temperature, K

g ¼ collection of geometric terms, dimensionless

t ¼ isothermal sintering time, s

DL ¼ change in length, m (convenient units: mm)

DL/L0 ¼ sintering shrinkage, dimensionless (convenient units: %)

V ¼ atomic volume, m3/mol

d ¼ activator phase width on the grain boundary, m (convenient units:nm or mm)

gSV ¼ solid–vapor surface energy, J/m2.

ACTIVATION ENERGY

See Arrhenius Relation.

ADSORPTION

See BET Specific Surface Area.

AGGLOMERATE STRENGTH (Pietsch 1984)

Powder that is wetted by a relatively small quantity of liquid or polymer will agglom-erate. If the fluid phase is not solidified or hardened, then the crush strength s for anagglomerated mass of powder depends on the fractional porosity 1 and the degree ofpore saturation S,

s ¼ 7SgLV

1� 1

D1

where the saturation S is the fraction of pore volume that is filled with liquid (often assmall as 0.01), gLV is the liquid–vapor surface energy, and D is the particle size.

AGGLOMERATE STRENGTH 5

Unless the agglomerate is wetted by a high-strength polymer, the strength of a typicalagglomerated powder is dominated by capillarity effects.


S ¼ degree of pore saturation, dimensionless fraction [0, 1]

1 ¼ fractional porosity, dimensionless [0, 1]

gLV ¼ liquid–vapor surface energy, J/m2

s ¼ strength of the agglomerate, Pa.

[Also see Capillarity.]

AGGLOMERATION FORCE

When a small powder is exposed to water or other condensable vapor, a liquid bridgecan form at the contact points between particles. Initially the liquid bridges are smalland do not merge, giving a structure termed the pendular state. As long as the liquid iswetting, then at low concentrations the resulting capillary bonds provide an attractiveforce. As an approximation, the attractive force F between contacting particles varieswith the liquid–vapor surface energy gLV, and particle size D, as follows:

F ¼ 3DgLV.


F ¼ attractive force between contacting particles, N

gLV ¼ liquid–vapor surface energy, J/m2.

AGGLOMERATION OF NANOSCALE PARTICLES

See Nanoparticle Agglomeration.

ANDREASEN SIZE DISTRIBUTION (Andreasen 1930)

Originally isolated in colloidal particle-packing studies, the Andreasen particle sizedistribution is applicable to all powders where a high packing density is desired.The cumulative particle-size distribution is expressed in terms of the weight fractionof particles F(D) given as the fractional weight of powders with a size less than par-ticle size D. The Andreasen size distribution is described as follows:

F(D) ¼ AD

DL

� �q

6 CHAPTER A

where A is a fitting parameter, DL is the largest particle size in the distribution, and q isthe distribution exponent. For the highest packing densities, it is observed that theexponent q tends to range near 0.6. As an alternative, the cumulative particle-size dis-tribution can be expressed with respect to a limiting size by defining a distributionconstant B ¼ A/DL

q, giving

F(D) ¼ BDq

A ¼ fitting parameter, dimensionless

B ¼ distribution constant, 1/mq

D ¼ particle size, m (convenient units: mm)

DL ¼ size of the largest particle, m (convenient units: mm)

F(D) ¼ cumulative weight-based particle-size distribution, dimensionless [0, 1]

q ¼ distribution exponent, dimensionless.

APPARENT DIFFUSIVITY (Porter and Easterling 1981)

In cases where both volume diffusion DV and grain-boundary diffusion DB are actingto induce sintering shrinkage, the data from shrinkage experiments only provide ameans to extract an apparent diffusivity, not an absolute diffusivity. For a constanttemperature, the combined or apparent diffusivity DA depends on the two contri-butions as follows:

DA ¼ DV þbd

GDB

where G is the grain size of the microstructure, d is the grain-boundary width (usuallyassumed to be 5 to 10 times the atomic size), and b is an adjustable parameter nearunity (typically ranges from 0.5 to 1.5). Both diffusivities are functions oftemperature.

DA ¼ apparent diffusivity, m2/s

DB ¼ grain boundary diffusivity, m2/s

DV ¼ volume diffusivity, m2/s

G ¼ grain size, m (convenient units: mm)

b ¼ adjustable parameter, dimensionless

d ¼ grain-boundary width, m (convenient units: mm or nm).

ARCHARD EQUATION (Archard 1957)

Sliding wear is commonly treated in terms of the loss of material as a function ofthe hardness, sliding distance, and normal load. The coefficient of friction between

ARCHARD EQUATION 7

the substrate and sliding component is a factor that can greatly change wear rates. TheArchard equation calculates the wear behavior by assuming asperity removal, where asingle circular cross section is acted upon by an intense wear event. Fragments form andcontribute to the mass loss based on the assumption that the hardness and yield strengthof the material are proportional. The resulting wear equation is given as follows:

Q ¼ kWL

H

where Q is the volume of material removed from the test or wear material, k is awear constant that provides a measure of the wear resistance, W is the normal(perpendicular to the surface) load causing wear, L is the total sliding length for thewear event, and H is the material hardness (assuming units of Pa or N/m2, where itis assumed the opposing material is much harder). The first derivative of this equationwith respect to time then says the wear rate (volume per unit time) is proportional to thesliding velocity.

H ¼ hardness, Pa (convenient units: MPa)

L ¼ sliding length, m (convenient units: mm)

Q ¼ wear volume, m3 (convenient units: mm3)

W ¼ normal load, N (convenient units: kN or MN)

k ¼ wear constant, dimensionless.

[Also see Friction and Wear Testing.]

ARCHIMEDES DENSITY

A standard means to determine the volume of an irregular shape is based on fluiddisplacement when the component is immersed in a fluid such as water. The measure-ment must prevent fluid intrusion into surface-connected pores to extract an accuratevolume. Combined with the dry mass determined prior to the test, a densitycalculation follows. First, the sample is weighed dry (W1), then again after oil impreg-nation of the evacuated pores (W2), and finally the oil-impregnated sample is immersedin water for the final weight (W3). Usually a wire is used to suspend the sample in thewater and its weight WW must be measured in water too. Then the actual or Archimedesdensity r can be calculated from the weight determinations as follows:

r ¼ W1fW

W2 � (W3 �WW )

where fW is the density of water in kg/m3, which is temperature dependent asgiven here,

fW ¼ 1001:7� 0:2315T

8 CHAPTER A

with T being the water temperature in 8C. Dividing the measured density by the theor-etical density gives the fractional density. One variant uses water impregnation insteadof oil to fill the pores, which still involves two immersion events, but there is no oiltrapped in the pores.

T ¼ water temperature, 8CW1 ¼ dry mass of the sample prior to testing, kg (convenient units: g)

W2 ¼ wet mass of the sample after filling pores with fluid, kg (convenientunits: g)

W3 ¼ mass of the component immersed in water, kg (convenient units: g)

WW ¼ mass of the suspension wire, kg (convenient units: g)

fW ¼ temperature-corrected density of water, kg/m3

(convenient units: g/cm3)

r ¼ component density, kg/m3 (convenient units: g/cm3).

[Also see Fractional Density.]

ARRHENIUS RELATION

The change in atomic motion due to a temperature increase or decrease duringsintering is described by an Arrhenius relation. It corresponds to an approxi-mation of the integral area under the tail of the Boltzmann energy distributionfor the higher energies. Inherently the Arrhenius relation gives the fraction ofatoms with an energy of Q or greater at any time. This integral determines thecumulative probability that an atom has more energy than that required tomove, as determined by the activation energy Q. For example, the volume-diffu-sion coefficient DV is determined from the atomic vibrational frequency D0, absolutetemperature T, universal gas constant R, and the activation energy Q, whichcorresponds to the energy required to induce atomic diffusion via vacancyexchange, giving,

DV ¼ D0 exp � Q

RT

� �

Variants on this relation exist for grain-boundary diffusion, surface diffusion, evapor-ation, creep, and other high-temperature processes.

D0 ¼ diffusion frequency factor, m2/s

DV ¼ volume-diffusion coefficient, m2/s

Q ¼ activation energy, J/mol (convenient units: kJ/mol)


T ¼ absolute temperature, K.

ARRHENIUS RELATION 9

ATMOSPHERE MOISTURE CONTENT

See Dew Point.

ATMOSPHERE-STABILIZED POROSITY

See Gas-generated Final Pores.

ATOMIC FLUX IN VACUUM SINTERING (Johns et al. 2007)

When sintering in a vacuum, a rate of gas impingement exists on any surface, andthat rate depends on the pressure and temperature in the sintering furnace. Thecorresponding atomic flux is the frequency at which gas molecules collide withthe surface. Considering an external surface (not inside the pore), the numberof gas molecules that strike the surface per unit time and per unit area is the flux Jestimated as,

J ¼ Pffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pkTmp

where P is the gas pressure, k is Boltzmann’s constant, T is the absolute temperature,and m is the molecular weight of the species. In a similar manner, oxide reduction in apartial pressure of hydrogen or vacuum surface carburization both depend on thissame flux. If the density of desired reaction sites is known for the exposed surface,then it is possible to estimate from the flux the time required for the desired effect;the characteristic time is the density of surface sites (number per unit area) dividedby the flux.

J ¼ flux, atom/(m2 . s) or molecule/(m2 . s)

P ¼ gas pressure, Pa


k ¼ Boltzmann’s constant, 1.38 . 10223 J/(atom . K)

m ¼ molecular weight, kg/atom or kg/molecule.

ATOMIC-SIZE RATIO IN AMORPHOUS METALS

The formation of a glassy metal, or bulk amorphous metal, depends on severalfactors, with the atomic-size ratio of the constituents being one of the importantfactors. If atoms are very different in size, as well as having differences in valenceand crystal structure, then it is difficult to crystallize a solid on cooling a

10 CHAPTER A

homogeneous liquid so formation of the amorphous state is favored. Accordingly, onefactor that helps in the formation of an amorphous metal is a large atomic-size ratio(RB/RA). This ratio is linked to the solute concentration CB needed to form an amor-phous phase as follows:

RB

RA

� �3

�1

��CB � 0:1

where RB is the solute (minor constituent) atomic radius and RA is the solvent (majorconstituent) atomic radius. Less solute additive is needed to access the amorphousstructure during cooling, as the atomic sizes are substantially different (such thatthe size ratio is significantly different from unity).

CB ¼ solute concentration to form an amorphous phase, m3/m3 (dimensionless)

RA ¼ atomic radius of the solvent phase, m (convenient units: nm or A)

RB ¼ atomic radius of the solute phase, m (convenient units: nm or A).

ATOMIZATION SPHEROIDIZATION TIME

See Spheroidization Time.

ATOMIZATION TIME

See Solidification Time.

AVERAGE COMPACTION PRESSURE

See Mean Compaction Pressure.

AVERAGE PARTICLE SIZE

See Mean Particle Size.

AVRAMI EQUATION (Avrami 1939)

The Avrami equation is used to describe the rate of phase transformation in a processthat first involves nucleation of the new phase followed by transformation with a

AVRAMI EQUATION 11

progressively slower rate as the source species for the reaction are exhausted. Asillustrated in Figure A4, the general shape is a lazy-S curve showing the fractionor percent transformed versus time. It is fit by an equation of the form:

y ¼ 1� exp(�Ktn)

where y is the fraction transformed, t is the time, n and K are constants for a givenreaction. Typically the parameter K is temperature dependent (Arrhenius temperaturedependence with an activation energy representative of the underlying mechanism)and n ranges from 1 to 4.

K ¼ temperature-dependent reaction rate, 1/sn

n ¼ time exponent, dimensionless

t ¼ reaction time, s

y ¼ fraction of phase transformed, dimensionless [0, 1].

Figure A4. A plot of the Avrami equation showing a typical fit to reaction kinetics using a timeexponent of unity.

12 CHAPTER A

B

BALL MILLING

See Jar Milling.

BEARING STRENGTH (MPIF 55 2007)

A crush test is applied to straight-wall cylindrical bearings to measure the strength;this test is also known as the radial crush-strength test. The strength measured thisway is sometimes termed the K-factor. As shown in Figure B1, it is measured bycrushing the bearing between two parallel platens. The bearing strength sK is deter-mined from the maximum load F encountered during crushing, giving the following:

sK ¼ Fd � t

lt2

where l is the cylinder length, d is the outer diameter, and t is the wall thickness. Formost materials this strength is not directly comparable to tensile, compressive, or otherstrength tests.

F ¼ fracture load, N (convenient units: kN or MN)

d ¼ cylinder outer diameter, m (convenient units: mm)

l ¼ cylinder length, m (convenient units: mm)

t ¼ cylinder wall thickness, m (convenient units: mm)

sK ¼ bearing strength, Pa (convenient units: MPa).

BELL CURVE

See Gaussian Distribution.


13

BENDING-BEAM VISCOSITY (Bollina and German 2004)

The end-supported bending-beam analysis, as adapted from glass testing, allows forthe calculation of the effective viscosity of a powder compact during sintering. Thetest is illustrated in Figure B2. Support is provided at the two ends for a simplerectangular powder compact. During sintering the midspan deflection is measured(usually by photography or video imaging) as a function of time and temperature.Most typically, the test is performed using a constant heating-rate cycle, so bothdensity and deflection change as a function of time and temperature during the test.In the case of uniform gravitational loading of elastic beams, the general deflectionequation giving the vertical motion as a function of the horizontal position x isexpressed as,

1

[1þ (dd=dx)2]3=2

d2d

dx2¼ M

EI

Figure B1. Outline of the bearing crush test as applied to sintered bearings, where the cylind-rical sample is placed on its side and crushed. The maximum load during crushing is used alongwith the bearing diameter, length, and wall thickness to calculate the strength.

14 CHAPTER B

where d is the deflection of the beam, and x is the distance from the nearest support.On the right-hand side, M is the bending moment, I is the moment of inertia, and E isthe elastic modulus of the material at the test temperature, which changes during thetest. Since the elastic modulus changes with both temperature and density, either anindependent measure is required, such as by ultrasonic testing, or it can be estimatedusing models of elastic modulus versus density and temperature. Most typically, thetest is performed on a rectangular cross section, and without an applied load the bodybends under its own weight, giving

M ¼ qx

2(l� x) and I ¼ bh3

12

where q is the distributed load due to the beam weight, l is the span length, b is thewidth, and h is the thickness of the sample. Note the cluster q is the distributed loaddue to the beam’s own weight and is defined as follows:

q ¼ rgbh

where g is the gravitational constant and r is the density of the sample, respectively. Insome testing variants an externally applied load is applied via a loading dilatometer sothat load is added to the gravitational load. For small deflections, that is, dd/dx ,

0.15, the second-order term (dd/dx)2 is usually neglected and the first relation isexpressed as follows:

d2d

dx2¼ M

EI

Figure B2. Sketch of the bending-beam viscosity test for determination of viscosity duringsintering.

BENDING-BEAM VISCOSITY 15

The maximum deflection dmax occurs at the middle of the span as

dmax ¼5rgl4

32Eh2

In sintering experiments, materials undergoing densification and distortion demon-strate viscous behavior rather than elastic behavior. Using the analogy betweenthese deformation modes, it is possible to replace the elasticity E by the uniaxialviscosity h, and deflection dmax by the deflection rate dmax. If pure Newtonianviscous flow is assumed, then the preceding equation is rearranged to give the analo-gous viscous form; where the in situ viscosity h, which varies with time, temperature,and density, is calculated as

h ¼ 5rgl4

32 _dmaxh2

Note that the viscosity changes continuously, so time-resolved images provide a traceof the viscosity during sintering. System tests show that viscosities in the 10- to100-GPa . s range are common during sintering densification, similar to stiff pastes.

E ¼ elastic modulus at the test temperature, Pa (convenient units: GPa)

I ¼ moment of inertia, m4

M ¼ bending moment, N . m

b ¼ sample width, m (convenient units: mm)

g ¼ gravitational acceleration, 9.8 m/s2

h ¼ sample thickness, m (convenient units: mm)

l ¼ span length, m (convenient units: mm)

q ¼ distributed load due to the beam weight, N/m

x ¼ distance from the nearest support, m (convenient units: mm)

d ¼ deflection of the beam, m (convenient units: mm)

dmax ¼ maximum deflection of the beam at the middle of the span, m(convenient units: mm)

dmax ¼ maximum deflection rate of the beam at the middle of the span, m/s

h ¼ powder-compact viscosity, Pa . s (convenient units: GPa . s)

r ¼ beam density, kg/m3 (convenient units: g/cm3).

[Also See Viscosity During Sintering.]

BENDING TEST (Meyers 1985; Morrell 1989; Green 1998)

One of several names given to standard tests used to measure the modulus of ruptureor transverse-rupture strength. The bending test is designed to measure the strength of

16 CHAPTER B

brittle materials, including green compacts. The two test situations are illustrated inFigure B3. In each, a rectangular sample supported on two lower rods is fracturedusing 3-point bending or 4-point bending. The test is invalid for ductile materialsbecause the calculation assumes that fracture initiates in the outer fiber, which is inpure tension, while deflection gives work hardening and a more complex stressstate. For 3-point loading, the bending strength sT is calculated from the specimengeometry and failure load F as follows:

sT ¼3FL

2wt2

where t is the thickness, w is the width, and L is the span distance between the lowersupport rods. A typical aspect ratio for the sample test dimensions is near 1: 2: 4—thickness to width to test length. For example, in many studies the 3-point testrelies on sample dimensions of t ¼ 6 mm, w ¼ 12 mm, and L ¼ 25 mm, but othersizes are allowed with similar dimensional ratios. For the 4-point test, the equivalent

Figure B3. Two forms of the bending strength test, also know as the transverse rupture test andthe modulus of rupture test, based on 3-point and 4-point loading to failure.

BENDING TEST 17

strength formulation is given as follows:

sT ¼3FL

4wt2

where the upper span is L/2 and the lower span is L. Recent practical data indicate thatthe 3-point test is more consistent, probably because there is less chance for fixturemisalignment. Based on many tests, generally the bending strength is proportionalto 25% of the compressive strength.


L ¼ span distance between support rods, m (convenient units: mm)

t ¼ sample thickness, m (convenient units: mm)

w ¼ sample width, m (convenient units: mm)

sT ¼ bending strength, Pa (convenient units: MPa).

BET EQUIVALENT SPHERICAL-PARTICLE DIAMETER

Gas absorption is a technique used to measure the surface area of a loose powder.When the surface area is measured by this approach, an equivalent spherical diameteris calculated by assuming that the particles are monosized spheres. This gives theBET equivalent spherical-particle diameter DBET as follows:

DBET ¼6

rT SBET

where SBET is the specific surface area of the powder, which is usually measured inm2/g, as determined using the BET gas-absorption process, rT is the theoreticaldensity of the powder, which is usually based on the pycnometer density, andDBET is the particle size whose conventional units are mm. In this form the propor-tionality factor of 6 reflects an assumed spherical shape. If the median particle sizeand surface area are measured independently, any departure from 6 is usually takenas an indication of nonspherical particles; however, such a particle-shape index isnot very reliable considering the typical errors in the measured parameters.

DBET ¼ particle size, m (convenient units: mm)

SBET ¼ specific surface area of the powder, m2/kg (convenient units: m2/g)

rT ¼ theoretical density of the powder, kg/m3 (convenient units: g/cm3).

BET SPECIFIC SURFACE AREA (Brunauer et al. 1938)

The BET technique, named after Stephen Brunauer, Paul Emmett, and Edward Teller,scrutinizes the surface of a powder or porous body using gas adsorption on a cold

18 CHAPTER B

sample. The approach assumes each gas molecule occupies a precise area.Low-pressure adsorption isotherms provide a means to deduct the mass of adsorptioncorresponding to a single molecule layer and from that calculate the surface area. TheBET surface area is calculated from the adsorption behavior using repeated measuresover a range of partial pressures. Letting P equal the measured partial pressure ofadsorbate, while P0 equals the saturation or equilibrium pressure of adsorbate,which depends on the gas and temperature, then with X equal to the mass of gasadsorbed at pressure P, a calculation is made of Xm as the adsorption capacity ofthe powder (the mass of gas necessary to form a saturated surface coating oneatomic layer thick), giving,

P

X(P0 � P)¼ 1

XmC1þ P

P0(C � 1)

� �

with C equal to a constant relating to the adsorption enthalpy. Formally, C is calcu-lated from the exponential of the difference in adsorption enthalpy for the outersurface layer versus an inner layer, which represents fully coordinated molecularbonding. This equation gives a linear relation between the term on the left of theequal sign and the partial pressure ratio P/P0 on the right side. It is known as theBET equation and is valid for measuring the surface area of a powder when thepressure range P/P0 is from 0.05 to about 0.35. The equation can be rewritten in ageneral reduced-parameter form as,

P

X(P� P0)¼ Bþ A

P

P0

where

Xm ¼1

Aþ B

where A is the slope and B is the intercept of the linear relation. From this simple formthe BET-specific surface area SBET is calculated as follows:

SBET ¼XmN0A0

WM

where M is the molecular weight of the adsorbate, A0 is the average occupational areaof an adsorbate molecule (for example, nitrogen is the most popular gas and it has anaverage occupational area of 16 . 10220 m2), N0 is equal to Avogadro’s number, andW is equal to the sample mass. In some cases the specific surface area, whose con-venient units are m2/g, is converted into an equivalent spherical diameter.

A ¼ slope of adsorption quantity versus partial-pressure ratio, 1/kg

A0 ¼ average occupational area of adsorbate molecule, m2/atom

B ¼ intercept of adsorption quantity versus partial-pressure ratio, 1/kg

BET SPECIFIC SURFACE AREA 19

C ¼ constant relating to the adsorption enthalpy, dimensionless

M ¼ molecular weight of the adsorbate, kg/mol (convenient units: g/mol)

N0 ¼ Avogadro’s number, 6.02 . 1023 atom/mol

P ¼ partial-pressure of adsorbate, Pa

P/P0 ¼ partial-pressure ratio, dimensionless

P0 ¼ equilibrium pressure of adsorbate, Pa

SBET ¼ specific surface area, m2/kg (convenient units: m2/g)

W ¼ sample mass, kg (convenient units: g)

X ¼ amount of gas adsorbed, kg (convenient units: g)

Xm ¼ adsorption capacity of the powder, kg (convenient units: g).

[Also see BET Equivalent Spherical-particle Diameter.]

BIMODAL POWDER PACKING

It is possible to improve the packing density of a powder by mixing in much smallerparticles that fill the voids between the large particles. Small quantities of smallerpowder improve the density, but there can be too much of the smaller powder. Asillustrated in Figure B4, the optimal composition corresponding to the highest attain-able density is calculated in terms of the weight fraction of large particles XL

�. The

Figure B4. Bimodal powder packing and the variation of large–small powder density withcomposition, showing the maximum fractional density and the composition of the maximumdensity.

20 CHAPTER B

packing density peaks at the optimal composition, since this condition corresponds toall of the voids between the large particles being filled with small particles, assumingthe smaller particles fit into these voids. Usually this requires the small particles to be15% of the large particles. The optimal density is calculated from the amount of voidspace between large particles, which equals 1 – fL, where fL is the fractional packingdensity of the large particles,

X�L ¼fLf �

with the fractional packing density at the optimal composition f � given as,

f � ¼ fL þ fS(1� fL)

with the fractional packing density for the small particles given as fS. For two spheri-cal powders with a large size difference but the same theoretical density, each with anideal fractional packing density of 0.64, the corresponding weight fraction of largeparticles for maximum packing is 0.734, or 26.6 wt % of the smaller particles. Theexpected fractional packing density would be 0.87 or 87%. Similar concepts can beemployed for trimodal or other multiple-mode mixtures.

fL ¼ large-particle fractional packing density, dimensionless [0, 1]

fS ¼ small-particle fractional packing density, dimensionless [0, 1]

f � ¼ fractional packing density of the optimal composition, dimensionless[0, 1]

XL� ¼ weight fraction of large particles at the maximum packing density,

dimensionless [0, 1].

BIMODAL POWDER SINTERING (German 1996)

With bimodal powders there is a packing benefit when small and large particles aremixed, but often the sintering response in terms of shrinkage or densification isdegraded by using a mixture of large and small powders. The sintering problem issolved using specific volume-fraction concepts, effectively the inverse of the frac-tional density V ¼ 1/f. The prediction of sintering shrinkage for mixtures of twodifferent particle sizes is calculated from the sintering behavior of the two individualpowders fired under the same heating cycle. At compositions rich in the smallerpowder there is a progressive decrease in sintered density as the volume fraction oflarge powder increases. During sintering the change in specific volume fraction DVvaries with composition as,

DV ¼ DVm � BXL

BIMODAL POWDER SINTERING 21

where DVm is the sintering specific volume-fraction change for the pure small-particlematrix powder under equivalent sintering conditions (green density, temperature,time, atmosphere, etc.), XL is the volume fraction of large powder, and B is a para-meter that reflects the retarding stress and displaced shrinkage volume associatedwith large-particle additions. For composites, the large powder might be a fiber orwhisker. For compositions with a large difference in particle size, below theoptimal packing condition XL , XL

�, the sintering shrinkage variation with compo-sition is given as follows:

Y ¼ DL

L0¼ YS � (YS � YL)

XL

X�L

where Y is used to indicate the shrinkage defined as the change in dimension dividedby the initial dimension. In this nomenclature, YS is the shrinkage of the small powderand YL is the shrinkage of the large powder. These shrinkages are for the same sinter-ing cycle and for the same green density using the pure small or pure large powders.The bimodal powder sintering shrinkage is calculated from these relations betweencomposition and the sintering behavior of the two powders. The sintered density rS

can then be calculated from the green density rG and shrinkage Y as follows:

rS ¼rG

(1� Y)3

B ¼ retarding stress parameter, dimensionless

L0 ¼ component size prior to sintering, m (convenient units: mm)

V ¼ specific volume fraction, dimensionless

XL ¼ volume fraction of large powder, dimensionless [0, 1]

XL� ¼ volume fraction of large powder at optimal packing, dimensionless [0, 1]

Y ¼ DL/L0 ¼ sintering shrinkage, dimensionless

YL ¼ sintering shrinkage of the large powder, dimensionless

YS ¼ sintering shrinkage of the small powder, dimensionless

f ¼ fractional density, dimensionless [0, 1]

DL ¼ change in component size on sintering, m (convenient units: mm)

DL/L0 ¼ sintering shrinkage, dimensionless

DV ¼ specific volume-fraction change on sintering, dimensionless

DVm ¼ specific volume-fraction change on sintering matrix powder,dimensionless

rG ¼ green density, kg/m3 (convenient units: g/cm3)

rS ¼ sintered density, kg/m3 (convenient units: g/cm3).

BINDER BURNOUT

See Polymer Pyrolysis.

22 CHAPTER B

BINDER (MIXED POLYMER) VISCOSITY

Most binders used in powder processing are formed by mixing polymers of differentstructures and molecular weights. As long as the polymers can be mixed, the binderviscosity hB can be approximated using the logarithmic additivity rule:

lnhB ¼Xn

i¼1

Wi lnhi

with the condition of

Xn

i¼1

Wi ¼ 1

where hi is the viscosity of the individual constituent in the binder at the same temp-erature, Wi is the weight fraction of each component (the sum of the weight fractionsequals unity), and n is the number of components. This says the binder mixtureviscosity depends on the weight fraction and viscosity of the constituent.

Wi ¼ weight fraction of ith binder constituent, kg/kg (dimensionless) [0, 1]

i ¼ dummy index for summation, dimensionless

n ¼ number of binder constituents, dimensionless

hB ¼ binder viscosity, Pa . s

hi ¼ ith binder constituent viscosity, Pa . s.

BINGHAM MODEL

See Viscosity Model for Injection-molding Feedstock.

BINGHAM VISCOUS-FLOW MODEL (Foong et al. 1995)

The rheology of particle–polymer mixtures depends on several factors, such asthe ratio of powder to polymer, temperature, applied stress, shear rate, and themixture viscosity. At high temperatures the pure binders used in powder processingare often Newtonian with no yield strength. However, with high solids loading thepowder–binder mixture tends toward Bingham behavior with an effective yieldstrength sY such that the deformation strain rate involves first a yielding event priorto undergoing Newtonian viscous flow,

d1

dt¼ s� sY

h

BINGHAM VISCOUS-FLOW MODEL 23

where 1 is the strain, t is the time, s is the stress, and h is the mixture viscosity. Such amodel is typical to the flow response of materials in binder-assisted shaping (injectionmolding, extrusion, and tape casting) and for semisolid materials undergoing liquid-phase sintering. Measures of the yield strength, when the binder is molten, generallygive values in the range of 1 to 1000 kPa.

d1/dt ¼ strain rate, 1/s

t ¼ time, s

1 ¼ strain, dimensionless

h ¼ mixture viscosity, Pa . s

s ¼ stress, Pa

sY ¼ yield strength, Pa (convenient units: kPa).

BOLTZMANN STATISTICS


BOND NUMBER

A dimensionless parameter used in the study of viscous response, such as in atomiza-tion and liquid-phase sintering. The Bond number Bo expresses the body forces as aratio to the liquid–vapor surface energy gLV,

Bo ¼ rR2g

gLV

where r is the fluid density, R is the characteristic length scale (typically the particlesize), and g is the gravitational acceleration. This parameter is a measure of theimportance of surface energy and in cases where it is much larger than unity, theassumption is that surface energy is of low importance. The alternative of a smallvalue suggests that surface energy dominates the response.

Bo ¼ Bond number, dimensionless

R ¼ characteristic length scale, m



r ¼ density, kg/m3 (convenient units: g/cm3).

BRAGG’S LAW

This relation is named for William Henry Bragg and William Lawrence Bragg, whoshared the Nobel Prize in physics in 1915. It provides a determination of angles where

24 CHAPTER B

there is constructive reinforcement in X-ray diffraction. The relation between thewavelength, diffraction angle, and material atomic structure is described by Bragg’slaw. If we assume a cubic crystal structure, then the formation of a diffractionpattern is given by the conditions that form high-intensity diffraction peaks basedon the spacing between parallel atomic planes. This is given as,

l ¼ 2dhkl sin u

where l is the X-ray wavelength, dhkl is the interplanar spacing, and u is the diffractionangle. The Miller indices are given by (h, k, l ) for a cubic system and determine thedistance between parallel planes. In this case, parallel planes with the (h, k, l ) indiceshave a separation distance dhkl that is calculated from the lattice constant (also knownas the lattice parameter) a as follows:

dhkl ¼affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

h2 þ k2 þ l2p

a ¼ cubic crystal lattice constant, m (convenient units: nm or A)

dhkl ¼ spacing between planes with Miller indices (h, k, l ), m (convenientunits: nm or A)

h, k, l ¼ Miller indices for lattice plane, dimensionless

l ¼ X-ray wavelength, m (convenient units: nm or A)

u ¼ diffraction angle, rad (convenient units: degree).

BRAZILIAN TEST (Frocht 1947)

The Brazilian test is a means for estimating the strength of a powder compact using aflat, circular disk loaded on the outer diameter until it fractures. As shown inFigure B5, the disk is loaded between two parallel platens and compressed untilrupture. This allows calculation of the breaking strength sB based on the peakbreaking load F, disk thickness t, and disk diameter d as follows:

sB ¼2F

p td

The samples tend to be 5- to 7-mm thick and the diameter tends to be two to fourtimes larger than the thickness (nominally 25 mm).


d ¼ disk diameter, m (convenient units: mm)

t ¼ disk thickness, m (convenient units: mm)

sB ¼ breaking strength, Pa (convenient units: MPa).

BRAZILIAN TEST 25

BREAKAGE MODEL (Berestycki 2003)

The random fracture events that occur during atomization, milling, grinding, emulsi-fication, or fracture of brittle particles or fibers is described by a cascade function.These fracture cascades lead to progressive reductions in particle size. The ratio offinal to initial size fk (size of k piece divided by initial size) follows a distributionas follows:

fk ¼1n

Xn

i¼k

1i

when there are n pieces. This outcome is illustrated in Figure B6 where the mean andmedian sizes are shown versus the number of fracture cycles. After many repeatedbreakage or fracture events, the size distribution becomes log-normal.

fk ¼ fractional size of k-piece divided by the initial size, dimensionless [0, 1]

i ¼ dummy index for summation, dimensionless

k ¼ number of pieces, dimensionless

n ¼ number of pieces, dimensionless.

[Also see Log-Normal Distribution.]

Figure B5. Illustration of the Brazilian crush test for strength where a thin disk is loaded on itsdiameter and crushed.

26 CHAPTER B

BRINELL HARDNESS

A spherical indenter is impressed into a flat test sample and the resistance to indenta-tion provides a measure of hardness. A hard material produces a small indentation.Specifically, the Brinell hardness test is based on forcing a 10-mm hardened steelor cemented carbide ball into a flat surface on the test material. After a hold timeof 10 to 30 s (longer for softer materials), the diameter of the indent is measured.From this the Brinell hardness number BHN (also designated HB) is calculatedas follows:

BHN ¼ HB ¼ 2F

pD2 1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� (d=D)2

ph i

where F is the test load, D is the ball diameter, and d is the impression diameter. In thetypical test the test force F is 3000 kgf (29,000 N), the ball diameter D is 10 mm, andthe mean diameter of the impression on the material surface d is measured in mm. Onthis basis, a value of 245 BHN is about the same as 100 HRB (Rockwell B) or 24HRC (Rockwell C) or 250 VHN (Vickers hardness number).

BHN ¼ HB ¼ Brinell hardness number, kgf/mm2 (or Pa)

D ¼ ball diameter, mm

F ¼ test load, kgf (or N)

HB ¼ Brinell hardness number, kgf/mm2 (or Pa)

Figure B6. Plot of the mean and median particle sizes versus the number of breakage eventsfor random fracture.

BRINELL HARDNESS 27

HRB ¼ Rockwell B hardness number, dimensionless

HRC ¼ Rockwell C hardness number, dimensionless

VHN ¼ Vickers hardness number, kgf/mm2 (convenient units of GPa)

d ¼ impression diameter, mm.

BRITTLE MATERIAL STRENGTH DISTRIBUTION

See Weibull Distribution.

BROADENING

Small grains will broaden the higher-angle X-ray diffraction peaks since the normalcancellation of offsetting wave patterns is less complete within small crystals. Thus,peak broadening in X-ray diffraction is used to estimate grain size or particle size,assuming no internal boundaries or strain in the crystals. The determination usesBT as the total measured broadening for the test powder; thus, the particle-sizecontribution to broadening B is calculated from the total using the difference ofthe squares,

B2 ¼ B2T � B2

S

where BS is the peak broadening for the standard at a similar diffraction angle u. Asillustrated in Figure B7, the peak broadening is measured at half the distance betweenthe maximum and the background intensities. Usually the broadening is used toestimate a typical grain size, but the reader should recognize that the techniqueonly produces an average size. The Scherrer formula is employed to make theconversion from broadening to crystal size. It gives the crystal or particle size Dbased on the diffraction angle u and X-ray wavelength l as follows:

D ¼ 0:9lB cos u

The accuracy increases for larger diffraction angles, corresponding to high (h, k, l )index planes.

B ¼ broadening used to measure crystal size, rad

BS ¼ broadening for a standard, rad

BT ¼ total measured broadening, rad

D ¼ crystal size or particle size, m (convenient units: nm or mm)

h, k, l ¼ Miller indices for lattice plane, dimensionless

l ¼ X-ray wavelength, m (convenient units: nm or A)

u ¼ Bragg’s law diffraction angle, rad (convenient units: degree).

28 CHAPTER B

BROWNIAN MOTION (Einstein 1956)

When small particles are dispersed in a fluid, unbalanced molecular impacts inducea random motion or jitter to the particle. The mean displacement and time betweencontacts is determined by the particle concentration and diffusional translationvelocity. The mean time per mol tB between contacts is estimated as follows:

tB ¼3pDhl2

2RT

where D is the effective size of the particle (which might be larger than the true size),h is the fluid viscosity, l is the mean separation distance between particles (edge-to-edge distance), R is the universal gas constant, and T is the absolute temperature.

D ¼ particle size, m (convenient units: nm)



tB ¼ mean time per mol between particle contacts, s/mol


l ¼ mean particle separation distance, m (convenient units: nm).

[Also see Stokes–Einstein Equation.]

Figure B7. X-ray diffraction peak broadening and the extraction of the half-heightbroadening, which is then used to estimate the crystal or grain or particle size.

BROWNIAN MOTION 29

BUBBLE POINT

See Washburn Equation.

BULK TRANSPORT SINTERING

See Sintering Shrinkage and Surface-area Reduction Kinetics.

30 CHAPTER B

C

CANTILEVER-BEAM TEST

See Bending-beam Viscosity.

CAPILLARITY

Capillary action, also known as capillarity, is the ability of a porous substance to drawa liquid into it. The standard reference is to a circular tube, but capillary action isreadily seen by the wicking action of a porous paper. Capillary action occurs whenthe intermolecular forces between the liquid and a solid are stronger than the cohesiveintermolecular forces inside the liquid. As a consequence, a concave meniscus formswhere the liquid is touching a vertical surface. This same effect causes porousmaterials to soak up liquids. For a circular tube of diameter dP, the capillary riseheight h for the liquid column is given by

h ¼ 4gLV cos u

rgdP

where gLV is the surface energy of the liquid–vapor interface, u is the contact angle, ris the density of liquid, and g is the acceleration due to gravity. This is essentially arearrangement of the Washburn equation, where the pressure head rgh is substitutedfor the pressure; this is the same pressure head used to quantify atmospheric pressureusing a mercury column.

dP ¼ diameter of tube, m (convenient units: mm)

g ¼ acceleration due to gravity, 9.81 m/s2

h ¼ the height of a liquid column, m (convenient units: mm)

g ¼ the liquid-air surface energy, J/m2

u ¼ contact angle, rad (convenient units: degree)

r ¼ density of liquid, kg/m3 (convenient units: g/cm3).


31

CAPILLARITY-INDUCED SINTERING

See Surface Curvature–Driven Mass Flow in Sintering.

CAPILLARY PRESSURE DURING LIQUID-PHASE SINTERING

See Mean Capillary Pressure.

CAPILLARY RISE


CAPILLARY STRESS

See Laplace Equation.

CASE CARBURIZATION

Like most diffusion problems, the penetration of interstitial carbon into the surface ofa steel component depends on the random walk diffusion process. Accordingly, for agiven hardness or carbon level, the depth of carburization X depends on the squareroot of hold time t for a given situation,

X ¼ Kffiffitp

The factor K is a rate constant that depends on the temperature (Arrhenius temperaturedependence) and material. If the steel is not fully dense, then a short-circuit diffusionprocess is possible where the carburizing gas is able to permeate into the structure viathe open porosity. Thus, when there is more than 5% porosity, the factor K furtherincludes a porosity term that increases the penetration rate, since vapor permeationin open pores is much faster than interstitial diffusion. By general convention, thecase depth is measured by the distance into the compact that achieves a hardnessof at least 50 HRC (Rockwell C hardness number).

K ¼ temperature-dependent rate constant, m/s1/2

X ¼ penetration depth, m (convenient units: mm or mm)

t ¼ isothermal hold time, s.

CASSON MODEL (Agote et al. 2001)

To describe the yield stress associated with the onset of viscous flow in a powder–polymer feedstock, a modified rheology model is applied that goes beyond the

32 CHAPTER C

Newtonian and Bingham approaches. In the Casson model, the shear stress t dependson the yield stress tY and shear strain rate dg/dt as follows:

t ¼ (tY )1=2 þ Cdg

dt

� �1=2 !2

where C is a system specific constant.

C ¼ feedstock constant, Pa . s

dg/dt ¼ shear strain rate, 1/s

t ¼ shear stress, Pa (convenient units: MPa)

tY ¼ feedstock yield stress, Pa.

CEMENTED-CARBIDE HARDNESS (Luyckx 2000)

The prediction of hardness for cemented carbides HCC involves both composition andmicrostructure terms, as lumped into an empirical expression of the following form:

HCC ¼ CWCHWCV þ HCo(1� CWCV)

where CWC is the contiguity of the carbide phase. This form gives the compositehardness as a function of the carbide hardness HWC and the cobalt hardness HCo,with V being the volume fraction of the WC phase. Both the carbide and cobalt hard-ness depend on the microstructure. For the tungsten carbide compound there is aninverse square root of the grain-size G effect,

HWC ¼ HO þKffiffiffiffiGp

This is often termed the Hall–Petch dependence. For the cobalt phase it has a hard-ness variation with the ligament size between grains measured in terms of the grainseparation l effect,

HCo ¼ HM þLffiffiffilp

CWC ¼ carbide contiguity, dimensionless

G ¼ carbide grain size, m (convenient units: mm)

HCC ¼ hardness of cemented carbide, Pa (convenient units: VHN (Vickers hard-ness number))

HCo ¼ cobalt hardness, Pa (convenient units: VHN)

HM ¼ inherent cobalt matrix hardness, Pa (convenient units: VHN)

CEMENTED-CARBIDE HARDNESS 33

HO ¼ inherent large-grain carbide hardness, Pa (convenient units: VHN)

HWC ¼ carbide hardness, Pa (convenient units: VHN)

K ¼ grain-size parameter, Pa . m1/2

L ¼ grain-separation parameter, Pa . m1/2

V ¼ carbide volume fraction, dimensionless

l ¼ grain separation or mean free path, m (convenient units: mm).

CENTRIFUGAL ATOMIZATION DROPLET SIZE (Jones 1960)

Centrifugal atomization relies on a liquid and a rotating disk to generate droplets thatsolidify into spherical particles. At high rotational speeds the droplets emerge from athin sheet of liquid thrown off the disk. The disintegration process has a few keyvariables, leading to a dimensionless cluster K that is a constant for a given atomizer,as follows:

K ¼ Dvdr

gLV

� �1=2

where D is the droplet diameter, which is a precursor to the solidified particle size, dis the rotating-disk diameter, r is the melt density, v is the disk angular rotation rate,and gLV is the melt liquid–vapor surface energy. The K parameter depends on thematerial being atomized.

D ¼ droplet diameter, m (convenient units: mm)

K ¼ atomization parameter, dimensionless

d ¼ rotating-disk diameter, m (convenient units: cm)


r ¼ melt density, kg/m3 (convenient units: g/cm3)

v ¼ disk rotation rate, 1/s (convenient units: revolutions per minute (rpm).

CENTRIFUGAL ATOMIZATION PARTICLE SIZE

The energy delivered to a melt during centrifugal atomization largely dictates the finalparticle size; more energy delivered to the melt gives a smaller median particle size.In centrifugal atomization the shear on the melt increases as the spinning rate ofthe atomizer increases, giving smaller particles. Thus, a relation is possiblebetween the median particle size and the centrifugal force, assuming a balancewith the surface tension force of the liquid on the rim of the rotating device. Thisrelation is expressed as the median particle size D50 with a functional dependence

34 CHAPTER C

on the operating parameters as follows:

D50 ¼A

v

� � ffiffiffiffiffiffiffiffiffigLV

rMd

r

where A is a process constant that depends on factors such as the melt friction with theatomizer, melt viscosity, and related processing parameters; v is the angular velocity;gLV is the liquid–vapor surface energy for the melt, rM is the density of the melt, andd is the diameter of the spinning disk or electrode. Besides the median particlesize, various efforts have been made to predict the particle-size distribution, butlargely the spread in the particle size is simply related to the median size. Thespread is analogous to the standard deviation and generally decreases as themedian size become smaller.

A ¼ process constant, dimensionless

D50 ¼ median particle size, m (convenient units: mm)

d ¼ spinning-disk diameter, m (convenient units: cm or mm)


rM ¼ melt density, kg/m3 (convenient units: g/cm3)

v ¼ disk revolution rate, 1/s (convenient units: revolutions per minute orRPM).

CHARLES EQUATION FOR MILLING (Herbst et al. 2003)

This is an empirical relation between the milling conditions in jar or ball milling andthe median particle size. It assumes that as more milling time or more milling energyis added to the process, the consequence is a smaller particle size. The relative energyW required to mill a brittle powder starting from an initial particle size of DI to a finalparticle size of DF is estimated by this empirical equation,

W ¼ g1

DaF

� 1Da

I

� �

with g being a constant that depends on the material, balls, mill design, and mill-operation parameters. Variants on this model have been proposed by Rittinger,Kick, and Bond, and all give an exponent a between 0.5 and 2. Data on severalmaterials show a typical value for the exponent a is near 2.

DF ¼ final particle size, m (convenient units: mm)

DI ¼ initial particle size, m (convenient units: mm)

W ¼ energy required to change the particle size from DI to DF, J

a ¼ milling exponent, dimensionless

g ¼ empirical constant, units depend on a, between J . m and J . m2.

CHARLES EQUATION FOR MILLING 35

CHEMICALLY ACTIVATED SINTERING

See Activated Sintering, Early-stage Shrinkage.

CLOSED-PORE PRESSURE

See Spherical-pore pressure.

CLOSED POROSITY

See Open-pore Content.

COAGULATION TIME (Kruis et al. 1993)

Very small particles will sinter bond and coagulate or fuse into larger agglomerates.With nanoscale particles the coagulation event occurs during synthesis, leading torapid size growth when the particles come into contact. Based on the classic neck-growth model for sintering, the neck size X divided by the particle size D dependson a cluster of material and temperature terms B (Arrhenius temperature dependence),

X

D

� �n

¼ Bt

Dm

where t is the sintering time (assumed isothermal). For complete sintering to form anagglomerate of small particles, it is assumed the neck size reaches the particle size,X ¼ D. Accordingly, regrouping gives a characteristic sintering time t for nano-particle coagulation as follows:

t ¼ Dm

B

The exponent m is known as the Herring scaling law exponent, and for mostnanoparticles is near 4, indicating a process dominated by surface diffusion orgrain-boundary diffusion.

B ¼ collection of material, mechanism, and temperature terms, m4/s

D ¼ particle size, m (convenient units: mm or nm)

X ¼ neck size, m (convenient units: mm or nm)

m ¼ mechanism dependent exponent, typically near 4, dimensionless

n ¼ mechanism dependent exponent, typically near 6 or 7, dimensionless

36 CHAPTER C


t ¼ characteristic sintering time, s.

[Also see Herring Scaling Law and Kuczynski Neck-growth Model.]

COALESCENCE

See Coagulation Time.

COALESCENCE-INDUCED MELTING OF NANOSCALEPARTICLES (Hendy 2005)

Nanoscale particles will undergo collisions that lead to coalescence and with thereduction in surface area there is a heat release as the surface energy is reduced.Assume the particles are of diameters D1 and D2, with D2 being larger. The tempera-ture increase DT for the coalesced mass is estimated as follows:

DT ¼ 6gSV

D2rCP

1þ z2½ � � 1þ z3½ �2=3

1þ z3

where the parameter z ¼ D1/D2 is the particle-size ratio. It is possible that thesurface-energy release when the clusters coalesce will induce heating abovethe melting point, especially considering that small particles exhibit melting temp-erature reduction that follows the general from,

TM ¼ TB 1� L

D

� �

Here TM is the melting temperature of the particle of size D and TB is the bulkmelting temperature. The parameter L depends on the material, and is approximately1 nm for lead.

CP ¼ solid heat capacity, J/(kg . K)

D ¼ particle size, m (convenient units: nm)

D1 ¼ smaller particle size, m (convenient units: nm)

D2 ¼ larger particle size, m (convenient units: nm)

L ¼ material scaling constant, m

TB ¼ bulk melting temperature, K

TM ¼ melting temperature of the nanoscale particle, K

z ¼ particle-size ratio, dimensionless

DT ¼ temperature increase due to coalescence, K

COALESCENCE-INDUCED MELTING OF NANOSCALE PARTICLES 37

gSV ¼ solid–vapor surface energy, J/m2

r ¼ solid density, kg/m3 (convenient units: g/cm3).

COALESCENCE OF LIQUID DROPLETS

See Liquid-droplet Coalescence Time.

COALESCENCE OF NANOSCALE PARTICLES

See Nanoparticle Agglomeration.

COBLE CREEP (Coble 1963)

Named for Robert Coble, this is also known as grain-boundary diffusion–controlledcreep. Grains change shape, elongating in the tensile orientation and shortening in thecompression orientation to accommodate deformation from the applied stress. Therate of deformation, when controlled by diffusion along the grain boundaries, istermed Coble creep. In grain-boundary diffusion–controlled creep the incrementallength change d(DL) per unit time dt is divided by the initial length L0 to give thecreep rate as follows:

1L0

d(DL)dt¼ 48dDBVPE

RTG3

where T is the absolute temperature, R is the gas constant, V is the atomic volume, Gis the grain size, PE is the effective pressure or stress, d is the grain boundary width(about five atoms wide), and DB is the boundary diffusivity. If the body is under com-pression, such as in hot isostatic pressing, then it densifies over time, and undertension, it lengthens in the loading direction over time. A key difference in powdercompacts versus bulk material is the role of pores. For a porous body undergoing den-sification by hot pressing, hot isostatic pressing, or another pressure-assisted process,the stress at the particle contacts is much larger than the mean or applied stress. Thisamplification is particularly large at high porosities and is treated via the effectivepressure concept.

DB ¼ grain-boundary diffusion coefficient, m2/s


L0 ¼ initial body length, m (convenient units: mm)

PE ¼ effective pressure or stress, Pa (convenient units: MPa)


38 CHAPTER C


t ¼ time, s

DL ¼ length change, m (convenient units: mm)

DL/L0 ¼ shrinkage, dimensionless


d ¼ grain boundary width, m (convenient units: nm).

[Also see Effective Pressure.]

COEFFICIENT OF THERMAL EXPANSION

See Thermal Expansion Coefficient.

COEFFICIENT OF VARIATION

This is a nondimensional parameter defined as the standard deviation divided by themean value. It might be expressed as a fraction or percentage, giving a measure ofuniformity or dispersion of an attribute. The coefficient of variation CV allows nor-malization of size variation date,

CV ¼s

XM

where s is the standard deviation, and XM is the mean value. This parameter is mul-tiplied by 100 to express the value as a percent. Often in statistical tests a coefficientof variation under 0.05 or 5% is required to accept the relation as being significant.Otherwise, the sampling needs to be extended to increase the accuracy of the results.

CV ¼ coefficient of variation, dimensionless

XM ¼ mean value, same units as standard deviation

s ¼ standard deviation, same units as mean value.

COERCIVITY OF CEMENTED CARBIDES

See Magnetic Coercivity Correlation in Cemented Carbides.

COLD-SPRAY PROCESS

See Spray Deposition.

COLD-SPRAY PROCESS 39

COLLOIDAL PACKING PARTICLE-SIZE DISTRIBUTION

See Andreasen Size Distribution.

COMBINED-STAGE MODEL OF SINTERING (Johnson 1969)

A combined-stage model for sintering that includes the initial, intermediate, and finalstages is possible if there are no phase transformation or abrupt shifts in mechanism—ideally, a pure material sintering by solid-state processes; for example, pure nickel. Byemploying the geometric factors and other assumptions, a general model for grain-boundary and volume diffusion is derived as follows:

1f

df

dt¼ 3gSVV

RT

dBDBGB

G4þ DVGV

G3

� �

where GB and GV are dimensionless geometric factors associated with grain-boundarydiffusion and volume diffusion, respectively. These geometric factors often changeduring sintering, depending on pore–grain morphology, so they need to be readjustedas sintering proceeds. Consequently, the two clusters GB and GV are either extractedfrom microstructural models of sintering or, more typically, determined byexperimentation.






f ¼ density, dimensionless fraction

t ¼ time, s

GB ¼ geometric factor for grain-boundary diffusion, dimensionless

GV ¼ geometric factor for volume-diffusion, dimensionless


dB ¼ grain-boundary width, m (convenient units: nm)


COMMINUTION

See Grinding Time.

COMMINUTION LAW

See Charles Equation for Milling.

40 CHAPTER C

COMPACTION-INDUCED BOND SIZE

See Contact Size as a Function of Density.

COMPACTION-INDUCED NECK SIZE

Pressure will rearrange and deform contacting particles, inducing an incipient neckbetween them. The size of that initial neck depends on both the material propertiesand applied pressure. If no pressure is applied, there is still a small degree ofelastic deformation, leading to a small insipient neck. The compaction-induced defor-mation at the particle contacts produces a circular contact that expands in size withhigher pressures. At high relative pressures compared to the material strength, assum-ing plasticity, the deformed particles will consist entirely of prismatic faces. Prior toformation of prismatic grains, the size of the contact faces can be approximated by acircle of diameter X. The fractional green density fG and contact size are relatedas follows:

X ¼ D 1� fGfA

� �2=3" #1=2

where D is the particle diameter, and fA is the fractional apparent density correspond-ing to X ¼ 0. In uniaxial die compaction, the applied pressure decays with depthbelow the punch. Accordingly, the compaction-induced initial neck size will varywith position in the green body.


X ¼ diameter of contact between pressed powders, m (convenient units: mm)

fA ¼ apparent density, dimensionless fraction [0, 1]

fG ¼ green density, dimensionless fraction [0, 1].

COMPACTION PRESSURE EFFECT ON GREEN DENSITY

See Green-density Dependence on Compaction Pressure.

COMPLEXITY

Complexity is a dimensionless means to assess component design compatibility withvarious production capabilities. The formal complexity definition derives from realiz-ing the need to broadly link cost to engineering design options such as the number oftoleranced features and the allowed dimensional variation on those features. This

COMPLEXITY 41

leads to a definition of complexity c as follows:

c ¼ 0:3Xn

i¼1

log101

CVi

where n is the number of specified dimensions, and CVi is the coefficient of variationcorresponding to the individual toleranced dimensions (standard deviation for thetolerance divided by the mean size, as a fraction). In this form, complexity is anondimensional reflection of the information needed to specify the component andthe difficulty anticipated in its fabrication. This provides a nondimensional meansto assess the interplay between the number of specified dimensions, their relativetolerances, and the implications with respect to cost and ease of production. Forthe given production process and component mass, the concept is that cost increaseslinearly with complexity.

CVi ¼ coefficient of variation for individual dimensions, dimensionless

n ¼ number of specified dimensions, dimensionless

c ¼ complexity, dimensionless.

COMPLEX VISCOSITY (Gadow et al. 2005)

In an oscillating or changing stress condition, simple viscosity relations fail to explaindamping and other time dependent and strain rate–specific behavior. Complexviscosity h� is a means to combine elastic (storage modulus or viscosity h0) andplastic (loss modulus or viscosity h00) behavior. Fundamentally, the mechanicalanalog is a spring–damper system, leading to the following definitions:

h0 ¼ t

gcos d

where t is the shear stress, g is the deformation strain, and d is the phase shift.

h00 ¼ t

gsin d

The complex viscosity comes from determination of the viscous behavior associatedwith both the elastic and plastic portions of the oscillating stress,

h�j j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih0ð Þ2þ h00ð Þ2

q��

The complex viscosity of a powder–binder feedstock, such as is used in powderinjection molding, is strongly influenced by its solid content, the amount and typeof thermoplastic binder, and the process temperature. High solid contents lead tohigh values of complex viscosity, which is related to high injection pressures and

42 CHAPTER C

can promote the formation of a subtle particle concentration or alignment change inthe flow field. The most common flow behavior of powder–binder feedstocks is shearthinning with a flow limit caused by the high solid content.

g ¼ deformation shear strain, m/m or dimensionless

d ¼ phase shift, rad (convenient units: degree)

h� ¼ complex viscosity, Pa . s

h0 ¼ storage viscosity, Pa . s

h00 ¼ loss viscosity, Pa . s

t ¼ shear stress, Pa (convenient units: MPa).

COMPOSITE DENSITY

One common error in mixing powders or powders and polymers is to assume themixture density is simply the rule of mixtures based on weight fractions. Proper cal-culation of the composite density is defined as: the density is the sum of the massesdivided by the sum of the volumes. If the volume ratio of each ingredient is known,then a volumetric rule of mixtures is proper. For example, in mixing powder–polymer feedstocks there is a mixture density rM determined by the theoreticalpowder density rP and binder density rB,

rM ¼ frP þ (1� f)rB

where f is the volumetric solids loading. If the mixture is able to flow, then the solidsloading is below the critical condition and there is an excess of binder, but typicallyno voids. However, if voids exist in the powder or feedstock mixture, then the mixturedensity will be lower, since the voids take up volume but add no mass. When twopowders are mixed or when the formulation is based on weight fractions, then thecomposite density is determined by the inverse rule of mixtures:

1rM¼ W1

r1þW2

r2

where W is the weight fraction and r is the density, and the subscript denotes thepowder, assuming here just two powders. This formula can be expanded as asimple series for more than two powders. Note the sum of the weight fractionsmust equal unity (for example, with two powders W1 þ W2 ¼ 1).

W1 ¼ weight fraction of first powder, kg/kg or dimensionless

W2 ¼ weight fraction of second powder, kg/kg or dimensionless

f ¼ volumetric powder solids loading, dimensionless fraction

r1 ¼ theoretical density of first powder, kg/m3 (convenient units: g/cm3)

r2 ¼ theoretical density of second powder, kg/m3 (convenient units: g/cm3)

rB ¼ theoretical binder density, kg/m3 (convenient units: g/cm3)

COMPOSITE DENSITY 43

rM ¼ mixture density, kg/m3 (convenient units: g/cm3)

rP ¼ theoretical powder density, kg/m3 (convenient units: g/cm3).

COMPOSITE ELASTIC MODULUS (Nakamura andGurland 1980; Green 1998)

A liquid-phase sintered material consists of a two-phase microstructure, one phasebeing liquid at the sintering temperature and the second being solid at the sinteringtemperature. The liquid phase solidifies on cooling to form a two-phase microstruc-ture. Usually there is a different elastic modulus for each phase. Accordingly, thereare varying levels of sophistication in modeling the composite elastic modulusfrom the constituent properties. An upper bound estimate of the composite elasticmodulus EC comes from assuming equivalent strains in the two phases:

EC ¼ E1 f1 þ E2 f2

where E1 is the elastic modulus of the first phase, E2 is the elastic modulus of thesecond phase, and f1 and f2 are the respective volume fractions. This is sometimestermed the rule of mixtures solution, and it tends to overestimate the measuredvalues. Note that this treatment assumes full density, so residual porosity requires afurther modification to account for the elastic modulus decrement due to pores. Ifthe two phases are interwoven, as is typical in liquid-phase sintering, then a lower-bound calculation of the composite elastic modulus EC result in the inverse ruleof mixtures:

1EC¼ f1

E1þ f2

E2

which tends to underestimate the measured values. The Hashin–Shtrikam bound isone such modification, where each dispersed particle is surrounded with an imaginarycoating used to fill space. Other models assume that various microstructure infor-mation is available, namely, the contiguity of the solid phase C (phase 2). This isa measure in the microstructure, as the fraction of the grain perimeter in contactwith a like phase as a trace is made around the grains. For example, assume atypical liquid-phase sintered microstructure at full density, then the compositeelastic modulus is estimated as follows:

EC ¼b2

1 þ b1b2

b1 þ b2(1� b3)where

b1 ¼ E1(1� fC)

b2 ¼ (E2 � E1)( f2 � fC)2=3

b3 ¼ ( f2 � fC)1=3

44 CHAPTER C

and

fC ¼ f2C

An example problem is in the elastic modulus of an extruded aluminum powderwith 40 vol % silicon carbide dispersed in the aluminum, so C ¼ 0. The elasticmodulus of the aluminum is 69 GPa, while the silicon carbide is 450 GPa. Therule-of-mixture predicted composite elastic modulus is 221 GPa, the inverse rule-of-mixture predicted value is 104 GPa, and the measured value is 166 GPa, whilethe calculation based on the interaction model is 154 GPa. This model says that anincreasing contiguity lowers the composite elastic modulus.

C ¼ contiguity of second phase, dimensionless [0, 1]

E1 ¼ elastic modulus of the first phase, Pa (GPa)

E2 ¼ elastic modulus of the second phase, Pa (GPa)

EC ¼ composite elastic modulus, Pa (GPa)

b1 ¼ calculation parameter, Pa (GPa)

b2 ¼ calculation parameter, Pa (GPa)

b3 ¼ calculation parameter, dimensionless

f1 ¼ volume fraction of first phase, dimensionless [0, 1]

f2 ¼ volume fraction of second phase, dimensionless [0, 1].

COMPOSITE THERMAL CONDUCTIVITY

A composite, such as is formed by sintering or hot consolidation of mixed powders,consists of two phases. One phase is connected to itself and the second might beeither connected or isolated. The microstructure has an influence on the thermal con-ductivity, but for first principles the behavior is dominated by the relative amount ofeach phase. Without detailed microstructure data, the upper-bound estimate of thecomposite thermal conductivity kC is derived by a simple rule of mixtures, assumingno pores:

kC ¼ k1 f1 þ k2 f2

where k1 is the conductivity of the first phase, k2 is the conductivity of the secondphase, and f1 and f2 are the respective volume fractions. When one phase is simplythe residual pores, then gas thermal conductivity is substituted. Interfacial disruptions(grain boundaries) and contaminants reduce the thermal conductivity. For mostmaterials the thermal conductivity and electrical conductivity are proportional, sothis formulation is equally valid for electrical conduction. However, when aceramic is part of the composite, the electrical conductivity relation breaks down,

COMPOSITE THERMAL CONDUCTIVITY 45

for example, aluminum nitride, silicon carbide, beryllium oxide, and similar speciesare thermally conductive but not electrically conductive.

f1 ¼ volume fraction of first phase, dimensionless

f2 ¼ volume fraction of second phase, dimensionless

kC ¼ composite thermal conductivity, W/(m . K)

k1 ¼ composite thermal conductivity, W/(m . K)

k2 ¼ composite thermal conductivity, W/(m . K).

COMPOSITE THERMAL EXPANSION COEFFICIENT(Fahmy and Ragai 1970)

Composites are fabricated using powder techniques, and most sintered products havea two-phase microstructure. Liquid-phase sintering is a common means to form suchparticulate composites. The prediction of the thermal expansion coefficient dependson the microstructure. For a simple composite, especially one with one phase dis-persed in the other so there is little long-range interaction, the simple rule of mixturesis a first basis for estimating the composite thermal expansion coefficient aC:

aC ¼ a1 f1 þ a2 f2

where a1 is the thermal expansion coefficient of the first phase, which is present at avolume fraction f1, and a2 is the thermal expansion coefficient of the second phase,which is present at a volume fraction f2. Here we assume no residual pores, so f1 þf2 ¼ 1. However, the differential strains between phases lead to interactions thatmake this form inaccurate, requiring a formulation that includes elastic propertiesof the two phases as follow:

aC ¼ a1 �E2

E1

3(a1 � a2)(1� v1)f22[1� 2v2f1 þ 2f2(1� 2v1)þ (1þ v1)]

where n indicates the Poisson’s ratio, and E indicates the elastic modulus, with thesubscripts denoting the respective phases.

E1 ¼ elastic modulus of the first phase, Pa (GPa)

E2 ¼ elastic modulus of the second phase, Pa (GPa)

f1 ¼ volume fraction of the first phase, m3/m3 or dimensionless

f2 ¼ volume fraction of the second phase, m3/m3 or dimensionless

n1 ¼ Poisson’s ratio of the first phase, dimensionless

n2 ¼ Poisson’s ratio of the second phase, dimensionless

a1 ¼ thermal expansion coefficient of the first phase, 1/K (convenient units:ppm/K or 1026/K)

46 CHAPTER C

a2 ¼ thermal expansion coefficient of the second phase, 1/K (convenientunits: ppm/K or 1026/K)

aC ¼ composite thermal expansion coefficient, 1/K (convenient units: ppm/Kor 1026/K).

COMPRESSION RATIO

The compression ratio applies to uniaxial die pressing. It is defined as the ratio of theheight of loose powder to the height of the compact. For a constant cross-sectioncompact, the compression ratio CR expresses the volume change or density changewith a standardized compaction pressure, say 400 MPa, and can be calculated fromthe density ratio, height ratio, or volume ratio:

CR ¼H

HO¼ VL

VC¼ rG

rA

where H is the pressed compact height, HO is the power fill height prior to compac-tion, VL is the volume of the loose powder, VC is the volume of the compactedpowder, rG is the green density, and rA is the apparent density. For a die with constantcross section, this ratio is simply the fill height divided by the pressed height. Thecompression ratio is used to calculate the fill setting in uniaxial die compaction.

CR ¼ compression ratio, dimensionless

H ¼ compacted powder height, m (convenient units: mm)

HO ¼ loose-powder fill height, m (convenient units: mm)

VC ¼ compacted-powder volume, m3 (convenient units: mm3)

VL ¼ loose-powder fill volume, m3 (convenient units: mm3)

rA ¼ loose-powder apparent density, kg/m3 (convenient units: g/cm3)

rG ¼ pressed green density, kg/m3 (convenient units: g/cm3).

CONDUCTIVE HEAT FLOW (Chung 1983)

The conductive heat flow in or out of a component per unit of surface area isdescribed in terms of the temperature difference DT between the componentand furnace,

Q ¼ �KDT

y

where Q is the conduction, y is the surface separation distance, and K is the thermalconductivity of the gas medium. The negative sign indicates heat flows from the hotto cold regions or opposite to the temperature gradient. During sintering the distance

CONDUCTIVE HEAT FLOW 47

is determined by the separation of the component from the heating elements or heatedmuffle and the gas thermal conductivity is determined by the process atmosphere,with hydrogen and helium being much more conductive versus argon and nitrogen.However, convection tends to dominate lower-temperature heat flow if the atmos-phere is moving and conduction tends to dominate if the atmosphere is stagnant.At high temperatures heat transfer is dominated by radiation and in a vacuum thereis no atmosphere to enable conduction. Determination of the heating rate dependson the component mass, heat capacity, and surface area, assuming no phase trans-formations, such as melting.

K ¼ thermal conductivity of the separation medium, W/(m . K)

Q ¼ conductive heat flow, W/m2

y ¼ surface separation distance, m (convenient units: mm)

DT ¼ temperature difference, K.

[Also see Convective Heat Transfer and Radiant Heating.]

CONDUCTIVITY (Luikov et al. 1968)

For porous metals the electrical and thermal conductivities k depends on the porosityas follows:

k ¼ k01� 1

1þ x12

where k0 is the conductivity of the fully densified material, and 1 is the porosity. In thecase of a ceramic or nonconducting powder, the model is still relatively accurate.Here the coefficient x expresses the sensitivity to pores. This equation lacks internalstructure-dependent parameters, but analysis of several sintered metal powdercompacts, representing a variety of pore sizes and shapes, gives a best-fit value of11 for x. In the low-porosity region, the relative conductivity follows a linearbehavior with porosity 1; thus,

k ¼ k0(1� v1)

where v is between 1 and 2. This second model is most appropriate at porosities lessthan 30% for either electrical or thermal conductivity.

k ¼ conductivity,

for thermal conductivity, units are W/(m . K)

for electrical conductivity, units are S/m

k0 ¼ conductivity for full-density material,

48 CHAPTER C

for thermal conductivity, units are W/(m . K)

for electrical conductivity, units are S/m

x ¼ pore sensitivity coefficient, dimensionless

1 ¼ porosity, dimensionless fraction [0, 1]

v ¼ pore sensitivity coefficient, dimensionless.

[Also see Electrical Conductivity.]

CONNECTIVITY (German 1996)

The connectivity is a microstructure parameter applicable to any two-phase system. Itis often applied to a two-dimensional microstructure such as seen in liquid-phase sin-tering. Connectivity is defined as the average number of grain–grain connections pergrain observed on a random two-dimensional cross section. It is related to the under-lying three-dimensional grain coordination number through the dihedral angle. In atypical sintered microstructure the connectivity per grain Cg from two-dimensionalsectioning relates to the dihedral angle f and the three-dimensional grain coordi-nation number NC as follows:

Cg ¼ 0:68 NC sinf

2

� �

This equation assumes a typical grain-size distribution and a random-sectionplane through the underlying three-dimensional structure. For example, the three-dimensional coordination number is often measured at 6 contacts for a soliddensity near 60 vol % with a dihedral angle of 608. This gives 1.6 contacts pergrain in two-dimensions, in agreement with experimental observations. However,the connectivity is less accurate as the grain density approaches 100%.

Cg ¼ connectivity or two-dimensional contacts per grain,dimensionless

NC ¼ three-dimensional grain coordination number, dimensionless

f ¼ dihedral angle, rad (convenient units: degree).

CONSTITUTIVE EQUATIONS FOR SINTERING

See Macroscopic Sintering Mode, Constitutive Equations.

CONSTRUCTIVE REINFORCEMENT IN X-RAY DIFFRACTION

See Bragg’s Law.

CONSTRUCTIVE REINFORCEMENT IN X-RAY DIFFRACTION 49

CONTACT ANGLE

Also known as the wetting angle, the contact angle is formed at the intersectionof liquid, solid, and vapor phases. When gravity is ignored, the contact angleu is defined by the horizontal equilibrium of surface energies, as illustrated inFigure C1. The general consensus is to measure the contact angle on a surfaceperpendicular to the gravity vector. When gravity is ignored the treatment is basedon what is known as Young’s equation,

gSV ¼ gSL þ gLV cos u

where gSV is the solid–vapor surface energy, gSL is the solid–liquid energy, and gLV

is the liquid–vapor surface energy. Wetting liquids are associated with contact anglesnear zero and nonwetting liquids are associated with contact angles over 908. Duringspreading or retraction of a liquid over a solid surface, the contact is not in equili-brium. Further, various corrections exist for the effect of surface roughness, sincefinely textured solid surfaces will induce wetting even though the contact angle pre-dicts nonwetting.



gSL ¼ solid–liquid surface energy, J/m2


CONTACT PRESSURE

See Effective Pressure.

Figure C1. Contact angle definition based on the sessile drop experiment, where the angle isdefined by the equilibrium vector resolution at the solid-liquid-vapor shape.

50 CHAPTER C

CONTACT SIZE AS A FUNCTION OF DENSITY(Moon and Choi 1985)

Let X represent the diameter of the contact (neck size) between particles of diameterD (assumed spheres). For sintering processes that involve particle sliding dueto shear,

X

D

� �2

¼ 1� fGfS

� �2=3

where fG is the green packing density at the formation of point contacts, and fS isthe sintered density that gives the contact of size X. For hydrostatic compactionwithout shear,

X

D

� �2

¼ 13

fs � fG1� fG

� �

Based on these concepts, the approximate relation between the effective pressure PE

and applied pressure PA is

PE ¼PA(1� fG)

f 2S ( fS � fG)

where fS is the density at contact size X, and fG is the starting fractional density.This relation gives an infinite effective pressure at the beginning of densification.The effective pressure equals the applied pressure when the compact is fullydensified ( fS ¼ 1).


PA ¼ applied pressure, Pa (convenient units: MPa)

PE ¼ effective pressure, Pa (convenient units: MPa)

X ¼ contact diameter, m (convenient units: mm)

fG ¼ packing density, dimensionless fraction [0, 1]

fS ¼ sintered density, dimensionless fraction [0, 1].

CONTACTS PER PARTICLE

See Coordination Number and Density.

CONTACTS PER PARTICLE 51

CONTAINER-SIZE EFFECT ON RANDOM-PACKINGDENSITY (Ayer and Soppet 1965)

A container wall induces local order into an otherwise random packing. This walleffect on packing-density is evident since the packing density increases as thecontainer size increases; there is less relative contact with the container wall. Thelow-density region induced by the wall propagates into the packed structure forabout 10 particle diameters. The overall decrease in fractional random-packingdensity with container size is expressed as a function of the container diameter asa ratio to the particle diameter as follows:

f ¼ f0 � a exp � bDC

D

� �

where f is the actual packing density for powder of particle diameter D in a containerof diameter DC. The parameters f0, a, and b depend on the particle characteristics,such as particle shape. For large monosized spherical particles packed to the denserandom packing condition, the parameter a ¼ 0.216, b ¼ 0.313, and f0 ¼ 0.635.For particles with flat faces, the container size has less effect on packing density.


DC ¼ container diameter, m (convenient units: mm)

a ¼ material parameter, dimensionless

b ¼ material parameter, dimensionless

f ¼ packing density, dimensionless fraction [0, 1]

f0 ¼ packing density for infinite-sized container, dimensionless fraction [0, 1].

CONTIGUITY (German 1996)

Contiguity CSS is that portion of a grain perimeter that is in contact with grains ofsimilar composition. It is measured from two-dimensional quantitative microscopeimages using contact counting on test lines by automated image-analysis devicesor by manual quantitative microscopy. The test lines are randomly overlaid on arandom microstructure cross section. The number of same-grain contacts NS anddifferent-grain contacts NX are counted for the phase of interest. Then the contiguityis determined as follows:

CSS ¼2 NS

2 NS þ NX

The factor of 2 arises because each same-grain contact is only counted once, yet isshared by two grains, and so should be counted twice—once for the left grain andonce for the right grain. Inherently, contiguity depends on the dihedral angle f

52 CHAPTER C

and three-dimensional grain coordination number NC. For a typical grain sizedistribution seen in a sintered compact, the contiguity increases with increasingsolid density or volume fraction of solid VS and dihedral angle f as follows:

CSS ¼ V2S (0:43 sinfþ 0:35 sin2f)

This empirical relation is valid for low solid densities where there is no grain-shapeaccommodation, so it is not accurate at the highest solid–volume fractions seen inmany liquid-phase sintered compositions.

CSS ¼ contiguity, dimensionless

NC ¼ three-dimensional grain coordination number, dimensionless.

NS ¼ number of intersections by test line with same-grain contacts,dimensionless

NX ¼ number of intersections by test line with different-graincontacts, dimensionless

VS ¼ volume fraction of solid, dimensionless


CONTINUUM THEORY OF SINTERING (Olevesky 1998)

Densification during sintering is predicted by a continuum model for the dimensionalchanges in response to the sintering stress and any external stresses. The response isgiven by the constitutive equation:

�sx ¼ AWm�1 w _1crx þ c� 13w

� �_1crx þ _1cry

� �� þ PL

where w and c are normalized shear modulus and bulk viscosity modulus dependingon porosity 1; 1crx and 1cry are components of the shrinkage rate corresponding to themechanism of power-law creep; PL is the effective sintering stress depending onporosity; A and m are power-law creep frequency factor and power-law creepexponent, respectively; and W is the equivalent effective strain rate, which in mostcases can be calculated as follows:

W ¼ 1ffiffiffiffiffiffiffiffiffiffiffi1� 1p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi23w _1crx � _1cry

� �2þc _1crx � _1cry

� �2

r

These equations provide the basis for the calculation of the grain-boundarydiffusion–controlled and dislocation creep–controlled densification kinetics.Constitutive modeling renders densification mechanism maps that reveal the

CONTINUUM THEORY OF SINTERING 53

dominant driving forces for the densification at different initial densities and averagegrain sizes. The developed model framework serves as a basis for the processoptimization.

A ¼ material constant frequency factor, Pa . sm

PL ¼ effective sintering stress, Pa

W ¼ effective strain rate, 1/s

m ¼ power-law creep exponent, dimensionless

w ¼ normalized shear modulus, dimensionless

c ¼ normalized bulk viscosity modulus, dimensionless


1crx ¼ x-direction shrinkage rate, 1/s

1cry ¼ y-direction shrinkage rate, 1/s.

CONTINUUM THEORY FOR FIELD-ACTIVATED SINTERING(Olevsky and Froyen 2006)

Based on the continuum theory of sintering [see Continuum Theory of Sintering],for pressing in a rigid die, characteristic of spark sintering, the deformation occursonly in the x direction, while 1cry ¼ 0. Therefore taking into account the negativesigns of the shrinkage rate and the compressive axial stress, and employing thefollowing relationships for w, c, and PL:

w ¼ f 2

c ¼ 23

f 3

(1� f )

and

PL ¼3gSV

2Gf 2

where gSV is the solid–vapor surface energy, and G is the grain size. These give

_1crx ¼ �3(1� f )

2

� �3=2 3gSV

2Gf 2 � �sx

� 1

Af 5=2

( )1=m

The power-law frequency factor can be written as

A ¼ A0 expQcr

RT

� �

54 CHAPTER C

In this framework, the total shrinkage rate during spark sintering is equal to thesuperposition of the shrinkage rates corresponding to the grain-boundary diffusionand power-law creep mechanisms:

_1x ¼ _1crx þ _1gbx

where 1gbx is the total strain rate due to field assisted sintering [see Field-activatedSintering].

A ¼ power-law creep frequency factor, Pa . sm – 1

A0 ¼ power-law creep frequency factor, Pa . sm – 1


PL ¼ effective sintering stress, Pa (convenient units: MPa)

Qcr ¼ activation energy for power-law creep, J/mol (convenient units: kJ/mol)


m ¼ power-law creep exponent, dimensionless


1x ¼ total shrinkage rate in the x direction, 1/s

1crx ¼ shrinkage rate from power-law creep in the x direction, 1/s

1gbx ¼ total strain rate due to field assisted sintering, 1/s

sx ¼ effective external stress in the x direction, Pa (convenient units: MPa)

w ¼ normalized shear viscosity modulus, dimensionless

c ¼ normalized bulk viscosity modulus, dimensionless.

[Also see Electromigration Contributions to Spark Sintering, Field-activatedSintering, and Micromechanical Model for Powder Compact.]

CONVECTIVE HEAT TRANSFER (Chung 1983)

The transfer of heat by convection depends on the relative velocity of the fluid phasesurrounding the component in a furnace. Under conditions of a stirred or movingatmosphere, convection is an important contributor to heating and cooling. Heattransport by convection varies with the temperature difference between the fluid(atmosphere) and the component. This temperature difference DT controls heat trans-fer as follows:

Q ¼ �hDT

where h is the convective heat-transfer coefficient, which depends on the gas and thegas velocity with respect to the component surface. The negative sign indicates heat

CONVECTIVE HEAT TRANSFER 55

flow if it is from the hot to cold surfaces. Natural convection and forced convectiongive widely differing heat-transfer coefficients, roughly 20 W/(m2 . K) versus100 W/(m2 . K). In processes such as gas atomization, the heat-transfer coefficientcan range up to 1000 W/(m2 . K).

h ¼ convective heat-transfer coefficient, W/(m2 . K)

Q ¼ heat transfer, W/m2

DT ¼ temperature difference, K or 8C.

COOLING RATE IN ATOMIZATION

See Newtonian Cooling Approximation, Gas Atomization Cooling Rate, andSecondary Dendrite Arm Spacing.

COOLING RATE IN MOLDING

In powder injection molding, the cold cavity is usually filled in a split second and thebulk of the molding cycle is dependent on the time to cool the feedstock to a temp-erature where sufficient strength exists for ejection. After mold filling, the rate ofcooling depends on the heat capacity of the feedstock and the heat flow rate Qgiven by,

Q ¼ �KAdT

dx

where K is the thermal conductivity, A is the cross-sectional area, and dT/dx is thetemperature gradient. The minus sign indicates that heat flows from the high tempera-ture feedstock to the low temperature mold. The time for the mixture to harden in themold varies with the square of the thickness.

A ¼ surface area for heat exchange, m2

K ¼ feedstock thermal conductivity, W/(m K)

Q ¼ heat flow rate, W or J/s

T ¼ temperature, K

dT/dx ¼ temperature gradient, K/m

x ¼ distance, m (convenient units: mm).

COOLING TIME IN MOLDING (German and Bose 1997)

In slurry casting or powder injection molding, the equipment cycle time dependsmostly on the cooling phase. Design requirements to minimize cooling times are

56 CHAPTER C

thin walls, high thermal conductivity tooling, and large temperature differencesbetween the feedstock and mold temperature, which is often induced by internalcooling channels in the mold,

tC ¼ �L2

2paln

p

4(TE � TM)(TC � TM)

� �

where tC is the cooling time after filling the die cavity with feedstock at temperatureTC when the ejection temperature is TE and the mold initially is at temperatureTM. The temperatures are related, as TC . TE . TM. The section thickness thatdictates cooling time is represented by L, and a is the thermal diffusivity of thefeedstock, which depends on specific heat CP, thermal conductivity K, and densityr as a ¼ K/(rCP).

CP ¼ feedstock specific heat, J/(kg K)

K ¼ feedstock thermal conductivity, W/(m K)

L ¼ maximum section thickness, m (convenient units: mm)

TC ¼ feedstock temperature on mold filling, K

TE ¼ ejection temperature, K

TM ¼ initial mold temperature, K

tC ¼ cooling time, s

a ¼ feedstock thermal diffusivity, m2/s

r ¼ feedstock density, kg/m3 (convenient units: g/cm3).

COORDINATION NUMBER AND DENSITY (German 1996)

Coordination number refers to the number of touching particles, atoms, or otherobjects in a packing. For ordered packings, such as atoms, the precise points relatingcoordination number and packing density are tabulated for many ordered packinggeometries (for example, a density of 74% corresponds to a coordination numberof 12). For random packings, such as powders in a die cavity, there is an impreciserelation, and the coordination number has a distribution within the structure.Assuming monosized spheres give a relation useful for estimating the coordinationnumber during sintering based on the shrinkage when the structure starts as loose par-ticles without compaction, such as in injection molding,

NC ¼ 7:3þ 43Y

where NC is the number of particle contacts induced by a fractional shrinkage of Y,starting from a density of 60%. Such a relation helps quantify how shrinkageinduces new contacts that further enhance sintering. Similar relations have beengenerated with the neck-size ratio. Another relation between the coordination

COORDINATION NUMBER AND DENSITY 57

number NC and solid density f is based on experimental measurements with mono-sized spheres,

NC ¼ 14� 10:4(1� f )2=5

where the powder starts sintering with approximately 7 contacts per particle andreaches 14 at full density. Compaction also increases the density, decreases the par-ticle separation distance, and increases the coordination number.

NC ¼ particle coordination number, dimensionless number per grain

Y ¼ shrinkage, dimensionless fraction

f ¼ density, dimensionless fraction [0, 1].

COORDINATION NUMBER AND GRAIN SIZE

See Grain-size Affect on Coordination Number.

COORDINATION NUMBER FOR ORDEREDPACKINGS (Gray 1968)

Various relations have been presented for the coordination number variation withpacking density for monosized spheres in an ordered packing. These packingsrange from a diamond structure, which has a coordination of 4 and fractionalpacking density of 0.3401, to a face-centered cubic structure, with a coordinationof 12 and fractional packing density of 0.7405. Between these two extremes thereis some ambiguity since, for example, there are two structures that have a coordinationof 8—orthorhombic with a fractional density of 0.6046 and body-centered cubic witha fractional density of 0.6802. A simple approximation for the coordination numberNC based on the fractional packing density f is as follows:

NC ¼p

1� f

This relation is of most value because of its simplicity. However, a better fit to themonosized sphere packing behavior is given as follows;

NC ¼ 2 exp(2:4f )

There are several similar relations, but these two provide the most accurate yetsimple fits.

NC ¼ coordination number, dimensionless

f ¼ fractional density, dimensionless [0, 1].

58 CHAPTER C

COORDINATION NUMBER FROM CONNECTIVITY

Usually, the three-dimensional coordination number is difficult to measure in apowder body, so it is usually estimated from the two-dimensional connectivity Cg

if the dihedral angle f is known,

NC ¼32

Cg

sin(f=2)

The connectivity is measured by preparing a two-dimensional microstructuresection and counting the number of contacting grains of the same phase for eachgrain. Observations over a range of conditions suggest the three-dimensional particlecoordination number NC is a function of pressure P as follows:

NC ¼ NC0 þ kP

where the coefficient k is estimated from experiments. As one example of the beha-vior, liquid-phase sintered compacts have a different coordination number from thetop to the bottom.

Cg ¼ connectivity, dimensionless

NC ¼ particle coordination number, dimensionless

NC0 ¼ loose-packing coordination number, dimensionless

P ¼ pressure, Pa (convenient units: MPa)

k ¼ empirical coefficient, 1/Pa

f ¼ dihedral angle, rad (convenient unit: degree).

COORDINATION NUMBER IN LIQUID-PHASE SINTERING(German 1996)

In liquid-phase sintering the solid–solid contacts between grains become stabilizedby a nonzero dihedral angle, leading to a rigid solid skeleton. By careful serial sec-tioning it is established that the mean number of contacts per grain exceeds two,corresponding to a continuous chainlike structure. Generally a grain coordinationof at least four is required to hold a particulate system rigid in a gravitational field(the grains will settle until the mean three-dimensional coordination reaches thislevel). The coordination number NC is related to the volume fraction of solid VS

(must differentiate from density, because pores might be present) and the dihedralangle f by an empirical relation,

VS ¼ �0:83þ 0:81 NC � 0:056 N2C þ 0:0018 N3

C � 0:36Aþ 0:008A2

COORDINATION NUMBER IN LIQUID-PHASE SINTERING 59

where the parameter A ¼ NC cos(f/2). Note the polynomial is an empirical fit to abroad range of data.

A ¼ calculation parameter, dimensionless


VS ¼ volume fraction of solid, dimensionless [0, 1]

f ¼ dihedral angle, rad (convenient unit: degree).

COSTING AND PRICE ESTIMATION (German 2005)

A part price estimate can be generated in the following five steps:

† Cost of raw material CM

CM ¼ PpM

† Cost of processing CP

CP ¼ MC0Fc

c0

� �M

Mi

� �0:6

† Cost of secondary operations CS

CS ¼ FSSC

† Cost of tooling and engineering CE

CE ¼cC1 NM

T B=B0ð Þ0:1þEC

B

† Additional costs for process yield, administrative expenses, profit, and riskaversion,

P ¼ 1þ G

Y(1� p� r)(CM þ CP þ CS þ CE)

where Pp is the powder price per unit mass; M is the component mass; C0 is thebase process conversion cost (approximately $4/kg); c is the part complexity;c0 is the characteristic complexity for each process; Mi is the optimal mass for atechnology; F is the cost scaling-factor characteristic of each technology; SC isthe secondary cost per unit time, mass, area or steps for operations such aselectroplating, machining, or polishing; FS is the secondary factor for a

60 CHAPTER C

component (mass, area, time, or number of steps); C1 is the cost of a singlecavity tool set; NT is the number of tool sets or cavities; M is a scaling factorfor extra cavities (typically near 0.75); B is the order or batch size; B0 is theoptimal batch size; EC is the engineering charge (ranges from $200 forsimple projects to $100,000 for large efforts involving complex tooling, auto-mation, and exhaustive qualification trials); P is the estimated selling price; Gis the general and administrative cost factor (usually from 0.08 to 0.20), Y isthe process yield (fraction of starting material successfully shipped, usuallynear 0.95 to 0.98 for metals, but is typically as small as 0.8 for ceramics); pis the fractional profit (and other marginal factors such as interest expenses,usually from 0.04 to 0.20); and r is the risk factor or safety margin (usuallyfrom 0.00 to 0.10).

In this approach any currency can be substituted into the calculations, $ used forillustration.

B ¼ batch size, number

B0 ¼ optimal batch size, number

C0 ¼ process conversion cost, $/kg

C1 ¼ cost of a single cavities tool set, $

CE ¼ cost of tooling and engineering, $/part

CM ¼ cost of raw material, $/kg

CP ¼ cost of processing, $/part

CS ¼ cost of secondary operations, $/part

EC ¼ engineering charge, $

F ¼ scaling factor, dimensionless

FS ¼ secondary factor for a part, based on mass, area, time, or number of steps

G ¼ general and administrative cost factor, dimensionless fraction

M ¼ component mass, kg

M ¼ tool or cavity scaling factor, dimensionless

Mi ¼ optimal mass, kg

NT ¼ number of tool sets or cavities, dimensionless

P ¼ price, $/part

Pp ¼ powder price, $/kg

SC ¼ secondary cost per unit time, mass, area, or steps, $/part

Y ¼ process yield, dimensionless fraction

p ¼ fractional profit, dimensionless

r ¼ risk factor, dimensionless fraction

c ¼ part complexity, dimensionless

c0 ¼ characteristic complexity for process, dimensionless.

COSTING AND PRICE ESTIMATION 61

COULOMB’S LAW FOR PLASTIC YIELDING(Adams and Briscoe 1994)

In 1773, Coulomb provided the criterion for the flow or strength of particulatematerials based on early experiments with masonry samples under compression.The failure curve was defined in terms of a Mohr’s circle of shear t and normal stres-ses s, as illustrated in Figure C2. For a stress state defined by these two factors, theMohr’s circle gives the major and minor principal stresses and s1 and s2. The criticalshear stress on the failure plane is defined as the sum of the cohesive strengthparameter c (sometimes called the stickiness, it corresponds to the strength at zeronormal force) as follows:

t ¼ cþ s tanf

where tan f is the corresponding friction coefficient and is sometimes simply givenas the coefficient of friction, but here it is captured by the angle f. The critical statecorresponds to the yield loci from the tangential to the Mohr’s circle, so the failureline combines the mixture of normal and shear stresses.

c ¼ cohesive strength parameter, Pa (MPa)

tan f ¼ friction coefficient, dimensionless

s ¼ normal stress, Pa (MPa)

s1 and s2 ¼ major and minor principal stresses, Pa (MPa)

t ¼ shear stress, Pa (MPa)

f ¼ failure plane angle, rad (degree).

Figure C2. A Mohr circle construction to illustrate the Coulomb stress state during compac-tion based on the major and minor stresses and the corresponding definition of the cohesionshear strength and friction terms for a powder mass.

62 CHAPTER C

COURTNEY MODEL FOR EARLY-STAGE NECK GROWTH INLIQUID-PHASE SINTERING (Courtney 1977a)

Thomas Courtney suggested that neck growth in liquid-phase sintering, prior toattaining the limiting neck size set by the dihedral angle, was by solution rep-recipitation through the liquid. Accordingly, the predicted neck size versus time isas follows:

X

G

� �n

¼ gDLCgLVVt

G3RT

where X is the neck size, G is the grain size, DL is the diffusion rate of the dissolvedsolid in the liquid, C is the solubility of the solid in the liquid, V is the atomic volumeof the solid, t is the isothermal time, R is the universal gas constant, and T is the absol-ute temperature. For short times, n ¼ 5 and g ¼ 60, but at long times, n ¼ 6 and g ¼144. Many of these factors depend on temperature. In most cases, neck growth occursby several simultaneous mechanisms involving coalescence, solid-state diffusion, andsolution reprecipitation, so this model tends to underestimate the overall kinetics.

C ¼ solid solubility in the liquid, m3/m3 (dimensionless)

DL ¼ diffusivity of the solid in the liquid, m2/s


R ¼ universal gas constant, 8.31 J/(mol K)


X ¼ neck size, m (convenient units: mm)

g ¼ geometric coefficient, dimensionless

n ¼ time dependence, dimensionless

t ¼ isothermal time, s

V ¼ solid atomic volume, m3/mol.

CREEP-CONTROLLED DENSIFICATION (Wilkinson andAshby 1978)

Creep-controlled densification gives the densification rate (change in density perunit time), df/dt in terms of the creep strain rate d1/dt (change in strain per unittime) as follows:

df

dt¼ A

f 1� fð Þ

1� 1� fð Þ1=Mh iM

d1

dt

where f is the fractional density, A is a geometric constant, and M is an empiricalfactor that depends on the powder characteristics. For a first pass, the creep strain

CREEP-CONTROLLED DENSIFICATION 63

rate varies with the absolute temperature T and stress s with a phenomenologicalequation as follows:

d1

dt¼ Csm exp � Q

RT

� �

where R is the gas constant, and C and m depend on the actual mass flow events,microstructure, and material.

A ¼ geometric constant, dimensionless

C ¼ material-dependent coefficient, (1/Pa)m

M ¼ empirical exponent, dimensionless


df/dt ¼ densification rate, 1/s

d1/dt ¼ creep strain rate, 1/s



m ¼ empirical exponent, dimensionless

t ¼ time, s

s ¼ stress, Pa (convenient units: MPa).

CRITICAL SOLIDS LOADING

See Solids Loading.

CROSS MODEL

During the flow of a powder–binder system there are wall effects with small channels,since particles tend to migrate away from the container wall. The particle depletionnear a wall depends on the shear strain rate. Accordingly, by testing viscosityover a range of shear strain rates the shear thinning behavior can be extracted togive a mixture viscosity h as a function of shear strain rate dg/dt as follows:

h� h1

h0 � h1

¼ 1

1þ 1C

dg

dt

� �M

where h1 is the limiting high shear rate viscosity (asymptotic value), and h0 is the zeroshear-rate viscosity corresponding to dg/dt ¼ 0. Typically, the exponent M is nearunity for powder–polymer mixtures. The parameter C is the critical shear rate

64 CHAPTER C

corresponding to the onset of shear thinning behavior, and it is often a very low value(on the order of 1025 to 1024 1/s).

C ¼ critical shear rate at onset of shear thinning, 1/s

M ¼ strain-rate sensitivity, dimensionless


t ¼ time, s

g ¼ shear strain, dimensionless


h1 ¼ asymptotic mixture viscosity at high shear rates, Pa . s

h0 ¼ zero shear-rate mixture viscosity, Pa . s.

[Also see Viscosity model for injection-molding feedstock.]

CURVED-SURFACE STRESS

See Neck Curvature Stress.

CYCLONE SEPARATION OF POWDER (Mular 2003)

A cyclone is used to separate particles based on size. Loose powder is fed at highvelocity into a tapered cyclone to induce radial separation of the particles in a rotatinggas flow. The vortex produces a velocity gradient from the core to the outside ofthe device. The tangential fluid velocity vt at a horizontal distance r from the coreis given as,

vt ¼C

rN

where C is a device dependent constant and the exponent N depends on laminar(N ¼ 0.8) or turbulent (N ¼ 0.5) flow. A spherical particle of diameter D that isfully involved in the vortex will experience a centrifugal force FC that allows forsize separation,

FC ¼vt

2pD3

6r(rM � rF)

where (rM 2 rF) is the density difference between the particle and fluid. In a centri-fuge the force is proportional to radial distance, but in a cyclone the force varies inver-sely with radial distance. This gives a higher centrifugal force near the center in acyclone than near the wall. As a consequence, larger particles near the core are

CYCLONE SEPARATION OF POWDER 65

driven outward while small particles near the wall will migrate toward the center,since they experience minor centrifugal force on the wall.

C ¼ device-dependent constant, typical units m3/2/s


FC ¼ centrifugal force, N

N ¼ dimensionless exponent, laminar (N ¼ 0.8) or turbulent (N ¼ 0.5)

r ¼ horizontal or radial distance from the cyclone core, m

vt ¼ tangential fluid velocity, m/s

rF ¼ density of the fluid or gas phase, kg/m3 (convenient units: g/cm3)

rM ¼ theoretical density of the particles, kg/m3 (convenient units: g/cm3).

CYLINDRICAL CRUSH STRENGTH

See Bearing Strength.

66 CHAPTER C

D

DARCY’S LAW (Scheidegger 1960)

There are several common laws in science and engineering that rely on a materialparameter to predict a system’s response to a gradient: Ohms’ law, where currentflow depends on the voltage, with the proportionality being the material’s electricalresistance, is the most famous. In a similar manner, the flow of fluid through a poroussintered material provides an index of the pore structure. This material index is thepermeability coefficient a and is determined using Darcy’s law, expressed here forgaseous flow as follows:

Q ¼ aA

hL

P21 � P2

2

2P2

� �

where Q is the flow rate in m3/s, A is the cross-sectional area of the material, L is thelength, h is the gas viscosity, and P1 and P2 are the upstream and downstream press-ures, respectively. Note that the right-hand parenthetical cluster is the change inpressure (P1 2 P2) times the average pressure (P1 þ P2)/2 normalized to the exitpressure P2. For liquids it is common to only used the (P1 2 P2) term. The flowrate Q is the standardized gas volume (at one atmosphere pressure) per unit time,and this can be converted into a superficial velocity (not the true velocity in thepores) by dividing Q by the cross-sectional area A. The permeability coefficient isan indirect way to characterize the porosity, since it varies with the amount of porosity1 and open-pore surface area S (which excludes closed pores). An additional factor isthe tortuosity t, which is a measure of the actual flow path for the fluid phase as a ratioto the geometric sample thickness L. For sintered materials the relation between theseparameters is given as follows:

a ¼ CS13

t

where C is the proportional constant (near 0.8), and 1 is the porosity. Although this isan empirical relation, it does provide a first estimate of the permeability for most


67

common man-made materials. For sintered materials used in filters the measuredpermeability tends to be a small number in the range of 10212 m2. Since sinteredstructures have closed pores at higher relative densities, this empirical predictionfor the permeability is generally focused on materials with more than 15% porosity.

A ¼ sample cross-sectional area, m2 (convenient units: mm2)

C ¼ proportional constant, dimensionless

L ¼ sample length, m (convenient units: mm)

P1 ¼ upstream pressure, Pa (convenient units: kPa)

P2 ¼ downstream pressure, Pa (convenient units: kPa)

Q ¼ flow rate, m3/s

S ¼ surface area, m2

a ¼ permeability coefficient, m2


h ¼ gas viscosity, Pa . s

t ¼ tortuosity, dimensionless.

DEBINDING

See Polymer Pyrolysis, Solvent Debinding Time, Thermal Debinding Time, VacuumThermal Debinding Time, and Wicking.

DEBINDING MASTER CURVE

See Master Decomposition Curve.

DEBINDING TEMPERATURE (Atre 2002; Van Krevelan 1990)

Thermal pyrolysis of a polymer used as a binder takes place over a narrow tempera-ture range for each polymer. The pyrolysis event is characterized by an onset temp-erature, peak burnout-rate temperature, and a maximum temperature. The details ofdebinding also depend on the material, atmosphere, and other factors, such as thecomponent size. The average decomposition temperature TD for a polymer is esti-mated as follows:

TD ¼P

NiYiPMi

where Ni is the amount of the ith group present in each repeating unit of the polymer,Yi is the contribution of the ith group to the decomposition of the binder, and Mi is the

68 CHAPTER D

molecular weight of the ith group present in the binder. Each common polymer givesan estimated decomposition temperature from the repeating unit using the tabulatedparameters given in Table D1 for various repeating units. As an example, polyethy-lene oxide has a repeating chemical structure of (22CH222CH222O22), and theaverage decomposition temperature is estimated as {2(9.5) þ 8}/0.044 or (twoCH2 groups at 9.5 K . kg/mol and one O group at 8 K . kg/mol, with a molecularweight of 2.12 þ 4.1 þ 1.16 ¼ 44 g/mol or 0.044 kg/mol, because of the twocarbon, four hydrogen, and one oxygen). This gives an estimated decompositiontemperature of 614 K. The soak temperature during thermal debinding is generallyset slightly lower to avoid generation of internal stresses during polymer burnout.Other examples are polyethylene at 680 K, polypropylene at 665 K, polystyrene at630 K, poly (vinyl acetate) at 545 K, poly (methyl methacrylate) at 605 K, andpoly (vinyl alcohol) at 535 K.

Mi ¼ molecular weight of the ith group, kg/mol

Ni ¼ number of the ith group present in the repeating unit, dimensionless

TD ¼ average decomposition temperature, K

Yi ¼ energy contribution for the ith group decomposition, K . kg/mol.

[Also see Polymer Pyrolysis.]

TABLE D1. Common Polymer Group and Associated Energy Contributionsfor Decomposition

Repeating-UnitGroup Name

Repeating-UnitChemical Structure

EnergyContribution,K . Kg/Mol

Ethylene 22CH222 9.5Propylene 22CH(CH3)22 18.5Styrene 22CH(C6H5)22 60Methyl acrylate 22CH(COOCH3)22 56.5Vinyl acetate 22CH(OCOCH3)22 42.5Methyl

methacrylate22C(CH3)(COOCH3)22 37.5

Vinyl fluoride 22CHF22 18Vinyl chloride 22CHCl22 23.5Acrylonitrile 22CH(CN)22 28Vinyl alcohol 22CH(OH)22 14Tetrafluoroethylene 22CF222 38.5Neoprene 22CH55CH22 18Oxide 22O22 8Sulfide 22S22 22Amine 22NH22 16Amide 22CO22NH22 22.5

DEBINDING TEMPERATURE 69

DEBINDING TIME

See Solvent Debinding Time, Thermal Debinding Time, Vacuum Thermal DebindingTime, and Wicking.

DEBINDING BY SOLVENT IMMERSION

See Solvent Debinding Time.

DEBINDING WEIGHT LOSS (German and Bose 1997)

Debinding is the removal of a polymer (binder) from a powder compact, usually bypyrolysis or solvent extraction. All models for debinding predict that the depth oramount of binder removed is proportional to the square root of the debinding time,assuming a homogeneous binder distribution in the component. The cumulativeweight loss is limited to the total binder content. At any point under isothermal con-ditions, the debinding rate, based on weight W and time t, can be expressed as

dW

dt¼ B

2W

which says the rate of debinding weight loss or mass loss is inversely dependent onthe amount of binder remaining. This leads to a parabolic relation between weightloss and time as follows:

W2 ¼ Bt

where B is determined by the porosity, temperature, and other factors, such as thebinder density. However, in most instances debinding is performed in a series ofsteps, often at different temperatures, so a simplified model of the weight removedversus square-root time is not accurate. Here an integral work approach with amaster debinding curve is more satisfactory.

B ¼ experimental constant, kg2/s

W ¼ binder weight loss, kg (convenient units: g)

t ¼ debinding time, s.

(Also see Master Decomposition Curve.)

DELUBRICATION


70 CHAPTER D

DENSIFICATION (Lenel 1980)

Although not directly useful in component design, the concept of densification ishelpful in dealing with sintering cycles. It is most beneficial when comparing sinter-ing cycles under conditions where the green density is not constant. Densification C

is defined as the change in density due to sintering, starting from the green fractionaldensity fG, divided by density change needed to attain a pore-free solid. An alterna-tive definition is the change in porosity divided by the initial porosity. In terms offractional sintered density, fS densification is given as follows:

C ¼ fS � fG1� fG

Since shrinkage can also be linked to these same parameters, it is possible to definedensification based on shrinkage.

fG ¼ fractional green density, dimensionless [0, 1]

fS ¼ fractional sintered density, dimensionless [0, 1]

C ¼ densification, dimensionless fraction [0, 1].

DENSIFICATION IN LIQUID-PHASE SINTERING

See Dissolution-induced Densification.

DENSIFICATION IN SINTERING

See Shrinkage-induced Densification.

DENSIFICATION RATE (Gupta 1971; Kang 2005)

Related to densification, the generalized densification rate df/dt represents the instan-taneous change in solid density with time during sintering. Although there are severalvariants, the generalized first order rate equation relies on how much porosity remainsand a material parameter B that scales with the rate of atomic transport,

df

dt¼ (1� f )Bg

g SV

l¼ 1Bg

g SV

l

In this equation f is the fractional solid density, so 1 ¼ (1 2 f ) is the remainingporosity, B collects material properties such as diffusivity (temperature-dependent)

DENSIFICATION RATE 71

and particle size, while g is a geometric term, gSV is the surface energy, and the para-meter l represents the scale of the microstructure. For example, in the final stageof sintering, l would be the pore diameter and g would equal 4. Because porosityis eliminated, the rate of densification declines to zero. For a typical engineeringmaterial, the solid–vapor surface energy gSV is in the range from 1 to 2 J/m2, andthe microstructure scale is often on the order of 0.1 to 20 mm. Consequently, the mag-nitude of the term ggSV/l is typically in the range from 1 to 20 MPa. This is termedthe sintering stress since it arises from the stress associated with curved surfaces ofthe particles and pores. In some computer simulations of sintering there is noeffort to calculate the sintering stress and it is simply set to a constant value, say 1MPa. The term B couples to this stress to determine the sintering densification rate.In hot pressing, hot isostatic pressing (HIP), or related approaches, an externalpressure is amplified at the particle contacts in the microstructure to supplementthe inherent sintering stress. This amplified pressure is termed the effective pressurePE. At low compact densities the effective pressure is several times higher than theapplied pressure. As a consequence, the densification rate is significantly increasedby the localized contact pressure:

df

dt¼ (1� f )B g

gSV

lþ PE

� �¼ 1B g

gSV

lþ PE

� �

In practice there is often a measurable densification rate gain from an appliedpressure of just 0.1 MPa (one atmosphere pressure), especially when the compact islow in density. However, there is a negative densification effect from gas trapped inthe pores. Once the pores close at approximately 92 to 95% density, the internal gaspressure in the pores continuously increases with densification. Most harmful areinsoluble gases, such as argon. If the gas remains in the pores, then the increasingpressure hinders densification, giving a further modification to the densificationrate equation,

df

dt¼ (1� f )B g

g SV

lþ PE � PP

� �¼ 1B g

g SV

lþ PE � PP

� �

where PP is the gas pressure in the pores. This generic model is a fruitful basis for ana-lyzing most time-dependent pressure-assisted sintering data. Surface energy providesan inherent sintering stress that is assisted by the external pressure, but degraded bytrapped gas. Densification rates then depend on the net stress times the thermally acti-vated rate of diffusional creep, as captured in the material parameter B. High tempera-tures soften the material and increase the diffusion rate, and both factors significantlyaid densification. Temperature is not directly evident in this form, but it plays animportant role because many material properties change with temperature, especiallydiffusion rates. In most cases the temperature dependence follows the Arrhenius

72 CHAPTER D

form, meaning that at typical consolidation conditions a small change in temperaturecan have a very large impact on the product densification.

B ¼ collection of material properties, 1/(Pa . s)


PP ¼ gas pressure in the pores, Pa (convenient units: MPa)


f ¼ fractional solid density, dimensionless [0, 1]

g ¼ geometric term, dimensionless

t ¼ time, s



l ¼ geometric scale of the microstructure, m (convenient units: mm).

[Also see Coble Creep, Effective Pressure, and Nabarro–Herring ControlledPressure-assisted Densification.]

DENSIFICATION RATIO (Blaine et al. 2006)

The densification ratio F is defined as the ratio of density difference between thecurrent density and the initial density as a ratio to total initial porosity 1 (where1 ¼ 1 – f) with a range of f0 , f , 1,

F ;f � f01� f

¼ f � f01

where f is the fractional density after sintering, and f0 is the initial fractionaldensity. The densification ratio is used to linearize data in the master sintering-curve approach to normalization of sintering data. Note that the relation betweendensification C and densification ratio F is 1/C ¼ 1 þ 1/F.

f ¼ fractional sintered density, dimensionless [0, 1]

f0 ¼ fractional green density, dimensionless [0, 1]

F ¼ densification ratio, dimensionless

C ¼ densification, dimensionless

1 ¼ fractional sintered porosity, dimensionless [0, 1].

(Also see Master Sintering Curve.)

DENSIFICATION RATIO 73

DENSITY CALCULATION FROM DILATOMETRY(Park et al. 2006)

A six-step process has been proposed for extracting information from dilatometry datato construct a master sintering curve. These steps are detailed as follows:

Step 1: Engineering strain from dilatometry data, usually taken from constantheating-rate experiments:

1d ¼L� L0

L0

Step 2: Elimination of any nonzero initial values in the dilatometry data:

10d ¼1d � d

1þ d

Step 3: Consideration of the effect of thermal expansion, especially for low shrink-age conditions:

100d ¼ 1d � am f 1=3 T � T0ð Þ

Step 4: Calculation of relative density from engineering strain data, assuming iso-tropic and homogeneous sintering with no mass change:

f ¼ f0

1þ 100d� �3

Step 5: Satisfaction of the basic requirement for a nondecreasing function:

f 0iþ1 ¼ fi if fiþ1 , fif 0iþ1 ¼ fiþ1 otherwise

Step 6: Requirement of maximum value of 1 at full density:

f 00 ¼ f 0 if f 0 , 1

f 00 ¼ 1 otherwise

In this procedure, 1d is the engineering strain or shrinkage from dilatometry data, Lis the instantaneous length of the specimen, L0 is the initial length of the specimen, 10dis the modified strain after nonzero initial-value treatment, d is the nonzero initialvalue from the dilatometry data, 100d is the modified strain after consideration of theeffect of thermal expansion, am is the thermal expansion coefficient of bulk material,f is the relative density, T is the temperature, T0 is the initial temperature of the dila-tometry test (usually room temperature), f0 is the initial relative density of the sample,i and i þ 1 represent the time steps used for data acquisition during the dilatometrytest, f 00 is the modified relative density after the nondecreasing function treatment,

74 CHAPTER D

and f 00 is the modified relative density after maximum relative-density treatment.Through this manipulation of the dilatometry data, the thermal expansion effectsare removed, thus yielding an accurate measure of relative density during sintering.In addition, this manipulation removes the effects of the following circumstanceson dilatometer shrinkage measurements: (1) a nonzero initial value, (2) nondecreas-ing function, and (3) maximum value of 1.0. Taking these factors into accountreduces the error in predicting relative density from experimental data.

L ¼ instantaneous length of specimen, m (convenient units: mm)

L0 ¼ initial length of specimen, m (convenient units: mm)

T ¼ temperature, K

T0 ¼ initial temperature of dilatometry test, K

f ¼ relative density, dimensionless [0, 1]

f0 ¼ initial relative density of specimen, dimensionless [0, 1]

f 0 ¼ relative density after nondecreasing function treatment, dimensionless[0, 1]

f 00 ¼ relative density after maximum density treatment, dimensionless [0, 1]

am ¼ thermal expansion coefficient of bulk material, 1/K

d ¼ nonzero initial value from dilatometry data, dimensionless

1d ¼ engineering strain or shrinkage from dilatometry data, dimensionless

10d ¼ modified strain after nonzero initial-value treatment, dimensionless

100d ¼ modified strain after consideration of thermal expansion, dimensionless.

DENSITY EFFECT ON DUCTILITY

See Sintered Ductility.

DENSITY EFFECT ON SINTERED NECK SIZE

See Neck-size Ratio Dependence on Sintered Density.

DENSITY EFFECT ON STRENGTH

See Sintered Strength.

DEW POINT

Dew point is a temperature at which air (or gas) becomes saturated and begins to con-dense moisture. Historically, a mirror was used to measure the water content in aprocess atmosphere, where quantification of the atmosphere quality was based on

DEW POINT 75

the temperature where moisture condensed to fog the mirror. Thus, the water contentin a process atmosphere is directly related to the dew-point temperature. Unfortunatelythe relation between the dew-point temperature and the volumetric content of watervapor is not simple. However, for historical reasons the dew-point is important todetermining the atmosphere oxidation-reduction potential during heat treatment andsintering. The parametric relationship between the volume percent of water in anatmosphere VH2O and the dew point TD is:

log10 (VH2O) ¼ �0:237þ 3:36 � 10�2TD � 1:74 � 10�4T2D þ 5:05 � 10�7T3

D

Note this empirical fit relies on temperature in Celsius and gives the water content inpercent by volume.

TD ¼ dew-point temperature, 8CVH2O ¼ volume of water, percent (dimensionless).

DIE-WALL FRICTION (Jones 1960)

The friction of the powder against the die wall during uniaxial compaction results in aloss of applied pressure in the powder with distance from the punch. The major impli-cation is a green density gradient in die-compacted powder. To determine the die-wallfriction effect, consider a cylindrical compact of diameter d and height h. As illus-trated in Figure D1, in a thin section of height Dh there is a small top-to-bottom

Figure D1. Calculation of the pressure decay in a powder bed with depth below the punchbased on die wall friction forces.

76 CHAPTER D

pressure difference. If the top pressure on this section is P, then at the bottom of thethin element there is a slightly lower pressure PB. The change in pressure of this thinsection is due to the normal force acting against the die wall, which creates counter-friction. On this thin section the balance of forces along the central axis can beexpressed as follows:

XF ¼ 0 ¼ A(P� PB)þ mFN

where FN is the normal force, m is the coefficient of friction between the powder andthe die wall, and A is the cross-sectional area. The normal force at the die wall is givenin terms of the applied top pressure P with a proportionality factor z; this factor rep-resents the pseudofluid character of a powder (for a liquid, z would be 1, and for asolid, z would be Poisson’s ratio),

FN ¼ pzPdDh

During compaction the die-wall friction force FF is determined by the normal forceacting against the die wall and the coefficient of friction m as,

FF ¼ pmzPdDh

Combining terms gives the pressure difference between the top and bottom of thepowder element DP ¼ P 2 PB as,

DP ¼ �FF

A¼ � 4mzPDh

d

Integration with respect to compact height leads to an expression for the pressure atany position x below the punch as follows:

P(x) ¼ PA exp � 4mzx

d

� �

where PA is the applied pressure at the punch, and x is the distance from the top punchinto the powder bed. Variants on this function are shown in Figure D2, The precedingsolution is for single-action compaction. Since there is friction on all tool members, aradial density gradient is also present, and when core rods and multiple height stepsare present in the compact, then the density gradients must be predicted by finiteelement analysis.

A ¼ compact cross-sectional area, m2 (convenient units: mm2)

F ¼ total force, N (convenient units: kN or MN)

FF ¼ friction force, N (convenient units: kN or MN)

FN ¼ normal force on die wall, N (convenient units: kN or MN)

DIE-WALL FRICTION 77

P ¼ top pressure, Pa (convenient units: MPa)

PA ¼ applied pressure at the punch, Pa (convenient units: MPa)

PB ¼ bottom pressure, Pa (convenient units: MPa)

P(x) ¼ pressure at distance x from top punch, Pa (convenient units: MPa)

d ¼ compact diameter, m (convenient units: mm)

h ¼ compact height, m (convenient units: mm)

x ¼ distance from top punch, m (convenient units: mm)

z ¼ radial pressure ratio to applied pressure, dimensionless

Dh ¼ incremental height, m (convenient units: mm)

DP ¼ pressure increment, Pa (convenient units: MPa)

m ¼ coefficient of friction, dimensionless.

DIFFUSION

See Vacancy Diffusion.

DIFFUSION-CONTROLLED GRAIN GROWTH INLIQUID-PHASE SINTERING

See Grain Growth in Liquid-phase Sintering, Diffusion Control at High SolidContents.

Figure D2. Plots of the pressure decay function for single action die pressing for three variantsof the cluster mzx/d.

78 CHAPTER D

DIFFUSIONAL NECK GROWTH

See Kuczynski Neck-growth Model.

DIFFUSIONAL HOMOGENIZATION IN SINTERING

See Homogenization in Sintering.

DIFFUSIONAL TRANSLATION

See Stokes–Einstein Equation.

DIHEDRAL ANGLE

The angle formed by a grain boundary where it intersects with another solid, pore, orliquid during sintering is described by a thermodynamic balance termed the dihedralangle. As illustrated in Figure D3, the dihedral angle f is determined by a vertical

Figure D3. The dihedral angle is defined based on the groove representing a balance of theinterfacial energies where two grains of one phase intersect with another phase (vapor,liquid or solid) of different composition. In this case emergence of a grain boundary into aliquid phase is represented by the vertical balance between the resolved solid-liquid surfaceenergies and the solid-solid or grain boundary energy.

DIHEDRAL ANGLE 79

surface energy balance. For the case of a grain boundary in contact with a liquidduring liquid-phase sintering, the vector balance gives,

gSS ¼ 2g SL cosf

2

� �

where gSS is the solid–solid interfacial energy (grain-boundary energy), and gSL isthe solid–liquid interfacial energy. Alternatively,

f ¼ 2 arccosgSS

2gSL

� �

In the case of a grain boundary in contact with the free surface, a thermal grooveforms and the dihedral angle f is determined by the solid–vapor surface energygSV. In materials that have been held at high temperature for a prolonged time thedihedral angle f is evident at all surfaces and exposed grain boundaries. Grain-boundary grooving on a free surface is a reflection of the dihedral angle. Since seg-regation changes surface energy, the dihedral angle will exhibit a time variation asdiffusion events deposit segregants to grain boundaries and free surfaces. The dihe-dral angle f is one factor that determines the contiguity, which in turn governs thestrength and ductility for a liquid-phase sintered material.


gSS ¼ solid–solid grain-boundary energy, J/m2



[Also see Fragmentation by Liquid.]

DIHEDRAL ANGLE–LIMITED NECK GROWTH

See Neck Growth Limited by Grain Growth.

DILATANT FLOW MOMENTUM MODEL

The rheology model for monosized spheres assumes that each rigid particle dispersedin a suspension moves at velocity set by the local fluid. Under laminar conditions, theflow of the fluid layer depends on the shear rate and each particle center is assumed toexist in the fluid without aggregation, Brownian motion, or rotational motion. If twoparticles exist in different but adjacent laminar flow layers, their collision is at a rela-tive velocity that is proportional to the shear rate and separation between layers. This

80 CHAPTER D

results in a transfer for momentum. The momentum difference between those twoparticles determines the energy transfer from the fluid layer, where the faster particleresides to the adjacent layer where the slower particle resides. The summation ofmomentum transfer through the interfacial area determines the momentum flux,resulting in a viscosity augmentation. Based on the model, the increased momentumflux t is expressed as

t ¼ krpD7L2 _g 2

where k is the parameter determined by experiment, rp is the particle density, D is theparticle diameter, L is the number density of particles, and _g is the shear rate.


L ¼ number density of particles per unit volume, 1/m3

k ¼ material parameter, m

_g ¼ shear rate, 1/s

rp ¼ particle density, kg/m3 (convenient units: g/cm3)

t ¼ momentum flux, Pa.

DILATANT FLOW VISCOSITY MODEL (Rahaman 1995)

For a powder–binder suspension, dilatant flow is a special rheological condition thattends to occur at high shear strain rates. The mixture viscosity changes with the flowconditions such that the mixture dilates (effectively changes volume) under stress. Indilatant flow, the shear stress t and the shear viscosity h increase with the shear strainrate _g ¼ dg=dt and are characterized as follows:

t ¼ h _g ¼ hdg

dtand

h ¼ K _g m�1 ¼ Kdg

dt

� �m�1

where K is a material viscosity parameter and m is greater than 1 for dilatant or shearthickening flow. Note, for comparison, pure liquids such as water exhibit Newtonianflow (m ¼1 for Newtonian flow) and pure polymers such as polyethylene exhibitsshear thinning flow (m , 1 for shear thinning flow).

K ¼ parameter related to the viscosity, Pa . sm (m is an experimental exponent)

m ¼ strain rate sensitivity exponent, dimensionless

t ¼ time, s

DILATANT FLOW VISCOSITY MODEL 81


_g ¼ dg/dt ¼ shear strain rate, 1/s

h ¼ shear viscosity, Pa . s.

t ¼ shear stress, Pa.

DILUTE SUSPENSION VISCOSITY (Tanner and Walters 1998)

In a suspension of spherical particles dispersed in a fluid at a low concentration, theviscosity increase with the addition of solid particles was treated by Einstein, giving

h ¼ hB(1þ 2:5f)

where h is the mixture viscosity, hB is the pure binder viscosity, and f is the feed-stock solids loading. This model only proves valid for dilute suspensions wherethe particle separation is large.


hB ¼ pure binder viscosity, Pa . s

f ¼ solids loading, dimensionless.

DIMENSIONAL CHANGE

See Sintering Shrinkage.

DIMENSIONAL VARIATION


DIMENSIONAL PRECISION AND GREEN MASS VARIATION

Powder-shaping processes are good at replicating the tool size such that the green sizeoften has a low scatter, in the range of a few micrometers; however, sintered com-ponents show a much larger dimensional variation. A relation between green massand sintered dimensional precision is possible, assuming isotropic sintering shrinkagebehavior. Let the subscript G designate the green condition and the subscript S des-ignate the sintered condition, with L being the mean dimension, DL being the dimen-sional change from green to sintered size, DL/LG being the sintering shrinkage, Mbeing the component mass, V being the component volume, and f being is the

82 CHAPTER D

fractional density. Ignoring the binder and lubricant, and assuming isotropic shrink-age, then the relation between sintering shrinkage DL/LG, fractional green density fG,and fractional sintered density fS is given as follows:

fS ¼fG

(1� (DL=LG))3

This equation can be rearranged to give shrinkage as a function of the fractional greendensity divided by the fractional sintered density,

DL

LG¼ 1� fG

fS

� �1=3

Since DL is LG 2 LS, the sintered size LS is calculated from the green size and densityratio as follows:

LS ¼ LGfGfS

� �1=3

Usually, the tooling and forming steps give close control of the green size, but thesintered size has more scatter. To determine controlling factors, a partial derivativegives

dLS ¼fGfS

� �1=3

dLG þLG

3fGfS

� ��2=3

dfG �LG

3fGfS

� �4=3

dfS

This shows that the variation in the sintered dimension dLS has three direct sources:the green size variation dLG, the green density variation dfG, and the sintered densityvariation dfS. Hard tooling makes the green-size variation small. Good sinteringimplies that dfS can be ignored, since grain growth or other microstructure factorsoften limit sintering densification in practice. Thus, sintered dimensional variationis dominated by the green density scatter. Since density is mass over volume, andgreen volume is controlled by the tooling,

fG ¼MG

VG

so for constant tool volume,

dfG ¼1

VGdMG

showing that the green density variation is directly linked to the green mass variation.The combination of equations provides a link between the green mass variation andthe dimensional precision of the sintered product. It says that to a first approximation

DIMENSIONAL PRECISION AND GREEN MASS VARIATION 83

the sintered dimensional scatter (normalized to the means size) is simply one-third themass variation normalized to the mean mass.



LG ¼ green dimension, m (convenient units: mm)

LS ¼ sintered dimension, m (convenient units: mm)

MG ¼ green mass, kg (convenient units: g)

VG ¼ green volume, m3 (convenient units: mm3)

DL ¼ dimensional change from green to sintered size, m(convenient units: mm)

DL/LG ¼ sintering shrinkage, dimensionless.

DIRECT LASER SINTERING

See Laser Sintering.

DISK CRUSH TEST

See Brazilian Test.

DISLOCATION CLIMB-CONTROLLED PRESSURE-ASSISTEDSINTERING DENSIFICATION (Ramqvist 1966)

Dislocation motion and plastic flow in a powder compact are fundamental to hotpressing, hot isostatic pressing, and other pressure-assisted sintering technologies.When both the stress and temperature are high, then the rate of densificationduring pressure-assisted sintering depends on the rate of dislocation climb. This isoften termed power law creep. The corresponding mathematical form is as follows:

d

dt

DL

L0

� �¼ bCUDV

RT

PE

U

� �n

where the shrinkage rate is given by d(DL/L0)/dt, b is the magnitude of Burger’svector, C a material constant, U the material’s shear modulus at the processing temp-erature, DV is the lattice or volume diffusion coefficient at the processing temperature,R is universal gas constant, T is the absolute temperature, PE is the effective pressureon the compact, and n is an exponent expressing the stress sensitivity. Although thisequation is empirical, it has been successful in explaining experimental hot consoli-dation data for powders under many conditions. However, it tends to be invalid forthe sinter–HIP (hot isostatic pressing) process, since pressure is applied late in the

84 CHAPTER D

consolidation cycle and most of the mobile dislocations have been annealed out of thematerial. A special situation occurs for superplastic flow when the material annealsduring deformation to allow very large deformations at slow strain rates.Superplastic flow is accomplished in pressure-assisted sintering cycles when thestress and creep strain rate are related by n ¼ 2. This is a special condition thatoccurs in a two-phase microstructure with a stable, small (below 1 mm) primarygrain size, such as with high carbon steels and ceramic-ceramic composites.

C ¼ material constant, 1/mol


L ¼ characteristic length, m (convenient units: mm)





U ¼ shear modulus, Pa (convenient units: GPa)

b ¼ Burger’s vector, m (convenient units: nm)

d(DL/L0)/dt ¼ shrinkage rate, 1/s

n ¼ stress-sensitivity exponent, dimensionless


DL/L0 ¼ shrinkage, dimensionless.

DISLOCATION GLIDE IN SINTERING

See Plastic Flow in Sintering.

DISPERSION FORCE

See London Dispersion Force.

DISSOLUTION INDUCED DENSIFICATION (Savitskii et al. 1980;Savitskii 1993)

One option in liquid-phase sintering is to form sufficient liquid during the heatingcycle so that the structure densifies instantly when the melt forms. (Note that thetime for melt formation is comparatively long, since heat must be supplied to formthe liquid and that depends on heat transport though the sintering body, a slowprocess compared to the chemical reactions and particle rearrangement steps.) Theminimum volume fraction of liquid forming additive needed for maximum densifica-tion (effectively zero porosity, 1 ¼ 0) by dissolution events in liquid-phase sintering

DISSOLUTION INDUCED DENSIFICATION 85

is estimated as Cm, where

Cm ¼10(1� CL)

CL � CL10 þ 10

where 10 is the initial or green porosity and CL is the volumetric concentration of solidthat can be dissolved into the liquid.

CL ¼ volumetric concentration of solid dissolved into the liquid, m3/m3

Cm ¼ minimum volume fraction of liquid-forming additive, m3/m3

1 ¼ porosity, dimensionless fraction

10 ¼ initial porosity, dimensionless fraction.

DORN TECHNIQUE (Bacmann and Cizeron 1968)

John Dorn suggested a novel means to extract the activation energy from creepexperiments using a step change in temperature. Since sintering shrinkage behavessimilarly to a creep process, the Dorn technique has been adapted to sintering andthe extraction of activation energies. In this approach, experiments are conductedby dilatometry, with changes in the rate of heating to identify the activation energyfor mass transport. The sintering rate is noted just before and just after the temperaturechange. The apparent process activation energy Q is calculated from the ratio ofshrinkage rate,

Q ¼ RT1T2

T1 � T2ln

n1

n2

� �

where R is the gas constant, T1 and T2 are the two absolute temperatures, and v1 and v2

are the instantaneous sintering rates taken from dilatometry. Note these are termedapparent activation energies, since processes such as evaporation–condensationand surface diffusion might be active, yet they do not contribute to densificationand are not directly measured with the Dorn technique. Sintering usually involvesmultiple mass-transport mechanisms, so the presumption of a single activationenergy is a simplification. Accordingly, the activation energy extracted using theDorn technique is not expected to match any of the diffusion activation energies astabulated in handbooks.



T1 ¼ first temperature, K

T2 ¼ second temperature, K

n1 ¼ sintering rate at first temperature, 1/s

n2 ¼ sintering rate at second temperature, 1/s.

86 CHAPTER D

DRAINAGE

See Wicking.

DROPLET COOLING IN ATOMIZATION

See Newtonian Cooling Approximation.

DUCTILITY VARIATION WITH DENSITY


DUCTILITY VARIATION WITH DENSITY 87

E

EFFECTIVE PRESSURE (Artz et al. 1983)

During pressure-assisted sintering, the local pressure acting at the individual particlecontacts is much higher than the bulk applied pressure, especially when the density islow. A relation between the applied pressure PA and the local or effective pressure PE,measured at the particle contact, is based on the instantaneous fractional density f andthe green fractional density fG as follows:

PE ¼ PA(1� fG)

f 2( f � fG)

Figure E1 plots the ratio of the effective pressure to the applied pressure (PE/PA) toshow the significant pressure amplification possible at low densities, in this case,assuming the green density is 0.60.




fG ¼ fractional green density, dimensionless.

EJECTION STRESS

See Maximum Ejection Stress.

ELASTIC BEHAVIOR

See Hooke’s Law.


89

ELASTIC-DEFORMATION NECK-SIZE RATIO (Zhu andAverback 1996)

When small particles come into contact there is an attractive force that causes insipi-ent neck growth due to localized elastic deformation. For larger particles the size ofthe initial contact is relatively small and is often ignored in sintering calculations,other than to avoid the mathematical complication from an infinite rate of sinteringwhen the neck size is zero. For many calculations the common solution is to setthe initial neck size to a value of 1% of the particle size (X ¼ 0.01D). However,the actual neck size depends on the particle size D, solid–vapor surface energygSV, dihedral angle f, and the shear modulus of the material m,

X

D

� �3

¼ gSV[1� cos (f=2)]2Dm

As the particle size decreases, this contribution to the presintering neck-size ratiobecomes significant. For many cases the assumption of X/D ¼ 0.01 is not a signifi-cant error when used to seed computer simulations. Also, when used to estimate thegreen strength of loose powders, often measured in the kPa range, this neck-size ratiogives a proper order-of-magnitude estimate for strength calculations.



Figure E1. A plot of the effective pressure at the particle contacts divided by the bulk appliedpressure, such as in hot isostatic pressing, versus the fractional density. This model assumesspherical, monosized powder with a starting density of 0.6.

90 CHAPTER E


m ¼ shear modulus, Pa (convenient units: GPa)


ELASTIC-MODULUS VARIATION WITH DENSITY (Bocchini 1985)

The elastic modulus is also known as Young’s modulus and is sometimes called thestiffness. For an isotropic, polycrystalline material the elastic modulus E varies withdensity f in a power-law relation,

E ¼ E0 f Y

where E0 is the full-density elastic modulus, and the exponent Y varies from 0.3 to 4,depending on the pore structure.

E ¼ elastic modulus, Pa (convenient units: GPa)

E0 ¼ full-density elastic modulus, Pa (convenient units: GPa)

Y ¼ density sensitivity exponent, dimensionless

f ¼ density, dimensionless fraction [0, 1].

ELASTIC-PROPERTY VARIATION WITH POROSITY (Panakkalet al. 1990)

Elastic properties primarily pertain to the elastic modulus, which is also known asYoung’s modulus, and are secondarily concerned with the shear modulus andPoisson’s ratio. For dense structures, with over 80% of theoretical density or lessthan 20% porosity, there are several models for the porous elastic modulus Ebased on the dense-parent-material elastic modulus E0, and common examplesinclude the following:

E ¼ E0 exp(�a1)

1E¼ 1

E0þ B1

f

and

E ¼ E0 exp(�b1� c12)

where f is the fractional density and 1 is the fractional porosity ( f ¼ 1 – 1), and a, B,b, and c are constants that are determined by experiments. Of these three relations, thelast generally provides the best fit to experimental data for sintered powders. Forexample, in sintered iron compacts, where E0 is 212 GPa, the finding is that b ¼1.68 and c ¼ 10.5, which gives the best fit to the elastic-modulus variation with

ELASTIC-PROPERTY VARIATION WITH POROSITY 91

porosity. In a similar manner, the shear modulus G follows the same form, with G0

equal to 80 GPa, b ¼ 1.73, and c ¼ 8.9. The Poisson’s ratio n is expressed as a func-tion of porosity as follows:

n ¼ n0(1� v1)

with n0 being the full-density Poisson’s ratio, and v being a pore sensitivity factor.The pore sensitivity factor for sintered iron is 0.8 when the porosity 1 is less than 0.2.

B ¼ adjustable material constant, dimensionless


E0 ¼ dense-material elastic modulus, Pa (convenient units: GPa)

G ¼ shear modulus, Pa (convenient units: GPa)

G0 ¼ dense-material shear modulus, Pa (convenient units: GPa)

a ¼ adjustable material constant, dimensionless

b ¼ adjustable material constant, dimensionless

c ¼ adjustable material constant, dimensionless


n ¼ Poisson’s ratio, dimensionless

n0 ¼ full-density Poisson’s ratio, dimensionless


v ¼ pore sensitivity factor, dimensionless.

ELECTRICAL-CONDUCTIVITY VARIATION WITH POROSITY(Koh and Fortini 1973)

Compared to bulk material, the electrical conductivity is reduced by pores or noncon-ducting phases (dispersed ceramic particles) or impurities. A model for the conduc-tivity variation with fractional density f is given as follows:

C ¼ C0f

1þ x12

where C is the measured conductivity, C0 is the conductivity of fully dense materialin the same condition (grain size, stress state), and 1 is the volumetric content of insu-lator phase or pores. This model assumes the second phase is the minor phase. Thecoefficient x expresses the sensitivity to the second phase or pores. At high densities,over about 90% of theoretical, the electrical conductivity is essentially a linear func-tion of density.


C ¼ measured conductivity, S/m

92 CHAPTER E

C0 ¼ conductivity of fully dense material, S/m

x ¼ porosity sensitivity coefficient, dimensionless

1 ¼ volume fraction of insulator phase or pores, dimensionless [0, 1].

ELECTROMIGRATION CONTRIBUTIONS TO SPARKSINTERING (Olevsky and Froyen 2006)

Several newer concepts in sintering composites, nanoscale particles, bulk amorphousmetals, and other advanced materials rely on simultaneous heating and pressurization,with the further option of electrical discharge through the powder compact. Sinteringmodels for hot compaction with a simultaneous electrical current flow are known byseveral names. One variant of these field-activated sintering technologies (FAST) isknown as spark plasma sintering (SPS) on spark sintering; other names include fieldeffect sintering and spark-activated sintering. For the typical case of grain-boundarydiffusion (also known as Coble creep), the flux vector of matter J caused by boundarydiffusion is determined by the two-dimensional Nernst–Einstein equation, includingthe chemical potential gradient along the grain boundaries due to both normal stressand electromigration:

J ¼ CEEþ Csrs

where E is the vector component of the electric field in the tangent plane of thegrain boundary, rs is the gradient of stresses normal to the grain boundary, CE isthe electrical diffusion parameter, and Cs is the stress diffusion parameter. Theelectrical diffusion parameter CE is determined by a formula attributed to Blech:

CE ¼dDBZe

VkT

where DB is the coefficient of the grain-boundary diffusion, d is the grain-boundarythickness, V is the atomic volume, k is Boltzmann’s constant, T is the absolute temp-erature, Z is the valence of a migrating ion, and e is the electron charge. The productZe is called the effective charge. The stress diffusion parameter Cs is determined bythe following equation:

Cs ¼dDB

kT

In electromigration, the grain-boundary diffusion flux is enhanced in the x directionJgb

x and this combined effect is given as follows:

Jgbx ¼

dDB

kT

ZeU

Vlþ @sy

@x

� �

ELECTROMIGRATION CONTRIBUTIONS TO SPARK SINTERING 93

Here U and l are the electronic potential and the characteristic length along the elec-tric field in the x direction. SPS is a process involving the hot deformation of a powderunder pressure. Under these conditions, power-law (dislocation) creep typicallycontributes to densification.

CE ¼ electrical diffusion parameter, (atom . C)/(J . s)

Cs ¼ stress diffusion parameter, atom/(Pa . s)


E ¼ (vector) electric-field tangent to the grain boundary, N/C or V/m

J ¼ (vector) two-dimensional flux of matter, atom/(m . s)

Jgbx ¼ grain-boundary diffusion flux enhancement in the x direction,

atom/(m2 . s)


U ¼ electric potential, V

Z ¼ ionic valence, 1/atom

Ze ¼ effective charge, C/atom

e ¼ electron charge, 1.60 . 10219 C

k ¼ Boltzmann’s constant, 1.38 . 10223 J/(atom.K)

l ¼ characteristic length, m

r ¼ (vector) gradient, 1/m

rs ¼ (vector) stress gradient normal to the grain boundary, Pa/m

V ¼ atomic volume, m3/atom

d ¼ grain-boundary thickness, m (convenient units: nm)

sy ¼ normal stress to the grain boundary in the y direction, Pa.

ELONGATION

Elongation is a measure of the ductility of a material determined by the plastic strainat the rupture in a tensile test. It is also called a break elongation or an ultimate tensileelongation. Elongation e is the increase in gauge length DL (measured after rupture)divided by original gauge length L as follows:

e ¼ DL

L

Higher elongation indicates higher ductility. Elongation cannot be used to predictbehavior of materials subjected to sudden or repeated loading.

L ¼ original gauge length, m (convenient units: mm)

e ¼ elongation, dimensionless (convenient units: %)

DL ¼ increase in gauge length, m (convenient units: mm).

94 CHAPTER E

ELONGATION VARIATION WITH DENSITY


ENERGY-GOVERNING EQUATION FOR POWDER INJECTIONMOLDING (Kwon and Ahn 1995)

In accordance with the Hele–Shaw approximation of the filling process in powderinjection molding (PIM), based on a standard coordinate system (x, y, z), theenergy equation can be simplified as follows:

rCP@T

@tþ u

@T

@xþ n

@T

@y

� �¼ k

@2T

@z2þ h _g 2

where r is the feedstock density, CP is the feedstock heat capacity, T is the tempera-ture, t is the time, u is the velocity component in the x direction, v is the velocity com-ponent in the y direction, k is the feedstock thermal conductivity, h is the feedstockviscosity, and g is the generalized shear rate defined as:

_g ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi@u

@z

� �2

þ @v

@z

� �2s

CP ¼ constant pressure–specific heat, J/(kg . K)

T ¼ temperature, K

t ¼ time, s

u ¼ velocity components in the x direction, m/s

v ¼ velocity components in the y direction, m/s

x ¼ coordinate, m

y ¼ coordinate, m

z ¼ coordinate, m

g ¼ generalized shear rate, 1/s

h ¼ shear viscosity, Pa . s

k ¼ thermal conductivity, W/(m . K)


ENERGY IN A PARTICLE

For a particle of diameter D the surface energy per unit volume E effectively dependson the surface energy gSV and the inverse of the particle size,

E ¼ 6g SV

D

ENERGY IN A PARTICLE 95

This relation assumes the particles are spheres and can be approximated by a singlesurface energy that is independent of crystallographic orientation.


E ¼ energy per unit volume, J/m3


ENHANCED SINTERING

See Activated Sintering, Early-stage Shrinkage.

EQUILIBRIUM CONSTANT (Silbey et al. 2005)

The equilibrium constant is used to treat the balance between reactants and productsin chemical reactions. It is frequently employed to understand atmosphere interactionsduring sintering. For example, the ease of oxide reduction is measured by the equili-brium constant. Assume the solids are of fixed composition, such that a metal M is inequilibrium with oxygen gas O2 and the oxide MxO2, where the subscript—represents the stoichiometry of the oxide (and s ¼ solid, g ¼ gas),

xM(s)þ O2(g) ¼ MxO2(s)

For this reaction, the equilibrium constant K is defined as follows:

K ¼ aMxO2

axMPO2

where a designates the thermodynamic quantity known as the activity. For the solidphase, the activity is unity (meaning there is plenty of each solid available for reac-tion) and the PO2 oxygen partial pressure is the only factor that determines which waythe reaction progresses. Thus, for a fixed temperature the equilibrium constant for theoxidation-reduction reaction depends only on the inverse partial pressure of oxygen,while lower partial pressures favor oxide reduction. In turn, the equilibrium constantreflects the standard free energy DG for the reaction,

DG ¼ �RT ln(K) ¼ RT ln(PO2 )

where R is the gas constant and T is the absolute temperature.

a ¼ activity, dimensionless

K ¼ equilibrium constant, dimensionless

PO2 ¼ oxygen partial pressure, dimensionless

96 CHAPTER E

R ¼ universal gas constant, 8.31 J/(mol.K)


DG ¼ free-energy change, J/mol.

EQUIVALENT PARTICLE SIZE BASED ON AREA

See BET Equivalent Spherical-particle Diameter.

EQUIVALENT SINTERING

See Temperature adjustments for equivalent sintering.

EQUIVALENT SPHERICAL DIAMETER (Allen 1997)

To calculate an effective particle size, it is common to measure some specific para-meter associated with the powder, such as the particle volume or particle surfacearea, and then convert that measure into a linear particle size, assuming sphericalparticles. The particle size calculated in this manner assumes spheres independentof actual particle shape and even agglomeration. If a particle has a projected areaA, then the equivalent spherical diameter based on projected area DA is calculatedby setting the measured projected area to the equivalent area of a circle, giving,

DA ¼4A

p

� �1=2

Alternatively, if the particle volume V is measured, then by a similar manipulation theequivalent spherical volume diameter DV is given as,

DV ¼6V

p

� �1=3

If the external surface area S is measured, then the equivalent spherical surface diam-eter DS is given as,

DS ¼S

p

� �1=2

A ¼ projected area, m2 (convenient units: mm2)

DA ¼ equivalent diameter based on projected area, m (convenient units: mm)

EQUIVALENT SPHERICAL DIAMETER 97

DV ¼ equivalent diameter based on volume, m (convenient units: mm)

DS ¼ equivalent diameter based on surface area, m (convenient units: mm)

S ¼ external surface area, m2 (convenient units: mm2)

V ¼ particle volume, m3 (convenient units: mm3).

ERROR FUNCTION FOR CUMULATIVE LOG-NORMALDISTRIBUTION (Aitchison and Brown 1963)

A cumulative distribution gives the fraction of objects smaller than a given size (suchas on a cumulative particle-size distribution). Frequently, powders, pores, and othergeometric parameters encountered in powder processing are best represented by alog-normal distribution. The cumulative version of the log-normal distribution ismeasured by the error function,

F(x) ¼ 12þ 1

2erf

ln x� mffiffiffi2p

s

� �

where F(x) is the cumulative fraction smaller than the dimensionless size x, andthe two parameters s and m are the distribution-shape parameter and median ordistribution-scale parameter, respectively. The error function is defined as follows:

erf(x) ¼ 2ffiffiffiffipp

ðx

0exp (�t2) dt ¼ 2ffiffiffiffi

pp

X1n¼0

(�1)nx2nþ1

(2nþ 1)n!

In some cases of low dispersion, the solution can be approximated by a series solutionallowing easy numerical solution. The first few terms of this expansion are givenas follows:

erf(x) ¼ 2ffiffiffiffipp

X1n¼0

(�1)nx2nþ1

(2nþ 1)n!¼ 2ffiffiffiffi

pp x� x3

3þ x5

10� x7

42þ x9

216� � � �

� �

More typically, the function is embedded in statistical, mathematical, or compu-tational tools, including spreadsheets. The error function is essentially identical tothe cumulative of the standard normal distribution function and only differs byscaling and translation. When a series of measurements is described by a normaldistribution with a standard deviation s, then erf(a=s

ffiffiffi2p

) is the probability that theerror of a single measurement lies between 2a and þa.

F(x) ¼ cumulative fraction smaller than the size x, dimensionless [0, 1]

m ¼ median size, dimensionless

n ¼ dummy counter, dimensionless

98 CHAPTER E

x ¼ size variable assumed to be a dimension, dimensionless

s ¼ distribution-shape parameter, dimensionless.

EULER RELATION (McNutt 1968)

The polygonal grain geometry found in a fully dense sintered microstructure satisfiesthe Euler relation:

f þ c� e ¼ 2

which links the number of grain faces f, grain corners c, and grain edges e. Forexample, the often assumed grain shape for a full-density sintered material is thetetrakaidecahedron, consisting of a mixture of 8 hexagonal faces and 6 squarefaces, 36 edges, and 24 corners (where 14 þ 24–36 is 2).

c ¼ integer number of grain corners, dimensionless

f ¼ integer number of grain faces, dimensionless

e ¼ integer number of grain edges, dimensionless.

EVAPORATION (Silbey et al. 2005)

For any material the equilibrium vapor pressure P depends on absolute temperatureT and follows an Arrhenius dependence, since the breaking of atomic bonds isthermally activated,

P ¼ P0 exp � Q

RT

� �

where P0 is a preexponential material constant, Q is the activation energy for evap-oration, and R is the gas constant. Materials with high vapor pressures at the sinteringtemperature tend to sinter by evaporation–condensation, giving round pores but nosintering densification. The addition of a transport species accelerates the evaporationprocess. Halide additions to hydrogen sintering atmospheres are most effective inthis regard.

P ¼ equilibrium vapor pressure, Pa

P0 ¼ preexponential material constant, Pa

Q ¼ activation energy for evaporation, J/mol (convenient units: kJ/mol)



EVAPORATION 99

EVAPORATION–CONDENSATION

See Initial-stage Neck Growth.

EXAGGERATED GRAIN GROWTH

See Abnormal Grain Growth.

EXOTHERMIC SYNTHESIS

See Self-propagating High-temperature Synthesis.

EXPANSION FACTOR FOR TOOLING

See Tool Expansion Factor.

EXPERIMENTAL SCATTER


EXPONENTIAL DISTRIBUTION FUNCTION (M. Evans et al. 1993)

The exponential distribution is encountered in some manufacturing events, where theprobability P of a dimensionless size x is given by the probability density functionas follows:

P(x) ¼ 1b

exp � x� m

b

� �

where b is the mean or scale parameter, and m is the distribution offset. The median isb ln(2), with a mode at 0. The cumulative fraction smaller than a given size x is thengiven as F(x) as follows:

F(x) ¼ 1� exp � x

b

� �

(Note: The units for x must be consistent, for example, if the size variation in a grind-ing process is measured in mm, then all of the parameters need to be in a consistentset of dimensions.)

100 CHAPTER E

F(x) ¼ cumulative distribution, dimensionless [0, 1]

P(x) ¼ probability density function for size x, dimensionless

m ¼ distribution offset, dimensionless

x ¼ size variable assumed to be a dimension, dimensionless

b ¼ mean size, dimensionless.

EXTRUSION CONSTANT (Bufferd 1972)

In both powder–binder (low-pressure, low-temperature) and pure metal-powderextrusion (high-pressure, high-temperature), the extrusion constant C provides ameasure of the difficulty in achieving the deformation and flow of the feedstock.The extrusion force F and extrusion constant are related as follows:

F ¼ CAS ln R ¼ CAS lnAS

AF

� �

where AS is the cross-sectional area of the feed material, and R is the reduction ratio orextrusion ratio. The reduction ratio equals the cross-sectional area of the billet dividedby the cross-sectional area of the product, R ¼ AS/AF. With consideration of a temp-erature effect, the extrusion pressure P can be obtained by the following equation:

P ¼ b

Tln

AS

AF

� �

where T is the temperature in 8C, and b is the extrusion constant with temperaturedependency, which depends on the alloy.

AS ¼ cross-sectional area at the die inlet, m2 (convenient units: mm2)

AF ¼ cross-sectional area of the die outlet, m2 (convenient units: mm2)

C ¼ extrusion constant, Pa (convenient units: MPa)

F ¼ extrusion force, N (convenient units: kN or MN)


R ¼ reduction ratio, dimensionless

T ¼ temperature, 8Cb ¼ extrusion constant with temperature dependency, Pa . 8C.

EXTRUSION CONSTANT 101

F

FEEDSTOCK FORMULATION

For powder injection molding, tape casting, extrusion, and similar powder–bindershaping processes the feedstock is formulated on a weight basis. Calculation of theweight fraction from the volume fraction depends on the theoretical densities ofthe constituents. The relation determining the weight fraction of powder WP whenthe solids loading or volume fraction of powder f is known relies on using the theo-retical densities of the constituents as follows:

WP ¼rPf

rPfþ rB(1� f)

where rP is the theoretical powder density (at 100% density), and rB is the theoreticalbinder density (with no voids). Since the weight fractions of powder and binder sumto unity, then WB ¼ 1 2 WP as a simple calculation.

WB ¼ binder weight fraction, dimensionless fraction [0, 1]

WP ¼ powder weight fraction, dimensionless fraction [0, 1]

f ¼ volume fraction of solid, dimensionless fraction [0, 1]

rB ¼ theoretical density of the binder, kg/m3 (convenient units: g/cm3)

rP ¼ theoretical density of the powder, kg/m3 (convenient units: g/cm3).

FEEDSTOCK VISCOSITY

See Suspension Viscosity and Viscosity Model for Injection-molding Feedstock.

FEEDSTOCK VISCOSITY AS A FUNCTION OF SHEAR RATE

See Cross Model.


103

FEEDSTOCK YIELD STRENGTH

See Yield Strength of Particle–Polymer Feedstock.

FIBER FRACTURE FROM BUCKLING (Salinas andPittman 1981)

In mixing fibers with powders and binders, shear will buckle the fiber and inducefracture. If sufficient loading occurs, the fiber will bend and eventually fracture.The criterion for fracture from end loading gives the critical fiber bending radiusRB criterion as follows:

RB ¼Ed

2s

where E is the elastic modulus of the fiber, d is the fiber diameter, and s is thefiber strength.

E ¼ elastic modulus of the fiber, Pa (convenient units: GPa)

RB ¼ bend radius for fracture, m (convenient units: mm)

d ¼ diameter of the fiber, m (convenient units: mm)

s ¼ fiber strength, Pa (convenient units: MPa).

FIBER-FRACTURE PROBABILITY

Fiber fracture is most probable for the larger fibers during the mixing and shaping ofparticulate composites. On each handling in the mixing step, the fiber size reduces,and after repeated fractures the fragment size distribution becomes log-normal.Since most fibers are brittle, Weibull statistical models are employed to predictbrittle failure and the evolution of the fragment size distribution. Such an approachgives a probability of fracture that depends on the fiber length L compared to a start-ing length L0 as follows:

F(L) ¼ L

L01� exp � L

L0

� ��

where F(L) is the cumulative probability of fracture for fibers of length L or smaller.The first derivative gives the probability-density function. At any point in the proces-sing, this says the probability of fracture is highest for the longer fragments and smal-lest for the shorter fragments. Figure F1 plots both the cumulative distribution (givenpreviously) and the probability density, emphasizing how larger fiber pieces are mostlikely to be damaged in processing.

104 CHAPTER F

F(L) ¼ cumulative fracture probability, dimensionless fraction [0, 1]

L ¼ fiber length, m (convenient units: mm)

L0 ¼ initial fiber length, m (convenient units: mm).

[Also see Weibull distribution and Log-normal distribution.]

FIBER PACKING DENSITY (Milewski and Davenport 1987)

The random packing density of nonspherical particles declines as the particle shapedeparts from that of a sphere. However, for ordered packings of monosized andequiaxed particles, such as cubes, it is possible that the packing density is increased.But long fibers do not pack efficiently. Assume the fibers are characterized by a lengthL that is much larger then the diameter D. The fractional packing density f decreasesas the length-to-diameter ratio L/D becomes large,

f ¼ 1

[1:98þ 0:038(L=D)2]1=2

For a cylinder-shaped particle, with D ¼ L, this equation predicts a fractional densitynear 0.70, which is better than that attained with monosized spheres.

D ¼ fiber diameter, m (convenient units: mm)

L ¼ fiber length, m (convenient units: mm)

L/D ¼ length-to-diameter ratio, dimensionless

f ¼ packing density, dimensionless fraction.

Figure F1. The cumulative fiber-fracture probability and the probability density for fiberfracture during processing, shown versus the relative fiber length. Both views of fiber fractureshow that larger fibers are most likely to be damaged during processing.

FIBER PACKING DENSITY 105

FICK’S FIRST LAW (Shewmon 1989)

Fick’s first law is fundamental to sintering. This law is a relation between the flow ofatoms and the change in atomic energy (or curvature) over distance. The resultingmathematical model says the equivalent of “water flows downhill” or, in sintering,atoms move from high-energy convex surfaces to lower-energy concave surfaces,which leads to the filling of the sinter neck over time. In one-dimensional form,Fick’s first law is given as follows:

J ¼ �DVdC

dx

where J is the flux in terms of atoms or vacancies per unit area per unit time, DV is thediffusivity, and dC is the vacancy concentration change over a distance dx. Severalvariants might be encountered. In a typical simplification, the dC/dx term is replacedby the linear change in concentration, with distance DC/Dx. For anisotropicsituations, both the diffusivity and the gradient vary with orientation, especially fornoncubic crystallographic systems. Three-dimensional forms exist with the diffusioncoefficient changing with orientation and even concentration, as would be expected inan anisotropic material.

C ¼ vacancy concentration, mol/m3

DV ¼ diffusion coefficient, m2/s

J ¼ atomic flux, mol/(m2 . s)

dC/dx � DC/Dx ¼ concentration gradient, mol/m4

x ¼ distance, m.

FICK’S SECOND LAW (Shewmon 1989)

This second relation attributed to Fick adds time dependence to the model fordiffusion-controlled events. The second law gives a time-based relation betweenthe changes in concentration as a function of the geometric concentration gradient,where the material diffusivity is involved in determining the rate of change:

@C

@t¼ DV

@2C

@x2

Here C is the concentration, x is the distance, t is the time, and DV is the temperature-dependent diffusivity. A full three-dimensional form involves the gradients in thex, y, and z directions. The solution to this equation requires knowledge of thediffusivity, composition effects, temperature effects, and any orientation dependence.During sintering simulations, it is important to combine the solution to Fick’slaws with time-dependent geometric changes. Like Fick’s first law, the second

106 CHAPTER F

law can be written in three-dimensional form, and it is expected that the materialsare anisotropic.

C ¼ concentration, mol/m3

DV ¼ diffusion coefficient, m2/s

t ¼ time, s

x ¼ distance, m.

FIELD-ACTIVATED SINTERING (Olevsky and Froyen 2006)

Field-activated sintering is a general name for spark sintering, spark-plasma sintering,and related electric-field consolidation processes. During densification, an electricfield is used to induce diffusion while an external stress provides plastic flow orcreep contributions. The densification strain rate 1gbx ( _1gbx should be negative incase of shrinkage in the x direction) in the orthogonal direction is expressedthrough a relation with the atomic flux:

_1gbx ¼ �VJgb

y

��y¼c

(aþ ap)(cþ cp)

where Jygb is the grain-boundary diffusion flux enhancement in the y direction, V is

the atomic volume, a is the grain semiaxis in the x direction, ap is the pore semiaxisin the x direction, and c and cP are the grain and pore measures in the y direction.The shrinkage kinetics follow from this micromechanical model for a porousmaterial. For spark-sintering processing conditions, pressing is in a rigid die, sothe axial component of the shrinkage rate equals the overall volumetric shrinkagerate, giving,

_1gbx ¼ �dDB

kT

V

(aþ ap)(cþ cp)Ze

V

U

lþ 3gSV

c

1rc� 1

csin

f

2

� �� sx

cþ cp

c2

� �

where d is the grain-boundary thickness, DB is the grain-boundary diffusion coeffi-cient, k is Boltzmann’s constant, T is the absolute temperature, Z is the valence ofa migrating ion, e is the electron charge, U is the electronic potential, l is the charac-teristic length along the electric filed in the x direction, gSV is the surface energy, rc isthe maximum curvature radius of the elliptical-pore contour in the y direction, f is thedihedral angle, and sx is the effective (far-field) external stress in the x direction. Therelationship accounts for the structure anisotropy and, for simplicity, it is assumedthat the pore–grain structure is homogeneous, giving:

_1gbx ¼ �dDB

kT

V

(Gþ rp)2

Ze

V

U

lþ 3gSV

G

1rp� 1

2G

� �� sx

Gþ rp

G2

� �

FIELD-ACTIVATED SINTERING 107

Here G ¼ a ¼ c is the grain size, rp ¼ ap ¼ cp ¼ rc is the pore radius, and the dihedralangle is assumed to be 608. Based on this equation, three factors control the overalldensification behavior. These are the shrinkage-rate components 1gbx

em , 1gbxst , 1gbx

dl dueto electromigration, surface energy, and the contribution from the external load sx

with respect to the diffusion, respectively:

_1emgbx ¼ �

dDB

kT

Ze

(Gþ rp)2

U

l

_1stgbx ¼ �

3dDB

kT

V

(Gþ rp)2

gSV

G

1rp� 1

2G

� �

and

_1dlgbx ¼ �

dgbDgb

kT

V

(Gþ rp)sx

G2

These expressions for the axial strain rate are valid for spark sintering or spark-plasmasintering.


G ¼ a ¼ c ¼ grain size, m (convenient units: mm)

Jygb ¼ grain-boundary diffusion flux enhancement in the y direction,

atom/(m2 . s)


U ¼ electric potential, V

Z ¼ ionic valence, 1/ion or 1/atom

a ¼ grain semiaxis in the x direction, m (convenient units: mm)

ap ¼ pores’ semiaxis in the x direction, m (convenient units: mm)

c ¼ grain semiaxis in the y direction, m (convenient units: mm)

cp ¼ pores’ semiaxis in the y direction, m (convenient units: mm)

e ¼ electron charge, 1.60 . 10219 C


l ¼ characteristic length, m

rc ¼ minimum pore radius in the y direction, m (convenient units: mm)

rp ¼ ap ¼ cp ¼ rc ¼ pore radius, m (convenient units: mm)



d ¼ grain-boundary thickness, m (convenient units: nm)

1gbx ¼ strain rate, 1/s

1gbxem ¼ shrinkage rate from electromigration, 1/s

1gbxst ¼ shrinkage rate from surface energy, 1/s

108 CHAPTER F

1gbxdl ¼ shrinkage rate from external load, 1/s

f ¼ dihedral angle, rad (convenient units: degree)

sx ¼ normal grain-boundary stress in the x direction, Pa (convenient units: MPa)

sx ¼ effective external stress in the x direction, Pa (convenient units: MPa).

[Also see Electromigration Contributions to Spark Sintering and MicromechanicalModel for Powder Compact.]

FILTRATION RATING (Hoffman and Kapoor 1976)

An empirical correlation exists between the filter rating for a porous sintered materialand the permeability as follows:

a ¼ B(1F)N

where the filtration rating F is usually given in terms of the smallest particle size (98%assurance of capture) that will not pass through the filter, 1 is the porosity, and a is thepermeability coefficient as determined through Darcy’s law. Both B and N are empiricalconstants, and they depend on the powder shape and processing cycle. This relationassumes a high porosity, so it is not valid for less than about 10% open pore space.

B ¼ empirical constant, m2 – N

F ¼ filtration rating, m (convenient units: mm)

N ¼ exponent, dimensionless


1 ¼ fractional porosity, dimensionless.

FINAL-STAGE DENSIFICATION (Markworth 1972)

Alan Markworth realized that fission-gas swelling in nuclear fuels provided theantithesis of final-stage sintering. Accordingly, his model for mass transport vialattice diffusion examines pore shrinkage, assuming closed pores with trapped gas.The model gives the densification rate as,

df

dt¼ 12DVV

kTG3

4gSV

dP� PG

� �

where f is the fractional density, t is the sintering time, V is the atomic volume, DV isthe volume diffusivity, k is Boltzmann’s constant, T is the absolute temperature, G isthe grain size, gSV is the solid–vapor surface energy, dP is the pore size, and PG is thegas pressure in the pore. Effectively densification stops when the pore capillary

FINAL-STAGE DENSIFICATION 109

pressure equals the internal pore-gas pressure. Both factors increase as the poreshrinks. As a practical solution for sintering in one atmosphere gas, this means thatthe limiting density is 98 to 99% of theoretical. For the case of no gas in thepores, corresponding to vacuum sintering, an empirical equation relates fractionalporosity 1 to sintering time t as,

1 ¼ 1F � BF lnt

tF

� �

where 1F and tF correspond the point where the pores become closed (beginning ofthe final stage). The term BF is a collection of material constants.

BF ¼ material constant, dimensionless

DV ¼ volume diffusion coefficient, m2/s


PG ¼ gas pressure in the pore, Pa


dP ¼ pore diameter, m (convenient units: mm)




tF ¼ time at pore closure, s




1F ¼ fractional porosity at pore closure, dimensionless [0, 1].

FINAL-STAGE LIQUID-PHASE SINTERING DENSIFICATION(German 1996)

During the final stage of liquid-phase sintering, the residual pores are isolated spheresdispersed in the matrix while the total porosity is less than 8%, giving a microstruc-ture of solid grains, interlaced liquid, and isolated spherical pores. The correspondingfinal-stage densification rate is given as follows:

df

dt¼ 12DLCV

RTG2

j

1þ j

4gLV

dP� PG

� �

where f is the density (liquid plus solid density), t is the time, DL is the diffusivity ofthe solid in the liquid, C is the solid concentration in the liquid that is close to thesolubility limit, V is the atomic volume, R is the gas constant, T is the absolute

110 CHAPTER F

temperature, G is the solid grain size, gLV is the liquid–vapor surface energy, dP is thepore size, and PG is the gas pressure in the pore. The dimensionless geometric termj depends on the grain size, pore size, and number of pores per unit volume NV

as follows:

j ¼ p

6NV G2dP

For the typical case of final-stage sintering, j approaches zero as the pores disappear.Several factors inhibit full densification, the chief one among them being trapped gasin the pores. Densification ceases when the increasing pore-gas pressure due to poreshrinkage equals the stress from the surface energy working over the curved poresurface,

PG ¼ 4gLV

dP

This leads to a limiting final porosity that depends on the solubility of the gas in thematerial being sintered. If the pores pinch closed with an insoluble atmosphere, suchas argon, at a pressure of P0 (typically, one atmosphere or 0.1 MPa) and a porosityof 10, then the minimum porosity 1m is determined by the balance point betweenthe capillary stress and the gas compression, assuming there is no change in thenumber of pores,

1m ¼10

8P0dP0

gLV

� �3=2

The porosity at pore closure (when the gas is trapped in the pores) is typically about0.08. Since the pore size scales with the particle size, a lower final porosity resultsfrom smaller particles.

C ¼ solid solubility in the liquid, m3/m3 (dimensionless)

DL ¼ solid diffusivity in the liquid, m2/s


NV ¼ number of pores per unit volume, 1/m3

P0 ¼ atmosphere pressure at pore closure, Pa

PP ¼ pore-gas pressure, Pa



dP ¼ pore size, m (convenient units: mm)

dP0 ¼ pore size at pore closure, m (convenient units: mm)

f ¼ density, dimensionless fraction [0, 1]

t ¼ isothermal hold time, s


FINAL-STAGE LIQUID-PHASE SINTERING DENSIFICATION 111


10 ¼ porosity at pore closure, dimensionless fraction

1m ¼ minimum sintered porosity, dimensionless

j ¼ geometric term, dimensionless.

FIRST-STAGE NECK GROWTH IN SINTERING

See Initial-stage Neck Growth.

FINAL-STAGE PORE SIZE (Coble 1961)

For final-stage sintering, Coble assumed spherical pores located in a periodic arrayabout each grain. His choice was a tetrakaidecahedron grain shape with one sphericalpore on each of the 36 corners shared by three grains. This approach relies on ageometric link between the fractional porosity 1, pore diameter dP (assumed to bea sphere), and tetrakaidecahedron grain-edge length L, assuming all of the poresare in ideal positions, yielding,

1 ¼ pffiffiffi2p dP

2L

� �

The grain-edge length L is proportional to the common measure of the grain size.

L ¼ grain-edge length, m (convenient units: mm)


1 ¼ porosity, dimensionless fraction [0, 1].

FINAL-STAGE PRESSURE-ASSISTED DENSIFICATION(Ramqvist 1966; Helle et al. 1985)

In the final stage of solid-state hot consolidation of crystalline particles, such as in hotisostatic pressing, creep models are used to explain the typical slow strain rate associ-ated with pressure-assisted sintering. The rate of pore closure by diffusion givesgeneralized densification equations that reflect the pressure enhancement due to theresidual porosity 1 and the diminishing densification rate due to the elimination ofporosity. Using DL/L0 as the shrinkage and t as the isothermal time,

_f ¼ df

dt¼ Af 1

(1� 11=M)M

� �d

dt

DL

L0

� �

where f is the fractional density, A is a geometric constant that is near unity, and Mreflects the effect of work hardening with deformation and the amplification of

112 CHAPTER F

stress around pores. A typical value for the exponent M is close to 3, but it can rangefrom 1 to 6, so it needs to be experimentally isolated for each situation. In mostinstances, several densification processes are simultaneously active. A linear combi-nation of the rates of sintering densification by each of the individual processes is onemeans to approximate the total densification rate. This is achieved in practice by acomputer solution to each of the individual rate equations and integration of thesummed rates over time.



M ¼ exponent reflecting deformation and work hardening,dimensionless



f ¼ df/dt ¼ densification rate, 1/s

t ¼ hold time, s


DL/L0 ¼ shrinkage, dimensionless.

FINAL-STAGE PRESSURE-ASSISTED VISCOUS FLOW

Final-stage sintering corresponds to the closure and collapse of the pores as full densityis attained. From a theoretical standpoint, assuming a material consisting of grains thatare all the same size, the calculated onset of final-stage sintering occurs with poreclosure at just slightly more than 8% porosity. However, since there is a distributionin particle sizes, packing defects, and pore sizes, pore closure occurs over a range ofdensities. Often the first closed pores are seen at 85% density, and all pores areclosed by 95% density. Pore closure occurs because the surface energy of a longpore is higher than a collection of spherical pores, so a cylindrical pore of length Land diameter dP will close into spherical pores when L � pdP, which corresponds tothe Raleigh instability criterion. For cylindrical pores occupying grain edges, thisinstability occurs at approximately 92% density. The pores become spheres with afinal diameter about 1.9 times the cylinder diameter, resulting in an apparent pore-size increase during final-stage sintering. If the solid is treated as a viscous system,then a relation emerges that links fractional porosity 1 and sintering time t,

ln1

10

� �¼ � 3PEt

4h

which says the porosity decays from an initial value of 10 with isothermal hold time t.The factor PE represents the effective pressure during sintering, and h is the effectiveviscosity. However, this assumes the effective pressure is constant and that the systemhas no dependence on microstructure. In reality, during densification the contactsbetween particles grow, so the effective pressure decays, making densification

FINAL-STAGE PRESSURE-ASSISTED VISCOUS FLOW 113

slower than anticipated from the use of a simple viscous flow model. Even so, such amodel provides a first basis for describing hot consolidation of powders.

L ¼ pore length or grain-edge length, m (convenient units: mm)

PE ¼ effective stress, Pa (convenient units: MPa)


t ¼ final-stage isothermal hold time, s


10 ¼ initial fractional porosity, dimensionless [0, 1]

h ¼ viscosity, Pa . s.


FINAL-STAGE SINTERING BY VISCOUS FLOW(Ristic and Milosevic 2006)

Depending on the assumed geometry during final-stage sintering of an amorphousmaterial, it is possible to estimate the time for pore elimination and the ensuingdensification rate. The time for pore closure when there is no gas trapped in thepores is estimated as follows:

tF ¼2hD

3gSV

where tF is the time for pore closure in the final stage of sintering, h is the viscosity ofthe material at the temperature where it is being sintered, D is the initial particlesize, and gSV is the solid–vapor surface energy. This assumes that the pores arewidely spaced and small compared to the particle size, with the initial particlesbeing spheres of the same size (monosized). As is evident from this equation,the general expectation is a linear change in density with time during the finalstage of sintering.


tF ¼ time for pore closure, s



FINAL-STAGE SINTERING GRAIN GROWTH ANDPORE DRAG (Kang 2005)

In the final stage of solid-state sintering, the pore surface area varies inversely withthe grain size. The corresponding rate of grain growth depends on the grain-boundary

114 CHAPTER F

mobility and any retarding effect from pore drag, assuming the pores and boundariesare coupled. This gives a grain-growth rate in final-stage sintering that is related to thepore mobility as follows:

_G ¼ dG

dt¼ 4KfgSVMP

G MP=MG þ 1ð Þ

where G is the grain size, t is the isothermal sintering time, gSV is the solid–vaporsurface energy, MG is the grain-boundary mobility, MP is the pore mobility, and Kf

is a geometric constant that depends on the pore spacing and the grain-boundarycurvature, and usually is near unity. Typically, the grains are more mobile than thepores, so the pore mobility MP is rate controlling. For conditions where the micro-structure undergoes coarsening with an unchanged pore population per grain, theratio of coarsening and densification rates determines whether 100% density isobtained during sintering. For the typical case of pore motion by surface diffusionand grain motion by a process that scales with grain-boundary diffusion, the coarsen-ing-to-densification ratio G is defined as,

G ¼ 1300

DS

DB

gSS

gSV

where DS is the surface diffusivity, DB is the grain-boundary diffusivity, and gSS is thegrain-boundary energy (solid–solid surface energy). When G is less than unity, it ispossible to achieve full density. On the other hand, when it is larger than unity, therate of grain growth makes sintering to full density quite difficult.


DS ¼ surface-diffusion coefficient, m2/s


G ¼ dG/dt ¼ grain-growth rate, m/s

Kf ¼ microstructure-dependent constant, dimensionless

MP ¼ pore mobility, m3/(s . N)

MG ¼ grain-boundary mobility, m3/(s . N)

t ¼ time, s

G ¼ coarsening-to-densification ratio, dimensionless



FINAL-STAGE SINTERING LIMITED DENSITY

In final-stage sintering, gas trapped in the pores inhibits densification. Thus, a pointoccurs where densification ceases no matter how long the compact is heated. Indeed,

FINAL-STAGE SINTERING LIMITED DENSITY 115

prolonged heating often leads to pore coarsening and compact swelling. The peak orlimiting density in final-stage sintering corresponds to a balance between the capillarypressure from the curved pore surface and the internal gas pressure,

4gSV

dP¼ PG

where gSV is the solid–vapor surface energy, dP is the pore diameter, and PG is thegas pressure in the pore. This condition is often encountered when sintering in aninert atmosphere. Assume that a compact is sintered in argon at an ambient pressureP1 with pore diameter of dP1 at pore closure, which occurs at the theoretical point, 8%porosity. The final porosity is calculated by recognizing that the mass of the gas in thepores is conversed. If the number of pores and the temperature remain constant with aspherical pore shape, then the pore volume and gas pressure are related,

P1V1 ¼ P2V2

The final pore size dP2 is given as follows:

dP2 ¼P1d3

P1

4gSV

!1=2

The final porosity 12 can be calculated from the porosity at pore closure 11 as follows:

12 ¼ 11dP2

dP1

� �3

For example, if 20-mm pores close at 0.08 porosity, this predicts that the peak sintereddensity would be 99% of theoretical. However, if pore coarsening occurs, then theterminal density will be lower. Indeed, de-densification occurs as pores coarsen.An alternative happens when the gas is soluble in the material being sintered.In that case, shrinkage of the pore and the corresponding pressurization of thegas lead to dissolution of the gas into the solid. Sievert’s law says that thesolubility of the gas in the solid increases as the gas pressure increases. Thus, aspores containing trapped gas shrink, the pressure increase results in a higher solubi-lity, and the gas dissolves into the solid to reduce the pore-gas pressure, enablingfurther densification.

P1 ¼ initial gas pressure, Pa

P2 ¼ final gas pressure, Pa

PG ¼ gas pressure in a closed pore, Pa

V1 ¼ initial pore volume, m3 (convenient units: mm3)

V2 ¼ final pore volume, m3 (convenient units: mm3)

116 CHAPTER F


dP1 ¼ initial pore diameter at pore closure, m (convenient units: mm)

dP2 ¼ final pore diameter, m (convenient units: mm)

11 ¼ porosity at pore closure, dimensionless fraction [0, 1]

12 ¼ final porosity, dimensionless fraction [0, 1]


FINAL-STAGE SINTERING PINNED GRAINS

See Zener Relation.

FINAL-STAGE SINTERING STRESS

By the final stage of sintering, the sintering stress takes on two contributions, oneattributed to the pores and the second attributed to the grains. The composite deter-mination of the sintering stress s acting during the final stage of sintering is thengiven as follows:

s ¼ 2gSS

Gþ 4gSV

dP

where G is the grain size, gSS is the solid–solid grain-boundary energy, gSV is thesolid–vapor surface energy, and dP is the pore size. The curvature of the pore deter-mines the sign of the pore-size term. Concave near spherical pores give a negativecurvature, and convex pores are positive.





s ¼ sintering stress, Pa (convenient units: MPa).

[Also see Sintering Stress in Final-Stage Sintering for Small Grains and FacetedPores, Sintering Stress in Final-stage Sintering for Spherical Pores Inside Grains,and Sintering Stress in Initial-stage Solid-state Sintering.]

FIRST-STAGE SHRINKAGE IN SINTERING

See Initial-Stage Shrinkage in Sintering.

FIRST-STAGE SHRINKAGE IN SINTERING 117

FIRST-STAGE SINTERING SURFACE-AREA REDUCTION

See Surface-area Reduction Kinetics.

FISHER SUBSIEVE PARTICLE SIZE

One means to estimate the equivalent spherical particle size is by air permeation usingthe Fisher Subsieve Sizer (often the particle size measured this way is abbreviated byFSSS). The device measures the air permeability using a packed bed of test powder toestimate the surface area. In turn, the surface area is converted into an equivalentspherical-particle diameter. At low-pressure differences, Darcy’s equation for flowthrough a particle bed gives the volumetric flow rate Q as a function of the pressuredrop DP ¼ PU 2 PL and the gas viscosity h as follows:

Q ¼ aADP

hL

where a is the permeability coefficient, L is the sample length, and A is cross-sectional area assuming a cylindrical geometry. At a constant and small pressuredifferential, the superficial gas velocity passing through the powder bed (velocitymeasured outside the pores, not the velocity in the pores) is given as

V ¼ aDP

hL

with V equal to the flow rate per unit area (Q/A). Based on an analysis by Kozeny andCarman, the specific surface area per unit mass S of the compact is related to thepermeability through the fractional porosity 1 as,

S ¼ 1r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13

5a(1� 1)2

s

with r equal to the theoretical density of the material. The device, known as the FisherSubsieve Sizer, converts the surface area into an equivalent spherical diameter D asfollows:

D ¼ 6Sr

The technique is typically applied to particles between 0.5 mm and 50 mm.




118 CHAPTER F

PL ¼ downstream pressure, Pa (convenient units: MPa)

PU ¼ upstream pressure, Pa (convenient units: MPa)

Q ¼ gas flow rate, m3/s

S ¼ specific mass surface area, m2/kg (convenient units: m2/g)

V ¼ superficial gas velocity, m/s

DP ¼ PU 2 PL ¼ pressure drop, Pa (convenient units: MPa)




r ¼ theoretical density, kg/m3 (convenient units: g/cm3).

FLATNESS

See Particle-shape Index.

FLAW EFFECT ON GREEN STRENGTH

See Green Strength Variation with Flaws.

FLOW GOVERNING EQUATION DURING POWDER INJECTIONMOLDING (Kwon and Ahn 1995)

A concentrated mixture of particles and polymer binder is used for powder injectionmolding. Such a suspension does not behave as a Newtonian fluid. In plasticinjection molding the inertial terms in the momentum conservation equation areoften assumed to be negligible. The flow is assumed to be quasi-steady state, and alubrication approximation can be used for modeling the flow behavior in the moldcavity. In this case, the Hele–Shaw model for a thin part can be employed. Withthe coordinate system (x, y, z) as shown in Figure F2, the resultant sets of equationscan be written as:

@ b�uð Þ@xþ @ b�vð Þ

@y¼ 0,

@P

@x¼ @

@zh@u

@z

� �

and

@P

@y¼ @

@zh@v

@z

� �

FLOW GOVERNING EQUATION DURING POWDER INJECTION MOLDING 119

where the continuity equation is expressed in terms of the averaged z-directionvelocity components u and v, where u and v are the respective velocity componentsin the x and y directions, b is the half thickness, h is the apparent shear viscosity, andP is the cavity pressure. When the no-slip boundary condition is employed at the wall,the velocity components u and v can be calculated from the last two equations asfollows:

u(z) ¼ @P

@x

ðb

z

t

hdt

and

v(z) ¼ @P

@y

ðb

z

t

hdt

Finally, the following equation to govern flow phenomena during injection moldingcan be obtained:

@

@xS@P

@x

� �þ @

@yS@P

@y

� �¼ 0

with the flow conductivity S as follows:

S ¼ðb

�b

z2

hdz

P ¼ cavity pressure, Pa (convenient units: MPa)

S ¼ flow conductivity, m3/Pa . s

b ¼ half thickness, m (convenient units: mm)

t ¼ dummy variable for integration, m

u ¼ velocity components in the x direction, m/s

Figure F2. The layout for defining the flow governing equations in injection molding.

120 CHAPTER F

u ¼ gapwise averaged velocity in the x direction, m/s

v ¼ velocity components in the y direction, m/s

�v ¼ gapwise averaged velocity in the y direction, m/s

x ¼ coordinate, m (convenient units: mm)

y ¼ coordinate, m (convenient units: mm)

z ¼ gapwise coordinate, m (convenient units: mm)

h ¼ apparent shear viscosity, Pa . s.

FLUIDIZED-BED PROCESSING

Fluid-bed processing relies on the upward flow of gas to lift and stir a powder forreduction, coating, or agglomeration. To achieve fluidization in a fluid bed, the criticalReynolds number gives the combination of conditions for a given particle size, whereRe ¼ DrfV/h, where D is the particle diameter, rf is the fluid density, V is the fluidvelocity, and h is the fluid viscosity. This is estimated as follows for fluidization in afluid bed of powder:

Re ¼ A13

f

1� 1f

where 1f is the fractional porosity of the powder bed at the onset of fluidization, whichis typically near 0.44. The factor A is known as the Archimedes number, and isgiven as,

A ¼rf g(rm � rf )

cD3h2

where rm is the material density of the powder, g is the gravitational acceleration, andc is a numerical constant estimated as 150.

A ¼ calculation parameter, dimensionless


Re ¼ Reynolds number, dimensionless

V ¼ fluid velocity, m/s

c ¼ numerical constant, dimensionless


1f ¼ fractional powder porosity at the onset of fluidization, dimensionless [0, 1]


rf ¼ fluid density, kg/m3 (convenient units: g/cm3)

rm ¼ theoretical density of powder material, kg/m3 (convenient units: g/cm3).

FLUIDIZED-BED PROCESSING 121

FORCE DISTRIBUTION IN RANDOMLY PACKED POWDER(Liu et al. 1995)

When a powder is randomly loaded into a container the distribution in contacts andthe force distribution over those contacts is not uniform. Inhomogeneous packingleads to a force that tends to follow strings of high concentration through the particlebed. When the distance from the surface of force application is at least five times theparticle size, then the probability distribution in force P(v) is as follows:

P(v) ¼ NN

(N � 1)!vN�1 exp(� Nv)

where v is the normalized vertical depth (v is the vertical force divided by the depthfrom the top surface), N is the number of particles in contact in the next layer. Formost packing structures N is 3.

N ¼ number of particle contacts below the particle, dimensionless

P(v) ¼ probability distribution to the force, dimensionless

v ¼ normalized force divided by vertical depth, N/m.

FOUR-POINT BENDING STRENGTH

See Transverse-Rupture Strength.

FRACTIONAL COVERAGE OF GRAIN BOUNDARIESIN SUPERSOLIDUS SINTERING

Compact softening is one of the requirements for rapid densification during super-solidus liquid-phase sintering. This softening occurs when sufficient liquid existsto penetrate most of the grain boundaries, leading to a transition from solid toviscous behavior. Calculation of the fraction of grain boundaries coated with aliquid helps predict the process conditions required for rapid densification. The frac-tional coverage of grain boundaries due to liquid penetration FC is approximated as

FC ¼(1� f)(1� FI)D3

dG2(0:8 NC þ 3nG)

realizing that the fractional coverage also affects the number of grains per particle nG.In this relation f is the relative volume fraction of the solid phase, so 1 – f is theliquid quantity, and NC is the three-dimensional particle-packing coordination thatis estimated from the fractional density. The liquid forms either as isolated poolsinside the grains or as a coating on the grain boundaries, FC denotes the fraction ofliquid on the grain boundaries, and FI denotes the fraction of liquid sitting in internal

122 CHAPTER F

pools. The particle size and grain size are given by D and G, and the final partition ofgrain boundary liquid depends on the number of particle contacts and the number ofgrains inside the particles.


FC ¼ fractional coverage of liquid on grain boundaries, dimensionless [0, 1]

FI ¼ fraction of liquid formed in internal pools, dimensionless [0, 1]


NC ¼ particle coordination, dimensionless

nG ¼ number of grains per particle, dimensionless

d ¼ width of the liquid layer on the grain boundary, m (convenient units: mm)

f ¼ volume fraction of the solid phase, dimensionless [0, 1].

FRACTIONAL DENSITY

The fractional density is the actual density divided by the density corresponding topore-free material of the same composition. The full density might be differentfrom a handbook density, if the material has impurities or a composition shift.Accordingly, fractional density f is usually normalized to either the pycnometerdensity or the calculated theoretical density rT for the composition,

f ¼ r

rT


r ¼ measured density, kg/m3 (convenient units: g/cm3)

rT ¼ theoretical density, kg/m3 (convenient units: g/cm3).

FRAGMENTATION BY LIQUID

During liquid-phase sintering a newly formed liquid spreads and penetrates thesolid–solid interfaces shortly after liquid formation, usually resulting in a dimen-sional change. The first penetration often results in swelling, where the amount ofswelling varies with the liquid flow into the surrounding pores. The liquid flow inthe pores is estimated as a function of hold time as follows:

x2 ¼ dPgLVt cos u

4h

where x is the depth of liquid penetration, dP is the pore size, gLV is the liquid–vaporsurface energy, u is the contact angle, t is the hold time, and h is the liquid viscosity.

FRAGMENTATION BY LIQUID 123

dP ¼ characteristic pore size, m (convenient units: mm)

t ¼ time after liquid formation, s

x ¼ liquid penetration distance, m (convenient units: mm)


h ¼ liquid viscosity, Pa . s

u ¼ contact angle, rad (convenient units: degree).

[Also see Fractional Coverage of Grain Boundaries in Supersolidus Sintering.]

FRAGMENTATION MODEL

See Breakage Model.

FREEFORM SPRAYING


FRENKEL MODEL

See Two-particle Viscous Flow Sintering.

FRICTION AND WEAR TESTING (Matsugi et al. 2007)

Sintered materials are commonly formulated to maximize wear life, and verificationof this behavior is performed using laboratory tests for friction and wear. A commontest involves a ring-on-disk configuration. In this situation the wear test is conductedat a constant applied force F using a rolling ring of diameter d and width w, as shownin Figure F3. The corresponding relation between depth h and length b of the weartrace is determined by ignoring elastic deformation,

2 h ¼ d �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2 � b2p

When the wear depth is small, it approximates to a simple form,

b ¼ 2ffiffiffiffiffidhp

For small wear depths, the wear amount W is calculated as follows:

W ¼ wb3

6d

124 CHAPTER F


W ¼ wear amount, m3 (convenient units: mm3)

b ¼ wear groove length, m (convenient units: mm)

d ¼ ring diameter, m (convenient units: mm)

h ¼ wear depth, m (convenient units: mm)

w ¼ wear groove width, m (convenient units: mm).

[Also see Archard Equation.]

FUNICULAR-STATE TENSILE STRENGTH (Keey 1992)

The funicular state corresponds to a liquid–particle mixture where the liquid contentis sufficient to be connected throughout the particle mass, but not sufficient to saturateall the void spaces between particles. Simply, a funicular structure corresponds totubular pores in the liquid that partly fills the voids between particles. This contrastswith the pendular state, where the liquid sits at the particle contacts and is notconnected. Figure F4 compares the three cases (saturation, funicular, and pendular).For the pendular state, the strength s depends on the pore size dP, surface energy gLV,fractional density f, and porosity 1 (1 ¼ 1 2 f ) as follows:

s ¼ 3fgLV

1dP

Figure F3. The wear test involving a ring on a disk, where the wear groove is used to quantifythe wear resistance or wear rate.

FUNICULAR-STATE TENSILE STRENGTH 125

As more liquid is added, the funicular state strength emerges with an additionalsaturation term, as follows:

s ¼ 8sfgLV

1dP

where s is the saturation or fraction of the void space filled with liquid. Pores totallyfilled with liquid correspond to s ¼ 1. Both relations assume the liquid is wetting withrespect to the solid particles.



s ¼ fractional saturation, dimensionless [0, 1]



s ¼ strength, Pa.

Figure F4. Comparison of the three levels of fluid in a pore structure, corresponding to (a) allof the voids between particles being filled with liquid at saturation, (b) the funicular state, wherethere is incomplete saturation but the liquid is connected within the structure, and (c) the pend-ular state, with separated liquid bonds, but where the bonds are disconnected from each other.

126 CHAPTER F

G

GAS-ABSORPTION SURFACE AREA

See BET Specific Surface Area.

GAS-ATOMIZATION COOLING RATE

In gas atomization, the molten droplets move quickly out of the atomization zoneand pass through relatively cool gas. Convective cooling dominates the heatrelease during this transition, although for high-temperature materials there is also sig-nificant radiant cooling. The calculated droplet cooling rate T ¼ dT/dt depends on thedroplet size, initial droplet temperature, and the ambient environmental temperature:

dT

dt¼

6DrmCp

b T � T1ð Þ þ ST1 T4 � T41

� ��

In this equation, assuming a sphere, D is the droplet diameter, rm is the melt density, Cp

is the heat capacity of the melt, T is the absolute temperature of the melt, T1 is theambient gas temperature, b is the convective heat-transfer coefficient, 1 is the emissiv-ity of the melt (often assumed at 0.8), and ST is the Stefan–Boltzmann constant. Theconvective heat-transfer coefficient is very dependent on operating parameters, includ-ing the process atmosphere. The dependence of cooling rate on the inverse particle sizeforces atomization to smaller particles, if metastable, novel, homogeneous, or amor-phous materials are desired.

Cp ¼ constant-pressure heat capacity, J/(kg . K)

D ¼ droplet size, m (convenient units: mm)

ST ¼ Stefan–Boltzmann constant, 5.67 . 1028 W/(m2 . K4)

T ¼ temperature, K

T ¼ dT/dt ¼ droplet cooling rate, K/s


127

T1 ¼ ambient gas temperature, K

t ¼ time, s

b ¼ convective heat-transfer coefficient, W/(m2 . K)

1 ¼ emissivity, dimensionless fraction

rm ¼ melt density, kg/m3 (convenient units: g/cm3).

GAS-ATOMIZATION MELT FLOW RATE (Lawley 1992)

There are several empirical models used to link atomizer operating parameters to theoutput particle size. For a given material and atomizer design, over a range of operat-ing parameters, the particle-size distribution is reasonably narrow and the focus isgenerally on determining the change in either the mean or median particle sizewith the operating parameters. The dominant feature is the ratio of the gas-to-meltflow rates, with smaller particles resulting from higher ratios. Other factors includethe gas exit pressure and gas velocity, but both depend on the gas and melt flowrates. The pressure of the atomizing gas tends to be very important, but most atomi-zers operate over a relatively narrow pressure range. The gas flow rate dMG/dt isgiven as follows:

dMG

dt¼ A 2

CV

CP

� �N

P

ffiffiffiffiffiffibg

RT

r

where A is the cross-sectional area of the gas nozzle at the exit, CP is the gas-specificheat at constant pressure, CV is the gas-specific heat at constant volume, P is the oper-ating gas pressure, T is the absolute temperature of the gas at the nozzle, g is the gravi-tational acceleration, and R is the gas constant. The exponent N depends on the gas,and for a common atomization gas of nitrogen it equals 1.6.

A ¼ gas-nozzle exit cross-sectional area, m2 (convenient units: mm2)

CP ¼ gas-specific heat at constant pressure, J/(kg . K)

CV ¼ gas-specific heat at constant volume, J/(kg . K)

MG ¼ gas mass, kg

N ¼ exponent that depends on the gas, dimensionless

P ¼ gas pressure in the nozzle, Pa


T ¼ absolute temperature of the gas in the nozzle, K

dMG/dt ¼ gas flow rate, kg/s


t ¼ time, s

b ¼ constant, 2 (kg . s2)/(m . mol).

128 CHAPTER G

GAS-ATOMIZATION PARTICLE SIZE

The particle size produced by gas atomization is determined by the transfer of gasexpansion energy into the creation of surface area in the melt. A key factor isthe ratio of the metal flow rate to gas flow rate (normalized by recording them askg/s). A simple relation applies to most gas atomizers,

D50 ¼ a_MM

_MG

N

where D50 is the median particle size; MM is the metal mass, so MM ¼ dMM/dt is themetal mass flow rate; and MG is the gas mass, so MG ¼ dMG/dt is the gas mass flowrate. Various estimates put the exponent N in the range from 0.5 to 1.0, and the vari-ation is probably a reflection of design differences and the efficiency of coupling gasexpansion with melt disintegration. Secondary factors captured in the a factor includethe nozzle design, gas type, metal temperature, and pressures, but the mass flow-rateratio is usually the most important factor. Close-coupled nozzle designs, where thegas is injected directly into the melt stream, are very efficient in the transfer of gasexpansion energy. As a refinement of the preceding discussion for close-coupleddesigns, the predicted median particle size D50 largely depends on the mass flowratio as follows:

D50 ¼ Kd 1þ_MM

_MG

1

We

hM

hG

� �

where K is an empirical constant, d is the melt-stream diameter, hM is the meltviscosity, hG is the gas viscosity, and We is the dimensionless Weber number. TheWeber number is a measure of the relative importance of the melt’s inertia comparedto its surface tension as follows:

We ¼rGV2d

2gLV

where rG is the gas density, V is the gas velocity, gLV is the liquid–vapor surfaceenergy for the melt. This empirical correlation typically estimates the median particlesize only within a factor of 2 of the measured size. Other relations that try to capturemore accuracy in the particle-size prediction require knowledge of the ligament sizescoming out of the atomizer.


K ¼ empirical constant, dimensionless

MG ¼ gas mass, kg

GAS-ATOMIZATION PARTICLE SIZE 129

MG ¼ dMG/dt ¼ gas mass flow rate, kg/s

MM ¼ metal mass, kg

MM ¼ dMM/dt ¼ metal mass flow rate, kg/s

N ¼ nozzle-design exponent, dimensionless

V ¼ gas velocity, m/s

We ¼ Weber number, dimensionless

d ¼ melt-stream diameter, m (convenient units: mm)

t ¼ time, s

a ¼ nozzle-design factor, m

gLV ¼ liquid-vapor (melt) surface energy, J/m2

hG ¼ gas viscosity, Pa . s

hM ¼ melt viscosity, Pa . s

rG ¼ gas density, kg/m3 (convenient units: g/cm3).

GAS-GENERATED FINAL PORES

A reaction that produces a gaseous by-product during final-stage sintering will resultin stable, spherical pores in the microstructure. Gas generation will proceed until oneof the reacting species is exhausted. If the reaction product is insoluble, this leads tostabilized spherical pores that resist elimination, even during prolonged sintering. Agood example is observed in sintering copper (which has dissolved oxygen) in ahydrogen atmosphere. The hydrogen and dissolved oxygen react to form steam(H2O) that stabilizes the residual pores. Since the steam is not soluble in copper, itremains in the pores and bloats the pores to eventually form blisters. Besidesoxygen and hydrogen reactions, other variants are seen with carbon and oxygen,carbon and hydrogen, and even some high vapor-pressure compounds or elements(zinc, aluminum, molybdenum oxide). Gas that stabilizes the pores will result in sin-tered densities below theoretical. The problem is easily identified by spherical poresin the final microstructure. Even after the reaction ceases, there is a fixed quantity ofgas in the pore, and from the ideal gas law the estimated equilibrium spherical poresize dP is,

dP ¼ 2 aNRT

gSV

� �1=2

where N is the molar concentration of contaminant causing pore growth, and R is thegas constant. The constant a ¼ 3p/8. Effectively this requires an assumption aboutthe amount of contaminant per pore, which relates to the number of pores per unitvolume and the total contaminant level. This relation assumes that the capillarypressure from the pore (solid–vapor) surface energy gSV is in equilibrium with theyield strength of the sintering material at the peak temperature.

130 CHAPTER G

N ¼ molar concentration of contaminant, mol


T ¼ temperature, K


a ¼ constant, 3p/8


GAS PERMEABILITY

See Kozeny–Carman Equation.

GATE STRAIN RATE IN INJECTION MOLDING

In molding or extrusion, the feedstock flows through a constriction known as a gate.For a cylindrical gate, the shear strain rate g ¼ dg/dt for molten feedstock flow isdetermined by the volumetric flow rate Q and gate diameter d,

d ¼32Q

p _g

� �1=3

In many molding operations the volumetric fill rate (velocity of feedstock timesthe gate area) through the gate might reach 10 cm3/s or higher, and the upper limitfor shear strain rate is near 104 s21, thus, the proper gate diameter is more than0.1 mm. For conventional tooling and typical fill times, the gate diameter would bedetermined as follows:

d ¼32W

prtF _g

� �1=3

where W is the part weight, r is the feedstock density, and tF is the fill time.

Q ¼ volumetric flow rate, m3/s

W ¼ component weight, kg

d ¼ gate diameter, m (convenient units: mm)

t ¼ time, s

tF ¼ fill time, s


g ¼ dg/dt ¼ shear strain rate for molten feedstock in the gate, 1/s

r ¼ feedstock density, kg/m3 (convenient units: g/cm3).

[Also see Poisueille’s Equation.]

GATE STRAIN RATE IN INJECTION MOLDING 131

GAUDIN–SCHUHMANN DISTRIBUTION (Mular 2003)

During milling or grinding, a particle-size distribution becomes self-similar, meaningthe size becomes progressively smaller, but the shape of the particle-size distributionremains the same. Various proposals exist to describe the resulting particle-size dis-tribution, ranging from log-normal to Rosin–Rammler distributions. The Gaudin–Schuhmann distribution builds from the early concept of Gates in 1915 and is oneof the simple forms that often fits particle-size data obtained from crushing,milling, grinding, or other brittle fracture processes. This is a cumulative massparticle-size distribution F(D) given as,

F Dð Þ ¼D

Dmax

� �W

As examples of this distribution, Figure G1 plots three variants where Dmax is set to1.5 and W is 0.5, 1.0, and 2.0.


Dmax ¼ maximum particle size, m (convenient units: mm)

F(D) ¼ cumulative mass particle-size distribution, dimensionless [0, 1]

W ¼ exponent also known as the distribution modulus, dimensionless.

Figure G1. Three variants of the Gaudin–Schuhmann cumulative particle-size distribution,corresponding to a normalized maximum particle size of 1.5 and modulus parameters ofW ¼ 0.5, 1.0, and 2.0.

132 CHAPTER G

GAUSSIAN DISTRIBUTION

The Gaussian distribution, also known as the normal distribution, is an importantfamily of continuous probability distributions, applicable in many fields. Eachmember of the family can be defined by two parameters, location and scale, alsoknown as the mean and standard deviation, respectively. A normalized version isobtained when the size is divided by the mean size. Carl Friedrich Gauss promotedthis distribution as a means to analyze the error in astronomical data. Today, in schoolit is often called the bell curve because the graph of its probability density resemblesa bell. The continuous probability density function of the normal distribution isas follows:

P x; m, sð Þ ¼1

sffiffiffiffiffiffi2pp exp �

x� mð Þ2

2s 2

¼

1s

Px� m

s

� �

where s . 0 is the standard deviation, the parameter m is the expected value ormean, and

P(x) ¼ P x; 0, 1ð Þ ¼1ffiffiffiffiffiffi2pp exp �

x2

2

is the probability density function of the standard normal distribution, that is, thenormal distribution with m ¼ 0 and s ¼ 1. The integral of P(x; m, s) is equal tounity. Further, unlike other distributions encountered in powder processing, theGaussian distribution is unique since the density function is symmetric about itsmean value m. In addition, the mean m is also its mode and median. The inflectionpoints for this distribution occur at one standard deviation away from the mean,that is, at m 2 s and m þ s. The cumulative distribution function is the probabilitythat a random variable X is less than or equal to x. The cumulative distribution func-tion F(x) for the Gaussian distribution is expressed as follows:

F(x) ¼1

sffiffiffiffiffiffi2pp

ðx

�1

exp �u� mð Þ

2

2s2

du

Figure G2 shows the example for the cumulative distributions at u ¼ 1 with s ¼ 0.15and 0.30. The standard normal cumulative distribution function is simply evaluatedwith the expected or mean value centered about m ¼ 0, with the normalized standarddeviation set to s ¼ 1:

F(x) ¼ F x; 0, 1ð Þ ¼1ffiffiffiffiffiffi2pp

ðx

�1

exp �u2

2

du

F(x) ¼ cumulative distribution function corresponding to size x,dimensionless

P(x) ¼ probability of occurrence for a size x, dimensionless

GAUSSIAN DISTRIBUTION 133

m ¼ median, same units as x

x ¼ size, units as defined by the situation (for example, mm)

m ¼ expected value or mean, same units as x

s ¼ standard deviation or distribution dispersion, same units as x.

GEL-DENSIFICATION MODEL (Philip et al. 2000)

In this concept, a gel structure is treated as a collection of cylindrical tubes. Duringsintering, it is assumed that the geometry of the cellular gel structure is preserved byviscous phase sintering while the process takes place. Let h be the viscosity of thesystem, and l, a be the length and radius of the cylinders, respectively; then energydissipated in viscous flow Ef ¼ dEf/dt is given by the following:

_Ef ¼dEf

dt¼

3pha2

l

dl

dt

� �2

The overdot indicates a derivative with respect to time. As the surface area is elimi-nated, the corresponding energy change Es leads to a surface-energy dissipation rategiven by the following:

_Es ¼dEs

dt¼ gSV

dS

dt

Figure G2. Two plots of the cumulative Gaussian distribution for mean values of 1.0 andstandard deviations of 0.15 and 0.30.

134 CHAPTER G

where gSV is the solid–vapor interfacial energy, S is the surface area, and t is the time.The energy balance requires the following condition:

_Ef þ _Es ¼dEf

dtþ

dEs

dt¼ 0

From the preceding, we can see that the rate of densification is given as follows:

g

hl0

1f0

� �1=3¼

1t � t0ð Þ

ðx

0

2

3px2 � 8ffiffiffi2p

x3� �1=3 dx

where x ¼ a/l represents the aspect ratio for the cylindrical pores. For a cubic calcu-lation cell, x is related to the cylinder volume fraction as

f ¼ 3pa

l

� �2� 8

ffiffiffi2p a

l

� �3

where f corresponds to the measured volume fraction of the pores in the gel, andt0 is the fictitious time at which the pores correspond to x ¼ 0. The cluster(g/hl0)(1/f0)1/3 is a constant for a given initial volume fraction f0. Indeed, f0

sets the initial cylinder length l0. When the ratio of cylinder radius to its length isequal to 0.5, the neighboring cylinders touch and the cell contains only closedpores. The corresponding theoretical density (volume fraction) of the samplewould be 0.942.

Ef ¼ energy dissipation, J

Ef ¼ dEf/dt ¼ energy dissipation rate, J/s

Es ¼ energy change from surface-area reduction, J

Es ¼ dEs/dt ¼ energy change rate from surface-area reduction, J/s


a ¼ radius of cylinder, m (convenient units: mm or nm)

l ¼ length of cylinder, m (convenient units: mm or nm)

l0 ¼ initial length of cylinder, m (convenient units: mm or nm)

t ¼ time, s

t0 ¼ the time at which x ¼ 0, s

x ¼ a/l ¼ aspect ratio of gel cylinder, dimensionless

f ¼ volume fraction of the gel, dimensionless

f0 ¼ initial volume fraction of the gel, dimensionless

gSV ¼ surface energy, J/m2


GEL-DENSIFICATION MODEL 135

GESSINGER MODEL FOR INTERMEDIATE-STAGELIQUID-PHASE SINTERING (Gessinger et al. 1973)

A fundamental concept associated with densification during liquid-phase sintering isthe relation between bulk dimensional change (shrinkage) and the corresponding flat-tening of the grain–grain contacts. The assumption is that the process is controlled bydiffusion of solid atoms dissolved in the liquid. Accordingly, the sintering shrinkageDL/L0 due to diffusion in the liquid-filled grain boundaries follows a law very similarto that introduced by Kingery,

DL

L0

� �3:1¼

192dVgLVCDLt

G4RT

where d is the thickness of the grain boundary layer between the grains; V is theatomic volume of the solid phase; gLV is the liquid–vapor surface energy; DL isthe diffusivity of the solid in the liquid at the process temperature, C is the solid con-centration dissolved in the liquid, which changes with temperature; t is the isothermalhold time; R is the universal gas constant; T is the absolute temperature; and G is thegrain diameter, which increases during intermediate-stage sintering, with G3 beingproportional to hold time t. In this form there is low sensitivity to the amount ofliquid, as long as sufficient liquid exists to coat the grain boundaries. Shrinkage isenhanced by the high solubility of the solid in the liquid, longer process times, andsmaller grains or particles. The role of temperature is primarily seen in the diffusivityof the solid in the liquid, which follows an Arrhenius relation. Typically, severalparameters shift with temperature, including the solid solubility in the liquid andliquid–vapor surface energy, but the diffusivity change with temperature is domi-nant. Efforts to extract an activation energy by measures of shrinkage variationwith temperature are clouded by the simultaneous changes in several parameters;thus, it is proper to term the resulting temperature sensitivity an apparent activationenergy for the system.

C ¼ solid solubility in the liquid, m3/m3



L0 ¼ initial dimension, m (convenient units: mm)



t ¼ time, s

DL ¼ change in dimension associated with sintering, m (convenient units: mm)


V ¼ solid atomic volume, m3/mol

d ¼ grain boundary layer thickness, m (convenient units: mm or nm)


136 CHAPTER G

GLASS VISCOSITY TEST

See Bending-beam Viscosity.

GRAIN ACCOMMODATION

See Grain-shape Accommodation.

GRAIN BONDING

See Contiguity.

GRAIN BOUNDARY–CONTROLLED CREEP

See Coble Creep.

GRAIN-BOUNDARY ENERGY AND MISORIENTATION ANGLE

The energy assigned to a solid–solid grain boundary gSS increases from zero as thegrain boundary misorientation angle increases, up to approximately a 108 misorienta-tion, after which there is less change. If a is the misorientation angle betweencontacting grains, then

gSS ¼ a A� B ln a½ �

where gSS is the grain-boundary energy (solid–solid interface), and A and B are con-stants relating to the dislocation energy. For small misorientation angles, the dihedralangle will tend toward 1808. For this reason, a small portion of the sintering grain–grain contacts will lack a dihedral groove during sintering. In such cases, grain coales-cence is observed, which is outside the normal assumptions of Ostwald ripening. As themisorientation increases beyond about 108 or 0.17 rad, the grain boundary energy isless predictable and depends on several parameters that include the crystal structure.

A and B ¼ constants relating to the dislocation energy, J/(rad . m2)

a ¼ grain-boundary misorientation angle, rad

gSS ¼ solid–solid grain-boundary energy, J/m2.

GRAIN-BOUNDARY FRACTION

In small grain structures, such as nanoscale materials, the fraction of atoms involvedin disrupted atomic bonding because of a proximity to the grain boundaries FB

GRAIN-BOUNDARY FRACTION 137

depends on the grain size G and the thickness d of the disrupted bonding around agrain boundary (usually assumed to be between 5 and 10 atom diameters) as follows:

FB ¼6dG

FB ¼ fraction of atoms involved associated with a grain boundary, dimensionless


d ¼ thickness of disrupted grain-boundary layer, m (convenient units: mmor nm).

GRAIN-BOUNDARY GROOVE

See Dihedral Angle.

GRAIN-BOUNDARY MISORIENTATION

See Grain-boundary Energy and Misorientation Angle.

GRAIN-BOUNDARY PENETRATION

See Fragmentation by Liquid.

GRAIN-BOUNDARY PINNING

See Zener Relation.

GRAIN-BOUNDARY WETTING

During liquid-phase sintering under slow heating rates, the newly formed liquid isaggressive and preferentially dissolves grain boundaries. At equilibrium, the dihedralangle f describes the grain-boundary energy balance between the solid and liquidphases; gSS is the solid–solid grain-boundary energy and gSL is the solid–liquidsurface energy. Normally, the dihedral angle is based on an assumed equilibriumthat occurs after each phase has reached chemical and thermal equilibrium. Prior toliquid formation, the solid–vapor surface energy sets a different dihedral angle.After a new liquid forms, there is a transition period where a rapid change takesplace in the dihedral angle as the solid dissolves into the liquid. Differentiation ofthe surface-energy definition for the dihedral angle f allows for analysis of the

138 CHAPTER G

dihedral-angle sensitivity to changes in solid–liquid surface energy,

DgSL

gSL

¼Df

f

f

2tan

f

2

� �

A momentary decrease in solid–liquid surface energy occurs when there is dissol-ution across the solid–liquid interface to bring the newly formed (unsaturated)liquid up to equilibrium or saturation. Because a free-energy change drives transportacross the interface, there is a simultaneous decrease in the solid–liquid surfaceenergy. This causes near spontaneous penetration of the grain boundary by thenewly formed liquid, leading to disintegration of the solid skeleton that formedduring heating. One symptom of this is the necklace microstructure formation ofsolidified islands of formerly liquid phase on the grain boundaries in the sinteredmaterial. The relative surface-energy change needed to totally disintegrate thesolid grain boundaries and weaken the solid skeleton to essentially zero strengthis estimated based from grain-boundary wetting. This condition corresponds toa reduction in the dihedral angle f to zero, or Df ¼ 2f. Accordingly, Df/f ¼ 21, giving a relation for the relative solid–liquid surface-energy decreaseneeded for grain-boundary penetration,

DgSL

gSL

¼ �f

2tan

f

2

� �

Larger decreases in solid–liquid surface energy are required to penetrate highdihedral-angle grain boundaries. For example, if the dihedral angle is 308 (0.524 rad),then a 7% decrease in the relative solid–liquid surface energy leads to liquid penet-ration of the grain boundaries. Alternatively, for a 608 (1.05 rad) dihedral angle, therequired surface energy reduction is close to 30%. The surface energy for a solid isgenerally less than 100 kJ/mol, so a small energy decrease associated with solvationis possible. Wetting systems with low dihedral angles (those with a high solid solu-bility in the liquid) offer the largest opportunity for grain-boundary penetration uponfirst-melt formation. These same characteristics are associated with activated sinter-ing. A high solid solubility in the liquid correlates with a low dihedral angleand melt penetration of grain boundaries. An empirical link observed between thedihedral angle f and the change in atomic solubility on melt formation,

f ¼ a� bDC

where DC is the solubility change in the solid in newly formed liquid as comparedwith the solid solubility in the additive. When the dihedral angle is in degrees, andDC is in kg/kg, then a ¼ 75 and b ¼ 638. A good example is iron with added tita-nium, which forms a liquid on heating that gives penetration of the iron-grain bound-aries with the newly formed titanium-rich liquid. During subsequent holds, the liquidsaturates and the dihedral angle increases to cause a pinching off of the liquid filminto a necklace microstructure.

GRAIN-BOUNDARY WETTING 139

a ¼ constant term, rad (convenient units: degree)

b ¼ solubility coefficient, rad (convenient units: degree)

DC ¼ change in solid solubility with newly formed liquid, kg/kg




GRAIN COORDINATION NUMBER IN LIQUID-PHASE SINTERING

See Coordination Number in Liquid-phase Sintering.

GRAIN DIAMETER BASED ON AN EQUIVALENT CIRCLE

For solid grains with noncircular shapes, the projected area A and perimeter Pmeasured from a random cross section through the grain provides a means to estimateits size. This is a single grain-size parameter known as the diameter of an equivalentcircle Ge, and it is calculated using the following equation:

Ge ¼4A

P

This calculated equivalent grain size might further be transformed from a two-dimensional to a three-dimensional equivalent grain size.

A ¼ projected grain area, m2 (convenient units: mm2)

Ge ¼ diameter of grain based on equivalent circle, m (convenient units: mm)

P ¼ projected grain perimeter, m (convenient units: mm).

GRAIN GROWTH

In the absence of pores, grain growth is rapid at temperatures typically used for sin-tering. For dense materials, the mean grain size G increases with time t according tothe volumetric law,

G3 ¼ G30 þ Kt

where t is the isothermal hold time, G0 is the initial grain size corresponding to timeequal to zero, and K is an Arrhenius thermally activated parameter that contains theactivation energy for grain growth. Effectively, this says the mean grain volumeincreases linearly with time. When pores are present, then the factor k is reduced

140 CHAPTER G

roughly in proportion to the fraction of grain boundaries intersected by pores.As densification occurs and pores are eliminated, k increases to allow for largegrains, which results in slower sintering.

G ¼ mean grain size, m (convenient units: mm)

G0 ¼ initial grain size, m (convenient units: mm)

K ¼ grain-growth rate constant, m3/s (convenient units: mm3/s)

t ¼ time, s.

[See also Ostwald Ripening.]

GRAIN GROWTH IN LIQUID-PHASE SINTERING, DIFFUSIONCONTROL AT HIGH SOLID CONTENTS (P. Lu and German 2001)

Models for grain growth in liquid-phase sintering often assume the solid grains areunrealistically far from one another. Actual liquid-phase sintering systems show ahighly connected solid skeletal structure with an interwoven liquid dispersed inthis skeleton of solid grains. Accurate determination of the grain-growth rate con-stant at high solids contents requires inclusion of the solid–solid, solid–pore, andsolid–liquid interfaces, as estimated by contiguity, porosity, and dihedral anglecorrections. The general case is for solution reprecipitation, where solid diffusionin the liquid phase is the rate-controlling event. In this case, units for the grain-growth rate constant are volume per unit time. For diffusion-controlled coarsening,the solid diffusivity in the liquid changes rapidly with an increase in temperature,for example,

DS ¼ D0 exp �QG

RT

� �

where DS is the solid diffusion rate in the liquid phase, D0 is a frequency factor, QG

is the activation energy for grain growth, R is the gas constant, and T is the absolutetemperature. Likewise, the solid solubility in the liquid changes with temperature;thus, the grain-growth rate constant K is very sensitive to temperature. For thecases of diffusion-controlled grain growth, the integral kinetic law is generalizedas follows:

G3 ¼ G30 þ Kt

where G is the mean grain size after sintering at temperature T for time t, with astarting mean grain size of G0. The parameter K depends on the way in which thegrains grow and scales with factors such as the activation energy for diffusion. Inhigh concentrations of a solid, typical to liquid-phase sintering, the rate constant K

GRAIN GROWTH IN LIQUID-PHASE SINTERING, DIFFUSION CONTROL 141

depends on the diffusion distance, which scales with the liquid content as follow:

K ¼ K0 þKL

V2=3L

where K0 þ KL is the infinite dilution rate constant applicable to pure liquid, KL isa parameter sensitive to the microstructure factors such as contiguity, and VL is theliquid fraction. In diffusion-controlled coarsening (solution-reprecipitation) duringliquid-phase sintering, with rounded grains, the grain size enlarges and the coar-sening rate increases as the amount of liquid decreases. This is due to the smallerdiffusion distances for grain growth as the liquid layers between grains becomethinner. A combined model for the grain-growth rate constant K that includessolubility, microstructure, temperature, and related factors such as solid volumefraction is given as follows:

K ¼ggSLVCDS

RT1þ

sin f=2ð Þ

V2=3L

" #

where g is a numerical constant near 6; gSL is the solid–liquid surface energy; Vis the atomic volume; C is the solid solubility in the liquid, which varies withtemperature and composition; k is Boltzmann’s constant; and f is the dihedralangle. Note that diffusivity, solubility, surface energy, and solid volume fractionvary with temperature.

C ¼ solid solubility in the liquid, m3/m3 or dimensionless

D0 ¼ frequency factor, m2/s

DS ¼ solid diffusivity in the liquid, m2/s


G0 ¼ grain size at the start of the isothermal period, m (convenient units: mm)

g ¼ numerical constant, near 6, dimensionless


K0 ¼ grain-growth rate constant intercept, m3/s (convenient units: mm3/s)

KL ¼ grain-growth rate constant liquid term, m3/s (convenient units: mm3/s)

QG ¼ activation energy for grain growth, J/mol (convenient units: kJ/mol)



VL ¼ volume fraction of liquid phase, dimensionless





142 CHAPTER G

GRAIN GROWTH IN LIQUID-PHASE SINTERING, DILUTESOLIDS CONTENTS (Voorhees 1992)

In the early models for grain growth during liquid-phase sintering, it was necessaryto assume dilute systems of widely dispersed solid grains in the liquid. The dissolvedsolid was assumed to be at a constant concentration in the liquid, independent of micro-structure. Accordingly, during liquid-phase sintering the solid grains either give or takemass from the surrounding liquid phase. Greenwood proposed the following equationfor the growth rate of an individual grain by Ostwald ripening, assuming the individualgrains are separated by a liquid at a mean concentration of dissolved solid:

dGi

dt¼

2DSCVgSL

RTG

1G�

1Gi

� �

Here Gi is the size of a specific grain, G is the mean grain size, t is the isothermal holdtime, DS is the diffusivity of the solid grain material in the matrix, C is the solubilityof the grain material in the matrix, V is the molar volume of the solid, gSL is thesolid–vapor or grain–matrix surface energy, R is the gas constant, and T is theabsolute temperature. This equation was derived assuming a diffusion-controlledprocess involving mass transport through the liquid between spherical grains, and agrain-volume fraction close to zero. Under these circumstances, a mean concentrationof solid material in the matrix is valid and small gradients are assumed to existbetween the grain surface and the liquid mean concentration. This is known as themean field assumption, but it is invalid for the high solid contents encountered inliquid-phase sintering. It is only valid when the grains are far apart such that theirdiffusion fields do not interact and the mean concentration of solid in the matrix isdetermined by the mean grain size. In this case, with no short-range grain–graininteractions, all grains smaller than the mean size shrink, while all grains largerthan the mean size grow. In contrast, observations on liquid-phase sinteringsystems show this is not necessarily true, since small grains in clusters of smallergrains grow, and vice versa. Lifshitz and Slyozov extended the dilute solution analy-sis to include predictions of the grain-size distribution, again for noncontactingspherical solid grains at infinite dilution. They suggested that, with sufficient time,the system reaches a steady state where the grain-size distribution, normalized bythe mean grain size, is invariant. Additionally, the mean grain size increases withtime as given in the following equation:

G3 ¼ G30 þ KLSW t

where G0 is the grain size at the beginning of the steady state, and KLSW is:

KLSW ¼649

DSCVgSL

RT

GRAIN GROWTH IN LIQUID-PHASE SINTERING, DILUTE SOLIDS CONTENTS 143

where LSW is the Lifshitz, and Slyozov [1961], and Wagner [1961] model. Thesituation in liquid-phase sintering systems is quite far from the conditions assumedin these theories. Usually, the solid-volume fraction is very high (over 95 vol% inmany cases), grains are in contact with one another, pores attach to the grains tokeep some of the surface from coarsening, the density is continuously changingduring the sintering cycle, and the mean field assumption for the solute concentrationin the matrix does not apply. When the solid-volume fraction increases, coarsening isaccelerated, because the diffusion distance of the solute across the liquid is reduced.Moreover, local concentration gradients make some small grains grow, while somelarge grains shrink. For the same reason, the presence of particle clusters in thegreen compacts also enhances overall coarsening, since they create local pockets ofshort diffusion distances. Another factor not included in the model is the contri-butions from grain coalescence, where grains rotate into coincidental orientationsand merge by the elimination of the grain boundary between grains. Finally, attentionmust be given to the pore shadowing effect that shields some portion of the solid fromparticipation in growth, an effect that decays as time advances and pores are elimi-nated. In spite of these drawbacks, the linear dependence of G3 on time is confirmedby many studies; however, experimental results show that the LSW rate constant islow and fails to account for the proximity of growing and shrinking grains.Further, the actual grain-size distribution in liquid-phase sintering is much broaderthan predicted by the LSW model.

C ¼ solubility of the solid-grain material in the matrix, m3/m3 ordimensionless

DS ¼ solid diffusivity in the matrix, m2/s


G0 ¼ grain size at the start of grain growth, m (convenient units: mm)

Gi ¼ grain size for a specific grain, m (convenient units: mm)

KLSW ¼ grain growth rate constant, m3/s (convenient units: mm3/s)




V ¼ molar volume of the solid, m3/mol

gSL ¼ solid–liquid or grain–matrix surface energy, J/m2.

GRAIN GROWTH IN LIQUID-PHASE SINTERING, INTERFACIALREACTION CONTROL (Kingery 1959)

In liquid-phase sintering, the usual rate-controlling step for grain growth is diffusionof the dissolved solid in the liquid. However, when the grains have flat faces, then themost likely condition is interfacial reaction control, where the limited availabilityof surface sites for dissolution or precipitation controls the growth rate. In this

144 CHAPTER G

case, the mean grain size increases with sintering time under isothermal condition, asfollows:

G2 ¼ G20 þ Kt

where G is the mean grain size, t is the isothermal hold time, G0 is the mean grain sizeat the beginning of the isothermal hold, and K is the grain-growth rate constant. Sincethe population of surface sites is thermally activated, the grain-growth rate constantexhibits an Arrhenius temperature dependence expressed as:

K ¼K0

Texp �

QG

RT

� �

where T is the absolute temperature, K0 is the associated preexponential factor, QG isthe apparent activation energy for grain growth, and R is the universal gas constant.Since the temperature term inside the exponential is dominant, the preexponentialtemperature term is often ignored.


G0 ¼ mean grain size at the start, m (convenient units: mm)


K0 ¼ preexponential factor, m3/s (convenient units: mm3/s)




t ¼ isothermal hold time, s.

GRAIN-GROWTH MASTER CURVE (S. J. Park et al. 2006)

Grain growth requires an atomic displacement across the boundary between the grainsor across the grain–matrix interface in two-phase systems. As long as there is abun-dance of dissolution and deposition sites, then growth is diffusion controlled.Diffusion depends on the temperature-controlled rate of atomic motion, so the oper-ating diffusivity DS is related to temperature by the Arrhenius law:

DS ¼ D0 exp �QS

RT

� �

where D0 is the preexponential factor, and QS is the activation energy for solid diffu-sion. Other factors involved in grain growth are also temperature-dependent, such assurface energies, porosity (due to sintering shrinkage), solid grain contiguity, and

GRAIN-GROWTH MASTER CURVE 145

solid solubility in the diffusive phase. As a result of this complex temperature sensi-tivity, the grain-growth rate constant K exhibits an overall exponential dependence onabsolute temperature that can be expressed as

K ¼ K0 exp �QG

RT

� �

where K0 is the associated preexponential factor, and QG is the apparent activationenergy, which includes the various temperature effects in a single term. As a gener-alized relation, the instantaneous rate of grain growth is expressed as follows:

dG

dt¼

K0

3G2exp �

QG

RT

� �

A master curve for grain growth is constructed based on the integral work during theheating cycle. In the master-curve treatment, terms related with the microstructure areseparated from those sensitive to temperature. Then, both sets of terms are integratedindependently assuming that: (1) microstructural evolution, both in grain size and inshape, is only a function of density; and (2) the apparent activation energy QG doesnot change during the sintering cycle. Point (1) is true when the same basic atomictransport mechanics are responsible for densification and microstructural evolution,which is most common. Point (2) is satisfied when the proportional contribution ofthese mechanisms does not change during the sintering cycle. When point (2) isnot true because of a change of basic mechanism during the sintering cycle, it isappropriate to divide the thermal cycle into consecutive stages that are integratedseparately. For example, grain growth before a liquid forms is treated separatelyfrom that after the liquid forms. An additional condition is that the powders, theirprocessing, and the compaction pressure are the same for all of the green compacts.This implies that the green density and microstructure are common in all of the com-pacts. When these conditions are met, integration leads to a means to batch differentthermal cycles:

G3 ¼ G30 þ

ðt

0K0 exp �

QG

RT

� �dt

which is transformed into the master-curve form as follows:

G ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiG3

0 þQ3

q

where

Q ¼

ðt

0K0 exp �

QG

RT

� �dt

Although the parameters K0 and QG can be estimated from diffusion concepts, theyare usually determined from experimental data. For a given thermal cycle involving

146 CHAPTER G

a functional relation between time t and temperature T, the grain size G dependsonly on the initial grain size G0, regardless of the thermal path. The conditionimposed in this variation is to minimize the mean residual, r:

r K0, QGð Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1n

XN

i¼1

GM�i

GE�i� 1

� �2vuut

where N is the number of experimental grain-size data measures, i is a dummy vari-able for summation, GM2i is the ith grain size predicted by the master-curve modelwith given K0 and QG values, and GE2i is the ith experimentally measured grainsize. The method is valid when all the grain-growth data from different sinteringcycles lie on a single master curve, the resulting activation energy QG has a reason-able value, and the mean residual term r is small. The practical advantage of themaster-curve approach is to predict grain growth during a complicated thermalcycle with a limited number of grain-size determinations.

D0 ¼ preexponential frequency factor, m2/s

DS ¼ atomic diffusivity, m2/s


G0 ¼ grain size at the start of isothermal growth, m (convenient units: mm)

GE2i ¼ ith experimentally measured grain size, m (convenient units: mm)

GM2i ¼ ith grain-size predicted by the master curve, m (convenient units: mm)

K ¼ growth rate constant, m3/s (convenient units: mm3/s)

K0 ¼ grain-growth rate frequency factor, m3/s (convenient units: mm3/s)

N ¼ number of experimental grain-size measures, dimensionless integer


QS ¼ activation energy for solid diffusion, J/mol (convenient units:kJ/mol)

R ¼ universal gas constant 8.31 J/(mol . K)


i ¼ dummy summation variable, dimensionless integer

r(K0, QG) ¼ mean residual, m3

t ¼ time, s

Q ¼ integral work of sintering, m3.

GRAIN-GROWTH MASTER CURVE, INTERFACIAL REACTIONCONTROL (Park et al. 2007)

In some liquid-phase sintering systems grain growth is by interfacial reactioncontrol. Here the differential form for grain size under nonisothermal conditions is

GRAIN-GROWTH MASTER CURVE, INTERFACIAL REACTION CONTROL 147

given as follows:

dG

dt¼

12G

K0

Texp �

QG

RT

� �

By rearrangement, it is possible to group grain-size terms separately from timeand temperature terms, allowing for the independent integration. This approachassumes grain size can be described by a mean size and that the grain-growthmechanism is unchanged over the temperatures range under study; namely, theapparent activation energy QG does not change during the thermal cycle.Integration then provides a means to link the mean grain size to the integralthermal work as follows:

G2 ¼ G20 þ

ðt

0

K0

Texp �

QG

RT

� �dt

This equation for interfacial reaction-controlled grain growth is transformed into amaster-curve form as

G ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiG2

0 þQ

q

where the parameter Q represents the integral work over the entire thermal cycle,

Q ¼

ðt

0

K0

Texp �

QG

RT

� �dt

The parameters K0 and QG are determined from experimental data. Note that themean grain size depends only on the integral thermal work and the initial grainsize of G0. A problem arises in the case of WC-Co hard metals, where compositionchanges during processing shift the rate constant, such as might result from carbonbalance changes or the spreading of grain-growth inhibitors.


G0 ¼ mean grain size at the start of the isothermal hold, m (convenientunits: mm)


K0 ¼ preexponential factor, m3/s (convenient units: mm3/s)





Q ¼ integral thermal work, m2.

148 CHAPTER G

GRAIN NUMBER CHANGES IN LIQUID-PHASE SINTERING

Several parallel microstructural changes occur as the grain size enlarges during liquid-phase sintering. The number of grains per unit volume decreases as the meansize grows larger. Assuming zero porosity and volume conservation, the cubicgrain-growth law leads to the conclusion that NV, the grain density per unitvolume, will vary as

NV � t�1

where t is the isothermal sintering time.

NV ¼ grain density per unit volume, number/m3

t ¼ time, s.

GRAIN PINNING BY PORES IN FINAL-STAGESINTERING (Ring 1996)

As final-stage sintering densification proceeds, the pores generally tend to be lower inmobility as compared to the grain boundaries. Thus, as the grain size enlarges, itinitially drags the pores. The drag effect retards grain growth, as long as the poresremain attached to the grain boundaries. However, as the pores shrink they provideless relative grain-boundary drag. Further, grain growth causes the pores located ongrain boundaries to distort into lenticular shapes. The resulting steady-state porevelocity depends on the mobility of the pore and is usually controlled by surfacediffusion. The pinning effect is determined by the pore mobility, dihedral angle,and the difference in curvature from the leading surface of the pore versus the trailingsurface. The velocity of the pore VP then determines the rate of grain growth, as longas the pore-boundary system is coupled. Assuming surface diffusion–controlled poremigration from the leading surface (nearly flat) to the trailing surface (concave),

VP ¼DSdgSVV

pG3RT

� �g

where DS is the surface diffusion coefficient (which depends on temperature), d is thethickness of the disrupted bonding layer on the surface, gSV is the solid–vaporsurface energy, V is the atomic volume, R is the gas constant, T is the absolute temp-erature, and G is the grain size. The parameter g is a geometric term that changesslightly with the pore shape and dihedral angle, but is approximately 6. Note thatthe natural variation in grain size within a sintering microstructure will induce a dis-tribution in grain-boundary velocities. Further, the distribution in grain misorientationin the sintering microstructure will add a variation in the dihedral angle. Thus, there isno single pore velocity, but a natural distribution of velocities and pinning conditions.

GRAIN PINNING BY PORES IN FINAL-STAGE SINTERING 149

In turn, pores are slow to move and become stranded if they are large compared to thegrain size. The combination of diminished pinning force and continued grain growtheventually leads to pore-boundary separation, followed by rapid grain growth. Up tothe point of separation, the pinning effect limits grain size G as dependent on the poresurface area per unit volume SV,

1G¼ aþ

b

SV

where b is a temperature-dependent term, and a is a constant linking the initial grainsize and surface area. Temperature plays a key role in the interaction of grain growthand pore attachment to the grain boundaries.

DS ¼ surface diffusion coefficient, m2/s


R ¼ universal gas constant, 8.31 J(mol . K)

SV ¼ surface area per unit volume, m2/m3


VP ¼ pore velocity, m/s (convenient units: mm/s)

a ¼ constant, 1/m

b ¼ constant, 1/m2

g ¼ geometric term near 6, dimensionless



d ¼ surface diffusion layer thickness, m (convenient units: nm)

u ¼ dihedral angle, rad (convenient units: degree).

GRAIN SEPARATION DISTANCE IN LIQUID-PHASE SINTERING

During liquid-phase sintering the solid grains bond at contact points and diffusionincreases those bonds up to a size that satisfies the dihedral angle. For two grainsof equal size G, bonded by a neck of size X, the grain-center separation distance yis given as

y ¼ G cosf

2

� �

where f is the dihedral angle.



y ¼ grain center-to-center separation distance, m (convenient units: mm)


150 CHAPTER G

GRAIN SEPARATION IN CEMENTED CARBIDES

See Mean Free Path, Carbide Microstructure.

GRAIN-SHAPE ACCOMMODATION (Kipphut et al. 1988;Lee and Kang 2001)

Grain-shape accommodation is a necessary condition for densification in low liquid-content compositions during liquid-phase sintering. During grain growth via solutionreprecipitation, grain-shape accommodation allows the solid grains to flatten orreshape in a way that allows closer packing of grains. Accordingly, the solid grainsthat fit together best release liquid that fills the residual pores, annihilating the associ-ated surface energy. In a system where the liquid wets the solid, the grains departfrom a spherical shape because the solid–liquid surface energy is lower than thesolid–vapor of liquid–vapor surface energy. The change in interfacial energy withlocal curvature can be related to the concentration of solid in the liquid C versusthe flat surface equilibrium. Here C0 is captured by the local concentration of solidin the liquid as driven by the grain curvature, thus,

C � C0 1þsV

RT

� �

where the stress s induced by surface energy and interfacial curvature is given by,

s ¼4gLV

G

where gLV is the liquid–vapor surface energy, G is the mean grain size of the solid, andV is the atomic volume. This assumes that the length of diffusion is much less than thegrain size. The characteristic diffusion distance is proportional to the average liquid-layer thickness between grains. In addition, the diffusional flux also depends on thesolid solubility in the liquid. Hence, in systems with a deficiency of liquid there is aneed to induce grain-shape accommodation to fill pores and reduce surface energy.The rate of densification and grain-shape accommodation are both dependent on thediffusivity times the solubility, and vary inversely with the grain size squared.Similar equations apply to the coalescence of solid grains during liquid-phase sinteringwhere the surface energy corresponds to the solid–liquid interface.

C ¼ local concentration, m3/m3 or dimensionless

C0 ¼ flat-surface equilibrium concentration, m3/m3 or dimensionless






s ¼ stress, Pa.

GRAIN-SHAPE ACCOMMODATION 151

GRAIN SIZE (Underwood 1970)

In opaque polycrystalline solids, the true grain size is not evident from typical cross-sectional microstructures, so the grain size is estimated from random two-dimensionalsections. This approach assumes no bias in the section plane with respect to themicrostructure. The grain size is estimated from the polished microstructure usingrandom intercept, equivalent circle, or outer touching circle techniques, but therandom intercept is the most common. The mean two-dimensional grain interceptsize G2D is determined from the total test-line length LL divided by the magnificationM and the number of boundary intercepts NB over the test line,

G2D ¼LL f

NBM

where f is the fractional density. Since a pore is an empty space that separates twograins, each pore intercept is equivalent to one grain boundary. In random sectionsthe grains appear smaller than their maximum dimension; hence, the three-dimensionalgrain size G3D is larger by an average factor of 1.5, giving G3D¼ 1.5G2D.Mathematical transformation of the intercept size to the true grain size requires assump-tion of a grain-size distribution. For the special case of monosized spheres, the three-dimensional grain size G3D is given by the number of intersections per unit lengthof test line NL and the number of features per unit cross-sectional area NA as follows:

G3D ¼4p

NL

NA

G2D ¼ two-dimensional grain intercept size, m (convenient units: mm)

G3D ¼ three-dimensional grain size, m (convenient units: mm)

LL ¼ test-line length, m (convenient units: mm or mm)

M ¼ magnification, dimensionless

NA ¼ number of features per unit cross-sectional area, 1/m2

NB ¼ number of boundary intercepts, dimensionless

NL ¼ number of intersections per unit test line, 1/m

f ¼ fractional density, dimensionless.

GRAIN-SIZE AFFECT ON COORDINATION NUMBER

The coordination number is the number of first nearest neighbors that are touching agiven grain or particle. In a full-density sintered microstructure, the grains are not ofequal size, so the coordination number NC depends on the grain size G relative to themean grain size Gm,

NC ¼ 2þ 11G

Gm

� �2

152 CHAPTER G

This relation says that the average grain will have 13 sides at full density; a value closeto the generally accepted 14 sides for a tetrakaidecahedron. When the structure con-tains porosity, the coordination number will be smaller in proportion to the porosity.


Gm ¼ mean grain size, m (convenient units: mm)

NC ¼ grain coordination number, dimensionless.

GRAIN-SIZE DISTRIBUTION FOR LIQUID-PHASE SINTEREDMATERIALS (German 1996)

In liquid-phase sintered materials, the cumulative two-dimensional grain-sizedistribution is usually measured by linear intercepts. When measured by the linear-intercept size, distribution in grain size is best described by a Raleigh distributionas follows:

F(L) ¼ 1� exp(�0:7L2)

where F(L) is the cumulative fraction of grains of normalized size L ¼ G/G50, whereG is the grain size and G50 is the median grain size, meaning that L is a normalizedgrain size. Figure G3 plots the distribution.

F(L) ¼ normalized cumulative grain-size distribution, dimensionless [0, 1]


Figure G3. The cumulative grain-size distribution based on linear intercepts for liquid-phasesintered materials, where the grain-intercept size is normalized to the median size.

GRAIN-SIZE DISTRIBUTION FOR LIQUID-PHASE SINTERED MATERIALS 153

G50 ¼ median grain size, m (convenient units: mm)

L ¼ normalized grain size (L ¼ G/G50), dimensionless.

GRAIN-SIZE DISTRIBUTION FOR SOLID-STATE SINTEREDMATERIALS (Aboav and Langdon 1969)

Most sintered materials are opaque; hence, determination of the true grain-sizedistribution is difficult. Since grain-size data are measured by two-dimensionalcross-section procedures, a transformation is one means to estimate the underlyingthree-dimensional distribution. In two dimensions, the grain-size distribution fora solid-state sintered material gives the relative frequency P(G) of grains of size Gas follows:

P(G) ¼ Pm exp �a

ffiffiffiffiffiffiffiG

Gm

r� 1

� �2" #

where Pm is the peak in the frequency distribution (the amount at the mode size), G isthe grain size, Gm is the mode grain size, and a is typically between 2 and 6.Figure G4 plots the frequency and cumulative distributions for a ¼ 5. In two-dimen-sions the average size of a grain varies linearly with the number of sides, where G isproportional to (n 2 2), with n being the number of sides. In other words, the larger

Figure G4. Grain-size distributions for solid-state sintered materials, showing both thefrequency distribution and cumulative distribution based on a median size of 1.0 and shapeparameter a ¼ 5.

154 CHAPTER G

grains have more sides. The number of grains with n sides varies with (n 2 2)1/2. Intwo dimensions, almost no grains have 12 sides and 97% of the grains have between4 and 8 sides. In some instances the cumulative grain-size distribution for solid-statesintering matches the distribution for liquid-phase sintering.


Gm ¼ mode grain size, m (convenient units: mm)

P(G) ¼ relative frequency of grain size G, dimensionless

Pm ¼ peak or mode value in the grain-size distribution, dimensionless

n ¼ number of sides, dimensionless

a ¼ exponent, dimensionless.

GRAIN-SIZE EFFECT ON STRENGTH

See Hall–Petch Relation.

GRAIN SIZE TO PORE SIZE IN FINAL-STAGE LIQUID-PHASESINTERING (H. H. Park et al. 1986)

During final-stage liquid-phase sintering pores remain stable up to the critical con-dition when they are filled by liquid. As grain growth occurs, the filling conditionoccurs when the grain size to pore size ratio is favorable,

G

dP¼

gSS

2gSV

¼ cosf

2

� �

where N is the dihedral angle, G is the grain size, and dP is the pore size. Recognizingthat grain size increases with time, it is easy to see that spontaneous liquid filling ofprogressively larger pores takes place in the latter stages of sintering, assuming thereis no change in pore size.



gSS ¼ grain-boundary energy, J/m2



GRANULATION FORCE

See Agglomerate Force.

GRANULATION FORCE 155

GRANULE STRENGTH

See Agglomerate Strength.

GREEN DENSITY DEPENDENCE ON COMPACTIONPRESSURE (Jones 1960)

In powder compaction the average green density depends on the average compactionstress. Since wall friction decreases the average stress, thicker compacts in thepressing direction will naturally have lower densities. On close scrutiny, there aredensity gradients within the compact. If it is complicated in shape, especially ifthere are multiple thicknesses, then the green density can be highly variable withinthe body. As an approximation, the average fractional green density fG will dependon the compaction pressure P approximately as follows:

dfGdP¼ �Q1 ¼ �Q 1� fGð Þ

where 1 is the fractional porosity (1 ¼ 1 2 fG), and Q is a constant that varies with thepowder. Rearranging and integrating gives,

ln1� fG1� fD

� �¼ �QP

where fD is the fractional density at the onset of deformation, which is often near thetap density. This equation does not include particle rearrangement, so the addition ofa term to include early-stage effects gives,

ln1� fG1� fD

� �¼ B�QP

where B is added to account for particle rearrangement. Modified expressions buildfrom this with terms for deformation and particle hardening. Accordingly, ageneric model linking green density to compaction pressure results,

fG ¼ 1� 1� fDð Þ exp(B�QP)

where fD can be approximated by the apparent or tap density. In some cases,a simplified version can be used to link fractional green density fG to compactionpressure P,

fG ¼ fA � A exp(�KP)

156 CHAPTER G

where fA is the apparent density of the powder, and A and K are constants that changewith each powder.

A ¼ material constant, dimensionless

B ¼ rearrangement constant, dimensionless

fA ¼ fractional apparent density, dimensionless

K ¼ constant, 1/Pa

P ¼ compaction pressure, Pa (convenient units: MPa)

fD ¼ fractional density at the onset of deformation, dimensionless

fG ¼ fractional green density, dimensionless

Q ¼ powder-dependent constant, 1/Pa

1 ¼ 1 2 fG ¼ fractional porosity, dimensionless.

GREEN DENSITY DEPENDENCE ON PUNCH TRAVEL

The green density rG in compaction depends on the apparent powder density rA,initial powder fill height H0, and final compacted height H as follows:

rG ¼ rAH0

H

The compacted height can be expressed as a function of the height change DH from theinitial height, which is the change in spacing between the upper and lower punches,

H ¼ H0 � DH

giving the pressed density as a simple function of the change in punch spacing,

rG ¼rAH0

H0 � DH

H ¼ final compact height, m (convenient units: mm)

H0 ¼ initial powder fill height, m (convenient units: mm)

DH ¼ height change from the initial height, m (convenient units: mm)

rA ¼ apparent density, kg/m3 (convenient units: g/cm3)

rG ¼ green density, kg/m3 (convenient units: g/cm3).

GREEN DENSITY FROM REPEATED PRESSING (Fu et al. 2002)

Repeated pressure pulses will give progressive, but diminishingly small green densityincreases. With repeated pressure cycles, it is possible in thousands of cycles to push

GREEN DENSITY FROM REPEATED PRESSING 157

the green density to nearly the theoretical density, and the approach to theoreticaldensity depends on the total energy input,

fG ¼ fA þ aW þ bW2

where fG is the fractional green density, fA is the apparent fractional density, W is thecompaction energy integrated over repeated pressurization cycles, and a and b arepowder-dependent constants. The density cannot exceed 100%, so there is a limit tothe compaction density that bounds the solution. Compaction energy is calculatedfrom the pressure-displacement curve; effectively, the integral of the compaction pressuretimes the change in height divided by the initial height, giving energy per unit volume.

W ¼ total compaction energy, J/m3

a ¼ powder-dependent constant, m3/J

b ¼ powder-dependent constant, m6/J2

fA ¼ apparent fractional density, dimensionless

fG ¼ fractional green density, dimensionless.

GREEN STRENGTH

For a deformable powder, the green strength depends on the bonding between the par-ticles as induced by the compaction pressure. In most cases, the green strength sG

varies with the fractional density as follows:

sG ¼ Cf mG

where C is a constant, fG is the fractional green density, and m is an empiricalexponent often observed to be near 6.

C ¼ strength constant, Pa (convenient units: MPa)


m ¼ empirical exponent, dimensionless

sG ¼ green strength, Pa (convenient units: MPa).

GREEN STRENGTH DISTRIBUTION


GREEN STRENGTH VARIATION WITH FLAWS (Cha et al. 2006)

The flaws in a green body concentrate stress when that body is loaded. For the trans-verse-rupture strength test, if the samples are flawed, then there is a relation between

158 CHAPTER G

the concentrated stress near the flaw, as normalized to the nominal stress without theflaw given by the following equation:

sM ¼ s0 1�2Dt

t

� �1�

2Dl

l

� �

where sM is maximum stress near the flaws, s0 is nominal stress without the flaws, t isthe sample thickness, l is the sample width, Dt is the distance between the flaws andthe tensioned surface, and Dl is the distance between the flaws and the loading axis.Failure in the transverse-rupture test occurs when the maximum stress exceeds thematerial green strength. Based on this concept, flaws located on the surface are ofthe greatest detriment to the green strength.

l ¼ transverse-rupture bar width, m (convenient units: mm)

t ¼ transverse-rupture bar thickness, m (convenient units: mm)

Dl ¼ distance between a flaw and the loading axis, m (convenient units: mm)

Dt ¼ distance between a flaw and the tensioned surface, m (convenient units: mm)

s0 ¼ nominal stress calculated for flaw-free material, Pa (convenient units: MPa)

sM ¼ maximum stress near a flaw, Pa (convenient units: MPa).

GRINDING TIME (Austin 1984)

The time t necessary to obtain a homogeneous product by attrition milling depends onthe agitator rotational speed N (in revolutions per unit time) as follows:

t ¼ Cd2

N1=2

where d is the grinding-ball diameter, and C is an empirical constant that changeswith the process details.

C ¼ empirical constant, s1/2/m2

N ¼ agitator rotational speed, 1/s (convenient units: 1/min or rpm)

d ¼ grinding-ball diameter, m (convenient units: mm)

t ¼ grinding time, s.

GROWTH

See Sintering Shrinkage.

GROWTH 159

H

HALL–PETCH RELATION (Meyers 1985)

The yield strength sY of a sintered or hot consolidated polycrystalline materialfollows an inverse dependence on the grain size, known as the Hall–Petch relation,

sY ¼ s0 þGffiffiffiffiGp

where G is the characteristic grain size with s0, and G represents the materialconstants. The Hall–Petch relation is reported to break down for grain sizes below10 nm for some materials, such as copper.


G ¼ material constant, Pa . m1/2 (convenient units: MPa . m1/2)

s0 ¼ material constant, Pa (convenient units: MPa)

sY ¼ yield strength, Pa (convenient units: MPa).

HARDENABILITY FACTOR (Saritas et al. 2002)

In the postsintering heat treatment of steel, the ability to form martensite in the componentinterior is related to a property known as the hardenability. Composition and porosityare key determinants of hardenability. For full-density steels, tables exist that show theimpact of different alloying elements on the heat-treatment response in terms ofhardenability equivalents. Residual porosity lowers the thermal conductivity, however,and significantly degrades hardenability; thus, powder-metallurgy steels have a poorresponse to heat treatment. The usual focus is on the bar diameter that will produce50% martensite at the core, termed d50, and calculated as follows for full-density steel:

d50 ¼ dCFNiFCrFMnFSiFMo


161

where dC is the core diameter, which depends on the grain size and carbon level, and theFi factors are determined from available tables for the individual element i. For example,the formulas for molybdenum and chromium are as follows:

FMo ¼ 1þ 3:2XMo

and

FCr ¼ 1þ 2:2XCr

where X indicates the alloying concentration in the wt% of molybdenum and chromiumin these examples. In terms of porosity, the decrement in thermal conductivity due topores greatly reduces the core diameter or the hardenability. Conversely, if the poresare all open, then quenching fluid penetration into the pores can increase the hardenabil-ity. In the porosity range from 0.0 to 0.2, the model for the thermal conductivity of theporous material K versus the thermal conductivity of the full-dense material K0 is asfollows:

K ¼ K01� 1

1þ x12

where 1 is the fractional porosity, and x is the pore sensitivity factor. For stainless steels,the pore sensitivity factor is 11, but for many other materials it is not determined. It isrecognized however, that for a heat treatable steel, the change from 10% porosity to14% porosity reduces the hardenability depth by a factor of 2.

Fi ¼ hardenability factor for the ith element, dimensionless

K ¼ thermal conductivity, W/(m . K)

K0 ¼ full-dense thermal conductivity, W/(m . K)

Xi ¼ alloying concentration for ith element, wt%

d50 ¼ diameter giving 50% martensite at the core, m (convenient units: mm)

dC ¼ core diameter, m (convenient units: mm)


x ¼ pore sensitivity factor, dimensionless.

HARDNESS (Sherman and Brandon 2000)

Hardness is a commonly used term to denote pressing an indenter into a test materialat a slow strain rate to measure the resistance to penetration. Often the indenter is asharp-tipped diamond or hardened ball that is loaded so that it pushes into thesurface. Resistance to the penetration is determined by the yield strength, work hard-ening, and ultimate strength. The depth of penetration, length of impression, or area

162 CHAPTER H

of impression is used in various hardness scales. In general, hardness H is defined asthe applied force F divided by the required area A,

H ¼ F

A

which gives units of strength, so it is often related to the yield strength of the materialsY. One common relation is given as follows:

H ¼ CsY

where C is the proportionality constant. Using the Vickers scale (VHN) the constantC is often approximated at 3. However, many hardness scales are in use, includingRockwell (HRA, HRB, and HRC), Brinell (BHN or HB), Knoop (KHN), as wellas the Vickers scale.

A ¼ area, m2 (convenient units: mm2 or mm2)

BHN ¼ HB ¼ Brinell hardness number, kgf/mm2 (or Pa)

C ¼ proportionality constant, dimensionless

F ¼ force, N (convenient units: kN)

H ¼ hardness, Pa (convenient units: GPa)

HRA ¼ Rockwell hardness number in A scale, kgf/mm2 (or GPa)

HRB ¼ Rockwell hardness number in B scale, kgf/mm2 (or GPa)

HRC ¼ Rockwell hardness number in C scale, kgf/mm2 (or GPa)

KHN ¼ Knoop hardness number, kgf/mm2 (or GPa)

VHN ¼ Vickers hardness number, kgf/mm2 (convenient units: GPa)

sY ¼ yield strength, Pa.

[Also see Brinell Hardness, Knoop Hardness, Rockwell Hardness, and VickersHardness Number]

HARDNESS VARIATION WITH GRAIN SIZE IN CEMENTEDCARBIDES (Luyckx 2000)

In WC-Co hard metals, also known as cemented carbides, the Vickers hardnessnumber VHN depends on the carbide grain size GWC in a manner that follows theHall–Petch relation, namely:

VHN ¼ 1178� 1326VCo þ654� 497VCoffiffiffiffiffiffiffiffiffiffi

GWC

p

HARDNESS VARIATION WITH GRAIN SIZE IN CEMENTED CARBIDES 163

where VCo is the volume fraction of cobalt. Note that the grain size is applied inmicrometers for this formulation.

GWC ¼ tungsten carbide grain size, mm

VCo ¼ volume fraction of Co, dimensionless [0, 1]

VHN ¼ Vickers hardness number, kgf/mm2 (convenient units: GPa).

HEATING-RATE EFFECT IN TRANSIENT LIQUID-PHASESINTERING (German 1996)

Heating rate has an effect on the liquid quantity and amount of densification duringtransient liquid-phase sintering. More swelling occurs at slower heating rates due tochemically driven diffusional homogenization. As the additive homogenizes by solid-state diffusion, there is a corresponding reduction in the quantity of liquid that formsat the peak temperature. The liquid quantity and its duration determine the net shrink-age, however, a loss of liquid because of diffusion during slow heating results indiminished densification. The relation between the volume fraction of liquid VL

that forms, additive concentration C, and heating rate dT/dt is given as follows:

1� VL

jC

� �1=3

¼ kTL

dT=dt

� �1=2

where j is the proportionality constant, k is the rate constant, TL is the liquid-formation temperature, and t is the time. Parameters j and k are system-specific con-stants. Fast heating suppresses solid-stage bonding, leading to better densification intransient liquid-phase sintering. Alternatively, in cases where controlled porosity isthe technical objective, slow heating is desired, since that reduces the liquid volume.

C ¼ liquid-additive concentration, m3/m3 or dimensionless

T ¼ temperature, K

TL ¼ liquid-formation temperature, K

VL ¼ liquid-volume fraction, dimensionless [0, 1]

dT/dt ¼ heating rate, K/s

t ¼ time, s

k ¼ rate constant, 1/s

j ¼ proportionality constant, dimensionless.

HEAT TRANSFER IN SINTERED MATERIALS (Hsu 2005)

A body with a high level of residual porosity presents a complex problem in deter-mining the heat transfer, since all the constituents contribute to heat flow. Furtherthe problem is complicated by the motion of the fluid phase in the pores. In some

164 CHAPTER H

instances heat transfer is improved by fluid mobility, such as in heat pipes. However,in the typical case it is assumed that the fluid is stagnant and in local thermal equili-brium with the solid porous structure. Here various mixture models are used to deter-mine the effective properties of the two-phase body. For example,

(rCP)M@T

@t¼ r � [krT]

where T is the temperature, k is the effective thermal conductivity of the porous bodywith saturated pores, CP is the heat capacity, r is the density, and t is time. The Msubscript indicates that this is the property for the mixture of fluid and porousbody, given by a simple volumetric rule of mixtures as follows:

(rCP)M ¼ 1rFCPF þ (1� 1)rSCPS

In this form 1 is the fractional porosity, r is the density, and CP is the constant-pressure heat capacity, with the F subscript indicating the stagnant fluid phase andthe S subscript indicating the porous solid phase. If there are unsaturated pores,then the problem requires inclusion of yet another term to reflect the voids. Also,if the fluid undergoes motion or convection in the pores, then the analysisbecomes more complicated.

CP ¼ constant-pressure heat capacity, J/(kg . K)

T ¼ temperature, K

t ¼ time, s



r ¼ density, kg/m3 (convenient units: g/cm3)

F subscript indicates fluid

M subscript indicates mixture

S subscript indicates solid.

HEAT-TRANSFER RATE IN MOLDING

See Cooling Rate in Molding.

HERRING SCALING LAW (Herring 1950)

A fundamental problem encountered in sintering is the determination of particle-sizeeffects on the response. One important link is between a change in particle size and

HERRING SCALING LAW 165

the required change in sintering time needed to reach an equivalent level of densifica-tion. This relation is known as the Herring scaling law. It makes several assumptionsbetween the particle size and sintering time based on the operational sintering mech-anism. Inherently the approach is inaccurate, since densification occurs from multiplemechanisms, yet the underlying assumption is that sintering is done by a single mech-anism. The Herring scaling law assumes that the time t1 required to sinter a particle ofdiameter D1 to achieve a sintered neck size of X1 is known. Then the effect of achange in particle size can be predicted. The sintering time t2 for a particle of sizeD2 to reach the same neck size ratio (X1/D1 ¼ X2/D2) is given as,

t2t1¼ D2

D1

� �m

where m is the scaling-law exponent. This scaling-law exponent varies with thesintering mechanism. In the original treatment, the m values were determined forseveral mechanisms as follows: m ¼ 1 for viscous flow and plastic flow, m ¼ 2 forevaporation–condensation, m ¼ 3 for volume diffusion, and m ¼ 4 for surfacediffusion and grain-boundary diffusion. Since many materials densify by grain-boundary-diffusion controlled sintering (m ¼ 4), the generalized Herring scalinglaw holds that a 2-fold increase in particle size requires a 16-fold increase in sinteringtime to achieve the same degree of sintering. This model fails to include latter-stagesintering microstructure coarsening, so it is invalid when applied beyond the firststage of sintering, namely, X/D , 0.3 is an upper limit.

D1 ¼ particle size for particle 1, m (convenient units: mm)

D2 ¼ particle size for particle 2, m (convenient units: mm)

m ¼ scaling-law exponent, dimensionless

t1 ¼ sintering time for particle 1, s

t2 ¼ sintering time for particle 2, s

X1 ¼ neck size for particle 1, m (convenient units: mm)

X2 ¼ neck size for particle 2, m (convenient units: mm)

X/D, X1/D1, X2/D2 ¼ neck-size ratio, dimensionless.

HERTZIAN STRESS

See Elastic Deformation Neck-size Ratio.

HETERODIFFUSION

See Mixed-Powder Sintering Shrinkage.

166 CHAPTER H

HETEROGENEOUS NUCLEATION (Porter and Easterling 1981)

Most treatments of nucleation assume homogenous conditions, but most practiceinvolves heterogeneous conditions. In powder fabrication from vapor phases orliquids, heterogeneous nucleation of the solid is favored because it has a lowerenergy barrier. Nucleation occurring on sites that reduce the energy of first-phasechange—container walls, other particles, impurities, or intentional seeds—isdefined as heterogeneous nucleation. Assuming some form of wetting onto theheterogeneous site, then the energy barrier is reduced. For the typical treatment ofa solid–liquid–vapor equilibrium, the wetting is measured by the contact angle u.Accordingly, the critical free-energy barrier for transformation DG� is reducedfrom the homogeneous case as follows:

DG� ¼ 16pg 3

3DG2S

S

where g is the surface energy (solid–vapor or solid–liquid, depending on the situ-ation) of the new phase against its environment, DGS is the free-energy associatedwith the solid formation (from either liquid or vapor), and S is a wetting shapefactor, which depends on the contact angle as follows:

S ¼ 14

(2þ cos u)(1� cos u)2

Because of substrate wetting, the critical nucleus size is reduced for heterogeneousversus homogeneous nucleation, meaning heterogeneous nucleation is greatly pre-ferred if the contact angle is small.

S ¼ wetting shape factor, dimensionless

DGS ¼ volumetric free-energy change with solid formation, J/m3

DG� ¼ critical free energy for the transformation, J

g ¼ surface energy, J/m2


HIGH SOLID-CONTENT GRAIN GROWTH

See Grain Growth in Liquid-phase Sintering, Diffusion Control at High SolidContents.

HOMOGENEITY

See Segregation Coefficient.

HOMOGENEITY 167

HOMOGENEITY OF A MICROSTRUCTURE (Sebastian andTendolkar 1979)

A bulk material tested for chemistry may not reflect the heterogeneity observed at asmaller scale in the sintered material. This is a special problem, since many sinteredmaterials are formed by mixing powders that diffusionally homogenize during sinter-ing. If the chemical testing is applied to smaller regions, then there is a size scale(scale of scrutiny) where a regional variation becomes apparent. A simple test formicrostructure homogeneity is attained via a homogeneity index H taken fromrepeated microhardness tests,

H ¼ 100S

HM

where S is the standard deviations, and HM is the mean microhardness determination.The Vickers hardness test is usually used, because of the small indentation, but otherhardness tests can be specified. The scale of scrutiny is inherently a part of homogen-eity, and when the testing is performed on many small test spots, the homogeneitydecreases.

H ¼ homogeneity index, dimensionless

HM ¼ mean hardness, Pa or kg/m2 (convenient units: GPa)

S ¼ standard deviation, Pa or kg/m2 (convenient units: GPa).

HOMOGENEITY OF MIXED POWDERS

See Mixture Homogenization Rate.

HOMOGENEOUS NUCLEATION (Porter and Easterling 1981)

The classic model for a phase transformation occurs when a liquid is chilled to atemperature below the melting temperature to form a homogeneous dispersion ofsolid nuclei in the undercooled liquid. For the transformation of liquid to solidbelow the equilibrium melting temperature, the free energy of the stable solid islower than the free energy of the undercooled liquid, hence, there is a free-energydifference per unit volume DGV that is negative (energy is released on solidification).It is often reasonable to assume that this free-energy difference varies linearly withthe undercooling DT (T is the temperature, and TM is the equilibrium melting temp-erature, so DT ¼ TM 2 T ), giving,

DGV ¼ DG0DT

168 CHAPTER H

where DG0 represents the change in free-energy with the temperature below the equi-librium solidification temperature TM. To form the solid nucleus requires the creationof solid–liquid surface energy gSL at the interface between the nucleus and the sur-rounding liquid. Thus, there is a total free-energy change DGT for a solid nucleus ofradius R to form in the undercooled liquid,

DGT ¼ 4pR2gSL þ43pR3DGV

The total system energy increases with the formation of a small nucleus.Consequently, transformation is unfavorable unless a nucleus larger than the criticalsize is formed. The critical-sized nucleus R� is derived using differentiation withrespect to the nucleus radius, giving

R� ¼ � 2gSL

DGV

The negative sign reflects the fact that this is an energy reduction, since the volumetricfree energy DGV is negative and increases in magnitude with undercooling. This saysthat it is easier to randomly form a solid nucleus with increasing undercooling. At thesame time, atomic motion slows as the temperature reduces, which eventually lowersthe transformation rate.

R ¼ radius of nucleus radius, m (convenient units: nm)

R� ¼ critical radius of nucleus, m (convenient units: nm)

T ¼ temperature, K

TM ¼ melting temperature, K

DG0 ¼ change in free energy with temperature below TM, J/(K . m3)

DGT ¼ total free-energy change, J

DGV ¼ free-energy change per unit volume, J/m3

DT ¼ undercooling, K

gSL ¼ solid–liquid surface energy, J/m2.

HOMOGENIZATION IN SINTERING (Masterller et al . 1975)

Mixed powders are used to form compounds or alloys during sintering by interdiffu-sion of the species. In the formation of alloys from mixed powders, the isothermaldegree of homogenization H is defined as the point-to-point chemistry variation.This homogeneity varies with the controlling diffusion rate as follows:

H � DV t

l2

HOMOGENIZATION IN SINTERING 169

where l is the scale of the microstructural separation, or segregation length, DV is thevolume diffusivity of the lower concentration species (which has an Arrhenius temp-erature dependence), and t is the hold time at temperature. The scale of segregation asmeasured by l primarily depends on the particle size, concentration, microstructure,and initial mixing of the powders. Thorough mixing and small particle sizes aid hom-ogenization by reducing l for any given composition. A higher temperature increasesthe diffusivity and increases homogeneity.

DV ¼ diffusivity, m2/s

H ¼ degree of homogenization, dimensionless

t ¼ hold time, s

l ¼ microstructure scale, m (convenient units: mm).

HOMOGENIZATION RATE IN POWDER MIXING

See Mixture Homogenization Rate.

HOOKE’S LAW (Meyers 1985)

Stress and strain are linearly proportional in the elastic region for crystalline materials.This proportionality results in a material constant termed the elastic modulus orYoung’s modulus. In the one-dimensional form, Hooke’s law is presented as,

s ¼ 1E

where s is the stress, 1 is the strain, and E is the elastic modulus, which is also calledYoung’s modulus. A three-dimensional matrix form is used in complex loadingsituations, and in some materials the elastic modulus depends on orientation, so itis not a single-valued material property.




HOT PRESSING IN THE PRESENCE OF A LIQUID PHASE

See Pressure-assisted Liquid-phase Sintering.

170 CHAPTER H

I

IMPREGNATION

See Infiltration Pressure.

INERTIAL-FLOW EQUATION (Scheidegger 1960)

The flow of a compressible fluid in a porous body generally exhibits energy losses athigher flow velocities due to inertial effects. Unlike a capillary tube, a sintered bodyhas nonuniform and crooked pores that cause the flowing gas to repeatedly changedirection. With each of these direction changes there is a loss in energy as the gastwists and turns inside the pore structure. Because the pore geometries in sinteredmaterials are considered to be complex in size and shape and tortuosity, Darcy’slaw generally overestimates flow rates at higher velocities and gas pressures. Toaccount for the energy loss associated with the tortuous flow path a second term isadded to Darcy’s law for the gas inertial effects,

P22 � P2

1

2P0¼ LhV

aþ

LrgV2

b

where a is the permeability coefficient, b is the inertial coefficient, rg is the gas densityat standard conditions (one atmosphere pressure and room temperature), h is the gasviscosity, V is the gas velocity passing through the test, and P0 is the atmosphericpressure. The upstream pressure P2 is larger than the downstream pressure P1, andthey are separated by the sample length L. The ratio b/a provides an index of the poreshape and tortuosity. This equation is generally valid for superficial velocities rangingfrom 0.1 m/s to 20 m/s, but it does not work in the choked-flow region.


P0 ¼ atmospheric pressure, 105 Pa (convenient units: MPa)


171

P1 ¼ downstream pressure, Pa (convenient units: MPa)

P2 ¼ upstream pressure, Pa (convenient units: MPa)



b ¼ inertial coefficient, m


rg ¼ gas density, kg/m3 (convenient units: g/cm3).

INFILTRATION DEPTH (Martins et al. 1988)

When liquid metal is infiltrated into a porous body, the melt feeds through thesurface-connected open pores. After the melt forms, the depth of infiltration hvaries with the square root of time t as follows:

h ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidPtgLV cos u

4h

s

where dP is the pore size, gLV is the liquid–vapor surface energy, u is the contactangle between the solid and liquid, and h is the liquid viscosity. Since viscosityand wetting properties generally decrease with higher temperatures, the depth of infil-tration improves as temperature increases.


h ¼ infiltration depth, m (convenient units: mm)

t ¼ time, s


h ¼ melt viscosity, Pa . s


INFILTRATION PRESSURE (Washburn 1921)

Capillarity has an effect equivalent to an external pressure with respect to liquid pen-etration of pores. In an infiltration event the capillary pressure pulls a wetting liquidinto the pores by a wicking action. The infiltration pressure derives from theWashburn equation for capillary rise. For a wetting liquid, the capillary pressurerise DP over ambient pressure is responsible for pulling the infiltrant into thepores; this varies with the inverse of the pore diameter dP as follows:

DP ¼ 4gLV cos u

dP

172 CHAPTER I

where gLV is the liquid–vapor surface energy, and u is the solid–liquid–vaporcontact angle. This model uses a cylindrical capillary tube for the pore shape.


DP ¼ capillary pressure rise, Pa



INFILTRATION RATE (Martins et al. 1988)

The flow of liquid into or out of a green or sintered porous body is usually modeledbased on an equation attributed to Poiseuille. Assuming the pores can be treated ascapillary tubes and the fluid is wetting, then a relation for the infiltration rates isgiven as follows:

dh

dt¼ dP

8hgLV

h� gr

4

� �

where h is the distance of infiltration, t is the time, dh/dt is the infiltration rate, dP isthe pore diameter, gLV is the liquid–vapor surface energy, h is the fluid viscosity, g isthe gravitational acceleration, r is the fluid density.


dh/dt ¼ infiltration rate, m/s


h ¼ infiltration distance, m (convenient units: mm)

t ¼ time, s



r ¼ fluid density, kg/m3 (convenient units: g/cm3).

INHIBITED GRAIN GROWTH

See Zener Relation.

INITIAL-STAGE LIQUID-PHASE SINTERING STRESS

See Sintering Stress in Initial-stage Liquid-phase Sintering.

INITIAL-STAGE LIQUID-PHASE SINTERING STRESS 173

INITIAL-STAGE NECK GROWTH (German 1996)

Initial-stage sintering models generally trace their origin to the calculations of Frenkelfor viscous-flow sintering of two equal-sized spheres. The Frenkel model suggested arelation between time and the size of the sinter bond. It was followed by theKuczynski model and many similar treatments of this core problem. Figure I1shows the difference between neck growth in the initial stage without shrinkage bysurface transport, and the particle center–center approach by bulk transport. Thelatter sintering geometry gives densification, where mass is removed from thecontact grain boundary and deposited to grow the neck. Although most solutionsare based on several approximations, still there is a consistent finding that theneck-size ratio X/D as a function of a kinetic term B, isothermal time t, and particlesize D is as follows:

X

D

� �n

¼ Bt

Dm

Figure I1. Two variants on the neck growth process during sintering of two spheres termedsurface transport and bulk transport. If the atomic flux to the neck is from surface sources,such as evaporation–condensation (E–C) or surface diffusion (SD), then there is neckgrowth but no shrinkage. On the other hand, if the atomic flux to the neck originates fromthe particle interior or grain boundary (GBD), such as by volume diffusion (VD), plasticflow (PF), or grain-boundary diffusion, then neck growth is accompanied by shrinkage.

174 CHAPTER I

The exponent n depends on the sintering mechanism, and typical values are tabulatedin the following list. The particle-size exponent m is known as the Herring scaling-law exponent. The neck size is given by the diameter X of the neck bonding theparticles together, D is the sphere diameter, t is the isothermal sintering time, B isthe kinetic term treated below where temperature T enters in an exponential formas associated with the mass-transport process, delivering neck growth,

B ¼ B0 exp � Q

RT

� �

Typically B0 is a collection of material, crystal structure, and geometric constants, R isthe gas constant, T is the absolute temperature, and Q is activation energy associatedwith the atomic-transport process. The activation energy varies with each of themechanisms. The values of n, m, and B also depend on the mechanism of masstransport, as described in the following list.

Mechanism n m B0

Viscous flow 2 1 3gSV/(2h)Plastic flow 2 1 9p(gSVDv/b2)(V/RT)Evaporation–condensation 3 2 (p/2)1/2(3PgSV/rT

2)(M/RT)3/2

Lattice (volume) diffusion 5 3 80DvgSVV/(RT)Grain-boundary diffusion 6 4 20dDbgSVV/(RT)Surface diffusion 7 4 56dDsgSVV/(RT)

B ¼ kinetic term, mm/s (the value of m is listed in the text)

B0 ¼ preexponent kinetic parameter, mm/s (the value of m is listed inthe text)


Db ¼ grain-boundary diffusivity, m2/s

Ds ¼ surface diffusivity, m2/s

Dv ¼ volume diffusivity, m2/s

M ¼ molecular weight, kg/mol (convenient units: g/mo)

P ¼ vapor pressure, Pa




X ¼ neck diameter, m (convenient units: mm)

X/D ¼ neck-size ratio, dimensionless


m ¼ Herring scaling-law exponent, dimensionless

n ¼ neck-growth exponent, dimensionless

INITIAL-STAGE NECK GROWTH 175


V ¼ molar (atomic) volume, m3/mol


d ¼ diffusion-layer width or thickness, m (convenient units: nm)

h ¼ viscosity, Pa . s


[See also Herring Scaling Law.]

INITIAL-STAGE SINTERING

See Surface Diffusion–Controlled Neck Growth.

INITIAL-STAGE SINTERING MODEL


INITIAL-STAGE SINTERING STRESS

See Sintering Stress in Initial-stage Solid-state Sintering.

INITIAL-STAGE SINTERING SURFACE-AREA REDUCTION

See Surface-area Reduction Kinetics.

INITIAL-STAGE SHRINKAGE IN SINTERING(Kingery and Berg 1955)

Shrinkage DL/L0 during the initial stage of sintering follows a kinetic law thatinvolves the same terms as initial-stage neck growth,

DL

L0

� �n=2

¼ Bt

2nDm

where n/2 is typically between 2.5 and 3, D is the particle diameter, and t is theisothermal time. The parameters B, n, and m are the same as described earlier forthe initial-stage neck growth during sintering. This relation is valid only forspheres in initial-point contact (no compaction) up to a total neck size ratio of0.3, corresponding to an approximate upper limit of 3% shrinkage. Since

176 CHAPTER I

surface diffusion and evaporation–condensation do not induce sintering shrink-age, this equation is only applicable to sintering dominated by viscous flow,plastic flow (including dislocation climb), volume diffusion, and grain-boundarydiffusion.

B ¼ kinetic term, mm/s (the exponent m varies with the transportmechanism)





t ¼ time, s


DL/L0 ¼ sintering shrinkage, dimensionless.

[See also Initial-stage Neck Growth during Sintering.]

INJECTION-MOLDING VISCOSITY


IN SITU SINTERING STRENGTH (Xu et al. 2002)

During the sintering of a porous structure, the strength varies with the square of theneck size X to particle size D ratio (X/D)2, fractional density f, and number of touch-ing grains via the coordination number NC, giving a measured in situ sinter strengths as follows:

s ¼ fNCs0

pK

X

D

� �2

where K is the stress amplification factor that reduces the test strength in proportionto the neck curvature. The inherent full-density strength for the material s0 istemperature-dependent, since all materials exhibit thermal softening where thestrength approaches zero at the melting temperature. From stress-concentrationconcepts, K is inversely proportional to the curvature at the base of the neck,which depends on the neck-size ratio (X/D), giving,

K ¼ 12

D

X

� �2

IN SITU SINTERING STRENGTH 177

The neck-size ratio X/D maximum is near 0.5, but in cases where the dihedral anglef is below 608 or 1.05 rad, the final neck size is limited as follows:

X

D¼ sin

f

2

� �

Since X/D peaks at 0.5, K effectively becomes a constant of 2 after the initial stage ofsintering.


K ¼ stress amplification factor, dimensionless

NC ¼ grain coordination number, dimensionless





s ¼ in situ strength, Pa (convenient units: MPa)

s0 ¼ full-density strength, Pa (convenient units: MPa).

INTEGRAL WORK OF SINTERING

See Master Sintering Curve.

INTERDIFFUSION

See Mixed-powder Sintering Shrinkage.

INTERFACE-CONTROLLED GRAIN GROWTH (Kang 2005)

In liquid-phase sintering there are two potentially rate-controlling grain-growth steps;diffusion through the liquid and/or reaction at the solid–liquid interface. Duringcoarsening due to interfacial reaction–controlled grain growth, the diffusion rate inthe liquid is faster in comparison to the reaction rate, often due to limited availabilityof reaction sites. This typically occurs when the grains are flat faced. The flat facelacks defective sites onto which atomic or molecular addition or dissolution mighttake place; fundamentally, this limited number of reaction sites is the cause forinterface-controlled grain growth. In these cases, the grain growth law gives themean grain size G versus time t as follows:

G2 ¼ G20 þ

256gSLCVkrt

81RT

178 CHAPTER I

where kr is the interfacial-reaction rate constant, and G0 is the initial mean grain size att ¼ 0. The solid–liquid surface energy is given by gSL, C is the solubility of the solidin the liquid, V is the molar volume, R is the universal gas constant, and T is theabsolute temperature. For a controlled reaction, the activation energy is usuallyhigh, resulting in high sensitivity to sintering temperature.






kr ¼ interfacial-reaction rate constant, m/s

t ¼ time, s



[See also Grain-growth Master Curve, Interfacial-reaction Control.]

INTERMEDIATE-STAGE LIQUID-PHASE SINTERING MODEL(Kingery 1959)

In liquid-phase sintering, most of the systems show densification, which is controlledby solid diffusion in the liquid phase, a process termed solution reprecipitation. Thedominance of diffusion control is evident if the grains are round in shape and followthe cubic coarsening law (G3 � t, where G is the grain size and t is the time) duringdensification. An expression given by Kingery on the diffusion-controlled shrinkageDL/L0 in the intermediate stage is as follows:

DL

L0

� �3

¼ 192dVgLVDLCt

G4RT

where d is the thickness of the liquid layer between the grains, V is the atomicvolume, gLV is the liquid–vapor surface energy, DL is the diffusivity of the solidin the liquid, C is the solid concentration in the liquid, t is the isothermal holdtime, R is the gas constant, T is the absolute temperature, and G is the grain diameter.Cases where the grains have flat faces will grow slowly, which is an indication oflimited sites for dissolution and precipitation. The process is termed interface reac-tion control, and the sintering shrinkage is described as follows:

DL

L0

� �2

¼ 16 kRVgLVDLCt

G2RT

INTERMEDIATE-STAGE LIQUID-PHASE SINTERING MODEL 179

with kR is defined as a reaction constant. Here diffusion is relatively fast, but thenumber of sites available for dissolution or precipitation is limited by the structure ofthe grain-face defect, so this reaction rate for dissolution or precipitation at thesolid–liquid interface captures the impediment in terms of the limited surface-siteavailability.

C ¼ solid concentration in the liquid, m3/m3 or dimensionless






kR ¼ reaction constant, 1/m

t ¼ time, s

DL ¼ change in initial length, m (convenient units: mm)



d ¼ liquid-layer thickness, m (convenient units: mm)


INTERMEDIATE-STAGE LIQUID-PHASE SINTERING SHRINKAGE

See Solution-reprecipitation-induced Shrinkage in Liquid-phase Sintering.

INTERMEDIATE-STAGE PORE ELIMINATION (Coble 1961a)

Sintering’s intermediate stage is associated with tubular-shaped pores occurringalong the grain edges as long, tubular voids. The rate of pore elimination dependson the diffusion of vacancies away from the pore, and thus in turn is dominated bythe pore size. The pore diameter is the fundamental determinant for the rate ofvacancy emission, since the pore length is typically large by comparison.Calculation of the change in fractional porosity 1 with sintering time t in the inter-mediate stages derives from the following:

d1

dt¼ �JANV

where N is the number of pores per unit volume, J is the atomic flux, A is the poresurface area, and V is the atomic volume. Vacancy migration from the pore resultsin pore shrinkage. In his solution, Coble assumed 12 pores per grain for N basedon a tetrakaidecahedron grain shape (each 14-sided grain has 36 edges, each

180 CHAPTER I

shared by 3 grains). The atomic flux depends on the diffusivity and concentrationgradient according to Fick’s first law. The negative sign indicates pore shrinkage.The concentration gradient between the grain boundary and pore surface at avacancy concentration C determines the atomic flux,

lnC

C0

� �¼ 2gSVV

RTdP

In this equation, gSV is the surface energy, dP is the pore diameter (assumed cylind-rical), R is the gas constant, and T is the absolute temperature. Although not fullyaccepted, the usual assumption is that the vacancy sink is located at the center ofthe grain boundary, with an equilibrium vacancy concentration of C0. This flatinterface equilibrium concentration depends only on temperature for a given material.However, the curved pore surface gives a higher vacancy concentration, so the sourceemitting the vacancies is at the pore surface and the annihilation of the vacancies isat the grain boundaries. As an approximation, the distance from the source to the sinkis assumed to be approximately G/6, where G is the grain size. Subsequent modelshave tried to set the location of C0 directly, independent of the grain size, but gener-ally the location is assumed to be proportional to the grain size. The vacancy fluxdepends on the change in concentration (C – C0) and the distance (G/6) times thediffusivity DV (assuming volume diffusion of the vacancies out of the pores). Thearea A over which mass flows also depends on the pore size and grain size, and isestimated as follows:

A ¼ p

3dPG

Consequently, the change in fractional density f versus time t gives a densification ratedf/dt that can be estimated as follows:

df

dt¼ ggSVVDV

RTG3

where g is a collection of geometric terms that is typically near 5, but it depends onvarious assumptions, including the means for measuring the grain size G (intercept,area, or volume).

A ¼ pore surface area, m2 (convenient units: mm2)

C ¼ vacancy concentration, 1/m3

C0 ¼ equilibrium vacancy concentration, 1/m3



J ¼ atomic flux, mol/(m2 . s)

INTERMEDIATE-STAGE PORE ELIMINATION 181

N ¼ number of pores per unit volume, 1/m3





d1/dt ¼ rate of pore elimination, 1/s


g ¼ geometric constant, dimensionless

t ¼ time, s



1 ¼ fractional porosity, dimensionless [0, 1].

[See also Tetrakaidecahedron.]

INTERMEDIATE-STAGE SINTERING-DENSITY MODEL(Beere 1976)

The intermediate stage of sintering corresponds to a neck-size to particle-size ratiothat is greater than 0.3, while the pores are still open. In the final stage of sinteringthe pores are closed and no longer connected to the process atmosphere. When sin-tering starts with loose powders, the intermediate-stage nominally corresponds to arange of sintered densities from 75 to 92% of theoretical. During this stage of sinter-ing, assuming bulk transport–controlled sintering, the densification rate df/dt isdetermined by the flux of vacancies and atoms (which depends on the pore sizeand temperature), the diffusion distance (which depends on the grain size), and thenumber of pores per volume. When set up and integrated, the density versus timemodel generally leads to an equation of the following form:

fS ¼ fI þ BI lnt

tI

� �

where fS is the fractional sintered density, fI is the fractional density at the beginningof the intermediate stage, and BI is a rate term that has an Arrhenius temperaturedependence. In this model tI is the time corresponding to the onset of the intermediatestage, and t is the total isothermal sintering time, where t is greater than tI. Typically,BI contains an inverse cube dependence on the grain size, reflecting the important roleplayed by the grain boundaries in intermediate-stage sintering densification.

BI ¼ rate constant, dimensionless


182 CHAPTER I

fI ¼ density at start of intermediate stage, dimensionless fraction [0, 1]

fS ¼ sintered density, dimensionless fraction [0, 1]

t ¼ sintering time, s

tI ¼ time at the start of the intermediate stage, s.

INTERMEDIATE-STAGE SURFACE-AREA REDUCTION(German 1978)

During the intermediate stage of sintering substantial surface area still remains in thepowder compact. Unlike the initial stage where sintering is driven by curvaturegradients, in the intermediate stage the pore structure is smoother, so the key driverfor sintering is the excess surface energy. The relation between the surface area(effectively the surface energy) and compact density leads to an ability to followintermediate-stage sintering. Sintering can be monitored using the specific surfacearea S versus the sintering time t, with the general observation that the rate ofsurface-area loss depends on the remaining surface area,

dS

dt¼ �BSa

where dS/dt is the rate of surface-area loss, a is a constant that depends on the trans-port mechanism (such as grain-boundary diffusion), and S is the remaining surfacearea. The parameter B depends on the material, temperature, and microstructure,especially the grain size.

B ¼ rate constant, (m2/kg)12a/s.

S ¼ specific surface area, m2/kg (convenient units: m2/g)

dS/dt ¼ rate of surface-area loss, m2/(kg . s)

t ¼ time, s

a ¼ transport-mechanism constant assumed to be near unity,dimensionless.

INTERRUPTED HEATING-RATE TECHNIQUE

See Dorn Technique.

INVERSE RULE OF MIXTURES

See Composite Density and Mixture Theoretical Density.

INVERSE RULE OF MIXTURES 183

J

JAR MILLING (Austin 1984)

One of the most common milling techniques for comminution of a material intoparticles is by ball milling or jar milling using a horizontal rotating cylinder filledwith tumbling balls. Jar milling is dependent on a proper balance of centrifugalforce and gravity, such that the balls are lifted but not held against the cylinderwalls. Optimal milling depends on the ball size, the mill rotational rate, and thevolume fill of the mill. With respect to the rotation rate, a critical condition isdescribed as follows:

NC ¼bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dM � dBp

where NC is the critical rotational rate for the mill, b is a rotation constant for optimalmilling that changes slightly with the volume fraction of balls in the mill (normallyequal to 0.7 per second), when the inside diameter of the mill is dM and the balls havea diameter dB. The ball diameter is often ignored in this calculation. The falling ballsimpact onto the bed of powder to fracture the particles into smaller pieces. If the millis about 45% filled will balls, this is optimal. During repeated impacts, defects formand grow to fracture the powder, and the efficiency is optimized by a ball diameterabout 30-fold larger than the particle diameter. Since the particle size decreases con-tinuously during jar milling, the efficiency varies over time. Particle fracture oftenoccurs on 458 slip lines, causing multiple fragments. Shear without compression ismost effective with respect to milling efficiency. The impact energy needed to frac-ture the powder increases with the material’s toughness, so jar milling and relatedtechnologies are best applied to brittle materials. Models for predicting the par-ticle-size distribution require input parameters for the material, mill, and operatingconditions. With respect to input mill energy to create a given particle size, aninverse square-root effect is observed. Starting with a particle size of DI, then the


185

energy E required to obtain a final particle size of DF is estimated as follows:

E ¼ gEI1ffiffiffiffiDp

F

� 1ffiffiffiffiDp

Z

� �

where g is a constant that depends on the initial material, and EI is a mill-specificparameter. Since energy delivery is from an electric motor that runs at nearly constantpower (J/s), the required milling time to obtain a target particle size is calculated fromthe mass of powder and the required energy.

DF ¼ final particle size, m (convenient units: mm)

DI ¼ initial particle size, m (convenient units: mm)

E ¼ milling energy, J

EI ¼ mill-specific energy constant, J

NC ¼ critical rotational rate, 1/s (convenient units: 1/min or rpm)

W ¼ energy required to change the particle size from DI to DF, J

dB ¼ ball diameter, m (convenient units: mm)

dM ¼ mill diameter, m

g ¼ material-specific constant, mm1/2

b ¼ milling-rate optimization parameter, m1/2/s.

JET MIXING TIME (Revill 1985)

Jet mixing of powders with liquids is used to form slurries in a continuous process.The mixing event takes place via coaxial flow under turbulent conditions with highReynolds numbers. This is effectively the same behavior as is used in plasma atomi-zation or plasma spraying. The central injection pipe is smaller in diameter and islocated inside a larger-diameter pipe that is used to transport the slurry. Thetypical Reynolds number is over 1000, and best mixing occurs when the outsidetube is operating at a Reynolds number of 5000 and the inside tube is operating ata Reynolds number of 2000. In this case, the time t required to properly mix thetwo streams is given as

t ¼ 150d

V

where d is the diameter of the outer tube and V is the exit velocity of the mixture.

V ¼ mixture exit velocity, m/s

d ¼ outer tube diameter, m (convenient units: mm)

t ¼ mixing time, s.

186 CHAPTER J

K

KAWAKITA EQUATION (James 1983)

A similar density–pressure relation is seen in both cold isostatic pressing and diecompaction. But in cold isostatic pressing there is no deviatoric stress. An equationwidely employed for linking pressed density and compaction pressure in coldisostatic pressing is known as the Kawakita equation. This equation gives thevolume reduction as a function of the pressure P over a wide range of materialsand particles as follows:

C ¼ V0 � V

V0¼ 1� f0

f¼ abP

1þ bP

where V is the volume of the powder after pressing; V0 is the volume of the loosepowder, which has a fractional packing density of f0; and f is the compacted fractionaldensity. The parameters a and b are constants. This equation is rearranged to give analternative form,

P

C¼ 1

abþ P

a

A plot of the term P/C versus P gives a straight line, where the slope is 1/a and theintercept on the P/C axis at zero pressure P is 1/ab. This is illustrated for threepowders in Figure K1. Generally, the slope a relates to the initial porosity, and brelates to the particle geometry and the plastic flow behavior of the material beingcompacted.

C ¼ volume reduction, dimensionless


V ¼ powder volume after pressing, m3

V0 ¼ loose-powder volume, m3


187

a ¼ slope constant, dimensionless

b ¼ plastic flow constant, 1/Pa (convenient units: 1/MPa)

f ¼ compacted fractional density, dimensionless [0, 1]

f0 ¼ initial fractional packing density, dimensionless [0, 1].

KELVIN EQUATION (Keey 1992)

The Kelvin equation applies to the vapor pressure for a curved surface, such as aliquid in a porous body. Depending on the contact angle, a wetting liquid willhave a vapor pressure P that is below the temperature-dependent equilibrium vaporpressure P0, since the meniscus is curved to wet the pores. The estimated vaporpressure associated with a liquid wetting the pores is given relative to the equilibriumvapor pressure as,

lnP

P0

� �¼ 4gLVV cos u

dPRT

where gLV is the liquid–vapor surface energy, V is the molecular volume, u is thecontact angle, dP is the pore diameter, R is the gas constant, and T is the temperature.When the pore size is in the submicrometer-size range the pressure change is

Figure K1. A plot of the Kawakita equation in terms of P/C versus P, showing the straight-line relation for cold isostatic pressing of iron, copper, and stainless steel powders.

188 CHAPTER K

measurable. Contrarily, for larger particles, the vapor-pressure reduction associatedwith the pores is effectively not measurable.

P ¼ vapor pressure over small pore, Pa

P0 ¼ equilibrium vapor pressure, Pa




V ¼ molecular volume, m3/mol



KELVIN MODEL

See Viscoelastic Model for Powder–Polymer Mixtures.

K-FACTOR

K-Factor is a term that is used in industry to describe a radial crush test applied tosleeve bearings. It is equivalent to the bearing strength test. A straight cylindricalbearing is tested upto the peak crushing load, and that load is used to calculate aneffective strength. The maximum load FB is recorded to give the K-factor as follows:

Kfactor ¼ FBd � l

lt2

� �

where l is the cylinder length, d is the cylinder outer diameter, and t is the wall thick-ness. The cylinder length tends to be 1.5 times the cylinder diameter; a commondiameter is 25 mm. Strength determined by this test is not directly comparable toother strength tests.

FB ¼ peak load during crushing, N (convenient units: kN or MN)

Kfactor ¼ radial crush strength, Pa (convenient units: MPa)

d ¼ cylinder outside diameter, m (convenient units: mm)

l ¼ cylinder length, m (convenient units: mm)

t ¼ cylinder wall thickness, m (convenient units: mm).

KINGERY INTERMEDIATE-STAGE LIQUID-PHASESINTERING MODEL

See Intermediate-stage Liquid-phase Sintering Model.

KINGERY INTERMEDIATE-STAGE LIQUID-PHASE SINTERING MODEL 189

KINGERY MODEL FOR PRESSURE-ASSISTED LIQUID-PHASESINTERING

See Pressure-assisted Liquid-phase Sintering.

KINGERY REARRANGEMENT SHRINKAGE KINETICS

See Rearrangement Kinetics in Initial-stage Liquid-phase Sintering.

KISSINGER METHOD (Aggarwal et al. 2007)

The decomposition of a polymer while heating is termed delubrication or thermaldebinding, and depends on an integral combination of time and temperature. Themaster decomposition curve approach allows for integration over a nonisothermalheating cycle to predict the polymer, binder, or lubricant decomposition curve. Theconcept is effective in accommodating any heating cycle. Calculation of the workof decomposition involves identification of an apparent activation energy Q. Thisactivation energy is reflective of the polymer decomposition process and is usuallydetermined from burnout data in a method attributed to Kissinger. It is found byidentification of the temperature Tmax at which the maximum rate of weight lossoccurs at various heating rates as follows:

d

dt� dW

dt

� �¼ 0

and

T ¼ Tmax

at the maximum rate of weight loss, where T is the absolute temperature, W is theweight, and dW/dt is the rate of weight loss. Under the condition of constantheating rate r, where dT/dt ¼ r, the decomposition can be expressed as

rQ

RT2max

¼ K0 exp � Q

RTmax

� �

where Q is the activation energy, R is the gas constant, K0 is the rate constant for thereaction, or via rearrangement and application of a logarithmic version,

lnr

T2max

� �¼ � 1

RTmax

� �Q� ln

Q

K0R

� �

190 CHAPTER K

Thus, a plot of ln[r/(Tmax)2] versus –1/RTmax from the weight loss versus temp-erature experiments with several constant heating rates gives the apparent activationenergy Q for the burnout reaction.

K0 ¼ rate constant, 1/s




Tmax ¼ temperature for maximum rate of weight loss, K

W ¼ weight, kg (convenient units: g)

dW/dt ¼ rate of weight loss, kg/s (convenient units: g/s)

r ¼ dT/dt ¼ heating rate, K/s

t ¼ time, s.

KNOOP HARDNESS (Sherman and Brandon 2000)

The knoop hardness is determined by a microhardness test that relies on a load F thatpresses an elongated diamond pyramid (length, L, is 7.11 times the width) into thesurface of the test material. A sketch of the indenter and the measurement is givenin Figure K2. Measurements are taken from the long diagonal of the impression

Figure K2. The Knoop hardness test indenter and how the resulting impression in the testsample is used to determine hardness.

KNOOP HARDNESS 191

to calculate the hardness KHN from the length of the impression based on thefollowing formula:

KHN ¼ 14:2F

L2

F ¼ load, N

KHN ¼ Knoop hardness number, Pa (convenient units: kgf/mm2)

L ¼ impression length, m (convenient units: mm or mm).

KNUDSEN DIFFUSION

See Vapor Mean Free Path.

KOZENY–CARMAN EQUATION (Scheidegger 1960)

Gas flow through a packed powder, or through the pores open to the external surfaceon a sintered porous body, depends on the pressure drop, gas viscosity, and otherfactors. Inherently, generalizations are required to link the pore structure tothe simple flow attributes as determined by bulk properties such as the porosity.The Kozeny–Carman equation is one that relates the specific surface area S of thecompact to the permeability a and the fractional porosity 1,

S ¼ 1rM

1

1� 1

1

5a

h i1=2

with rM equal to the theoretical density of the material. Normally, the permeability ismeasured by bulk size and flow using Darcy’s law.


a ¼ gas permeability, m2


rM ¼ theoretical density of the material, kg/m3 (convenient units: g/cm3).

[Also see Darcy’s Law and Fisher Subsieve Particle Size.]

KUCZYNSKI NECK-GROWTH MODEL (Kuczynski 1949)

Initially introduced by Frenkel in 1945, the two-particle sintering model shown inFigure K3 treats the growth of a bond or neck between contacting particles as a

192 CHAPTER K

time-dependent process. The size of the neck is controlled by mass transport from theparticle surface (surface transport) or the particle interior (bulk transport) to fill in thesaddle surface at the interparticle neck. Surface transport gives neck growth, but nodensification. Kuczynski relied on the two-particle concept to derive rate equationsfor the neck-size versus time. The resulting approximate laws give the neck-sizeratio as being proportional to time with an Arrhenius temperature dependence. Theintegral form gives the isothermal neck-size ratio X/D as follows:

X

D

� �n

¼ Bt

Dm

where X is the neck diameter, D is the particle diameter, t is the isothermal sinteringtime, B is the kinetic term, m is the Herring scaling-law exponent, and n is themechanism-dependent exponent. The values of parameters n, m, and B depend onthe mechanism as treated by several theoretical derivations. Temperature T appearsin an exponential form,

B ¼ B0 exp � Q

RT

� �

where B0 is a collection of material, crystal structure, temperature, and geometric con-stants; R is the gas constant; T is the absolute temperature; and Q is the activationenergy associated with the atomic transport process. The Kuczynski model is validfor a neck-size ratio X/D below 0.3. In some situations there is an accompanyingshrinkage, but it is not necessary for shrinkage to accompany neck growth.

B ¼ kinetic term, mm/s (the units depend on the mechanism)

B0 ¼ collection of material constants, mm/s (the units depend on themechanism)

Figure K3. A sketch of the Frenkel two-particle sintering model initially generated forviscous flow sintering and subsequently employed by Kuczynski for diffusion processes incrystalline particles where a bond grows between the contacting particles, but there is nodihedral angle at the root of the neck.

KUCZYNSKI NECK-GROWTH MODEL 193







t ¼ time, s


n ¼ mechanism-dependent exponent, dimensionless.

[See Also Initial-Stage Neck Growth.]

194 CHAPTER K

L

LAMINAR FLOW SETTLING

See Stokes’ Law.

LAPLACE EQUATION (Heady and Cahn 1970)

The Laplace equation gives the stress s associated with a curved surface as,

s ¼ g1

R1þ 1

R2

� �

where g is the energy associated with the curved surface (for example, solid–liquid,solid–vapor, or liquid–vapor surface energy), and R1 and R2 are the principal andorthogonal radii of curvature for the surface. For a sphere, both radii are the sameand equal to the radius of the sphere, but during sintering the radii are often oppositein sign. At the saddle surface corresponding to the sintering neck, one radius islocated in the pore, while the second and larger radius is located inside the solid.By convention a surface that is concave (radius is outside the solid) is in compressionand is being forced outward toward a flat surface. Conversely, a surface that isconvex (radius is inside the solid) is being pulled inward and is in tension. Asaddle surface, such as is frequently encountered in sintering, is a mixture ofconvex and concave surfaces of opposite sign. Thus, the Laplace equation helps tospecify the surface stress at each position, and when combined with atomicmotion, gives the basis for sintering and the elimination of curved surfaces.

R1 and R2 ¼ principal radii of curvature, m


s ¼ stress, Pa.

[Also see Vacancy Concentration Dependence on Surface Curvature.]


195

LAPLACE NUMBER

See Suratman Number.

LASER SINTERING (Nelson et al. 1993)

Laser sintering is encountered in stereolithography, the serial building of a three-dimensional object in layers. In a common variant, the desired three-dimensionalstructure is grown out of a layered powder bed via selective laser scanning. Thelaser traces over the powder bed in a pattern corresponding to the sliced computerimage of the component at the current build height. The laser beam locally heatsthe powder, sticking the particles together. Local heating can bond the particlestogether by melting a wetting polymer, or with high-power lasers, can heat the par-ticles to their sintering temperature. In another variant, the powder is fed into the laserbeam on a moving two-dimensional positioning head. The latter approach is useful inperforming repairs. For a given beam power, heat transfer near the beam is mathemat-ically simplified by assuming radial symmetry,

N ¼ BSRB

a

where N is a figure of merit, BS is the beam scanning speed, RB is the beam radius, anda is the thermal diffusivity of the powder bed. Often the problem is simply treated inone dimension due to the poor thermal diffusivity of powders. The degree of particlebonding is linked to the figure of merit through experiments. In some forms of lasersintering, polymer particles are used and they only require a small heat input, while inother forms the particles are metallic and require a much higher heat input.

BS ¼ beam scanning speed, m/s

N ¼ figure of merit, dimensionless

RB ¼ laser-beam radius, m

a ¼ thermal diffusivity of the powder bed, m2/s.

LATTICE DIFFUSION


LIFSCHWIZ, SLYOZOV, WAGNER MODEL (Voorhees 1992)

Lifshitz, Slyozov, and Wagner describe Ostwald ripening or grain coarsening indilute solutions when growth is controlled by the rate of diffusion through thematrix phase. In liquid-phase sintering, diffusional growth is observed during the

196 CHAPTER L

solution-reprecipitation process, but at a much higher solids content and withcomplications, including coalescence and solid-solid sintering. The Lifshitz,Slyozov, and Wagner treatment, often designated as the LSW model, treats the limit-ing case of infinitely separated solid grains. In this model, diffusion-controlled growthgives the mean grain size G cubed when it is coarsening from an initial mean grainsize G0 over time t as follows:

G3 ¼ G30 þ

329

Kt

where the parameter K is the grain-growth rate constant. Fundamentally the rateconstant relates to the diffusion rate of dissolved solid in the liquid; thus, it is inherentlysensitive to changes in temperature since diffusion, surface energy, and solubilitychange with temperature. A prediction from this model is that the grain-size distributionremains fairly narrow. Under steady-state conditions the maximum grain size (based ongrain volume) should only be 50% larger than the mean grain size. Testing of that pre-diction has generally shown it is invalid when dealing with the high concentrations ofgrains encountered in sintering, precipitation hardening, and particulate systems.


G0 ¼ initial grain, m (convenient units: mm)

K ¼ rate constant, m3/s (convenient units: mm3/s)

t ¼ time, s.

LIGAMENT PINCHING

See Raleigh Instability.

LIMITING NECK SIZE

During sintering the bond between particles or grains in contact with each otherenlarges until the neck size X reaches a limit dictated by the dihedral angle f andthe grain size G, where

X ¼ G sinf

2

� �

Once sintering achieves a bond size as defined by the equilibrium dihedral angle,then further neck growth depends on the rate of grain growth. In diffusion-controlledsituations, which are most common in sintering, the grain size increases with the cuberoot of time, so by implication the neck size will grow with the cube root of time. Forsolid-state sintering, the dihedral angle depends on the relative grain boundary and

LIMITING NECK SIZE 197

solid–vapor surface energies. For liquid-phase sintering the dihedral angle dependson the relative grain boundary and solid-liquid surface energies.




LIMITING SIZE FOR SEDIMENTATION ANALYSIS (Bernhardt 1994)

Particle-size analysis done by sedimentation (Stokes’ law settling) is widelyapplied to small particles, but there are two limits to the technique. The upper limitcomes from turbulence if the particles are large in size, and the lower limit comesfrom Brownian motion if the particles are small in size. For the turbulence case, thelimiting particle size is obtained by combining Stokes’ law with the maximumallowed Reynolds number RN. The Reynolds number depends on the particle velocityV, particle diameter D, fluid density rF, and fluid viscosity h:

RN ¼VDrF

h

If the Reynolds number is large, then turbulence occurs and the measurement isinvalid. In the conservative limit, the Reynolds number should be set at 0.2.Solving both Stokes’ law and the Reynolds number equation for velocity and thenequating the two velocities gives,

Dmax ¼18RNh

2

grF(rM � rF)

� �1=3

where Dmax is the maximum particle size that can be measured using the specific sedi-mentation parameters, RN is the limiting Reynolds number prior to turbulence(suggested value of 0.2), and g is the gravitational constant. For example, if the exper-iment is performed in air, then the fluid viscosity is 1.8 . 1025 Pa . s, the gravitationalconstant g is 9.8 m/s2, and the fluid density is 1.2 kg/m3. In the case of a commonceramic powder such as alumina with a density of nearly 4000 kg/m3, the resultingmaximum size is 29 mm.

At the other end of the applicable particle-size range, Brownian motion givesrandom particle motion that interferes with the assumed viscous settling. The meanvelocity due to Brownian motion VB for a sphere depends on particle size D asfollows:

VB ¼18kT

pD3rM

� �1=2

198 CHAPTER L

where k is Boltzmann’s constant, and T is the absolute temperature. At allparticle sizes there is a Brownian motion contribution, but it becomes larger asparticle size decreases. Once a relative error is set in using Stokes’ law, then alower-limit particle size can be calculated. For example, using the aluminasettling in an air situation, the point where the Brownian motion velocity isequal to the Stokes’ law velocity occurs at about 2 mm. These upper and lowerbounds suggest that Stokes’ law particle-size data for alumina settling in air areonly valid between 2 and 29 mm. However, this allows for a substantial Brownianmotion contribution at the lower end, so a narrower size range would be mostappropriate.


Dmax ¼ maximum particle size, m (convenient units: mm)

RN ¼ Reynolds number, dimensionless


V ¼ particle velocity, m/s

VB ¼ Brownian motion velocity, m/s

g ¼ gravitational constant, 9.8 m/s2

k ¼ Boltzmann’s constant, 1.38 . 10223 J/K


rF ¼ fluid density, kg/m3 (convenient units: g/cm3)

rM ¼ material density, kg/m3 (convenient units: g/cm3).

[Also see Stokes’ Law Particle-Size Analysis.]

LIQUID AND SOLID COMPOSITIONS IN PREALLOYPARTICLE MELTING

For supersolidus liquid-phase sintering, and other semisolid treatments involvingreheating of prealloyed powders, rapid microstructure coarsening occurs when theliquid first forms. Assuming a linear relation between melting temperature and com-position makes it possible to estimate the solid-volume fraction as a simple functionof temperature: The solidus and liquidus temperatures (TS and TL) change linearlywith alloy composition XA as follows:

TL ¼ TM þ AXA

and

TS ¼ TM þ BXA

where TM is the baseline melting temperature, XA is the alloying content on a weightbasis, and A and B are the slopes. In turn, the compositions at the liquidus and solidus

LIQUID AND SOLID COMPOSITIONS IN PREALLOY PARTICLE MELTING 199

lines (XL and XS, respectively) for any composition and temperature are given as,

XL ¼T � TM

Aand

XS ¼T � TM

B

The tie line between these two compositions allows calculation of the solid-massfraction MS in the particle at a given sintering temperature T,

MS ¼XL � XA

XL � XS

In turn, the volume fraction of solid F depends on the solid-mass fraction and thedensities of the solid rS and liquid rL phases as follows:

F ¼ MS=rS

MS=rS þ (1�MS)=rL

Thus, the solid-volume fraction inside the alloy particles can be calculated from theliquid and solid densities.

A ¼ liquidus slope, K

B ¼ solidus slope, K

MS ¼ solid-mass fraction, dimensionless [0, 1]

T ¼ temperature, K

TL ¼ liquidus temperatures, K

TM ¼ melting temperature, K

TS ¼ solidus temperature, K

XA ¼ alloy composition, fractional weight or kg/kg

XL ¼ liquidus composition, fractional weight or kg/kg

XS ¼ solidus composition, fractional weight or kg/kg

F ¼ volume fraction of solid, dimensionless [0, 1]

rL ¼ liquid density, kg/m3 (convenient units: g/cm3)

rS ¼ solid density, kg/m3 (convenient units: g/cm3).

LIQUID DISTRIBUTION IN SUPERSOLIDUS LIQUID-PHASESINTERING (German 1997)

Above the solidus temperature a prealloyed particle undergoes incipient meltingand rapidly densifies by supersolidus liquid-phase sintering. With respect to a

200 CHAPTER L

single spherical particle of diameter D, the liquid fraction is tied up in three forms sothe solid-volume fraction is given by the subtraction of the liquid volumes as follows:

F ¼ 1� 6p

VB þ VN þ VI

D3

� �

where VB, VN, and VI designate the liquid volumes at the boundary, neck, and graininterior, and F is the solid volume fraction. The liquid film on the grainboundaries inside the particles is assumed to be relatively small compared to thegrain size G; thus,

VB ¼ SGd

2

� �FCnG

where SG is the surface area per grain, d is the grain-boundary width (d/2 is the widthof the grain boundary film assigned to each grain), FC is the fractional grain-boundarycoverage by liquid (0 � FC � 1), and nG is the number of grains per particle. Thefractional coverage is determined by the quantity of liquid, the grain size, andthe liquid-film thickness on grain boundaries. Initially, before liquid formation, thenumber of grains per particle approximately depends on the cube of the grain sizeG to particle size D ratio,

nG �D

G

� �3

The approximate grain volume is assumed to be VG ¼ G3/2. An approximate solu-tion for the volume of liquid and solid associated with each grain VG is given as,

VG ¼(Gþ FCd)3

2� (G3 þ 3G2FCd)

2

Accordingly, the number of grains becomes,

nG �D3

G2(Gþ 3FCd)

A substitution of SG ¼ 3G2 gives,

VB ¼ 1:5FCdG2nG

as the volume of boundary liquid per particle. The volume of liquid per particlelocated at the necks between particles depends on the neck size, as measured by

LIQUID DISTRIBUTION IN SUPERSOLIDUS LIQUID-PHASE SINTERING 201

its diameter X; thus,

VN ¼p

4d

2X2FCNC

where NC is the particle-packing coordination. The dimension of the liquid filmlocated in the necks is usually the same as on the grain boundaries. Hence, theneck size approximates to the grain size, giving,

VN ¼p

4d

2X2FCNC

The particle-packing coordination number varies with the fractional density f and canbe empirically estimated for sintering structures as follows:

NC ¼ 14� 10:4(1� f )0:38

Because the particles remain essentially spherical, but the grains are shape accommo-dated, there is a difference in the coordination numbers. The quantity of liquid at thegrain interior is assumed to remain a constant fraction of the total liquid VL

VI ¼ FIVL

where FI is the fraction of liquid at the grain interior (0 � FI � 1). Whether thisoccurs depends on the details of the powder microstructure as dictated by the com-position and atomization process. This gives the liquid volume as follows:

VL ¼ FIVL þ G2dFC(0:4NC þ 1:5nG)

or

VL ¼G2dFC

1� FI(0:4NC þ 1:5nG)

This gives a relation between the liquid volume forming inside a particle on heating,as estimated from the phase diagram, the microstructural features, and the systemviscosity. It is the system viscosity that induces densification based on these micro-structural parameters.


FC ¼ fractional grain-boundary coverage by liquid, dimensionless [0, 1]

FI ¼ fractional liquid at the grain interior, dimensionless [0, 1]


202 CHAPTER L

NC ¼ particle-packing coordination, dimensionless

VB ¼ volume of liquid at the intergrain boundary, m3 (convenient units:mm3)

VG ¼ grain volume, m3 (convenient units: mm3)

VI ¼ volume of liquid in the grain interior, m3 (convenient units: mm3)

VL ¼ total volume of liquid, m3 (convenient units: mm3)

VN ¼ volume of liquid at the particle neck, m3 (convenient units: mm3)

SG ¼ surface area per grain, m2 (convenient units: mm2)

X ¼ sinter neck diameter, m (convenient units: mm)

f ¼ density, dimensionless fraction

nG ¼ number of grains per particle, dimensionless

F ¼ solid volume fraction, m3/m3

d ¼ grain-boundary film width, m (convenient units: mm).

LIQUID-DROPLET COALESCENCE TIME (Hendy 2005;Hawa and Zachariah 2005)

For two spheres of equal size there is a characteristic time t associated with coalesc-ence by viscous flow. Such coalescence is seen in atomization. This time is deter-mined by the viscosity h of the liquid, particle size D, and liquid–vapor surfaceenergy gLV as follows:

t ¼ hD

gLV

This relation is derived by applying viscous-flow sintering equations with the necksize X set equal to the particle size D.



t ¼ coalescence time, s


h ¼ liquid viscosity, Pa . s.

LIQUID-DROPLET VISCOUS FLOW

See Viscous Flow of a Liquid Droplet.

LIQUID EMBRITTLEMENT

See Fragmentation by Liquid.

LIQUID EMBRITTLEMENT 203

LIQUID PENETRATION OF GRAIN BOUNDARIES

See Melt Penetration of Grain Boundaries on Liquid Formation.

LIQUID-PHASE SINTERING GRAIN GROWTH

See Grain Growth in Liquid-phase Sintering, Interfacial Reaction Control,and Grain Growth in Liquid-phase Sintering, Diffusion Control at High SolidContents.

LIQUID-PHASE SINTERING GRAIN-SIZE DISTRIBUTION

See Grain-size Distribution for Liquid-phase Sintered Materials.

LIQUID-PHASE SINTERING INTERMEDIATE-STAGE SHRINKAGE

See Gessinger Model for Intermediate-stage Shrinkage in Liquid-phase Sintering.

LIQUID-PHASE SINTERING NECK GROWTH

See Neck Growth Early in Liquid-phase Sintering and Courtney Model for Early-stage Neck Growth in Liquid-phase Sintering.

LIQUID-PHASE SINTERING RHEOLOGICAL MODEL

See Rheological Model for Liquid-phase Sintering.

LIQUID-PHASE SINTERING SURFACE-AREA REDUCTION

See Surface-area Reduction During Liquid-phase Sintering.

LIQUID VELOCITY IN ATOMIZATION (Jones 1960)

Water atomization relies on pressurized converging jets to generate small droplets.The water jets vent into a chamber with a residual gas pressure of P0 and impartenergy to the molten-metal stream to form melt droplets that become particles. Thehigher the velocity V of the water jet, the smaller the atomized powder. This velocity

204 CHAPTER L

depends on the injection pressure PI as follows:

V ¼ CPI � P0

r

� �1=2

where C is a parameter that depends on the nozzle design, and r is the fluid density.

C ¼ atomization nozzle-design parameter, dimensionless

P0 ¼ chamber gas pressure, Pa (convenient units: MPa)

PI ¼ injection pressure, Pa (convenient units: MPa)

V ¼ jet velocity, m/s


LOGARITHMIC VISCOSITY RULE

See Binder (Mixed Polymer) Viscosity.

LOG-NORMAL DISTRIBUTION (Aitchison and Brown 1963)

The log-normal distribution arises from many small, multiplicative random events. Itis most useful for describing particle size, pore size, and other packing attributesassociated with powders. The probability density for a log-normal distribution isgiven as P(x), where x is the dimensionless-size metric,

P(x) ¼ 1

xsffiffiffiffiffiffi2pp exp � [ ln(x=m)]2

2s2

� �

where s is the measure of the dispersion, and m is the median. There are severalimportant relations used in treating distributions that derive from the median and vari-ation by letting w ¼ exp(s2), then,

mean ¼ m exp(0.5s2)

median ¼ m

variance ¼ m2w(w 2 1)

mode ¼ m/w

coefficient of variation ¼ (w 2 1)1/2

At a small dispersion the log-normal distribution converges to a Gaussiandistribution. The cumulative log-normal distribution is obtained by integrating theprobability density from 0 to the size x, and this integral is calculated by using theerror function. An equivalent form of the log-normal probability density function

LOG-NORMAL DISTRIBUTION 205

can be generated by separating the parameter m into a separate term,

P(x) ¼ 1

xsffiffiffiffiffiffi2pp exp � [ ln x� m]2

2s2

� �

where the median is exp(m) ¼ m or ln(m) ¼ m. When a property is log-normal, theconfidence interval is not plus–minus, but is multiply–divide. For example,there is a recognized probability of Alzheimer’s disease forming by the age of 60years. This probability has a dispersion factor of 1.16; in other words, there is a68% probability for contacting Alzheimer’s in the age range from 52 to 70years (60 times 1.16 and 60 divided by 1.16). The 95.5% probability rangewould be 2s, or 60 years multiply–divide by 2.32, giving an age range from 46 to81 years.

P(x) ¼ probability density function of occurrence for a size x, dimensionless

m ¼ median, dimensionless

w ¼ calculation parameter, dimensionless

x ¼ size, dimensionless (0, þ1)

m ¼ logarithmic value of m, dimensionless

s ¼ distribution dispersion, dimensionless [0, þ1].

LOG-NORMAL SLOPE PARAMETER

For the cumulative log-normal particle-size distribution, a linear plot results when thestandard deviations from the cumulative distribution are plotted against thelogarithm of the particle size. This is illustrated in Figure L1. The slope parameterprovides a measure of the particle-size distribution width. The particle sizes at the90% and 10% cumulative points on the distribution correspond to D90 and D10

and differ by 2.56 standard deviations. Since the size is expressed on a logarithmicbasis of the slope, better known as the size distribution width SW, this is givenas follows:

SW ¼2:56

log10 (D90=D10)

If the particle-size distribution is very narrow, then D90 and D10 are close and thecumulative distribution is steep, giving a high SW.

D10 ¼ particle size at 10% cumulative point, m (convenient units: mm)

D90 ¼ particle size at 90% cumulative point, m (convenient units: mm)

SW ¼ distribution width, dimensionless.

206 CHAPTER L

LONDON DISPERSION FORCE (J. R. G. Evans 1993)

The London dispersion force provides for attraction between particles based oninstantaneous synchronized electron polarization in neighboring molecules. This isa short-range force active over separations of 20 nm or less, so it is most activewith nanoscale particles and molecules. The interaction energy between sphericalparticles from coupled electronic oscillators is proportional to the separation distancel, and the force of attraction F is given as,

F ¼ AD

2l2

where A is the Hamaker constant. For touching particles the separation isusually set to 0.2 nm, resulting in an attractive force near 1 mN between particlesof 1-mm size.

A ¼ Hamaker constant, 10218 J


F ¼ attractive force between particles, N

l ¼ separation distance between particles, m (convenient units: nm).

LOW-SOLID-CONTENT GRAIN GROWTH

See Grain Growth in Liquid-phase Sintering, Dilute-solids Content.

Figure L1. The log-normal distribution gives a normal curve when the size scale is presentedon a logarithmic basis, as illustrated here for a small stainless steel powder.

LOW-SOLID-CONTENT GRAIN GROWTH 207

LUBRICANT BURNOUT


LUBRICANT CONTENT

See Maximum Lubricant Content.

208 CHAPTER L

M

MACROSCOPIC SINTERING MODEL CONSTITUTIVEEQUATIONS (Olevsky 1998)

A constitutive model used to predict component size and shape in sintering relies onthe rheological behavior of the porous structure during the heating cycle. Theapproach is a macroscopic view, since it is applied by constitutive equations infinite element analysis, starting with the green body and its density gradients.Details of evolving features such as grain size, pore size, and grain–pore interactionare not required. To determine the parameters in the underlying viscoplastic constitu-tive laws requires calculation of the rheological response of the porous continuum.For solid-state sintering, the viscosity modulus G, bulk modulus K, and sinteringstress sS are expressed as follows:

G ¼ f 2h

K ¼ 43

f 3

1h

sS ¼6gSV

Df 2

where f is the fractional density, 1 ¼ 1 2 f is the fractional porosity, h is viscosity ofthe material at the sintering temperature, gSV is the solid–vapor surface energy, and Dis the particle size, assuming spherical particles. For most materials, the solid isdeformable at high temperatures and the system viscosity h varies with temperaturein an exponential manner,

h ¼ h0 expQ

RT

� �

where Q is the activation energy for viscous flow, T is the absolute temperature, R isthe gas constant, and h0 is the reference viscosity of the material at an equivalent


209

strain rate (which is usually low for sintering problems). Such an approach requiresonly a few parameters, so it is easy to implement in numerical simulations.Although this is a macroscopic model, still micromechanical concepts are used toestimate the constitutive parameters by consideration of different mass-transportmechanisms as follow:

G ¼ f0f

� �2=3 f � f01� f0

� �2 f D3

720DE

K ¼ f0f

� �2=3 f � f01� f0

� �2 f D

432DE

and

sS ¼16gSV

D

f0f

� �1=3

f

where f0 is the initial relative density of the sintering body, and DE is the effectivediffusivity term. As an example, for grain-boundary diffusion, a common densifica-tion mechanism in sintering, an effective diffusivity is calculated as,

DE ¼VdBDBo exp (�QB=RT)

kT

where QB is the activation energy for grain-boundary diffusion, and DBo is the grain-boundary diffusion frequency factor. The width of the grain boundary is dB. Most ofthe relevant parameters can be extracted from dilatometry experiments. Experimentsbased on free sintering and uniaxial loading in a dilatometer allow extraction of theuniaxial strain rate �_1

vpz . Then the uniaxial viscosity hz and the viscous Poisson’s ratio

v are determined by the following expressions:

hz ¼�_1

vpzsz

and

v ¼ ��_1

vpz

�_1vpr

where sz is any external stress applied on the material. By the analogy of linearelastic theory, the shear-viscosity modulus G and bulk-viscosity modulus K arecalculated as follows:

G ¼ hz

2(1þ v)

210 CHAPTER M

and

K ¼ hz

3(1� 2v)


DBo ¼ grain boundary diffusion frequency factor, m2/s

DE ¼ effective diffusivity, m5/(N . s)

G ¼ shear-viscosity modulus, Pa . s

K ¼ bulk-viscosity modulus, Pa . s

Q ¼ viscous flow-activation energy, J/mol(convenient units: kJ/mol)

QB ¼ grain-boundary diffusion-activation energy, J/mol(convenient units: kJ/mol)




f0 ¼ initial fractional density, dimensionless [0, 1]


vvp ¼ viscous Poisson’s ratio, dimensionless



dB ¼ width of the grain boundary

1 ¼ fractional porosity, dimensionless�_1

vpz¼ uniaxial strain rate, 1/s

h ¼ viscosity of the wrought material at temperature, Pa . s

h0 ¼ viscosity of wrought materials at room temperature and equivalentstrain rate, Pa . s

hz ¼ uniaxial viscosity, Pa . s

sS ¼ sintering stress, Pa

sz ¼ applied external stress during dilatometry, Pa(convenient units: MPa).

MAGNETIC COERCIVITY CORRELATIONIN CEMENTED CARBIDES

For sintered tungsten carbides cemented with cobalt (WC-Co), known as hard metalsor cemented carbides, it is convenient to estimate the average grain size by testing ofmagnetic properties. In this system, an empirical relation links the magnetic coercivity

MAGNETIC COERCIVITY CORRELATION IN CEMENTED CARBIDES 211

K to bulk composition and carbide grain size GWC (measured as a linear interceptsize), as follows:

K ¼ (c1 þ d1WCo)þ (c2 þ d2WCo)1

GWC

Out of convenience, the typical relation relies on coefficients tuned to grain sizemeasured in mm and cobalt content WCo measured in weight percent. The relation isgenerally valid over the range from 6 to 25 wt % cobalt. The parameters c1, c2, d1,and d2 are constants that vary with composition and other manufacturing details. Thisequation relies on stoichiometric WC compositions consolidated by liquid-phase sinter-ing, so it is not necessarily valid for other compositions or consolidation routes.

GWC ¼ tungsten carbide grain size, m (convenient units: mm)

K ¼ magnetic coercivity, A/m

WCo ¼ weight fraction of cobalt, kg/kg or dimensionless(convenient units: wt%)

c1 and d1 ¼ material constants, A/m

c2 and d2 ¼ material constants, A.

MASS FLOW RATE IN ATOMIZATION

See Gas-atomization Melt Flow Rate.

MASTER DECOMPOSITION CURVE (Enneti et al. 2006)

In the thermal degradation of a polymer the decomposition or pyrolysis event is inte-grated into a single curve that combines many different possible heating and holdcycles. The integrated form shows that pyrolysis takes the following form:

�ða

0

da

a¼ � lna ¼

ðt

0k0 exp

�Q

RT

� �dt ¼ k0Q

where Q is the work of decomposition and is defined as follows:

Q(t, T ; Q) ;ðt

0exp � Q

RT

� �dt

The remaining weight fraction of the polymer a is related to the work of decompo-sition Q as follows:

a(Q; k0) ¼ exp[�k0Q]:

212 CHAPTER M

Q ¼ activation energy for polymer degradation, J/mol (convenient units:kJ/mol)



k0 ¼ rate constant frequency factor, 1/s

t ¼ time, s

Q ¼ work of decomposition, s

a ¼ remaining weight fraction of polymer, dimensionless.

MASTER SINTERING CURVE (Su and Johnson 1996)

The master sintering curve combines temperature and time into a single sinteringparameter, which is similar to how time and temperature are combined to formthe Larsen–Miller parameter for stress rupture or creep-life predictions. Once con-structed, a master sintering curve allows for interpolation of sintered material proper-ties (density, grain size, and distortion have been demonstrated) under variousproposed heating cycles. Assuming no change in sintering mechanism over therange of processing conditions (no phase transformation or melting), the integralwork of sintering Q(T, t) is represented by the following equation:

Q(T , t) ¼ð

1T

exp � Q

RT

� �dt

where T is the absolute temperature, t is the time, R is the gas constant, and Q is theapparent activation energy for sintering. In this form the units of Q(T, t) are s/K.Usually for sintered density, densification, or shrinkage the behavior is simplifiedusing a sigmoid equation. For parameters such as sintered density, then densificationC is expressed as a function of the work of sintering Q(T, t) as follows:

C ¼ f � f01� f0

¼ 1

1þ exp � lnQ(T , t)� a

b

� �

where f0 is the fractional green density at the start of sintering, and f is the fractionalsintered density. The parameters a and b vary with the powder. In cases where there isno upper-limit bound, such as in grain growth (or distortion, pore coarsening, damageaccumulation), then the bounded sigmoid is not the proper fit. In these cases, theresponse parameter is simply a polynomial fit to the work of sintering and scales con-tinuously with the integral work of sintering.

Q ¼ apparent activation energy, J/mol (convenient units: kJ/mol)


MASTER SINTERING CURVE 213


a and b ¼ powder-specific curve-fitting constants, s/K

f ¼ sintered density, dimensionless fraction [0, 1]

f0 ¼ initial density, dimensionless fraction [0, 1]

t ¼ time, s

Q(T, t) ¼ integral work of sintering, s/K

C ¼ densification, dimensionless.

MASTER SINTERING CURVE FOR GRAIN GROWTH

See Grain-growth Master Curve.

MAXIMUM DENSITY IN PRESSURE-ASSISTED SINTERING

To avoid trapped gas in a powder compact it is desirable to vacuum sinter into theclosed-pore condition prior to applying external pressure. The technology istermed pressure-assisted sintering or sinter-HIP (hot isostatic pressing). In thecase of atmosphere sintering to the closed-pore condition, it is possible to estimatethe final pressurized system density. One suggestion is that after pressurization, themaximum final fractional density fm could be estimated as follows:

fm ¼b

1þ b

where

b ¼ PA

P0

fF1� fF

� �

where fF is the sintered fractional density when the pores close at the onset of the finalstage of sintering, PA is the applied gas pressure for densification, and P0 is the gaspressure in the pores at pore closure. The fractional sintered density at pore closure istheoretically set at 0.92, but in practice it ranges from 0.85 to 0.95 due to the initialparticle-size distribution. As an example, if a compact is sintered in one-atmospheregas (0.1 MPa) to a closed-pore condition ( fF ¼ 0.92) and subsequently pressurized at20 MPa, the final limiting density is 99.96%. But if the pressurization is 10 atmos-pheres (about 1 MPa), then the limiting density is about 99.1%. This concept isimportant when internal gas pressure is high or processing pressure is low.

P0 ¼ gas pressure in the pores at closure, Pa (convenient units: MPa)

PA ¼ applied gas pressure, Pa (convenient units: MPa)

fF ¼ density when pores close, dimensionless fraction [0, 1]

214 CHAPTER M

fm ¼ maximum density, dimensionless fraction [0, 1]

b ¼ calculation parameter, dimensionless.

MAXIMUM EJECTION STRESS (Jones 1960)

After uniaxial die compaction, pressure is required to strip the compact out of the die.Empirically, the maximum ejection stress sX is estimated as follows:

sX ¼ gj(1:27j þ 1)

where the stress factor g depends on the component shape and the cluster j ¼ uzh/d,where u is the friction coefficient with the wall, z is the axial–radial force ratio, h isthe compact height, and d is the compact diameter. Usually ejection stresses are lowenough there is not a serious concern, but as the j parameter increases, the maximumejection stress can exceed the green strength, especially in longer compacts thatrequire higher ejection stresses.



g ¼ component shape-dependent stress factor, dimensionless

u ¼ powder–die friction coefficient, dimensionless fraction

z ¼ axial–radial force ratio, dimensionless

sX ¼ maximum ejection stress, Pa (convenient units: MPa)

j ¼ compaction parameter, dimensionless.

MAXIMUM GRAIN SIZE IN SINTERING (Olgaardand Evans 1986)

Usually, dispersoids retard early sintering densification, but there is an offsettingbenefit from dispersoids, since they also retard grain growth and enable continuedsintering densification. During sintering the retarded grain growth from a dispersiongenerally gives a relation between grain size G and the volume fraction of dispersoidsVd as follows:

Gmax ¼ Cddq

D Vmd

where Gmax is the maximum expected grain size during sintering, D is the initial par-ticle size, dd is the dispersoid diameter, and q is a measure of the dispersoids pinningeffectiveness and is typically between 1 and 2. Similar relations have a combinedeffect from the dispersoid content and porosity, resulting in a declining effectiveness

MAXIMUM GRAIN SIZE IN SINTERING 215

in pinning grain boundaries to retard grain growth as densification proceeds.Experiments have quantified this decline in pinning effectiveness at high densitiesfor a few materials.

C ¼ proportionality factor, dimensionless

Gmax ¼ maximum expected grain size, m (convenient units: mm)

Vd ¼ dispersoids volume fraction, m3/m3 or dimensionless

dd ¼ dispersoid diameter, m (convenient units: mm or nm)

m ¼ experimental exponent, dimensionless

q ¼ pinning effectiveness, dimensionless.

MAXIMUM LUBRICANT CONTENT

When the desired green density in a pressed-powder compact is known, the maximumvolume fraction of the lubricant is defined by the remaining void space. This assumesa saturated structure where all voids between particles are filled with lubricant. Thesaturation condition determines the maximum lubricant content in terms of aweight fraction WL. Its calculation depends on the theoretical densities of thepowder rP and lubricant rL. The calculation of the weight fraction of lubricant WL

comes from the fractional solid density f as,

WL ¼(1� f )rL

rL(1� f )þ f rP

WL ¼ maximum lubricant content, weight fraction (kg/kg)

f ¼ compact density, dimensionless fraction [0, 1]

rL ¼ theoretical lubricant density, kg/m3 (convenient unit g/cm3)

rP ¼ theoretical powder density, kg/m3 (convenient unit g/cm3).

MAXWELL MODEL

See Viscoelastic Model for Powder–Polymer Mixtures.

MEAN CAPILLARY PRESSURE

Capillary pressure determines the liquid migration and rearrangement in liquid-phasesintering. In the intermediate stage of liquid-phase sintering solution reprecipitationcauses dissolution of the solid into the liquid at the grain contacts. Mass is movedby diffusion in the liquid, from grain contacts under compression to regions under

216 CHAPTER M

tension. The following equation provides an estimate of the mean capillarypressure P:

P ¼ 5:2gLV cos u

D(DL=L0)

where D is the particle diameter, gLV is the liquid–vapor surface energy, u is thecontact angle of the liquid on the solid, DL/L0 is the sintering shrinkage given bythe compact length change divided by the initial length. Typical initial values forP are the order of a few MPa (for a gLV of 2 J/m2 and particle size of 100 mm,the mean capillary pressure goes from 10 to 1 MPa as the linear shrinkage goesfrom 0.01 to 0.1).



P ¼ mean capillary pressure, Pa





MEAN COMPACTION PRESSURE (Jones 1960)

During uniaxial die compaction the applied force is on one or both faces of thecompact. Die-wall friction leads to a decrease in pressure with distance from thepoint of force application. For single-ended pressing, the mean compaction pressurePM depends on the applied pressure P, and is estimated as,

PM ¼ P 1� 2uzh

d

� �

where u is the friction coefficient between the compressed powder and the die wall(which varies with density); z is a proportionality factor, which determines the rela-tive radial pressure or stress based on the applied axial stress; h is the compact height;and d is the compact diameter. For double-action compaction, where there is anapplied force from both the top and bottom punches, there is more homogeneouspressurization of the powder. In this case, the mean compaction pressure is approxi-mated as,

PM ¼ P 1� uzh

d

� �

The mean pressure is a key predictor of the green density and is always less than theapplied pressure. The mean pressure depends on the geometry by the factor h/d

MEAN COMPACTION PRESSURE 217

(height to diameter, assuming simple cylinder), axial-to-radial pressure distributionby the factor z, and die-wall friction by the factor u.

P ¼ applied pressure, Pa (convenient units: MPa)

PM ¼ mean compaction pressure, Pa (convenient units: MPa)



u ¼ friction coefficient with the die wall, dimensionless fraction [0, 1]

z ¼ axial–radial proportionality factor, dimensionless fraction [0, 1].

MEAN FREE DISTANCE

See Pore-separation Distance.

MEAN FREE PATH, CARBIDE MICROSTRUCTURE (Luyckx 2000)

In a sintered tungsten carbide with cobalt as the cement, the space between carbidegrains is called a ligament and the mean size of that ligament is called the meanfree path. The estimation of the mean free path l depends on the volume fractionof carbide grains V, carbide contiguity CCC, and the carbide grain size G,

l ¼ G(1� V)V(1� CCC)

CCC ¼ carbide contiguity, dimensionless

G ¼ carbide grain size, m (convenient unit mm)

V ¼ volume fraction of carbide grains, dimensionless

l ¼ mean free path, m (convenient units: mm).

[Also see Contiguity.]

MEAN FREE PATH IN LIQUID-PHASE SINTERING

See Grain Separation Distance in Liquid-Phase Sintering.

MEAN FREE PATH, SINTERING ATMOSPHERE (Johns et al. 2007)

The processing atmosphere works to sustain thermal equilibrium between the con-tainer walls, heating elements, and the sintering work. Even in a vacuum there is suf-ficient molecular motion to transport heat and the residual small pressure is important

218 CHAPTER M

for oxidation, reduction, carburization, and related mass-transfer processes. The meanfree path in the atmosphere is the average distance a gas molecule travels withoutcolliding with another molecule. It determines the heating rate, reaction rate, andgeneral transfer of energy in a vacuum sintering furnace. Calculation of the meanfree path l relies on the molecule size d and the density of the molecules n,

l ¼ 1ffiffiffi2p

pd2n

where the molecular density n varies with the gas pressure as,

n ¼ P

kT

with P is the gas pressure, k is Boltzmann’s constant, and T is the absolute temper-ature. This approach provides a means to estimate the mean-free-path variationwith furnace atmosphere pressure. When sintering occurs in a vacuum furnace, thetypical mean free path is large and can range up to 1 m. In such cases collisionswith the furnace walls or sintering parts will happen more frequently than collisionswith other molecules. Inside the sintering compact, the pore size is very small,and here the molecule collides with the solid more frequently than it encountersother molecules,



d ¼ molecule size or diameter, m (convenient units: nm)


n ¼ molecular-gas density, atom/m3

l ¼ mean free path, m/atom (convenient units: nm/atom).

MEAN PARTICLE SIZE (Bernhardt 1994; Allen 1997)

The average value from the particle-size distribution is also known as the meanparticle size. Normally, size distribution data are available based on the particlemass, volume, projected area, or surface area, so there are multiple measures.Accordingly, the basis for measuring the mean particle size must be specified. Forparticle-size data in a histogram form (typical for screen analysis where theamount is given for each size interval), the approximate arithmetic mean size DA

and geometric mean size DG are calculated as follows:

DA ¼1N

XN

i

yiDi

MEAN PARTICLE SIZE 219

and

log (DG) ¼ 1N

XN

i

yi log(Di)

where Di is the midpoint size for each interval, yi is the frequency of occurrence ineach size interval, and N is the total number of occurrences, that is, N is the sumof yi over all size intervals.

DA ¼ arithmetic mean particle size, m (convenient units: mm)

DG ¼ geometric mean particle size, m (convenient units: mm)

Di ¼ midpoint size for a size interval, m (convenient units: mm)

N ¼ total number of occurrences, dimensionless

yi ¼ frequency or population for a size interval, dimensionless fraction.

MEAN TIME BETWEEN PARTICLE CONTACTS

See Brownian Motion.

MEASURE OF SINTERING

See Sintering Metrics.

MELTING-TEMPERATURE DEPRESSION WITH PARTICLE SIZE

See Nanoparticle Melting-point Depression.

MELT PENETRATION OF GRAIN BOUNDARIES (Aksay et al. 1974)

In most liquid-phase sintering systems, the newly formed liquid is initially not atchemical equilibrium. At equilibrium, the dihedral angle f describes the balanceof surface energies between solid and liquid phases. The dihedral angle is definedby the balance between the solid–solid grain-boundary energy gSS and the solid–liquid interfacial energy gSL as follows:

gSS ¼ 2gSL cosf

2

� �

Differentiation of this equation allows analysis of the dihedral angle sensitivity to anychange in solid–liquid surface energy gSL,

dgSL

gSL

¼ df

f

f

2tan

f

2

� �

220 CHAPTER M

A nonequilibrium decrease in the solid–liquid surface energy comes from thereduction in free energy during solid dissolution across the interface into newlyformed, yet unsaturated liquid. Substantial drops in the solid–liquid surface energyare possible in reactive systems. Calculation of the relative system free-energychange needed to break up the grain boundaries and weaken the solid skeleton pre-dicts whether rearrangement events are expected at melt formation. Total dissolutionof the boundary corresponds to a reduction in the dihedral angle to 0, or df ¼2f.Accordingly, df/f ¼21, giving the solid–liquid surface-energy decrease neededfor grain-boundary penetration,

� dgSL

gSL

¼ f

2tan

f

2

� �

When this solid–liquid surface-energy change is satisfied, then the spreading newlyformed liquid penetrates the solid–solid interfaces, usually within a few seconds afterliquid formation. An empirical link is observed between the dihedral angle f indegrees and the change in atomic solubility on melt formation,

f ¼ 75� 638DkA

where DkA is the atomic solubility change for the solid in the newly formed liquid(mol of solute per mol of solvent) as compared with the solid solubility in the addi-tive. This gives a large initial dimensional change that often first appears as swelling.This swelling depends on the penetration rate, which is estimated as follows:

x2 ¼ dPgLVt cos u

4h

where x is the depth of liquid penetration along the grain boundary, dP is the pore size,gLV is the liquid–vapor surface energy, u is the contact angle (which approaches zeroduring penetration), t is the isothermal time, and h is the liquid viscosity. A smalldihedral angle is needed for the liquid to remain connected once it has penetratedinto the grain boundaries. Otherwise, a necklace microstructure results when theliquid film forms discrete lens-shaped islands.


t ¼ time, s

x ¼ depth of liquid penetration, m (convenient units: mm)

DkA ¼ change in atomic solubility on liquid formation,dimensionless mol/mol

gLV ¼ liquid–vapor interfacial energy, J/m2

gSL ¼ solid–liquid interfacial energy, J/m2


MELT PENETRATION OF GRAIN BOUNDARIES 221




MERCURY POROSIMETRY


MESH SIZES

See Sieve Progression.

MICROHARDNESS

See Vickers Hardness Number.

MICROMECHANICAL MODEL FOR POWDER COMPACT(Olevsky et al. 2005)

A simplified micromechanical model is used to represent a powder compact consist-ing of simply packed, rectangular grains. Figure M1 shows the calculation cell, whichconsists of coordinate axes a and c for the elliptical pores located at the four corner

Figure M1. A sketch of the computation arrangement for modeling the pore shrinkage foranisotropic pore shapes during sintering, consisting of grains with elliptical pores on thegrain corners.

222 CHAPTER M

junctions on each grain. The maximum and minimum curvature radii ra and rc of theelliptical pores are defined as:

ra ¼c2

p

ap

and

rc ¼a2

p

cp

where ap and cp are the minor radius and major radius for the pores. The followingrelation describes the stress sx in the x direction:

sx ¼ a1rc�1

csin

f

2

� �� sx cþ cp

� �c

3y2

2c2�a

21rcþ3

csin

f

2

� �� 3

2

sx cþ cp

� �c

where a is the surface tension, f is the dihedral angle, a is the grain semiaxis; sx isthe effective (far-field) external stress in the x direction (compressive sx is negative).Parameter sx(cþ cp)=c is a local stress on the grain boundary; (cþ cp)=c is thestress-concentration factor.

a ¼ grain radius in the x direction, m (convenient units: mm)

ap ¼ pore radius in the x direction, m (convenient units: mm)

c ¼ grain radius in the y direction, m (convenient units: mm)

cp ¼ pore radius in the y direction, m (convenient units: mm)

cþ cp

� �=c ¼ stress-concentration factor, dimensionless

ra ¼ maximum pore-curvature radius in the x direction, m(convenient units: mm)

rc ¼ minimum pore-curvature radius in the y direction, m(convenient units: mm)



sx ¼ stress in the x direction, Pa (convenient units: MPa)

sx ¼ effective external stress in the x direction, Pa(convenient units: MPa)

sx cþ cp

� �=c ¼ local stress on the grain boundary, Pa (convenient units: MPa).

MICROSTRUCTURE HOMOGENEITY

See Homogeneity of a Microstructure.

MICROSTRUCTURE HOMOGENEITY 223

MICROWAVE HEATING

Microwave heating is one means to rapidly heat and sinter a powder compact. Thetypical microwave frequency is 2.45 GHz (2.45 . 109 1/s), a resonance frequencyfor water. In microwave sintering, the depth of penetration x varies with theinverse square-root of the microwave frequency n,

x ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffipnmSp

where S is the material conductivity, and m is the magnetic permittivity. The numberof modes in the microwave cavity determines the uniformity of heating, so thisrelation is only an approximation.

S ¼ material conductivity, S/m ¼ (s3 . A2)/(kg . m3)

v ¼ microwave frequency, 1/s

m ¼ magnetic permeability, H/m ¼ (kg . m)/(s2 . A2)

x ¼ depth of penetration, m (convenient units: mm).

MIGRATION OF PARTICLES (Kainuma et al. 2003)

Dispersed particles in a solid that is undergoing grain growth pin the grain bound-aries, assuming the particles are slow moving with respect to the grain-boundarymigration rate. Moving grain boundaries, in turn, exert a force on the particles andsweep the particles along with the moving boundary. The curvature of the gainboundary associated with a grain of size G and surface energy gSS determines thedriving force for migration in terms of an effective pressure P as follows:

P ¼ 4gSS

G

Assuming the pressure is acting normal to the grain boundary, the driving force actingon a particle determines its velocity of migration as a function of the diffusion processcontrolling the particle motion. If it is assumed that the particle is migrating byvolume diffusion, then the particle velocity V is given as

V ¼ 8DVVP

RTD3 N

where DV is the volume diffusion coefficient for the particle solute in the matrix, V isthe atomic volume of the matrix, R is the gas constant, T is the temperature, D is theparticle diameter, and N is the number of particles per unit of grain-boundary area.Similar relations are generated for grain-boundary diffusion control where DV/D3

224 CHAPTER M

is replaced by dDB/D4, where d is the grain-boundary width, and DB is the grain-boundary diffusion coefficient.





N ¼ number of particles per unit grain-boundary area, 1/m2

P ¼ effective pressure on the grain boundary, Pa




gSS ¼ grain-boundary energy, J/m2

d ¼ grain-boundary width, m (convenient units: nm).

MILLING ENERGY

See Charles Equation for Milling.

MILLING TIME

See Grinding Time.

MIXED GRAIN BOUNDARY AND LATTICE DIFFUSION

See Apparent Diffusivity.

MIXED LATTICE AND GRAIN-BOUNDARY DIFFUSION

See Apparent Diffusivity.

MIXED-POWDER SEGREGATION (Harnby 1985a)

In a powder system formed by mixing different particle chemistries, or otherwise dis-tinctive particles or binder phases, homogeneity is a means to monitor mixing andhandling. For example, in mixed powders that tend to separate due to density differ-ences, there will be a top versus bottom concentration difference. Segregation isdetermined by the minor phase concentration versus position. The segregation

MIXED-POWDER SEGREGATION 225

coefficient CS is calculated as follows:

CS ¼XT � XB

XT þ XB

where XT is the measured fraction of the minor phase in the top half of the containerand XB is the fraction of the minor phase in the bottom half on the container. Forexample, if small particles and large particles of the same composition are mixedtogether, then X would correspond to the weight fraction of large particles, whichcould be measured at the top and bottom by sieving or other techniques. A homogeneousmixture will have a segregation coefficient approaching zero. In a modified form, attrib-uted to Lacy, the variance from many repeat tests is used to measure the homogeneity.Each sample is analyzed for the weight fraction of the components. Mixture homogeneityH is determined by the variance in powder concentration between multiple samples S2,compared to the variance anticipated for perfectly mixed but random powder samples Sr

2,and the variance for the initial mixture S0

2 as follows:

H ¼ S20 � S2

S20 � S2

r

Homogeneity varies from 0 to 1, with unity representing an ideal mixture. Notethis measure of homogeneity depends on the scale of scrutiny. On a macroscale, com-position tests will show the mixture is homogeneous, but a variation arises as smallersamples are taken, and if only one particle is tested, then a great deal of variation willexist. Hence, homogeneity tests for powder mixtures depend on the sample size. For apowder–lubricant mixture, the initial state is a totally segregated system that has aninitial variance given as,

S20 ¼ XP(1� XP)

where XP is the weight fraction of the powder component. The final variance for afully mixed, randomly sampled system should approach zero, or Sr

2 ¼ 0 in theideal. For the general case, this has a simplified form given as,

H ¼ 1� S2

S20

CS ¼ segregation coefficient, dimensionless

H ¼ mixture homogeneity, dimensionless fraction

S ¼ standard deviation from test samples, kg/kg

S0 ¼ standard deviation of segregated mixture, kg/kg

Sr ¼ standard deviation in perfectly mixed samples, kg/kg weight fraction

X ¼ weight fraction, kg/kg or dimensionless fraction [0, 1]

XB ¼ minor phase at the bottom of the container, dimensionless fraction [0, 1]

XT ¼ minor phase at the top of the container, dimensionless fraction [0, 1]

226 CHAPTER M

XP ¼ weight fraction of the powder, kg/kg dimensionless fraction [0, 1].

MIXED-POWDER SINTERING SHRINKAGE (German 1996)

When two powders are mixed, the sintering behavior depends on the response of eachpowder to the sintering cycle, as well as to the overall composition. The dimensionalchange during sintering for the mixture is predicted from the behavior of the two purepowders and the overall composition. The linear sintering shrinkage increases withthe volume fraction of the additive,

DL

L0¼ YAAV2

A þ YBBV2B þ 2YABVAVB

where VA is the volume fraction of the additive powder, VB is the volume fraction ofthe base or majority powder, YA is the sintering shrinkage of the pure additivepowder, YB is the sintering shrinkage of the pure base or major powder, and YAB isthe sintering shrinkage for a 50–50 mixture of additive and base powders.Although this requires experimentation to determine YAB, it does provide insightabout the composition effects on sintering shrinkage for mixed powders.Alternatively, depending on the percolation conditions for the powder mixture, theshrinkage-volume fraction limits can be estimated without experimentation.However, this, too, requires assessment of the green particle structure. In situationswhere a reinforcing fiber is added to a small powder to create a sintered composite,the fiber often has no sintering shrinkage, while the powder has a large sinteringresponse. In that case, the YA term will be zero and the fiber constraint effect willmake YAB small; thus, even with a high shrinkage for the small powder this simplemodel shows the inhibited sintering expected with composites.


VA ¼ volume fraction of additive powder, m3/m3 or dimensionless fraction

VB ¼ volume fraction of base powder, m3/m3 or dimensionless fraction

YA ¼ sintering shrinkage of pure additive powder, dimensionless fraction

YAB ¼ sintering shrinkage of 50–50 mixture, dimensionless fraction

YB ¼ sintering shrinkage of pure base powder, dimensionless fraction


DL/L0 ¼ sintering shrinkage, dimensionless.

[Also see Bimodal Powder Sintering.]

MIXED-POWDER SWELLING DURING SINTERING

See Swelling Reactions during Mixed-powder Sintering.

MIXED-POWDER SWELLING DURING SINTERING 227

MIXING OPTIMAL ROTATIONAL RATE

See Optimal Mixer Rotational Speed.

MIXTURE HOMOGENIZATION RATE (German and Bose 1997)

During mixing of powders with polymers, the initial homogeneity is low, butimproves with mixing until a point is reached where the rate of mixing equals therate of segregation. Prior to this point, the homogeneity H varies exponentiallywith mixing time,

H ¼ H0 þ a exp(kt þ C)

where H0 is the initial mixture homogeneity; t is the mixing time; and a, C, and k areconstants that depend on specific conditions, such as the mixer design, mixer operat-ing parameters, powder characteristics, degree of initial powder agglomeration, andthe surface condition of the powder. It is common for homogeneity to reach anasymptotic value less than 1, reflecting the balance of segregation and mixingevents as determined by the parameter a.

H ¼ mixture homogeneity, dimensionless fraction

H0 ¼ initial mixture homogeneity, dimensionless fraction

k ¼ mixing rate parameter, 1/s

t ¼ mixing time, s

a ¼ mixing parameter, dimensionless.

MIXTURE THEORETICAL DENSITY

The theoretical density of a powder mixture, or a powder–polymer mixture, is calcu-lated by the inverse rule of mixtures. Fundamentally, density is equal to mass dividedby volume, so when two powders are mixed the sum of the mass is divided by thesum of the volume to calculate the mixture theoretical density. This is not the ruleof mixtures, but what is better termed the inverse rule of mixtures. Weight fractionsare the most typical means for performing the mixture theoretical density calculation.Consider the case of a powder mixed with a lubricant, then

1rT¼ WL

rLþWP

rP

where rT is the theoretical density of the mixture, consisting of WL, the weight frac-tion of lubricant with density rL, and WP, the weight fraction of powder with a theor-etical density of rP. A similar form applies to determining the theoretical density oftwo or more powders.

228 CHAPTER M

WL ¼ weight fraction of lubricant, kg/kg or dimensionless

WP ¼ weight fraction of powder, kg/kg or dimensionless

rT ¼ mixture theoretical density, kg/m3 (convenient units: g/cm3)

rL ¼ theoretical density of the lubricant, kg/m3 (convenient units: g/cm3)

rP ¼ theoretical density of the powder, kg/m3 (convenient units: g/cm3).

MODULUS OF RUPTURE

See Bending Test and Transverse Rupture Strength.

MOISTURE CONTENT

See Dew Point.

MOLECULAR MEAN FREE PATH

See Mean Free Path and Sintering Atmosphere.

MULTIPLE-MECHANISM SINTERING

Particle sintering takes place because mass transport at the particle contact regionleads to the growth of the interparticle bond. Several different, simultaneous andcomplementary mechanisms are usually active. For example, grain-boundary diffu-sion will produce a hillock where the grain-boundary discharges mass onto an exter-nal surface. If mass redistribution is not supported by a cooperative mechanism, thenthe resulting hillock will slow continued sintering. Conversely, if surface diffusionacts simultaneously, then the discharge from the grain boundary is removed andthe surface smoothed. The determination of the sintering rate during multiple-mechanism sintering involves the addition of the instantaneous mass fluxes arrivingat the bond between the particles. When summed, the contribution gives an instan-taneous rate of sintering,

dX

dt

��total

¼X dX

dt

��i

where X is the instantaneous neck size, t is the time, and dX/dt is the instantaneousrate of neck growth. This concept assumes that each of the individual mechanisms,represented by the subscript i, provides an independent contribution to the totalneck growth. In computer simulations of sintering, the newly added neck volume(size) is used to calculate the relative contribution by each mechanism, and a cycleof time progression is used to integrate over the thermal cycle. Since surface-transport

MULTIPLE-MECHANISM SINTERING 229

events (such as surface diffusion or evaporation–condensation) do not contribute toshrinkage during sintering, this summation approach is invalid for predicting totalshrinkage. Indeed, since surface-transport events consume surface energy and reducethe driving force on sintering, the net result is a decrease in shrinkage for a givenneck size. For most materials, grain-boundary diffusion or volume diffusion dominatesdensification. The following model provides a general means to predict densification,

13

df

dt¼ gSVVGdD0

RTGnexp � Q

RT

� �

For volume diffusion–controlled sintering, n is equal to 3 and d ¼ 1, while for grain-boundary diffusion-controlled sintering, n is 4 and d is approximately five timesthe atomic size. In this model, f is the fractional density, t is the sintering time, gsv

is the solid–vapor surface energy, V is the atomic volume, R is the gas constant,T is the absolute temperature, G is a collection of geometric factors, D0 is the diffu-sivity frequency factor, G is the grain size, and Q is the apparent activation energy fordensification.

D0 ¼ diffusion frequency factor, m2/s





X ¼ instantaneous neck diameter, m (convenient units: mm)


dX/dt ¼ rate of neck growth, m/s

f ¼ relative density, dimensionless fraction [0, 1]

i ¼ subscript indicating each independent mechanism, dimensionless

n ¼ mechanism-dependent integer, dimensionless

t ¼ time, s

G ¼ geometric factor, dimensionless



d ¼ 1 for volume diffusion, approximately 1 nm for grain-boundarydiffusion.

MULTIPLE-STAGE MODEL OF SINTERING

See Combined-stage Model of Sintering.

230 CHAPTER M

N

NABARRO–HERRING CREEP-CONTROLLED PRESSURE-ASSISTEDDENSIFICATION (Swinkels et al. 1983)

In the situations where an external stress is applied during sintering, such as hotpressing or hot isostatic pressing, the densification process is controlled by creep.Initially, the concentrated stress at the particle contacts might induce plastic flow.As the bonds between the particles grow, the local stress will be insufficient toinduce further plastic deformation, and the densification rate is then controlled bycreep. Nabarro–Herring creep occurs by atomic-vacancy motion and is alsotermed volume diffusion-controlled creep. In this process, particle contacts in theporous microstructure under compressive stress are mass sources, while neighboringregions under tensile stress are mass sinks. The diffusion process is thereby coupledto the microstructure by the stress gradients. For transport by volume diffusioncontrol, the shrinkage—rate change in compact size or length DL, normalized tothe original size or length L0, over the change in time t—is given as follows:

d

dt

DL

L0

� �¼ 13DVVPE

RTG2

where T is the absolute temperature, R is the gas constant, V is the atomic volume, DV

is the lattice or volume diffusivity, G is the grain size, and PE is the effective pressure.Early in consolidation, the applied pressure is amplified at the small-particle contacts,so the effective pressure is much larger than the applied pressure. As densificationprogresses, the effective pressure converges to the applied pressure at full density.



L0 ¼ original length, m (convenient units: mm)

PE ¼ effective pressure, Pa



231


t ¼ time, s


V ¼ atomic volume, m3/mol.


NANOPARTICLE AGGLOMERATION (Tsantilis et al. 2001)

Often nanoparticles are formed by vapor-phase chemical reactions, with simul-taneous coagulation and even sintering during synthesis. Clusters form due to thisnear-spontaneous bonding. In the absence of convective currents, the model fornanoparticle agglomeration during synthesis gives the particle-number concentrationN (number of particles per unit volume in the gas phase) as a decreasing function oftime t due to coagulation as follows:

dN

dt¼ � 1

2bN2rg

where rg is the gas density, and b is the collision frequency function for Browniancoagulation, assuming the particles are monosized. The effect of the aggregate struc-ture is included in the b parameter, since it replaces the primary particle diameter Dwith the collision diameter. The rate of change of the average aggregate particlevolume V is given as follows:

1V

dV

dt¼ � 1

N

dN

dt

while the rate of change of the average aggregate area A is given by a related rateequation,

dA

dt¼ � 1

N

dN

dtA� 1

tS(A� AS)

where AS is the surface area of a completely fused (spherical) aggregate of volume V,and tS is the characteristic sintering time (time needed to reduce by approximately63% the excess surface area of an aggregate over that of a sphere of equal mass).The fused surface-area term AS is determined by the aggregate volume V and theinitial-particle volume V0 using the surface area of the initial particle A0,

AS ¼ A0V

V0

� �2=3

232 CHAPTER N

The surface area of an aggregate particle increases by coagulation and decreases bysintering, assuming a process dependent only on collisions. For spherical particles theprimary particle diameter D and number of primary particles per aggregate np aregiven by the following two relations:

D ¼ 6V

Aand

np ¼A3

36pV2

A ¼ average aggregate area, m2 (convenient units: nm2)

A0 ¼ surface area of the smallest entity, m2 (convenient units: nm2)

AS ¼ surface area of a completely fused aggregate, m2 (convenient units: nm2)

D ¼ primary particle diameter, m (convenient units: nm)

N ¼ nanoparticle concentration, 1/m3

V ¼ average aggregate-particle volume, m3 (convenient units: nm3)

V0 ¼ smallest-entity volume, m3 (convenient units: nm3)

np ¼ number of primary particles per aggregate, dimensionless

t ¼ time, s

b ¼ particle collision frequency, m6/(kg . s)

rg ¼ gas density, kg/m3 (convenient units: g/cm3)

ts ¼ characteristic sintering time, s.

NANOPARTICLE MELTING-POINT DEPRESSION (Buffat andBorel 1976; Lewis et al. 1997)

Small particles have high surface energy per unit volume, and the excess surfaceenergy increases as particle size decreases. The excess surface energy depressesthe melting temperature. In computer simulations of nanoparticle sintering andagglomeration, it is observed that the outer rim forms a liquid prior to particlemelting, a phenomenon termed premelting. The depressed melting temperaturewith small particle size is accounted for by a formulation as follows:

TP

TB¼ 1� 4gLV

DHFrS

gSV

gLV

� rS

rL

� �2=3" #

where TP and TB are the absolute melting temperatures of the particle with its size Dand the bulk material, respectively, HF is the latent heat of fusion, rS and rL are the

NANOPARTICLE MELTING-POINT DEPRESSION 233

solid and liquid densities, and gSV and gLV are the solid–vapor and liquid–vaporsurface energies. The values used in this equation are at the melting temperature.This negates any secondary effect of particle size on the surface energy at eitherthe liquid–vapor or solid–vapor interface.

D ¼ particle diameter, m (convenient units: nm)

HF ¼ latent heat of fusion, J/kg

TB ¼ melting temperature of the bulk material, K

TP ¼ melting temperature of the particle, K



rL ¼ liquid density, kg/m3 (convenient units: g/cm3)

rS ¼ solid density, kg/m3 (convenient units: g/cm3).

NANOSCALE PARTICLE-AGGLOMERATE SPHEROIDIZATION(Hawa and Zachariah 2006)

Nanoparticles have a strong tendency to form a cluster and agglomerate, and ifallowed to come into contact, a bond will form in a large cluster. Loss of surfacearea is associated with sintering and agglomeration. Thus, a phenomenologicaldescription of the nanoscale spheroidization process is possible by analysis ofthe surface area. The excess surface area is defined as the difference between thecurrent particle surface area and the terminal surface area for the agglomeratebased on the surface area of a sphere of equivalent volume. Coalescence or agglo-meration kinetics are expressed in terms of the rate of surface-area change,

dS

dt¼ � 1

tCS� SCð Þ

where S is the surface area of the cluster of nanoparticles, tC is a characteristiccoalescence time that varies with the material and temperature, SC is the surfacearea for the final coalesced cluster based on the volume of agglomerated particles.Effectively, SC is the surface area of the sphere of equivalent volume. This modelassumes the surface energy is isotropic. In the case where the surface energy isanisotropic the final particle shape is polygonal. As a first approximation, the coalesc-ence time tC depends on the temperature, surface energy, and diffusivity as follows:

tC ¼3kTNP

64pgSVDV

where k is Boltzmann’s constant, T is the absolute temperature, NP is the averagenumber of atoms in each particle, gSV is the solid–vapor surface energy at thetemperature where agglomeration occurs, and DV is the diffusivity for the material

234 CHAPTER N

at the temperature where agglomeration occurs. The number of atoms per particle iscalculated from the nanoparticle volume divided by the atomic volume.


NP ¼ number of atoms per particle, atom

S ¼ surface area of the cluster, m2

SC ¼ final cluster surface area, m2




tC ¼ characteristic coalescence time, s.

NANOSCALE PARTICLE-SIZE EFFECT ON SURFACE ENERGY

See Surface-energy Variation with Droplet Size.

NECK-CURVATURE STRESS (Gessinger et al. 1968; Schatt et al. 1983)

Classic sintering models assume a simple curved contact between two spheres, a geo-metry called a saddle surface. The neck geometry for equal-size spheres, assumingthere is no dihedral angle where the grain-boundary emerges to the surface, isshown in Figure N1. Because of the curved surface there is a stress, termed thesintering stress s, that is a function of the neck diameter X, particle diameter D,and solid–vapor surface energy gLV as follows:

s ¼ 4gLV

X

12� D

X

� �

Figure N1. The simplified saddle geometry associated with calculation of the sintering stressacting on the neck.

NECK-CURVATURE STRESS 235

This assumes the surface energy is isotropic. Various simulations, observations, andcalculations have been put forward to show that this stress is sufficient to exceed thelocal yield strength of the heated material, leading to dislocation flow or plastic flowas an initial-stage sintering mechanism. At the start of sintering the neck-curvaturestress is high when the neck size is small, so dislocation motion is possible, butthe stress decays over time as the neck size increases and the dislocation populationis annihilated by annealing and dislocation motion, so plastic flow is only a transientcontribution to sintering during periods of rapid heating.



gLV ¼ solid–vapor surface energy, J/m2

s ¼ neck-curvature stress, Pa.

NECK GROWTH EARLY IN LIQUID-PHASE SINTERING(Courtney 1977a; German 1996)

Prior to attaining the balance of neck size to grain size as set by the dihedral angle, thequantity of liquid does not significantly influence the early neck growth rate, as longas there is sufficient liquid to cover the neck. The result is early-stage liquid-phasesintering growth of the neck X with time t, as follows:

X

G

� �n

¼ Bt

where G is the grain size, and the exponent n ranges from 6 to 7. This is very similarto initial-stage solid-state sintering, suggesting that the liquid fundamentally doesnot induce a new mechanism. From experimental observations, a value of n equalto 6.22 is a best approximation. The rate factor B is a collection of material factorsthat is given as follows:

B ¼ gdDLgLVV

G4RT

where g is a numerical constant, d is the width of the grain-boundary liquid film, DL

is the diffusivity of the solid in the liquid, gLV is the liquid–vapor surface energy, Vis the molar volume, R is the gas constant, and T is the absolute temperature. Becauseof solidification and solubility changes, the liquid-film width and composition changeduring cooling. Hence, measurements based on postsintering observations are oftenin error with respect to the neck size, grain-boundary composition, and similarattributes. Generally, when samples are quenched from the sintering temperature awidth near five atomic layers is typical. The diffusivity has a strong Arrheniustemperature dependence with an activation energy reflective of the diffusion rate of

236 CHAPTER N

the liquid, assuming the solid atoms dissolved in the liquid are swept along with themobile liquid phase.

B ¼ material-specific rate constant, 1/s


G ¼ grain diameter, m (convenient units: mm)




g ¼ numerical constant, dimensionless


t ¼ time, s


d ¼ the grain-boundary liquid-film width, m (convenient units: nm)


NECK GROWTH–INDUCED SHRINKAGE

See Shrinkage Relation to Neck Size.

NECK GROWTH LIMITED BY GRAIN GROWTH (Readey 1990)

During sintering the contact bond between touching particles enlarges until the neckdiameter X encounters the limit dictated by the dihedral angle f and particle size D orgrain size G, where

X ¼ G sinf

2

� �

During initial sintering the neck size ratio is small and is generally expressed bythe parameter X/D. Once sintering achieves a bond-size ratio X/G as defined bythe equilibrium dihedral angle, then any further neck growth depends on the rateof grain growth. Since grain growth is often based on the grain-volume change,which is linear with time (G3 � t), naturally X3 � t is a consequence. Often thislatter-stage liquid-phase sintering neck growth is interpreted as representing specificmechanisms, while the fundamental limitation is induced by the thermodynamicbalance associated with the grain-boundary emergence at the surface. The dihedralangle f is determined by the surface-energy balance. For solid–liquid systemsencountered in liquid-phase sintering, the equilibrium is given as follows:

gSS ¼ 2gSL cosf

2

� �

NECK GROWTH LIMITED BY GRAIN GROWTH 237

with gSS being the solid–solid interfacial energy (grain-boundary energy), and gSL

being the solid–liquid interfacial energy. An analogous version applies to the situ-ation where the grain surface is in contact with a vapor phase:

gSS ¼ 2gSV cosf

2

� �

with gSV being the solid–vapor surface energy.

G ¼ grain diameter, m (convenient units: mm)


X/G ¼ bond-size ratio, dimensionless



gSS ¼ solid–solid surface energy, J/m2



NECK-GROWTH MODEL


NECKLACE MICROSTRUCTURE

See Fragmentation by Liquid Penetration.

NECK-SIZE RATIO DEPENDENCE ON SINTERED DENSITY

During the early stage of sintering, prior to impingement of neighboring necks,especially for low initial densities, the neck size X to particle size D ratio increaseswith the sintered fractional density f based on the initial fractional density f0 as follows:

X

D

� �2

¼ 4 1� f0f

� �1=3" #

with the constraint that X/D not exceed 0.5. Fundamentally, this relation assumes nocompaction of the powder prior to sintering, so it is inherently based on starting from aloose-powder green body of monosized spheres.



238 CHAPTER N

f ¼ sintered fractional density, dimensionless [0, 1]

f0 ¼ initial fractional density, dimensionless [0, 1].

NEWTONIAN COOLING APPROXIMATION

A droplet formed during atomization passes through a cold gas that extracts heat. Ifthe enthalpy of solidification is ignored, then the droplet or particle cooling rate dT/dtcan be estimated based on the Newtonian approximation:

dT

dt¼ �kA T � T0ð Þ

where T is the droplet or particle temperature, t is the time, A is its surface area, T0 isthe ambient temperature, and k represents the rate of heat transfer per unit area. Thisrelation is used in atomization, with the usual assumption of a spherical particle.Since there is often a substantial transformation enthalpy, however, it can only beconsidered as a first approximation.

A ¼ particle surface area, m2

T ¼ particle temperature, K

T0 ¼ ambient temperature, K

dT/dt ¼ cooling rate, K/s

t ¼ time, s

k ¼ heat-transfer coefficient, 1/(m2 . s).

NEWTONIAN FLOW (Tanner and Walters 1998)

In the Newtonian flow model, which is used to describe a fluid or paste flowing in acapillary tube, the required stress to maintain flow is proportional to the shear strainrate. Shear stress t is measured in terms of the force per unit area that causes the fluid(or powder–binder mixture) to flow in a die. The shear strain g is the relative motionof the fluid over the surface. The shear strain rate dg/dt, or the shear rate, isthe change in shear strain divided by time. Fluid resistance to shearing is termedviscosity h, and it links the shear strain rate to the shear stress t,

h ;t

dg=dt¼ h0

dg

dt

� �m�1

where h0 is the viscosity parameter, and m is equal to unity for Newtonian fluids. Thisis a special case that applies to liquids like water, but in powder–binder processing itis inaccurate. Most powder–polymer mixtures used in slurry, paste, or similar forms,

NEWTONIAN FLOW 239

have a more complicated behavior that involves a yield strength and shear strainrate sensitivity.


m ¼ strain-rate sensitivity exponent, dimensionless

g ¼ shear strain, m/m or dimensionless


h0 ¼ inherent viscosity parameter, Pa . sm21


NORMAL CURVE


NUCLEATION FREQUENCY IN SMALL PARTICLES

For small particles the nucleation frequency during solidification follows a Poissondistribution, where Mi is the number of heterogeneous nucleation sites per unitsurface area for droplets with area Ai. The fraction of droplets free of nucleants isdesignated as Xi and can be expressed as

Xi ¼ exp �MiAið Þ

The character of the Poisson distribution is such that for a large number of particles(as encountered in gas atomization) the nucleation events are not the same.Consequently, there is a probability that for any given particle size some of theparticles will solidify without crystallization. Smaller sizes have smaller areas Ai,thereby increasing the probability of forming an amorphous particle.

Ai ¼ droplet area, m2

Mi ¼ number of heterogeneous nucleation sites per unit area, 1/m2

Xi ¼ fraction of droplets free of nucleants, dimensionless.

NUCLEATION RATE (Hirth 1978; Turnbull 1986)

Atomization involves the transformation of undercooled droplets to solid particles.Vapor-phase condensation to produce nanoscale particles involves a similar trans-formation, but from vapor to solid. The rate of nucleation is a concern during the for-mation of nanoparticles from saturated vapors or solids from supercooled liquids.Most of the treatments in particulate materials processing look at particle solidifica-tion during atomization. Here the rate of nuclei formation has dimensions of nuclei

240 CHAPTER N

per unit volume per unit time. The rate varies with several factors, including theliquid viscosity and temperature. As an approximation, the nucleation rate I canbe expressed as follows:

I ¼ I0D2 exp �QL

RT

� �exp � WM

TDT2

� �

where I0 represents the number of attempted nucleation events per unit area and perunit time, D is the particle or droplet diameter, QL is an activation energy for atomicdiffusion through the liquid, R is the gas constant, T is the absolute temperature, andDT is the undercooling. The parameter WM collects several material properties into asingle term.


I ¼ nucleation rate, 1/s

I0 ¼ attempted nucleation events per unit area and per unit time, 1/(m2 . s)

QL ¼ activation energy for solid diffusion through the liquid, J/mol



WM ¼ material property, K3

DT ¼ undercooling, K.

NUMBER OF FEATURES

See Complexity.

NUMBER OF PARTICLES IN AGGLOMERATES

A powder tends to agglomerate, so most automated particle-size analyzers report theagglomerate size, not the discrete particle size. The number of particles making up anagglomerate NA is given as,

NA ¼ FrDA

4:310�6:9=S2

� �3

where F is the packing fraction in the agglomerate, which often can be estimatedat 0.6, D is the mass median agglomerate size as measured by a typical particle-size analyzer (screening, laser scattering, sedimentation), S is the slope ofthe log-normal particle-size distribution, A is the independently measured gasabsorption–specific surface area, and r is the theoretical density of the powder.For typical material parameters and conventional measurement units this can be

NUMBER OF PARTICLES IN AGGLOMERATES 241

simplified to give,

NA ¼rDAð Þ3

2620

when the following units are employed: agglomerate size D is in mm, specific surfacearea A is in m2/g, and theoretical density r is in g/cm3.

A ¼ gas absorption–specific surface area, m2/kg (convenient units: m2/g)

D ¼ median agglomerate size, m (convenient units: mm)

F ¼ fractional packing density in the agglomerate, dimensionless [0, 1]

NA ¼ number of particles in an agglomerate, dimensionless

S ¼ slope of the log-normal particle-size distribution, dimensionless

r ¼ theoretical density, kg/m3 (convenient units: g/cm3).

NUMBER OF PARTICLES PER UNIT WEIGHT

In a collection of particles, it is often convenient to characterize the particle size basedon either the number of particles or the weight of particles. In the case where data arecollected based on the weight of particles versus size, a transformation to the numberof particles is often necessary. In such a case, the number of particles n knowing theweight W and theoretical material density rM is given as follows:

n ¼ 6W

prMD3

where D is the characteristic particle diameter. In most automated particle-size ana-lyzers this transformation is performed in reverse, since for historical reasons mostconventional particle-size data are given based on a weight distribution, while mostsize analyzers collect data by measuring the number of particles versus their size.


W ¼ mass of the particles, kg (convenient units: g)

n ¼ number of particles, dimensionless


242 CHAPTER N

O

OPEN-PORE CONTENT

Open pores are connected to the external surface and allow for fluid flow in and out ofthe porous structure. The Archimedes water-immersion technique for measuringdensity can be adapted to determine the amount of open porosity. That approachrequires dry-sample weight in air W1, after oil impregnation W2, and immersed inwater W3. From these data the fractional open porosity is calculated as:

1O ¼W2 �W1ð ÞrW

W2 �W3ð ÞrO

where 1O is the open porosity, rW is the density of water, and rO is the oil density.

W1 ¼ dry-sample weight, kg (convenient units: g)

W2 ¼ sample weight after oil impregnation, kg (convenient units: g)

W3 ¼ sample weight immersed in water, kg (convenient units: g)

1O ¼ fractional open porosity, dimensionless [0, 1]

rO ¼ oil density, kg/m3 (convenient units: g/cm3)

rW ¼ water density, kg/m3 (convenient units: g/cm3).

OPTIMAL PACKING PARTICLE-SIZE DISTRIBUTION

See Andreasen Size Distribution.

OPTIMAL MIXER ROTATIONAL SPEED (Harnby 1985b)

In mixing a powder, when the mixer rotates too rapidly during the mixing process, thecentrifugal force exceeds the gravitational force and the powder remains pinned to thecontainer wall. On the other hand, when the mixer rotates too slowly, the powder


243

slides on the wall and there is minimal mixing. Optimal mixing occurs when the par-ticles are lifted and then fall across the mixer’s central axis. This optimal rotationalspeed N0 in revolutions per minute (rpm) for a mixer is given as follows:

N0 ¼bffiffiffidp

where b ¼ 32 m1/2/min, and d is the outer-container arc diameter in meters.

N0 ¼ optimal rotational speed, 1/min or rotations per minute, rpm

d ¼ rotational diameter of the container, m (convenient units mm)

b ¼ optimization parameter, m1/2/s.

ORDERED PACKING (Jernot et al. 1981)

In most applications involving powders, good packing characteristics are desired. Loosepacking of powders gives a low apparent density. Higher packing densities are possibleby adjustment of the particle size, shape, and size distribution. The higher packingcoordination number generally corresponds to a higher observed powder-packingdensity. The coordination number is the number of touching neighbors that any particlehas in a powder aggregate, and is synonymous with the number of nearest neighbors.Small particles will have more interparticle friction, because of a lower number ofnearest neighbors. This means the smaller the median particle size, the lower the appar-ent density. For larger particles, where surface effects are not controlling (over 50mm),the relation between coordination number NC and apparent density f is as follows:

f ¼ 1� 1NC

� �3

For a larger spherical powder, the loose packing density is near 0.60. Typically,the coordination number exhibits the distribution around a mean value. For loose-packed, monosized spheres, the mean coordination number is near 7. From mono-sized spheres the best packing density obtainable is 0.74 (close-packing particle incoordination number of 12). This packing can be improved by mixing selectivesizes to fill interstices between particles. Careful combination of such-sized particlescan in theory result in densities as high as 0.98.


f ¼ fractional density, dimensionless [0, 1].

[Also see Bimodal Powder Packing.]

OSPREY PROCESS


244 CHAPTER O

OSTWALD RIPENING (Voorhees 1992)

Named for a Nobel Prize–winning chemist, Ostwald ripening is a grain-growthprocess whereby the large grains grow at the expense of the small grains. It wasinitially postulated for dilute systems, but it has been subsequently extended intothe range encountered in liquid-phase sintering. In most derivations the volume ofthe mean particle increases linearly with time. However, the classic model predictionsof the grain-size distribution are wrong when extended to more than about 15 vol %solid content; the maximum grain size is predicted to be 150% of the mean size, whilein practice the largest size is much larger. However, the time dependence is correct.Thus, the grain size cubed (grain volume) enlarges at a rate given as,

G3 ¼ G30 þ Kt

where G is the grain size (any of several possible measures), G0 is the initial grainsize, and t is the isothermal time. The rate constant K represents a temperature andcomposition-dependent parameter, for liquid-phase sintering is often in the rangeof 1 mm3/s, but the actual value is very sensitive to impurities. Models forOstwald ripening assume the grains are far from one another, which is not typicallyobserved in powder systems except during precipitation from a solution. Moreinvolved treatments include nearest-neighbor interaction terms and many other com-plications needed to explain coarsening in concentrated systems such as in liquid-phase sintering.



K ¼ kinetic rate constant, m3/s (convenient units: mm3/s)

t ¼ time, s.

[Also see Lifschwiz, Slyozov, Wagner Model.]

OXIDE REDUCTION

Oxide reduction assumes that the solids are of fixed composition, such that a metalM (s) is in equilibrium with oxygen gas O2 (g) and the oxide MxO2 (s), where thesubscript x represents the stoichiometry of the oxide,

xM(s)þ O2(g) ¼ MxO2(s)

The metal and oxide are solids, as indicated by the notation of (s), and the oxygen is agas (g). For this reaction, there is an equilibrium constant K defined as follows:

K ¼ aMxO2

axMPO2

OXIDE REDUCTION 245

where ai designates the thermodynamic activity for the subscripted species i. For thesolid phase, the activity is 1 (meaning there is plenty of each solid available for reac-tion). In many situations it is appropriate to substitute the gas fractional partialpressure Pi of the subscripted species i for the activity, assuming ai ¼ Pi. With thissubstitution, the PO2 oxygen fractional partial pressure is the only factor that deter-mines which way the reaction progresses. Thus, the equilibrium constant for theoxidation-reduction reaction depends only on the inverse partial pressure ofoxygen, where lower partial pressures favor oxide reduction. In turn, the equilibriumconstant reflects the standard Gibbs free energy DG for the reaction,

DG ¼ �RT ln K ¼ RT ln PO2

where R is the gas constant, and T is the absolute temperature. The free energy forsuch reactions can be determined from tables or charts available in standard metallur-gical references.

K ¼ equilibrium constant, dimensionless

Pi ¼ gas fractional partial pressure for species i, dimensionless [0, 1]



ai ¼ activity for species i, dimensionless [0, 1]

DG ¼ standard Gibbs free energy, J/mol (convenient units: kJ/mol).

246 CHAPTER O

P

PACKING DENSITY FOR LOG-NORMAL PARTICLES (Sohn andMoreland 1968; Dexter and Tanner 1972)

Naturally formed particles usually exhibit a log-normal size distribution.Investigations show that the random packing density of a powder varies with thespread of the log-normal distribution. The distribution spread s is estimated bytwo points from the cumulative particle distribution, for example D84 and D50 (thelatter is the median particle size),

s ¼ lnD84

D50

� �

which is effectively one deviation on the log-normal distribution, where D84 indicatesthe particle size where 84% of the particles are smaller. For spheres, the randompacking density varies with the spread of the log-normal distribution, with wide dis-tributions allowing a better fitting together of the large–small particles. When thevalues of the distribution spread s are small, the packing density is a linear function,

f ¼ aþ bs

where a and b are constants related to the powder size, agglomeration, and shape.Since the fractional packing density is limited to f , 1, there is a need for a modifiedform for very broad-size distributions,

f ¼ c� d

s

which then gives an asymptotic packing density for very broad distributions. Again cand d are empirical constants that vary with each powder.

D50 ¼ median particle size, m (convenient units: mm)

D84 ¼ particle size for 84% on cumulative distribution, m (convenient units: mm)


247

a ¼ empirical constant, dimensionless

b ¼ empirical constant, dimensionless

c ¼ empirical constant, dimensionless

d ¼ empirical constant, dimensionless

f ¼ packing density, dimensionless fraction [0, 1]

s ¼ distribution spread, dimensionless.

PARTICLE COOLING IN ATOMIZATION

See Newtonian Cooling Approximation

PARTICLE COORDINATION NUMBER

See Coordination Number and Density

PARTICLE DIFFUSION IN MIXING (Bridgwater 1994)

When small and large particles are mixed, there is a percolation flow of the smallparticles between the large particles during vibration or agitation. Small-particlemotion in the interstitial voids follows a classic diffusion law,

dC

dt¼ DZ

d2C

dz2

where C is the probability of finding a small particle at a distance z, which is afunction of mixing or vibration time t, and DZ is the particle diffusion coefficient.Similar to other diffusion problems, this equation is solved with appropriate boundaryconditions, for example, assuming the small particles are added to position z ¼ 0at t ¼ 0. The particle probability as a function of mixing time and position is givenas follows:

C ¼ 1

2ffiffiffiffiffiffiffiffiffiffiffiDZtpp exp � z2

4DZt

� �

which is similar to diffusion laws for gases, liquids, and solids.

C ¼ small-particle probability at a distance z, 1/m

DZ ¼ particle diffusion coefficient, m2/s

t ¼ mixing time, s

z ¼ distance, m.

[Also see Mixture Homogenization Rate.]

248 CHAPTER P

PARTICLE FRACTURE IN MILLING (Koch 1998)

During milling the stress required to fracture a particle is often much lower than thematerial strength. This is because a defect population in the particles enables brittlefracture. The Griffith theory says that the stress needed to propagate a crackdepends on the inverse square-root of the crack size. Accordingly, for a brittle particleto fracture during milling requires the ball impact stress s exceed the resistance tocrack propagation. Since the resistance to crack propagation is dominated by thecrack size, most of the factors can be ignored, leading to the following:

s ¼ A

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigSVE

L(1� n2)

r

where E is the elastic modulus of the material, gSV is the surface energy, L is thelength of the crack, and n is Poisson’s ratio. The geometric factor A is near unity.In the limit the crack-tip radius is the atomic size and the crack length is a fractionof the particle size, so the primary factor is the length of the preexisting cracks orthe formation of such cracks during milling. When the impact stress exceeds this con-dition, the crack propagates and the material fails. This model is not accurate forductile materials. For example, copper and gold will deform into flakes, but willnot fracture like a glass or ceramic particle during milling.



L ¼ crack length, m (convenient units: mm)



s ¼ stress to fracture a particle, Pa (convenient units: MPa).

PARTICLE PACKING (Yu et al. 1997)

Particles pack to progressively lower densities as the ratio of interparticle frictionexceeds the bulk particle force from gravity. The interparticle friction dependson the surface area and particle shape, while the gravitational force depends on theparticle mass, which scales with the particle volume. For powders over approximately1-mm packing under normal gravitational conditions, the packing porosity dependson particle size as follows:

1 ¼ 10 þ (1� 10) exp(�aDb)

where D is the median particle size, 1 is the fractional porosity, and 10 is the limitingfractional porosity corresponding to either the tap or apparent density for a largepowder. The parameters a and b depend on the powder shape and the technique

PARTICLE PACKING 249

used for preparing the packing. For monosized glass spheres the packing densitychange with particle size is shown in Figure P1, where a is 0.5 mm20.33 and b is0.33 for 10 set to 0.36.


a ¼ powder-shape factor, m2b (convenient units: mm2b)

b ¼ particle-shape factor, dimensionless


10 ¼ limiting fractional porosity, dimensionless [0, 1].

[Also see Bimodal Powder Packing and Coordination Number and Density]

PARTICLE-SHAPE INDEX (Keey 1992; Bernhardt 1994)

Various problems arise in powder characterization due to irregular particle shapes.Both size and shape measures need to be specified. The sphericity c is defined asthe square of the ratio of equivalent volume to equivalent area diameters.

c ¼ DV

DA

� �2

where DV is the equivalent spherical diameter based on the measured volume, and DA

is the equivalent spherical diameter based on measured surface area. If the particle is asphere, then the sphericity c ¼ 1, but for other shapes c , 1. The particle size by

Figure P1. Fractional porosity versus particle size on a logarithmic scale showing how smallparticles exhibit inhibited packing.

250 CHAPTER P

sedimentation analysis DW is estimated from the sphericity and equivalent sphericaldiameter based on volume, as follows:

DW ¼ DVc1=4

Besides sphericity, there are other means to convey particle-shape information.The aspect ratio is the largest dimension divided by the smallest dimension, butthis fails to differentiate between a fiber and a flake. Two other indices are moreuseful, as illustrated for the case of a rectangular polygon with sides of size x, y,and z, where x . y . z. The formal definition of particle elongation E and particleflatness F is given as follows:

E ¼ x

yand

F ¼ y

z

Needle-shaped particles will be greatly elongated, and platelets will be veryflat. From these definitions Haywood suggested in 1947 that the size of a particlethat would pass through a sieve of opening size M is a function of the flatness as follows:

y

M¼ 2F2

F2 þ 1

� �1=2

This predicts that sieve analysis will have a large error when the particle is flat, since itwill pass diagonally through a mesh opening. Likewise, the equivalent particle dia-meter for sieve analysis departs from the mesh opening size as the flatness increases.

DA ¼ surface area–based equivalent spherical diameter, m (convenientunits: mm)

DV ¼ equivalent spherical diameter based on volume, m (convenientunits: mm)

DW ¼ spherical diameter based on sedimentation, m (convenient units: mm)

E ¼ particle elongation, dimensionless

F ¼ particle flatness, dimensionless

M ¼ mesh opening size, m (convenient units: mm)

x ¼ particle length, m (convenient units: mm)

y ¼ particle breadth, m (convenient units: mm)

z ¼ particle thickness, m (convenient units: mm)

c ¼ sphericity, dimensionless.

PARTICLE SIZE

See Equivalent Spherical Diameter and Mean Particle Size

PARTICLE SIZE 251

PARTICLE-SIZE ANALYSIS

See Sieve Progression

PARTICLE SIZE AND APPARENT DENSITY (German et al. 2006)

Data for the apparent density over a wide particle-size range shows that the apparentdensity falls as the particle size decreases, especially for particles below 1 mm in size.The interparticle friction depends on the surface area, while the particle mass thatinduces better settling and packing depends on the cube of the particle size. Fornanoscale powders, agglomeration dominates packing behavior, and apparent den-sities of 4 to 5% of theoretical are observed. A model used to map apparentdensity into the nanoscale-size range is as follows:

log10( fA) ¼ log10( f0)þ a log10(D=DR)

where fA is the fractional apparent density, D is the particle size, DR is a reference par-ticle size set to 1 mm, and f0 is the fractional packing density at DR. As an example, fortungsten powders over a particle-size range from 20 nm to 60 mm the parameters areDR ¼ 1 mm, a ¼ 0.2, and f0 ¼ 0.143.


DR ¼ reference particle size, m (convenient units: mm)

a ¼ particle-size ratio slope coefficient, dimensionless


f0 ¼ fractional packing density at size of DR, dimensionless.

PARTICLE SIZE BY VISCOUS SETTLING

See Stokes’ Particle Diameter.

PARTICLE-SIZE CONTROL IN CENTRIFUGAL ATOMIZATION

See Centrifugal Atomization Particle Size.

PARTICLE-SIZE DISTRIBUTION

See Andreasen Size Distribution, Gaudin–Schuhman Distribution, Log-NormalDistribution, and Rosin–Rammler Distribution.

PARTICLE-SIZE EFFECT ON INITIAL-STAGE SINTERING

See Herring scaling law.

252 CHAPTER P

PARTICLE-SIZE EFFECT ON PACKING DENSITY

See Particle Packing.

PARTICLE-SIZE EFFECT ON SOLUBILITY

See Solubility Dependence on Particle Size.

PARTICLE-SIZE IN ATOMIZATION

See Water Atomization Particle Size.

PARTICLE-SIZE MEASUREMENT ERROR (Allen 1998)

The error in measuring the particle-size distribution depends on several factors. Theseinclude human bias, sampling error, machine calibration errors, and even the size ofthe sample compared to the bulk mass of the powder lot. An estimate of the particle-size measurement error is possible based on sampling effects. This is possiblebecause N small samples, each of mass w, are taken from the large lot of totalmass W. Typically, the sum of these several small samples is a total mass that isstill small compared to the total lot mass, say one part in a million. If the bulkpower is homogeneous, the many samples will give a similar particle-sizedistribution. If the powder lot is inhomogeneous, however then a variation inparticle-size distributions will occur among the samples. The relative error in theparticle-size measured by the standard deviation SE is estimated from the samplingconditions (N and w), lot size W, and homogeneity H as follows:

SE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHrD3

L1

Nw� 1

W

� �s

where r is the material theoretical density, and DL is the largest particle size in thedistribution and is typically set to D95, corresponding to the size at the 95% pointon the cumulative particle-size distribution. The homogeneity ranges from 0 to 1,and for most powder values are near 0.9. Accordingly, particle-size measurementerrors increase with larger particles and smaller sample sizes, especially for inhomo-geneous powder lots.

D95 ¼ particle size for 95% of cumulative distribution, m (convenient units: mm)

DL ¼ largest particle size, m (convenient units: mm)

H ¼ powder-lot homogeneity, dimensionless [0, 1]

N ¼ number of samples, dimensionless

SE ¼ relative gauge of sampling error, dimensionless

PARTICLE-SIZE MEASUREMENT ERROR 253

W ¼ total powder-lot mass, kg

w ¼ mass of each sample, kg (convenient units: g)

r ¼ material theoretical density, kg/m3 (convenient units: g/cm3).

PEAK BROADENING

See Broadening.

PEAK STRESS FOR EJECTION

See Maximum Ejection Stress

PENDULAR-BOND CAPILLARY FORCE

There is an attractive force in the case of a wetting liquid between two particles. Usingthe geometry shown in Figure P2, the attractive force pulling the particles togetherdepends on the meniscus diameter X as follows:

F ¼ p

4X2DPþ pgLV cosc

where DP is the capillary pressure or pressure difference due to the curved surface, cis an angle associated with the liquid meniscus on the sphere, and gLV is the liquid–vapor surface energy. At equilibrium the energy of the configuration must be at aminimum. This gives a liquid profile that is described by a mathematical curvetermed a nodoid. Particle agglomeration from condensed moisture occurs when the

Figure P2. Two particles with a pendular bond, illustrating the calculation approach based onthe liquid meniscus inducing a capillary attractive force.

254 CHAPTER P

attractive force exceeds the gravitational body force FG given as,

FG ¼p

6rgD3

.


F ¼ capillary force, N

FG ¼ gravitational-body force, N

X ¼ bond size or meniscus diameter, m (convenient units: mm)

g ¼ gravitation acceleration, 9.8 m/s2

DP ¼ pressure difference across the interface, Pa


r ¼ material density, kg/m3 (convenient units: g/cm3)

c ¼ angle from sphere contact to meniscus contact, rad (convenientunits: degree)

[Also see Capillarity and Funicular-state Tensile Strength.]

PERCOLATION IN SEMISOLID PARTICLES

Percolation concepts are important to determining the onset of viscous flow forsemisolid particles during liquid-phase sintering, supersolidus sintering, and spraydeposition. If a high level of solid grain bonding exists, then the system is rigid andresists densification. Alternatively, if no bonding occurs, there is loss of rigiditywith sintering compact distorting during sintering. The ideal situation occurs withpartial rigidity. The relation between the microstructural connectivity and the onsetof viscous flow is given by the critical condition for loss of a percolated solid structure,

NGPC ¼ CN

where CN is the critical number of connections, NG is the grain coordination number,and PC is the probability of a connection between two contacting grains. It is assumedthe relation between the fractional grain-boundary coverage and contact probability isas follows:

PC ¼ 1� FC

For formation of a semisolid structure, CN is approximately 1.5. The intermediatecase between rigid and semisolid is termed the mushy condition, and this occurswhen CN is 2.4. Depending on the degree of solid connections, the structure canrange from a viscous liquid (no connections), to a high viscosity mixture, to a solid(fully connected). Sintering densification in the presence of a liquid phase occurswhen the fractional coverage is high but less than unity.

PERCOLATION IN SEMISOLID PARTICLES 255

CN ¼ critical number of connections, dimensionless

FC ¼ fractional coverage of the grain boundary by liquid, dimensionless

NG ¼ grain coordination number, dimensionless

PC ¼ grain connection probability, dimensionless.

PERCOLATION LIMITS (Kirkpatrick 1973)

If an ordered packing of monosized spheres is created, but some of the spheresare conductors and others are nonconductors, then a critical concentration ofconductors is required for the packing to exhibit conduction. The percolationlimit generally requires that from 20 to 30 volume of the particles be conduc-tors. This varies for situations where the particles are not monosized andwhere the particles are not spherical. Percolation behavior is rationalized tothe particle-packing density and particle-size ratio (conductor and nonconductormay differ in size). Near the composition corresponding to the percolation limitthe mixture changes conduction rapidly, with small composition changes. Inthree-dimensional monosized sphere packings, the critical point of network for-mation that expresses conduction is given by an average of 1.5 conductor–conductor contacts per sphere. This is the critical number of contacts per particlegiven as CP. It is calculated as a function of the composition based on thecoordination number NC as,

CP ¼ NCP

where P is the probability that a neighboring particle will be of the same com-position at the critical condition. For example, for the face-centered cubicpacking of monosized spheres the fractional packing density is 0.74 and thecoordination number is 12. This structure requires P to be 0.125 for CP toequal 1.5. This means that about 13% or more of the particles must be conduc-tors. Conversely, if the simple cubic packing is used with six contacts per par-ticle and a fraction packing density of 0.52, then 25% or more of the particlesneed to be conductors to form a conductive mixture. Randomly packed mono-sized spheres have a coordination number that averages 7, so conduction isexpected when the conductor concentration is over 20 vol % of the particles.

CP ¼ critical number of contacts per particle, 1.5


P ¼ probability that a neighboring particle will have a similar compositiondimensionless [0, 1].

PERIMETER-BASED PARTICLE SIZE

See Grain Diameter Based on Equivalent Circle.

256 CHAPTER P

PERMEABILITY-BASED PARTICLE SIZE

See Fisher Subsieve Particle Size.

PERMEABILITY COEFFICIENT (Ergun 1952; Hsu 2005; Civanand Nguyen 2005)

The permeability is a fundamental parameter for fluid flow in porous bodies. It pro-vides the linkage between the pressure gradient and the volumetric flow rate based onDarcy’s law. Various means to estimate permeability a exist based on the assumptionof a tubular or capillary pore structure. For example, based on Poiseuille’s law and theassumption that pores are equivalent to circular cross-section tubes, the permeabilityis given as follows:

a ¼ dP12

32

where dP is the diameter of the assumed tubular pore, and 1 is the fractional porosity.For loose powders or pore structures that are only lightly sintered, the most accurate ofthe relations is based on the particle size,

a ¼ a13D2

f 2

where a is an adjustable constant, D is the median particle diameter, and f is thefractional density, where fractional density and fractional porosity are given asf ¼ 121. In one concept, the constricted throat size dT (which is smaller thanthe generally accepted pore size dP) is used instead of the pore size. Thethroat size is useful in determining the size of the debris that might passthrough a filter.


a ¼ adjustable constant to account for pore-shape effects, dimensionless





[Also see Darcy’s Law.]

PHASE TRANSFORMATION

See Nucleation Rate.

PHASE TRANSFORMATION 257

PLASMA-SPRAYING PARTICLE SIZE

The difficulty in melting a material during plasma spraying limits the largest usefulparticle size. If the particles do not melt during the passage through the torch andplasma plume, then they will bounce off the substrate. The relation between thelargest useful particle size DL, material thermal diffusivity a, and dwell time in theplasma tD is approximately give by

DL ¼ 3ffiffiffiffiffiffiffiatDp

where the dwell times are on the order of 1024 s. This sets an upper particle size in the100-mm range. Conversely, small particles will evaporate during passage through theplasma gun. Consequently, often particles between 40 and 80 mm prove most suitablefor plasma spraying.

DL ¼ particle size, m (convenient units: mm)

tD ¼ dwell time in the plasma, s

a ¼ material thermal diffusivity, m2/s.

PLASTIC FLOW IN HOT COMPACTION (Swinkels et al. 1983;Artz et al. 1983)

Densification by plastic flow continues as long as the contact stress between particlesexceeds the material’s yield strength. Since the yield strength falls as temperatureincreases, plastic flow is more evident at higher temperatures. The fractional densityf attainable by plastic flow is determined from the applied pressure PA as follows:

f ¼ PA 1� fGð Þ1:3sY

þ f 3G

� �1=3

where fG is the fractional green density, and sY is the material’s yield strength at temp-erature. If the applied pressure is set equal to the yield strength at the process tempera-ture, then for a 60% green density the final fractional density will be about 0.8. Thisrelation is generally valid for final fractional densities less than 0.9, corresponding toa ratio of the applied stress divided by a yield strength of 1.7. At higher densities(over 0.9), the relation between applied pressure, yield strength, and density simplifiesto become as follows:

f ¼ 1� exp � 3PA

2sY

� �

For example, if the final fractional density is set to 0.99, then the required appliedpressure must be three times the yield strength at the compaction temperature.



258 CHAPTER P



sY ¼ yield strength, Pa (convenient units: MPa).

PLASTIC FLOW IN SINTERING (Schatt et al. 1987)

The dislocation climb contribution to pore elimination during solid-state sintering isestimated as follows:

d1

dt¼ seVDV

kTL2

where 1 is the fractional porosity, t is the time, se is the effective sintering stress(which is determined by the surface energy and pore geometry), V is the atomicvolume, DV is the volume-diffusion coefficient, k is Boltzmann’s constant, T is theabsolute temperature, and L is the mean distance between dislocations. Note thatthe effective stress is compressive, or negative, giving a decrease in porosity due todislocation climb. The diffusion distance from a dislocation to a free surface issmaller than from the particle contact zone to the free surface. As a consequence,plastic flow coupled with volume diffusion gives sintering rates far higher than antici-pated from just volume diffusion from pores to free surfaces. As long as the activedislocation density is at least 2 . 1012 m22, then plastic flow can contribute up to a10-fold and even as much as a 100-fold increase in neck-growth rate. However,since dislocation annihilation occurs at the high temperatures associated with sinter-ing, plastic flow in sintering is usually only observed during rapid heating, and losesimportance during isothermal sintering.


L ¼ mean distance between dislocations, m (convenient units: nm)


d1/dt ¼ porosity rate of change, 1/s


t ¼ time, s


1 ¼ fractional porosity, dimensionless

se ¼ effective surface stress, Pa (convenient units: MPa).

[Also see Neck-curvature stress.]

PLASTIC WORKING

See Strain Hardening.

PLASTIC WORKING 259

POISEUILLE’S EQUATION

Under laminar flow conditions Poiseuille’s equation gives the fluid volumetric flow rate Qas a function of the applied pressure P and fluid or powder–binder viscosityhM as follows:

Q ¼ P

hMK

where the flow resistance K depends on the mold geometry. For a cylindrical or tubeshape, the parameter K is calculated from the cylinder length L and the cylinder diameterd as follows:

K ¼ 128L

pd4

Alternatively, for a rectangular cross section, the relation for K in terms of the widthW and thickness t is given as follows:

K ¼ L

Wt3

K ¼ geometric flow-resistance term, 1/m3

L ¼ cylinder length, m (convenient units: mm)

P ¼ applied pressure, Pa (convenient units: MPa)

Q ¼ volumetric fluid-flow rate, m3/s

W ¼ rectangular width, m (convenient units: mm)

d ¼ cylinder diameter, m (convenient units: mm)

t ¼ rectangular thickness, m (convenient units: mm)

hM ¼ fluid viscosity, Pa . s.

POISSON’S RATIO (Haynes 1981)

Porosity degrades most material properties, and this is true for elastic behavior too.Poisson’s ratio n is given as follows:

n ¼ � 1Y

1X

where 1X is the axial strain, and 1Y is the perpendicular or radial strain. When poresare present Poisson’s ratio depends on the fractional density f, and experiments onsintered iron and steel give the following behavior:

n ¼ 0:068 e1:37f

as the density approaches 100%, Poisson’s ratio approaches the handbook value.

f ¼ fractional density, dimensionless


1X ¼ axial strain, dimensionless

1Y ¼ radial strain, dimensionless.

260 CHAPTER P

POLYMER-BLEND VISCOSITY

See Binder (Mixed-polymer) Viscosity.

POLYMER PYROLYSIS

Depolymerization of polymeric binders during delubrication, dewaxing, or debindingis described by first-order reaction kinetics. In this form, the remaining weight frac-tion of a polymer a is expressed as a time function:

da

dt¼ �Ka

where t is the time, and K is the rate constant for thermal degradation that follows theArrhenius equation:

K ¼ K0 exp � Q

RT

� �

in which K0 is the rate-constant frequency factor, Q is the apparent activation energyfor thermal degradation, R is the gas constant, and T is the absolute temperature,respectively.

K ¼ pyrolysis rate constant, 1/s

K0 ¼ rate-constant frequency factor, 1/s

Q ¼ apparent activation energy for degradation, J/mol (convenient units: kJ/mol)



t ¼ time, s

a ¼ weight fraction of unreacted polymer, dimensionless [0, 1].

[Also see Debinding Temperature.]

PORE ATTACHMENT TO GRAIN BOUNDARIES(Patterson et al. 1990)

There is a natural affinity between the pores and grain boundaries that gives a highprobability that a pore will remain attached to a grain boundary. At high sintereddensities, the pores are mostly associated with the largest grains. The increasedprobability of pore–grain boundary attachment ranges up to 5.7 times that forrandom contact. Consequently, the relation between grain size G, pore diameter dP,and fractional porosity 1 is given as,

G

dP¼ K

R1

PORE ATTACHMENT TO GRAIN BOUNDARIES 261

where R expresses the ratio of attached pores to randomly placed pores, and K is ageometric constant. Values of R range from 1.7 to 5.7 for various sintering materials.The degree of boundary–pore contact remains essentially constant during most of thesintering cycle.


R ¼ ratio of attached pores to randomly placed pores, dimensionless

K ¼ geometric constant, dimensionless



PORE CLOSURE

During intermediate-stage sintering the pores form a tubular network that is attachedto the grain boundaries. If densification occurs, the pores shrink while simultaneousgrain growth stretches the pores. As this continues, eventually the elongated and thin-ning pores pinch off into closed spherical pores, a process termed pore closure. Acalculation of the instability of a cylindrical pore of length l and diameter dP givesthe critical condition for closure into separate pores as follows:

l � dPp

For a cylinder-pore geometry occupying the edges of tetrakaidecahedron grains thisinstability occurs at a porosity of approximately 8%. In reality, due to distributions inthe initial particle sizes, the instability that induces pore closure occurs over a broadrange of densities from 85% to 95%.


l ¼ length of cylindrical pore, m (convenient units: mm).

[Also see Raleigh Instability.]

PORE DRAG

See Migration of Particles.

PORE FILLING IN LIQUID-PHASE SINTERING (Kang et al. 1984)

If a pore is much larger than the grain size, then it remains stable and exhibits delayedfilling during liquid-phase sintering. This is because of preferential capillary wettingby the liquid on the smaller intergrain spaces. Due to grain growth during prolongedheating, however, a large pore will eventually be filled by liquid. The favorable

262 CHAPTER P

condition for refilling is determined by the curvature at the solid–liquid–vaporinterface. The liquid meniscus radius rm at the pore–liquid–grain contact is given as,

rm ¼G

2

� �1� cosa

cosa

where G is the grain size, and a is the angle from the connector between grain centersto the solid–liquid–vapor contact point on the grain surface as illustrated inFigure P3. The meniscus radius increases in proportion to grain size. Eventually,the capillary pressure associated with the meniscus induces liquid flow into thepore to give a lower liquid pressure. For the ideal case of a zero-contact angle,pore filling occurs when the pore radius and the meniscus radius are equal. If thecontact angle is greater than zero, then the meniscus radius must exceed the grainradius for pore filling.


rm ¼ liquid-meniscus radius, m (convenient units: mm)

a ¼ angle defined by the connector between grain centers and the solid–liquid–vapor contact point on the grain surface, rad (convenient units: degree).

Figure P3. Pore filling during liquid-phase sintering depends on the liquid wetting the grainsand grain growth, where the spreading liquid meniscus eventually leads to liquid flow into thelarger residual pores.

PORE FILLING IN LIQUID-PHASE SINTERING 263

PORE-FREE COMPOSITE OR MIXED-PHASE DENSITY

Composite density is calculated by the inverse rule of mixtures for two materialsdesignated as A and B, with the mass of material A being WA and the mass of Bbeing WB. The theoretical densities of each material are designated as rA and rB.The density of the mixture is found by dividing the total mass by the total volume.The total mass WT is,

WT ¼ WA þWB

The volume of each material is the mass divided by the density,

VA ¼ WA=rA

andVB ¼ WB=rB

The total volume VT is the sum of VA þVB. Hence, the theoretical density for themixture rT is given as the total weight divided by the total volume,

rT ¼WT

VT¼ WA þWB

(WA=rA)þ (WB=rB)

VA ¼ volume of material A, m3 (convenient units: mm3)

VB ¼ volume of material B, m3 (convenient units: mm3)

VT ¼ total volume, m3 (convenient units: mm3)

WA ¼ mass of material A, kg (convenient units: g)

WB ¼ mass of material B, kg (convenient units: g)

WT ¼ total mass, kg (convenient units: g)

rA ¼ theoretical density of material A, kg/m3 (convenient units: g/cm3)

rB ¼ theoretical density of material B, kg/m3 (convenient units: g/cm3)

rT ¼ theoretical density for the mixture, kg/m3 (convenient units: g/cm3).

PORE MOBILITY DURING SINTERING (Kang 2005)

During the latter stages of sintering the grain size increases, progressively reducingthe energy associated with grain boundaries. The extension of mobile grain bound-aries through the porous microstructure leads to pore motion with the grain bound-aries. Pores are slower moving, so the pore mobility determines the rate of graingrowth up to pore-boundary separation. Pore mobility generally scales with theinverse square of the pore size, but can vary with the inverse fourth power ofthe pore size. The details depend on the dihedral angle and pore size, but mostlyare influenced by the mass-transport process whereby the pore moves. The net

264 CHAPTER P

driving force of boundary-migration is a combination of the boundary-motion drivingforce, which largely relates to the grain size and the retarding effect from the pores.If the pores remain attached to the grain boundary, then it is often the case that thepore mobility M limits grain growth. Pore mobility during sintering is given as,

M ¼ B

dNP

where dP is the pore size, B is a collection of kinetic terms that depend on the mass-transport mechanism controlling pore motion, and N depends on the mechanism—forsurface diffusion, N ¼ 4, for volume diffusion N ¼ 3, for evaporation–condensationN ¼ 2. When the pore is filled with gas that can form a reactive vapor species, there isan additional mechanism termed gas diffusion, where N ¼ 3. The form of the term Bvaries with the mechanism. For example, for the common processes of surface andvolume diffusion it is given as follows:

B ¼ 16DSdV

pRTand

B ¼ 8DVV

pRT

while for the evaporation–condensation and gas-diffusion processes, the forms areas follows:

B ¼ 4PV2ffiffiffiffiffiffiffi2 mp 1

pRT

� �3=2

and

B ¼ 4DGPffiffiffiffiffiffi2pp V

RT

� �2

In each term there are factors that exhibit an Arrhenius temperature dependence, soeach process has a corresponding activation energy. These equations are only nomin-ally accurate, since they assume spherical pores. If there is a dihedral angle associatedwith the grain-boundary intersection with the pore surface, then further mathematicalcomplications arise. Note that these same types of equations are evident in initial-stage sintering models, where similar simplification assumptions and mechanismsare common.

B ¼ mechanism-dependent kinetic term, dimensions vary with mechanism

DG ¼ gas-diffusion coefficient, m2/s

DS ¼ surface-diffusion coefficient, m2/s


M ¼ pore mobility, m3/(N . s)

PORE MOBILITY DURING SINTERING 265

N ¼ mechanism-dependent exponent, dimensionless





m ¼ molecular mass, kg/mol

V ¼ atomic or molecular volume, m3/mol

d ¼ surface-defective layer thickness, m (convenient units: nm).

PORE PINNING OF GRAIN BOUNDARIES

See Grain Pinning by Pores in Final-stage Sintering and Zener Relation

PORE-SEPARATION DISTANCE

The mean separation l (edge-to-edge) between pores is measured from two-dimensional section planes using quantitative microscopy. The separation is givenas follows:

l ¼ f

NL

where f is the fractional density, and NL is the number of pores (or phase of interest)intercepted by a random test line. The test-line length is corrected for the magnifi-cation; for example, if a 10-mm line is used at 100 magnifications, then the testline is 0.1 mm or 100 mm, and NL is then the number of pores per 100 mm.

NL ¼ number of pores per unit test-line length, 1/m (convenient units: 1/mm)

f ¼ fractional density, dimensionless

l ¼ mean separation between pores, m (convenient units: mm).

PORE SEPARATION FROM GRAIN BOUNDARIES

During grain growth, the critical fractional density fC for pore separation from thegrain boundary is a concern. Various efforts have suggested that a critical conditionoccurs based on the initial breadth of the particle-size distribution:

fC ¼ 0:6þ 0:4RG

where RG is the ratio of the average particle size to the maximum particle size. Onlyfor monosized powders where RG ¼ 1 is separation avoided.

266 CHAPTER P

fC ¼ critical fractional density for pore-boundary separation, dimensionless [0, 1]

RG ¼ ratio of the average particle size to the maximum particlesize, dimensionless.

PORE SIZE AND GRAIN SIZE IN FINAL-STAGE SINTERING

See Grain Size to Pore Size in Final-stage Liquid-phase Sintering.

PORE SIZE AND GRAIN SIZE IN INTERMEDIATE-STAGESINTERING

In the intermediate stage of sintering, the grain is assumed to be the fourteen-sided tetrakaidecahedron with cylindrical pores occupying the grain edges. Forthat geometry, the pore diameter dP, grain size G, and fractional porosity 1 arerelated as,

1 ¼ pdP

G

� �2

As long as the grain boundary remains attached to the pores, this relation says thegrain size will increase as pores coalesce (increasing pore size) or as porosity is elimi-nated (decreasing porosity).




[Also see Final-stage Sintering Grain Growth and Pore Drag Effect.]

PORE SIZE IN FINAL STAGE SINTERING

A constraint on sintering results from gas trapped in the pores. This limit applies topressure-assisted sintering as well, where the capillary force originating from surfaceenergy is supplemented by the external pressure. For densities over approximately92% of theoretical, the tubular pores found in the intermediate stage of sinteringpinch closed to form spherical pores with a diameter dP that depends on the grainsize G and fractional density f as follows:

dP ¼ G1� f

6

� �1=3

These spherical pores will continue to densify to a critical point where densificationceases. At that point in the final stage of solid-state sintering, the solid–vapor

PORE SIZE IN FINAL STAGE SINTERING 267

surface energy gSV of the curved spherical pore is balanced by the internalgas pressure Pg,

4gSV ¼ PgdP

If a compact is sintered in an inert gas such as argon at a pressure P1 with a pore sized1 at pore closure, then the limiting porosity is calculated by recognizing that the massof the gas in the pores is conserved. If the number of pores and temperature remainconstant with a spherical pore shape, then pressure and volume before and afterclosure is given as follows:

P1V1 ¼ P2V2

The final pore size d2 is estimated as follows:

d2 ¼12

ffiffiffiffiffiffiffiffiffiffid 3

1 P1

gSV

s

If the gas is soluble in the material being sintered, then the pressure caused by den-sification of the pore will allow eventual pore elimination.


P1 ¼ gas pressure in the pore at closure, Pa

P2 ¼ gas pressure in the end of densification, Pa

Pg ¼ gas pressure inside the pore, Pa

V1 ¼ pore volume at pore closure, m3 (convenient units: mm3)

V2 ¼ pore volume at the end of densification, m3 (convenient units: mm3)

d1 ¼ pore size at pore closure, m (convenient units: mm)

d2 ¼ pore size at the end of densification, m(convenient units: mm)




PORE SIZE IN VISCOUS-FLOW FINAL-STAGE SINTERING

Viscous flow is the operative sintering mechanism for amorphous materials such asglass and polymers. During viscous-flow sintering, the pore size decreases linearlywith time,

dP ¼ dP0 �gSVt

h

268 CHAPTER P

where dP is the pore size as it reduces from the initial pore size dP0 at the beginning ofthe final stage, t is the time of sintering in the final stage, gSV is the surface energy,and h is the viscosity.


dP0 ¼ initial pore size at start of final stage, m (convenient units: mm)

t ¼ sintering time during the final stage, s


h ¼ viscosity of the material, Pa . s.

POROSIMETRY

See Washburn Equation

POROSITY

See Fractional Density

POROSITY EFFECT ON DUCTILITY

See Sintered Ductility

POROSITY EFFECT ON ELASTIC BEHAVIOR (Green 1998)

Because of the lack of texture in most sintered materials, only average parameters areneeded to describe the elastic properties. Pores reduce the average elastic properties.One model for treating the elastic modulus E variation with density in sintered bodies,where only closed pores exist, is attributed to MacKenzie. It relies on a polynomialas follows:

E ¼ E0 1� a1 þ b12� �

where E0 is the full-density isotropic elastic modulus, 1 is the fractional porosity(less than approximately 0.08), and a and b are constants, estimated at 1.9 and 0.9,respectively. At the other end of the porosity range, for very high porosities over0.7 corresponding to open cell structures, the elastic-modulus dependence on porosityis approximated as follows:

E ¼ E0 1� 1ð Þ2

The shear modulus follows the same functional behavior with respect to porosity.



POROSITY EFFECT ON ELASTIC BEHAVIOR 269

a ¼ constant, dimensionless

b ¼ constant, dimensionless


POROSITY EFFECT ON SONIC VELOCITY

See Ultrasonic Velocity.

POROSITY EFFECT ON STRENGTH


POROSITY EFFECT ON THERMAL CONDUCTIVITY

See Thermal-conductivity Dependence on Porosity.

POROSITY IN SWELLING SYSTEMS WITH LIMITED SOLUBILITY

Liquid-phase sintering relies on a mixture of two or more powders to form a liquidbetween solid grains during heating. The porosity resulting from the melting ofone constituent depends on the chemical interactions and several processingfactors. Although most liquid-phase sintering results in densification, a few materialcombinations give swelling. This most typically occurs when the solid has a low solu-bility in the liquid, and becomes most pronounced when the liquid is soluble in thesolid. In these cases, the porosity 1 increases from the initial porosity 10 as a functionof the additive concentration C and the fraction b, which has reacted as follows:

1 ¼ 10 þ bC 1� 10ð Þ

which shows the effect of an initially increasing porosity or additive concentration onthe final porosity.

C ¼ additive concentration, kg/kg or dimensionless

b ¼ fraction of additive that reacts, dimensionless fraction


10 ¼ initial porosity, dimensionless fraction [0, 1].

POWDER-FORGING HEIGHT STRAIN AND DENSIFICATION

Densification in powder forging approximately relates to the height strain 1,

df ¼ �f 1� 2nð Þ d1

270 CHAPTER P

where f is the fractional density, and n is the actual Poisson’s ratio, which changeswith density. The minus sign accounts for the fact that compression, a negativestrain, gives a density increase.



1 ¼ height strain, dimensionless.

POWDER INJECTION-MOLDING FEEDSTOCK VISCOSITY


POWER-LAW CREEP (Artz et al. 1983; Helle et al. 1985)

Power-law creep models for hot consolidation combine diffusive transport and stressbias to link densification to the applied pressure in hot pressing and hot isostatic press-ing. When both the stress and temperature are high, the shrinkage rate depends ondislocation climb, which is described by a power-law creep equation,

1L0

d DLð Þdt¼ CbmDV

kT

PE

m

� �n

where T is the absolute temperature, k is Boltzmann’s constant, DV is the lattice orvolume diffusivity, PE is the effective pressure, b is the Burger’s vector or crystaloffset associated with a dislocation, C is a material constant (units per atom), m isthe shear modulus, and n is an exponent expressing the stress sensitivity, whichoften in powder consolidation is between 2 and 4.

C ¼ material constant, 1/atom


L ¼ length, m (convenient units: mm)





d(DL)/dt ¼ rate of change in length, m/s


n ¼ exponent, dimensionless

t ¼ time, s


m ¼ shear modulus, Pa (convenient units: GPa).

[Also see Dislocation Climb-controlled Pressure-assisted Sintering Densification.]

POWER-LAW CREEP 271

PREALLOYED-PARTICLE SINTERING

See Liquid and Solid Compositions in Prealloy-particle Melting.

PREALLOYED-POWDER LIQUID-PHASE SINTERING

See Supersolidus Liquid-phase Sintering Shrinkage Rate.

PRECISION

Precision is linked to the process capability as a ratio to the total tolerance range,

P ¼ XU � XL

s

where P is an index of precision, with XU is the upper allowed size or property, XL isthe lower allowed size or property, and s is the standard deviation measured for theprocess. Processes with a P ratio greater than 8 are designated as high precision, whileprocesses with P below 6 are deemed low precision.

P ¼ precision index, dimensionless

XL ¼ lower allowed size, m (convenient units: mm)

XU ¼ upper allowed size, m (convenient units: mm)

s ¼ standard deviation, m (convenient units: mm).

PRESSURE-ASSISTED LIQUID-PHASE SINTERING(Kingery et al. 1963)

In the case of hot pressing or other pressure-assisted sintering processes where aliquid exists, the semisolid mixture is treated as a viscous system. If the solid issoluble in the liquid, then densification is enhanced by the simultaneous solution pre-cipitation events in the liquid responding to the external pressure. A modified form ofthe intermediate-stage sintering-densification model is applicable when the control-ling mechanism is diffusion though the liquid phase. Accordingly, the liquid phasesintering model includes the external stress as a supplemental driving force, givingthe isothermal sintering shrinkage DL/L0 dependence on operating parameters as,

DL

L0

� �3

¼ gdDLCVt

G3RT

� �PE þ

2gLV

dP� PP

� �

In this equation, g is a geometric constant; d is the intergranular liquid-film thickness;DL is the diffusion rate of the dissolved solid in the liquid, and it has an exponential

272 CHAPTER P

temperature dependence; C is the solid solubility in the liquid; V is the atomicvolume; t is the sintering time; G is the grain size; R is the universal gas constant;T is the absolute temperature, gLV is the liquid–vapor surface energy; dP is thepore diameter; PE is the effective pressure; and PP is the gas pressure in the pores.The effective pressure is calculated from the applied pressure and the fractionaldensity. Because of the assumed semisolid behavior, the viscous body has a densifi-cation rate that is inversely related to solid diffusivity in the liquid—a high diffusivitycorresponds to a low viscosity. In the final stage of densification, where the residualpores are spherical and isolated, the rate of densification is then dominated by theexternal pressure. With the elimination of porosity, the effective pressure andapplied pressure converge.


DL ¼ diffusivity of the solid in the liquid, m2/s

G ¼ grain size, m (convenient units mm)


PE ¼ effective pressure, Pa

PP ¼ gas pressure in the pores, Pa




dP ¼ pore diameter, m (convenient units mm)

g ¼ geometric term, dimensionless

t ¼ time, s





d ¼ liquid-film thickness, m (convenient units nm).


PRESSURE-ASSISTED SINTERING MAXIMUM DENSITY

See Maximum Density in Pressure-assisted Sintering.

PRESSURE-ASSISTED SINTERING SEMISOLID SYSTEM

See Viscous Flow in Pressure-assisted Sintering.

PRESSURE-ASSISTED SINTERING SEMISOLID SYSTEM 273

PRESSURE EFFECT ON FEEDSTOCK VISCOSITY(Hausnerova et al. 2006)

Because of differences in bulk compressibility between the powder and binder, feed-stock for extrusion or injection molding had a sensitivity to the pressure, where theviscosity increases as the bulk pressure increases. The sensitivity is expressedas follows:

h ¼ h0 exp(bP)

where h is the viscosity of the mixture at pressure P, and h0 corresponds to theviscosity at the same temperature, solids loading, and shear strain rate at atmosphericpressure. Effectively, the parameter b reflects the increase in solids loading andparticle–particle friction from the compression of the lower bulk-moduluspolymer binder.

P ¼ pressure, Pa (convenient units MPa)

b ¼ pressure sensitivity parameter, 1/Pa

h ¼ mixture viscosity under pressure, Pa . s

h0 ¼ mixture viscosity at atmospheric pressure, Pa . s.

PRESSURE EFFECT ON FINAL-STAGE SINTERING BYVISCOUS FLOW

Final-stage sintering corresponds to the closure and collapse of the pores as full densityis attained. For a microstructure consisting of tetrakaidecahedron grains that are allthe same size, the calculated onset of final-stage sintering is 8.25% porosity.However, since there is a distribution in grain size and pore size, pore closure occursover a range of densities. Often the first closed pores are seen at 85% density and allpores are usually closed by 95% density. Pore closure occurs because the surfaceenergy of a long pore is higher than a collection of spherical pores, so a cylindricalpore of length L and diameter dP will close into spherical pores when L � pdP,corresponding to the Raleigh instability criterion. The pores become spheres with afinal diameter of more than 1.5 times the cylinder diameter, resulting in an increasein pore-size as final-stage sintering occurs. If the solid is treated as a viscous system,then a relation emerges that links fractional porosity 1 and sintering time t,

ln1

10

� �¼ � 3PEt

4h

which says the porosity decays from an initial value (at the onset of the final stage ofsintering) of 10 over time t. The factor PE represents the effective stress during sintering,andh is the effective viscosity. However, this assumes the effective pressure is constantand the system has no dependence on microstructure. In reality, during densification the

274 CHAPTER P

effective pressure decays, making densification slower than anticipated from thissimple viscous-flow model. Further, gas in the pores resists densification. Even so,this model provides a first basis for describing hot consolidation of powders.

L ¼ pore length or grain-edge length, m (convenient units: mm)

PE ¼ effective stress, Pa (convenient units: MPa)


t ¼ time, s


10 ¼ initial fractional porosity, dimensionless [0, 1]


PRESSURE-GOVERNING EQUATION IN POWDER-INJECTIONMOLDING (Kwon and Ahn 1995)

Figure P4 is used to schematically represent a mold cavity with a local coordinatesystem (x, y, z) as used in flow and heat-transfer analysis. The thickness of aninjection-molded part is usually small compared with the other dimensions.In such a case, the momentum equations is approximated by the Hele–Shaw approxi-mation described as follows:

0 ¼ � @P

@xþ @

@zh@u

@z

� �or

h@u

@z¼ @P

@xz

0 ¼ � @P

@yþ @

@zh@v

@z

� �

Figure P4. The layout for calculating the heat and mass flow in a mold cavity with a localcoordinate system (x, y, z) where the cavity is thin when compared to the width and length.

PRESSURE-GOVERNING EQUATION IN POWDER-INJECTION MOLDING 275

or

h@v

@z¼ @P

@yz

where P is the pressure, u and v are the velocity components in the x and y directions,h is the viscosity as a function of a generalized shear rate g and temperature. In thisapproximation, the generalized shear rate g is defined by

_g ¼ @u

@z

� �2

þ @v

@z

� �2" #1=2

Integration of the momentum equations in the z direction with an assumed symmetryto the velocity profile along the centerline results in the following equations:

u(x, y, z) ¼ � @P

@x

ðb

z

�z

hd�z

and

v x, y, zð Þ ¼ � @P

@y

ðb

z

�z

hd�z

The average velocity components u and v across the thickness are obtained byintegrating to yield

�u x, yð Þ ¼ � S

b

@P

@xand

v x, yð Þ ¼ � S

b

@P

@y

where the flow conductivity constant S is defined as

S ¼ð b

0

z2

hdz

Mass conservation is given as follows:

@u

@xþ @v@yþ @w

@z¼ 0

This equation is integrated in the z-direction to yield

@�u

@xþ @v@y¼ 0

276 CHAPTER P

Substituting the terms u and v in the pressure-governing equation results in thefollowing:

@

@xS@P

@x

� �þ @

@yS@P

@y

� �¼ 0


S ¼ flow conductivity, m3/(Pa . s)

b ¼ half-gap thickness of injection-molded part, m (convenient units: mm)

u, v ¼ velocity components in the x and y directions, m/s

u, v ¼ average in the z direction of velocity components in thex and y directions, m/s

x, y, z ¼ coordinate, m

z ¼ integrand variable for the z direction, m



PRESSURE-GOVERNING EQUATION FOR POWDER INJECTIONMOLDING WITH SLIP LAYER (Kwon and Ahn 1995)

In the slip-layer model for powder injection molding the flow conductivity Ssl isobtained by calculating both the bulk feedstock-core behavior and the thin pure-binder region at the mold wall. This leads to the following calculation,

Ssl ¼ðc1

z2

hmdzþ

ðb

c

z2

hbdz

The parameters hm and hb are the viscosities of the bulk powder–binder mixture andthe pure-binder system in the slip layer, respectively. The subscript sl represents theslip layer. As shown in Figure P5, the upper bound of the first integral c ¼12 d

(where d is the slip-layer thickness and b is the half-gap thickness of the injection-molded part), and the lower bound of the first integral term, 1, is the thickness ofthe region where no yield takes place. In other words, the shear stress at z ¼ 1 isthe same as the yield stress, ty. Substituting terms results in the following pressure-governing equation:

@

@xSsl@P

@x

� �þ @

@ySsl@P

@y

� �¼ 0

Once the pressure field P(x, y) is obtained from the solution, it is possible to findthe local velocity distribution from the generalize shear rate. It should be noted that Sdepends on the distribution of the viscosities hm, hb, which in turn depend upon theshear rate g with respect to the velocity field. In this respect the solution is a nonlinear

PRESSURE-GOVERNING EQUATION FOR POWDER INJECTION MOLDING 277

partial differential equation of the pressure field, which requires an iteration approachto obtain a convergent numerical solution.


Ssl ¼ flow conductivity with slip-layer model, m3/(Pa . s)




d ¼ slip-layer thickness, m

1 ¼ half-gap thickness of plug flow, m (convenient units: mm)

hb ¼ pure-binder viscosity, Pa . s

hm ¼ feedstock viscosity, Pa . s

ty ¼ yield stress of feedstock, Pa

c ¼ 12d ¼ y coordinate at slip-layer boundary, m.

PRESSURE-GOVERNING EQUATION IN 2.5 DIMENSIONS FORPOWDER INJECTION MOLDING WITH SLIP VELOCITY (Kwonand Ahn 1995)

With a slip-boundary condition at the wall, integration of the momentum equationyields local velocity components as follows:

u x, y, zð Þ ¼ us �@P

@x

ðb

z

�z

hd�z

and

v x, y, zð Þ ¼ vs �@P

@y

ðb

z

�z

hd�z

Figure P5. Slip-layer formation in the filling of a mold cavity shown in cross section from thecenterline to the mold wall, where the velocity profile corresponds to a region that is effectivelypure binder and at the mold surface; that is, the slip layer. The central plug corresponds to theregion where there is no yield.

278 CHAPTER P

where us and vs denote slip-velocity components, which are determined as follows:

us ¼ Vs� @P=@xð Þrpj j

and

vs ¼ Vs� @P=@yð Þrpj j

where Vs is the magnitude of the slip velocity. The average velocity componentsacross the thickness are obtained by integrating to give,

�u x, yð Þ ¼ � Ssv

b

@P

@xþ us

and

�v x, yð Þ ¼ � S

b

@Psv

@yþ ys

where the flow conductivity S for the slip-velocity model is defined by

Ssv ¼ðb

1

z2

hmdz

A combination of results gives the pressure-governing equation as follows:

@

@xSsv

@P

@x� bus

� �þ @

@ySsv

@P

@y� bvs

� �¼ 0

This is a nonlinear partial differential equation of the pressure field.


Ssv ¼ flow conductivity with slip-velocity model, m3/(Pa . s)

Vs ¼ magnitude of the slip velocity, m/s


u, v ¼ velocity components in the x and y directions, m/s

us, vs ¼ slip-velocity components in the x and y directions, m/s

u, v ¼ average in the z direction of velocity components in the x and ydirections, m/s


z ¼ integrand variable for the z direction, m

r ¼ gradient, 1/m

1 ¼ half-gap thickness of plug flow, m



PRESSURE-GOVERNING EQUATION IN 2.5 DIMENSIONS 279

PRESSURE GRADIENTS IN COMPACTION

See Die-wall Friction.

PRESSURE-INDUCED NECK FLATTENING

See Compaction-induced Neck Size.

PRICE ESTIMATION

See Costing and Price Estimation.

PROCESS CAPABILITY

Denoted as Cp, the process capability measures the ratio of the process spread basedon three standard deviations and compares this process spread with the tolerance orallowed spread,

Cp ¼UM � UA

3s

where UM is the maximum (or minimum) control limit, UA is the average, and s is thestandard deviation. If the mean value is centered between the upper and lowerbounds, then the process is allowed to have the largest variation. However, if theprocess skews toward one end or the other (upper or lower bound), then less variationis allowed. (NOTE: The units for this calculation must be consistent; for example, allmetric, such as kg or m.)

Cp ¼ process capability, dimensionless

UM ¼ maximum or minimum control limit, consistent units

UA ¼ average, consistent units

s ¼ standard deviation, consistent units.

PROJECTED AREA–BASED PARTICLE SIZE

See Equivalent Spherical Diameter.

PROOF TESTING


280 CHAPTER P

PYCNOMETER DENSITY

The gas pycnometer is used to measure the theoretical density of a loose powderbased on gas infiltration into the pores between the particles. Pressurized gas, oftenhelium, is used to measure the volume of open pores in a powder of known mass,in a sample chamber of known volume VS, but unknown powder volume VP.Initially, the chamber is at a pressure P1. An initially evacuated calibrationchamber with volume VC is connected to the sample volume. After the connectingvalve is opened, the pressure decreases to P2. Applying the ideal gas law gives,

P1 VS � VPð Þ ¼ P2 VS � VP þ VCð Þ

The powder volume VP as the only unknown,

VP ¼ VS þVC

1� P1=P2ð Þ

The powder mass divided by its volume gives the pycnometer density.

VC ¼ calibration chamber with volume, m3

VP ¼ powder volume, m3

VS ¼ sample chamber volume, m3

P1 ¼ initial chamber pressure, Pa

P2 ¼ final chamber pressure, Pa.

PYCNOMETER DENSITY 281

Q

QUANTITATIVE-MICROSCOPY DETERMINATION OF SURFACEAREA

See Surface Area by Quantitative Microscopy.

QUASI-3-DIMENSIONAL ENERGY-GOVERNING EQUATIONFOR POWDER INJECTION MOLDING

See Energy-governing Equation for Powder Injection Molding.

QUASI-3-DIMENSIONAL PRESSURE-GOVERNING EQUATION FORPOWDER INJECTION MOLDING

See Pressure-governing Equation in Powder Injection Molding.

QUASI-3-DIMENSIONAL PRESSURE-GOVERNING EQUATION FORPOWDER INJECTION MOLDING WITH SLIP-LAYER MODEL

See Pressure-governing Equation in Powder Injection Molding with Slip-layer Model.

QUASI-3-DIMENSIONAL PRESSURE-GOVERNING EQUATION FORPOWDER INJECTION MOLDING WITH SLIP-VELOCITY MODEL

See Pressure-governing Equation in 2.5 Dimensions for Powder Injection Moldingwith Slip-velocity Model.


283

R

RADIAL CRUSH STRENGTH

See Bearing Strength

RADIANT HEATING (Chung 1983)

In sintering, the heat flow by radiant heating Q varies with the temperature gradientbetween the furnace walls and the component based on the following formula:

Q ¼ F1s T4f � T4

� �

where F is the viewing factor, which represents the angular orientation of the compactwith respect to the heating elements and is usually assumed to be near 0.9; 1 is theemissivity, and for powder compacts with rough surfaces it is usually near 0.6;s is the Stefan–Boltzmann constant; Tf is the furnace temperature; and T is thecompact surface temperature (both temperatures are on the absolute scale). In prac-tice, radiant heating is not effective until higher temperatures are reached, usuallyrequiring furnace temperatures in the 773 to 873 K (500 to 6008C) range.

F ¼ viewing factor, dimensionless

Q ¼ heat flow, J/(m2 . s)

T ¼ absolute temperature of the component surface, K

Tf ¼ absolute temperature of the furnace, K

1 ¼ emissivity, dimensionless

s ¼ Stefan–Boltzmann constant, 5.7 . 1028 W/(m2 . K4).


285

RALEIGH INSTABILITY

Fluid structures change from elongated ligaments into spheres if the surface energyof the spheres is lower than that of the ligament. This leads to instability in long,ligamental or tubular structures, as is seen in pore pinching during sintering. Herethe pore diameter decreases due to sintering densification, while the pore lengthincreases due to grain growth. In the same way, during atomization the primaryliquid ligaments formed by the atomizer inherently undergo this same Raleigh instabil-ity, transforming ligaments to droplets. Consider a ligament formed from a cylinder offluid material, for example, molten metal during atomization. Let the cylindrical dia-meter be d and length be L, where L is much larger than d. The initial volume andsurface area of the cylinder are pLd2/4 and pdL (ignoring end effects), respectively.This cylinder spontaneously decomposes into spheres during atomization. For thattransformation from a cylinder to sphere to occur, the system energy cannot increaseand volume must be conserved. Assume that surface energy is the only importantenergy term and that surface area is proportional to surface energy. If atomizationproduces N spheres with diameter of D, then two equations with two unknowns arepossible. The first equation states that there will be no change in volume,

p

6ND3 ¼ p

4Ld2

The second equation states that the surface area (energy) must be preserved (or evenreduced), giving the following relation if the ends of the cylinder are ignored:

pND2 ¼ pLd

The solution to these two equations gives the number and diameter of theresulting spherical droplets (which become particles) in terms of the original ligamentdiameter,

D ¼ 32

d

and

N ¼ 4L

9d

Thus, the final particle size is about 1.5 times the original ligament diameter, andthe number of spherical particles per ligament depends on the starting lengthover the diameter ratio. One implication is that the formation of small particles inatomization requires first attention to the production of small-diameter ligaments.With respect to final-stage sintering, the result shows an increase in the pore diameterwhen the pores close, which typically occurs near 92 to 95% density, but in somecases at 85% density.

D ¼ diameter of sphere, m (convenient units: mm)

L ¼ length of fluid cylinder, m (convenient units: mm)

286 CHAPTER R

N ¼ number of spheres, dimensionless

d ¼ diameter of fluid cylinder, m (convenient units: mm).

RANDOM PACKING DENSITY (Scott 1960; McGeary 1961)

A random packing is constructed by a sequence of events that are not correlated withone another. When a powder is poured into a container, the particles bounce, tumble,and settle to produce such a random structure. An ordered structure occurs whenobjects are placed systematically into periodic positions, such as are seen in theatomic structure of crystals. Random structures lack long-range repetition, and typi-cally exhibit lower packing densities. For monosized spheres the maximum packingdensity occurs in an ordered close-packed array with a coordination number of 12 anda density of 74%. With respect to most powders, unless placed one at a time, thepacking is random and the highest packing density is less than ideal; monosizedspheres poured into a container usually pack between 60 and 64%. Tap density,the highest-density random packing, occurs when the particles have been vibratedwithout introducing long-range order or deformation. Random loose packingresults when particles are poured into a container without agitation or vibration;this type of packing is also commonly called the apparent density. For the highestdensities, it is appropriate to vibrate the powder to eliminate bridging, large voids,or other defects. For this reason, the tap density provides the best first measure of par-ticle packing and proves relevant to many forming operations. The measurementdepends on the material, vibration amplitude, vibration direction, applied pressure,vibration frequency, particle density, shear, and test apparatus. During vibration thedensity varies with the number of vibrations by an exponential function,

f ¼ fT þ ( fT � fA)exp �K

N

� �

where K is a constant that depends on the device, height of fall, and velocity; N is thenumber of vibration cycles; fT is the fractional tap density; f is the fractional densityafter N cycles; and fA is the fractional apparent density. Generally, the more irregularthe particle shape, the greater the packing benefit from vibration. Powders will reachthe dense random packing limit more rapidly as the particle size increases. Further,various models link the coordination number and packing density. Unfortunately,there is no exact relation between packing density and particle coordinationnumber NC, but a simple model is,

NC ¼ 2e2:4f

For the monosized spheres, the dense random packing possible by vibration (tapdensity) is 63.7%, which equals 2/p. As a point of comparison, the packing densityof a random loose array of monosized spheres is 60%, with approximately six

RANDOM PACKING DENSITY 287

contacts per sphere. The dense random packing (tap density) and loose randompacking (apparent density) are fairly similar for large spherical particles. As theparticle shape becomes more rounded (spherical), the packing density increases.Consequently, spherical particles of sizes greater than approximately 100 mmundergo only a small densification during vibration. In contrast, smaller spheresand nonspherical particles exhibit a greater difference between the apparent andtap densities. These particles undergo a greater increase in packing density withvibration and exhibit higher packing densities in the presence of fluids, surfactants,and pressure. The packing density and coordination number decrease as the particleshape departs from that of a sphere. The difference between random and orderedpacking densities increases as the particle shape becomes nonspherical.

K ¼ device constant, dimensionless

N ¼ number of vibration cycles, dimensionless



fA ¼ fractional apparent density, dimensionless [0, 1]

fT ¼ fractional tap density, dimensionless [0, 1].

RANDOM PACKING RADIAL-DISTRIBUTION FUNCTION(Mason 1968)

The radial-distribution function provides information on the population density ofneighboring spheres versus the distance form a central coordinate site. Randompackings tend to have oscillations in the number of neighbors versus distance.Clearly, there is a high density of neighboring particles at just one particle diameter,corresponding to the touching spheres. However, the larger the radial distance, theless predictable is the occurrence of another sphere. Two descriptions exist—theradial-distribution function showing the probability of encountering the center ofanother sphere versus position, and the cumulative distribution function, whichgives the total number of sphere centers encountered versus distance. The cumulativedistribution function relies on a concentric contact sphere centered on a selectedsphere and plots the neighbors of neighbors versus the radius of the contact sphere.It is given as the neighboring sphere centers versus radial distance as approximatedas follows:

G(r) ¼ 7:3þ 15:5r

D� 1

where G(r) is the cumulative distribution of the number of neighboring sphere centersper unit radial distance r, with D being the monosized sphere diameter. This modelis only applicable to larger spheres and is invalid for small particles that tend toagglomerate or have high interparticle friction that resists packing.

288 CHAPTER R

D ¼ sphere diameter, m (convenient units: mm)

G(r) ¼ the cumulative distribution function, dimensionless

r ¼ radial distance, m (convenient units: mm).

REACTION-CONTROLLED GRAIN GROWTH

See Grain-growth Master Curve, Interfacial Reaction Control, and Interface-controlled Grain Growth.

REACTION-RATE EQUATION

See Avrami Equation.

REACTIVE SYNTHESIS (Frade and Cable 1992)

A suitable kinetic model for diffusion-controlled reactions between mixedpowders is,

1� (1� b)1=3 ¼ Gffiffitp

where b is the fraction transformed, t is the isothermal time, and G is a rate constantthat depends on temperature T as follows:

G ¼ G0 exp � Q

RT

� �

where Q is the activation energy, and k is Boltzmann’s constant, and G0 is theexperimentally determined frequency factor.



T ¼ temperature, K

t ¼ time, s

G ¼ constant, 1/s1/2

b ¼ fraction transformed, dimensionless.

[Also see Self-propagating High-temperature Synthesis.]

REACTIVE SYNTHESIS 289

REARRANGEMENT KINETICS IN LIQUID-PHASE SINTERING(Huppmann and Riegger 1975)

During liquid-phase sintering there is a rapid burst of dimensional change when aliquid that is associated with particle rearrangement first forms. This burst is due tothe spreading and wetting by the newly formed liquid, which contributes a capillaryforce that induces particle clustering and rearrangement, even for insoluble systems.At low green densities the rearrangement shrinkage at liquid formation DL/L0 varieswith the capillary force F as

DL

L0� (F � F0)

where F0 represents an inherent resistance to rearrangement, effectively a yield point.The rate of densification is rapid during rearrangement, occurring in the split secondafter melt formation. The shrinkage rate d(DL/L0)/dt is estimated as follows:

d

dt

DL

L0

� �¼ DPw

Dh

where DP is the capillary pressure from the wetting liquid, D is the solid-particle size,w is the liquid thickness, and h is the liquid viscosity. However, the observed shrink-age rates are lower than expected from this model because heat transport and meltformation are slow steps by comparison. This initial burst of densification contributesslightly to shrinkage, mostly if the green compact is low in density. Kingery firstsuggested a rearrangement event, but measurements of the rate of densificationproved elusive. The problem traces to the fact that for high green densities theliquid penetration between the particles leads to swelling. Shrinkage shows a depen-dence on time t after liquid formation in the early stage of liquid-phase sinteringas follows:

DL

L0� t1þy

D

where D is the particle diameter, and 1 þ y is slightly larger than unity. The exponentof 1 þ y corrects for changes in viscosity and capillary force during the rearrange-ment stage. In concept y should be 0, but because heat transport is the slow stepduring the heating of a powder compact, not all liquid forms at the same time, sothe onset of liquid formation varies from the heat source with distance in the body.Since bulk dimensions are used to model rearrangement, the factor 1 þ y helpscorrect the onset time error. Experiments show this model is substantially correctwith the value of the exponent 1 þ y at 1.1 to 1.6. Probably a value of 1.3 is a

290 CHAPTER R

good approximation for most systems. It must be anticipated that y will vary withcompact size.


F ¼ capillary force, N

F0 ¼ resistance to rearrangement, N

L0 ¼ initial size, m (convenient units: mm)


t ¼ time, s

w ¼ liquid thickness, m (convenient units: mm)

y ¼ viscous-flow correction factor, dimensionless

DL ¼ size change, m (convenient units: mm)


DP ¼ capillary pressure, Pa

h ¼ liquid viscosity, Pa . s.

RECALESCENCE TEMPERATURE

During molten droplet atomization, heat is extracted as the particle passes through thecool surrounding gas. As the liquid transforms into solid, heat is released inside theparticle that reheats the particle in proportion to its heat capacity CP,

DTR ¼DHS

CP

where DTR is the recalescence temperature rise (increase in temperature from the pointof nucleation), and DHS is the solidification enthalpy. In some cases, the recalescencetemperature rise is all the way back to the solidus temperature, but in other cases, theparticle is rapidly solidified below the solidus.


DHS ¼ solidification enthalpy, J/kg

DTR ¼ recalescence temperature rise, K.

REDUCTION IN AREA

The reduction in area RA during tensile testing is a measure of the ductility. It ismeasured based on the difference between the original test sample cross-sectionalarea and the area of the fracture surface divided by the original area:

RA ¼DA

A0

REDUCTION IN AREA 291

where DA is the cross-section area change from the original A0 or initial area of thetensile sample.

A0 ¼ initial cross-sectional area, m2

RA ¼ reduction in area, dimensionless (convenient units: %)

D A ¼ change in cross-section area, m2.

REDUCTION OF OXIDES

See Oxide Reduction.

REDUCTION RATIO IN EXTRUSION

See Extrusion Constant.

REYNOLDS NUMBER IN POROUS FLOW (Carman 1938)

Generally, permeable flow of fluids in porous structures shows a dependence on thefluid compressibility and the relative velocity in the pores. At very low pressures, suchas in a vacuum, the flow is by molecular diffusion. At higher pressures, the gas meanfree path approaches the pore size, and slip flow occurs and it is essentially a hybrid ofmolecular diffusion and viscous flow. At higher pressures, such as are typicallyencountered near one-atmosphere pressure, the flow is laminar. As pressure andvelocity increase, however, there is an inertial energy loss that is not accounted forby Darcy’s law for laminar flow. The Reynolds number Re is a dimensionlessparameter for determination if laminar flow is expected, and a modification applicableto fluid flow in porous media is given as follows:

Re ¼ rV

hS

where r is the fluid density, V is the superficial fluid velocity (volumetric flow ratedivided by the cross-sectional area), h is the fluid viscosity, and S is the surfacearea per unit volume for the porous structure, assuming open pores. This is not theactual Reynolds number in the pores, since the velocity is based on the flow rate allo-cated over the whole sample area. Generally, it is safe to assume laminar flow if theReynolds number is below 4 when calculated this way; however, the approach is notprecise because of several assumptions.


S ¼ surface area per unit volume, m2/m3

V ¼ superficial-fluid velocity, m/s

292 CHAPTER R



[Also see Darcy’s Law]

RHEOLOGICAL MODEL FOR LIQUID-PHASE SINTERING

The kinetic equation for liquid-phase sintering useful in computer simulations ofcomponent size and shape is deduced from a general rheological conception.The densification rate can be interpreted as the rate of porosity decreasing(df/dt ¼ 2d1/dt). According to the viscous-flow model, that is the underpinningfor the semisolid rheological theory of sintering,

d1

1dt¼ � 9

2gLV

Gh

where gLV is the liquid–vapor surface energy that drives the liquid-phase motion,G is the average solid grain size, t is the isothermal sintering time, and h is theeffective shear viscosity of the sintering body. Note that the viscosity depends ontemperature, density, and the relative liquid-volume fraction. When the solid-to-liquid ratio is high, as is typical for liquid-phase sintering, the effective viscositydepends on several microstructural details, but can be estimated as follows:

1h¼ gDLCVLV

RTG2

where g is a numerical constant that is estimated to be near 100, VL is the relativevolume fraction of the liquid phase, DL is the diffusion rate of the solid in theliquid phase (which has an Arrhenius temperature dependence), C is the solubilityof the solid in the liquid phase, V is the atomic volume, and G is the average grainsize. Thus,

d1

1dt¼ � 9

2gDLCgLVVLV

RTG3

When solution reprecipitation is the controlling mechanism of grain growth, then thegrain shape is rounded and the average grain size grows according to a cubicpower law:

G3 ¼ G30 þ Kt

where G0 is the grain size at the onset of isothermal conditions corresponding to timet ¼ 0, and K is the grain-growth rate constant. This grain growth rate constant has a

RHEOLOGICAL MODEL FOR LIQUID-PHASE SINTERING 293

form that depends on the liquid-volume fraction VL as follows:

K ¼ K0 þKL

V2=3L

where K0 is the intercept corresponding to solid-state grain growth, and KL is thesensitivity to liquid content.

C ¼ solid solubility in liquid, m3/m3 or dimensionless

DL ¼ rate of solid diffusion in the liquid, m2/s




K0 ¼ solid-state grain-growth rate constant, m3/s (convenient units: mm3/s)

KL ¼ liquid-sensitivity grain-growth rate term, m3/s (convenientunits: mm3/s)


VL ¼ liquid-volume fraction, dimensionless [0, 1]


d1/dt ¼ porosity elimination rate, 1/s


g ¼ numerical constant, dimensionless





h ¼ shear viscosity, Pa . s.

RHEOLOGICAL RESPONSE

See Complex Viscosity.

ROCKWELL HARDNESS

In the common Rockwell hardness tests (A, B, and C scales, denoted HRA, HRB, andHRC) two types are indenters are used, either a 1208 diamond cone with a 0.2-mmradius spherical tip or a ball with a 1.6-mm diameter. These indenters are pressedinto the surface of the test piece. As illustrated in Figure R1, the impression takesplace in two steps: an initial or preliminary test force is applied to seat the indenter,followed by a higher total test force to create an impression. The initial test forceis maintained for up to 3 seconds, and an indenter depth reading is recorded.

294 CHAPTER R

The increase in force to the final test force then occurs in between 1 and 8 seconds.This force is maintained for 4+2 s, and the additional test force is then removed.While the initial test force is still applied, a second depth reading is made after ashort stabilization period. The Rockwell hardness number (value) is calculated as:

HR ¼ N � h

S

where h is the permanent increase in penetration depth in mm at the preliminary testforce (which is typically 100 kgf), S is a constant set to 0.002 mm, and N is a constantgiven below. There are additional Rockwell scales based on differing loads that mightbe applied to lower strength materials such as polymers (see Table R1)

HR ¼ Rockwell hardness number, dimensionless

HRA ¼ Rockwell A hardness number, dimensionless

HRB ¼ Rockwell B hardness number, dimensionless

HRC ¼ Rockwell C hardness number, dimensionless

N ¼ constant, dimensionless

Figure R1. A depiction of the Rockwell hardness test. First, an indenter is impressed into thesurface using an initial test load. Then the load is increased to include both the initial and testloads. When the test load is removed, yet while the initial load is still applied, the indentationdepth is measured and used to calculate the Rockwell hardness.

TABLE R1. Additional Rockwell Scales

Initial Test

Scale N Indenter force, N force, N Application

HRA 100 Diamond cone 98.07 588.4 Sheet steel, shallow case-hardened

HRB 130 1.587-mm ball 98.07 980.7 Copper and aluminum alloys,annealed low-carbon steels

HRC 100 Diamond cone 98.07 1471 Hardened steels, cast irons, deepcase-hardened

ROCKWELL HARDNESS 295

S ¼ constant, 0.002 mm

h ¼ increase in penetration depth, mm.

ROSIN–RAMMLER DISTRIBUTION (Hogg 2003)

One of the characteristic size distributions applied to powders, especially milledceramic and mineral powders, it given in terms of a cumulative mass particle-sizedistribution F(D) as follows:

F Dð Þ ¼ 1� exp � D

DC

� �M" #

where D is the particle size, DC is a characteristic size for the distribution, and M is themodulus for the distribution. It is common to fit experimental sieve-analysis data tothis distribution by taking a double natural logarithm to form a linear equation,

ln ln1

1� F Dð Þ

� �� ¼ M ln D�M ln DC

An alternative is to use of base-10 logarithms. A plot of the double logarithms on theleft-hand side versus ln D results in a slope of M and an intercept of M ln DC. Thecharacteristic size corresponds to 63.21% on the cumulative-size distribution and,if known, it allows direct calculation of the modulus as,

M ¼ 2log DC � log D1

where D1 corresponds to the size where F(D) ¼ 0.01 (1% point on the cumulativeparticle-size distribution).


D1 ¼ particle size at 1% on the cumulative distribution, m(convenient units: mm)

DC ¼ characteristic size for the particle-size distribution, m(convenient units: mm)

F(D) ¼ cumulative distribution, dimensionless [0, 1]

M ¼ distribution modulus, dimensionless.

296 CHAPTER R

S

SADDLE-SURFACE STRESS

See Neck-curvature Stress.

SCHERRER FORMULA (Langford and Wilson 1978)

In X-ray diffraction, small crystals such as nanoscale particles or small grains, lead toless than full destructive interference or signal cancellation of angles away from oneof the Bragg conditions. This leads to a broadening of the diffraction peaks in amanner not anticipated by Bragg’s law. This peak broadening is treated by theScherrer formula. It relates the crystal or particle size D in to the peak broadeningB, diffraction angle u, and X-ray wavelength l as follows:

D ¼ 0:9lB cos u

Usually the width of the diffraction peak is measured in terms of the angular spread atone-half the peak height, but that measure needs to be corrected for instrumentationcontributions. For practical reasons the estimation of grain or particle size via X-raydiffraction peak broadening is easier when large diffraction angles are used, corre-sponding to high (h, k, l ) planes, and when longer wavelengths are used.

D ¼ crystal size or particle size, m (convenient units: nm)

B ¼ peak broadening, rad (convenient units: degree)

u ¼ Bragg’s law diffraction angle, rad (convenient units: degree)

l ¼ X-ray wavelength, m (convenient units: nm).


297

SCREEN SIZES

See Sieve Progression.

SECONDARY DENDRITE–ARM SPACING (Joly and Mehrabian 1974)

Dendrite arms are residual markers of the liquid–solid transformation in a castmaterial. The side dendrites, known as secondary dendrites, have sizes inversely pro-portional to the cooling rate during solidification. Thus, they are used to estimatecooling events by examination of the atomized powder. During atomization, thesolid nucleates within the liquid droplet and progressively changes the liquid tosolid. For a spherical particle of diameter D that is cooling by convection, implyingthat atomization involves a fast-moving particle passing through a cool gas, thesecondary dendrite–arm spacing l varies as follow:

l ¼ CDn

where C is a collection of material and process constants, and n is an exponent thatranges between 0.5 and 1, depending on the alloy composition and solidificationmode.

C ¼ material and process constants, m1/2 to m


l ¼ secondary dendrite–arm spacing, m (convenient units: mm)

SECONDARY RECRYSTALLIZATION

See Abnormal Grain Growth.

SECOND-STAGE LIQUID-PHASE SINTERING MODEL

See Intermediate-stage Liquid-phase Sintering Model.

SECOND-STAGE SINTERING DENSIFICATION

See Intermediate-stage Sintering-density Model.

SECOND-STAGE SINTERING PORE ELIMINATION

See Intermediate-stage Pore Elimination.

298 CHAPTER S

SECOND-STAGE SINTERING SURFACE-AREA REDUCTION

See Intermediate-stage Surface-area Reduction.

SEDIMENTATION PARTICLE-SIZE ANALYSIS

See Stokes’ Law Particle-size Analysis.

SEGREGATION COEFFICIENT

Mixed powders with differing sizes, shapes, or densities have a tendency to separateduring motion or vibration. In a powder lot, the segregation coefficient CS is calcu-lated as follows:

CS ¼XT � XB

XT þ XB

where XT is the fraction of large particles in the top half of the container, and XB is thefraction of large particles in the bottom half of the container.

CS ¼ segregation coefficient, dimensionless

XB ¼ fraction of large particles in the bottom half of the container, dimensionless

XT ¼ fraction of large particles in the top half of the container, dimensionless.

SEGREGATION OF MIXED POWDERS

See Mixed-powder Segregation.

SEGREGATION TO GRAIN BOUNDARIES DURINGSINTERING (Kang 2005)

Solutes segregate to grain boundaries during sintering if there are lower energysites in comparison to the bulk. This can be seen in many ceramics and somemetals. Let the ratio of solute to bulk atoms be X0 (ratio of atomic concen-trations), then on a grain boundary at the sintering temperature the ratio XB

will be as follows:

XB ¼ X0 exp � E

RT

� �

where T is the temperature, R is the gas constant, and E is the energy differenceassociated with the solute located at the grain boundary versus the solute locatedin the bulk. It is effectively the driving force for segregation, and if negative,then segregation is favored.

SEGREGATION TO GRAIN BOUNDARIES DURING SINTERING 299

E ¼ difference in solute-atom substitution at a grain boundary versus bulk,J/mol



XB ¼ atomic ratio of solute atoms on the grain boundary, dimensionless

X0 ¼ atomic ratio of solute atoms, dimensionless.

SELF-PROPAGATING HIGH-TEMPERATURE SYNTHESIS (Munirand Anselmi-Tamburini 2000)

A reactive synthesis process relies on the propagation of a combustion wave throughmixed and compacted powders to produce a compound. The compound is stable, andhence in forming liberates heat in a near-spontaneous exothermic reaction.Sometimes termed SHS (self-propagating high-temperature synthesis), the reactionpropagation is controlled by heat transfer into the unreacted material from theexothermic reaction event,

rCP@T

@t¼ k

@2T

@x2þ rQ

@f

@t� b

W(T � TO)� s

W(T4 � T4

O)

where r is the product density, CP is the heat capacity, T is the temperature, TO is theambient temperature, t is the time, k is the thermal conductivity of the unreactedcompact, x is the reaction coordinate (direction for the advancing reaction), Q isthe reaction heat, f is the fraction of the reaction completed, s is the Stefan–Boltzmann constant, b is the convective heat-transfer constant, and W is thesample width. As the reaction progresses, heat is conducted forward to the unreactedmaterial where the net energy accumulation depends on the heat loss and heatevolution. The reaction rate generally follows this form,

@f

@t¼ K(1� f )N exp � E

RT

� �

where K is a frequency factor, N is the reaction order, R is the gas constant, and E isan activation energy for the process.

CP ¼ heat capacity, J/(kg . K)

E ¼ activation energy, J/mol (convenient units: kJ/mol)

K ¼ reaction frequency factor, 1/s

N ¼ reaction order, dimensionless

Q ¼ reaction heat, J/kg (convenient units: kJ/kg)


T ¼ temperature, K

300 CHAPTER S

TO ¼ ambient temperature, K

W ¼ sample width, m (convenient units: mm)

f ¼ fraction reacted, dimensionless

t ¼ time, s

x ¼ reaction coordinate, m

b ¼ convective heat-transfer constant, W/(m2 . K)


r ¼ product density, kg/m3 (convenient units: g/cm3)

s ¼ Stefan–Boltzmann constant, 5.6704 1028 kg/(s3 . K4).

SEMISOLID-SYSTEM VISCOSITY

See Viscosity of Semisolid Systems.

SHAPIRO EQUATION (Jones 1960)

One of the first models for green density as a function of compaction pressure wasdeveloped in the 1944 Ph.D. thesis by Shapiro at the University of Minnesota,Minneapolis, St. Paul. This model assumes that the porous-powder mass behavesas if it were a solid under isostatic compression. The resulting equation for fractionalgreen density fG as a function of compaction pressure P is then,

fG ¼ 1� 1� fAð Þ exp(�kP)

where k is a powder-compressibility factor, and fA is the powder fractional apparentdensity.




k ¼ powder-compressibility factor, 1/Pa (convenient units: 1/MPa).

SHEAR MODULUS

In engineering materials the shear modulus G (which is sometimes denoted S or m) isalso referred to as the modulus of rigidity. It is defined as the ratio of shear stress to theshear strain:

G ¼ t

g¼ F=A

Dx=h

SHEAR MODULUS 301

where t is the shear stress, F is the force that acts on area A, g is the shear strain withinitial length h, and the transverse displacement is Dx. Like the elastic modulus, theshear modulus is usually expressed in units of GPa.

A ¼ area, m2 (convenient units: mm2)

F ¼ force, N (convenient units: kN or MN)

G ¼ shear modulus, Pa (convenient units: GPa)

h ¼ initial length, m (convenient units: mm)


Dx ¼ transverse displacement, m (convenient units: mm)

t ¼ shear stress, Pa (convenient units: MPa).

SHEAR-RATE EFFECT ON VISCOSITY

See Cross Model.

SHRINKAGE

Sintering shrinkage is expressed as the percentage change in compact size divided bythe initial compact size, which is actually negative during most sintering treatments.Often the negative sign is dropped and shrinkage is simply given as the positive valuefor DL/L0, where DL is the change in dimension from the size L0 prior to sintering.Because of shrinkage during sintering, the compact densifies from the green densityrG to the sintered density rS according to the following relation, which expresses theshrinkage as a positive fraction,

rS ¼rG

1� DL=L0ð Þ3

This assumes no mass loss during sintering, so in cases where a polymer addition isincluded in the green mass a correction is required.


DL ¼ change in a dimension from the size L0, m (convenient units: mm)


rS ¼ sintered density, kg/m3 (convenient units: g/cm3)

SHRINKAGE FACTOR IN INJECTION-MOLDING TOOL DESIGN

Sintering shrinkage requires that tooling be dilated to accommodate the size changebetween forming and sintering. Proper dilation ensures the final component is withinthe specified size range. In powder injection molding the shrinkage factor Y is

302 CHAPTER S

calculated from the feedstock solids loading f and sintered fractional density fS,

Y ¼ 1� f

fS

� �1=3

Y ¼ shrinkage factor, dimensionless fraction

fS ¼ sintered density, dimensionless fraction

f ¼ feedstock fractional solids loading, dimensionless.

SHRINKAGE-INDUCED DENSIFICATION

Sintering shrinkage is a permanent strain resulting form the elimination of porosity. Itis expressed as the positive value of DL/L0, where DL is the change in a dimensionfrom the initial size L0. The compact densifies from the fractional green density fG tothe fractional sintered density fS using a positive shrinkage,

fS ¼fG

1� DL=L0ð Þ3

Densification C is another means to express the effect of shrinkage. It is used whenthere is a variation in the green density that might directly cloud comparisons ofshrinkage or final density. Densification is defined as the change in porosity (greento sinter) divided by the initial porosity:

C ¼ D1

10¼ fS � fG

1� fGWhen there is no mass loss, it is common to calculate densification from shrinkagewith the assumption of isotropic sintering.

L0 ¼ initial dimension prior to sintering, m (convenient units: mm)



DL ¼ change in a dimension from sintering, m (convenient units: mm)

DL/L0 ¼ fractional sintering shrinkage, dimensionless

D1 ¼ fractional change in porosity from sintering, dimensionless

C ¼ densification, dimensionless

10 ¼ initial fractional porosity, dimensionless [0, 1].

SHRINKAGE IN INTERMEDIATE-STAGE LIQUID-PHASE SINTERING

See Solution-reprecipitation-induced Shrinkage in Liquid-phase Sintering.

SHRINKAGE IN INTERMEDIATE-STAGE LIQUID-PHASE SINTERING 303

SHRINKAGE IN SINTERING

See Sintering Shrinkage (Generic Form).

SHRINKAGE RATE FOR SUPERSOLIDUS LIQUID-PHASESINTERING


SHRINKAGE RELATION TO NECK SIZE (Kingery and Berg 1955)

When starting with loose powders with no compaction or other deformation at theparticle contacts, then the approach of the particles during sintering can be relatedto the neck-size ratio X/D (where X is the neck diameter and D is the particle dia-meter). This assumes that mass transport is by a bulk mechanism (volume diffusion,viscous flow, grain-boundary diffusion, dislocation climb, or plastic flow). During theinitial stage of sintering, the direct relation is as follows:

DL

L0¼ 1

4X

D

� �2

where the shrinkage DL/L0 is the compact-length change divided by the initiallength. This is an approximate relation that is invalid if there is much contributionfrom surface diffusion or other mechanisms that contribute to neck growth bymass transfer from the pores. The relation is valid for small neck sizes, generallylimited to about 2 to 3% sintering shrinkage. If sintering is by a surface-transportmechanism, then there is no shrinkage and this relation is not applicable.




X/D ¼ neck-size ratio, dimensionless fraction


DL/L0 ¼ fractional sintering shrinkage, dimensionless.

SIEVE PROGRESSION (Bernhardt 1994)

Screens for sieving particles are selected according to a geometric progression insizes. The standard step in both particle size analysis and particle classification isbased on a progression of openings with a linear size ratio equal to a constantfactor. Most standards rely on the quarter-root of two, meaning the nominal mesh-size

304 CHAPTER S

ratio can be described as,

21=4 ¼ 1:19 ¼ Mesh opening i

Mesh opening iþ 1

� �

This idealized step size of 21/4 between each sieve mesh corresponds to an increase inthe opening by a factor of 1.19, or a mass ratio between size classes of nearly 1.7. Inreality, the steps only approximate to this size. For example, a 325-mesh screen has anominal 45-micrometer linear opening, and a 400-mesh screen has a 38-micrometerlinear opening. In that case, the ratio is 1.18 (a step of 1.19 would give 45.2micrometers at the ideal 325-mesh step size). Note that the allowed tolerance onthese openings is about 5% of the nominal size, yet the maximum allowedopening is about 50% of the nominal size.

SINTERED DUCTILITY (Haynes 1977)

For sintered metals, during tensile testing pores act to nucleate premature failureversus what is observed in a full-density material tested in the same thermal con-ditions. The decrease in ductility is sensitive to many microstructure and processingfactors, but is dominated by the sintered density. The relative ductility as a function ofthe fractional sintered density fS can be approximated as follows:

Z ¼ f 3=2S

1þ c12ð Þ1=2

where c is an empirical constant that relates to the sensitivity to pores, and 1 is thesintered fractional porosity. The relative ductility factor Z is defined as the ductilityof the porous material divided by the ductility of equivalently processed full-densematerial in the same microstructural condition (grain size, heat treatment).

Z ¼ ductility factor or ratio of porous-to-dense ductility, dimensionless

c ¼ material constant relating to pore size and shape, dimensionless

fS ¼ fractional sintered density, dimensionless fraction [0, 1]

1 ¼ sintered fractional porosity (12 rS), dimensionless [0, 1].

SINTERED STRENGTH (Haynes 1981)

In the absence of microstructural defects, sintered strength s will vary with the frac-tional sintered density fS as follows:

s ¼ s0Kf MS

where s0 is the strength of the same material at full density in the same condition(grain size and heat treatment), K is a constant that depends on the test geometryand processing details (effectively a stress-concentration factor), and M is the

SINTERED STRENGTH 305

exponential dependence on fractional density; it tends to range from 3 to 6, but it isinfluenced by the processing conditions. From a simple view, the pore reduction inthe cross-sectional area would suggest M ¼ 1, but in reality the higher valuescome from the stress concentration and premature fracture seen where fracturefollows a path of least resistance, propagating from pore to pore. As a consequence,the fracture surface shows a much higher level of porosity than would be anticipatedfrom a random cross section through the material. For example, a heat-treated sinteredsteel with 13.5% porosity has a tensile strength that is just 57% of the full-densitytensile strength.

K ¼ processing, material, and testing-dependent constant, dimensionless

M ¼ sensitivity exponent, dimensionless

fS ¼ fractional sintered density, dimensionless fraction [0, 1]

s ¼ sintered strength, Pa (convenient units: MPa)

s0 ¼ full-density strength, Pa (convenient units: MPa).

SINTERING ATMOSPHERE-FLUX EQUATION

See Vacuum Flux in Sintering.

SINTERING GRAIN SIZE

See Maximum Grain Size in Sintering.

SINTERING METRICS

One of the fundamental measures of sintering is the neck diameter X, which increaseswith higher sintering temperatures and longer sintering times. Many other monitorsof sintering are related to the kinetics of neck growth. For uncompacted powder,the relation between the fractional density ratio (green and sintered) and neck-sizeratio X/D is estimated as follows:

X

D¼ 4 1� fG

fS

� �1=3" #1=2

where fG is the initial loose-powder fractional packing density (at X ¼ 0) and fS is thefractional packing density after sintering. This relation is only valid for initial-stagebulk transport–controlled sintering starting with loose powders (X/D , 0.3), so itis limited in applicability. However, based on mass conservation it can be used to esti-mate the contact size between pressed powders using the loose and pressed densities.Likewise, the mean particle-coordination number Nc, defined as the number of

306 CHAPTER S

touching neighbors, can be estimated from the fractional solid density as follows:

Nc ¼ 7þ 17:5( fS � 0:6)

This expression is less accurate at higher sintered densities, but proves accuratebetween fractional densities from 0.6 to 0.9. These three measures—density, necksize, and packing coordination—provide a description of the sintered body that isthe basis for predicting many material properties and the compact response duringsintering.


Nc ¼ packing-coordination number, dimensionless


X/D ¼ neck-size ratio, dimensionless fraction.


fS ¼ fractional sintered density, dimensionless.

SINTERING SHRINKAGE (GENERIC FORM) (Mackenzie andShuttleworth 1949; Coble 1961b; Olevsky 1998)

Using 36 different sintering models, Olevsky showed that most of the formulationsfollow the character established in the 1940s by Mackenzie and Shuttleworth. Overthe years, this basic sintering model has been refined by several studies, includingthe important treatment by Coble on grain-boundary diffusion. Inherent in these for-mulations is the realization that smaller powders, smaller grains, and smaller porescontribute to a higher interfacial energy. In turn, high interfacial energy drivesfaster sintering at all temperatures. In those cases where shrinkage and densificationoccur, such as grain-boundary diffusion or volume diffusion, the sintering shrinkagemodels take on a form as follows:

1L0

dL

dt¼ � Am

hGmf VSð ÞSn

where L0 is the original length, L is the instantaneous length, A is a combination ofmaterial and geometric constants, t is the isothermal sintering time, h is the sinteringsystem viscosity, G is the mean grain size (or other controlling microstructurefeature), the grain size exponent m is typically 3, and the stress exponent n is oftennear 1. The density amplification function f (VS) relates the actual (local) stress tothe bulk (or applied) stress through various functions of the fractional solid VS

content or fractional solid density, which is the fractional density for solid-state sin-tering, or the fractional solid content for liquid-phase sintering. The stress function S

includes several factors that enhance or retard densification and deformation during

SINTERING SHRINKAGE (GENERIC FORM) 307

sintering, including the capillary stress from the particles, applied stresses (hot press-ing or hot isostatic pressing are examples), gravity, trapped gas, or rigid inclusions.These ideas are similar to creep concepts, to the point where they have beenapplied to the high-temperature diffusional creep of sintered materials.

A ¼ geometric and numerical constant, variable units


L ¼ length, m (convenient units: mm)


VS ¼ fractional solid density at the sintering temperature, dimensionless

f (VS) ¼ stress amplification factor relating local to bulk stress, dimensionless

m ¼ grain-size exponent, dimensionless

n ¼ stress exponent, dimensionless



S ¼ stress, Pa (convenient units: MPa).


SINTERING SHRINKAGE FOR MIXED POWDERS

See Mixed-powder Sintering Shrinkage.

SINTERING SHRINKAGE IN SUPERSOLIDUSLIQUID-PHASE SINTERING


SINTERING STRESS, BULK

The bulk sintering stress sB (local contact stress resolved over the sample cross-sectional area) depends on the sintered fractional density fS, packing coordinationNC, neck size to particle size ratio X/D, and sintering stress s as follows:

sB ¼ s fSNC

p

X

D

� �2

At full density ( fS ¼ 1), the average neck size to particle size ratio X/D ranges near0.5, depending on the assumed grain shape (at full density the grains are mostly dode-cahedrons or tetrakaidecahedron with coordination numbers NC ¼ 12 or 14). Mostsintering models assume a terminal grain shape of a tetrakaidecahedron with fourteen

308 CHAPTER S

sides. This grain shape gives an equivalent neck-size ratio X/D of 0.56, which is nearthe average experimental value of 0.53. Thus, it is appropriate to set the upper limitfor X/D, attainable at full density at 0.5.


NC ¼ particle or grain-packing coordination, dimensionless




s ¼ sintering stress, Pa (convenient units: MPa)

sB ¼ bulk sintering stress, Pa (convenient units: MPa).

[Also see Neck-curvature Stress.]

SINTERING STRESS IN FINAL-STAGE SINTERING FOR SMALLGRAINS AND FACETED PORES

When the grain size is small and the pores are faceted, then the final-stage sinteringstress has a high sensitivity to the dihedral angle and pore–grain structure. In this casethe sintering stress is given as follows:

s ¼ gSV

dP

ffiffiffiffiffiffiNP

pcos

f

2

� ��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiNP � 4

pcos

f

2

� �þ 2 sin

f

2

� ��

where s is the sintering stress, gSV is the solid–liquid surface energy associated withpores of diameter dP, f is the mean dihedral angle associated with the grain-boundaryintersection at the pore surface, and NP is the number of pores per grain.

NP ¼ number of pores per grain, dimensionless

dP ¼ typical pore size, m (convenient units: mm)

gSV ¼ solid–liquid surface energy, J/m2

s ¼ sintering stress, Pa (convenient units: MPa)

f ¼ dihedral angle for grain boundary and pore, rad (convenientunits: degree).

SINTERING STRESS IN FINAL-STAGE SINTERING FORSMALL GRAINS AND ROUNDED PORES

In solid-state, final-stage sintering the structure consists of a nearly dense grainstructure and dispersed spherical pores that are not necessarily attached to the grainboundaries. The pores are closed and have no communication with the processing

SINTERING FOR SMALL GRAINS AND ROUNDED PORES 309

atmosphere. If sintering is in a vacuum at this stage, then only vapor products from thematerial contributes to trapped gas in the pores so there should be little inhibiteddensification. In this case, the sintering stress arising from small pores is estimatedas follows:

s ¼ � 2gSS

Gþ 4gSV

dP

where s is the sintering stress that is acting to remove the pores, gSS is the solid–solidgrain-boundary energy associated with a grain size of G, and gSV is the solid–vaporsurface energy associated with the surfaces of pores of diameter dP. Both surface ener-gies depend on crystal orientation, so there will be variations within the microstructure.Most materials contain a trapped atmosphere or vapor products in the pores that gener-ate a resistance to densification. Hence, the net sintering stress can reach zero with sub-stantial residual porosity (for example, when sintering in argon), and the material willnever reach full density. Another densification impediment occurs when reaction pro-ducts or evaporation products fill the pores. In some cases, the pore-filling event willeventually exceed the sintering stress, causing the compact to swell (this is the oppositeof densification from the sintering stress just shown). An intermediate case is when thereaction product or process atmosphere is soluble in the material, leading to a slow final-stage densification as the pore pressure is reduced over time.

G ¼ typical grain size, m (convenient units: mm)

dP ¼ typical pore size, m (convenient units: mm)


gSV ¼ solid–liquid surface energy, J/m2


SINTERING STRESS IN FINAL-STAGE SINTERING FORSPHERICAL PORES INSIDE GRAINS

In the final-stage of solid-state sintering, for the case where the remaining pores arestranded inside the grains, little densification occurs. The stress on the pores thatmight sustain continued sintering densification is estimated as follows:

s ¼ 4gSV

dP

where s is the sintering stress, dP is the spherical-pore diameter, and gSV is the solid–vapor surface energy. This stress leads to compression of trapped atmospheres orvapors in the pores, but as the pore collapses the internal pore pressure increases toa point where the two factors balance and pore closure comes to an end.




310 CHAPTER S

SINTERING STRESS IN INITIAL-STAGE LIQUID-PHASESINTERING

The initial stage of liquid-phase sintering is characterized by small necks wetted byliquid in the form of pendular bonds between the particles. For this case the sinteringstress s depends on the liquid–vapor–solid contact angle u, liquid–vapor surfaceenergy gLV, particle diameter D, and sintering shrinkage DL/L0:

s ¼ 5:2gLV cos u

D(DL=L0)



DL ¼ dimensional change in sintering, m(convenient units: mm)



u ¼ liquid–vapor–solid contact angle, rad(convenient units: degree)


SINTERING STRESS IN INITIAL-STAGE SOLID-STATE SINTERING

In the initial stage of solid-state sintering, the neck surface forms a saddle surface witha sharp curvature at the root. The typical assumptions include isotropic surface energyand spherical particles, without a dihedral angle in the neck. At the smaller neck sizesthe sintering stress s is given as follows:

s ¼ gSV �2Xþ 4 D� Xð Þ

X2

� �

where X is the neck diameter, D is the particle (sphere) diameter, and gSV is thesolid–vapor surface energy. This relation is invalid if X/D . 0.3.





SINTERING SWELLING WITH MIXED POWDERS

See Swelling Reactions during Mixed-powder Sintering.

SINTERING SWELLING WITH MIXED POWDERS 311

SINTERING VISCOUS FLOW

See Viscosity during Sintering.

SIZE DISTRIBUTION

See Andreasen Size Distribution, Gaudin–Schuhman Distribution, GaussianDistribution, Log-Normal Distribution, and Rosin–Rammler Distribution.

SLENDERNESS

Slenderness S is defined as the wall thickness t divided by the square root of thecomponent area A perpendicular to the thinnest wall,

S ¼ t=ffiffiffiAp

It is a geometric term initially developed to characterize plastic designs and has beenadapted as a means to identify good candidates for powder injection-molding tech-nologies. The typical slenderness encountered in powder injection molding is 0.15.

A ¼ projected component area perpendicular to wall, m2

(convenient units: mm2)

S ¼ slenderness, dimensionless

t ¼ wall thickness, m (convenient units: mm).

SLIP CHARACTERIZATION OF POWDER–BINDER MIXTURES(Kwon and Ahn 1995)

For powder–binder mixtures there is a substantial difference in flow rheology versuspure polymers. One factor is associated with slip during powder–polymer flow alongstationary surfaces. There are two models for characterizing the slip phenomena:(1) the velocity model, and (2) the slip-layer model. Figure S1 contrasts the conceptsof the slip velocity and the slip-layer models with a normal velocity profile. A rheo-logical characterization of powder–binder mixtures involves the determination ofviscosity as a function of the shear rate and temperature and determination of theslip velocity or the slip-layer thickness as a function of the relevant dependentvariables. Mooney proposed an expression to determined the slip velocity Vs for aconstant wall shear stress tw,

Vs ¼18@ 32Q=pD3ð Þ@ 1=Dð Þ

� �tw

312 CHAPTER S

To make use of the preceding formula, it is necessary to obtain the relation betweenthe flow rate Q and wall shear stress, which is possible with a capillary rheometerusing several different capillary-tube diameters D. Various attempts to determinethe slip-layer thickness have been reported, and the most successful uses the formula,

d ¼ @ 32Q=pD3ð Þ@ 1=Dð Þ

18 g_2w � g_1wð Þ

� �tw

again at a constant wall shear stress, where g1w and g2w represent the wall shear rateof the binder material in the slip layer and the bulk mixture, respectively, whensubjected to the wall shear stress. Figure S2 shows the relationship between theslip-velocity and slip-layer models. Both the slip-velocity and the slip-layer thicknesscan be curve fitted by a power-law function involving the wall shear stress tw. OnceVs has been determined as a function of tw, the bulk shear viscosity as a function ofshear rate can be determined by the Rabinowitsch correction, taking into account thetrue shear rate after the slip effect. The non-Newtonian viscosity obtained in this way

Figure S1. Contrast of three variants on the feedstock flow near the wall. Although simplein concept, the velocity profile associated with no slip in (a) is not valid for powder–bindermixtures. In contrast, (b) shows the slip velocity concept, and (c) gives the slip-layer concept.

SLIP CHARACTERIZATION OF POWDER–BINDER MIXTURES 313

fits a modified Cross model with a yield stress ty given as follows:

h( _g, T) ¼ h0

1� (t�=h0 _g )n�1 þty

_g

with the usual assumption of a thermally activated viscosity,

h0 ¼ B expTb

T

� �

B ¼ preexponential reference viscosity, Pa . s

D ¼ diameter of capillary, m (convenient units: mm)

Q ¼ flow rate, m3/s (convenient units: mm3/s)

T ¼ temperature, K

Tb ¼ reference temperature for the viscosity model, K

Vs ¼ slip velocity, m/s

n ¼ shear thinning exponent, dimensionless

d ¼ slip-layer thickness, m

g ¼ shear rate, 1/s

g1w ¼ wall shear rate of the binder material in the slip layer, 1/s

g2w ¼ wall shear rate of the bulk mixture, 1/s

h ¼ feedstock viscosity, Pa . s

h0 ¼ zero shear rate viscosity of feedstock, Pa . s

tw ¼ wall shear stress, Pa

ty ¼ yield stress, Pa

t� ¼ transition stress from Newtonian flow to shear thinning flow, Pa.

Figure S2. A detailed layout for the slip-layer concept associated with powder–binder flow.

314 CHAPTER S

SLIP FLOW IN PORES (Evans et al. 1961)

When a gas is passing through a porous structure with a pore size and molecular meanfree path near the same dimension, then a mixture of viscous and diffusion flowoccurs in the pores, termed slip flow. A single permeability parameter is used tobridge the range of mixed behavior from diffusion to viscous flow. Since the gas mol-ecule mean free path varies with pressure, slip flow and the corresponding materialpermeability will vary with pressure, as well as with pore shape and tortuosity.The general solution is to apply Darcy’s law, with a slip permeability aS that ispressure dependent,

aS ¼ a 1þ b

P

� �

where a is the Darcy’s law permeability applicable to laminar flow in the pores, b isan adjustable experimental factor, and P ¼ 1

2(P1 þ P2) is the mean gas pressure. Thenthe flow of gas in the pores is treated with a modified version of Darcy’s law,

Q ¼ aSA

hL

P21 � P2

2

2P2

� �

where Q is the volumetric gas flow rate, A is the cross-sectional area of the material, Lis the bulk length for the flow in the sample, h is the gas viscosity, and P1 and P2 arethe upstream and downstream pressures, respectively. The flow rate Q is the standar-dized gas volume, measured at one-atmosphere pressure, passing though the bodyper time.



P ¼ mean gas pressure, Pa (convenient units: MPa)

P1 ¼ upstream pressure, Pa (convenient units: MPa)

P2 ¼ downstream pressure, Pa (convenient units: MPa)

Q ¼ flow rate, m3/s (convenient units: mm3/s)

b ¼ adjustable factor, Pa


aS ¼ slip-corrected permeability coefficient, m2

h ¼ gas viscosity, Pa . s.

SLOPE OF THE LOG-NORMAL DISTRIBUTION

See Log-Normal Slope Parameter.

SLOPE OF THE LOG-NORMAL DISTRIBUTION 315

SMALL PARTICLE–INDUCED X-RAY LINE BROADENING

See Scherrer Formula.

SOLIDIFICATION TIME

In atomization, the solidification time t for a droplet of diameter D (assumedspherical) is calculated based on the heat balance to cool the droplet to the meltingtemperature and to remove the solidification enthalpy, as follows:

t ¼ Drm

6bCp ln

TM � T0

TS � T0

� �þ DHS

TS � T0

� �

where T0 is the ambient temperature of the gas, TM is the melt temperature, TS isthe melt solidus temperature, DHS is the heat of fusion, and Cp and rm are the heatcapacity and density of the melt, respectively. The convective heat-transfer coefficientb increases with the gas thermal conductivity and Reynolds number of the system.This form assumes cooling is dominated by convective cooling, which is not thecase with vacuum atomization.

Cp ¼ heat capacity at constant pressure, J/(kg . K)


T0 ¼ ambient gas temperature, K

TM ¼ melt temperature, K

TS ¼ alloy solidus temperature, K

t ¼ solidification time, s

DHS ¼ heat of fusion, J/kg (convenient units: kJ/kg)

b ¼ convective heat-transfer coefficient, W/(m2 . K)

rm ¼ density of the melt, kg/m3 (convenient units: g/cm3).

SOLIDS LOADING (J. R. G. Evans 1993)

Feedstocks for polymer-assisted forming processes are described by the relativepowder-volume fraction in the mixture. The critical solids loading gives the concen-tration of powder at the condition where the mixture shows infinite viscosity andcannot be formed in conventional equipment. When starting with molten binder,the viscosity increases as powder is added to the binder. This gives the mixtureviscosity hM as a function of the pure-binder viscosity hB (at the same temperature)in terms of the solids loading f normalized to the critical solids loading fC,

hM ¼hB

1� ffC

� �2

316 CHAPTER S

In some instances the exponent is reported to be different from 2, and even morecomplicated forms are listed below:

hM ¼ hB 1þ 1:25f1� f=fC

� �2

hM ¼ hB 1þ 0:75f=fC

1� f=fC

� �2

These forms have been attributed to several authors—Cross, Kreiger, Chong, Eilers,and others. The solids loading f is the volumetric ratio of solid powder to the totalvolume of powder and binder,

f ¼ WP=rP

(WP=rP)þ (WB=rB)

where WP and WB are the weight fraction of powder and binder, respectively. Theseare calculated as the mass of powder divided by the mass of powder and binder, andthe mass of binder divided by the mass of powder and binder. The parameters rP andrB are the densities of the powder and binder, respectively.

WB ¼ binder weight fraction, dimensionless [0, 1]

WP ¼ powder weight fraction, dimensionless [0, 1]

hB ¼ binder viscosity, Pa . s

hM ¼ mixture viscosity, Pa . s

rB ¼ theoretical binder density, kg/m3 (convenient units g/cm3)

rP ¼ theoretical powder density, kg/m3 (convenient units g/cm3)

f ¼ volume fraction of powder or solids loading, dimensionlessfraction [0, 1]

fC ¼ critical solids loading, dimensionless fraction [0, 1]

SOLUBILITY DEPENDENCE ON PARTICLE SIZE

There is a particle-size effect on solid solubility in a liquid. This solubility increaseonly becomes significant when the particles become small, extending into the nano-scale. For an assumed spherical particle, the relation is as follows:

lnC

C0

� �¼ 4gSLV

DRT

where V is the atomic volume, gSL is the solid–liquid surface energy, D is the particlediameter, C is the solubility of the particle, and C0 is the equilibrium solubilitycorresponding to a flat surface.

SOLUBILITY DEPENDENCE ON PARTICLE SIZE 317

C ¼ the solubility of the particle, m3/m3 or kg/m3

C0 ¼ equilibrium solubility, m3/m3 or kg/m3



T ¼ temperature, K



SOLUBILITY RATIO

The solubility ratio is a concept important to predicting swelling versus shrinkage in thesintering of mixed-powder systems. The solubility ratio SR is determined by the tempera-ture-dependent mutual solute and solvent solubilities on an atomic basis; effectively, theamount of the base material that dissolves into the additive CB (base solubility in theminor phase) divided by how much of the additive dissolves into the base CA,

SR ¼CB

CA

During heating this ratio might change considerably, depending on the phasediagram. If the solubility ratio is larger then 1, then shrinkage is expected, but whenthe solubility ratio is smaller than 1, the system is expected to swell during sintering.The latter situation is commonly seen in transient liquid-phase sintering where theadditive dissolves into the base, leaving a pore behind where the additive melted priorto dissolving into the base material.

CA ¼ additive solubility in the base phase, m3/m3 or kg/m3

CB ¼ base solubility in the additive phase, m3/m3 or kg/m3

SR ¼ solubility ratio, dimensionless.

SOLUTION-REPRECIPITATION-CONTROLLED LIQUID-PHASESINTERING

See Dissolution-induced Densification.

SOLUTION-REPRECIPITATION-INDUCED SHRINKAGEIN LIQUID-PHASE SINTERING (Kingery 1959)

During the intermediate stage of liquid-phase sintering, the pores are still open andthe bonds between the solid grains are growing. Accordingly, as long as the solidis soluble in the wetting liquid, the sintering shrinkage DL/L0 depends on the

318 CHAPTER S

diffusional transport of solid from small grains (dissolution) and reprecipitation onlarge grains. This is termed solution-reprecipitation. An inherent assumption is thatshrinkage occurs due to perpendicular mass transport away from the contact regionbetween the grains, a process known as contact flattening. The sintering shrinkageDL/L0 during this stage depends on solid solubility in the liquid C, solid grainsize G, and other factors,

DL

L0

� �3¼ bCdVgSVt

RTG4D0 exp � Q

RT

� �

where b is a geometric constant related to the liquid film of thickness d, C is thesolid solubility in the liquid, gSV is the solid–vapor surface energy, t is the isothermalsintering time, G is the grain size (which is often much larger than the particlesize due to grain growth), Q is the activation energy for solid diffusion in theliquid, D0 is a diffusion frequency factor, R is the gas constant, and T is the absolutetemperature. A related expression is associated with the case where dissolutionof small grains is the rate-limiting step, resulting in a form similar to that forcontact flattening,

DL

L0

� �3

¼ 48CdVgSVt

RTG3D0 exp � Q

RT

� �

These forms are usually correlated to experimental data to extract any unknownterms. In most instances, the experimental findings generally best correspond tothe solution-reprecipitation concept, where grain-shape accommodation (contact flat-tening) takes place by dissolution of the small grains, and grain growth occurs byreprecipitation of that dissolved solid on the large grains with grain center-to-center shrinkage motion.

C ¼ solid solubility in the liquid, m3/m3

D0 ¼ diffusive frequency factor, m2/s

G ¼ solid grain size, m (convenient units: mm)


Q ¼ activation energy for diffusion, J/mol (convenient units: kJ/mol)




DL ¼ dimensional change, m (convenient units: mm)



b ¼ geometric constant, dimensionless

SOLUTION-REPRECIPITATION-INDUCED SHRINKAGE IN LIQUID-PHASE 319


d ¼ grain-boundary liquid-layer thickness, m (convenient units:mm or nm).

SOLVENT DEBINDING TIME (German and Bose 1997)

Binder removal from an injection-molded body immersed in a solvent involves the pro-gressive transport of dissolved binder from the compact interior to the exterior surface.Usually, the reverse diffusion of fresh solvent from the compact surface into the bodyvia the pore structure is assumed to be fast and not rate controlling. In contrast, thereverse diffusion flux of the binder moving to the compact exterior is slow; thus, it isthe rate-controlling step. As debinding progresses, the permeation distance in thepores increases, making the process progressively slower. Thus, the time t for debind-ing depends on the section thickness h, and absolute temperature T as follows:

t ¼ h2

bln

VB

1� f

� �exp

Q

RT

� �

where VB is the fraction of binder to be removed (some of the binder may not solvate),and b depends on the binder solubility in the solvent. The initial binder level is 1 2 f,where f is the fractional solids loading. The quantity Q is an activation energy associ-ated with dissolution of the binder into the solvent, and R is the universal gas constant.

Q ¼ solvation-activation energy, J/mol (convenient units: kJ/mol)



VB ¼ volume fraction of the binder to be removed, dimensionless [0, 1]

h ¼ thickness of the largest section, m (convenient units: mm)

t ¼ debinding exposure time, s

b ¼ system-specific frequency factor that depends on the binder andsolvent, m2/s

f ¼ fractional volumetric-solids loading, dimensionless.

SOUND VELOCITY

See Ultrasonic Velocity.

SPARK SINTERING

See Field-activated Sintering.

320 CHAPTER S

SPECIFIC SURFACE AREA

The specific surface area is the surface area of a powder divided by the mass of thepowder, typically expressed in units of m2/g. For monosized spheres, the area persphere A and the volume per sphere V are given as,

A ¼ pD2

V ¼ p

6D3

Thus, the area per unit volume is 6/D. The mass M of a spherical particle is calculatedfrom the particle volume and theoretical material density rM

M ¼ rMV

Consequently, the surface area per unit mass (S ¼ A/M ) is given as

S ¼ 6rMD

where S is the specific surface area. For a powder with a variety of sizes, usually themedian particle size from the population-based size distribution provides a first esti-mate for use in the preceding equations. [Note: The convenient units for particle size,specific surface area, and theoretical density are mm, m2/g, and g/cm3, respectively.When these units are used, the conversion factors cancel.]

A ¼ area of a spherical particle, m2

D ¼ diameter of a spherical particle, m (convenient units: mm)

M ¼ mass of a spherical particle, kg (convenient units: g)


V ¼ volume of a spherical particle, m3 (convenient units: mm3)


[See also BET Specific Surface Area.]

SPHERICAL PORE PRESSURE

For a spherical pore, or even a spherical droplet, the surface curvature is constant onall surface locations. In turn, this curvature of the surface leads to a pressure that isinversely dependent on the pore size. The pressure difference DP associated withthe pore or droplet then depends on the surface energy g—either gLV, the liquid–vapor energy for droplets, or gSV, the solid–vapor surface energy for pores:

DP ¼ 4gD

SPHERICAL PORE PRESSURE 321

with D equal to the droplet or pore diameter; the pressure inside a spherical droplet isgreater than the external pressure by a value of DP.

D ¼ droplet or pore diameter, m (convenient units: mm)

DP ¼ difference in pore or droplet pressure due to surface curvature, Pa

gLV ¼ liquid–vapor surface energy for droplets, J/m2

gSV ¼ solid–vapor surface energy for pores, J/m2.

SPHERICITY

The sphericity is a parameter used to provide a semiquantitative description of theparticle shape. The sphericity c depends on the equivalent volume diameter DV.The sphericity is defined as the ratio of the surface area of a sphere with the samevolume as the particle divided by the actual surface area,

c ¼ pD2V

S

Independent measures of the particle volume and particle-surface area are required.

DV ¼ equivalent volume diameter, m (convenient units: mm)

S ¼ surface area, m2 (convenient units: mm2)

c ¼ sphericity parameter, dimensionless.

[Also see Equivalent Spherical Diameter.]

SPHEROIDIZATION OF NANOSCALE PARTICLES

See Nanoscale Particle-agglomerate Spheroidization.

SPHEROIDIZATION TIME (Lawley 1992)

During atomization of a melt, the surface energy of the droplet acts to minimizesurface area by forming a sphere, but the cooling droplet is viscous and takes timeto respond. The calculation for how much time is required to spheroidize thedroplet assumes that the melt is first broken into ligaments and that the ligamentsthen pinch off to form spherical droplets. The time for a droplet to spheroidize byviscous flow under the action of the surface energy is termed the spheroidizationtime tS. It is estimated as follows:

tS ¼12p 2h

gLVVD4 � d4� �

322 CHAPTER S

where h is the viscosity of the liquid (which is a function of temperature), D is thesphere diameter, d is the diameter of the ligament that is a precursor to thesphere, V is the volume of the ligament, and gLV is the liquid–vaporsurface energy. Usually the diameter of the sphere is about 1.5 times the ligamentdiameter.

D ¼ droplet diameter, m (convenient units: mm)

V ¼ ligament volume, m3 (convenient units: mm3)

d ¼ diameter of ligament prior to droplet formation, m(convenient units: mm)

tS ¼ spheroidization time, s


h ¼ liquid or solid–liquid melt viscosity, Pa . s.

[Also see Solidification Time.]

SPOUTING VELOCITY (Epstein and Grace 1984)

A cylindrical vessel filled with loose (uncompacted) powder will exhibit fountainflow or spouting behavior when gas is injected at the bottom of the vessel. Theonset of spouting depends on the operating parameters, as captured by the velocityV at the onset of spouting as follows:

V ¼ D

d

� �dI

d

� �1=3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2gh(rA � rF)

rF

s

where D is the characteristic particle size, d is the cylindrical-vessel diameter, dI isthe fluid inlet-tube diameter at the bottom center of the vessel, g is the gravitationalacceleration, h is the height of the powder bed prior to spouting, rA is the apparentdensity of the powder, and rF is the fluid density. If the particles are not spherical,then the value used for particle size is the minimum cross-sectional diameter relevantto fluid flow.

D ¼ mean particle size, m (convenient units: mm)

V ¼ minimum flow velocity for spouting, m/s

d ¼ cylindrical vessel diameter, m (convenient units: mm)

dI ¼ fluid inlet diameter, m (convenient units: mm)


h ¼ powder-bed height, m (convenient units: mm)

rA ¼ apparent powder density, kg/m3 (convenient units: g/cm3)

rF ¼ fluid density, kg/m3 (convenient units: g/cm3).

SPOUTING VELOCITY 323

SPRAY DEPOSITION (Delplanque et al. 2007)

Spray forming by atomization, plasma, or flame techniques relies on the deposit of asemisolid particle on the substrate or target surface. Typically, the function of thecarrier gas is to condition the particles prior to impact by bringing them to a semisolidthermal state defined by a combination of the solid content and droplet velocity. Highvelocities allow sticking by mostly solid particles, but low velocities require a greaterproportion of liquid to adhere the particles onto the substrate. In spray processes,deposition is induced by the gas flow used to entrain the semisolid particle. Theflow of a single-component incompressible fluid is governed by mass conservation:

r � V ¼ 0

where V is the velocity field, given by the Navier–Stokes equation:

rDV

Dt¼ �rPþ hr2Vþ F

where D/Dt ¼ @/@t þ V .r is the total derivative, r is the fluid density at the oper-ating temperature, h is the fluid viscosity at the operating temperature, P is thepressure field, and F represents externally applied forces per unit volume (e.g., rgfor gravity). The thermal energy transport equation is

rCv

DT

Dt¼ kr2T þ hFv

where T is the temperature field, Cv is the fluid heat capacity at constant volume, k isthe fluid thermal conductivity, and Fv is the viscous dissipation function. The historyof a particle during spraying is primarily determined by the conditions experienced inthe flame or plasma. Therefore, to ensure sticking and densification on the targetrequires accurate determination of the particle trajectory. The particle behavior isapproximately described by the following equation:

mPdVP

dt¼ 1

2rAkV� VPk(V� VP)CD þ mP 1� r

rP

� �gþ FP

where VP is the particle velocity vector; mP is the particle mass; rP is the particledensity; CD is the drag coefficient for the particle, typically expressed as a functionof the particle Reynolds number,

Re ¼ rkV� VPkD=h

for a sphere of diameter D, where A is the cross-sectional area used to define the dragcoefficient; and FP represents the other forces that can influence the particle trajectoryin specific configurations, such as electromagnetic forces.

324 CHAPTER S

A ¼ cross-sectional area, m2 (convenient units: mm2)

CD ¼ particle drag coefficient, dimensionless

Cv ¼ fluid heat capacity at constant volume, J/(kg . K)

D ¼ sphere diameter, m (convenient units: mm)

F ¼ externally applied forces per unit volume, N/m3

FP ¼ other forces that influence the particle trajectory, N

P ¼ pressure, Pa


T ¼ temperature field, K

V ¼ velocity, m/s

VP ¼ particle velocity vector, m/s


mP ¼ particle mass, kg

Fv ¼ viscous dissipation function, K/s2

r ¼ fluid density, kg/m3 (convenient units: g/cm3)

k ¼ fluid thermal conductivity, W/(m . K)


rP ¼ particle density, kg/m3 (convenient units: g/cm3).

SPRAY FORMING


SPREADING

A newly formed liquid will spread over a solid surface if there is a reduction in thefree energy. For spreading to occur there is an increase in the liquid–vapor andsolid–liquid surface areas and a decrease in the solid–vapor surface area. Thus,the energy change associated with spreading requires that the following inequalitybe satisfied:

gSV . gSL þ gLV

where g is the surface energy and the subscripts denote the interface (SL is the solid–liquid interface, LV is the liquid–vapor interface, and SV is the solid–vapor interface).




SPREADING 325

STANDARD DEVIATION

A measure of the dispersion around the mean, calculated by summing the squares ofthe differences of each observation from the mean and dividing that sum by thenumber of observations less one,

s ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1n� 1

Xn

i¼1

Xi � XMð Þ2s

where n is the number of observations, XM is the mean, and Xi is the observed value.The Table S1 shows the standard deviations and the associated percentage points on acumulative (normal) distribution.

XM ¼ mean of the observations, consistent units

Xi ¼ observed value, consistent units

n ¼ number of observations, dimensionless

s ¼ standard deviation, consistent units.

STIFFNESS

See Elastic-modulus Variation with Density.

STOKES–EINSTEIN EQUATION (Einstein 1956)

For very small particles dispersed in a fluid there is a random, thermally inducedmotion known as Brownian motion. The Stokes–Einstein equation relates the particle

TABLE S1. Percentage Points and Standard Deviations on aCumulative Distribution

Deviations Percentage

22.0 2.2821.5 6.6821.0 15.8720.5 31.85

0.0 50.000.5 69.151.0 84.131.5 93.322.0 97.72

326 CHAPTER S

diameter D and the translational diffusivity DT as follows:

D ¼ kT

3phDT

where k is Boltzmann’s constant, T is the absolute temperature, and h is the fluid vis-cosity. For very small particles in a fluid such as water, Brownian motion becomessignificant for particle sizes of 100 nm or smaller. As such, Stokes’ law settling tech-niques are often invalid in this range. Convective currents also tend to be a larger errorsource for small-particle settling experiments.


DT ¼ translational diffusivity, m2/s



h ¼ fluid viscosity, Pa . s.

STOKES’ LAW (Bernhardt 1994; Allen 1997)

Particle sedimentation in a fluid allows for size classification and particle-size distri-bution analysis. Under laminar flow conditions, the terminal velocity v for a particleof size D in a fluid of viscosity h depends on the acceleration g (usually 9.8 m/s2 orone gravity, but can be supplemented by centrifugal forces), fluid density rF, andmaterial density rM. At the terminal velocity, the forces are balanced, so a solutionattributed to Stokes provides a means to measure particle size. The gravitationalsettling force FG equals mass times acceleration,

FG ¼ grmpD3

6

The buoyancy force FB is determined by the volume of fluid displaced by the particle,

FB ¼ grFpD3

6

where rF is the fluid density. Finally, the viscous drag force FV is given as,

FV ¼ 3pDvh

where v is the terminal velocity, and h is the fluid viscosity. For sedimentation, thevelocity v is calculated from the settling height h and time t to settle to the bottom of ameasuring tube. Combining equations gives,

v ¼ gD2 rM � rF

18h

STOKES’ LAW 327

for the terminal velocity, which is Stokes’ law. Particle-size measurements based onStokes’ law often rely on a known settling height h while measuring the time forsetting t. In such cases, the particle size is calculated from the settling time as follows:

D ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

18 hh

gt(rM � rF)

s

This relation is valid for spheres settling without turbulence and without significantBrownian motion with minimal thermal convective currents in the fluid phase.


FB ¼ buoyancy force, N

FG ¼ gravitational settling force, N

FV ¼ viscous drag force, N

g ¼ acceleration, m/s2

h ¼ settling height, m (convenient units: mm)

t ¼ settling time, s

v ¼ terminal velocity, m/s (convenient units: mm/s)


rM ¼ theoretical density of the test material, kg/m3 (convenient units: g/cm3)


[Also see Acceleration of Free-settling Particles and Limiting Size forSedimentation Analysis.]

STOKES’ LAW PARTICLE-SIZE ANALYSIS (Hogg 2003)

Two techniques are generally employed to measure particle-size distributions usingStokes’ law. The first is the line-start method, where all of the particles start at thesame height h, such as at the top of a column. In practice, this means a thin layerof particles mixed with fluid is added to the top. Over time t the larger particlessettle to the bottom first, while the small particles arrive at the bottom much later.If the quantity or concentration at the bottom is measured over time, such as by abalance, then the time-dependent mass arrival C(t) gives the volumetric cumulativeparticle-size distribution F(D) as follows:

F(D) ¼ C(t)CO

where CO is the total amount accumulated at the detection height h, and D is thecalculated particle size based on,

D ¼ 18hg rM � rFð Þ

ffiffiffih

t

r

328 CHAPTER S

The alternative technique is to create a homogeneous initial distribution of particlesin the container prior to the start of sedimentation. In this case, the largest particleswill settle first, while the smaller particles remain in suspension. Thus, the cumulativesize distribution is given by

F(D) ¼ 1� C(t)þ dC(t)dt

CO ¼ total concentration accumulated, kg

C(t) ¼ time-dependent arrival of particles at a depth h, kg


F(D) ¼ cumulative volumetric size distribution, dimensionless

g ¼ acceleration, m/s2

h ¼ settling height, m

t ¼ settling time, s


rM ¼ theoretical density of the test material, kg/m3 (convenient units: g/cm3)


STOKES’ PARTICLE DIAMETER (Hogg 2003)

Stokes’ law provides a means to measure particle size under the assumption that theparticle is a sphere. If the particle is nonspherical, then the Stokes’ law measurementresult will differ from the particle size as measured using other techniques, such aslaser scattering or sieving. If the equivalent spherical diameter is known based onparticle-volume measurements DV and the projected area equivalent spherical dia-meter is known DA, then the Stokes’ law particle size DS is given as follows:

DS ¼

ffiffiffiffiffiffiffiD 3

V

DA

s

DA ¼ equivalent spherical diameter from projected area, m (convenientunits: mm)

DS ¼ Stokes’ law particle size, m (convenient units: mm)

DV ¼ equivalent spherical diameter from volume, m (convenient units: mm).

[Also see Equivalent Spherical Diameter.]

STRAIN HARDENING (Straffelini 2005)

In a porous ductile material, such as a sintered metal, the pores are dispersed ina ductile matrix. Pores cannot carry load and act to concentrate stress locally.

STRAIN HARDENING 329

The strain-hardening behavior of the deformable matrix is given by the followingrelation:

sm ¼ Km1nm

where sm and 1m are the average stress and strain in the matrix, and Km and n are thestrength and strain-hardening coefficients of the matrix. The actual stress s and theexternally measured stress sm are connected as follows

s ¼ zFsm

where F is the fraction of the load-bearing section, and z is a constant, larger thanunity, which accounts for the local notch-strengthening effect exerted by pores.The pores are considered as being internal notches. The actual strain 1 and measuredstrain 1m are connected by the following equation

1 ¼ F3=21m

These can be combined to generate the following constitutive relation,

s ¼ zKmF1�(3=2)nm1nm

The fractional cross section able to carry a load F depends on the porosity andincreases with tensile plastic straining. It is estimated by the following relation:

F ¼ E

E0

� �2

where E is the elastic modulus of the porous alloy, and E0 is that of the matrix. Thefollowing relation is used to estimate the elastic modulus ratio E/E0

E

E0¼ 1� Kp(1� f )

where f is the fractional density, and Kp is a constant depending on the pore mor-phology: Kp is about 2 for spherical pores, and increases as the pore shapebecomes sharper and typically varies between 2 and 3.5. The volume fraction ofpores during plastic deformation depends on the initial porosity and on the voidgrowth rate, usually as a function of the strain.



Km ¼ strength-hardening coefficient of the matrix, Pa

Kp ¼ constant dependent on the pore morphology, dimensionless


m ¼ strain-hardening coefficient of the matrix, dimensionless

F ¼ fraction of the load-bearing section, dimensionless

330 CHAPTER S

1 ¼ actual strain, dimensionless

1m ¼ average strain in the matrix, dimensionless

z ¼ constant, dimensionless

s ¼ actual stress, Pa (convenient units: MPa)

sm ¼ average or measured stress, Pa (convenient units: MPa)

STRAIN RATE IN INJECTION MOLDING

See Gate Strain Rate in Injection Molding.

STRENGTH


STRENGTH DISTRIBUTION


STRENGTH EVOLUTION IN SINTERING

See In Situ Sintering Strength.

STRENGTH-EVOLUTION MODEL (Xu et al. 2002)

For a porous body undergoing sintering, the strength initially increases as sinterbonds form, but usually decreases as thermal softening and annealing occur.Sintered density is the dominant factor and both the yield and tensile strengthsincrease with density. Let the yield strength be sY and the ultimate tensilestrength be sU:

sY ¼ sOf

KC

and

sU ¼ sOf

1þ as(KC � 1)(1� f )

where f is the fractional density, as is a constant, sO indicates the strength of thewrought material at the same test temperature. The factor KC reflects the stress inten-sity that favors fracture at the particle contacts when the neck size is small (smallnecks have a sharp notch character that leads to frequent fracture at the necks).Generally, KC is near 2. For evaluating the in situ sintering strength, the

STRENGTH-EVOLUTION MODEL 331

thermal-softening effect must be considered, and a functional relation is required,expressed as:

sO ¼ �sOg(T)

where sO is the strength of the wrought materials at room temperature, and g(T ) is thethermal-softening factor. As a simple model, the thermal-softening factor is adecreasing linear function from unity at room temperature to zero at the melting temp-erature. However, for many materials thermal softening in the sintering temperaturerange is better approximated by the following form:

g(T) ¼ a

1þ exp[(T � b)=c]

where a, b, and c are adjustable material constants. Generally a is near unity. A fewexamples of these constants for some steels during sintering are as listed in Table S2.From a more refined view, the growth of an interparticle bond, even without densifi-cation, contributes substantially to the compact strength, where the sintering strengthis given as,

s ¼ sOfNC

p

X

D

� �2

where NC is the particle-packing coordination, and X/D is the neck-size to particle-size ratio. At full density, the average neck-size to particle-size ratio X/D is near 0.5(in practice, it ranges from 0.47 to 0.53).


K ¼ geometric and processing constant, dimensionless

KC ¼ stress-intensity factor, dimensionless

NC ¼ particle-packing coordination, dimensionless

X ¼ neck diameter, m, dimensionless (convenient units: mm)

a, b, c ¼ material constants, dimensionless

as ¼ constant

TABLE S2. Material Constants of the Thermal-softening Factor

Material a b (K) c (K)

M2 tool steel 1.02 1102 69.14316L stainless steel 0.953 1003 138.3Fe–2Ni–1B (Fe–2Ni–0.7Cr) 1.019 764.6 112.5Fe–10Cr–0.5B (Fe–9Cr–1.5Mo) 1.064 787.7 172.5

332 CHAPTER S


g(T ) ¼ thermal-softening factor, dimensionless

m ¼ exponential dependence on density, dimensionless

s ¼ sintered strength, Pa (convenient units: MPa)

s ¼ wrought material strength at room temperature, Pa (convenientunits: MPa)

sO ¼ yield or ultimate wrought strength, Pa (convenient units: MPa)

sU ¼ ultimate tensile strength, Pa (convenient units: MPa)

sY ¼ yield strength, Pa (convenient units: MPa)

STRENGTH OF PRESSED POWDER

See Green Strength.

STRESS CONCENTRATION AT A PORE (Green 1998)

There is a stress mismatch at a pore or inclusion. In the case of a pore, the missingmatter has no ability to carry stress and gives a local concentration to the stressthat is largest at the pore tip. As shown in Figure S3, we assume the pore is ovoidin shape, where the pore length is 2c, and the radius of curvature at the pore tip is

Figure S3. A description of the ovoid pore orientation with respect to the stress axis used inthe calculation of maximum stress concentration.

STRESS CONCENTRATION AT A PORE 333

p. Let the bulk applied tensile stress be s. Then for the worst case, with the elongatedpore aligned perpendicular to the stress axis, the maximum stress sM at the tip of thepore is given as,

sM ¼ s 1þ 2ffiffiffic

p

r� �

This peak stress declines rapidly with distance squared from the crack tip. For acircular-shaped pore, the peak stress is 3s.

c ¼ half the pore length, m (convenient units: mm)

p ¼ pore-tip radius of curvature, m (convenient units: mm)

s ¼ applied tensile stress, Pa (convenient units: MPa)

sM ¼ maximum stress at the tip of the pore, Pa (convenient units: MPa).

STRESS IN LIQUID-PHASE SINTERING

See Sintering Stress in Initial-stage Liquid-phase Sintering.

STRIPPING STRESS

See Maximum Ejection Stress.

SUBSIEVE PARTICLE SIZE

See Kozeny–Carman Equation.

SUPERPLASTIC FORMING (Kear and Mukherjee 2007)

A small grain size in a two-phase microstructure, such as obtained with nanoscalecomposites, enables the superplastic densification of the powder or consolidatedbody. The constitutive relation for the operative conditions is as follows:

d1

dt¼ ADB

Gb

kT

b

D

� �Ps

G

N

which links the strain rate to the stress and grain-boundary diffusivity DB. Thisequation says that the strain in forming depends on the material properties andstress s, which is very sensitive to the grain or particle size D. Diffusion on thegrain boundary depends on temperature by the grain-boundary diffusivity, which is

334 CHAPTER S

thermally activated and exhibits an exponential Arrhenius temperature dependence.Without a second phase to retard grain growth, the system response will show pro-gressively slower creep as the grain size enlarges. Generally, the stress exponent Nis near 2 and the microstructure-scale exponent P is determined experimentally.

A ¼ material constant, 1/atom

D ¼ grain size or particle size, m (convenient units: mm)

DB ¼ grain-boundary diffusivity, m2/s

G ¼ shear modulus at the operating temperature, Pa (convenient units: MPa)

N ¼ stress exponent, dimensionless

P ¼ microstructure-scale exponent, dimensionless


b ¼ Burgers vector, m (convenient units: nm)

d1/dt ¼ creep strain rate, 1/s


t ¼ time, s

1 ¼ strain, m/m or dimensionless


SUPERSOLIDUS LIQUID-PHASE SINTERINGLIQUID DISTRIBUTION

See Liquid Distribution in Supersolidus Liquid-phase Sintering.

SUPERSOLIDUS LIQUID-PHASE SINTERING SHRINKAGE RATE(German 1997)

Supersolidus liquid-phase sintering gives rapid densification by heating a preal-loyed powder to a semisolid condition. As the particle melts it has a dramaticstrength loss and becomes soft enough to undergo rapid viscous-flow densification.Based on an analog to Frenkel’s viscous flow-sintering model, the shrinkage duringsupersolidus liquid-phase sintering under isothermal conditions is given as follows:

DL

L0¼ 3gLVt

4Dh

where gLV is the liquid–vapor surface energy, h is the solid–liquid viscosity, t isthe sintering time, DL/L0 is the sintering shrinkage, and D is the particle diameter.This assumes viscous flow and is generally valid for shrinkages less than approxi-mately 6%. During heating, once the liquid forms in the powder, densificationdepends on two nondimensional parameters, a and fc. The latter is called thecritical solids-volume fraction, and it corresponds to the point where sufficient

SUPERSOLIDUS LIQUID-PHASE SINTERING SHRINKAGE RATE 335

liquid exists to initiate viscous flow in response to the interparticle capillaryforces. The term a is a nondimensional energy dissipation term,

a ¼ gLVt

Dh0and

fc ¼ 1� gvdFc

2(1� Fi)gvG

where Fc is the fractional coverage of grain boundaries by liquid, Fi is the fractionof liquid inside the grains, G is the grain size, gv and gs are geometric constantsthat depend on the grain shape, t is the isothermal sintering time, D is the particlediameter, and h0 is the viscosity of the pure liquid. The link between solid–liquidviscosity h and solid fraction f is,

h ¼ h0

1� f

fc

� �2

Consequently, the sintered fractional density f varies as follows with respect to thegreen fractional density fG as:

f ¼ fG

1� 34 a 1� f

fc

� �2" #3

with the solid fraction f being a function of composition and temperature. It isestimated from the phase diagram or experimental data.


Fc ¼ fractional coverage of liquid on grain boundaries, dimensionless

Fi ¼ fraction of liquid inside the grains, dimensionless


a ¼ nondimensional parameter

f ¼ sintered density, dimensionless fraction

fG ¼ green density, dimensionless fraction

gv, gs ¼ geometric constants that depend on the grain shape, dimensionless


a ¼ energy dissipation term, dimensionless


h ¼ solid–liquid viscosity, Pa . s

h0 ¼ viscosity of pure liquid phase, Pa . s

f ¼ solid fraction, dimensionless

fc ¼ critical solids level, dimensionless.

336 CHAPTER S

SURFACE AREA–BASED PARTICLE SIZE


SURFACE AREA BY GAS ABSORPTION

See Specific Surface Area.

SURFACE AREA BY QUANTITATIVE MICROSCOPY(Underwood 1970)

The surface area can be determined for a porous sintered material by quantitativeanalysis performed on polished cross sections. The measurement relies on countingthe pore intersections as observed on the two-dimensional cross-section image.The number of pore–solid intersections per unit-test line length NL provides ameasure of the surface area S per unit volume,

SV ¼ 2NL

f

where f is the fractional solid density, which can be determined by simultaneouspoint-counting or pixel-counting steps. The surface-area calculation is usually per-formed using automated image-analysis devices.

f ¼ fractional solid density, dimensionless [0, 1]

NL ¼ number of pore–solid intersections per unit-test line length, 1/m

SV ¼ surface area per unit volume, m2/m3 ¼ 1/m.

SURFACE-AREA REDUCTION DURING LIQUID-PHASESINTERING (Courtney 1977b)

The grain–liquid surface area per unit volume depends on the grain size duringliquid-phase sintering (assuming the neck-size to grain-size ratio is stabilized bythe dihedral angle). Accordingly, the surface area per unit volume SV will varyapproximately as,

SV ¼ pNV G2

with G being the grain size. Since the grain population decreases with inverse timeand the mean grain size increases with time to the 1/3 power, the interfacialsurface area decreases with time to the 21/3 power.

SURFACE-AREA REDUCTION DURING LIQUID-PHASE SINTERING 337


SV ¼ surface area per unit volume, m2/m3 ¼ 1/m

NV ¼ constant, 1/m3.

SURFACE AREA FOR BROAD PARTICLE-SIZE DISTRIBUTIONS(Santamairna et al. 2001)

Simple relations exist for monosized spheres that give the surface area from theinverse of the particle diameter. However, the small particles make a disproportionalcontribution to the total surface area for broad particle-size distributions of sphericalparticles. To calculate the surface area for such a case, let the breadth of the sizedistribution be expressed by two points on the cumulative mass particle-sizedistribution,

C ¼ D60

D10

then the specific surface area S is estimated as follows:

S ¼ 3(C þ 7)4rD

where D is the median particle size (50% point on the cumulative particle-sizedistribution), and r is the material density. For monosized spheres this becomesthe anticipated formula where the specific surface area equals 6 divided by thematerial density and particle diameter.

C ¼ measure of the distribution breadth, dimensionless



r ¼ material theoretical density, kg/m3 (convenient units: g/cm3).

SURFACE-AREA REDUCTION KINETICS

Surface-area loss occurs during sintering, even if a powder fails to shrink. The growthof the interparticle sinter bond reduces the surface energy and surface area. In the firststage of sintering there is a correlation between the neck-size ratio X/D and the loss ofsurface area DS/S0,

DS

S0¼ ks

X

D

� �M

338 CHAPTER S

where DS/S0 is the change in surface area divided by the original surface area prior tosintering. In the initial stage of sintering for particles that are not compacted, thesurface-area reduction relates to the sintering mechanism as follows:

DS

S0

� �V¼ CSt

where CS is a kinetic term that includes mass-transport constants and otherparameters, t is the isothermal sintering time, and V is an exponent that dependson the mechanism, and is roughly 2.5 for volume diffusion, 3.0 for grain boundarydiffusion, and 3.5 for surface diffusion.

CS ¼ kinetic parameter, 1/s


ks ¼ rate term, dimensionless

M ¼ mechanism exponent, dimensionless

S0 ¼ initial surface area, m2

V ¼ mechanism-dependent exponent, dimensionless



DS ¼ change in surface area, m2.

SURFACE CARBURIZE

See Case Carburize.

SURFACE CURVATURE–DRIVEN MASS FLOW INSINTERING (German 1996)

At any point on the neck there is a surface contour defined by the function S(v), wherev is a parameter describing the relation between the x-y coordinate system along thesurface. Consider surface transport–controlled sintering, where there is no densifica-tion, then any point on the surface can be defined, and from that definition of pos-itions it is possible to minimize energy. Specifically,

dS

dv¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidx

dv

� �2

þ dy

dv

� �2s

During sintering, the instantaneous change in the surface function defining the neckprofile depends on the normal motion at the surface that is described as,

dS

dv

� ��1 dx

dvY � dy

dvX

� �

SURFACE CURVATURE–DRIVEN MASS FLOW IN SINTERING 339

where X and Y are the unit axis vectors. At the point v on the neck profile the principalradii of curvature R1 and R2, which are perpendicular or orthogonal, are given asfollow:

R1 ¼ ydS=dv

dx=dv

and

R2 ¼dS=dv

d2ydv2

� �dxdv

� d2x

dv2

dydv

This allows determination of the local curvature k,

k ¼ 1R1þ 1

R2

It is the curvature gradient determined by performing this calculation from point topoint that drives the surface diffusion flux of atoms during initial-stage sintering.Comparing the curvature at a position v and neighboring positions v 2 1 and v þ 1gives a basis for calculation of the diffusion flux. The accuracy of the calculationincreases with the number of steps along the surface, but the calculation timeincreases as well. Accordingly, the flux depends on the curvature gradient at eachpoint and the mobility of atoms, meaning that the neck-volume change depends onthe arrival rate for mass at the sinter bond,

dV

dt¼ JAV

where J is the atomic flux, A is the bond area over which the new mass is distributed,and V is the volume of a single atom or molecule. Normally, this equation is numeri-cally integrated using small time steps to calculate the deposited volume at each pos-ition. In turn, the reshaped surface profile is parameterized and smoothed to ensuremass or volume conservation. It is the change in curvature with position along thesurface profile that gives the chemical gradient that directs mass flow into thesinter bond between the contacting particles. The deposited or removed atomschange the neck size and shape, and most importantly, remove the curvature gradientto reduce the system energy. In turn, the new neck shape and curvature gradient deter-mine the atomic flux in the next iteration. High temperatures promote faster masstransport, and thereby contribute to faster neck growth. In early numerical solutionsthere was a loss of mass due to numerical round-off errors, so volume conservation isan important metric to corroberate calculations.

A ¼ bond area for distribution of new mass, m2

J ¼ atomic flux, atoms/(m2 . s)

340 CHAPTER S

R1, R2 ¼ orthogonal radii of curvature, m (convenient units: mm)

S(v) ¼ surface-contour function, m

X, Y ¼ unit axis vectors for x and y

v ¼ position parameter relating surface position to the x-y coordinatesystem, m

x ¼ horizontal coordinate, m

y ¼ vertical coordinate, m


k ¼ local curvature, 1/m.

SURFACE DIFFUSION–CONTROLLED NECK GROWTH(Djohari and Derby 2003)

Initial-stage solid-state sintering is often dominated by surface diffusion, since thesurface area is high and grain boundaries at the particle contacts are not developeduntil the neck grows. Thus, grain-boundary diffusion is subservient to surface diffu-sion. A first model for the surface-diffusion process was given by Kuczynski and wassubsequently clarified by various computer simulations. To a large degree, the com-puter simulations verify that the neck-size ratio X/D is an approximate function ofisothermal sintering hold time t as follows:

X

D

� �7¼ Bt

D4

The neck-size exponent of 7 and particle-size exponent of 4 are consistent withseveral studies (including the Herring scaling law), and clearly show the importanceof a small particle size to promote surface diffusion–controlled initial sintering. Indetailed studies, these exponents are found to vary some with time, and are also func-tions of the dihedral angle, a factor largely ignored in most treatments. Note that thereis no shrinkage during surface diffusion–controlled sintering. The parameter B con-tains the surface diffusivity, surface energy, atomic volume, temperature, and similarmaterial-specific parameters. It is dominated by the Arrhenius temperature depen-dence, where temperature enters in an exponential form as associated with themass-transport process delivering neck growth,

B ¼ B0 exp� Q

RT

� �

where R is the gas constant, T is the absolute temperature, and Q is an activationenergy associated with the atomic transport process. The frequency factor B0 isgiven as

B0 ¼56dD0gSVV

RT

SURFACE DIFFUSION–CONTROLLED NECK GROWTH 341

where d is the width of the diffusion zone on the surface and is usually assumed to be2 to 5 times the atom size, D0 is the surface-diffusion frequency factor, V is theatomic volume, gSV is the solid–vapor surface energy, R is the gas constant, and Tis the absolute temperature. Again, this form is a simplification that proves reasonableover a range of materials during surface diffusion–controlled sintering.

B ¼ material constant, m4/s

B0 ¼ preexponential material constant, m4/s


D0 ¼ surface-diffusion frequency factor, m2/s

Q ¼ activation energy for surface diffusion, J/mol(convenient units: kJ/mol)







d ¼ surface-diffusion zone width, m (convenient units: nm).

[Also see Herring Scaling Law and Kuczynski Neck-growth Model.]

SURFACE-ENERGY VARIATION WITH DROPLET SIZE(Tolman 1949)

The surface energy g depends slightly on the surface curvature. Any departure fromthe equilibrium value corresponding to a flat surface only becomes significant forvery small droplets and particles. Because the atomic packing density on thesurface varies with size D, Tolman shows the following approximation,

g

gO¼ 1

1þ 4d=D

where gO is the equilibrium or flat-surface energy, d is the superficial density ofmatter on the surface divided by the difference in densities from inside to outside(for example, liquid–vapor) across the surface,

d ¼ G

rI � rO

where G is the superficial density of matter at the boundary between the two phases, rI

is the density inside (such as liquid density), and rO is the density outside (such as thevapor density). The parameter d is near the molecular spacing, so any measurable

342 CHAPTER S

change to surface energy is only detectable as the droplet or particle size approachesthe molecular size.

D ¼ droplet or particle diameter, m (convenient units: nm)

G ¼ superficial density at the interface, kg/m3 (convenient units: g/cm3).


gO ¼ equilibrium surface energy, J/m2

d ¼ interfacial density parameter, dimensionless

rI ¼ density inside the droplet, kg/m3 (convenient units: g/cm3)

rO ¼ density outside the droplet, kg/m3 (convenient units: g/cm3).

SURFACE-TRANSPORT SINTERING

See Surface Area–Reduction Kinetics.

SURATMAN NUMBER

The Suratman number is also known as the Laplace number. It is a dimensionlessratio of the surface energy to the momentum or dissipation in a fluid, such as isencountered in liquid-phase sintering or liquid atomization. The Suratman numberSu is given as follows:

Su ¼ gLVrR

h

where r is the liquid density, R is the characteristic length scale and for most cases istaken as the particle size, gLV is the liquid–vapor surface energy, and h is the fluidviscosity.

Su ¼ Suratman number, dimensionless





SUSPENSION VISCOSITY

Feedstock for powder forming is a mixture of powder and polymer or other liquid. Theviscosity h of such a crowded mixture of particles in a fluid binder gives a responsethat is dominated by the volume fraction of solid f as,

h ¼ ghL(G=G0)2

(d1=dt)n(1� cf)2

SUSPENSION VISCOSITY 343

where g is a geometric term that includes a reference shear rate, hL is the liquid viscosity,G is the grain size or particle size (grain size is during sintering and particle size ifthere is no chemical interaction between the solid and liquid), G0 is a reference grainsize or particle size (such as 1mm), n is typically between 0.5 and 1.0, and c is typicallybetween 1.2 and 2.0. This latter parameter is effectively the inverse of the critical solidsloading. Further, liquid viscosity depends on composition and temperature. Overlimited temperature ranges, above the liquid formation temperature, yet below anydecomposition temperature, the viscosity variation with temperature can be estimatedusing the Arrhenius dependence:

hL ¼ h0 expQ

RT

� �

where Q is an apparent activation energy for viscous flow, h0 is an inherent viscosityterm, R is the gas constant, and T is the absolute temperature. Since several factorschange with an increase in temperature, often the combined effect is lumped into thisexpression; thus, the apparent activation energy differs from that measured independentlyfor the pure liquid.

G ¼ grain size or particle size, m (convenient units: mm)

G0 ¼ reference grain size or particle size, m (convenient units: mm)

Q ¼ activation energy for viscous flow, J/mol (convenient units: kJ/mol)



c ¼ material constant (typically between 1.2 and 2.0), dimensionless


g ¼ geometric term, units of 1/s to 1/s1/2

n ¼ material constant (typically between 0.5 and 1.0), dimensionless

t ¼ time, s


f ¼ volume fraction of solid, m3/m3 or dimensionless

h ¼ viscosity of powder–fluid suspension, Pa . s

h0 ¼ inherent viscosity term, Pa . s

hL ¼ liquid viscosity, Pa . s.

SWELLING

See Shrinkage and Shrinkage-induced Densification.

SWELLING REACTIONS DURING MIXED-POWDERSINTERING (Savitskii 1993)

Chemical reactions between mixed powders during sintering often have larger free-energy changes than the weak surface-energy reduction effects that normally drive

344 CHAPTER S

sintering. As a consequence, early during the sintering of mixed powders the strongchemical gradients drive mass flow, often with an initial period of swelling duringheating. The initial porosity and the volumetric solubility determine the extent ofswelling. For cases where the newly formed liquid is soluble in the solid there willbe a swelling event on first liquid formation. After a swelling reaction the porosity1 depends on the initial porosity 10 and the volumetric concentration of additive C,

1 ¼ 10 þ FC(1� 10)

where F is the fraction of liquid that has gone into solution as follows:

F ¼ CL(1� 10)1� C � CL

where CL is the volumetric concentration of solid dissolved in the liquid. This predictsa linear relation between expansion and concentration, giving a higher final porositywith a higher initial porosity. If there were no interaction between the powders, thenthe final porosity would equal the initial porosity.

C ¼ concentration of additive, m3/m3 or dimensionless

CL ¼ concentration of solid dissolved in liquid, m3/m3 or dimensionless

F ¼ fraction of additive dissolved in the solid matrix, dimensionless [0, 1]


10 ¼ initial fractional porosity, dimensionless [0, 1].

SWELLING REACTIONS DURING MIXED-POWDER SINTERING 345

T

TAP DENSITY

See Vibration-induced Particle Packing.

TEMPERATURE ADJUSTMENTS FOR EQUIVALENT SINTERING

When the sintering time is constant and a reduction in sintering temperature is desiredfor an equivalent degree of sintering (typically measured by the same neck-size ratio),then the new temperature corresponding to a change in particle size is given as,

1T2¼ 1

T1� R

Qln

D2

D1

� �

where T1 and T2 are the sintering temperatures, D1 and D2 are the particle sizescorresponding to temperatures T1 and T2, R is the universal gas constant, and Q isthe activation energy. This model assumes there is no change in the sintering mech-anism. If particle size D2 is smaller than D1, then there is a reduction in the sinteringtemperature, where T2 , T1. A lower sintering temperature is beneficial in certainmaterials, especially those that evaporate or decompose at high temperatures.Nanoscale powders (dimensions measured in nm or smaller than 0.1 mm) exhibitlarge reductions in the sintering temperature, but some of the controlling thermo-dynamic factors are unchanged by particle size, so thermochemical reactions mightbe altered by a lower temperature.

D1 ¼ particle size corresponding to temperature T1, m (convenient units: mm)

D2 ¼ particle size corresponding to temperature T2, m (convenient units: mm)



T1, T2 ¼ sintering temperature, K


347

TEMPERATURE DEPENDENCE


TERMINAL DENSITY

See Final-stage Sintering Limited Density.

TERMINAL NECK SIZE

See Neck Growth Limited by Grain Growth.

TERMINAL NECK SIZE IN SINTERING

See Limiting Neck Size.

TERMINAL PORE SIZE

See Final-stage Pore Size.

TERMINAL SETTLING VELOCITY

See Stokes’ Law.

TERMINAL SINTERING

See Trapped-gas Pore Stabilization.

TERMINAL VELOCITY

See Acceleration of Free-settling Particles.

TETRAKAIDECAHEDRON (Smith 1964)

In sintering, hot isostatic pressing, and powder compaction, a 14-sided polyhedron isused to represent the grain shape in a powder compact as it approaches full density.Although the typical loose powder only has about 7 contacts per particle, in the dense

348 CHAPTER T

condition the polyhedral grains average 14 faces. This structure is illustrated inFigure T1. The tetrakaidecahedron was introduced as a grain-shape model by CyrilStanley Smith and subsequently employed in sintering models by Robert Coble.The tetrakaidecahedron is also known as a truncated octahedron. It is composedof 8 hexagonal faces and 6 square faces. The number of edges is 36 and thenumber of corners is 24. If the length of an edge segment on the polyhedron is L,then the volume V, surface area S, and grain size G are related as follows:

V ¼ffiffiffiffiffiffiffiffi128p

L3 ¼ 11:31L3

S ¼ffiffiffiffiffiffiffiffi432p

þ 6� �

L2 ¼ 26:78L2

and

G ¼ffiffiffi8p

L ¼ 2:83L


L ¼ edge-segment length, m (convenient units: mm)

S ¼ surface area, m2 (convenient units: mm2)

V ¼ volume, m3 (convenient units: mm3).

Figure T1. The tetrakaidecahedron is a favorite grain shape for modeling final-stage sintering,forging, hot pressing, and hot isostatic pressing. It consists of 14 sides, 8 being hexagons and 6being squares, with 24 corners and 36 edges. In the intermediate stage of sintering the edges areassumed to be tubular pores, and in the final stage of sintering the corners are assumed to bespherical pores.

TETRAKAIDECAHEDRON 349

THEORETICAL DENSITY FOR MIXED POWDERS

See Mixture Theoretical Density.

THERMAL CONDUCTION

See Conductive Heat Flow.

THERMAL CONDUCTIVITY (Speyer 1994)

Most typically the thermal diffusivity l is measured on a material by pulsedheating, such as with laser flash heating. The thermal conductivity k is calculatedfrom these data using the experimental diffusivity data. The relation betweenthe two parameters involves the material density r and the constant pressure heatcapacity CP,

k ¼ lCPr


k ¼ thermal conductivity, W/(K . m)

l ¼ thermal diffusivity, m2/s

r ¼ density of the sample, kg/m3 (convenient units: g/cm3)

THERMAL CONDUCTIVITY DEPENDENCE ON POROSITY(Luikov et al. 1968; Koh and Fortini 1973)

The thermal conductivity decreases with porosity and generally follows a structurethat is independent of the pore size and shape. In the higher-density region, overabout 70% of theoretical density, thermal conductivity k follows a linear dependenceon fractional porosity 1:

k ¼ k0(1� v1)

where k0 is the inherent conductivity for the material, and v is the sensitivity coeffi-cient, usually ranging between 1 and 2. A few isolated, spherical pores have a smallimpact, while a high level of porosity makes the material an insulator. Later Koh andFortini also proposed a semiempirical relationship as follows

k ¼ k01� 1

1þ 1112


k ¼ thermal conductivity of porous material, W/(m . K)

350 CHAPTER T

k0 ¼ thermal conductivity for the dense material, W/(m . K)

v ¼ sensitivity coefficient, dimensionless.

THERMAL CONDUCTIVITY FROM ELECTRICAL CONDUCTIVITY

For metals, the thermal conductivity is quite high and those metals that are thebest electrical conductors are also the excellent thermal conductors, since electrontransport explains both events. At a given temperature, the thermal and electricalconductivities of metals are proportional, but raising the temperature decreases theelectrical conductivity. Estimates of the thermal conductivity k are possible fromthe electrical conductivity s using the Wiedemann–Franz–Lorenz relation (1853to 1872),

k ¼ LsT

where L is the Lorenz number [(L ¼ p2/3 (k/e)2, where k is Boltzmann’s constantand e is the electron charge], and T is the temperature. Qualitatively, this relationshipis based on the fact that the heat and electrical transport both involve the free electronsin the metal. For the cases where conduction also has a significant lattice-vibrationcontribution, the relation is modified to include a second vibration term k0,

k ¼ k0 þ LsT

For materials such as diamond and graphite, the lattice term is dominant, while forsemiconductors both terms are important, and in metals the electron transport termis dominant. The thermal conductivity increases with the average carrier velocity,but higher temperatures induce more collisions in the lattice to reduce electronenergy transport. The ratio of thermal to electrical conductivity depends upon theaverage velocity squared, which is proportional to the kinetic temperature.

L ¼ Lorenz number, 2.45 . 1028 J2/(C2 . K2) [or 2.45 . 1028 W/(S . K2)]

T ¼ temperature, K

k ¼ thermal conductivity, W/(K . m)

s ¼ electrical conductivity, S/m.

THERMAL CONVECTION

See Convective Heat Transfer.

THERMAL DEBINDING

See Polymer Pyrolysis and Vacuum Thermal Debinding.

THERMAL DEBINDING 351

THERMAL DEBINDING MASTER CURVE

See Master Decomposition Curve.

THERMAL DEBINDING TIME (German 1987)

When a powder compact is heated to a temperature where the polymer decomposes,the time to remove the polymer is determined by the section thickness and severalmicrostructure and operating parameters. Assuming there are no defects from rapidheating, the thermal debinding time t under the isothermal condition is approximatedas follows:

t ¼ 22:5 h2f2Ph

(P2 � P20)(1� f)3D2F

where h is the compact thickness, f is the fractional solids loading, P is thegas pressure in the pores, P0 is the ambient pressure, h is the vapor viscosity, D isthe particle diameter, and F is the volume change associated with burnout ofthe binder.


F ¼ volume change for binder on evaporation, dimensionless

P ¼ gas pressure for decomposition products in the pores, Pa

P0 ¼ gas pressure for decomposition product outside the compact, Pa

h ¼ compact thickness, m (convenient units: mm)

t ¼ debinding time, s

f ¼ fractional solids loading, dimensionless [0, 1]

h ¼ vapor viscosity, Pa . s.

THERMAL DIFFUSIVITY

See Thermal Conductivity.

THERMAL EXPANSION COEFFICIENT (Tuchinskii 1983)

A porous component consists of pores with near-zero thermal expansion and a solidthat changes volume on heating. The porous structure behavior is lower in thermalexpansion than the dense material. The proportionality between length change andtemperature change is given by the thermal expansion coefficient which is alsoknown as the coefficient of thermal expansion (CTE or TEC). As an approximation,

352 CHAPTER T

the thermal expansion coefficient a depends on the density r as follows:

a ¼ a0r

rT

� �1=3

where a0 is the bulk thermal expansion coefficient for the material corresponding to atheoretical density of rT.

a ¼ porous thermal expansion coefficient, 1/K (convenient units: 1026/K)

a0 ¼ dense thermal expansion coefficient, 1/K (convenient units: 1026/K)

r ¼ density, kg/m3 (convenient units: g/cm3)


THERMALLY ACTIVATED


THERMAL SHOCK RESISTANCE (T. J. Lu and Fleck 1998)

The thermal shock resistance for a material, designated TR, depends on the tempera-ture change and thermal expansion coefficient CT, thermal conductivity k, and elasticmodulus E:

TR ¼sk

ECT

where s is the strength at the test temperature.

CT ¼ thermal expansion coefficient, 1/K (convenient units: ppm/K or1026 1/K)


TR ¼ thermal shock resistance, N/s


s ¼ strength, Pa (convenient units: MPa).

THETA TEST (Morrell 1989)

The theta test is a strength test based on a flat-sample geometry that has two horizontalcutouts that are shaped like half-moons. A generalized view of the test geometry isgiven in Figure T2. This test geometry is used to measure the tensile strength of

THETA TEST 353

ceramics or other brittle materials by applying a compressive force that generates atensile force inside the sample. A preferred outside-diameter to thickness ratio is30, and for the standard 75-mm-diameter disk this gives a 2.5-mm thickness. Thesample is loaded perpendicular to the crosspiece, and the outward deflection leadsto a tensile stress s in the central ligament calculated as follows:

s ¼ KF

dt

where F is the fracture load, d is the diameter of the sample, and t is the thickness. Thedimensionless parameter K depends on the sample geometry and it usually taken tobe 13.8 when the sample follows the 30-fold diameter to thickness ratio [one estimateis that K ¼ 25–336 (t/d ) to account for different geometries].


K ¼ sample geometry parameter, dimensionless

d ¼ sample outer diameter, m (convenient units: mm)

t ¼ sample thickness, m (convenient units: mm)

s ¼ tensile strength, Pa (convenient units: MPa).

THIRD-STAGE SINTERING DENSIFICATION

See Final-stage Densification.

Figure T2. The theta test is used to measure tensile strength by applying compression to abrittle sample. The radial expansion perpendicular to the loading axis generates a tensilestress on the central crosspiece.

354 CHAPTER T

THIRD-STAGE SINTERING STRESS

See Final-stage Sintering Stress, Sintering Stress in Final-stage Sintering for SmallGrains and Faceted Pores, Sintering Stress in Final-stage Sintering for Small Grainsand Rounded Pores, and Sintering Stress in Final-stage Sintering for Spherical PoresInside Grains.

THREE-POINT BENDING STRENGTH

See Transverse-rupture Strength.

THREE-POINT BENDING TEST

See Bending Test.

TIME FOR THERMAL DEBINDING

See Thermal Debinding Time.

TIME TO SOLIDIFY IN ATOMIZATION

See Solidification Time.

TIME TO SPHEROIDIZE IN ATOMIZATION

See Spheroidization Time.

TOOL EXPANSION FACTOR

To account for shrinkage in sintering, the forming tool needs to be positively dilatedto account for the sintering shrinkage (in the case of swelling, there is a need to do theopposite: make the tool smaller than the final size). For the common shrinkage case,the tool expansion factor Z is multiplied times the final component size to give thetool size. Assuming isotropic dimensional change, the expansion is estimated fromthe shrinkage Y as follows:

Z ¼ 11� Y

TOOL EXPANSION FACTOR 355

As a consequence, the tool dimension L0 is enlarged from the final or specifiedcomponent dimension Lf by the expansion factor,

L0 ¼ Lf Z

L0 ¼ tool dimension, m (convenient units: mm)

Lf ¼ final or specified component dimension, m (convenient units: mm)

Y ¼ sintering shrinkage, dimensionless

Z ¼ tool expansion factor, dimensionless.

[Also see Shrinkage Factor in Tool Design.]

TORTUOSITY

See Darcy’s Law.

TRANSFORMATION KINETICS

See Avrami Equation.

TRANSIENT LIQUID-PHASE SINTERING

Transient liquid-phase sintering occurs when a liquid forms during heating, yet thatliquid is soluble in the solid at the peak sintering temperature. Consequently, duringheating the liquid disappears over time as it forms an alloy; thus, the time theliquid exists depends on the heating rate. Such a reaction between mixed powdersis a three-dimensional diffusion-controlled process. The degree of reaction followsa kinetic law as shown below,

[1� (1� a)1=3]2 ¼ Gt

where a is the fraction of phase that has been reacted, t is the isothermal hold time,and G is the rate constant. The rate constant is sensitive to temperature since it followsthe Arrhenius relation, described by

G

G0¼ exp � Q

RT

� �

where G0 is the preexponential parameter, T is the absolute temperatures, R is the gasconstant, and Q is the activation energy associated with the rate-limiting step, whichis usually diffusion of the liquid into the solid. As illustrated in Figure T3, the liquidinitially disappears quickly and the impact of a higher temperature is to shorten

356 CHAPTER T

the time. As an approximation, the liquid will persist for a period of approximately1/G. These equations are differentiated and numerically solved for cases involvingnonisothermal sintering. Sintering involving a reaction, especially with liquid, issensitive to many processing conditions, especially the initial particle size andheating rate. Transient liquid-phase sintering is evident in some common mixed-powder sintering systems, such as Cu–Sn and Fe–P.




t ¼ time, s

G ¼ rate constant, 1/s

G0 ¼ preexponential parameter, 1/s

a ¼ fraction of a phase that has reacted, dimensionless [0, 1].

[Also see Heating Rate Effect on Liquid Quantity in Transient Liquid-PhaseSintering.]

TRANSVERSE-RUPTURE STRENGTH (Sherman and Brandon 2000)

Strength is measured on brittle materials in three-point or four-point bending. Theresulting property is known as the transverse-rupture strength or sometimes themodulus of rupture. A typical test is illustrated in Figure T4, where fracture occurs

Figure T3. Reaction kinetics illustrated for transient liquid-phase sintering, where the fractionof liquid reacted is plotted versus dimensionless time for two temperatures. For this plot theactivation energy is set to 200 kJ/mol, the frequency factor to 106 1/s, and the lower tempera-ture is 1200 K and the higher temperature is 1300 K.

TRANSVERSE-RUPTURE STRENGTH 357

on the surface opposite the upper loading rod. The transverse rupture strength sT iscalculated from the specimen geometry and failure load F as,

sT ¼3FL

2WT2

where T is the thickness, W is the width, and L is the distance between the lowersupport rods. Although there is a great deal of allowed variation in absolute dimen-sions, typical values are T ¼ 6 mm, L ¼ 25 mm, and W ¼ 12 mm. Should the mid-point deflection before failure exceed 4% of the initial thickness, then thecalculated strength should not be accepted (for 6-mm thickness this limit is a deflec-tion of 0.24 mm). For brittle materials, the transverse-rupture test gives a significantlylower value when compared to the cylinder compressive test. Over a broad array ofceramics, the ratio of transverse strength to compressive strength is about 25%.When compared to the tensile strength as measured for metals s, the three-pointtransverse-rupture strength sT tends to be 60 to 80% higher; the exact ratiodepends on the Weibull modulus m as follows:

sT ¼ s [2(mþ 1)2]1=m

For low-ductility materials, the ratio is typically 1.6 (the tensile strength is signifi-cantly lower), corresponding to a Weibull modulus near 13 or a typical scatter inmeasured strength near 10% of the mean value. Likewise, for a brittle material, thethree-point and four-point transverse-rupture strengths are related as follows:

sT ¼ s412

(mþ 2)

� �1=m

Figure T4. A three-point test is applied to green compacts to measure the transverse-rupturestrength.

358 CHAPTER T

F ¼ failure load, N (convenient units: kN or MN)

L ¼ distance between lower support rods, m (convenient units: mm)

T ¼ sample thickness, m (convenient units: mm)

W ¼ sample width, m (convenient units: mm)

m ¼Weibull modulus, dimensionless

s ¼ tensile strength, Pa (convenient units: MPa)

s4 ¼ four-point transverse-rupture strength, Pa (convenient units: MPa)

sT ¼ transverse-rupture strength, Pa (convenient units: MPa).

[Also see Bending Test and Weibull Distribution.]

TRAPPED-GAS PORE STABILIZATION (Markworth 1972)

In final-stage sintering, gas trapped in the pores will stop densification based on abalance between the capillary pressure from the curved pore surface and the internalgas pressure,

4gSV

dP¼ PG

where gSV is the solid–vapor surface energy, dP is the pore diameter, and PG is thegas pressure in the pore. Assume a compact is sintered in argon at a furnace pressureP1, with a pore diameter of dP1 at pore closure, which occurs near 8% porosity. Thefinal porosity is calculated by recognizing that the mass of the gas in the pores isconserved since argon is insoluble. If the number of pores and temperature remainconstant with a spherical pore shape, then

P1V1 ¼ P2V2

where P2 is the final pore pressure, V1 is the initial pore volume, and V2 is the finalpore volume. The final pore size dP2 is given as follows:

dP2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiP1d3

P1

4gSV

s

The final fractional porosity 12 is estimated from the fractional porosity at pore closure11 as follows:

12 ¼ 11dP2

dP1

� �3

P1 ¼ initial pore pressure, Pa

P2 ¼ final pore pressure, Pa

PG ¼ gas pressure in the pore, Pa

TRAPPED-GAS PORE STABILIZATION 359

V1 ¼ initial pore volume, m3 (convenient units: mm3)

V2 ¼ final pore volume, m3 (convenient units: mm3)

dP1 ¼ initial pore diameter, m (convenient units: mm)

dP2 ¼ final pore diameter, m (convenient units: mm)

12 ¼ final fractional porosity, dimensionless [0, 1]

11 ¼ fractional porosity at pore closure, dimensionless [0, 1]


TRUNCATED OCTAHEDRON

See Tetrakaidecahedron.

TWO-DIMENSIONAL GRAIN CONTACTS

See Connectivity.

TWO-PARTICLE SINTERING MODEL


TWO-PARTICLE VISCOUS-FLOW SINTERING (Ross et al. 1981;Ristic and Milosevic 2006)

In the corrected version of Frenkel’s derivation for viscous-flow sintering of twoequal-size spheres, the model assumes an existing contact and evaluates thebalance between surface-area reduction and the viscous-flow energy consumption.By equating the energy reduction due from neck growth to the viscous flowenergy, the neck-growth-rate equation results. Assume two spheres of equal sizein contact with an existing neck, as shown in Figure T5. The angle u allowslinkage of the particle size D and the neck size X as follows:

X ¼ D sin u

Diffusivity in a material is generally related to an effective viscosity, so the linkagebetween viscosity and diffusivity allows generalization of this model to crystallinematerials. In crystalline materials, however, the random contact between the particlesleads to a grain boundary with an associated energy. The retarding effect from theincreasing grain-boundary energy during neck growth is missing in many sinteringmodels. The extension of the grain boundary during neck growth reduces the

360 CHAPTER T

energy available from surface-area reduction. Further, a dihedral angle will limit neckgrowth. Ignoring those factors, as is appropriate for an amorphous material, the geo-metry of the two-particle sintering model leads to a volume-conservation equationthat says the region of overlapping spheres due to shrinkage results in a positivedilation of the two spheres to give a new sphere diameter of DN,

p

6D3 ¼ p

24D3

N(2þ 3 cos u� cos3 u)

For small angles u (small degrees of neck growth) the relation between the angle u

and the surface-area change DS and new particle size is given as,

DS ¼ 2pD2Nu

2

Note that this assumes the neck size is small. Unfortunately, many studies in recentyears, especially those modeling the coalescence of two spheres by computer simu-lations, have forgotten this limitation and extended the Frenkel model to unreasonableneck sizes. The balance of energy reduction from loss of surface area S and the workof viscous deformation gives,

�gSV

dS

dt¼ p

2gSVD2

N

d(u2)dt

where t is the time, gSV is the surface energy (assumed to be a solid–vaporinterface, but if the particle is a liquid, then it should be a liquid–vapor surfaceenergy). It is assumed the relation between the angle u and viscosity h is givenas follows:

u ¼ 3gSVt

phDN

Figure T5. The two-particle geometry employed for the first viscous-flow sintering modelby Frenkel.

TWO-PARTICLE VISCOUS-FLOW SINTERING 361

When substituted into the energy-balance equation, this gives the first-stage sinteringneck-growth equation as follows:

X2 ¼ 3p

DNtgSV

h

There are several points in this equation to note. First, different assumptions andapproximations lead to differences in the numerical constant (p/3), and various treat-ments have adjusted this value. The consensus is that it is a numerical term close tounity. This largely depends on the range used to limit application of the equation.Computer simulations of viscous-phase sintering consistently show that this modelis only accurate for small neck sizes, but it is a reasonable approximation up to aneck size ratio nearing unity. Often the neck size is given as a ratio to the initial par-ticle size, expressed as X/D. If there is a grain boundary, then the neck growth islimited by the interfacial energy, which is ignored here. Extensions of this modelto the intermediate and final stages of sintering are possible, with some cautions asto accuracy, especially for crystalline materials.


DN ¼ new sphere diameter, m (convenient units: mm)




DS ¼ surface-area change, m2



u ¼ angle, rad (convenient units: degree).

362 CHAPTER T

U

ULTRASONIC VELOCITY (Mukhopadhyay and Phani 2000)

The velocity V of an ultrasonic signal in a porous body departs from the full-densityvelocity V0 in a manner that depends on the fractional porosity 1 and the scatteringdue to pores. The underlying behavior shows a velocity that depends on the squareroot of the ratio of the elastic modulus E divided by the theoretical density. Whenmanipulated, these two factors give the ultrasonic velocity V as a function of frac-tional porosity 1 as follows:

V ¼ V0(1� a1)N

where the parameter a is a pore-shape factor that is generally in the range from 1 to 2,and the exponent N is generally near 0.4. In some higher-density steels, a is near 2

Figure U1. A plot of the sound velocity versus pressed density for water-atomized ironpowder, demonstrating the nonlinear influence of residual porosity.


363

and N is slightly larger than unity. In turn, the compaction pressure dominates thegreen density and porosity. Thus, as shown in Figure U1, the sonic velocity forsintered iron increases with compaction pressure.


N ¼ exponent, dimensionless

V ¼ velocity of the ultrasonic signal in a porous body, m/s (convenientunits: km/s)

V0 ¼ velocity of the ultrasonic signal in a full-density body, m/s (convenientunits: km/s)

a ¼ pore-shape factor, dimensionless


364 CHAPTER U

V

VACANCY CONCENTRATION DEPENDENCE ON SURFACECURVATURE (Kuczynski 1949)

A curved surface has a stress that results in a vacancy concentration that departs fromthe equilibrium concentration. Generally a flat surface that is free of stress is con-sidered to be at equilibrium. In sintering processes, microstructure curvature drivesmass flow to restore equilibrium by taking both the concave and convex surfacestoward a flat state. Mass from the convex surface moves to fill in the concavity.The vacancy concentration C under a curved surface depends on the local curvature,

C ¼ C0 1� gV

kT

1R1þ 1

R2

� ��

where C0 is the equilibrium vacancy concentration associated with a flat surface at thesame temperature, g is the surface energy (either solid–liquid or solid–vapor), V isthe atomic volume, k is Boltzmann’s constant, and T is the absolute temperature. Notethat the equilibrium concentration also increases on heating. As shown in Figure V1,there are two perpendicular arcs that pass through at any point on the surface. Thesearcs have radii of curvature designated as R1 and R2. The more highly curved thesurface, the smaller R1 and R2. Accordingly, the smaller radii lead to a greater depar-ture from equilibrium. For a concave surface, the vacancy concentration is higher thanequilibrium; for a convex surface it is lower; thus, atomic flow is from regions ofvacancy deficiency—convex—to regions of vacancy excess—concave. The conven-tion is that when a radius of curvature is located inside the solid it is deemed negative,while a radius located outside the solid is positive. A concave surface is effectively asource of vacancies that leads to a counterflow of atoms that work to fill the concavity.A convex surface is a sink for vacancies, so atoms flow away from a convex surface,effectively removing the asperity over time.

C ¼ vacancy concentration, 1/m3

C0 ¼ equilibrium vacancy concentration, 1/m3


365

R1, R2 ¼ perpendicular radii of curvature, m (convenient units: mm)




g ¼ surface energy, J/m2.

[Also see Kelvin Equation.]

VACANCY DIFFUSION (Shewmon 1989)

Atomic motion (usually termed lattice diffusion or volume diffusion) depends onatomic exchange with neighboring vacancies. For diffusion to occur, an atom musthave sufficient energy, QB, to break existing bonds with neighboring atoms andthen additional energy to exchange its position with a neighboring vacant site. Theprobability of a neighboring atomic site being vacant depends on the vacancy-formation energy QN. In other words, lattice diffusion requires both the formationof a vacancy and the provision of sufficient energy to break an atom free so that it

Figure V1. Capillary forces and vacancy concentrations are calculated for any point ona surface using two perpendicular arcs passing through that point. Shown here as R1 and R2,these two arcs then define the tension or compression or departures from equilibriumvacancy concentrations. Concave surfaces have an excess of vacancies and convex surfaceshave a deficiency.

366 CHAPTER V

can jump into the vacant site. As an approximation to the rate of atomic diffusion, theArrhenius equation gives the relative number of active atoms NA compared with thetotal number of atoms N0 as follows:

NA ¼ N0 exp �QB þ QN

RT

� �

where R is the gas constant and T is the absolute temperature. Most typically, the rateof atomic diffusion is termed the diffusivity, which depends on several parameters,including the frequency of atomic vibration, crystal class, lattice parameter, andsimilar factors. The resulting form for the diffusion coefficient is an Arrheniusequation,

D ¼ D0 exp � Q

RT

� �

with D being the diffusivity, and D0 being the frequency factor. The activation energyQ is the sum QN þ QB. In turn, for a given crystal structure both activation energiescan be rationalized to the number of atomic bonds that must be broken to form avacancy and the number of atomic bonds that must be broken to move an atom.Many handbooks compile data diffusion data as D0 and Q, which allows calculationof D at any temperature.

D ¼ diffusion coefficient, m2/s

D0 ¼ frequency factor, m2/s

N0 ¼ total number of atoms, dimensionless

NA ¼ number of active atoms moving at any time, dimensionless


QB ¼ activation energy to break bonds with neighboring atoms, J/mol(convenient units: kJ/mol)

QN ¼ activation energy for vacancy formation, J/mol(convenient units: kJ/mol)



VACUUM DEBINDING

See Vacuum thermal debinding.

VACUUM DISTILLATION RATE (Jones 1960)

When a powder is heated to a high temperature in a vacuum, the vapor pressureallows the distillation of the powder into the vacuum chamber at a rate G thatdepends on the molecular weight M of the vapor species, the equilibrium vapor

VACUUM DISTILLATION RATE 367

pressure P (which depends on temperature), and the absolute temperature T, asfollows:

G ¼ P

ffiffiffiffiffiffiM

RT

r

This assumes that there is no vapor accumulation in the chamber, implying that there is acold wall orothercondensation site in the furnace. Accordingly, from the distillation rate itis possible to estimate the time it takes to evaporate a particle if its mass is known.

G ¼ vacuum distillation rate, kg/(m2 . s)

M ¼ molecular weight, kg/mol (convenient units: g/mol)




VACUUM FLUX IN SINTERING (Johns et al. 2007)

When sintering is in a vacuum, a gas-impingement rate exists on any surface thatdepends on the pressure and temperature in the sintering furnace. The correspondingatomic flux is the frequency at which gas molecules collide with the surface.Considering an external surface, the number of gas molecules that strike thesurface per unit time and per unit area is the flux J, estimated as,

J ¼ Pffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pkTmp

where P is the gas pressure, and m is the molecular weight of the species. Examplesthat depend on this flux are oxide reduction in a partial pressure of hydrogen andvacuum surface carburization in a partial pressure of methane. If the density of thedesired reaction sites is known for the surface, then it is possible to estimate fromthe flux the time required for the desired effect; the characteristic time is thedensity of surface sites (number per unit area) divided by the flux.

J ¼ flux, atom/(m2 . s)




m ¼ molecular weight, kg/atom (or kg/molecule).

VACUUM THERMAL DEBINDING (German and Bose 1997)

Thermal binder removal in a vacuum depends on pyrolysis and evaporation of thepolymer, with subsequent diffusion through the open, surface-connected pores.

368 CHAPTER V

Usually the diffusion step is rate controlling, and in such a situation the debindingtime t under idealized, isothermal conditions varies as follows:

t ¼ h2

2DDP(1� f)2

(MWRT)1=2

VM

where h is the compact wall thickness, MW is the molecular weight of the burnoutproduct, R is the gas constant, T is the absolute temperature, f is the solidsloading or fractional solid in the body (excluding binder), D is the particle diameter,DP is the pressure gradient from where the vapor is formed inside the compact to thecompact surface, and VM is the molecular volume of the vapor. Usually, the molecularweight of the burnout product is smaller than the molecular weight of the polymer,and various studies tend to show that the burnout species typically range from frag-ments with 8 to 10 carbon atoms. In cases of rapid heating, the slow diffusion controlstep will lead to vapor accumulation inside the body, resulting in blistering.


MW ¼ molecular weight of evaporating species, kg/mol (convenientunits: g/mol)



VM ¼ vapor molecular volume, m3/mol

h ¼ compact thickness, m (convenient units: mm)

t ¼ debinding time, s

DP ¼ vapor-pressure gradient from evaporation site to compact surface, Pa

f ¼ particle solids loading, dimensionless.

VAPOR MEAN FREE PATH (Neale and Nader 1973)

At a low atmospheric pressure, the distance traveled by a gaseous molecule betweencollisions with other molecules is often greater than the pore dimensions. As aconsequence, vapor transport rates in a porous body are lower than expected basedon gaseous diffusion. The critical determinant is the mean free path, or averagedistance traveled between collisions. This geometric parameter varies with the gaspressure, molecular mass, and temperature. Comparison of the vapor mean freepath to the pore size determines if atomic diffusion in the pores is operating or ifimpeded flow is the controlling mechanism. For a vapor species, the mean freepath l depends on gas density M as follows:

lM ¼1ffiffiffi

2p

pMd2

where lM indicates the inherent molecular mean free path, and d is the moleculardiameter. For example, water vapor at 323 K (508C) in vacuum has an equilibrium

VAPOR MEAN FREE PATH 369

vapor pressure of 12 kPa, with a corresponding mean free path of 200 nm. For aporous body, the alternative calculation of the mean free path comes from the physicallimitation on molecule travel imposed by the pore size, and is estimated as follows:

lP ¼41S

where lP indicates the pore size–controlled mean free path, 1 is the fractional poros-ity, and S is the surface area per unit volume. Vapor transport through a porous struc-ture via vapor diffusion is slower than free diffusion if collisions with the pore wallsare more frequent than collisions with other molecules. The pores are assumed tobe open and the molecular collisions with the pore walls to be more frequent thancollisions with other molecules, effectively corresponding to the condition wherethe mean free path in the vapor is much larger than that possible in the pores,lM . lP. Thus, a comparison of the mean free path with that induced by the physicallimits of the pore size is a first step in the analysis. Vapor diffusion in the pores isknown as Knudsen diffusion, where the flux J depends on the Knudsen diffusivityK and pressure gradient DP/L (DP is the change in pressure over the distance L),

J ¼ �KDP

RTL

where R is the gas constant, and T is the absolute temperature. The minus sign indi-cates the flow is from higher pressure to lower pressure. Vapor diffusion reflects theflux J in terms of the quantity of gas per unit area per unit time. The ratio of diffu-sivity in the pores to the inherent molecular diffusivity, relevant to the determinationof K, was treated by Maxwell for uniform pores, giving the ratio as G,

G ¼ 212þ f

where f is the fractional density, and 1 is the fractional porosity. This approach isidealistic and requires that the inherent gas diffusion be known and then discountedfor the restrictions from the pores. More recent formulations predict K directlyas follows:

K ¼ dP1

3t

ffiffiffiffiffiffiffiffiffi8RT

pm

r

where m is the molecular mass, dP is the pore size, and t is the tortuosity, which is thedistance the gas travels in the pore space versus the bulk pore-structure dimension.Generally, the effective diffusivity in a porous structure is far less than the inherentmolecular diffusivity, and for 20% porosity the estimate would be G ¼ 0.14 orless, depending on the pore shape and pore-size distribution.

J ¼ molecular flux, mol/(m2 . s)

K ¼ Knudsen diffusivity, m2/s

370 CHAPTER V

L ¼ thickness, m

M ¼ gas density or number of molecules per unit volume, 1/m3


S ¼ pore surface area per unit volume, m2/m3




m ¼ molecular mass, kg/mol

DP ¼ pressure change, Pa

DP/L ¼ pressure gradient, Pa/m

G ¼ ratio of diffusivity to the inherent molecular diffusivity, dimensionless

d ¼ molecular diameter, m (convenient units: nm)


l ¼ mean free path, m (convenient units: mm or nm)

lM ¼ inherent molecular mean free path, m (convenient units: mm or nm)

lP ¼ pore size determined the mean free path, m (convenient units: mm or nm)

t ¼ tortuosity, dimensionless.

VAPOR PRESSURE (Silbey et al. 2005)

For any material, the equilibrium vapor pressure P over a flat surface depends on theabsolute temperature T with an Arrhenius dependence (thermally activated). Duringsintering, the difference in curvature between convex (outward curved particlesurfaces) and concave (inward curved neck and pore surfaces) leads to evapor-ation–condensation. The convex surfaces have a vapor pressure above equilibrium,and the concave surfaces have a vapor pressure below equilibrium. Surface-curvaturegradients that naturally exist during sintering lead to pressure gradients that produceneck growth, compact strengthening, but no densification. The activation energyassociated with the evaporation or sublimation generally scales with the meltingpoint of the material and varies with temperature as follows:

P ¼ P0 exp � Q

RT

� �

where P0 is a preexponential material constant, Q is the activation energy forevaporation, and R is the gas constant, which equals Boltzmann’s constant timesAvagadro’s number. In sintering there might be a secondary factor associated withatmosphere interactions that promote evaporation, for example, the evaporation ofmolybdenum is sensitive to the partial pressure of oxygen.


P0 ¼ preexponential material constant, Pa

VAPOR PRESSURE 371

Q ¼ activation energy for evaporation, J/mol (convenient units: kJ/mol)



VIBRATION-INDUCED PARTICLE PACKING (Scott and Kilgour 1969;Barker 1994)

Loose powder will bridge and not pack efficiently when simply poured into a containerwith nominal gravitational forces. Particles formed from a low-density material, such asa plastic, have a lower apparent density when compared to a high-density material, suchas lead or steel. The tap density represents the asymptotic packing density possible viarepeated vibrations. It is also called dense random packing. For nonspherical particles,a large density difference will exist between the apparent density and the tap density.During vibration the density change from the fractional apparent density fA to the frac-tional tap density fT is described by an exponential function,

fN � fA ¼ (fT � fA) exp �K

N

� �

where fN is the fractional packing density after N vibration cycles (N is larger than 1),and K represents a device-specific constant that depends on the acceleration, amplitude,and frequency of the vibration. Another form of this equation relies on vibration timewith a similar functional behavior,

fN ¼ fT � (fT � fA) exp (�at)

where t is the vibration time, and a is a constant related to the vibration conditions.

K ¼ device-specific constant, dimensionless

N ¼ number of vibration cycles, dimensionless integer

a ¼ constant related to vibration conditions, 1/s

fA ¼ fractional apparent density, dimensionless [0, 1]

fN ¼ fractional packing density after N vibration cycles, dimensionless [0, 1]

fT ¼ fractional tap density, dimensionless [0, 1]

t ¼ vibration time, s.

VICKERS HARDNESS NUMBER (Meyers 1985)

The Vickers hardness test is the most universal of the hardness scales. It is also knownas a microhardness number. As illustrated in Figure V2, the Vickers scale depends onthe size of an indent from a 1368 diamond pyramid pressed by a known load P. After

372 CHAPTER V

the load is removed, a microscope is used to measure the size of the impression in thetest material. As illustrated, the diagonal of the impression w is measured and theVickers hardness number (VHN), in kgf/mm2 or HV in MPa or GPa, is calculatedfrom the indent size or area as follows:

VHN ¼ 1:854P

w2

where P is the load in kg, and w is the diagonal of the indent impression in mm. TheVickers hardness number is typically expressed in either of two different units. Oneis VHN, which corresponds to units of kgf/mm2 and is equivalent to 106 kgf/m2.The second is determined by multiplying the VHN by the acceleration of gravity(9.8 m/s2) to get force over area, which is the same as stress, often expressed asMPa or GPa (note 1 VHN ¼ 1 MPa). This is convenient for comparison withtensile or fracture strength. Softer materials have larger indentations and lower hard-ness values. The load applied during the test changes the apparent hardness,especially for lighter loads, so in some cases, it is appropriate to indicate theapplied load with the hardness (1, 2, 5, . . . , 100 kgf). For example, 290 VHN5 isone way to designate the hardness and load. Brittle materials can crack, and it is poss-ible to use the crack length to estimate the fracture toughness. For example, soft steelswill have a VHN of 135, which corresponds to 1323 MPa, while a sintered technicalceramic might be tenfold harder.

P ¼ load, kgf

VHN ¼ Vickers hardness number, kgf/mm2 (convenient units: GPa)

w ¼ diagonal of the indent impression, mm (convenient units: mm).

VISCOELASTIC MODEL FOR POWDER–POLYMER MIXTURES(Green 1998)

The mechanical response of a powder–polymer mixture is often between that of aliquid (viscous) and a solid (elastic), and is termed viscoelastic. In an elasticsolid, the stress and strain are proportional and essentially simultaneous duringloading. In a viscous material, there is a time-dependent deformation. Viscoelastic

Figure V2. The Vickers hardness test is based on impressing a 1368 diamond pyramid intoa surface and then measuring the tip distances to calculate the hardness. The harder thematerial, the smaller the resulting indent size.

VISCOELASTIC MODEL FOR POWDER–POLYMER MIXTURES 373

response applies to a wide range of deformation properties and is very evident inpowder–polymer mixtures; the viscous polymer phase is slow to respond in compari-son with the elastic particle phase. A simple model that explains this behavior isknown as the Maxwell model, where strain rate d1/dt depends on the applied stresss and material properties such as the elastic modulus E and viscosity h. The resultingresponse model is given as follows:

d1

dt¼ 1

E

ds

dtþ s

h

where 1 is the strain. This equation can be integrated by assuming that a constantstrain applied at time t ¼ 0 results in an initial peak stress s0. In turn, this definesthe stress relaxation behavior,

s ¼ s0 exp � t

tR

� �

where tR is the stress relaxation time constant, given as the ratio of viscosity to elas-ticity (h/E), assuming isothermal conditions. The relaxation time constant corre-sponds to the time for 63.2% of the initial stress to be lost even though the strainthat initially produced the stress is constant. Likewise, if a constant stress isapplied at t ¼ 0, then a strain model results as,

1 ¼ s1Eþ t

h

� �

where 1 is the strain. These models are known as the Kelvin or Voigt models.



t ¼ time, s

tR ¼ stress relaxation time constant, s


h ¼ tensile fluid viscosity, Pa . s

s ¼ stress, Pa (convenient units: MPa)

s0 ¼ initial peak stress, Pa (convenient units: MPa).

VISCOELASTIC RESPONSE (Rasteiro et al. 2007)

Suspensions of particles in a binder fluid exhibit a response to applied stress termedviscoelastic, implying a combination of viscous and elastic behaviors. Oscillatoryrheometry is used to extract the rheology behavior versus attributes such as solidsloading, particle size, particle shape, binder chemistry, surface-active agents, temp-erature, time (aging), and surface chemistry. When an oscillatory stress is applied,

374 CHAPTER V

often as a sine wave, the deformation response is measured as a function of time. Thisresulting phase lag between stress and strain is a characterization tool used to quantifythe powder–binder combination and its behavior in various forming devices. Asshown in Figure V3, when the applied stress follows a sine wave, the mixed elasticand viscous aspects of the strain response are out of phase. The time difference,or phase lag, between the two functions is designated w, and ranges from totallyin-phase (w ¼ 0) for a purely elastic solid to totally out of phase (w ¼ p/2 or 908)for a purely viscous liquid. Most particle suspensions are of mixed behavior, whatis termed viscoelastic. Characterization of the viscoelastic response comes from anoscillatory test where the shear stress t is defined as,

t ¼ t0 cos(vt)

where t0 is the peak shear stress, t is the time, and v is the angular frequency. Inresponse to the oscillatory shear stress, the shear strain g is given as,

g ¼ g0 cos(vt � w)

In this form, g0 is the peak shear strain, and w is the lag between stress and strain. Therate of deformation is determined from the shear strain rate, dg/dt as follows:

dg

dt¼ vg0 cos vt � w� p

2

� �

Since the applied stress is a sinusoidal function in oscillatory rheometry, complexstress and complex strain t� and g� are defined as,

t� ¼ t0 exp(ivt)

Figure V3. Complex viscosity involves analysis under a cyclic load to extract the phase lagbetween stress and strain. When the stress is applied in a sine wave, the resulting strain responseis out of phase. The time difference, or phase lag, between the two functions is designated w. Asystem that is in-phase and corresponds to a purely elastic solid.

VISCOELASTIC RESPONSE 375

and

g� ¼ g0 exp[i(vt � w)]

with the standard notation that i is the square root of 21 i ¼ffiffiffiffiffiffiffi�1p

. The ratioof complex stress and complex strain gives the complex modulus G�, which then con-sists of real and imaginary parts,

G� ¼ t�

g�¼ GS þ iGL

The complex modulus consists of a real part GS, called the storage modulus, whichrepresents the elastic behavior, since it is the in-phase behavior, and an imaginary partGL, known as the loss modulus, which reflects the viscous properties. Generally, thesetwo terms are functions of the angular frequency and have values that help characte-rize the suspension.

G� ¼ complex modulus, Pa . s

GS ¼ storage modulus, Pa . s

GL ¼ loss modulus, Pa . s


i ¼ffiffiffiffiffiffiffi�1p

¼ the square root of 21, dimensionless

t ¼ time, s


g0 ¼ peak shear strain, m/m or dimensionless

g� ¼ complex shear strain, 1/s

w ¼ phase lag, rad (convenient units: degree)

t ¼ shear stress, Pa

t0 ¼ peak shear stress, Pa

t� ¼ complex shear stress, Pa

v ¼ angular frequency, 1/s.

VISCOSITY (Jinescu 1974)

For ideal or Newtonian fluids the viscosity is the proportionality between the shearstress t and applied shear strain rate dg/dt;

t ¼ hdg

dt

where h is the viscosity. This form is attributed to Newton, and the viscosity calcu-lated from measurements of the shear strain rate and shear stress is termed theNewtonian viscosity. For example, honey has a viscosity that is 100 times

376 CHAPTER V

higher than water at room temperature (water at room temperature has a viscosity of1023 Pa . s). An older set of units is still encountered in the field, where viscosityis expressed as Poise (P) or centipoise (cP), where 1023 Pa . s is equal to 1 cP.Suspensions encountered in powder processing include paints, slurries, putties, andfeedstocks for extrusion and injection molding. The relation between stress andstrain for suspensions is often more complex and might involve a yield strength,elastic modulus, and viscous modulus that lead to a complex viscoelastic response.A typical viscosity for a polymeric binder used in powder forming depends ontemperature as follows:

h ¼ h 0 expE

R

1T� 1

T0

� ��

where R is the gas constant, T is the absolute temperature, h0 and T0 are referenceviscosity and temperature, and E is the apparent activation energy.

E ¼ apparent activation energy, J/mol (convenient units: kJ/mol)



T0 ¼ reference temperature, K


t ¼ time, s



h0 ¼ reference viscosity, Pa . s


[Also see Newtonian Flow.]

VISCOSITY DEPENDENCE ON SHEAR RATE

See Cross Model.

VISCOSITY DURING SINTERING (Mohanram et al. 2004)

During the densification portion of a sintering cycle, it is possible to estimate the sin-tering viscosity from the sample deflection. One test for viscosity during sinteringrelies on measuring the sagging behavior of an unsupported beam. Based onbending-beam viscosity tests as applied to glasses, the deflection of an elasticbeam, such as is encountered in a cantilevered geometry during sintering, is relatedto the viscosity. The analysis for effective system viscosity usually relies on a flatcompact with end supports, where the downward deflection y is measured as a

VISCOSITY DURING SINTERING 377

function of the horizontal x-position from the end support, giving a deflectiondescribed as follows:

1

[1þ (dy=dx)2]3=2

d2y

dx2¼ Mz

EIz

where Mz is bending moment about the z axis, E is elastic modulus, and Iz isa moment of inertia about the z axis. For a beam with a rectangular cross section,Mz and Iz are calculated by the following expressions:

Mz ¼18

q(L2s � 4x2)

and

Iz ¼112

bh3

where Ls is the unsupported span distance of the beam. For a typical sinteringexperiment, the span space is about 40 mm. The specimen cross section isdescribed by b and h, which are the width and thickness; they tend to be afew millimeters thick and about 6 to 12 mm wide. The unit load due togravity, q ¼ ragbh causes midpoint sagging. Here ra is the apparent density(not the theoretical density), and g is gravitational acceleration. When there isa small amount of deflection, the term (dy/dx)2 can be ignored. Accordingly,the deflection is expressed as follows:

d2y

dx2¼ 3rag(L2

s � 4x2)2Eh2

Applying the boundary conditions y ¼ 0 at x ¼+Ls/2, the solution of thepreceding equation yields the following result:

y ¼ 3rag

2Eh2� x4

3þ L2

s x2

2� 5L4

s

48

� �

At the central position, x ¼ 0, the deflection takes the maximum value,expressed as:

d ¼ �ymax ¼5ragL4

s

32Eh2

where d is measured midway from the end supports at the middle of the beam.With the viscous–elastic analogy, the deflection d is replaced by the deflectionrate d, and the elastic modulus E is replaced by the uniaxial viscosity h, giving

_d ¼ dd

dt¼ 5ragL4

s

32hh2

378 CHAPTER V

In the beam-bending tests during sintering, the deflection d and deflection rate d canbe measured. So the uniaxial viscosity of the sintering body is determined as

h ¼ 5ragL4s

32 _dh2

Note that the viscosity changes constantly during sintering due to densification andtemperature changes.


Iz ¼ inertial moment about the z axis, m4

Ls ¼ beam-span distance between supports, m (convenient units: mm)

Mz ¼ bending moment about the z axis, N . m

b ¼ sample width measured in the z direction, m (convenient units: mm)


h ¼ sample thickness measured in the y direction, m (convenient units: mm)

q ¼ distributed gravitational load per unit length, N/m

x ¼ horizontal position in the x direction, m (convenient units: mm)

y ¼ vertical position in the y direction, m (convenient units: mm)

ymax ¼ maximum deflection, m (convenient units: mm)

z ¼ lateral position in the z direction, m

d ¼ maximum deflection, m

d ¼ dd/dt¼maximum deflection rate, m/s

h ¼ uniaxial viscosity, Pa . s

ra ¼ the sample density, kg/m3 (convenient units: g/cm3).

[Also see Bending-Beam Viscosity.]

VISCOSITY MODEL FOR INJECTION-MOLDING FEEDSTOCK(Najmi and Lee 1991)

Powder–binder mixtures have a viscous response that is far from the idealizedmodels, since they have yield strengths and shear thinning. A viscosity model deve-loped for powder injection-molding feedstock incorporates the yield stress into amodified Cross model as follows:

hm( _g, T) ¼ h0

1þ (h0 _g=t�)1�n þty

_g

where

h0 ¼ B expTb

T

� �

VISCOSITY MODEL FOR INJECTION-MOLDING FEEDSTOCK 379

In the preceding equation, the subscript m indicates the powder–binder mixtures. Thezero shear rate viscosity h0 corresponds to the Newtonian viscosity in the low shear-rate region and ignores the yield stress. This viscosity has a complex temperaturedependence, since the polymer viscosity decreases on heating and the solid–polymer ratio also changes due to a large difference in thermal expansion. Thestress parameter t� corresponds to the approximate stress needed to induce a transitionfrom Newtonian behavior into shear thinning behavior. The exponent n is themeasure of shear-thinning viscosity behavior. A term ty/g is added to the modifiedCross model for the yield stress.

B ¼ preexponent of temperature dependency, Pa . s


Tb ¼ reference temperature, K

n ¼ exponent of shear thinning, dimensionless


h0 ¼ zero shear-rate viscosity, Pa . s

hm ¼ viscosity of powder–binder mixture, Pa . s

ty ¼ feedstock yield stress, Pa

t� ¼ transition stress parameter, Pa.

VISCOSITY OF SEMISOLID SYSTEMS (Flemings 1991)

The viscosity h of a pore-free solid–liquid suspension, such as is encountered duringthixomolding or liquid-phase sintering, depends on the degree of intergrain bonding.These semisolid systems have a dramatic dependence on the amount of liquid and its dis-tribution in the compact. In the lowest-viscosity case, where there is no intergrainbonding, the viscosity depends on the solid–volume fractionf approximatelyas follows:

h ¼ ghLG2(ffiffiffitp� ffiffiffiffiffi

tYp

)

_g n(1� cf)2

where g is a geometric term, hL is the viscosity of the pure liquid at the equivalent temp-erature, G is the mean solid grain size, t is the applied stress, tY is the apparent yieldstrength, g ¼dg/dt is the shear rate, the value of the exponent n is typically between0.5 and 1.0 for slow strain rates, and c is typically near 1.6 for spherical grains, but canrange from 1.2 to 2.0, depending on the grain shape. In turn, the liquid viscositydepends on composition and temperature. Over limited temperature ranges, viscositymight vary with the temperature squared or, more commonly, is expressed with anArrhenius temperature dependence,

hL ¼ h0 expQ

RT

� �

380 CHAPTER V

where Q is an activation energy for viscous flow, h0 is an inherent viscosity term,R is the gas constant, and T is the absolute temperature. Pores influence the viscosity.Initially, the pores liberate liquid to reduce the viscosity. In sintering this increases thedensification rate, because there is also a capillary force associated with the pores;however, with pore closure there is an increase in resistance to rearrangement becausethat same capillary force creates solid–solid bonds that increase the viscosity. This beha-vior change determines how the semisolid system responds via the combination of capil-lary force and lubricated grain motion. Over time, grain–grain bonds form and this leadsto a time-dependent viscosity increase. This latter effect is accounted for by the yieldstress and linkages to the neck size between contacting grains. Thus, the formation ofa solid skeleton during processing quickly increases the in situ resistance to flow. Priorto the formation of a solid skeleton, viscous flow of the solid–liquid–vapor mixtureallows rapid densification. But, once the solid skeleton forms, further pore eliminationrelies on diffusion-controlled solution reprecipitation.


Q ¼ activation energy for viscous flow, J/mol (convenient units: kJ/mol)



c ¼ inverse critical solids loading, m3/m3 or dimensionless

g ¼ geometric constant, 1/(N1/2 . m . sn)

n ¼ the value of the exponent is from 0.5 to 1.0, dimensionless

t ¼ time, s

f ¼ volume fraction of solid, m3/m3 or dimensionless [0, 1]


g ¼ dg/dt ¼ shear strain rate, 1/s


h0 ¼ inherent viscosity, Pa . s

hL ¼ liquid viscosity, Pa . s

t ¼ applied stress, Pa

tY ¼ yield strength, Pa.

VISCOSITY OF SUSPENSION

See Suspension Viscosity.

VISCOSITY VARIATION WITH HYDROSTATIC PRESSURE

See Pressure Effect on Feedstock Viscosity.

VISCOSITY VARIATION WITH HYDROSTATIC PRESSURE 381

VISCOUS FLOW IN PRESSURE-ASSISTED SINTERING(Scherer 1986)

For glass, or other amorphous materials, sintering densification from an externalpressure is by viscous flow. A liquid cannot sustain a shear stress; thus, a viscousmaterial with no yield strength will densify in proportion to the effective stress,

df

dt¼ 3PE1

4h

where f is the fractional density, PE is the effective pressure, h is the system viscosity,and 1 is the volume fraction of porosity (1 ¼ 1 – f ). The effective pressure is calcu-lated based on the force concentration at the points of contact between particles. If theeffective pressure is high, then the inherent sintering densification rate is not signifi-cant: contrarily, if the effective pressure is low, then the net densification rate is thecombination of the pressure term and the inherent viscous-phase sintering rate.




t ¼ time, s




VISCOUS FLOW OF A LIQUID DROPLET (Clift et al. 1978)

During atomization of a melt, the liquid droplet exiting the atomizer is treated as aviscous liquid. The motion of an incompressible fluid that behaves as an idealizedNewtonian system is bounded by a free surface. The governing equations are takento be the Navier–Stokes momentum balance,

r@ v

@tþ rv � rv ¼ rgþr � T

with a continuity equation

r � v ¼ 0

In the preceding equations, r is the fluid density, v is the velocity vector, t is time,r is the divergence operator, g is the gravitational acceleration vector, and T is thetotal stress tensor. The total stress tensor as a function of pressure and velocity

382 CHAPTER V

gradients using the constitutive equation for a Newtonian fluid is given as follows:

T ¼ �PIþ h(rvþrvT )

where P is the pressure, I is the identity tensor, h is the fluid viscosity, and the super-script T denotes the transpose operator. For the cases where the fluid is bounded bya free surface such that,

n � v� @ xs

@t

� �¼ 0

with a parallel-force balance,

n � Tþ n(Pg � gLVk) ¼ 0

In the preceding equation, n denotes an outward-pointing unit vector normal to thefluid surface, @ x s/@t is the time derivative of the surface-position vector xs, Pg isthe pressure of the gas phase surrounding the fluid, gLV is the surface energy, andk is the mean curvature of the fluid surface. For moving-boundary problems with sig-nificant capillary effects, an appropriate nondimensionalization of the momentumequation can be obtained by normalizing the spatial coordinate system x with acharacteristic length R, the components of the stress tensor T and pressure P withgLV/R, the fluid-velocity field v with gLV/h, time t with Rh/gLV, and the surfacecurvature k with 1/R. Now the momentum equation takes on the following form:

Su@v�

@t�þ v� � r�v�

� �¼ Bo

g

gþr� � T�

where the asterisk denotes dimensionalized or normalized parameter, Su ¼ rgLVR/h2

is the Suratman number and Bo ¼ rR2g/gLV is the Bond number.


I ¼ identity tensor, dimensionless

P ¼ pressure, Pa

Pg ¼ pressure of the gas phase surrounding the fluid, Pa

P� ¼ P/(gLV/R) ¼ dimensionalized or normalized pressure, dimensionless

R ¼ characteristic atomization length scale, m


T ¼ superscript denoting transpose operator, dimensionless

T ¼ total stress tensor, Pa

T� ¼ T/(gLV/R) ¼ dimensionalized or normalized total stress tensor,dimensionless


g ¼ gravitational acceleration vector, 9.8 m/s2

n ¼ outward-pointing unit vector normal to the fluid surface, dimensionless

t ¼ time, s

VISCOUS FLOW OF A LIQUID DROPLET 383

t� ¼ t/(Rh/gLV) ¼ dimensionalized or normalized time, dimensionless

v ¼ velocity vector, m/s

v� ¼ v/(gLV/h)¼ dimensionalized or normalized velocity vector, dimensionless

x ¼ spatial vector coordinates, m

x s ¼ surface-position vector coordinates, m



k ¼ mean curvature of the fluid surface, 1/m

k� ¼ Rk ¼ normalized mean curvature of the fluid surface, dimensionless

r ¼ density of the liquid, kg/m3 (convenient units: g/cm3)

r ¼ divergence operator, 1/m

r� ¼ Rr ¼ normalized divergence operator, dimensionless.

VISCOUS FLOW SINTERING (Frenkel 1945)

The flow of a soft porous body, such as a heated glass powder, occurs in response toboth an internal and external stress. In sintering the internal stress comes from surfaceenergy, as driven by curvature variations in the microstructure. Over a limited temp-erature range, an amorphous material (polymer, glass, or metallic glass) has a vis-cosity h that varies with temperature dependence approximately as follows:

h ¼ h0 expQ

RT

� �

where Q is the apparent activation energy, h0 is the proportionality coefficient, T isthe absolute temperature, and R is the gas constant. Early during isothermalviscous-flow sintering the neck diameter X between particles of diameter D growsin proportion to the square root of the sintering time, according to the Frenkel relation,

X

D

� �2

¼ 3gSVt

Dh

where gSV is the surface energy, and t is the isothermal sintering time. According tothe viscous-flow concept, along with neck growth there is shrinkage during sinteringDL/L0, given as,

DL

L0¼ 3gSVt

4Dh

If the structure is isotropic, then the predicted sintered density rS for a compactstarting at a green density rG is given as,

rS ¼rG

(1� DL=L0)3

384 CHAPTER V

Here the final result is given in terms of density, but the expression is equally valid ifbased on fractional density.


L0 ¼ initial length, m






DL ¼ change in length, m

DL/L0 ¼ sintering shrinkage, m/m or dimensionless



h0 ¼ material constant or reference viscosity Pa . s


rS ¼ sintered density, kg/m3 (convenient units: g/cm3).

VISCOUS FLOW SINTERING OF GLASS

It is possible to further simplify the viscous-flow sintering model for the caseof glassy particles that do not have a yield stress. In the simplification, thecharacteristic length scale (which is usually assumed to be the particle size) is setto R and is often on the order of 1027 m to 1024 m. The glass-phase density r ison the order of 103 kg/m3, while the viscosity at the sintering temperature h is onthe order of 106 to 109 Pa . s, and the liquid–vapor surface energy gLV is on theorder of 0.1 J/m2. These conditions have a Suratman number, Su ¼ rgLVR/h2 andBond number Bo ¼ rR2g/gLV that can be approximated as Su � 10223 to 10210

and Bo � 1029 to 1023. Since Su� 1 and Bo� 1, it is possible to ignore the inertialand gravitational terms. Accordingly, the momentum equation becomes simply theStokes equation

r� � T� ¼ 0with

T� ¼ �P�Iþ [r�v� þ (r�v�)T ]

Assume an incompressible fluid and ignore the external gas pressure (effectivelyassuming vacuum sintering), then the force balance at the surface simplifies to give

n � T� � k�n ¼ 0

Note that all terms of the preceding equations are nondimensional. The capillary-driven flow during viscous sintering is then described by these three equations.

VISCOUS FLOW SINTERING OF GLASS 385


I ¼ identity tensor, dimensionless

P ¼ pressure, Pa

P� ¼ P/(gLV/R) ¼ normalized pressure, dimensionless



T ¼ total stress tensor, Pa

T� ¼ T/(gLV/R) ¼ normalized total stress tensor, dimensionless


v ¼ velocity vector, m/s

v� ¼ v/(gLV/h) ¼ normalized velocity vector, dimensionless



k ¼ mean curvature of the fluid surface, 1/m

k� ¼ Rk ¼ normalized mean curvature of the fluid surface, dimensionless

r ¼ density of the liquid, kg/m3 (convenient units: g/cm3)

r ¼ divergence operator, 1/m

r� ¼ Rr ¼ normalized divergence operator, dimensionless.

[Also see Viscous Flow of a Liquid Droplet.]

VISCOUS-PHASE SINTERING

See Viscosity of Semisolid Systems.

VISCOUS SETTLING

See Stokes’ Law.

VISCOUS SINTERING, VISCOUS-PHASE SINTERING

See Two-particle Viscous-flow Sintering.

VOIGT MODEL

See Viscoelastic Model for Powder–Polymer Mixtures

386 CHAPTER V

VOLUME-BASED PARTICLE SIZE


VOLUME DIFFUSION


VOLUME DIFFUSION–CONTROLLED CREEP DENSIFICATION

See Nabarro–Herring Creep-Controlled Pressure-assisted Densification.

VOLUME DIFFUSION–CONTROLLED CREEP DENSIFICATION 387

W

WASHBURN EQUATION (Washburn 1921; Zhmud et al. 2000)

Wetting liquids penetrate small pores and capillary tubes due to the meniscus pressuregradient. An understanding of the relation between pressure, penetration, and poresize is used in models for infiltration, impregnation, and composite fabrication.Also, mercury porosimetry derives the relationship between pressure and pore-sizedistribution from the Washburn equation. This relationship gives the capillarypressure change DP associated with a small tube of diameter dP as follows:

DP ¼ � 4gLV cos u

dP

In practice, this equation is used to relate measured pressure to the pore sizeand contact angle u, based on the solid–liquid surface tension gLV. The pore isassumed to be a uniform-diameter capillary tube. Many corrections, variations, andrefinements have been published since this model was first delineated in 1921.


DP ¼ capillary pressure, Pa

gLV ¼ liquid–vapor surface tension, J/m2


WATER-ATOMIZATION PARTICLE SIZE

High water pressure, or high water velocity, causes a decrease in the median particlesize D produced during water atomization. A simple empirical relation betweenatomization conditions and the particle size is expressed as follows:

D ¼ �b ln P=P0ð ÞV sina


389

where b is a constant that incorporates both material and atomizer effects, P is thewater pressure at the outlet, P0 is a reference water pressure, V is the water velocity,anda is the angle between the melt stream and the water jets. Typically, thewater velocityincreases as pressure is increased, yielding a smaller particle size at higher pressures, butthere is a lower limit at about the 1-mm size range for many metallic systems.


P ¼ water pressure, Pa (convenient units: MPa)

P0 ¼ reference water pressure (one atmosphere), Pa (convenient units: MPa)

V ¼ water velocity, m/s

a ¼ angle between the melt stream and the water jets, rad (convenient units:degree)

b ¼ kinetic constant, m2/s.

WATER IMMERSION DENSITY

See Archimedes Density.

WEBER NUMBER

One dimensionless parameter used in powder production by gas atomization is theWeber number We. This number can be thought of as a measure of the relativeimportance of the fluid’s inertia compared to its surface tension. The Webernumber We depends on the gas velocity V, gas density rG, surface energy of themelt gLV, and melt ligament diameter as discharged from the atomizer dL,

We ¼ rGV2dL

2gLV

Most gas atomization is performed at Weber numbers We below 1000.


We ¼ Weber number, dimensionless

dL ¼ melt ligament diameter, m (convenient units: mm or mm)

gLV ¼ liquid–vapor melt surface energy, J/m2

rG ¼ gas density, kg/m3 (convenient units: g/cm3)

WEIBULL DISTRIBUTION (Weibull 1951; Morrell 1989; Green 1998)

In the fracture of brittle materials, especially sintered ceramics, carbides, and greenmetal powder compacts, the Weibull distribution provides a representation of the vari-ations in strength. When the compact lacks ductility, fracture strength has a variation

390 CHAPTER W

that reflects the underlying distribution in defects. The statistical treatment advancedby Weibull assumes there is a characteristic strength for the material and there is adistribution to that strength that has its origin in the random-defect population.Often the defects result from the manufacturing operation. The treatment of strengthdata according to the Weibull distribution involves determination of both the Weibullmodulus m and the characteristic strength s0. Accordingly, the cumulative probabilityof failure F at a stress s is given by,

F ¼ 1� exp � V

V0

s� sU

s0

� �m� �

where V is the actual volume of the sample, s is the applied stress over that volume V,sU is a lower-limit stress needed to cause failure (known as the proof stress, oftenassumed to be zero), V0 is the volume used in the testing employed to measure theWeibull modulus m, and s0 is the characteristic strength. The exponent m is ameasure of the distribution width and inherently the process variability. If N testsare performed, then the failure-stress results can be ranked in ascending order fromlowest failure stress to highest, and each is then assigned a probability of failure.Typically, the preceding equation is rearranged and presented on a double logarithmplot of 1/(1 2 F ) versus the logarithm of s, and the slope is used to calculate m,assuming sU is zero. Figure W1. is a plot of the double logarithm of the failure prob-ability versus strength, showing the Weibull distribution for glass and aluminasamples. As a rule of thumb, accurate results require at least 40 samples for thestrength distribution. Typically, a narrow range of failure strengths is desirable,

Figure W1. The Weibull distribution is illustrated in terms of the cumulative failure prob-ability and the failure strength using data from glass and alumina ceramics.

WEIBULL DISTRIBUTION 391

which gives a high Weibull modulus. Because of the volume term, higher apparentstrengths are obtained from smaller samples, since the probability of a large flawexisting in the sample decreases with the sample volume. This says that smallobjects are always stronger when compared to large objects. Accordingly, the appar-ent strength of brittle sintered samples can appear to increase simply by testingsmaller samples.

F ¼ cumulative probability of failure, dimensionless [0, 1]

N ¼ number of tests, dimensionless

V ¼ actual sample volume, m3 (convenient units: mm3)

V0 ¼ volume used in measuring the Weibull modulus m, m3 (convenientunits: mm3)

m ¼ Weibull modulus, dimensionless

s ¼ failure stress, Pa (convenient units: MPa)

s0 ¼ characteristic strength, Pa (convenient units: MPa)

sU ¼ proof stress, Pa (convenient units: MPa).

WETTING ANGLE (Liu and German 1996)

The wetting angle, u, is also known as the contact angle. As shown in Figure W2,the wetting angle is defined by the horizontal equilibrium based on treating surfaceenergies as vectors,

gSV ¼ gSL þ gLV cos u

where gSV is the solid–vapor surface energy, gSL is the solid–liquid surface energy,and gLV is the liquid–vapor surface energy. Wetting is generally observed as thewetting or contact angle decreases. This equation is also known as Young’s equation,and is actually invalid since it fails to include gravitational flattening and spreadingeffects. For example, advancing and receding contact angles differ, and because of

Figure W2. The description of the wetting angle in terms of the three phase equilibriumbetween a liquid droplet resting on a horizontal solid surface. The definition of the wettingangle is based on the horizontal resolution to the three surface interfacial vectors.

392 CHAPTER W

gravity the contact angle varies with droplet size. There is no resolution of the verticalforces. Even so, it is a widely applied concept used in treating wetting problems inliquid-phase sintering and similar areas.




u ¼ wetting angle, rad (convenient units: degree).

WICKING (German 1987)

One means of debinding a component, such as is formed by powder injectionmolding, is by embedding it in packing powder of a smaller particle size. Whenthe packed component is heated, the binder melts and flows into the packingpowder due to capillarity. Although not all of the binder can be removed bywicking, it is a very effective means of accelerating the first stage of debinding.Likewise, wicking phenomena are applicable to other situations of fluid flowinginvolving porous components, such as impregnation and infiltration. Assuming thepores can be treated as small capillary tubes allows calculation of the wicking timeas a function of the geometric and viscous attributes. The depth of fluid flow bywicking h depends on the pore diameter dP, fluid viscosity h, and time t as follows:

h ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffigLVtdP

4h

s

where gLV is the liquid–vapor surface energy. Higher viscosity binders and smallerpores require more time for wicking. In the opposite situation, where an externalpolymer is being impregnated into the porous solid, the surface energy is supple-mented by the external pressure.


h ¼ depth of wicking, m (convenient units: mm)

t ¼ time, s


h ¼ fluid viscosity, Pa . s.

WORK HARDENING

See Strain Hardening.

WORK OF SINTERING

See Master Sintering Curve.

WORK OF SINTERING 393

X

X-RAY LINE BROADENING

See Scherrer Formula.


395

Y

YIELD STRENGTH IN VISCOUS FLOW

See Bingham Viscous-flow Model.

YIELD STRENGTH OF PARTICLE–POLYMER FEEDSTOCK(D. M. Liu and Tseng 1998; Flatt and Bowen 2007)

A powder dispersed in a binder or fluid phase, often a polymer, is called a feedstock.Most models for injection molding, tape casting, or extrusion assume the feedstockrheology can be described by a simple Newtonian fluid model. However, mostfeedstocks have a yield strength tY, otherwise, they would flow and fail to holdshape. The yield strength depends on the solids loading f as follows:

tY ¼ C1f

A� f� C2 ¼ C1R� C2

where A, C1, and C2 are constants that depend on the polymer, powder, and tempera-ture. The cluster R ¼ f/(A 2 f) is known as a flow-resistance parameter. Whenpowder injection-molding feedstock is measured at the molding temperature,the observed yield strength is often under 1 kPa. A related derivation includesthe solids loading at the percolation threshold fP and the critical solids loadingfC as follows:

tY ¼ C3f(f� fP)2

fC(fC � f)

where C3 is a constant that depends on the polymer, powder, and temperature.

A ¼ constant, dimensionless

C1, C2, and C3 ¼ material system-specific constants, Pa


397

R ¼ f/(A 2 f) ¼ flow-resistance parameter, dimensionless

f ¼ fractional solids loading, dimensionless [0, 1]

fC ¼ critical fractional solids loading, dimensionless [0, 1]

fP ¼ fractional solids loading at the percolation limit, dimensionless[0,1]

tY ¼ yield strength, Pa.

[Also see Bingham Viscous-flow Model.]

YOUNG’S EQUATION

See Contact Angle and Wetting Angle.

YOUNG’S MODULUS

See Elastic Modulus.

398 CHAPTER Y

Z

ZENER RELATION (Harun et al. 2006)

It is typical for grains to grow at high temperatures. However, an otherwise movinggrain boundary can be restrained by a dispersion of pores or inclusions. Thispinning effect is most pronounced with a low dihedral angle that induces propor-tionally more coverage of the brain boundary for a given volume fraction of thesecond phase or pores. Under the isothermal conditions involved in normalgrain growth, the grain size and inclusion or pore size are related by the Zenerrelation. This relation is applied to the final stage of sintering densification as ameans to account for the role of residual pores on grain growth, assuming thepores remain at grain boundaries. The pinning force is calculated in terms of theexcess grain-boundary area, due to grain-boundary bowing, needed to breakaway from the pore or inclusion. Figure Z1 plots two-dimensional variants, onewith a smaller dihedral angle and the second with no dihedral angle. Mostmodels for grain-boundary pinning assume a spherical pore or inclusion, althoughthis is not often valid. Even so, the concept attributed to Zener shows how pores orinclusions apply a pinning force to a moving grain boundary. Assuming a randomdispersion of the second phase consisting of spherical pinning agents attached tothe grain boundaries, then the number of inclusions or pores per area of grainboundary N2 is given as,

N2 ¼6V2

pd22

where V2 is the fractional volume of pores or second phase, and d2 is the size ofthe pore or second-phase inclusion. The resulting pinning stress sP on the grainboundary is determined by the force needed to pull away from the pores andthe population of grain-boundary pores,

sP ¼3V2gSV

2d2


399

where gSV is the solid–vapor surface energy. Equating the driving stress for graingrowth to the pinning stress gives the generalized Zener relation,

G ¼ 3d2

8V2

where G is the grain size. With respect to final-stage sintering, this relation inher-ently assumes that the pores are coupled to the grain boundaries. Some pore–boundary combinations (depending on grain-boundary orientation) are not stableand the pores separate from the grain boundaries during sintering, leading to aloss of effectiveness in retarding grain growth. Hence, the Zener relation is abest-case condition for final-stage sintering, and the grain size is often largerthan predicted.

Figure Z1. The Zener relation describes the pinning effect from a pore or inclusion and thegrain-boundary structure. The cases for final-stage sintering are deposited where the pore inthe upper figure has no dihedral angle and the pore in the lower picture has a small dihedralangle. If the grain boundary breaks away from the pore, then a spherical pore forms and thegrain boundary is no longer impeded with respect to grain growth.

400 CHAPTER Z


N2 ¼ number of inclusions or pores per grain-boundary area, 1/m2

V2 ¼ fractional volume of second phase, m3/m3 or dimensionless [0,1]

d2 ¼ size of the second phase, m (convenient units: mm)

gSV ¼ solid-vapor surface energy, J/m2

sP ¼ pinning stress, Pa.

ZETA POTENTIAL (Fuerstenau and Somesundaran 2003)

A nonconductive particle will build up a charged surface when suspended in asolution. When the solution is polarized, the charged particle will move.Alternatively, when the solution is moving through a packing of particles that cancharge, a voltage is generated. The surface charge on the particle divided by the dis-tance between the surface and the shear plane for the solution’s local “ionic atmos-phere” is termed the zeta potential. In practice, the zeta potential is measured usinga streaming potential to determine the surface charge on nonconductive particles in anelectrolyte. The test solution is pumped through a porous plug composed of the par-ticles, and the potential developed across the plug is measured. This electricpotential E is termed the streaming potential and is directly related to the zeta poten-tial z as follows:

z ¼ 4phEl

xP

where h is the fluid viscosity, l is the specific conductivity, x is the permittivity, andP is the pressure across the powder bed. The streaming approach is not suitable formeasuring the absolute value of the zeta potential, but it provides a test to evaluatehow various species can alter the powder behavior.

E ¼ streaming potential, V

P ¼ pressure across the bed, Pa

x ¼ permittivity, F/m


l ¼ specific conductivity, 1/(V . m)

z ¼ zeta potential, V.

ZETA POTENTIAL 401

APPENDIXConstants and Conversion Factors:System of International Units

Prefixes

c (centi) ¼ 1022

G (giga) ¼ 109

k (kilo) ¼ 103

m (milli) ¼ 1023

M (mega) ¼ 106

n (nano) ¼ 1029

m (micro) ¼ 1026

Fundamental Units

Amount ¼ mol (mole)

Current ¼ A (ampere)

Length ¼ m (meter)

Luminous intensity ¼ cd (candela)

Mass ¼ kg (kilogram)

Plane angle ¼ rd (radian)

Solid angle ¼ sr (steradian)

Temperature ¼ K (Kelvin)

Time ¼ s (second)

Derived Units

V (ohm) ¼ V/A

bar (bar) ¼ 0.1 MPa


403

C (coulomb) ¼ A.s

F (farad) ¼ A.s/V

h (hour) ¼ 3600 s

H (henry) ¼ V.s/A

Hz (hertz) ¼ 1/s

J ( joule) ¼ N.m ¼ kg.m2/s2

L (liter) ¼ 1023 m3

lm (lumen) ¼ cd.sr

min (minute) ¼ 60 s

N (newton) ¼ kg.m/s2

Pa (pascal) ¼ N/m2 ¼ kg/(m.s2)

S (siemens) ¼ 1 A/V

t (ton) ¼ 1000 kg

T (tesla) ¼ V.s/m2

V (volt) ¼ J/C

W (watt) ¼ J/s

Wb (weber) ¼ V.s

8C (Celsius) ¼ K 2 273

Important Constants

Acceleration of gravity ¼ 9.8 m/s2

Atomic mass unit ¼ 1.661.10224 g

Avogadro’s number ¼ 6.022.1023 molecules

Bohr magneton ¼ 9.27.10224 A.m2

Boltzmann’s constant ¼ 1.381.10223 J/K

Electric permittivity of vacuum ¼ 8.854.10212 C/V

Electron mass ¼ 9.11.10228 g

Elementary charge ¼ 1.602.10219 C

Faraday’s constant ¼ 9.65.104 C/mol

Gas volume at standard temperature and pressure ¼ 0.0224 m3

Lorenz number ¼ 2.45.1028 J2/(C2 .K2)¼ 2.45.1028 W/(S .K2)

Permeability of vacuum ¼ 7.958.105 H/m

Planck’s constant ¼ 6.626.10234 J .s

Speed of light ¼ 2.998.108 m/s

Stefan–Boltzmann constant ¼ 5.67.1028 J/(m2.s .K4)

Universal gas constant ¼ 8.314 J/(mol .K)

404 APPENDIX

Length Conversions

1 m ¼ 39.4 in (inch) 1 inch ¼ 0.0254 m

1 m ¼ 3.28 ft (foot) 1 foot ¼ 0.3048 m

1 m ¼ 1.09 yd (yard) 1 yard ¼ 0.914 m

1 cm ¼ 0.394 in (inch) 1 inch ¼ 2.54 cm

1 mm ¼ 0.0394 in (inch) 1 inch ¼ 25.4 mm

1 mm ¼ 39.4 min 1 minch ¼ 0.0254 mm

1 nm ¼ 10 A (angstrom) 1 angstrom ¼ 0.1 nm

Area and Volume Conversions

1 cm2 ¼ 0.155 in2 (square inch) 1 in2 ¼ 6.45 cm2

1 m2 ¼ 1550 in2 (square inch) 1 in2 ¼ 0.000645 m2

1 cm3 ¼ 0.061 in3 (cubic inch) 1 in3 ¼ 16.38 cm3

1 m3 ¼ 35 ft3 (cubic foot) 1 ft3 ¼ 0.0283 m3

1 L ¼ 1000 cm3 (cubic centimeter) 1 cm3 ¼ 0.001 L

1 L ¼ 0.264 gal (gallons) 1 gal ¼ 3.79 L

1 L ¼ 1.06 qt (quart) 1 qt ¼ 0.946 L

Angle Conversions

1 rad ¼ 57.3 deg (degree) 1 deg ¼ 0.0174 rad

1 rad ¼ 0.159 rev (revolutions) 1 rev ¼ 6.28 rad

Amount of Substance Conversion

1 mol ¼ 6.022 . 1023 molecules 1 molecule ¼ 1.66 . 10224 mol

Density Conversions

1 Mg/m3 ¼ 1 g/cm3 1 g/cm3 ¼ 1 Mg/m3

1 g/cm3 ¼ 0.0361 lb/in3 (pound per inch3) 1 lb/in3 ¼ 27.68 g/cm3

1 g/cm3 ¼ 0.578 oz/in3 (ounce per inch3) 1 oz/in3 ¼ 1.73 g/cm3

1 kg/m3 ¼ 1023 g/cm3 1 g/cm3 ¼ 1000 kg/m3

Temperature Conversion

To convert K to 8F (Fahrenheit), multiply by 1.8 then subtract 459.48FTo convert 8F (Fahrenheit) to K, add 459.48F then multiply by 0.555

To convert 8C to 8F (Fahrenheit), multiply by 1.8 then add 328FTo convert 8F (Fahrenheit) to 8C, subtract 328F then multiply by 0.555

APPENDIX 405

Heating- and Cooling-rate Conversions

To convert 8C/min to 8F/min, multiply by 1.8

1 K/s ¼ 18C/s ¼ 1.88F/s

1 K/min ¼ 1.88F/min

Mass Conversions

1 g ¼ 5 ct (carat) 1 ct ¼ 0.2 g

1 g ¼ 0.03215 tr oz (troy ounce) 1 tr oz ¼ 31.103 g

1 g ¼ 0.035 oz (ounce) 1 oz ¼ 28.349 g

1 kg ¼ 2.2 lb (pound) 1 lb ¼ 454 g

1 Mg ¼ 1.1 ton (2000 pounds) 1 ton ¼ 907.2 kg

Force Conversions

1 N ¼ 0.1020 kg force 1 kg force ¼ 9.807 N

1 N ¼ 105 dyne 1 dyne ¼ 1025 N

1 N ¼ 0.225 lb force (pound force) 1 lb force ¼ 4.44 N

Pressure, Stress, Strength Conversions

1 Pa ¼ 0.0075 torr (mm of mercury) 1 torr ¼ 133 Pa

1 kPa ¼ 4.015 inch of water 1 in H2O ¼ 0.249 kPa

1 kPa ¼ 0.295 inch of mercury 1 in Hg ¼ 3.386 kPa

1 Pa ¼ 10 dyne/cm2 1 dyne/cm2 ¼ 0.1 Pa

1 kPa ¼ 0.0102 kg/cm2 1 kg/cm2 ¼ 98.07 kPa

1 kPa ¼ 0.145 psi (pounds per in2) 1 psi ¼ 6.895 kPa

1 MPa ¼ 9.87 atm (atmosphere) 1 atm ¼ 0.1020 MPa

1 MPa ¼ 145 psi (pounds per in2) 1 psi ¼ 0.006895 MPa

1 MPa ¼ 0.145 kpsi (1000 psi) 1 kpsi ¼ 6.895 MPa

1 GPa ¼ 145 kpsi 1 kpsi ¼ 0.006895 GPa

1 GPa ¼ 0.145 Mpsi 1 Mpsi ¼ 6.895 GPa

Energy Conversions

1 J ¼ 9.48.1024 btu (British thermal unit) 1 btu ¼ 1055 J

1 J ¼ 0.737 ft . lb (foot pound) 1 ft . lb ¼ 1.356 J

1 J ¼ 0.239 cal (calorie) 1 cal ¼ 4.187 J

1 J ¼ 107 erg 1 erg ¼ 1.1027 J

1 J ¼ 2.8 .1027 kW.h (kilowatt hour) 1 kW.h ¼ 3.6 .106 J

406 APPENDIX

1 J ¼ 6.24.1018 eV (electron volt) 1 eV ¼ 1.60.10219 J

1 J ¼ 3.725.1027 hp.h (horsepower hour) 1 hp.h ¼ 2.686.106 J

1 J ¼ 1 W.s (watt second) 1 W.s ¼ 1 J

1 J ¼ 1 V.C (volt coulomb) 1 V.C ¼ 1 J

1 kJ ¼ 0.239 kcal (kilocalorie) 1 kcal ¼ 4.186 kJ

Power Conversions

1 W ¼ 0.737 ft . lb/s (foot pound per s) 1 ft . lb/s ¼ 1.356 W

1 W ¼ 1.34.1023 hp (horsepower) 1 hp ¼ 0.746 kW

1 W ¼ 1 V.A (volt amp) 1 V.A ¼ 1 W

1 W ¼ 3.412 btu/h (British thermal unit/h) 1 btu/h ¼ 0.291 W

Thermal Conversions

1 J/(kg .K) ¼ 2.39.1024 btu/(lb .8F) (British thermal unit per pound per degreeFahrenheit)

1 btu/(lb .8F) ¼ 4184 J/(kg.K)

1 J/(kg .K) ¼ 2.39.1024 cal/(g.8C) (calorie per gram per degree Celsius)

1 cal/(g.8C) ¼ 4184 J/(kg.K)

1 W/m2 ¼ 0.860 kcal/(m2.h) (kilocalorie per square meter per hour)

1 kcal/(m2.h) ¼ 1.163 W/m2

1 W/m2 ¼ 0.317 btu/(ft2 .h) (British thermal units per square foot per hour)

1 btu/(ft2 .h) ¼ 3.155 W/m2

1 W/(m.K) ¼ 0.578 btu/(ft .h .8F) (British thermal unit per foot per hour perdegree Fahrenheit)

1 btu/(ft .h .8F) ¼ 1.73 W/(m.K)

1 W/(m.K) ¼ 2.39.1023 cal/(cm.s .8C) (calorie per centimeter per second perdegree Celsius)

1 cal/(cm.s .8C) ¼ 418.4 W/(m.K)

Viscosity Conversions

1 Pa.s ¼ 1 kg/(m.s) 1 kg/(m.s) ¼ 1 Pa.s

1 Pa.s ¼ 10 P (poise) 1 P ¼ 0.1 Pa.s

1 Pa.s ¼ 103 cP (centipoise) 1 cP ¼ 1023 Pa.s

Stress-intensity Conversion

1 MPa. ffiffiffiffimp¼ 0.91 kpsi .

ffiffiffiffimp

(kilopounds per square inch times square-root inch)

1 kpsi .ffiffiffiffimp¼ 1.1 MPa. ffiffiffiffi

mp

APPENDIX 407

Electrical Conversions

1 J ¼ 6.24.1018 eV (electron volt) 1 eV ¼ 1.60.10219 J

1 MJ ¼ 0.2778 kW.h 1 kW.h ¼ 3.6 MJ

1 W ¼ 1 V.A 1 V.A ¼ 1 W

Magnetic Conversions

1 T ¼ 104 G (gauss) 1 G ¼ 1.1024 T

1 A/m ¼ 1.257.1022 Oe (oersted) 1 Oe ¼ 79.55 A/m

1 Wb ¼ 108 Mx (maxwell) 1 Mx ¼ 1.1028 Wb

Computer Conversions

1 bit ¼ 0.125 byte 1 byte ¼ 8 bit

1 Mbyte ¼ 1.04.105 byte 1 Gbyte ¼ 1.07.109 byte

408 APPENDIX

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Handbook of Mathematical Relations in Particulate Materials Processing

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