Principles of Solidification

Martin Eden Glicksman

Principles of Solidification

An Introduction to Modern Castingand Crystal Growth Concepts

123

Martin Eden GlicksmanMaterials Science and Engineering DepartmentUniversity of FloridaGainesville, FL 32611-6400, USAmglic@mse.ufl.edu

ISBN 978-1-4419-7343-6 e-ISBN 978-1-4419-7344-3DOI 10.1007/978-1-4419-7344-3Springer New York Dordrecht Heidelberg London

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To my wife, Lucinda

Preface

How many have truly benefited from my classes I cannotjudge. But I have the consolation that one person has learnedmuch thereby, and that is me.

Ludwig Boltzmann, Vienna, 1905 [1]

This book is a distillation based on more than three decades of teaching universitycourses and providing professional seminars on solidification, casting and weldingof metals, and crystal growth. The courses themselves were offered for many yearsat Rensselaer Polytechnic Institute and, most recently, at the University of Florida.Their purpose, consistently, was to present to a variety of engineering and sciencestudents a logical progression of the essential elements of materials science relevantto molten phases and processes leading to their crystallization. This text providesa comprehensive survey of scientific and engineering fundamentals for understand-ing crystallization processes, dwelling especially on applications to pure materi-als, metallic alloys, oxides, semiconductors, and polyphase systems. The didacticapproach adhered to derives in large measure from the author’s personal lecturenotes that were prepared annually for one-semester courses populated by advancedundergraduate and graduate students in materials science, chemical and mechanicalengineering, geology and physics. This book is designed for teaching this group,as well as for professionals interested in specific topics or exposure to integratedaspects of solidification and crystal growth.

The development of most chapters included herein favors, where appropriate, aquantitative approach, augmented by scientific descriptions anchored in materialsscience and condensed matter physics. Familiarity with the calculus, ordinary andpartial differential equations, and at least introductory chemical thermodynamics isassumed, all of which will prove helpful to most readers. Due care has been takento provide the reader comprehensive, logical developments supported by citationsof pertinent research papers and helpful reviews. Although the literature on solid-ification and crystal growth still lacks a consistent, canonical terminology, I haveendeavored, with some personal biases, to apply best practices throughout, as Iobserved their application by other researchers and thoughtful authors attentive tothis field.

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The text’s present author, in various ways, has also attempted to mirror theapproach and logical structure so successfully used in Professor Bruce Chalmers’shighly regarded monograph by the same title, published almost 50 years ago [2],while Dr. Chalmers was Gordon Mackay Professor of Metallurgy at Harvard Uni-versity. I used Professor Chalmers’s book when first teaching the subject of solid-ification and crystal growth; his book was reprinted for my classes in later yearsthrough the gracious permission of his estate. That work remained our backgroundtext, as I expanded some topics using my personal notes to up-date them or to intro-duce new ones. The present book, as already mentioned, resulted from those serialteaching efforts, through steady improvements based on feedback from students andcolleagues, and progress in the field. This book is written with the express desirethat it recaptures at least some of the spirit and purpose originally stated by BruceChalmers, namely,

. . . to provide a critical review of the state of knowledge and understanding of the processof solidification, defined for this purpose as the discontinuous change of state from liquidto crystalline solid.

Professor Chalmers’s personal charm, his many research contributions, his uniquetechnical and expository style, and, of course, his book itself, were mainstays inthe early development of solidification science. It is the author’s further hope thatthe current monograph will continue to assist students and practicing scientists andengineers in obtaining a firmer grasp of this interesting and important field, whichhas continued its development at a brisk pace ever since the publication of ProfessorChalmers’s, Principles of Solidification.

I owe thanks to former students for their feedback each year that I taught this sub-ject, which collectively helped shape the present form and content of this volume.My deepest appreciation extends to my dear friend Professor Markus Rettenmayr,Friedrich Schiller University, Jena, Germany, who provided numerous useful com-ments, alternative perspectives, and detailed suggestions for improving the clarityand consistency of the book, and who was willing to read critically an early draftof this work; to my friend and collaborator Professor Paulo Rios, Universidad Fed-eral de Volta Redonda, Brazil, for his encouragement, discussions, and unflagginginterest in my writing this book; and to my friend and colleague Professor DiranApelian, Worcester Polytechnic Institute, who successfully located out-of-print ref-erence materials that proved so useful.

Support provided to the author through the Florida twenty-first Century ScholarProgram, and the warm encouragement and interest always received from ProfessorKevin Jones, Chairman, Department of Materials Science & Engineering, and fromthe College of Engineering, University of Florida, are gratefully acknowledged andwere much appreciated during many months of writing.

Finally, my heartfelt appreciation and love extend to my wife, Lucinda, whoseunderstanding, forbearance and tolerance of the lengthy and often intrusive process

Preface ix

of book writing, along with her provisioning countless nourishing snacks and mealsthat kept me going through the writing of this work.

Gainesville, FL Martin Eden Glicksman

References

1. E. Broda, Ludwig Boltzmann, Man · Physicist · Philosopher, Ox Bow Press, Woodbridge, CT,1983, p. 102.

2. B. Chalmers, Principles of Solidification, Wiley, New York, NY, 1964.

Contents

Part I Introductory Aspects

1 Crystals and Melts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Melts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Viscosity of Melts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Comparison of Crystalline and Melt Structures . . . . . . . . . . . . . . . 8

1.3.1 RDFs and Local Atomic Densities . . . . . . . . . . . . . . . . 91.3.2 Radial Distribution Functions (RDFs) . . . . . . . . . . . . . . 10

1.4 Structure of Melts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.1 RDFs in Monatomic Melts . . . . . . . . . . . . . . . . . . . . . . . 101.4.2 RDFs in Molecular Melts . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Theoretical Estimates of RDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.5.1 Atomic Coordination in Melts . . . . . . . . . . . . . . . . . . . . 151.5.2 Atomic Packing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.5.3 Bonding in Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.5.4 Influence of Temperature on RDFs . . . . . . . . . . . . . . . . 19

1.6 Theories of Melting and the Liquid State . . . . . . . . . . . . . . . . . . . . 201.6.1 Lattice Vibrational Instability . . . . . . . . . . . . . . . . . . . . . 201.6.2 Hole Theories of Liquids . . . . . . . . . . . . . . . . . . . . . . . . 21

1.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Thermodynamics of Crystal-Melt Phase Change . . . . . . . . . . . . . . . . . . 272.1 Enthalpy: Heat and Work Exchanges in Solid-Liquid

Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Comparison of Terms in ΔH f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3 Co-existence of Solids and Melts . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.1 Solid-Melt Critical Point . . . . . . . . . . . . . . . . . . . . . . . . . 342.4 Thermal Supercooling and Metastability . . . . . . . . . . . . . . . . . . . . 34

2.4.1 Characteristic Melt Supercooling . . . . . . . . . . . . . . . . . . 372.5 Free Energy Changes During Freezing and Melting . . . . . . . . . . . 38

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2.5.1 Reversible Solidification . . . . . . . . . . . . . . . . . . . . . . . . . 382.5.2 Irreversible Solidification . . . . . . . . . . . . . . . . . . . . . . . . 39

2.6 Principal Types of Binary Alloy Solidification . . . . . . . . . . . . . . . 412.7 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3 Thermal Concepts in Solidification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.1 Near-Equilibrium Freezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.1.1 Preliminary Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 533.1.2 Energy Balances Within a Freezing System . . . . . . . . . 563.1.3 Energy Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2 Energy Balances at Crystal-Melt Interfaces . . . . . . . . . . . . . . . . . . 603.2.1 Stefan’s Interface Energy Balance . . . . . . . . . . . . . . . . . 603.2.2 Fourier’s Law of Heat Conduction . . . . . . . . . . . . . . . . . 623.2.3 Thermal Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.2.4 Newton’s Law of Cooling: Heat Transfer Coefficients 653.2.5 Interfacial Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4 Solidification of Pure Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.1 Quasi-static Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.1.2 Analysis of 1-Dimensional Freezing . . . . . . . . . . . . . . . 714.1.3 Quasi-static Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.1.4 Solidification of a Circular Cylinder . . . . . . . . . . . . . . . 76

4.2 Moving Boundary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.2.2 Conduction-Advection . . . . . . . . . . . . . . . . . . . . . . . . . . 814.2.3 Neumann’s Solution: Semi-infinite Freezing . . . . . . . . 85

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Part II Macrosegregation

5 Solute Mass Balances: Macrosegregation . . . . . . . . . . . . . . . . . . . . . . . . . 1015.1 Local and Global Interfacial Equilibrium . . . . . . . . . . . . . . . . . . . . 101

5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.1.2 The Distribution Coefficient, k0 . . . . . . . . . . . . . . . . . . . 103

5.2 Solute Rejection at the Solid–Liquid Interface . . . . . . . . . . . . . . . 1065.2.1 Interfacial Solute Balance . . . . . . . . . . . . . . . . . . . . . . . . 106

5.3 Gulliver–Scheil Macrosegregation Theory . . . . . . . . . . . . . . . . . . . 1095.3.1 Determination of the Distribution Coefficient . . . . . . . 110

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5.4 Macrosegregation with Solid–State Diffusion . . . . . . . . . . . . . . . . 1135.4.1 Solid-State Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.4.2 Mass Balances with Solid–State Diffusion . . . . . . . . . . 1145.4.3 Time Dependence of Fraction Solidified . . . . . . . . . . . . 1165.4.4 Solutal Fourier Number . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.5 Limits of the Brody–Flemings Solute Balance . . . . . . . . . . . . . . . 1205.6 Binary Alloy Segregation Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.6.1 Gulliver–Scheil Segregation: Solubility Limits . . . . . . 1215.6.2 Influence of the Fourier Number . . . . . . . . . . . . . . . . . . 122

5.7 Purification via Freezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.7.1 Fractional Crystallization . . . . . . . . . . . . . . . . . . . . . . . . 1235.7.2 Chemical Purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.7.3 Why Pure Materials? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.7.4 Cyclic Unidirectional Solidification . . . . . . . . . . . . . . . 125

5.8 Zone Refining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.8.1 Single Zone Pass: Pfann’s Equation . . . . . . . . . . . . . . . 1295.8.2 Multipass Zone Refining . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6 Plane-Front Solidification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1456.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.1.1 Steady-State Macrosegregation . . . . . . . . . . . . . . . . . . . 1466.1.2 Steady-State Plane-Front Freezing . . . . . . . . . . . . . . . . 1466.1.3 Solute Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.2 Transient Macrosegregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1546.2.1 Initial Transients in Infinite Systems . . . . . . . . . . . . . . . 1556.2.2 Transients in Finite Systems . . . . . . . . . . . . . . . . . . . . . . 1586.2.3 Final Segregation Transient . . . . . . . . . . . . . . . . . . . . . . 159

6.3 Numerical Studies in Finite Systems . . . . . . . . . . . . . . . . . . . . . . . 1606.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

7 Composition Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1637.1 Convection in Freezing Melts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.1.1 Mixing Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1637.1.2 The BPS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1667.1.3 Solution to the BPS Transport Equations . . . . . . . . . . . 167

7.2 Effective Distribution Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 1717.2.1 Definition and Role of Effective Distribution

Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1717.2.2 Results from BPS Theory . . . . . . . . . . . . . . . . . . . . . . . . 173

7.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

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Part III Solid–Liquid Interfaces: Capillarity, Stability, Nucleation

8 Crystal-Melt Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1818.1 Capillarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

8.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1818.2 Planar and Curved Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

8.2.1 Surface Patches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1848.2.2 Curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1858.2.3 Kinematics of Interfacial Deformation . . . . . . . . . . . . . 1878.2.4 Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1938.2.5 Interfacial Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

8.3 Gibbs–Thomson Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1998.3.1 Equilibrium at Curved Interfaces . . . . . . . . . . . . . . . . . . 1998.3.2 Chemical Potentials at Curved Interfaces . . . . . . . . . . . 2028.3.3 Interfacial Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2048.3.4 Kelvin’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

9 Constitutional Supercooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2139.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2139.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2149.3 Decanting Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

9.3.1 Interfacial Instabilities in Single Crystals . . . . . . . . . . . 2159.3.2 Interfacial Instabilities in Polycrystals . . . . . . . . . . . . . 216

9.4 Constitutional Supercooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2219.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2219.4.2 The Interfacial Solute Mass Balance . . . . . . . . . . . . . . . 2219.4.3 Constitutional Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . 2239.4.4 Stable Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2249.4.5 Unstable Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2259.4.6 Marginal Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2269.4.7 Bulk Crystal Growth: Limiting Forms for Stability . . . 229

9.5 Verification of Constitutional Supercooling . . . . . . . . . . . . . . . . . . 2309.5.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

9.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

10 Linear Morphological Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23710.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23710.2 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

10.2.1 Stability of Planar Interfaces . . . . . . . . . . . . . . . . . . . . . 23910.3 Steady-State Plane-Front Freezing . . . . . . . . . . . . . . . . . . . . . . . . . 240

10.3.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

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10.3.2 Transport Field Equations . . . . . . . . . . . . . . . . . . . . . . . . 24110.3.3 Perturbation Analysis: The Concept . . . . . . . . . . . . . . . 24110.3.4 The Transport Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 24210.3.5 Local Interfacial Equilibrium . . . . . . . . . . . . . . . . . . . . . 24610.3.6 Linearization of the Curvature . . . . . . . . . . . . . . . . . . . . 24710.3.7 Interfacial Flux Balances . . . . . . . . . . . . . . . . . . . . . . . . 24710.3.8 Solutions to the Field Equations . . . . . . . . . . . . . . . . . . 248

10.4 Stability Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24910.4.1 Amplitude Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25010.4.2 Criteria for Interfacial Stability . . . . . . . . . . . . . . . . . . . 25110.4.3 Marginal Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

10.5 Low Wavenumber Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25410.6 General Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

10.6.1 Marginal Wavenumber . . . . . . . . . . . . . . . . . . . . . . . . . . 25610.7 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25710.8 Absolute Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25810.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

11 Non-linear Stability Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26311.1 Limitations of Linear Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26311.2 Interfacial Patterns Just Beyond Marginal Stability . . . . . . . . . . . 26411.3 Patterns Further from the Margin of Stability . . . . . . . . . . . . . . . . 26411.4 Morphological Stability Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26611.5 Absolute Stability in the Non-linear Regime . . . . . . . . . . . . . . . . . 26811.6 Plan-Forms of Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26911.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

12 Nucleation Catalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27312.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27312.2 Fluctuation-Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27412.3 Heterogeneous Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

12.3.1 Free Energy Changes for Nucleation . . . . . . . . . . . . . . . 27712.4 Homogeneous Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

12.4.1 Free Energy Barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27912.5 Heterogeneous Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

12.5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28312.5.2 Spherical Cap Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28412.5.3 Heterophase Fluctuations on a Substrate . . . . . . . . . . . . 289

12.6 Grain Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29012.6.1 Inoculants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29112.6.2 Substrate Area, Contact Angle, and Phase Spreading . 29312.6.3 Accommodation Strains . . . . . . . . . . . . . . . . . . . . . . . . . 294

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12.6.4 Athermal Heterogeneous Nucleation . . . . . . . . . . . . . . . 29512.6.5 Melt Flow Control and Cavitation . . . . . . . . . . . . . . . . . 298

12.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

Part IV Microstructure Evolution

13 Dendritic Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30513.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

13.1.1 Some Early History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30613.2 Dendrites in Metals Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

13.2.1 Observations and Simulation of Dendritic Growth . . . 30713.2.2 Initiation of Dendritic Growth . . . . . . . . . . . . . . . . . . . . 30813.2.3 Directional Solidification . . . . . . . . . . . . . . . . . . . . . . . . 309

13.3 Dendrites in Castings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31113.4 Dendritic Growth Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

13.4.1 Ivantsov’s Transport Model . . . . . . . . . . . . . . . . . . . . . . 31313.4.2 Ivantsov’s Transport Solution . . . . . . . . . . . . . . . . . . . . . 317

13.5 Dendritic Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32013.5.1 Boundary Layer Thickness . . . . . . . . . . . . . . . . . . . . . . . 32013.5.2 Boundary Layers at Small Péclet Numbers . . . . . . . . . 32113.5.3 Boundary Layers at Large Péclet Numbers . . . . . . . . . 322

13.6 Dendritic Capillarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32413.7 Marginal Stability Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

13.7.1 Estimating v and Rtip . . . . . . . . . . . . . . . . . . . . . . . . . . . 32713.8 Dendritic Growth Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

13.8.1 Velocity Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 32813.8.2 Gravitational Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32813.8.3 Stability Constants for Dendritic Growth . . . . . . . . . . . 330

13.9 Stochastics and Determinism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33213.9.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33213.9.2 Anisotropic Capillarity . . . . . . . . . . . . . . . . . . . . . . . . . . 33313.9.3 Energy Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33413.9.4 Shape Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33413.9.5 Deterministic Dynamics: Two Dimensions . . . . . . . . . . 33813.9.6 Deterministic Dynamics: Three Dimensions . . . . . . . . 340

13.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

14 Microsegregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34514.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34614.2 Cellular Microsegregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

14.2.1 Key Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

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14.2.2 Intercellular Solute Mass Balance . . . . . . . . . . . . . . . . . 34714.3 Dendritic Microsegregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

14.3.1 Microsegregation in Mushy Zones . . . . . . . . . . . . . . . . 35614.4 Influence of Solid-State Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 358

14.4.1 Solute Mass Balances in Mushy Zones . . . . . . . . . . . . . 35814.5 Structure of Castings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36514.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

15 Interface Structure and Growth Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . 36915.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

15.1.1 Faceted Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37015.1.2 Non-faceted Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 372

15.2 Two-Layer Interface Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37415.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37415.2.2 Energy Changes for Interfacial Configurations . . . . . . 374

15.3 Kinetic Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38215.3.1 Atomically Rough Interfaces . . . . . . . . . . . . . . . . . . . . . 38315.3.2 Molecularly Smooth Interfaces . . . . . . . . . . . . . . . . . . . 387

15.4 Kinetic Roughening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38915.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38915.4.2 Interface Diffuseness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39015.4.3 Roughening Transition . . . . . . . . . . . . . . . . . . . . . . . . . . 39015.4.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

15.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

16 Polyphase Solidification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39716.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39716.2 Eutectics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

16.2.1 Thermodynamics: Polyphase Solidification . . . . . . . . . 39816.2.2 Classification of Eutectics . . . . . . . . . . . . . . . . . . . . . . . . 40016.2.3 Importance of Eutectics . . . . . . . . . . . . . . . . . . . . . . . . . 40216.2.4 Nucleation of Eutectics . . . . . . . . . . . . . . . . . . . . . . . . . . 40316.2.5 Growth of Eutectics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40416.2.6 Tiller’s Theory of Eutectic Growth . . . . . . . . . . . . . . . . 40516.2.7 Eutectic Phase Spacing . . . . . . . . . . . . . . . . . . . . . . . . . . 412

16.3 Hunt–Jackson Theory of Eutectics . . . . . . . . . . . . . . . . . . . . . . . . . 41416.4 Eutectic Microstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

16.4.1 Lamellar Eutectics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41516.4.2 Rod Eutectics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41616.4.3 Lamellar-to-Rod Transition . . . . . . . . . . . . . . . . . . . . . . 41816.4.4 Crystallography of Lamellar Eutectics . . . . . . . . . . . . . 420

16.5 Computation of Eutectic Microstructures . . . . . . . . . . . . . . . . . . . . 421

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16.6 Directional Freezing of Polyphase Alloys . . . . . . . . . . . . . . . . . . . 42116.7 Cast Iron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42416.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

17 Rapid Solidification Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42717.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42717.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

17.2.1 Splat Quenching of Alloys . . . . . . . . . . . . . . . . . . . . . . . 42817.3 Early Research in Rapid Solidification . . . . . . . . . . . . . . . . . . . . . . 429

17.3.1 Non-equilibrium Phase Diagrams . . . . . . . . . . . . . . . . . 42917.3.2 Hypercooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42917.3.3 Thermodynamic Limits . . . . . . . . . . . . . . . . . . . . . . . . . . 434

17.4 Solute Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43417.4.1 Interface Non-equilibrium . . . . . . . . . . . . . . . . . . . . . . . 43417.4.2 Laser Melting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

17.5 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43617.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43617.5.2 Continuous Growth Models . . . . . . . . . . . . . . . . . . . . . . 43717.5.3 Response Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43817.5.4 Quantitative RSP Experiments . . . . . . . . . . . . . . . . . . . . 439

17.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

Part V Appendices

A Thermodynamic Functions and Legendre Transforms . . . . . . . . . . . . . 449

B Grain Boundary Grooves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

C Deterministic Simulation of Dendritic Growth . . . . . . . . . . . . . . . . . . . . 467References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

D Directional Solidification Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

E Hunt–Jackson Theory of Eutectics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499