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Solid State Sintering
Shantanu K Behera
Dept of Ceramic EngineeringNIT Rourkela
CR 320 CR 654
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Chapter Outline
1 Sintering Mechanisms
2 Scaling Law
3 Stages of Sintering
4 Initial Stage
5 Intermediate Stage Sintering
6 Final Stage Sintering: Geometrical Model
7 Sintering with Externally Applied Pressure
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Sintering Mechanisms
3 Particle Model
Figure : Fig 2.1, Sintering of Ceramics, Rahaman, pg. 46
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Sintering Mechanisms
Sintering Mechanisms and Routes
Mechanisms Source Sink DensifyingSurface Diffusion Surface Neck NoLattice Diffusion Surface Neck No
GB Diffusion GB Neck YesLattice Diffusion GB Neck YesVapor Transport Surface Neck No
Plastic Flow Dislocations Neck Yes
Note that mechanisms that extend the GB region (solid-solid interface) aredensifying mechanisms. That keep the solid-vapor interface arenon-densifying mechanisms.
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Sintering Mechanisms
3 Particle Model
Calculate the free energy (surface related) difference between a set ofparticles, and the same set of particles when sintered.
Note that the net reduction in energy would be equal to the total grainboundary energy less the total surface (solid-vapor) energy.
4Ed = As(γgb
2− γsv)
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Sintering Mechanisms
Curvature
Figure : Curvature in solids, and their effect of vacancy concentration
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Sintering Mechanisms
Vacancy under a Curved SurfaceChemical potential of atoms in a crystal can be written as
µa = µoa + pΩa + kBT ln Ca
Similarly, chemical potential of vacancies in a crystal can be written as
µv = µov + pΩv + kBT ln Cv
Chemical potential of vacancies under a curved surface can be written as
µv = µov + (p + γsvκ)Ω + kBT ln Cv
where κ = 1R1
+ 1R2
Accordingly, the equilibrium vacancy concentrationbeneath a curved surface
Cv = Co,ve−γsvκΩ
kBT
For γsvκΩ << kBT, this reduces toCv
Co,v= 1− γsvκΩ
kBT
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Sintering Mechanisms
Vapor Pressure over a Curved Surface
Figure : Curvature in solids, and their effect on vapor pressure
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Sintering Mechanisms
Vapor Pressure over a Curved Surface
Vapor pressure over a curved surface can be defined as
Pvap = P0eγsvκΩ
kBT
This simplifies to:
Pvap = P0
[1 +
γsvκΩ
kBT
]
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Sintering Mechanisms
Diffusional Flux Equations
The general expression for flux:
J =−DiCkBT
dµdx
Flux of atoms:Ja =
−DaCa
ΩkBTd(µa − µv)
dx
Flux of vacancies and atoms are opposite to each other:
Ja = −Jv
Flux of vacancies:Jv =
−DvCv
ΩkBTdµv
dx=−Dv
Ω
dCv
dx
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Scaling Law
Herring’s Scaling Law
Length scale is an important parameter in sintering.How does the change of scale (e.g. particle size) influence the rate ofsintering?The law is based on simple models and assumptions.Particle size remains the same.Similar geometrical changes in different powder systems.Similar composition.
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Scaling Law
Herring’s Scaling Law
Define λ as the numerical factorSay, λ = a2
a1, where a is the radius of the particle
Similarly, λ = X2X1
, where X is the neck dimension of the two particle system.
Time required to produce a certain change by diffusional flux can be written as
4t =VJA
Comparing two systems, we can write
4t2
4t1=
V2J1A1
V1J2A2
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Scaling Law
Scaling Law for Lattice DiffusionWhile comparing two spherical particles of sizes, a1 and a2, we can say thatthe volume of matter transported is V1 ∝ a3
1, and V2 ∝ a32. And since λ = a2
a1,
we can write V2 = λ3Va.
Similarly A2 = λ2A1
Again, flux (J) is ∝ the gradient in chemical potential (i.e. 5µ)
µ varies as 1r , Therefore, J ∝ 5 1
r , Or J ∝ 1r2
Therefore, J2 = J1λ2
Summary: the parameters for lattice diffusion are:V2 = λ3V1; A2 = λ2A1; J2 = J1
λ2
Comparing two systems, we can write
4t2
4t1= λ3 =
[a2
a1
]3
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Scaling Law
Scaling Law for Other Mechanisms
In a general form, we can write as:
4t2
4t1= λn =
[a2
a1
]3
where m is the exponent that depends on the mechanism of sintering. Someof the exponents for different mechanisms are as follows.
Sintering Mechanisms ExponentSurface Diffusion 4Lattice Diffusion 3
GB Diffusion 4Vapor Transport 2
Plastic Flow 1Viscous Flow 1
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Scaling Law
Relative Rates of Mechanisms
For a given microstructural change tha rate is inversely proportional to thetime required for the change. Therefore,
Rate2
Rate1= λ−n
If grain boundary diffusion is thedominant mechanism; thenRategb = λ−4
If evaporation-condensation is thedominant mechanism; thenRateec = λ−2
Figure : Relative rates of sintering for GBand EC as a function of length scale
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Stages of Sintering
Generalized Sintering Curve
Figure : Schematic of a sintering curve of a powder compact during three sinteringstages.
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Stages of Sintering
Sintering Stages
SinteringStage
Microstructural Fea-tures
RelativeDensity
Idealized Model
Initial Interparticle neckgrowth
Up to0.65
Spheres in contact
Intermediate Equilibrium poreshape with continu-ous porosity
0.65 -0.9
Tetrakaidecahedronwith cylindricalpores of the sameradius along edges
Final Equilibrium poreshape with isolatedporosity
≥0.9 Tetrakaidecahedronwith spherical poresat grain corners
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Stages of Sintering
Sintering Stage Microstructures (Real)
Initial stage (a)rapid interparticle growth (variousmechanisms), neck formation,linear shrinkage of 3-5%.Intermediate stage (b)Continuous pores, porosity isalong grain edges, pore crosssection reduces, finally porespinch off. Up to 0.9 of TD.Final stage (c)Isolated pores at grain corners,pores gradually shrink anddisappear. From 0.9 to TD.
Figure : Examples of real microstructureswith various sintering stages.
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Stages of Sintering
Schematic of Intermediate and Final Stage Models
Figure : Idealized models of grains during (a) intermediate, and (b) final stage ofsintering. After R L Coble
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Initial Stage
Geometrical Model for Initial Stage
Figure : Geometrical models for the initialsintering stage; (a) non-densifying, and (b)densifying mechanism.
Non-densifying
Parameter Densifying
r = X2
2a Radius ofNeck
r = X2
4a
r = π2X3
a Area ofNeckSurface
A = π2X3
2a
r = πX4
2a Volumeinto Neck
r = πX4
8a
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Initial Stage
Kinetic Equations
Flux of atoms into the neckJa =
Dv
Ω
dCv
dxVolume of matter transported to neck per unit time
dVdt
= JaAgbΩ
Note that Agb = 2πXδgb Therefore,
dVdt
= Dv2πXδgbdCv
dx
Assuming that the vacancy concentration between surface and neck remainsconstant dCv
dx = CvX Therefore,
4Cv = Cv − Cvo =CvoγsvΩ
kBT
[1r1
+1r2
]
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Initial Stage
Kinetic Equations Contd..If we take r1 = r and r2 = −X, and assuming X >> r, we have
dVdt
=2πDvCvoδgbγsvΩ
kBTr
Using dVdt from geometrical model, and Dgb = DvCvo,
πX3
2adXdt
=2πDgbδgbγsvΩa2
kBT
[4aX2
]On simplification
X5dX =16DgbδgbγsvΩa2
kBTdt
Upon integrating
X6 =96DgbδgbγsvΩa2
kBTt
We can write in another form:
Xa
=
[96DgbδgbγsvΩa2
kBTa4
] 16
t16
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Initial Stage
Kinetic Equations Contd..
Xa
=
[96DgbδgbγsvΩ
kBTa4
] 16
t16
This expression tells you that the ratio of neck radius to the sphere radiusincreases as t
16 . For densifying mechanisms the shrinkage can be measured
as the change in length over original length.
4ll0
= − ra
= − X2
4a2
Therefore4ll0
=
[3DgbδgbγsvΩ
kBTa4
] 13
t13
The shrinkage is therefore predcited to increase as t13
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Initial Stage
Kinetic Equations for Viscous Flow
Rate of energy dissipation by viscous flow should equal to rate of energygained by reduction in surface area.
The final expression looks like
Xa
=
[3γsv
2ηa
] 12
t12
How would the expression for shrinkage by viscous flow look like?
4ll0
=
[3γsv
8ηa
]t
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Initial Stage
Generalized Expressions
There can be general expressions for neck growth and densification asfollows: [
Xa
]m
=
[Han
]t
[4ll0
] m2
= −[
H2man
]t
m, and n are numerical exponents that depend on sintering mechanisms.H contains geometrical and material parameters.A range of values for m and n can be obtained.
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Initial Stage
Summary: Initial Sintering Stage
Mechanism m n H♥
Surface diffusion♦ 7 4 56DsδsγsvΩ/kBTLattice diffusion from sur-face♦
5 3 20DlγsvΩ/kBT
Vapor transport♦ 3 2 3P0γsvΩ/(2πmkBT)1/2kBTGB diffusion 6 4 96DgbδgbγsvΩ/kBTLattice diffusion from GB 4 3 80πDlγsvΩ/kBTViscous flow 2 1 3γsv/2η
♦ - non-densifying mechanism♥ - Diffusion coefficients and constants with usual meanings.If you recall, the exponent n here is same as the Herring’s Scaling Lawexponent.Also note that, for nondensifying mechanisms m is an odd number.
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Intermediate Stage Sintering
Intermediate Sintering Stage
If you recall, the intermediate stage is characterized by continuous pores,porosity is along grain edges, pore cross section reduces, with finally pinchingoff of pores.
Figure : Coble’s geometrical model for intermediate stage (a), and final stage (b).
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Intermediate Stage Sintering
Geometrical Model
Geometrically, sintering can be achieved as per the following two points:
Minimization of total interfacialarea (intfc tension eqlb.)Filling of space without voidsIn 2 dimensions, this can beachieved by a hexagonal array
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Intermediate Stage Sintering
Geometrical Model Contd..
In 3D tension equilibriumrequirement: 6 planes (grainboundaries) and 4 lines (grainedges) meet.So, the number of corners that areneeded for a grain to be inequilibrium is 22.8.Two possible structures:pentagonaldodecahedron andtetrakaidecahedron.
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Intermediate Stage Sintering
Tetrakaidecahedron
Figure : Formation of a Tetrakaidecahedron from an octahedron; Source: Rahaman
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Intermediate Stage Sintering
Geometrical Model Contd..
Figure : Tetrakaidecahedron, 6 Squares, 8Hexagons, 24 Corners
Figure : Pentagonaldodecahedron, 12Pentagons, 20 Corners
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Intermediate Stage Sintering
Tetrakaidecahedron
Figure : Model of a piece of crystalline material with TKD units
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Intermediate Stage Sintering
Geometrical Model for Sintering
Space-filling array of equal sized tetrakaidecahedron, each of it describingone particle. Cylindrical channel pores at TKD edges. Volume oftetrakaidecahedron
Vt = 8√
2l3pwhere lp is the edge length of the TKD. Total porosity (with r as the radius ofthe pore)
Vp =13
36πr2lp
Therefore, porosity of the unit cell:
Vt
Vp= Pc =
3π2√
2
[r2
l2p
]
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Intermediate Stage Sintering
Sintering Equations
For Lattice Diffusion:1ρ
dρdt
=10DlγsvΩ
ρG3kBT
Densification rate at a fixed density scales inversely with the cube of grain size(Check Herring’s law).
For Grain Boundary Diffusion:
1ρ
dρdt
=43
[DgbδgbγsvΩ
ρ(1− ρ)1/2G4kBT
]
Densification rate at a fixed density scales inversely with the fourth power ofgrain size.
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Final Stage Sintering: Geometrical Model
Final Sintering Stage
Cylindrical pore channels pinch offPores become isolatedPores at 4 grain junctions
Average density can be defined as:
ρ = 1−[ r
b
]3
Number of pores per unit volume
N =3
4π
[1− ρρr3
] Figure : Pore radius and improvement ofdensity
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Final Stage Sintering: Geometrical Model
Final Stage Sintering Equations
Porosity at time t:
Ps =6π√
2
[DlγsvΩ
l3kBT
](tf − t)
For diffusion of atoms occurring by lattice diffusion:[dρdt
]LD
=288DlγsvΩ
G3kBT
For diffusion occurring by grain boundary diffusion:[dρdt
]GBD
=735DgbδgbγsvΩ
G4kBT
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Final Stage Sintering: Geometrical Model
Phenomenological Sintering Equation
In this approach, empirical equations are developed to fit experimental data(ρ ∼ t)
ρ = ρ0 + K ln[
tt0
]where K is a temperature dependent parameter.
For Coble’s lattice diffusion model:
dρdt
=ADlγsvΩ
G3kBT
where A is a constant that relates to the sintering stage.
If grain coarsening occurs by (say) cubic law:
G3 − G30 = Kt
where G,G0 are grain sizes at time t and 0, and if, G3 G30, then
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Final Stage Sintering: Geometrical Model
Phenomenological Sintering Equation
densification can be written as:
dρdt
=K′
t; K′ =
ADlγsvΩ
KG3kBT
This equation is expected to be valid for both intermediate and final stagesintering.
When grain growth is limited, shrinkage can be fitted to the following form:
4ll0
= Kt1β
where K is a temperature dependent parameter, and β is an integer.
See that the above equation has a form similar to the initial sintering stagemodel.
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Sintering with Externally Applied Pressure
Hot PressingSimultaneous application of pressure and temperature.
Figure : Schematic of a Hot Press Unit
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Sintering with Externally Applied Pressure
Analytical Model for Hot Pressing
Coble’s model can be changed with an additional stress term.
4Cv,neck =Cv,∞γsvΩ
kBTκ
where Pe is External Pressure= φPa;φ is the stress intensification factor, Pa isthe applied pressure. Therefore,
4Cv,boundary = −Cv,∞γsvPe
kBT= −Cv,∞γsvφPa
kBT
For the initial stage:
4C = 4Cv,neck −4Cv,boundary =Cv,∞Ω4a
kBTx2
[γsv +
Paaπ
]
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Sintering with Externally Applied Pressure
Creep
Creep: Deformation due todiffusion of atoms frominterfaces subjected to acompressive stress (higherchemical potential) to thosesubjected to a tensile stress.
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Sintering with Externally Applied Pressure
Nabarro-Herring Creep
Lattice Diffusionε =
dlldt
=403
DlΩPa
G2kBT
Orε ∝ G−2
Intermediate Stage
1ρ
dρdt
=403
[DlΩ
G2kBT
] [Paφ+
γsv
r
] Final Stage
1ρ
dρdt
=403
[DlΩ
G2kBT
] [Paφ+
2γsv
r
]
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Sintering with Externally Applied Pressure
Coble Creep
Grain Boundary Diffusion
ε =952
DgbδgbΩPa
G3kBT
Orε ∝ G−3
Intermediate Stage
1ρ
dρdt
=952
[DgbδgbΩPa
G3kBT
] [Paφ+
γsv
r
] Final Stage
1ρ
dρdt
=403
[DgbδgbΩPa
G3kBT
] [Paφ+
2γsv
r
]
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Sintering with Externally Applied Pressure
Dislocation Creep
Application of higher stress induces matter transport by dislocation motion.
ε =ADµb
kBT
[Pa
µ
]n
Orε ∝ Pn
a
Intermediate Stage
1ρ
dρdt
= A[
DµbkBT
] [Paφ
µ
]n
Final Stage
1ρ
dρdt
= B[
DµbkBT
] [Paφ
µ
]n
A,B are numerical constants.
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Sintering with Externally Applied Pressure
Densification rate in Hot Pressing
Since in the hot press, one of the dimension stays fixed, densification rate isproportional to the rate of change in the thickness of the compact.
11l
dldt
=1d
d(d)
dt=
1ρ
dρdt
So, simply, linear strain represents the densification rate. Can be obtained bythe travel distance of the hot press ram (plunger).
The driving force for sintering in hot press is the two different forces addedtogether: DF due to curvature and DF due to applied pressure.
DF = Pe + γsvκ = Paφ+ γsvκ
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Sintering with Externally Applied Pressure
Hot Pressing Mechanisms
1ρ
dρdt
=HDφn
GmkBTPn
a
where H is a numerical constantD is the diffusion coefficientφ is the stress intensification factorG is the grain sizem is the grain Size exponentn is the stress exponent.
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Sintering with Externally Applied Pressure
Hot Pressing Mechanisms
Mechanism m n Diffusion Coeffi-cient
Lattice diffusion 2 1 DlGB diffusion 3 1 DgbPlastic deformation 0 ≥3 DlViscous flow 0 1 -Grain boundary sliding 1 1 or 2 Dl or Dgb
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