lecture-solid state sintering

Solid State Sintering

Shantanu K Behera

Dept of Ceramic EngineeringNIT Rourkela

CR 320 CR 654

Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 1 / 47

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


Sintering Mechanisms

3 Particle Model

Figure : Fig 2.1, Sintering of Ceramics, Rahaman, pg. 46



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.



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)



Curvature

Figure : Curvature in solids, and their effect of vacancy concentration



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



Vapor Pressure over a Curved Surface

Figure : Curvature in solids, and their effect on vapor pressure



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

]



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


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.


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


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


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


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


Stages of Sintering

Generalized Sintering Curve

Figure : Schematic of a sintering curve of a powder compact during three sinteringstages.


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


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.


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


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


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

]


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


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


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


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.


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.


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).



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



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.



Tetrakaidecahedron

Figure : Formation of a Tetrakaidecahedron from an octahedron; Source: Rahaman



Geometrical Model Contd..

Figure : Tetrakaidecahedron, 6 Squares, 8Hexagons, 24 Corners

Figure : Pentagonaldodecahedron, 12Pentagons, 20 Corners



Tetrakaidecahedron

Figure : Model of a piece of crystalline material with TKD units



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

]



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.


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



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



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



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.


Sintering with Externally Applied Pressure

Hot PressingSimultaneous application of pressure and temperature.

Figure : Schematic of a Hot Press Unit



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

]



Creep

Creep: Deformation due todiffusion of atoms frominterfaces subjected to acompressive stress (higherchemical potential) to thosesubjected to a tensile stress.



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

]



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

]



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.



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



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.



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
