cr320 lecture solid state sintering

Solid State Sintering

Sudip Dasgupta

Dept of Ceramic Engineering NIT Rourkela

CR 212

Sudip Dasgupta (NIT Rourkela) SINTERING CR 320 CR 654 1 / 26

Chapter Outline

1 Sintering Mechanisms

2 Scaling Law

3 Stages of Sintering

4 Initial Stage


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) are densifying mechanisms. That keep the solid-vapor interface arenon-densifying mechanisms.



3 Particle Model

Calculate the free energy (surface related) difference between a set of particles, and the same set of particles when sintered.

Note that the net reduction in energy would be equal to the total grain boundary energy less the total surface (solid-vapor) energy.

γgb∆Ed = As( 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 = 1 + 1 Accordingly, the equilibrium vacancy concentrationR1

R2

beneath a curved surface Cv = Co,ve

− γ svκΩ

kBT

For γsvκΩ << kBT, this reduces to

Co,v

Cv = 1 γsvκΩ−

kB

TSudip Dasgupta (NIT Rourkela) SINTERING CR 320 CR 654 10 / 26


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 asγ svκ Ω

Pvap = P0e kBT

This simplifies to:

Pvap = P0

.1 + Bk

T

γsvκΩ .



Diffusional Flux Equations

The general expression for flux:

J = −DiC dµ

kBT dxFlux of atoms:

Ja = − DaCa d(µa − µv)ΩkBTdxFlux of vacancies and atoms are opposite to each other:

Ja = −Jv

Flux of vacancies:

Jv = − DvCv

dµv= −Dv

dCvΩkBT dx Ωdx


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 of sintering?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 factora1

Say, λ = a2 , where a is the radius of the particle

X1Similarly, λ = X2 , 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

V∆ t = JA

Comparing two systems, we can write ∆t2 = V2J1A1

∆ t1 V1J2A2


Scaling Law

Scaling Law for Lattice DiffusionWhile comparing two spherical particles of sizes, a1 and a2, we can say that 1 1 2

3 3a

2

2a1

the volume of matter transported is V ∝ a , and V ∝ a . And since λ =,we can write V2 =

λ3Va. Similarly A2 =

λ2A1Again, flux (J) is ∝ the gradient in chemical potential (i.e. dµ/dr)

r11 rµ varies as , Therefore, J ∝ ∇ Or J ∝ 1

r2

Therefore, J2 = J1

λ2

Summary: the parameters for lattice diffusion are:V2 = λ3V1; A2 = λ2A1; J2 = J1

λ2

Comparing two systems, we can write

∆ t1 a

1

∆t2 = λ3 = . a2

.

3


Scaling Law

Scaling Law for Other Mechanisms

In a general form, we can write as:

∆ t1

a1

∆t2 = λm = . a2

3

where m is the exponent that depends on the mechanism of sintering. Some of 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 the time required for the change. Therefore,

Rate2 = λ− m

Rate1

If grain boundary diffusion is the dominant mechanism; then Rategb = λ− 4

If evaporation-condensation is the dominant mechanism; thenRateec = λ− 2

Figure :Relative rates of sintering for GB and 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 sintering stages.


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 continuous porosity

0.65 -0.9

Tetrakaidecahedronwith cylindrical pores of the same radius along edges

Final Equilibrium poreshape with isolated porosity

≥0.9 Tetrakaidecahedronwith spherical pores at grain corners


Stages of Sintering

Sintering Stage Microstructures (Real)

Initial stage (a)rapid interparticle growth (various mechanisms), neck formation, linear shrinkage of 3-5%.Intermediate stage (b) Continuous pores, porosity is along grain edges, pore cross section reduces, finally pores pinch off. Up to 0.9 of TD.Final stage (c)Isolated pores at grain corners, pores gradually shrink and disappear. From 0.9 to TD.


Tetrakaidecahedron


Stages of Sintering

Schematic of Intermediate and Final Stage Models

Figure :Idealized models of grains during (a) intermediate, and (b) final stage of sintering. After R L Coble


Initial Stage

Geometrical Model for Initial Stage

Figure :Geometrical models for the initial sintering stage; (a) non-densifying, and (b) densifying mechanism.

Non-densifying

Parameter Densifying

2r = X2a Radius of

Neck

2r = X4a

2 3

A = π Xa Area

of Neck Surface

2 3

A = π X2a

4

V = πX2a Volume

into Neck

4

V = πX8a


Initial Stage

Kinetic Equations

Flux of atoms into the neck

Ja = ΩDv

dCvdxVolume of matter transported to neck per unit time

dVa gb

= J A Ωd

t

Note that Agb = 2πXδgb Therefore,dVdt

= D 2πXδ

dCv

v gb

dxAssuming that the vacancy concentration between surface and neck remainsdx Xconstant d Cv = ∆Cv Therefore,

∆Cv = Cv − Cvo

= BCvoγsvΩ .

1 k Tr

+r1

2

1 .


Initial Stage

Kinetic Equations Contd..If we take r1 = r and r2 = −X, and assuming X >> r, we have

dV = 2πDvCvoδgbγsvΩdtkBTr

dt

Using dV from geometrical model, and Dgb = DvCvo, πX3 dX

2a dt

=kB

TX2

2πDgbδgbγsvΩa2 . 4a .

On simplification5X dX

=16D δ γ Ωa

2gb gb

sv

kBTdt

Upon integrating6 gb gb

sv96D δ γ Ωa

2

kB

T

tX =We can write in another form:

Xa =

kBTa4

.

96DgbδgbγsvΩa2 .

16

1

t 6


Initial Stage

Kinetic Equations Contd..

Xa =

kBTa4

1.

96DgbδgbγsvΩ .

6

1

t 6

1This expression tells you that the ratio of neck radius to the sphere radius increases as t 6 . For densifying mechanisms the shrinkage can be measured as the change in length over original length.

∆l = − r = − X2

l0 a4a2

Thereforel02kBTa4

∆ l = . 3DgbδgbγsvΩ

.

13 1

t 3

1The shrinkage is therefore predcited to increase as t 3


Initial Stage

Kinetic Equations for Viscous Flow

Rate of energy dissipation by viscous flow should equal to rate of energy gained by reduction in surface area.

The final expression looks like

X =a2ηa

1. 3γsv .

2

1

t 2

How would the expression for shrinkage by viscous flow look like?

l08ηa

∆ l = . 3γsv t


Initial Stage

Generalized Expressions

There can be general expressions for neck growth and densification as follows:

=. X . m . H .

a an

t

m. ∆ l . 2

l0

= −

H2ma

n

. .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 Stages

BT

Mechanism m n H ♥

Surface diffusion♦ 7 4 56DsδsγsvΩ/kBTLattice diffusion from sur-face♦

5 3 20DlγsvΩ/kBT

Vapor transport♦ 3 2 3P0γsvΩ/(2πmkBT)1/2k

GB 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 theHerring’s Scaling Law exponent.Also note that, for nondensifying mechanismsm is an odd number.


cr320 lecture solid state sintering

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