lecture-solid state sintering

Solid State Sintering
Shantanu K Behera
Dept of Ceramic Engineering
NIT 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 Densifying
Surface Diffusion Surface Neck No
Lattice Diffusion Surface Neck No
GB Diffusion GB Neck Yes
Lattice Diffusion GB Neck Yes
Vapor 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 are
non-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.
Ed = As(
γgb
2
− γsv)

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

Vacancy under a Curved Surface
Chemical 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
R1
+ 1
R2
Accordingly, the equilibrium vacancy concentration
beneath a curved surface
Cv = Co,ve
−γsvκΩ
kBT
For γsvκΩ << kBT, this reduces to
Cv
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 = P0e
γsvκΩ
kBT
This simpliﬁes to:
Pvap = P0 1 +
γsvκΩ
kBT

Diffusional Flux Equations
The general expression for ﬂux:
J =
−DiC
kBT
dµ
dx
Flux of atoms:
Ja =
−DaCa
ΩkBT
d(µa − µv)
dx
Flux of vacancies and atoms are opposite to each other:
Ja = −Jv
Flux of vacancies:
Jv =
−DvCv
ΩkBT
dµv
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 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 factor
Say, λ = a2
a1
, where a is the radius of the particle
Similarly, λ = X2
X1
, 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
t =
V
JA
Comparing two systems, we can write
t2
t1
=
V2J1A1
V1J2A2

Scaling Law
Scaling Law for Lattice Diffusion
While comparing two spherical particles of sizes, a1 and a2, we can say that
the volume of matter transported is V1 ∝ a3
1, and V2 ∝ a3
2. And since λ = a2
a1
,
we can write V2 = λ3
Va.
Similarly A2 = λ2
A1
Again, ﬂux (J) is ∝ the gradient in chemical potential (i.e. µ)
µ varies as 1
r , Therefore, J ∝ 1
r , Or J ∝ 1
r2
Therefore, J2 = J1
λ2
Summary: the parameters for lattice diffusion are:
V2 = λ3
V1; A2 = λ2
A1; J2 = J1
λ2
Comparing two systems, we can write
t2
t1
= λ3
=
a2
a1
3

Scaling Law
Scaling Law for Other Mechanisms
In a general form, we can write as:
t2
t1
= λn
=
a2
a1
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 Exponent
Surface Diffusion 4
Lattice Diffusion 3
GB Diffusion 4
Vapor Transport 2
Plastic Flow 1
Viscous 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
Rate1
= λ−n
If grain boundary diffusion is the
dominant mechanism; then
Rategb = λ−4
If evaporation-condensation is the
dominant mechanism; then
Rateec = 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
Sintering
Stage
Microstructural Fea-
tures
Relative
Density
Idealized Model
Initial Interparticle neck
growth
Up to
0.65
Spheres in contact
Intermediate Equilibrium pore
shape with continu-
ous porosity
0.65 -
0.9
Tetrakaidecahedron
with cylindrical
pores of the same
radius along edges
Final Equilibrium pore
shape with isolated
porosity
≥0.9 Tetrakaidecahedron
with 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.

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
r = X2
2a Radius of
Neck
r = X2
4a
r = π2
X3
a Area of
Neck
Surface
A = π2
X3
2a
r = πX4
2a Volume
into Neck
r = πX4
8a

Initial Stage
Kinetic Equations
Flux of atoms into the neck
Ja =
Dv
Ω
dCv
dx
Volume of matter transported to neck per unit time
dV
dt
= JaAgbΩ
Note that Agb = 2πXδgb Therefore,
dV
dt
= Dv2πXδgb
dCv
dx
Assuming that the vacancy concentration between surface and neck remains
constant dCv
dx = Cv
X Therefore,
Cv = Cv − Cvo =
CvoγsvΩ
kBT
1
r1
+
1
r2

Initial Stage
Kinetic Equations Contd..
If we take r1 = r and r2 = −X, and assuming X >> r, we have
dV
dt
=
2πDvCvoδgbγsvΩ
kBTr
Using dV
dt from geometrical model, and Dgb = DvCvo,
πX3
2a
dX
dt
=
2πDgbδgbγsvΩa2
kBT
4a
X2
On simpliﬁcation
X5
dX =
16DgbδgbγsvΩa2
kBT
dt
Upon integrating
X6
=
96DgbδgbγsvΩa2
kBT
t
We can write in another form:
X
a
=
96DgbδgbγsvΩa2
kBTa4
1
6
t
1
6

Initial Stage
Kinetic Equations Contd..
X
a
=
96DgbδgbγsvΩ
kBTa4
1
6
t
1
6
This expression tells you that the ratio of neck radius to the sphere radius
increases as t
1
6 . For densifying mechanisms the shrinkage can be measured
as the change in length over original length.
l
l0
= −
r
a
= −
X2
4a2
Therefore
l
l0
=
3DgbδgbγsvΩ
kBTa4
1
3
t
1
3
The shrinkage is therefore predcited to increase as t
1
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
a
=
3γsv
2ηa
1
2
t
1
2
How would the expression for shrinkage by viscous flow look like?
l
l0
=
3γsv
8ηa
t

Initial Stage
Generalized Expressions
There can be general expressions for neck growth and densiﬁcation as
follows:
X
a
m
=
H
an
t
l
l0
m
2
= −
H
2man
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/kBT
Lattice diffusion from sur-
face♦
5 3 20DlγsvΩ/kBT
Vapor transport♦
3 2 3P0γsvΩ/(2πmkBT)1/2
kBT
GB diffusion 6 4 96DgbδgbγsvΩ/kBT
Lattice diffusion from GB 4 3 80πDlγsvΩ/kBT
Viscous 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 Law
exponent.
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 pinching
off 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 interfacial
area (intfc tension eqlb.)
Filling of space without voids
In 2 dimensions, this can be
achieved by a hexagonal array

Geometrical Model Contd..
In 3D tension equilibrium
requirement: 6 planes (grain
boundaries) and 4 lines (grain
edges) meet.
So, the number of corners that are
needed for a grain to be in
equilibrium is 22.8.
Two possible structures:
pentagonaldodecahedron and
tetrakaidecahedron.

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

Geometrical Model Contd..
Figure : Tetrakaidecahedron, 6 Squares, 8
Hexagons, 24 Corners
Figure : Pentagonaldodecahedron, 12
Pentagons, 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 describing
one particle. Cylindrical channel pores at TKD edges. Volume of
tetrakaidecahedron
Vt = 8
√
2l3
p
where lp is the edge length of the TKD. Total porosity (with r as the radius of
the pore)
Vp =
1
3
36πr2
lp
Therefore, porosity of the unit cell:
Vt
Vp
= Pc =
3π
2
√
2
r2
l2
p

Sintering Equations
For Lattice Diffusion:
1
ρ
dρ
dt
=
10DlγsvΩ
ρG3kBT
Densification rate at a fixed density scales inversely with the cube of grain size
(Check Herring’s law).
For Grain Boundary Diffusion:
1
ρ
dρ
dt
=
4
3
DgbδgbγsvΩ
ρ(1 − ρ)1/2G4kBT
Densification rate at a fixed density scales inversely with the fourth power of
grain size.

Final Stage Sintering: Geometrical Model
Final Sintering Stage
Cylindrical pore channels pinch off
Pores become isolated
Pores 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 − ρ
ρr3
Figure : Pore radius and improvement of
density

Final Stage Sintering Equations
Porosity at time t:
Ps =
6π
√
2
DlγsvΩ
l3kBT
(tf − t)
For diffusion of atoms occurring by lattice diffusion:
dρ
dt LD
=
288DlγsvΩ
G3kBT
For diffusion occurring by grain boundary diffusion:
dρ
dt GBD
=
735DgbδgbγsvΩ
G4kBT

Phenomenological Sintering Equation
In this approach, empirical equations are developed to ﬁt experimental data
(ρ ∼ t)
ρ = ρ0 + K ln
t
t0
where K is a temperature dependent parameter.
For Coble’s lattice diffusion model:
dρ
dt
=
ADlγsvΩ
G3kBT
where A is a constant that relates to the sintering stage.
If grain coarsening occurs by (say) cubic law:
G3
− G3
0 = Kt
where G, G0 are grain sizes at time t and 0, and if, G3
G3
0, then

Phenomenological Sintering Equation
densification can be written as:
dρ
dt
=
K
t
; K =
ADlγsvΩ
KG3kBT
This equation is expected to be valid for both intermediate and final stage
sintering.
When grain growth is limited, shrinkage can be fitted to the following form:
l
l0
= Kt
1
β
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 stage
model.

Sintering with Externally Applied Pressure
Hot Pressing
Simultaneous 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.
Cv,neck =
Cv,∞γsvΩ
kBT
κ
where Pe is External Pressure= φPa; φ is the stress intensiﬁcation factor, Pa is
the applied pressure. Therefore,
Cv,boundary = −
Cv,∞γsvPe
kBT
= −
Cv,∞γsvφPa
kBT
For the initial stage:
C = Cv,neck − Cv,boundary =
Cv,∞Ω4a
kBTx2
γsv +
Paa
π

Creep
Creep: Deformation due to
diffusion of atoms from
interfaces subjected to a
compressive stress (higher
chemical potential) to those
subjected to a tensile stress.

Nabarro-Herring Creep
Lattice Diffusion
˙ =
dl
ldt
=
40
3
DlΩPa
G2kBT
Or
˙ ∝ G−2
Intermediate Stage
1
ρ
dρ
dt
=
40
3
DlΩ
G2kBT
Paφ +
γsv
r
Final Stage
1
ρ
dρ
dt
=
40
3
DlΩ
G2kBT
Paφ +
2γsv
r

Coble Creep
Grain Boundary Diffusion
˙ =
95
2
DgbδgbΩPa
G3kBT
Or
˙ ∝ G−3
Intermediate Stage
1
ρ
dρ
dt
=
95
2
DgbδgbΩPa
G3kBT
Paφ +
γsv
r
Final Stage
1
ρ
dρ
dt
=
40
3
DgbδgbΩPa
G3kBT
Paφ +
2γsv
r

Dislocation Creep
Application of higher stress induces matter transport by dislocation motion.
˙ =
ADµb
kBT
Pa
µ
n
Or
˙ ∝ Pn
a
Intermediate Stage
1
ρ
dρ
dt
= A
Dµb
kBT
Paφ
µ
n
Final Stage
1
ρ
dρ
dt
= B
Dµb
kBT
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 is
proportional to the rate of change in the thickness of the compact.
1
1
l
dl
dt
=
1
d
d(d)
dt
=
1
ρ
dρ
dt
So, simply, linear strain represents the densification rate. Can be obtained by
the travel distance of the hot press ram (plunger).
The driving force for sintering in hot press is the two different forces added
together: DF due to curvature and DF due to applied pressure.
DF = Pe + γsvκ = Paφ + γsvκ

Hot Pressing Mechanisms
1
ρ
dρ
dt
=
HDφn
GmkBT
Pn
a
where H is a numerical constant
D is the diffusion coefﬁcient
φ is the stress intensiﬁcation factor
G is the grain size
m is the grain Size exponent
n is the stress exponent.

Hot Pressing Mechanisms
Mechanism m n Diffusion Coefﬁ-
cient
Lattice diffusion 2 1 Dl
GB diffusion 3 1 Dgb
Plastic deformation 0 ≥3 Dl
Viscous ﬂow 0 1 -
Grain boundary sliding 1 1 or 2 Dl or Dgb

lecture-solid state sintering

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