Cr320 grain growth-lectureslides

Grain Growth

Shantanu K Behera

Dept of Ceramic Engineering
NIT Rourkela

CR 320 CR 654

Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 1 / 27

Chapter Outline
1 General Features

Grain Growth and Coarsening

NGG and AGG
2 Ostwald Ripening

3 Normal Grain Growth

Burke and Turnbull Model

Topology

4 Abnormal Grain Growth
5 Boundary Mobility

Solute Drag

Particle Inhibited Grain Growth

General Features

Introduction

Engineering properties of materials are influenced by:
Microstructure
Shape and size of grains
Porosity, Pore size, and their distribution
Second phases, and their distribution
The first step is to analyze grain growth in fully dense single phase
ceramics/materials.
This method lets you study only grain growth without other effects such
as that of porosity, second phases, impurities, solutes, dopants etc.
Subsequently, the influence of pores and second phases can be studied
to design fabrication parameters.


General Features NGG and AGG

Grain Growth

Grain Growth Coarsening
is generally referred to the is generally referred to the
increase in the average grain size simultaneous growth of grains as
of a dense compact (either single well as pores (of course, in a
phase, or containing a second porous solid).
phase particle/precipitate. Both pores and grains increase in
Grains grow at the expense of size, and decrease in number.
other grains (Imagine a king Complex in nature.
extending his empire by winning
smaller states).
Relatively simple analysis.



Occurrence of Grain Growth

˚
General width of a GB is 5A.
Atoms from the convex side of the
grain move to the concave side of
the grain surface.
Resulting atomic ﬂux induces the
boundary to move towards its
center of curvature.
Chemical potential difference
across the two surfaces is
responsible.

Figure : Fig 3.1, Sintering of Ceramics,
Rahaman, pg. 106



Normal and Abnormal Grain Growth




NGG: Self similar microstructural
development.
Only scaling dependence.
AGG: Time invariant distribution is
lost.
Some grains grow at the expense
of others, causing bimodal
distribution.
At a later stage, these large grains
impinge to make a unimodal
distribution again, but with much
larger average grain size.




Figure : Normal and abnormal grain growth in alumina; Dillon,Behera,Harmer, Lehigh
University



Importance of Grain Size

1
Strength varies with G as G− 2
Dielectric breakdown strength
(ZnO) increases as G−1 .
Dielectric constant increases with
decreasing G (upto ∼ 1 µ.
Densification decreases with
Figure : Fig. 3.5, MN Rahaman, pg. 110; increase in grain size. [ ρ dρ =
1
dt
K
Gm ]
Densificaiton mechanism for pores But, creep deformation increases
attached to a GB, and in the bulk, Arrows with decreasing grain size.
indicate possible diffusion paths. Pores
when detached from the GB can become
difficult to be removed, thus limiting
density. Therefore, keeping a low grain size
is key to attainment of high density.


Ostwald Ripening

Ostwald Ripening
Ostwald ripening refers to the coarsening of particles or precipitates in a solid
or liquid medium. Features of grain and pore growth are similar to O.R.
Chemical potential on the surface of the
particle with radius a is
2γΩ
µ = µ0 +
a
The solute concentration dependence can be
written as

Figure : Fig. 3.6, MN Rahaman, C 2γΩ
kT ln = µ − µ0 =
pg. 111; Coarsening of particles C0 a
due to materials transport from the
C C
smaller particles to the larger Since ln C0 = C0 , therefore
ones.
C 2γΩ
=
C0 kTa


Ostwald Ripening

Ostwald Ripening Mechanisms

OR controlled by Interface Reaction OR controlled by Diffusion

αT C0 γΩ2 8DC0 γΩ2
[a ]2 − [a0 ]2 = t [a ]3 − [a0 ]3 = t
kT 9kT
αT is a transfer constant. a is the critical radius that neither
Follows a parabolic growth law. grows nor shrinks.
Interface reaction is rate Follows a cubic growth pattern.
controlling. Diffusion of the solutes is rate
Rate is independent of the volume controlling.
fraction. Volume fraction of the media
affects the rate.


Ostwald Ripening

An Example

Figure : Fig. 15.8, Sintering, SJL Kang, pg. 219; Growth of a spinel crystal (MgAl2 O4 )
from a glass melt. Diffusion controlled? Or..Interface controlled?


NGG Burke and Turnbull Model

Burke-Turnbull Model
Transfer of atoms across the grain boundary under the driving force of the
pressure difference between the two internal interfaces.
The grain boundary energy (γb ) is considered isotropic.
The boundary width (δb is assumed to be constant.

The grain boundary velocity, therefore, can be deﬁned as:
dG
vb =
dt
where G is the average grain size.

Additionally, the boundary velocity can also be deﬁned in terms of the drag
force (Fb ), which is essentially the results of difference in curvature) and an
additional term called, boundary mobility (Mb , with units m.N−1 .s−1 ).
vb = Fb Mb

So, boundary mobility is the velocity of the boundary per unit drag force.


Burke-Turnbull Model Contd.
The pressure difference is

1 1 α
P = γb + = γb
r1 r2 G
where α is a geometrical constant.

Force, as the gradient of chemical potential over the boundary width, can be
written as
dµ 1 1 Ωγb α
Fb = = [Ω P] =
dx dx δb G

Atomic ﬂux is
Da dµ Da Ωγb α
J= . =
ΩkT dx ΩkT δb G
dG Da Ω γ b α
vb = = ΩJ =
dt kT δb G



Burke-Turnbull Model Contd.
Boundary mobility can be deﬁned as

Da Ω
Mb =
kT δb

Therefore,
dG γb α
vb = = Mb
dt G

Upon integration, we have
G2 − G2 = Kt
0

where K = 2αγb Mb

This is called the parabolic law for grain growth, quite similar to the interface
reaction-controlled Ostwald ripening. This expression generally describes the
growth of grain in a pure material (metal or ceramic) that is not inﬂuenced by
any solutes, segregants, pores, second phases etc.


Activation Energy of Grain Growth

The rate constant (or growth factor, as it is called sometime), K, has Arrhenius
dependence, and can be written as
Qa
K = K0 e kT

where, Qa is called the activation energy of grain growth. This can give
information on the type of diffusion. For example, in an ionic solid (generally a
ceramic) the rate controlling species (either the cation or the anion) will have
its diffusion activation energy, similar to that of the A.E of grain growth. Here, it
is the slowest moving species.

The boundary mobility (Mb ) in pure materials is called the intrinsic boundary
moblity, and the Da in the mobility expression represents the diffusion
coefﬁcient of the rate limiting species.



Deviation from Burke-Turnbull Model

In practice, however, normal grain growth doesn’t follow the parabolic growth
law in many ceramics, and in some metal systems. The growth law, therefore,
is generalized as:
Gm − Gm = Kt
0

where m (called the grain growth exponent) can take any value from 2 to 4.
The value of m = 3 is widely reported in ceramics. This is the cubic grain
growth law.

The deviation from m = 2 is generally explained as the effect of solutes and
impurities.


Boundary Mobility

Intrinsic Boundary Mobility

Figure : Fig. courtesy: Shen Dillon. There is difference in the calculated and
experimental intrinsic boundary mobility of alumina by orders of magnitude.


Boundary Mobility Solute Drag

Solute Drag

Dopants and/or impurities still can change the boundary mobility. Dopants
and impurities, if dissolved in the matrix, can cause solute drag.

If the grain boundary has an interaction potential with the solute, which may
result from elastic strain energy considerations (size mismatch between the
host and dopant cations) or from electrostatic potential energy (due to the
charge effect; eg. if the host and the dopant have different valencies).

This could lead to a distribution of the solute across the grain boundary, which
can become asymmetric once the boundary starts moving.



Solute Drag: Schematic

Figure : Distribution of the solutes across the boundary (a); Asymmetric distribution
due to a moving boundary (b); Left out solute cloud and boundary break away event
approaching the mobility of a clean boundary (c).



Solute Drag: Cahn Model
Solute drag, Fs is analytically deﬁned as
αC∞ vb
Fs =
1 + β 2 vb
where, α can be deﬁned as the solute drag per unit velocity per unit dopant
concentration (in the low velocity limit), and 1/β is the drift velocity with which
the solute atom/ion moves across the grain boundary.

The total drag force is
αC∞ vb vb
F = Fs + Fb = 2v
+
1+β b Mb

In the low velocity limit, we can neglect β 2 v2 . Therefore,
b

F
vb = 1
Mb + αC∞



Solute Drag: Boundary Velocity vs Force

Figure : The relationship between driving force and velocity for boundary migration
controlled by solute drag. Individual components of the intrinsic drag and the solute
drag, as well as combined drag on the boundary are indicated. Note that when the drift
velocity β −1 is is comparable to the boundary velocity, the dominance of solute drag
decreases (this refers to the boundary break-away event).



Effective Mobility

The effective boundary mobility can be deﬁned in terms of the intrinsic
component Mb and the solute drag component Ms :

eff 1 1 −1
Mb = +
Mb Ms
1
where Ms = αC∞ . For conditions where the solute segregates to the grain
boundary core, the centre of the boundary contributes heavily to the drag
effect.. Here, α can be approximated as:

4Nv kTδb Q
α=
Db

where Q is the partition coefﬁcient for the dopant distribution between the
boundary region and the bulk region (i.e the solute concentration in the
boundary region is QC∞ ).



Adsorbate Drag

The mobility due to solute drag is, therefore,

Db
Ms =
4Nv kTδb QC∞

Some principles can be outlined for the selection of dopants that are most
effective in reducing boundary mobility:

When the diffusion coefﬁcient of the rate limiting species (Db ) is low.
which means that the oversized dopant ions (bigger than the host) can be
effective since the bigger ions possess lower diffusivity in general.
When the segregated solute concentration (QC∞ ) is high.
which means that highly segregating dopants can be effective.

Some examples in ceramics for grain growth control: Host(solute): Al2 O3 (Mg,
Y, Zr), BaTiO3 (Nb, Co), ZnO(Al), Y2 O3 (Th), CeO2 (Y, Nd, Ca)


Boundary Mobility Particle Inhibited Grain Growth

Zener Model
Assumptions: Monosized, spherical, insoluble, immobile, and randomly
distributed particles in a polycrystalline matrix.

The driving force (per unit area) of a grain boundary with principal radii of
curvature a1 and a2 :
1 1 αγb
Fb = γb + =
a1 a2 G
where α is a geometrical factor (2 for spherical grains), γb is the boundary
energy, G is the grain size.

Figure : The zener model for particle inhibited grain growth.



Zener Model
When the boundary meets the particle, extra work is required for its motion.
Therefore, the retarding force exerted is:

Fr = γb Cosθ(2πrSinθ)

Maximum retarding force is applied when θ = 45◦ . Thus

Fmax = πrγb
r

For NA inclusions, the total force is

Fmax = NA πrγb
d

If the volume fraction of the inclusions is f , the number of inclusions per unit
3f
volume is Nv = 4πr3 . Therefore, the total drag due to the particles is:

3f γb
Fmax =
d
2r


Zener Model
Therefore, the net driving force per unit boundary area is:

α 3f
Fnet = Fb − Fmax = γb
d −
G 2r

When Fnet = 0, boundary motion is ceased. The intrinsic drag is balanced by
the drag exerted by the particle. For this condition a limiting grain size can be
deﬁned:
2αr
GL =
3f
This is called the Zener Relationship. GL is proportional to the (second phase)
particle radius, and inversely proportional to the fraction of the second phase
precipitates/particles. Further grain growth could occur if

the inclusion coarsens by Ostwald ripening,
the inclusion dissolves and goes into solid solution,
if AGG occurs.

Cr320 grain growth-lectureslides

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Cr320 grain growth-lectureslides