Chapter 7
Grain boundary effect in the coarsening of
polycrystalline solid films
7.1 Introduction
In this chapter, we investigate the impact of grain boundary diffusivity on the kinetics of
spinodal decomposition and coarsening in polycrystalline solid films. These films are used
extensively in the electronic, magnetic, chemical, photonic and microelectromechanical devices and systems. In films, as in bulk materials, the grain boundaries act as fast diffusion
paths due to their high density of defects. Here, we include these fast diffusion paths by
tying mobility to a specific phase field variable used to distinguish grain boundaries.
There exists a great deal of literature for modelling phase transformation with either
constant mobility or mobility that depends on composition or order-parameter [81, 82,
22, 23, 24, 25]. In this chapter, we investigate a variable mobility field owing to grain
boundaries. We consider a free-standing polycrystalline solid film with both fixed and
migrating grain structures. The mobility in the CH model is assumed to be a function of the
grain structure, where the magnitude of mobility is much higher on the grain boundaries
than in the grain.
The organization of this chapter is as follows. In Sections 7.2, the governing equations
for composition evolution and grain boundary migration are given. In Section 7.3, the
numerical method as well as an analysis of the relative time scales for grain boundary migration and composition diffusion is presented. Simulation results are presented in Section
7.4.
7.2 Governing equations
We adopt a phase field or a generalized Cahn-Hilliard (CH) equation for phase separation.
The general form of this model for a binary alloy is given in Eqn. (2.13) as
µ
µ
¶¶
∂f (C) ∂Eel
∂C
= ∇ · B(C, Ψ(x, t))∇
+
− β∆C
,
∂t
∂C
∂C
(7.1)
7.2
Governing equations
119
where C is the composition, B(C, Ψ(x, t)) is the mobility of the atomic species described by
C, f (C) is the stress-free chemical free energy density, Eel is the elastic energy density, and
β is a measure of interfacial penalty for the rapid variation of C. We choose f (C) to be a
multiwelled function. Also, β is assumed to be constant.
If the surface energy is composition-independent and the body has no mass exchange
with the environment, then we have the following boundary conditions:
∂C
= 0,
∂n
where
JCH = −ρB(C, Ψ(x, t))∇
(7.2)
JCH · n = 0,
and
µ
∂f (C) ∂Eel
+
− β∆C
∂C
∂C
¶
(7.3)
is a flux derived from the gradient of effective diffusion potential and ρ is an atomic density. Also, n is the outward normal of the body surface. In a periodic case, these boundary
conditions need to be adjusted appropriately.
In the expression for the mobility B(C, Ψ(x, t)) in Eqn. (7.1), we have used Ψ(x, t) (possibly a vector) to describe a general phase field (order parameter) variable. In this chapter,
Ψ(x, t) will be used to describe the grain boundary structure. Note that in the literature,
special forms of composition dependent mobility, such as B = 1 − αC 2 (Langer et al. [81])
or B = C(1 − C)(Cm1 + (1 − C)m2 ) (Zhu et al. [24]), have been considered independent of
grain boundary structure. However, in our model we consider a dependence of the mobility on the crystallographic grain boundaries (as described by a particular choice of Ψ(x, t))
and assume the mobility is independent of the composition profile C(x, t).
The grain structure in a polycrystalline body is associated with a distribution of diffu-
sivity, where the diffusivity on grain boundary, DGB , is orders of magnitude greater than
the diffusivity in the grain, DL especially at temperatures below 0.6Tm , where Tm is the
melting point of matrix. Kaur et al. [83] gives the ratio DGB /DL as at least 105 . Further, the
mobility B in the CH equation can be shown to be connected with the diffusivity D via a
simple model as follows. Ignoring the elastic and interfacial effects, the two fluxes: J CH in
the CH model (see Eqn. (7.3)) and JF from Fick’s law are
JCH ≈ −ρB∇
µ
∂f (C)
∂C
¶
≈ −ρB
·
¸
∂ 2 f (C)
∇C,
∂C 2
(7.4)
and
JF ≈ −D∇C.
(7.5)
7.2
Governing equations
120
Since these two fluxes, JCH and JF both describe the rate of change of composition, B
and D must be connected by a certain scaling. Here, since we are not interested in the
composition dependence of B (or D), we may assume that B follows D in its dependence
on the grain structure. That is, for fixed grain boundaries, we assume BGB /BL = DGB /DL .
For the case where grain boundaries are allowed to migrate, a grain boundary structure is
projected from Ψ(x, t) and the mobility B is set as in the fixed grain boundary case. The
migrating grain boundary case is further discussed in Section 7.2.1.
7.2.1
Grain boundary migration
Grain boundary migration (grain growth) occurs because of a decrease of the energy owing
to shrinking the grain boundaries. Grain boundary migration is common in polycrystals
and has been treated by a number of authors using a variety of theoretical and experimental approaches [84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94]. There has also been much work done
to numerically simulate grain boundary motion. Kinetic Monte Carlo Potts model (see [55]
for a review), Phase field models have also been used to simulate the evolution of grain
boundaries. For example, Chen et al. performed 2D simulations of the grain-growth with
a large number of nonconserved order parameters [56, 57]. Fan and Chen [95] studied the
effect of ratios of grain boundary energies to interfacial energy on the microstructure in the
isotropic case. Here, isotropic means that the grain boundary energy and the mobility for
the migration are both independent of the crystallographic orientation of the grain and the
boundary orientation. Kazaryan et al. [58, 59] investigated the dynamics and morphology
of grain growth for anisotropic grain boundary energy and anisotropic grain boundary
mobility using a generalized phase-field model.
In this work, we consider only the case of isotropic grain boundaries. We denote the
crystallographic orientations of the polycrystals, using a total number of p = 36 order
parameters ηi (i = 1..p). Roughly, this means that we divide the 2D orientation space of
the crystal into 10◦ intervals. We assume the following free-energy density functional [56]:
f P (η1 , η2 , · · · , ηp ) =
p X
p
p h
X
X
c1
c2 i
ηi2 ηj2 ,
− ηi2 + ηi4 + c3
2
4
i=1
(7.6)
i=1 j>i
where ‘P’ means phase field and c1 , c2 , c3 are phenomenological parameters. We choose
the following values: c1 = 1.0, c2 = 1.0, c3 = 1.0. The energy in Eqn. (7.6) then ensures
that for each (x, t), at most only one element in ηi (i = 1..p) has value close to ±1 and the
other elements are all close to 0.
With the free-energy density in Eqn. (7.6), the kinetics of the evolution of the order
7.2
Governing equations
121
parameters are given by the Allen-Cahn equation [56]:


p
X
dηi
= −µi −c1 ηi + c2 ηi3 + 2c3 ηi
ηj2 − λi ∇2 ηi 
dt
j6=i
(i = 1, 2, · · · , p, no sum),
(7.7)
where the µi are the kinetic coefficients and the λi are energy penalties owing to grain
boundaries. Note in Eqn. (7.7), no summation rule is employed. Since we consider the
isotropic case, we set µi = µ̄ and λi = λ̄ for all i, where µ̄ and λ̄ are reference values. In
[56], a grain structure (described by Ψ(x, t)) based on the ηi is given as
Ψ(x, t) =
p
X
ηi2 (x, t).
(7.8)
i=1
The variable Ψ(x, t) has low values on the grain boundaries and is close to 1 inside grains.
Hence we propose the following form of the mobility for our CH model:
M (Ψ(x, t)) ≡ B/B0 =

R,
1,
|1 − Ψ(x, t)| ≥ Th (on grain boundaries)
|1 − Ψ(x, t)| < Th (inside grains),
(7.9)
where B0 is a reference mobility and is chosen to be the mobility in the grain BL , R is
set to BGB /BL (assumed a constant) and Th is some threshold (typically taken to be 0.2).
Equations (7.8) and (7.9) qualitatively describe the features of the grain boundaries and
their effect on the mobility in the CH model.
7.2.2
Contribution of elasticity
The term ∂Eel /∂C in the evolution equation (7.1) is associated with an elastic energy contribution arising from composition dependence of the lattice parameter of the film. This
term has been considered extensively in both film and bulk phase transformations. However, the elasticity problem associated with the polycrystal structure is very complicated
because of the inhomogeneity of elastic moduli introduced by the grain boundaries. While
it is possible to assume isotropic elasticity for the film (so grain boundary rotation does
not lead to inhomogeneity), or to use a homogenized equivalent medium, neither option
seems particularly physical. Hence we choose to neglect elasticity.
7.3
Numerical analysis
122
7.3 Numerical analysis
7.3.1
Nondimensionalization
The nondimensionalization of Eqn. (7.1) follows as in the previous chapter (Section 6.2.1).
We introduce length scale L̄ and time scale T̄ , and nondimensionalize all spatial and temporal variables. We also introduce ² = β/β0 and M = B/B0 (see also Section 7.2.1) as the
nondimensional measures of interfacial penalty and mobility respectively, where β 0 and
B0 are reference values. We assume the stress-free chemical free energy f is the same as in
Eqn. (6.8). By introducing Γ = (2C − Cα − Cγ )/(Cγ − Cα ), f in Eqn. (6.8) can be nondimensionalized as f = W/K 4 f¯, where f¯ = (Γ2 − 1)2 and K = 2/(Cγ − Cα ). By substituting all
the nondimensional variables into Eqn. (7.1) and setting
β0 B0 T̄
= 1,
L̄4
and
W L̄2
= 1,
K 2 β0
(7.10)
we get the length scale L̄ and the time scale T̄ as:
L̄ =
µ
β0
W
¶1/2
K,
and
T̄ =
β0
K 4.
B0 W 2
(7.11)
The nondimensional governing equation and boundary conditions then can be written as

∂ f¯

Q = ∂Γ
− ²∆Γ,



 J = −M ∇Q,
∂Γ

= −∇ · J,

∂τ


 ∂Γ = 0,
J·n=0
∂n
(7.12)
on boundary,
where Γ, Q, J are nondimensional composition, effective diffusion potential and flux respectively. Note also that periodic boundary conditions can be assumed instead of those
given in Eqn. (7.12).
We then consider the evolution of grain orientation order parameters. Using the time
and length scales given in Eqn. (7.11), Eqn. (7.7) is nondimensionalized to


dηi
= Rτ λL ∇2 ηi − −c1 ηi + c2 ηi3 + 2c3 ηi
dτ
p
X
j6=i
where
Rτ = µ̄T̄ ,
and

ηj2 
λL =
(i = 1..p, no sum on i) (7.13)
λ̄
.
L̄2
(7.14)
7.3
Numerical analysis
123
Note µ̄ is of dimension [T −1 ] hence Rτ is dimensionless. In the next subsection, we
will show that Rτ is a measure of the relative mobility of grain boundary migration to
composition diffusion. Also, Rτ is estimated based on experimental data.
7.3.2
Determination of Rτ
Fan and Chen [96] show that for isotropic grain boundary energy and relaxation constants,
within the diffuse-interface description, a grain boundary moves at velocity V due to its
mean curvature κ as
V = µ̄λ̄κ,
(7.15)
where µ̄ and λ̄ are reference values for µi and λi introduced in Section (7.2.1).
For a half-loop constant driving force technique [93], Gottstein and Shvindlerman
showed
V =
2Ab
,
a
(7.16)
where a is the diameter of the half-loop and Ab is a reduced mobility which can be measured experimentally. In this half-loop technique, κ in Eqn. (7.15) is 1/a, so
µ̄λ̄ = 2Ab .
(7.17)
Introducing Sτ ≡ Rτ λL , and using Eqns. (7.14), (7.17) and (7.11), we have
Sτ = µ̄λ̄
T̄
K2
=
2A
.
b
B0 W
L̄2
(7.18)
Using Eqns. (7.4) and (7.5), we have a rough relation between B0 and the bulk diffusivity
D0 as in Eqn. (6.23):
B0 ≈
D0 K 2
,
W Rd
(7.19)
where Rd is the mean value of the second derivative of the stress free chemical energy to
composition at its wells. In our model, Rd is roughly 8. With Eqn. (7.19), Eqn. (7.18) leads
to
Sτ ≈ 2Rd
Ab
Ab
≈ 16 .
D0
D0
(7.20)
By Eqn. (7.20), Sτ (hence Rτ ) measures the relative mobility of grain boundary migration
and diffusion. From [93], Ab generally has a wide range of values depending on factors
such as species, orientations, etc. The typical range for Ab is 10−16 ∼ 10−8 m2 /s. From
standard texts on diffusion (such as [97]), the typical range for D0 is 10−16 ∼ 10−13 m2 /s.
Thus, we expect that Sτ can have a range of values from 10−2 ∼ 109 . For specific materials
7.3
Numerical analysis
124
and environments, one can determine a narrower range for Sτ .
In this work, we choose λ̄ such that λL = h2 /2 (see Eqn. (7.14)), where h is the uniform
nondimensional spatial step; hence Rτ in Eqn. (7.13) is Rτ = Sτ /λL = 2Sτ /h2 . A typical
range is Rτ ∈ (1, 1010 ). Note that by Eqn. (7.20), if Ab ≈ D0 , one expects that the grain
migration and bulk diffusion have roughly the same time scale. In this situation, S τ ≈ 16
and Rτ ≈ 800 with h = 1/5 used in our simulation. The role of Rτ in the coupled process
of diffusion and migration is shown in the simulation results in Section 7.4.3.
7.3.3
Numerical scheme
The time marching scheme for Eqn. (7.12) follows as in the previous chapter (Section 6.2.3).
However, we introduce three sets of grid in the physical domain. To simplify the description, suppose the spatial step in x1 , x2 , x3 are all h. The physical domain is discretized as
(i, j, k)h, where i = 0..L1 /h, j = 0..L2 /h, and k = −Hf /h..Hf /h. The grid for the flux coincides with the this physical grid; while the composition Γ and potential Q are computed
on a grid (i + 21 , j + 21 , k + 21 )h. Hence the Laplacian operator ∆ is taken on the Γ grid, the
gradient operator ∇ is taken on the Q grid and generates flux on the J grid, and the diver-
gence operator ∇· is taken on the J grid and produces results on the Γ (Q) grid. Since the
gradient and divergence operators have different object size, we denote their components
e i·
(as in x1 , x2 and x3 direction) as follows: the divergence operator ∇· is denoted as ∇
e
e i (i = 1, 2, 3). Then, the following
(i = 1, 2, 3), and the gradient operator ∇ is denoted as ∇
time marching scheme is used:
µ
¶
¡
¢
Γn+1 − Γn
e
n+1
2
n+1
n
ei · M∇
e i [4(Γ
=∇
) + Ac − ²D∆ ]Γ
+R
∆τ
(i is summed), (7.21)
where D∆ is the discrete Laplacian operator operating on the composition Γ, and R n =
−(4 + Ac )Γn is the contribution from time level n. Note the nondimensional mobility M
should be evaluated on the same grid as J.
If the Γ grid points are gathered into a vector by a lexicographic ordering of the nodes,
e
e i, ∇
e i are gathered into matrix form (still denoted by
and similarly the operators D∆ , M , ∇
the same symbol), the scheme in Eqn. (7.21) can be written as
h
I − ∆τ [DM ][diag(4(Γn+1 )2 ) + Ac I − ²D∆ ]
i£
¤
Γn+1 = [Γn ] + ∆τ [DM ]Rn ,
(7.22)
where DM is a mobility dependent pseudo Laplacian operator:
e
e iM ∇
ei
DM = ∇
(i is summed).
(7.23)
7.4
Simulation results
125
The grain boundary migration system in Eqn. (7.13) is integrated explicitly in time using a standard fourth order Runge-Kutta marching scheme. Periodic boundary conditions
for ηi (i = 1..p) are used.
7.4 Simulation results
In this section, simulation results are presented to show how enhanced grain boundary
diffusion affects phase separation and coarsening in solid films. We consider first a fixed
grain boundary structure in two dimensions. Then, we consider a fixed three-dimensional
columnar grain boundary structure obtained by extending a two-dimensional one into the
third dimension. We then introduce migrating grain boundaries in two dimensions.
The following nondimensional parameters apply to all cases: ² = 0.1 and unless otherwise stated the magnitude of the ratio MGB /ML of grain boundary to bulk mobility is
taken to be in 103 (recall from Section 7.2 that this ratio can be as high as 105 ). This choice
qualitatively describes the phenomena occurring in the evolution without sacrificing numerical performance. Specifically, choosing too high a value of MGB /ML may cause some
numerical issues due to the large range of time scales. The simulations are performed on
a computational domain of 51.2 × 12.8 (with a grid 256 × 64) for two-dimensional cases
and 12.8 × 12.8 × 3.2 (with a grid 64 × 64 × 16) for three-dimensional cases. All plots
are rendered with grain boundary structure and composition overlapped. The white and
black areas represent the phases of Γ ≈ +1 and Γ ≈ −1 respectively. The underlying grain
boundaries are shown in gray and can be easily seen.
7.4.1
Results in two dimensions with fixed grain boundaries
We first consider a two-dimensional case with fixed grain structure Rτ = 0. The two phases
are distributed in such a way that one of the phase (Γ ≈ +1) is on the grain boundaries,
while the other (Γ ≈ −1) resides in the grains, as shown in Figure 7.1 (the gray-scale
scheme is shown in Figure 7.2). The average composition is Γ0 = −0.8062. The background
grain boundary structure is taken from a binary image of an experimental picture from a
graphics website. It is slightly modified to be periodic in the horizontal direction and also
thickened digitally so that the boundaries themselves are a few grid points wide.
Figure 7.2(a) shows a case with the relative mobility ratio MGB /ML = 1000. The Γ ≈
+1 phase on the grain boundary diffuses immediately toward the triple junctions and
coarsens. To compare, Figure 7.2(b) depicts a case with MGB /ML = 1. It is clear that
in Figure 7.2(a), the grain boundaries are the main paths for coarsening and the resulting
particles are retained in the triple junctions, with a shape conformal to the junction. On the
7.4
Simulation results
126
Figure 7.1: An initial composition profile for 2D simulation. The two phases are distributed
in such a way that one of the phases (Γ ≈ +1) is on the grain boundaries, while the other
(Γ ≈ −1) resides in the grains. The gray-scale scheme is shown in Figure 7.2.
(a)
(b)
Figure 7.2: Composition evolution for a 2D case with initial profile in Figure 7.1. (a) The
relative mobility ratio MGB /ML = 1000. The Γ ≈ +1 phase diffuses immediately toward
the triple junctions and coarsens. (b) The mobility ratio MGB /ML = 1. Standard coarsening occurs.
7.4
Simulation results
127
(a)
(b)
Figure 7.3: Two grain structures with the same initial composition profile for a simulation
with Rτ = 1. (a) A relative large grain structure; (b) A very large grain structure. The
gray-scale scheme is shown in Figure 7.4.
(a)
(b)
Figure 7.4: The evolution of a particle system in a polycrystalline solid film with large
grains. MGB /ML = 1000. The grain boundary migration is very slow, with Rτ = 1,
thus emulating a ‘fixed’ grain boundary. Coarsening occurs with preference on the grain
boundary. Note the shaded color indicates channels with fast diffusion rate. (a) A relative
large grain structure; (b) A very large grain structure.
7.4
Simulation results
128
contrary, standard coarsening behavior occurs in Figure 7.2(b) and there is no preferential
coarsening direction. Also, note that coarsening with grain boundary diffusion is much
faster than without grain boundary diffusion.
We also consider a case in the migrating grain scenario (Section 7.4.3) but with R τ = 1,
so the grain boundary is essentially fixed. Figure 7.4 depicts two such cases for large grain
films. The initial composition profile and grain structures are shown in Figure 7.3. Note
that in the figures (also in the following figures in this chapter), the part of the particles
with Γ ≈ +1 over the grain boundary is shaded due to the contrast of colors. These
shaded regions hence can be treated as an indicator of channels with fast diffusion rate.
One observes that coarsening is accelerated along the grain boundaries. For phases inside
a grain, there are only very small variations of composition due to the low bulk mobility.
Γ
1.10
τ = 0.0000000
Γ
1.10
0.55
0.55
0.00
0.00
-0.55
-0.55
-1.10
-1.10
τ = 0.0000000
Figure 7.5: A three-dimensional initial profile with small range random composition
around Γ = 0. The average composition is Γ0 = 0. The plot is overlapped with a threedimensional fixed grain boundary. The grain boundary structure is extended from a twodimensional grain boundary profile. On the left is a top view, on the right a side view.
7.4.2
Results in three dimensions with fixed grain boundaries
A columnar grain boundary structure in three-dimensions is obtained by extending the
two-dimensional fixed grain boundary structure into the third dimension (thickness direction). Though more complex grain boundary structure can be generated, for example, by
Voronoi tessellation or other methods, we don’t consider them in this work.
We investigate a case with a random initial composition profile with a small range
around the critical composition Γ = 0, as depicted in Figure 7.5. The average composition
is Γ0 = 0. The relative mobility ratio MGB /ML = 1000. The evolution process is shown in
Figure 7.6 using both a top view and a side view.
Figure 7.6 shows that, as in the two-dimensional case, the grain boundary and its junctions serve as fast channels for phase separation and coarsening. Spinodal decomposition
7.4
Simulation results
Γ
Γ
Γ
Γ
1.10
τ = 0.0000700
129
Γ
1.10
0.55
0.55
0.00
0.00
-0.55
-0.55
-1.10
-1.10
1.10
τ = 0.0007100
Γ
1.10
0.55
0.55
0.00
0.00
-0.55
-0.55
-1.10
-1.10
1.10
τ = 0.0036300
Γ
1.10
0.55
0.55
0.00
0.00
-0.55
-0.55
-1.10
-1.10
1.10
τ = 0.0079500
Γ
1.10
0.55
0.55
0.00
0.00
-0.55
-0.55
-1.10
-1.10
τ = 0.0000700
τ = 0.0007100
τ = 0.0036300
τ = 0.0079500
Figure 7.6: Composition evolution in a free-standing film with a fixed columnar grain
structure in three dimensions. MGB /ML = 1000. Spinodal decomposition occurs in the
plane and along the thickness direction on the grain boundary. The initial composition
profile is in Figure 7.5. On the left is a top view, on the right a side view.
7.4
Simulation results
130
occurs first along the grain boundaries, followed by coarsening.
7.4.3
Results in two dimensions with migrating grain boundaries
Grain boundaries and phase interfaces are distinct, i.e., in one grain, there can be two
phases; while in a single phase, there can be two crystallographic orientations. Hence
grain boundaries can migrate at the same time that the composition evolves. As described
in Eqn. (7.9), the migrating grain boundaries (Ψ(x, t)) generate a time-dependent mobility
M for composition diffusion, hence depending on the magnitude of the relative mobility
Rτ , migrating grain boundaries can have a strong effect on phase separation. This can
be viewed as follows. As grains grow, fast diffusion channels are established or removed
between adjacent particles. When the channel is present, coarsening occurs at a faster rate;
when it is not present, the coarsening rate drops significantly. The interaction between
these two mechanisms gives complicated behavior for the composition evolution.
An initial composition profile is generated for a polycrystalline grain structure as shown
in Figure 7.7. The average composition is taken to be Γ0 = 0. Figures 7.8(a) and 7.8(b) depict two cases with the same initial conditions except Rτ = 1K and 10K respectively. In
both cases, MGB /ML = 1000 and ² = 0.1. As expected, the grain boundary structure
moves faster as Rτ increases.
Figure 7.7: A particle initial composition profile in a polycrystalline solid film. The grayscale scheme is shown in Figure 7.8.
Figure 7.8(a) depicts the evolution for Rτ = 1K. As stated earlier, this corresponds
to the situation where the grain migration and bulk diffusion occurs at roughly the same
rate. The grain structure shows noticeable changes over the elapsed time. For such a rate,
enough time is allowed for particles to coarsen and the topology of the shaded regions
shows smooth transitions. Recall that the shaded region is the part of phase Γ ≈ +1 which
stays over the grain boundary (hence is subject to a fast coarsening rate).
When Rτ increases, grain evolution occurs faster than the composition evolves. Figures
7.8(b) depicts the case for Rτ = 10K. Since the fast diffusion channels move through the
domain quickly, the composition is essentially “frozen”. Compared to Figure 7.8(b), the
topology of shaded regions in Figure 7.8(a) shows smoother transitions.
7.4
Simulation results
(a)
131
(b)
Figure 7.8: The evolution of a particle system in a polycrystalline solid film. For both cases,
MGB /ML = 1000. (a) Rτ = 1K. Noticeable change of the grain boundary structure occurs.
(b) Rτ = 10K. Grain boundaries move faster. The fast growing grains freeze some of the
composition when the grain boundary moves away quickly.
7.5
Summary and discussion
132
7.5 Summary and discussion
In this chapter, we use a CH model to investigate how variable mobility owing to grain
boundary structure affects phase separation and coarsening in polycrystalline solid films.
The mobility, which is roughly proportional to the diffusivity, is taken to be much greater
on the grain boundaries than in the grain. Both fixed and migrating grain boundary structure are considered. For grain boundary migration, a phase field model is adopted and a
grain structure is projected from the order parameters. In this case, the relative mobility of
the grain boundary migration to bulk diffusion is found to be a key factor.
Numerical simulations are performed in two dimensions for a fixed grain boundaries
and in three dimensions for a fixed columnar grain structure. The grain boundaries are
observed to accelerate the coarsening, especially at the triple junctions. Simulations of migrating grain boundaries are performed in two dimensions. If the relative mobility of grain
migration is large enough, then the migrating grain structure can freeze the composition
on the time scale of grain boundary motion.
For the migrating grain boundary cases, we assume specific forms for both the phase
field model describing the grain structure, and the dependence of the mobility on the grain
structure. For the fixed grain structures, the width of the grain boundary is taken to be
relatively thick, and the ratio of grain boundary diffusivity to bulk diffusivity is limited by
the numerical implementation. Finally, other grain boundary related mechanisms, such as
grooving, pinning, solute segregation, etc., are not considered. Some of these features will
be addressed in future investigations.