Axisymmetric finite-element simulation of grain growth behaviour

Abstract

An axisymmetric finite-element method is developed for predicting the evolution behaviour of microstructures by evaporation–condensation. The formulation of the method is conducted on the basis of the energy principle during the interface motion and the numerical procedure is described in detail. The accuracy and potential of the method have been demonstrated by a good agreement of the theoretical solutions with the numerical simulation of a grain evolution process induced by both surface tension and the free-energy difference between the two phases in bulk. Finite-element simulations show that the grain growth behaviour is not only sensitive to its initial shape, but also influenced by the environment. As the initial aspect ratio of the penny-shaped grain increases, both the spheroidization time and the volume shrinkage time increase continuously. The volume shrinkage process of the penny-shaped grain can be greatly promoted with an increase in the free-energy difference. In addition, we find that the evolution time is a linear function of the aspect ratio or the free-energy difference when the free-energy difference or the aspect ratio is small, respectively.

Full text

INSTITUTE OF PHYSICS PUBLISHING MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING
Modelling Simul. Mater. Sci. Eng. 11 (2003) 41–55 PII: S0965-0393(03)55408-0
Axisymmetric finite-element simulation of grain
growth behaviour
Peizhen Huang1,3, Jun Sun1and Zhonghua Li2
1State Key Laboratory for Mechanical behaviour of Materials, School of Materials Science and
Engineering, Xi’an Jiaotong University, 710049, Xi’an, People’s Republic of China
2School of Civil Engineering and Mechanics, Shanghai Jiaotong University, 200240, Shanghai,
People’s Republic of China
E-mail: pzhuang@mail.xjtu.edu.cn
Received 9 July 2002, in final form 9 October 2002
Published 22 November 2002
Online at stacks.iop.org/MSMSE/11/41
Abstract
An axisymmetric finite-element method is developed for predicting
the evolution behaviour of microstructures by evaporation–condensation. The
formulation of the method is conducted on the basis of the energy principle
during the interface motion and the numerical procedure is described in detail.
The accuracy and potential of the method have been demonstrated by a good
agreement of the theoretical solutions with the numerical simulation of a grain
evolution process induced by both surface tension and the free-energydifference
between the two phases in bulk. Finite-element simulations show that the grain
growth behaviour is not only sensitive to its initial shape, but also influenced
by the environment. As the initial aspect ratio of the penny-shaped grain
increases, both the spheroidization time and the volume shrinkage time increase
continuously. The volume shrinkage process of the penny-shaped grain can be
greatly promoted with an increase in the free-energy difference. In addition,
we find that the evolution time is a linear function of the aspect ratio or the
free-energy difference when the free-energy difference or the aspect ratio is
small, respectively.
1. Introduction
Solid-state phase transformations, very often, produce second-phase precipitate particles
coherently embedded in the parent phase matrix, followed by particle coarsening during which
large particles grow at the expense of smaller ones. The morphology of the precipitate particles,
i.e. the sizes, shapes, volume fraction, crystallographic relations with the matrix, and the mutual
arrangement of the particles, plays an important role in determining mechanical, electric and
magnetic properties of a wide variety of high technology alloys [1].
3Author to whom correspondence should be addressed.
0965-0393/03/010041+15$30.00 © 2003 IOP Publishing Ltd Printed in the UK 41

42 P Huang et al
There have been a number of analytical studies on the relative stabilities of different shapes
of a coherent particle [2–6], and on the temporal evolution during growth or coarsening by
numerical simulations with phase-field models [1, 7, 8]. Significant progress has been made in
modelling material behaviour, and it can be said that the earlier efforts have enabled us to relate
many aspects of the macroscopic behaviour of engineering materials to their microstructures
and the underlying physical processes. However, real material systems are often complex,
and various, often unrealistic, assumptions have to be made in the material models in order
to solve the mathematical equations [9]. Following the classic work of Herring [10], the
finite-difference methods have been used to analyse the microstructure evolution, including
grain boundary grooving [11, 12], the spheroidization of rod-shaped particles of finite length
[13], grain boundary cavitation [14], powder sintering [15] and grain-boundary void growth
and shrinkage [16–18]. The methods based on finite-difference are conceptually simple and
relatively fast. They require, however, very smooth representation of the surface because of
the higher order differentiation. They are inconvenient when other energetic forces and rate
processes are present in the problem, or when the surface forms facets and corners due to, e.g.
surface tension anisotropy or elastic field concentration [19].
Recently, a weak statement, which combines the energetics and the kinetics for interface
motion has been formulated [20–22]. It has weaker requirements on the smoothness of surface
[21–23]. On the basis of the weak statement of evaporation–condensation, a two-dimensional
finite-element formulation for grain boundary movement was carried out by Sun et al [24],
and applied to grain growth behaviour [25]. Then, it was extended to three-dimensional finite-
element schemes to simulate the migration of interfaces in materials under linear kinetics
[26]. Kim and Kishi [27] developed the method to simulate the Zener pinning behaviour
involving a three-dimensional effect without considering the free-energy difference between
the two adjacent phases in the total Gibbs free energy of the system. Numerical examples
have demonstrated that the method can capture intricate details in transient motion and readily
includes multiple energetic forces and rate processes. Important physical conclusions were
drawn from these numerical studies [26–31]. However, the numerical technique has certain
limitations, for instance, the axisymmetrical cases involving the free-energy difference have
not been reported in the literature up to now.
In this paper, an axisymmetric finite-element program is developed based on the framework
demonstrated above and applied to the study of the morphological evolution of the precipitate
particle controlled by the evaporation–condensation kinetics, which can also be explored to
simulate the morphology change of the penny-shaped microcracks. The driving force for the
microstructure evolution is the reduction of the excess free energy of the system and determined
by the interfacial energy and the free-energy density difference between the matrix and the
particle. Atomic mobility during the evolution of the solid particle can be nonuniform due to,
for example, crystalline anisotropy or temperature nonuniformity caused by the Joule heating.
The effect of surface energy anisotropy has been modelled in two- and three-dimensional finite-
element methods [24, 26]. However, it is of limited use to include anisotropic properties in
an axisymmetric model. Consequently, we assume that surface properties are independent of
crystallographic orientations, namely surface energy and interface migration rate are isotropic.
2. Axisymmetric finite-element formulation
When a solid particle is in contact with an environment (a vapour or a liquid solution), or an
isolated precipitate surrounded by a matrix, the solid may gain mass from, or lose mass to, the
environment, both causing the interface to move. We assume that the long-range heat and mass
diffusion are absent or rapid, so that the interface process limits its velocity. At atomic scale,

Simulation of grain growth behaviour 43
the interface migration occurs by a thermally activated atomic jump across the interface. The
driving force, p, for the atomic jump is supplied by a curvature-induced energy difference and
the free-energy difference between the two phases in bulk [21]. When pis small compared
with the average thermal energy per atom, the normal velocity of the interface motion, vn,is
proportional to the driving force p[32], namely
vn=mp (1)
Here, mis the specific rate of evaporation–condensation and depends on temperature in the
usual way, m=m0exp(−q/kT), where m0is the frequency factor, qis the activation energy,
kis the Boltzmann’s constant, and Tis the temperature. The values of m0and qvary with the
properties of the different materials. For a certain material, mhas a corresponding value. The
linear kinetic law (equation (1)) is a special case of a more general law introduced by Hilling
and Charles [33] and is valid when the driving force is small compared to the average thermal
energy [31].
A virtual displacement, δrn, of the interface is a motion in the direction normal to the
interface and varies arbitrarily along the curved interface. It need not obey any kinetic law.
Associated with the virtual motion, the total free energy of the system varies by an amount
δG. Using the kinetic law (equation (1)), a weak statement can be written as [21]
vnδrn
mdA=−δG (2)
The actual velocity distribution, vn, satisfied equation (2) for arbitrary distributions of the
virtual displacement. When the normal velocity of the interface motion, vn, is obtained, the
displacement of the point on the surface can be updated for a small time increment. Repeating
the procedure for many time increments, the evolution of the microstructures can be traced.
An axisymmetric surface is generated by rotating a plane curve around an axis lying on
the same plane. Following Sun et al [24], we divide the generating curve into many small
straight elements. The motions of the nodes describe the motion of the surface. Each node
on the plane curve represents a circle on the surface in three dimensions. Figure 1 shows one
frustum element with two nodes (x1,y
1)and (x2,y
2)in two dimensions. The element has
length land slope θ, which relate to the coordinates of the two nodes as
x2−x1=lcos θ, y2−y1=lsin θ(3)
The local coordinate, s, is measured from the mid-point of the element. The surface area of
the element is π(x1+x2)l, and the initial free energy, G0,is
G0=γπ(x
1+x2)l +gV (4)
where γis the surface energy per unit area (or surface tension), g is the free-energy difference
per unit volume of atoms between the solid and the vapour phases (i.e. the chemical potential
of the solid minus that of the environment) [21, 24] and V is the volume of the frustum.
For each surface element, the virtual motions of the nodes are
[δxj]e=[δx1δy1δx2δy2]T(5)
where the superscript e represents the value of the element, while T denotes transposed matrix.
Let ˙
x1,˙
y1,˙
x2and ˙
y2be the nodal velocities of the element. The generalized velocities are
[˙
xj]e=[˙
x1˙
y1˙
x2˙
y2]T(6)
At a point, which is at a distance s, its virtual displacement δrnand normal velocity vnhave
the following relations:
δrn=N1δx1+N2δy1+N3δx2+N4δy2
vn=N1˙
x1+N2˙
y1+N3˙
x2+N4˙
y2
(7)

44 P Huang et al
The interpolation coefficients are given by
N1=−1
2−s
lsin θN
2=1
2−s
lcos θ
N3=−1
2+s
lsin θN
4=1
2+s
lcos θ
(8)
For each frustum element, the left-hand side of equation (2) becomes, through the
integration extending over the element surface,
[δxj]eT[Hij ]e[˙
xj]e(9)
where the viscosity matrix of the element
[Hij ]e=πl
6m(x1+x2)



2 sin2θ−2 sin θcos θsin2θ−sin θcos θ
−2 sin θcos θ2 cos2θ−sin θcos θcos2θ
sin2θ−sin θcos θ2 sin2θ−2 sin θcos θ
−sin θcos θcos2θ−2 sin θcos θ2 cos2θ




+πl2cos θ
6m




−sin2θsin θcos θ00
sin θcos θ−cos2θ00
0 0 sin2θ−sin θcos θ
00−sin θcos θcos2θ




(10)
In the axisymmetric linear element, as shown in figure 1, γacts on each node in two
directions. One is the parallel direction to the boundary, and the other is the normal direction
to the plane of the figure, whereas only the parallel force acts in the two-dimensional case.
Due to the existence of the axisymmetric curvature of radius R2, the displacement of the node
Figure 1. A frustum element.

Simulation of grain growth behaviour 45
can be divided into two components [27]. One, δl, is parallel to the element, and the other,
δr, is normal to the element as shown in figure 1. When the two nodes move by l1and l2,
both the surface area and the volume of the frustum element change, and the corresponding
free energy varies by δGe
l. Neglecting the terms of second order, δGe
lcan be represented as
δGe
l=2γπx
1δl1−2γπx
2δl2+πg
3{(y2−y1)cos θ[δl1(2x1+x2)
+δl2(2x2+x1)]+(x2
1+x2
2+x1x2)(δl2−δl1)sin θ}(11)
The variation of the free energy caused by the displacement of the nodes in the normal
direction (i.e. δr1and δr2)is
δGe
r=γπlsin θδr1+γπlsin θδr2+πg
3{−(y2−y1)sin θ[δr1(2x1+x2)
+δr2(2x2+x1)]+(x2
1+x2
2+x1x2)(δr2−δr1)cos θ}(12)
Then, the total variation of the free energy can be expressed in terms of virtual motion of
the nodes:
δGe=−f1δx1−f2δy1−f3δx2−f4δy2(13)
The force components acting on the two nodes due to the element surface tension and
free-energy difference g are
[fi]e=γπ



2x1cos θ−lsin2θ
2x1sin θ+lsin θcos θ
−2x2cos θ−lsin2θ
−2x2sin θ+lsin θcos θ




+πg
3




(2x1+x2)(y2−y1)
−(x2
1+x2
2+x1x2)
(2x2+x1)(y2−y1)
(x2
1+x2
2+x1x2)




(14)
Assemble the coordinates of all nodes into a column x, and the forces on all nodes into f,
and the virtual motion of all nodes into δx. Equation (2) then becomes
(δx)T·(H˙
x)=(δx)T·f(15)
Since the equations are to hold for any variation δx, we can obtain the controlling equation
of the finite-element method
H˙
x=f(16)
The matrix Hrelates the surface velocity to the applied surface, which is a symmetric matrix
and is usually positive definite by the second law of thermodynamics [23]. However, if part
of the surface becomes flat, Hbecomes singular since a node on a flat surface may move
either normal to the surface, or along the surface. The latter motion does not change the
surface shape and has no physical significance [24]. To circumvent the difficulty, we add
small positive numbers of the diagonal element of Hto prevent it from becoming singular.
These small numbers physically correspond to additional viscosities of the nodes. Their exact
values are unimportant, but should be large enough to stabilize the matrix, and small enough
not to affect noticeably the solution accuracy. In practice, we have found values of the order
10−3–10−6Lm−1are adequate, where Lis a representative length of the system under study,
such as the grain size.
Since the viscosity matrix Hdepends on the current position, equation (16) is a set of
nonlinear ordinary differential equations. In the implementation of the numerical method, they
are solved by Gaussian elimination and the nodal positions are updated by integration using
an explicit fifth-order Runge–Kutta scheme [34], with adaptive step-size control to monitor
accuracy. As the interface of the microstructure moves, some elements shorten and others
elongate. A very short element substantially reduces the time step; a very long element poorly
models a curved geometry. We prescribe a minimum and a maximum element, and eliminate
a very short element, and divide a very long one.

46 P Huang et al
3. Numerical simulation and discussion
It is known from energetic theory that the surface tension and the difference in the free energy of
the two phases in the evolution process of microstructures constitute a thermodynamic force,
causing microstructures, such as grains, microcracks and so on, to evolve. In this section,
the behaviour of grain growth controlled by evaporation–condensation is analysed by the
axisymmetric finite-element method developed in this paper. First, we consider the behaviour
of the spherical particles. This problem has been examined by Suo [21], and we are able to
compare our results directly with his analytical solutions in order to check the reliability of the
numerical technique. Then, the finite-element method is applied to examine the effects of the
initial shape and the environment on the grain evolution. Finally, we focus on the evolution of
the penny-shaped grains. For convenience, we introduce the dimensionless time ˆ
t=tmγ R−2
0,
where R0is the initial radius, and the surface tension is assumed to be isotropic.
3.1. Spherical particle growth
Consider a spherical particle immersed in a large mass of a vapour or solution. The system has
only one degree of freedom, R, the radius of the sphere. Within the environment, molecular
mobility is so large that the free energy is taken to be uniform. The solid and the environment,
however, are not in equilibrium with each other: the solid loses mass to the environment by
dissolution, and the driving force on the surface of the spherical particle pis −(2γ/R)−g.
The total free energy can be written as
G=4πR2γ+4
3πR3g (17)
Here, γis always positive, but g can be either positive or negative. For a positive g,
the volume term (in equation (17)) reduces the free energy when the particle shrinks. In the
case of g < 0, a critical radius Rccorresponding to maximal free energy is obtained by
setting dG/dR=0, giving
Rc=−2γ
g (18)
Imagine a particle of radius R= Rc. Thermodynamics requires that the particle change
size to reduce G.IfR>R
c, the volume term in equation (17) becomes increasingly important,
and the particle will expand to a larger and larger sphere to reduce Gbecause p>0. In contrast,
if R<R
c, the particle shrinks to a sphere and disappear.
From equation (1), the analytical solution for the evolution of the spherical particle from
an initial radius R0is given by [21]
(R −R0)+Rcln 



R−Rc
R0−Rc


=−mgt (19)
The particle radius as a function of time, R(t), is shown by the solid line in figure 2. The
predictions based on the present FEM scheme are also plotted in figure 2 as dots. They agree
very well with the analytical solutions.
Hence, the present axisymmetric finite-element formulation is valid in simulating the
behaviour of grain growth controlled by evaporation–condensation.
3.2. Grain growth in different environments
We now consider the environmental effects on the evolution of a single-crystal particle. The
initial shape of the particle is taken to be ellipsoidal with the ratios of its three principal axes

Simulation of grain growth behaviour 47
Figure 2. Evolution of the spherical grains driven by the isotropic surface tension and the
free-energy difference.
(a,band c) being 5:5:3asshowninfigure 3(a). By equating the volume between an
ellipsoid and a sphere, an equivalent radius of the particle can be defined as R0=3
√abc.
When the grain evolution is only driven by surface diffusion (i.e. g =0, Rc=∞),
the volume of the particle will shrink because atoms evaporate to the vapour as shown in
figure 3(b)atˆ
t=1.4×10−4. On the other hand, the isotropic surface tension serves to change
the initial non-spherical shape to a spherical one and the spherical particle will eventually
disappear.
When R0/Rc=0.67, the particle gradually takes on a spherical shape and shrinks at the
same time because the effect of surface tension is dominant in the process of grain growth
(see figure 3(c)at ˆ
t=3.0×10−4). When the effect of the free-energy difference increases,
especially when it overruns the effect of the surface tension, the particle will expand and also
take on a spherical shape as shown in figure 3(d)(R0/Rc=1.02, ˆ
t=1.6×10−3).
From above, the surface tension serves to retard grain growth and to change the initial
non-spherical shape to a spherical one.
3.3. Bifurcation in evolution caused by different initial shapes
We next consider the evolution of two crystals of the same material, with the same initial
volume and the same initial free energy, but different initial shapes. The initial shape
of one crystal is a sphere and the other is a cylinder as shown in figures 4(a) and (c),
respectively.
For the case of isotropic surface tension, the critical radius of the spherical particle, Rc
is −2γ/g. Using similar calculations, the equivalent critical radius R
cfor the cylinder is
−3
√12γ/gand its equivalent radius can be written as 3
√12r/2. Here, ris the underside radius
and its height his assumed to be 2r(see figure 4(c)). We take an initial radius Rc=1.1R0
in the numerical simulation, which is between Rcand R
c. The spherical particle gradually
expands as shown in figure 4(b). For the initially cylindrical particle, the particle gradually

48 P Huang et al
(b) (c)
(a)
(d)
Figure 3. Shape evolution for an ellipsoidal grain driven by the isotropic surface tension and the
free-energy difference under different conditions: (a) Initial geometry with the ratio of the three
semi-axes: 5:5:3;(b)Rc=∞, the particle shrinks to a spherical particle at ˆ
t=1.4×10−4;
(c)R0/Rc=0.67, the particle shrinks to a spherical particle at ˆ
t=3.0×10−4;(d)R0/Rc=1.02,
the particle expands to a spherical particle at ˆ
t=1.6×10−3.
2r
h0
(c) (d)
(b)
(a)
Figure 4. Effect of initial shapes on particle evolution: (a) a spherical single-crystal particle at
ˆ
t=0, R0/Rc=1.1; (b) the particle expands to a spherical particle at ˆ
t=0.9; (c) initial shape
of a cylindrical particle for h=2r;(d) the cylindrical particle shrinks to a spherical shape at
ˆ
t=0.1.

Simulation of grain growth behaviour 49
takes a spherical shape (see figure 4(d)atˆ
t=0.1), but shrinks at the same time since the effect
of surface tension overruns the effect of the free-energy difference.
3.4. Evolution of penny-shaped grains
Since a penny-shaped particle, which is one of the most representative states, is susceptible
to shape instabilities as a result of curvature-induced processes which lead to a diminished
interfacial surface area per unit volume [35], we now consider the evolution of the penny-
shaped particle driven by interface tension and the free-energy difference between the grain and
the matrix. For convenience, we, introduce, here, the dimensionless parameters: ˆ
t=tmγ /h2
0,
ˆg=gh0/γ , where the thickness of the grain h0is taken as 0.2µm in all the calculations.
The initial shape of the particle is similar to that shown in figure 4(c)at ˆ
t=0, which is
characterized by the aspect ratio βdefined as the ratio of the diameter 2rto the thickness h0.
We assume that the equivalent radius is lower than its equivalent critical radius and the surface
mobility mis also assumed to be isotropic. At a fixed temperature, the shape of the penny-
shaped particle evolves because the curvature at the underside or top plane is not the same
as that at the side face, which causes the variations of the atom driving force at the interface
points and leads to spheroidization gradually.
Let us first examine the effect of the free-energy difference ˆgon the particle shape.
Figure 5 shows the simulation of a penny-shaped particle under the condition of β=10.
For simplicity, the shapes are drawn in two dimensions, i.e. figure 5 shows the cross section
of the particles. The particle changes shape from a plate to a sphere and the mass in the
grain leaves the grain, crosses the grain boundary, and joins the matrix, leading to its volume
shrinkage. This shape evolution process can be easily identified in many natural phenomena.
When ˆgis very small or even zero, the process of evolving into a sphere is very obvious.
However, it is easy for the particle to keep its penny-shaped shape with increasing ˆg(see
the case of ˆg=0.5 in figure 5) because the volume term gradually becomes dominant in
equation (17).
Figure 5. A shape comparision of the penny-shaped grains for three values of ˆgfor
β=10.

50 P Huang et al
As we know, a microstructure due to capillarity-induced surface diffusion (such as an
internal crack or a second-phase precipitate particle) first blunts its edge, from which finger-like
channels emerge, and the channels, depending upon the initial aspect ratio of the crack, either
spheroidize or break into strings of isolated voids similar to the Rayleigh instability dominated
by surface diffusion without matter exchange between the particle and the environment [36].
In order to determine whether there could exist the Rayleigh instability in the morphological
evolution of a penny-shaped particle under evaporation–condensation with matter exchange
between the particle and its environment, two limited cases are investigated. Considering
the critical aspect ratios of microstructural instability under surface diffusion (18 in three
dimensions and 133 in two dimensions), a penny-shaped particle with ˆg=0 and β=150,
which is much larger than 18, is first simulated. The other case is ˆg=1.0 and β=150
referring the results of the shape evolution with ˆg(see figure 5). The numerical results
indicate that there is no bulge formed on the particle surface because of the absence of the
atom diffusion along the surface and the particle gradually shrinks into a sphere (the figures are
omitted) dominated by evaporation–condensation. This can also be due to the kinetic law of
evaporation–condensation. In evaporation–condensation, the atom driving force can be given
by p=−γκ −g [21] and equation (1) can be rewritten as
vn=−m(γ κ +g ) (20)
Here, κis the curvature of the particle surface, which is positive for a convex bulge.
It is obvious that the atoms move away from the centre of the curvature. That is, the
matter diffuses from the particle to the environment around, leading to the shrinkage of the
particle.
By plotting the spheroidization time of the penny-shaped grains as a function of the original
aspect ratio β(figure 6), we find that the spheroidization time increases with increasing β
because the transportation path of the mass is longer for larger β. Of course, the spheroidization
time ˆ
tsdoes not only depend on the aspect ratio βbut also on the free-energy difference ˆg,
as shown in figure 6. When ˆg0.02, ˆ
tsis a linear function of βfor the given ˆg, i.e. ˆ
tsis
linearly proportional to βwhen the effect of ˆgis very small. But the relation between βand
ˆ
tsbecomes nonlinear gradually with ˆgincreasing.
Figure 6. The spheroidization time of the penny-shaped grains driven by surface tension and
free-energy difference as a function of βand ˆg.

Simulation of grain growth behaviour 51
Figure 7. The spheroidization time ˆ
tsof the penny-shaped grains for four values of βas a function
of ˆg.
Figure 7 shows the ˆgdependence of the spheroidization time for four cases of βin detail.
It is obvious that the evolution time decreases with ˆgincreasing, while the shrinkage rate
increases continuously. The slope of these curves at any point represents the magnitude of the
dependence. When β=12, the ˆgdependence of the spheroidization time is the greatest.
When βis small, the grain evolution appears to mainly depend on ˆgand the surface tension
becomes the weak driving force. This is the reason why the effect of ˆgon ˆ
tsis roughly linear
when β=2 as shown in figure 7.
Figure 8 shows the relative volume shrinkage Vs/V0as a function of time ˆ
tfor different
values of β(when ˆg=0 and 0.2, respectively), where V0is the initial volume of the penny-
shaped grain and Vsis the volume shrinkage. Vs/V0=1 means that the grain completely
vanishes. As can be seen in figure 8, the grain vanishing time ˆ
tvconsistently increases as
βincreases. The slope of these curves at any point represents the shrinkage rate of the grain
volume. The curves in figure 8 show that, for a given penny-shaped grain, its volume shrinkage
rate is greater at the beginning of the evolution process and then it becomes smaller gradually.
The curves also show that the volume shrinkage rate decreases with increasing β. Comparing
figures 8(a) and (b), we can find that the βdependence of the volume shrinkage is not sensitive
with increasing ˆg, which also can be seen obviously from figure 9. Moreover, as shown in
figure 9, it is obvious that the aspect ratio dependence of the volume shrinkage rate is relatively
weakened with increasing ˆg.
Now, let us discuss the variation in the volume shrinkage rate with ˆg. The volume
shrinkage is shown in figure 10 as a function of time for various values of ˆgin the case of
β=12. The grain shrinkage time ˆ
ttaken to attain a certain value (Vs/V0)decreases with
increasing ˆg, while the shrinkage rate increases continuously. This means that the increase in
ˆgaccelerates the shrinkage rate of the penny-shaped grains and it is obvious that the volume
shrinkage rate is also faster at the beginning of the evolution process.
4. Conclusions
In this paper, an axisymmetric finite-element method is developed and applied to study the
grain growth behaviour controlled by evaporation–condensation in three dimensions. It was

52 P Huang et al
Figure 8. Penny-shaped grain volume shrinkage—time behaviour as a function of βfor ˆg=0(a)
and ˆg=0.2(b).
demonstrated that the axisymmetric finite-element method is robust, accurate and efficient. It
can capture intricate details in the transient motion and other kinetic processes, such as diffusion
on interfaces; range elastic and electric fields can be included in the same framework. Our
aim is to concentrate on incorporating realistic models (e.g. nonlinear kinetics laws, elastic–
plastic solid models, etc). Actually, for the morphological evolution process of the particle,
whether the Rayleigh instability could occur might depend on whether surface diffusion
or evaporation–condensation is dominant. In this simulation, the particle shrinkage rate
under evaporation–condensation, related to the free-energy difference and the aspect ratio,
is so great that the aspect ratio of the particle is much reduced, leading to the Rayleigh
instability under surface diffusion to be inhibited. That is, whatever the particle aspect ratio is,
Rayleigh instability is suppressed in the morphological evolution of the penny-shaped particle
under evaporation–condensation and the particle shrinks gradually. Further investigation
is needed to clarify the alternatively morphological evolution process of the penny-shaped
particle.

Simulation of grain growth behaviour 53
Figure 9. Influence of βon the shrinkage time ˆ
ttaken to attain the grain shrinkage of a certain
value (Vs/V0)for ˆg=0 and ˆg=0.2, respectively.
Figure 10. Penny-shaped grain volume shrinkage–time behaviour as a function of ˆgfor
β=12.
Acknowledgments
The financial support from the National Natural Science Foundation of China under Grant
no 19972053, 10272075 and 59889101 is gratefully acknowledged. One of the authors, Jun Sun,
wishes to express his special thanks for the support of National Outstanding Young Investigator
Grant of China through Grant no 59925104.
References
[1] Li D Y and Chen L Q 1999 Shape evolution and splitting of coherent particles under applied stresses Acta Mater.
47 247–57
[2] Khachaturyan A G 1983 Theory of Structural Transformations in Solids (New York: Wiley)

54 P Huang et al
[3] Johnson W C and Cahn J W 1984 Elastically induced shape bifurcations of inclusions Acta Metal. 32
1925–33
[4] Johnson W C and Voorhees P W 1987 Elastic interaction and stability of misfitting cuboidal inhomogeneities
J. Appl. Phys.61 1610–19
[5] Khachaturyan A G, Semenovskaya S V and Morris J W 1988 Theoretical analysis of strain-induced shape
changes in cubic precipitates during coarsening Acta Metal. 36 1563–72
[6] Miyazaki T and Doi M 1989 Shape bifurcations in the coarsening of precipitates in elastically constrained
systems Mater. Sci. Engng. A110 175–85
[7] Li D Y and Chen L Q 1997 Computer simulation of morphological evolution and rafting of γprime particles in
Ni-based superalloys under applied stresses Scripta Mater.37 1271–77
[8] Li D Y and Chen L Q 1999 Computer Aided Design of High Temperature Materials ed A Pechenik, R K Kalia
and P Vashishta (Oxford: Oxford University Press) at press
[9] Pan J, CocksACFandKucherenko S 1997 Finite element formulation of coupled grain-boundary and surface
diffusion with grain-boundary migration Proc. R. Soc. Lond. A453 2161–84
[10] Herring C 1951 The Physics of Powder Metallurgy ed W E Kingston (McGraw-Hill, New York) 143–79
[11] Binh V T, Moulin Y, Uzan R and Dreschler M 1976 Grain boundary groove evolutions by surface self-diffusion
on a plane and on a wire Surf. Sci. 57 184–204
[12] Binh V T, Moulin Y, Uzan R and Dreschler M 1979 Grain boundary grooving under the influence of evaporation
(or corrosion) Surf. Sci. 79 133–56
[13] Nichols F A 1976 On the spheroidization of rod-shaped particles of finite length J. Mater. Sci. 11
1077–82
[14] Pharr G M and Nix W D 1979 A numerical study of cavity growth controlled by surface diffusion Acta Metall.
27 1615–31
[15] Hsueh C H, Evans A G and Coble R L 1982 Microstructure development during final/intermediate stage
sintering—i. Pore/grain boundary separation Acta Metall. 30 1269–79
[16] Martinez L and Nix W D 1981 A numerical study of cavity growth controlled by coupled surface and grain
boundary diffusion Metall. Trans. A13 427–37
[17] Takahashi Y, Takahashi K and Nishiguchi K 1991 A numerical analysis of void shrinkage processes controlled
by coupled surface and interface diffusion Acta Mater. 39 3199–216
[18] Takahashi Y, Inoue K and Nishiguchi K 1993 Indentification of void shrinkage mechanisms Acta Mater. 41
3077–84
[19] Sun B and Suo Z 1997 A finite element method for simulating interface motion—II. Large shape change due to
surface diffusion Acta Mater. 45 4953–62
[20] Carter W C, Taylor J E and Cahn J W 1997 Variational methods for microstructural-evolution theories JOM-J.
Min. Met. Mat. S. 49 30–6
[21] Suo Z 1997 Motions of microscopic surfaces in materials Adv. Appl. Mech. 33 193–294
[22] Sun B, Suo Z and Evans A G 1994 Emergence of cracks by mass transport in elastic crystals stressed at
temperatures J. Appl. Phys. Solids 42 1653–77
[23] Yang W and Suo Z 1996 Global view of microstructural evolution: energetics, kinetics and dynamical systems
Acta Mech. Sinica 12 144–57
[24] Sun B, Suo Z and Yang W 1997 A finite element method for simulating interface motion—i. migration of phase
and grain boundaries Acta Mater. 45 1907–15
[25] Kim B N and Kishi T 1997 2-dimensional simulation of grain boundary migration by finite element method
J. JPN. I. Met. 61 1347–51
[26] Huang J M and Yang W 1999 Three-dimensional evolution of interfaces under evaporation–condensation
kinetics: a finite-element simulation Model. Simul. Mater. Sci. Eng. 787–105
[27] Kim B N and Kishi T 1999 Finite element simulation of Zener pinning behavior Acta Mater.47
2293–301
[28] Kim B N, Hiraga K, Sakka Y and Ahn B W 1999 A grain-boundary diffusion model of dynamic grain growth
during superplastic deformation Acta Mater. 47 3433–9
[29] Kim B N 2000 Analyses of grain growth behavior in particle-dispersed materials by finite element simulation
J. JPN. I. Met. 64 21–6 (in Japanese)
[30] Kim B N 2001 Modeling grain growth behavior inhibited by dispersed particles Acta Mater. 49 543–52
[31] Prevost J H, Baker T J, Liang J and Suo Z 2001 A finite element method for stress-assisted surface reaction and
delayed fracture Inter. J. Solids and Structures 38 5185–203
[32] Mullins W W 1957 Theory of thermal grooving J. Appl. Phys. 28 333–9
[33] Hilling W B and Charles R J 1965 Surfaces, stress-dependent surface reactions, and strength High Strength
Materials ed V F Zakay (Wiley, New York) 682–705

Simulation of grain growth behaviour 55
[34] Cash J R and Karp A H 1990 ACM Trans. Math. Software 16 210–22
[35] Lee J K and Courtney T H 1989 Two-dimensional finite difference analysis of shape instabilities in plates Metall.
Tran. A20 1385–94
[36] Huang P Z, Li Z H and Sun J 2001 The splitting and cylinderization processes of damage microcracks analyzed
by finite element method Model. Simul. Mater. Sci. Eng. 9193–206