Acta Materialia 52 (2004) 1365–1378
www.actamat-journals.com
Large-scale simulations of Ostwald ripening in elasticallystressed solids. II. Coarsening kinetics and particle size distribution
K. Thornton a,*, Norio Akaiwa b, P.W. Voorhees a
a Department of Materials Science and Engineering, Northwestern University, 2220 Campus Drive, Evanston, IL 60208, USAb National Research Institute for Metals, Tsukuba, Japan
Received 6 January 2003; received in revised form 17 November 2003; accepted 19 November 2003
Abstract
Ostwald ripening of misfitting second-phase particles in an elastically anisotropic solid is studied by large-scale simulations. The
coarsening kinetics for the average particle size are described by a t1=3 power law with a rate constant equal to its stress-free value
when the particles are fourfold symmetric. However, the rate constant increases when the elastic stress is sufficient to induce a large
number of twofold-symmetric particles. We find that interparticle elastic interactions at a 10% area fraction of particles do not affect
the overall coarsening kinetics. A mean-field approach was used to develop a theory of Ostwald ripening in the presence of elastic
stress. The simulation results on the coarsening kinetics agree well with the theoretical predictions. The particle size distribution
scaled by the average particle size is not time invariant, but widens slightly with an increasing ratio of elastic to interfacial energies.
No time-independent steady state under scaling is found, but a unique time-dependent state exists that is characterized by the ratio
of elastic energy to interfacial energy.
� 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Coarsening; Coherent precipitates; Phase transformations; Alloys
1. Introduction
Predicting the growth of the average size of precipi-
tates in multiphase mixtures during thermal treatment is
of technological interest as the particle size controls
many material properties. In coherent solids, elastic
stress plays a significant role in determining the micro-structure, as discussed in [1] (hereafter, Paper I). This
elastic stress can come from many sources, such as the
internal stress due to misfit between the particles and the
matrix or an applied stress. We consider the most ge-
neric source of elastic stress, that coming from the misfit,
or the difference between the lattice parameters of the
matrix and the precipitate, which is present in virtually
all two-phase coherent solids.To characterize a system in which interfacial and
elastic energies compete, we use the dimensionless
* Corresponding author. Tel.: +1-847-491-7818; fax: +1-847-491-
7820.
E-mail address: [email protected] (K. Thornton).
1359-6454/$30.00 � 2003 Acta Materialia Inc. Published by Elsevier Ltd. A
doi:10.1016/j.actamat.2003.11.036
parameter that is a measure of the relative importance
of elastic and interfacial energies in the system [2],
L ¼ �2lC44=r; ð1Þwhere � is the particle–matrix misfit, l is a characteristiclength, e.g., an equivalent radius of a particle, C44 is an
elastic constant, which is used for non-dimensionaliza-
tion of other elastic constants, and r is the interfacial
energy. For a system with many particles, hLi ¼�2hliC44=r, where hf i indicates the average of f over all
particles. L is a ratio of a characteristic elastic energy,
�2C44l3 (�2C44l2 in 2D), to a characteristic energy due to
the presence of interfaces, rl2 (rl in 2D). L can beconsidered a dimensionless radius, as it is linearly pro-
portional to the characteristic length of a particle.
In the absence of stress the coarsening process is
driven by the reduction in interfacial energy. As a result
the kinetics of ripening are described by hRðtÞi3 ¼ Kt,where hRðtÞi is the average particle size at time t [3,4].The dynamics of coarsening in stress-free two-phase
systems, such as liquid–liquid and solid–liquid two-phase mixtures, have been studied both experimentally
ll rights reserved.
1366 K. Thornton et al. / Acta Materialia 52 (2004) 1365–1378
and theoretically; for reviews see [5,6]. After sufficiently
long coarsening, it is predicted that coarsening systems
become self-similar under scaling of length by the av-
erage particle size. The form of the spatial correlation
function, particle size distribution (PSD) and rate con-stant, K, have been predicted [5,6].
In contrast, in solid–solid systems wherein the parti-
cles possess a different lattice parameter than the matrix
and are coherent, the coarsening process is driven by the
decrease in the sum of the elastic and interfacial energies.
These particles engender long-ranged elastic strain fields
that affect particle shapes, spatial correlations and can
even cause smaller particles to grow at the expense oflarger particles. Understanding the effects of elastic
stress on the coarsening process has been hampered by
the strong dependence of the elastic field on the shape of
a particle. Therefore, it is necessary to allow particle
shapes to evolve in a manner that is consistent with these
long-ranged elastic and diffusion fields. It is thus not
surprising that most theoretical studies have employed
numerical simulations using a variety of methods fromthe phase-field method [7–13] to Ising models [14–16].
For reviews see [5,16,17]. A range of behavior has been
reported from these studies, from no change in the
power law exponent for elastically homogeneous sys-
tems [12,18,19] to a prediction of a different exponent
[13,20]. Other studies indicate that elastic inhomogeneity
causes stabilization against coarsening [19,21–24]. Ana-
lytically, a change of the exponent in the power-law ispredicted if the system evolves in a self-similar manner
at the limit elastic energy dominates the system [25].
In this paper we examine the kinetics of ripening as
measured by several different length scales. We develop
a new analytical theory for coarsening based upon the
Gibbs–Thomson equation using the insight provided by
the simulations. Elastic interactions and its effects on
coarsening kinetics are examined in detail, as well as theevolution of particle size distribution. Together with the
characterization of particle morphology and the spatial
distribution in Paper I, this paper provides a complete
analysis of the results of numerical simulations of Ost-
wald ripening in an elastically homogeneous anisotropic
system.
2. Formulation
We have discussed the advantages and disadvantages
of existing numerical approaches that can be applied to
a study of coarsening systems in Paper I. We have
chosen to use a sharp interface description of the coars-
ening process, which provides excellent resolution of
the interfaces, combined with algorithms for solvingboundary integral equations [26], non-stiff time stepping
[27], and a fast multipole method for anisotropic
elasticity [28].
We consider a system of misfitting particles (b) in a
matrix phase (a) that is elastically homogeneous with
coherent interfaces. The misfit strain, �, is taken to be
purely dilatational. The elastic constants are assumed to
be that of pure Ni and are c11 ¼ 1:98 and c12 ¼ 1:18 afternon-dimensionalization by C44 (c44 ¼ 1 by this defini-
tion). Lower case variables are dimensionless throughout.
We approximate an infinitely large system by a compu-
tational cell repeated periodically to fill all space, thus
eliminating edge effects. Therefore, we impose periodic
boundary conditions around the computational cell.
The system is governed by the following equations.
The dimensionless concentration, c, in the matrix phasesatisfies the steady-state diffusion equation
r2c ¼ 0: ð2ÞThe concentration in the matrix at the particle–
matrix interface is given by the Gibbs–Thomson
equation [29],
cðxÞ ¼ jþ L1
2st � ~et
�� ta � set
�; ð3Þ
where j is the dimensionless interfacial curvature, t is
the dimensionless elastic stress, ~e is the scaled elastic
strain, e is the scaled total strain, and sf t ¼ f b � f a for
a quantity f at the interface. Here, c is given by [30,31]
c ¼ ðC � Ca1Þls
lcCa1
; ð4Þ
and the dimensionless time, t, is
t ¼ T=~t; ð5Þwhere
~t ¼ fl2sD
; ð6Þ
f ¼ ðCb1 � Ca
1ÞlslcCa
1; ð7Þ
C is the mole fraction of solute, C1 is the value of C at a
flat interface in a stress-free system, superscripts a and bdenote the matrix and particle phases, respectively, ls isthe length scale used in non-dimensionalization, lc is thecapillary length, T is the dimensional time, and D is the
diffusion coefficient. The characteristic length l that
appears in L is taken to be the dimensional circularly
equivalent radius, R ¼ffiffiffiffiffiffiffiffiffiA=p
p, where A is the dimen-
sional area of the particle. At the beginning of a simu-
lation we assign the ratio hL0i, the initial value of Laveraged over all particles. Since L is proportional to the
equivalent radius of the particle, hLi averaged over allparticles increases with time. Therefore, the elastic ef-
fects become more important during coarsening. Eq. (2)
is solved with periodic boundary conditions around the
computational cell, the interfacial boundary condition,
Eq. (3), and a global mass conservation condition,
K. Thornton et al. / Acta Materialia 52 (2004) 1365–1378 1367
XNj¼1
Zcj
V ðxÞdsj ¼ 0; ð8Þ
where V is the normal velocity, x is a point on the in-
terface, dsj is the arc length element of the interface cj ofthe jth particle and N is the total number of particles.
The evolution of the interfaces is computed using an
interfacial mass balance,
V ðxÞ ¼ ocon
; ð9Þ
where n is the coordinate in the direction normal to the
interface. We assume that the lattice parameter is not a
function of concentration. This, along with elastic ho-
mogeneity, enables us to obtain the stress and strain
fields at the interfaces through an integral over the in-
terfaces only [32]. For details, see Paper I and [28].
Fig. 1. Microstructure and interfacial concentration: from the top,
t ¼ 2:34, 22.6, and 215.3, corresponding to hLi ¼ 2:0, 4.0, and 7.1, with
hL0i ¼ 1:5. The variations of the interfacial concentration in the matrix
are shown along the interfaces. The left column shows the total in-
terfacial concentration, while the right shows the contribution by in-
terparticle elastic interaction. A quarter area of the computational
domain is shown.
3. Interfacial concentration and evolution of microstruc-
ture
Our simulations are started with 4000 particles withvarious values of hL0i. Since hLi is proportional to the
average particle size, the simulations with different hL0iexplore various ranges of hLi. The area fraction of
particles is 10%. The runs are repeated two to four times
with different initial spatial distributions of particles to
minimize the effects of the initial configuration of the
particles.
The interfacial concentration in the matrix is a usefulquantity to examine in order to understand both the
global and local evolution of a coarsening system. In
Fig. 1, the interfacial concentration is shown on the
particle interfaces at hLi ¼ 2:0, 4.0, and 7.1. When dis-
cussing the multiparticle simulation results, we have
chosen the length scale in non-dimensionalization as
ls ¼ hL0i�2C44
r
� ��1
: ð10Þ
Thus, the initial dimensionless average particle size,
hr0i ¼ 1, and the dimensionless interfacial concentration
of a circular particle of size hr0i is one if there is no
elastic stress. A prime on c is used when this length scale
is employed. Only a quarter area of the computational
domain is displayed in Fig. 1 so that the details are ev-ident. A description of the microstructure evolution is
given in Paper I for the same set of microstructures.
Both interfacial energy and elastic energy affect the
interfacial concentration. The left column of Fig. 1
shows the total concentration. As it is shown later, the
total interfacial concentration for a given particle (cðxÞin Eq. (3)) is largely inversely proportional to the par-
ticle size. Fig. 1.1a–c clearly show that the smaller par-ticles have higher concentration, and therefore larger
particles generally grow at the expense of smaller par-ticles. This is similar to the case where elastic stress is
absent. However, examining the shapes and alignment
of the particles, it is clear that elastic stress plays a role
in this system. While the particles would generally be
circular at this area fraction if elastic stress is absent, the
particles are more like square or rectangular in Fig. 1.
At later times, it is also evident that particles become
spatially correlated. Because the interfacial concentra-tions, Eq. (3), drive the evolution, Fig. 1 allows us to
understand qualitatively the effects of elastic stress on
the coarsening process, and to examine how the differ-
ence changes the kinetics of the system.
The concentration in Fig. 1, c0, is 1 for a stress-free
circular particle of the initial average particle size since
non-dimensionalization where hr0i ¼ 1 is used. The total
concentration shown in Fig. 1 is greater than that givenby 1=hri ¼ 0:75 because there are additional contribu-
tions to the interfacial concentration due to the presence
of elastic stress. The right column of Fig. 1 shows the
1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20
Fig. 2. A magnification of a chain is shown with total concentration.
1368 K. Thornton et al. / Acta Materialia 52 (2004) 1365–1378
contribution of the interparticle elastic interactions to
the interfacial concentration,
c0IeðxÞ ¼ c0ðxÞ � c0selfðxÞ; ð11Þ
where c0selfðxÞ is the concentration that would exist in the
absence of other particles, given its current morphology.
We find that the elastic interaction portion of the con-
centration does not vary much in time compared to the
total concentration (cf. Fig. 1.2a–c). The total change in
the concentration over the entire color bar is the same
on both columns to show this effect. From hLi ¼ 2:0 to7.1, hc0Iei, c0Ie averaged over the interface of a particle,
then averaged over all particles, changes by only about
15% of the corresponding change in the total concen-
tration, hc0i. Therefore, particle alignment indeed lowers
the total energy, but most of the total energy change
during the evolution is due to the interfacial energy re-
duction for the range of hLi examined. However, as the
elastic stress become more important and the particlesalign, the spatial variation in c0Ie increases. In particular,
particles out of alignment with other particles tend to
have a higher concentration than those in alignment due
to higher c0Ie. Thus, they tend to dissolve to feed the
growth of particles in a chain. Many of the regions of
lower concentration in Fig. 1.2b are already forming
chains, and they are thus more likely to survive to the
time shown by Fig. 1.2c.It is often asked which of the two mechanisms is re-
sponsible for particle alignment: particle migration or
survival of particles that are in alignment. Some of the
particle formations seen in Fig. 1.2b–c give us a clue. In
Fig. 1.2b, a chain that is slightly misaligned in the
middle is present (see the red arrow on the figure).
Elastic interactions help it to survive, as indicated by the
lower concentration due to interparticle elastic interac-tions (blue) of the particles in the chain. Indeed, the
chain persists and is still present in Fig. 1.2c (see the red
arrow). Interestingly, much of the misalignment still
remains, indicating that particle migration is not effec-
tive in aligning the chain in this case. However, in an-
other situation (see the black arrow) where a particle in
the middle of a chain is displaced compared to the chain,
particle migration seems to be effective. Thus, particlemigration seems to be relatively short ranged, and
therefore most of chains must first be established
through selective survival of particles that are aligned. It
should also be noted that it is difficult to separate par-
ticle migration in the strict sense (source of mass is the
particle itself) and preferential growth/shrinkage to-
wards a certain direction (source of mass is the sur-
rounding particles).Once some alignment is established, it is clear that
particle migration does lead to further alignment in
many cases. A magnified view of a chain noted by the
blue arrow in Fig. 1.1c is shown in Fig. 2 with total
concentration variation along the interfaces. The parti-
cle in the middle has a lower interfacial concentration at
the bottom than at the top, and therefore it is expected
to move down to align itself with the one on the left. The
particle on the left is expected to move upwards as well.
Also, the distance between the two left particles is notoptimal, and the interfaces are moving closer to each
other. This is indicated by the lower concentration on
the interfaces that are close to the particles. Thus, the
dominant mass flow is from the top and the bottom of
the particles into the region between the interfaces, and
not from one particle to another across the small gap. If
the particles survive long enough, they will orient
themselves at a distance similar to that of an isolatedsystem of two particles in [33]. However, the smallest
particle clearly has a higher concentration and thus is
shrinking.
4. Kinetics
4.1. Simulation results
The average particle size is given by
hri ¼ 1
N
XNi¼1
ri; ð12Þ
where ri is the dimensionless circularly equivalent radius
defined by ri ¼ffiffiffiffiffiffiffiffiffiai=p
p, ai is the dimensionless area of the
ith particle, and N is the number of particles in the
system. In the simulation, hri increases as small particles
(r < 0:05hri) are removed from the system and N de-
creases. To improve statistical accuracy, we also take thetime average of the averaged quantities such as hri overa time interval of about 30.
The data shown in Fig. 3 fit very well to hri3 ¼ Ktwith K ¼ 0:443, 0.442, and 0.453, for hL0i ¼ 0:125, 0.25,and 0.5, respectively. These K �s are very close to the
stress-free value, K � 0:45 [34]. This is expected for
the hL0i ¼ 0:125 case because the system never reaches
the stage where elastic energy becomes important (hLi att ¼ 550 is approximately 0.78). However, even in the
case of hL0i ¼ 0:5, ripening proceeds as if there is little
or no effect from elastic stress, despite the significant
0 100 200 300 400 5000
50
100
150
200
250
<r>
3
<L0>=0.125: K=0.443
<L0>=0.25 : K=0.442
<L0>=0.5 : K=0.453
Time
Fig. 3. hri3 as a function time for low hL0i values, taken from [36].
500
K. Thornton et al. / Acta Materialia 52 (2004) 1365–1378 1369
amount of elastic energy in the system near the end of
the calculation where hLi � 3:1 at t ¼ 550.In contrast, Fig. 4 shows that with hL0i ¼ 1:0 and 1.5
there is an increase in the coarsening rate from that of a
stress-free system (a solid line with slope 0.45). The in-
crease is greater in the hL0i ¼ 1:5 case than in the hL0i ¼1:0 case. We assume now that after sufficient ripening,
simulation results can be characterized by a single pa-
rameter hLi, independent of the initial conditions. This issupported by a careful examination of our results in theshape factor and Fourier analysis of the shapes, as well
as the particle correlations discussed in Paper I, and of
the PSD in Section 5. Such phenomena have also been
suggested from experiments [24]. Then, with a proper
scaling of time, a run with a smaller hL0i can be con-
sidered an earlier part of a run with a larger hL0i. We
assume that coarsening kinetics are described by
hri3 ¼ Ks; ð13Þ
0 100 200 300 400 5000
50
100
150
200
250
<r>
3
<L0>=1.0
<L0>=1.5
Slope = 0.45Slope = 0.733
Time
Fig. 4. Same as Fig. 3, but for larger hL0i values, taken from [36].
that is, ignore the possible effects of elastic stress. Here, sis the time measured such that hrijs¼0 ¼ 0, and K is the
rate constant for hri3. Rewriting Eq. (13) in the form
hri3 ¼ ðhLi=hL0iÞ3 ¼ Kðt þ t0Þ; ð14Þwhere t0 is introduced to offset the fact that the simu-
lation time t is measured such that hrijt¼0 ¼ hr0i ¼ 1.
For hLiK 5, all cases follow Eq. (14) with single value
of K and t0. Therefore,
hLi3 ¼ Kðt þ t0ÞhL0i3: ð15ÞThis indicates that the proper form of the scaled time is
t̂ ¼ ðt þ t0ÞhL0i3; ð16Þresulting in
hLi3 ¼ Kt̂: ð17ÞThis equation, a priori, is only valid for self-similar
evolution where elastic effects are negligible. However, if
the time-dependent state is characterized by hLi and if
the simulation is given an initial condition such that its
early evolution follows Eq. (17), then the scaling of time,
Eq. (16), can be applied even when elastic effects play a
significant role in coarsening kinetics at a later time.
However, the form of the equation describing the ki-netics of coarsening may change at this later time. Using
this definition of the scaled time, the curves of hLi vs.
scaled time with various hL0i fall onto a single curve, as
shown in Fig. 5. Since the hL0i ¼ 0:5 and 1.0 cases fol-
low closely on the initial part of the hL0i ¼ 1:5 case, the
small increase in the coarsening rate of the hL0i ¼ 1 case
in Fig. 4 is an early part of the gradual increase in the
coarsening rate in the hL0i ¼ 1:5 case. In short, thesimulations using different initial conditions result in a
single evolving system characterized by hLi; the choice ofhL0i only changes the range in the scaled time that is
0 200 400 600 800 1000
(t+t0) <L0>3
0
100
200
300
400
<L>
3
<L0>=0.5
<L0>=1.0
<L0>=1.5
Fig. 5. hLi3 is plotted against the scaled time, which is defined as
ðt þ t0ÞhL0i3, taken from [36]. A small number, t0, is determined by
extrapolating the best fit line of hri3 vs. t to hri3 ¼ 0 for runs with
negligible elastic stress. Only a partial range of data is shown for
hL0i ¼ 1:5.
1370 K. Thornton et al. / Acta Materialia 52 (2004) 1365–1378
simulated. We therefore conclude that the proper pa-
rameter to examine the coarsening kinetics is hLi, whichis a single-valued function of scaled time t̂, rather than
the particle size, hri.The resulting equation describing the kinetics is
similar to that of a coarsening system in the absence of
elastic stress, but there is a major difference. In the
presence of elastic stress, the ratio of elastic energy to
interfacial energy increases linearly with the particle size.
Therefore elastic stress becomes increasingly more
important with time, and we no longer expect a self-
similar evolution. To develop a theory to describe this
evolving system, we must include the effects of the elasticstress.
4.2. Gibbs–Thomson equation for isolated particle
The mass flux from a particle to another is driven by
the difference in the interfacial concentrations in the
matrix. Thus in formulating a theory, we start by ex-
amining the interfacial concentration of an isolatedparticle in equilibrium as given by the Gibbs–Thomson
equation. This must be determined numerically since the
equilibrium particle shape is not a simple geometric
shape and changes with L. We examined the change of
dimensionless concentration of a particle in equilibrium
0 2 4 6 8 10L
0
2
4
6
8
10
12
14
c
Before bifurcationFitAfter bifurcationFit
Fig. 6. The dimensionless concentration for an isolated particle for
c11 ¼ 1:98, c12 ¼ 1:18, and c44 ¼ 1:0, taken from [36]. The linear fit and
the equilibrium shapes at various L are also shown.
Table 1
The constants of the linear fits to the equilibrium concentration (c12 ¼ 1:3)
Ar Lc Fourfold symmetric
Constant (a) Slop
1.3 13.8 1.012 1.22
2.5 5.30 1.008 0.88
4.5 4.09 1.010 0.66
as a function of L. In this section, we chose ls ¼ R, whereR is the dimensional particle size. Using dimensionless
elastic constants appropriate for Ni (c44 ¼ 1 by defini-
tion, c11 ¼ 1:98, and c12 ¼ 1:18), the interfacial con-
centration in the matrix as a function of L is calculated.Fig. 6 shows that, for the most part, two separate lines
fit the value of the interfacial concentration as a function
of L very well; that is, cðLÞ ¼ aþ bL with two sets of aand b. The changes in the slope and intercept occur near
a critical value Lc, where the symmetry of the equilib-
rium particle shape bifurcates from a fourfold symmet-
ric shape to twofold symmetric shapes. Lc ¼ 5:6 in this
case [2]. Within the range of L of interest, the linearrelationship between the interfacial concentration and Lholds as long as the symmetry of the equilibrium shape,
but not the actual shape, is fixed. The fits are given by
cðLÞ ¼ 1þ 0:87L for fourfold symmetric shapes;
ð18Þ
cðLÞ ¼ 1:654þ 0:76L for twofold symmetric shapes:
ð19ÞThe linear approximation of cðLÞ works well for other
choices of elastic constants. We examined c vs. L for
anisotropy ratios of Ar ¼ 2c44=ðc11 � c12Þ ¼ 1:3, 2.5 and
4.5 for c12 ¼ 1:3 and c44 ¼ 1:0. This range of Ar com-
prises most metals, except those with Ar < 1:0. As Ar
decreases, the bifurcation point, Lc, increases for Ar > 1.
This is given by an analytical fit, Lc ¼ 3:18Ar=ðAr � 1Þ.For all cases, the two-segment-linear fit with different
values of a and b predicts the value of cðLÞ quite well.The slope of the linear segments clearly depends on the
values of the elastic constants (cf. Table 1). The slope
decreases with increasing Ar for both the twofold sym-
metric region and the fourfold symmetric region, but the
change is greater for the twofold symmetric case. The
crossover between the two linear sections is sharp for
low Ar cases; it is imperceptible for Ar ¼ 1:3 and very
small for 2.5. However, more nonlinear behavior is ev-ident in the case of Ar ¼ 4:5, which clearly exhibits a
gradual crossover between the two linear sections over
the range of L from about 4 to 8. Fig. 7 shows cðLÞ andtwo linear fits for this case. The corresponding figures
are omitted for Ar ¼ 1:3 and 2.5 as the behavior is
similar to that in Fig. 6. In general, the two-segment
linear approximation of the interfacial concentration of
Twofold symmetric
e (b) Constant (a) Slope (b)
3 2.190 1.157
78 1.992 0.7382
57 2.042 0.4838
0 2 4 6 8 10 12 14L
0
2
4
6
8
10
12
c
Before bifurcationFit: 1.010 + 0.6657 LAfter bifurcationFit: 2.042 + 0.4839 L
Fig. 7. The dimensionless concentration for an isolated particle as a
function of L with Ar ¼ 4:5.
K. Thornton et al. / Acta Materialia 52 (2004) 1365–1378 1371
an isolated particle works well for L�s and Ar�s found in
many two-phase alloys.
When a particle remains fourfold symmetric past thebifurcation point, cðLÞ continues to increase linearly
with the slope given by that for fourfold symmetric
shapes below the bifurcation point. Therefore, in very
low volume fraction systems where interparticle inter-
actions may not cause twofold shape transitions beyond
Lc, the coarsening rate may not change from that of a
stress-free system, as discussed below. For the Ar ¼ 1:3case, the concentration for fourfold symmetric shapes isgiven by a linear function of L well beyond L ¼ 2Lc.However, the Ar ¼ 4:5 case exhibits a very small de-
parture from the linear approximation even for L < 2Lc.The Ar ¼ 2:5 case is similar, but the deviations occur
only for L about 2Lc and greater. Figures showing cðLÞfor fourfold symmetric shapes where L > Lc are omitted.
4.3. Mean-field theory
A theory of coarsening is based upon a solution to a
diffusion equation with boundary conditions as given by
the Gibbs–Thomson equation and global mass conser-
vation. The particle growth rates are then determined by
the interfacial mass balance condition. Below, we de-
velop each of these equations to describe coarsening in
an elastically stressed solid. We show that the equationsare identical to those used by Marqusee [35]. We assume
a mean-field description of the diffusion field, in which
the surrounding particles are considered an effective
medium. A mean-field description is reasonable in this
case, despite the obviously strong spatial correlations,
because the diffusional screening distance, or the dis-
tance over which a particle will interact diffusionally
with another particle, is much larger than the spatialcorrelations. Direct calculations indicate that for this
screened logarithmic potential, the screening distance is
approximately 15hri [34]. This is significantly larger thanthe interparticle correlation length (see Paper I, Fig. 9).
Thus, the diffusion field is described by
ðr2 � n�2ÞðCðrÞ � C1Þ ¼ 0; ð20Þ
where n is the screening distance that can be determined
self-consistently [35], r is the position vector, and C1 is
the bulk concentration. In the absence of elastic stress,
the particle shape is well approximated by a circle, and
thus the solution to Eq. (20) can be obtained analyticallyin terms of the zeroth modified Bessel function. In the
case where elastic stress affects the particle morphology,
the particles are no longer circular. We therefore further
approximate a particle as a point sink for which the
concentration profile can be estimated by the solution to
Eq. (20) with radial symmetry and an appropriate
boundary condition that includes the effects of elastic
stress at R, the circularly equivalent radius of the par-ticle. This approximation is valid at low area fraction of
the particle phase. Thus, for a particle located at the
origin, the total mass flow into the particle is given by
integrating the flux along a circle of radius R� PR that
contains the point sink;
JT ¼ZcDrCðvÞjv¼R� � nds
¼ 2pDðR�=nÞK1ðR�=nÞK0ðR=nÞ
½C1 � CðvÞjv¼R�; ð21Þ
where c is the contour of a circle of radius R� centered at
the origin, v is the radial coordinate, n is the unit normalon c, K0 and K1 are the zeroth and the first modified
Bessel functions, respectively, and CðvÞjv¼R, the dimen-
sional interfacial concentration in the matrix, is pro-
vided by the Gibbs–Thomson boundary condition. The
last equality is obtained using Marqusee�s result for thesolution for Eq. (20). K1ðxÞ / 1=x for sufficiently small x,and therefore ðR�=nÞK1ðR�=nÞ � ðR=nÞK1ðR=nÞ. Thus,
JT � 2pDðR=nÞK1ðR=nÞK0ðR=nÞ
½C1 � CðvÞjv¼R�; ð22Þ
which is then identical within the approximations to JTused in [35].
A mass balance equation for the area of a non-
circular particle is given by
dðA=AmÞdt
¼ JT; ð23Þ
where A is the area of the particle and Am is the molar
area inside of the particle, which is assumed constant.
We use A ¼ pR2 to be consistent with the definition of
radius used in the simulation and in the Gibbs–Thom-
son equation. Therefore,
dRdt
¼ JTAm
2p; ð24Þ
which is identical to the corresponding equation in [35].
1372 K. Thornton et al. / Acta Materialia 52 (2004) 1365–1378
The global mass conservation provides
C0 ¼ Cað1� /Þ þ Cb/; ð25Þwhere C0 is the overall composition, Ca and Cb are the
concentration in the matrix and the particle, respec-tively, and / is the area fraction of the particle phase.
We assume that R � lc and R � Llc so that
C0 � Ca1 þ ðCb
1 � Ca1Þ/: ð26Þ
Thus, the usual global mass conservation employed in
theories of Ostwald ripening is satisfied at this limit.
Finally, the interfacial boundary condition is given by
the Gibbs–Thomson equation. In absence of stress, theboundary condition is given by c ¼ 1 in dimensionless
form for a circular particle with ls ¼ R, or in dimen-
sional form,
C ¼ CðvÞjv¼R ¼ Ca1 þ lcCa
1R
; ð27Þ
at the interface of a particle with radius R. The first termis constant for all particles, and does not influence thecoarsening rate. The second term is the dimensional
form of the first term in Eq. (3) that is proportional to
the interfacial curvature. The dependence of this term on
the particle size drives the coarsening, where large par-
ticles grow at the expense of small particles. When
elastic stress is present, the dimensionless Gibbs–
Thomson equation for an isolated particle with the
equilibrium morphology is modified as shown in Eqs.(18) and (19). Furthermore, in a multiparticle system,
the Gibbs–Thomson equation differs from that of an
isolated particle with the equilibrium morphology as a
result of interparticle elastic and diffusional interactions
and the resulting non-equilibrium morphology. The
Gibbs–Thomson equation for a generally shaped parti-
cle in a multiparticle system with a circularly equivalent
radius of R in dimensional form is,
CðxÞ ¼ Ca1 þ lcCa
1bLþ lcCa1aR
þ CI xð Þ; ð28Þ
where x is a point on the interface,L ¼ �2C44=r, CI xð Þ isthe dimensional concentration due to interparticle elas-
tic interactions and deviations in morphology from the
equilibrium shape of the particle, and a and b are the
intercept and slope, respectively, of the Gibbs–Thomson
equation for an isolated particle with the equilibrium
shape. The first three terms follow directly from di-
mensionalizing Eqs. (18) or (19). The last term includes
any deviations from the equilibrium concentration. Wenow take the average of CðxÞ over the interface and the
statistical average over all particles of size R. Then,
hCiR ¼ Ca1 þ lcCa
1bLþ lcCa1aR
þ hCIiR; ð29Þ
where hf iR is the statistical average of f over all particles
of size R, and f is the average of f over the interface of a
particle. The first two terms are constants. We now as-
sume that hCIiR is not a strong function of R and can be
considered constant on the average. The validity of this
assumption will be examined at the end of this section.
The boundary condition is then given by
hCiR � G ¼ lcCa1aR
; ð30Þ
where G is a constant given by
G ¼ Ca1 þ lcCa
1bLþ hCIiR: ð31ÞThis is identical to the boundary condition for C at the
interface in the absence of stress, Eq. (27), and used by
Marqusee, except for the presence of a in the right-hand
side and the definition of the constant G. Thus, in this
statistically averaged sense, elastics stress can alter the
magnitude of the surface energy term in the classicalGibbs–Thomson equation. Consequently, the depen-
dence of the statistically averaged boundary condition
on R and the other defining equations, Eqs. (22) and
(24), are identical to those used by Marqusee. We thus
follow his analysis directly and by enforcing mass con-
servation and stability of the scaled PSD, we find that in
the limit T ! 1,
hRi3 ¼ DAmf /ð ÞlcCa1a
� �T ; ð32Þ
where R and T are the dimensional radius and time,respectively, and f ð/Þ gives the dependence of the
coarsening rate on the area fraction of particles [35]. (In
[36], Am was inadvertently dropped in the corresponding
equation to Eq. (32).) Consequently, in this elastically
homogeneous system, elastic stress will not modify the
exponent of the temporal power law. Furthermore, since
a change in the value of a is mathematically equivalent
to changing the value of the interfacial energy, the shapeof the PSD is predicted to be identical to that in the
absence of elastic stress at the given area fraction in the
limit that all particle have only one symmetry of shapes.
In addition, as long as the shapes remain fourfold
symmetric, where a ¼ 1, the amplitude of the power law
will remain unchanged; elastic stress will not influence
the coarsening kinetics even though the elastic stress
modifies the morphology of the particles and their spa-tial distribution. Only when a majority of the particles
are twofold shaped, where a 6¼ 1, will the coarsening
kinetics change.
The results from numerical simulations are in excel-
lent agreement with this theory. As shown in Figs. 3 and
4, elastic stress has no effect on the exponent or ampli-
tude of the temporal power law for hLi < 4:5. This is
consistent with the theory since many particles are stillapproximately fourfold symmetric in this range due to
the fact that their L values are below the bifurcation
point. Any twofold symmetry stems purely from elastic
and diffusional interactions, which remain small. In
contrast, for hLi > 8 the majority of the particles are
twofold symmetric. Thus, from Eq. (32) with a ¼ 1:654,we expect the exponent on hri to remain three and K to
2 4 6 8 10 12L
0.0
0.2
0.4
0.6
0.8
aver
age
c Ie’
<L> = 7.1
Fig. 8. The interfacial concentration due to interparticle elastic inter-
actions averaged over the interface of each particle is plotted at
hLi ¼ 7:1, for a simulation with hL0i ¼ 1:5.
2 4 6 8 10 12L
1.8
2.0
2.2
2.4
2.6av
erag
e c’
<L> = 7.1
Fig. 9. Scatter plot of concentration of particles, averaged over each
interface, at hLi ¼ 7:1, for a simulation with hL0i ¼ 1:5. The concen-
tration predicted by the theory is given by the curve (see text).
K. Thornton et al. / Acta Materialia 52 (2004) 1365–1378 1373
increase by a factor of 1.654 compared to the limit where
the particles are fourfold symmetric (K ¼ 0:443), re-
sulting in a new rate constant, K2 ¼ 0:733. The theo-
retically predicted slope is in excellent agreement with
that of the upper part of the hri3 curve for hL0i ¼ 1:5;see the dotted line in Fig. 4 with a slope given by
K2 ¼ 0:733. Linear regression of the curve in the range
t ¼ 300–420, where the particles are mostly twofold
symmetric, gives a slope of 0.730, compared to 0.733
predicted, with an estimated error of �0.028.
A recent study by Vaithyanathan and Chen [12] ex-
amined similar coarsening systems using a phase-field
method. For systems with various area fractions, theyfind that the coarsening kinetics are described by a single
power-law with a time exponent of approximately 1/3. It
appears from their results for 20% area fraction that the
fit was obtained at a sufficiently high hLi, such that most
of the particles are beyond the bifurcation point, and
therefore the single power-law result is consistent with
ours. Although a direct comparison may not be appro-
priate, the rate constant for their 10% area fraction caseappears to be close to our result for the twofold sym-
metric shape case as well.
In the above analysis, we have assumed that CI does
not bias the coarsening process, that is, the value of CI is
not systematically higher or lower depending on the size
of the particles. CI has two components: CIe due to in-
terparticle elastic interactions and CIs due to shape de-
viations from the equilibrium shape. Using thecalculated interfacial concentration in a multiparticle
system, we examine c0Ie, the dimensionless interfacial
concentration due to interparticle elastic interactions,
averaged over the interface of each particle. As intro-
duced in Section 3, a prime on c indicates that it is non-dimensionalized such that c0 ¼ 1 for a circular particle
of size hr0i ¼ 1 in the absence of elastic stress. For each
particle, c0Ie, the portion of c0I due to elastic stress, isplotted against L in Fig. 8 for hLi ¼ 7:1. As expected, c0Ievaries from particle to particle in the system. However,
the average of c0Ie over a given particle-size range is, to
within the scatter, a constant or at most a very weak
function of L (and thus r). As a result, while c0Ie is re-
sponsible for the development of spatial correlations
and the small changes in particle morphology, it will not
affect hrðtÞi, since c0Ie is nearly independent of r. That is,the elastic interaction term does not on average strongly
bias the growth rates of small particles compared with
large particles.
In Fig. 9, we plot c0, the total interfacial concentra-
tion averaged over the interface of each particle, as a
function of L for hLi ¼ 7:1. As discussed in Paper I, the
shapes of particles in general differ from those of the
equilibrium state of an isolated particle due to elasticand diffusional interactions. Thus, c0 includes variation
in concentration due to elastic interaction and due to
deviations from the equilibrium morphology, as well as
the dependence on the particle size as given by the
Gibbs–Thomson equation. Fig. 9 also shows for com-
parison the concentration predicted by the theory and
used in the analysis for coarsening kinetics,
hc0ðrÞir ¼arþ g; ð33Þ
where g corresponds to G in Eq. (31), g ¼ bhL0i þ hc0Iir,and hf ir is the statistical average of f over all particles of
size r. To obtain the curve, a ¼ 1 and b ¼ 0:87 are used
for L < Lc as given by Eq. (18) (shown with dotted
curve), while a ¼ 1:654 and b ¼ 0:76 is employed for
L > Lc (Eq. (19), dashed curve). hc0Ii is calculated di-rectly from the simulation results. The result clearly
shows the trend in c0 from the simulation follows the
Gibbs–Thomson equation despite the large scatter. (hc0Iiris assumed to be independent of r and thus taken equal
hc0Ii).
0.0 0.5 1.0 1.5 2.0ρ
0.0
0.5
1.0
1.5
f(ρ)
<L0>=0.5<L0>=0.125Initial Condition
Fig. 11. The PSDs for hL0i ¼ 0:125 and 0.5. Each PSD is produced by
using the results in a range where coarsening has tripled to quadrupled
the average size of the particles (i.e., 0:375K hLiK 0:5 for hL0i ¼ 0:125
and 1:5K hLiK 2:0 for hL0i ¼ 0:5). They are nearly identical, indi-
cating that elastic stress does not significantly affect the PSD below
hLi � 2:0.
1374 K. Thornton et al. / Acta Materialia 52 (2004) 1365–1378
5. The particle size distribution
We first establish that the initial condition we adop-
ted is appropriate. Fig. 10 shows the evolution of the
PSD for a system with very small elastic stress,hL0i ¼ 0:125, with the initial PSD shown by solid circles.
The initial PSD is obtained from the steady-state result
of 2D coarsening simulation in the absence of elastic
stress by Akaiwa and Meiron [34], which also agrees
well with an analytical theory prediction [35]. The curves
correspond to times at which hri has increased by a
factor of two (dotted line), three (dashed line), and four
(solid line). When hLi is small, the system should evolveapproximately as if there is no elastic stress present.
Therefore, we expect the PSD to reach a time indepen-
dent form when scaled by the average particle size,
identical to that of stress-free PSD. Indeed, Fig. 10
shows an agreement between the initial PSD and the
PSD at various hLi < 0:5 to a good approximation.
The same initial PSD is used regardless of hL0i. Thiscan be problematic if the PSD is sensitive to a change inhLi below all hL0i because it may cause a long transient
evolution in simulations. Therefore, we compare the
PSDs in Fig. 11 when the systems have coarsened by a
factor of three to four for hL0i ¼ 0:125 and hL0i ¼ 0:5with the initial PSD. To reduce statistical noise, the PSD
from four runs are averaged in the range 0:375KhLiK 0:5 and 1:5K hLiK 2 for hL0i ¼ 0:125 and 0.5,
respectively. The PSD for hL0i ¼ 0:5 case is almostidentical to that for the hL0i ¼ 0:125 case. Therefore, the
elastic stress does not affect the PSD up to hLi � 2:0.The result is consistent with the theory at hLiK 2:0,which predicts that if the particles remain fourfold
symmetric, the coarsening proceeds as if the elastic stress
is not present, even if the elastic stress is large enough to
0.0 0.5 1.0 1.5 2.0ρ
0.0
0.5
1.0
1.5
f(ρ)
<L>~0.25<L>~0.375<L>~0.5Initial Condition
Fig. 10. The PSDs in a system with very small elastic stress,
hL0i ¼ 0:125 at various times at which hLi ¼ 0:25 (dot), 0.325 (dash),
and 0.5 (solid). The initial condition is shown by filled circles. Here, qis the scaled radius, q ¼ r=hri, and f ðqÞ is the scaled frequency such
that the area under the curve is 1.
change the morphologies of particles into non-circular
shapes. The PSD at hLi � 1:5, the maximum value of
hL0i, is very similar to the initial PSD. Thus, the initial
PSD is acceptable for all values of hL0i we employed.
In Paper I, we examined the existence of an evolvingattractor state that is solely characterized by the value of
hLi. For example, the average shape factor evolves
rapidly to a value that depends only on the value of hLi,or that of the attractor state. Although the approach to
the attractor state is slower than that of the average
shape factor, the anisotropic pair-correlation functions
are also consistent with the existence of an evolving state
that is only a function of hLi. The analysis of the PSDsindicates a similar conclusion. Fig. 12 compares the PSD
obtained at hLi � 5 from the two runs with different
0.0 0.5 1.0 1.5 2.0ρ
0.0
0.5
1.0
1.5
f(ρ)
<L0>=1.0<L0>=1.5
Fig. 12. The PSDs at hLi � 5 for hL0i ¼ 1:0 and 1.5. The two cases are
nearly identical, and it indicates that the value of hLi uniquely char-
acterizes the system after the transient evolution passes.
K. Thornton et al. / Acta Materialia 52 (2004) 1365–1378 1375
initial conditions, hL0i ¼ 1:0 and 1.5. They are in
agreement to within statistical fluctuations.
It is then reasonable to study the evolution of PSD
with hLi. We gather all data from various runs, taking
care not to include those that may have transient effects,and plot the results at hLi � 1:5, 3.0, 6.0 and 8.0 in
Fig. 13. Note that the statistical quality decreases with
increasing hLi due to the decrease in the total number of
particles in the system. The plot shows a noticeable
change in the PSD as hLi increases. As we discussed
previously, the change in the PSD is negligible at
hLiK 1:5, and therefore the PSD at hLi � 1:5 can be
considered identical to the steady-state PSD in the ab-sence of elastic stress. On the other hand, the PSDs at
greater values of hLi show significant variations. At
hLi � 3, the peak is lower and the distribution is very
slightly widened. The trend continues, and by hLi � 8,
the peak is much lower and the distribution wider to-
ward the large particle sizes than the PSD at hLi ¼ 1:5.The data clearly show that the statistical significance is
smaller at hLi ¼ 8:0, where some raggedness in the PSDis evident, since only about 500 particles are used in
obtaining the PSD. However, the consistency of the
trend – the decrease in the height and the increase in the
width of the distribution – supports the claim that this
trend is a result of increasing elastic stress in the system,
not from statistical fluctuations. Thus, for hLi > 1:5, thePSD as measured by the circularly equivalent radius
scaled by the average particle size is not self-similar.Cho and Ardell�s [37] coarsening experiment in a Ni–
Si alloy reports no systematic change of the standard
deviation of the scaled PSD on the coarsening time at
long times. On the other hand, Miyazaki and Doi report
continuous decrease in the standard deviation of the
PSD (that is, narrowing of the PSD) in a Ni–Mo alloy at
long times [38]. The direct comparison between these
experimental results and our simulation and theory is
0.0 0.5 1.0 1.5 2.0ρ
0.0
0.5
1.0
1.5
f(ρ)
<L>~1.5<L>~3<L>~6<L>~8.0
Fig. 13. PSD evolution. The data is gathered from all available runs.
The distribution widens slightly and its height decreases as the elastic
effects increase.
not appropriate due to the differences between our as-
sumptions and the physical properties of these model
alloys. Factors such as the dimensionality, the volume
fraction, and the differences in elastic constants of the
precipitates and the matrix may affect the simulationresults significantly. However, a recent experiment by
Lund and Voorhees using three dimensional measure-
ment of the volume of c0 particles in a c–c0 system also
found widening of the PSD compared to that in the
absence of elastic stress [39], a qualitative agreement
with the simulation result.
The theory does not take into account changes in the
PSD. We also ignore the fact that even at a sufficientlyhigh hLi, there are particles at the small end of the PSD
that have L < Lc; thus, the theory is only exact as
L ! 1 if no other assumptions are violated at this limit.
In particular, the shape of the PSD predicted by the
theory does not agree with the simulations for hLiJ 3.
This difference may be a remnant of the transition in the
average morphology of the particles from fourfold to
twofold with increasing L. During this transition, therewill be a mixture of particles with a significant fraction
of both two- and fourfold shapes, thus violating the
assumption of the theory that particles of only one
symmetry are present. We conjecture that the change in
the PSD from that predicted by theory is transient in
nature and associated with this morphological transition
region. We thus expect with longer coarsening that the
distribution will return to its shape at lower hLi, as-suming that no new phenomena, such as coalescence,
occur at these larger values of hLi. The reason why the
non-steady-state distribution does not alter the rate
constant from that predicted by theory is that there
appears to be a rather weak coupling between the value
of the rate constant and the shape of the distribution
function for coarsening processes in 2D [34].
5.1. Other measures of kinetics
We examine other measures of the evolution of the
system. Here, we have chosen three quantities: (1) thetotal interfacial length per unit area, (2) the average size
of particles along the major axis of the particles, and (3)
the average size of particles along the minor axis of the
particles. The definition of the particle size, r, used
above is based on a measure of the area of the particle,
regardless of its shape. The three measures we examine
here differ from r in that they depend on the shape of the
particle, as well as on its area.The interfacial length of a particle is influenced by
both its area and shape. In the absence of stress, parti-
cles are nearly circular (in 2D) except at high area
fractions, and the total surface length per unit area is
inversely proportional to the average particle radius (or
the average circularly equivalent radius if not circular).
However, when elastic stress is important, the particle
1
2
2a
2a
Fig. 15. Schematic of an elongated particle. The sizes along the major
axis, a1, and along the minor axis, a2, are shown.
1376 K. Thornton et al. / Acta Materialia 52 (2004) 1365–1378
shapes deviate from circles significantly due to the
competition between the surface energy and elastic en-
ergy. In this case, the interfacial length per unit area
increases compared to that in the absence of elastic
stress for a given area fraction of particles and a givenaverage circularly equivalent radius.
In principle, it is best to separate the kinetics of rip-
ening and the shape evolution, but it is not always
possible in experiments. Measuring the volume of each
particle involves identifying individual particles, and
thus requires three-dimensional information. Therefore,
Lund and Voorhees [40] chose to measure the total in-
terfacial area per unit volume of c0 particles in a c–c0
mixture. A corresponding quantity in 2D is the total
surface length per unit area, SA, and it is a decreasing
quantity with time. Lifshitz–Slyzov–Wagner theory [3,4]
predicts S�3A / t. In Fig. 14, S�3
A is plotted against time
using hL0i ¼ 1:5 (compare with the dash line in Fig. 4).
The effect of an increasing coarsening rate constant due
to the presence of twofold shaped particles is not evident
here. If the rate constant increases while the shapes ofparticles remain nearly circular, we expect a similar in-
crease in the rate of change in S�3A . However, as the
particles elongate with increasing hLi, the surface lengthsof these particles become greater than those of nearly
circular particles with the same areas. Thus, the increase
in the coarsening rate as measured by hri is compensated
by the increase in SA due to shape change. The ampli-
tude of the power law, KS in S�3A ¼ KSt, is approximately
constant over the range of hLi studied, except for the
very early times, tK 80. This result is in agreement with
the experimental study by Lund and Voorhees [40]. KS is
about 15% less than that found in a stress-free system.
The particle size may be measured by a side length.
We employ the size along the major axis of the particle,
a1, and the size along the minor axis, a2. A schematic
0 100 200 300 400Time
0
5.0•103
1.0•104
1.5•104
2.0•104
2.5•104
SA-3
Fig. 14. S�3A , (the surface length per unit area)�3, vs. time for the
hL0i ¼ 1:5 case shown in Fig. 4. Although S�1A is a measure of length
like hri, it exhibits a very different evolution in time. Most significantly,
the increase in coarsening rate seen in hri for hLiJ 7 is mostly offset by
the increase in surface length due to elongation.
diagram is given in Fig. 15. These quantities have been
studied by Finel [13] for his numerical study using the
phase-field method. As in SA, a1 and a2 measure both
the growth of the particle and the particle morphology
evolution. Fig. 16 shows the evolution of hri, ha1i, andha2i, normalized to hr0i ¼ 1, with time. Due to the
elongation of the particles, ha1i increases more rapidly
than hri, while ha2i lags hri in growth. It has been sug-
gested in [13] that ha1i follows t1=2 power law, while ha2iis described by a t1=4 power law. To examine our results
from this viewpoint, we have plotted ha1i2 against time
in Fig. 17. (The corresponding figure for ha2i4 is omit-
ted.) Since Finel�s theory predicts these power laws onlywhen particles are twofold shaped, a poor linear fit at
early time is expected. Therefore, only late-time data
should yield a straight line. The power law for ha1i is notinconsistent with a t1=2 dependence, and in fact, a good
0 100 200 300 400
Time
0
2
4
6
8
10
<si
ze>
<r><a1><a2>
Fig. 16. Three measures of sizes, ha1i, ha2i, and hri, normalized to hr0i,are plotted against time for hL0i ¼ 1:5. As expected, the size along the
major axis increases faster than hri, and the size along the minor axis
increases slower than hri.
0 100 200 300 400
Time
0
20
40
60
80
<a 1
>2
DataStraight line
Fig. 17. ha1i2 is plotted against time for hL0i ¼ 1:5 with solid line. The
straight dashed line is given to emphasize the approximate linearity of
the curve at the late time.
K. Thornton et al. / Acta Materialia 52 (2004) 1365–1378 1377
linear fit is obtained for t > 200 or hLiJ 7:0. Similarly,
for ha2i4 we find a good linear fit t > 270, or hLiJ 7:8.Although we do not have an explanation, we find thatthe linear fit for ha2i4 is good at 0 < t < 270 which in-
cludes a regime where the particles are fourfold sym-
metric and a crossover where the number of twofold
shaped particles increases significantly. Since these data
are obtained by post-processing the simulation data, the
statistical accuracy is not as good as the data for hri vs.time. Given the statistical uncertainty and the limited
range of time where the power law is valid, we concludethat our results neither conclusively agree nor disagree
with those of Finel�s.The difference in elastic constants between the matrix
and the particles may alter the results presented here
[21–23,41]. In particular, elastic inhomogeneity may
modify the concentration-L relation in Eqs. (18) and (19)
significantly, or strengthen the interparticle interactions.
In systems with larger hLi than those considered here,other effects may also lead to different ripening behavior.
For example, although there is an elastic repulsion be-
tween the interfaces at short distances, particles aligned
along a chain can be squeezed together as the particles
elongate at high area fractions and high L. Thus, coa-lescence may occur [42]. We recently developed a hybrid
method to handle coalescence in our simulations, and
we plan to study high area fraction and/or high L sys-tems in the near future.
6. Conclusion
We studied the development of microstructures in
elastically anisotropic solids. In this paper, we focused
on the kinetics of coarsening and the particle size dis-tribution in systems with a 10% area fraction of parti-
cles. We find:
1. The t1=3 power law for the average circularly equiva-
lent radius, hri, and coarsening rate constant is given
by that in the absence of stress when elastic stress
is sufficiently small that the morphologies of the
majority of the particles are fourfold symmetric, spe-cifically, when the parameter, hLi, which is approxi-
mately the ratio of elastic to interfacial energies, is
below 4.5. For hLiJ 5, the coarsening rate increases
above the value for a stress-free system.
2. We find that the Gibbs–Thomson equation in the
presence of elastic stress for an isolated particle
of a fixed area in equilibrium is a sole function of
L ¼ �2C44R=r for given elastic constants, andthe form of the function depends only on the symme-
try of the equilibrium particle shape to a good
approximation.
3. A theory based on a mean-field approach and interfa-
cial concentrations that includes elastic stress effects
predicts that the coarsening kinetics are described by
the t1=3 power law, with the same coarsening rate con-
stant as in the absence of elastic stress when particlesare fourfold symmetric.When the system is dominated
by twofold symmetric particles the theory predicts the
same power law but with a larger rate constant. The
simulation results are in agreement with the theory.
4. Since the microstructure is not self-similar, the kinet-
ics of coarsening depend on the quantity used as a
length scale. The increase in coarsening rate found
for hri is not evident when the surface length per unitarea is employed. The rate constant for S�1=3
A de-
creases slightly compared to its stress-free value.
5. The particle size distribution scaled with average par-
ticle size is not self-similar, but changes with hLi. It isfound that the elastic stress has no effect on the scaled
particle distribution for hLiK 2:0. However, a signif-
icant change occurs above hLi � 3:0. As the average
particle size increases, the effects of elastic energy in-crease, and the height of the distribution decreases
while the width increases.
6. Using the results of Paper I and those in this paper,
we conclude that the microstructure can be character-
ized solely by the value of hLi.
Acknowledgements
We thank W.C. Carter, A. Finel, and M. Brenner for
stimulating discussions. This project was supported by
the National Science Foundation under Grant No.DMR-9707073.
References
[1] Thornton K, Akaiwa N, Voorhees PW. Acta Mater 2004, doi:
10.1016/j.actamat.2003.11.037.
1378 K. Thornton et al. / Acta Materialia 52 (2004) 1365–1378
[2] Thompson ME, Su CS, Voorhees PW. Acta Metall Mater
1994;42:2107.
[3] Lifshitz IM, Slyozov VV. J Phys Chem Solids 1961;19:35.
[4] Wagner C. Z Elektrochem 1961;65:581.
[5] Voorhees PW. Ann Rev Mater Sci 1992;22:197.
[6] Ratke L, Voorhees PW. Growth and coarsening: Ripening in
materials science. Berlin: Springer; 2002.
[7] Wang Y, Chen LQ, Khachaturyan AG. Acta Metall et Mater
1993;41:279.
[8] Onuki A. J Phys Soc Jpn 1989;58:3069.
[9] Khachaturyan AG. Theory of structural phase transformations in
solids. New York: John Wiley; 1983.
[10] Leo PH, Lowengrub JS, Jou HJ. Acta Mater 1998;46:2113.
[11] Wang Y, Chen LQ, Khachaturyan AG. J Am Ceram Soc
1993;78:657.
[12] Vaithyanathan V, Chen LQ. Acta Mater 2002;50:4061.
[13] Finel A. In: Turchi PEA, Gonis A, editors. Phase transformations
and evolution in materials. Warrendale, PA: TMS; 2000. p. 371.
[14] Lee JK. Metall Trans A 1996;27:1449.
[15] Fratzl P, Penrose O. Acta Metall Mater 1995;43:2921.
[16] Fratzl P, Penrose O, Lebowitz J. J Stat Phys 1999;95:1429.
[17] Hou TY, Lowengrub JS, Shelley MJ. J Comput Phys
2001;169:302.
[18] Fratzl P, Penrose O. Acta Mater 1996;44:3227.
[19] Sagui C, Orlikowski D, Somoza AM, Roland C. Phys Rev E
1999;58:R4096.
[20] Nishimori H, Onuki A. Phys Rev B 1990;42:980.
[21] Onuki A, Nishimori H. Phys Rev B 1991;43:13649.
[22] Nishimori H, Onuki A. Phys Lett A 1992;162:323.
[23] Onuki A, Furukawa A. Phys Rev Lett 2001;86:452.
[24] Paris O, F€ahrmann M, Fratzl P. Phys Rev Lett 1995;75:3458.
[25] Leo PH, Mullins WW, Sekerka RF, Vinals J. Acta Metall et
Mater 1990;38:1573.
[26] Greenbaum A, Greengard L, McFadden GB. J Comput Phys
1993;105:267.
[27] Hou TY, Lowengrub JS, Shelly MJ. J Comput Phys 1994;114:312.
[28] Akaiwa N, Thornton K, Voorhees PW. J Comp Phys 2001;173:61.
[29] Leo PH, Sekerka RF. Acta Metall 1989;37:3119.
[30] Voorhees PW, McFadden GB, Boisvert RF, Meiron DI. Acta
Metall 1988;36:207.
[31] Voorhees PW, McFadden GB, Johnson WC. Acta Metall Mater
1992;40:2979.
[32] Mura T. Micromechanics of defects in solids. Dordrecht, The
Netherlands: Kluwer Academic Publishers; 1987.
[33] Su CH, Voorhees PW. Acta Mater 1996;44:2001.
[34] Akaiwa N, Meiron DI. Phys Rev E 1995;51:5408.
[35] Marqusee JA. J Chem Phys 1984;81:976.
[36] Thornton K, Akaiwa N, Voorhees PW. Phys Rev Lett
2001;86:1259.
[37] Cho JH, Ardell AJ. Acta Mater 1997;45:1393.
[38] Miyazaki T, Doi M. Mater Sci Eng A 1989;110:175.
[39] Lund A, Voorhees PW. Phil Mag 2003;83:1719.
[40] Lund AC, Voorhees PW. Acta Mater 2002;50:2085.
[41] Paris O et al. Zeitschrift fur Metallkunde 1995;86:860.
[42] Wang Y, Banerjee D, Su CC, Khachaturyan AG. Acta Mater
1998;46:2983.