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: k-thornton@northwestern.edu (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.

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