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Acta Materialia 56 (2008) 2585–2591

Impact of diffusion on concentration profiles around near-criticalnuclei and implications for theories of nucleation and growth

Jiankuai Diao a, Rafael Salazar a,c, K.F. Kelton b, Lev D. Gelb a,*

a Department of Chemistry and Center for Materials Innovation, Washington University in St. Louis, St. Louis, MO 63130, USAb Department of Physics and Center for Materials Innovation, Washington University in St. Louis, St. Louis, MO 63130, USA

c Universidad Nacional Mayor de San Marcos, Fac. Fısica, AP 14-0149, Lima 14, Peru

Received 3 September 2007; received in revised form 14 January 2008; accepted 26 January 2008Available online 10 March 2008

Abstract

In order to better understand the interplay of diffusion and interfacial processes in nucleation phenomena we have performed kineticMonte Carlo simulations of a lattice gas model with realistic but generic microscopic dynamics. These simulations are used to probe thecomplete dynamic range extending from diffusion-limited through interface-limited kinetics. Most phenomenological theories describingnucleation, growth and/or coarsening focus on either the diffusion-limited or interface-limited regime. Calculations are performed oninitially monodisperse clusters placed into solutions of uniform concentration. In agreement with predictions, our simulations showthe appearance of regions of enhanced solute concentration around clusters smaller than the critical size, and of solute depletion aroundclusters larger than the critical size. The range and magnitude of these effects are largely controlled by the ratios of the rate of free dif-fusion to those of interfacial attachment and detachment processes. Furthermore, these simulations show that the rate of cluster growthdepends strongly on the diffusion rate and correlates well with the local solute concentration at the cluster surface, over the entiredynamic range studied. In ‘‘diffusion-limited” phenomenological models the solute concentration at the cluster surface is assumed tobe determined by the radius of the cluster, through a local-equilibrium condition. Our results indicate that in the intermediate regimein which the rates of diffusion and interfacial processes are similar, such assumptions are qualitatively incorrect and so models thatassume either fully diffusion-limited or fully interface-limited growth and coarsening should not be used. We show, in particular, thatthe recently proposed ‘‘coupled-flux model” correctly and naturally describes the underlying physics over the complete dynamic rangeand therefore is generally preferable.� 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Phase transformations; Nucleation and growth; Precipitation; Monte Carlo simulation; Multicomponent diffusion

1. Introduction

Most first-order phase transformations of interest inphysics, chemistry, biology and materials science are initi-ated by a process of nucleation. Polymorphic transforma-tions, such as freezing and condensation, for which thechemical compositions of the initial and final states areidentical, can usually be described well using the classicalBecker–Doring [1] theory of nucleation [2,3]. In such casesthe evolution of cluster size is ‘‘interface-limited”, that is,

1359-6454/$34.00 � 2008 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2008.01.044

* Corresponding author. Tel.: +1 314 935 5026; fax: +1 314 935 4481.E-mail address: gelb@wustl.edu (L.D. Gelb).

determined only by the rates of monomer attachment anddetachment (excluding coagulation events). However, thisis not the case for partitioning transformations where theinitial and final phases have different composition, suchas solid-state precipitation [4,5], void formation in irradi-ated metals [6] and nanocrystal formation from metallicglasses [7,8]. In such cases, the transport of material toand from growing and shrinking clusters must be correctlydescribed or else one obtains an incorrect functional formfor the growth behavior of large clusters [4,5].

‘‘Diffusion-limited” kinetics are assumed in many earlytheories, including the Ham model for growth from a super-saturated medium [9] and the Lifshitz–Slyozov–Wagner

rights reserved.

n, and k n,(k+ )

n, n,( and )

ClusterClusterClusterClusterClusterClusterClusterClusterClusterClusterCluster

Parent Phase

ShellInterface attachment rates

Cluster

Fig. 1. Schematic illustration of the processes considered in the coupled-flux model.

2586 J. Diao et al. / Acta Materialia 56 (2008) 2585–2591

(LSW) [10,11] theory of coarsening. Such models assume alocal chemical equilibrium condition at cluster surfaceswhich implies that, for spherical clusters, the solute concen-tration at the surface is determined entirely by the clusterradius. This condition is equivalent to assuming that theprocesses at the cluster surface occur very much faster thandiffusion through the surrounding medium. The LSWapproach has been augmented by both spatial inhomogene-ity [12] and nucleation terms [13] in more recent work.

A more recent approach that assumes neither very fastnor very slow diffusion is the ‘‘coupled-flux” model[14,15], itself an extension of an earlier method fordescribing the dynamic diffusion field near forming nuclei[16]. While a proper account of the coupling betweenthese two stochastic fluxes is extremely difficult, as a firstapproximation the varying concentration in the long-range diffusion field can be modeled using a shell of var-iable composition around each cluster [16,14,15]. Atomsare transported to and from the shell by diffusion in theparent phase. At the interface the normal attachment/detachment kinetics assumed in the classical theory areused.

In the coupled-flux model the distribution of clusters isconsidered to be a function of both the cluster size n andthe number of atoms of the correct type that are in thenearest neighbor shell to the cluster, q. Cluster growthis based on the relative rates of exchange of atoms withthe parent phase, and with the cluster. The distributionfunction, N(n,q), is time-dependent and evolves in the dis-crete (n,q) phase space according to the followingequation:

dNðn; qÞdt

¼ In�1 � In þ Jq�1 � Jq ð1Þ

where

In�1 ¼ Nðn� 1; qþ 1Þkþðn� 1; qþ 1Þ � Nðn; qÞk�ðn; qÞð2Þ

In ¼ Nðn; qÞkþðn; qÞ � Nðnþ 1; q� 1Þk�ðnþ 1; q� 1Þ ð3ÞJq�1 ¼ Nðn; q� 1Þaðn; q� 1Þ � Nðn; qÞbðn; qÞ ð4ÞJq ¼ Nðn;qÞaðn; qÞ � Nðn; qþ 1Þbðn; qþ 1Þ ð5Þ

The rate constants are defined as in Fig. 1. k+ is the rateof attachment of a monomer from the shell to the cluster,and k� is the rate for the reverse process, i.e. detachment.a is the rate of transfer of monomers from the shell to thebulk and b is the rate for the reverse process. To solvethese equations we first consider the equilibrium casewhere all fluxes are zero; the problem can then be rewrit-ten as

Rðn; qÞ � N eqðnþ 1; q� 1ÞN eqðn; qÞ ¼ kþðn; qÞ

k�ðnþ 1; q� 1Þ ð6Þ

Sðn; q� 1Þ � N eqðn; qÞN eqðn; q� 1Þ ¼

aðn;q� 1Þbðn; qÞ ð7Þ

One possible solution of these equations is a symmetricone

kþðn;qÞ ¼ q6D0

k2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiRðn; qÞ

pð8Þ

k�ðnþ 1; q� 1Þ ¼ q6D0

k2

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiRðn; qÞ

p ð9Þ

aðn; q� 1Þ ¼ nqD

k2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSðn; q� 1Þ

pð10Þ

bðn; qÞ ¼ nqD

k2

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSðn; q� 1Þ

p ð11Þ

where D0,D are the diffusion constants in the shell andbulk, respectively, k is the jump distance, and n is a correc-tion constant [14]. The following approximation for theequilibrium cluster distribution is applied [14]:

N eqðn; qÞ ¼ N 0 expð�W n=kBT Þanqm

n !

q!ðqmn � qÞ!

x1� x

� �q

ð12Þwhere N0 is the initial number of monomers, x is the mono-mer concentration or bulk density, qm

n ¼ 4n2=3 is the maxi-mum number of monomers in the shell of a cluster of size n,and an is a normalization constant. The work of cluster for-mation is determined from the free-energy profile with re-spect to the cluster size, and is here taken to be the sameas in the classical theory

W n ¼ ndlþ An2=3 ¼ W � �2nn�

� �þ 3

nn�

� �2=3� �

ð13Þ

where A ¼ ð36pÞ1=3�v2=3r for a spherical cluster shape, �v isthe atomic volume, and r is the surface tension. dl is thedifference in chemical potential per atom of the new phaseless that of the initial phase, which can be calculated fromthe equation of state of the precipitating phase; for lowconcentration c an ideal-gas approximation givesdl = kBTln(c/ccoex), where ccoex is the concentration atcoexistence. Finally, the critical size n* and the correspond-ing critical energy W* can be calculated

n� ¼ 2A3jdlj

� �3

ð14Þ

W � ¼ A3ðn�Þ2=3 ð15Þ

J. Diao et al. / Acta Materialia 56 (2008) 2585–2591 2587

In this work, we use an Euler-type first-order integrationalgorithm to obtain the time evolution of N(n,q) from Eq.(1) under the other approximations given above. A moredetailed presentation of the coupled-flux model and morecomplex numerical solutions have appeared previously[14,15]; here we have found that casting of the R and S quan-tities in terms of the equilibrium distribution, Neq(n,q), sim-plifies the presentation and implementation considerably.

This model has been previously applied to precipitationtransformations in solids [4,5]. The predicted time-depen-dent nucleation behavior is similar to that observed inmany polymorphic transformations. In cases where diffu-sion is rapid, the nucleation rate scales with the interfacialattachment rate, as described by classical theory. However,as long-range diffusion becomes more important, the nucle-ation rate becomes smaller and the induction time becomeslarger than calculated from the classical theory, sometimesby several orders of magnitude. As the diffusion rate isincreased (or the concentration of the initial phase becomesmore similar to that of the new phase), the nucleation rateand the induction times approach their classical theoryvalues.

The coupled-flux model and other models that explicitlytreat diffusion predict that the density (concentration) ofsolute near clusters smaller than the critical size will beenhanced, because these clusters should be (on average)shrinking and thus enriching the regions around them. Thiscan affect phase formation and stability and has beeninvoked to explain nanocrystal formation during the crys-tallization of some metallic glasses [7]. This key predictionwas first made over 50 years ago [17] but has never beendirectly observed experimentally, because the range of theenrichment is only of the order of nanometers.

Here, we report the results of a simulation study of dif-fusion-limited cluster growth and shrinkage in a lattice-based model with conservative and physically reasonablekinetics. By averaging over many widely separated, nearlymonodisperse clusters, we can specifically address theeffects of different diffusion rates on cluster growth andshrinkage kinetics and on the concentration profiles of sol-ute around the clusters.

2. Lattice gas calculations

Computer simulations have been used to study nucle-ation and growth in many systems, including partitioningtransformations such as vacancy clustering [18] and solid-state precipitation [19,20], as well as crystallization fromsolution [21,22] and phase transformations in lattice gasmodels [23–28]. However, the interplay of diffusion andinterfacial attachment rates and its effects on cluster growthand shrinkage dynamics have not been studied under con-trolled conditions. We consider a simple three-dimensionalcubic lattice gas model [29,30]

H ¼ �JX<ij>

rirj ð16Þ

where each lattice site is either occupied or empty as de-noted by r = 1 or 0, J is an attractive interaction betweenneighboring particles and the summation runs over allpairs of nearest neighbors. This model exhibits a high-den-sity/low-density phase transition, the equilibrium proper-ties of which are very well-known [30,29]; it can also beinterpreted as describing atomic impurities or vacanciesin a crystal, where these ‘‘solutes” interact via an effectiveattractive potential.

In order to study non-equilibrium and kinetic phenom-ena in a lattice gas one must introduce a microscopicdynamics amenable to computer simulation. Many possi-ble dynamic schemes can be constructed [28], differing inwhich microscopic processes are ‘‘enabled” and their rates.Here we develop a dynamics with only a single adjustablequantity that nevertheless allows for a systematic studyof diffusion-related effects. We consider only three pro-cesses: free diffusion (D), interfacial attachment (IA) andinterfacial detachment (ID). These describe motion of aparticle between adjacent lattice sites where neither hasoccupied neighbors (D), where only the final site has occu-pied neighbors (IA), and where the initial site has occupiedneighbors (ID). Note that the third case includes cluster-restructuring processes such as translation of a particlealong a surface or step-edge. Arrhenius-type reaction ratesare given by

kD ¼ A expð�dD=kBT Þ ð17Þ

kIA ¼ A expð�dI=kBT Þ ð18Þ

kID ¼ A exp � dI þ JX

j=i

rj

!=kBT

" #ð19Þ

where dD and dI are the potential energy barriers associatedwith diffusion and interfacial attachment/detachment,respectively, kB is Boltzmann’s constant, T is the tempera-ture, and the summation in the kID expression runs over thenearest neighbors j of the particle at i. In order to detachfrom a cluster or translate along its surface, a particle mustovercome both the interfacial barrier dI and its bindingenergy to the cluster. These dynamics satisfy microscopicreversibility. We choose the unit of time to be k�1

IA (equiva-lent to dividing through by Aexp(�dI/kBT)), and J as thenatural unit of energy. In these ‘‘reduced” units the temper-ature and energy difference are T* = kBT/J and(dD � dI)* = (dD � dI)/J, and the rates become

k�D ¼ exp½�ðdD � dIÞ�=T �� ð20Þk�IA ¼ 1 ð21Þ

k�ID ¼ exp �X

j=i

rj

!=T �

" #ð22Þ

kD/kIA, or equivalently, (dD � dI)*, is a new parameter thatcontrols the rate of diffusion relative to the rate of interfa-cial attachment, and is the single ‘‘adjustable” dynamicquantity within the reduced scheme. Of course, this choiceof dynamics determines only the time-dependent properties

2588 J. Diao et al. / Acta Materialia 56 (2008) 2585–2591

of the system; the equilibrium properties of the model aredetermined by T* alone.

In order to generate representative trajectories followingthese dynamics we have implemented three equivalentkinetic Monte Carlo algorithms: the first reaction method(FRM) [31,32], the variable step size method (VSSM)[31,32] and the random selection method (RSM) [32].Under the conditions studied here the RSM was found tobe the most efficient and was used in this work. For a singleRSM ‘‘move”, one first randomly selects a particle withprobability 1/N (N is the number of particles) and one ofits six nearest neighbor sites with a probability 1/6. A reac-tion i is then chosen with probability ki=

Pjkj; this can be

one of diffusion, interfacial attachment or detachment.Detachment from sites with different binding energies arecounted separately, as they have different rate constants.Finally, if the reaction i is possible in the present configu-ration, it is made to happen. The time index is always incre-mented by a fixed value dt ¼ 1=½6NðkD þ kIA þ

PkID�,

where the summation runs over all types of detachmentevents. dt is typically between 10�4 and 10�6 reduced timeunits. This sequence is repeated and the system evolvesstochastically.

3. Results and discussion

Our object in this work is the controlled study of theevolution of clusters and their local environment andhow this is influenced by diffusion and interfacialattachment rates. Since these effects will depend on clus-ter size, it is desirable to study monodisperse samples ofclusters in order to obtain size-selected data. Further-more, large samples are required in order to obtainhigh-quality averages. Therefore, rather than generateclusters by quenching from a high-temperature configu-ration (which would generate a broad distribution ofcluster sizes), we instead ‘‘seed” the system with manyspherical clusters of the same size. The initial clusterseeds are generated simply by templating a sphere ontothe cubic lattice.

Simulations were all done at T* = 0.4 (the model criticaltemperature is 1.1277 [29]). In different simulations, we havesimulated clusters of initial radii of each of r0 = 3,4,5,6 and7, at initial background densities c0 of 0.1% and 0.2%. Thelow-density phase boundary at this temperature, estimatedfrom Monte Carlo calculations, is approximately 0.057%,so these densities are both slightly supersaturated. Simula-tions were repeated at kD/kIA ratios of 0.00764,0.105, 1,9.49 and 42.5, corresponding to reduced energy bar-rier differences (dD � dI)* of 1.9,0.9, 0,�0.9 and �1.5.

These supersaturations are small enough that, althoughpossible, no nucleation of new clusters is observed on thetimescales accessible to this type of simulation.

For each set of conditions we typically averaged over 64independent simulations, each with 125 clusters arranged ina cubic lattice in a single large periodic cell; the datareported are therefore averaged over 8000 clusters of the

same initial size. The shortest distances between clusterswere 50 lattice sites for clusters of r0 = 3 or 4 and 100 lat-tice sites for clusters of r0 = 5–7.

The critical cluster size depends on both the backgroundconcentration and the temperature. On average, clusterslarger than the critical size grow and clusters smaller thanthe critical size shrink [2]. At T* = 0.4 the critical radiusis approximately 4.5 lattice sites at 0.1% background con-centration and approximately 2.5 lattice sites at 0.2% back-ground concentration; this was determined by measuringthe growth rates of clusters of various sizes and interpolat-ing for zero growth rate.

Fig. 2a shows the evolution of the size distribution D(n)of seeded clusters of r0 = 7 (1419 particles, above the criti-cal size) for kD/kIA = 1 at c0 = 0.1%. Some of these clustersgrow and some shrink, but the average size increases withtime. The size distributions are Gaussian, with widths thatincrease approximately linearly with time. Fig. 2c showsthe evolution of the normalized radial distribution func-tion, g(r), of the background particle concentration (iso-lated solute particles) around these clusters. Note thatg(r) measures the background concentration around theclusters relative to a homogeneous ideal-gas reference state,and thus decays to a value near 1 at large distances. Deple-tion is observed near the clusters, and grows in both rangeand severity as clusters deplete their local environmentsand additional particles slowly diffuse in from fartheraway. Fig. 2b and d shows the corresponding quantitiesfor r0 = 3 (123 particles, below the critical size) with kD/kIA = 0.105 at c0 = 0.1%. Here the clusters shrink (on aver-age), and the region around them is enriched in particles.This occurs first only at short range, but as particles slowlydiffuse away from the shrinking clusters the effect extendsto longer range. For particles that are, on average, grow-ing, and those that are shrinking, g(r) tends towards a lim-iting form at long times in both cases. Similar behavior wasobserved in all cases considered, except for those when thediffusion rate was extremely high (see below). These effectsare predicted both by LSW-type theories and coupled-fluxmodels, but not by the classical theory of nucleation.

Fig. 3a and b shows the evolution of the average size hnifor r0 = 3 at different diffusion rates for (a) c0 = 0.1% and(b) c0 = 0.2%. These data display brief transients occurringfor very short times due to a rapid equilibration process.The structures produced by the initial templating proce-dure are not perfectly equilibrated at the conditions simu-lated, and during these transients the clusters undergosurface rearrangements to achieve more favorable configu-rations. Over the time-period shown, the growth andshrinking kinetics are consistent with the classical analysisof Ham [9] for growth of clusters from a uniform supersat-urated medium, which predicts a short transient, as theconcentration gradient in the local environment is estab-lished, followed by growth of the form dhni/dt / t3/2, orequivalently, as shown in Fig. 3, (dhni/dt)2/3 / t. We notedthat these data also appear quite linear even when plottedsimply as dhni/dt vs. t (not shown). We therefore calculated

1410 1420 1430 1440 1450n

0

0.1

0.2

0.3

0.4

)n(D

t=13t=51t=204t=815t=2273

110 115 120 125 130n

0

0.1

0.2

0.3

0.4

0.5

)n(D

t=13t=51t=202t=808t=3158

8 10 12 14r (lattice spacings)

0

0.2

0.4

0.6

0.8

1

)r(g t=13t=51t=204t=815t=2273

4 6 8 10 12 14r (lattice spacings)

0

0.5

1

1.5

2

2.5

)r(g

t=13t=51t=202t=808t=3158

a b

c d

Fig. 2. (a) Size distributions D(n) of clusters of initial radius r0 = 7 (1419 particles) for kD/kIA = 1 at c0 = 0.1%. (b) Size distributions D(n) of clusters ofinitial radius r0 = 3 (123 particles) for kD/kIA = 0.105 at c0 = 0.1%. (c) Normalized radial distribution functions g(r) [33] of the background particleconcentration corresponding to plot (a). (d) Normalized radial distribution functions g(r) of the background particle concentration corresponding to plot(b). Here n is the number of particles in a cluster and D(n) is the fraction of clusters of size n.

0 10000 20000 30000t

20

22

24

> n<

3 /2)t (

0.006740.10519.4942.5

0 10000 20000 30000t

24

26

28

>n<

3/ 2)t(

0.006740.10519.4942.5

0.01 0.1k

D/k

IA

1e-05

0.0001

0.001

0.01

0.1

|td />n

<d |

0.1%, r0=3, shrink

0.2%, r0=3, grow

0.1%, r0=6, grow

0.1%, r0=7, grow

Ham (1958)LSW

a b

c

1 10 100

Fig. 3. Evolution of average size hni with varying kD/kIA for clusters (a) ofr0 = 3 at c0 = 0.1%, (b) of r0 = 3 at c0 = 0.2%, and (c) the growth/shrinking rate jdhni/dtj of clusters of different sizes at different initialdensities as a function of kD/kIA. t is the time index, in units reduced bykIA. The dashed line in (c) illustrates a Ham-type dependence of jdhni/dtjon kD/kIA, and the dotted line illustrates a linear (LSW-type) dependence,both shown for comparison.

J. Diao et al. / Acta Materialia 56 (2008) 2585–2591 2589

growth and shrinking rates by linear regression of dhni/dtvs. t in the reduced time interval 5000–22,500. These dataare collected in Fig. 3c. The shrinking and growth rates

become independent of kD/kIA for large values (fast diffu-sion), corresponding to interface-limited kinetics. For verylow values of kD/kIA, the shrinking and growth rates are asmuch as two orders of magnitude smaller. This corre-sponds to diffusion-limited kinetics, and even slower ratescould probably be achieved by further reduction of kD/kIA.

LSW-type kinetics suggest a linear dependence ofgrowth rate on the diffusion constant D for small D, whileHam’s analysis predicts that the rate should be propor-tional to D3/2, or equivalently (dhni/dt)2/3 / D. The variousdata in Fig. 3c do approach power-law dependences (lin-ear, on the double-logarithmic scale of the Fig. 3c) at smallkD/kIA. The corresponding power-law exponent for thegrowing particles with r0 = 7 lattice spacings is near to 1,in accord with LSW-type coarsening predictions. Sincethere are many particles (all of similar size) in the simula-tion cell, is possible for coarsening behavior to be observedin all these simulations, and this is expected at very longtimes when growing particles have sufficiently depletedthe surrounding medium. In the duration simulated, how-ever, only systems with particles of initial radius r0 = 7 dis-played this behavior. For the other systems the power-lawexponents observed at small kD/kIA are all much lower.The r0 = 3, c0 = 0.2% and r0 = 6, c0 = 0.1% systems arein reasonable agreement with the Ham prediction of 2/3,while the r0 = 3, c0 = 0.1% system follows a power-lawwith an exponent somewhat lower than this.

Normalized radial distribution functions g(r) for clustersof r0 = 3 with kD/kIA = 0.105 and kD/kIA = 9.49 are shownin Fig. 4a at c0 = 0.1%, where the clusters are shrinking,and in Fig. 4b at c0 = 0.2%, where they are growing. These

3 4 5 6 7 8 9 10r (lattice spacing)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

)r(g

kD

/kIA

=0.105

kD

/kIA

=9.49

3 4 5 6 7 8 9 10r (lattice spacing)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

)r(g

kD

/kIA

=0.105

kD

/kIA

=9.49

ba

Fig. 4. The steady-state normalized radial distribution g(r) of backgroundparticle concentration for clusters of r0 = 3 at (a) c0 = 0.1% (shrinkage)and (b) c0 = 0.2% (growth) for two different diffusion rates (kD/kIA = 0.105 and 9.49). The data are averaged over the time interval5000–22,500.

0.1 0.16 0.18c (%)

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

( td/>n

<dx

013 )

Bulk densityLocal density, shrinkLocal density, grow

0.006740.105

1

9.4942.5

0.00674

0.105

1

9.4942.5

0.12 0.14 0.2

Fig. 5. The growth rate dhni/dt of clusters of r0 = 3 as a function of thebulk concentration at high diffusion rate kD/kIA = 42.5 (circles), and thegrowth rates of clusters of r0 = 3 at initial background densities of 0.1%(shrinking) and 0.2% (growing) at different diffusion rates, plotted as afunction of the local particle concentration. The numbers are the values ofkD/kIA corresponding to each point (triangles).

2590 J. Diao et al. / Acta Materialia 56 (2008) 2585–2591

g(r) data are accumulated over a long time interval duringwhich the particles are growing or shrinking. Distances aremeasured relative to the instantaneous center of mass ofthe cluster rather than to its surface, but the very smallchange in particle radii over the time interval (less that5% in these systems) does not lead to significant blurring;short-time averages within this interval give essentiallyidentical curves. At the higher diffusion rate little enrich-ment or depletion is visible, consistent with the interface-limited growth kinetics of the classical theory of nucleation,while at low diffusion rates the local distributions of parti-cles show steady-state enrichment or depletion, consistentwith the predictions of diffusion-limited theories.

Slow diffusion clearly causes enrichment of solute parti-cles around clusters smaller than the critical size. Due tothis enrichment, shrinking clusters experience a higher localsolute concentration and therefore shrink more slowly;likewise, depletion around growing clusters retards theirgrowth. Since a higher bulk solute concentration wouldalso retard shrinkage, and a lower solute bulk concentra-tion would also retard growth, it is possible that the localconcentration of solute particles around clusters underconditions of slow diffusion determines their rates ofgrowth or shrinking in the same way that the bulk concen-tration determines these rates under interface-limited con-ditions. This hypothesis is tested using three series ofsimulations of particles with r0 = 3, shown in Fig. 5.

In the first series, all performed at kD/kIA = 42.5 butwith c0 varied from 0.1% to 0.2%, very little enrichmentor depletion was observed (as in Fig. 4) and the local con-centration was always very close to the bulk value, as onewould expect for interface-limited conditions. In these‘‘bulk density” data, at concentrations below approxi-mately 0.162%, clusters shrink, and above 0.162% theygrow; this concentration therefore corresponds to the crit-ical concentration for clusters of this size.

The second two series of simulations, labeled ‘‘local den-sity” in Fig. 5, are performed at either 0.1% or 0.2% bulkconcentration and different values of kD/kIA. Since forlow kD/kIA values the local concentration differs from the

bulk value, these data are displayed as growth rates vs.concentration at the clusters’ surfaces, estimated from themeasured radial distribution functions at a distance ofr = 4. Since a concentration of 0.1% is below critical, clus-ters in this group of simulations (downward-pointing trian-gles) shrink, on average, while those simulated at 0.2%monomer concentration (upward-pointing triangles),which is supercritical, grow.

Comparison of the growth rate data from the two seriessimulations clearly shows that the local concentrationaround clusters correlates strongly with their growth rateover the entire range of diffusion rates considered. Thatis, under conditions that produce a local concentration ofsome c, one observes nearly the same growth rate as underinterface-limited (fast diffusion) conditions at a bulk con-centration c. Interestingly, in the extreme limit of kD = 0,diffusion is entirely suppressed and clusters can only estab-lish an equilibrium with their local environment; the localconcentration will then be that at which the cluster size isthe critical size, and the two sets of data should cross atdhni/dt = 0, which appears consistent with the data.

In Fig. 6 we plot the results from calculations on thecoupled-flux model for the systems considered in Fig. 5.In order to apply the coupled-flux model to the systemstudied in this paper, we use a mean-field theory to estimatedl and critical sizes extracted from the kinetic Monte Carlosimulations mentioned above, r0 = 4.5 at 0.1% bulk den-sity, and r0 = 2.5 at 0.2% bulk density. For comparisonof rates, we note that the ratio nD/6D0 plays the same rolein the coupled-flux model that the ratio kD/kIA does in thesimulations. The main features of Figs. 5 and 6 are quitesimilar, showing that the coupled-flux model correctlydescribes the behavior of the local density around growingand shrinking clusters. Similar comparisons fail for bothdiffusion-limited and interface-limited theories. Diffusion-limited (e.g. LSW) models assume that the concentration

0.1c (%)

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

( td/>n

<dx

013 )

Bulk densityLocal density, shrinkLocal density, grow

0.010.1

0.51

2

0.010.1

1

10

40

0.5

2

1040

0.12 0.14 0.16 0.18 0.2

Fig. 6. Coupled-flux model calculations for the growth rate dhni/dt ofclusters of r0 = 3 as a function of the bulk concentration at high diffusionnD/6D0 = 40 (circles), and the growth rates of clusters of r0 = 3 at initialbulk densities of 0.1% (shrinking) and 0.2% (growing) at different diffusionrates, plotted as a function of the average local shell density hq=qm

n i. Thenumbers are the values of kD/kIA corresponding to each point (triangles).

J. Diao et al. / Acta Materialia 56 (2008) 2585–2591 2591

at a cluster surface is determined only by the cluster radius,through a local-equilibrium condition. Since the radii of allthe clusters in these figures are approximately the same,such theories will predict no dependence of growth andshrinking rates on the local density, and therefore fail toexplain the effects observed. Likewise, in interface-limitedclassical nucleation theory the bulk density is assumed uni-form and there is therefore no difference between local andbulk density. Finally, we note that, while the coupled-fluxmodel certainly captures the important physics of the sim-ulated system, the agreement of actual predicted rates isonly semi-quantitative, which is probably due to the useof a classical-theory-type model for Wn (Eqs. (13–15)). Atthe relatively low reduced temperature simulated here somefaceting of clusters is expected to occur [25], and so theclassical assumptions of spherical particles and uniformsurface tension are likely to prove wanting.

4. Conclusions

In summary, we have used a simple but general latticegas model to study certain aspects of the growth andshrinking of seeded clusters in a supersaturated medium.Growing clusters, larger than the critical size, are foundto deplete the particle density near to them, while shrinkingclusters, smaller than the critical size, enrich the particledensity near to them. Such effects are predicted by well-known theories of diffusion-limited growth and coarseningthrough the use of local-equilibrium conditions at the clus-ter surface, but we have found that such approximationsare not sufficient to fully explain the dependence of thelocal densities and growth kinetics on the diffusion andinterfacial attachment and detachment rates. In particular,the strong dependence of cluster growth (and shrinkage)rates on the ratio of diffusion rate to interfacial attachmentrate, kD/kIA, is explained through the establishment of anon-equilibrium local density different from the bulk den-

sity. These results suggest that for a proper phenomenolog-ical description of cluster growth and shrinkage underconditions where kD/kIA is neither extremely high norextremely low, one should not impose the condition oflocal equilibrium at the cluster surface. Instead, fullydynamic approaches such as the coupled-flux model shouldbe used.

Acknowledgements

This work was supported by the Washington UniversityCenter for Materials Innovation. L.D.G. also acknowl-edges support from the National Science Foundation(Grant CHE-0241005). K.F.K. acknowledges supportfrom the National Science Foundation (Grant DMRDMR-06-06065).

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