7940 DOI: 10.1021/la900613p Langmuir 2009, 25(14), 7940–7953Published on Web 06/01/2009

Shape Selection in Diffusive Growth of Colloids and Nanoparticles

Vyacheslav Gorshkov,†,‡ Alexandr Zavalov,† and Vladimir Privman*,§

†Institute of Physics, National Academy of Sciences, 46 Nauky Avenue, Kiev 680028, Ukraine, ‡NationalTechnical University of Ukraine “KPI,” 37 Peremogy Avenue, Building 7, Kiev-56, 03056 Ukraine, and §Centerfor Advanced Materials Processing, Department of Physics, Clarkson University, Potsdam, New York 13699

Received February 18, 2009. Revised Manuscript Received March 30, 2009

We report numerical investigations of a 3Dmodel of diffusive growth of fine particles, the internal structure of whichcorresponds to different crystal lattices. A growing cluster (particle) is immersed in and exchanges monomer buildingblocks with a surrounding medium of diffusing (off-lattice) monomers. On-surface dynamics of the latter is accountedfor by allowing, in addition to detachment, monomer motion to the neighboring vacant crystal sites, according toprobabilistic rules mimicking local thermalization. The key new feature of our model is the focus on the growth ofa single cluster, emerging as a crystalline core, without development of defects that can control large-scale growthmodes. This single, defect-free core growth is imposed by the specific dynamic rules assumed. Our results offer a possibleexplanation of the experimentally observed shape uniformity (i.e., fixed, approximately evenly sized proportions) in thesynthesis of uniform colloids and nanoparticles. We demonstrate the basic principles of well-defined particle shapeemergence in such growth. Specifically, several shapes are possible for a given crystal structure. The formation of shapesthat follow the crystal symmetry and are uniform can be a result of the nonequilibrium nature of the growth process. Theshape of a growing particle can be controlled by varying the relative rates of kinetic processes as well as by adjusting theconcentration of monomers in the surrounding medium.

1. Introduction

A theoretical understanding of the mechanisms of the growthof well-defined (uniform) particles in aqueous and other suspen-sions has been an important challenge1-24 for colloids and,more recently, for nanoparticles. Uniform colloid particles, ofmicrometer and submicrometer sizes synthesized in solution, havefound numerous applications and were extensively studied

experimentally, including a large body of recent work,1-19,25-57

with new developing emphasis on smaller, nanosized particles.Solution synthesis is an important approach because it avoids

*Corresponding author. E-mail: privman@clarkson.edu. Phone: +1-315-268-3891.(1) Halaciuga, I.; Robb, D. T.; Privman, V.; Goia D. V. Proc. Conf. ICQNM

2009; IEEE Comp. Soc. Conf. Publ. Serv.: Los Alamitos, CA 2009; p 141.(2) Privman, V. J. Optoelectron. Adv. Mater. 2008, 10, 2827.(3) Robb, D. T.; Privman, V. Langmuir 2008, 24, 26.(4) Robb, D. T.; Halaciuga, I.; Privman, V.; Goia, D. V. J. Chem. Phys. 2008,

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Posted as e-print 0902.3243 at www.arxiv.org

DOI: 10.1021/la900613p 7941Langmuir 2009, 25(14), 7940–7953

Gorshkov et al. Article

caging or templating the growing particles, thus allowingfor better uniformity in their composition.

Quantitative models that identify diffusional growth mechan-isms that can yield a narrow particle size distribution in solutionare rather recent. Burst nucleation of nanocrystals, driven by thediffusional transport of atom-size building blocks, was shown2,3

to yield size distributions that are narrow because the smaller-particle side of the distribution, for particle (cluster) sizes belowthe Gibbs free-energy barrier, is eroded by cluster dissolution andthe larger, over-the-barrier particles grow irreversibly.

The uniformity of colloid particle size has been explained2,9,18

by a model involving the kinetic interplay of burst nucleation onthe nanoscale with further diffusional aggregation of the nano-sized primary particles to form polycrystalline colloids. The latterapproach yields a good description of the colloid-particle sizeselection1,2,4,7-9,12,16,18 by focusing on the master equations forthe secondary particle concentrations for s-subunit particlescontaining s nanocrystals. The dynamics of this distribution islargely determinedby the diffusional transport of building blocks:single- and few-nanocrystal particles to the growing aggregates.

The aforementioned approaches to uniform size (narrow sizedistribution) emergence in the synthesis of colloids or nano-particles, although successful in certain regimes, have limitations.Specifically, burst nucleation per se applies only to very smallclusters. Growth processes of all but the tiniest nanoparticlesinvolve different and/or additional mechanisms that actuallybroaden the size distribution2,3,58 as compared to the predictionsof the simplest burst-nucleation model. In the synthesisof colloids, the two-stage growth model has been quantitativelysuccessful in explaining1,2,4,7-9,12,16,18 the growth of sphericalpolycrystalline particles. However, this model’s applicabilityand utility in the synthesis of nonspherical particles in not certain.It is likely that such particles are formed by more than a singlepossible mechanism and in many cases the process might involvethe growth of polycrystalline colloids directly from atom-sizedmatter rather than by the secondary aggregation of primary-particle nanosized-crystal building blocks.

Whereas quantitative modeling of particle size selection hasbeen partially successful, the challenge of explaining the unifor-mity of the particle shape and,more generally, the morphology inmany growth experiments has remained virtually unanswered.One exception is the imperfectly oriented attachment mecha-nism6,43,45,59 identified as the persistence in successive nanocrystalattachment leading to the formation of uniform short chains ofaggregated nanoparticles. However, the bulk of the presentexperimental evidence for the uniformity of particle shapes insolution synthesis has been collected for particles of colloid size,from the submicrometer to a couple of micrometers, and remainslargely unexplained quantitatively or even qualitatively.

In fact, colloid particles synthesized and used in applicationscan assume a plethora of shapes and structures, depending ontheir growth conditions. Some particles are grown as singlecrystals. In many other situations, the growth does not yielda well-defined size and morphology (structure) distribution at all.However, with properly chosen experimental conditions, there is

a large recent body of experimental evidence1-57,59,60 for growthofwell-defined uniform-shape particles. It is the latter growth thatwill be addressed in the present work.

Whereas in most situations the flat faces of the formed shapesand the particle proportions follow the crystal structure of theunderlying material, it is experimentally well-established that theparticles are almost always polycrystalline, consisting of andfrequently (though not always) growing by the aggregationof smaller, nanosize crystal subunits. We point out that in manycases the particles assume shapes that, although uniform, arenot those of standard, equilibrium single-crystal growth. We willrefer to well-defined fixed-shape particles as those with anarrowdistribution of shapes and proportions but not necessarilymonocrystalline.

Shape evenness is another experimentally observed propertythat refers to the tendency of polycrystalline, colloid-size particlesto be formed with shapes of fixed and in many cases relativelyeven proportions. There are, of course, many examples of thegrowth of rods and platelets that are not evenly shaped.However,they are still uniform in that they usually have similar proportionsas well as surfaces and internal structure (morphology).

Furthermore, in most cases uniformly shaped particles arealso grown with relatively narrow size distributions. However,it is likely that the latter experimentally documented tendency isrelated to the fact that such particles are of interest in applicationsand also are easier to characterize. There is no compellingevidence that a very narrow size distribution (which, for mostapplications, would mean no more than about 10% spread inparticle modal dimensions) is directly related to particle shapeuniformity. Uniformly shaped particles can have rather wide sizedistributions, and it is likely that the mechanisms of size selectionand shape selection are not in one-to-one correspondence.

Indeed, in modeling the size, morphology, and shape selection,we have to consider several processes and their competition.Various sets of processes and their interrelations will control theresulting particle size, shape, and other structural features.The challenging aspect of the modeling is the large numberof processes that compete to yield the final particle structure.In addition to diffusional transport of the atoms (ions andmolecules) to form nanocrystals or that of nanocrystalline build-ing blocks to the particle surface and their attachment to formcolloids, these atoms/blocks can detach and reattach. They canalso move and roll on the surface as well as, for nanoparticlesas building blocks, restructure and further grow diffusivelyby capturing solute species.

We know from experimental evidence for (primarily spherical)colloids, for instance, that the arriving nanocrystals eventually get“cemented” in a dense structure but retain their unique crystallinecores.2,9,18 The mechanism for this is presently not well under-stood. In fact, diffusional transport without restructuring wouldyield a fractal structure.58,61 On the experimental side, character-ization of the various processes occurring on the surface andinternally, as time-dependent snapshots during particle growth,has been rather limited, which represents another challengefor modeling because the results can be compared only tothe measured distributions of the final particles as well as to theresults of their structural analysis.

In this work, we address aspects of uniform shape selection inparticle growth in solutions. Numerical difficulties always imposesimplifications and make it impractical to explore all of the

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size-, shape-, and structure-distribution features of growingparticles in a single realistic model. Therefore, approaches areneeded that focus on subsets of kinetic processes that controla particular property to be studied. One of the main difficulties inmodeling particle shapes numerically58,62,63 has been, in fact, theestablishment of the crystalline (for nanoparticles) or dense(for colloids) stable core on top of which the growth of thestructure then continues. In modeling the shape selection, wethus find it practical to focus on the growth of a single (colloidor nanosize) particle as a seed rather than on a distribution,which would be the main object to consider in studying sizeselection.

Therefore, we will adopt an approach focusing on modelingsingle, defined-core particle growth with emphasis on the dy-namics of its surface and morphological features. The initialformation of the amorphous cluster in nanocrystal growth hasbeen estimated2,15 to occur for up to roughly 25 atoms/molecules,followed by its restructuring into a crystalline core. The latter cangrow into a nanoparticle containing on the order of 105-108

atoms. Typically, it takes 104-107 nanoparticles to aggregate intocolloids. The time scales range from fractions ofmicroseconds forthe initial nucleation stages of diffusionally transported atoms totens of seconds for the final colloid aggregation.

OurMonte Carlo (MC) approach reported in the present workwas originally designed to address the diffusional aggregation ofnanocrystals into colloids but is likely equally suitable to describeaspects of the formation ofnanoparticles in the diffusional growthstage past their initial burst nucleation (as well as the growth stagefrom on the order of 102 to 108 molecules per crystal in proteincrystallization5,64 after the initial small-cluster formation butbefore the onset of the macroscopic growth modes).

We report MC modeling of diffusing building block particles/atoms, with the core defined by the rule that the particlesattaching to the growing, initially small seedwill always “register”with the lattice distances/vectors originally assigned to the seedstructure. This approach, although still requiring substantialnumerical resources, has the flexibility of allowing one toexplicitly control the processes of particles (or atoms) “rolling”on the surface and detachment/reattachment: We use thermal-type, (free-)energy-barrier rules. The diffusional transport occursin 3D space, without any lattice. However, the registered attach-ment rule prevents the growing cluster from developing defectsand thus again ensures the maintenance of a well-defined core.We can then focus on the emergence of the surface and shapemorphological features. Our results indicate that there are threeregimes of particle growth.

The first regime corresponds to very slow growth rates, forinstance, when the concentration of externally diffusing buildingblocks, to be termed atoms, is low. In this case, the time scale ofthe diffusion of already attached atoms on the cluster surface,τd, is much smaller than the time scale of the formation of newmonolayers, τlayer. Then the shape of the growing cluster is closeto the Wulff construction configuration.65-67

The second regime, τlayer , τd, corresponds to fast growth andto the development of instabilities of the growing cluster surface.

It is expected that the dynamics of the cluster shape is stronglycorrelated with the spatial density of the diffusional flux,ΓB(rB), near the cluster: ΓB(rB) is maximal near the highest-curvatureregions of the surface. As a result, small-scale perturbations of thesurface due to random fluctuations are accompanied by theincreased diffusional flux of atoms to surface protrusions, whichthen grow further provided that near such protrusions the influxof atoms overwhelms the outflow due to on-surface diffusion.Eventually the cluster assumes the form of a clump of sub-structures of smaller sizes.

The third, most interesting regime corresponds to τd ≈ τlayer,with the cluster in a nonequilibrium growth mode, but asdemonstrated in this work, it can have an evenly shaped formwith well-developed faces that correspond to the underlyingcrystal structure defined by the seed and by the attachment rules.In this work, we explore the regularities and shape selection in thisnonequilibrium growth regime. Specifically, we consider simplecubic (SC), body-centered cubic (BCC), face-centered cubic(FCC), and hexagonal close-packed (HCP) crystal lattices. Forinstance, we demonstrate that for the SC case a cubic particleshape can be only grown in the nonequilibrium growth regime.There are several possible cluster shapes for a given type of crystalsymmetry, with the realization of each determined by the growth-process parameters.

Our model is simplified, but it apparently captures the keyingredients required for well-defined shape formation in thenonequilibrium growth regime: It suppresses the formation ofmacroscopically persisting defect structures, and for numericaltractability, it focuses on the growth of a single particle ratherthan a distribution. Thus, we believe that it captures the followingimportant feature in both the colloid and nanoparticle growth:In the nonequilibrium growth regime, the model corresponds tothe situation when the dynamics of the growing particle’s facesis not controlled by extended defects, which is a well-knownmechanism that can determine growth modes in nonequilibriumcrystallization.64,68,69 Apparently, this property allows theevolving surface features to overwhelm imperfections, even forcolloids that are formed from aggregating nanocrystallinesubunits rather than just from the flux of atomic matter ontothe surface. The growing cluster faces then evolve to result inwell-defined particle shapes and proportions. In fact, we find thatthe most densely packed, low-index crystal-symmetry faces,which dominate the equilibrium crystal shapes, also emergein our nonequilibrium regime. However, generally the particleshapes, faces present, and proportions are not the same asin equilibrium.

Obviously, this nonequilibrium growth mode cannot last forvery large clusters. However, colloids and nanoparticles are nevergrown indefinitely. When numerical simulations are carried outfor a realistic number of constituent atoms, our results success-fully reproduce many experimentally observed particle shapes.Our numerical approach is detailed in section 2. Section 3 reportsthe results of our nonequilibrium growth model as well asauxiliary results for growth close to equilibrium, under steady-state conditions to be defined in sections 2 and 3. We also offerdiscussion of whywe consider the studied nonequilibrium growthregime to be appropriate for fine-particle synthesis processes.Additional results and considerations and a summarizing discus-sion are offered in section 4.

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cules and Glasses. Cambridge University Press: Cambridge, U.K., 2003.(64) Winn, D.; Doherty, M. F. AIChE J. 2000, 46, 1348.(65) Herring, C. In Structure and Properties of Solid Surfaces; Gomer, R.,

Smith, C. S., Eds.; University of Chicago Press: Chicago, 1953; Chapter 1.(66) Mackenzie, J. K.; Moore, A. J. W.; Nicholas, J. F. J. Phys. Chem. Solids

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2. Numerical Model

Manyof the heuristic assumptions and expectations outlined inthe preceding section are not new. The novel features of ourapproach are really threefold. First, we focus on growth from asingle core, which is initially placed (Figure 1) near the center of a3Dregion,Q: such a nanosized seed is illustrated inFigure 1a. Theattachment rules, to be defined shortly, do not allow for theformation of extended defects. Second, we identify the dynamicprocesses to include in a simplified model, as well as the appro-priate nonequilibriumgrowth regimeand the range of cluster sizesfor which the shapes of interest are found. Third, and equallyimportant, to reach these sizes, the numericalMCsimulations hadto be carried out for a very large (as compared to reachable inearlier simulations) number of constituent building blocks (to betermed atoms): up to the order of two million. The latter wasfacilitated by the first two assumptions, especially our focus on asingle seed and the proper selectionof the dynamic processes. Still,the actual simulations were large-scale in terms of moderncomputational facilities. For example, each cluster-growth reali-zation required between 100 and 400 CPU hours on a 3.2 GHzworkstation. In addition, a careful approach to designing thesimulation itself was needed, as detailed in this section.

Free atoms randomly move inQ and can attach to the growingcluster. The latter thus grows because of the diffusional fluxof freeatoms to the surface. The numerical simulations consist of twointerrelated parts: modeling the motion of the atoms externalto the cluster and those presently at the surface of the cluster.2.1. Dynamics of Free Atoms. In this subsection, we will

describe in detail the modeling of the free-atom motion (bydiffusion). Actually, this is a rather straightforward part of settingup the modeling approach. However, we offer a detailed pre-sentation for two reasons. First, if not properly programmed,free-atom diffusion can be numerically taxing and can limit theefficiency of the simulation. As a result, while the atomsmove in acontinuum space, modeling their diffusion has to be properlydiscretized. This leads us to the second reason for carefullyaccounting for the details: once discretized, we want to be surethat the process of diffusion per se does not introduce any latticefeatures into the system’s dynamics.

Isotropic, off-lattice diffusion of individual atoms can berealized as hopping along randomly oriented unit-length vectors,eBf, with step lengths, l, randomly taken from a selected distribu-tion. An atom at rB1 at time t1 will hop to rB2 = rB1 + leBf at t2 =t1 + Δt. The total number of such hopping events per singlecluster-growth simulation in this work has reached the order of1012. For continuously varying l and eBf, the calculation of thecomponents of the displacement vector leBf, involving sines andcosines of directional angles, noticeably affects the overall simula-tion time. Thus, discretization is warranted. The time steps anddisplacements are made constant, Δt = τ and l = l0, andsufficiently small. Because the spatial distribution of the atomflux, ΓB(rB, t), near the cluster surface depends on its shape, thehopping length l0 should be comparable to (or smaller than) thelattice spacing in the cluster. In what follows, l0 and τ are used asunits of distance and time. In these units, l = l0 = 1 and Δt =τ = 1, whereas the volume Q (Figure 1) was 500 � 500 � 500 inour simulations and was aligned with the x, y, z spatial axis.

Similarly, the continuum of {eBf} was replaced by a discrete set,{eBf

(k)}, k=1, 2, 3, ...,K, and the components of the displacementvectors {leBf

(k)} were precomputed. Specifically, in the sphericalcoordinate system the angle θ with the z axis was discretized70

according to θj = arccos(1 - (2j - 1)/Nθ), with j = 1, 2, ..., Nθ,which mimics the distribution in 0 e θ e π with weight p(θ) =(sinθ)/2. The azimuthal anglejwas equally spaced in 0ej< 2πaccording to ji = 2π(i - 1)/Nj with i = 1, 2, ..., Nj. In oursimulations, we usedNθ = 20 and Nj = 40. Thus, the set {eBf

(k)}consisted of K = 800 different unit-vector directions. However,we also ran several selective simulations with continuum {eBf} toconfirm that the results were unchanged. Such runs were at leastseveral times longer than those with discretized hopping.

Consider a sequence of M vectors eBm, with M . K = NθNj,randomly generated according to the above rules from the set ofvalues {eBf

(k)}. For a fixed unit vector eB0, the mean of the squaredprojection, Æs12æ � Æ(eB0 3 eBm)

2æ = Σm=1M (eB0 3 eBm)

2/M, shouldapproach the continuum value 1/3. In our simulations, we hadmax[Æs12(eB0)æ- 1/3]= 8� 10-4. Because for fixed hopping lengthl = 1 the quantity Æs12æ equals the mean-squared displacementalong an arbitrary direction for time stepΔt=1, then it gives thediffusion constant of the free atoms, D= Æs12æ/2Δt = 1/6, in ourdimensionless units.

The boundary of the region Q was defined as the outermostlayer of the 5003 box, of thickness 3 units. The concentrationof free atoms in that layer, n0(t), was kept constant as a function

Figure 1. (a) System geometry: the initial seed is placed in the 3Dregion Q. (b) Illustration of the cluster growth dynamics, here forsimplicity shown in twodimensions.Theattached cluster atomsareexactly registered with the lattice structure of the cluster, definedstarting from the initial seed (which is not specially marked): thecenters of the shaded squares. Each atom interacts with its nearestneighbors, here along the lattice directions of type (10). Thus, themaximum number of interactions is four for our square-latticeillustration, and each interaction is assigned an energy of ε<0; seethe text. The numbers in the cells labeled b, c, and d enumerate newinteractions for the atom presently in cell a if it hops to therespective location (b, c, or d). Free atoms are captured in thesurface vacancies that are nearest neighbors to at least one of thecluster atoms,markedbyopen circles, once theyhop into the centerpart of a vacant lattice cell at a distance of less than a half hoppinglength to that cell’s center. The captured atom is then instanta-neously registered with the lattice by positioning exactly at thatcell’s center.

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of time. Specifically, the total number of atoms, Nb, in this outerlayer of volume Vb ≈ 4.45 � 106 was monitored, and free atomswere added at random positions (in the outer layer) wheneverNb was found to be below Nb0 = n0Vb. These numbers varied inour simulations, but a representative typical value, used in severalruns (section 3) was Nb0 ≈ 8000, corresponding to density n0 ≈1.8� 10-3 (i.e., near the boundary ofQ, the density was one freeatom per cube of linear size ∼8).

Finally, we point out that the time for an atom to driftdiffusively a distance L ≈ 200 from the boundary toward thecentral region ofQwas on the order of td≈ L2/(2D) = 1.2� 105,and this number also gives the count of the hopping stepsinvolved.2.2. Dynamics at the Cluster Surface. In this subsection,

we detail the dynamic rules for atoms that are at the surface of thegrowing cluster. The initial seedwas typically a compact structurecontaining up to ∼15000 atoms that were positioned accordingto the displacement vectors of one of the standard 3D crystallinestructures with respect to each other. A key feature of our modelwas that free atoms adsorbing at the cluster due to diffusionalways attached as the centers of the lattice structureWigner-Seitzcells thus continuing to build up the same ideal structure; seeFigure 1b. Asmentioned, themodel focuses on late enough stagesof growth, when the initial seed is already formed. Thus, we avoidthe issue of how the initial seed is formed in the initial growthstages. The formation of the initial seed and its assuminga compact, defect-free crystal structure for nanocrystals orcompactification to a bulk-density polycrystal for colloids havebeen discussed in the literature,2,3,5 but the process is not wellunderstood primarily because of the difficulty in obtaining anyrelevant experimental data to test theoretical expectations.

For the HCP lattice, we assumed an interatomic distance of 1(= l). For the SC and BCC lattices, the cubic lattice spacing,a, was taken as a= l=1 inour simulations, whereas for theFCC,we took the cubic lattice spacing of a = 2l = 2. The diffusivehopping distance, 1, was found to be small enough for all of thesecases, allowing the free atoms to explore the lattice cavities (pits)at the outer surface of the cluster. However, the occupied latticecells were marked so that any attempt by a free atom to hop intotheir full volumes was rejected, and another hopping directionwas then generated for that atom. This prevented atoms fromdiffusing deeply into the cluster. We note that the diffusing freeatoms (section 2.1) were treated instead as pointlike and non-interacting. This distinction is in line with our focus on thedynamics of a single central cluster, whereas possible interactions(and thus additional cluster formation) of the free atoms areotherwise ignored.

In most of our simulations, the seed was defined by the latticecells within a sphere of radius r0 = 15. This choice was madebecause the independent-particle approximation is correct only aslong as the particles do not meet each other, which means thattheir density is low and they diffuse slowly (which becomes truefor large enough particles). In reality, the suspension will have adistribution of multiatom clusters of various sizes. The processesof cluster-cluster aggregation and of cluster dissolution/breakupdefinitely do not favor crystal shapes but rather lead to rando-mization (i.e., spherical shapes as long as we assume that someof the dynamic processes going on also compact the particlestructures). Thus, the whole isolated-particle approximationshould be simulated starting only from some large enough sizes,below which the clusters are best regarded as spherical.

Inour nonequilibriumgrowth simulations, the seed atomswerefully immobile. This assumptionwasmade to save run time on thebasis of preliminary run observations in which the seed in such

simulations rarely evolved much from its original shape anddensity. Thus, only the atoms later adsorbed at the seed and thegrowing cluster underwent the dynamic motion described in thefollowing paragraphs. However, this rule was not applied to somesteady-state growth runs to be defined in the next subsection.Indeed, if all of the arriving atoms, after attaching to the seed/cluster, remained immobile, then we would grow fractal struc-tures.61,71 In reality, surface atoms should be allowed to roll onthe growing cluster surface (for instance, in the case of thermalequilibration, to minimize the local free energy) as well as detachunder random agitation. These processes will combine with thecluster growth to yield the formation of a compact particle. In ournumerical model, such processes were incorporated as follows.

During the time intervals Δt= τ= 1 of the system dynamics,the algorithm of the simulation included an analysis of possiblemoves for each atom presently attached to the cluster, with theexclusion of those atoms in the original seed and those that arefully surrounded by filled crystal cells. For simplicity, we outlinethe procedure for the 2D example shown in Figure 1b. (In reality,our program treated the 3D lattice structures enumerated earlier.)For each surface atom, such as a in Figure 1b, possible dynamicmoves were considered.

First,we calculate the probability,Pa,mov, for atom a to attemptto change its position, defined according toPa,mov= (p0)

ka, where0< p0< 1 is a system parameter. ka is the number of the nearest-neighbor cluster atoms (with which atom a interacts): ka = 2, inFigure 1b, which corresponds to two out of four maximumpossible contacts for the 2D square lattice in our example. Wethen generate a random number r (0e re 1). If r > Pa,mov, thenatom a is left alone and the calculation switches to the next atom(in random order).

If r e Pa,mov, then we assumed that the considered atom canattempt hoping to each of its unoccupied nearest- and next-nearest-neighbor lattice cells (the set of such possible moves wasmodified in some cases; see section 3). Specifically, in Figure 1b,atom a can hop to surface cells b or d or detach from the clustersurface, by hopping to cell c, or even remain in its originalposition, cell a. In location b, atom a will have one interaction,kb=1. If hoping to d, thenkd=2.Obviously, kc=0for hoppingto c. The probability, pξ, of each of the possible final positions,here ξ= a, b, c, d, was given by the expression of the form pξ =C exp(Rkξ), with positive parameterR andwith the normalizationconstant C determined via Σξ=a,b,c,d pξ = 1.

If we identify R = |ε|/kBT, with kB being the Boltzmannconstant, then the dependence on a free-energy-like parameter,ε< 0, is suggestive of thermal-equilibration rules for this part ofthe dynamics. However, we point out that processes of formationof nanoparticles, and more so for colloid-sized particles, are inmost situations irreversible. Therefore, this free-energy designa-tion is questionable. The dynamic rules here rather reflect theexpectation that allowed moves of atoms on the surface are stillapproximately controlled by energy-like considerations (via theirbinding to the cluster, counted by their number of contacts): therelative probabilities to end up in various possible locationsare determined by factors of the form λkξ-kξ0. However, theidentification of the constant λ � exp R> 1 with the Boltzmannweight and in particular the resulting temperature dependence,λ = exp(-ε/kBT), are at best suspect.

Thus, the dynamics should depend on the physical and likelychemical conditions of the cluster and the surrounding suspen-sion, including the temperature, but the latter dependence cannot

(71) Family, F.; Vicsek, T. Dynamics of Fractal Surfaces; World Scientific:Singapore, 1991.

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be assumed to be equilibrium-like without further experimentalor theoretical substantiation. Furthermore, our treatment of theatom-atom/atom-surface interactions generally ignores thepossible directionality of the bonding energies as well as longer-range forces such as the double layer, which will be important toaccount for in more elaborate treatments.2.3. Technical Aspects of theNumerical Approach. In this

subsection, our comments apply to the 3D simulations, the resultsof which are detailed in the next section: we provide additionaldetails of our numerical approach. For each 3D simulation, weselected the type of crystal lattice for the initial seed and thusfor the resulting cluster. We also defined the set of lattice vectors,{eBint}, along which the atoms interact with each other for theaforementioned determination of the weights Pξ,mov. Typically,these will be the nearest-neighbor vectors (e.g., the six type-(100)vectors for the SC lattice). The set of possible lattice vectorsfor atom moves, {eBmov}, resulting in on-surface displacement ordetachment, was typically larger than the set {eBint} and usuallyalso included the next-nearest neighbors, such as the 12 diagonallattice vectors of type (110) for the SC case.

The quantities p0 and R, and the concentration of the freeatoms, n0, in the boundary layer of the volume Q were thenselected as the parameters for each simulation run, as were theshape and size of the initial seed as described earlier. Ournumerical “experimentation” suggested that the range of valuesfor the parameter R for which one can seek interesting none-quilibrium growthmust be fine tuned by preliminary steady-stateruns that are described presently. This steady-state-like regime onits own can also result in interesting particle shapes. However, weargue in section 3 that, unlike the nonequilibrium regime, thissteady-state particle shape evolution is not applicable to practicalexperimental situations.

Initially, a (spherical) seedof a rather large radius, up toR≈ 50,was placed in the center of Q. Its evolution with time wascalculated for varying values of R in the absence of any flux offree atoms from the boundary ofQ, whichwasmade reflective forthese particular runs. Note that for these preliminary simulations,the atoms in the seed were not fixed because there were no otheratoms for the dynamics. Ifwe took less than a certain cutoff value,R < Rcr, then the cluster fully disintegrated. However, for R >Rcr, after some time a steady state developed, with the remainingcluster exchanging atoms with the surrounding dilute solution ofatoms with density approximately 2 orders of magnitude lowerthan our aforementioned typical values of n0 = 1.8� 10-3 takenfor the boundary concentration for our later, nonequilibriumcluster-growth runs. We comment (see section 3 for details) thatthe resulting forms of the steady-state clusters, illustrated for theSC lattice in Figures 2 and 3, show some resemblance to theequilibrium (free-)energy-defined configurations. We found thatthe parameter p0 primarily determined the time scale of approachto the steady state. The value of Rcr only weakly depended on p0.The change in the behavior at the cutoff Rcr is obviously atransition form disintegration to apparent steady state behavioron the time scales of the simulations. We do not offer a verdicton whether this is a sharp dynamic phase transition: We simplydid not explore the relevant quantities, such as the system- andseed-size dependences of the time scales involved.

For nonequilibrium cluster-growth simulations (with newatoms externally supplied), our numerical experiments indicatethat nontrivial cluster shapes should be sought (but are not alwaysfound) with values of R taken in the range from somewhat lessthan Rcr to up to 30-50% larger than Rcr. Indeed, for thesesimulations (which, as described in section 2.2, bypassed theearly stage spherical-shape growth by starting the large-cluster

simulations from the initial seed of radius r0≈ 15), we found thatthe cluster would remain nearly spherical and not much largerthan the (frozen) seed as a result of evaporation of atoms if wetake R much smaller than Rcr. If R is too large, then the cluster isdestabilized. The SC results are exemplified in Figure 4, which isrevisited in section 3.1.Note that for drawingswithmore thanoneparticle shape shown, the image sizes in this work were generallynot depicted in proportion to the actual linear dimensions butrather were scaled to fit the figure.

The number of atoms in the seed of size r0 ≈ 15 is on the orderof 1.5 � 104. In our cluster-growth simulations, the final clusterstypically contained on the order of 5 � 105 to 1.5 � 106

(and sometimes even a larger number of) atoms. This means thatthe seed is overgrown with at least on the order of 30 monolayersof atoms. Initially, the flux of atoms onto the surface of the clusteris maximal, Γ≈Dn0/r0= 2� 10-5 (recall that we use dimension-less time and distances). Thus, the time of formation of a singlemonolayer is τlayer ≈ 1/Γ = 5 � 104, which also counts thenumber of time-step cycles for dynamic diffusion and on-surface/detachment hopping attempts per particle while a monolayer

Figure 2. Steady-state SC lattice simulations for variant A of thedisplacements/detachments for surface atoms. The simulation de-tails and parameter values are given in section 3.1. The resultingparticle shape is shown for the cluster of 3.8� 105 atoms,whichwasin the steady state with a dilute solution of 2.4 � 103 free atoms.Also shown is the projection of the cluster shape onto the xy planeas well as a shape formed by lattice planes of the types (100), (110),(111) by an equilibriumWulff construction (assuming that they allhave equal interfacial free energies).

Figure 3. Steady-state SC lattice simulations for variant B of thedisplacements/detachments for surface atoms. The dynamic ruleswere the same as for variantA, presented in Figure 2 (and detailedin section 3.1). Also shown is the projection of the cluster shapeonto the xy plane. The dilute solution contained 1.3 � 103 freeatoms in this case.

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is forming.We note that the number of free atoms in the system ison the order of 2 � 105 and active surface atoms number severaltimes 104 to several times 105. Multiplication of all these countersindicates the large-scale nature of the present simulations,typically requiring at least on the order of 1012 hopping-eventcalculations per grown cluster.

Finally, let us address some technical issues related to thecluster structure. Whereas the atoms in the initial seed are frozen,some later-attached atoms, when hopping, can cause pieces(consisting of several atoms) to break off (become detached from)the main cluster. In our simulations, this connectivity issue wasaddressed in twoways. First, if amiddle atom in a small “sticking-out” structure hopped in such a way that one of more isolatedatoms became detached, then these single atoms were made partof the free-atom solution. However, larger “floater” structureswere left to evolve according to the system dynamics. Further-more, a simplified version of the cluster-labeling algorithm frompercolation theory72 was used to classify the cluster connectivityat occasional time instances during the simulation.We found thatthe floaters usually were reattached to the main (seed-connected)cluster and never became large in all but two or three of thecluster-growth realizations (which were then discarded from ourresults sample).

We also point out that the algorithm as formulated, althoughnot allowing the free atoms to diffuse into the cluster structure(as described earlier), does allow the trapping of internal cavitiesthat will then undergo their own dynamics involving internallytrapped pockets of free atoms. We did not attempt a study of the

detailed dynamics of such cavities or their effect on the clustercompactness or density because we were primarily interested inthe emergence of the cluster shape. For the same reason,we did not explore in detail the cluster surface morphology(such as roughness). All of these topics could be the subject ofa future study.

It is important to note that all of the empirical observationssummarized in this subsection referred to 3D simulations. Eventhough we used the 2D case for illustration in Figure 1, no actual2D simulationswere carried out. It is well known that the effect offluctuations may be more significant in lower dimensions. There-fore, our conclusions cannot be simply extrapolated from 3D to2D without a separate numerical study. For each 3D lattice andparameter value, we actually ran a number of simulations(typically, at least three) with different random numbersequences. The cluster shape and size were generally fully repro-ducible for regimes of interest in our present study (i.e., those ofrelatively stable and well-defined shapes). For the longer-time ordifferent-parameter-value less-stable growth regimes, exploredonlymarginally in the present work (sections 3 and 4), the chaoticcluster shape does vary significantly from one run to another.

3. Results

This section reports the results of our simulations for theselected 3D lattice structures. The most detailed study of thesteady-state regime (reported first) was carried out for the SClattice, followed by a subsection on the nonequilibrium SC-latticegrowth. Results for the other three lattice structures (BCC, FCC,and HCP) are reported in later subsections.3.1. SC Lattice: Steady State. In the SC-lattice case, each

atom that is attached to the cluster can have up to six bonds withits nearest neighbors, described by the set {eBint} of six latticedisplacements of type (1,0,0). The set of displacements/detachments for surface atoms, {eBmov}, was defined in twodifferent ways. Case A: In this variant, {eBmov}A included boththe set {eBint} and also the 12 next-nearest-neighbor displacementsof type (1,1,0) of length lj =

√2. Case B: Here {eBmov}B = {eBint}.

In the latter variant, the dynamics of the surface atoms is slower.Let us first describe the steady-state regime results, as defined

earlier in section 2.3. The initial large spherical seed in this casehad a radius of R= 45, and its dynamics was simulated withoutany external supply of free atoms (with reflecting boundaryconditions at the outer periphery of the box Q in Figure 1). Theshape of the cluster changes in time, as a result of atom evapora-tion and surface drift. However, with a proper selection of theparameters, here exemplified by p0 = 0.8 and R = 3.5, a steadystate develops for the cluster with the surrounding solution ofatoms for extended time intervals of the simulation. Note that thevalue of parameter R, defined in section 2, is here well above ourestimate of Rcr ≈ 2.7 for the SC lattice type-A dynamics. For thetype-B variant of the dynamics, we estimated Rcr ≈ 2.5. Atomevaporation and surface drift result in a gradual change in thecluster shape. However, as already mentioned, detached freeatoms are reflected at the boundary of the simulation volume Qin this case, and as a consequence, in time a steady state developsand seems to persist for the longest times of our runs.

Figure 2 illustrates the resulting steady-state particle shape forvariant A of the SC simulation. We also show a schematic thatillustrates that the resulting cluster shape is formed with the type(100), (110), and (111) lattice planes, which also happen to be thedensely packed, low-index faces that dominate the low-tempera-ture Wulff construction for the SC lattice.65-67 However,this superficial similarity with the equilibrium Wulff shape is

Figure 4. Nonequilibrium SC-lattice cluster shapes in variantA (detailed in section 3.1). Here n0 = 1.8 � 10-3, and the radiusof the initial seed is r0= 15. (a) Projection onto the xy plane of theinitial atoms in the seed (gray circles) and in the growing cluster(green circles), with parameter values of p0 = 0.6 and R = 2.5,shown at t=3� 105, illustrating the emergence of the cubic shape.(b) The same cluster at a later time, t = 2.5 � 106, containing4.5 � 105 atoms with a cube edge length of 77. (c) Cluster grownwith different parameter values, p0 = 0.8 and R = 3.5, shown attime t=5.2� 106, containing 1.8� 106 atoms andof characteristicsize 125.

(72) Hoshen, J.; Kopelman, R. Phys. Rev. B 1976, 14, 3438.

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misleading. Indeed, our system dynamics does not correspond tothermal equilibration. The resulting shape is thus dependent onthe dynamic rules. Specifically, Figure 3 shows the shape obtainedfor the same system but with variant B for the displacements/detachments, which imposes slower surface dynamics.

The evolution of the cluster shape can be facilitated by twomechanisms. The first one is themigration of atoms on the clustersurface. The second process corresponds to the preferentialdetachment of atoms from some parts of the cluster surface andthe preferential attachment of free atoms in other parts. To shedsome light on the role of the second mechanism in variant Bdynamics, we carried out the variant of the calculation with nodetachment at all. If a selected surface atom move correspondedto detachment from the main cluster, then that move was notcarried out: the atomwas left in its original lattice position. In thismodified model, there were no atoms in the dilute solution. Wefound that the cluster shape no longer evolved into the one shownin Figure 3. Instead, it reverted to the shape of Figure 2, found forvariant A. It transpires that the detachment-reattachment me-chanism dominates for dynamic rule B as opposed to A. Wespeculate that this is actually because the first mechanism;theon-surface diffusion;is relatively slower for B than for A, whencompared with evaporation/reattachment, for the parametervalues and time scales of the simulations.

It is interesting that whereas the evaporation/attachmentmechanism has a profound effect on the cluster shape the actualnumber of atoms in the dilute solution (quoted in the captions forFigures 2 and 3) is surprisingly small as compared to none-quilibrium simulations reported in the next subsection, for whichthe concentration of free atoms is nearly 2 orders of magnitudelarger, with ∼(1-2) � 105 atoms in simulation box Q.

The above results lead to several interesting observations. First,the particle shape is not universal73 in the sense expected formanyprocesses that yield macroscopic behavior in statistical me-chanics: the microscopic details of the dynamic rules do matter.In practical terms, this makes it unlikely that particle shapes canbe predicted on the basis of generalized arguments such as theminimization of some free-energy-like quantity. The secondconclusion is that the surrounding medium canmediate processesthat profoundly affect the particle shape. The growth processshould thus be considered in a self-consistent formulation thatincludes the particle’s interactionswith and the resulting transportof matter to and from its environment.

Another interesting observation is that well-defined particleshapes can be obtained in the present steady-state regime. Thereader may then ask why not stop at this point? Why cannot thisregime be a candidate for predictable (within the present model)and well-defined particle shape selection mechanisms? The an-swer is in the observed extreme sensitivity to the density of andtransport to and from the very dilute surrounding medium.Indeed, in this regime the isolated cluster assumption breaksdown: Other clusters (particles) will compete for the atoms(solutes) in the dilute solution, and growth mechanisms74 thatinvolve the exchange of matter between clusters will becomeimportant (Ostwald ripening). Because such effects are not inthemodel, we cannot consider it to be presently predictive for verydilute solution (late-stage growth) situations.

Let us expand on the observation/speculation that the inter-relation between the form of the cluster and the spatial distri-bution of the flux of free atoms to its surface could provide

a early-stage catalyzingmechanism for the particle shape selectionin the later growth stages. An evaluation of the steady-statedistribution of the diffusive flux of atoms, ΓB(rB), near an irrever-sibly absorbing surface, is analogous to the problemof calculatingthe surface electric field, EBsurf, at the surface, which is consideredto be charged and conducting. Obviously, EBsurf is larger in thelarge-curvature regions of the surface. This means that, generallyspeaking, protrusions appearing dynamically at the particle sur-facewill be also regions of largerΓB(rB). This provides amechanismfor the instability of the particle surface. Processes such as surfacediffusion as well as preferential detachment of atoms, the latterassociated with the same regions of larger surface curvature,provide stabilizing mechanisms.75 Our simulations reported inthis subsection indicate that these three mechanisms combinedsuffice to offer growth regimes with, on one hand, the particleshape that is obviously not thermally equilibrated and, on theother hand, stability in that no uncontrollable distortions of theshape arise.3.2. SC Lattice: Nonequilibrium Growth.We now turn to

our main, nonequilibrium growth regime, with free atoms atconcentration n0 6¼ 0 supplied at the outer layer of simulation boxQ; see section 2. Apparently, the conclusion of the precedingparagraph still applies: the interplay of the detachment andattachment of atoms, as well as their surface motion, suffices toyield in some (but not all) cases well-defined particle shapes.A striking example is presented in Figure 4: growth of a cubiccluster (dynamics variant A). We will describe this case in detailpresently.

At early times, the flux of atoms is uniform at the (spherical)cluster surface. However, the uniformity of the deposited layer ontop of the initial seed is lost rather rapidly, as pointed out inFigure 4a. Atoms deposited at the type-(100) faces drift on thesurface and fill up vacancy sites with larger binding energies at theedges (e.g., atom 1 in Figure 4a). The reverse processes, thosedecreasing the approximately flat type-(100) faces, occur with asmaller probability (e.g., atom 2). A consideration of 3D config-urations in our simulations suggests that the latter processes haveto go over a potential barrier of sorts, even though there is noequilibration in the regime considered.

Empirical observations also suggest that at later times the cubicshape (Figure 4b) is maintained by the interplay of severalprocesses that, in possible continuum descriptions, are likelyhighly nonlinear.64,68 Specifically, the diffusional fluxes of freeatoms indeed favor the edges and corners of the cubic shape.However, excessive growth of these is prevented by the on-surfacedrift of atoms toward the centers of the flat faces. The time scale ofthis drift, τd, increases as the square of the characteristic clustersize, R (i.e., τd ≈ R2), whereas the time scale of formation of newlayers, τlayer, grows linearly with the size, τlayer ≈ R/n0 (strictlyspeaking, this result becomes accurate only in the limit of fullyadsorbing particles, i.e., no detachment).

If the concentration of free atoms at the boundary of thesimulation box is constant, which corresponds to an unlimitedsupply of building blocks, then the balance between the compet-ing processes of atom arrival (with preference to sharp features ofthe cluster) and surface drift directed away from these features caneventually be violated. An unstable regime is then obtained atlarger times, with sharp features destabilizing as illustrated inFigure 4c. This can lead to a “dendritic” instability, wherebysecondary structures are formed hierarchically, or to destabiliza-tion leading to the growth of rods, platelets, or other highly

(73) Privman, V. In Encyclopedia of Applied Physics; Trigg J. L., Ed.; AmericanInstitute of Physics: New York, 1998; Vol. 23, p 31.(74) Voorhees, P. W. J. Stat. Phys. 1985, 38, 231.

(75) Sevonkaev, I.; Privman, V. World J. Eng. 2009, in press (e-print 0903.2841at www.arxiv.org).

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uneven shapes. Our simulations were generally not long enoughto observe such late-stage growth. However, we give one exampleof the onset of a rod-shape formation for a particularly largesimulation in section 4.

We point out that the regime of plentiful supply of the buildingblock matter, which leads to fast nonequilibrium growth, is quitecommon both in nanoparticle nucleation and in colloid growth.However, this available matter;atoms, molecules, ions, andsometimes whole nanoparticles in colloid growth;is rapidlydepleted. Results such as those presented in Figure 4 suggest thatshapes including simple ones (a cube), bounded by low-indexplanes with the symmetry of the material’s crystal structure, canbe obtained and then practically frozen if the surroundingbuilding-block matter is consumed at an appropriate rate so thatthe effective times of fast growth are just right. In many nano-particle and colloid growth experiments, the appropriate systemparameters are frequently determined by empirical trial-and-errorapproaches.

We also note that the main difference between the presentnonequilibrium regime and the steady-state regime described insection 3.1 is that the former corresponds to a fast, dominantgrowth process. Other processes, such as those involving theexchange ofmatter with other clusters or cluster-cluster aggrega-tion, are simply slower and therefore do not play a significant roleover relevant time scales, as opposed to the situation for thesteady-state regime. Thus, shapes found to emerge over inter-mediate time scales in the fast-growth nonequilibrium regimesimulations are likely to correspond to experimentally realizablesituations, though the actual connection between experimentalquantities and our MC simulation parameters may not bestraightforward to establish.

Let us now comment specifically on the parameter R. Wealready reported our empirical finding that in order to seek well-defined nonequilibrium shapes the value of R should not bechosen too much below Rcr ≈ 2.7 (all values presently are forthe A-variant SC lattice dynamics). Figure 4a,b illustrates thesituation with R = 2.5 < 2.7. In such cases, well-defined shapescan be formed.However, ifR is too small then the growing clusterremains spherical. In nonequilibrium-regime simulations we alsoobserved that if R is increased well over Rcr, under the conditionsof a constant supply of atoms (fixed n0 at the volume boundary),then the particle shape destabilizes. In fact, for the largest valuessimulated, R≈ 4, we observed completely random particle shapesduring SC-lattice growth. Thus, there is an approximate rangefor the R values, 2 < R < 3.5 for the SC lattice, for whicheven-shaped, low-index-plane particles are obtained for a range ofintermediate times.

Illustrative results for variant B are shown in Figure 5, whichactually corresponds to the choice R = 2.5 and was also ourestimate for Rcr, though we have not explored the issue of thesharpness of the definition of the latter quantity from steady-statesimulations. There is a transient simulation time interval of smallenoughR, for this dynamics as well, during which the cluster sizeis in the regime of surface drift (τd≈R2)which proceeds faster thatthe formation of additional layers (τlayer ≈ R). The cluster shapeagain assumes a form bounded by low-index planes. However, atlater times the diffusive flux redistributes to amplify the instabilityof the vertices of the obtained octahedral shape. The resultingdistortion yields a particle with negative surface curvature insome regions; see Figure 5b. For a larger n0, the atom flux isgenerally larger, and the instability is more pronounced andappears for smaller particle sizes; see Figure 5c, which alreadyshows the onset of dendritic instability (emergence of secondarystructures).

In summary, the variation of quantities such as the availabilityof building block material (i.e., the time dependence of n0(t)),possibly also the building blocks’ diffusion constant and some oftheir on-surface dynamics parameters (if these parameters can becontrolled), and the attachment/detachment probabilities (whichcan be modified by varying pH and/or ionic strength) can go along way toward stopping the process in the well-defined particleshape regime.A careful balance is needed to, onone hand, achievethe fast-growth nonequilibrium conditions and, on the otherhand, not to push the system into the unstable-growth regime.3.3. BCC Lattice. For the BCC lattice simulations, the set

{eBint} consisted of eight nearest-neighbor vectors along the type-(111) directions; see Figure 6.However, the surface atomhoppingset {eBmov} included not only these eight vectors {eBint} but also sixadditional type-(100) next-nearest-neighbor vectors. (The vectorsare defined in terms of theCartesian cubic cell, whichwas of lineardimensions 1 in the hopping length units.)

The BCC lattice (Figure 6) has 12 planes of type (110) and6 planes of (100) type, which will dominate the equilibriumWulff

Figure 5. Nonequilibrium SC cluster shapes in variant B. Here,p0=0.6,R=2.5, n0=1.8� 10-3, and the radiusof the initial seedr0= 15. (a) Projection onto the xy plane for three different times: t=105, 2� 106, and 3� 106. (b) Shape found for time t=7� 106,with 1.18 � 106 atoms in the cluster, which can be inscribed in asphere of diameter 210. (c) Shape for a larger availability of freeatoms: n0=3� 10-3, with 3.8� 105 atoms in the cluster (t=2.4�106), which can be inscribed in a sphere of diameter 175.

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construction at low temperatures.65-68 Assuming that all haveequal interfacial free energies, the Wulff construction for thesefaces is shown in Figure 7a. However, even for our steady-statesimulations, for the BCC lattice we never obtained particles withthese proportions, even though the faces just identified diddominate the growing cluster shapes.

A steady-state (n0 = 0, reflecting boundaries, large seed withnonfixed atoms, etc.) simulation result is summarized inFigure 7b,and it demonstrates that the type-(100) faces dominate. Itmight berelated to the property that free atoms attaching “next-layer” tosuch fully filled faces are bound to four neighbors whereas for thetype-(110) faces the number of neighbors is two. However, theevaporation of anatom from inside a fully filled (100) face requiresthe breaking of six bonds whereas from (110) only four bondsmust be broken. Thus, kinetically these faces substantially differ invarious energy values involved (in dynamic moves).

An interesting phenomenon encountered for this lattice is thatRcr, determined as described in the Introduction, for the initial

seed radius of 45was estimated asRcr≈ 1.6. Thiswas the value forwhich the cluster did not rapidly disintegrate. However, itassumed well-defined shapes only for larger R values. We usedR = 2 for Figure 7b and for the nonequilibrium simulationsbelow. Even forR=1.8, the steady-state simulation cluster shapedid not have well-defined flat faces, though traces of the emergingstructure depicted in Figure 7b could be guessed once one knewthe larger-R results.

The noticed nonequivalence of the dominant low-index planespersists in nonequilibrium simulations as well and can lead to avariety of particle shapes obtained on intermediate time scales.These are illustrated in Figure 8. Specifically, the initially sphericalseed first evolves into a rhombic dodecahedron shape; seeFigure 8a. Apparently, the formation of the pyramidal vertexesof this shape is driven by the increased diffusional flux to sharpfeatures. However, at later times the overall diffusional fluxdecreases; recall that ÆΓ(x, y, z)æ≈ 1/R(t). This results in the vertexesof the rhombic dodecahedron flattening out as a consequence ofenhanced detachment and diffusion of atoms away from thesesharp features (Figures 8b,c), and the shape shows the tendency toevolve toward the one found in the steady-state simulation reportedearlier. In fact, the shape in Figure 8c is actually even closer to thehypothetical shape inFigure 7a, not actually realized in steady-statesimulations, but of course the shape in Figure 8c is not obtained inany thermal equilibrium or in a steady state.

The shape at these intermediate times can be varied byadjusting the system parameters, as illustrated in Figure 8d(nearly a cube) and Figure 8e (a rhombic dodecahedron survivingfrom shorter times). Note that for the parameter valuesof Figure 8e the diffusional flux is increased enough not only tomaintain the vertexes of the shape on the time scales of oursimulation but actually to show the onset of the destabilization:the piling up of attached atoms, marked by arrows in the Figure,similar to that found earlier for a different lattice (and differentshape); see Figure 5.3.4. FCC Lattice. The FCC-lattice vectors will be defined in

terms of the Cartesian cubic cell of linear dimension 2 in our

Figure 6. Eight nearest neighbors and six next-nearest neighborsof an atom in the BCC lattice.

Figure 7. (a) Wulff construction based on the assumption of thedominance of the type-(100) and -(110) faces, with equal interfacialfree energies, for the BCC lattice. (b) The actual cluster shapeobtained in the steady-state regime simulation, with p0= 0.7, R=2, and R(t= 0) = 45.

Figure 8. Nonequilibrium regime for the BCC lattice. (a) Para-meter values r0 = 15, R = 2, p0 = 0.6, and n0 = 1.8 � 10-3:formation of a rhombic dodecahedron from the type-(110) planesin the initial growth stages, with the caliper (maximal) clusterdimension equal to 42. The inset shows the xy-plane projectionof the initial seed (gray) and the formed cluster. (b, c) Evolution ofthe cluster shape at later times for cluster sizes 59 and 68, respec-tively, measured along the x, y, z directions. The time sequence fora-c was t = 5 � 105, 2 � 106, and 3 � 106. (d) Parameter valueschanged top0=0.7,n0=2.25� 10-4: cluster shapewhengrown tosize 68 (t = 3 � 107). (e) Parameter values changed to p0 = 0.6,n0 = 4.5 � 10-3: rhombic dodecahedron of size 69 (t= 106). Thearrows point to regions in which one can see the local piling up ofthe attached atoms (the onset of destabilization).

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hopping-length units. Still, the (111) direction refers to the vectorwith these cubic-cell coordinates and collectively refers to planesperpendicular to it and to similar lattice vectors in the otherdirection. This notation is self-explanatory, and we do not treatthe lattice structure in the formal notation of crystallography nordowe use theMiller indices for lattice plane designation.We notethat for the FCC case Rcr ≈ 0.9.

For the FCC lattice, the densely packed, low-index faces thattend to be present65-68 in equilibrium shapes are (111) as well as(100) and (110). It is therefore natural to expect (some of ) theseto showup in our simulations.Whereas each surface atomat suchfaces on average has a binding energy of 6ε, for dynamic processeswe also have to consider the binding of attaching atoms on topof these faces. The first atom settling on (100) and starting a newlayer has a binding energy of 4ε, and for (111), we get 3ε.

The corrugated (ridged) for FCC, type-(110) faces were notseen in our nonequilibrium simulations, reviewed shortly, eventhough the first extra-atombinding energy is 5ε. It seems that eachfragment of such an overlayer, once formed, acts as a sink for fastadditional atomattachment and little detachment, thus becomingthe base for a growing wedge bounded by type-(111) faces.A posteriori this seems to be a general mechanism, encounteredearlier for the SC lattice: The binding energy of an atom at a type-(110) SC face is larger than that at a type-(100) face. For none-quilibrium growth, the (110) faces are not obtained but ratherprovide “bases” for emerging wedges involving type-(100) faces,resulting in a cube: Figure 4b instead of the shapes shown, forinstance, in Figure 2 or 3. In fact, the expectation that the slowestgrowing faces are the surviving ones is quite accepted in the crystalgrowth literature,69 though obviously the detailed behaviordepends on the specific dynamic rules. We remark that we didnot measure rates of growth of various faces in our simulations;the subject of a possible future project;because this wouldrequire an algorithm for a dynamic, in-process identification ofthe crystal structure (for the program to decide where and whichare the main growing cluster faces), which represents a nontrivialpattern-recognition programming issue.

This mechanism seem to facilitate shape selection in the FCCcase: Even for relatively slow nonequilibrium growth, as illu-strated inFigure 9, a shape is obtained that involves only the (100)and (111) faces and resembles the proportions of the equilibriumWulff shape, also shown, that would be obtained for equalinterfacial free energies of the two types of faces.

Let us now consider fast nonequilibrium growth for the FCCcase. As illustrated in Figure 10, such growth involves nonlinearprocesses, as discussed earlier and, as seen for other latticesymmetries, can yield well-defined shapes for intermediate times(Figure 10a) with the leading growth mode involving vertexesadvancing along type-(100) directions. Less stable growth ispossible for other parameter choices (e.g., Figure 10b), includingthe situations where more than one unstable direction is observed(Figure 10c: (100) and (111)), yielding shapes ranging fromdistorted to ultimately chaotic.3.5. HCP Lattice. Recall that for the HCP lattice we

assumed that the nearest-neighbor distance from a central atomto its 12 neighbors, shown in Figure 11, is 1, in units of the free-atom hopping steps, l. For this lattice, we estimated Rcr≈ 0.8.Wewill use the Cartesian coordinates and “self-explanatory” nota-tion for vector and plane orientation here as well; see Figure 11.

The minimal value of the surface energy, 6ε, is achieved foratoms in the following locations (for the lattice orientation shown

Figure 9. Relatively slow-growth nonequilibrium FCC-latticesimulation: R = 1, p0 = 0.7, and n0 = 2.25 � 10-4. The particledimensions along the coordinate axes are 127 (63 lattice constants),and the total number of atoms in it is 6.82 � 105. The run time ist=2.2� 107. The inset shows theWulff construction shape basedon the type-(100) and -(111) faces, assuming equal interfacial freeenergies for both.

Figure 10. NonequilibriumFCC-lattice simulation: (a) ParametervaluesR=1, p0=0.7, and n0=1.8� 10-3. The dimensions alongthe coordinate axes are 198, and the total number of atoms is 7 �105 (t = 3.6 � 106). (b) Parameter values R = 1.4, p0 = 0.7, andn0=1.8� 10-3. The dimensions along the coordinate axes are 233,and the total number of atoms is 1 � 106 (t = 3.6 � 106). (c)Parameter values R = 3.5, p0 = 0.8, and n0 = 9 � 10-4. Thedimensions along the coordinate axes are 214, and the totalnumber of atoms is 1 � 106 (t= 7 � 106). The gray circles denoteprotrusions grown in the (111) directions. The inset gives theprojection of this shape onto the xy plane.

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in Figure 11): Two horizontal planes (0 0 1) and (0 0 -1). Twelvesurfaces that are at θ ≈ 62� with respect to the horizontal plane;each of these surfaces is actually corrugated (ridged); see contourI in Figure 11. Six vertical corrugated surfaces: see contour V inFigure 11. Figure 12 illustrates a relatively slow growth simulationas well as the would-be Wulff shape had the particle been inthermal equilibrium and with all of these faces having equalinterfacial free energies.

It is obvious that the faces identified are not equivalent indynamic growth. Indeed, a single atom attaching to (001) hasa binding energy of 3ε. The type-I faces have 4ε, whereas thevertical faces (type V) have 6ε. For slow growth, τd < τlayer, thecluster shape is not far from the Wulff shape (Figure 12), thoughV faces are practically not formed because of the fast adsorptionof free atoms on the “equator” of the particle, where the bindingenergy is large. Instead, wedges made of type-I faces are present.

For faster nonequilibrium growth, the cluster shapes canbe quite different; see Figure 13. Here, τd is initially small ascompared to τlayer, and the shape is close to that in Figure 12. Theshape then evolves as shown as a result of the dynamics of thespatially distributed diffusional flux, Γ(x, y, z, t). The shape inFigure 13a has its sharpest features close to the bases, at the edgesof faces (0 0 1) and (0 0-1), where “-1” is self-explanatory: recallthat we are not using the Miller index notation. Therefore,diffusional flux is preferential in these regions. In addition, ina nonequilibrium regime the probability of an atommoving fromtype-(001) faces to type-I faces is larger than the opposite move(because the latter faces bindmore strongly to single atoms). As aresult, a prism is obtained; see Figure 13b.Weagain encounter theshape-transformation mechanism found earlier for other lattices.

It is interesting that the growth is faster in the horizontaldirections (Figure 13). In part, this could be attributed to the

packing of atoms along the z direction (Figure 11). However, thelarger the ratio d^/h becomes, the more of the growth asymmetrythat can be assigned to that the diffusional flux to the sides ofthe growing cluster, which becomes (somewhat) larger than thatto the horizontal faces: Γside > Γz.

Furthermore, for longer times (Figure 13c), the regions near theedges of the bases of the prism constitute preferential attachmentlocations. Apparently, in these regions τd > τlayer; therefore,particle outflow along the surface is ineffective: the shape beginsto bulge at the edges in a manner not dissimilar to the late-timedistortionof the vertexes of the cube inFigure 4c.However, in thiscase the evolution is more orderly.

If n0 is not too large (in real experiments, this quantity isactually a decreasing function of time), then the on-surface driftof atoms between regions such a those marked by a and b inFigure 13c can prevent destabilization and yield the growth ofa hexagonal platelet shape. Otherwise, protruding features suchas a and b in Figure 13c will become the bases for the destabili-zation and formation of secondary structures. Figure 14 illus-trates the latter type of growth. For the selected parametervalues, the system is not controllable. However, the shape inFigure 14 did not yet become fully chaotic: each of the protruding

Figure 11. Features of the HCP lattice: The structure is shownassuming that the straight chains of atoms are positioned along thex-axis direction. The central atom (blue) interacts with 12 nearestneighbors (shown in green). Contours V and I denote corrugatedsurfaces for which εhsurf = -6ε is minimal. (See the text.)

Figure 12. HCP lattice simulation for slow growth, with para-meter values of p0 = 0.7, R= 0.9, and n0 = 6� 10-4. The clustercontains 1.122 � 106 atoms (t = 4.7 � 106). The inset shows theWulff shape based on all of the faces with strong binding (see thetext), which would be obtained under the condition of equalinterfacial free energies for all of them.

Figure 13. HCP lattice simulation for faster growth, with para-meter values of p0 = 0.6, R = 0.9, and n0 = 1.8 � 10-3. (a) Theinitial stage of the growth: Shown is the projectionof the shape intothe yz plane (t=5� 105). (Only the atoms added to the initial seedas a “coating” are shown.) (b) Formation of a right hexagonalprismof heighth=66,abaseof size (diameter of the circumscribedcircle) d^=83, a shape ratio of d^/h=1.26, and 3.77� 105 atomsin the particle (t = 3 � 106). (c) Cluster shape at a later time:h= 96, d^ = 134, shape ratio d^/h= 1.4, and 1.271 � 106 atoms(t= 6.5 � 106).

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secondary-structure “blobs” can be loosely associated with oneof the 18 corners in configurations such as Figure 12 or 13c.

4. Discussion and Additional Considerations

Our numerical simulations reported thus far suggest thefollowing key conclusions. Under certain specific conditions,straightforward processes that include the arrival of singletbuilding blocks (our “atoms”) to the growing cluster, detachmentof these singlets, and also their drift on the cluster’s surface canaccount for the formation of particle shapes defined by a set of themost densely packed, low-index crystal-symmetry faces.

The most important condition is that no large-scale defects arepresent that would dominate the singlet kinetics and the overallrates of the growth dynamics. Interestingly, colloids, given theirgenerally polycrystalline nature, developed by the aggregationof nanocrystalline singlets or by other internal restructuringprocesses that are presently notwell studied or understood, satisfythis condition. The constituent nanocrystalline subunits areapparently not correlated in their crystal-structure orientationor defect continuity: to the extent that evidence is available, thesubunits are likely separated by amorphous interlayers.Of course,the actualmechanismof the singlet attachment, surface dynamics,overall cluster compactification, and ultimately singlets getting“cemented in place” to become part of the growing structure,buried by later arriving singlets, should not be expected to bareany resemblance to our simple model of forced ideal crystalformation. However, the net result is apparently the same, assuggested by ample experimental evidence.

Another condition has been the use of dynamic rules thatmimic thermal, free-energy-driven relaxation. Colloid synthesisgenerally involves processes far from equilibrium or steady-statedynamics. However, energy-related transition rates of the typedefined in section 2, are not that far fetched for the attachment oflarger than atomic-size singlets, as long as the explicit temperaturedependence used inBoltzmann-formweights is not taken literally.

The applicability of our observations for nanocrystals isalso conditional on them having no dynamics-controlling,global defect structures and should probably be verified on acase-by-case basis.

The most densely packed, low-index crystal-symmetry faces,some of which dominate the generated crystal shapes, also figurein the thermal-equilibrium Wulff construction in typical situa-tions. However, our nonequilibrium-growth particles assumesimilar proportions at best only when the growth is slow (or whenwe consider the steady-state growth as defined in section 2.3).

For faster growth, some of the faces are not present and theparticle proportions are different, and we have identified some

rule of thumb regularities connected to the binding energies ofatoms within a flat surface and those attaching as singlets onsuch a filled surface. Some of the observed regularities have beenassociated with the distribution of the diffusional fluxes to sharpsurface features. In some situations, direct identification is pos-sible for sharp surface features, which are expected to cause, anddo lead to, distortions and instabilities and serve as regions on topof which new structures develop.

It would be interesting to extend our simulations to study theformation of other, longer-time shapes as well as other particle-growth processes. However, as pointed out in section 2, thesimulationswere large-scale: resource- andCPUtime-consuming.Therefore, we leave such explorations to future work. Further-more, the applicability of the presentmodel to larger-scale growthhas not been established.

We did run some simulations whereby an attempt was madeto introduce protruding features to control the cluster distortion.For an illustration, let us point out that in much experimentalwork colloid particles shaped as ellipsoids of revolution weresynthesized. Therefore, we attempted an SC simulation forwhich the initial seed was ellipsoidal, with the large axis made1.5 times longer than the short axis.Although a guess can bemadethat the two large-axis ends will serve as the starting regionsfor fast growth, in particular, collecting larger diffusional flux,it turns out to be an oversimplification: In addition to elongation,secondary distortions develop (Figure 15) along directionstypical for the SC symmetry. We attribute this to the formationof fragments of the type-(111) faces due to fluctuations.These regions have large atom-binding energies and thereforeserve as “seed regions” for faster growth and, ultimately, possibleinstability.

We know experimentally that in many systems the growth oflong rods and other structures of uneven proportions is quitecommon. Can our approach reproduce such large-time regimes?The simulations would be formidable and beyond our presentnumerical capabilities. However, we did find one manageable(by a lengthy run) example, shown in Figure 16.

For the BCC lattice (Figure 16), the rhombic dodecahedronshape (Figure 8) is initially symmetric with respect to the x, y, zdirections. The distortion along the z axis for later times (Fig-ure 16) is a result of random fluctuations. Specifically, thetendency to elongate is already seen in Figure 8e, in whichlx = 75 and lz = 69, so that there is preferential growth in thex direction. The rodlike distortion seems to grow without devel-oping instabilities, though we did not attempt to follow thedynamics on the time scales of possible long rod formation(because of computational resource limitations). Furthermore,the reported result is not fully reproducible: this growth regime is

Figure 14. Onset of multimode unstable growth: HCP latticesimulation with p0 = 0.5, R = 1, and n0 = 1.8 � 10-3. Here thecluster contains 1.278 � 106 atoms (t= 6 � 106).

Figure 15. SC lattice simulation (type-A dynamics) for an initiallyelongated (in the horizontal direction) seed.Herep0=0.5,R=2.3,n0=1.8� 10-3, and t=8� 106, and the cluster contains 1.9� 106

atoms.

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too close to chaotic. The emergence of a rodlike shape did nothappen for a fraction of the runs. Instead, for some runs clustersranging from evenly shaped to chaotic were obtained, starting

from those shown in Figure 8, with details dependent on themodel parameters.

In summary, we hope that our present numerical results willsomewhat demystify the long-standing openproblemof the originof shape selection in colloid and nanoparticle synthesis and shedlight on the mechanisms and conditions for obtaining particleshapes sought after for applications. The status of this researchfield is still far from predictive, and additional theoretical studiesand a comparison of the modeling results with experimental dataare needed. Our results suggest an emphasis on experimentalprobes of the morphological features on the scales of the particleas a whole in order to test the conclusion/conjecture that theabsence of persistent defect structures, which could globallyinfluence the face growth dynamics, is crucial to shape selectionin fine-particle synthesis.

Acknowledgment. We gratefully acknowledge instructivediscussions and collaboration with G. P. Berman, D. V. Goia,I. Halaciuga, S. Libert, E. Matijevi�c, and I. Sevonkaev as well asresearch funding by theU.S. National Science Foundation (grantDMR-0509104).

Figure 16. Onset of rodlike shape growth in a long-run none-quilibriumBCC lattice simulation.Here p0=0.6,R=2, and n0=1.125� 10-3. (a) Time dependence of the transverse sizes, ly(≈ lx),and of the longitudinal size, lz. The inset shows the shape of thecluster for t = 1.75 � 107. (b) t = 5.2 � 107, at which time thecluster contained ∼5 � 106 atoms, lz = 240, and lx = 140.

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