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Colloids, Nanocrystals, and Surface Nanostructures of Uniform Size and Shape: Modeling of Nucleation and Growth in Solution Synthesis Vladimir Privman Complex-shaped Metal Nanoparticles: Bottom-Up Syntheses and Applications, First Edition. Edited by Tapan K. Sau and Andrey L. Rogach. Ó 2012 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2012 by Wiley-VCH Verlag GmbH & Co. KGaA. 7.1 Introduction 239 7.2 Burst Nucleation Model for Nanoparticle Growth 242 7.3 Colloid Synthesis by Fast Growth 247 7.4 Improved Models for Two-Stage Colloid Growth 251 7.5 Particle Shape Selection in Solution Synthesis 254 7.6 Applications for Control of Morphology in Surface Structure Formation 261 7.7 Summary 263 References 264 Abstract We review models that explain mechanisms of obtaining narrow distribution of sizes and shapes in the solution synthesis of nanosize crystals, certain surface structures, and polycrystalline colloid particles. We consider dynamic selection of geometrical features and morphology in processes ranging from nucleation to growth by aggregation, for kinetics involving diffusional transport of matter in solution and restructuring of the growing particle surfaces, yielding well-defined structures and particles. Keywords: aggregation; catalysis; cluster; colloid; crystal; deposition; detachment; diffusion; growth; morphology; nanocluster; nanocrystal; nanoparticle; nanopillar; nanosize; nucleation; surface; symmetry. bstract Keywords: bstract Keywords:

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Page 1: Colloids, Nanocrystals, and Surface Nanostructures of ...vprivman/240.pdf · Colloids, Nanocrystals, and Surface Nanostructures of Uniform Size and Shape: Modeling of Nucleation and

Colloids, Nanocrystals, and Surface Nanostructuresof Uniform Size and Shape: Modeling of Nucleationand Growth in Solution SynthesisVladimir Privman

Complex-shaped Metal Nanoparticles: Bottom-Up Syntheses and Applications, First Edition.Edited by Tapan K. Sau and Andrey L. Rogach.� 2012 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2012 by Wiley-VCH Verlag GmbH & Co. KGaA.

7.1 Introduction 2397.2 Burst Nucleation Model for Nanoparticle Growth 2427.3 Colloid Synthesis by Fast Growth 2477.4 Improved Models for Two-Stage Colloid Growth 2517.5 Particle Shape Selection in Solution Synthesis 2547.6 Applications for Control of Morphology in Surface Structure

Formation 2617.7 Summary 263

References 264

Abstract

We review models that explain mechanisms of obtaining narrow distribution of sizes and shapes in the solution synthesis of nanosize crystals, certain surface structures, and polycrystalline colloid particles. We consider dynamic selection of geometrical features and morphology in processes ranging from nucleation to growth by aggregation, for kinetics involving diffusional transport of matter in solution and restructuring of the growing particle surfaces, yielding well-defined structures and particles. Keywords: aggregation; catalysis; cluster; colloid; crystal; deposition; detachment; diffusion; growth; morphology; nanocluster; nanocrystal; nanoparticle; nanopillar; nanosize; nucleation; surface; symmetry.

bstract

Keywords:

bstract

Keywords:

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7Colloids, Nanocrystals, and Surface Nanostructuresof Uniform Size and Shape: Modeling of Nucleationand Growth in Solution SynthesisVladimir Privman

7.1Introduction

In most applications of microscopic particles and synthetic surface structures,uniformity of their sizes, shapes, and other morphological features is paramount.Kineticmechanisms for emergence of narrow size and shape distributions in growthfrom solutions of the constituent matter depend on the structure grown. We refer tocolloids as suspensions of few-micron down to submicron size particles, whereassuspended nanoparticles and on-surface nanostructures are objects of smaller sizes,typically below few 0.01 mm(10 nm). Not only size and shape but also the uniformity/control of composition, internal structure/morphology, and surface properties areimportant.

Multiscale kinetic processes are involved in synthesis by transport of matter insolution/suspension: nucleation, growth, aggregation, surface interactions, andthese involve atoms (ormolecules or ions, as well as possibly their chemical reactionswith each other and with the solution species), already formed nanosize objects, andgrowing particles/structures. A realistic modeling approach, therefore, will beplagued by numerical difficulties unless a subset of kinetic processes can be singledout in a �cartoon� description that captures themechanisms of size, shape, and otherfeature uniformity of experimental relevance. We review our development ofapproaches to and results on studies [1–17] of size selection for burst nucleationof nanocrystals in solution, for a secondary process of their diffusional aggregationinto polycrystalline colloids, and aspects of shape selection innonequilibriumgrowthin solution and on substrates.

Synthesis of particles of narrow size distribution has been of importance in colloidchemistry for a long time [1, 18, 19]. The field has faced new challenges fornanotechnology because size and shape selection mechanisms at the nanoscale canbe different from those at the microscale, and the experimental data obtainable,especially those on the time dependence of the particle growth, are more limited fornanoparticles/nanostructures.

Complex-shaped Metal Nanoparticles: Bottom-Up Syntheses and Applications, First Edition.Edited by Tapan K. Sau and Andrey L. Rogach.� 2012 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2012 by Wiley-VCH Verlag GmbH & Co. KGaA.

j239

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In solution synthesis, the particles and constituent units from which they areformed are suspended/dissolved in aqueous or another medium under controlledchemical and physical conditions and the transport of solute/suspension matter ofrelevant sizes is assumed to be diffusional. The constituent units, also termedmonomers or singlets, in nanoparticle/nanostructure synthesis are solute species:atoms, ions, and molecules. For polycrystalline colloids, singlets can be nanocrystal-line precursor �primary� particles, the supply of which is in itself a growth process,typically by burst nucleation. In all cases, in addition to the in-solution/suspensionkinetics, the monomers can also be introduced externally as a means to control thegrowth kinetics, which in turn can be initiated in variousways ranging fromchemicalreaction release of atoms, etc., to seeding and on-surface growth.

Size distribution of clusters (particles), with the size defined as the number ofconstituent units, s, in each cluster, with the desirable property of having a relativelynarrow peak at large sizes, speak, is shown in Figure 7.1. Kinetic processes can causethe average particle size at the peak, speak(t), to grow with time t. However, they alsobroaden the peak. This is a generic expectation for cluster–cluster aggregation orcluster ripening due to exchange of monomers because generally larger particleshave bigger surface area for capturing small matter (monomers, etc.). Furthermore,their surfaces have less curvature, fraction of edges, corners and therefore offer, onaverage, somewhat better binding and thus less detachment probability for mono-mers and so on. Thus, the larger s side of the peak �outruns� the smaller s side (seeFigure 7.1), resulting in broadening during growth. Similarly, particle growth is alsoaccompanied by surface destabilization that results in random/nonuniform shapes,which will be further discussed later.

Ns speak(t )

C(t )

Shoulder Peak

4321 s

Figure 7.1 Desirable peaked particle sizedistribution, NsðtÞ, with the peak position,speakðtÞ, growing with time while the peakremaining relatively narrow. Depending on thenature of the synthesis, the peak growth isdriven by the chemically released, or nucleated,or physically added monomer �singlets�

maintained at concentration CðtÞ ¼ N1ðtÞ.Separate values shown for particles of sizess ¼ 1; 2; 3; 4 monomers draw attention that s isactually discrete. For large enough s, NsðtÞ canbe considered a function of continuous0 � s < ¥.

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Approaches to obtain relatively narrow size (and shape) distributions can vary. Forexample, one could limit the extent to which the larger s side of the peak can grow byforming the particles inside objects, such as micelles or inverse micelles [20, 21].Seeding by growth on top of separately prepared uniform size (and shape)templates [22] is another technique of general utility. Here, we discuss approachesbased on the specifics of the growth kinetics in situation when uniform (narrowdistribution) size and shape selection can be achieved. Specifically, in Section 7.2 weconsider [9–11] the process of burst nucleation of nanocrystals rapidly growing in asupersaturated solution of constituent atoms, molecules, or ions. Size selectionoccurs by the smaller s side of the peak (Figure 7.1), being fast eroded compared tothe peak broadening due to its growth on the larger s size by consumption ofmonomers. Additional coarsening processes broaden the distribution after theinitial nucleation burst, limiting this mechanism to the truly nanosize crystalgrowth stage.

A two-stage colloid growthmechanism [1, 6, 10, 11] for polycrystalline colloids canproduce narrow particle size distributions on a relative scale. Primary particleprecursor nanocrystals of concentration CðtÞ (see Figure 7.1) are burst nucleated,and their availability as monomers to aggregate into colloids allows the peak to growto large sizes fast enough so that it is not significantly broadened. A proper control ofCðtÞ is needed to avoid buildup of a significant few-monomer �shoulder� (Figure 7.1).For nanoparticle growth, there have also been approaches [23, 24] based on stepwiseaddition of batches of atomic size monomers, similar to the aforementioned controlof CðtÞ. To be more specific, let NsðtÞ denote the density of particles containing ssinglets at time t. Except for very small s values, the distribution is treated as a functionof continuous s variable (Figure 7.1). The singlet (monomer) concentration, however,is separately controlled

CðtÞ � N1ðtÞ; ð7:1Þfor instance, through the introduction of additional species that can be also synthe-sized by another process at the rate rðtÞ per unit volume, while consumed byaggregation into small clusters in the �shoulder,� as well as by the growing largerclusters including those in the main peak.

The initial emergence of the peak is also an interesting problem to address. Fornanostructures, it is formed naturally in burst nucleation (Section 7.2). For colloids,the initial peak formation can be a by-product of cluster–cluster aggregation at theearly growth stages. This will be discussed in Section 7.4, where some of the technicalmatters relevant to modeling of uniform colloid growth are addressed. This isdiscussed further in Section 7.5. Seeding is, of course, a useful mechanism toinitiate the peaked size distribution.

Control of particle shape distribution for uniformity is much less understood forfast, nonequilibrium growth than control of size. It is likely that shape selectionmechanisms are not unique and depend on specific situations. We have developed amodel [14] (reviewed in Section 7.5), suggesting that growth without development oflarge internal defect structures can lead to well-defined shapes with �faces� similar tothose dominating the equilibrium crystal form, but of different aspect ratios.

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These ideas have also been successfully applied [17] to study the morphology ofcertain growing nanostructures on surfaces synthesized for catalysis applications(Section 7.6). Finally, conclusions are drawn in Section 7.7.

7.2Burst Nucleation Model for Nanoparticle Growth

Burst nucleation growth [9–11, 25, 26] is driven by a large supersaturation ofmonomers in solution. The model can realistically be used for growth of onlynanosize particles consisting of nmonomers. Gibbsian assumptions for nucleationare made. A critical cluster size, n > nc, is introduced (defined shortly), such thatparticles with n > nc grow by irreversibly capturing diffusing solutes (monomers):atoms, ions, or molecules, whereas small subcritical embryos in the shoulder, forn < nc (see Figure 7.2), are instantaneously thermalized.

Burst nucleation is initiated by externally supplied or chemical reaction producedmonomers of concentration cðtÞ, well exceeding the equilibrium value c0. Innucleation theory, thermal fluctuations are assumed to cause formation of smallembryos, with their surface free energy resulting in a free energy barrier peaked at nc.The restructuring dynamics of few-atom embryos in solution is not well understood.Instead, these processes are assumed to be fast enough so that the n < nc embryosizes are approximately thermally distributed according to theGibbs free energy of ann-atom (n-solute) embryo,

P(n,t )

nnc

nnc(t )

t

Figure 7.2 Nanoparticle size distributionwithin the burst nucleation approximation. Theactual distribution, shownby the dotted line, willbe steep but smooth at nc. Time variation of the

critical cluster size, nc, is shown in the inset:initial induction period precedes the �burst,�while larger time linear growth typically has avery small slope.

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DG n; cð Þ ¼ � n�1ð ÞkT ln c=c0ð Þþ 4pa2 n2=3�1� �

s; ð7:2Þ

where k is theBoltzmann constant,T is the temperature, ands is the effective surfacetension. The effective radius, a, is defined in a way that the radius of an n-atomembryo is an1=3. It can be estimated by requiring that 4pa3=3 equals the unit cellvolume per singlet (including the surrounding void volume) in the bulk material.This free energy (Figure 7.3) attains maximum (the nucleation barrier) at n ¼ nc,

nc cð Þ ¼ 8pa2s3kT ln c=c0ð Þ

� �3: ð7:3Þ

The first, �bulk,� term in Eq. (7.2), proportional to n (we ignore small n correctionsfor the moment), is negative (for c > c0), favoring growth. The logarithmic depen-dence on the monomer concentration, c(t), follows from the entropy of mixing ofnoninteracting solutes lost when they are bound, whereas the c0 denominator in thelogarithm originates from the reference energy gained by binding of solutes. Thesecond, �surface free energy cost,� term is proportional to n2=3 (i.e., to the surfacearea) and positive, thus disfavoring growth of clusters. Competition of two freeenergy contributions results in the nucleation barrier (Figure 7.3). Embryos withn < nc are assumed to be instantaneously thermally distributed. The kinetics ofclusters for n > nc is assumed to correspond to fast irreversible capture of solutes.Both assumptions are typical for homogeneous nucleation. The unique property ofthe burst nucleation mechanism is that the bulk term is explicitly dependent on c(t)and therefore varies with time. Therefore, the critical cluster size, ncðcðtÞÞ, and theheight of the nucleation barrier are also time dependent.

nnc

ΔG(n)

Figure 7.3 Schematic of the free energyfunction given in Eq. (7.2). Embryos with sizesup to the nucleation barrier (peak) value at nc areassumed to be thermally distributed, whereas

larger clusters (particles) are assumed to growirreversibly and their size distribution is notcontrolled by the free energy values.

7.2 Burst Nucleation Model for Nanoparticle Growth j243

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In nucleation theories of the type discussed here, for an approximate evaluation ofthe growth of the size distribution within a theoretically tractable approach, thedistribution of particle shapes is ignored. A �representative� spherical form is usedinstead for calculation of surface area,monomer transport rate, and other properties.The shape distributions, including surface details and, for crystals, symmetry faces,edges, corners, and so on, affect all the properties that determine matter transportand binding in the structure. Even for spherical surface portions, the effective surfacetension, for instance, depends on the radius via surface curvature. These geometry-and morphology-dependent corrections are usually ignored in the nucleation theoryprimarily because of the computational difficulties of treatingmultivariable (in termsof feature sizes and shapes of the particles) distributions. However, another impor-tant reason has been that various properties, such as the effective surface tension ofapproximately spherical nanoparticles, s, and generally nanostructure size/shape-related properties are currently understood and experimentally quantifiable only to avery limited extent [27]. Specifically, s in Eqs. (7.2) and (7.3) is typically not knownfrom independent nanosize ormicrosize particle propertymeasurements. Instead, ithas been treated [1, 5, 7, 8] as an adjustable parameter or set to the measured bulkmaterial value, sbulk, even though the latter might not always be the correct estimatefor fine particles, especially those smaller than 5–10 nm.

Most of the reported experimental approaches have allowed to obtain – for colloidsand to a much more limited extent for nanoparticles – only size and other propertydistribution information for the final products by direct observation/counting. Timedependence information is rather limited [8], and even the final distribution data arescarce in the case of nanoparticles. Therefore, certain simplifiedmodel assumptionshave to be used to �translate� the microscopic dynamical behavior into certainlumped properties that can be measured for the system as the whole, in order to usethe data thus obtained for model validation. Recently, such an approach, whichallows model validation via multiscale numerical calculations for industrial-scaleprocesses for which data taking is possible, has been developed and applied inRefs [28, 29].

In nucleation, c=c0 decreases from its initial value cð0Þ=c0 � 1, approaching itslarge time value 1. The term �burst nucleation� refers to the fact that the logarithmic(entropic) term in the free energy (Eq. (7.2)) thendecreases inmagnitude, resulting ina larger barrier for nucleatingnew supercritical clusters. The particle size distributionthen evolves into the late-stage form [9–11] shown in Figure 7.2. Subcritical embryosare thermalized on timescales faster than other dynamical processes,

Pðn < nc; tÞ ¼ cðtÞ exp �DGðn; cðtÞÞkT

� �; ð7:4Þ

Pðn; tÞdn gives the number concentration per unit volume of embryos with sizesin dn.

Not only is nc ¼ ncðcðtÞÞ time dependent, but also the approximate (but of coursenot the actual) particle size distribution in burst nucleation is discontinuous at nc(seeFigure 7.2). The growth of clusters that �go over the barrier� at n ¼ nc occurs at therate rðtÞ per unit time, per unit volume, modeled [1] as

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rðtÞ ¼ KnccP nc; tð Þ ¼ Kncc2 exp

�DG nc; cð ÞkT

� �: ð7:5Þ

Here, we define

Kn ¼ 4pan1=3D; ð7:6Þ

where cKn is the Smoluchowski rate for the irreversible capture of diffusingmonomers by spherical clusters of sizes n � nc � 1 andD is the diffusion coefficientfor monomers in a dilute solution of viscosity g, which is estimated as � kT=6pga,up to geometrical factors relating the effective radius a to the hydrodynamic radius.

Growth for n > nc is irreversible according [9] to the kinetic equation:

qPðn; tÞqt

¼ cðtÞ�c0ð Þ Kn�1Pðn�1; tÞ�KnPðn; tÞð Þ; ð7:7Þ

where cðtÞ�c0 is introduced in place of cðtÞ in the Smoluchowski rate in order toensure that the growth stops as cðtÞ approaches c0. This approximately accounts [2] forthe detachment of matter if we ignore curvature and other surface shape andstructure effects. The latter include variation of the surface tension with particleradius, resulting in a variable effective �equilibrium concentration� and, oncepossible monomer detachment is considered, in Ostwald ripening driven byexchange ofmonomers between larger clusters. This and other additional coarseningprocesses, such as cluster–cluster aggregation [30, 31], are ignored here becausethey are typically slower than burst nucleation [1, 9–11]. The large time lineargrowth [9–11] of ncðtÞ, shown in the inset in Figure 7.2, has a very small slope[Sevonkaev, I. (2008) private communication]: the particle growth would practicallystop. However, for later times other coarsening processes will dominate and alsobroaden the particle size distribution compared to the burst nucleation alone.

Dynamical processes involving the surrounding solution and solutematter controlthe evolution of the particle size distribution and at the same time their internalrestructuring. Understanding of the latter for nanosize clusters is not well devel-oped [32, 33]. Without restructuring, clusters would grow as fractals [30, 31] ratherthan nanocrystals. Even for larger particles, density measurements and X-raydiffraction data for colloids aggregated from burst nucleated nanocrystals indicatethat while they typically have polycrystalline structure, their density is close to that ofthe bulk material [1, 18]. There is both experimental and indirect modeling evi-dence [1, 4, 5, 7, 8] that in fast irreversible growth colloid synthesis, internalrestructuring leads to compact particles with relatively smooth surfaces.

The n > ncðtÞ particles in burst nucleation are assumed to grow by capturingsolutes. At the same time, sizes of n < ncðtÞ embryos are redistributed by fastthermalization. The cutoff at ncðtÞ is monotonically increasing, as shown in Fig-ure 7.2. This sharp change in the dynamics is, of course, an approximation of thetheory. The short-time formof the particle size distribution as a function ofndependson initial conditions. One can generally expect [9–11] that for large times the sizedistribution will attain maximum at n ¼ nc, and while thermal for n < ncðtÞ

7.2 Burst Nucleation Model for Nanoparticle Growth j245

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(see Eq. (7.4)), will assume the shape of a truncated Gaussian for n > ncðtÞ, where thepeak of the Gaussian curve (not shown in Figure 7.2) is actually to the left of nc.

The asymptotic properties of the particle size distribution described previouslywere derived [9–11] and then numerically verified by calculating time-dependentdistributions for several sets of parameters and initial conditions, by utilizing a novelefficient numerical integration scheme [9]. Here, we survey the analytical derivationfor large times. In that limit, the kinetic equation has a Gaussian solution,

PGðn; tÞ ¼ zðtÞc0 exp � aðtÞð Þ2 n�MðtÞð Þ2h i

; ð7:8Þ

for n > ncðtÞ and the �peak offset� ncðtÞ�MðtÞ is a positive quantity. The continuous nformof Eq. (7.7), keeping termsup to the secondderivative inn in order to capture thediffusive nature of the peak broadening, is

qPqt

¼ c�c0ð Þ 12q2

qn2� qqn

� �KnPð Þ

� �: ð7:9Þ

For large times, t, the peak is always narrow [9], we can further approximate

Kn � Knc ¼k ncðtÞð Þ1=3

c0; ð7:10Þ

with

k � 4pc0aD: ð7:11ÞEquations (7.3) and (7.6) then imply that the product of the coefficients, c�c0ð ÞKnc ,in Eq. (7.9) becomes a constant,

qPqt

¼ z2

212q2

qn2� qqn

� �P; ð7:12Þ

where

z2 � 64p2a3sc0D3kT

: ð7:13Þ

Equation (7.12) is solved by the Gaussian of Eq. (7.8), with

aðtÞ � 1ffiffiffiffiffiffiz2t

p ; MðtÞ � z2t2

; zðtÞ � Vffiffiffiffiffiffiz2t

p : ð7:14Þ

The prefactorV is fixed by the overall height of the distribution, that is, by the initialconditions, and therefore cannot be calculated from the large time analysis alone: itshould be determined from the conservation of matter. Additional detailed math-ematical considerations [9] yield ncðtÞ�MðtÞ /

ffiffiffiffiffiffiffiffiffit ln t

p(with a positive coefficient) for

the �peak offset.� Since MðtÞ is linear in time, theffiffiffiffiffiffiffiffiffit ln t

p�offset� is subleading,

ncðtÞ � z2t=2: ð7:15Þ

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Since the width of the truncated Gaussian is proportional to 1=a / ffiffit

p, the final

conclusion is that the relative width decreases according to� t�1=2. This is the sensein which the size distribution of the nucleated (supercritical) particles in burstnucleation can be regarded as narrow: not in absolute terms, but relative to the meanparticle size. One can also show [9] that the difference cðtÞ�c0 approaches zero(� t�1=3) as expected.

Numerical tests have shown [9] that the Gaussian shape offers a good approxi-mation for the burst nucleationmodel particle size distribution also for intermediatetimes, including the initially seeded distributions. Experimentally, it has beenchallenging to quantify distributions of nucleated nanocrystals because of theirnonspherical shapes and tendency to aggregate. The distribution is usually moreevenly two sided around the peak, the peak itself is broader than the burst nucleationprediction, and the final particles stop growing after a certain time. These propertiesare at odds with the predictions of the simplest burst nucleation model. Some of thedifferences can be clearly associated with the assumptions of instantaneous ther-malization of clusters of all the sizes below the critical or with ignoring other growthmechanisms. The latter include cluster–cluster aggregation, or a fuller accounting forthe effects of a possible monomer detachment, beyond the replacement, followingthe ideas of Ref. [2], of the prefactor: c! c�c0, in the Smoluchowski rate expressionin writing the kinetic equation (7.7). Monomer detachment, competing with theircapture, and structure (curvature) related surface free energy differences betweenparticles are the ingredients for the process of Ostwald ripening.

Thestructural properties anddynamicsof verysmall clustersareparticularlydifficultto observe experimentally. However, up to sizes tentatively estimated [6–8, 34, 35],[Goia, D.V. (1999) private communication] to correspond tonth � 15�25 �monomers�(atoms, ions, molecules, subclusters), such particles should evolve rapidly enough forthe assumptionof fast thermalization tobe fully justified. Larger clusters thendevelop abulk-like core and their internal restructuring slows down. Thus, for ncðtÞ � nth, the�classical� nucleation model should be regarded as approximate. Modifications of themodel have been contemplated [9, 36, 37]. This, however, requires introduction ofadditional dynamical parameters, which are not as well understood as those of thealready reviewed model.

7.3Colloid Synthesis by Fast Growth

Nanosize particles, of sizes typically up to a couple of 10nm, burst nucleating and tosome extent also further growing by other slower coarsening mechanisms can at thesame time aggregate, themselves becoming the building block �singlets� for theformation of polycrystalline colloids in a two-stage mechanism [1], as shown inFigure 7.4. The two processes should be properly related in order to have narrow sizedistribution colloids thus synthesized, typically with average sizes in the range from afraction to a couple of micrometers. Syntheses of (nearly) uniform colloid particles ofvarious chemical compositions and shapes have been reported [1, 7, 8, 18, 38–61],

7.3 Colloid Synthesis by Fast Growth j247

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with structural properties consistent with the current considered growth mechanism.SphericalcolloidshadpolycrystallineX-raycharacteristics, includingZnS[40],CdS[7,8,39], Fe2O3 [38], Au, Ag, and other metals [1, 26, 55–57, 59, 61]. Generally, experimentsconfirmthatmany (nearly)monodispersedinorganiccolloidsconsistofnanocrystallinesubunits [1, 7, 8, 18, 38–57, 59–61]. These subunits [1, 53, 55] have the same sizes as thesizesof theprecursorsinglets formedinsolutions.Compositeparticlestructurehasalsobeen observed for certain uniform nonspherical colloids [38, 47, 49, 51, 60], but thesefindings have not been conclusive to definitively support the present two-stage growthmechanism presented in Figure 7.4. Here, we present a model in the minimalformulation, with simplifications that allow us to avoid introduction of unknownmicroscopic parameters. In the next section, we describe improvements that allow abetter agreement with experimental observations. Additional information and resultscan be found in Refs [1, 5, 7, 8, 12, 13, 28, 29].

The particles are assumed to grow primarily by irreversible capture of singlets.This is appropriate as a description of the evolution of an already well-developed peak(see Figure 7.1), with the role of the few particles in the �shoulder� (Figure 7.1) beingminimal. The emergence of the initial peak is discussed in Section 7.4. We use rateequations with Cs denoting the rate constants for singlet capture by the s � 1aggregates:

dNs

dt¼ ðCs�1Ns�1�CsNsÞC; s > 2; ð7:16Þ

dN2

dt¼ 1

2C1C�C2N2

� �C; ð7:17Þ

dCdt

¼ r�X¥s¼2

sdNs

dt¼ r�C1C

2�CX¥s¼2

CsNs: ð7:18Þ

Here, we ignore cluster–cluster aggregation: the s > 1 aggregates capture onlysinglets at the rate proportional to the concentration of the latter, GsC (Eq. (7.1)).This assumption is generally accepted in the literature [1, 5, 6, 62–64], and we willdescribe possible elaborations later: Processes such as cluster–cluster aggrega-tion [30, 31], detachment [2, 4] and exchange of singlets (ripening), and so on alsocontribute to andmodify the pattern of growth, andmost of thembroaden the particlesize distribution. However, they are usually slower than the singlet-driven growth.

Supersaturatedsolution

Burst–nucleatedprecursornanocrystals

Aggregatescompactifyinginto colloids

Figure 7.4 Two-stage synthesis of narrow size distribution colloids as aggregates of precursornanocrystals burst nucleating in a supersaturated solution.

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Equations (7.16)–(7.18) do not account for possible particle shape andmorphologydistribution. These are not as well understood and are difficult to model(see Section 7.5) compared to size distribution. Experimentally, it has beenobserved [1, 41–43, 48, 50, 55] that the growing aggregates rapidly restructure intocompact bulk-like particles of typically, thoughnot always, spherical shape for the two-stage aggregation. Without such restructuring, the clusters would be fractal [31, 65].

For the singlet supply-driven growth, if the singlets are supplied constantly, thenthe size distribution develops a large shoulder at small aggregates, with no pro-nounced peak at s � 1. If the supply of singlets is limited except initially, then onlysmall aggregates will be formed (no growth). In studies of colloid synthesis [1, 6], itwas concluded that there are protocols of singlet supply, at the rate rðtÞ that is aproperly decreasing function of time, that yield growth of relatively narrow peaked(at large s) distributions. The primary process of burst nucleated nanocrystallineprecursors growing past the nucleation barrier and further coarsening, naturally�feeds� the secondary process of these precursor nanoparticles aggregating to formcolloids, just at a desired rate.

Growth of the secondary particles in particular must be facilitated by the appro-priate chemical conditions in the system set by the ionic strength and pH. Surfacepotential should be close to zero (the isoelectric point) and the electrostatic screeningshould be strong, to avoid electrostatic barriers, in order to promote fast irreversibleprimary particle attachment [1, 41–43, 48, 50, 55]. Clusters of s singlets are thenaggregated in solution with the volume number densities Ns¼1;2;3;...ðtÞ. The count sis the number of the incorporated primary particles (nanocrystalline domains);the domains themselves (originating from singlets) may somewhat vary. We takethe initial conditions Ns¼1;2;3;...ð0Þ ¼ 0. The simplest choice for rate constants is theSmoluchowski expression (cf. Eq. (7.6))

Gs � 4pRpDps1=3: ð7:19Þ

Here,Rp andDp are theeffectiveprimaryparticle radiusanddiffusionconstant,whicharediscussedlater.Theapproximatesignisusedbecausepossible improvementtothisexpression can be offered. A numerical calculation result for a model of the typeoutlinedhere isshowninFigure7.5.It illustrateskeyfeaturessuchastheemergenceofthe peak and then �size selection,� that is, the practical �freezing� of the growth, evenwhen looked at in exponentially increasing time intervals (shown in steps 10).

For rðtÞ in Eq. (7.18), we use the rate of production of supercritical clusters (seeEq. (7.5)), the calculation of which requires cðtÞ. The following [1] approximaterelation can then be used to get an equation for cðtÞ,

dcdt

¼ �ncr; ð7:20Þ

this is combined with Eqs. (7.3), (7.5) and (7.6). Complicated steps [9] are required toderive the expression (not reported here) for dc=dt in burst nucleation, even withoutthe added secondary aggregation. If the burst nucleated, growing supercriticalparticles are in addition consumed by the secondary aggregation, even more

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complicated considerations are involved. The solute species (of concentration cðtÞ inthe supersaturated solution) are also partly stored in the n> 1 subcritical embryos, aswell as in the supercritical primary particles and in the secondary aggregates. Theycan be captured, but also detach back into the solution. Equation (7.20) is anapproximation that offers tractability by ignoring the possible rebalancing of the�recoverable� stored solute species in various parts of the particle distributions. Itfocuses on the loss of the solute species availability due to mostly unrecoverablestorage in secondary particles of sizes s ¼ 1; 2; 3; . . .. Here, the s ¼ 1 particles are thesinglets – the nucleated supercritical nuclei; s > 2 corresponds to their aggregates.The right-hand side of Eq. (7.20), used with Eqs. (7.3), (7.5) and (7.6), also ignoresfurther capture by and detachment from larger particles. Thus, we get closedequations for the rate rðtÞ and the availability of singlets for the secondary aggre-gation, starting with the initial supercritical concentration cð0Þ � c0,

dcdt

¼ � 214p5a9s4Dac2

ð3kTÞ4½ln ðc=c0Þ4exp � 28p3a6s3

ð3kTÞ3½ln ðc=c0Þ2( )

; ð7:21Þ

rðtÞ ¼ 25p2a3sDac2

3kT ln ðc=c0Þ exp � 28p3a6s3

ð3kTÞ3½ln ðc=c0Þ2( )

: ð7:22Þ

We denoted the diffusion constant of the solutes by Da (cf. Dp for the primaryparticles).

Let us now comment on the choice of model parameters. In the next section,possible modifications of the model are considered, and in fact Figure 7.5 was basedon one of the sets of the parameter values used for modeling formation of uniform

Figure 7.5 Growing particle radial size distribution for several times, t, calculated with modelparameters for spherical gold colloids [1, 5].

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spherical Au particles, and it already includes some of those improvements [5]. Ifthe assumption s � 1 is not made throughout, then the full Smoluchowski rateexpression should be used. For aggregation of particles of sizes s1 and s2, onencounters due to their diffusional motion, we use

Gs1;s2 ! s1 þ s2 ’ 4p Rp s1=31 þ s1=32

� �h iDp s�1=3

1 þ s�1=32

� �h i; ð7:23Þ

where for singlet capture s1 ¼ s and s2 ¼ 1. This introduces nontrivial factors forsmall s, compared to Eq. (7.19). Equation (7.23) also includes the assumption that thediffusion constant of s singlet, dense particles is inversely proportional to the radius,that is, s�1=3, which might not be accurate for very small few-singlet aggregates.

Equations (7.19) and (7.23) also assume that the radius of a representative s singlet,dense particle can be estimated as Rps1=3. Primary particles actually have a distri-bution of radii, and they can also grow/coarsen before their capture by andincorporation into the structure of secondary particles. Regarding the size distribu-tion of the singlets, it can be argued that since their capture rate by the largeraggregates is approximately proportional to their radius times their diffusionconstant, this rate will not be that sensitive to that specific particle�s size, becausethe diffusion constant for each particle is inversely proportional to its radius. Thus,the product is well approximated by a single typical value.

The primary particle �aging� before capture has been approximately accounted forby using the experimentally determined typical primary particle linear size, 2Rexp,instead of attempting to calculate it dynamically as a function of time. The radius ofthe s singlet particle, in the first factor in Eq. (7.23), in the sum ofRps1=3 terms shouldbe then calculated with the replacement

Rps1=3 ! 1:2Rexps

1=3: ð7:24ÞHere,we added the factor ð0:58Þ�1=3 � 1:2,where 0.58 is thefilling factor of a randomclose packing of spheres [66], to approximately account for that as the growingsecondary particle compactifies by restructuring, and not all its volume will becrystalline. A fraction will consists of amorphous �bridging regions� between thenanocrystalline subunits.

The approximations described in connection with Eq. (7.20) and the use of theuniform singlet radii (see Eq. (7.24)) both can lead to nonconservation of the totalamount of matter. This can be largely compensated for [1] by renormalizing the finaldistributions so that the formed secondary particles contain the correct amount ofmatter. This effect seems not to play a significant role in the dynamics. Someadditional technical matters are addressed in Refs [1, 3, 5, 7, 8, 12, 13].

7.4Improved Models for Two-Stage Colloid Growth

Themodel of polycrystalline colloid synthesis has been applied for a semiquantitativedescription (without adjustable parameters) of the processes of formation of

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spherical colloids of metals Au [1, 3, 5, 7, 12, 13], [Gorshkov, V., Libert, S., andPrivman, V. (2003) unpublished], Ag [12, 13], generally metals [28, 29], a salt CdS [7,8], as well as used to qualitatively explain the synthesis of microspheres of an organiccolloid insulin [58]. There have been studies on improving the two-stage model forquantitative agreement with experimental results for CdS [7, 8], Au [Gorshkov, V.,Libert, S., and Privman, V. (2003) unpublished], and Ag [12, 13]. For CdS, sphericalparticle radius distribution was measured several times during the process and forvarying protocols of introducing the solutes into the system. When solute ions (oratoms/molecules) are not released as a �batch� or externally supplied, we have to addto the model the rate equations for their production in chemical reactions. Unfor-tunately, for many common colloid synthesis protocols, both experimental identi-fication and evenmore somodeling of the chemical kinetics of various solute speciesare not well researched.

Numerical simulations have suggested that the physical properties of the primarynucleation, the effective surface tension and the equilibrium concentration, if variedas adjustable parameters mostly affect the timescales of the onset of �freezing� of thesecondary process. In fact, using the measured bulk values for these parametersyields results consistent with the experimentally observed �freezing� times. Theparameters of the kinetics of the secondary process have been found to primarilycontrol the size of thefinal products. Sizes obtainedwith the �minimal�model [1, 3, 5,7, 8, 12, 13], [Gorshkov, V., Libert, S., and Privman, V. (2003) unpublished] were of thecorrect order of magnitude, but consistently smaller than the experimentallyobserved ones. Thus, the kinetics assumed for the aggregation gives too manysecondary particles that grow to sizes smaller than those observed.

Improvement of the model has been considered along the lines of revisiting theaggregation kinetics assumptions. The first approach [5, 12] argues that for thesmallest �secondary� aggregates, those consisting of one or few primary particles,the spherical particle diffusional expressions for the rates, which are ambiguous fortiny clusters, should be modified. In order to avoid introduction of many adjustableparameters, the rate G1;1! 2 (cf. Eq. (7.23)) was multiplied by a �bottleneck� factorf < 1. This accounts for that merging of two singlets (and, in fact, other very smallaggregates)may involve substantial restructuring. This reduces the rate of successfulformation of a bicrystalline entity: The two nanocrystals may instead unbind anddiffuse apart, ormerge into a single larger nanocrystal, effectively contributing to therate G1;1! 1, and this process is not in the original model. Data fits [5, 7, 12] yieldvalues of order 10�3 or smaller for f. Note that for the simplest model and theimproved one (but with more parameters), numerical simulations require substan-tial computational resources. Simulation speedup techniques valid for the kinetics oflarger clusters have been devised [6, 12, 13].

The second approach to improving the simplest model [7, 8], [Gorshkov, V., Libert,S., and Privman, V. (2003) unpublished] starts with the observation that thisminimalmodel already assumes a bottleneck for particle merger, because only singlet captureis allowed: the rates in Eq. (7.23) with both s1 > 1 and s2 > 1 are set to zero. Thiscorresponds to the experimental observation that such colloidal size particles werenot observed to pairwise �merge� in solution. This is motivated by the expectation

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that restructuring processes that cause compactification of the growing colloidsmediate the incorporation of primary particles, but not of larger aggregates, in theevolving structure, and the incorporated particles mostly retain their crystalline coreto yield the final polycrystalline colloids. The improved model allows sufficientlysmall clusters, up to certain sizes, smax > 1, to also be rapidly incorporated intogrowing aggregates. In addition to the monomer–cluster aggregation, the modelincludes cluster–cluster (i.e., s1;2 > 1) aggregation with rates given by Eq. (7.23), butonly as long as at least one of the sizes, s1;2, does not exceed smax (see Refs. [7, 8] fordetails). This sharp cutoff is obviously an approximation, but it offers the convenienceof a single new adjustable parameter. Indeed, data fits for CdS and Au sphericalparticles have yielded quantitative agreement with experiments, as shown in Fig-ure 7.6, with values of smax ranging [7, 8], [Gorshkov, V., Libert, S., and Privman, V.(2003) unpublished] from �15 for Au to �25 for CdS. These values are not onlyreasonable as defining �small� aggregates, but also remind us of a similar concept ofthe cutoff value n, discussed in Section 7.2, beyond which atomistic aggregatesdevelop a �bulk-like� core. Indeed, a numerical estimate of such a quantity insolution [35], for AgBr nanoaggregates, suggests that n is comparable to or somewhatlarger than�18. Finally, the added cluster–cluster aggregation at small sizes offers amechanism for the formation of the initial peak in the secondary particle distribution(for more details, see Section 7.5).

The three approaches referenced – singlet-only attachment, the same with theadded bottleneck factor, and small cluster attachment – are all modifications of the

0.2 0.4 0.6 1.2Radius (μm)

1.4

700 s

Siz

e di

strib

utio

n (a

.u.)

500 s

Figure 7.6 Particle size distributions: calculated (solid lines) and experimentally measured (barhistograms) for two different times, for synthesis of spherical polycrystalline CdS colloids [8], withthe model parameter smax ¼ 25.

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rates of the diffusional transport-driven irreversible capture (Eq. (7.23)). Allowingcluster–cluster aggregation and the need to fit additional parameter values haverequired large-scale numerical effort and consideration of efficient algorithmictechniques for simulations (not reviewed here), including conversion of the discretes equations to continuumones, with the adaptive grid (re)discretization both in time tand in cluster size s [7, 8].

7.5Particle Shape Selection in Solution Synthesis

When synthesized according to �cookbook� protocols to get uniform products,colloids and nanoparticles can in many cases assume various nonspherical shapesand have specific morphologies useful in applications. They can be single crystals orpolycrystalline entities, as well as amorphous or compound structures. Experimentalevidence is available [22, 23, 38, 47, 50, 51, 59, 60, 67–73] for uniform (size, shape)nonspherical particle growth under properly chosen conditions. The challenge ofexplaining uniformity of shape and, more generally, morphology has not beensuccessfully addressed. Similar to size selection, instead of seeking a single, all-encompassing approach that would anyway be numerically intractable, a morerealistic goal can be set for identifying specific growth conditions and situationsthat can promote uniform shape selection. One suchmechanismhas been identifiedrecently and is reviewed in this section [14].

One of the ingredients of this approach has been encountered in studies of�imperfect-oriented attachment� [74–77]: Persistence in successive nanocrystalattachments causes them to form uniform short-chain aggregates, as well as certainother shapes [18, 77] for a selected range of aggregate sizes. Generally, �persistence�manifests itself in that for many growth conditions, microscopic particles are simplynot sufficiently large – they do not contain enough constituent building blocks – todevelop shape destabilizing growing surface morphology features. The latter couldbe �dendritic instability� of growing side branches followed by branches on branches,fractal growth, and so on. Small enough synthetic particles simply do not haveenough �phase space� for their growth to develop the full range of surface morpho-logical fluctuations large enough to control their shape.

Thus, as long as they contain not too many singlets/atoms/monomers, part of themechanism for shape selection in both colloid and nanoparticle growth can beidentified as being self-templating (seeding their own further growth) for a range ofsizes [14–16]. However, the emergence of specific shapes requires additionalmechanism considerations. Recall that, as described previously, uniform sizeselection already requires the growth to be fast, which means that the shape/surfaceformation follows a locally nonequilibrium dynamics. For instance, nanocrystalscannot be expected to assume the equilibriumWulff construction shapes appropriatefor their lattice symmetries and surface energies.

Numerous processes are involved in particle growth. Diffusional transport ofmatter allows attachment of atoms (ions, molecules) and possibly larger clusters,

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specifically, whole nanocrystalline building blocks in growth of polycrystallinecolloids. Atoms and other entities can also detach or move/roll on the surface.Furthermore, attached clusters and nanoparticles can restructure, merge, andfurther grow by capturing solute species before they are overgrown by otherdepositing matter. Modeling all these processes at once would be a formidablenumerical challenge. Experimental evidence for the detailed kinetics of all theseprocesses is quite limited. One experimental observation [1] primarily for sphericalcolloids has suggested that the constituent nanocrystals eventually get settled in adense structure, but retain their crystalline cores. It is also known that diffusionaltransport with attachment without such restructuring would yield a fractal struc-ture [30, 65]. Quantitative experimental data on the time dependence of the fastprocesses involved in uniform nanoparticle and colloid growth have been especiallyscarce [8, 78], and therefore modeling results can at best be compared only with themeasured distributions and other data for the final particles.

Fast particle synthesis processes are initiated at large supersaturations. Shapeselection for each particle is therefore not controlled by surface equilibration, thoughthe actual shapes frequently display some of the crystallographic faces of thematerial�s crystal structure. The difficulties in modeling particle shapes [30, 79,80] have included the challenge to describe the initial formation and structure of thecrystalline or compact (for colloids) stable �core,� on top of which the growth ofthe structure then continues. As described in Section 7.4, the core is formed duringthe early growth, which is the least understood stage of the structure emergencein multicluster processes. At the later stages, the clusters in the growing peak(Figure 7.1) are sufficiently dilute to treat them as isolated, independently capturing�monomer� solute matter.

A recent kinetic Monte Carlo study [14] assumed that the seed was a compactparticle, approximately spherical, with an already formed well-defined internalcrystalline order without any large, size-spanning defects. This is a natural assump-tion for the early growth when all the clusters are still small, consisting of fewmonomers, and due to the fact that the monomers are initially significantly over-saturated, cluster–cluster aggregation is dominant in the dynamics, while the shapeat least for the smallest clusters initially will be globular, restructuring/compactify-ing, and ultimately developing the core with crystalline/bulk characteristics. Asemphasized, this latter process is currently not well understood.

As the cores develop, the distinct peak in the size distribution (Figure 7.1) formsand from a certain time on, we can consider growth of each particle in the peak aspractically independent, �interacting� with the growth of other such larger particlesonly via their consumption of the remaining monomer solute matter. Each corecaptures diffusing �atoms� preferentially attached in positions locally defined by thelattice symmetry of the structure. In slower protein crystallization [81, 82], the growthstage from �102 to �108 molecules per crystal, after the initial small clusterformation but before the onset of the really macroscopic growth modes, is alsoconsistent with a single ordered core approach. While motivated by (nano)crystalgrowth, this approach can also shed some light on the formation of those colloidswhose faces follow the underlying material symmetry. It is possible that the main

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singlet nanocrystal in the seed for polycrystalline colloid growth dominates theemergence of the surface faces. However, another scenario, suggested by a recentpreliminary dark field and bright field TEM experimental evidence [78] for cubic-shaped polycrystalline neighborite (NaMgF3) colloids, is that the leading crystalstructure is formed by the process of outer constituent singlets in the shell of theparticle recrystallizing to form an effectively continuous single crystal on top of apolycrystalline core. In the latter situation, the core does not therefore define thecrystal pattern based on which the final larger particle shape will emerge.

We focus on particle growth in situations when the earlier formed core structuredominates shape selection, and we attempt to formulate �atomistic� rule that imitatesthe effect of persistency for a range of particle sizes. Evenwith all these simplifications,substantial numerical resources [14] are required. However, we were able to includethe processes of atoms moving on the surface and their detachment/reattachment,governed by thermal-type, (free) energy barrier rules. The diffusional transport occursin the three-dimensional space without any lattice. However, a �registered� latticeattachment rule was enforced starting from the seed, which prevents the growingmoderate size clusters fromdevelopingmacroscopic (size-spanning) defects and thusensures the maintenance of the crystal symmetry imposed by the core. We can thenfocus on the emergence of the surface and shape morphological features.

The results [14] have suggested that this type of growth can yield interesting shapeselection provided a certain size core (seed) of approximately 30 crystal cells indiameter indeed first emerges by the initial fast aggregation, followed/accompaniedby its forming a well-defined crystalline structure. Modeling results have allowed toidentify three possible modes of further particle growth. For slow growth, thediffusional transport-driven supply of building block �atoms� is low, and thetimescale, td, of surface motion, including hopping to neighbor sites and detach-ment/reattachment, all contributing to effective on-surface diffusion of atoms, ismuch smaller than the timescale of the addition of new monolayers, tlayer. Theshape of the growing cluster is then close to, but not identical with, the Wulffconstruction configuration [83–86]. The opposite regime is that of fast growth,tlayer � td, and corresponds to the development of surface instabilities. Thedynamics of the particle shape is then driven by the local random fluctuations ofthe surface, which are amplified due to diffusional flux nonuniformities, and thecluster assumes a shape of a random clump.

The intermediate regime corresponds to td � tlayer. It has been found [14] that inthis nonequilibrium growth, particles can maintain an even-shaped form with well-defined faces corresponding to the underlying crystal structure imposed by the seedand atom attachment rules. This numerically identified shape selectionwas observedonly for a certain range of particle sizes. Beyond such sizes, growth modes involvingbulges, dendritic structures, and other irregularities can be supported and arerealized. This pattern of shape selection in the nonequilibrium steady-state growthregime have been explored [14] for the simple cubic (SC), body-centered cubic (BCC),face-centered cubic (FCC), and hexagonal close-packed (HCP) crystal lattices.Possible shapes have been identified for each symmetry, with their selectiondetermined by growth parameters.

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Let us illustrate the results [14] by considering the three-dimensional SC latticesymmetry. As mentioned, preliminary numerical experiments have suggested thatthe seed should be defined by lattice cells within a sphere with radius of 15 latticeconstants. Furthermore, the seed atoms can be assumed fully immobile – anassumption also made based on preliminary simulations – because the seed atomsrarely moved enough to change its initial structure. Only the atoms later adsorbed atthe growing structure underwent the dynamicalmotion. The dynamics of atoms [14],supplied at a boundary of a large simulation box, has involved off-lattice diffusion inspace, capture at the vacant sites neighboring the cluster growing from the seed, aswell as detachment and on-surface hopping. Thermal-like free energy barrier ruleswere used for variousmove rates, and,most important, the on-surface atompositionswere exactly �registered� at the sites of the SC lattice originally defined by the seed.The latter �registration� rule is not realistic: It is used as an artificial tool to avoidformation of defects spanning large parts of the growing structure to begin dom-inating relative growth rates of some of the crystal faces. In reality, we expect that it isthe already addressed �persistency� for clusters, which are not too large, thatmakes itimprobable for most of them to develop such �macroscopic� defects.

In addition to the kinetic parameters [14], the model can also include differentmotion rules. For example, for the SC symmetry, each atomattached to the cluster canhave up to six bonds pointing to nearest neighbors – the set ~eintf g of six latticedisplacements of the type ð100Þ. The set of displacements/detachments for surfaceatoms, ~emovf g, was defined in two different ways: (1) variant A, ~emovf gA included boththe set ~eintf g and also the 12 next-nearest-neighbor displacements of the type ð110Þ oflength

ffiffiffi2

p; (2) variant B, ~emovf gB ¼ ~eintf g. Thus, for variant B, the dynamics of the

surface atoms is more limited (effectively slower).With parameter choices for the slow growth �steady-state� regime (tlayer � td),

Figure 7.7 illustrates the resulting steady-state particle shape for the variant A of theSC simulation. We also show a schematic that illustrates the cluster shape formedwith the type ð100Þ, ð110Þ, ð111Þ lattice planes, which happen to be those dense-packed, low-index faces that dominate the low-temperature Wulff constructionfor the SC lattice [83–85]. The superficial similarity of the �steady-state� and trueequilibrium shapes is misleading. Indeed, our system�s dynamical rules do notcorrespond to true thermal equilibration. The resulting shape is thus dependent onthe dynamics. As an example, Figure 7.7 also shows the shape obtained for the samesystem but with variant B for the displacements/detachments. We conclude that theparticle shape is not universal even in the steady state, in the sense expected [87] ofmany processes that yield macroscopic behavior in statistical mechanics: the micro-scopic details of the dynamical rules do matter. In practical terms, this makes itunlikely that nonequilibrium particle shapes can be predicted based on argumentssuch as minimization of some free energy-like quantity.

We emphasize that while well-defined particle shapes can be obtained in thesteady-state regime within the present model, this does not offer a predictable andwell-defined particle shape selection mechanism in practical synthesis situations.This occurs because themodel itself is not fully valid. Indeed, we observed [14] strongsensitivity of the results to the density of the monomer matter and its redistribution

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Figure 7.7 The top shape exemplifies resultsof the steady-state SC lattice simulations forvariant A of the displacements/detachments forsurface atoms. (The white lines were added forguiding the eye.) The schematic shows the SCshape formed by lattice planes of the typesð100Þ, ð110Þ, and ð111Þ for the equilibriumWulff

construction assuming that all these faces haveequal interfacial free energies. The bottomshape exemplifies steady-state growth forvariant B of the surface dynamics. The detailsandparameter values for each simulation canbefound in Ref. [14].

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by transport to and from the surrounding medium. Therefore, the isolated clusterassumption breaks down.Other clusters (particles) compete for the �atoms� (solutes)in the dilute solution, and, as a result, additional growth mechanisms that involveexchange of matter between clusters (Ostwald ripening) cannot really be ignored (asdone in our approximate model).

The main difference between nonequilibrium and steady-state regimes is that theformer corresponds to a fast growth process fully dominated by capture of singletmatter from a dilute solution. Other processes, such as those involving exchange ofmatter with other clusters, are slower. For nonequilibrium growth, the cluster shapescan be quite different. For example, for the SC lattice, a cubic shape, shown inFigure 7.8, was found only in the nonequilibrium regime [14] with the kinetictransition rates [14] for atom intake versus surface dynamics corresponding totd � tlayer. Other less symmetrical shapes have also been found, one of which isshown in Figure 7.9. Regular shapes obtained for nonequilibrium growthwith latticesymmetries other than SC are exemplified in Figure 7.8. Several shapes obtained fordifferent symmetries are cataloged in Ref. [14], as are examples of the onset ofunstable growth (see Figure 7.9 and other interesting growth modes, further

Figure 7.8 Examples of regular shapes obtained in the nonequilibrium growth regime for variouslattice symmetries.Other shapes and faceproportions are possible [14] in this regime, depending onthe growth dynamics parameter values. (The white lines were added for guiding the eye.)

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exploration of which has been limited by the demands on numerical resourcesrequired for simulating large particles.

In summary, the key ingredients for one possible shape selection mechanismidentified in this section, in the experimentally relevant nonequilibrium growth

Figure 7.9 Steady-state growth for parametervalues and times resulting in several shapesdiscussed in Section 7.5. The equilibrium Wulffshape for FCC is schematically drawn, assumingthat the (111)- and (100)-type faces have thesame interfacial energies. It illustrates that thesteady-state FCC shape shown in Figure 7.8 forcertain growth conditions can well approximatethe Wulff construction shape. However, for asomewhat faster but still steady-state growth,effective corner truncations disappear and onlythe (111)-type faces remain, as shown here. Theshown FCC shape also exhibits the first signs ofthe tendency to destabilize for fast enough/largetime growth: the onset of bulging at the growingcorners. The three FCC shapes here and inFigure 7.8 are similarly oriented and theCartesian axes shown define the lattice-face

orientations (100) and others in a self-explanatory notation based on the cubic latticecell for FCC. The steady-state SC cluster shownwas grown with parameters yielding fastergrowth, such that the onset of destabilization ismore profound, and therefore bulges haveemerged, compared to the nearly cubic shapeshown for slower steady-state growth inFigure 7.8, though each bulge is still smallenough to show tendency for retaining thecubic-shaped faces. Finally, the HCP clustershown corresponds to an even faster/relativelylonger time growth and shows bulging/onset ofbranching, but retains no tendency to form theHCP symmetry faces on the overall or localsubstructure scales. The resulting shape hasonly a vague resemblance to slower growthHCPshapes with hexagonal bases [14].

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regime, include �persistence,� allowing the clusters to avoid the formation ofmacroscopic defect structures. The growth of the particle�s faces is then notcontrolled by such defects – which is a well-known mechanism [82, 86] that candetermine growth modes in crystallization. The densest packed, low-index crystalsymmetry faces, which dominate the equilibrium crystal shapes, also play role in thenonequilibrium growth, but their proportions can vary and are not the same as inequilibrium.

7.6Applications for Control of Morphology in Surface Structure Formation

The model discussed in Section 7.5 has recently been applied [17] to growth ofnanoclusters and nanopillars in surface deposition of building blocks (atoms)diffusionally transported from solution to the forming surface structure. Emergenceof nanosize morphology in surface growth deposits formed by atoms, ions, andmolecules is an active field [17, 88–93] of research and applications. Processes ofsurface restructuring are also accounted for in the model, which then yieldsmorphologies of interest in catalysis applications. The kineticMonteCarlo numericalapproach has been used to explore the emergence of FCC symmetry surface featuresin Pt-type metal nanostructures. Reported results [17] exemplify evaluation of thefraction of the resulting active sites with desirable properties for catalysis, such as anapproximate (111) face coordination, and suggest optimal growth regimes. Indeed, ithas been experimentally found [94–96] that nanoclusters and nanopillars can beformed in surface growth for Pt and similar deposits and be grownwith a substantialfraction of (111)-like symmetry faces.

The primary goal of this study is to understand how surface structures can begrown with well-defined, preferably uniform morphology of nanoclusters or largernanopillars resulting from the kinetics of the constituent building blocks: atoms,ions, or molecules. The approach [17] is based on an earlier developedmodel [14] forthe unsupported (off-surface) growth of nanoparticles of well-defined shapes dis-cussed previously. Shape selection results from the competition of several dynamicalprocesses: transport of matter, on-surface restructuring, and atom detachment/reattachment. We do not consider the physical or chemical properties of surfacestructures relevant for their use once synthesized, for example, their catalytic activity.Rather, we focus on their synthesis with desirable morphologies. As a prototypesystem of interest, we have selected the crystal structure of metal Pt and thepreference for (111)-type crystalline faces.

It is important to realize that we are not interested in large surface layer growthwith significant fluctuations developing, as traditionally studied in statisticalmechanics, but only an overgrowth of the initial substrate with a finite quantity ofdeposited matter of nanosize average thickness. The growth mechanisms haverelative rates described in Section 7.5, and it is expected that shape selection foron-surface nanostructures occurs in the nonequilibrium growth regime identifiedearlier. To illustrate the results [17], numerically grownnanoclustermorphologies are

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shown in Figure 7.10. These depict a succession of surface structures developing,with the initial on-surface islands forming and coarsening, and then serving as basesfor the emergence of pyramidal nanoclusters, which grow into nanopillars, and thelatter eventually destabilize.

Similar to our discussion in Section 7.5, of the origin of the seed in isolated particlegrowth, here also the selection of a proper substrate for on-surface growth isimportant for obtaining catalytically active and more or less uniform surfacestructures. The crystallographic plane and patterning of the underlying surface, thelatter when seeding/templating is used, affect themorphology of the formed deposit.We assumeFCC symmetry of the lattice and also of the substrate, the latter aflat (100)lattice plane. Indeed, as shown in Figure 7.9, for FCC the Wulff form involves the(100)- and (111)-type faces. For nonequilibriumgrowth, however, regimes exist when(111) will dominate (Figure 7.9). Generally, such considerations suggest that (100)and (111) are naturally complementary lattice faces in nonequilibrium FCC sym-metry growth, and therefore (100) consistent substrates are a good choice for growing(111). Octahedral shapes made of (111)- and (100)-type faces have indeed beenobserved in experiments for on-surface Pt nanoclusters [97].

Figure 7.10 Nonequilibrium growth of FCCsymmetry deposit. Shown are 200 200sections of simulations for initially flat500 500 (100) substrates. Only the �active�(those that can move/detach at the timeinstance shown) growing surface atoms areshown. The parameter values and various

definitions can be found in Ref. [17] and thesimulation times are increasing as given (interms of the Monte Carlo system sweeps [17]),illustrating the initial isolated islands, then theemergence of pyramidal nanoclusters, andfinally the formation of competing nanopillars,with the eventual onset of irregular growth.

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Another technical difference of the surface versus isolated particle simulation is thebox size in relation to the flux of matter. An isolated cluster distorts only the diffuserdensity profile at distances from its surface comparable to its own linear dimensions.Thus, a boxexceeding amultiple of the largest size cluster dimensions during a growthsimulationwith fixed concentration at its faces suffices to represent an isolated clustergrowing in the background of diffusing atoms of that concentration. On the otherhand, an on average planar, (partially) absorbing surface, say, at the bottom of thesimulation box, depletes the concentration of the diffusers (atoms), causing the latterto rapidly assumea linear profile up to the topof thebox atwhich the concentration canbe controlled [17]. Despite the possibility of detachment, in the present study thesurface structures on average constantly grow (Figure 7.10). Control of the flux ofmatter to the surface as one of the importantmodel parameters thenmust be carefullyconsidered and checked by numerical simulations, as alluded to in Ref. [17] andreported in greater detail in Ref. [98]. Indeed, the gradient of the distribution of theconcentration of solutes in the box is not only nonuniform for a short transient time,but is also at least somewhat time dependent on larger timescales.

Let us assume timescales tðislandsÞ < tðnanoclustersÞ < tðnanopillarsÞ < tðunstableÞ for thegrowth stages illustrated in Figure 7.10, that is, the coarsening of the on-surfaceislands, then the three-dimensional nanocluster structures, their protrusion awayfrom the substrate as developing nanopillars, and finally their destabilization. Theactual growth time, t, for applications will be selected to correspond to useful surfacestructures of well-defined properties, which means that usually t < tðunstableÞ. The�transient� time for the establishment of a linear concentration distribution,tðtransientÞ, should be tðtransientÞ < tðislandsÞ, whereas once the linear distribution of thediffuser concentration from the �roof� of the deposit to the top of the box isestablished, its slope (determining the flux) should remain approximately constantor otherwise controlled for timescales tðcontrolled fluxÞ exceeding the desirable growthtime: t < tðcontrolled fluxÞ. A good selection of the simulation parameters, including thebox size, corresponds to all the already shown �less than� requirements betweentimes, being realized at least as �a small fraction of� relations between the pairs oftimescales involved.

With proper parameter selection, a sizable fraction of the growing nanoclusterfaces are the desirable (111) lattice planes (see Figure 7.10), while the nanoclustershapes (and in fact also the tops of thenanopilars) resemble halves of the isolated FCCshapes such as the those shown in Figure 7.9, or those with approximately (111)slopes, but truncated by small approximately (100)-type �bold spots� (cf. Figure 7.8and theWulff shape in Figure 7.9). Numericalmodeling can suggest optimal regimesfor catalysis applications [17].

7.7Summary

In summary, we reviewed models of particle/cluster size, shape, and certainmorphology feature selection in colloid and nanoparticle synthesis, as well as in

7.7 Summary j263

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on-surface growth. Numerical simulations are required to obtain quantitative resultsand gain qualitative insights into themodel predictions. The field is still wide open tofuture research. Indeed, theoretical understanding is currently at the stage when wehave tentatively identified certain specific mechanisms and conditions that seem toallow growth of structures of relatively uniform size and shape. However, theuniqueness of thesemechanisms, the degree of their �universality,� and a possibilityof alternative explanations and other mechanisms for different growth modes havenot been ruled out. Most of the experimental data are limited to observation of thefinal products. Experimental results for time-dependent kinetic processes, as well asdetailed morphological data, would benefit future modeling research.

Acknowledgments

The author thanks his colleagues P. B. Atanassov, D. V. Goia, V. N. Gorshkov, I.Halaciuga, R. Irizarry, S. Libert, E. Matijevi�c, D. Mozyrsky, J. Park, D. Robb, I.Sevonkaev, Y. Shnidman, and O. Zavalov for rewarding scientific interactions andcollaboration and acknowledges funding by the U.S. ARO under grant W911NF-05-1-0339.

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268j 7 Colloids, Nanocrystals, and Surface Nanostructures of Uniform Size and Shape

Vladimir PrivmanClarkson UniversityDepartment of PhysicsCenter for Advanced MaterialsProcessingPotsdam, NY 13699USA