Magic Cluster Sizes in Nucleation of Crystals - American Chemical

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Magic Cluster Sizes in Nucleation of Crystals Dimo Kashchiev* Institute of Physical Chemistry, Bulgarian Academy of Sciences, ul. Acad. G. Bonchev 11, Sofia 1113, Bulgaria ABSTRACT: In an ensemble of nanoscale crystalline clusters, those of magic sizes are evidenced by their greater number. The respective cluster size distribution is multipeaked and cannot be described by the classical nucleation theory. Employing the atomistic nucleation theory, we present expressions for the cluster size distribution in stationary nucleation of two- or three-dimensional crystals. We show that this distribution is multipeaked at magic cluster sizes and compare these sizes with those obtained from computer simulations of cluster evolution on square Kossel−Stranski or Ising lattice and from experiments on formation of Si islands on Si(111).

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INTRODUCTION Clusters of a small number of atoms or molecules or, more generally, building blocks, are of great interest to various branches of nanoscience and nanotechnology because their properties usually differ considerably from those of the corresponding bulk material.1−5 In particular, the cluster stability with respect to growth and decay does not change monotonically with the cluster size but is highest at certain socalled magic sizes.1−20 These sizes are readily revealed by the positions of a series of peaks in the cluster size distribution.1−7,9,11,12,14,17,18,20,21 Many authors connect this fact to local minima in the cluster binding energy, an idea that is also followed in the present study. When computer simulations for searching global minimum of cluster potential energy with realistic interatomic interactions are applied, magic clusters are discovered which usually have regular shapes but internal structure often different from that of bulk crystals.1−3 One of the well-known examples are the Mackay icosahedra with magic sizes of 13, 55, 147, etc. atoms and noncrystalline pentagonal symmetry, which are typically formed during nucleation from the gas phase. The noncrystalline clusters may attain crystalline structure at a size which is usually significantly larger than that of the clusters involved in the nucleation process. An additional feature of the clusters is the possible instability of their shape. For instance, being exposed to intense electron-beam irradiation, a gold cluster on SiO2-covered silicon substrate was observed to change its shape approximately every few tenths of a second.22 It should be noted as well that kinetic effects may also be responsible for the appearance of magic-size clusters: the formation of large icosahedral clusters of C60 molecules is an example.3,14 The kinetic effects become important when the clusters do not have sufficient time to reach the corresponding energy minima and thus remain trapped in metastable configurations. Clearly, already the above-mentioned peculiarities concerning the © 2012 American Chemical Society

cluster structure, shape, and growth kinetics make the theoretical prediction of the magic cluster sizes very difficult. As nucleation usually involves clusters of less than a few scores of atoms or molecules, it is clear that the smallest magic cluster sizes may have a significant effect also on this process. Hitherto, however, this effect has rarely been taken into account in nucleation theory.17,18,23,24 In fact, the role of the magic cluster sizes at the initial stage of cluster formation can be understood solely in the scope of the atomistic nucleation theory (ANT),25−33 because unlike the classical nucleation theory (CNT),32,34 ANT allows for the atomic scale irregularities of the cluster surface. Recently,33,35 it has been shown that ANT describes well the nucleation rate of KosselStranski crystals. It can therefore be expected that ANT will also be successful in treating the problem of magic cluster sizes. The objective of the present study is (i) to employ ANT for derivation of a formula for the cluster size distribution in stationary nucleation of single-component two-dimensional (2D) or three-dimensional (3D) crystals, and (ii) to demonstrate that this formula predicts the existence of magic cluster sizes. Our considerations are limited to stable crystalline or noncrystalline clusters which evolve along a specified sequence of shapes corresponding to cluster energy minima.

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GENERAL We recall here three general formulas of nucleation theory that are needed for the analysis to follow. According to this theory (e.g., ref 32), the stationary size distribution of the smallest clusters during nucleation of a single-component 2D or 3D phase can be presented generally and exactly as (n = 1, 2, 3, ...) Received: March 25, 2012 Revised: April 30, 2012 Published: May 15, 2012 3257

dx.doi.org/10.1021/cg300394c | Cryst. Growth Des. 2012, 12, 3257−3262

Crystal Growth & Design

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Here ε, given by

M−1

X n = Cn

∑m = n (fm Cm)−1 M−1

∑m = 1 (fm Cm)−1

ε=

(1)

In this formula, n (or m) is the number of atoms in a cluster, Xn is the stationary concentration of n-sized clusters, Cn is the respective equilibrium concentration of such clusters, f n (s−1) is the frequency of atom attachment to an n-sized cluster, and M is the number of atoms in a cluster that is sufficiently bigger than the nucleus (in single-component nucleation the nucleus, called also the critical nucleus, is that particular cluster which requires maximum work for its formation). Equation 1 is valid in the scope of the Szilard−Farkas model of nucleation, according to which the clusters change size solely by attaching and detaching single atoms and have the same shape for a given size.32 Here and in what follows, the term atom is used to denote also a molecule or, more generally, a building block of the clusters. In general, the equilibrium cluster size distribution Cn can be expressed as32,36,37 (n = 1, 2, 3, ...) Cn = C1 exp(w1 − wn)

κd 0 kT

(5)

is the dimensionless cluster specific edge energy, and c2 is a numerical shape factor (e.g., c2 = 4 for square-shaped clusters). As to the frequency f n, it depends on the particular mechanism of atom attachment. We shall consider here only the case of direct attachment of atoms to the cluster periphery. Then f n is proportional to ln (ref 32) and with the aid of the above formula for ln, it can be written as (n = 1, 2, 3, ...)

fn = f1 n1/2

(6)

where f1 is the atom-to-atom attachment frequency. Thus, using eqs 2−4 and 6, from eq 1 we obtain (n = 1, 2, 3, ...) X n = C1e

s(n − 1) − c 2ε(n1/2 − 1)

M−1

1/2

M−1

1/2

∑m = n m−1/2e−sm + c2εm ∑m = 1 m−1/2e−sm + c2εm

(7)

This formula represents the CNT cluster size distribution Xn when stationary nucleation of 2D crystals with the shape of regular polygon occurs on its own substrate, that is, on a substrate of the same material. It is readily applicable to 2D nucleation of such crystals on a foreign substrate upon replacing s by the effective dimensionless supersaturation sef given by32

(2)

Here C1 is the actual concentration of single atoms in the metastable 2D or 3D phase, wn ≡ Wn/kT, Wn is the work to form an n-sized cluster, k is the Boltzmann constant, and T is the absolute temperature. Equation 2 is obtained by regarding the single atoms as the smallest clusters so that in it w1 is the value of wn at n = 1. Importantly, this equation is compatible with the law of mass action and is self-consistent in the sense that at n = 1 it returns the equality C1 = C1. Most generally, the dimensionless work wn for cluster formation is given by32

sef = s −

aef Δσ kT

(8)

Here aef is the effective area occupied by an atom on the substrate, Δσ ≡ σ + σcs − σs, and σ, σcs and σs are the specific surface energies of the crystal/old-phase, crystal/substrate, and substrate/old-phase interfaces, respectively. For 2D nucleation on own substrate one has σcs = 0 and σs = σ so that then Δσ = 0 and, hence, sef = s. Physically, the energy parameter Δσ accounts for the “wetting” of the substrate by the 2D crystal.32 The curve in Figure 1 displays Xn/C1 calculated numerically from eq 7 at c2 = 4 (square clusters), M = 54 (larger M values do not affect the calculation when n ≤ 50), ε = 5.7 and s = 10.

Φn (3) kT where s  Δμ/kT is the dimensionless supersaturation (Δμ is the dimensional one), and Φn is the cluster excess energy. To a good approximation, for condensed-phase clusters Φn is the total peripheral or surface energy of the n-sized 2D or 3D cluster, respectively. Both CNT and ANT are based on eq 3 (e.g., ref 32). The principal difference between these theories is in the way they express the dependence of Φn on n. It will be seen below that this difference results in different cluster size distributions in nucleation of either 2D or 3D crystals. The considerations to follow are confined to crystalline or noncrystalline nanoclusters which preserve their atomic structure during growth and follow a single sequence of minimum-energy shapes corresponding to regular polygon or polyhedron whose edges or faces have the same specific edge or surface energy. wn = −sn +

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2D CRYSTALS CNT Cluster Size Distribution. In nucleation of 2D crystals, according to CNT,32 Φn is the product of the cluster specific edge energy κ and the length d0ln of the cluster periphery. Thus, Φn = κd0ln, where d0 is the distance occupied by an atom on the cluster periphery, and ln is the number of such distances, that is, the number of atomic sites on the periphery of the n-sized cluster. CNT treats the 2D clusters of all sizes as having the same shape and as then ln = c2n1/2, Φn takes the form32 (n = 1, 2, 3, ...) Φn = c 2εkTn1/2

Figure 1. Size distribution of 2D clusters with square lattice: curve and solid-bar histogram − Xn/C1 from eq 7 of CNT and eq 11 of ANT, respectively, at ε = 5.7 and s = 10; open-bar histogram − simulation data17 (the reported cluster numbers are divided by 1000). The polygon illustrates a cluster of 53 atoms (the squares schematize atoms, and the numbers indicate the sequence of atom incorporation into the cluster). The sequential cluster shapes are closest to squares because of the requirement for minimum cluster peripheral length.

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We observe that Xn is a monotonically decreasing function of n. This monotonic decrease remains unchanged for any other cluster shape and ε or s value, which means that CNT does not predict the existence of magic nanocluster sizes. ANT Cluster Size Distribution. Similar to CNT, ANT also uses the relation (n = 1, 2, 3, ...)

Φn = κd0ln = εkTln

these sizes remain the same at different supersaturations (result not shown) and, understandably, they correspond to clusters with atomically completely built-up edges. Indeed, as these clusters are trapped in local energy minima (see line 2D-KS in Figure 2), they resist growth or decay and thus become more

(9)

for determination of the cluster total peripheral energy. Unlike CNT, however, ANT accounts for the atomically irregular shapes of the differently sized 2D clusters. This is done by counting the number ln of atomic sites on the periphery of an nsized cluster with shape corresponding to the minimum of the cluster peripheral length. Thus, ln is not an analytical function of n, and ε has the physical meaning of effective dimensionless broken-bond energy per atomic site on the cluster periphery.33 The polygon in Figure 1 illustrates a 53-atom crystalline cluster with square lattice and minimal peripheral length, and the numbers indicate a succession of atom incorporation. Direct counting of the number of atomic sites on the cluster periphery corresponding to the successive cluster shapes yields33 ln = 4, 6, 8, 8, 10, 10, 12, 12, 12, 14, 14, 14, 16, 16, 16, 16, 18, 18, 18, 18, 20, 20, 20, 20, 20, 22, 22, 22, 22, 22, 24, 24, 24, 24, 24, 24, 26, 26, 26, 26, 26, 26, 28, 28, 28, 28, 28, 28, 28, 30, 30, 30, and 30 at, respectively, n = 1, 2, 3, ..., 52 and 53. In ANT, f n is also considered as proportional to ln when atoms are directly attached to the cluster periphery, so that33 (n = 1, 2, 3, ...) l fn = f1 n l1

Figure 2. Size dependence of the work for cluster formation: line 2DKS − eqs 3 and 9 with ε = 5.7 and s = 10 for 2D Kossel-Stranski clusters; line 2D-H − eqs 3 and 9 with ε = 6 and s = 5.8 for 2D honeycomb-structured clusters; line 3D-KS − eqs 3 and 16 with ω = 5 and s = 13 for 3D Kossel-Stranski clusters. The global maximum of each line is at the nucleus size n*: n*=3, 7, and 5 for lines 2D-KS, 2DH, and 3D-KS, respectively.

numerous than the clusters of neighboring sizes. We note as well that the CNT cluster size distribution (the curve in Figure 1) is in both qualitative and quantitative disagreement with the ANT one. We can now juxtapose Xn of ANT with the cluster size distribution obtained in computer simulation of Ostwald ripening of square-lattice clusters with ε = 5.7 (Figure 4a in ref 17). This distribution is displayed by the open-bar histogram in Figure 1 upon dividing the obtained cluster numbers by 1000. The distribution corresponds to a postnucleation stage of the clustering process and cannot be directly compared with Xn. Qualitatively, however, we observe that the magic cluster sizes during Ostwald ripening are around or at those existing during nucleation. Hence, the magic sizes remain the same throughout the overall process of phase transformation provided the clusters preserve their structure during the process. The size distribution of 2D Si islands in homoepitaxial growth of Si at T = 725 K (ref 11) offers a possibility for a qualitative comparison of Xn from eq 11 with experimental data. The islands are on a reconstructed 7 × 7 Si(111) surface and are triangular because of the triangular shape of the half unit cell (HUC) of this surface. The building block of the islands is a HUC fully occupied by 49 deposited Si atoms.11,12 Thus, islands appear and grow by attachment/detachment of HUCs (see the polygon in Figure 3 in which the numbers indicate a succession of HUC incorporation into an island of 58 HUCs when all islands in the succession have a minimal peripheral length). The open-bar histogram in Figure 3 represents the Si island size distribution obtained experimentally,11 the reported island numbers being divided by 200. The curve and the solid-bar histogram display Xn/C1 from eq 7 of CNT and from eq 11 of ANT, respectively (n is the number of HUCs constituting the island). The calculation is done with c2 = 3 (triangular islands),

(10)

As f n is in fact proportional to the number l′n of attachment sites at the cluster periphery, the proportionality of f n to ln is an approximation which, however, has practically no effect on the cluster size distribution, because l′n is either equal to ln (for clusters with kinkless periphery) or only slightly smaller than ln (for clusters with kinked periphery). For example, the 53-atom cluster in the inset of Figure 1 has l′n = 29 (because of the presence of one kink site) and ln = 30. Combining now eqs 2, 3, 9, and 10, from eq 1 we readily obtain (n = 1, 2, 3, ...) M−1

X n = C1e s(n − 1) − ε(ln− l1)

∑m = n lm−1e−sm + εlm M−1

∑m = 1 lm−1e−sm + εlm

(11)

This is the sought ANT formula for the cluster size distribution in stationary nucleation of 2D crystals with shape corresponding to that of regular polygon. Similar to eq 7, eq 11 is applicable to nucleation on a foreign substrate provided in it s is replaced by sef from eq 8. We emphasize that eq 11 is highly reliable, because it is based on the exact eq 1. The solid-bar histogram in Figure 1 depicts Xn/C1 from eq 11 at l1 = 4 (square lattice), ε = 5.7, and s = 10. The sums are calculated numerically with the aid of the above ln values and the large enough M = 54. As seen in Figure 1, Xn possesses wellexpressed maxima at cluster sizes n = 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, and 49, that is, at n = i2 and n = i(i + 1) where i = 2, 3, 4, .... These namely are the magic sizes of the 2D nanocrystals with square lattice and they coincide with those found by computer simulation of nanoscale 2D Kossel-Stranski crystals17 and 2D Ising ferromagnets.18 ANT is thus able to predict the existence of magic sizes of nanoscale 2D crystals. Importantly, 3259



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σa0an, where a0 is the area occupied by an atom at the cluster surface, and an is the number of such areas, that is, the number of atomic sites on the surface of the n-sized cluster. Since CNT treats the 3D clusters of all sizes as identically shaped, an is given by an = c3n2/3 so that Φn takes the form32 (n = 1, 2, 3, ...)

Φn = c3ωkTn2/3

(12)

Here the dimensionless cluster specific surface energy ω is specified by σa ω= 0 (13) kT and c3 is a numerical shape factor (e.g., c3 = 6 for cubic crystals). For direct attachment of atoms to the cluster surface, the frequency f n is proportional to the cluster surface area,32 that is, to n2/3. Hence, f n can be represented as (n = 1, 2, 3, ...)

Figure 3. Size distribution of 2D clusters with honeycomb lattice: curve and solid-bar histogram − Xn/C1 from eq 7 of CNT and eq 11 of ANT, respectively, at ε = 6 and s = 5.8; open-bar histogram − experimental data11 for Si/Si(111) (the reported island numbers are divided by 200). The polygon illustrates a Si island of 58 HUCs (the triangles schematize HUCs, and the numbers indicate the sequence of HUC incorporation into the island). The sequential cluster shapes are closest to triangles because of the requirement for minimum cluster peripheral length.

fn = f1 n2/3

(14)

With the help of eqs 2, 3, 12, and 14, from eq 1 we thus find that during stationary homogeneous nucleation of crystals with the shape of regular polyhedron, the CNT size distribution is given by (n = 1, 2, 3, ...)

l1 = 3 (honeycomb lattice), ε = 6, s = 5.8 and the sufficiently large M = 59. Other s values not too smaller than the ε value do not change Xn qualitatively, and following Voigtländer et al.,11 the value of ε is chosen to correspond to strong enough binding energy of nearest-neighbor HUCs. The ln values employed in eq 11 are obtained by directly counting the number of HUC sites at the island periphery and are as follows: ln = 3, 4, 5, 6, 7, 8, 9, 8, 9, 10, 11, 12, 11, 12, 11, 12, 13, 14, 15, 14, 15, 14, 15, 14, 15, 16, 17, 18, 17, 18, 17, 18, 17, 18, 17, 18, 19, 20, 21, 20, 21, 20, 21, 20, 21, 20, 21, 20, 21, 22, 23, 24, 23, 24, 23, 24, 23, and 24 at, respectively, n = 1, 2, 3, ..., 57 and 58. These ln values correspond to the succession of island shapes illustrated in Figure 3. As seen in Figure 3, while CNT does not predict magic island sizes, ANT is able to do that rather accurately. Indeed, theoretically, the most numerous Si islands are of 4, 8−10, 15− 17, 24−26, 35−37, and 48−50 HUCs, which compares with the experimental finding of most numerous islands of 8−10, 14−19, 24−28, 34−35, and 50 HUCs. The ANT island size distribution cannot be compared quantitatively with the experimental one, because the latter corresponds to a deposition process in which the Si island nucleation is not stationary. Nonetheless, the ANT prediction of magic island sizes n = 8, 15, 24, 35, and 48 (i.e., n = i2 − 1 at i ≥ 3) is quite successful and is also in line with Voigtländer et al.’s11 computer simulation result that the magic sizes are n = 4, 9, 16, 25, 36, 49, and 64 (i.e., n = i2 at i ≥ 2). Possibly, the ANT prediction could be improved if it is taken into account that prior to attachment while a given HUC may not have a stacking fault relative to the substrate, its neighboring HUC is faulted.11,12 We note as well that the ANT-predicted magic island sizes correspond to the local minima in the dependence of wn on n (line 2D-H in Figure 2).

X n = C1e

s(n − 1) − c3ω(n2/3 − 1)

M−1

2/3

M−1

2/3

∑m = n m−2/3e−sm + c3ωm ∑m = 1 m−2/3e−sm + c3ωm

(15)

Curves 13 and 16 in Figure 4 exhibit Xn from eq 15, calculated numerically with the large enough M = 54, c3 = 6 (cubic

Figure 4. Stationary size distribution of 3D clusters with simple cubic lattice: curves 13 and 16 − eq 15 of CNT at ω = 5 and s = 13 and 16, respectively; solid- and open-bar histograms − eq 18 of ANT at ω = 5 and s = 13 and 16, respectively (for clearer presentation, (Xn/C1)1/5 rather than Xn/C1 is shown). The inset illustrates the first 10 smallest clusters with minimal surface area (the cubes schematize atoms, and the successive cluster shapes are closest to that of cube because of the requirement for minimum cluster surface area).

crystals), ω = 5, and s = 13 and 16, respectively. As seen, Xn is a monotonically decreasing function of the cluster size n. Therefore, like in 2D nucleation, CNT does not predict magic nanocrystal sizes in 3D nucleation either. ANT Cluster Size Distribution. ANT relaxes the CNT approximation of fixed shape of the clusters of all sizes. Instead, it takes into account the atomic irregularity of the cluster surface. This irregularity is illustrated in the inset of Figure 4 which shows the first 10 smallest 3D crystalline clusters with simple cubic lattice and minimal cluster surface area. ANT thus also uses the relation

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3D CRYSTALS CNT Cluster Size Distribution. We limit our analysis to the case of homogeneous nucleation. In this case CNT expresses Φn as the product of the cluster specific surface energy σ and the area a0an of the cluster surface.32 Thus, Φn = 3260



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Φn = σa0an = ωkTan

become more numerous than the irregularly shaped nanocrystals of neighboring sizes.

(16)

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but now the number an of atomic sites on the cluster surface is not an analytical function of n, and ω has the physical meaning of effective dimensionless broken-bond energy per atomic site on the cluster surface. The values of an are determined by direct counting of the atomic sites on the surface of the irregularly shaped clusters. For the cluster sequence schematized in the inset of Figure 4 and extended up to n = 53, these values are33 an = 6, 10, 14, 16, 20, 22, 24, 24, 28, 30, 32, 32, 36, 38, 40, 40, 42, 42, 46, 48, 50, 50, 52, 52, 54, 54, 54, 58, 60, 62, 62, 64, 64, 66, 66, 66, 70, 72, 74, 74, 76, 76, 78, 78, 78, 80, 80, 80, 84, 86, 88, 88, and 90 at, respectively, n = 1, 2, 3, ..., 52 and 53. As to f n, in ANT it is also considered as proportional to an when atoms are directly attached to the cluster surface, so that33 (n = 1, 2, 3, ...) fn = f1

an a1

CONCLUSION Equations 11 and 18 are central results in this study. They reveal that, in contrast to CNT, ANT is able to predict the existence of magic cluster sizes. Equations 11 and 18 are applicable to 2D or 3D nanocrystals with a given symmetry, which evolve by the direct-attachment mechanism and have minimum-energy shapes that correspond, respectively, to regular polygon or polyhedron with edges or faces with the same effective broken-bond energy ε or ω. These equations apply also to nanoclusters with noncrystal (e.g., icosahedral) symmetry which satisfy the above requirements for atom attachment, regular shape and broken-bond energy. More complicated cluster shapes and atom attachment kinetics can be treated with the aid of the general eqs 1−3 with appropriately determined cluster total peripheral or surface energy Φn and atom attachment frequency f n. However, the particular mechanism of atom attachment to the clusters can hardly have a strong effect on the stationary cluster size distribution because of the rather weak dependence of f n on the cluster size n for the commonly encountered attachment mechanisms. We note as well that eqs 11 and 18 are applicable when ε and ω are sufficiently greater than unity, as then in line with the Szilard− Farkas cluster evolution model, a single sequence of cluster shapes plays the leading role in the nucleation process. When the value of C1 is known and the dependence of ln or an on n is determined on the basis of an assumed cluster shape sequence, in eq 11 or 18 ε or ω is the sole free parameter, because the supersaturation s is experimentally controllable, for example, by the pressure or the concentration of the nucleating vapor or solution. Therefore, fitting Xn from eq 11 or 18 to experimental size distributions of 2D or 3D nanoclusters could provide information about the cluster shape evolution and about the value of ε or ω and, thereby, of the specific edge energy κ or surface energy σ. Finally, we note that with l1 = c2 = 4 and a1 = c3 = 6, the results obtained are relevant to the 2D and 3D Ising models of ferromagnets with nearest-neighbor coupling, because the coupling constant Js and the external field H correspond to (ε/2)kT or (ω/2)kT and to (s/2)kT, respectively. Thus, Figures 1 and 4 refer to Js = 5.7kT/2 and H = (10/2)kT for Figure 1 and to Js = 5kT/2 and H = (13/2)kT or H = (16/2)kT for Figure 4.

(17)

Similar to 2D nucleation, f n is actually proportional to the number a′n of attachment sites at the cluster surface. The proportionality of f n to an is thus an approximation which, however, is reasonable, because a′n is either equal to an (for clusters with kinkless surface) or not too smaller than an (for clusters with kinked surface). For instance, the trimer in the inset of Figure 4 has a′n = 13 (because of the presence of one kink site) and an = 14, the tetramer has a′n=an = 16 (because it is kinkless), and the pentamer has a′n = 18 (because of the presence of two kink sites) and an = 20. Using now eq 1 in combination with eqs 2, 3, 16, and 17, we arrive at the expression (n = 1, 2, 3, ...) M−1

X n = C1e s(n − 1) − ω(an − a1)

∑m = n am−1e−sm + ωam M−1

∑m = 1 am−1e−sm + ωam

(18)

This is the sought ANT formula for the cluster size distribution in stationary homogeneous nucleation of crystals with shape corresponding to that of regular polyhedron. We stress that, being based on the exact eq 1, eq 18 is also highly reliable. The solid- and open-bar histograms in Figure 4 illustrate Xn from eq 18 at a1 = 6 (simple cubic lattice), ω = 5, and s = 13 and 16, respectively. The sums are calculated numerically with the help of the above an values and the large enough M = 54. We observe in Figure 4 that Xn has strongly manifested maxima for clusters of size n = 4, 8, 12, 18, 27, 36, and 48. These namely are the magic sizes of 3D nanocrystals with simple cubic lattice, and they coincide with almost all of those seen in the cluster size distribution obtained by Wonczak et al.21 in computer simulation of nucleation of 3D Ising ferromagnet (Figure 5 in ref 21). The above magic sizes coincide as well with the experimentally found magic sizes of TiN crystalline clusters9 provided the building block of these clusters is chosen to be the cube formed by the centers of four Ti and four N nearestneighbor atoms in the clusters (see Table 1 and Figure 5 in ref 9). Thus, as for 2D nucleation, ANT conforms well to the simulation and experimental observations of magic sizes of nanoscale crystalline clusters. Again, the magic sizes remain supersaturation-independent and correspond to 3D nanocrystals with atomically completely built-up faces. In full analogy with the 2D case, these nanocrystals are in local energy minima (see line 3D-KS in Figure 2) and by withstanding growth or decay via attachment or detachment of single atoms, they

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AUTHOR INFORMATION

Corresponding Author

*Phone: (+3592)9792557; fax: (+3592)9712688; e-mail: [email protected]. Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS The author is grateful to Prof. Vladimir Kaganer for helpful correspondence concerning his simulation data presented in Figure 1. Thanks are also due to an anonymous reviewer for valuable recommendations.

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REFERENCES

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Magic Cluster Sizes in Nucleation of Crystals - American Chemical

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