Proc. Natl. Acad. Sci. USA Vol. 88, pp. 5067-5071, June 1991

Clusters of clusters: Self-organization and self-similarity in the intermediate stages of growth of Au-Ag supraclusters (fractal) BOON K. TEO AND HONG ZHANG Department of Chemistry, University of Illinois at Chicago, Chicago, IL 60680 Communicated by Lawrence F. Dahl, January 7, 1991 (receivedfor review September 10, 1990)

ABSTRACT A systematic structural investigation ofa new systems where the 13- icosahedral cluster acts as a basic series of high-nuclearity Au-Ag clusters containing 25, 37, 38, building block. and 46 led to the description of these clusters as Primary Clusters. Fig. 1 shows the early stages of cluster "clusters of clusters" based on vertex-sharing icosahedra as growth based on the pioneering work of Hoare and Pal (29) building blocks. Based on the observed structures, a growth and Briant and Burton (30). This particular growth sequence sequence is proposed here for the formation of these secondary (30) starts with one atom and adds one atom at a time. The clusters (clusters ofclusters) from a single 13-atom icosahedron cluster c(n) grows via an atom-by-atom mechanism where n to a 127-atom icosahedron of icosahedra via successive addi- is the nuclearity. We shall refer to these clusters, c(n), as tions of vertex-sharing icosahedral units. This cluster-of- primary clusters (17, 18). Numerous examples of known clusters growth mechanism parallels the atom-by-atom growth structures can be found in the literature for the first six pathway for the primary clusters from a single atom to a members c(1)-c(6) of the series shown in Fig. 1. Some 13-atom icosahedron. It is hypothesized that the formation of examples are: monomeric, c(l), Fe(CO)5; dimeric, c(2), these clusters of clusters is a manifestation of the spontaneous Fe2(CO)9; trimeric, c(3), Fe3(CO)12; tetrahedral, c(4), self-organization and self-similarity processes often observed in Fe4(CO)13 (33); trigonal bipyramidal, c(5), Os5(CO)16 (34); nature. It is conceivable that the concept of cluster of clusters and bicapped tetrahedral, c(6), Os6(CO)18 (35). A few struc- may be important in the intermediate stages of some cluster turally characterized examples are known for the pentagonal growth as exemplified by the polyicosahedral growth ofAu-Ag bipyramidal c(7) and the icosahedral c(13) structures as supraclusters. exemplified by [Au7(PPh3)7]+ (36) and [Au13Cl2(PMe2Ph)10]3+ (37), respectively. (Note that dimerization oftwo pentagonal- High-nuclearity clusters are often formed by fusing together bipyramidal c(7) clusters via sharing of one apical atom smaller cluster units (1-8). Indeed, this modular or building produces the icosahedral cluster c(13)-namely, N = 2 X 7 block approach is a highly promising route to clusters of - 1 = 13.) increasing nuclearity. Recently we reported the syntheses To date, no examples are known for the structures c(8)- and structures of a new series of high-nuclearity Au-Ag c(12) depicted in Fig. 1. However, a recent elegant work by clusters Fayet et al. (38) of carbonyl clusters in molecular containing 25 (9), 37 (10), 38 (11), and 46 (B.K.T., X. beams provides some indirect experimental evidence for the Shi, and H.Z., unpublished data) metal atoms. The metal existence of these clusters. These authors mass-selected configuration of these "supraclusters" can be visualized on individual Ni+ = the basis of (n 1-13) cluster and studied their vertex-sharing 13-atom Au-centered icosahedra reactions with to give Nin(CO)t in the gas as building blocks (13-18) (Fig. 1). We refer to these supra- phase, where n = 1-13 when 1 varies as a function of cluster clusters as "clusters of clusters" (13-18) (Fig. 2). We also size. Mingos and Wales (39) interpreted the structures of the developed atom- and electron-counting schemes for rational- latter members of the series as due to successive face izing or predicting the structure and bonding of these and cappings of the pentagonal bipyramidal cluster c(7) c(8) -* related supraclusters (15-18) (see Appendix 1). ... **c(13) as originally envisioned by Briant and Burton (30) as well as Hoare and Pal (29) (see Fig. 1). SELF-ORGANIZATION AND SELF-SIMILARITY Secondary Clusters. Although Briant and Burton (30) con- PRINCIPLE sidered further growth of the 13-atom cluster to form a 33-atom pentagonal dodecahedral cluster to a 45-atom cluster A comparison of the structures of supraclusters Sn(N) of to a 55-atom v2 icosahedral cluster, we propose that the nuclearity N = 13-127 (Fig. 2), where nuclearity is the formation of the 13-atom icosahedral cluster, c(13), may number of metal atoms, with those of primary clusters c(n) signify the "end" of the "early stages" of cluster growth for of nuclearity n = 1-13 (Fig. 1) reveals a significant degree of some system such as the Au-Ag supraclusters considered similarities (1-26). Indeed, the existence of these clusters of here. Further growth can take on many different pathways, clusters may be a manifestation of the spontaneous self- depending upon the kinetics and thermodynamics of the organization and self-similarity processes often observed in system. Two distinct pathways (among others) can be iden- nature (27,28). Ifthe structures ofclusters c(n) (n = 1-13) can tified. For inert gas clusters (40-44), a growth sequence be considered as models for the early stages ofcluster growth based on the v,, icosahedra with magic numbers 13, 55, 147, or particle formation, as envisioned by Hoare and Pal (29), ... [the so-called Mackay sequence (45, 46)] has been Briant and Burton (30), and others (31, 32), then the struc- observed. This particular growth sequence may be described tures of the Au-Ag supraclusters, Sn(N) of nuclearity (N) as the "layer-by-layer" growth mechanism. For the Au-Ag ranging from 13 to 127, may be considered as the "interme- supraclusters, on the other hand, the growth sequence with diate stages" of the cluster growth for Au-Ag supracluster the magic numbers 13 (ref. 37), 25 (ref. 9), 36 [actually, only the related 37- and 38-atom clusters (refs. 10 and 11, respec- The publication costs of this article were defrayed in part by page charge tively) have been observed so far], 46 (B.K.T., X. Shi, and payment. This article must therefore be hereby marked "advertisement" H.Z., unpublished data),... are based on a vertex-sharing in accordance with 18 U.S.C. ยง1734 solely to indicate this fact. polyicosahedral growth pathway. This latter growth se- 5067 Downloaded by guest on September 28, 2021 5068 Chemistry: Teo and Zhang Proc. Natl. Acad. Sci. USA 88 (1991) ters. We shall now discuss the structural characteristics ofthe cluster-of-clusters growth pathway as exemplified by the Au-Ag supraclusters (refs. 9-11, and B.K.T., X. Shi, and 0 H.Z., unpublished data). In analogy to the primary clusters c(n) (where n = 1-13) based on the Briant and Burton (30) growth pattern shown in c(l) Fig. 1, Fig. 2 depicts the corresponding supraclusters Sn(N) (where N denotes the nuclearity of the supracluster) formed by n-centered icosahedra sharing vertices (n = 1-13). Here each atom in c(n) is replaced by an icosahedron in Sn(N) with the nuclearity N given by 13n minus the number of shared vertices (15). Instead of adding one atom at a time, the supracluster "grows" by adding one icosahedron at a time, resulting in the formation of the secondary clusters Sn(N). As depicted in Fig. 2, the "intermediate stage" of cluster c(4) c(/ ) growth starts with a 13-atom-centered icosahedral cluster unit, sl(13). Adding one icosahedral unit via sharing of one vertex produces the 25-atom cluster, s2(25), since 13 + 13 - 1 = 25, as exemplified by [(p-Tol3P)j0Au13Ag12Br8]+ (where Tol = tolyl; ref. 9) and [(Ph3P)10Au13Ag12Br8]+ (12). (Note that these two clusters differ in the relative orientation of the two icosahedral units. Only the latter cluster is portrayed in Fig. 2.) In Fig. 2, each "added" icosahedron is represented by heavy bonds. All radial bonds from the central atom (filled c(6) c(7) circle) of each icosahedron are omitted for clarity. Adding a third icosahedron to s2(25) via sharing of two vertices gives rise to a 36-atom cluster, s3(36). This supra- / cluster can also be formed by three icosahedra sharing three vertices since 3 x 13 (three icosahedra) - 3 (sharing three vertices) = 36. Though this structure is not yet known, the closely related 37-atom [(p-Tol3P)12Au18Ag19Brj1]2+ (10) and 38-atom [(p-Tol3P)12Au18Ag20Cl14] (11) clusters, containing one and two exopolyhedral atoms (vide supra), respectively, c(8) have recently been synthesized and structurally character- ized (10, 11). One interesting stereochemical characteristic of this 36- atom cluster is that it has a central equilateral triangle, which serves as anchoring point for additional 13-atom icosahedral units. The nuclearity of the resulting cluster should increase by 13 - 3 (sharing three vertices) = 10 for each additional icosahedral unit via sharing of three vertices (see ref. 16 for structural rules for vertex-sharing polyicosahedral supraclus- ters). Indeed, a 46-atom [(Ph3P)12Au24Ag22Cl10] cluster, c(10) S4(46), has recently been synthesized and structurally char- acterized (B.K.T., X. Shi, and H.Z., unpublished data). In describing the structure and bonding ofthese supraclus- ters, it is advantageous to define a "superpolyhedron" formed by the centers (filled circles in Fig. 2) ofthe individual icosahedral units (15). In general, the shared vertices are located at the midpoints ofthe edges (including hidden edges) of the superpolyhedron. For example, the superpolyhedron of the s4(46) supracluster is a supertetrahedron. We can now refer to the addition of an icosahedron to a supracluster as capping of one of the "faces" of a superpolyhedron. Thus, C(12) c(13) monocapping of one of the four supertriangular faces of the 46-atom supracluster, s4(46), gives rise to the trigonal- FIG. 1. Growth sequence of cluster c(n) via an atom-by-atom bipyramidal supracluster, s5(56). Here the nine shared ver- growth pathway to form a 13-atom icosahedron where n is the tices form two octahedra sharing a triangular face. Further nuclearity. The point group symmetries of these primary clusters capping of one of the six supertriangular faces of s5(56) c(n) are: C1, Dah, D3h, Td, D3h, C2,, D5h, Cs, C5, Cs, Cs, C5v, and produces the bicapped-tetrahedral supracluster s6(66) (16, Ih for n = 1-13, respectively. 17). The s6(66) supracluster can "grow" by capping a third quence may be described as the cluster-of-clusters growth supertriangular face of the bicapped supertetrahedron with a mechanism (13-18). We shall designate clusters formed by 13-atom icosahedron by sharing three vertices, resulting in a either the layer-by-layer or the cluster-of-clusters growth 76-atom (66 + 13 - 3 = 76) cluster (17) (not shown). As shown pathways as secondary clusters (17, 18). in figure 1 of ref. 16, successive cappings of the supertetra- It is obvious that primary clusters are of prime importance hedron give rise to mono-, bi-, tri-, and tetracapped tetrahe- in that they can serve either as the "nucleation core" for the dra, corresponding to the vertex-sharing icosahedral supra- layer-by-layer growth or as the "building blocks" for the clusters s4(46), s5(56), s6(66), s7*(76), and sg*(86), respec- cluster-of-clusters growth, giving rise to the secondary clus- tively. (For models of s6(66), s7*(76), and s8*(86), see figures Downloaded by guest on September 28, 2021 Chemistry: Teo and Zhang Proc. Natl. Acad. Sci. USA 88 (1991) 5069 15 and 16 a and b of ref. 17.) The asterisks designate stellated superpolyhedra. This growth sequence in general, and the existence ofthe structures of s7*(76) and S8*(86) in particular, am though probable, are not considered here because they do not to the ultimate formation of the icosahedron of icosa- hedra, s13(127). In contrast, placing the third "capping" icosahedron on the "butterfly" side (see figure 4 of ref. 16) S1(13) of the bicapped supertetrahedron by sharing four vertices gives rise to a 75-atom (66 + 13 - 4 = 75) supracluster, s7(75). S2(25) / S (36) The centers of the seven icosahedral units thus form an idealized super-pentagonal-bipyramid of D5h symmetry (see also Appendix 2). As in Fig. 1, further growth of the supracluster can occur by successive cappings (17) of the five (upper) triangular faces of the super-pentagonal-bipyramid. This successive addition of five icosahedra (Fig. 2) to one side of the supra- cluster s7(75) results in s8(85), s9(94), slo(103), sil(112), and s12(120), where the nuclearity increments AN of 10, 9, and 8 atoms represent cappings of triangular, butterfly, and trap- ezoidal sites of the superpolyhedron, respectively (16). Fi- S4(46) nally, addition of a 7-atom pentagonal-bipyramidal cluster completes the last icosahedron [via capping of a pentagonal- pyramidal site (16)], producing the 127-atom supracluster s13(127). The latter can be called an "icosahedron of icosa- hedra" which has Ih symmetry (see Appendix 3). It is apparent from the above discussion that there is a striking similarity between the early stages of cluster growth with nuclearities < 13 and the intermediate stages with nuclearities ranging from 13 to 127 if one replaces each of the added atoms in the early stages (Fig. 1) by a 13-atom icosahedron in the intermediate stages (Fig. 2). The parallel construction of c(n) (from n = 1 for a single atom to n = 13 for an icosahedron) and sn(N) (from n = 1 for a single icosahedron to n = 13 for an icosahedron of icosahedra) is conceptually pleasing, although to date experimental struc- tural data are available for the early members [s,(N) where n s 4] of the series only. In this paper, we consider only the geometrical design of clusters of clusters based on vertex-sharing icosahedra. The issues of electronic requirements have been addressed else- where (15-18). Since an icosahedron has two-, three-, and fivefold symmetries (but not fourfold symmetry), we expect the supraclusters, Sn(N), to have the same symmetry require- ments. Indeed, as shown in the captions of Figs. 1 and 2, with the exceptions ofthe first two members ofthe series, c(n) and Sn(N) have the same point group symmetries for the corre- sponding n values. Furthermore, we have considered in this paper only those sn(N) structures that correspond to the primary clusters c(n). Many other structures ofa given sn(N), with different nuclearities, N, depending upon the geometry and symmetry, are also possible. Nevertheless, while a different metal- combination may follow different clus- ter design rules, the self-organization and self-similarity principle illustrated herein for the vertex-sharing icosahedral Au-Ag supraclusters may be applicable to a wide variety of cluster systems (provided, of course, that the electronic and steric effects are satisfied). Fractal. Since many ofthe structurally known clusters and supraclusters are formed by spontaneous self-assembly (1- 26), it is concluded that this similarity is indeed a manifes- tation of the self-organization and self-similarity principle-

time. The nuclearity (in parenthesis) increases by 13 minus the number of the shared vertices. Each "added" icosahedron is rep- S12(120) S13(127) resented by heavy bonds. All radial bonds from the central atom (filled circles) of each icosahedron are omitted for clarity. The point group symmetries ofthese secondary clusters sj(N) are: Ih, D5h, D3h, FIG. 2. Cluster-of-clusters growth sequence from icosahedron to Td, D3h, C2v, DMh, C5, C5, C~, C, C5v, and Ih for n = 1-13, icosahedron of icosahedra. The supraclusters s,(nuclearity) of n respectively. Note that with the exceptions of the first two members vertex-sharing icosahedra "grow" by adding one icosahedron at a of the series, the point symmetries are the same for c(n) and sn(N). Downloaded by guest on September 28, 2021 5070 Chemistry: Teo and Zhang Proc. Natl. Acad. Sci. USA 88 (1991) often found in nature (27, 28). The principle of self- form the pentagonal bipyramid, c(7). Here, the pentagonal- organization and self-similarity to supramolecular bipyramidal cluster c(7) has a hidden bond connecting the assembly. For clusters, self-organization means spontaneous two apical atoms. assemblage of atoms to form energetically stable clusters of relatively efficient packing and reasonably high symmetry. The buildup of primary clusters [e.g., from c(1) to the icosahedral c(13)] or secondary clusters [e.g., from the sl(13) to s13(127)] from their respective building blocks is an exam- ple of such spontaneous self-organization. Self-similarity means that the resulting cluster looks more or less alike when primary: c'(6) c(7) examined at different levels of magnification. Indeed, the secondary: s6'(68) s7(75) secondary supraclusters sn(N) look very much like the cor- responding primary clusters c(n) when viewed at roughly half The corresponding capping ofa super-pentagonal-pyramid of the magnification. Such underlying geometric similarity is S6'(68) gives rise to the super-pentagonal-bipyramid of s7(75) sometimes called "scale invariance," and the resulting pat- with an increment in nuclearity of AN = 13 - 6 (sharing 6 terns are often referred to as "fractal" (27, 28). Studies of vertices) = 7. For s7(75), each atom in c(7) is replaced by a fractal pattern formation have allowed a better understanding 13-atom icosahedron, each of which in turn shares a vertex of the random aggregation of particles into clusters, the (located at the midpoint of the super-pentagonal-bipyramid) growth process of crystals, etc. with the neighboring icosahedra. Since there are 15 edges on In this paper, we have explored the possible application of the surface of the super-pentagonal-bipyramid and one hid- the concept of fractal geometry (27, 28) to cluster growth in den edge [in analogy to the primary cluster c(7)], the total general and to cluster-of-clusters growth in particular. The number of shared vertices in s7(N) is 16, giving rise to a spontaneous self-organization of clusters into clusters of nuclearity N of 7 x 13 - 16 = 75. clusters (as exemplified by the self-assembly of icosahedra via vertex-sharing to form, ultimately, an icosahedron of icosahedra for the Au-Ag supraclusters) is, in fact, symmetry APPENDIX 3 across scale, pattern within pattern. The property of self- Another useful concept in the description of primary and similarity is most strikingly clear when one replaces each secondary clusters is the concept of hole(s) as exemplified atom in the primary clusters, c(n), by an icosahedron to give below by s~l'(113), s12(120), and s13(127). the secondary clusters, Sn(N). It is conceivable that the concept of cluster of clusters may be important in the intermediate stages of some cluster growth as exemplified by the polyicosahedral growth of Au-Ag supraclusters consid- ered in this paper.

APPENDIX 1 In a previous publication (15), we considered the formation of supraclusters sn(N) (where N is the nuclearity) in which the n centers of the 13-atom icosahedral units form triangu- Sio'(110) sll'(113) sll''(118)-->S12(120) S12'(126)-+S13(127) lated superpolyhedra (or superdeltahedra) analogous to that observed in the cluster BH2-. Here each The super-pentagonal-antiprism of a slo'(110) (open cir- boron atom is conceptually replaced by a 13-atom centered cles) has a pentagonal-antiprismatic hole, which can accom- icosahedron. In this particular series, supraclusters sn(N) modate three more atoms (A& N = 3), thereby forming a new formed by n vertex-sharing centered icosahedra of 13 atoms icosahedron in the center (filled circle), resulting in s~l'(113). each (as building blocks) are predicted to have nuclearities N Likewise, filling the capped-pentagonal-antiprismatic hole in of 13, 25, 36 (parent of 37 and 38 atom clusters), 46, 56, 75, s1l"(118) with two atoms (namely, AN = 2) results in sl2(120). and 127 for n = 1, 2, 3, 4, 5, 7, and 13, respectively. And, finally, filling the icosahedral hole of s12'(126) with one Geometrical considerations of close-packing of 13-atom cen- atom (namely, AN = 1) gives rise to s13(127). tered icosahedral units via vertex-sharing revealed that the resulting supraclusters cannot possess fourfold rotational We thank the National Science Foundation (CHE-8722339) for symmetry. Hence, square (D40), octahedral (Oh), or triangu- financial support. lar-dodecahedral (D2d) arrays of vertex-sharing icosahedral supraclusters will have severe geometrical constraints. In 1. You, J.-F., Snyder, B. S. & Holm, R. H. (1988) J. Am. Chem. other of BH2- where n = Soc. 110, 6589-6591. words, Sn(N) analogs 6, 8, 9, 10, 2. Freeman, M. J., Green, M., Orpen, A. G., Salter, I. D. & 11 are unlikely to occur without significant distortions. The Stone, F. G. A. (1983) J. Chem. Soc. Chem. Commun., 1332- n = 12 case deserves some comments. Presumably, the S12 1334. analog of B12H2 can be built from 12 centered icosahedra 3. Doyle, G., Eriksen, K. A. & Van Engen, D. (1985) J. Am. sharing 30 vertices. This gives rise to a supracluster s12'(126) Chem. Soc. 107, 7914-7920. with a nuclearity of 126. However, the icosahedral cage 4. Adams, R. D., Dawoodi, Z., Forest, D. F. & Segmuller, B. E. created by the 12 icosahedra can be filled with one additional (1983) J. Am. Chem. 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