Proc. Natl. Acad. Sci. USA Vol. 88, pp. 5067-5071, June 1991 Chemistry Clusters of clusters: Self-organization and self-similarity in the intermediate stages of cluster 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-atom icosahedral cluster acts as a basic series of high-nuclearity Au-Ag clusters containing 25, 37, 38, building block. and 46 metal atoms 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 nickel 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 ions and studied their vertex-sharing 13-atom Au-centered icosahedra reactions with carbon monoxide 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. §1734 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.