Proc. Natl. Acad. Sci. USAVol. 88, pp. 5067-5071, June 1991Chemistry

Clusters of clusters: Self-organization and self-similarity in theintermediate stages of cluster growth of Au-Ag supraclusters

(fractal)

BOON K. TEO AND HONG ZHANGDepartment of Chemistry, University of Illinois at Chicago, Chicago, IL 60680

Communicated by Lawrence F. Dahl, January 7, 1991 (received for review September 10, 1990)

ABSTRACT A systematic structural investigation of a newseries of high-nuclearity Au-Ag clusters containing 25, 37, 38,and 46 metal atoms led to the description of these clusters as"clusters of clusters" based on vertex-sharing icosahedra asbuilding blocks. Based on the observed structures, a growthsequence is proposed here for the formation of these secondaryclusters (clusters of clusters) from a single 13-atom icosahedronto a 127-atom icosahedron of icosahedra via successive addi-tions of vertex-sharing icosahedral units. This cluster-of-clusters growth mechanism parallels the atom-by-atom growthpathway for the primary clusters from a single atom to a13-atom icosahedron. It is hypothesized that the formation ofthese clusters of clusters is a manifestation of the spontaneousself-organization and self-similarity processes often observed innature. It is conceivable that the concept of cluster of clustersmay be important in the intermediate stages of some clustergrowth as exemplified by the polyicosahedral growth ofAu-Agsupraclusters.

High-nuclearity clusters are often formed by fusing togethersmaller cluster units (1-8). Indeed, this modular or buildingblock approach is a highly promising route to clusters ofincreasing nuclearity. Recently we reported the synthesesand structures of a new series of high-nuclearity Au-Agclusters containing 25 (9), 37 (10), 38 (11), and 46 (B.K.T., X.Shi, and H.Z., unpublished data) metal atoms. The metalconfiguration of these "supraclusters" can be visualized onthe basis of vertex-sharing 13-atom Au-centered icosahedraas building blocks (13-18) (Fig. 1). We refer to these supra-clusters as "clusters of clusters" (13-18) (Fig. 2). We alsodeveloped atom- and electron-counting schemes for rational-izing or predicting the structure and bonding of these andrelated supraclusters (15-18) (see Appendix 1).

SELF-ORGANIZATION AND SELF-SIMILARITYPRINCIPLE

A comparison of the structures of supraclusters Sn(N) ofnuclearity N = 13-127 (Fig. 2), where nuclearity is thenumber of metal atoms, with those of primary clusters c(n)of nuclearity n = 1-13 (Fig. 1) reveals a significant degree ofsimilarities (1-26). Indeed, the existence of these clusters ofclusters may be a manifestation of the spontaneous self-organization and self-similarity processes often observed innature (27,28). If the structures ofclusters c(n) (n = 1-13) canbe considered as models for the early stages of cluster growthor particle formation, as envisioned by Hoare and Pal (29),Briant and Burton (30), and others (31, 32), then the struc-tures of the Au-Ag supraclusters, Sn(N) of nuclearity (N)ranging from 13 to 127, may be considered as the "interme-diate stages" of the cluster growth for Au-Ag supracluster

systems where the 13-atom icosahedral cluster acts as a basicbuilding block.Primary Clusters. Fig. 1 shows the early stages of cluster

growth based on the pioneering work of Hoare and Pal (29)and Briant and Burton (30). This particular growth sequence(30) starts with one atom and adds one atom at a time. Thecluster c(n) grows via an atom-by-atom mechanism where nis the nuclearity. We shall refer to these clusters, c(n), asprimary clusters (17, 18). Numerous examples of knownstructures can be found in the literature for the first sixmembers c(1)-c(6) of the series shown in Fig. 1. Someexamples are: monomeric, c(l), Fe(CO)5; dimeric, c(2),Fe2(CO)9; trimeric, c(3), Fe3(CO)12; tetrahedral, c(4),Fe4(CO)13 (33); trigonal bipyramidal, c(5), Os5(CO)16 (34);and bicapped tetrahedral, c(6), Os6(CO)18 (35). A few struc-turally characterized examples are known for the pentagonalbipyramidal c(7) and the icosahedral c(13) structures asexemplified by [Au7(PPh3)7]+ (36) and [Au13Cl2(PMe2Ph)10]3+(37), respectively. (Note that dimerization oftwo pentagonal-bipyramidal c(7) clusters via sharing of one apical atomproduces the icosahedral cluster c(13)-namely, N = 2 X 7- 1 = 13.)To date, no examples are known for the structures c(8)-

c(12) depicted in Fig. 1. However, a recent elegant work byFayet et al. (38) of nickel carbonyl clusters in molecularbeams provides some indirect experimental evidence for theexistence of these clusters. These authors mass-selectedindividual Ni+ (n = 1-13) cluster ions and studied theirreactions with carbon monoxide to give Nin(CO)t in the gasphase, where n = 1-13 when 1 varies as a function of clustersize. Mingos and Wales (39) interpreted the structures of thelatter members of the series as due to successive facecappings of the pentagonal bipyramidal cluster c(7) c(8) -*...**c(13) as originally envisioned by Briant and Burton (30)as well as Hoare and Pal (29) (see Fig. 1).Secondary Clusters. Although Briant and Burton (30) con-

sidered further growth of the 13-atom cluster to form a33-atom pentagonal dodecahedral cluster to a 45-atom clusterto a 55-atom v2 icosahedral cluster, we propose that theformation of the 13-atom icosahedral cluster, c(13), maysignify the "end" of the "early stages" of cluster growth forsome system such as the Au-Ag supraclusters consideredhere. Further growth can take on many different pathways,depending upon the kinetics and thermodynamics of thesystem. Two distinct pathways (among others) can be iden-tified. For inert gas clusters (40-44), a growth sequencebased on the v,, icosahedra with magic numbers 13, 55, 147,... [the so-called Mackay sequence (45, 46)] has beenobserved. This particular growth sequence may be describedas the "layer-by-layer" growth mechanism. For the Au-Agsupraclusters, on the other hand, the growth sequence withthe magic numbers 13 (ref. 37), 25 (ref. 9), 36 [actually, onlythe related 37- and 38-atom clusters (refs. 10 and 11, respec-tively) have been observed so far], 46 (B.K.T., X. Shi, andH.Z., unpublished data),... are based on a vertex-sharingpolyicosahedral growth pathway. This latter growth se-

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c(/ )

c(7)

/

c(8)

c(10)

C(12) c(13)

FIG. 1. Growth sequence of cluster c(n) via an atom-by-atomgrowth pathway to form a 13-atom icosahedron where n is thenuclearity. The point group symmetries of these primary clustersc(n) are: C1, Dah, D3h, Td, D3h, C2,, D5h, Cs, C5, Cs, Cs, C5v, andIh for n = 1-13, respectively.

quence may be described as the cluster-of-clusters growthmechanism (13-18). We shall designate clusters formed byeither the layer-by-layer or the cluster-of-clusters growthpathways as secondary clusters (17, 18).

It is obvious that primary clusters are of prime importancein that they can serve either as the "nucleation core" for thelayer-by-layer growth or as the "building blocks" for thecluster-of-clusters growth, giving rise to the secondary clus-

ters. We shall now discuss the structural characteristics ofthecluster-of-clusters growth pathway as exemplified by theAu-Ag supraclusters (refs. 9-11, and B.K.T., X. Shi, andH.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 inFig. 1, Fig. 2 depicts the corresponding supraclusters Sn(N)(where N denotes the nuclearity of the supracluster) formedby n-centered icosahedra sharing vertices (n = 1-13). Hereeach atom in c(n) is replaced by an icosahedron in Sn(N) withthe nuclearity N given by 13n minus the number of sharedvertices (15). Instead of adding one atom at a time, thesupracluster "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

growth starts with a 13-atom-centered icosahedral clusterunit, sl(13). Adding one icosahedral unit via sharing of onevertex produces the 25-atom cluster, s2(25), since 13 + 13 -1 = 25, as exemplified by [(p-Tol3P)j0Au13Ag12Br8]+ (whereTol = tolyl; ref. 9) and [(Ph3P)10Au13Ag12Br8]+ (12). (Notethat these two clusters differ in the relative orientation of thetwo icosahedral units. Only the latter cluster is portrayed inFig. 2.) In Fig. 2, each "added" icosahedron is representedby heavy bonds. All radial bonds from the central atom (filledcircle) 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 threevertices since 3 x 13 (three icosahedra) - 3 (sharing threevertices) = 36. Though this structure is not yet known, theclosely related 37-atom [(p-Tol3P)12Au18Ag19Brj1]2+ (10) and38-atom [(p-Tol3P)12Au18Ag20Cl14] (11) clusters, containingone and two exopolyhedral atoms (vide supra), respectively,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, whichserves as anchoring point for additional 13-atom icosahedralunits. The nuclearity of the resulting cluster should increaseby 13 - 3 (sharing three vertices) = 10 for each additionalicosahedral unit via sharing of three vertices (see ref. 16 forstructural rules for vertex-sharing polyicosahedral supraclus-ters). Indeed, a 46-atom [(Ph3P)12Au24Ag22Cl10] cluster,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 individualicosahedral units (15). In general, the shared vertices arelocated at the midpoints ofthe edges (including hidden edges)of the superpolyhedron. For example, the superpolyhedronof the s4(46) supracluster is a supertetrahedron. We can nowrefer to the addition of an icosahedron to a supracluster ascapping of one of the "faces" of a superpolyhedron. Thus,monocapping of one of the four supertriangular faces of the46-atom supracluster, s4(46), gives rise to the trigonal-bipyramidal supracluster, s5(56). Here the nine shared ver-tices form two octahedra sharing a triangular face. Furthercapping of one of the six supertriangular faces of s5(56)produces the bicapped-tetrahedral supracluster s6(66) (16,17).The s6(66) supracluster can "grow" by capping a third

supertriangular face of the bicapped supertetrahedron with a13-atom icosahedron by sharing three vertices, resulting in a76-atom (66 + 13 - 3 = 76) cluster (17) (not shown). As shownin figure 1 of ref. 16, successive cappings of the supertetra-hedron give rise to mono-, bi-, tri-, and tetracapped tetrahe-dra, corresponding to the vertex-sharing icosahedral supra-clusters s4(46), s5(56), s6(66), s7*(76), and sg*(86), respec-tively. (For models of s6(66), s7*(76), and s8*(86), see figures

0

c(l)

c(4)

c(6)

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S1(13)

am

S2(25) / S (36)

S4(46)

S12(120) S13(127)

FIG. 2. Cluster-of-clusters growth sequence from icosahedron toicosahedron of icosahedra. The supraclusters s,(nuclearity) of nvertex-sharing icosahedra "grow" by adding one icosahedron at a

15 and 16 a and b of ref. 17.) The asterisks designate stellatedsuperpolyhedra. This growth sequence in general, and theexistence of the structures of s7*(76) and S8*(86) in particular,though probable, are not considered here because they do notlead 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)of the bicapped supertetrahedron by sharing four verticesgives rise to a 75-atom (66 + 13 - 4 = 75) supracluster, s7(75).The centers of the seven icosahedral units thus form anidealized super-pentagonal-bipyramid of D5h symmetry (seealso Appendix 2).As in Fig. 1, further growth of the supracluster can occur

by successive cappings (17) of the five (upper) triangularfaces of the super-pentagonal-bipyramid. This successiveaddition of five icosahedra (Fig. 2) to one side of the supra-cluster s7(75) results in s8(85), s9(94), slo(103), sil(112), ands12(120), where the nuclearity increments AN of 10, 9, and 8atoms represent cappings of triangular, butterfly, and trap-ezoidal sites of the superpolyhedron, respectively (16). Fi-nally, addition of a 7-atom pentagonal-bipyramidal clustercompletes the last icosahedron [via capping of a pentagonal-pyramidal site (16)], producing the 127-atom supraclusters13(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 astriking similarity between the early stages of cluster growthwith nuclearities < 13 and the intermediate stages withnuclearities ranging from 13 to 127 if one replaces each of theadded atoms in the early stages (Fig. 1) by a 13-atomicosahedron in the intermediate stages (Fig. 2). The parallelconstruction of c(n) (from n = 1 for a single atom to n = 13for an icosahedron) and sn(N) (from n = 1 for a singleicosahedron to n = 13 for an icosahedron of icosahedra) isconceptually pleasing, although to date experimental struc-tural data are available for the early members [s,(N) wheren s 4] of the series only.

In this paper, we consider only the geometrical design ofclusters of clusters based on vertex-sharing icosahedra. Theissues of electronic requirements have been addressed else-where (15-18). Since an icosahedron has two-, three-, andfivefold symmetries (but not fourfold symmetry), we expectthe supraclusters, Sn(N), to have the same symmetry require-ments. Indeed, as shown in the captions of Figs. 1 and 2, withthe exceptions of the first two members of the series, c(n) andSn(N) have the same point group symmetries for the corre-sponding n values. Furthermore, we have considered in thispaper only those sn(N) structures that correspond to theprimary clusters c(n). Many other structures ofa given sn(N),with different nuclearities, N, depending upon the geometryand symmetry, are also possible. Nevertheless, while adifferent metal-ligand combination may follow different clus-ter design rules, the self-organization and self-similarityprinciple illustrated herein for the vertex-sharing icosahedralAu-Ag supraclusters may be applicable to a wide variety ofcluster systems (provided, of course, that the electronic andsteric effects are satisfied).

Fractal. Since many of the structurally known clusters andsupraclusters 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 thenumber of the shared vertices. Each "added" icosahedron is rep-resented by heavy bonds. All radial bonds from the central atom(filled circles) of each icosahedron are omitted for clarity. The pointgroup symmetries ofthese secondary clusters sj(N) are: Ih, D5h, D3h,Td, D3h, C2v, DMh, C5, C5, C~, C, C5v, and Ih for n = 1-13,respectively. Note that with the exceptions of the first two membersof the series, the point symmetries are the same for c(n) and sn(N).

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often found in nature (27, 28). The principle of self-organization and self-similarity leads to supramolecularassembly. For clusters, self-organization means spontaneousassemblage of atoms to form energetically stable clusters ofrelatively efficient packing and reasonably high symmetry.The buildup of primary clusters [e.g., from c(1) to theicosahedral 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-similaritymeans that the resulting cluster looks more or less alike whenexamined at different levels of magnification. Indeed, thesecondary supraclusters sn(N) look very much like the cor-responding primary clusters c(n) when viewed at roughly halfthe magnification. Such underlying geometric similarity issometimes called "scale invariance," and the resulting pat-terns are often referred to as "fractal" (27, 28). Studies offractal pattern formation have allowed a better understandingof the random aggregation of particles into clusters, thegrowth process of crystals, etc.

In this paper, we have explored the possible application ofthe concept of fractal geometry (27, 28) to cluster growth ingeneral and to cluster-of-clusters growth in particular. Thespontaneous self-organization of clusters into clusters ofclusters (as exemplified by the self-assembly of icosahedravia vertex-sharing to form, ultimately, an icosahedron oficosahedra for the Au-Ag supraclusters) is, in fact, symmetryacross scale, pattern within pattern. The property of self-similarity is most strikingly clear when one replaces eachatom in the primary clusters, c(n), by an icosahedron to givethe secondary clusters, Sn(N). It is conceivable that theconcept of cluster of clusters may be important in theintermediate stages of some cluster growth as exemplified bythe polyicosahedral growth of Au-Ag supraclusters consid-ered in this paper.

APPENDIX 1In a previous publication (15), we considered the formationof supraclusters sn(N) (where N is the nuclearity) in whichthe n centers of the 13-atom icosahedral units form triangu-lated superpolyhedra (or superdeltahedra) analogous to thatobserved in the boron hydride cluster BH2-. Here eachboron atom is conceptually replaced by a 13-atom centeredicosahedron. In this particular series, supraclusters sn(N)formed by n vertex-sharing centered icosahedra of 13 atomseach (as building blocks) are predicted to have nuclearities Nof 13, 25, 36 (parent of 37 and 38 atom clusters), 46, 56, 75,and 127 for n = 1, 2, 3, 4, 5, 7, and 13, respectively.Geometrical considerations of close-packing of 13-atom cen-tered icosahedral units via vertex-sharing revealed that theresulting supraclusters cannot possess fourfold rotationalsymmetry. Hence, square (D40), octahedral (Oh), or triangu-lar-dodecahedral (D2d) arrays of vertex-sharing icosahedralsupraclusters will have severe geometrical constraints. Inother words, Sn(N) analogs ofBH2- where n = 6, 8, 9, 10,11 are unlikely to occur without significant distortions. Then = 12 case deserves some comments. Presumably, the S12analog of B12H2 can be built from 12 centered icosahedrasharing 30 vertices. This gives rise to a supracluster s12'(126)with a nuclearity of 126. However, the icosahedral cagecreated by the 12 icosahedra can be filled with one additionalatom, giving rise to s13(127) (see Appendix 3).

APPENDIX 2It is helpful to invoke the concepts of hidden bond (edge) andhidden vertex in describing the structures of primary andsecondary clusters, respectively.The hidden bond (edge) in a primary cluster is exemplified

by the capping of the pentagonal-pyramidal cluster c'(6) to

form the pentagonal bipyramid, c(7). Here, the pentagonal-bipyramidal cluster c(7) has a hidden bond connecting thetwo apical atoms.

primary:secondary:

c'(6)s6'(68)

c(7)s7(75)

The corresponding capping of a super-pentagonal-pyramid ofS6'(68) gives rise to the super-pentagonal-bipyramid of s7(75)with an increment in nuclearity of AN = 13 - 6 (sharing 6vertices) = 7. For s7(75), each atom in c(7) is replaced by a13-atom icosahedron, each of which in turn shares a vertex(located at the midpoint of the super-pentagonal-bipyramid)with the neighboring icosahedra. Since there are 15 edges onthe surface of the super-pentagonal-bipyramid and one hid-den edge [in analogy to the primary cluster c(7)], the totalnumber of shared vertices in s7(N) is 16, giving rise to anuclearity N of 7 x 13 - 16 = 75.

APPENDIX 3Another useful concept in the description of primary andsecondary clusters is the concept of hole(s) as exemplifiedbelow by s~l'(113), s12(120), and s13(127).

Sio'(110) sll'(113) sll''(118)-->S12(120) S12'(126)-+S13(127)

The super-pentagonal-antiprism of a slo'(110) (open cir-cles) has a pentagonal-antiprismatic hole, which can accom-modate three more atoms (A& N = 3), thereby forming a newicosahedron in the center (filled circle), resulting in s~l'(113).Likewise, filling the capped-pentagonal-antiprismatic hole ins1l "(118) with two atoms (namely, AN = 2) results in sl2(120).And, finally, filling the icosahedral hole of s12'(126) with oneatom (namely, AN = 1) gives rise to s13(127).

We thank the National Science Foundation (CHE-8722339) forfinancial support.

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