摘要:
What can calculations employing empirical potentials teach us about baretransition-metal clusters?David J. Wales,* Lindsey J. Munro and Jonathan P. Ii. DoyeUniverxitj. Cllmiic-ul Laboratories, LensJield Road, Cambridge CB2 I E W, UKThe implications for transition-metal clusters of theoretical results for systems containing 10-1 48 atoms boundby empirical potentials have been considered. The effects of the range of the interatomic pair potential andanisotropy on the potential-energy surface are now quite well understood. For example, as the range decreasesthe favoured morphology changes from icosahedral to decahedral and then to cuboctahedral. Since strainincreases with size the crossover between electronic and geometrical ‘magic numbers’ exhibited by alkali-metalclusters can be rationalised.Calculations employing specific potentials designed to represent face-centred-cubictransition metals enable the study of changes in morphology and surface migrations in clusters of theseelements. Single-step mechanisms exist for highly co-operative rearrangements between different structures, butthe associated barriers scale as the total number of atoms. Hence, at larger size the same mechanisms aremediated by a series of transition states. The barriers for surface processes are comparable to those deducedexperimentally and theoretically for bulk surfaces. It is predicted that icosahedral order is ‘frozen in’ atrelatively small size and Mackay icosahedra grow uiu anti-Mackay and then Mackay overlayers.In this contribution we consider some of the ways in whichcalculations based upon empirical potentials can provide newinsight into the structure and dynamics of bare transition-metalclusters. Very large uh initio calculations on systems such asbare and ligated Au, clusters (including relativistic effects)and large fullerenes (using Yang’s ‘divide-and-conquer’approach) have recently become feasible. However, to survey amultidimensional potential-energy surface (PES) in detail andcharacterise low-lying minima, transition states and reactionpathways for a cluster containing tens of transition-metal atomswould be extremely computationally expensive.Since thenumber of minima appears to scale exponentially with thenumber of atoms ’ even finding the global potential-energyminimum rapidly becomes a difficult task.We therefore consider two simple potentials which each haveonly one free parameter corresponding to anisotropy andrange, respectively, and two classes of empirical many-bodypotentials fitted to specific face-centred-cubic (f.c.c.) transitionmetals.Our results are most relevant to molecular-beam studiessuch as the recent flow-tube reactor experiments of Parks andco-workers.X ’ In this work the number of active binding sitesof different kinds, and hence the structure of the cluster, isdeduced from the characteristics of ligand uptake. Comparisonsof our results with previous calculations 1 4 . 1 5 and, whereappropriate. with results for surface processes are also possible.The original aspects of this paper are primarily our com-parisons of the trends found for the different potentials andour analysis of nickel clusters in the light of recent flow-tuberesults (third section).Details of the geometry optimisationsand reaction-path calculations are omitted but may be foundelsewhere. ’A Model Anisotropic PotentialThe Lennard-Jones l 7 (LJ) potential has the form (1) where rijv = 4E i < = j [(:)12 - (31t Basis of the presentation given at Dalton Discussion No. 1, 3rd-5thJanuary 1996. University of Southampton, UK.is the separation of atoms i and j , E the pair-potential well depthand (T the separation where the pair interaction goes throughzero. However, once E and (T are chosen as the units of energyand distance there are no adjustable parameters. Furthermore.pairwise additive potentials are well known to be inadequate inproviding a description of transition metals which do notgenerally obey the Cauchy relation, for example.’’ A simpleanisotropic form is obtained ’ by adding the Axilrod-Teller 2o(AT) tripledipole three-body term, equation (2), where 6,, 6,1 + 3 cos 6, cos 0, cos e3-~ -1 (2) .I[ i < j < k ( ri j r i k r j k )and 63 are the internal angles of the triangle Auk and Z is theparameter which specifies the magnitude of the three-bodyterm. If we use a reduced unit system in which E is the unit ofenergy and (T the unit of length then there is just one adjustableparameter left, namely Z* = Z O ~ / E . Previous applications ofthis potential have been made to both main-group and metalclusters and solids.We have previously considered the effect of Z* on thetopography of the PES systematically for small clusters 2 1 andfor some larger metal clusters.22 Negative values of Z*reinforce the LJ term and do not result in a qualitative change ofthe PES.However, for positive Z* the three-body termdestabilises triangles and favours linear arrangements of atoms.For sufficiently large values of Z* the LJAT potential supportsrings and chains.2 A systematic survey for six-atom clusters ”showed that dozens of new stationary points occur as Z* variesfrom zero to 3.0. Fig. 1 shows all the non-planar minima and thetransition states which link them to other minima that werefound in this study.The use we envisage for this result is toprovide a first guess at how newly characterised clusters mightrearrange. This entails finding a range of Z* over which theknown geometry is stable for the LJAT potential and thenconsidering the rearrangements that the model clusterundergoes. In effect this procedure fits the value of Z* so thatthe experimental geometry is known to be supported as aminimum (this is a more stringent condition than is generallyappreciated). Most of the observed mechanisms can beJ . Cheni. SOC., Dalton Trans., 1996, Puges 6II-623 61Fig. Istructures do not all generally exist for a given value of Z*Interconnections between all the non-planar minima and transition states found for a six-atom cluster bound by the LJAT potential.Theseclassified as diamond-square-diamond (DSD) type 2 3 [Fig.2(a)] or edge-bridging 24 [Fig. 2(b)], although some others werealso found [see, for example, Fig. 2 ( ~ ) ] . ~ ’The structural properties of 13-, 55-, 147- and 309-atomMackay icosahedra,25 Ino decahedra 26 and cuboctahedra havealso been analysed for the LJAT potential.22 At each of thesesizes, known as ‘magic numbers’, high-symmetry structuresbelonging to point groups I,, D,, and 0, are completed, asshown in Fig. 3. The Ino decahedron is obtained by truncating apentagonal bipyramid to give five new { 100)-type square faces.Of course, this means that the structure now has more than tenfacets, but it is usual to call such clusters ‘decahedral’ in view ofthe aforementioned construction.For each size the icosahedronlies lowest in energy, followed by the decahedron. However,these structures contain five-fold rotation axes, and thereforecannot pack indefinitely without incurring excessive strain,hence crossovers in stability must occur with increasing size.Our general result is that as Z* increases these crossovers occurat smaller size.22 This is easily rationalised by considering thenumber of (111)- and [100)-type surface facets in eachstructure. If we attempt to fit bulk properties using the singlefree parameter our results are generally in the range0.1 < Z* < 0.35, although accurate fits are certainly notFig. 2 Mechanisms found for six-atom LJAT clusters: (a) diamond-square-diamond process, (b) edge-bridging process and (c) edge-bridged-terminal-edge-bridged processpossible.Transmission electron microscope results for smallmetal particles on supports, summarised in ref. 22, suggest that612 J . Chem. SOC., Dalton Trans., 1996, Pages 61 1 6 2 (a 1 (b 1Fig. 3 561 -Atom clusters: (a) f.c.c. cuboctahedron, (b) Ino decahedron and (c) Mackay icosahedron0.4 -0.2 -0.0 -: -0.2 -xt!J-0.4 - W-0.6 --0.8 --1.0-Table 1 The Morse range parameter po for various fitsMetalNaKRb csMgCaSrBaCrMo wFeRhNiPtc uAgAuA1PbNeArKrXeBulkfit"3.153.173.103.143.683.683.534.334.494.283.953.953.894.273.794.42-.-~-~Diatomicf i t h~~-~~4.164.70--~-~--~~3.133.684.18-~2.055.726.8 16.67Sutton-Chen' Murrell-Mottram'(W (MM)- -6.005.206.335.20 5.506.00 6.006.33 7.00---" Obtained by fitting to the experimental vaporisation energy, latticeconstant and compressibility.28 Obtained by fitting to spectroscopicdata for diatomic molecules.29 From analytical calculations.smaller values of Z* are appropriate for Au and Ag,intermediate values for Pd and Cu, and relatively large valuesfor Pt, where the cuboctahedral morphology appears todominate for particles of diameter 10-100 A. We also findsurface contractions of a reasonable magnitude (compared toresults for bulk surfaces) for the same range of Z* values.22The Morse PotentialWe now consider clusters bound by the Morse potential2,which may be written as in equation (3) where E = 1 and r , = 1define the units of energy and length respectively, po = pro andrlij denotes the distance between atoms i a n d j in these reduced0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0rjj I r,Fig.4 Plots of the Morse potential for a range of po valuesunits. Morse clusters will be denoted M,. Compared to thesimple LJ potential we again have one free parameter, po, whichdetermines the range of the attractive part of the potential. Asshown in Fig. 4, decreasing po increases the range of theattractive part of the potential and softens the repulsive wall,widening the potential well. For comparison, p, = 6 matchesthe curvature of the LJ potential at the bottom of the well, andat this value the Morse and LJ PES's are very similar.Someother best-fit values are given in Table 1 from various sources;note that the relative values in each column are in reasonableagreement.In previous work Braier et ~ 1 . ~ ' made a systematic search ofthe PES for M, and M, as a function of the range. They foundthat the PES became more complex, i.e. supported more minimaand transition states, as the range of the poteMial decreased.This is easily understood since the short range of the potentialmeans that distant atoms are far less sensitive to one-another'sprecise positions, and has been noted by other ~ o r k e r s . ~ , ~ ~ . ~ The simplification of the PES at long range led Stillinger andStillinger33 to suggest that this approach might be used forglobal minimisation by hypersurface deformation, 34 but thisnow seems unlikely in the light of more systematic surveys ofhow the PES changes with p,.Previous studies have alsoconsidered the effect of the range of the potential. Stave andDePristo l4 analysed their results for nickel and palladiumclusters in terms of the range and Bytheway and Kepert35showed that icosahedral packing becomes less favourable thanstructures with higher co-ordination numbers for sufficientlylong-ranged potentials.Girifalco's model intermolecular potential for C,, isisotropic and exceptionally short-ranged relative to theJ. Chem. SOC., Dalton Trans., 1996, Pages 611423 61equilibrium pair ~eparation;~, the Morse potential whichmatches the curvature at the bottom of the well3, has po =13.62.Not surprisingly, (C60)n clusters exhibit rather differentproperties from those of LJ clusters 16*38.3y and the bulk liquidphase is predicted to have only marginal stability4’ or none atall.41 Here we consider M, clusters with 7 < N < 25 and a fewadditional sizes in the range 35 < N d 80. We search for theglobal minimum as a function of N and pol and drawcomparisons with previous work on bare metal clusters. Ofcourse, since this is still a pair potential, we cannot expect toachieve quantitative agreement for any particular metal.We partition the Morse potential into three contributions,equation (4), where the number of nearest-neighbour contacts(4)is given by equation ( 5 ) with xij = r‘ij - 1 and x0 is a nearest-neighbour cut-off.The strain energy, Estrain, is given byequation (6) and the contribution to the energy from non-nearest-neighbours, En,,, by equation (7). The most importantterms are nnn and Eslralnr and the formation of stable structuresinvolves competition between the two since larger strainsgenerally admit higher co-ordination numbers.42 For givennon-zero values of xLJ, which may be identified as the strains,E,trJln increases rapidly with po, i.e. as the range decreases,because the potential well becomes narrower (Fig. 4). The effectof the range upon the relative stabilities of the icosahedron,decahedron and cuboctahedron is easily explained. In each casethe interior atoms are twelve-co-ordinate, and so differences innnn arise from differences in the co-ordination of surface atoms.The icosahedron has only 1 1 1 )-type faces and the cubocta-hedron has the most 100) type, and therefore the smallest nnn.At intermediate range the icosahedron therefore lies lowest ofthe three, and as p, increases first the decahedron and then thecuboctahedron become more favourable.The smallest cluster for which we have found the globalminimum to change as a function of range42 is M,.We havecollected together all the structures that we believe to be globalminima for some range of p, in Figs. 5-8 (except for the verysmallest, which are well known). A detailed discussion of thesestructures is provided e1sewhe1-e;~~ here we will concentrate oncomparing them with previous results.Some of the structures we find at low po are very similar tothose reported by Stave and DePristo l 4 in their studies ofnickel and palladium clusters. In particular, their lowest-energyminima between 7 and 16 atoms correspond to our 7A-9A,IOB, 1 lC, 12B, 13A-16A (Figs.5 and 6). The ranges of po forwhich these structures are global minima42 are generally inreasonable agreement with the value of po 2 3.95 for Ni inTable 1 , suggesting that geometrical considerations, as opposedto electronic, play a significant role in determining the moststable structures even in this size range. Structures 8A-10A(Fig. 5 ) have relatively small values of nnn but have favourablecontributions from En,, because the next-nearest-neighbourshell is significantly closer than for structures based onicosahedra.These structures are roughly spherical and have thesame shape as the boron skeletons of B,H,’-, B,H,’- andB,,H,,2 . Most of these minima are not based upon anyidentifiable regular structure, making prediction of low-energystructures at low po rather hard.For N > 13 growth can occur by capping the 13-atomicosahedron in two distinct ways, as shown in Fig. 9. The anti-Mackay overlayer 44 leads to the 45-atom rhombic triconta-hedron 45 with I,, point-group symmetry,,, and the Mackayoverlayer leads to the 55-atom Mackay i c ~ s a h e d r o n . ~ ~ TheMackay sites continue the f.c.c. close packing in each of the 20distorted tetrahedra from which the icosahedron is constructed.Other authors have referred to the anti-Mackay overlayer aspolyi~osahedral,~~ because the growth sequence includesstructures with interpenetrating icosahedra such as the double(19A) and triple (23A) icosahedra (Fig.6). It has also beencalled the face-capping ~ v e r l a y e r . ~ ~ In LJ clusters the anti-Mackay overlayer is initially adopted in the series of globalminima, and then for N 3 31 the Mackay overlayer lies lowerin energy.44 The anti-Mackay overlayer results in more nearestneighbours but greater Estrdin and hence the crossover to theMackay overlayer occurs at smaller size as the range of thepotential is decreased. For example, M24 and M,, are morestable with a Mackay overlayer for po = 10. In contrast, alkalimetals are expected to correspond to relatively long-rangedpotentials (Table I ) , and this is borne out in uth initiocalculations for lithium clusters 48 where the anti-Mackayoverlayer is lowest in er,ergy up to N = 45.To predict likely ‘magic numbers’ as a function of p, we haveconsidered4, the second difference of the energy, A2E =E(N + 1) + E(N - 1 ) - 2E(N).For po = 6 the pattern is thesame as for the LJ potential, with magic numbers at N = 7, 10,13, 19 and 23. The last three values correspond to the single,double and triple icosahedron, respectively. For smaller po therelative stability of M, and M disappears, and at still larger pothe magic number character of the icosahedron is also lost. Forpo = 14 our results should be appropriate to C,, clusters andwe predict a magic number at n = 23 corresponding to thecomplete decahedron.We have also considered M,,, M,,, M,,, M,, and M,,,although for these sizes we cannot be confident that we havefound the global minimum in each case.42 For M,, at low poour lowest minimum is distorted and highly strained.As poincreases structures based on icosahedra with anti-Mackayoverlayers become most favourable; at shorter range, clusterswith Mackay overlayers lie lower, but are never the lowestbecause the truncated octahedron shown in Fig. 10(n) is theglobal minimum over a wide range of po. It is noteworthy thatthe crossover from decahedral to f.c.c.-based global minima,based on extrapolation for magic number LJ clusters,49 occursat around N z lo5.Our lowest minimum for M,, at long rangeis based on the 45-atom rhombic tricontahedron, but has anextra atom in the outer shell [Fig. 10(h)]. As p, increasesour lowest minimum changed from an incomplete Mackayicosahedron to an incomplete decahedron and finally to astructure based on the 3 1 -atom truncated tetrahedr~n.,~As discussed above, the 55-atom cluster is a magic numberspecies for various atomic y * 5 0 - 5 3 and molecular systems. 1*54The relative energies of Mackay icosahedra, Ino decahedra 26and cuboctahedra have previously been compared for variouspotentials. 22.5 5.56 However, this may not provide the mostuseful comparison, because the Ino decahedron and the cub-octahedron are not necessarily the lowest-energy decahedraland f.c.c.-type structures.In both cases it is possible toconstruct less-symmetrical clusters of the same size with asmaller proportion of [ 100; -type facets. Marks’ decahedra,with re-entrant [ 1 1 1 ; faces between the edges of the (100)faces,57 and truncated octahedra or species with furtherf a ~ e t t i n g , ~ ~ appear to be more favourable morphologies. ForM,, the Ino decahedron and the cuboctahedron are never thelowest-energy decahedral or f.c.c. 55-atom clusters.42 The mostfavourable f.c.c. clusters are instead based on the 31-atomtruncated tetrahedron. As expected the lowest-energy minima614 J. Clzem. Soc., Dalton Trans., 1996, Puges 61142Fig. 5MATHEMATICA 43 using a cut-off of 1.2 for triangulationGlobal minima associated with low values of po.Each structure is labelled by the symbol given in Table 1. The graphics were produced withchange from being based on icosahedra to decahedra and thento f.c.c.-type structures as the range decreases.Similar patterns are seen for M70, M,, and M79. The mostfavourable structures based upon decahedra may be constructedfrom the 75-atom Marks’ decahedron57 shown in Fig. lO(c).The crossover from icosahedra to decahedra based on the magicnumber sequence for LJ clusters49 is at N z 1600. Hence it isinteresting that the 75-atom Marks’ decahedron seems to be theglobal minimum for LJ,,. The f.c.c. clusters are based on the79-atom truncated octahedron [Fig. 10(d)] which is close to thecorresponding Wulff polyhedron. * We expect ‘magic numbers’for potentials of appropriate range for the sequence of Marks’decahedra with N = 75,192,389, .. ., the sequence of truncatedoctahedra with hexagonal faces and N = 38,201,586, . . ., andperhaps for truncated octahedra with irregular hexagonal facesand N = 79, 140, 314.have made detailed studies of the structureof small nickel clusters using chemical probes. Nitrogen is aparticularly useful probe, as it can be used to ascertain thenumber of surface atoms with a particular range of co-ordination number. From these data they have deduced theParks et ( I / .structure of bare nickel clusters with 3 to 28 and 49 to 7 1 atoms.Although the interpretation of the experimental results is notwithout ambiguity, they provide convincing evidence thatthe dominant morphology in these size ranges is icosahedral.However, there is some disparity between their results and thecalculations of Stave and DePristo. l4 The experimental resultsare consistent with a shorter-ranged potential than thatemployed by the latter authors.Parks et al. find that growth ontheir 13- and 55-atom icosahedra starts in the anti-Mackaysites, but for Ni,, the surface has changed to a Mackayoverlayer. This is a surprisingly small size for the Mackayoverlayer to appear; for the LJ potential (p,, = 6) the crossoverin stability occurs at LJ,Our results suggest alternative structures for Nil, and Nil,that were not considered by Parks et al. First, for Nil, theirproposal of a hollow, bicapped hexagonal antiprism l 1 seemsunlikely, as this structure is never a true minimum for the Morsepotential at any range.A capped icosahedron with 12 surfacesites that bind one N, molecule relatively strongly and one site(the cap) that binds two seems more likely to us. For Ni, ,, Parkset al. dismiss one structure based upon icosahedral order (ourJ . Chem. SOC., Dalton Trans., 1996, Pages611423 61Fig. 6 Global minima based on icosahedral packing17C) but did not consider isomers such as our 17B and 17D apparently been seen in sodium clusters as the size increases.59(Fig. 6) and instead proposed a non-icosahedral structure. Temperature-dependent spectra indicate that electronic magicSince our structural predictions here are based entirely on numbers are seen for liquid-like clusters, and geometric magicempirical potentials, with no explicit consideration of electronic numbers for solid-like clusters.60 A similar temperaturestructure or Jahn-Teller distortions, we must also ask when dependence has also been observed for aluminium clusters.61geometric rather than electronic effects may dominate. A This behaviour matches that expected for clusters bound by acrossover from electronic to geometric 'magic numbers' has long-ranged Morse potential. For small clusters the lowest-616 J.Chem. Soc., Dalton Trans., 1996, Pages 61 1-62Fig. 7 Global minima based on decahedral packinFig. 8 Global minima based on close packingAnti-Mackay MackayThird ShellFig. 9 Atomic positions for the two possible overlayers of theicosahedron, anti-Mackay (left) and Mackay (right).These are shownfor a single face of the icosahedronFig. 10 (a) The M3* truncated octahedron. (6) The M,, minimumbased on the 45-atom rhombic tricontahedron. ( c ) The M,, Marks'decahedron. ( d ) The M,, truncated octahedronenergy minima are amorphous and only a very low temperatureis required for melting as the liquid-like band of minimais approximately continuous in energy. In this size rangethe clusters would exhibit electronic shell structure at alltemperatures. However, as the size increases the liquid-like stateshould become less stable, until a critical size is reached at whichthe icosahedron becomes the ground state. Above this size thecluster would exhibit geometric shell structure as long as thetemperature is below the melting point.For metals withshorter-ranged potentials, geometric rather than electronicmagic numbers are likely to dominate at relatively small size.On the basis of Table 1 we tentatively suggest that this may bethe case for nickel clusters.Sutton-Chen and Murrell-Mottram PotentialsWe now consider the rearrangement mechanisms of cappedmagic number clusters bound by the many-body Sutton-Chen 6 2 and the two- plus three-body Murrell-Mottrampotential^,^^ parameterised for f.c.c. transition metals.64 Theresults provide insight into the most favourable morphology ofeach cluster and also into the competition between dynamicsand thermodynamics. In fact, fluctuations between cubocta-hedral and icosahedral morphologies have been reported byelectron m i c r ~ s c o p i s t s .~ ~ - ~ ~ Sawada and Sugano 68 investigatedthe corresponding energetics of Au,, and Au,,, using a many-body Gupta potential.69 They concluded that some additionaleffect is needed to explain the observation of cuboctahedra, andthat this is most likely to be the interaction with the substrate.Although we have found that capping can have a similar effectwe concur with this conclusion.The Sutton-Chen (SC) potential has the form (8) where pi =1 (a/rij)", c is a dimensionless parameter, E a parameter withdimensions of energy, a the lattice constant and rn and n arepositive integers with n > rn. We use the parameters given bySutton and Chen6' for the metals Ni, Ag and Au; Cu has thesame scaled parameters as Ni, Rh the same as Ag and Pt thesame as Au, so the corresponding results for these metals cansimply be obtained from their partners by rescaling, asmentioned above. For Ni and Cu n = 9 and n? = 6, for Ag andRh n = 12 and m = 6 and for Au and Pt n = 10 and rn = 8.The Sutton-Chen potential provides a reasonable descriptionof various bulk proper tie^,^^." with an approximate many-body representation of the delocalised metallic bonding.However, it does not include any directional terms, which arelikely to be important for transition metals with partiallyoccupied d shells.The two- plus three-body Murrell-Mottram (MM) poten-tial 6 3 , 7 1 may be written as in equation (9).In this case up toj # iaround 15 parameters are fitted to bulk properties such asphonon frequencies, elastic constants, bulk cohesive energy,lattice constant and vacancy-formation energy.63.71-73,** The c1 parameter for Cu in ref.7 2 has the wrong sign.',618 J . Chem. SOC., Dalton Trans., 1996, Pages 61142One obvious question which must be asked is how wellpotentials like these, which have been fitted to bulk properties,are likely to perform for clusters where most of the atoms findthemselves in the surface.75 Some clues are available fromsurface calculations. The SC potentials considerably underesti-mate surface energies, and give surface relaxations which aretoo large but of the right form.76 The change in mechanism ofthe surface migration from bulk Pt (which occurs by surfaceexchange) to Rh (which occurs by simple migration) is correctlyreproduced.’ O However, Hammonds 77 has concluded that theSC potentials do not support surface reconstructions of thetop-layer contraction type, because the surfaces are too stablewith respect to the bulk. On the other hand, SC potentials havebeen shown to reproduce step-roughening phenomena. 78 TheFig. I I ( ( I ) The 6DSD process for SC Ag,,. (6) The 5DSD process forSC Ag,,. The clusters were triangulated using a distance cut-offcriterion and the transition vector (i.e. the displacements Correspondingto the normal mode with the unique imaginary frequency) issuperimposed on the transition stateinclusion of directional terms in the MM potentials improvesresults for the bulk over those for SC and gives better surfaceenergies, but surface relaxations are underestimated.7 2For the present results direct comparison with experimentalmechanisms is not usually possible. However, we can compareour calculated rate constants with the lifetimes observed forcuboctahedral and icosahedral gold clusters using electronmicroscopy. We can also test some of our results againstcalculations using the embedded-atom model 79 and effectivemedium theory.80 Where we identify rearrangements whichcorrespond to surface diffusion, it is often possible to makecomparisons with experimental or theoretical results for anal-ogous processes on bulk surfaces. We have found clusteranalogues of terrace diffusion, exchange processes anddiffusion over edges which are important in understandinghomoepitaxial growth, island formation and diffusion-limitedaggregation.1-84For the ‘magic number’ 13-, 55- and 147-atom clusters wehave identified a number of highly co-operative processes.64For 13 atoms the cuboctahedron and decahedron are bothtrue transition states for degenerate rearrangements of theicosahedron for all potentials, with the decahedron lower inenergy. (A degenerate rearrangement is one in which the twominima differ only by permutations of atoms of the sameelement.85)The MM decahedra are higher index saddles for the largersizes, whilst the corresponding SC decahedra are minima. ForSC Ag,, and Au,, the icosahedron is the global minimum ofenergy according to previous geometry optimisations 22 andsystematic quenching from molecular dynamics trajectories.86Mackay probably first described the direct geometricaltransformation of a cuboctahedron into an ico~ahedron.~~ Inthis process one of the diagonals of each square face iscontracted and the faces are folded along the same diagonal togive two equilateral triangles.The rearrangement of eachsquare face is related to Lipscomb’s diamond-square-diamond(DSD) mechanism2, [Fig. 2(a)], which was first proposed inthe context of borohydride chemistry. Mackay’s transformationFig. 12 The 6DSD process ( a ) in SC Au,, via a T,, transition state and ( h ) in SC Au,,, via a C, transition stateJ. Chem. Soc., Dalton Trans., 1996, Puges 61 I 623 61Fig. 13 Degenerate rearrangements of SC,, clusters capped on a square face.(a) Surface dilfusion via a vertex-exchange process for Ag. (b) The6DSD process in which the capping atom is essentially a spectator for Ni. (c) Mechanism in which a capped gold cluster finishes at a minimum wheretwo atoms essentially share a vertex sitefrom one icosahedron to another via a cuboctahedron cantherefore be described in terms of a concerted sextuplediamond-square-diamond (6DSD) mechanism [Fig. 1 I (a)].There is another highly co-operative mechanism which canpermute icosahedra in a different way. In this case the tran-sition state is a decahedron (D5h symmetry) and the concertedprocess corresponds to a quintuple DSD process (SDSD)[Fig. 1 l(b)].For the potentials and the sizes considered here theicosahedron is always lower in energy than the cuboctahedron.This is consistent with the calculations for Ni by Cleveland andLandman ” using an embedded-atom potential and with theelectron microscopy results of Doraiswamy and Marks ’’ whostudied small gold particles on an SiO support.Our resultsare also consistent with those of Vlachos et al.” who alsoconsidered the relative energetics of cuboctahedra andicosahedra. We find that MM,, and SC Ni,, cuboctahedra aretransition states, whilst SC Ag,, and Au,, cuboctahedra aretrue minima which can rapidly transform to icosahedra via Thsymmetry transition states.64 The stability of the icosahedralmorphology in this size range is also in general agreement withthe chemical probe results of Parks and co-workers mentionedabove.*-’ 3,90 However, for SC Au,, quenching has previouslyshown that the lowest-energy minima generally have littlesymmetry and lie below both the icosahedron and thecuboctahedron.86 This is due to the short range of thisFor the larger SC clusters, the single-step co-operativemechanisms are replaced by two-step processes except for Ni,,.In such cases the decahedron or cuboctahedron becomes aminimum, and there must be at least one transition state oflower symmetry in between.For the cuboctahedron this changein morphology can occur via a single Th transition state [Fig.12(a)], as found previously3’ for (c60)55, or via a C, transitionstate [Fig. 12(b)].The barrier to the 6DSD process scales roughly as thenumber of atoms for each MM potential.However, withincreasing size we expect the cuboctahedron to become aminimum and the overall Oh to I,, transformation to occur via asequence of transition states, as for the larger SC clusters. Forboth classes of potential, our calculated rate constants indicatethat cuboctahedra should not be observable at experimentally620 J. Chem. SOC., Dalton Trans., 1996, Pages 61 162Fig. 14 Rearrangements of SC,,, clusters showing surface migration of the cap via an edge-bridging transition state in (a) Ag,,, and ( h ) Ni148.In fact these are really different views of the same mechanismrelevant temperatures for Au,, and Au147. These results do notagree with the experimental observations; Sawada and Suganoreached the same conclusion and were therefore led to suggestthat interaction with the substrate might preferentially stabilisethese cu boc tahedra.For capped magic number clusters the mechanisms are notalways clear-cut, but may broadly be divided into rearrange-ments that are surface migrations of capping atoms and co-operative rearrangements of the underlying cluster where thecapping atom is essentially a spectator. For MM clusters inparticular, the latter processes d~rninate,,~ except for Cu.Forthe large SC clusters, on the other hand, we mostly foundsurface migrations after capping. Some examples are shown inFig. 13. In some cases capping turns a structure which was atransition state into a minimum, and the reaction path mustsplit into two, each part mediated by a new transition state.Capping always reduces the barrier to the 6DSD process inMM,, clusters and in SC Ni,,; the associated frequency factorsdo not change much from the magic number rearrangements.For the 148-atom clusters we found only surface migrationsfor SC AU,,,, only 6DSD processes for MM Ag,,, and Au,,,and both surface and co-operative processes for the others.Twoexamples are shown in Fig. 14; in each case an edge-bridgingprocess 24 results in migration over the same { 100)-type face ofa cuboctahedron. We have also found mechanisms where atomsmigrate between faces of the cluster, either by edge-bridging(Fig. 15) or exchange-type processes.We have compared our results with experiment and previoustheoretical calculations as far as possible, exploiting theanalogous migration processes for bulk surfaces wherepossible.64 It seems unlikely that a single capping atom couldstabilise the Au,, and Au,,, cuboctahedra sufficiently toaccount for the experimental observations discussed above.Sawada and Sugano’s suggestion that the substrate interactionis responsible still seems the most plausible.We note that surface-diffusion processes generally havelower barriers and frequency factors than do the co-operativerearrangements. Hence we expect most of the surfacemigrations to have significant rate constants at roomtemperature.In contrast, changes in morphology from theicosahedron via concerted rearrangements are very unlikely tobe seen at room temperature.This suggests that icosahedralorder will be ‘frozen in’, and hence kinetic rather thanthermodynamic products may be observed in molecular beamexperiments for clusters of more than around lo3 atoms. Ingeneral our results agree quite well with those for similarprocesses in bulk systems as detailed elsewhere.64ConclusionIn this contribution we have considered the structure andrearrangement mechanisms predicted for clusters bound byfour classes of empirical interatomic potential. The modelanisotropic LJAT potential can support structures rangingfrom close packed to rings and chains as a function of a singleparameter. Matching an experimental structure to a givenminimum for this potential can then provide a first guess of thesorts of rearrangements the cluster may undergo.We alsopredict that cuboctahedra will become more energeticallyfavourable as the magnitude of the anisotropy increases.The Morse potential has a single adjustable parameter whichgoverns the range of the atomic interaction. Very long-rangepotentials, which appear to be appropriate for alkali metals,result in amorphous global minima. As the range decreases firsticosahedral, then decahedral and finally f.c.c. morphologies arethe most favourable. These crossovers, as well as the relativeimportance of geometric versus electronic effects in determiningstructure, can all be understood in terms of the strain energy. Ashort-ranged potential destabilises strained geometries such asthe icosahedron.Comparison with experiment suggests that arelatively short-ranged potential is appropriate for Ni.Two classes of potential specifically parameterised for f.c.c.J . Chem. Soc., Dalton Trans., 1996, Pages 611423 62Fig. 15 Surface rearrangement of an SC Ag,,, cluster in which the cap moves between facets via an edge-bridging transition statetransition metals have also been considered. In both casesicosahedra generally have lower energy than cuboctahedra andhighly co-operative rearrangements exist, especially for smallerclusters. As the nuclearity increases the single-step co-operativepathways break down into multiple steps, each one mediated bya new transition state. Both potentials predict that AuS5 andAu,,, cuboctahedra should not be observable experimentally,in contradiction to results from electron micro~copy.~~*~’ Thediscrepancy is probably due to a substrate interaction, assuggested by Sawada and Sugano.68Both classes of potential suggest that co-operative changes inmorphology will not occur at an appreciable rate for the largersizes considered here under a wide range of experimentalconditions.Hence we predict that icosahedral order is likely tobe frozen in to bigger clusters which grow from such nuclei.Surface migrations, on the other hand, are likely to haveappreciable rates at reasonable temperatures. 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ISSN:1477-9226
DOI:10.1039/DT9960000611
出版商:RSC
年代:1996
数据来源: RSC