atom ordering in cuboctahedral ni–al nanoalloys

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Inorganica Chimica Acta 359 (2006) 3649–3658

Atom ordering in cuboctahedral Ni–Al nanoalloys

Nicholas T. Wilson, Mark S. Bailey 1, Roy L. Johnston *

School of Chemistry, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK

Received 2 November 2005; accepted 23 February 2006Available online 3 March 2006

This article is dedicated to Mike Mingos with many thanks for his help and inspiration.

Abstract

Energy calculations have been carried out on high-symmetry cuboctahedral Ni–Al nanoalloy clusters, of varying composition, withthe interatomic interactions modelled by the Gupta many-body potential. Relaxations of cuboctahedral fragments cut from the bulklattice of Ni3Al, with 13–561 atoms, were undertaken, as were relaxations of high symmetry clusters with 55 and 147 atoms. The low-est energy isomers were found to be dominated by three factors: the tendency toward mixing due to the favourable energy of mixing,DmixE; the size difference between nickel and aluminium; and the higher cohesive and surface energy of nickel compared to aluminium.The latter two factors favour Al-segregation to the surface. The most stable Ni:Al composition approaches 3:1 for larger clusters.� 2006 Elsevier B.V. All rights reserved.

Keywords: Atom ordering; Clusters; Geometric shells; Gupta potential; Nanoalloys; Nanoparticles; Nickel–aluminium alloys; Segregation; Simulation

1. Introduction

Clusters are aggregates of between a few and many mil-lions of atoms or molecules. They may consist of identicalatoms, or molecules, or two or more different species. Clus-ters are formed by most of the elements in the periodic tableand can be studied in a number of media such as molecularbeams, the vapour phase, in colloidal suspensions and iso-lated in inert matrices or on surfaces [1]. Interest in clustersarises, in part, because they constitute a new type of materialwhich may have properties which are distinct from those ofdiscrete molecules or bulk matter and also because theirproperties often vary significantly as a function of size [1,2].

1.1. Nanoalloy clusters

The range of properties of metallic systems can begreatly extended by taking mixtures of elements to generate

0020-1693/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.ica.2006.02.029

* Corresponding author. Tel.: +44 121 414 7477; fax: +44 121 414 4403.E-mail address: [email protected] (R.L. Johnston).

1 Present address: Materials Science Division, Argonne National Lab-oratory, 9700 S. Cass Ave., Argonne, IL 60439, USA.

intermetallic compounds and alloys [3]. In many cases,there is an enhancement in specific properties upon alloy-ing, due to synergistic effects. The rich diversity of compo-sitions, structures and properties of metallic alloys, has ledto widespread applications in electronics, engineering andcatalysis. The desire to fabricate materials with welldefined, controllable, properties and structures, on thenanometre scale, coupled with the flexibility afforded byintermetallic materials, has generated interest in bimetallicalloy clusters, or ‘‘nanoalloys’’. Such clusters have beenstudied in colloidal solutions, in the solid state, on solidsupports and in molecular beams [4–8].

One of the major reasons for interest in nanoalloy par-ticles is the fact that their chemical and physical propertiesmay be tuned by varying the composition and atomicordering, as well as the size of the clusters. Their surfacestructures, compositions and segregation properties [9]are of interest as they are important in determining chem-ical reactivity, and especially catalytic activity [10,11].Nanoalloy clusters are also of interest as they may displaystructures and properties which are distinct from those ofthe pure elemental clusters. Finally, there are examples ofpairs of elements (such as Fe and Ag) which are immiscible

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Table 1Number of atoms in the kth shell, S(k), and total number of atoms, N(k),in a k-shell cuboctahedral cluster (for k = 1–5-shells)

k 1 2 3 4 5

S(k) 12 42 92 162 252N(k) 13 55 147 309 561

3650 N.T. Wilson et al. / Inorganica Chimica Acta 359 (2006) 3649–3658

in the bulk phase but which readily mix in finite clusters[12].

1.2. Ni–Al nanoalloy clusters

Ni–Al clusters have been the subject of several previoustheoretical studies [8,13–19] and have become useful mod-els for investigating those factors which are responsible fordetermining the atomic ordering or segregation in nanoal-loy clusters. Ni–Al clusters are also of importance in heter-ogeneous catalysis: for example synergistic effects havebeen observed in the reductive dehalogenation of aliphaticand aromatic halides and polychlorinated arenes by sub-nanometre sized Ni–Al particles [20].

1.3. Homotops

On going from pure metal clusters to bimetallic nanoal-loys, there is an increase in complexity, due to the presenceof two different types of atoms, which leads to the possibil-ity of isomers based on the permutation of unlike atoms, aswell as the regular geometrical isomers (with different skel-etal structures). Jellinek has introduced the term ‘‘homo-tops’’ to describe AaBb alloy cluster isomers, with a fixednumber of atoms (a + b) and composition (a/b ratio),which have the same geometrical arrangement of atoms,but differ in the way in which the A and B-type atomsare arranged [8,13–16,21]. As the number of homotopsrises combinatorially with cluster size, global optimization(in terms of both geometrical isomers and homotops) is anextremely difficult task.

Fig. 1. Subshells (sets of symmetry-equivalent atoms indicated by thesame colouring) in the outer shells of (a) 2-shell (55 atom) and (b) 3-shell(147 atom) cuboctahedral clusters.

1.4. Geometric shells and subshells

In this study, we have focused on high symmetry cuboc-tahedral ‘‘geometric shell’’ clusters which are composed ofcentred concentric polyhedral shells of atoms [21]. Cuboc-tahedral structures are commonly observed for elementalmetal clusters [1]. The cuboctahedron has an fcc arrange-ment of atoms, with cubic close packing (ccp) – as in thestoichiometric ordered Ni3Al phase (Cu3Au-type L12 struc-ture). Other ordered Ni–Al phases include: NiAl (bcc B2structure, with the Ni and Al atoms adopting a CsCl-typearrangement); and NiAl3 (orthorhombic, DO11 structure)[3]. In this study, however, we will consider cuboctahedralfcc-structures with Ni:Al compositions close to 3:1, 1:1 and1:3.

The number of atoms in the kth shell, S(k), and the totalnumber of atoms (including the central atom), N(k), of a k-shell cuboctahedral shell cluster are given by

SðkÞ ¼ 10k2 þ 2 ð1aÞ

NðkÞ ¼ 1

3ð10k3 þ 15k2 þ 11k þ 3Þ ð1bÞ

S(k) and N(k) values for 1–5-shell cuboctahedral clustersare listed in Table 1.

To calculate the energies of all possible homotops fornanoalloy clusters with several hundred atoms would becomputationally infeasible, so in this study we have consid-ered only those homotops where all the atoms forming asymmetry-equivalent set (i.e., a geometric ‘‘subshell’’ [21–23]) are constrained to be of the same element. This hasthe advantages of reducing the number of isomers to amanageable level and maintaining the symmetry of the

Table 2Subshell labels (subshell type: v, vertex; e, edge; f, face) and number ofatoms in each subshell (in parentheses) for the kth shell of 1–3-shellcuboctahedral clusters

k

1 1v (12)2 2f (6) 2e (24) 2v (12)3 3f24 (24) 3f8 (8) 3e (48) 3v (12)

Within each shell, the subshells are listed in order of increasing distancefrom the centre of the cluster.

Table 3Parameters defining the Gupta potential for Ni–Al clusters [24]

Parameter Ni–Ni Ni–Al Al–Al

A (eV) 0.0376 0.0563 0.1221f (eV) 1.070 1.2349 1.316P 16.999 14.997 8.612Q 1.189 1.2823 2.516r0 2.4911 2.5222 2.8637

N.T. Wilson et al. / Inorganica Chimica Acta 359 (2006) 3649–3658 3651

cluster, leading to greater efficiency in the calculation of thepotential energy. Fig. 1 shows the subshells of the outer-most shells of 2-shell (55 atom) and 3-shell (147 atom) cub-octahedral clusters, with the different subshells indicated bydifferent colours.

1-, 2- and 3-shell cuboctahedral clusters consist of 1, 4and 8 subshells, respectively. The number of atoms in eachsubshell for cuboctahedral clusters with 1–3 shells are listedin Table 2. Within each shell, the subshells are listed inorder of increasing distance from the centre of the cluster.The different types of subshells are labelled according towhich topological sites the atoms occupy in the clusterpolyhedron: v, vertex; e, edge; f, face.

A convenient way of defining the high-symmetry homo-tops (with all atoms in each subshell being of the sametype) is as a string of atomic labels (in the present case‘‘Ni’’ and ‘‘Al’’) indicating the atom type of each subshell– where the subshells are ordered as in Table 2. For a3-shell cuboctahedral cluster, the ordering is therefore

0c : 1v : 2f ; 2e; 2v : 3f24; 3f8; 3e; 3v

where the number is the index of the shell (using 0 to rep-resent the centre of the cluster), c denotes the central atomand v, e and f denote vertex, edge and face-localized atoms,respectively, as in Table 2. The terms ‘‘3f24’’ and ‘‘3f8’’ rep-resent two different sets of symmetry-equivalent face-local-ized atoms, comprising 24 and 8 atoms, respectively (seeTable 2). The string Al:Ni:Al, Ni, Al, for example, identi-fies a 2-shell, 55-atom icosahedron with a central Al atom,12 Ni atoms in the first shell and on the edges of the secondshell and Al atoms on the vertices and face sites of the sec-ond shell.

2. Computational details

2.1. The Gupta potential

Since, for large clusters (of hundreds or thousands ofatoms) ab initio calculations covering large regions of con-formation space are still, at present, infeasible, there hasbeen much interest in developing empirical atomisticpotentials for the simulation of such species. In this work,we have adopted the empirical Gupta many-body potential[24]. Empirical potentials, such as the Gupta potential, arederived by fitting experimental data to an assumed func-tional form.

The Gupta potential is defined in terms of repulsive (Vr)pair and attractive many-body (Vm) terms, which areobtained by summing over all (N) atoms of the cluster

V clus ¼XN

i

ðV ri � V m

i Þ ð2aÞ

V ri ¼

XN

j 6¼i

Aða; bÞ exp �pða; bÞ rij � r0ða; bÞr0ða; bÞ

� �� ð2bÞ

V mi ¼

XN

j6¼i

f2ða; bÞ exp �2qða; bÞ rij � r0ða; bÞr0ða; bÞ

� �� " #12

ð2cÞ

In Eqs. (2b) and (2c), rij is the distance between atoms i

and j in the cluster and the parameters A, r0, f, p and q arefitted to experimental values of the cohesive energy, latticeparameters and independent elastic constants for the refer-ence crystal structure at 0 K. For NixAly nanoalloy clus-ters, the parameters take different values for each of thedifferent types (Ni–Ni, Ni–Al and Al–Al) of interaction.Therefore, in Eqs. (2b) and (2c), a and b are the atom labelsfor atoms i and j, respectively. The homonuclear parame-ters were derived by fitting to the pure metals and are takento be unchanged in the alloys. The heteronuclear (Ni–Al)parameters were obtained by fitting to crystalline Ni3Al.The Gupta parameters used in this study are listed in Table3 [24]. Using this potential, Cleri and Rosato have correctlypredicted the structural changes occurring in Ni3Al uponintroduction of chemical disorder [24]. Our own calcula-tions have verified that this potential does indeed predictthe stability of the Ni3Al phase [25].

From the total cluster potential energy, Vclus, the aver-age binding energy for an N-atom cluster is defined asthe positive quantity

Eb ¼�V clus

Nð3Þ

2.2. Radial relaxation of cuboctahedral Ni–Al nanoalloys

As Ni–Ni and Al–Al bond lengths are different (themetallic radii are: r(Ni) = 1.25 A and r(Al) = 1.43 A [26]),starting with a pure Ni cluster and introducing Al atoms(or vice versa) will lead to strained structures, which mustbe relaxed (energy minimized). Radial relaxation of thecuboctahedral shell clusters is accomplished by minimizingthe energy as a function of the radii of all of the subshells[21], using a numerical minimization algorithm. For atomswhich do not lie on rotation axes of symmetry, tangential


relaxation (i.e., motion perpendicular to the radial vector)is also possible, though previous calculations have shownthat such relaxations are small and lead to very smallchanges in cluster binding energy [23].

In this study, we have performed radial relaxations oncuboctahedral shell clusters of the following types:

(i) 1–3-shell cuboctahedral composed of subshells whichare filled entirely with either Ni or Al atoms. A subsetof these structures have shells which are filled com-pletely with Ni or Al atoms.

(ii) 1–5-shell cuboctahedral fragments of the bulk Ni3Al(fcc) phase. For these ‘‘Ni3Al’’ clusters, there aretwo possible types of cuboctahedron, one in whichthe central atom is Al and one in which it is Ni. Asthese structures are finite, and the bulk ratio onlyapplies in the limit of infinite nuclearity, the exactcomposition of these structures is not a multiple ofthe bulk 3:1 Ni:Al ratio, though it will tend to thisvalue as the nuclearity increases.

2.3. Mixing energy

The change in cluster binding energy upon mixing,DmixE, is defined as [21]

DmixE ¼ EABb � F A � EA

b � F B � EBb ð4Þ

where EABb is the binding energy (per atom) of the N-atom

AB nanoalloy cluster, EAb and EB

b are the cohesive energiesof the pure AN and BN clusters and FA and FB (=1 � FA)are the fractions of A and B atoms in the nanoalloy clus-ters. A positive value of DmixE corresponds to energeticallyfavoured mixing.

For the bulk alloys, DmixE is essentially the negative ofthe enthalpy of mixing, which Cleri and Rosato used inthe fitting of the Ni–Al Gupta potential parameters[24,27a]. For the Ni3Al bulk fcc phase, using bulk cohesiveenergies of 4.44 eV (Ni [26]), 3.39 eV (Al [26]) and 4.54 eV(Ni3Al [24]), gives a mixing energy of approximately0.36 eV/atom, corresponding to a strongly exothermicenthalpy of mixing of �35 kJ mol(atoms)�1 [27a]. Thus,Ni–Al mixing is strongly favoured thermodynamically forthe bulk alloys and also (see below) for Ni–Al nanoalloys.

3. Results and discussion

3.1. Cuboctahedral shell clusters

The binding energies of energy-minimized 2-shell (55atom) and 3-shell (147 atom) Ni–Al cuboctahedral shellclusters (of types (i) and (ii) – as defined above) are plottedagainst the fraction of Ni atoms (FNi) in Fig. 2. In the fig-ure, certain structures, including those clusters cut from thebulk, are highlighted. ‘‘Ni3Al, fcc,(Al)’’ is a cuboctahedralfragment cut from the Ni3Al lattice with an aluminium cen-tral atom; ‘‘NiAl, (fcc), (Al)’’ is an Al-centred cuboctahe-

dral fragment of the hypothetical fcc NiAl phase (withthe ordered CuAu structure – rather than the bcc orderedNiAl structure); ‘‘Layered (Al)’’ is a cluster whose shellsare alternately filled with Al and Ni, with an Al atom atthe centre.

It is apparent from Fig. 2 that the points occur as clo-sely spaced pairs, corresponding to clusters which areidentical except for the choice of central atom and that,therefore, have FNi values (and also average binding ener-gies, Eb) that differ by very little. The figure also showsthat DmixE is positive (corresponding to favourable alloy-ing) for all of the isomers – i.e., the points lie above theline connecting the pure Al (FNi = 0) and the pure Ni(FNi = 1) clusters and that the most stable composition(corresponding to the maximum in the plot of Eb againstFNi) moves closer to the bulk composition of highestcohesive energy (Ni3Al, with FNi = 0.75 ) on increasingthe cluster size from 55 to 147 atoms. It should also benoted that the cluster with the maximum Eb does nothave the maximum value of DmixE. The greatest increasein binding energy on mixing (i.e., the maximum heightabove the line joining the pure Al and Ni clusters) is actu-ally quite similar for the most stable clusters in theapproximate range FNi = 0.25–0.75.

Considering the highlighted structures, we can see thatthe stability of the layered structures depends on the clus-ter size and the nature of the central atom. Thus, for the55 atom clusters (Fig. 2a), the ‘‘Layered (Al)’’ structure is(for its composition) relatively stable (i.e., has a high Eb

value), because the Al–Ni–Al layered structure (includingthe central Al atom) ensures that the element with thelowest surface energy (Al) is on the surface of the cluster.However, the ‘‘Layered (Ni)’’ structure has the oppositeNi–Al–Ni arrangement, with the higher surface energyelement (Ni) on the surface, which is destabilizing. Forthe 147 atom clusters (Fig. 2b), however, the situation isreversed, with the Ni–Al–Ni–Al configuration being rela-tively stable and Al–Ni–Al–Ni being destabilized by hav-ing Ni at the surface. Finally, for the cuboctahedron cutfrom the fcc Ni3Al and NiAl lattices, there is not sucha large variation in relative stability between Al- andNi-centred clusters and not much change in ordering ongoing from 2 to 3 shells. This is because, for these struc-tures, there is not a large change in surface compositionon going from 2 to 3 shells.

Calculated values of Eb and DmixE are listed in Table 4for 2- and 3-shell cuboctahedral clusters of the pure ele-ments and for the most stable high-symmetry cuboctahe-dral type (i) nanoalloys of approximate compositionsNiAl3, NiAl and Ni3Al. The corresponding values for thebulk fcc Ni3Al phase and the hypothetical bulk fcc NiAl3phase are also included in the Table. Eb and DmixE forthe nanoalloys can be seen to approach the bulk valueswith increasing size. The higher mixing energy for Ni3Al,as compared with NiAl3, should be noted – though itshould be remembered that the latter value is based on acalculated enthalpy of formation [27b].

Fig. 2. Binding energies (Eb) of energy-minimized 2-shell (a), and 3-shell (b). Ni–Al cuboctahedral shell clusters plotted against the fraction of Ni atoms(Fni). (See the text for definitions.)

Table 4Binding energies, Eb, and mixing energies, DmixE, for 2-shell (55-atom) and3-shell (147 atom) cuboctahedral Ni–Al clusters, composed of subshellswhich are comprised of all Ni or all Al atoms – i.e., type (i)

N FNi Eb (eV) atom�1 DmixE (eV)

55 0 2.859755 0.2364 3.2467 0.271055 0.5454 3.4515 0.324155 0.7636 3.4569 0.222455 1 3.3505147 0 2.9983147 0.2517 3.4723 0.2937147 0.5 3.6379 0.2814147 0.75 3.7944 0.2589147 1 3.7146Bulk Al 0 3.39 [26]‘‘Bulk NiAl3’’ (fcc)a 0.25 3.9b 0.3Bulk Ni3Al 0.75 4.54 [24,27a] 0.36Bulk Ni 1 4.44 [26]

For each composition, the Eb and DmixE values listed are for the moststable homotop.

a The hypothetical L12 fcc structure.b Calculated from a computed value for the enthalpy of formation [27b].


In order to obtain a better understanding of Fig. 2, theshell structures of 3-shell (147 atom) cuboctahedral Ni–Alnanoalloys with FNi = 0.2925 (Ni43Al104) and FNi =0.7075 (Ni104Al43) are decomposed in Figs. 3a and b,respectively. These two compositions were chosen, because,amongst their high-symmetry homotops, they includestructures which exhibit shell-layering, as postulated byMolenbroek et al. for Cu–Pd clusters [10]. The growthand dynamics of such ‘‘onion-like’’ layered structures havealso been studied theoretically, for a number of alloy sys-tems by Ferrando and co-workers [28,29]. Of course, inour study, the structures (shown in Fig. 3) are constrainedto have all of the atoms in a given subshell of the sametype. It is possible, that other, lower-symmetry, homotops(where some subshells contain both Ni and Al atoms) maybe more stable – but these are beyond the scope of thisstudy. Finally, it should be noted that none of these struc-tures correspond to fragments of fcc Ni3Al or NiAl3.

Considering the clusters with FNi = 0.2925, Fig. 3ashows that the most stable homotop (configuration:


Ni1:Al12:Ni42:Al92, Eb = 3.5471 eV/atom) has good inter-shell Ni–Al mixing, because of having alternating Ni–Al–Ni shells – though there are is no intra-shell Ni/Al mixing.On the other hand, the least stable homotop (configura-tion: Ni1:Al12:Ni6Al36:Ni24Al8Al48Ni12, Eb = 3.3732 eV/atom) has a relatively small degree of Ni–Al mixing, with48 Al atoms in the cluster core (the inner 2 shells) and a rel-atively high number (36) of Ni atoms on the surface. Onthe basis of maximizing the coordination of the Ni atoms,placing Ni atoms at the vertices of the outer shell should beless stable than placing them in the centres of the (100-like)square faces, but the sites adopted are constrained by thecomposition and maintaining high symmetry.

Considering the clusters with FNi = 0.7075 (Fig. 3b),similar arguments can be used to explain the relative stabil-ities of the homotops, though now there are more Ni thanAl atoms. Interestingly, for this composition, the most sta-ble homotop (configuration: Al1:Ni12:Al6Ni36:Al24Ni8Ni48-Al12, Eb = 3.7811 eV/atom) has the inverse configuration

Fig. 3. Decomposition of the shell structures of (a) Ni43Al104 and (b) Ni1interpretation of the references to color in this figure legend, the reader is refe

(i.e., interchanging Ni and Al atoms) of the least stablehomotop for FNi = 0.2925. In this case, the enhanced sta-bility is because this homotop maximizes the number ofsurface Al atoms. The least stable homotop forFNi = 0.7075 (configuration: Al1:Al12:Al30Ni12:Ni92, Eb =3.5613 eV/atom) has the greatest number of surface Niatoms, having an Al13 inner core and a complete Ni92 outershell Even the second shell has the 30 Al atoms (in the 2fand 2e sites) closer to the centre of the cluster than the12 Ni atoms (2v). In this case, however, the least stablehomotop for Ni104Al43 is not the inverse of that forNi43Al104, as it does not have alternating shells of Ni andAl atoms. In fact, the Al1:Ni12:Al42:Ni92 homotop has abinding energy of 3.6718 eV/atom, which is almost exactlyhalf-way between the most and least stable homotops. Inthis case, even though the alternating homotop has anouter Ni92 shell, the large number of inter-shell Ni–Alinteractions lead to stabilization relative to the Al(core)–Ni(shell) arrangement.

04Al43 cuboctahedral clusters (red atoms, Ni; yellow atoms, Al). (Forrred to the web version of this article.)

Fig. 3 (continued )


3.2. Comparison of cluster and bulk energetics

The binding energies of 1–5-shell cuboctahedral nanoal-loys cut out of the bulk fcc Ni3Al structure – type (ii)according to the definition in Section 2.2 – are listed inTable 5, along with those of the pure Ni and Al clustersand the corresponding solids. For these ‘‘Ni3Al’’ clusters,there are two possible structures for each number of shells:one with Ni at the centre, Ni3Al(Ni) and one with Al at thecentre, Ni3Al(Al).

Following the approach of Northby et al. [30], the aver-age cluster binding energies (Eb) of pseudo-spherical clus-ters can be fitted to the following expression

Eb ¼ aþ bN�13 ð5Þ

The fitted coefficients a and b represent different contribu-tions to the overall cluster potential energy: a represents thecontribution from the volume of the cluster and should be

equal to the bulk cohesive energy (Ecoh); and b representsthe destabilization introduced by creating a surface and isrelated to the surface energy. By fitting a straight line toa plot of Eb against N�

13 (see Fig. 4), a is obtained as the

intercept (N�13 ! 0 as N!1) and b is the gradient.

For the elemental clusters, all points (1–5 shells) essen-tially lie on a straight line, as shown in Fig. 4. ElementalNi has a higher bulk cohesive energy than Al (Ecoh(Ni) =4.44 eV; Ecoh(Al) = 3.39 eV) [26], but a higher surfaceenergy (average Esurf(Ni) = 149 meV A�2 [31]; averageEsurf(Al) = 71–75 meV A�2 [32]), which is consistent withthe higher intercept and gradient of the Ni line, as com-pared with the Al line, in Fig. 4.

For the larger ‘‘Ni3Al’’ clusters (2–5 shells), Fig. 4 showsthat the data points are also well represented by a linear fit,and the Ni-centred and Al-centred structures have approx-imately equal binding energies (getting closer in energywith increasing nuclearity), as their compositions are

Table 5Binding energies, Eb, and mixing energies, DmixE, for 1–5 shell cubocta-hedral Ni, Al and ‘‘Ni3Al’’ clusters

N Structure FNi Eb (eV atom�1) DmixE (eV)

13 Ni 1 2.603313 Al 0 2.547413 Ni3Al(Ni) 0.692 2.7528 0.166713 Ni3Al(Al) 0.923 2.6683 0.069355 Ni 1 3.350555 Al 0 2.859755 Ni3Al(Ni) 0.782 3.4704 0.226955 Ni3Al(Al) 0.655 3.4871 0.3059

147 Ni 1 3.7146147 Al 0 2.9983147 Ni3Al(Ni) 0.728 3.8206 0.3008147 Ni3Al(Al) 0.816 3.8084 0.2256309 Ni 1 3.9254309 Al 0 3.0728309 Ni3Al(Ni) 0.767 4.0326 0.3059309 Ni3Al(Al) 0.699 4.0364 0.3676561 Ni 1 4.0624561 Al 0 3.1225561 Ni3Al(Ni) 0.736 4.1577 0.3434561 Ni3Al(Al) 0.791 4.1520 0.2860

Bulk Ni 1 4.44 [26]Bulk Al 0 3.39 [26]Bulk Ni3Al 0.75 4.54 [24,27a] 0.3625

Type (ii) ‘‘Ni3Al’’ clusters are cut out of bulk Ni3Al and may be Ni- or Al-centred.

Table 6Comparison of a and b values obtained from a linear fit of the bindingenergies for 1–5 shell Ni, Al and ‘‘Ni3Al’’ clusters plotted against N�

13 (see

Fig. 4)

Cluster system a (eV) Ecoh (eV) b (eV) Esurf (meV A�2)

Ni 4.667 4.44 [26] �5.014 149 [31]Al 3.347 3.39 [26] �1.850 71–75 [32]‘‘Ni3Al’’a 4.735 4.54 [24,27a] �4.797 150–200b

a Ni- and Al-centred cuboctahedral clusters cut from the fcc Ni3Alstructure.

b Calculated values reported in Ref. [33].


approximately the same. For the smallest (1-shell) clusters,however, the Ni-centred and Al-centred clusters have quitedifferent compositions: (Ni-centred = Ni9Al4; Al-cen-tred = Ni12Al) and the binding energies lie either side ofthe straightline plot. Inspection of Table 5 reveals that,despite the higher bulk cohesive energy of Ni comparedwith Al (and the higher Eb values for pure Ni clusters),

0 0.1 0.2

N

2

2.5

3

3.5

4

4.5

5

Eb /

eV

Fig. 4. Binding energies (Eb) of energy-minimized 1–5 cuboctahedral shell Ni, Aplotted against N�

13. Also included are lines showing the best linear fit of Eb t

for each nuclearity, the ‘‘Ni3Al’’ clusters with the smallernumber of Ni atoms (smaller FNi) have the higher bindingenergy. This is because, for these relatively small clusters, alarge proportion of the atoms lie on the surface. For eachshell size, the larger FNi value corresponds to the greaternumber of surface Ni atoms so this isomer is destabilizedmore by the higher surface energy contribution of Ni thanit is stabilized by the increased bulk contribution.

The a and b values obtained from the linear fits for Ni,Al and ‘‘Ni3Al’’ clusters (shown in Fig. 4) are listed, alongwith experimental bulk cohesive energies and surface ener-gies in Table 6. The Gupta potential can be seen to slightlyoverestimate the stability of bulk Ni (a > Ecoh) and tounderestimate that of Al. This may be due to our not hav-ing included a cut-off in the potential. The lack of such aconstraint causes the range of the binding term to be toolarge in comparison to that of the repulsive term. Hence,at long range, the repulsion dies out much more quicklythan the attraction. It is also apparent from Table 6 thatthe calculated ratio of surface energies is too high: b(Ni)/b(Al) = 2.71, compared with an experimental ratio of

0.3 0.4 0.5-1/3

NiAlNi

3Al(Ni)

Ni3Al(Al)

Ni fitAl fitNi

3Al fit

l and Al- and Ni-centred ‘‘Ni3Al’’ (cut out of the fcc Ni3Al lattice) clusterso Eq. (5).


Esurf(Ni)/Esurf(Al) � 2. Although with many body poten-tials there is not always a simple linear relationshipbetween cohesive and surface energies, the overestimationof Esurf(Ni) is consistent with the overestimation ofEcoh(Ni).

Table 6 shows that the calculated bulk cohesive energyof the fcc Ni3Al alloy phase is slightly overestimated (byapproximately 0.2 eV – i.e., under 5%). This is to beexpected since the alloy has 75% Ni and Ecoh(Ni) is knownto be overestimated. The b value for the ‘‘Ni3Al’’ clusters is,therefore, also likely to be an overestimate – in fact theaverage surface energy for Ni3Al, calculated from our b

value is 230 meV A�2, which is higher than the range ofvalues (150–200 meV A�2) previously calculated for low-index surfaces of Ni3Al [33].

4. Summary and conclusions

Energy calculations (with radial relaxation) have beencarried out on 2- and 3-shell cuboctahedral Ni–Al nanoal-loy clusters, of varying composition, with the interatomicinteractions modelled by the Gupta many-body potential.It has been found that the peak of the binding energy ver-sus fraction of Ni atom moves towards the composition ofthe most stable alloy (Ni3Al) on increasing the size of theclusters and that the relative stabilities can be explainedin terms of the degree of Ni–Al mixing and the nature ofthe surface atoms. Extrapolation of the cluster bindingenergy for 1–5-shell clusters cut from the fcc Ni3Al struc-ture leads to a slight overestimation of the bulk cohesiveenergy of Ni3Al and a larger overestimation of the surfaceenergy.

Previous computational studies of nanoalloy clustershave shown that homotop stability – i.e., whether there issegregation or ordered/random mixing of the unlike atoms– is determined by a number of factors, which (dependingon the geometry, size and composition of the cluster andthe nature of the component elements) may oppose or rein-force each other [8,13–19,21,28,29,34–42]. For Ni–Al nano-alloys, these factors have been shown to be [16,17,19,34,35]:

(i) Maximization of the number of Ni–Al interactions(since the Ni–Al interaction is stronger than boththe Ni–Ni and the Al–Al interactions) – this favoursNi–Al mixing and ordering.

(ii) Minimization of the cluster surface energy – thisfavours segregation, with the cluster surface becom-ing richer in the element (Al) which has the lower sur-face energy.

(iii) Minimization of bulk strain – this favours the loca-tion of the smaller atom (Ni) at the centre of icosahe-dral clusters.

In our previous study of Ni–Al nanoalloys with up to 55atoms, we found the lowest energy structures of Ni–Alclusters to be both size and composition dependent [19].For these small clusters, icosahedral structural motifs dom-

inate, though there are regions of stability of other struc-tures, such as those based on the local fcc packing (as inthe cuboctahedral clusters studied here). A tendency forNi–Al mixing was also observed (consistent with the strongexothermic mixing in the alloys) although some segregationof Al atoms to the surface was noted (due to the lower Alsurface energy and the larger size of the Al atoms[16,17,19,34,35]). In clusters with the approximate compo-sition ‘‘Ni3Al’’, the minority Al atoms were rarely situatednext to each other, thereby maximizing the number ofstrong Ni–Al interactions, as was previously noted by Jel-linek and co-workers [8,16] and by Rey et al. [17].

The present study has confirmed that these factors arealso important in determining the homotop stability forcuboctahedral Ni–Al nanoalloys. Our results are also con-sistent with recent atomistic simulations by Polak andRabinovich, using the statistical mechanical free-energyconcentration expansion method (FCEM) [43]. They havestudied the interplay of ordering and Al surface segregationfor cuboctahedral Ni–Al clusters with 13–923 atoms (1–6shells), predicting that the order–disorder temperature risesfrom 1030 K for Ni36Al19 to 1450 K for Ni205Al104 (with apredicted value of 1580 K for bulk Ni3Al).

Acknowledgements

R.L.J. is grateful to EPSRC, the Leverhulme Trust andthe University of Birmingham for financial support. Theauthors are grateful to Ben Curley and Graham Cox forhelp in preparing some of the figures. R.L.J. also acknowl-edges and thanks Mike Mingos for the inspiration andexample that he has shown throughout his research careerand for his friendship.

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atom ordering in cuboctahedral ni–al nanoalloys

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