Structures and Chemical Ordering of Small CuâAg Clusters - The

J. Phys. Chem. C 2010, 114, 13255–13266

13255

Structures and Chemical Ordering of Small Cu-Ag Clusters S. Nuń˜ez† and R. L. Johnston*,‡ Departamento de Fı´sica Teo´rica, UniVersidad de Valladolid, Valladolid 47011, Spain, and School of Chemistry, UniVersity of Birmingham, Edgbaston, Birmingham, U.K. B15 2TT ReceiVed: May 26, 2010; ReVised Manuscript ReceiVed: June 28, 2010

The structures and chemical ordering (segregation properties) of Cu-Ag clusters with 34, 38, and 98 atoms have been studied using a combination of a genetic algorithm (GA) global optimization technique coupled with the Gupta semiempirical many-body potential. A detailed analysis of Cu-Ag structural motifs and segregation effects is presented. For the 38-atom cluster, the effect of varying the potential parameters is studied, where the heteronuclear Cu-Ag parameters in the Gupta potential are derived as weighted averages of the Cu-Cu and Ag-Ag parameters. The structural motifs predicted by the Gupta potential, for selected compositions, have been compared to previously published Gupta and density functional theory (DFT) calculations for 38-atom Cu-Ag clusters. From this study, we have found that the lowest energy arrangement is the CucoreAgshell configuration. For the 98-atom clusters, it was seen that Leary tetrahedron isomers are higher in energy than the isomers obtained by the GA. 1. Introduction Bimetallic alloy nanoclusters (or “nanoalloys”) are the subject of increasing interest, both from the viewpoint of basic science and for their practical applications, ranging from catalysis to magnetism and optics.1,2 The interest in nanoalloys arises because they allow the range of properties and compositions afforded by intermetallic materials to be combined with nanoscale control of size, structure, composition, and chemical ordering (e.g., core-shell segregation vs mixing), which are often different to those seen in the bulk limit. The Cu-Ag binary system is attractive for study because of the fact that silver and copper are immiscible (0.1% solid solution at 298 K) in the bulk, but not on the nanoscale. To date, Cu-Ag clusters have not been extensively studied experimentally. Previous studies have shown, however, that for particles in the nanometer size range the system preferentially adopts the CucoreAgshell arrangement.3,4 Recently Langlois et al.5 have analyzed the growth and structural properties of Cu-Ag nanoparticles and have observed that the core-shell arrangement is the most favorable for Cu core diameters of less than 12 nm. For larger sizes, however, it appears that layered structures predominate. Theoretical studies performed on the Cu-Ag nanoalloy system have mainly been global optimizations, in order to search for the most stable isomers. Global optimization studies, using the Gupta potential model, have been performed for all compositions for 34 and 38 atoms.6,7 These studies showed that the most stable structures are generally core-shell polyicosahedra with a Cu core embedded in a Ag shell. The special stability of core-shell Cu-Ag nanoalloys has been confirmed by density functional theory (DFT) calculations6,8-10 and originates from the interplay of size mismatch, bond order-bond length correlation, and the tendency for surface segregation of Ag, due to the lower surface energy of Ag.1 These classical geometric arguments can be reinforced by specific quantum

effects in those clusters which have closed electronic shells.8 A very nice example of the interplay of geometric and electronic effects has been found by Barcaro et al.11 for 40 atoms, which is a magic jellium size. The structure of this paper is as follows. In section 2 we briefly present technical details of the theory and the search method employed. Section 3 describes the cluster structures obtained for the three systems under study: the 34-, 38-, and 98-atom Cu-Ag clusters, including a detailed study of the sensitivity of the lowest energy structures and segregation properties to the potential parameters used to model the nanoalloy interactions and presenting the results of DFT calculations for 38-atom clusters. As an example of the application of the GA technique to larger clusters, the specific case of the 98-atom cluster will be considered in section 4. Because finding the global minimum structure becomes more difficult with increasing size, symmetry constraints are applied to restrict the size of the problem. Section 5 draws conclusions from the main results obtained. 2. Computational Methods 2.1. Gupta Potential. The atom-atom interactions in the Cu-Ag nanoalloys studied here are modeled by a semiempirical many-body potential of the form proposed by Gupta12 and inspired by the approach of Ducastelle.13 This interatomic potential is derived within the second-moment approximation to the tight-binding model. The contribution to the total potential energy of atom i is made up of a many-body (nonlinear) bonding term

Vm(i) ) ζ(s, w)

∑ ∑

(

exp -2q(s, w)

j*i s,w)A,B

))

rij -1 r0(s, w)

and a repulsive Born-Mayer pair term N

Vr(i) ) A(s, w)

∑ ∑

j*i s,w)A,B

* To whom correspondence should be addressed. † Universidad de Valladolid. ‡ University of Birmingham.

(

N

(

(

exp -p(s, w)

))

rij -1 r0(s, w)

rij is the distance between the atoms at sites i and j; since we are considering a bimetallic system, s, w are the two atomic

10.1021/jp1048088  2010 American Chemical Society Published on Web 07/16/2010

13256

J. Phys. Chem. C, Vol. 114, No. 31, 2010

species that form the nanoalloy, and we need to distinguish between the homoatomic and the heteroatomic interactions. r0(s, s) is the nearest-neighbor distance in the pure metal s ) Cu, Ag; if the interaction is heteroatomic, i.e., s * w, r0(s, w) is obtained as

r0(s, w) )

r0(s, s) + r0(w, w) 2

The parameters A(s, w), ζ(s, w), p(s, w), q(s, w) used in this study are listed in Table 1.14 The homoatomic interactions were fitted to several bulk experimental values, such as the cohesive energy Ecoh, the lattice parameter, the bulk modulus B, and the elastic constants C44 and C′. For heteroatomic interactions, A(s, w) and ζ(s, w) were fitted to the solubility energy, and p(s, w) and q(s, w) were taken as averages of the values of the pure constituents. It should be noted that the interaction potential has been used without a distance cutoff, though the potential was originally fitted14 by imposing a cutoff between the secondneighbor Ag distance of Ag and the third-neighbor Cu distance. Eliminating the cutoff may change isomer ordering in the cases where energy differences fall below 0.1 eV, but for 34- and 38-atom clusters we do not expect this to affect the results qualitatively. 2.2. Density Functional Theory Calculations. Density functional theory calculations were performed using the NWChem (version 5.1) quantum chemistry package30 and the Perdew-Wang exchange-correlation functional (PW91).31,32 Spherical Gaussiantype orbital basis sets of double-ζ quality were used for Ag and Cu, combined with effective core potentials (ECP): 19 valence electrons were used for both Ag and Cu. Charge density fitting was used to speed up the evaluation of the Coulombic contributions. All calculations were performed spin-restricted (geometry optimization for singlet spin states; after optimization, a single-point triplet spin state calculation is performed). 2.3. Genetic Algorithm Optimization. The genetic algorithm (GA)15-17 is an optimization technique based on the principles of natural evolution. It is inspired by the natural selection process, in a competitive survival environment. The GA belongs to the class of evolutionary algorithms, which also includes evolution strategies, differential evolution, and genetic programming.18 The Birmingham cluster genetic algorithm19,20 operates as follows: 1. The energy function is defined, along with the crossover and mutation schemes and the GA parameters. 2. The initial population is generated randomly. 3. Each individual from the initial population is evaluated and sorted according to its fitness, such that low-energy structures have high fitness. 4. For each generation (a) parents are chosen with a probability depending on their fitness; (b) offspring are generated from the selected parents using an appropriate crossoVer scheme; (c) mutation is carried out on the set of offspring, with a certain probability, to increase the diversity of the population; (d) the population is ranked according to fitness; (f) some of the old individuals are replaced by new individuals, depending on their relative fitnesses. 5. This process is repeated until a convergence criterion is reached or a maximum number of generations is reached. 6. The individuals in the last generation are stored. We acknowledge that not including explicit pairwise exchange in the GA (e.g., as a mutation operation) lowers the probability of finding the absolute lowest energy homotop, but experience

Nuń˜ez and Johnston TABLE 1: Gupta Potential Parameters Used in This Studya parameters

Cu-Cu

Ag-Ag

Cu-Ag

A (eV) ζ (eV) p q r0 (Å)

0.0894 1.2799 10.55 2.43 2.556

0.1031 1.1895 10.85 3.18 2.8921

0.0980 1.2274 10.70 2.8050 2.72405

a

Ref 14.

has shown that atom displacement mutation, plus local minimization, tends to find low-energy (if slightly suboptimal) homotops for low-lying geometrical isomers. We emphasize here the importance of finding the lowest energy structure type and the type of segregation which is favored. The GA parameters adopted for this study were population size ) 40 clusters; crossover rate ) 0.8; crossover type ) onepoint weighted; selection ) roulette; mutation rate ) 0.1, mutation type ) atom displacement; maximum number of generations ) 400; number of GA runs for each composition ) 50. For the case of 98-atom clusters we increased the population size to 60 and the maximum number of generations to 500, knowing that the probability of finding the global minimum decreases as the cluster size increases. 2.4. Shell Optimization Routine. For high-symmetry polyhedral cluster geometries, a substantial reduction in the search space is obtained if all sets of symmetry-equivalent atoms (which we have termed atomic shells) in a particular structure are constrained to be of the same chemical species.21 These atomic shells are also known as orbits of the cluster point group.22 For a given geometrical structure of a bimetallic cluster, this reduces the number of inequivalent compositional and permutational isomers (homotops23) from 2N to 2S, where N is the number of atoms and S is the number of atomic shells. With the use of the shell optimization routine, it is possible to conduct a systematic investigation of all high-symmetry chemical arrangements for a given structural motif and then carry out a local minimization on all of them, with greatly reduced computational effort. In this way, benchmark high-symmetry homotops can be studied and compared with the lowest energy isomers found by the GA search for the same compositions. In this study, we have used our shell optimization routine to investigate highly symmetric 98-atom Leary tetrahedron (LT) structures. The LT structure (which has ideal Td symmetry) was found to be the global minimum (GM) for the 98-atom LennardJones cluster by Leary and Doye.24 The LT consists of a central 20-atom tetrahedron (with fcc packing), where each of the four (111) faces has a truncated (fcc) tetrahedron built on it, forming a 56-atom truncated stellated tetrahedron. Finally, six puckered centered hexagonal patches are located over the edges of the original tetrahedron, decorating the close-packed surfaces of the stellated tetrahedron. In terms of the shell model, the LT has S ) 9 shells (in order of increasing distance from the center of the cluster, these shells have 4:12:12:12:4:6:12:12:24 atoms), resulting in 29 ) 512 Td-symmetry LT isomers.25 2.5. Energetic Analysis. The relative stabilities of clusters, calculated using the Gupta potential, were analyzed in terms of the excess energy (or mixing energy) ∆NGupta, which, for binary nanoalloys with fixed size (N atoms) but variable composition, is defined as

∆NGupta

)

ENGupta(AnBm)

ENGupta(AN) ENGupta(BN) -n -m N N

where n + m ) N and ENGupta is the sum of the pair and manybody contributions to the Gupta potential energy, summed over

Structures and Chemical Ordering of Cu-Ag Clusters

J. Phys. Chem. C, Vol. 114, No. 31, 2010 13257

Figure 1. Excess energy for 34-atom Cu-Ag clusters, modeled by the Gupta potential, as a function of the number of Ag atoms (n).

all N atoms in the cluster. Negative values of ∆NGupta correspond to energetically favorable (exothermic) mixing. The corresponding excess energy at the DFT level of theory, ∆DFT N , is calculated in a similar manner. 3. Results and Discussion 3.1. 34-Atom Cu-Ag Clusters. Figure 1 shows a plot of Gupta ∆34 against the number of Ag atoms (n), for the Gupta potential GM for all compositions Cu34-nAgn. The lowest value Gupta was found for the composition Cu7Ag27, indicating of ∆34 that this is a relatively stable GM, with maximum exothermic Cu-Ag mixing, while having a surface entirely composed of the element (Ag) having the lowest surface energy. This GM is identical to that reported by Ortigoza and Rahman, based on DFT calculations, for the same composition.10 Regarding the structural motifs found (see Figure 2), at the extremes of the sequence, pure Cu34 and Ag34 clusters have different GM structures. Ag34 is characterized as a fragment of the Mackay 55-atom icosahedron (Ih55), whereas Cu34 has some local fivefold symmetry but cannot be classified as decahedral or icosahedral. Considering Ag-rich compositions, the GM for n ) 33 has the same incomplete Mackay arrangement as Ag34. For n ) 32-30, the GM have low-symmetry polyicosahedral (pIh) structures, as do those for n ) 29 and 28. The GM for n ) 30 and n ) 27-10 all adopt the same pIh structure, known as the fivefold pancake (pIh5). Cu7Ag27, which has the minimum excess energy, is a perfect core-shell cluster with D5d symmetry, having a pentagonal bipyramidal core of Cu encased in a monolayer thick Ag shell. Every Cu atom in this cluster is in the center of a 13-atom icosahedron. CucoreAgshell configurations are favored by the lower surface energy of the surface atoms (Ag) and the smaller size and higher cohesive energy of the interior atoms (Cu). [Cohesive energies: -3.49 eV (Cu); -2.95 eV (Ag).26 Surface energies: 2130 mJ m-2 (Cu); 1210 mJ m-2 (Ag).27 The atomic radius of Ag is 13% larger than that of Cu.26] On continuing to replace Ag atoms by Cu, the additional Cu atoms occupy surface sites of the cluster, leading to an increase in the excess energy. For the particular case of n ) 21, the Cu atoms form a perfect Ih13 unit which is asymmetrically surrounded by Ag atoms, resulting in a so-called “ball and cup”28 arrangement. The remainder of the Cu-rich compositions (n ) 9-1) have low-symmetry pIh structures, as for the Cu34 cluster.

It is evident that at low Cu concentrations, Cu atoms tend to aggregate together in the core, which is driven by the stronger Cu-Cu bonding. For low Ag concentrations there is also a tendency for the surface Ag atoms to be linked. In this case, however, the driving force is in part the retention of a higher number of the stronger Cu-Cu bonds which results from Ag aggregation. When our results are compared with those obtained by Rapallo et al.,7 the agreement is good in general, but for certain compositions we did not find the same isomer, sometimes not even a homotop belonging to the same structural family. For example, for Ag34 we find an icosahedral structure which is distinct from the incomplete decahedron found by Rapallo et al.7 3.2. 38-Atom Cu-Ag Clusters. 3.2.1. Gupta Potential Calculations. Figure 3 shows a plot of ∆Gupta against the number 38 of Ag atoms (n), for the Gupta potential GM for all compositions was found Cu38-nAgn (see Figure 4). The lowest value of ∆Gupta 38 for the composition Cu9Ag29. Our calculations predict that the 38-atom GM structure is a perfect truncated octahedron (TO) for both Ag38 and Cu38. Starting from Ag38, however, a single Cu impurity (n ) 37) is sufficient to change the structure from TO to an incomplete Mackay icosahedron, where the Cu atom occupies the interior of the cluster (the same happens for n ) 36). As the number of Cu atoms increases, there is a series of pIh structures made up of six elementary Ih13. This structure, which will be referred to as the sixfold pancake (pIh6) as in ref 7, is the global minimum from Cu3Ag35 to Cu6Ag32, which has a perfect CucoreAgshell configuration. There are low-symmetry pIh structures for n ) 31 and 30, which are the last examples of perfect core-shell clusters, because thereafter the number of Ag atoms is not sufficient to completely cover the Cu core. GM in the wide composition range from n ) 29 to 18 (with the exception of n ) 24, where a pIh was found) adopt the same structural motif: an incomplete anti-Mackay Ih45, with all the Ag atoms occupying surface sites. For n ) 17, the GM is again a pIh, whereas a modified pIh6 structure, with two Ag atoms moved from the external ring to cap the pancake, was found from n ) 16 to 14. For Cu-rich compositions from Cu25Ag13 to Cu30Ag8, the GM structure is an incomplete anti-Mackay Ih45. From n ) 7 to 5 the structures are incomplete Mackay icosahedra, and to

13258


Nuń˜ez and Johnston

Figure 2. GM found (for the Gupta potential) for 34-atom Cu-Ag clusters of all compositions. Cu and Ag atoms are shown in dark red and light gray, respectively.

Figure 3. Excess energy for 38-atom Cu-Ag clusters, modeled by the Gupta potential, as a function of the number of Ag atoms (n).

complete the sequence TO structures were found from n ) 4 until Cu38, with the Ag atoms clustering together on the surface. In summary, many of the global minima belong to the family of perfect or defective pIh structures. Structures of enhanced stability (relatively low excess energy) are found when the pIh structural arrangement is associated with a perfect core-shell chemical ordering. In this case, all of the larger Ag atoms are on the cluster surface, forming a single-layer shell, while all the smaller Cu atoms lie inside the cluster. If we compare our results with those of Rapallo et al.,7 we observe the same general trends, but, as for the 34-atom clusters, for some compositions there are discrepancies. We, therefore, decided to carry out calculations for the configurations reported

in ref 7 at these compositions, when it was possible for us to reproduce those arrangements of atoms, in order to see how the energies compared. The results are shown in Table 2. It should be noted that all of our GA results are lower in energy than those previously reported,7 perhaps due to more exhaustive searching. 3.2.2. DFT Calculations. In order to test the accuracy of the semiempirical Gupta potential, we decided to reminimize a number of the 38-atom Cu-Ag structures found by the GA (using the original fitted Gupta potential parameters14) at the DFT level. We chose to reminimize compositions Cu38-nAgn around the minimum of the excess energy plot (see Figure 4sfrom n ) 26 to 32); the 1:1 composition (n ) 19); and the



Figure 4. GM found (for the Gupta potential) for 38-atom Cu-Ag clusters of all compositions.

TABLE 2: Comparison of the Potential Energies for Some “GM” for 38-Atom Cu-Ag Clusters from This Study and Those Reported in Ref 7 Vcluster/eV no. of Ag atoms

this work

ref 7

1 8 37

-113.2428 -110.9531 -97.6738

-113.0733 -110.5796 -97.5799

Cu-rich (n ) 0, 1) and Ag-rich (n ) 37, 38) extremes. These include two compositions (n ) 1 and n ) 37) for which our GA found different isomers than those reported previously.7 For each of these compositions, we selected the four lowest energy isomers from the Gupta GA results (where possible: sometimes the GA consistently converged on only one or two isomers) to use as the initial configurations for DFT reminimization. Gupta DFT and ∆38 ) are The Gupta and DFT excess energies (∆38 compared in Table 3. For each composition, the isomers are ordered according to decreasing ∆Gupta values. It is apparent from 38 Table 3 that the DFT excess energies are usually (but not always) more negative than those for the Gupta potential. Considering the pure metal clusters, we find that for Cu38 and Ag38 the TO structure is more stable than the Mackay Ih isomer. By definition, the most stable pure metal isomer is defined as Gupta DFT and ∆38 . having zero excess energy, both for ∆38 The lowest energy minima at the Gupta and DFT levels are compared in Table 4 for those mixed compositions for which we have reminimized more than one Gupta-generated isomer at the DFT level. It should be noted that we have not performed a search of the DFT energy surface so the “lowest energy” DFT structure referred to here corresponds to the lowest energy

isomer resulting from reminimization of the low-energy isomers found using the GA to search the Gupta energy surface. The same energy ordering is found at the Gupta and DFT levels for Cu37Ag1, with the lowest energy structure being a TO with the single Ag atom occupying a surface square face (100) site. We also find the same lowest energy isomer for Cu1Ag37 (incomplete Mackay Ih, with the single Cu atom occupying a core icosahedral vertex site) and Cu12Ag26 (incomplete anti-Mackay Ih, with the 12 Cu atoms adopting icosahedral vertex sites in the core). For Cu7Ag31 and Cu19Ag19, the lowest energy Gupta and DFT isomers are not the same, corresponding to distinct low-symmetry pIh structures for Cu7Ag31 and to homotops23 of the same incomplete anti-Mackay Ih for Cu19Ag19. For Cu6Ag32 the Gupta GM is a pIh6 structure (with a hexagonal Cu6 core and ideal D6h symmetry), whereas the lowest energy DFT isomer is a low-symmetry pIh structure (the second lowest isomer found at the Gupta level). The DFT calculations confirm the energetic preference (predicted by the Gupta potential calculations) for Ag atoms to occupy surface sites and Cu atoms to go into the core of small Cu-Ag clusters, i.e., for CucoreAgshell segregation, as reported previously.6,8-10 Core-shell inversion (by swapping all Cu and Ag atoms and then reminimizing the energy) leads to a large increase in energy for Cu19Ag19 at both the Gupta and DFT levels, yielding positive excess energies of 4.39 eV (Gupta) and 2.15 eV (DFT). Finally, we note that for the two examples that we have studied (Cu1Ag37 and Cu37Ag1) where the GA found lower energy isomers (at the Gupta level) than those repeated previously,7 the DFT calculations confirm that our structures are more stable. 3.3. Effect of Varying the Gupta Potential Parameters. In order to determine the sensitivity of the GM structural motifs and chemical ordering on the Gupta potential parameters, we

13260


TABLE 3: Excess Energies for Selected Compositions of 38-Atom Cu-Ag Clusters Calculated at the Gupta Potential and DFT Levelsa Ag38 Cu1Ag37

Cu6Ag32

Cu7Ag31

Cu12Ag26

Cu19Ag19

Cu37Ag1 Cu38

∆Gupta (eV)

∆DFT (eV)

0.1791 0.0000 -0.0081 -0.0398 -0.0544 -0.1020 -1.4285 -1.4290 -1.5482 -1.8370 -1.7703 -1.7770 -1.8709 -1.8878

0.1367 0.0000 -0.0349 -0.0040 -0.0507 -0.2969 -1.5975 -1.6417 -2.0369 -1.4600 -2.2947 -1.6098 -1.9512 -1.4629

∆Gupta (eV)

∆DFT (eV)

-2.1808 -2.1817 -2.2145 -2.2612 -1.4127 -1.4166 -1.4203 -1.4244 0.1528 0.1181 -0.0167 0.1731 0.0000

-2.1487 -3.1213 -2.9480 -3.2353 -2.5570 -2.5612 -2.3988 -2.3970 0.1999 0.1437 0.0081 0.1947 0.0000

a The lowest excess energies found at the Gupta and DFT levels are indicated in bold.

TABLE 4: Comparison of Lowest Energy Isomers for Selected Compositions of Cu38-nAgn at the Gupta and DFT Levels

have carried out a series of GA calculations for 38-atom Cu-Ag clusters (for all compositions), using the same Gupta potential homonuclear parameters as before, but modifying the heteronuclear Cu-Ag parameters. The methodology follows that adopted for previous studies of Pd-Pt28 and Pd-Au clusters.29 In this work, the Cu-Ag parameters are calculated by taking weighted averages of the Cu-Cu and Ag-Ag parameters as follows:

P(Cu - Ag) ) wiPi(Ag - Ag) + (1 - wi)Pi(Cu - Cu) (1) where Pi represents a particular Gupta potential parameter (described previously) and wi is the weighting factor (0 e wi e 1; ∆wi ) 0.1).

Nuń˜ez and Johnston Following the recent study of Pd-Au nanoalloys,29 three weighting schemes have been adopted (the resulting parameters are listed in Table 5), Parameter set I: all parameters (A, ζ, p, q, and r0) are varied in the same sense (w(A) ) w(ζ) ) w(p) ) w(q) ) w(r0) ) w). Parameter set II: only the energy scaling parameters (A and ζ) are varied, again in the same (“symmetric”) sense (w(A) ) w(ζ) ) ws). Parameter set III: the energy scaling parameters are varied in the opposite (“antisymmetric”) sense (w(A) ) 1 - w(ζ) ) wa). Figure 5 shows excess energy plots, as a function of composition and weighting factor, for parameter sets I (top), II (middle), and III (bottom). The excess energy curve for the fitted potential (see Figure 3) is included in each plot. The plots for parameter sets I and II are similar in shape and energy range, with both exhibiting large regions of positive excess energy (indicating unfavorable Cu-Ag mixing) for high w values, i.e., where the Cu-Ag interaction is close to the weaker Ag-Ag interaction. A similar finding was previously reported for Pd-Pt clusters,28 though the curves for Pd-Au are quite different.29 For parameter set III, the excess energy plot looks quite different (though the curves for w ) 0.5 are, by definition, identical in each case) and the range of energies (approximately 4 eV) is significantly smaller than for sets I (9 eV) and II (8 eV). The minima in the excess energy curves occur for Ag-rich compositions for all potentials, typically lying in the range of n ) 24-30 Ag atoms, with some change of the shape of the curves and position of the minima upon changing w. The excess energy curve for the fitted Cu-Ag potential lies very close to the w ) 0.6 curve for parameter set I, and it is almost exactly superimposed on the ws ) 0.6 curve for set II. This degeneracy between the excess energies for the fitted potential and the II (ws ) 0.6) potential occurs because the Cu-Ag parameters in the II (ws ) 0.6) potential (Table 5) are almost exactly the same as those in the fitted potential (Table 1). The better match for potential II (ws ) 0.6), compared to I (w ) 0.6), occurs because (as for parameter set II) the fitted potential takes average (i.e., w ) 0.5) values for the Cu-Ag p and q parameters, while both the fitted A and ζ values turn out to be approximately weighted averages with w ≈ 0.6. Considering the excess energy plot for parameter set III, Figure 5 shows that the curve for the fitted potential lies close to the wa ) 0.3 curve, presumably because, as shown in Table 5, the slightly lower Cu-Ag A parameter (lower pair repulsion), is approximately compensated for by the slightly lower ζ parameter (lower many-body attraction). The structural motifs found as putative GM for Cu38-nAgn clusters are mapped out, as a function of composition (n) and weighting factor (w) in Figure 6 for parameter sets I (Figure 6a), II (Figure 6b), and III (Figure 6c). Those found using the fitted potential14 (see section 3.2.1) are shown in Figure 6d. The following structural motifs have been identified: truncated octahedron (TO), incomplete Mackay Ih, sixfold polyicosahedral pancake (pIh6), polyicosahedron (pIh), incomplete anti-Mackay Ih, capped sixfold pancake, and distorted structures. Figure 6 shows that the structural maps for parameter sets I and II are qualitatively similar, with blocks of stability of various structural motifs occurring in roughly the same regions of (n, w) space. This is as expected, since both these sets involve symmetrical weighting of the parameters, and has previously been observed in a study of Pd-Au potentials. Also as expected, the structural map for parameter set III (which involves



TABLE 5: Potential Parameters for Parameter Sets: I (Symmetric Weighting (w) of All Parameters), II (Symmetric Weighting (ws) of A and ζ), and III (Antisymmetric Weighting (wa) of A and ζ) Iw A ζ p q r0 II ws A ζ p q r0 III wa A ζ p q r0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.089 1.280 10.55 2.43 2.556

0.091 1.271 10.57 2.49 2.589

0.092 1.262 10.61 2.57 2.622

0.093 1.253 10.63 2.65 2.657

0.095 1.244 10.67 2.72 2.690

0.096 1.235 10.69 2.80 2.724

0.098 1.226 10.73 2.87 2.757

0.099 1.217 10.75 2.94 2.790

0.100 1.208 10.79 3.02 2.825

0.102 1.198 10.81 3.10 2.858

0.103 1.190 10.85 3.18 2.892

0.5

0.6

0.0

0.1

0.2

0.3

0.4

0.7

0.8

0.9

1.0

0.089 1.280

0.091 1.271

0.092 1.262

0.093 1.253

0.095 0.096 0.098 1.244 1.235 1.226 10.70 (fixed at average) 2.8050 (fixed at average) 2.724 (fixed at average)

0.099 1.217

0.100 1.208

0.102 1.198

0.103 1.190

0.0

0.1

0.2

0.3

0.4

0.7

0.8

0.9

1.0

0.089 1.190

0.091 1.198

0.092 1.208

0.093 1.217

0.095 0.096 0.098 1.226 1.235 1.244 10.70 (fixed at average) 2.8050 (fixed at average) 2.724 (fixed at average)

0.099 1.253

0.100 1.262

0.102 1.271

0.103 1.280

antisymmetric weighting of A and ζ) shows greater differences to the other parameter sets, though it should be noted that there are still significant similaritiessmuch more so than was previously observed for Pd-Au.29 The reason for the similarities of the structural maps for parameter sets I, II, and (to a lesser extent) III can be seen from Table 5, where it is evident that the range of parameter values (with the exception of q) is fairly small. This is because most of the homonuclear Cu-Cu and Ag-Ag parameters (see Table 1) are quite similar, differing by between 3% (for p) and 15% (for A), whereas the q values differ by 30%. The differences in homonuclear parameters are significantly larger for Pd-Au.29 When Figures 5 and 6 are compared, we can see that for all three weighting schemes the composition range of lowest excess energy (most exothermic Cu-Ag mixing: n ≈ 24-30) occupies a region of (n, w) space which is dominated by incomplete antiMackay Ih structures, with some competition (especially for higher n and w) from pIh structures. This is consistent with the discussion (see section 3.2.1) of the composition-dependent GM structures for 38-atom Cu-Ag clusters using the fitted potential (see also Figures 4 and 6d). When the GM structural motifs found with the fitted potential14 are compared to those found using weighted parameters, Figure 6 shows that for parameter set I (weighting all parameters), the greatest structural similarity is found for w ) 0.5, i.e., for the average potential. For parameter set II (symmetric weighting of A and ζ), however, the best agreement occurs for ws ) 0.6 and is a closer match than for set I. This is consistent with the degeneracy of the excess energy plots for the fitted potential and the II (ws ) 0.6) potential, discussed above, and is to be expected since the parameters of these two potentials are virtually identical. The fact that the best structural agreement between the type-I and the fitted potential is for w ) 0.5, rather than w ) 0.6 (which gave closer excess energies), illustrates how small changes in the potential can give rise to qualitative changes in the lowest energy structure types. For parameter set III (antisymmetric weighting of A and ζ), the best match with the composition-dependent structures found for the fitted Cu-Ag potential occurs at wa ) 0.3, which is consistent with this weighting giving rise to an excess energy curve close to that for the fitted potential. However, the structural match is not as good as for potentials I (w ) 0.5) and II (ws )

0.5

0.6

0.6). This again shows that the lowest energy structures can be more sensitive than energies to small changes in the potential parameters. Figure 7 shows the isomers of Cu38-nAgn corresponding to the compositions with minimum (most negative) excess energy, for parameter sets I (Figure 7a), II (Figure 7b), and III (Figure 7c), as a function of the weighting factor (w, ws, and wa, respectively). In those cases (w ) 0.9, 1.0 Issee Figure 5) where the excess energies are positive for all values of n (apart from the extremes of n ) 0 and 38, where the excess energy is zero by definition), the isomer shown corresponds to the deepest dip in the excess energy plot. Figure 7d shows the minimum excess energy isomer found for the fitted (exp-fit) Cu-Ag potential14 (which occurs at composition Cu9Ag29). Also shown are the putative GM with approximate Cu/Ag compositions of 1:2, 1:1, and 2:1 (see Figure 4 for the GM for the fitted potential across the whole composition range). The w-dependent change in the excess energy minimum composition (seen in Figure 5) is apparent in Figure 7, as is the dominance of incomplete anti-Mackay Ih and pIh structures (see also Figure 6). The type and degree of Cu-Ag segregation in the 38-atom clusters is also affected by the change of Gupta parameters, but the effect is less significant than found for other systems, such as Pd-Pt28 and Pd-Au.29 This is again presumably due to the relatively small differences in the homonuclear parameters of Cu and Ag. As shown in Figure 7, the general preference is for CucoreAgshell segregation. In order to see how the segregation is modified by varying the Gupta potential parameters we will analyze in more detail what happens in the case of parameter set I (symmetrical weighting of all parameters). For w ) 0.0, there is a degree of Cu-Ag mixing across a wide composition range due to the fact that in this case the heteroatomic Cu-Ag interaction is as strong as the strongest homonuclear interaction (Cu-Cu). This increased mixing also shows up in more negative excess energies, as shown in Figure 5. There is also Cu-Ag mixing for w ) 0.1 but with the Cu atoms tending to cluster together more. On increasing w, the minimum in the excess energy becomes less negative. From w ) 0.2 to w ) 0.6 the lowest energy structures consist of a Cu core surrounded by (either a complete or partial) shell of Ag atoms. For w ) 0.7-0.8 the Ag atoms tend to cluster together

13262



Figure 5. Excess energy plots for 38-atom Cu-Ag clusters, as a function of composition (n) and weighting parameter (w), including a comparison with the fitted parameters (ref 7): (top) parameter set I; (middle) parameter set II; (bottom) parameter set III. For parameter sets I and II, w (or ws) increases from bottom to top, whereas for set III wa increases from top to bottom.

across a wide composition range (though this is not evident from the minimum excess energy structures shown in Figure 7, since

these only have eight Cu atoms) rather than forming an even coverageswhat we have previously termed “ball-and-cup”



Figure 6. Map of the GM structural motifs found for 38-atom Cu-Ag clusters, as a function of composition (number of Ag atoms, n) and weighting factor (w). Different colors indicate particular structural motifs: TO (white), incomplete Mackay Ih (green), sixfold pancake (magenta), pIh (blue), incomplete anti-Mackay Ih (yellow), capped sixfold pancake (red), and distorted (orange). Each square represents a specific composition, with n increasing from 0 to 38 from left to right: (a) parameter set I; (b) parameter set II; (c) parameter set III; (d) fitted parameters (ref 14).

Figure 7. Examples of GM for 38-atom Cu-Ag clusters using parameter sets I (a), II (b), III (c) and the parameters fitted in ref 14 (d).

segregation.28,29 Finally, for w ) 0.9-1.0 the Cu and Ag atoms adopt a layer-like (“spherical cap”) segregation pattern, as previously found for Pd-Pt and Pd-Au upon weakening the heteronuclear bonding.28,29 This layering indicates that Cu-Ag mixing is unfavorable, which is consistent with the positive excess energies seen in Figure 5 for these values of w. 4. 98-Atom Cu-Ag Clusters As the Leary tetrahedron has been proposed as the lowest energy geometry adopted by Ag98 using the Sutton-Chen potential,24 we have used the GA to search for the GM for all compositions of 98-atom Cu-Ag clusters, using the original fitted Gupta potential.14 Gupta Figure 8 shows a plot of ∆98 against Ag content n for all compositions, obtained by the GA search. The curve is not as smooth as for smaller sizes, which can be attributed to difficulties in finding low-energy homotops for some composi-

tions, at this relatively large cluster size. The lowest excess energy values are found in the composition range of n ) 58-64. A number of different structural motifs have been identified as putative GM for Cu98-nAgn. These structures, all of which show a certain degree of structural regularity and CucoreAgshell segregation, are shown in Figures 9 (for n ) 98-50) and 10 (for n ) 49-0). Analyzing the structural motifs found by the GA, we can classify the structures as follows: Ag98 is an incomplete Marks decahedron, then from n ) 97 to 84 the GM is an incomplete 147-atom Mackay Icosahedron (Ih147). From n ) 83 to 74 the GM structures are pIh. Anti-Mackay Ih dominate in the range of n ) 73-59, although for some cases, such as n ) 71, 70, 69, 68, 65, and 59, some of the outer atoms are in Mackay and others in anti-Mackay sites over the icosahedral core. Finally, from n ) 58 until pure copper Cu98 incomplete Mackay Ih are observed.

13264



Figure 8. Plot of the excess energy, ∆Gupta 98 , as a function of Ag content (n), modeled by the Gupta potential. Only the lowest energy LT shell structure values are shown.

Figure 9. Putative GM found by the GA for 98-atom Cu-Ag clusters with Ag content n ) 98-50.

It is noticeable that, in contrast to previous work on 98-atom Pd-Pt25 and Pt-Au clusters,33 no Leary tetrahedra structures were found by the GA for 98-atom Cu-Ag clusters. However, in a recent study of 98-atom Pd-Au clusters, it was discovered that, for certain compositions, LT isomers generated by the shell optimization program were lower in energy than the non-LT isomers found by the GA.34 We therefore used the shell optimization program to generate Td-symmetry LT Cu-Ag isomers to compare with the putative GM from the GA search. The excess energies for the lowest energy Td-symmetry LT isomers (where more than one Td homotop is possible for a given

composition) of Cu98-nAgn are shown as isolated points in Figure 8. It is clear that most LT isomers have positive excess energies and in all cases the lowest energy LT isomer has a higher excess energy (i.e., is less stable) than the putative GM from the GA search. The LT isomers with excess energies closest to the putative GM at particular compositions are shown in Figure 11. The surface segregation of Ag atoms is again noticeable. Finally, it should be noted that, in contrast to the Sutton-Chen potential,24 the GM for Ag98 is not an LT for the Gupta potential. The lack of stability of the LT structure for 98-atom Cu-Ag clusters (at the Gupta potential level) is in particular contrast



Figure 10. Putative GM found by the GA for 98-atom Cu-Ag clusters with Ag content n ) 49-0.

Figure 11. Representative examples of the lowest Td-symmetry LT isomers generated by the shell optimization program, including the pure Cu and Ag LT clusters.

to the situation for Pd-Pt, where there is a wide composition range of stability.25 This difference is probably due to the large size mismatch (13%) between Cu and Ag, (Pd and Pt have almost identical atomic radii26) and is consistent with the failure to observe the 34-atom hybrid decahedral close-packed “Dhcp(DT)” structure (observed for Pd-Pt clusters35) for Cu-Ag. 5. Conclusions We have reported a detailed study, based on the Gupta manybody potential, of the structures and chemical ordering, as a function of composition, for Cu-Ag nanoalloy clusters with 34, 38, and 98 atoms, using a GA to search for the putative GM and plotting the excess (mixing) energy against composition. The GM are found to adopt CucoreAgshell configurations for all compositions. This type of arrangement is consistent with the lower cohesive and surface energies of Ag and the smaller size of Cu, all of which favor segregation of Ag to the cluster surface. In order to test the reliability of the semiempirical Gupta potential, DFT calculations (geometry reminimizations) were also performed for some low-energy 38-atom structures at specific compositions. It was found that the lowest energy DFT isomer is generally either identical to or is a homotop belonging to the same structural family as the Gupta potential GM. We have also analyzed the effect of varying the Gupta potential parameters on the structures and chemical ordering of 38-atom clusters, where the parameters describing the

heteroatomic Cu-Ag interactions were obtained as weighted averages of the homoatomic Cu-Cu (stronger bonding) and Ag-Ag (weaker bonding) parameters. The GM isomers were found to be less sensitive to the relative parameter weighting than was previously observed for Pd-Pt and Pd-Au clusters. When the Cu-Ag interaction is almost as large as the Cu-Cu interaction, there is significant Cu-Ag mixing. In the intermediate weighting range, the GM structures have CucoreAgshell ordering. Weakening of the Cu-Ag interaction results in clustering together of Ag atoms, leading to “ball-and-cup” and layered “spherical cap” configurations. Although we are conscious of the decrease in the probability of the GA finding the GM as the cluster size increases, we calculated the excess energy plot for 98-atom Cu-Ag clusters. All the putative GM structures were again found to have CucoreAgshell arrangements. In contrast to previous studies of Pd-Pt, Pd-Au, and Pt-Au clusters, however, Leary tetrahedral structures, which were investigated using a shell optimization program, were not found to be GM at the Gupta level for any composition. Acknowledgment. S.N. is grateful to the University of Valladolid for the award of a Ph.D. scholarship. S.N. is grateful to Ramli Ismail for his help during her visit to Birmingham. All the calculations presented here were performed on the University of Birmingham’s BlueBEAR linux cluster.36

13266


References and Notes (1) Ferrando, R.; Jellinek, J.; Johnston, R. L. Chem. ReV. 2008, 108, 846. (2) Johnston, R. L.; Ferrand, R. Nanoalloys: From Theory to Applications. Faraday Discuss. 2008, 138. (3) Cazayous, M.; Langlois, C.; Oikawa, T.; Ricolleau, C.; Sacuto, A. Phys. ReV. B 2006, 73, 113402. (4) Koretsky, G. M.; Kerns, K. P.; Nieman, G. C.; Knickelbein, M. B.; Riley, S. J. J. Phys. Chem. A 1999, 103, 1997. (5) Langlois, C.; Alloyeau, D.; Le Bouar, Y.; Loiseau, A.; Oikawa, T.; Mottet, C.; Ricolleau, C. Faraday Discuss. 2008, 138, 375. (6) Rossi, G.; Rapallo, A.; Mottet, C.; Fortunelli, A.; Baletto, F.; Ferrando, R. Phys. ReV. Lett. 2004, 93, 105503. (7) Rapallo, A.; Rossi, G.; Ferrando, R.; Fortunelli, A.; Curley, B. C.; Lloyd, L. D.; Tarbuck, G. M.; Johnston, R. L. J. Chem. Phys. 2005, 122, 194308. (8) Ferrando, R.; Fortunelli, A.; Rossi, G. Phys. ReV. B 2005, 72, 085449. (9) Jiang, Z.-Y.; Lee, K.-H.; Li, S.-T.; Chu, S.-Y. Phys. ReV. B 2006, 73, 235423. (10) Ortigoza, M. A.; Rahman, T. S. Phys. ReV. B 2008, 77, 195404. (11) Barcaro, G.; Fortunelli, A.; Rossi, G.; Nita, F.; Ferrando, R. J. Chem. Phys. 2006, 110, 23197. (12) Gupta, R. P. Phys. ReV. B 1981, 23, 6265. (13) Ducastelle, F. J. Phys. (Paris) 1970, 31, 1055. (14) Baletto, F.; Mottet, C.; Ferrando, R. Phys. ReV. B 2002, 66, 155420. (15) Holland, J. Adaptation in Natural and Artificial Systems; University of Michigan Press: Ann Arbor, MI, 1975. (16) Goldberg, D. E. Genetic Algorithms in Search, Optimization and Machine Learning; Addison-Wesley: Reading, MA, 1989. (17) Mitchell, M. An Introduction to Genetic Algorithms; MIT Press: Cambridge, MA, 1998. (18) Cartwright, H. M. Struct. Bonding (Berlin) 2004, 110, 55–94. (19) Johnston, R. L.; Roberts, C. Birmingham Cluster Genetic Algorithm; University of Birmingham: Birmingham, U.K., 1999.

Nuń˜ez and Johnston (20) Johnston, R. L. Dalton Trans. 2003, 4193. (21) Wilson, N. T.; Johnston, R. L. J. Mater. Chem. 2002, 12, 2913. (22) Wales, D. J. Energy Landscapes; Cambridge University Press: Cambridge, U.K., 2003. (23) Jellinek, J.; Krissinel, E. B. Chem. Phys. Lett. 1996, 258, 283. (24) Leary, R. H.; Doye, J. P. K. Phys. ReV. E 1999, 60, R6320. (25) Paz-Borboń, L. O.; Mortimer-Jones, T. V.; Johnston, R. L.; PosadaAmarillas, A.; Barcaro, G.; Fortunelli, A. Phys. Chem. Chem. Phys. 2007, 9, 5202. (26) Kittel, C. Introduction to Solid State Physics, 6th ed.; Wiley: New York, 1986. (27) Vitos, L.; Ruban, A. V.; Skriver, H. L.; Kollar, J. Surf. Sci. 1998, 411, 186. (28) Paz-Borboń, L. O.; Gupta, A.; Johnston, R. L. J. Mater. Chem. 2008, 18, 4154. (29) Ismail, R.; Johnston, R. L. Phys. Chem. Chem. Phys. 2010; DOI: 10.1039/c004044d. (30) Kendall, R. A.; Apra`, E.; Bernholdt, D. E.; Bylaska, E. J.; Dupuis, M.; Fann, G. I.; Harrison, R. J.; Ju, J.; Nichols, J. A.; Nieplocha, J.; Straatsma, T. P.; Windus, T. L.; Wong, A. T. Comput. Phys. Commun. 2000, 128, 260. (31) Perdew, J. P.; Wang, Y. Phys. ReV. B 1986, 33, 8800. (32) Perdew, J. P.; Chevary, J. A.; Vosko, S. H.; Jackson, K. A.; Pederson, M. R.; Singh, D. J.; Fiolhais, C. Phys. ReV. B 1992, 46, 6671. (33) Logsdail, A.; Paz-Borboń, L. O.; Johnston, R. L. J. Comput. Theor. Nanosci. 2009, 6, 857. (34) Paz-Borboń, L. O. Computational Studies of Transition Metal Nanoalloys. Ph.D. Thesis, University of Birmingham, Birmingham, U.K., 2009. (35) Paz-Borboń, L. O.; Johnston, R. L.; Barcaro, G.; Fortunelli, A. J. Phys. Chem. C 2007, 111, 2936. (36) The University of Birmingham Environment for Academic Research (BEAR); http://www.bear.bham.ac.uk.

JP1048088

Structures and Chemical Ordering of Small CuâAg Clusters - The

Recommend Documents

Structures and Chemical Ordering of Small CuâAg Clusters - The