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Notes Monte Carlo Simulation of Cu-Ni Nanoclusters: Surface Segregation Studies Daniela S. Mainardi and Perla B. Balbuena* Department of Chemical Engineering, University of South Carolina, Columbia, South Carolina 29208 Received October 10, 2000. In Final Form: January 11, 2001
Introduction Bimetallic catalysts are widely used in the automobile and chemical industries1,2 due to their enhanced activity, selectivity, and stability in comparison with monometallic systems. Novel technologies for nanoparticle fabrication offer new opportunities to understand, control, and design catalysts for particular applications.3 To achieve these goals, new insights into the physics and chemistry of nanoclusters are needed. For bimetallic catalysts, an important problem is the assessment of surface composition, particularly on active sites, containing less than a hundred atoms, which are responsible for catalysis phenomena. Theoretical4-12 and experimental13-16 work on Cu-Ni macroscopic alloys have determined that surface segregation of Cu atoms is driven toward the lowest coordination sites, such as those on corners, edges, and surfaces. This is because, energetically, three competing factors are known to influence surface segregation: differences in surface energies, mixing energies as a function of composition, and entropies of mixing. Since Cu has lower surface energy, σ(111)Cu ) 69.5 kJ/mol, than Ni, σ(111)Ni ) 80 kJ/mol,17 Cu atoms are more likely to segregate to the lowest coordination sites causing Cu surface enrichment. Also, Cu-Ni experimental studies show small positive mixing energies over the complete range of compositions.17 Positive mixing energies favor ensembles with neighbors of the same type, whereas negative mixing energies favor neighbors of a different type. Thus, in bulk * To whom correspondence should be addressed: balbuena@ engr.sc.edu. (1) Jacobson, R. L.; Kluksdahl, H. E.; McCoy, C. S.; Davis, R. W. Proc., Am. Pet. Inst., Div. Refin. 1964, 49, 504. (2) Campbell, C. T. Annu. Rev. Phys. Chem. 1990, 41, 775. (3) Schilderberger, M.; Prins, R.; Bonetti, Y. J. Phys. Chem. B 2000, 104, 3250. (4) Burton, J. J.; Hyman, E.; Fedak, D. G. J. Catal. 1975, 37, 106. (5) Sundram, V. S.; Wynblatt, P. Surf. Sci. 1975, 52, 569. (6) Donnelly, R. G.; King, T. S. Surf. Sci. 1978, 74, 89-108. (7) King, T. S.; Donnelly, R. G. Surf. Sci. 1984, 141, 417-454. (8) Foiles, S. M. Phys. Rev. B 1985, 32, 7685. (9) Yang, L.; DePristo, A. E. J. Catal. 1994, 149, 223. (10) Zhu, L.; DePristo, A. E. J. Chem. Phys. 1995, 102, 5342. (11) Ruban, A. V.; Skriver, H. L.; Norskov, J. K. Phys. Rev. B 1999, 59, 24. (12) Pourovskii, L. V.; Skorodumova, N. V.; Vekilov, Y. K.; Johansson, B.; Abrikosov, I. A. Surf. Sci. 1999, 439, 111-119. (13) Ng, Y. S.; Tsong, T. T.; McLane, S. B. Surf. Sci. 1979, 84, 31. (14) Kuijers, F. J.; Ponec, V. Surf. Sci. 1977, 68, 294-304. (15) Brongersma, H. H.; Sparnaay, M. J. Surf. Sci. 1978, 71, 657678. (16) Webber, P. R.; Rojas, C. E.; Dobson, P. J.; Chadwick, D. Surf. Sci. 1981, 105, 20-40. (17) Zhu, L.; DePristo, A. E. J. Catal. 1997, 167, 400.
Cu-Ni alloys, the differences in surface energies and the positive mixing energies favor segregation of the atoms with the lowest surface energy to the lowest coordination sites. In addition, the alloy entropy of mixing in the random approximation18 always favors mixing of atoms of different types. However, very little is known about segregation phenomena in nanoscopic systems, where besides the thermodynamic effect, other variables such as geometry and cluster size may play a significant role. The objective of this work is to investigate the role of thermodynamic (surface energies, mixing energies, and entropies) and geometric (nanocluster size, crystallographic faces) variables on the mixing of Cu and Ni atoms in nanoclusters. The BOS-mixing model17 is used to study surface segregation on Cu-Ni nanostructures as a function of temperature, size, and composition of the cluster using Monte Carlo simulations.19,20 Minimum energy configurations are found using the Metropolis algorithm19 for Ni-Cu nanoclusters containing from 64 to 8000 atoms arranged on cubic stacks of layers corresponding to a facecentered cubic (fcc) crystal, with exposed (111), (110), and (331) faces. The overall composition ranges from 5% to 95% of atomic Cu content. Force Field for Monte Carlo Simulations For a system of N atoms, consisting of two types “A” and “B”, the BOS-mixing17 interaction energy can be expressed as a sum of the site energies N
∆E({Rk}) )
∑ R Z ,M k)1 k
k
k
of βk
(1)
where Rk and βk are either A-type or B-type atoms. N is the total number of atoms in the system; Zk and Mk are the coordination number (i.e., number of nearest neighbors) and the number of unlike atoms around a central atom of type Rk, respectively. The site energy for a R-type atom surrounded by M atoms of β-type and (Z-M) atoms of R-type is given by
RZ,M of β ) ZR + Mβ ∆EβZ,R-β +
Mβ(Mβ - 1) R λ Z,R-β 2 (2)
where ZR is the interaction energy for a Z-coordinated R-type atom with all R-type neighbors, ∆ERZ,R-β is the energy change of the first R-β vs R-R bond, and λRZ,R-β is the incremental variation in R-β bond.17 The dependence of ZR on site coordination Z is linear as indicated in eq 3. (18) Hill, T. L. Introduction to Statistical Thermodynamics; AddisonWesley Publishing Co.: Reading, MA, 1962. (19) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: Oxford, 1990. (20) Gubbins, K. E.; Quirke, N. E. Molecular Simulation and Industrial Applications: Methods, Examples, and Prospects; Gordon and Breach Science Publishers: Amsterdam, 1996; Vol. 1.
10.1021/la0014306 CCC: $20.00 © 2001 American Chemical Society Published on Web 02/21/2001
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R
Z
{
Z-1 1eZe9 8 ) Z-9 9R + (12R - 9R) 9 e Z e 12 3 1R + (9R - 1R)
Notes Table 1. Site Energies of Homogeneous Systems in eV
(3)
When Z ) 1, 1R ) -D0/2, where D0 is the dimer dissociation energy. For Z ) 9, 9R ) -Ecoh + σ(111)(31/2/4)a02, where Ecoh is the bulk cohesive energy, σ(111) is the (111) surface energy at 0 K, and a0 is the lattice constant.17 Finally, for Z ) 12, 12R ) -Ecoh. The parameters needed in eq 2 are the site energies of the pure systems, ZA and ZB, given by eq 3, and the mixing parameters, ∆EAZ,A-B, ∆EBZ,A-B, λAZ,A-B, and λBZ,A-B. Input to eq 3 includes the experimental cohesive energies for the pure metals, the dimer dissociation energies that can be obtained very precisely with density functional theory (DFT) calculations,21 the lattice constants, and the surface energies that can be obtained from extended DFT approaches.22 The site energies of monometallic systems used in the calculations presented here are given in Table 1. The four remaining mixing parameters shown in Table 2 (∆EA12,A-B, ∆EB12,A-B, λA12,A-B, and λB12,A-B) were taken from published values calculated on the basis of experimental mixing energies for the bulk systems.17 Equations 1 to 3 are used in a Monte Carlo program to find the minimum energy configuration for the systems under study.
A-B
1A
1B
9A
9B
12A
12B
Ni-Cu -0.9730a -0.9545a -3.81917 -3.00417 -4.44023 -3.49023 a Calculated in this work using density functional theory, B3PW91/LANL2DZ.24
Table 2. Mixing Parameters in eV A-B
∆EA12,A-B
∆EB12,A-B
λA12,A-B
λB12,A-B
Ni-Cu
0.0442
-0.0348
-0.0012
-0.0002
Systems and Monte Carlo Procedure Face-centered cubic Cu-Ni alloys containing a total of 64, 125, 216, 343, 512, 729, 1000, and 8000 atoms were investigated, corresponding to clusters of 4, 5, 6, 7, 8, 9, 10, and 20 atomic layers (NL), respectively. Stacks of fcc (111) planes were built according to an ABC pattern repeated in the vertical direction. The side surfaces of this fcc stack correspond to pairs of (110) and (331) faces. Calculations were done for eight alloy compositions of Ni-Cu systems, with 5, 15, 25, 30, 40, 50, 75, and 95 atomic percent of Cu. Every system was studied at five temperatures: 10, 300, 500, 923, and 1073 K. The number of Monte Carlo steps for each simulation was fixed at 200 times the total number of atoms N, which proved sufficient to ensure convergence. Averages were taken over 99 simulations performed with different random number initializations in order to avoid bias produced by a particular initial configuration. Results and Discussion An absolute definition of surface segregation is dependent on the cluster size and overall Cu concentration. At each overall Cu concentration xCu, and cluster size NL3, there is a maximum fraction of Cu atoms that can be accommodated on the surfaces, including all the nanocluster-exposed faces. We name this fraction yCumax ) (maximum number of surface Cu atoms at xCu and NL3)/ (total number of surface sites). The ratio R of the surface fraction, yCu ) (calculated number of Cu surface atoms)/ (total number of surface sites), over yCumax, for a given Cu overall concentration xCu and cluster size NL3 is displayed in Figure 1 at several temperatures and at three values of xCu. When R ) 1, there is total segregation of Cu atoms to the surface. As Figure 1 indicates, R is strongly (21) Seminario, J. M. Ed. Recent Developments and Applications of Density Functional Theory; Elsevier: Amsterdam, 1996; Vol. 4. (22) Sinnott, S. B.; Stave, M. S.; Raeker, T. J.; DePristo, A. E. Phys. Rev. B 1991, 44, 8927.
Figure 1. Ratio (R) of calculated average surface Cu atomic fraction (yCu) over maximum Cu atomic fraction (yCumax) at a given overall Cu concentration (xCu) and nanocluster size (NL), at three temperatures.
dependent on overall composition, cluster size, and temperature. We found that for the Cu-Ni system, R lies in the range 0.75 < R < 1. At low overall Cu concentration, xCu ) 0.05 (Figure 1, top), there is a slight variation of Cu segregation with cluster size and with temperature. Most Cu atoms are located on surface sites, and surface segregation (practically greater than 93%) is almost independent of temperature and cluster size. At intermediate overall Cu concentration, xCu ) 0.50 (Figure 1, middle), total Cu surface segregation is observed for the smallest nanoclusters (NL ) 4 and 5), but the effect is less pronounced as the number of available surface sites increases with NL, and a minimum is observed at NL ) 10, at all temperatures. At high overall Cu concentration, xCu ) 0.75 (Figure 1, bottom), a similar pattern is observed, but the minimum segregation shifts to lower cluster size, between NL ) 6 and 7, depending on temperature. At the lowest calculated temperature, T ) 10 K (not shown), segregation is total, and it is practically independent of cluster size and overall composition. Further discussion demonstrates that the cluster size dependence of R is related to the dispersion, defined as
Notes
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Figure 2. Calculated distribution of Cu (dark) and Ni (light) atoms on a NL ) 7 nanocluster (73 atoms) at 300 K and at three overall atomic fractions xCu. The evolution of the Cu surface segregation pattern is observed on the different crystallographic faces.
the ratio of surface atoms over total number of atoms in a given volume. Details of the microscopic mixing can be visualized in Figure 2 that depicts nanoclusters containing 743 atoms at 300 K at three Cu overall concentrations. At xCu ) 0.05, Cu atoms occupy corner and edge sites preferentially. At xCu ) 0.25, Cu atoms occupy almost totally the “side” (110) surfaces and partially the lowest coordination sites of the (331) surfaces. Other sites on (331) surfaces become occupied as the Cu overall concentration increases (xCu > 0.50), until all the surfaces are covered by Cu atoms. Thus, surface segregation is strongly dependent on the crystallographic face, and nanoclusters of less than a thousand atoms are much more susceptible to these variations. Since corner and edge sites are shared by all faces converging to these sites, we have separated the effects by computing the average fraction of Cu atoms present at three different locations: corners and edges, lateral (110) and (331) faces, and (111) faces. This is illustrated in Figure 3, where the vertical axis is the average Cu fraction on one of these three locations, yCu(hkl) ) {average number of Cu atoms on (hkl) faces}/ {total number of surface sites on (hkl) surfaces}. The local surface Cu composition is clearly dependent on the overall Cu concentration, cluster size, and crystallographic face. Corners and edge sites are definitely preferred by Cu atoms, at all Cu overall concentrations and for all cluster sizes. For xCu < 0.50 the preferred occupation sequence for Cu atoms is on corners and edge sites, on lateral (110) and (331) faces, and, last, on the (111) surfaces. The same behavior is observed at a Cu overall concentration of 0.50, for nanoclusters of NL < 6. At NL ) 6, there is a switching in the behavior, and after corners and edges, the (111) face is preferred over the lateral faces at all overall compositions. At NL ) 7, the behavior comes back to that observed for NL < 6. But, at NL g 8 the Cu occupation sequence becomes equal to the one found for NL ) 6. To understand these features, note that we have grouped (110) and (331) surfaces as “lateral” surfaces. When the overall Cu concentration is sufficiently high, the (111) sites start to be preferred over the (331) sites. This is evident by analyzing the (331) surface structure, which contains three rows of atoms located on successive planes. Thus, the “interior-row” atom sites are less favorable than the (111) sites, and therefore, they remain occupied by Ni atoms, whereas the excess of Cu atoms locate on the (111) faces. At even higher Cu concentrations, xCu g 0.75 the (111) surface is preferred to the (331) for all cluster sizes,
Figure 3. Calculated average Cu atomic fraction (yCu(hkl)) on corners and edge sites (triangles), (111) fcc sites (circles), and lateral (110) + (331) fcc sites (squares) as a function of the nanocluster size (NL) and overall atomic fraction (xCu): (a) at 300 K; (b) at 923 K.
which makes the (111) fraction higher than the combined (110) and (331) surface fraction (Figure 3a). Similar features are observed at high temperatures, but much less pronounced, since high temperatures favor micromixing. At 923 K (Figure 3b), at xCu ) 0.50 the change of behavior detected between NL ) 6 and 7 at 300 K is shifted to a higher cluster size, between NL ) 7 and NL ) 8, where after corners and edges, the (111) faces are preferred over the lateral faces. The observed cluster size dependence may be explained in terms of the dispersion, defined as the ratio of surface
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Figure 4. Total dispersion function, defined as the ratio of surface atoms over total number of atoms in a (NL)3 volume, and partial dispersion functions associated with surface atoms on corner and edges, (111) sites, and (110) + (331) sites.
atoms, NS, to total number of atoms, NTotal.25 For the fcc clusters, the total dispersion given by eq 4 decreases when NL increases, as it is shown in Figure 4.
NS 2(NL)2 + [2(NL) + 2(NL - 2)](NL - 2) ) (4) NTotal (NL)3 The right-hand side of eq 4 can be separated into three terms: the first one corresponding to the (111) faces, the second to lateral (331) and (110) faces, and the third term from corner and edge sites:
NS (NL - 2)2 (4NL - 8)(NL - 2) )2 + + NTotal NL3 (NL)3 12NL - 16 (5) NL3 The partial dispersions are also shown in Figure 4. While the corners and edges contribution to dispersion is a monotonically decreasing function of the cluster size, the other two partial dispersion functions show maxima at about NL ) 6. Nanoclusters can be classified into three groups according to their dispersion behavior. The first group is for NL < 6, where the corners and edges dispersion (23) Yang, L.; Raeker, T. J.; DePristo, A. E. Surf. Sci. 1993, 290, 195. (24) Foresman, J. B.; Frisch, A. Exploring Chemistry with Electronic Structure Methods, 2nd ed.; Gaussian, Inc.: Pittsburgh, PA, 1993. (25) Polak, M.; Rubinovich, L. Surf. Sci. Rep. 2000, 38, 127-194.
Notes
term is the dominant contribution. The second set is for 6 e NL e 9 where the three terms are comparable. Finally, a third group is that of nanoclusters with 103 or more atoms, where the dispersion term due to corners and edges becomes negligible compared to those from the surface planes and the total dispersion tends to the bulk behavior. The three groups can be identified from Figure 1 at low, intermediate, and high overall concentrations and from Figure 3 in relation to segregation behavior on the several sites associated with the partial dispersion functions. Note that eqs 4 and 5 are written for a fcc stack as described in this study, and they are system-independent. If the suggested dispersion analysis were the only effect ruling surface segregation, a transition should take place at NL between 6 and 9 for any bimetallic system. However, due to the intrinsic interatomic interaction energies, this threshold value can be shifted to higher or lower cluster sizes, depending on the bimetallic system. For the Cu-Ni system, we have shown that for nanoclusters containing less than about 1000 atoms, the surface segregation behavior depends on crystallographic face, cluster size, overall concentration, and temperature, whereas a tendency to the bulk behavior is starting to be shown in the largest nanocluster investigated (8000 atoms). Threshold values between NL ) 6 (216 atoms) and NL ) 9 (729 atoms) were manifested in the several properties investigated (Figures 1 and 3). Conclusions The Cu surface segregation pattern found for Cu-Ni nanoclusters from 64 to 8000 atoms results from a competition between thermodynamic and geometric (size, shape) effects. The surface sites are sequentially occupied by Cu atoms, which locate on corners and edges, (110), (111), and finally, at high overall compositions (g75% Cu), they fill the (331) inner sites. Transitions in the segregation behavior on the different crystallographic faces are found for a nanocluster size between NL ) 6 (216 atoms) and NL ) 9 (729 atoms), with a composition threshold detected at 50% atomic fraction of Cu. Geometric effects dominate on nanoclusters up to about 1000 atoms. Above this size, thermodynamic effects result in the segregation behavior reported for macroscopic Cu-Ni systems. Acknowledgment. This work is supported by the National Science Foundation (Career Award Grant CTS9876065), and by the Army Research Office Grant No. DAAD19-00-1-0087. Supercomputer resources from NCSA and NERSC are gratefully acknowledged. LA0014306
