Size-Dependence of Adsorption Properties of Metal Nanoparticles: A

Dec 4, 2008 - ... Christian Glinsvad , Frank Abild-Pedersen , Jeff Greeley , Karsten W. Jacobsen , and ... George Pal , Georgios Lefkidis and Wolfgang...
0 downloads 0 Views 291KB Size
Download PDF
J. Phys. Chem. C 2008, 112, 20269–20275

20269

Size-Dependence of Adsorption Properties of Metal Nanoparticles: A Density Functional Study on Palladium Nanoclusters Ilya V. Yudanov,†,‡ Manuela Metzner,† Alexander Genest,† and Notker Ro¨sch*,† Department Chemie, Theoretische Chemie, Technische UniVersita¨t Mu¨nchen, 85747 Garching, Germany, and BoreskoV Institute of Catalysis, 630090 NoVosibirsk, Russian Federation ReceiVed: August 25, 2008

Interatomic distances in metal nanoparticles are reduced from their values in the bulk. We studied computationally how this size-dependent geometry change (from the bulk) relates to the size-dependence of other properties of large metal clusters, including their reactivity. For this purpose, using an all-electron scalarrelativistic density-functional approach, we calculated structures and binding energies for the example of CO adsorption on 3-fold hollow sites at the center of (111) facets of cuboctahedral nanoscale clusters Pdn (n ) 55-260). The average nearest-neighbor Pd-Pd distance of optimized structures is 4-7 pm (2-3%) shorter than the extrapolated limit of the lateral distance within an infinite (111) surface. In consequence, the energy of CO adsorption on a cluster of ∼100 atoms is ∼15 kJ mol-1 smaller than the extrapolated limit. On the basis of these results, we suggest a strategy for modeling particles of larger size, e.g. of 1000 atoms and more, with the help of smaller model particles of up to ∼300 atoms where one keeps the core of a model cluster fixed at the bulk structure and restricts the structure optimization to the outermost shell of cluster atoms. 1. Introduction Clusters constitute an intermediate state of matter between isolated molecular species and solids.1,2 Their physical and chemical properties are size-dependent3 and thus tunable. This effect is particularly important for the synthesis of nanostructured materials2,4,5 and for heterogeneous catalysis by metal particles.6,7 Modern experimental techniques permit one to design and characterize model catalysts in the form of metal nanoparticles of well-defined structure, grown on oxide surfaces.8-15 For instance, palladium nanoparticles supported on alumina and silica films and their characteristics for adsorption and oxidation of CO were intensively studied.6,10-13 To explore catalytic properties of nanosized metal catalysts, we developed an approach based on three-dimensional symmetric model clusters of fcc metals terminated by low-index surfaces.16,17 This nanocluster strategy allowed us to calculate the electronic structure of particles and catalytic reactions on model clusters with a diameter of 1 nm and beyond at an accurate DFT level.16,17 With a CO probe on cuboctahedral Pd particles we demonstrated that these symmetric three-dimensional cluster models of ∼100 metal atoms afford an essentially quantitative description of various adsorption properties of a single crystal (111) surface or of (111) facets of model catalyst particles that are usually more extended than the model clusters.17 Furthermore, we studied the stabilization of carbonaceous CHx species (x ) 0-3) at different sites of Pd nanoparticles,18-20 including subsurface sites for atomic carbon.19 Very recently, we employed this modeling strategy to explore how the C-O bond breaks during methanol decomposition on nanocrystallites of palladium catalysts.21 * Corresponding author. † Department Chemie, Theoretische Chemie, Technische Universita¨t Mu¨nchen. ‡ Boreskov Institute of Catalysis.

An important issue of this modeling technique is the choice of the “lattice” parameters to be employed in the cluster models. Previously, we used mainly two variants for describing the structure of the substrate cluster:16-21 (i) a bulk-terminated geometry where Pd-Pd nearest-neighbor distances were fixed at 275 pm, the experimental value of Pd bulk,22 and (ii) full optimization of the cluster structure within selected symmetry constraints. Already at a cluster size of ∼80 atoms the bulkterminated approach yields adsorption energies on (111) facets of clusters close to those obtained for an extended surface.16 In contrast, the optimized clusters of the series Pd79, Pd116, Pd140 exhibit considerably reduced average nearest-neighbor distances dj(Pd-Pd); also, the adsorption energy of CO on these clusters is ∼15 kJ mol-1 lower than on clusters of bulk-terminated structure.16 Ultimately, our models are intended to describe model catalyst particles that comprise several thousand metal atoms,17 i.e., systems much larger than the clusters for which an accurate DF treatment is feasible, but still far from the bulk limit. The local structure of adsorption sites and the reactivity of metal particles is expected to reflect the size-dependence of the nearestneighbor distance. In the present work we studied these issues with the specific goal to design suitable models for estimating adsorption properties of such large particles. Here we intend to operate in the size range where deviations of particle properties from values of the bulk phase scale with particle size. Only the smallest cluster model considered in the present work, Pd55, according to its adsorption properties, does not fall into this size range; instead, it belongs to the class of particles that comprise only few dozens of atoms and thus are small enough, so that, due to quantum size effects, their (adsorption) properties can no longer be deduced or extrapolated from those known for larger sizes (the so-called “non-scalable size regime”23). Previously,20 we applied an intermediate approach to a Pd79 cluster where only the outermost shell of the cluster (60 surface atoms) was relaxed while the structure of the “core”, a Pd19

10.1021/jp8075673 CCC: $40.75  2008 American Chemical Society Published on Web 12/04/2008

20270 J. Phys. Chem. C, Vol. 112, No. 51, 2008

Yudanov et al. TABLE 1: Structural Characteristics of Cuboctahedral Pdn Cluster Models n a

Nun NaVb nICc nOSd Rpe

55

79

116

140

260

5 7.85 13 42 0.58

6 8.51 19 60 0.65

7 8.90 38 78 0.74

8 9.09 44 96 0.79

12 9.60 116 144 0.97

a

Number of atoms that are inequivalent by Oh symmetry. Average coordination number of the atoms. c Number of atoms in the core of the cluster. d Number of atoms in the surface (outer shell) of the cluster. e Effective cluster radius (in nm); see the Appendix. b

Figure 1. Sketches of the cuboctahedral clusters Pd55, Pd79, Pd116, Pd140, and Pd260.

octahedron, was kept fixed at the truncated bulk geometry. In the present work we justify this latter approach and develop it further: using a series of cluster models Pdn (n ) 55, 79, 116, 140, 260) we consider in detail how local structural parameters of larger clusters (e.g., the distance between Pd nearest neighbors) depend on the cluster nuclearity and, subsequently, how these structural parameters affect the calculated adsorption energy of a CO probe molecule. Finally, we systematically explore the “frozen-core” approach and discuss to what extent such models are applicable when describing properties of particles in the range of 103-104 atoms. 2. Computational Details and Cluster Models We carried out all-electron calculations using the linear combination of Gaussian-type orbitals fitting-functions DF (LCGTO-FF-DF) method24 as implemented in the parallel code PARAGAUSS.25,26 We used the scalar relativistic variant of the LCGTO-FF-DF method which employs a second-order Douglas-Kroll transformation of the Dirac-Kohn-Sham equation.27,28 We optimized the structures of Pdn clusters with the local density approximation (LDA-VWN parametrization)29 of the exchange-correlation (xc) functional, to ensure accurate bond lengths. Note that DF calculations with a generalized-gradient approximation (GGA) of the xc functional significantly (by ∼10 pm) overestimate Pd-Pd distances.30,31 The resulting LDA structure of a Pdn cluster was kept fixed when we subsequently calculated the adsorption of CO: only the positions of the atoms C and O at the cluster surface were optimized when determining the adsorption energy Eads. These latter calculations were carried out with a GGA xc functional, in most cases BP86,32,33 but we also used PW9134 and RPBE.35 The resulting Eads value was corrected for the basis set superposition error with the counterpoise technique.36 For other technical details, in particular the Gaussian-type orbital and auxiliary basis sets, we refer the reader to a previous investigation.16 We studied the adsorption of CO on a series of cuboctahedral clusters: Pd55, Pd79, Pd116, Pd140, and Pd260 (Figure 1, Table 1). To preserve Oh symmetry, we deposited one CO molecule at

the central 3-fold hollow site, 3f, of each (111) facet, 8 CO molecules per cluster. When modeling adsorption on bridge sites of cluster edges, CO molecules were symmetrically deposited at the central bridge sites on each edge of the cluster Pd140, i.e., 12 CO molecules in total. We used three types of cluster models, differing in how the cluster geometry was established: (i) the nearest-neighbor distances d(Pd-Pd) were fixed uniformly throughout the whole cluster; three values were used to span the range from clusters of ∼102 atoms to the bulk limit: 265, 270, and 275 pm; (ii) the cluster structure was fully optimized under restrictions of Oh point group; (iii) a fixed-core approach was invoked, according to which the distances d(Pd-Pd) were fixed uniformly throughout the “inner core” of the cluster (at 265, 270, or 275 pm) while the outermost cluster shell was relaxed. The number of atoms in the core region and the outer shell of each cluster model are specified in Table 1. As all geometry optimizations in approaches (ii) and (iii) were performed for bare clusters only (see above), adsorbate-induced substrate relaxation was neglected in these models. 3. Results and Discussion 3.1. Size Dependence of Bond Lengths in Pdn Nanoparticles. The contraction of the lattice parameter of small metal particles with decreasing size was first detected experimentally for Ag particles37 and since that time was proved for many other metals, including Pd.38 The contraction of bond lengths in metal clusters with decreasing size has been rationalized as a result of decreasing coordination number Nav ∼ Rp (Table 1) with their higher surface-to-volume ratio.30,31,40 It is now well-established by computational studies of various metals, Ni,39 Au,40 and Pd,30,31,41 that relaxed structures of metal clusters exhibit shorter interatomic distances than bulk crystals. This effect was first demonstrated computationally by relaxing the (totally symmetric) breathing mode of clusters,30,39-41 i.e. the uniform nearest-neighbor distance was optimized as a single parameter of the cluster. Later, the same trend was observed for Pd clusters with independently optimized positions of nonequivalent Pd centers.16,31 The average calculated Pd-Pd nearest-neighbor distance, dj, increases linearly with decreasing value of n-1/3 (Figure 2; note the inverted scale of the abscissa), where n is the number of atoms in the cluster:

dj ) d0 - τn-1/3 1/3

(1)

The value of n is proportional to the linear dimension of a bulky nanoparticle, e.g. the effective radius Rp of a quasispherical particle (Table 1 and Appendix). Although the data points in Figure 2 are extended compared to our previous study,16 by results for the larger cluster with n ) 260 and a series of smaller clusters with n ) 6, 13, and 19, extrapolation of the average Pd-Pd bond distance to infinite cluster size, n-1/3

Palladium Nanoclusters

J. Phys. Chem. C, Vol. 112, No. 51, 2008 20271 TABLE 3: Adsorption Energies of CO, kJ mol-1, and Selected Geometry Characteristics in pm of Pdn Clusters with Different Treatment of Cluster Geometry: Fixed Structure (f) with Specified d(Pd-Pd); Fixed Core (fc) and Optimized Outer Shell n

da

djb fc

djOSc fc

Eadsd fc

Eadse f

55

265.0 270.0 275.0 265.0 270.0 275.0 265.0 270.0 275.0 265.0 270.0 275.0 265.0 270.0 275.0

266.1 267.2 268.4 266.6 267.9 269.4 266.6 268.5 270.5 266.9 268.8 270.8 266.7 269.4 272.1

265.1 266.3 267.6 265.7 267.0 268.3 265.8 267.5 269.3 266.0 267.7 269.5 266.0 268.5 271.1

116.0 120.8 125.8 155.1 158.9 162.6 155.1 161.8 168.0 150.9 157.6 164.0 154.0 162.3 169.8

114.3 133.1 151.7 141.6 160.1 176.9 145.9 162.1 176.8 141.4 156.1 170.0 146.9 158.9 170.3

79 116 Figure 2. Average bond distances dj(Pd-Pd) of Pdn clusters as a function of n-1/3 calculated at LDA level. Exrapolation to infinite cluster size yields dj(Pd-Pd) ) 273.1 pm. The experimental value of bulk Pd, 275 pm, is indicated by the dashed horizontal line.

TABLE 2: Calculated Distances (pm) of Pdn Clusters Optimized at the LDA Level with Adsorption Characteristics of CO (pm and kJ mol-1) Determined at the GGA Level dj(Pd-Pd)a djIC(Pd-Pd)b djOS(Pd-Pd)c dmin,OS(Pd-Pd)d dmax,OS (Pd-Pd)d dPd3,OS(Pd-Pd)e dmin(Pd-Pd)f dmax(Pd-Pd)f Eads(CO)g d(Pd-C)h d(C-O)h

Pd55

Pd79

Pd116

Pd140

Pd260

266.2 265.4 265.1 263.4 266.9 266.9 261.3 270.9 116.4 206.5 118.7

267.4 268.1 266.4 263.7 270.0 270.0 263.7 270.4 158.5 205.3 118.9

267.8 268.1 266.8 264.5 269.7 269.7 263.5 270.9 159.5 207.1 118.6

268.2 268.5 267.1 263.4 270.6 270.6 263.4 271.5 156.6 207.8 118.5

268.8 268.9 267.9 264.8 270.1 270.1 263.9 271.4 160.6 208.7 118.3

a Average nearest-neighbor distance of the whole cluster. Average nearest-neighbor distance in the core of the cluster. c Average nearest-neighbor distance in the surface of the cluster. d Minimum (dmin) and maximum (dmax) Pd-Pd distances in the surface of the cluster. e Distance between three Pd atoms which form a 3-fold hollow site in the center of (111) facet. f Minimum (dmin) and maximum (dmax) Pd-Pd distances in the cluster. g Energy of CO adsorption at the 3f position of (111) facet of cluster. h Geometry characteristics of CO adsorption complex at the 3f site in the center of (111) facets. b

) 0, yields almost the same limit d0 ) 273.1 pm, that is quite close to the experimental characteristics of bulk Pd, dexp ) 275 pm.22 A least-squares fit to these data yields τ ) 25.3 pm (eq 1). For particles with sizes of 100, 1000, and 10000 atoms, this relationship translates to dj that are 5.5, 2.5, and 1.2 pm reduced, respectively, compared to the extrapolated bulk limit. 3.2. Adsorption Energy of CO and Lengths of Metal-Metal Bonds in Pdn Nanoparticles. Previously, we demonstrated16 that the energy of CO adsorption at 3f sites of (111) cluster facets varies considerably with the nearest-neighbor distances of the metal cluster. Clusters with a bulk terminated geometry, i.e. with all d(Pd-Pd) ) 275 pm, show systematically higher adsorption energies of CO than clusters with optimized structures.16 For both series, bulk-terminated or optimized clusters with n > 55 (Pd79, Pd116, Pd140, and Pd260), we calculated the energy Eads of CO adsorption on 3f sites to vary within narrow margins: between 170 and 177 kJ mol-1 for clusters with bulkterminated geometry and between 158 and 161 kJ mol-1 for optimized clusters. The results for the largest cluster studied in the present work, Pd260 with a diameter of about 2 nm, also fall into these ranges (Tables 2 and 3): Eads ) 170.3 kJ mol-1 for the cluster with bulk-terminated structure [d(Pd-Pd) ) 275 pm] and Eads ) 160.6 kJ mol-1 for the geometry optimized under

140 260

a Fixed nearest-neighbor Pd-Pd distance fixed either for the whole cluster (f) or the cluster core (fc). b Nearest-neighbor Pd-Pd distance averaged over the whole cluster with fixed core (at a given d in the core) and optimized outer shell. c Nearest-neighbor Pd-Pd distance averaged over the optimized outer shell of a cluster with fixed core. d Energy of CO adsorption at the 3f site of a cluster with fixed core (at a given d in the core) and optimized outer shell. e Energy of CO adsorption at the 3f site of clusters with a fixed uniform nearest-neighbor distance d.

Oh symmetry constraints. Apparently, an energy difference of 10-15 kJ mol-1 results when the cluster structure is compressed compared to the bulk metal. This finding also agrees with the principle of bond order conservation:42,43 in the optimized species, one expects weaker adsorption bonds and stronger binding within the adsorbate as a result of better saturated valences of the substrate atoms. In this context, one can mention the effect of strain on the reactivity of metal surfaces,44 where the activity of the surface, in particular adsorbate binding energies, have been shown to increase with increasing lattice constants. The strain effect in overlayers is associated with an upward shift of the valence d band with increasing lattice parameter.44 According to our calculations of Pd116 its vertical ionization potential, IP, decreases by 0.07 eV when the lattice parameter d(Pd-Pd) is extended from 265 to 275 pm, while the electron affinity, EA, remains essentially unchanged. Concomitantly, the optimized clusters Pd79, Pd116, Pd140, Pd260 exhibit IPs up to 0.09 eV higher than their congeners with bulk terminated structure (d ) 275 pm). This change of electronic properties corresponds to the upward shift of cluster d levels in clusters with higher value of dj(Pd-Pd) in agreement with the findings for overlayers.44 Figure 3a shows for the example of Pd79 how the adsorption energy of CO at a 3f site varies with the calculated structural constants: the values of Eads increase linearly with increasing Pd-Pd distance, again in obvious analogy to the strain effect found for metal overlayers.44 However, in contrast to the strain effect where interaction with the support or impurities induce an expansion of the lattice, the contraction of the structure with decreasing particle size is an intrinsic property of the (pure) metals (Figure 2) which entails weaker binding capabilities of adsorption sites of finite metal species compared to (infinite) metal surfaces, e.g. at 3f sites on (111) facets. Such effects are not restricted to (111) facets, but reflect a general trend that also holds for adsorption at other features of

20272 J. Phys. Chem. C, Vol. 112, No. 51, 2008

Yudanov et al. produce the best agreement with the experimental adsorption energy of 142-149 kJ mol-1.46,47 To reveal a general relationship between adsorption energy and lattice parameter of a cluster, we established a linear regression of the BP86 data of energies Eads, calculated for CO adsorption at 3f sites of (111) facets of Pd79, Pd116, Pd140, and Pd260 with d(Pd-Pd) ) 265, 270, and 275 pm (Table 3):

Eads ) E0 - ε(d0 - d)

(2)

The fit yields  ) 2.96 kJ (mol pm)-1. Assuming that results with bulk-terminated geometry oscillate around the infinite surface limit,16 we define E0 ) 173.5 kJ mol-1 as average over results for d(Pd-Pd) ) 275 pm. To deal with optimized clusters the experimental value d0 ) 275 pm can be replaced in eq 2 by the computationally extrapolated value 273 pm (eq 1, Figure 2). Substituting dj from eq 1 for d, we predict a relationship between Eads and the size n of the cluster: Figure 3. Correlation between CO adsorption energy, Eads, and a uniformly fixed structural constant d(Pd-Pd) of the model cluster Pdn: (a) Eads calculated with PW91 (circles), BP86 (squares) and RPBE (triangles) xc functionals for the 3f site of Pd79; (b) Eads at the BP86 level for 3f sites (squares) of Pd140 and the corresponding bridge sites at edges (rhombs).

the particles. As evidence, we compare in Figure 3b calculated energies Eads of CO adsorption at bridge sites on the edges of the cluster Pd140 to the results for the central 3f sites of the (111) facet of the same cluster. Obviously, both types of Eads values exhibit similar trends, although the increment of 2.03 kJ (mol pm)-1 for bridge sites is notably smaller than for hollow coordination, 2.86 kJ (mol pm)-1. Adsorption at low-coordinated sites in general is stronger than interaction with regular facets; we showed this in a previous computational study of CO adsorption at different sites of Pdn clusters.17 As the concentration of low-coordinated sites roughly changes with the surfaceto-volume ratio, i.e. with the inverse of the effective radius of a particle, nanoparticles and clusters exhibit a higher activity than extended low-index surfaces.16 An interesting, often addressed16,20,21 issue is the correspondence between adsorption properties calculated on cluster models with bulk-terminated geometry (Pd-Pd fixed at 275 pm) and results of slab model calculations. This body of work provided evidence that in most cases nanocluster models of ∼100 metal atoms (e.g., Pd79, Pd116), terminated by low-index planes, yield adsorption energies on a (111) facet that are close to the values calculated with slab models, as long as the same xc functional was used. However, in that approach, we neglected the fact that in slab model calculations geometries are usually optimized at the GGA level, yielding somewhat longer Pd-Pd distances, ∼280 pm, than experiment and LDA calculations, 275 pm. As shown in Figure 3a, expanding the structural constant from 275 to 280 pm increases the CO adsorption energy by about 15 kJ mol-1. Thus, for a fully consistent, quantitative comparison between cluster and slab models, one should describe the substrate by choosing the same structural constant and the same xc functional. Indeed, the CO adsorption energies calculated for models with d(Pd-Pd) ) 280 pm and the RPBE functional are quite close if one takes the differences in computational methodology into account (e.g., plane-wave vs Gaussian-type basis sets, pseudopotentials vs all-electron method): 164 kJ mol-1 from Pd79 and 152 kJ mol-1 with a slab model.45 Also the RPBE results for the bulk-terminated geometry (d ) 275 pm), 149 kJ mol-1 for Pd79 (cf. 142 kJ mol-1 for Pd140),16

Eads ) E0 - ετn-1/3

(3)

The slope is τ ) 74.9 kJ mol-1. With the substitution just mentioned, we postulated a correspondence between the average nearest-neighbor distance dj of optimized cluster structures and the uniformly fixed parameter d. Next, we will validate eq3 by calculating CO adsorption on Oh model clusters with a fully relaxed structure. 3.3. CO Adsorption on Pdn Nanoparticles with Optimized Structure. Table 2 summarizes the characteristics of the clusters Pd55, Pd79, Pd116, Pd140, and Pd260 for optimized structures. Besides the Pd-Pd distance averaged over the whole cluster, dj (see section 3.1), we also present nearest-neighbor distances averaged separately over the “core” region of the cluster, djIC, and its outer shell, djOS. To illustrate the dispersion of these distances we also list minimum and maximum values specific to the outer shell and for the whole cluster. Furthermore, we characterize the 3f adsorption site in the center of the (111) facets by the corresponding nearest-neighbor distances, dPd3,OS (Table 2). Not unexpectedly, the average nearest-neighbor distance of the core region is systematically larger than that of the outer shell. In view of the discussions in section 3.1, this finding is straightforward to rationalize: all atoms of the core region have 12 nearest neighbors, hence their coordination shell is saturated. In contrast, atoms of the outer shell, namely the surface atoms, have rather low coordination numbers. On the other hand, differences of the calculated characteristics between different clusters with n > 55 are rather small, due to the slow convergence of dj(Pd-Pd) as displayed in Figure 2. Indeed, the values of n-1/3 of the clusters Pd79 to Pd260 all fall within a narrow range of only 0.1. Figure 4 shows the energies of CO adsorption on Pd79, Pd116, Pd140, and Pd260 as a function of n-1/3. Remarkably, the Eads values for optimized clusters fall within a very narrow interval, from 157 to 161 kJ mol-1. A trend to larger adsorption energies for larger clusters is obvious (Figure 4) when one compares the smallest and the largest clusters, Pd79 and Pd260, or in the pairs Pd79-Pd116 and Pd140–Pd260, (the (111) facets are very similar within these pairs, Figure 1). However, this trend is not monotonic as Eads calculated for the rather large cluster Pd140 is smaller than that for Pd79 and Pd116 (Figure 4). Thus we, are lead to conclude that differences in cluster shape affect the adsorption properties in a rather strong fashion. In this range of cluster nuclearities, this influence is comparable to the size effect. In fact, from eq 3, one would expect Eads to increase by ∼7 kJ mol-1 on going from Pd79 to Pd260, while the actual

Palladium Nanoclusters

Figure 4. Adsorption energies Eads (at the BP86 level) of CO at 3f sites of Pdn clusters (n ) 79, 116, 140, 260) with optimized geometries (triangles) as a function of n-1/3. The estimate for an infinite (111) surface, E0 ) 173.5 kJ mol-1, indicated by the dashed line, is obtained as average over clusters with uniformly fixed distances d(Pd-Pd) ) 275 pm (circles). The solid line represents the correlation established by eq 3.

increase is only 2 kJ mol-1. Oscillations of the Eads values of up to 10 kJ mol-1, induced by variations of the cluster shape, were observed also for clusters with bulk-terminated geometry;16 see the results for clusters with uniformly fixed nearest-neighbor distances of 275 pm, also shown in Figure 4. Therefore the expected regular increase of adsorption energies due to growing cluster size is significantly modulated by effects due to variations in the shape of the clusters. Nevertheless, the linear relationship defined by eq 3, with E0 ) 173.5 kJ mol-1 for the infinite size limit (section 3.2), nicely falls into the region of the data points of the fully optimized structures of Pd79 to Pd260 (Figure 4). Indeed, this finding confirms that the optimized structures do not contradict the trend of growing Eads, despite their strong scattering. On the basis of these results, we may use eq 3 to estimate Eads of CO on 3f sites of (111) facets for clusters of any size in the range n > 260. Moreover, one may conjecture that the procedure, used in section 3.2 to arrive at eq 3, can be generalized to other adsorption sites and adsorbates. 3.4. Fixed-Core Approach for Pdn Clusters. Finally, we tested the approach to cluster models of adsorption where the core of cluster is kept fixed at a uniform Pd-Pd distance and only the structure of the surface shell is optimized. Table 1 shows the numbers of core and surface atoms for each of the clusters considered here. Obviously, the ratios between the number of core and surface atoms increases with cluster size; hence, one anticipates a stronger effect for larger cluster models. The calculated characteristics of the frozen-core clusters Pd79, Pd116, Pd140, and Pd260 are shown in Table 3 and compared to the results for the corresponding fully optimized clusters. As above for the clusters of fixed geometry, we assigned the values of 265, 270, and 275 pm to the structural constant of the frozen cores. The average nearest-neighbor distances in the core of the fully optimized clusters Pd79 to Pd260 range from 268 to 269 pm (Table 2). Thus, the fixed-core parameter d(Pd-Pd) ) 265 pm appears to fall outside the physically interesting range. Indeed, the average nearest-neighbor distance in the optimized outer shell is larger than the fixed distance of the frozen core, 265 pm. This situation seems unphysical as in fully optimized structures nearest-neighbor distances in the outer shell are always shorter than in the cluster core; cf. values djOS and djIC in Table 2. On the other hand, the frozen-core distance of 270 pm is only slightly, by 1-2 pm, longer than the average nearestneighbor distance determined for the cores of the optimized structure, as shown by the data of Table 2 for the cluster Pd79 to Pd260. This minor expansion of the core region induces a

J. Phys. Chem. C, Vol. 112, No. 51, 2008 20273

Figure 5. Adsorption energies Eads (at the BP86 level) of CO on 3f sites of Pdn clusters (n ) 79, 116, 140, 260) as a function of the average distance dj(Pd-Pd) from three computational strategies: (i, circles) cluster geometry fully optimized with Oh symmetry constraints; geometry of the outer shell (surface) optimized with nearest-neighbor Pd-Pd distance of the cluster core fixed at (ii, squares) 270 pm and (iii, triangles) 275 pm. The solid line results from a least-squares fit of the entire data set. The dashed line indicates the same limiting value as in Figure 4.

slight expansion of the surface shell: for the fixed-core models djOS is 0.6-0.7 pm longer than in the fully optimized structures (Table 3). Also, the adsorption energies of CO on clusters with a fixed-core parameter of 270 pm are 0.5-2.3 kJ mol-1 larger than on the Pdn congeners with fully optimized structures (cf. Eads in Tables 2 and 3). As expected, the most pronounced effects in these fixed-core models are observed when a strong perturbation of the cluster core is introduced by fixing its structure to that of Pd bulk, i.e., with d ) 275 pm. In this case the values of djOS are 1-3 pm larger than in the corresponding cluster with fully optimized structures. We determined the largest expansion, by 3.2 pm, for the largest cluster in series, Pd260. This is not accidental because for this cluster the number ratio nIC/nOS ) 0.80 of the atoms in the core and the surface regions is considerably larger than for other clusters of the series (Table 1). Such a notable expansion of the structure entails a significant increase of the CO adsorption energy (Table 3). Concomitantly, for Pd260, we calculated the largest adsorption energy, Eads ) 169.8 kJ mol-1. This value is quite close to the BP86 limit value, 173.5 kJ mol-1, estimated above. Figure 5 presents the calculated adsorption energies of CO on the clusters Pd79, Pd116, Pd140, and Pd260 as a function the nearest-neighbor distance dj(Pd-Pd) averaged over the entire cluster. Obviously, all series show similar linear trends, irrespective of how the structure was determined: (i) fully optimized, or optimized for a fixed core with the parameter (ii) 270 pm and (iii) 275 pm. A joint least-squares fit of all data points from the three series yields the extrapolated value of Eads ) 177 kJ mol-1 for an infinite structure, in reasonable agreement with the above estimate of 173.5 kJ mol-1 (dashed line in Figure 5). For clusters with a fixed-core parameter of 270 pm, similar CO adsorption energies were calculated as for the corresponding particles with a fully optimized structure. In contrast, the results for clusters with the distances of the core fixed at 275 pm in general fall into a range that is expected for Pdn species with n > 260, if their structure were fully optimized. To quantify this observation, one may derive an effective particle size neff by inverting eq 1

neff ) [τ/(d0 - dj)]3

(4)

using the calculated average bond lengths dj(Pd-Pd) for results from clusters with a fixed-core parameter 275 pm. According

20274 J. Phys. Chem. C, Vol. 112, No. 51, 2008

Yudanov et al.

a ) 2√2rat

to eq 4, the fixed-core clusters Pd79, Pd116, Pd140, and Pd260 reproduce the adsorption properties of particles with about 300, 900, 1300, and 15 300 atoms, respectively. Thus, this “fixedcore” approach provides an efficient tool for filling the gap which exists in computational studies on nanoparticles:16 these “fixed-core” models allow one to treat particles with sizes anywhere beyond 200-300 atoms.

If one requires the joint volume of four atomic spheres, with an effectiVe atomic radius reff, to match the volume of the unit cell,

4. Conclusions

one obtains:

Using a DF approach, we studied with cuboctahedral nanoclusters how structural and adsorption properties of metal nanoparticles scale with particle size. In line with results of a previous study on Pdn clusters,16 we found a linear correlation between the average bond length and n-1/3 ∼ Rp-1 where Rp designates the radius of the nanoparticle. In the all-electron calculations of this work, n ranged from 6 to 260. In addition, we determined that the adsorption energies of CO at standard positions [3-fold hollow site at (111) facets, bridge sites at cluster edges, etc.] increase linearly with the average nearest-neighbor distance of the metal cluster. This latter finding is reminiscent of strain effects known from slab model calculations.44 Based on these two correlations we established a linear relationship between the size of a particle and the corresponding adsorption energy of CO, which is valid for a cluster size as small as ∼80 atoms and converges to the correct infinite surface limit. The average Pd-Pd distance increases by ∼6 pm (2%) when the particles grow from about 100 atoms to essentially infinite size; simultaneously the CO adsorption energy on (111) facets strengthens by ∼20 kJ mol-1 (12%). We expect that a similar relationship between adsorption energies and the size of nanoclusters can be found for other adsorbates and substrates. The present study provided a rich set of data to judge in detail the accuracy of such scaling relationships. In the range of the present work with nanoparticles of about 80 to about 250 atoms, often scaling of properties is expected23 and from one point of view, the present study confirmed this expectation. However, the computational results also showed that, if inspected at sufficient detail, properties, e.g. binding energies of adsorbates, may exhibit shape effects even within a series of relatively similar particles of about 100-200 atoms. Finally, we proposed a “fixed-core” approach which extends our technique of modeling adsorption properties of nanoparticles16 and fills the gap that existed for n ranging from 103 to 104. In these model clusters, the structural parameters of the bulk metal are fixed for the core of cluster and only positions of atoms in the surface of the cluster are optimized. For instance, as judged by calculated CO adsorption energies, with this approach one is able to estimate properties of species with ∼1000 atoms using computational results for the nanocrystallites Pd116 and Pd140, while Pd260 is suitable for modeling particles of about 15000 atoms, with diameters of 7-8 nm. Acknowledgment. We thank S. Kru¨ger for stimulating discussions. I.V.Y. is grateful to the Deutsche Forschungsgemeinschaft (DFG) for making possible his visit to the TU Mu¨nchen. M.M. was supported by Elitenetzwerk Bayern within the program NanoCat. This work also benefited from financial support by the Russian Foundation for Basic Research (06-0333020), Deutsche Forchungsgemeinschaft, and Fonds der Chemischen Industrie (Germany). Appendix: Effective Radius Rp of a Particle The cubic unit cell of a face-centered cubic (fcc) lattice with edge a contains four atoms of radius rat and one has:

a3 ) 4

4π 3 r 3 eff

reff ) (3√2 ⁄ π)1/3rat ≈ 1.11rat

(5)

(6)

(7)

In the same spirit, using a spherical approximation for the cluster shape, one may define an effective radius Rp of a particle of n atoms by comparing volumes:

4π 3 4π R ) n reff3 w Rp ≈ 1.11n1/3rat 3 p 3

(8)

Table 1 shows the effective radii Rp ≈ n1/3 × 152 pm for Pdn clusters that result from the atomic radius rat ) 0.5 × 275 pm. References and Notes (1) Haberland, H., Ed. Clusters of Atoms and Molecules; Springer Series in Chemical Physics 52; Springer: Berlin, 1994; Vol. 1. (2) Braunstein, P., Oro, L. A., Raithby, P. R., Eds. Metal Clusters in Chemistry; Wiley-VCH: Weinheim, Germany, 1999; Vols. 2 and 3. (3) Ro¨sch, N.; Pacchioni, G. In Clusters and Colloids-From Theory to Applications; Schmid, G., Ed.; Verlag Chemie: Weinheim, Germany, 1994; p 5. (4) Teranishi, T; Miyake, M. Chem. Mater. 1998, 10, 594. (5) Xiong, Y. J.; Xia, Y. N. AdV. Mater. 2007, 19, 3385. (6) Ba¨umer, M.; Freund, H.-J. Prog. Surf. Sci. 1999, 61, 127. (7) Henry, C. R. Surf. Sci. Rep. 1998, 31, 231. (8) Dellwig, T.; Rupprechter, G.; Unterhalt, H.; Freund, H.-J. Phys. ReV. Lett. 2000, 85, 776. (9) Frank, M.; Ba¨umer, M. Phys. Chem. Chem. Phys. 2000, 2, 3723. (10) Libuda, J.; Meusel, I.; Hoffmann, J.; Hartmann, J.; Piccolo, L.; Henry, C. R.; Freund, H.-J. J. Chem. Phys. 2001, 114, 4669. (11) Unterhalt, H.; Rupprechter, G.; Freund, H.-J. J. Phys. Chem. B 2002, 106, 356. (12) Rupprechter, G.; Unterhalt, H.; Morkel, M.; Galletto, P.; Hu, L.; Freund, H.-J. Surf. Sci. 2002, 502-503, 109. (13) Giorgi, J. B.; Schroeder, T.; Ba¨umer, M.; Freund, H.-J. Surf. Sci. 2002, 498, L71. (14) Wo¨rz, A. S.; Judai, K.; Abbet, S.; Heiz, U. J. Am. Chem. Soc. 2003, 125, 7964. (15) Freund, H.-J. Top. Catal. 2008, 48, 137. (16) Yudanov, I. V.; Sahnoun, R.; Neyman, K. M.; Ro¨sch, N. J. Chem. Phys. 2002, 117, 9887. (17) Yudanov, I. V.; Sahnoun, R.; Neyman, K. M.; Ro¨sch, N.; Hoffmann, J.; Schauermann, S.; Joha´nek, V.; Unterhalt, H.; Rupprechter, G.; Libuda, J.; Freund, H.-J. J. Phys. Chem. B 2003, 107, 255. (18) Neyman, K. M.; Inntam, C.; Gordienko, A. B.; Yudanov, I. V.; Ro¨sch, N. J. Chem. Phys. 2005, 122, 174705. (19) Yudanov, I. V.; Neyman, K. M.; Ro¨sch, N. Phys. Chem. Chem. Phys. 2004, 6, 116. (20) Yudanov, I. V.; Neyman, K. M.; Ro¨sch, N. Phys. Chem. Chem. Phys. 2006, 8, 2396. (21) Yudanov, I. V.; Matveev, A. V.; Neyman, K. M.; Ro¨sch, N. J. Am. Chem. Soc., 2008, 130, 9342. (22) CRC Handbook of Chemistry and Physics, 77th ed.; Lide D. R., Ed.; CRC Press, Boca Raton, FL, 1996. (23) Landman, U.; Yoon, B.; Zhang, C.; Heiz, U.; Arenz, M. Top. Catal. 2007, 44, 145. (24) Dunlap, B. I.; Ro¨sch, N. AdV. Quantum Chem. 1990, 21, 317. (25) Belling, T.; Grauschopf, T.; Kru¨ger, S.; Mayer, M.; No¨rtemann, F.; Staufer, M.; Zenger C.; Ro¨sch, N. In High Performance Scientific and Engineering Computing, ; Bungartz, H.-J., Durst, F., Zenger, C., Eds.; Lecture Notes in Computational Science and Engineering 8; Springer: Heidelberg, Germany, 1999; p 439. (26) Belling, T.; Grauschopf, T.; Kru¨ger, S.; No¨rtemann, F.; Staufer, M.; Mayer, M.; Nasluzov, V. A.; Birkenheuer, U.; Hu, A.; Matveev, A. V.; Shor, A. M.; Fuchs-Rohr, M. S. K.; Neyman, K. M.; Ganyushin, D. I.; Kerdcharoen, T.; Woiterski, A.; Gordienko, A. B.; Majumder, S.; Ro¨sch, N. ParaGauss, Version 3.1, Technische Universita¨t Mu¨nchen: Munich, Germany, 2008. (27) Ha¨berlen, O. D.; Ro¨sch, N. Chem. Phys. Lett. 1992, 199, 491.

Palladium Nanoclusters (28) Ro¨sch, N.; Kru¨ger, S.; Mayer, M.; Nasluzov, V. A. In Recent DeVelopments and Applications of Modern Density Functional Theory; Seminario J. M., Ed.; Theoretical and Computational Chemistry Series 4; Elsevier: Amsterdam, 1996; p 497. (29) Vosko, S. H.; Wilk, L.; Nusair, M. Can. J. Phys. 1980, 58, 1200. (30) Kru¨ger, S.; Vent, S.; No¨rtemann, F.; Staufer, M.; Ro¨sch, N. J. Chem. Phys. 2001, 115, 2082. (31) Nava, P.; Sierka, M.; Ahlrichs, R. Phys. Chem. Chem. Phys. 2003, 5, 3372. (32) Becke, A. D. Phys. ReV. A 1988, 38, 3098. (33) Perdew, J. P. Phys. ReV. B 1986, 33, 8622. Perdew, J. P. Phys. ReV. B 1986, 34, 7406. (34) 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. (35) Hammer, B.; Hansen, L. B.; Nørskov, J. K. Phys. ReV. B 1999, 59, 7413. (36) Boys, S. F.; Bernardi, F. Mol. Phys. 1970, 19, 553.

J. Phys. Chem. C, Vol. 112, No. 51, 2008 20275 (37) Berry, C. R. Phys. ReV. 1952, 88, 596. (38) Lamber, R.; Wetjen, S.; Jaeger, N. I. Phys. ReV. B 1995, 51, 10968. (39) Pacchioni, G.; Chung, S.-C.; Kru¨ger, S.; Ro¨sch, N. Chem. Phys. 1994, 184, 125. (40) Ha¨berlen, O.; Chung, S.-C.; Stener, M.; Ro¨sch, N. J. Chem. Phys. 1997, 106, 5189. (41) Kru¨ger, S.; Vent, S.; Ro¨sch, N. Ber. Bunsen-Ges. Phys. Chem. 1997, 101, 1640. (42) Shustorovich, E.; Sellers, H. Surf. Sci. Rep. 1998, 31, 5. (43) van Santen, R. A.; Kramer, G. J. Chem. ReV. 1995, 95, 637. (44) Mavrikakis, M.; Hammer, B.; Nørskov, J. K. Phys. ReV. Lett. 1998, 81, 2819. (45) Lopez, N.; Nørskov, J. K. Surf. Sci. 2001, 477, 59. (46) Bradshaw, A. M.; Hoffmann, F. M. Surf. Sci. 1978, 72, 513. (47) Guo, X.; Yates, J. T., Jr J. Chem. Phys. 1989, 90, 6761.

JP8075673