Binding Energy of Metal Oxide Nanoparticles - ACS Publications

Mar 3, 2009 - particle diameter for spherical clusters or, equivalently, on the inverse of the cube root of the number of particles in the cluster. We...
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J. Phys. Chem. C 2009, 113, 4786–4791

Binding Energy of Metal Oxide Nanoparticles R. D. Parra*,† and H. H. Farrell‡ Department of Chemistry, DePaul UniVersity, Chicago, Illinois 60614, and Idaho National Laboratory, MS 2211, P.O. Box 1625, Idaho Falls, Idaho 83415 ReceiVed: August 07, 2008; ReVised Manuscript ReceiVed: December 23, 2008

Current models for the binding energy of nanoparticles generally predict a linear dependence on the inverse particle diameter for spherical clusters or, equivalently, on the inverse of the cube root of the number of particles in the cluster. We have tested this assumption on the simple oxides MgO, CuO, and TiO2 and for the complex oxide FeTiO3. We find that a linear dependence is not observed for these materials despite the fact that it generally occurs for simple metals. Instead, using first principles, density functional theory calculations to calculate the binding energy of these materials, we find an approximately quadratic dependence on the inverse of the particle size, in direct contradiction to current assumptions. This nonlinear dependence of the binding energy has substantial implications on physical properties that depend on the cohesive energy, particularly in the small particle regime. Ebn ) E∞b [1 - c′/n]

Introduction Though generally studied as solids, nanoparticles are a form of matter that is intermediate between the gaseous or molecular state and the bulk solid state. Consequently, they have physical properties that differ from those of the corresponding bulk materials. Although the borders between the nano and the bulklike regimes depend on the property being studied, most properties of nanoparticles with diameters less than about 10 nm appear to be very different from those from the bulk case.1 It is the purpose of this work to explore the binding energies in metal oxide clusters as a function of the size of the clusters. The results of this work will then help us understand better the fundamental physics and chemistry of the nanoparticle state. It is known that the internal cohesive energy or binding energy of nanoparticles or clusters is reduced relative to the bulk counterpart because their large surface to volume ratio decreases the number of nearest neighbor bonds. The binding energy is defined as the energy required to disassemble a whole into separate parts. Because a bound system has a lower potential energy than its constituent parts, the usual convention is to report binding energies as positive quantities. Current theories predict that the binding energy of clusters will be inversely proportional to the particle size, at least for low-aspect-ratio nanoparticles.2-5 In fact, Vanithakumari and Nanda make the statement that “According to all theoretical models, the cohesive energy of free nanoparticles decreases linearly with the inverse of the particle size.”2 While this is somewhat of a generalization, it is true that the current models, while they differ somewhat in detail, all predict that the per-atom binding energy, Ebn, for spherical nanoparticles having n3 ) N atoms all scale with size or n as either

Ebn ) E∞b [1 - c/r]

(1a)

(1b)

where Eb∞ is the bulk binding energy, c and c′ are materialdependent constants, and r is the radius of the cluster.2-5 Note that, for spherical nanoparticles, where the radius, r, is equal to nr0 and r0 is the effective radius for one atom or molecular unit, eq 1a and 1b are essentially equivalent. While a number of experimental studies on nanoparticles do exist,6-8 there are often difficulties in interpreting the experimental results, particularly for very small particles. These difficulties can involve a number of factors including inadvertent oxidation or reduction, size changes due to sintering or sublimation, substrate effects and artifacts, or uncertainties introduced by the measurement technique. To supplement these experimental studies, we have engaged in a series of theoretical studies of the binding energies of isolated nanoparticles. In an earlier work, we found that for many metals an inverse dependence on the nanoparticle size is actually a good approximation for describing the binding energy. However, for metals such as Pb and Sn, we found that the binding energy increases much more rapidly than 1/r or 1/n. 9 In addition, our work on the elemental semiconductors, C, Si, and Ge, also showed a nonlinear dependence of the binding energy on inverse particle size.9 We also found that the cohesive energy for small particles of AlP, GaAs, ZnSe, ZnS, and GaN varies nonlinearly in the small size regime.10,11 In this work, we have extended our studies to metal oxide clusters. We have considered four different materials, MgO, CuO, TiO2, and FeTiO3. The monoxides allow us to compare the refractory, highly ionic MgO with the softer, more covalent CuO. The selection of TiO2 extends this study to dioxides and to a material where the metal has a higher oxidation number. The complex oxide FeTiO3 gives us the opportunity to study an oxide with two different metal atoms with different metal-oxygen bond strengths. Method of Calculation

or * Towhomcorrespondenceshouldbeaddressed.E-mail:[email protected]. † DePaul University. ‡ Idaho National Laboratory.

First principles, density functional theory (DFT) calculations in the generalized gradient approximation (GGA) including spinpolarization were performed on a variety of different clusters

10.1021/jp807070a CCC: $40.75  2009 American Chemical Society Published on Web 03/03/2009

Binding Energy of Metal Oxide Nanoparticles

Figure 1. Binding energies for the (CuO)m clusters, where m ranges from 1 to 30, are shown as a function of 1/n, where n3 ) m. Here, we take the zero of energy to be the m ) 1 unit, and thus, in the graph, EB ) E1 - Em. The straight line is what would be expected based on a linear dependence of the binding energy to the average particle diameter. The curved, quadratic line fits the data according to the equation y ) 4.45x2 - 0.34x - 4.36 (R2 ) 0.98).

to obtain binding energies. The modeling program that was used for calculating the geometry and energetics of the clusters was the commercially available “DMol4.1” code.12,13 Unless otherwise noted, with this program, we used an all electron, relativistic, real-space numerical basis set (DND). In this code, the GGA is implemented with the Perdew-Wang-91 parametrization of the correlation energy.14 For the bulk reference materials, we first used the companion CASTEP program to calculate geometries and then used the DMol program to calculate binding energies. As a final point before turning to our calculations for the oxides, we would like to note that, as a check on the general validity of our computations, in our work on the group IV elements, we also performed calculations on a variety of clusters using both relativistic and nonrelativistic basis sets and within both the LDA and the GGA approximations.9 In all cases, a nonlinear dependence of Eb on 1/n was observed. In fact, for these materials, to a good approximation, a quadratic dependence was obtained. We take this result as one indication that this quadratic behavior is not an artifact of our computational methods. A number of authors have carefully studied the variations in binding energy as a function of the detailed geometry or configuration of oxide nanoclusters with a given number of atoms. These include, for example, the work by Chen et al. on BaO clusters,15 by Wang et al. on ZnO clusters,16 by Shen et al. on one-dimensional ZnO nanostructures,17 by Qu and Kroes on TiO2 clusters,18 and by Bawa and Panas on MgO, SrO, and BaO nanoclusters,19 among others. It should be noted that while we generally sought the most stable configuration for a given number of atoms, it was not our purpose to make a careful study of the variation in binding energy as a function of detailed configuration for a given cluster size. Rather, we sought the overall trend in binding energy as we varied the size from the smallest chemical unit (e.g., a diatomic or triatomic molecule) up to particles with on the order of 100 atoms (e.g., Mg50O50). Results and Discussion CuO. The binding energies for the (CuO)m clusters, where m ranges from 1 to 30, are shown as a function of 1/n, where n3 ) m, in Figure 1. Here, we take the zero of energy to be the m ) 1 unit. The point on the far right corresponds to the m ) 1 diatomic CuO unit. The point on the far left corresponds to m ) ∞, that is, the bulk material. The stable configurations for the clusters with relatively small numbers of atoms tended to

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Figure 2. Selected low-energy CuO clusters. Cu atoms are shown in blue, and oxygen in red.

be ringlike in form (e.g., square or hexagonal) as shown in Figure 2. However, a very interesting transformation occurred for larger units where the stable configurations appear to adopt the structure of “barrels” or truncated nanotubes as also shown in Figure 2. We have explored the formation of these novel, cylindrical structures in great detail, and the results will be published elsewhere.20 In brief, these structures appear to be the result of the nature of the bonding of the Cu and the O atoms. In the bulk, the CuO is characterized by a square coordination around the Cu atoms. This is a result of the fact that most of the Cu 3d orbitals are full and are therefore unavailable for hybridization and bonding, leaving only the dx2-y2 orbital (where z has been assumed to be perpendicular to the cylinder axis). This favors the formation of CuO sheets that, if enough units are present, can fold or curve over to form the cylindrical units in much the same fashion that the graphene sheets that comprise graphite can be “rolled up” to form carbon nanotubes. Returning to Figure 1, as can been seen, the binding energy is not linearly dependent on 1/n for clusters or particles in this size range. The nonlinearity behavior is in marked contrast to simple metal particles in the same regime and, interestingly enough, is reminiscent of the behavior of the semiconductor clusters.9-11 Here, possibly, the reason for this deviation from linearity can be attributed to the covalent nature of the relevant chemical bonds. However, this explanation becomes less satisfactory for other oxides, particularly in the case of the more ionic oxides such as MgO, to be discussed in the next section. MgO. The binding energies for the (MgO)m clusters are shown as a function of 1/n, where n3 ) m, in Figure 3. As before, for convenience, we take the zero of energy to be the m ) 1 unit. Unlike CuO, and even TiO2, where the geometry of the nanoclusters seems to be largely determined by partial covalent bonding between the hybridized 3d orbitals on the metal atom, the geometry of the most stable MgO nanoclusters appears to be derived primarily from relatively simple electrostatic interactions. Correspondingly, the shapes of the larger units are closer to what one might expect based on the structure of bulk MgO. The most stable configuration for the m ) 2 unit is an approximately square arrangement of two diametrically opposed Mg atoms and two O atoms. The most stable m ) 3 unit was not coplanar in a similar fashion to the results of Chen et al. for BaO.15 The structures for various larger clusters are shown in Figure 4. We found that the most stable clusters for these larger units were generally regular in shape (cubes, or rectan-

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Figure 3. Binding energies for the (MgO)m clusters, where m ranges from 1 to 50, are shown as a function of 1/n, where n3 ) m. Here, we take the zero of energy to be the m ) 1 unit and EB as in Figure 1. Open circles are for vacancy containing structures. Open triangles are for those structures formed by removing corner atoms. The curved, quadratic line fits the data according to the equation y ) 6.76x2 + 0.25x - 7.22 (R2 ) 0.99). In the fitting, we have ignored clusters with an odd number of atoms. Straight line as in Figure 1.

Figure 4. Selected low-energy MgO clusters. Mg atoms are shown in green, and oxygen in red.

gular solids), also consistent with the work of Chen et al. on BaO clusters.15 Specifically, the cubic cluster for m ) 4 has 8 atoms, symmetrically arranged, and forms the simplest threedimensional cluster. The m ) 32 cluster has 64 symmetrically arranged atoms. Note that cubes with odd numbers of atoms are less stable. This point will be discussed in more detail below. Of the rectangular solids, m ) 6, 18, and 40 have (2 × 2 × 3), (3 × 3 × 4), and (4 × 4 × 5) atoms, respectively. Similarly, the m ) 24 and 50 rectangular solids have (4 × 4 × 3) and (5 × 5 × 4) atoms, respectively. We also considered systems with less regular shapes. For example, an m ) 36 nanocluster was constructed from the m ) 40 cluster by removing the 8 (4 Mg and 4 O) corner atoms. This structure was less stable than either the parent m ) 40 or the slightly smaller m ) 32 clusters. Similarly, we formed an m ) 28 structure by removing the corner atoms (again 4 Mg and 4 O) from the m ) 32 cluster. The new m ) 28 structure was less regular than the similarly formed m ) 36 structure. The remaining surface atoms showed a tendency to form a slightly more cylindrical shape than the parent m ) 32 cluster. Finally, we studied several clusters with an odd number of atoms including the cube with 27 atoms. In the case where we had

Parra and Farrell

Figure 5. Binding energies for the (TiO2)m clusters, where m ranges from 1 to 20, are shown as a function of 1/n, where n3 ) m. Here, we take the zero of energy to be the m ) 1 unit and EB as in Figure 1. Open circles are from the calculations of Qu and Kroes.18 The curved, quadratic line fits our data according to the equation y ) 5.36x2 0.27x - 5.34 (R2 ) 0.97). Straight line as in Figure 1.

more O atoms than Mg atoms (14 versus 13), the binding energy was 0.035 eV higher than its most stable (MgO)13 relative. Several (MgO)13 units were formed from the cube containing 27 atoms. In the case of O vacancy formation, it is found that the variant with a face centered vacancy was the most stable. The variant with a corner vacancy was 0.53 eV less stable, and that with the O vacancy positioned at the cube center was 0.030 eV less stable. Perhaps the most significant fact about Figure 3 is that the binding energies for the most stable nanoclusters are not linear in 1/n. Rather, they are closer to the parabolic (1/n)2 dependence seen for the more covalent CuO. As before, this behavior is in contrast to that seen in the simple metals, and predicted by current theories, and is consistent with that found for the group IV elements. This is despite the fact that the bonding in this case is mostly ionic rather that covalent, and implies that our earlier explanation (that covalency governed the deviation from linearity) needs to be revisited.9 TiO2. The structure of small and intermediate sized clusters of TiO2 has been studied by several groups. Perhaps the most systematic study as a function of particle size is that by Qu and Kroes.18 These authors did an exhaustive study of (TiO2)m series for m ) 1-9 to determine the lowest free energy configuration for each cluster size. We followed their work in constructing our clusters in this size range. The smallest m ) 1 structure has a bent configuration with an angle of 110.2° between the two Ti-O bonds. The m ) 2 configuration includes a rhombohedral Ti2O2 unit and two nonplanar singly bonded O atoms. The m ) 3 structure is the first fully three-dimensional cluster with a Ti3O3 ring capped with an O atom bonded to all three Ti atoms, and two singly bonded O atoms attached to the ring as shown by Qu and Kroes.18 The higher order structures become increasingly complex, but all appear to be based on the anatase structure rather than the rutile structure. All are three-dimensional with one or two terminal, singly bonded O atoms and a cagelike TixOy core. In our calculations, the resulting structures matched closely those of Qu and Kroes.18 Both their binding energies and ours are shown in Figure 5, and the corresponding structures are shown in Figure 6. As can be seen in Figure 5, the binding energy is not linear in 1/n but follows a higher order in this size range. We also included larger m ) 17 and m ) 20 clusters with 51 and 60 atoms, respectively. These clusters were based on the rutile structure and had diameters somewhat larger than 1 nm. While we did not perform a global analysis to determine if these were

Binding Energy of Metal Oxide Nanoparticles

Figure 6. Selected low-energy TiO2 clusters. Ti atoms are shown in gray, and oxygen in red.

the lowest free energy configurations for these cluster sizes, and they almost certainly are not, as can be seen from Figure 5, they still have sufficient stability to fall below the 1/n line. As with MgO and CuO, a roughly quadratic dependence appears to be a better fit for the binding energy of the TiO2 clusters that we investigated. FeTiO3. As noted in the Introduction, not much is known experimentally about the very early growth pattern of even simple oxide clusters when the particle sizes are very small. The situation is even more difficult in the case of complex oxides such as FeTiO3. We do know from the computational work of Qu and Kroes18 that, in the case of TiO2, the structure of very small particles is apparently anatase-like rather than rutile-like, even though rutile is the more stable bulk structure. The reason for this may be similar to that found in ionic compound semiconductors11 where wurtzite-like architectures are favored for small particles even when the stable bulk structure is zinc blende. The reason for this is that nanoparticles based on a wurtzite-like scaffolding will generally have fewer “dangling” orbitals (broken bonds) than those based on a zinc-blende-like construction. It seems reasonable that one of the organizational driving forces for oxides as well as for compound semiconductors is to minimize the number of nonbonding or “dangling” orbitals in order to obtain the lowest free energy bonding state. This driving force along with the one to maximize the number of the strongest metal-oxygen bonds are only two of the criteria that must be satisfied during the formation of stable nanoparticles of complex oxides. Again, given the difficulties of doing experiments in this regime, it is fortunate that we have first principles calculations to help us understand the nature of these complexities. In order to study the behavior of the binding energy as a function of particle size (or n) for the more complex oxides, we selected FeTiO3 as our example. FeTiO3 has the ilmenite structure in the bulk where all of the Ti atoms are surrounded by a distorted octahedron of O atoms, and the Fe atoms have 8 O nearest neighbors. Because of the increased number of atoms in each formula unit, 5, there are many more a priori configurations that could be considered for each cluster size. Thus, this problem is considerably more complex than that of the monoxides, MgO and CuO, or even of TiO2. Therefore, unlike Qu and Kroes,18 we did not perform an exhaustive study to determine the structure having the global minimum for each value of m corresponding to clusters having the formula (FeTiO3)m. Instead, we selected several possible model paradigms and investigated the binding energy for m ) 1, 2, 3, and 4. For m ) 1, the stable cluster appears to be a unit with the 3

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Figure 7. Binding energies for the (FeTiO3)m clusters, where m ranges from 1 to 4, are shown as a function of 1/n, where n3 ) m. Here, we take the zero of energy to be the m ) 1 unit and EB as in Figure 1. The curved, quadratic line is simply a guide to the eye. Straight line as in Figure 1.

Figure 8. Selected low-energy FeTiO3 clusters. Ti atoms are shown in gray, Fe in yellow, and oxygen in red.

O atoms bonded to both the Fe and the Ti atoms, forming a 3-fold prism. As one paradigm, we tried constructing the larger clusters with combinations of this simple and symmetric unit. For all of the cases that we considered, we were able to construct stable clusters using this building block. As a second paradigm, we constructed clusters from the (TiO2)m clusters, with even values of m, from the work of Qu and Kroes18 by transforming half of the Ti atoms to Fe atoms and discarding m O atoms. Noting that the Ti-O bonds are stronger than the Fe-O bonds, care was taken to ensure that the average number of Ti-O bonds equaled or exceeded the average number of Fe-O bonds in each cluster. Again, we were able to form stable clusters for all of the values of m under study. Their binding energies are shown as a function of 1/n, with n3 ) m, in Figure 7. These structures were consistently more stable than those formed from the FeTiO3 prisms, and they are shown in Figure 8. As can be seen in Figure 7, even though a smaller number of cluster sizes were considered for FeTiO3, our data definitely indicate that the binding energy increases more rapidly than would be expected based on the assumption of an inverse linear dependence on size or n. Although our study of the size dependence of the binding energy of the nanoclusters for this material is not exhaustive, we have established that the FeTiO3 clusters are more stable than would be expected based on a 1/n dependence. Further, based on the limited number of structures studied here, the dependence appears to be roughly quadratic in 1/n. This dependence appears to be similar for predominantly ionic (MgO), as well as for the partially covalent (CuO) monoxides, and for the dioxide (TiO2), as well as for the more complex

4790 J. Phys. Chem. C, Vol. 113, No. 12, 2009 binary metal oxide (FeTiO3). The consistency of this result of such a wide range of different materials appears to indicate that this nonlinear dependence of binding energy on cluster size is probably a universal property of oxides in general. Overall, attempts to fit the data in Figures 1, 3, 5, and 7 over the entire range of 0 e 1/n e 1 with linear equations result in R2 values between 0.84 (TiO2) and 0.90 (FeTiO3). In sharp contrast, the quadratic fittings exhibit R2 values between 0.97 (TiO2) and 0.99 (MgO). It should be noted that the calculated nonlinearity has a solid theoretical basis as explained in the next section. Moreover, for Figures 1, 3, and 5, it is tempting to consider a linear fit over two distinct intervals, with one covering values of 1/n between 0 and about 0.6, and the other between about 0.6 and 1.0. The former interval has linear fittings with R2 values of 0.93, 0.96, and 0.91 for Figures 1, 3, and 5 respectively. The corresponding R2 values for the latter interval are 0.99, 0.99, and 0.98. However, this approach lacks the generality of having one single fit over the entire range of 1/n from 0 to 1. In fact, in all cases, the linear equation that fits one region rather well fails badly to fit the data of the other region. Perhaps, a more interesting result would be to have a nonlinear equation to fit the data for all the metal oxide clusters studied in this work. Such fitting can be accomplished by normalizing the energies represented in Figures 1, 3, 5, and 7 and plotting in a single graph the resulting energies as a function of 1/n. The normalized energies for any given metal oxide cluster of size m can be defined as

EB )

|

E1 - Em E1 - Ebulk

|

The graph (not shown) of such normalized EB versus 1/n over the entire interval 0 e 1/n e 1, with n3 ) m, can be fit by the quadratic equation y ) 0.940x2 - 0.0144x - 0.973, with R2 ) 0.98. Again, the relatively high value of R2 suggests that the nonlinear dependence of binding energy on cluster size is probably a property of metal oxides in general. Again, it is important to realize that there is a solid physical reason for the nonlinearity, as shown in the next section, and that the quadratic nature is an approximation that comes out of the long-range interactions, not a perfect dependence in and of itself. Long Range Ionic Interactions Finally, let us consider why the metal oxide clusters and nanoparticles have a nonlinear dependence of their binding energies on the inverse of the particle size. As noted above, this behavior is in sharp contrast to that of metal clusters and small nanoparticles and is closer to that of semiconductor particles. It is intuitively obvious that this behavior is strongly linked to the nature of the bonding in these different materials. In metals, the valence electrons are largely delocalized and the residual units of the nucleus and core electrons are well shielded from one another. As a consequence, it is primarily the nearest neighbor bonding that contributes to the binding energy with the result that, for the simple cubic structure, for example, the per particle binding energy scales as 3n2(n - 1) and the per atom energy as 3n2(n - 1)/n3 or 3(1 - 1/n), from which the 1/n or 1/r dependence can be seen. The case for semiconductors is somewhat more complex. Though their bonding is primarily covalent and nearest neighbor in nature, they also tend to reconstruct on the surface of the particle to minimize the number of partially occupied dangling orbitals. The details of this reconstruction vary between different

Parra and Farrell semiconductors, but often result in increasing the number of bonds between nearest neighbors on the surface of a particle involving rehybridization of the atomic orbitals relative to their bulk configuration. As a result, the binding energy of these surface atoms is often higher for semiconductor particles than it would be for comparable metals. As the surface to volume ratio increases with decreasing size, this effect is most pronounced for semiconductor clusters and small nanoparticles that, consequently, have higher binding energies than predicted by a simple nearest neighbor model. However, in the case of ionic materials such as MgO, for example, covalent bonding is largely irrelevant and longer range electrostatic forces become significant. Unlike metals where the nuclei and core electrons are largely shielded from those of nonadjacent ions, the cations and anions in ionic materials interact not only with nearest neighbors but also with second nearest neighbors and those of higher order. As the order of the neighboring shells increases, the distance increases, resulting in smaller interactions. However, this is balanced by an increase in the number of ions within each shell with the result that the longer range interactions, per shell, fall off very slowly, resulting in an analytically complex situation. This is the origin of the Madelung treatment of bulk binding energies for ionic salts and oxides. We can use a similar approach for the total nanoparticle binding energy which illustrates the analytical complexity of the situation. Following Kittel,21 for a particle with 2N ions, we can write the total lattice energy as

U0)Nφ

(1)

where the per ion pair binding energy, φ, is the sum of pairwise interactions, φij, where we count each pair interaction only once. If we assume, to a first approximation, that φij is the sum of a central field repulsive potential varying as rνij and a Coulomb potential, then

φij ) λ/rijν ( q2e2 /rij

(2)

Here, rij is the separation between the ith and the jth ions, q is the charge transfer between positive and negative ions, the (+) sign is to be taken for interactions between like charges, and the (-) sign for those between unlike charges. Introducing the quantity pij such that

rij)pijR

(3)

where R is the nearest neighbor separation, we can write

φ ) λAn /Rν ( R′q2e2 /R

(4)

where An ) ∑j′pνij and the nanoparticle equivalent of the Madelung constant, R′, is defined as

R′ ) Σj′(()pij-1

(5)

The term An can be eliminated by noting that, at the equilibrium separation R0, ∂φ/∂R ) 0. This results in an expression for the total lattice energy

Binding Energy of Metal Oxide Nanoparticles

U0 ) [NR′q2e2 /R](1 - 1/ν)

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(6)

where U0 is proportional to R′, the nanoparticle Madelung constant. Just as for the bulk materials, the nanoparticle Madelung constant varies according to the underlying crystal structure. For ionic materials that have nanoparticles based on the sodium chloride structure, such as MgO, the per ion-pair binding energy scales with the nanoparticle Madelung constant as

R′ ≈ [3n2(n - 1) - 6n(n - 1)2 /√2 + 4(n - 1)3 /√3 6n2(n - 2)/√4...] /n3(7) where the first several terms of the slowly converging series are shown. The first term, 3(1 - 1/n), is the nearest neighbor term and is the same as that for a simple cubic metal (which one would obtain if the ions in MgO were replaced with metal atoms). However, this term is not dominant. In general, the second nearest neighbor term is of the same order of magnitude as the first, and the higher order terms fall off rather slowly. This complexity leads to the nonlinear 1/n behavior of the ionic oxides and reflects the long-range nature of the unscreened Coulombic interactions in these materials. For the less ionic metal oxides, such as CuO and FeTiO3, this effect will be less important, but it will be supplemented by the nonlinear effect accompanying the increased covalency of the bonds which is often marked by an increased tendency toward reconstruction of the atomic arrangements in clusters and small nanoparticles relative to the bulk structure. Conclusion Our work indicates that current theoretical models describing the binding energy of nanoparticles based on the assumption that the binding energy is proportional to the number of nearest neighbors are not applicable to metal oxides in this size range. For low-aspect-ratio clusters, the prediction that the binding energy is inversely proportional to the size of the particle (or to n, the cube root of the number of basic units in a particle) does not hold for these materials. Instead, an approximately quadratic functionality occurs such that the binding energy is inversely proportional to the square of the particle size (or n2). This is true for highly ionic oxides, such as MgO, for more

covalent oxides, such as CuO, for refractory dioxides, such as TiO2, and for more complex metal oxides, such as FeTiO3. Similarly, physical properties that are dependent on the binding energy will also be closer to the bulk value than would be expected by a 1/n dependence. Finally, we have demonstrated analytically how the long-range electrostatic interactions in ionic materials result in a nonlinear 1/r or 1/n dependence of the binding energy of clusters and nanoparticles in the small particle regime. Acknowledgment. This research was supported in part by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences, under Contract No. DE-AC0799ID13727. R.D.P. would also like to thank the Faculty Staff Exchange Program at Idaho National Laboratory for travel support that made the work at INL possible. References and Notes (1) Wautelet, M.; Duvivier, D. Eur. J. Phys 2007, 28, 953. (2) See, for example, Vanithakumari, S. C.; Nanda, K. K. J. Phys. Chem. B 2006, 110, 6985, and references therein. . (3) See, for example, Qi, W. H.; Wang, M. P.; Zhou, M.; Hu, W. Y. J. Phys. D: Appl. Phys. 2005, 38, 1429, and references therein. . (4) See, for example, Sun, C. Q.; Bai, H. L.; Li, S.; Tay, B. K.; Li, C.; Chen, T. P.; Jiang, E. Y. J. Phys. Chem. B 2004, 108, 2162, and references therein. . (5) See, for example, Wautelet, M.; Dauchot, J. P.; Hecq, M. J. Phys.: Condens. Matter 2003, 15, 3651, and references therein. . (6) See, for example, Alivisatos, A. P. J. Phys. Chem. 1996, 100, 13226, and references therein. (7) Peters, K. F.; Cohen, J. B.; Chung, Y.-W. Phys. ReV. B 1998, 57, 13430. (8) Zhang, M.; Efremov, M. Yu.; Schiettekatte, F.; Olson, E. A.; Kwan, A. T.; Lai, S. L.; Wisleder, T.; Greene, J. E.; Allen, L. H. Phys. ReV. B 2000, 62, 10548. (9) Farrell, H. H.; Van Siclen, C. D. J. Vac. Sci. Technol., B 2007, 25, 1441. (10) Farrell, H. H.; Van Siclen, C. D.; Ginosar, D. M.; Petkovic L. M.; Parra, R. D. J. Vac. Sci. Technol., A, submitted for publication. (11) Farrell, H. H. J. Vac. Sci. Technol., B 2008, 26, 1534. (12) Delley, B. J. Chem. Phys. 1990, 92, 508. (13) Delley, B. J. Chem. Phys. 2000, 113, 7756. (14) Perdew, J. P.; Wang, Y. Phys. ReV. B 1992, 45, 13244. (15) Chen, G.; Liu, Z. F.; Gong, X. G. J. Chem. Phys. 2004, 120, 8020. (16) Wang, B.; Nagase, S.; Zhao, J.; Wang, G. J. Phys. Chem. C 2007, 111, 4956. (17) Shen, X.; Allen, P. B.; Muckerman, J. T.; Davenport, J. W.; Zheng, J.-C. Nano Lett. 2007, 7, 2267. (18) Qu, Z.-W.; Kroes, G.-J. J. Phys. Chem. B 2006, 110, 8998. (19) Bawa, F.; Panas, I. Phys. Chem. Chem. Phys. 2002, 4, 103. (20) Parra, R. D.; Farrell, H. H., to be published. (21) Kittel, C. Introduction to Solid State Physics, 6th ed.; John Wiley & Sons, Inc.: New York, 1986.

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