Origin of Nanoscale Phase Stability Reversals in Titanium Oxide

4240

2009, 113, 4240–4245 Published on Web 02/20/2009

Origin of Nanoscale Phase Stability Reversals in Titanium Oxide Polymorphs Daniel R. Hummer,*,† James D. Kubicki,† Paul R. C. Kent,‡ Jeffrey E. Post,§ and Peter J. Heaney† Department of Geosciences, The PennsylVania State UniVersity, UniVersity Park, PennsylVania 16802, Center for Nanophase Materials Sciences, Computer Science and Mathematics DiVision, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, Tennessee 37831-6494, and Department of Mineral Sciences, Smithsonian Institution, Washington, D.C. 20560 ReceiVed: December 22, 2008; ReVised Manuscript ReceiVed: February 7, 2009

We have monitored the hydrothermal crystallization of titania nanoparticles by in situ X-ray diffraction (XRD). Using the refined average structures from the XRD measurements, we calculated potential energy variations with particle size on periodic bulk structures using density functional theory (DFT). These variations cannot account for the enthalpy required to stabilize anatase relative to rutile. Thus, the hypothesis that the strain of the surface structure of nanoparticles accounts for the stabilization of anatase is not applicable to the growth of titania in water. DFT calculations on model nanoparticles do generate lower surface energies for anatase than for rutile that are large enough to explain the stability reversal in nanoparticles relative to the bulk phase. Rather than arising from two-dimensional surface structure alone, as previously thought, the total surface energies are critically dependent upon defects associated with edges and corners of nanocrystals at particle sizes e3 nm (i.e., during the nucleation process). As the particles grow, the bulk free energy becomes relatively more important, causing rutile to become stable at larger particle sizes. This study quantifies for the first time the critical role of edge and vertex energies in determining the relative phase stabilities of TiO2 nanoparticles. Introduction Controlling the incipient stages of crystal growth is of vital importance in many nanotechnologies since nanoscale materials can possess novel properties that are not shared by macroscale materials with ostensibly the same structures. In particular, nanosized titanium oxides exhibit enhanced electrical and optical behaviors relative to bulk titania, and nanotitania is applied widely in pigments, sunscreens, and food additives.1,2 In addition, TiO2 nanomaterials are critical to alternative energy approaches and are being developed as catalysts for cost-efficient hydrogen fuel3 and as dielectric materials in dye-sensitized solar cells.4-6 Recently the electrical properties of doped, nanoparticulate TiO2 were found to provide a fourth fundamental circuit element known as the memristor,7 whose existence previously had been theoretically postulated.8 Engineering titania nanoparticles for such applications necessitates a deep understanding of their crystallization mechanisms and thermodynamic behavior at small particle sizes. Nevertheless, the factors that control phase stability and crystal shape for this intensively studied system are still not completely understood at the nanoscale. Although R-TiO2 (rutile) is predicted to be thermodynamically stable at all temperatures and pressures, anatase dominates nanosized samples in both natural9,10 and laboratory11 settings. To explain this observation, * To whom correspondence should be addressed. E-mail: dhummer@ geosc.psu.edu. Phone: (814)-321-8859. † The Pennsylvania State University. ‡ Oak Ridge National Laboratory. § Smithsonian Institution.

10.1021/jp811332w CCC: $40.75

many scientists12 have suggested that the high surface-to-volume ratios of nanoparticles give rise to a significant surface contribution to free energy. Subsequently, several studies have offered calorimetric evidence to support the inference that the energies of anatase surface structures are lower than those for rutile, thereby stabilizing anatase nanoparticles.13-18 Here, we report the results of time-resolved X-ray diffraction (XRD) measurements and quantum mechanical calculations that reveal that the summation of surface energies for the major crystal faces of anatase and rutile nanoparticles cannot account for all of the energy that reverses phase stability with particle size. Instead, critical energetic contributions arise from the structural defects associated with the junctions of different crystal faces. We propose that the structural features that stabilize bulk rutile with respect to bulk anatase reduce the capacity of nanorutile to accommodate the energetic costs associated with those surface, edge, and corner defects that are so significant at the nanoscale. Methods Experimental Methods. Time-resolved diffraction experiments were carried out at beamline X7B of the National Synchrotron Light Source, Brookhaven National Laboratories. Solutions of 0.5 and 1.0 M TiCl4 were prepared by dissolving reagent grade TiCl4 dropwise into chilled deionized water and diluting to volume. Solutions were loaded into a glass, 1 mm outer diameter capillary, sealed with heat resistant epoxy, and placed in the beam path. A forced air heater below the sample controlled with an Omega 3200 controller was used to heat the  2009 American Chemical Society

Letters

Figure 1. Stacked diffraction patterns of TiO2 crystallization from a 0.5 M TiCl4 solution at 100 °C. Pink peaks (labeled “A”) represent anatase, blue peaks (labeled “R”) represent rutile. In this experiment, anatase peaks appear ∼15 min after heating begins, and rutile peaks appear ∼10 min after the first appearance of anatase.

solution to experimental temperatures of 100, 150, and 200 °C, and temperature was monitored with a chromel-alumel thermocouple adjacent to the sample. Powder diffraction patterns were collected with a MAR345 image plate at a time interval of four minutes during a total of ten hours of crystallization per experiment. Images were integrated into intensity versus 2-θ plots using the program Fit2D19 using a polarization factor of 0.93. Structural refinements were carried out using the general structure analysis system (GSAS) developed by Larson and Von Dreele.20 Backgrounds were fitted graphically using a Chebychev polynomial of 15-18 terms, and peak profiles were modeled using a pseudo-Voigt function described by Van Laar and Yelon,21 adjusted for asymmetry attributable to axial divergence by Finger et al.22 and including terms for anisotropic microstrain broadening by Stephens.23 After convergence of peak and background parameters, unit-cell dimensions were refined. All initial values of unit-cell parameters were taken from Cromer and Herrington.24 This was followed by refinement of atomic positions, isotropic temperature factors, and in some cases, additional background terms. Computational Methods. The anatase and rutile lattice parameter data derived from our Rietveld analyses as a function of particle size were used in density functional theory (DFT) calculations to assess whether the transition from a nanoparticle strained structure to the normal bulk structure was sufficient to account for stability reversal between rutile and anatase (i.e., strain energy > 6 kJ/mol). All calculations were performed using the Vienna ab-initio simulation package (VASP)25 using an energy cutoff of 400 eV. Soft pseudopotentials utilizing the Perdew-Burke-Ernzerhof (PBE) exchange and correlation functionals26 and the projector-augmented wave (PAW) method27 were used in all energy calculations. Periodic boundary conditions were employed to calculate (with full structural relaxation) the energies of the bulk crystal structures and of planar surfaces (rutile {110} and {101} and anatase {001}, {101} and {100}) with at least four atomic layers and a vacuum gap of 15 Å above each surface. Anatase structures were modeled with a supercell containing 3 × 3 × 1 unit cells and 108 atoms; rutile was modeled with supercells having 2 × 2 × 3 unit cells and 72 atoms. Reciprocal space was sampled with a Monkhorst-Pack 2 × 2 × 1 grid28 for all supercells. Lattice parameters for these bulk structure calculations were fixed at the experimental values obtained from

J. Phys. Chem. C, Vol. 113, No. 11, 2009 4241

Figure 2. Phase abundance vs time during TiO2 crystallization from a 0.5 M TiCl4 solution at 100 °C. Phase abundances are refined using Rietveld analysis with the general structure analysis system and are normalized to the final total amount of precipitate. Pink ) anatase, blue ) rutile, brown ) total.

Figure 3. Rietveld refined lattice parameters from time-resolved diffraction data of TiO2 precipitation from aqueous TiCl4 solutions. (A) Anatase c axis length vs crystallization time; orange diamonds ) 0.5 M TiCl4 at 100 °C, red circles ) 1.0 M TiCl4 200 °C. (B) Rutile c axis length vs crystallization time; blue diamonds ) 0.5 M TiCl4 at 100 °C, purple circles ) 1.0 M TiCl4 200 °C.

our Rietveld analysis of the diffraction data. Although lattice parameter relaxations were not performed in this study, similar calculations result in excellent agreement with observed crystal structures and surface energies.29 Models of anatase and rutile nanoparticles (average diameters 1, 2, and 3 nm) were constructed using the Cerius2 software package (Accelrys Software Inc.) by faceting an anatase supercell containing 5 × 5 × 5 unit cells (1.9 × 1.9 × 4.8 nm) with the {001} and {101} faces, and a rutile supercell containing 6 × 6 × 6 unit cells (2.8 × 2.8 × 1.8 nm) with the {110} and {101} faces, consistent with the minimum energy morphologies predicted by Barnard et al.30-32 Excess surface atoms were removed until the models were stoichiometric and electrically neutral. The 1 nm particles contain 120 and 105 atoms, the 2 nm particles 441 and 483 atoms, and the 3 nm particles 627 and 816 atoms for rutile and anatase structures, respectively. The anatase 1 and 2 nm particles were placed in a 30 × 30 × 43 Å3 supercell, the rutile 1 and 2 nm in a 43 × 43 × 30 Å3

4242 J. Phys. Chem. C, Vol. 113, No. 11, 2009

Letters

Figure 4. Energy minimized model TiO2 nanoparticles. (A) 1 nm anatase, (B) 2 nm anatase, (C) 3 nm anatase, (D) 1 nm rutile, (E) 2 nm rutile, (F) 3 nm rutile. Blue ) titanium; red ) oxygen. Retention of the apical, three-coordinated Ti atoms on the 1 nm anatase particle was necessary to preserve electrical neutrality.

supercell, and the anatase and rutile 3 nm in a 40 × 40 × 55 Å3 supercell. The structures were first optimized using the General Utility Lattice Program (GULP)33 with a force field appropriate for the Ti-O-H system,34 and then all ions in the nanoparticle were relaxed to a potential energy miniumum with VASP using periodic boundary conditions and the same methodology employed for the bulk and planar surface calculations. The total surface free energy of a particle or surface slab was calculated by subtracting the energy of an equal number of atoms of fully relaxed, three-dimensional (3D) periodic bulk structure from the total energy of a nanoparticle or surface of the same phase. Results/Discussion A sequence of time-resolved XRD patterns for a representative experiment is reproduced in Figure 1. The patterns, with a time resolution of ∼ 4 min, reveal that anatase was always the first phase to nucleate, appearing within 20-30 min of the start of the experiments. Diffraction peaks corresponding to rutile appeared shortly (within 10 min) after the appearance of the first anatase peaks. In all experiments, both anatase and rutile initially increased in mass abundance until reaching a threshold, beyond which anatase content decreased as rutile continued to form (Figure 2). Kinetic modeling and observations by transmission electron microscopy provided no evidence for the generation of rutile through a solid-state transformation of anatase, as will be detailed in a separate paper. In particular, the observation of initial rutile crystallites with average diameters of 3 nm (as measured by Scherrer analysis of diffraction peak widths) strongly suggests that rutile coprecipitates with anatase

Figure 5. Total DFT calculated surface energies of model 1, 2, and 3 nm anatase and rutile particles. Orange represents the contribution from the sum of the particle’s constituent crystallographic surfaces. Yellow represents the remainder, which is attributed to edges and defects.

in aqueous environments, rather than arising purely through the structural rearrangement of aggregated, postcritical anatase crystallites.35 The less common TiO2 phase, brookite, was never observed during these hydrothermal syntheses. The collection of time-resolved diffraction data allowed for measurements of anatase and rutile structures as they evolved from surface-dominated nanoparticles to volume-dominated microcrystals. Rietveld analysis revealed systematic variations in unit-cell parameters as a function of particle size, but the effects were extremely subtle. Both anatase and rutile experienced an expansion in all lattice parameters as particles

Letters

J. Phys. Chem. C, Vol. 113, No. 11, 2009 4243

TABLE 1: DFT Calculated Energies of Anatase and Rutile Surfaces surface

energy (J/m2) (This work)

{001} {100} {101}

0.96 0.51 0.41

anatase 0.90a 0.53a 0.44a

0.51 0.39 0.35

{110} {101}

0.69 1.12

rutile 0.89b 1.40b

0.47 0.95

a

energy (J/m2)

energy (J/m2) Barnard and Zapol31

Lazerri et al.50,51 b Ramamoorthy et al.52

coarsened from 3 to 15 nm (as determined by Scherrer analysis of diffraction peak widths), and the increase was most pronounced along the c-axis for both phases (Figure 3). Nevertheless, the volumetric expansion coefficients were on the order of 0.8% or less, suggesting that the degree of surface relaxation for these closest-packed oxides is not dramatic. Although several studies have measured the in situ coarsening of titania nanoparticles in air36-40 and in supercritical fluids,41 to the authors’ best knowledge this study represents the first time-resolved measurements of TiO2 structures during homogeneous, in situ precipitation from an aqueous medium. Previous studies of nanocrystalline TiO2 lattice constants as a function of particle size have utilized XRD measurements of dried powders. The unit-cell volume of anatase in these studies was found to expand with increasing particle size,42 while that of rutile contracts.43 Our data, which show an expansion in the unit-cell volumes of both phases as the particles coarsen, is therefore consistent with the reported behavior of anatase but conflicts with that of rutile. The apparent contradiction with previous results for rutile can be attributed to the much higher degree of surface solvation in our in situ experiments. Many researchers have measured the enthalpy for the transition from anatase to rutile (∆Hanatasefrutile)15,44-47 as ranging from -1 to -10 kJ/mol with the value recommended in the JANAF thermochemical tables being -6.025 kJ/mol.48 Because the measured molar entropies of anatase and rutile (49.9 ( 0.3 and 50.6 ( 0.6 J/mol K, respectively) are identical within experimental error,49 the enthalpy difference of ∼6 kJ/mol is equivalent to the Gibbs free energy difference at low and moderate temperatures. Thus, for a surface-related energy term to reverse the relative stabilities of the bulk structures, its value must exceed ∆Hanatasefrutile ≈ 6 kJ/mol. DFT potential energy calculations based on the bulk anatase and rutile structures refined from our XRD data (i.e., DFT energies with the lattice parameters constrained to the observed

values at various times during the experiment) showed that rutile and anatase experienced respectively a 0.6 and 0.4 kJ/mol decrease in energy during the transition from a strained nanoparticle structure toward the bulk crystal structure as the particles coarsened and lattice constants expanded to their bulk values. This negligible decrease in energy (compared to ∆Hanatasefrutile ≈ 6 kJ/mol) indicates that the deviation from the bulk structure observed in the early formed anatase and rutile nanoparticles cannot account for the observed reversal in stability. The DFT calculations on model nanoparticles (Figure 4) did exhibit marked differences in surface energies. The calculated surface energies for 1 nm particles of anatase and rutile were 1.29 and 1.69 J/m2, respectively. Calculations of surface energies for 2 nm particles yielded 0.84 and 1.39 J/m2 for anatase and rutile, respectively, whereas the 3 nm anatase and rutile particles yielded 0.77 and 0.88 J/m2, respectively (Figure 5). Thus, the surface energies diminished with larger particle size, but the surface energy for anatase fell below that of rutile for each model nanoparticle (by 9.5 kJ/mol for 1 nm particles, by 12.8 kJ/mol for 2 nm particles, and by 8.5 kJ/mol for 3 nm particles). These values indicate that differences in nanoparticle surface energy are large enough in magnitude (i.e., >6 kJ/mol for the bulk transition) to account for the stability reversal between the anatase and rutile polymorphs for sufficiently small particles, consistent with previous modeling32 and calorimetric15,17 results. To assess the contributions to the total surface energy from the component surfaces, we compared the surface energies of the whole nanoparticles to those calculated using our surface slab models (calculated by subtracting the energy of the relaxed, 3D periodic bulk structure from that of the 2D surface slab). The DFT-calculated surface energies for each surface are shown in Table 1. The results for anatase surfaces agree well with those of Lazzeri et al.50,51 and reasonably well with those of Barnard and Zapol31 with the exception of anatase {001}. We attribute the discrepancy to the higher energy cutoff (400 eV) used in our calculations. (Note that if this previously reported lower energy for the anatase {001} surface were used in the following analysis, it would only serve to decrease the energetic contribution of this face to the total surface energy and strengthen our main conclusion.) The results for rutile lie between those of Barnard and Zapol31 and those of Ramamoorthy et al.52 Table 2 shows bond distances in the energy minimized surface structures, as well as the analogous bond distances in the energy minimized bulk structure. Overall, bonds between surface Ti and overlying bridging O (as well as O atoms in the underlying layer) contracted significantly relative to the bulk structure, while bonds to three-coordinated surface O (parallel to the surface)

TABLE 2: DFT Calculated Bond Distances in the Top Trilayer of Atoms on Anatase and Rutile Surfaces and the Corresponding Bond Distances in the DFT Calculated Bulk Structurea anatase {001} b

anatase {100}

anatase {101}

rutile {101}

rutile {110}

bond

surface

bulk

surface

bulk

surface

bulk

surface

bulk

surface

bulk

TiV-Obr TiV-Osurf TiV-Osub TiVI-Obr TiVI-Osurf TiVI-Osub

1.960 1.934 1.962 NA NA NA

1.941 1.941 1.992 NA NA NA

1.842 1.962 2.060 1.831 1.863 NA NA

1.992 1.941 1.992 1.941 1.941 NA NA

1.853 1.975 2.027 1.813 1.850 2.003 1.929 2.053

1.992 1.941 1.992 1.941 1.941 1.992 1.941 1.992

1.830 1.915 2.009 2.103 1.874 NA NA NA

1.955 1.980 1.956 1.982 1.956 NA NA NA

NA 1.949 1.870 1.844 1.865 2.067 2.110 2.061

NA 1.956 1.982 1.955 1.955 1.980 1.956 1.956

a An entry of “NA” indicates that no bond matching the description is present in the surface’s top trilayer of atoms. Two entries indicate that two symmetry distinct bonds matching the description are present in the surface’s top trilayer. b TiV ) five-coordinated surface titanium; TiVI ) six-coordinated surface titanium; Obr ) two-coordinated (bridging) surface oxygen; Osurf ) three-coordinated surface oxygen; Osub ) three-coordinated underlying oxygen.

4244 J. Phys. Chem. C, Vol. 113, No. 11, 2009 expanded slightly. These trends are in good agreement with previous DFT calculations of these surfaces.31 Interestingly, these calculations revealed a significant discrepancy between the total calculated particle surface energies and the summed energies of the constituent faces (i.e., the predicted surface energies of the nanoparticles based on the 2D periodic surface energies in Table 1). For example, 1 nm rutile particles were calculated to have a surface energy of 1.69 J/m2. However, a summation of the energies of its constituent {101} and {110} surfaces yielded a value of 1.06 J/m2, 63% of the total surface energy. Similar inconsistencies were observed for each of the other five particles (Figure 5). The large discrepancies can be attributed to the contributions of the nanoparticle edges and corners, which exist only on the nanocrystals and not within idealized 2D periodic surfaces. Because of their undercoordinated nature, these edge and corner energies contribute substantially to the surface energy that inverts the relative stabilities of anatase and rutile when the surface contributions dominate, as is the case with nanoparticles. Including both surface and bulk terms, the transition enthalpy (∆Htrans) for the nanoanatase to nanorutile transition is 3.5, 6.8, and 2.5 kJ/mol for 1, 2, and 3 nm particles, respectively. Thus, the overall trend is for nanoanatase and nanorutile to reach a state of phase equilibrium (∆Htrans ) 0) at a particle size not greatly larger than 3 nm. However, the contribution of edges and vertices to the total surface energy decreases with particle size, while the change in the contribution from 2D surface structure is more modest. As a consequence, if edge and corner energies were removed from consideration, the total transition enthalpy for nanoanatase to nanorutile would become 25.3, 11.9, and 13.2 kJ/mol for 1, 2, and 3 nm particles, respectively, resulting in a much slower convergence to phase equilibrium. Thus, although edge and corner energy constitutes a sizable fraction of the surface energy that reverses phase stability, it also has the effect of limiting the stability field of anatase as a function of particle size from what it would otherwise be (including only 2D surface structure). Conclusions Our DFT calculations point to a large, previously unreported discrepancy between the total particle surface energy and the sum of the energies of their constituent surfaces. This excess energy, which can comprise up to 68% of the particle’s total surface energy, arises from the undercoordinated Ti and O atoms on the edges and corners of nanoparticles and appears to be a significant factor governing the thermodynamics of nanoparticles. The structural features that stabilize rutile in its bulk form (e.g., highly regular, corner-sharing TiO6 coordination polyhedra) are less accommodating of the defects than are those in anatase, which has a lower density (Fan ) 3.89 vs Frut ) 4.25 g/cm3) and more highly distorted, edge-sharing TiO6 octahedra. Consequently, surfaces, edges, and vertices induce a much higher energetic penalty in rutile than in anatase, and since these defects are relatively more abundant when particle sizes are small, they stabilize anatase at the nanoscale. It seems likely that suites of surface defects, including edges and vertices, play a critical role in polymorphic stability reversals observed in other systems, such as zirconia and alumina. Acknowledgment. We thank Jon Hanson at NSLS for his assistance with instrumentation at beamline X7B. Use of the National Synchrotron Light Source, Brookhaven National Laboratory, was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under

Letters contract No. DE-AC02-98CH10886. This research used resources of the National Energy Research Scientific Computing Center, National Center for Computational Sciences, and the Center for Nanophase Materials Sciences, which are sponsored by the respective facilities divisions of the Advanced Scientific Computing Research and Basic Energy Sciences of the U.S. Department of Energy. We also thank Andrei Bandura and Jorge Sofo for their assistance with the GULP and VASP calculations utilized in this study. This work was made possible by National Science Foundation Grant EAR04-17714, EAR07-45374, and by the Center for Environmental Kinetics Analysis (CEKA), an NSF- and DOE-sponsored Environmental Molecular Science Institute (NSF CHE-0431328). References and Notes (1) More, B. D. Indian J. Dermatol. Venerol. 2007, 202, 80–85. (2) Singh, S.; Nalwa, H. S. J. Nanosci. Nanotechnol. 2007, 7, 3048– 3070. (3) Yoshida, H.; et al. J. Phys. Chem. C 2008, 112, 5542–5551. (4) Lagref, J.-J.; Nazeeruddin, M. K.; Gratzel, M. Inorg. Chim. Acta 2004, 361, 735–745. (5) Sebastian, P. J.; Olea, A.; Campos, J.; Toledo, J. A.; Gamboa, S. A. Sol. Energy Mater. Sol. Cells 2004, 81, 349–361. (6) Santa-Nokki, H.; Kallioinen, J.; Kololuoma, T.; Tuboltsev, V.; Korppi-Tommola, J. J. Photochem. Photobiol., A 2006, 182, 187–191. (7) Strukov, D. B.; Snider, G. S.; Stewart, D. R.; Williams, R. S. Nature 2008, 453, 80–83. (8) Chua, L. IEEE Trans. Circuit Theory 1971, CT18, 507–519. (9) Schaefer, C. E. G. R.; Fabris, J. D.; Ker, J. C. Clay Miner. 2008, 43, 137–154. (10) Schroeder, P. A.; Shiflet, J. Clays Clay Miner. 2000, 48, 151–158. (11) Wang, J.-y.; Liu, Z.-h.; He, Z.-k.; Cai, R.-x. Prog. Chem. 2007, 19, 1495–1502. (12) Gribb, A. A.; Banfield, J. F. Am. Mineral. 1997, 82, 717–728. (13) Zhang, H.; Banfield, J. F. J. Mater. Chem. 1998, 8, 2073–2076. (14) Zhang, H.; Banfield, J. F. J. Phys. Chem. B 2000, 104, 3481–3487. (15) Ranade, M. R.; et al. Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 6476– 6481. (16) Li, W.; Ni, C.; Lin, H.; Huang, C.-P.; Ismat Shah, S. J. Appl. Phys. 2004, 96, 6663–6668. (17) Levchenko, A. A.; Li, G.; Boerio-Goates, J.; Woodfield, B. F.; Navrotsky, A. Chem. Mater. 2006, 18, 6324–6332. (18) Navrotsky, A. Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 12096–12101. (19) Hammersley, A. P.; Svensson, S. O.; Hanfland, M.; Fitch, A. N.; Hausermann, D. High Pressure Res. 1996, 14, 235–248. (20) Larson, A. C.; Von Dreele, R. B. General Structure Analysis System (GSAS), (Report LAUR 86-748); Los Alamos National Laboratory: Los Alamos, NM, 2000. (21) Van Laar, B.; Yelon, W. B. J. Appl. Crystallogr. 1984, 17, 47–54. (22) Finger, L. W.; Cox, D. E.; Jephcoat, A. P. J. Appl. Crystallogr. 1994, 27, 892–900. (23) Stephens, P. W. J. Appl. Crystallogr. 1999, 32, 281–289. (24) Cromer, D. T.; Herrington, K. J. Am. Chem. Soc. 1955, 77, 4708– 4709. (25) Kresse, G.; Furthmuller, J. Phys. ReV. B 1996, 54, 11169–11186. (26) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. ReV. Lett. 1996, 77, 3865–3868. (27) Blochl, P. E. Phys. ReV. B 1994, 50, 17953–17979. (28) Monkhorst, H. J.; Pack, J. D. Phys. ReV. B 1976, 13, 5188–5192. (29) Bandura, A. V.; Kubicki, J. D.; Sofo, J. O. J. Phys. Chem. B 2008, 112, 11616–11624. (30) Barnard, A. S.; Zapol, P.; Curtiss, L. A. J. Chem. Theory Comput. 2005, 1, 107–116. (31) Barnard, A. S.; Zapol, P. Phys. ReV. B 2004, 70, 235403. (32) Barnard, A. S.; Erdin, S.; Lin, Y.; Zapol, P.; Halley, J. W. Phys. ReV. B 2006, 73, 205405. (33) Gale, J. D. J. Chem. Soc., Faraday Trans. 1997, 93, 629–637. (34) Bandura, A. V.; Kubicki, J. D. J. Phys. Chem. B 2003, 107, 11072– 11081. (35) Koparde, V. N.; Cummings, P. T. ACS Nano 2008, 2, 1620–1624. (36) Fernandez-Garcia, M.; Belver, C.; Hanson, J. C.; Wang, X.; Rodriguez, J. A. J. Am. Chem. Soc. 2007, 129, 13604–13612. (37) Fernandez-Garcia, M.; Wang, X.; Belver, C.; Hanson, J. C.; Rodriguez, J. A. J. Phys. Chem. C 2007, 111, 674–682. (38) Nicula, R.; Stir, M.; Burkel, E. Mater. Sci. Forum 2004, 467470, 1307–1312. (39) Rivallin, M.; Benmami, M.; Gaunand, A.; Kanaev, A. Chem. Phys. Lett. 2004, 398, 157–162.

Letters (40) Lee, G. H.; Zuo, J.-M. J. Am. Ceram. Soc. 2004, 87, 473–479. (41) Jensen, H.; et al. Angew Chem., Int. Ed. 2007, 46, 1113–1116. (42) Li, G.; Li, L.; Boerio-Goates, J.; Woodfield, B. F. J. Am. Chem. Soc. 2005, 127, 8659–8666. (43) Li, G.; Boerio-Goates, J.; Woodfield, B. F.; Li, L. Appl. Phys. Lett. 2004, 85, 2059–2061. (44) Margrave, J. L.; Kybett, B. D. Thermodynamic and Kinetic Studies of Oxides and Other Refractory Materials at High Temperature, Tech. Rep. No. AFMO-TR-65 1965; Air Force Materials Laboratory, Research and Technology Division, Air Force Systems Command: Wright-Patterson Air Force Base, OH, 1965; p 123. (45) Navrotsky, A.; Kleppa, O. J. J. Am. Ceram. Soc. 1967, 50, 626. (46) Mitsuhashi, T.; Kleppa, O. J. J. Am. Ceram. Soc. 1979, 62, 356–357.

J. Phys. Chem. C, Vol. 113, No. 11, 2009 4245 (47) Rao, C. N. R. Can. J. Chem. 1961, 39, 498–500. (48) Chase, M. W. NIST-JANAF Thermochemical Tables, 4th ed. J. Phys. Chem. Ref. Data 1998, Monograph 9. (49) Robie, R. A.; Hemingway, B. S. U.S. Geol. SurV. Bull. 1995, 2131. (50) Lazzeri, M.; Vittadini, A.; Selloni, A. Phys. ReV. B 2001, 63, 155409. (51) Lazzeri, M.; Vittadini, A.; Selloni, A. Phys. ReV. B 2002, 65, 119901. (52) Ramamoorthy, M.; Vanderbilt, D.; King-Smith, R. D. Phys. ReV. B 1994, 49, 16721.

JP811332W