Structures of Defect Clusters on Ceria - American Chemical Society

Article pubs.acs.org/JPCC

Structures of Defect Clusters on Ceria {111} Surface Jia Le Ma,† Fei Ye,*,†,‡ Ding Rong Ou,§ Lin Lin Li,† and Toshiyuki Mori⊥ †

School of Materials Science and Engineering, Dalian University of Technology, 2 Linggong Road, Dalian, Liaoning 116024, China Key Laboratory of Materials Modification by Laser, Ion, and Electron Beams (Dalian University of Technology), Ministry of Education, 2 Linggong Road, Dalian, Liaoning 116024, China § Laboratory of Fuel Cells, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, 457 Zhongshan Road, Dalian, Liaoning 116023, China ⊥ Fuel Cell Materials Group, Battery Materials Unit, National Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan ‡

S Supporting Information *

ABSTRACT: Defect clusters containing oxygen vacancies and Ce3+ cations on ceria (CeO2) surface dominate the electronic and chemical properties of the surface. However, the structures of the clusters, especially the arrangements of the oxygen vacancies in the clusters, have not been explained consistently. In this work, atomistic simulation based on energy minimization has been used to investigate the cluster structures on ceria {111} surface. It was found out that the oxygen vacancies are energetically favorable to be at the second-neighbor sites to their associated Ce3+ cations. Moreover, the subsurface oxygen vacancies on the third layer are essential for the arrangement of the surface oxygen vacancy clusters. Due to the existence of the subsurface oxygen vacancies, the adjacent surface oxygen vacancies tend to be separated by ⟨110⟩/2, and the linear surface clusters are more energetically favorable than the triangle ones. Then, the structure development with cluster size is discussed.

1. INTRODUCTION Catalytic materials based on ceria (CeO2) have been widely used in the production and purification of hydrogen, the purification of exhaust gases in three-way automotive catalytic converters, and other catalytic applications.1−3 All these applications are largely attributed to their high oxygen transport and storage capacities due to the remarkable ability of rapid redox cycles by releasing and storing oxygen.1 It is widely accepted that these properties of the ceria are dominated by the arrangement of the oxygen vacancies on the ceria surface. Therefore, a comprehensive knowledge of the structure of the vacancy clusters on the surface could provide means for tailoring the reactivity of ceria-based catalysts. The oxygen vacancy arrangements on the {111} surface, which is the most thermodynamically stable surface orientation of ceria, have mainly been investigated by the dynamic force microscopy (DFM) and scanning tunneling microscopy (STM).4−9 The first atomically resolved images were produced by Nörenberg and Briggs,4 in which oxygen vacancy clusters were on the top layer and the dominant cluster type is triangle shaped containing three surface oxygen vacancies. Upon annealing to more than 1000 °C, the linear surface oxygen vacancy clusters are the most widely observed.5 In the study of Namai et al.6,7 and Esch et al.,8 it was also observed that after annealing at 900 °C the linear surface oxygen vacancy clusters are the most abundant and the triangle surface clusters are the next abundant. These results indicate that the linear surface clusters are probably the most thermodynamically stable. Moreover, Esch et al. proposed that the subsurface oxygen © 2012 American Chemical Society

vacancies are essential for the nucleation of the linear surface clusters.8 To further understand the nature of the surface clusters, computer simulation has been performed.10−18 However, most of the simulations concentrated on the isolated oxygen vacancy,10−17 which are not enough to explain the structures of the surface clusters. Recently, Zhang et al. have systematically studied larger surface clusters by first-principles simulation. They suggested that triangular surface vacancy clusters are more stable than the double linear surface vacancy clusters containing subsurface vacancies.18 However, this result is only consistent with the earlier observation of Nörenberg and Briggs,4 but disagrees with other results.5−9 Upon the creation of an oxygen vacancy by the reduction of the ceria surface, two excess electrons are left which results into two Ce3+ cations due to the requirement of electric neutrality. It has been found out that the location of Ce3+ is vital in interpreting the stability of the clusters containing Ce3+ cations and oxygen vacancies, while the localization of the excess electrons is still under debate. In the simulation of the bulk ceria,19 it was concluded that an oxygen vacancy prefers a second-neighbor site common to the two associated Ce3+ cations. In the case of ceria surface, Ce3+ cations located at the first-neighbor sites4−14,18 and at the second-neighbor sites15−17 to the associated oxygen vacancies were both Received: July 6, 2012 Revised: November 14, 2012 Published: November 19, 2012 25777

dx.doi.org/10.1021/jp306699r | J. Phys. Chem. C 2012, 116, 25777−25782

The Journal of Physical Chemistry C

Article

suggested. Therefore, the structures of surface defect clusters should be studied carefully by taking the arrangements of Ce3+ cations into account. To further understand structures of surface clusters containing Ce3+ cations and oxygen vacancies, in this work, atomic computer simulation is employed, and the relative locations of Ce3+ cations and oxygen vacancies are addressed. Moreover, the essential role of the subsurface oxygen vacancy in the cluster structures is further examined. Then, based on the structures of defect clusters, the structure development of the clusters containing from only one oxygen vacancy to four vacancies is discussed, and an imaginary structure for larger cluster is given.

Table 1. Short-Range Potential Parameters species

A (eV)

ρ (Å)

C (eV Å6)

Ce −O Ce3+−O2− O2−−O2−

1809.68 2010.18 9547.96

0.3547 0.3449 0.2192

20.40 23.11 32.0

4+

2−

Table 2. Shell Model Parameters species 4+

Ce O2−

ΔE =

2. METHODOLOGY The surface cluster energy is studied by the two-region approach coded in the GULP program,20 which incorporates much of the functionality of MARVIN code. In this approach, the crystal is divided into region 1, which contains the surface and all layers of atoms below it that exhibit a significant atomic relaxation, and region 2, which contains the rest of the crystal where it is assumed that no displacements from the threedimensional crystal structure are induced. The {111} surface of ceria is stacked by repeating units of O−Ce−O trilayers (Figure 1). In this study, the size of the

k/eV Å−2

Y/e

177.84 6.3

−0.20 −2.04

∑ Eisolated − Ecluster

(2)

where Ecluster is the formation energy of a cluster and ∑Eisolated is the sum of the formation energy for isolated defects. The formation energies of isolated defects in different layers are considered respectively. Various arrangements of oxygen vacancies and Ce3+ cations have been examined to find out the most stable configuration.

3. RESULTS 3.1. Relative Location between Oxygen Vacancies and Ce3+ Cations. The electronically neutral clusters containing one oxygen vacancy and two Ce3+ cations are first studied, as shown in Figure 2. In these clusters, the surface and

Figure 1. Model of CeO2 {111} surface with oxygen-layer termination. The color coding of ions and defects on different layers is depicted, which will be used in the following figures. Figure 2. Defect clusters of (2CeCe′VO··)×. The oxygen vacancies are located on the first layer in (a)−(c), or the third layer in (d) and (e). The relative locations of oxygen vacancies to the Ce3+ cations are labeled. To clearly show the subsurface atoms, this and the following figures are viewed from a direction deviated from the ⟨111⟩ direction.

super cell containing the ceria surface is 4 × 4. Region 1 consists of four repeating units and is about 12.5 Å in thickness. Region 2 consists of three repeating units and is about 9.3 Å. The atomic interactions are based on the Born model of an ionic crystal,21 which include a long-range term to represent Coulombic interaction and a short-range term to describe the electronic repulsion and attractive van der Waals force between electron clouds. The short-range term is described by a Buckingham potential form Sij = A exp( −rij/ρ) − Crij−6

subsurface oxygen vacancies locate at the nearest-neighbor (NN) sites and next nearest neighbor (NNN) sites to the associated cations. By comparing the binding energies of these clusters, it can be seen that as in bulk ceria19 no matter which layer the oxygen vacancy locates on, it always prefers the NNN sites common to the associated cations. The binding energy of clusters with oxygen vacancies located at the third-neighbor sites to the associated cations is lower than that at the NNN sites. This result is consistent with the recent computer simulations based on the DFT+U approach.15−17 It can also be seen that, for the clusters (2CeCe′VO··)×, the binding energy of the cluster in which the oxygen vacancy is on the first layer (Figure 2b) is very close to that in which the oxygen vacancy is on the third layer (Figure 2e). This result is

(1)

where A, ρ, and C are three adjustable parameters, and a cutoff of 20 Å is set here. The shell model is employed to Ce4+ and O2− ions for describing the electronic polarizability.22 These parameters are listed in Tables 1 and 2.19,23,24 They have been successfully used to simulate the cluster structure in ceria and surface structure of ceria.19,23,24 To compare the stability of the clusters, the binding energies are calculated by 25778



Article

clusters are almost the same. When the two oxygen vacancies in the cluster are separated further, the binding energy is much lower. Then the cluster size is increased to (6CeCe′3VO··)× (Figure 4). When the oxygen vacancies are separated by ⟨110⟩/2 and

form linear cluster or triangle cluster, it can be seen that the linear surface vacancy cluster in Figure 4a is more stable than the triangular cluster. Moreover, of the two variants of triangular clusters, only the one with an oxygen ion at the center of the triangle is stable (Figure 4b). If two oxygen vacancies in the cluster (6CeCe′3VO··)× are separated by ⟨112⟩/ 2 (Figure 4d), the binding energy is a bit lower than the triangular and linear clusters. These results are consistent with most of the previous observations.5−9 However, it should be noticed that the difference in the binding energies of these clusters in Figure 4, a, b, and d, is not significant. Therefore, the calculation results cannot explain why the linear surface oxygen vacancy clusters are the dominant clusters.5−9 As suggested by Esch et al.,8 the subsurface oxygen vacancies play an important role in the cluster structure. In the following study, the effect of subsurface oxygen vacancy on the cluster structures is investigated. 3.3. Clusters Containing a Subsurface Oxygen Vacancy. When a subsurface oxygen vacancy is involved in the clusters (4CeCe′2VO··)×, the clusters in which two oxygen vacancies are separated by ⟨100⟩/2, ⟨111⟩/2, and ⟨012⟩/2 are considered (Figure 5). It can be seen that the cluster in which oxygen vacancies are separated by ⟨111⟩/2 with a Ce4+ cation half way between is the most energetically favorable. Then the clusters (6CeCe′3VO··)× with a subsurface oxygen vacancy are studied. When the two surface vacancies are separated by ⟨110⟩/2 and the subsurface oxygen vacancy is at a ⟨111⟩/2 site to one of the two surface vacancies, there are three possible cluster structures as shown in Figure 6. It can be seen

Figure 4. Defect clusters of (6CeCe′3VO··)×, in which oxygen vacancies are on the first layer and the adjacent oxygen vacancies are separated by ⟨110⟩/2. The formation energy of the cluster in (c) cannot be converged.

Figure 6. Defect clusters of (6CeCe′3VO··)×, in which two oxygen vacancies are on the first layer and one is on the third layer. Two surface oxygen vacancies are separated by (a)−(c) ⟨110⟩/2 and (d) ⟨112⟩/2.

consistent with the observation of Esch et al. that single surface and subsurface vacancies present similar distribution.8 In the following study, the clusters in which the oxygen vacancies are at the NNN sites common to their associated Ce3+ cations are considered. Moreover, it was noticed that the binding energies of clusters with different structures are not sensitive to the detailed locations of Ce3+ cations if Ce3+ cations are located at NNN sites. Therefore, the arrangements of the oxygen vacancies will mainly be discussed. 3.2. Arrangements of Oxygen Vacancies on Ceria Surface. When two oxygen vacancies are on the surface layer and separated by ⟨110⟩/2 and ⟨112⟩/2 respectively as shown in Figure 3, it can be seen that the binding energies of the two

Figure 3. Defect clusters of (4CeCe′2VO··)×, in which oxygen vacancies are on the first layer. The two oxygen vacancies are separated by (a) ⟨110⟩/2 and (b) ⟨112⟩/2, respectively.

Figure 5. Defect clusters of (4CeCe′2VO··)×, in which one oxygen vacancy is on the first layer and the other one is on the third layer. The two oxygen vacancies are separated by (a) ⟨100⟩/2, (b) ⟨012⟩/2, and (c) ⟨111⟩/2, respectively. 25779



Article

Furthermore, the energy difference between linear surface clusters (8CeCe′4VO··)× with or without a subsurface oxygen vacancy is 0.53 eV (Figure 7, a and c). This difference is only 0.21 eV for the vacancy trimer, indicating that the formation possibility of the clusters with subsurface vacancy increases with the cluster size. However, more subsurface vacancies do not mean a higher stability. When the cluster (8CeCe′4VO··)× contains two subsurface vacancies separated by ⟨110⟩/2 (Figure 7d), the binding energy is lower than that with one oxygen vacancy (Figure 7a). Therefore, the subsurface vacancies in a cluster also have a certain arrangement.

that the cluster in Figure 6a has the highest binding energy, in which the subsurface vacancies are at the ⟨111⟩/2 and ⟨012⟩/2 sites to the two surface vacancies, respectively. This arrangement of the subsurface vacancy is consistent with the stable arrangements in the smaller clusters (4CeCe′2VO··)× (Figure 5). The clusters (6CeCe′3VO··)× with two surface vacancies separated by ⟨112⟩/2 and a subsurface vacancy are also considered (Figure 6d); since the (4CeCe′2VO··)× cluster with two surface vacancies separated by ⟨112⟩/2 has relatively high binding energy as shown in Figure 3, it can be seen that the stability of this cluster is much lower than the cluster (6CeCe′3VO··)× with two surface vacancies separated by ⟨110⟩/2 (Figure 6a). Therefore, the existence of subsurface oxygen vacancies urges the adjacent surface oxygen vacancies to be separated by ⟨110⟩/2. Comparing the binding energy of the clusters (6CeCe′3VO··)× shown in Figure 6a with the one in Figure 4a, it can be seen that the binding energy increases 0.21 eV when a subsurface vacancy is involved. Therefore, it can be concluded that the existence of subsurface oxygen vacancies increases the stability of the surface clusters. In spite of this, the linear surface vacancy trimers without subsurface vacancies are also possible to form since this energy increase is not so significant. In the case of larger clusters (8CeCe′4VO··)×, the linear surface vacancy cluster with a subsurface oxygen vacancy as shown in Figure 7a is the most stable. Comparing this cluster

4. DISCUSSION 4.1. Development of the Defect Clusters. According to the stable clusters with different sizes, the development of the defect clusters can be seen as shown in Figure 8. For the clusters (2CeCe′VO··)×, the subsurface and surface vacancies with nearly the same ΔE have the same formation possibility. When the clusters develop into the vacancy dimers, the binding energies of the surface clusters (Figure 3) and the cluster with a subsurface vacancy (Figure 5b,c) are similar to each other, indicating they have the same formation possibility. Then, the clusters grow into vacancy trimers, in which the adjacent surface vacancies tend to be separated by ⟨110⟩/2 and the cluster with a subsurface oxygen vacancy is more stable. When the clusters develop to (8CeCe′4VO··)×, the binding energy of the linear surface cluster with one subsurface vacancy is higher than others, indicating it has higher formation possibility. Because the situation becomes more complicated for larger clusters, the clusters with more than four vacancies are not calculated in this work. However, the structure of larger clusters can be deduced based on the development from monomer to tetramer. It is reasonable to suggest that the linear surface clusters with subsurface vacancies become the dominant type when the clusters grow even larger. Moreover, if larger clusters are formed by a combination of the stable trimers in Figure 6a, the subsurface vacancies also linearly arranged along ⟨110⟩ direction. Then, an imaginary structure for larger cluster is given in Figure 8, which is formed by a combination of the most stable vacancy trimers or the most stable vacancy tetramers. The distribution of the subsurface oxygen vacancies in the imaginary structure can be supported by the observation of Torbrügge et al.9 In their work, it was found out that the subsurface oxygen vacancies arrange linearly along ⟨110⟩ direction and the adjacent vacancies are separated by ⟨110⟩. The structure derived in this work is the same as this observation. 4.2. Comparisons of Previous Observations and Our Simulation. From the calculations, it can be seen that the linear oxygen vacancy cluster is more stable than the triangular cluster when the clusters contains three surface oxygen vacancies and a subsurface oxygen vacancy. Moreover, the subsurface oxygen vacancies can improve the stability of the clusters. These results are consistent with the previous observations.5−9 The structure of stable clusters including dimers to tetramers and imaginary structure for larger cluster can be supported by the experimental fact of the ratio of surface vacancies to subsurface vacancies. In the imaginary structure as shown in Figure 8, which is the combination of the smaller calculated

Figure 7. Defect clusters of (8CeCe′4VO··)×. (a) The most stable linear surface oxygen cluster with a subsurface oxygen vacancy. The structures in the rectangles are the same as the stable trimer as shown in Figure 6a. (b) The triangle surface cluster with a subsurface oxygen vacancy. (c) Linear surface cluster without subsurface oxygen vacancy. (d) Clusters with two subsurface oxygen vacancy separated by ⟨110⟩/2.

with that shown in Figure 6a, it can be seen that the structure of this cluster is a combination of that in Figure 6a. In other words, the structure of the cluster in Figure 6a is probably a structure unit for larger cluster. The energy difference between the linear surface cluster (Figure 7a) and triangular surface cluster involving a subsurface vacancy (Figure 7b) is about 0.52 eV. This difference is more significant than that without subsurface vacancy as shown in Figure 4a,b (only 0.04 eV). Therefore, the existence of subsurface oxygen vacancy makes the linear surface vacancy clusters much more stable than the triangle surface clusters. 25780



Article

Figure 8. Development of the clusters. The binding energies (eV) of the clusters are also given. The possible paths for the cluster development are indicated by arrows.

atomic displacements are shown by the coordinates of relaxed O atoms in Figure 9. In the case of an isolated surface vacancy (Figure 9 a), most of the six surface O atoms surrounding the vacancy rise 0.01−0.07 Å and shift about 0.10−0.13 Å laterally and outward. However, the displacements of the other two O atoms indicated by arrows do not protrude out or shift outward. The discrepancy is possibly because in the cluster structure of this work the Ce3+ cations are at the NNN sites to the vacancies. In the case of an isolated subsurface vacancy (Figure 9b), three surface O atoms at the vertexes of the triangle rise 0.10−0.13 Å and shift 0.08−0.09 Å outward, which are well consistent with the previous work.8,9,11 Then, the atomic displacements around the cluster shown in Figure 6a are also compared with the previous work.8 In the work of Esch et al.,8 each LSVC (linear surface vacancy clusters) is characterized by a pair of rim O atoms that face each other, one appearing 0.1 Å below and the other 0.1 Å above the unperturbed surface. Both O atoms are relaxed laterally toward the inside of the defect. In our work, the existence of a subsurface oxygen vacancy makes a contraction in the lattice and five of the six surrounding oxygen atoms sink 0.09−0.21 Å. The surface O atom in the middle that sinks 0.09 Å is higher than the surrounding O atoms, which will result in its brightness in the STM image and the darkness of its opposite atom.8 However, the middle pair of O atoms do not shift inward in the surface vacancies, which is not consistent with the previous observation.8 The reason for this discrepancy remains unclear.

stable clusters, the ratio of surface vacancies to subsurface vacancies is about 2:1, which is the same as that in the stable trimer unit (Figure 6a). In the previous observation,8 the fractions for all subsurface vacancies and single ones are 3% and 2.2%, respectively; i.e., 0.8% of subsurface vacancies are in clusters. For the surface vacancies, the fractions are 3% and 1.5%, respectively; i.e., 1.5% of surface vacancies are in clusters. Therefore, the ratio of surface vacancies in clusters to the subsurface vacancies in clusters is 1.5:0.8 in their observation, which is close to 2:1. Therefore, our results are consistent with the previous observation.8 The cluster structures can also be compared with the previous observations8,9 and DFT calculations8,11 of the atomic displacements, as shown in Figure 9. In previous works,8,9 it

5. CONCLUSIONS The stability and structure of defect clusters containing oxygen vacancies and Ce3+ cations on ceria {111} surface have been studied by computer simulation. This study has shown that, in the energetically preferred configurations of surface clusters, (1) the oxygen vacancies locate at the second-neighbor sites common to their associated Ce3+ cations; (2) the subsurface oxygen vacancies increase the stability of the surface clusters; and (3) the existence of subsurface oxygen vacancy urges the adjacent surface oxygen vacancies separated by ⟨110⟩/2 and makes the linear surface vacancy clusters much more stable than the triangle surface clusters.

Figure 9. Schematic representations of the displacement of the oxygen atoms surrounding (a) the cluster with a surface vacancy, (b) the cluster with a subsurface vacancy, and (c) the cluster shown in Figure 6a. The cations are omitted for simplification. The dotted circles are the O atom sites before lattice relaxation and are the origins for the coordinates of the O atom after lattice relaxation. The unit of the coordinates is Å.

was found that the six O atoms surrounding an isolated surface vacancy protrude out of the unperturbed surface. Moreover, these atoms shift 0.08−0.09 Å laterally and outward.8 In the case of an isolated subsurface vacancy, three surface O atoms at the vertexes of the triangle in Figure 9b protrude out of the unperturbed surface.9,11 The displacement was 0.20 Å determined in one work9 or 0.19 Å in the other.11 Moreover, these atoms shift 0.19 Å laterally and outward.8 In our work, the

■

ASSOCIATED CONTENT

S Supporting Information *

Because it is not easy to fully understand the cluster structures in Figures 2−7, the original images drawn in the software CaRIne Crystallography 3.1 and side views of these figures are 25781



Article

provided. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*Tel./Fax: +86-411-84709230. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS Support from Scientific Research Foundation for the Returned Overseas Chinese Scholars, Ministry of Education, and from Open Project of Key Laboratory of Materials Modification by Laser, Ion, and Electron Beams (Dalian University of Technology), Ministry of Education, is gratefully acknowledged.

■

REFERENCES

(1) Trovarelli, A. Catalysis by Ceria and Related Materials; Imperial College Press: London, 2002. (2) Park, S.; Vohs, J. M.; Gorte., R. J. Nature 2000, 404, 265−267. (3) Deluga, G. A.; Salge, J. R.; Schmidt, L. D.; Verykios, X. E. Science 2004, 303, 993−997. (4) Nörenberg, H.; Briggs, G. A. D. Phys. Rev. Lett. 1997, 79, 4222− 4225. (5) Nörenberg, H.; Briggs, G. A. D. Surf. Sci. 1999, 424, L352−L355. (6) Namai, Y.; Fujui, K.; Iwasawa, Y. J. Phys. Chem. B 2003, 107, 11666−11673. (7) Namai, Y.; Fujui, K.; Iwasawa, Y. Catal. Today 2003, 85, 79−91. (8) Esch, F.; Fabris, S.; Zhou, L.; Montini, T.; Africh, C.; Fornasiero, P.; Comelli, G.; Rosei, R. Science 2005, 309, 752−756. (9) Torbrügge, S.; Reichling, M.; Ishiyama, A.; Morita, S.; Custance, Ó . Phys. Rev. Lett. 2007, 99, 056101. (10) Nolan, M.; Grigoleit, S.; Sayle, D. C.; Parker, S. C.; Watson, G. W. Surf. Sci. 2005, 576, 217−229. (11) Fabris, S.; Vicario, G.; Balducci, G.; de Gironcoli, S.; Baroni, S. J. Phys. Chem. B 2005, 109, 22860−22867. (12) Herschend, B.; Baudin, M.; Hermansson, K. J. Chem. Phys. 2007, 126, 234706. (13) Sayle, T. X. T.; Parker, S. C.; Catlow, C. R. A. J. Phys. Chem. 1994, 98, 13625−13630. (14) Conesa, J. C. Surf. Sci. 1995, 339, 337−352. (15) Li, H. Y.; Wang, H. F.; Gong, X. Q.; Guo, Y. L.; Guo, Y.; Lu, G. Z.; Hu, P. Phys. Rev. B 2009, 79, 193401. (16) Ganduglia-Pirovano, M. V.; DaSilva, J. L. F.; Sauer, J. Phys. Rev. Lett. 2009, 102, 026101. (17) Jerratsch, J. F.; Shao, X.; Nilius, N.; Freund, H. J.; Popa, C.; Ganduglia-Pirovano, M. V.; Burow, A. M.; Sauer, J. Phys. Rev. Lett. 2011, 106, 246801. (18) Zhang, C. J.; Michaelides, A.; King, D. A.; Jenkins, S. J. Phys. Rev. B 2009, 79, 075433. (19) Minervini, L.; Zacate, M. O.; Grimes, R. W. Solid State Ionics 1999, 116, 339−349. (20) Gale, J. D.; Rohl, A. L. Mol. Simul. 2003, 29, 291−341. (21) Born, M.; Huang, K. Dynamical Theory of Crystal Lattices; Clarendon Press: Oxford, UK, 1954. (22) Dick, B. G.; Overhauser, A. W. Phys. Rev. 1958, 112, 90−103. (23) Vyas, S.; Grimes, R. W.; Gay, D. H.; Rohl, A. L. J. Chem. Soc., Faraday Trans. 1998, 94, 427−434. (24) Binks, D. J.; Grimes, R. W. J. Am. Ceram. Soc. 1993, 76, 2370− 2372.

25782


Structures of Defect Clusters on Ceria - American Chemical Society

Recommend Documents