18128
J. Phys. Chem. 1996, 100, 18128-18132
Simulated Annealing Study of the Structure and Reducibility in Ceria Clusters H. Cordatos, D. Ford, and R. J. Gorte* Department of Chemical Engineering, UniVersity of PennsylVania, Philadelphia, PennsylVania 19104 ReceiVed: April 17, 1996; In Final Form: August 6, 1996X
The structure and reducibility of ceria clusters was investigated using simulated annealing and interionic potentials, given by Sayle and co-workers (Sayle, T. X. T.; Parker, S. C.; Catlow, C. R. A. Surf. Sci. 1994, 316, 329), which include terms for ionic and van der Waals interactions, overlap repulsions, and ion polarizability. For CenO2n, the lattice energies, pair-distribution functions, and angular-distribution functions were determined for the crystals at 0 K for n ) 2-20 and n ) 50. As expected, the lattice energy decreased with n. Only for the largest cluster was the fluorite structure clearly observed. This cluster also exhibited (111) surface facets, indicating that this is the most stable surface. The reducibility of the clusters was determined from the difference in lattice energies of CenO2n and CenO2n-1, where charge neutrality was maintained by changing two Ce4+ ions to Ce3+. The energy required to reduce the cluster generally increased with increasing cluster size, but large fluctuations were also observed. These results generally agree with the experimental observations that reduction of ceria is structure sensitive and that larger crystals are more difficult to reduce.
Introduction Cerium oxide is an important component in automotive, emissions-control catalysts. While ceria probably plays a number of roles, its primary function is the storage and release of oxygen under cyclic oxidizing and reducing environments, a property resulting from the ability of cerium ions to switch between the +3 and +4 states.1-4 However, the ease with which ceria undergoes oxidation and reduction changes with time in the exhaust stream, and this is a significant problem for the synthesis of catalysts which can meet future requirements.5 Questions remain as to why the oxygen storage properties are lost upon catalyst aging. There is considerable evidence that the redox properties of ceria depend on the structure of the oxide,3,6-8 and it has been observed that loss of ceria activity is accompanied by an increased crystallite size. The loss cannot be accounted for by loss in surface area.9 Therefore, deactivation must be the result of removing defects,10,11 stabilizing less active crystallographic surfaces,12,14 or some other factor associated with small crystals. Computer simulations have been used to examine the structures and relative stabilities of several surfaces of bulk CeO2, as well as the formation of anion vacancies in reduced phases (Ce2O3 and CeO2-x).12-14 These calculations started by assuming that ceria exists in its bulk fluorite structure and then examined a segement of that structure. It was first shown that a simple pair potential could be used to calculate the lattice energy, lattice parameters, and dielectric constants of bulk ceria to a good approximation, probably due to the highly ionic nature of the material.12,15 The calculations were then used to show that the (111) surface should be the most stable of the simple surfaces and that the (100) surface should be easier to reduce than the (111).12,14 However, actual catalytic materials can consist of ceria crystallites which are so small that the assumption of bulk properties may not be valid. For example, Bensalem and co-workers have reported the presence of ceria clusters in the 10-20 Å range on their catalytic materials and demonstrated that the electronic spectra for these catalysts differed from those for the bulk materials.30,31 X
Abstract published in AdVance ACS Abstracts, November 1, 1996.
S0022-3654(96)01110-0 CCC: $12.00
An alternative approach to investigating the properties of ceria crystallites is to minimize the potential energy of small clusters, analogous to what has been done with metals. It has been known for a number of years that small metal clusters form structures which differ from those of bulk metals and exhibit interesting properties. For example, clusters with specific numbers of atoms (so called “magic numbers”) have been found in both experimental16 and calculational17 studies to be much more stable than other clusters of similar size. The energy for removing an atom from a cluster is therefore not a monotonic function of the cluster size, and the transition from noncrystalline to crystalline structures is not distinct.17 Of additional interest is the fact that the calculations which led to a reasonable understanding of these special stabilities for certain cluster sizes used relatively simple interatomic potentials. In the present paper, we report our results for the “simulated annealing” of small ceria clusters, using the potentials previously reported in the literature for bulk ceria.14 We will first discuss the structure of ceria clusters. The smallest clusters do not crystallize into the bulk, fluorite structure; on larger clusters, the fluorite structure is found, along with evidence for special stabilities for certain crystallographic surface orientations. Then we will discuss the ease of reducing the ceria clusters. In agreement with experimental observations, ease of reduction is found to decrease with cluster size; however, reducibility is not a monotonic function of cluster size. Methodology Interatomic Potentials. The potentials used for the total lattice energy calculations have been described in detail by Conesa.14 In brief, the expression used for the lattice energy of the clusters is as follows:
Elatt ) Ecoul + Esr + Esh
(1)
The first term, Ecoul, accounts for the purely electrostatic part of the energy which, on average, makes the largest contribution. Due to the finite size of the crystallites, there was no need to introduce an Ewald cutoff parameter. The second term, Esr, introduces the effects of overlap repulsion between ions, © 1996 American Chemical Society
Structure and Reducibility in Ceria Clusters
J. Phys. Chem., Vol. 100, No. 46, 1996 18129
TABLE 1: Parameters Used for Short-Range and Shell Model Interactions, from Ref 12 4+
Ce Ce3+ O2-
A (eV)
F (Å)
C (eV Å6)
k1 (eV/Å)
k2 (eV/Å)
Y (e)
1986.8 1731.6 22764.3
0.3511 0.3535 0.1490
20.4 14.43 43.83
291.75 291.75 419.87
0.0 0.0 10000.0
7.7 7.7 -6.1
dispersion and covalence effects, collectively described as “short-range interactions”. It has the form of the Buckingham potential which includes a repulsive (Born) term and an attractive (van der Waals) term:
{ ( ) }
Esr ) ∑ Aij exp i,j
-rij Fij
-
Cij
rij6
(2)
where rij are the interionic distances. Since cerium oxide is a strongly ionic material, such a pair potential is a reasonable approximation. For more covalent materials, three-body contributions, such as bond-angle dependences, cannot be neglected. The third term, Esh, simulates the effects of electronic polarizability via a simple, mechanical, shell model.18 Each ion is represented by a point core and a spherical shell, coupled by a harmonic spring. The contribution to the lattice energy is
Esh ) ∑Kiδi2
(3)
i
where δi is the distance between core and shell centers for a particular ion and Ki the corresponding force constant. The values for each of the constants in the above potentials are listed in Table 1. An extensive description of the assumptions and limitations of the above model can be found elsewhere.14 In summary, the model assumes distinct ionic species (Ce4+, Ce3+, O2-) with each ion having the formal total charge. Furthermore, the excess electrons associated with reduction of CeO2 are considered as being totally localized, giving Ce3+ rather than several Cez+ ions with 3 < z < 4. The above assumptions are nontrivial but have been shown to provide a reasonable starting point for understanding the properties of bulk ceria.14 Calculation of Minimum Energies. The problem of finding the equilibrium structure of ceria clusters at 0 K is equivalent to finding the global minimum of the potential energy (eq 1) of the lattice. This is not a simple task, since the number of local minima, as well as the CPU time required for the calculations, increases exponentially with the total number of ions in the cluster.20 In order to bypass the local minima (isomers) and isolate the global minimum, we have used the method of “simulated annealing”,21 complemented by a standard Metropolis walk.22 This approach has proven very efficient for other minimization problems of a similar nature.17,23 For CeO2 clusters, the starting point is a random ensemble of Ce4+ and O2-, at least 2.5 Å apart, and at an initial temperature of 8 00010 000 K. For the reduced crystallites, the ensemble contained two Ce3+ ions and one less oxygen. Then, a Monte Carlo (MC) NVT simulation was performed at that temperature. The ions were constrained to a spherical volume 45 Å in diameter using reflecting boundary conditions; the periodic boundary conditions typical of bulk material studies were not employed, since the particles being modeled were of finite dimension. A 50% acceptance of attempted displacement moves was maintained during the simulation by adjusting the maximum random displacement. The simulation proceeded at the given temperature until the potential energy, measured in block averages over 600 MC cycles, converged to a constant value, within 0.5%. Then the temperature was lowered according to Tnew ) gTold,
Figure 1. Lattice energy of CeO2 clusters per formula unit at 0 K as a function of CeO2 formula units. The squares show values calculated without the shell model incorporated into the potential.
where g ) 0.999, and a series of MC cycles was run at the new temperature. These steps were repeated until the temperature was reduced to 10 K. During the simulated annealing procedure, only the Coulombic and short-range interactions were taken into account, since incorporation of the shell model would have doubled the number of species (cores + shells), incurring a dramatic increase in the CPU time required, without adding to the reliability of the resulting configuration. Once the equilibrium configuration was found, the shell model was incorporated and the structure corresponding to the global minimum of the lattice energy obtained using the method of “steepest descent”.24 In order to verify that the global minimum of the energy was actually obtained, we repeated the calculation at least twice for each cluster from 4 to 14 formula units, and for the cluster of 20 formula units, starting from different initial ensembles and temperatures, and using different parameters for the random number generator. The resulting structures were inspected visually and analyzed via pair-separation and angular-distribution functions.25 In all cases, the results for a given cluster were identical in each run. The fact that the same local minimum was found in each simulation may indicate that the minima for these ionic clusters are deep. The calculations were performed on three workstations, a Silicon Graphics “Power Challenge”, a Silicon Graphics “IRIS 4000”, and a DEC “Alpha 3000”, using a Fortran code developed in this study. Results Structure of Small Clusters. Using the method of simulated annealing and the potentials described earlier, the equilibrium structures at 0 K, corresponding to the global minimum of the lattice energy, were found for a series of crystallite sizes. The values for the lattice energy per CeO2 formula unit, relative to the ions at infinite separation, are plotted in Figure 1 as a function of the number of CeO2 formula units in the crystallite. The corresponding values obtained without incorporating the shell model are shown for comparison. Due to the exponential increase of CPU time with the number of ions in the ensemble, the largest cluster that we modeled had a total of 50 CeO2 formula units, or 150 ions. For this size, the simulation yielded an almost spherical particle with an average diameter of 13.8 Å. Each of the clusters was stable by an amount ranging from -97 to -112 eV/formula unit, as expected. From Figure 1 it is obvious that the stability of the small clusters increases rapidly with the number of ions, but changes more slowly for the larger ones. In order to quantify the structure of the clusters, we calculated the angular-distribution functions (ADF) and pair-distribution
18130 J. Phys. Chem., Vol. 100, No. 46, 1996
Cordatos et al.
Figure 4. Shapes of the CeO2 cluster containing 50 formula units. The (111) plane is highlighted.
Figure 2. Pair-distribution functions (PDF) of cerium-cerium distances for clusters containing (a) 4, (b) 10, (c) 20, and (d) 50 formula units. The Dirac delta functions corresponding to the theoretical fluorite structure are presented with dotted lines.
Figure 3. Angular-distribution functions (ADF) of Ce4+ triplets with ionic centers within one lattice parameter (5.411 Å) for clusters containing (a) 4, (b) 10, (c) 20 and (d) 50 formula units. The Dirac delta functions corresponding to the theoretical fluorite structure are presented with dotted lines.
functions (PDF) for each of the crystallites examined. Those functions corresponding to 4, 10, 20, and 50 formula units are shown in Figures 2 and 3 and are compared with the bulk fluorite structure. The distribution functions are simply normalized histograms of the angles between ions and ion separations in the final configuration.25 For the ADF, only the ionic centers within one lattice parameter (5.411 Å) were considered in every triplet. Figure 2 shows the PDF for cerium-cerium distances, and Figure 3 shows the ADF for cerium-cerium-cerium
angles. The angles and separations for the bulk structure are Dirac delta functions, plotted as vertical dotted lines. From both figures, it can be seen that the features associated with the bulk crystalline structure grow in prominence as the size of the crystallite increases. Only the 50 formula unit particle exhibited ADF and PDF features indicative of the fluorite structure. It appears that the transition to crystallinity occurs gradually with the cluster size, although we were unable to examine every cluster size between 15 and 50 formula units due to computertime limitations. First, a gradual transition was observed for small metallic clusters modeled with the embedded atom potential.17 Second, it appears that the PDF and ADF curves in Figures 2 and 3 are evolving to a fluorite-like structure. The peaks at 60, 90, and 135° in the ADF of the theoretical fluorite structure, prominent in the cluster of 50 formula units, are also distinguishable at the 10 and 20 formula units. A picture of the 50 formula unit cluster, the only one exhibiting the fluorite structure, is shown in Figure 4, with the radii of the ions, as well as the interionic distances, drawn to scale. It is particularly interesting that the only well-defined, surface facet on this cluster is the compact (111) plane. This agrees with results from other calculations, which have also shown that this crystallographic surface should be the most stable.12-14 Reduced Clusters. The equilibrium structures of the reduced crystallites (i.e., those missing one oxygen ion) at 0 K, corresponding to the global minimum of the lattice energy, were found using exactly the same procedure as for the CeO2 crystals. As stated earlier, two Ce4+ ions were changed to Ce3+ to maintain charge neutrality with the removal of the oxygen. A similar transition to structures approaching that of fluorite was observed for the reduced clusters via visual inspection and analysis by ADF and PDF (not shown). It was possible to detect differences between the reduced and oxidized crystallites only for the smallest sizes (below 6 formula units). An example of the “reduced” and “oxidized” clusters for 10 Ce ions is shown in Figure 5, again with ionic radii and interatomic distances drawn to scale. For this cluster and all others investigated, the Ce3+ ions were located at the surface of the crystallites, possibly because it is energetically more favorable to remove an oxygen ion from the surface rather than from the bulk.26 More likely, the total energy of the cluster is minimized by maintaining ions with the smallest charge at the surface, so that ions with a higher charge can interact with more oxygen anions. The Ce3+ ions are not paired in clusters of all
Structure and Reducibility in Ceria Clusters
J. Phys. Chem., Vol. 100, No. 46, 1996 18131 between the reducibilities in Figure 7 and the easily measured structural factors, such as the Ce3+-Ce3+ distances in Figure 6, although structure must be responsible for the effect. Discussion
Figure 5. Shape of (left) “oxidized” and (right) “reduced” ceria cluster (i.e., missing one oxygen ion) for a cluster containing 10 Ce ions.
Figure 6. Distances between Ce3+-Ce3+ ions in the reduced clusters as a function of CeO2 formula units.
Figure 7. Change in lattice energy, En - En-, for removing one oxygen ion from the cluster, as a function of CeO2 formula units.
sizes, and the variation in the Ce3+-Ce3+ distances is shown in Figure 6. The variations in Figure 6 cannot be attributed to isomers with similar energies, since multiple simulations with different initial conditions reproduced both the equilibrium configurations and the Ce3+-Ce3+ distances. The change in lattice energy for removing an oxygen from the cluster, En - En-, is plotted in Figure 7 as a function of n, the number of CeO2 formula units in the cluster. For the range of cluster sizes that we modeled, there is a clear trend toward a larger difference in energy between the oxidized and reduced clusters. This implies that removal of an oxygen ion becomes more difficult for the larger clusters, in agreement with experimental results.18 What is surprising is that the “reducibility” does not decrease monotonically with size. For example, removing an oxygen ion from a cluster containing four formula units requires almost 9 eV more energy than removing an oxygen ion from a cluster of three formula units. Since the reproducibilities in the energies of the clusters were within
