J. Phys. Chem. B 2006, 110, 16591-16599
16591
Thermodynamic Equilibrium Compositions, Structures, and Reaction Energies of PtxOy (x ) 1-3) Clusters Predicted from First Principles Ye Xu,† William A. Shelton,† and William F. Schneider*,‡ Computer Science and Mathematics DiVision, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, and Department of Chemical and Biomolecular Engineering and Department of Chemistry and Biochemistry, UniVersity of Notre Dame, 182 Fitzpatrick Hall, Notre Dame, Indiana 46556 ReceiVed: March 8, 2006; In Final Form: June 21, 2006
As synthetic nanocatalysis strives to create and apply well-defined catalytic centers containing as few as a handful of active metal atoms, it becomes particularly important to understand the structures, compositions, and reactivity of small metal clusters as a function of size and chemical environment. As a part of our effort to better understand the oxidation chemistry of Pt clusters, we present here a comprehensive set of density functional theory simulations combined with thermodynamic modeling that allow us to map out the T-pO2 phase diagrams and predict the oxygen affinity of PtxOy clusters, x ) 1-3. We find that the Pt clusters have a much stronger tendency to form oxides than does the bulk metal, that these oxides persist over a wide range of oxygen chemical potentials, and that the most stable cluster stoichiometry varies with size and may differ from the stoichiometry of the stable bulk oxide in the same environment. Further, the facility with which the clusters are reduced depends both on size and on composition. These models provide a systematic framework for understanding the compositions and energies of redox reactions of discrete metal clusters of interest in supported and gas-phase nanocatalysis.
Introduction In recent years metal nanoparticles (metal aggregates a few nanometers in size and smaller) have been recognized to possess distinctly different properties from extended phases of the same metals. In particular, research in the field of heterogeneous catalysis has revealed unique types of catalytic activity at the nanoscale. The archetypical example is Au, which when dispersed at the nanoscale shows dramatically enhanced activity for a number of important reactions, including CO oxidation, the partial oxidation of alkenes, and hydrogenation of unsaturated hydrocarbons.1-4 More recently, several groups have reported that oxide-supported clusters exhibit strongly sizedependent activity, sometimes due to just one more or fewer atom. For example, Heiz et al. have demonstrated that MgOsupported Pt5-Pt20 clusters have very different CO oxidation activities.5 Gates and co-workers have demonstrated that Ir4 clusters appear to have consistently higher hydrogenation activity than Ir6 clusters (for toluene on MgO6 and for ethene on γ-Al2O3,7 respectively). Lee et al. have reported the size dependence of the CO oxidation activity of TiO2-supported Au1-4,7 clusters.8 And Flytzani-Stephanopolous has provided evidence supporting the role of individual supported metal atoms in Aucatalyzed water-gas shift chemistry.9 To fully exploit these developments, it is crucial to be able to predict and explain the inherent reactivity of metal clusters. For instance, gas-phase studies of discrete Au and Ag clusters have contributed substantially to the understanding of their heterogeneous catalytic activity.10 One key aspect of this intrinsic reactivity that has not been well explored is the response * Author to whom correspondence should be addressed. E-mail:
[email protected]. † Oak Ridge National Laboratory. ‡ University of Notre Dame.
of clusters to their chemical environment.11 Under oxidizing conditions the active surface phases of nominally metallic transition metal catalysts have been shown, through combined experimental and theoretical efforts, likely to be metal oxides and not metals. Prominent examples include Ru for CO oxidation,12,13 Ag for ethylene epoxidation,14-16 Pt for CO oxidation,17-21 and Pd for CH4 catalytic combustion.22 Because of their smaller size, nanoparticles are expected to be only more susceptible to the environment than larger, bulklike particles. Yet knowledge about how the environment affects their properties and how the effect varies with particle size is scarce. To shed light on the coupling between particle size and chemical environment and its impact on reactivity, we have begun a modeling effort to explore the oxidation properties of platinum clusters, i.e., nanoparticles consisting only of a handful of Pt atoms. Pt is one of the most versatile and also one of the most expensive transition metals in catalysis and is a key component in many important environmental and energy applications, such as the automotive three-way catalyst, the lean NOx trap,23,24 and low-temperature fuel cells.25 In a previous article we investigated the oxidation of Pt clusters containing up to 10 Pt atoms using density functional theory (DFT) calculations and demonstrated systematic variation in the structure and oxidation energetics with respect to size.26 In this report we take one step closer to the actual catalyst system and present a first-principles thermodynamic model that incorporates the effect of an O2 atmosphere at finite temperature (T) and partial pressure (pO2). By coupling the Pt clusters to the O2 atmosphere, this model allows us to determine the extent of their oxidation at given T and pO2 from first principles. While the reactions of isolated Pt clusters with oxygen are of interest in their own right,27 it is relevant to heterogeneous catalysis to consider the modifying influence of a support. Because there are many types of metal oxide supports with
10.1021/jp0614446 CCC: $33.50 © 2006 American Chemical Society Published on Web 08/02/2006
16592 J. Phys. Chem. B, Vol. 110, No. 33, 2006 functionally and structurally disparate surfaces, the consideration of a specific oxide both complicates the modeling efforts and reduces the generality of the insights that can be gained. We choose to address the problem in a general way first, by introducing into our thermodynamic model a hypothetical support with no other structural or chemical properties other than the ability to restrict the motion of the clusters. The results serve as a base reference against which the chemical effects of specific oxide supports can be compared. They can also be thought of as representing a weak cluster-support interaction limit. The geometry and energetics of Pt clusters on a relatively unreactive oxide surface can be expected to be close to those presented below. In the following sections, we first report a DFT-based search of the Pt1-3Oy composition/structure space and then construct T-pO2 phase diagrams for the three Pt cluster sizes. The results indicate a surprising stability for higher oxide as well as nonstoichiometric clusters (relative to the stable bulk platinum oxides) over a broad range of conditions of practical interest. Further, we calculate the oxygen affinity of these clusters, showing that not only the composition but also the energetics of the redox reactions of the clusters are strongly size- and environment-dependent. Therefore, even more so than for reactions on extended metal surfaces, environmental effects must be accounted for in the simulations of atomic-scale metal aggregates for catalytic applications. Methods DFT Total Energy Calculations. Periodic spin-polarized DFT calculations are performed using the Vienna Ab initio Simulation Package (VASP)28-31 in the generalized gradient approximation (GGA). The exchange-correlation energy and potential are described by the PW91 functional.32 Recent results suggest that pure density functionals such as PW91 provide the best overall compromise in accuracy for both the structures and the energetics of inorganic systems.33 The ionic cores are described by Vanderbilt’s ultrasoft pseudopotentials.34 The Kohn-Sham valence states are expanded in a plane wave basis up to a kinetic energy of 396 eV. To accelerate convergence, electronic states are smeared using a Gaussian scheme (kBT ) 0.01 eV). The final electronic states all have unit occupancies, as required for discrete molecular clusters. As the intent is to simulate isolated clusters, k space is sampled at the Γ point only. Clusters are placed within an 18 × 18.2 × 18.4 Å3 unit cell, which is sufficiently large so that the electrostatic interaction among periodic images does not affect the total energy of the system to more than a few millielectronvolts. Reciprocal space integration is done on a 120 × 120 × 120 FFT grid to avoid wrap-around error. The ground-state spin multiplicity of each cluster is verified. The most stable structures of small Pt oxide clusters (those containing fewer than 4-5 atoms) are found by relaxing various manually constructed initial guesses. For larger clusters, we use DFT-based Nose´-Hoover molecular dynamics (MD) to sample greater portions of the configuration space than is possible manually (for more detail see ref 26). Multiple trajectories from different initial configurations are generated for each PtxOy composition, and candidate structures are extracted and relaxed until the maximum force on any atom is less than 0.01 eV/Å. The lowest in energy of the candidate structures is taken as the equilibrium structure. The systematic trends observed in the structures and energetics lend confidence that structures with significantly lower energy have not been overlooked in any of the compositions.
Xu et al.
Figure 1. Schematic showing the limiting cases of stationary (left) and two-dimensional ideal-gas (right) clusters.
We define the formation energy (FE) of a given PtxOy cluster as
FEPtxOy ) EPtxOy - EPtx - 1/2 yEO2 + ∆EZPE
(1)
ZPE ZPE - E Pt - yE OZPE ∆EZPE ) EPt xOy x 2
Here EPtxOy is the total energy of a PtxOy cluster, EPtx is the total energy of the pure Pt cluster of size x in its most stable geometry, EO2 is the total energy of an O2 molecule, and ∆EZPE is the difference in the zero-point energies (EZPE) of the species. The EZPE of a molecule depends on the vibrational frequencies (νi) of the normal modes of the molecule, which are calculated in the harmonic approximation by diagonalizing the mass-weighted Hessian matrix. For a linear cluster, there are 3n - 5 normal modes, where n ) x + y. For a nonlinear cluster, there are 3n 6 normal modes. If we define ΘVi ) hνi/kB as the characteristic vibrational temperature of each normal mode i, then
E ZPE )
1 2
∑i kBΘν
i
(2)
Here kB is the Boltzmann constant. ∆EZPE is less than 0.05 eV/O for all the Pt oxide clusters in this study. The PW91-calculated O-O stretching frequency is 1571 cm-1, in very good agreement with the 1580 cm-1 listed by the National Institute of Standards and Technology (NIST). As reported previously,26 we have calculated the FEs for PtO and PtO2 molecules and for bulk PtO and β-PtO2. The results agree with the tabulated standard heats of formation to within 0.2 eV; deviations for intermediate cluster sizes could be larger, but the DFT results presented here provide the best available estimates to date. Constructing Temperature-Pressure Phase Diagrams. The DFT results correspond to electronic ground-state energies. The effect of nonzero temperature and O2 chemical potential on the equilibrium oxygen content of a given Pt cluster (Ptx) can be determined by calculating and comparing the appropriate free energies of PtxOy. We envision the Ptx clusters to be immobilized by interactions with the aforementioned idealized support surface. The clusters do not interact with each other and are in thermal and chemical equilibrium with an infinite reservoir of O2 ideal gas. The clusters cannot leave the surface, and there is no external source of Pt. Two limiting cases of cluster mobility can be envisioned (Figure 1): In one limit the clusters are anchored to the support and can vibrate but possess neither translational nor rotational degrees of freedom. In the other limit the clusters are free to move about on the surface and form a two-dimensional (2-D) ideal gas. In reality, the mobility of supported clusters is limited compared to that of gas-phase clusters, so the phase behavior of supported Pt clusters is expected to be close to the stationary limit. The 2-D ideal-gas limit is included to illustrate the thermodynamic effect of cluster mobility and yields results similar to a 3-D gas. At constant Pt content x, temperature T, and oxygen chemical potential µO, the grand potential Ω governs chemical equilibrium. The grand potential of this ideal surface containing N
Compositions, Structures, and Thermodynamics of PtxOy
J. Phys. Chem. B, Vol. 110, No. 33, 2006 16593
identical clusters can be written in terms of the potential ω of an individual cluster
ωPtxOy(T, a, µO, x) ) Ω/N ) PtxOy - TsPtxOy - yµO
(3)
Here a is the area occupied by the cluster and and s are the internal energy and entropy of the cluster. In this formulation, the thermodynamic equilibrium composition PtxOy is that which minimizes ω at given external conditions. The three terms on the right-hand side of eq 3 are evaluated as follows. The internal energy of an immobile cluster is written as
PtxOy ≈ EPtxOy + Evib
(4)
Here the total electronic energy (EPtxOy) is the DFT-calculated value and is assumed to be independent of temperature, and Evib includes EZPE and a finite-temperature contribution according to standard statistical mechanics.35 To simplify the model we allow the clusters to access all the vibrational modes available in the gas phase. Evib of an explicitly supported cluster would be somewhat smaller, but the exact values of Evib have little impact on the final results. For instance, for the largest cluster in this study, Pt3O9, Evib is less than 5% of EPt3O9 even at 1000 K. The entropy per cluster s of N immobile and thus distinguishable clusters is calculated as follows35
∂ S s ) ) kB ln qeqvib + kB T ln qeqvib N ∂T
(5)
note that, by ignoring excited electronic states, the electronic entropy accounts for 9% and 7% of the total entropy (electronic plus vibrational) at 200 and 1000 K, respectively. At the same temperatures, the total entropy is 0.4% and 8% of EPt3O9, whereas the yµO term is 75-85% of EPt3O9. In general, the first excited state of a small cluster is several tenths of an electronvolt or more above the ground state, but the spacing decreases to ca. 0.1 eV or less for a cluster containing more than 8-10 atoms. One can imagine that, for example, all excited states of the Pt3O9 cluster are triplets with each lying 0.1 eV above the next-highestenergy state. The resulting sum ∑∞i 3e-(0.1eV×i/kBT) (see eq 6) does not exceed 0.01 at 200 K or 1.4 at 1000 K, compared to the first term η0 ) 3. By a similar argument, conformational isomerization will introduce entropic contributions of a similar magnitude to that of the electronic degree of freedom. Thus electronic and conformational entropies are not expected to have a major impact on the relative magnitudes of the grand potentials of Pt or Pt oxide clusters. To treat the clusters as a 2-D ideal gas, the translational and rotational contributions to energy and entropy need to be included in the formulation of ω. The entropy s is now calculated for N indistinguishable clusters35
s)
using Stirling’s approximation for large N, where e is the base of the natural logarithm. In the temperature range of interest here the translational and rotational partition functions are35
with only the electronic and vibrational degrees of freedom contributing to the entropy. The electronic and vibrational q’s are
qe ) η0 + η1e-∆e1/kBT + η2e-∆e2/kBT + ‚‚‚ qvib )
∏i
-Θνi/2T
e
1 - e-Θνi/T
(6)
where ηi is the degeneracy of the ith electronic state, ∆ei is the energetic separation between the ith state and the ground state, and Θνi is as described in eq 2. As explained below, qe is approximated as η0 in this study. Finally, to connect µO to O2 pressure we follow the example of Bollinger et al.36 and calculate µO2 first, taking O2 to be an ideal gas
µO2 ) ∆hO2(T, p0) +
()
E OZPE + EO2 - TsO2(T, p0) + kBT ln 2
pO 2 p0
(7)
In this formulation, the pressure dependence of µO2 enters through the pO2/p0 ratio, where the reference O2 pressure p0 is arbitrarily set to 1 bar. ∆hO2 is h(T, p0) - h(T ) 0 K, p0) for O2 gas. The DFT total energy of O2 is used for EO2. Since O atoms in the clusters are in chemical equilibrium with the gas-phase O2, i.e., PtxOy T 1/2yO2(g) + Ptx, then µO ) 1/2 µO2. The values of the enthalpy, entropy, and zero-point energy of gas-phase O2 as tabulated by NIST are used.37 For comparison, the entropies of gas-phase O2 at several temperatures and pressures are calculated using eq 5, and the results agree very closely with the tabulated values. To illustrate the potential contribution of electronic entropy to the grand potential, we again take the Pt3O9 cluster as an example (see “Equilibrium Structures” subsection below). We
qeqtrqrotqvibe qeqtrqrotqvibe S ∂ ) kB ln + kBT ln (8) N N ∂T N
qtr )
{
(
2πmkBT h2
)
A
(9)
1 T for a linear molecule σr Θ r qrot ) π1/2 3/2 T σr (Θ Θ Θ )1/2 for a nonlinear molecule rx ry rz Here h is Planck’s constant, m and σr are the mass and the symmetry factor of the cluster, A is the total surface area of the support, and Θri ) h2/8π2kBIi, where Ii is the ith moment of inertia. Note that three-dimensional qrot is used here as the upper limit of the rotational motion of the clusters. What remain to be specified are the surface area A and the total number of Ptx clusters, N. Since the typical surface area of catalyst supports is ca. 100 m2/g and the typical Pt loading ca. 1% (w/w), we set A ) 100 m2 to obtain a total number of ca. 3 × 1019 Pt atoms at 1% loading. Because the total amount of Pt is held constant, N is equal to 3 × 1019/x. ω/x Phase Diagram. We can also construct ω/x versus µO phase diagrams to assess the stability of a PtxOy cluster with respect to “phase separation” into smaller (fragmentation) or larger (agglomeration) clusters. It can be shown that if
ωPtm+nOi m+n
