- 8:30 AM

Stochastic Simulation of Catalytic Surface Reactions in the Fast Diffusion Limit

Ethan A. Mastny, University of Wisconsin - Chemical and Biological Engineering, 1415 Engineering Drive, Madison, WI 53706-1691, Eric L. Haseltine, Department of Chemical Engineering, California Institute of Technology, Division of Chemistry & Chemical Engineering 210-41, Pasadena, CA 91125, and James B. Rawlings, Department of Chemical and Biological Engineering, University of Wisconsin-Madison, 1415 Engineering Drive, Madison, WI 53706.

For many catalytic surface reaction mechanisms diffusion events occur more than eight orders of magnitude faster than reaction events. This time scale separation makes simulation of these surface reactions intractable with direct kinetic Monte Carlo (KMC) methods. Additionally, in master equation approaches of surface reactions each state represents an independent spatial configuration. Tracking the probability of being in each spatial configuration is not feasible.

Other researchers have struggled with this time scale separation problem. These researchers often use KMC methods where diffusion rates are set to a sufficiently high values (but not too high), to equilibrate the lattice between reaction events [1,2]. Another approach is net-event KMC where fast reversible events are lumped into single events [3]. We have developed a different approach that involves applying the reaction equilibrium assumption to the fast diffusion events in the master equation. This work extends our previous examination of the equilibrium assumption for well-mixed systems [4] to the lattice problem. With this assumption we have shown that the spatial configurations can be dropped out of the master equation, without resorting to the mean field assumption. For example a system with two surface species on a 10x10 lattice has 10^48 spatial configurations but only 5050 possible species coverages. The reduced dimension master equation tracks the probability of being in these 5050 possible coverages through time.

The solution of this reduced master equation requires information that can be obtained through short KMC simulations run at each coverage. These simulations need to be run only once, however, because the data can be stored and used to in simulations with any set of kinetic parameters and initial conditions.

We are also able to show that this reduced master equation becomes a set of closed differential equations for the coverage evolution in the thermodynamic limit. These differential equations also require information that can be obtained from KMC simulations. Again the data from the KMC simulations are stored for use in future deterministic simulations. This method allows us to determine the correct differential equations for the coverages without appealing to higher-order moments of the coverage (i.e. pairs, triplets, ...).

We have applied this method to CO oxidation with strong lateral interactions. This system displays checkerboard like surface patterns at high CO coverages. For small lattices, we will show examples of how the coverage probability evolves in time using the reduced master equation. For large lattices, we will demonstrate the behavior of the deterministic rate equations for this system and compare it to the mean field equation behavior.

[1] A. P. J. Jansen. Monte Carlo simulations of temperature-programmed desorption spectra. Phys. Rev. B, 69:035414, 2004.

[2] S. Raimondeau and D. G. Vlachos. Recent developments on multiscale hierarchial modeling of chemical reactors. Chem. Eng. J., 90:323, 2002.

[3] M. A. Snyder, A. Chatterjee, and D. G. Vlachos. Net-event kinetic Monte Carlo for overcoming stiffness in spatially homogeneous and distributed systems. Comput. Chem. Eng., 29:701712, 200.5

[4] Eric L. Haseltine and James B. Rawlings. On the origins of approximations for stochastic chemical kinetics. J. Chem. Phys., 123:164115, October 2005.