A general stochastic model of surface reactions

In this Chapter we introduce a stochastic ansatz which can be used to model systems with surface reactions. These systems may include mono-and bimolecular steps, like particle adsorption, desorption, reaction and diffusion. We take advantage of the Markovian behaviour of these systems using master equations for their description. The resulting infinite set of equations is truncated at a certain level in a small lattice region we solve the exact lattice equations and connect their solution to continuous functions which represent the behaviour of the system for large distances from a reference point. The stochastic ansatz is used to model different surface reaction systems, such as the oxidation of CO molecules on a metal (Pt) surface, or the formation of NH3. 

Surface reactions are very important in both theoretical and applied research. Experimental information on the individual reaction steps is difficult to obtain and the interpretation of the data is not easy. However, investigation of individual steps of a surface reaction can be obtained by using theoretical models, where discrete lattices are used to represent the surface. Depending on the number of different particles and on the adsorption and reaction steps, the models are classified as A + A- 0, A- -B 0, A + 62 0, 

More complex systems which model real systems cannot be solved using purely analytical methods. For this reason we want to introduce in this Chapter a novel formalism which is able to handle complex systems using analytical and numerical techniques and which takes explicitly structural aspects into account. The ansatz can be formulated following the theory described below. In the present stochastic ansatz we make use of the assumption that the systems we will handle are of the Markovian type. Therefore these systems are well suited for the description in terms of master equations. 

In such a representation of an infinite set of master equations for the distribution functions of the state of the surface and of pairs of surface sites (and so on) will arise. This set of equations cannot be solved analytically. To handle this problem practically, this hierarchy must be truncated at a certain level. In such an approach the numerical part needs only a small amount of computer time compared to direct computer simulations. In spite of very simple theoretical descriptions (for example, mean-field approach for certain aspects) structural aspects of the systems are explicitly taken here into account. This leads to results which are in good agreement with computer simulations. But the stochastic model successfully avoids the main difficulty of computer simulations the tremendous amount of computer time which is needed to obtain good statistics for the results. Therefore more complex systems can be studied in detail which may eventually lead to a better understanding of such systems. 

The simulation takes place on a discrete lattice with coordination number z. Each lattice site is given a lattice vector 1. The state of the site I is represented by the lattice variable au which may depend on the state of the catalyst site (e.g., promoted or not) and on the coverage with a particle. Let us assume we deal with the simple case in which all catalyst sites are equal 

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