Surface reactions mean-field approximation

R. D. Vigil, F. T. Willmore. Oscillatory dynamics in a heterogeneous surface reaction Breakdown of the mean-field approximation. Phys Rev E 54 1225-1231, 1996. 

Note that the rates of product formation and reactant conversion indeed have the dimensions of mol per unit of time, and that these rates are proportional to the number of sites, or, in fact, the amount of catalyst present in the reactor. Also, in the case of a second order reaction, e.g. betv een adsorbed species A and B, we write the rate in the form r = Nk0j 0 by applying the mean-field approximation. Here the rate is proportional to both the total number of sites on the surface and the probability of finding a species A adjacent to a species B on the surface, the latter being proportional to the coverages of A and B. In the mean-field approximation A and B are distributed randomly over the N available sites this only tends to be valid when the adsorbents repel each other. Thus the rate is not r= k(N0/ )(N02,) since the reactants need to be on adjacent sites. Another important consideration is that we want the rate to be linearly proportional to the amount of catalyst in the reactor, in accordance with r = Nk0A0B for a second order surface reaction. 

CO Stripping Chronoamperometiy Before discussing experimental results, let us examine what the LH mechanism predicts for the chronoamperometric response of an experiment where we start at a potential at which the CO adlayer is stable and we step to a final potential E where the CO adlayer will be oxidized. We will also assume that the so-called mean field approximation applies, i.e., CO and OH are well mixed on the surface and the reaction rate can be expressed in terms of their average coverages dco and qh- The differential equation for the rate of change of dco with time is... 

In recent years, there have been many attempts to combine the best of both worlds. Continuum solvent models (reaction field and variations thereof) are very popular now in quantum chemistry but they do not solve all problems, since the environment is treated in a static mean-field approximation. The Car-Parrinello method has found its way into chemistry and it is probably the most rigorous of the methods presently feasible. However, its computational cost allows only the study of systems of a few dozen atoms for periods of a few dozen picoseconds. Semiempirical cluster calculations on chromophores in solvent structures obtained from classical Monte Carlo calculations are discussed in the contribution of Coutinho and Canuto in this volume. In the present article, we describe our attempts with so-called hybrid or quantum-mechanical/molecular-mechanical (QM/MM) methods. These concentrate on the part of the system which is of primary interest (the reactants or the electronically excited solute, say) and treat it by semiempirical quantum chemistry. The rest of the system (solvent, surface, outer part of enzyme) is described by a classical force field. With this, we hope to incorporate the essential influence of the in itself uninteresting environment on the dynamics of the primary system. The approach lacks the rigour of the Car-Parrinello scheme but it allows us to surround a primary system of up to a few dozen atoms by an environment of several ten thousand atoms and run the whole system for several hundred thousand time steps which is equivalent to several hundred picoseconds. 

In mean-field approximation, the surface adsorbates are assumed to be uniformly distributed over the catalyst surface. The state of the catalyst surface is described by the surface temperature T and the fractional coverages of the adsorbates 0. Fractional coverage is the fraction of the surface covered by the surface adsorbed species k. Furthermore, it is assumed that the adsorption is limited to a mono atomic layer, and an uncovered surface is treated as the s th surface species. This means, there are only Ks surface adsorbed species. Assuming the surface temperature and coverages can be averaged over microscopic fluctuations, a chemical reaction can be defined in a way similar to gas-phase reactions. 

The following three sections describe the Bohmian quantum-classical approach [22,23] that uniquely solves the quantum back-reaction branching problem, the stochastic mean-field approximation [20] (SMF) that both resolves the back-reaction problem and incorporates the quantum decoherence and Franck-Condon overlap effects into NA-MD, and the quantized mean-field method [21] (QMF) that takes into account ZPE. The Bohmian and QMF approaches are illustrated by a model designed to capture some features of the O2 dissociation on a Pt surface. The concluding section summarizes the features of the methods and discusses further avenues for development and consideration. 

A detailed surface-reaction mechanism was used for the example system presented here, the partial oxidation of methane over rhodium (Deutschmann et al., 2001). The mechanism includes 20 species and 38 reactions. It can be imported into FLUENT in the format of the CHEMKIN database. The mean-field approximation was applied for modeling the surface chemistry (Section 2). In these simulations, the influence of the internal mass transfer—in contrast to the SFR models discussed above— was not directly covered by the employed CFD code but estimated by adapting the active catalytic surface area by an effectiveness factor. No gas-phase reaction mechanism was employed, as it was already shown in the literature that reactions in the gas phase can be neglected for the given operational conditions (Beretta et al., 2011 Bitsch-Larsen et al., 2008 Deutschmann and Schmidt, 1998 Veser and Frauhammer, 2000). 

Abstract. In this chapter we discuss approaches to solving quantum dynamics in the condensed phase based on the quantum-classical Liouville method. Several representations of the quantum-classical Liouville equation (QCLE) of motion have been investigated and subsequently simulated. We discuss the benefits and limitations of these approaches. By making further approximations to the QCLE, we show that standard approaches to this problem, i.e., mean-field and surface-hopping methods, can be derived. The computation of transport coefficients, such as chemical rate constants, represent an important class of problems where the QCL method is applicable. We present a general quantum-classical expression for a time-dependent transport coefficient which incorporates the full system s initial quantum equilibrium structure. As an example of the formalism, the computation of a reaction rate coefficient for a simple reactive model is presented. These results are compared to illuminate the similarities and differences between various approaches discussed in this chapter. 

It is well known that the rate constant of surface reactions depends on the local environment. Although the kinetic methods work in the mean-field approach, the local environment dependence is considered approximately by including the dependence of the effective activation energy and the preexponential factor on the average composition of the entire surface or on the specific surface in the given reactor section... 

Despite its severe limitations, the model shows interesting behavior, including kinetic phase transitions of two types continuous (second order) and discontinuous (first order). These phenomena are observed in many catalytic surface reactions. For this reason, the ZGB model has been widely studied and serves as a starting point for many more realistic models. This forms the first reason why we discuss the ZGB model in this section. The second reason is that MC simulations and mean-field (MF) solutions for this model give different results. Cluster approximations to the MF solutions offer a better agreement between the two methods, and then only small discrepancies remain. The ZGB model is therefore a nice example to illustrate the differences between the two approaches. 

Dickman [60], who was the first to study the mean-field description of the ZGB model, used a different approach in order to circumvent the problem of AB pairs. He split up the adsorption reactions in the ZGB model, thereby differentiating between adsorption adjacent to different surface species. For example, when an A molecule adsorbs next to a Bads> immediate reaction will occur. Thus, the reaction becomes A(g) -I- - - Bads —> AB(g) - - 2. This leads in the MF approximation to differential equations with fourth-order terms for the surface coverages of A and B. The infinite rate constant kr is absent from these equations. 

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