Mixing Monte Carlo technique

Simulation of a mixing process by using a computer provides a better alternative. Monte Carlo simulation techniques are often utilized. Monte Carlo techniques are numerical methods that involve sampling from statistical distributions, either theoretical or empirical, to approximate the real physical phenomena without reference to the actual physical systems. For the general discussion of Monte Carlo methods, readers are referred to Hammersley and Handscomb [19] and Tocher [20]. 

Spielman and Levenspiel (1965) appear to have been the earliest to propose a Monte Carlo technique, which comes under the purview of this section, for the simulation of a population balance model. They simulated the model due to Curl on the effect of drop mixing on chemical reaction conversion in a liquid-liquid dispersion that is discussed in Section 3.3.6. The drops, all of identical size and distributed with respect to reactant concentration, coalesce in pairs and instantly redisperse into the original pairs (after mixing of their contents) within the domain of a perfectly stirred continuous reactor. Feed droplets enter the reactor at a constant rate and concentration density, while the resident drops wash out at the same constant rate. Reaction occurs in individual droplets in accord with nth-order kinetics. 

There are basically two different computer simulation techniques known as molecular dynamics (MD) and Monte Carlo (MC) simulation. In MD molecular trajectories are computed by solving an equation of motion for equilibrium or nonequilibrium situations. Since the MD time scale is a physical one, this method permits investigations of time-dependent phenomena like, for example, transport processes [25,61-63]. In MC, on the other hand, trajectories are generated by a (biased) random walk in configuration space and, therefore, do not per se permit investigations of processes on a physical time scale (with the dynamics of spin lattices as an exception [64]). However, MC has the advantage that it can easily be applied to virtually all statistical-physical ensembles, which is of particular interest in the context of this chapter. On account of limitations of space and because excellent texts exist for the MD method [25,61-63,65], the present discussion will be restricted to the MC technique with particular emphasis on mixed stress-strain ensembles. 

The next section gives a brief overview of the main computational techniques currently applied to catalytic problems. These techniques include ab initio electronic structure calculations, (ab initio) molecular dynamics, and Monte Carlo methods. The next three sections are devoted to particular applications of these techniques to catalytic and electrocatalytic issues. We focus on the interaction of CO and hydrogen with metal and alloy surfaces, both from quantum-chemical and statistical-mechanical points of view, as these processes play an important role in fuel-cell catalysis. We also demonstrate the role of the solvent in electrocatalytic bondbreaking reactions, using molecular dynamics simulations as well as extensive electronic structure and ab initio molecular dynamics calculations. Monte Carlo simulations illustrate the importance of lateral interactions, mixing, and surface diffusion in obtaining a correct kinetic description of catalytic processes. Finally, we summarize the main conclusions and give an outlook of the role of computational chemistry in catalysis and electrocatalysis. 

Finally, we stress that the quantum chemical method presented here has the advantage over DFT-based techniques that it also furnishes wavefunctions that can be used to perform computations of spectra, and therefore have a better contact with the experiment. Another advantage of this approach is that, unlike the diffusion Monte-Carlo method, it can coherently be applied to studies of fermion and mixed boson/fermion doped clusters. An example can be found in our recent work on the Raman spectra of (He)w-Br2(X) clusters [27,28]. 

Whereas in mixed MD/MC simulations, some of the atoms are moved by pure MD, and other particles are moved by pure MC, it is also possible to constmct algorithms in which the displacement itself is determined in part by a deterministic factor and in part by a stochastic factor. In this class, we can further distinguish essentially three techniques Langevin or stochastic dynamics, hybrid Monte Carlo, and force bias Monte Carlo and related techniques. 

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