Spatio-temporal pattern formation

D. Walgraef. Spatio-Temporal Pattern Formation. New York Springer Verlag, 1997, pp. 1-301. 

R. J. Gelten et al. Monte Carlo simulation of a surface reaction model showing spatio-temporal pattern formations and oscillations. J Chem Phys 705 5921-5934, 1998. 

K. Asakura, J. Lanterbach, H.H. Rothermund, and G. Ertl, Spatio-temporal pattern formation during catalytic CO oxidation on a Pt(100) surface modified with submonolayers of Au, Surf. Sci. 374, 125-141 (1997). 

Figure 2.16. Computer simulation of spatio-temporal pattern formation in CO oxidation on a surface. [Adapted from R.J. Celten, A.P.J. Jansen, R.A. van Santen, j.j. Lukkien, j.P.L Segers and P.A.j. Hilbers,j. Chem. Phys. 108 (1998) 5921.]...

The kinetics of chemical reactions on surfaces is described using a microscopic approach based on a master equation. This approach is essential to correctly include the effects of surface reconstruction and island formation on the overall rate of surface reactions. The solution of the master equation using Monte Carlo methods is discussed. The methods are applied to the oxidation of CO on a platinum single crystal surface. This system shows oscillatory behavior and spatio-temporal pattern formation in various forms. 

In the following, we will discuss DMC simulations on the CO oxidation on the Pt(lOO) surface, that were done in our laboratories. The simulations show oscillations in the CO2 production rate as well as several types of spatio-temporal pattern formation. In essence, it is an extension of the ZGB model with desorption and diffusion of A, finite reaction rates and surface reconstruction. We will discuss it to illustrate the complexity of the models with which DMC simulations can be done nowadays. For clarity, we will stick to the A and B2 notation employed in the previous section. Species A corresponds to CO and B2 corresponds to 02- Furthermore, we will speak in terms of reaction rates instead of relative reaction probabilities. This terminology is entirely justified in the DMC approach that we used. 

In section 3.2.2 we have discussed four regimes of front generation. Spatio-temporal pattern formation is observed in two of these regimes. The type of pattern formation that is observed in the third regime was already discussed above. Figure 6A shows another example of the cellular patterns. 

Under conditions where the front generation is slow, i.e. in the second regime of section 3.2.2, spatio-temporal pattern formation is observed in several forms. Target patterns, rotating spirals and turbulent structures are the observed forms. When turbulent patterns are present, sometimes small fragments of reaction fronts exhibit solitonic behavior. Figure 6 shows the four main forms of pattern formation that we have observed in our simulations. 

Figure 6. Four examples of spatio-temporal pattern formation in our simulations (A) cellular structures, (B) target patterns, (C) a double rotating spiral, (D) turbulent patterns.

We have also cast the DMC model in a set of ordinary differential equations, thus translating it to a mean-field approach with the site-approximation. Only the kinetic oscillations can be modeled in this way. To model the spatio-temporal pattern formations, diffusion terms would have to be added to the mean-field description, in order to account for the spatial dependence of the reactant concentrations. 

The main application that was discussed was a microscopic model for the oxidation of CO, catalyzed by a Pt(lOO) single crystal surface. The simulations show kinetic oscillations as well as spatio-temporal pattern formation in the form of target patterns, rotating spirals and turbulent patterns. Finally, mean-field simulations of the same model were compared with the Monte Carlo simulations. When diffusion is fast and the simulation grids are small, the results of Monte Carlo simulations approach those of the mean-field simulations. 

R. J. Gelten, A. P. J. Jansen, R. A. van Santen, J. J. Lukkien, J. P. L. Segers, and P. A. J. Hilbers, Monte Carlo simulations of a surface reaction model showing spatio-temporal pattern formations and oscillations, J. Chem. Phys., 108 (1998) 5921. 

The experimental observation of concentrical wave fronts has been ascribed to structural defects in the surface . On these defects, oxygen dissociates faster, and they therefore act as a periodic pulse generator for reaction fronts. With each pulse, a new front is generated, which then grows continuously. This behavior is indeed reproduced by our simulations when such a defect is included. Nonetheless, concentrical circles can also be simulated on a perfectly homogeneous surface. A circular reaction wave front is initiates spontaneously, e.g. by oxygen adsorption on a site that is vacated by CO desorption, at a position we will call the primary center. Inside this front, at a position different from the primary center, a new front is initiated. Structural defects form a stabilizing, rather than a necessary factor for spatio-temporal pattern formation in the form of concentrical circles. 

Systematic studies on thermokinetic wave propagation have been reported for various reactions, and different approaches have been reported to implement nonisothermal effects in the theoretical modeling [53]. Even if the details of the reaction mechanism are less well understood, the basic features of spatio-temporal pattern formation with such systems can often be modeled successfully, because the decisive effects can be approximated by a heat balance equation in which the chemistry is reduced to single variable and surface diffusion of adsorbates is neglected [54,55]. 

Walgraef, D. Spatio-Temporal Pattern Formation. Springer, New York (1996)... 

ASA Asakura, K., Lauterbach, J., Rotermund, H., Ertl, G. Spatio-temporal pattern formation... 

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