Lattice diffusion model

Chapter 8 describes a number of generalized CA models, including reversible CA, coupled-map lattices, quantum CA, reaction-diffusion models, immunologically motivated CA models, random Boolean networks, sandpile models (in the context of self-organized criticality), structurally dynamic CA (in which the temporal evolution of the value of individual sites of a lattice are dynamically linked to an evolving lattice structure), and simple CA models of combat. 

We now sketch a simple deterministic lattice gas model of diffusion that becomes exactly Lorentz invariant in the continuum limit. We follow Toffoli ([toff89], [tofiSOb]) and Smith [smithm90]. 

We review Monte Carlo calculations of phase transitions and ordering behavior in lattice gas models of adsorbed layers on surfaces. The technical aspects of Monte Carlo methods are briefly summarized and results for a wide variety of models are described. Included are calculations of internal energies and order parameters for these models as a function of temperature and coverage along with adsorption isotherms and dynamic quantities such as self-diffusion constants. We also show results which are applicable to the interpretation of experimental data on physical systems such as H on Pd(lOO) and H on Fe(110). Other studies which are presented address fundamental theoretical questions about the nature of phase transitions in a two-dimensional geometry such as the existence of Kosterlitz-Thouless transitions or the nature of dynamic critical exponents. Lastly, we briefly mention multilayer adsorption and wetting phenomena and touch on the kinetics of domain growth at surfaces. 

In this subsection, we are not concerned with simulations which study the motion of single adsorbate atoms for realistic choices of the corrugation potential, but again restrict attention to simplified lattice gas models, where diffusion events are modelled by stochastic hops of adatoms from one lattice site to the next The description of the dynamics hence again is done... 

Fig. 8. The water-proton spin-lattice relaxation rates vs. magnetic field strength plotted as the Larmor frequency at 282 K for hexacyanochromate(II) ion ( ), trioxalatochromate(III) ion ( ), and trimalonatochromate(III) ion (A). The lines were computed using translational diffusion models developed by Freed with and without the inclusion of electron spin relaxation effects 54,121).

Watson E.B. and Eiang Y. (1995) A simple model for sector zoning in slowly grown crystals implications for growth rate and lattice diffusion, with emphasis on accessory minerals in crustal rocks. Am. Mineral. 80, 1179-1187. 

For example, the lattice diffusion theory of Torrey (16) was used (20) to calculate the theoretical relaxation times plotted in Figure 2 for two different models. The model for the upper dashed curves is that of diffusion of rapidly rotating SF6 molecules with respect to each other that for the lower set is that of diffusion of rapidly rotating SFfi molecules with... 

The lattice gas model is used to elucidate the "diffusion order-disorder transition on catalyst surfaces [92-102]. Finally, as has been mentioned already, this model is important in decoding thermodesorption spectra. 

In order to account for an observed global ferromagnetism, Zener (1951) had to make the further assumption that these pairs do not require an activation energy to diffuse through the lattice. This model leads to a charge-carrier mobility... 

Chapter 8 provides a unified view of the different kinetic problems in condensed phases on the basis of the lattice-gas model. This approach extends the famous Eyring s theory of absolute reaction rates to a wide range of elementary stages including adsorption, desorption, catalytic reactions, diffusion, surface and bulk reconstruction, etc., taking into consideration the non-ideal behavior of the medium. The Master equation is used to generate the kinetic equations for local concentrations and pair correlation functions. The many-particle problem and closing procedure for kinetic equations are discussed. Application to various surface and gas-solid interface processes is also considered. 

The kinetic equations for the volume phase of the solid body are equations of the diffusion type (63). Much attention has been given to them in the literature [154,155], therefore here will be reminded only those aspects of the theory of mass transfer for which the lattice-gas model has been used. These are problems involved in the construction of expressions for the diffusion the coefficients and boundary conditions of the diffusion equations. 

These models consider the mechanisms of formation of oscillations a mechanism involving the phase transition of planes Pt(100) (hex) (lxl) and a mechanism with the formation of surface oxides Pd(l 10). The models demonstrate the oscillations of the rate of C02 formation and the concentrations of adsorbed reactants. These oscillations are accompanied by various wave processes on the lattice that models single crystalline surfaces. The effects of the size of the model lattice and the intensity of COads diffusion on the synchronization and the form of oscillations and surface waves are studied. It was shown that it is possible to obtain a wide spectrum of chemical waves (cellular and turbulent structures and spiral and ellipsoid waves) using the lattice models developed [283], Also, the influence of the internal parameters on the shapes of surface concentration waves obtained in simulations under the limited surface diffusion intensity conditions has been studied [284], The hysteresis in oscillatory behavior has been found under step-by-step variation of oxygen partial pressure. Two different oscillatory regimes could exist at one and the same parameters of the reaction. The parameters of oscillations (amplitude, period, and the... 

Several groups have developed cellular automata models for particular reaction-diffusion systems. In particular, the Belousov-Zhabotinsky oscillating reaction has been examined in a number of studies.84-86 Attention has also been directed at the A + B —> C reaction, using both lattice-gas models 87-90 and a generalized Margolus diffusion approach.91 We developed a simple, direct cellular automaton model92 for hard-sphere bimolecular chemical reactions of the form... 

The BCF volume diffusion model has already been discussed in Sect. 5.1.2. The classical approach is to consider the diffusion of lattice ions from the... 

The first application of hierarchical SA for parameter estimation included refinement of the pre-exponentials in a surface kinetics mechanism of CO oxidation on Pt (a lattice KMC model with parameters) (Raimondeau et al., 2003). A second example entailed parameter estimation of a dual site 3D lattice KMC model for the benzene/faujasite zeolite system where benzene-benzene interactions, equilibrium constants for adsorption/desorption of benzene on different types of sites, and diffusion parameters of benzene (a total of 15 parameters) were determined (Snyder and Vlachos, 2004). While this approach appears promising, the development of accurate but inexpensive surfaces (reduced models) deserves further attention to fully understand its success and limitation. 

V.P. Zhdanov, General equations for description of surface diffusion in the framework of the lattice-gas model. Surface Sci 749 L13 (1985). 

The kinetics of CO oxidation on a pyramidal supported catalyst particle (Fig. 4) was simulated [11,27] by assuming CO adsorption to cause restructuring of the top (100) facet. The restructuring was described on the basis of the lattice-gas model, predicting phase separation in the overlayer. CO diffusion was much faster compared to other steps. Oscillatory and chaotic kinetic regimes were found in the simulations. One of the reasons of irregular oscillatory kinetics was demonstrated to be the interplay of the reactions on the (100) and (111) facets. 

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