Kinetic Monte Carlo method described

It is clear that a detailed knowledge of the structure of the catalyst under reaction conditions, in particular FTS reaction in harsh conditions (230-300 °C, 2-5 MPa) is required for any theoretical study to he successful described. Recent studies are combining DFT with simple thermodynamics to obtain phase diagrams and predict the thermodynamically favored catalyst structure under different conditions. This approach would be appreciable for the elucidation of surface structure of Co-based catalysts for FTS reaction. In addition. Kinetic Monte Carlo methods allow bridging the pressure gap from the vacuum conditions of DFT to real conditions, leading to the understanding, optimization and design of new catalysts. 

The Monte Carlo method as described so far is useful to evaluate equilibrium properties but says nothing about the time evolution of the system. However, it is in some cases possible to construct a Monte Carlo algorithm that allows the simulated system to evolve like a physical system. This is the case when the dynamics can be described as thermally activated processes, such as adsorption, desorption, and diffusion. Since these processes are particularly well defined in the case of lattice models, these are particularly well suited for this approach. The foundations of dynamical Monte Carlo (DMC) or kinetic Monte Carlo (KMC) simulations have been discussed by Eichthom and Weinberg (1991) in terms of the theory of Poisson processes. The main idea is that the rate of each process that may eventually occur on the surface can be described by an equation of the Arrhenius type ... 

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. 

It Is to be remarked that the process described by the Infinite set of kinetic (coagulation) equations can be simulated by Monte-Carlo methods ( ). The Information on the number of molecules of the respective size Is stored In the computer memory and weighting for selection of molecules Is applied given by the number and reactivity of groups In the respective molecule. 

Calculations [36] with the help of the Monte Carlo method, have shown (see, for example, Fig. 2) that, in the case most unfavourable for the theory, n(0) = A1(0), eqns. (27) describe the kinetics of electron tunneling at 50 and 80% of reagents decay with an accuracy of 2 and 4%, respectively [that is, 2% of the initial concentration, n(0) = 7V(0)]. Thus, eqns. (27) give a quantitative description of the kinetics of electron tunneling reactions up to a decay of 80%. 

Kinetic Monte Carlo and hyperdynamics methods have yet to be applied to processes involved in thermal barrier coating failure or even simpler model metal-ceramic or ceramic-ceramic interface degradation as a function of time. A hindrance to their application is lack of a clear consensus on how to describe the interatomic interactions by an analytic potential function. If instead, for lack of an analytic potential, one must resort to full-blown density functional theory to calculate the interatomic forces, this will become the bottleneck that will limit the size and complexity of systems one may examine, even with multiscale methods. 

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. 

Recently, Pyun et al. applied a kinetic Monte Carlo (KMC) method to explore the effect of phase transition due to strong interaction between lithium ions in transition metal oxides with the cubic-spinel structure on lithium transport [17, 28, 103]. The group used the same model for the cubic-spinel structure as described in Section 5.2.3, based on the lattice gas theory. For KMC simulation in a canonical ensemble (CE) where all the microstates have equal V, T, and N, the transition state theory is employed in conjunction with spin-exchange dynamics [104, 105]. 

In a staged multi-scale approach, the energetics and reaction rates obtained from these calculations can be used to develop coarse-grained models for simulating kinetics and thermodynamics of complex multi-step reactions on electrodes (for example see [25, 26, 27, 28, 29, 30]). Varying levels of complexity can be simulated on electrodes to introduce defects on electrode surfaces, composition of alloy electrodes, distribution of alloy electrode surfaces, particulate electrodes, etc. Monte Carlo methods can also be coupled with continuum transport/reaction models to correctly describe surfaces effects and provide accurate boundary conditions (for e.g. see Ref. [31]). In what follows, we briefly describe density functional theory calculations and kinetic Monte Carlo simulations to understand CO electro oxidation on Pt-based electrodes. 

Chemical reactions are treated by a rule that reflects the mass action kinetics. For example, for a bimolecular reaction oi A + B C, when an A particle and a B particle occupy the same lattice point, they have a probability of being converted to a C particle that depends on the rate constant of the reaction and the time scale. The procedure used to determine what reactions occur in a given interval is identical to that described in section 7.2 for the Monte Carlo method. 

As an alternative, this chapter describes methods for predicting small-molecule diffusivity that are based on transition-state theory (TST) [82-84] and kinetic Monte Carlo (KMQ [85,86]. These methods capitalize on the proposed penetrant jump mechanism. TST was described in Chapter 1 and is typically used to estimate the rates of chemical reactions from first principles here we use TST to calculate the rate of characteristic jumps for each penetrant in a host polymer matrix. The collection of jump rates can be combined with the penetrant jump topology and KMC to obtain the penetrant diffusion coefficient. Other results obtainable from these simulations are physical aspects related to the jump mechanism the sizes and shapes of voids accessible to penetrant molecules [87], enthalpic and entropic contributions to the penetrant jump rate [88,89], the extent and characteristics of chain motions that accompany each jump [90], and the shape and structure of the jump network itself [91]. 

The general solution of the model can be obtained using kinetic Monte Carlo (kMC) simulations. This stochastic method has been successfully applied in the field of heterogeneous catalysis on nanosized catalyst particles (Zhdanov and Kasemo, 2000,2003). It describes the temporal evolution of the system as a Markovian random walk through configuration space. This approach reflects the probabilistic nature of many-particle effects on the catalyst surface. Since these simulations permit atomistic... 

The relative fluctuations in Monte Carlo simulations are of the order of magnitude where N is the total number of molecules in the simulation. The observed error in kinetic simulations is about 1-2% when lO molecules are used. In the computer calculations described by Schaad, the grids of the technique shown here are replaced by computer memory, so the capacity of the memory is one limit on the maximum number of molecules. Other programs for stochastic simulation make use of different routes of calculation, and the number of molecules is not a limitation. Enzyme kinetics and very complex oscillatory reactions have been modeled. These simulations are valuable for establishing whether a postulated kinetic scheme is reasonable, for examining the appearance of extrema or induction periods, applicability of the steady-state approximation, and so on. Even the manual method is useful for such purposes. 

An overreaching theme of the present chapter, besides broken ergodicity, has to do with the fact that most of the enhanced sampling methods that we shall discuss address situations in which one cannot clearly identify a reaction coordinate that can be conveniently used to describe the kinetic evolution of the system of interest. While methods for enhanced sampling are designed to yield accurate results faster than regular molecular dynamics or Monte Carlo (MC) methods, it is our belief that there is no perfect method, but that, rather, there are methods that perform better for particular applications. Moreover, it should be noted that, while in instances when a proper reaction coordinate can be identified methods described in other chapters are probably more efficient, they could still benefit by sampling in conformational directions perpendicular to the reaction coordinate. 

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