Monte Carlo kinetic

The alternative to using the conventional Metropolis MC algorithm to simulate the evolution of a system toward equilibrium is the kinetic Monte Carlo (KMC) method [44-46] (also sometimes called dynamic Monte Carlo [47]). In this method, the correct dynamics are ensured by basing the transition probabilities on physical rate processes, usually through invocation of activated state theory to generate the appropriate kinetics  

This requires a priori knowledge of the number n of allowed transitions at any step of the simulation, typically prescribed by a set of rules for allowed transitions, in order to compute the probability of the next step in the simulation (and the waiting time associated with it). For this reason, early uses of KMC were called the n-Fold Way [44]. The time associated with a given transition out of configuration i at step a is 

In principle, an exhaustive enumeration of allowed transitions would generate not only the correct kinetics, but also the elapsed time associated with each step. This is often feasible for lattice models, but it presupposes some knowledge of the allowed transitions and a good kinetic model to describe the rates associated with each transition. On the other hand, full enumeration of allowed transitions in off-lattice models can be prohibitive, and the n-Fold Way KMC cannot strictly be applied. Importantly, the KMC method reduces to the conventional MC method for special cases where in Equation (6.7) and the denominator of Equation (6.8) are both constants, independent of configuration. Here again, appropriate selection of moves that satisfy these characteristics in an MC simulation is essential to obtain realistic dynamics. For two recent reviews of these methods, the reader is referred to References [48] and [49]. 

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C. C. Battaille, D. J. Srolvitz, J. E. Butler. A kinetic Monte Carlo method for the atomic-scale simulation of chemical vapor deposition application to diamond. J App Phys 52 6293, 1997. 

R. W. Smith. A kinetic Monte Carlo simulation of fiber texture formation during thin-film deposition. J Appl Physics 57 1196, 1997. 

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 ... 

FIGURE 35.7 Results of a kinetic Monte Carlo simulation. Number of islands as a function of the coverage degree for various deposition rates v of Ag on Au(lOO). (From Gimenez et al., 2002, with permission of The Electrochemical Society.)... 

Ab initio methods allow the nature of active sites to be elucidated and the influence of supports or solvents on the catalytic kinetics to be predicted. Neurock and coworkers have successfully coupled theory with atomic-scale simulations and have tracked the molecular transformations that occur over different surfaces to assess their catalytic activity and selectivity [95-98]. Relevant examples are the Pt-catalyzed NO decomposition and methanol oxidation. In case of NO decomposition, density functional theory calculations and kinetic Monte Carlo simulations substantially helped to optimize the composition of the nanocatalyst by alloying Pt with Au and creating a specific structure of the PtgAu7 particles. In catalytic methanol decomposition the elementary pathways were identified... 

Kieken LD, Neurock M, Mei DH. 2005. Screening by kinetic Monte Carlo simulation of Pt-Au(lOO) surfaces for the steady-state decomposition of nitric oxide in excess dioxygen. J Phys Chem B 109 2234-2244. 

Van Bavel, A. P., Hermse, C. G. M., Hopstaken, M. J. P. et al. (2004) Quantifying lateral adsorbate interactions by kinetic Monte-Carlo simulations and density-functional theory NO dissociation on Rh(100) , Phys. Chem. Chem. Phys., 6, 1830. 

Figure 2.3 Kinetic Monte-Carlo simulations of Au growth on N/Cu(l 0 0) for three different temperatures. From left to right, 7= 180 K, 7=240 K, and 7= 300 K. The coverage is 0.11 ML. (Reproduced with permission from Ref. [24].)...

Voter, A.F. Introduction to the kinetic Monte Carlo method, In Radiation Effects in Solids (eds K.E. Sickafus, E.A. Kotomin and B.P. Uberuaga), Springer, NATO Publishing Unit, Dordrecht, The Netherlands, 2006, pp. 1-24. 

Shim, Y., Amar, J.G. Semirigorous synchronous sublattice algorithm for parallel kinetic Monte Carlo simulations of thin film growth. Phys. Rev. B 2005, 71, 1254321-1-14. 

The dynamics of systems such as this one can be followed using a method called kinetic Monte Carlo (kMC) The idea behind this method is straightforward If we know the rates for all processes that can occur given the current configuration of our atoms, we can choose an event in a random way that is consistent with these rates. By repeating this process, the system s time evolution can be simulated. 

The kinetic Monte Carlo (kMC) algorithm we have described is straightforward to derive and implement but is not necessarily the most efficient of the various algorithms that exist. For a careful review of the properties and efficiency of different kMC algorithms, see J. S. Reese, S. Raimondeau, and D. G. Vlachos, J. Comput. Phys. 173 (2001), 302 321. 

A review of using DFT methods in combination with kinetic Monte Carlo for modeling active sites on metal catalysts, see M. Neurock, J. Catalysis 216 (2003), 73 88. 

A final caveat that must be applied to phase diagrams determined using DFT calculations (or any other method) is that not all physically interesting phenomena occur at equilibrium. In situations where chemical reactions occur in an open system, as is the case in practical applications of catalysis, it is possible to have systems that are at steady state but are not at thermodynamic equilibrium. To perform any detailed analysis of this kind of situation, information must be collected on the rates of the microscopic processes that control the system. The Further Reading section gives a recent example of combining DFT calculations and kinetic Monte Carlo calculations to tackle this issue. 

K. Reuter and M. Scheffler, First-Principles Kinetic Monte Carlo Simulations for Heterogeneous Catalysis Application to the CO Oxidation at RuO2(110), Phys. Rev. B 73 (2006), 045433. 

In this section, the mechanics ofthe kinetic Monte Carlo algorithm (KMC) are compared to the Metropolis algorithm. ... 

For each configuration of molecules at the surface, there are a number of possible events. The events occur randomly with a characteristic rate for each type of configuration. This model is the kinetic Monte Carlo simulation. 

Kinetic Monte Carlo Simulations. The approximations of the previous sections generally make it rather easy to interpret the results of a kinetic model. Their drawback is that it is very difficult to assess their accuracy. KMC simulations do not have this drawback. For a given reaction model the results of a kMC simulation are exact. 

CO/Rh(100). - This system forms an example where we have determined lateral interactions by fitting temperature-programmed desorption spectra that were simulated using kinetic Monte Carlo to experimental spectra. For coverages below 5ML CO adsorbs at top sites, which form a square grid. CO desorption has a rate constant... 

M. Biehl, F. Much and C. Vey. Off-lattice kinetic Monte Carlo simulations of strained heteroexpitaxial growth. http //arXiv.org/, paperno. cond-mat/0405641, 2004. 

Figure 2 illustrates major modeling methods, i.e., ab initio molecular dynamic (AIMD), molecular dynamic (MD), kinetic Monte Carlo (KMC), and continuum methods in terms of their spatial and temporal scales. Models for microscopic and macroscopic components of a PEFC are placed in the figure in terms of their characteristic dimensions for comparison. While continuum models are successful in rationalizing the macroscopic behaviors based on a few assumptions and the average properties of the materials, molecular or atomistic modeling can evaluate the nanostructures or molecular structures and microscopic properties. In computational... 

Figure 2. Multiscale modeling hierarchy. AIMD ab initio molecular dynamics, MD molecular dynamics, KMC kinetic Monte Carlo modeling, and FEA finite element analysis.

Fig. 7 The correlation function (tiT2)d as obtained from kinetic Monte Carlo simulations for the polypeptide model (green). The normalized correlations functions (ki 2fc "2) (red) and (R1R2) (black) are also shown for the sake of comparison. All correlation function are normalized so that their initial value is equal to 1. The following parameters were used under different conditions (1) Folded state ko = = 2000,...

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