Kinetic Monte Carlo simulation model

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. 

M. Biehl, Lattice gas models and kinetic Monte Carlo simulations of expitaxial growth. http //arXiv.org/, paperno. cond-mat/0406707, 2004. 

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,...

Contents 1. Introduction 176 2. Static NMR Spectra and the Description of Dynamic Exchange Processes 178 2.1. Simulation of static NMR spectra 178 2.2. Simulation of DNMR spectra with average density matrix method 180 3. Calculation of DNMR Spectra with the Kinetic Monte Carlo Method 182 3.1. Kinetic description of the exchange processes 183 3.2. Kinetic Monte Carlo simulation of DNMR spectra for uncoupled spin systems 188 3.3. Kinetic Monte Carlo simulation of coupled spin systems 196 3.4. The individual density matrix 198 3.5. Calculating the FID of a coupled spin system 200 3.6. Vector model and density matrix in case of dynamic processes 205 4. Summary 211 Acknowledgements 212 References 212... 

The purpose of this chapter is to selectively summarize recent advances in the molecular modeling of anode and cathode electrocatalytic reactions employing different computational approaches, ranging from first-principles quantum-chemical calculations (based on density functional theory, DFT), ab initio and classical molecular dynamics simulations to kinetic Monte Carlo simulations. Each of these techniques is associated with a proper system size and timescale that can be adequately treated and will therefore focus on different aspects of the reactive system under consideration. 

Figure 16.8. CO stripping peak potential Ep from kinetic Monte Carlo simulations as a function of the Ru fraction of the Pt Ru model surface, for three different surface diffusion rates D =0,1, and 1000 s . For details, see Ref. [78].

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. 

K. Reuter, First-principles kinetic Monte Carlo simulations for heterogeneous catalysis concepts, status and frontiers , in Modeling and Simulation of Heterogeneous Catalytic Reactions, ed. by Olaf Deutschmann, Wiley-VCH, Weinheim, 201L... 

The article has briefly considered the role of Monte Carlo and kinetic Monte Carlo simulations in understanding dissolution and selective dissolution processes that can occur spontaneously in the natural environment and under directed control in laboratories. Algorithms for both Metropolis Monte Carlo and KMC models were discussed, and some results from an implementation of the KMC algorithm were shown as examples. Last, the article surveyed several areas where KMC models have been used to study corrosion processes and where they can contribute in engineering applications. 

The resulting model was used internal in a kinetic Monte Carlo simulation in order to provide predictions of the lateral interactions along with site specificity as the reaction progressed. 

A simulation model was presented in order to describe the influence of dealumination process on the structural properties of mordenite [lOBl]. Using kinetic Monte-Carlo simulations, dealumination was described as a multistep process consisting of the removal of the framework A1 as well as the self-healing of silanol nests by Si atoms. 

Makeev, A.G., D. Maroudas, A.Z. Panagiotopoulos, and LG. Kevrekidis. 2002b. Coarse bifurcation analysis of kinetic Monte Carlo simulations a lattice gas model with lateral interactions. Journal of Chemical Physics 117(18) 8229-8240. 

The balance equations for 0, 9oh, and 6co were formulated and solved with two approaches a mean-field model with nucleation processes on active sites and kinetic Monte Carlo simulations, as illustrated in Figure 3.9. 

FIGURE 3.9 Approaches used to find the solution of the active site model of surface activity. The main distinction is made based on surface mobility. For the general case, the full interplay between on-site reactivity and extremely low COad surface diffusivity unfolds. All processes, including nucleation of active sites (rate constant kj ), forward and reverse rates of OH d formation (kf, kb), surface diffusion of COad (kdiff), and oxidative removal of COad (kox), are important for the overall kinetics. The solution for the general case, demands kinetic Monte Carlo simulations, where evolution of the system is described stochastically and positions of adsorbed COad and are relevant. Modeling is substantially simplified in the limit of... 

However, the MF approach is insufficient for particles with 3.3 nm size, if the potential exceeds 0.8 Nshe, that is, in the potential range with rapid kinetics of surface reactions. The MF approach fails for particles with sizes in the range of 1.8 nm. In these cases, it is necessary to account for the finite surface diffusivity of CO and, thus, solve the active site model with the kinetic Monte Carlo simulation approach. Figure 3.11a shows typical results of current transients for particles with mean size of 3.3 nm that are matched closely with the model. Analysis of the data with the kinetic model allows important structural and dynamic parameters of the catalytic system to be extracted and analyzed. 

Kinetic Monte Carlo simulations can be parameterized from DFT calculations and used to model surface processes. We shall demonstrate how they can be used to predict the rate at which NH3 molecules desorb from a surface during a temperature programmed desorption experiment, assuming there is no readsorption of molecules to the surface. 

Graham and Olmsted [166,167] used coarse-grained kinetic Monte Carlo simulations to simulate anisotropic nucleation based on the chain configurations obtained from a molecular flow model, the Graham-Likhtman and Milner-McLeish (GLaMM) model [168,169].These simulations confirm the power law with exponent 4 up to reasonably high shear rates (molecular stretch up to 3 to 4). They actually found an exponential dependence on the square of the molecular stretch. A practical problem of such an expression is that it contains an extra parameter besides the prefactor for the stretch, there is a prefactor for the exponential function as a whole, which gives the quiescent sporadic creation rate. Since quiescent nucleation is predominantly athermal, this parameter cannot be determined for common melts. 

By virtue of their simple stnicture, some properties of continuum models can be solved analytically in a mean field approxunation. The phase behaviour interfacial properties and the wetting properties have been explored. The effect of fluctuations is hrvestigated in Monte Carlo simulations as well as non-equilibrium phenomena (e.g., phase separation kinetics). Extensions of this one-order-parameter model are described in the review by Gompper and Schick [76]. A very interesting feature of tiiese models is that effective quantities of the interface—like the interfacial tension and the bending moduli—can be expressed as a fiinctional of the order parameter profiles across an interface [78]. These quantities can then be used as input for an even more coarse-grained description. 

Recently a cellular automata version of the DD model has been studied [87]. The reported results are in qualitative agreement with Monte Carlo simulations [83,84]. Also, mean-field results [87] are in agreement with those early obtained in [85]. Very recently, simulations of the kinetic behavior of the DD model have been reported [88]. 

In this review we put less emphasis on the physics and chemistry of surface processes, for which we refer the reader to recent reviews of adsorption-desorption kinetics which are contained in two books [2,3] with chapters by the present authors where further references to earher work can be found. These articles also discuss relevant experimental techniques employed in the study of surface kinetics and appropriate methods of data analysis. Here we give details of how to set up models under basically two different kinetic conditions, namely (/) when the adsorbate remains in quasi-equihbrium during the relevant processes, in which case nonequilibrium thermodynamics provides the needed framework, and (n) when surface nonequilibrium effects become important and nonequilibrium statistical mechanics becomes the appropriate vehicle. For both approaches we will restrict ourselves to systems for which appropriate lattice gas models can be set up. Further associated theoretical reviews are by Lombardo and Bell [4] with emphasis on Monte Carlo simulations, by Brivio and Grimley [5] on dynamics, and by Persson [6] on the lattice gas model. 

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. 

Monte Carlo simulation, an iterative technique which derives a range of risk estimates, was incorporated into a trichloroethylene risk assessment using the PBPK model developed by Fisher and Allen (1993). The results of this study (Cronin et al. 1995), which used the kinetics of TCA production and trichloroethylene elimination as the dose metrics relevant to carcinogenic risk, indicated that concentrations of 0.09-1.0 pg/L (men) and 0.29-5.3 pg/L (women) in drinking water correspond to a cancer risk in humans of 1 in 1 million. For inhalation exposure, a similar risk was obtained from intermittent exposure to 0.07-13.3 ppb (men) and 0.16-6.3 ppb (women), or continuous exposure to 0.01-2.6 ppb (men) and 0.03-6.3 ppb (women) (Cronin et al. 1995). 

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 ... 

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