KMC simulation

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 ... 

The results shown in this study are limited to the KMC algorithm. In principle, to model a realistic system the set k, can be found using molecular dynamics simulations, or other similar techniques these can then be used as input into KMC. Here, our purposes are more general. A square lattice is examined, and there are two simplified rate laws of interest. The first is often used in KMC simulations of deposition, and is termed i-kinetics. A second set represents a kinetics that is analogous to the hopping probabilities used in the Metropolis simulations. We term the latter A/-kinetics. 

The main progress in the theoretical study of the effect of lateral interactions on the kinetics of surface reactions in recent years has been in kMC simulations and in density-functional theory (DFT) calculations of lateral interactions. We deal with the latter in Section 3.4, and with the former in Section 3.1. Section... 

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. 

The third way derives the kMC simulations from the master equation ... 

Analytical expressions for interactions can be regarded either as mathematical in origin or with a physical origin.The former have often a simple form and they hold for many if not all systems, but they may have many parameters. The latter are based on the mechanism that leads to the interactions, they have seldom a simple mathematical form, they generally hold only for a limited set of systems, but they may have only few parameters.An expansion in terms two-, three-, four-, and more-particle interactions (see Sections 3.2.1 and 3.2.2) is an example of a mathematical model for lateral interactions. As an example of a physical model we only present the bond-order-conservation model in Section 3.2.3 as this is the model that has been used most extensively in kMC simulations. 

The explanation for the reduced energy of additional bonds is very similar to the through-the-surface mechanism of lateral interactions (see Section 2.1). The first bond changes the electronic structure of the atom. As a consequence the electronic structure is less favorable for forming another bond, so that such a bond yields a smaller energy gain. Because this mechanism is so similar to the one for lateral interactions, it is natural to use BOC or UBI EP to describe such interactions. This has been done for kMC simulations first by Lombardo and Bell and more recently by Baranov et al. and by Hansen and Neurock. 2a, 26... 

If we allow the lateral interactions to vary over a too large range during the optimization, then we occasionally get adlayer structures in the kMC simulations that differ from those found experimentally. This does not mean necessarily that very different temperature-programmed desorption spectra are... 

Another problem with kMC simulations is that they use a lattice-gas model. Even if we have a single-crystal surface, this may not be appropriate when the... 

In these cases, the result of the KMC simulation is the system s time-dependent concentration profile. In DNMR, the system simulated with the KMC method is in macroscopic equilibrium, and the microscopic changes of a single species are simulated. 

In this work, we have shown that the effects of the exchange processes and the scalar couplings can be separated in the simulation. The spin interactions are described by quantum mechanics while the dynamic effects are characterised by statistical methods. The easiest way to handle the latter one is by means of the KMC simulation. 

The microscopic processes occurring in a system, along with their corresponding transition probabilities per unit time, are an input to a KMC simulation. This information can be obtained via the multiscale ladder using DFT,... 

TST, and/or MD simulations (the choice depends mainly on whether the process is activated or not). The creation of a database, a lookup table, or a map of transition probabilities for use in KMC simulation emerges as a powerful modeling approach in computational materials science and reaction arenas (Maroudas, 2001 Raimondeau et al., 2001). This idea parallels tabulation efforts in computationally intensive chemical kinetics simulations (Pope, 1997). In turn, the KMC technique computes system averages, which are usually of interest, as well as the probability density function (pdf) or higher moments, and spatiotemporal information in a spatially distributed simulation. 

Below, the various types of multiscale simulation are elaborated and various examples are provided. The presentation on coarse graining is mainly focused on stochastic (KMC) simulations to provide the underlying foundations and ideas in some depth. Coarse graining of other atomistic, e.g., MD, and mesoscopic tools will be covered in a forthcoming communication. Some excellent reviews on coarse graining in soft-matter physics problems are available (e.g., Kremer and Muller-Plathe, 2001 Muller-Plathe, 2002, 2003 Nielsen et al., 2004). 

Fig. 7. Schematic illustrating the coupling of a fluid-phase mass transfer model with a discrete, particle model, such as KMC, through the boundary condition. The continuum model passes the external field and the KMC simulation computes spatial and temporal rates that are needed in the boundary condition of the continuum model.

In a similar spirit, Alkire, Braatz and co-workers developed coupled hybrid continuum-KMC simulations to study the electrodeposition of Cu on flat surfaces and in trenches (Drews et al., 2003b, 2004 Pricer et al., 2002a, b). A 3D KMC simulation accounted for the surface processes as well as diffusion in the boundary layer next to the surface, whereas a ID or 2D continuum model (with adaptive mesh) was applied to simulate the boundary layer farther away. In... 

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