Effective Dynamics for a Kinetic Monte–Carlo Model with Slow and Fast Time Scales

We consider several multiscale-in-time kinetic Monte Carlo models, in which some variables evolve on a fast time scale, while the others evolve on a slow time scale. In the first two models we consider, a particle evolves in a one-dimensional potential energy landscape which has some small and some large barriers, the latter dividing the state space into metastable regions. In the limit of infinitely large barriers, we identify the effective dynamics between these macro-states, and prove the convergence of the process towards a kinetic Monte Carlo model. We next consider a third model, which consists of a system of two particles. The state of each particle evolves on a fast time-scale while conserving their respective energy. In addition, the particles can exchange energy on a slow time scale. Considering the energy of the first particle, we identify its effective dynamics in the limit of asymptotically small ratio between the characteristic times of the fast and the slow dynamics. For all models, our results are illustrated by representative numerical simulations.     

Ag/Si(111)-(7X7) Heteroepitaxy—STM Experiment and KMC Simulations

Heteroepitaxial growth of Ag on Si(111)-(7X7) surface at various conditions was experimentally studied by scanning tunneling microscopy. A growth model based on experiments was used for kinetic Monte Carlo (KMC) simulations. The simulations of nucleation and island growth at low coverage and fitting experimental data provided basic growth parameters. Further growth—formation of a discontinuous transition film (wetting layer)—was implemented into the basic model. The suggested growth mechanism was successfully tested using the KMC simulations. The choice of experiments, the role of minimizing processes and parameters in the model, and efficiency of the used approach is demonstrated and discussed.     

Simulation of proliferation and differentiation of cells in a stem-cell niche

Stem-cell niches represent microscopic compartments formed of environmental cells that nurture stem cells and enable them to maintain tissue homeostasis. The spatio-temporal kinetics of proliferation and differentiation of cells in such niches depend on the specifics of the niche structure and on adhesion and communication between cells and may also be influenced by spatial constraints on cell division. We propose a generic lattice model, taking all these factors into account, and systematically illustrate their role. The model is motivated by the experimental data available for the niches located in the subventricular zone of adult mammalian brain. The general conclusions drawn from our Monte Carlo simulations are applicable to other niches as well. One of our main findings is that the kinetics under consideration are highly stochastic due to a relatively small number of cells proliferating and differentiating in a niche and the autocatalytic character of the symmetric cell division. In particular, the kinetics exhibit huge stochastic bursts especially if the adhesion between cells is taken into account. In addition, the results obtained show that despite the small number of cells present in stem-cell niches, their arrangement can be predetermined to appreciable extent provided that the adhesion of different cells is different so that they tend to segregate.     

Mechanism of associative oxygen desorption from Pt(111) surface

Mechanism of the associative desorption of oxygen from the Pt(111) surface has been studied on atomic level by means of DFT/GGA calculations and kinetic Monte Carlo simulations. It has been found that two oxygen adatoms can occur, with sufficient probability, in neighboring on-top sites, which is essential for formation and subsequent evaporation of the oxygen molecule. Monte Carlo simulations have demonstrated effectiveness of this channel for O₂ formation on Pt(111) and strongly support the suggested model of associative desorption from transition metal surfaces.     

随机反应扩散格气模型上钙动力学的粗粒化模拟

发展了一种基于随机格气模型的粗粒化方法,该方法能有效模拟内质网表面钙动力学信息. 首先将相邻的微观节点合并成粗粒化节点,再根据局域平均场近似推导出粗粒化反应速率,然后执行粗粒化动力学蒙特卡洛模拟. 发现粗粒化动力学蒙特卡洛模拟结果和微观模拟结果非常吻合. 有趣的是,存在一个最佳的粗粒化比m,使得粗粒化模拟与微观模拟的相变点偏差最小. 固定m,发现临界点随体系尺度增加而单调增加,而且相变点的偏差与体系尺度存在一个标度关系.此外,该粗粒化方法大大地加快了蒙特卡洛模拟速率,并且与微观模拟直接相关. 该方法可以广泛用来研究体系尺度效应,而节省大量计算时间.     

Size distribution evolution equations in space-competing domain growth systems

A model of microstructure development for phase transformations driven by nucleation and growth is presented. The model is based on a set of Fokker-Planck-like equations which allows us to compute the particle grain density distribution at any time during the transformation, provided that the nucleation and growth dependences on time and/or grain radius are known. The model is applicable to any kind of nucleation and growth protocols fulfilling the Johnson–Mehl–Avrami–Kolmogorov conditions, namely spatially random nucleation and isotropic growth. Comparison with stochastic (Monte Carlo) simulations is presented, giving quantitative agreement in all cases. This work shows the relationship between kinetic parameters and microstructure evolution as well as the accuracy of the developed model.     

Fast calculation of the density of states of a fluid by Monte Carlo simulations

Two related methods are proposed to calculate the density of states of a fluid from Monte Carlo simulations. In contrast to previous approaches, which require that histograms be accumulated in a stochastic manner, the methods proposed here rely on evaluation of the instantaneous temperature. In the first method, the temperature is calculated from the gradient of the forces. In the second, it is estimated from the kinetic contribution to the total energy. The validity and usefulness of the new approaches are demonstrated by presenting results from simulations of a Lennard-Jones fluid. It is shown that the new methods are considerably faster than previously available techniques.     

Boundary Mobility and Energy Anisotropy Effects on Microstructural Evolution During Grain Growth

We have performed mesoscopic simulations of microstructural evolution during curvature driven grain growth in two-dimensions using anisotropic grain boundary properties obtained from atomistic simulations. Molecular dynamics simulations were employed to determine the energies and mobilities of grain boundaries as a function of boundary misorientation. The mesoscopic simulations were performed both with the Monte Carlo Potts model and the phase field model. The Monte Carlo Potts model and phase field model simulation predictions are in excellent agreement. While the atomistic simulations demonstrate strong anisotropies in both the boundary energy and mobility, both types of microstructural evolution simulations demonstrate that anisotropy in boundary mobility plays little role in the stochastic evolution of the microstructure (other than perhaps setting the overall rate of the evolution. On the other hand, anisotropy in the grain boundary energy strongly modifies both the topology of the polycrystalline microstructure the kinetic law that describes the temporal evolution of the mean grain size. The underlying reasons behind the strongly differing effects of the two types of anisotropy considered here can be understood based largely on geometric and topological arguments.     

A kinetic Monte Carlo method on super-lattices for the study of the defect formation in the growth of close packed structures

A modified Kinetic Lattice Monte Carlo model has been developed to predict growth rate regimes and defect formation in the case of the homo-epitaxial growth of close packed crystalline structures. The model is an improvement over standard Monte Carlo algorithms, which usually retain fixed atom positions and bond partners indicative of perfect crystal lattices. Indeed, we extend the concepts of Monte Carlo growth simulations on super-lattices containing additional sites (defect sites) with respect to those of the reference material. This extension implies a reconsideration of the energetic mapping, which is extensively presented, and allows to describe a complex phenomenology that is out of accessibility of standard stochastic approaches. Results obtained using the Kawasaki and the Bond-Counting rules for the transition probability of the Monte Carlo event are discussed in details. These results demonstrate how the defect types (local or extended), the formation mechanisms and the defect generation regimes can be characterized using our approach.     

A kinetic Monte Carlo method on super-lattices for the study of the defect formation in the growth of close packed structures

A modified Kinetic Lattice Monte Carlo model has been developed to predict growth rate regimes and defect formation in the case of the homo-epitaxial growth of close packed crystalline structures. The model is an improvement over standard Monte Carlo algorithms, which usually retain fixed atom positions and bond partners indicative of perfect crystal lattices. Indeed, we extend the concepts of Monte Carlo growth simulations on super-lattices containing additional sites (defect sites) with respect to those of the reference material. This extension implies a reconsideration of the energetic mapping, which is extensively presented, and allows to describe a complex phenomenology that is out of accessibility of standard stochastic approaches. Results obtained using the Kawasaki and the Bond-Counting rules for the transition probability of the Monte Carlo event are discussed in details. These results demonstrate how the defect types (local or extended), the formation mechanisms and the defect generation regimes can be characterized using our approach.     

Kinetic Monte Carlo simulations of Pd deposition and island growth on MgO(1 0 0)

The deposition and ripening of Pd atoms on the MgO(1 0 0) surface are modeled using kinetic Monte Carlo simulations. The density of Pd islands is obtained by simulating the deposition of 0.1 ML in 3 min. Two sets of kinetic parameters are tested and compared with experiment over a 200-800 K temperature range. One model is based upon parameters obtained by fitting rate equations to experimental data and assuming the Pd monomer is the only diffusing species. The other is based upon transition rates obtained from density functional theory calculations which show that small Pd clusters are also mobile. In both models, oxygen vacancy defects on the MgO surface provide strong traps for Pd monomers and serve as nucleation sites for islands. Kinetic Monte Carlo simulations show that both models reproduce the experimentally observed island density versus temperature, despite large differences in the energetics and different diffusion mechanisms. The low temperature Pd island formation at defects is attributed to fast monomer diffusion to defects in the rate-equation-based model, whereas in the DFT-based model, small clusters form already on terraces and diffuse to defects. In the DFT-based model, the strong dimer and trimer binding energies at charged oxygen vacancy defects prevent island ripening below the experimentally observed onset temperature of 600 K.     

Two-scale modeling of adsorption processes at structured surfaces

We present an algorithm for the simulation of vicinal surface growth. It combines a lattice gas anisotropic Ising model with a phase-field model. The molecular behavior of individual adatoms is described by the lattice gas model. The microstructure dynamics on the vicinal surface are calculated using the phase-field method. In this way, adsorption processes on two different length scales can be described: nucleation processes on the terraces (lattice gas model) and step-flow growth (phase field model). The hybrid algorithm that is proposed here, is therefore able to describe an epitaxial layer-by-layer growth controlled by temperature and by deposition rate. This method is faster than kinetic Monte Carlo simulations and can take into account the stochastic processes in a comparable way.