Monte Carlo simulation of the two-dimensional Potts model using nonextensive statistics

The standard Potts model is investigated in the framework of nonextensive statistical mechanics. We performed Monte Carlo simulations on two-dimensional lattices with linear sizes ranging from 16 to 64 using the Metropolis algorithm, where the classical Boltzmann–Gibbs transition probabilities were modified for the nonextensive case. We found that the Potts model undergoes a phase transition in the nonextensive scenario. We established the order of the phase transition and we computed the critical temperature for different values of the Tsallis entropic index.     

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.     

两相钛合金再结晶退火组织与织构演变的蒙特卡罗模拟

蒙特卡罗(MC)方法被广泛应用于模拟金属材料在退火过程中的静态再结晶行为. 在已有两相材料晶粒长大MC模型基础上, 引入形核阶段, 综合考虑再结晶晶粒吞并形变晶粒和再结晶晶粒竞争长大两种情况, 建立了退火时两相合金再结晶MC模型.结合电子背散射衍射所测 初始晶粒形貌、相成分、晶体学取向及应变储能相对值, 该模型被应用于TC11钛合金退火过程中的微观组织及织构演变模拟.结果表明, 所建模型能够较好体现退火过程中两相晶粒的形核及晶粒长大行为. 与β相相比较, α相具有较低的再结晶速率和较高的晶粒长大速率, 前者主要归结于α相较低的初始应变储能, 后者则体现了该条件下初始组织形貌、分布及两相比例对晶粒长大具有重要影响; 由于非均匀形核的影响, 模拟得到的再结晶速率变化与 假设均匀形核的Johnson-Mehl-Avrami-Kolmogorov 再结晶方程存在明显差异.同时, 两相的基本织构特征在退火过程中无明显变化, 但织构强度增加. 关键词： 两相钛合金 再结晶 蒙特卡罗方法 织构     

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.     

The kinetic process of non-smooth substrate thin film growth via parallel Monte Carlo method

A Monte Carlo simulation model of thin film growth based on parallel algorithm is presented. Non-smooth substrate with special defect mode is introduced in such a model. The method of regionalizing is used to divide the substrate into sub-regions. This method is supposed to be modulated according to the defect mode. The effects of surface defect mode and substrate temperature, such as the nucleation ratio and the average island size, are studied through parallel Monte Carlo method. The kinetic process of thin film growth in the defect mode is also discussed. Results show that surface defect mode contributes to crystal nucleation. Analyzing parallel simulation results we find that density defect points, substrate temperature and the number of processors contribute decisively to the parallel efficiency and speedup. According to defect mode we can obtain large grain size more feasibly and the parallel algorithm of this model can guide the non-smooth substrate simulation work.     

Parallel three-dimensional Monte Carlo simulations for effects of precipitates and sub-boundaries on abnormal grain growth of Goss grains in Fe–3%Si steel

Using parallel three-dimensional Monte Carlo simulations, we investigated the effects of precipitates and sub-boundaries on abnormal grain growth (AGG) of Goss grains based on real orientation data of primary recrystallized Fe–3%Si steel. The simulations showed that AGG occurred in the presence of precipitates which inhibited the grain growth of matrix grains, whereas it did not in the absence of precipitates. The role of precipitates in enhancing AGG is to maintain a relatively high fraction of high energy boundaries between matrix grains, which increases the probability of sub-boundary-enhanced solid-state wetting of an abnormally growing grain. The microstructure evolved by the simulation could reproduce many realistic features of abnormally growing grains, such as the formation of island and peninsular grains and merging of abnormally growing grains which appeared to be separated initially on the cross-section.     

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.     

A phase-field/Monte-Carlo model describing organic crystal growth from solution

In this paper work we present a phase-field/Monte-Carlo hybrid algorithm for the simulation of solutal growth of organic crystals. The algorithm is subsequently used for an investigation of diffusion effects on the growth mechanisms. This method combines a two-scale phase-field model of the liquid phase epitaxial growth and a Monte-Carlo algorithm of the 2D nucleation and thus is faster than previous purely Monte Carlo simulations of crystal growth. The inclusion of supersaturation and diffusion in the method allows the study of crystal growth under various growth conditions. Parameters used in the hybrid algorithm are bound to the energetic parameters of crystal faces, which can be estimated from a detailed study of the actual crystal structure based on a connected nets analysis, which allows the prediction of the shape and morphology of real crystals. The study of the diffusion effect is carried out based on an example of a hydroquinone crystal, which grows from the water solution at various supersaturations. The dependencies of the growth rate and the nucleation rate on the supersaturation indicate the change of the growth mechanism from spiral growth to 2D nucleation. The difference in the growth rate for various faces is in agreement with the crystal morphologies derived from the attachment energy method and observed experimentally. The main result of the simulation is the evaluation of engineering limits for choosing appropriate external process conditions.     

Phase diagram of a 2D Ising model within a nonextensive approach

In this work we report Monte Carlo simulations of a 2D Ising model, in which the statistics of the Metropolis algorithm is replaced by the nonextensive one. We compute the magnetization and show that phase transitions are present for q ≠ 1. A q - phase diagram (critical temperature vs. the entropic parameter q) is built and exhibits some interesting features, such as phases which are governed by the value of the entropic index q. It is shown that such phases favors some energy levels of magnetization states. It is also shown that the contribution of the Tsallis cutoff is capital to the existence of phase transitions.     

Scaling of submonolayer island growth with reversible adatom exchange in surfactant-mediated epitaxy

Surfactant-mediated epitaxial growth is studied with a realistic model, which includes three main kinetic processes: diffusion of adatoms on the surfactant terrace, exchange of adatoms with their underneath surfactant atoms, and reexchange in which an exchanged adatom resurfaces to the top of the surfactant layer. The scaling behavior of nucleus density and island size distributions in the initial stage of growth is investigated by using kinetic Monte Carlo simulations. The results show that the temperature dependence of nucleus density and island size distributions governed by the reexchanging-controlled nucleation at high temperatures exhibits similar scaling behavior to that obtained by the standard diffusion-mediated nucleation at low temperatures. However, at intermediate temperatures, the exchanging-controlled nucleation leads to an increase of nucleus density with temperature, while the island size distribution scales to a monotonically decreasing function, showing nonstandard scaling behavior.     

Monte Carlo study of the step flow to island nucleation transition for close packed structures

Using joined super-lattice Kinetic Monte Carlo and continuous simulations we study the transition between step flow and two-dimensional island nucleation growth on stepped surfaces for close packed crystalline structures. The numerical analysis is performed in terms of misorientation cut, deposition rate and temperature. We compare the results of the atomistic approach with the predictions of the standard and generalized Burton-Cabrera-Frank (BCF) continuous model. The generalization consists in the explicit inclusion in the theory of the formation and dissolution of mobile dimers on the terraces. We show that the BCF-like continuous theories break down for low temperatures, large off-angle cuts and high deposition rates. In view of these results we critically discuss the basic assumptions of the continuous models.     

Monte Carlo simulations of luminescence processes under quasi-equilibrium (QE) conditions

Previous researchers have carried out Monte Carlo simulations of thermoluminescence (TL) phenomena by considering the allowed transitions of charge carriers between the conduction band, electron traps and recombination centers. Such simulations have demonstrated successfully the effect of trap clustering on the kinetics of charge carriers in a solid, and showed that trap clustering can significantly change the observed luminescence properties. While such Monte Carlo simulations have been carried out for TL, there has been no such trap clustering studies for optically stimulated luminescence phenomena (OSL). This paper presents a simplified method of carrying out Monte Carlo simulations for TL and linearly modulated optically stimulated luminescence (LM-OSL) phenomena, based on the General One Trap (GOT) model, which is a special case of the one trap one recombination center model (OTOR) when quasi-equilibrium conditions (QE) hold. The simulated results show that the presence of small clusters consisting of a few traps in a solid can lead to multiple peaks in both the TL and LM-OSL signals. The effects of retrapping and degree of trap filling are simulated for such multi-peak luminescence signals, and insight is obtained into the mechanism producing these peaks. The method presented in this paper can be easily generalized for other types of luminescence solids in which the recombination probability varies with time.     

多晶材料晶粒生长的计算机模拟研究

多晶固体材料显微结构的演化与诸多因素密切相关 ,是一复杂的过程。运用适当的计算机模拟方法进行模拟 ,实现对晶粒生长的预测 ,可为材料研究提供新的方法和重要依据。此文以晶粒生长动力学为基础 ,建立晶粒生长模型 ;分析了晶粒生长过程的计算机模拟理论和方法 ;以MonteCarlo方法为基础 ,提出快速Q statepotts算法 ,简化计算量 ,在PC机上编程实现了正常晶粒生长过程的计算机模拟。得到的显微结构图逼真度较好。通过对结果进行分析处理 ,得到的生长因子为 0 .4 7。     

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.     

Non-arrhenius behavior of the island density in metal heteroepitaxy: Co on Cu(001)

We present a combined theoretical and experimental study of island nucleation and growth in the deposition of Co on Cu(001)-a prototype for understanding heteroepitaxial growth involving intermixing. Experimentally, ion scattering is employed. Using density-functional theory, we obtain energy barriers for the various elementary processes and incorporate these into a kinetic Monte Carlo program to simulate the heteroepitaxial growth. Both the simulations and the experiments show a unique N-shape dependence of the island density on temperature that stems from the interplay and competition of the different processes involved.     

Nucleation barriers for the liquid-to-crystal transition in simple metals: Experiment vs. simulation

Crystal nucleation in the one-component metals Ni and Au is investigated using a combination of differential thermal analysis (DTA) experiments and Monte Carlo (MC) simulations. A novel experimental methodology allows to measure nucleation rates J over a range of 8 orders of magnitude. Evidence is given that these rates correspond to homogeneous nucleation. From the nucleation rates, free energy nucleation barriers ΔG ^? are extracted using an ansatz obtained in the framework of classical nucleation theory (CNT). The latter ansatz is rationalized by MC simulations that directly yield estimates for the temperature dependence of ΔG ^?. The values of ΔG ^?, as determined from the simulation, are in very good agreement with those extracted from the experiments. The simulations indicate that in the range where experiments are available the corrections to CNT are relatively small, thus justifying the application of CNT. We also discuss how the conditions for heterogeneous nucleation on a flat or structured wall can be obtained from computer simulations.     

Nonequilibrium wetting transition in a nonthermal 2D Ising model

Nonequilibrium wetting transitions are observed in Monte Carlo simulations of a kinetic spin system in the absence of a detailed balance condition with respect to an energy functional. A nonthermal model is proposed starting from a two-dimensional Ising spin lattice at zero temperature with two boundaries subject to opposing surface fields. Local spin excitations are only allowed by absorbing an energy quantum (photon) below a cutoff energy E _c . Local spin relaxation takes place by emitting a photon which leaves the lattice. Using Monte Carlo simulation nonequilibrium critical wetting transitions are observed as well as nonequilibrium first-order wetting phenomena, respectively in the absence or presence of absorbing states of the spin system. The transitions are identified from the behavior of the probability distribution of a suitably chosen order parameter that was proven useful for studying wetting in the (thermal) Ising model.