The Monte Carlo Method

This chapter is aimed at describing the Monte Carlo method for the simulation of grain growth and recrystallization. It has also been extended to phase transformations and hybrid versions (Monte Carlo coupled with Cellular Automaton) of the model can also accommodate diffusion. If reading this chapter inspires you to program your own version of the algorithm and try to solve some problems, then we will have succeeded! The method is simple to implement and it is fairly straightforward to apply variable material properties such as anisotropic grain boundary energy and mobility. There are, however, some important limitations of the method that must be kept in mind. These limitations include an inherent lattice anisotropy that manifests itself in various ways. For many purposes, however, if you pay attention to what has been found to previous work, the model is robust and highly efficient from a computational perspective. In many circumstances, it is best to use the model to gain insight into a physical system and then obtain a new theoretical understanding, in preference to interpreting the results as being directly representative of a particular material. Please also keep in mind that the “Monte Carlo Method” described herein is a small subset of the broader use of Monte Carlo methods for which an excellent overview can be found in the book by Landau and Binder (2000).

[1]  E. M.,et al.  Statistical Mechanics , 2021, Manual for Theoretical Chemistry.

[2]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[3]  W. Mullins.  Two‐Dimensional Motion of Idealized Grain Boundaries , 1956 .

[4]  鈴木 増雄.  Time-Dependent Statistics of the Ising Model , 1965 .

[5]  D. Aboav,et al.  The shape of grains in a polycrystal , 1969 .

[6]  A. B. Bortz,et al.  A new algorithm for Monte Carlo simulation of Ising spin systems , 1975 .

[7]  J. W. Martin,et al.  Stability of microstructure in metallic systems , 1976 .

[8]  O. Hunderi,et al.  Computer simulation of grain growth , 1979 .

[9]  H. Bunge.  Texture Analysis in Materials Science , 1982 .

[10]  Samuel A. Safran,et al.  Kinetics of ordering in two dimensions. II. Quenched systems , 1983 .

[11]  P. S. Sahni,et al.  Computer simulation of grain growth-III. Influence of a particle dispersion , 1984 .

[12]  P. S. Sahni,et al.  Computer simulation of grain growth—II. Grain size distribution, topology, and local dynamics , 1984 .

[13]  David J. Srolovitz,et al.  Computer simulation of recrystallization-I. Homogeneous nucleation and growth , 1986 .

[14]  Srolovitz,et al.  Effects of diffusing impurities on domain growth in the Ising model. , 1987, Physical review. B, Condensed matter.

[15]  Boris D. Lubachevsky,et al.  Efficient parallel simulations of dynamic Ising spin systems , 1988 .

[16]  Anthony D. Rollett,et al.  Computer simulation of recrystallization—II. Heterogeneous nucleation and growth , 1988 .

[17]  M. Jessell.  Simulation of fabric development in recrystallizing aggregates—I. Description of the model , 1988 .

[18]  Michael P. Anderson,et al.  Computer simulation of recrystallization in non-uniformly deformed metals , 1989 .

[19]  David J. Srolovitz,et al.  Computer simulation of normal grain growth in three dimensions , 1989 .

[20]  A. Rollett,et al.  Abnormal grain growth in three dimensions , 1990 .

[21]  E. Holm,et al.  Effects of particle size on inhibited grain growth , 1990 .

[22]  Brian Derby,et al.  The dependence of grain size on stress during dynamic recrystallisation , 1991 .

[23]  Anthony D. Rollett,et al.  Computer simulation of recrystallization—III. Influence of a dispersion of fine particles , 1992 .

[24]  Elizabeth Ann Holm.  Modeling microstructural evolution in single-phase, composite and two-phase polycrystals. , 1992 .

[25]  Elizabeth A. Holm,et al.  A fast serial algorithm for the finite temperature quenched Potts model , 1993 .

[26]  P. Peczak,et al.  A Monte Carlo study of influence of deformation temperature on dynamic recrystallization , 1995 .

[27]  M. Novotny,et al.  A new approach to an old algorithm for the simulation of Ising-like systems , 1995 .

[28]  J. Szpunar,et al.  Modeling the effect of coincidence site lattice boundaries on grain growth textures. , 1995, Physical review. B, Condensed matter.

[29]  Thomas Zacharia,et al.  Simulation of curvature-driven grain growth by using a modified monte carlo algorithm , 1995 .

[30]  A. Rollett,et al.  On the growth of abnormal grains , 1997 .

[31]  M. A. Novotny,et al.  Parallelization of a Dynamic Monte Carlo Algorithm: a Partially Rejection-Free Conservative Approach , 1998, ArXiv.

[32]  David J. Srolovitz,et al.  Atomistic Simulation of Curvature Driven Grain Boundary Migration , 1998 .

[33]  T. Zacharia,et al.  Modeling the kinetics and microstructural evolution during static recrystallization—Monte Carlo simulation of recrystallization , 1998 .

[34]  Mark Miodownik,et al.  On boundary misorientation distribution functions and how to incorporate them into three-dimensional models of microstructural evolution , 1999 .

[35]  David P. Landau,et al.  Phase transitions and critical phenomena , 1989, Computing in Science & Engineering.

[36]  T. Baudin,et al.  Simulation of the anisotropic growth of Goss grains in Fe3%Si sheets (grade HiB) , 1999 .

[37]  Mark Miodownik,et al.  Highly parallel computer simulations of particle pinning: zener vindicated , 2000 .

[38]  Dierk Raabe,et al.  A hybrid model for mesoscopic simulation of recrystallization , 2001 .

[39]  J. Humphreys.  Texture Analysis: Macrotexture, Microtexture and Orientation Mapping. , 2001 .

[40]  A. Rollett,et al.  Measuring relative grain boundary energies and mobilities in an aluminum foil from triple junction geometry , 2001 .

[41]  M. Miodownik,et al.  On misorientation distribution evolution during anisotropic grain growth , 2001 .

[42]  H. Wenk,et al.  Texture and Anisotropy , 2004 .

[43]  P. Dawson,et al.  Mesoscale Modeling of Microstructure and Texture Evolution During Deformation Processing of Metals , 2002 .

[44]  Mark Miodownik,et al.  On abnormal subgrain growth and the origin of recrystallization nuclei , 2002 .

[45]  A. Rollett,et al.  Grain growth in a low-loss cold-rolled motor-lamination steel , 2003 .

[46]  A. Kuprat,et al.  Bridging simulations and experiments in microstructure evolution. , 2003, Physical review letters.

[47]  A. Rollett.  Crystallographic texture change during grain growth , 2004 .

[48]  A. Rollett,et al.  Statistically representative three-dimensional microstructures based on orthogonal observation sections , 2004 .

[49]  K. Binder,et al.  A Guide to Monte Carlo Simulations in Statistical Physics: Preface , 2005 .