A stochastic grain growth model based on a variational principle for dissipative systems

A stochastic model for the evolution of a cellular network driven by dissipative forces is presented. The model is based on a variational formulation for the dissipated power, from which we obtain an expression for the transition-rate generating function to be used in kinetic Monte Carlo simulations. The model canonical variables are the positions and velocities of the network vertices where cell walls meet. We apply such a model to the study of grain growth in two dimensions, in which the network represents a cross-section of a polycrystalline microstructure and the cell walls represent grain boundaries. The results of the stochastic grain-growth model for relevant statistical quantities are compared to deterministic model results and analytic theories.

[1]  H. Atkinson.  Overview no. 65 , 1988 .

[2]  Jing-Song Pan,et al.  Computer simulation of superplastic deformation , 1993 .

[3]  E. Dantsker,et al.  On a stochastic theory of grain growth—II , 1990 .

[4]  J. Cahn,et al.  A microscopic theory for antiphase boundary motion and its application to antiphase domain coasening , 1979 .

[5]  R. W. Balluffi,et al.  Interfaces in crystalline materials , 2009 .

[6]  A. Mocellin,et al.  A Topological model for plane sections of polycrystals , 1981 .

[7]  A. Cocks,et al.  Variational principles, numerical schemes and bounding theorems for deformation by Nabarro-Herring creep , 1996 .

[8]  D. A Aboav,et al.  The arrangement of grains in a polycrystal , 1970 .

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

[10]  Hassan Aref,et al.  Numerical experiments on two-dimensional foam , 1992, Journal of Fluid Mechanics.

[11]  Jing-Song Pan,et al.  Finite element formulation of coupled grain-boundary and surface diffusion with grain-boundary migration , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[12]  Denis Weaire,et al.  Some remarks on the arrangement of grains in a polycrystal , 1974 .

[13]  S.P.A. Gill,et al.  A variational approach to two dimensional grain growth-I. Theory , 1996 .

[14]  Katsuya Nakashima,et al.  Vertex models for two-dimensional grain growth , 1989 .

[15]  E. Dantsker,et al.  On a stochastic theory of grain growth—III , 1991 .

[16]  Denis Weaire,et al.  Computer simulation of a two-dimensional soap froth , 1983 .

[17]  Jing-Song Pan,et al.  A numerical technique for the analysis of coupled surface and grain-boundary diffusion , 1995 .

[18]  A. Thorvaldsen.  Grain growth as a stochastic process , 1993 .

[19]  P. S. Sahni,et al.  Computer simulation of grain growth—I. Kinetics , 1984 .

[20]  F. T. Lewis,et al.  The correlation between cell division and the shapes and sizes of prismatic cells in the epidermis of cucumis , 1928 .

[21]  L. Shvindlerman,et al.  Computer simulation of grain growth in two dimensions , 1985 .

[22]  John R. Rice,et al.  Plastic creep flow effects in the diffusive cavitation of grain boundaries , 1980 .

[23]  On a stochastic theory of grain growth , 1987 .

[24]  Mats Hillert,et al.  On the theory of normal and abnormal grain growth , 1965 .

[25]  C. Thompson,et al.  The effect of nucleation conditions on the topology and geometry of two-dimensional grain structures , 1987 .

[26]  J. Stavans,et al.  The evolution of cellular structures , 1993 .

[27]  F. J. Humphreys,et al.  Modelling mechanisms and microstructures of recrystallisation , 1992 .

[28]  J. Kermode,et al.  Computer simulation of a two-dimensional soap froth II. Analysis of results , 1984 .

[29]  N. P Louat.  On the theory of normal grain growth , 1974 .

[30]  W. Mullins.  Grain growth of uniform boundaries with scaling , 1998 .

[31]  O. Hunderi,et al.  On the analytic description of normal grain growth , 1989 .

[32]  S. Jones,et al.  Variational methods in nonconservative phenomena , 1989 .

[33]  C. V. Thompson,et al.  A two-dimensional computer simulation of capillarity-driven grain growth: Preliminary results , 1988 .

[34]  Glazier,et al.  Soap froth revisited: Dynamic scaling in the two-dimensional froth. , 1989, Physical review letters.

[35]  James A. Glazier,et al.  The kinetics of cellular patterns , 1992 .