Universal grain size distribution, most chaotic microstructures and tessellation condition

In this paper a statistical model of grain size distribution is proposed. It is su¢ ciently simple to be studied analytically. The resulting grain size distribution does not contain …tting parameters and describes reasonably well the experimental data. The statistical model is based on the suggested in the paper tessellation condition, the necessary and su¢ cient condition for a set of grains to …ll the space without gaps and overlapping. The tessellation condition is written in the form of an in…nite chain of constraints, and the stochastic model considered employs only a few of these constraints. Another issue considered in the paper is the notion of most chaotic microstructures as applied to polycrystal grains. Keywords: grain size distribution, microstructure entropy, grain growth, recrystallization, tessellation I. INTRODUCTION Three themes are interwoven in this paper. The one is a theory of universal grain size distribution. The idea on the existence of a universal grain size distribution was suggested …rst, perhaps, by Smith (1952) who mentioned that the grain size distribution in a normal grain growth should tend to a universal function of relative grain sizes (the ratio of the grain size and the average grain size). The average grain size evolves while the distribution of grain sizes remains the same. Experiments (Beck 1954, Feltham 1957, Krill et al. 1998, Zhang et al. 2004) and computer simulations (Anderson et al. 1984, Chen et al. 1994, Wakai et al. 2000, Wang et al. 2009, Pande et al. 2010) widely support this idea. The universal grain size distribution is well …t by a lognormal distribution for a certain choice of the two parameters of lognormal distributions, and in the literature the grain size distribution is often called the lognormal law. The basic model of the theory of grain growth is extremely complex. The grain boundary structure is envisioned as a “closed cell foam”with cell boundaries evolving to reduce the surface energy. This is an in…nite-dimensional dissipative dynamical system possessing an attractor. The attractor seems having a …nite number of isolated points of attraction. Each attraction point corresponds to one grain that “ate”all other grains. Depending on the initial conditions, the system approaches to one or another point of attraction. A universal grain