Universal grain size distribution, most chaotic microstructures and tessellation condition

In this paper the probability distribution of grain sizes in the self-similar regime of grain growth is derived from the assumption that the grain boundary structure is the most chaotic. The grain size distribution obtained in this way does not contain fitting parameters and describes reasonably well the experimental data. The derivation is based on the suggested in the paper tessellation condition, the necessary and sufficient condition for a set of grains to fill the space without gaps and overlapping. The tessellation condition yields an infinite chain of constraints on the geometrical parameters of the grains. The derivation of the most chaotic grain size distribution employs only a few of these constraints, and, thus, the resulting distribution is approximate. It is checked that the incorporation of a few more constraints does not change the resulting distribution. Another aspect of the derivation is the notion of most chaotic microstructures as applied to polycrystal grains. This notion is introduced in accord with the Laplace principle of insufficient reasoning. Satisfactory description of experimental data indicates that the grain boundary structures developed in the self-similar regime of grain growth can be viewed as the most chaotic.