Statistical crystallography Structure of random cellular networks

Abstract

The methods of statistical mechanics are applied to the structure of random, space-filling, cellular structures (foams, metallurgical grain aggregates, biological tissues). Microreversibility of the structural properties under elementary transformations is demonstrated. Detailed balance correlates the shapes of neighbouring cells, in the form of Aboav's law, derived here in two or three dimensions. Maximum entropy inference under a few constraints yields structural equations of state, relating the size of cells to their topological shape. The ‘ideal’ structure corresponds to the minimal number of constraints, and has Lewis's law as its equation of state. Metallurgical aggregates are influenced by an additional constraint, and have a different equation of state. The relevant most probable distributions of cell sizes and shapes are given.