Distribution of Topological Types in Grain-Growth Microstructures

An open question in studying normal grain growth concerns the asymptotic state to which micro-structures converge. In particular, the distribution of grain topologies is unknown. We introduce a thermodynamiclike theory to explain these distributions in two- and three-dimensional systems. In particular, a bendinglike energy E_i is associated to each grain topology t_i, and the probability of observing that particular topology is proportional to [1/s(t_i )]e⁻β^Ei, where s(t_i) is the order of an associated symmetry group and β is a thermodynamiclike constant. We explain the physical origins of this approach and provide numerical evidence in support.