Distribution of Topological Types in Grain-Growth Microstructures

An open question in studying normal grain growth concerns the asymptotic state to which microstructures 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^{-βE_{i}}, 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.