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 thermodynamic-like theory to explain these distributions in two- and three-dimensional systems. In particular, a bending-like energy $E_i$ is associated to each grain topology $t_i$, and the probability of observing that particular topology is proportional to $\frac{1}{s(t_i)}e^{-βE_i}$, where $s(t_i)$ is the order of an associated symmetry group and $β$ is a thermodynamic-like constant. We explain the physical origins of this approach, and provide numerical evidence in support.

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