Phase-field approximation of sharp-interface energies accounting for lattice symmetry

We present a phase-field approximation of sharp-interface energies defined on partitions, designed for modeling grain boundaries in polycrystals. The independent variable takes values in the orthogonal group $\mathrm{O}(d)$ modulo a lattice point group $\mathcal{G}$, reflecting the crystallographic symmetries of the underlying lattice. In the sharp-interface limit, the surface energy exhibits a Read-Shockley-type behavior for small misorientation angles, scaling as $\theta \mathrm{\mid }\log \theta \mathrm{\mid }$. The regularized functionals are applicable to grain growth simulation and the reconstruction of grain boundaries from imaging data.