2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/28567We propose a two dimensional frame-invariant phase field model of grain impingement and coarsening. One dimensional analytical solutions for a stable grain boundary in a bicrystal are obtained, and equilibrium energies are computed. We are able to calculate the rotation rate for a free grain between two grains of fixed orientation. For a particular choice of functional dependencies in the model the grain boundary energy takes the same analytic form as the microscopic (dislocation) model of Read and Shockley.4 pages, 2 figuresMaterials Sciencetext