A continuum model of grain boundaries

A two-dimensional frame-invariant phase field model of grain boundaries is developed. One-dimensional analytical solutions for a stable grain boundary in a bicrystal are obtained, and equilibrium energies are computed. With an appropriate choice of functional dependencies, the grain boundary energy takes the same analytic form as the microscopic (dislocation) model of Read and Shockley [W.T. Read, W. Shockley, Phys. Rev. 78 (1950) 275]. In addition, dynamic (one-dimensional) solutions are presented, showing rotation of a small grain between two pinned grains and the shrinkage and rotation of a circular grains embedded in a larger crystal.