A 2-dimensional frame-invariant phase-field model was developed for grain boundaries. One-dimensional analytical solutions for a stable grain boundary in a bicrystal were obtained, and equilibrium energies were calculated. After making a suitable choice of functional dependences, the grain-boundary energy took the same analytical form as that in the microscopic Read-Shockley (dislocation) model. Dynamic 1-dimensional solutions were also presented which revealed the rotation of a small grain between 2 pinned grains, plus shrinkage and rotation of a circular grain which was embedded in a larger crystal.

A Continuum Model of Grain Boundaries. R.Kobayashi, J.A.Warren, C.W.Carter: Physica D, 2000, 140[1-2], 141-50