Phase-field model of crystal grains

Abstract

We introduce a two-dimensional phase field model designed to describe the dynamics of crystalline grains. The phenomenological free energy is a function of two order parameters. They reflect the degree of orientational order as well as the predominant local orientation of the crystal. Solutions to the gradient flow of this free energy can be interpreted as ensembles of grains (in which the phase of the order parameter is approximately constant in space) separated by mobile grain boundaries. We study the dynamics of the boundaries as well as the rotation of the grains. In the limit of the infinitely sharp interface, the normal velocity of the boundary is proportional to its curvature and its energy. The mobility of a grain boundary has a strong divergence in the limit of the small orientation mismatch. We derive and solve approximate equations describing a circular grain embedded in a fixed matrix.