  We analyze a two-dimensional phase field model designed to describe the
dynamics of crystalline grains. The phenomenological free energy is a
functional of two order parameters. The first one reflects the orientational
order while the second reflects the predominant local orientation of the
crystal. We consider the gradient flow of this free energy. Solutions can be
interpreted as ensembles of grains (in which the phase of the order parameter
is approximately constant in space) separated by 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 both its curvature and its energy. We obtain explicit formulas
for the interfacial energy and mobility and study their behavior in the limit
of a small misorientation. We calculate the rate of rotation of a grain in the
sharp interface limit and find that it depends sensitively on the choice of the
model.
