  Phase field models have been applied in recent years to grain boundaries in
single-component systems. The models are based on the minimization of a free
energy functional, which is constructed phenomenologically rather than being
derived from first principles. In single-component systems the free energy is a
functional of a ``phase field'', which is an order parameter often referred to
as the crystallinity in the context of grain boundaries, but with no precise
definition as to what that term means physically. We present a derivation of
the phase field model by Allen and Cahn from classical density functional
theory first for crystal-liquid interfaces and then for grain boundaries. The
derivation provides a clear physical interpretation of the phase field, and it
sheds light on the parameters and the underlying approximations and limitations
of the theory. We suggest how phase field models may be improved.
