  While in principle, finite temperature density functional theory (ftDFT)
should be a powerful tool for the study of crystallization, in practice this
has not so far been the case. Progress has been hampered by technical problems
which have plagued the study of the crystalline systems using the most
sophisticated Fundamental Measure Theory models. In this paper, the reasons for
the difficulties are examined and it is proposed that the tensor functionals
currently favored are in fact numerically unstable. By reverting to an older,
more heuristic model it is shown that all of the technical difficulties are
eliminated. Application to a Lennard-Jones fluid results in a demonstration of
power of ftDFT to describe crystallization in a highly inhomogeneous system.
First, we show that droplets attached to a slightly hydrophobic wall
crystallize spontaneously upon being quenched. The resulting crystallites are
clearly faceted structures and are predominantly HCP structures. In contrast,
droplets in a fully periodic calculational cell remain stable to lower
temperatures and eventually show the same spontaneous localization of the
density into 'atoms' but in an amorphous structure having many of the
structural charactersitics of a glass. A small change of the protocol leads, at
the same temperature, to the formation of crystals, this time with the FCC
structure typical of bulk Lennard-Jones solids. The FCC crystals have lower
free energy than the amorphous structures which in turn are more stable than
the liquid droplets. It is demonstrated that as the temperature is raised, the
free energy differences between the structures decreases until the solid
clusters become less stable than the liquid droplets and spontaneously melt.
The presence of energy barriers separating the various structures is therefore
clearly demonstrated.
