  In this paper we formulate a Stefan problem appropriate when the
thermophysical properties are distinct in each phase and the phase-change
temperature is size or velocity dependent. Thermophysical properties invariably
take different values in different material phases but this is often ignored
for mathematical simplicity. Size and velocity dependent phase change
temperatures are often found at very short length scales, such as nanoparticle
melting or dendrite formation; velocity dependence occurs in the solidification
of supercooled melts. To illustrate the method we show how the governing
equations may be applied to a standard one-dimensional problem and also the
melting of a spherically symmetric nanoparticle. Errors which have propagated
through the literature are highlighted. By writing the system in
non-dimensional form we are able to study the large Stefan number formulation
and an energy-conserving one-phase reduction. The results from the various
simplifications and assumptions are compared with those from a finite
difference numerical scheme. Finally, we briefly discuss the failure of
Fourier's law at very small length and time-scales and provide an alternative
formulation which takes into account the finite time of travel of heat carriers
(phonons) and the mean free distance between collisions.
