  In this paper a one-phase Stefan problem with size-dependent thermal
conductivity is analysed. Approximate solutions to the problem are found via
perturbation and numerical methods, and compared to the Neumann solution for
the equivalent Stefan problem with constant conductivity. We find that the
size-dependant thermal conductivity, relevant in the context of solidification
at the nanoscale, slows down the solidification process. A small time
asymptotic analysis reveals that the position of the solidification front in
this regime behaves linearly with time, in contrast to the Neumann solution
characterized by a square root of time proportionality. This has an important
physical consequence, namely the speed of the front predicted by size-dependant
conductivity model is finite while the Neumann solution predicts an infinite
and, thus, unrealistic speed as $t\rightarrow0$.
