  The role of thermal relaxation in nanoparticle melting is studied using a
mathematical model based on the Maxwell--Cattaneo equation for heat conduction.
The model is formulated in terms of a two-phase Stefan problem. We consider the
cases of the temperature profile being continuous or having a jump across the
solid-liquid interface. The jump conditions are derived from the
sharp-interface limit of a phase-field model that accounts for variations in
the thermal properties between the solid and liquid. The Stefan problem is
solved using asymptotic and numerical methods. The analysis reveals that the
Fourier-based solution can be recovered from the classical limit of zero
relaxation time when either boundary condition is used. However, only the jump
condition avoids the onset of unphysical `supersonic' melting, where the speed
of the melt front exceeds the finite speed of heat propagation. These results
conclusively demonstrate that the jump condition, not the continuity condition,
is the most suitable for use in models of phase change based on the
Maxwell--Cattaneo equation. Numerical investigations show that thermal
relaxation can increase the time required to melt a nanoparticle by more than a
factor of ten. Thus, thermal relaxation is an important process to include in
models of nanoparticle melting and is expected to be relevant in other rapid
phase-change processes.
