Modelling ultra-fast nanoparticle melting with the Maxwell–Cattaneo equation

Abstract

• New models of nanoparticle melting based on the Maxwell–Cattaneo equation are proposed. • Consider temperature profiles that are continuous or have a jump across the solid–liquid interface. • The jump condition for the temperature is derived by taking a sharp-interface limit of a phase-field model. • The jump condition is the most suitable condition to use in phase-change models based on the Maxwell–Cattaneo equation. • Thermal relaxation is shown to substantially slow the melting process. 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.