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

TL;DR

This paper investigates how thermal relaxation affects nanoparticle melting using a Maxwell-Cattaneo heat conduction model, revealing that incorporating a jump condition at the interface prevents unphysical results and significantly impacts melting times.

Contribution

It introduces a two-phase Stefan problem with jump conditions derived from a phase-field model for Maxwell-Cattaneo heat conduction, demonstrating the importance of the jump condition over the continuity condition.

Findings

Thermal relaxation can increase melting time by over ten times.

The jump condition avoids unphysical superluminal melting speeds.

The classical Fourier solution is recovered in the zero relaxation limit.

Abstract

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.