# The Stefan problem with variable thermophysical properties and phase change temperature

## TL;DR

This paper develops a generalized Stefan problem model incorporating variable thermophysical properties and size or velocity-dependent phase change temperatures, with applications to nanoparticle melting and dendrite formation, validated through numerical simulations.

## Contribution

It introduces a novel formulation of the Stefan problem that accounts for property variations and dynamic phase change temperatures, addressing limitations in existing models.

## Key findings

- The model accurately predicts nanoparticle melting behavior.
- Finite difference simulations validate the theoretical approach.
- Highlighting errors in previous literature on phase change modeling.

## Abstract

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.