Non-isothermal diffuse interface model for phase transition and interface evolution

TL;DR

This paper develops a thermodynamically consistent non-isothermal diffuse interface model for phase transitions, linking it to classical Stefan problems through asymptotic analysis and simulations.

Contribution

It introduces a novel non-isothermal diffuse interface model integrating thermodynamics and energetic variational principles, and connects it to classical Stefan problems.

Findings

The non-isothermal Allen-Cahn equation converges to a Stefan type problem in the sharp interface limit.

The interface motion is governed by mean curvature in the short term.

The model provides a physical justification for classical Stefan problems.

Abstract

In this paper, we derive a thermodynamically consistent non-isothermal diffuse interface model for phase transition and interface evolution involving heat transfer. This model is constructed by integrating concepts from classical irreversible thermodynamics with the energetic variational approach. By making specific assumptions about the kinematics of the temperature, we derive a non-isothermal Allen-Cahn equation. Through both asymptotic analysis and numerical simulations, we demonstrate that in the sharp interface limit, the non-isothermal Allen-Cahn equation converges to a two-phase nonlinear Stefan type problem, under a certain scale of the melting/freezing energy. In this regime, the motion of the liquid-solid interface and the temperature interface coincide and are governed by the mean curvature, at least for a short time. The result provides a justification for the classical Stefan problem within a certain physical regime.