A one-phase Stefan problem with size-dependent thermal conductivity

TL;DR

This paper investigates a one-phase Stefan problem incorporating size-dependent thermal conductivity, revealing that nanoscale effects slow solidification and lead to a finite front speed, contrasting with classical models.

Contribution

It introduces a novel analysis of the Stefan problem with size-dependent thermal conductivity, providing approximate solutions and highlighting physical implications at the nanoscale.

Findings

Size-dependent conductivity slows solidification.

Solidification front moves linearly with time at small scales.

Finite front speed contrasts with classical infinite speed prediction.

Abstract

In this paper a one-phase Stefan problem with size-dependent thermal conductivity is analysed. Approximate solutions to the problem are found via perturbation and numerical methods, and compared to the Neumann solution for the equivalent Stefan problem with constant conductivity. We find that the size-dependant thermal conductivity, relevant in the context of solidification at the nanoscale, slows down the solidification process. A small time asymptotic analysis reveals that the position of the solidification front in this regime behaves linearly with time, in contrast to the Neumann solution characterized by a square root of time proportionality. This has an important physical consequence, namely the speed of the front predicted by size-dependant conductivity model is finite while the Neumann solution predicts an infinite and, thus, unrealistic speed as $t\rightarrow0$.