Asymptotic analysis of the Guyer-Krumhansl-Stefan model for nanoscale solidification

TL;DR

This paper analyzes nanoscale solidification by coupling the Guyer-Krumhansl equation with the Stefan condition, revealing multiple thermal regimes and recovering Fourier's law at large times, with implications for controlling nanotech processes.

Contribution

It introduces an asymptotic analysis of the GK model in nanoscale solidification, highlighting non-classical thermal effects and multiple time regimes.

Findings

Multiple thermal regimes identified during solidification.

Recovery of Fourier's law at large times.

Model captures change in effective thermal conductivity.

Abstract

Nanoscale solidification is becoming increasingly relevant in applications involving ultra-fast freezing processes and nanotechnology. However, thermal transport on the nanoscale is driven by infrequent collisions between thermal energy carriers known as phonons and is not well described by Fourier's law. In this paper, the role of non-Fourier heat conduction in nanoscale solidification is studied by coupling the Stefan condition to the Guyer--Krumhansl (GK) equation, which is an extension of Fourier's law, valid on the nanoscale, that includes memory and non-local effects. A systematic asymptotic analysis reveals that the solidification process can be decomposed into multiple time regimes, each characterised by a non-classical mode of thermal transport and unique solidification kinetics. For sufficiently large times, Fourier's law is recovered. The model is able to capture the change in the effective thermal conductivity of the solid during its growth, consistent with experimental observations. The results from this study provide key quantitative insights that can be used to control nanoscale solidification processes.