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

• Published in: Applied Mathematical Modelling Vol. 61; pp. 1 - 17
• Main Authors: Hennessy, Matthew G., Calvo Schwarzwälder, Marc, Myers, Timothy G.
• Format: Journal Article
• Language: English
• Published: New York Elsevier Inc 01.09.2018; Elsevier BV
• Subjects: Asymptotic analysis; Asymptotic methods; Conduction heating; Conductive heat transfer; Fourier analysis; Fourier law; Freezing; Guyer-Krumhansl equation; Nanoscale phase change; Nanotechnology; Phonons; Solidification; Stefan problem; Thermal conductivity; Thermal energy; Transport; Guyer-Krumhansl equation; Nanoscale phase change; Stefan problem; Asymptotic analysis
• ISSN: 0307-904X; 1088-8691; 0307-904X
• DOI: 10.1016/j.apm.2018.03.026

•Model coupling Guyer–Krumhansl equation to the Stefan problem.•Asymptotic analysis elucidates the relationship between non-classical transport mechanisms and solidification.•Demonstrate that the initial transport of thermal energy is governed by a form of Fourier’s law with an effective thermal conductivity.•Demonstrate that non-Fourier heat conduction can alter the dynamics of solidification on the nanoscale. 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.