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

Abstract

Nanoscale solidification is becoming increasingly relevant in applications involving ultrafast 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. IntroductionAdvances in the field of nanotechnology are improving the efficiency, functionality, and costeffectiveness of modern devices. Nanowire-based solar cells, for instance, offer several advantages over traditional wafer-based and thin-film technologies [1]. Furthermore, the unique physical properties of carbon nanotubes have enabled the fabrication of new electrochemical biosensors [2]. Nanotechnology is also playing an increasing role in biology and medicine [3], where it finds applications in drug and gene delivery [4], protein detection [5], and tissue engineering * Published in Applied Mathematical Modelling, https://doi.

The one-dimensional Stefan problem with non-Fourier heat conduction

“…For large times, the classical solidification kinetics, s ∼ t 1/2 , are recovered. Similar time regimes were observed in Hennessy et al [33], although in their case the magnitude of the flux for small times is not as large as here due to their use of a Newton condition at x = 0 rather than a fixed-temperature condition.…”

“…This has been used in the modelling of melting of nanoparticles [44] and nanowires [45], and leads to a significant increase of the melting times. The cooling condition has also been used in a recent study by Hennessy et al [33], where a thorough asymptotic analysis of the Guyer-Krumhansl-Stefan problem is performed. The same authors also considered a cooling condition accounting for memory effects when studying the melting behaviour of nanoparticles using the MCE [29].…”

“…Other authors have studied the Stefan problem with Maxwell-Cattaneo conduction from a point of view of mathematical analysis [30][31][32]. Recently, Hennessy et al [33] performed a detailed asymptotic analysis of the one-dimensional Stefan problem with Guyer-Krumhansl conduction, where it is shown that non-classical effects can lead to important differences in the solidification kinetics with respect to Fourier's law.…”

The Stefan problem with variable thermophysical properties and phase change temperature

“…Hence a form of the jump conditions given in [1, Ch. 2, eqs [25][26][27] is easily retrieved. The momentum condition (9b) highlights a problem with this derivation form.…”

“…When Fourier's law is replaced by the Maxwell-Cattaneo equation, this temperature jump is needed to ensure that the speed of the interface is less than the finite speed at which thermal energy can be delivered to it. For example in the one-dimensional experimental setup described in [27] the temperature jump is given by…”