Non-classical thermal transport and phase change at the nanoscale

Premi Extraordinari de Doctorat, promoció 2018-2019. Àmbit de CiènciesFor 200 years, Fourier’s law has been used to describe heat transfer with excellent results. However, as technology advances, more and more situations arise where heat conduction is not well described by the classical equations. Examples are applications with extremely short time scales such as ultra fast laser heating, or very small length scales such as the heat conduction through nanowires or nanostructures in general. In this thesis we investigate alternative models which aim to correctly describe the non-classical effects that appear in extreme situations and which Fourier’s law fails to describe. A popular approach is the Guyer-Krumhansl equation and the framework of phonon hydrodynamics. This formalism is particularly appealing from a mathematical point of view since it is analogous to the Navier-Stokes equations of fluid mechanics, and from a physical point of view, since it is able to describe the physics in a simple and elegant way. In the first part of the thesis we use phonon hydrodynamics to predict the size-dependent thermal conductivity observed experimentally in nanostructures such as nanowires or thin films. In particular, we show that the Guyer-Krumhansl equation is suitable to capture the dependence of the thermal conductivity on the size of the physical system under consider- ation. During the modelling process we use the analogy with fluids to incorporate a slip boundary condition with a slip coefficient that depends on the ratio of the phonon mean free path to the characteristic size of the system. With only one fitting parameter we are able to accurately reproduce experimental observations corresponding to nanowires and nanorods of different sizes. The second part of the thesis consists of studying the effect of the non-classical fea- tures on melting and solidification processes. We consider different extensions and in- corporate them into the mathematical description of a solidification process in a simple, one-dimensional geometry. In chapter 5 we employ an effective Fourier law which replaces the original thermal conductivity by a size-dependent expression that accounts for non-local effects. In chapter 6 we use the Maxwell-Cattaneo and the Guyer-Krumhansl equations to formulate the Maxwell-Cattaneo-Stefan and the Guyer-Krumhansl-Stefan problems respec- tively. After performing a detailed asymptotic analysis we are able to reduce both models to a system of two ordinary differential equations and obtain excellent agreement with the cor- responding numerical solutions. In situations near Fourier resonance, which is a particular case where non-classical effects in the Guyer-Krumhansl model cancel each other out, the solidification kinetics are very similar to those described by the classical model. However, in this case we see that non-classical effects are still observable in the evolution of the heat flux through the solid, which suggests that this is a quantity which is more convenient to determine the presence of these effects in phase change processes.La llei de Fourier ha estat una peça clau per a descriure la conducció de calor des de que fou proposada fa gairabé 200 anys. No obstant, a mesura que avança la tecnologia ens hi trobem més sovint amb situacions on les equacions clàssiques perden la seva validesa. En aquesta tesi investiguem alguns models alternatius que tenen com a objectiu descriure la conducció de calor en situacions on la llei de Fourier no és aplicable. Un model que s'ha aconseguit establir com un extensió vàlida de la llei de Fourier és l'equació de Guyer i Krumhansl i el marc de la hidrodinàmica de fonons derivat d'aquesta. Es tracta d'un model particularment interessant, ja que les equacions són anàlogues a les equacions per a fluids dins de la hidrodinàmica clàssica. A la primera part de la tesi considerem aquesta equació per a descriure la conducció de calor estàtica per nanofibres de seccions transversals circulars i rectangulars. En particular, calculem una conductivitat tèrmica efectiva i trobem que és possible reproduïr els resultats experimentals amb un sol paràmetre d'adjust. En el cas de nanofibres cilíndriques, no és necessari cap paràmetre d'adjust si es consideren unes certes condicions de vora per al flux. Una conseqüència d'haver de considerar extensions per a la llei de Fourier és que s'ha d'estudiar l'efecte que tenen aquests canvis en la descripció de processos de canvi de fase. En la segona part de la tesi investiguem els efectes que tenen diversos models sobre la solidificació d'un líquid unidimensional. Al capítol 5 estudiem el cas en el que considerem la conductivitat tèrmica com a una funció de la mida del sòlid i que incorpora característiques que són importants quan el tamany del sòlid és comparable a les longituds característiques dels fonons, mentres que al capítol 6 considerem l'equació de Guyer i Krumhansl dels capítols anteriors. En ambdós casos, un anàlisi asimptòtic ens permet reduïr la complexitat del problema i proposar models reduïts formats per un parell de'equacions diferencials ordinàries.Award-winningPostprint (published version

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

We investigate the one-dimensional growth of a solid into a liquid bath, starting from a small crystal, using the Guyer-Krumhansl and Maxwell-Cattaneo models of heat conduction. By breaking the solidification process into the relevant time regimes we are able to reduce the problem to a system of two coupled ordinary differential equations describing the evolution of the solid-liquid interface and the heat flux. The reduced formulation is in good agreement with numerical simulations. In the case of silicon, differences between classical and non-classical solidification kinetics are relatively small, but larger deviations can be observed in the evolution in time of the heat flux through the growing solid. From this study we conclude that the heat flux provides more information about the presence of non-classical modes of heat transport during phase-change processes.Comment: 29 pages, 6 figures, 2 tables + Supplementary Materia

The Stefan problem with variable thermophysical properties and phase change temperature

In this paper we formulate a Stefan problem appropriate when the thermophysical properties are distinct in each phase and the phase-change temperature is size or velocity dependent. Thermophysical properties invariably take different values in different material phases but this is often ignored for mathematical simplicity. Size and velocity dependent phase change temperatures are often found at very short length scales, such as nanoparticle melting or dendrite formation; velocity dependence occurs in the solidification of supercooled melts. To illustrate the method we show how the governing equations may be applied to a standard one-dimensional problem and also the melting of a spherically symmetric nanoparticle. Errors which have propagated through the literature are highlighted. By writing the system in non-dimensional form we are able to study the large Stefan number formulation and an energy-conserving one-phase reduction. The results from the various simplifications and assumptions are compared with those from a finite difference numerical scheme. Finally, we briefly discuss the failure of Fourier's law at very small length and time-scales and provide an alternative formulation which takes into account the finite time of travel of heat carriers (phonons) and the mean free distance between collisions.Comment: 39 pages, 5 figure

Non-Fourier heat conduction: The Maxwell-Cattaneo equations

We study the derivation, properties and solution methods of the classical and the hyperbolic heat transfer equations. Some results to approximate the hyperbolic model with the classical model for small values of the thermal relaxation time are given. Finally, two examples of application of the hyperbolic model are studied and discussed

A mathematical model for the energy stored in green roofs

A simple mathematical model to estimate the energy stored in a green roof is developed. Analytical solutions are derived corresponding to extensive (shallow) and intensive (deep) substrates. Results are presented for the surface temperature and energy stored in both green roofs and concrete during a typical day. Within the restrictions of the model assumptions the analytical solution demonstrates that both energy and surface temperature vary linearly with fractional leaf coverage, albedo and irradiance, while the effect of evaporation rate and convective heat transfer is non-linear. It is shown that a typical green roof is significantly cooler and stores less energy than a concrete one even when the concrete has a high albedo coating. Evaporation of even a few millimetres per day from the soil layer can reduce the stored energy by a factor of more than three when compared to an equivalent thickness concrete roof.Peer ReviewedPostprint (published version

Effective thermal conductivity of rectangular nanowires based on phonon hydrodynamics

A mathematical model is presented for thermal transport in nanowires with rectangular cross sections. Expressions for the effective thermal conductivity of the nanowire across a range of temperatures and cross-sectional aspect ratios are obtained by solving the Guyer--Krumhansl hydrodynamic equation for the thermal flux with a slip boundary condition. Our results show that square nanowires transport thermal energy more efficiently than rectangular nanowires due to optimal separation between the boundaries. However, circular nanowires are found to be even more efficient than square nanowires due to the lack of corners in the cross section, which locally reduce the thermal flux and inhibit the conduction of heat. By using a temperature-dependent slip coefficient, we show that the model is able to accurately capture experimental data of the effective thermal conductivity obtained from Si nanowires, demonstrating that phonon hydrodynamics is a powerful framework that can be applied to nanosystems even at room temperature

Non-Fourier heat conduction: The Maxwell-Cattaneo equations

We study the derivation, properties and solution methods of the classical and the hyperbolic heat transfer equations. Some results to approximate the hyperbolic model with the classical model for small values of the thermal relaxation time are given. Finally, two examples of application of the hyperbolic model are studied and discussed

Non-Fourier heat conduction: The Maxwell-Cattaneo equations

We study the derivation, properties and solution methods of the classical and the hyperbolic heat transfer equations. Some results to approximate the hyperbolic model with the classical model for small values of the thermal relaxation time are given. Finally, two examples of application of the hyperbolic model are studied and discussed

The mathematical model of a soap film. Theory of minimal surfaces

This thesis is an introduction to the theory of minimal surfaces. After introducing the basic concepts of functional analysis, variational calculus and theory of surfaces, a theorem existence of nonparametric minimal surfaces and the more general Douglas-Courant-Tonelli method are given. The Weierstrass-Enneper representation theorems link our minimal surfaces with complex analysis and potential theory