Non-Fourier Heat Conduction. From Phase-Lag Models to Relativistic and Quantum Transport 9783031259722, 9783031259739

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Alexander I. Zhmakin

Non-Fourier Heat Conduction From Phase-Lag Models to Relativistic and Quantum Transport

Non-Fourier Heat Conduction

Alexander I. Zhmakin

Non-Fourier Heat Conduction From Phase-Lag Models to Relativistic and Quantum Transport

Alexander I. Zhmakin Department of Numerical Simulation Ioffe Institute Saint Petersburg, Russia Soft-Impact, Ltd. Saint Petersburg, Russia

ISBN 978-3-031-25972-2 ISBN 978-3-031-25973-9 (eBook) https://doi.org/10.1007/978-3-031-25973-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The Fourier heat conduction model is valid for most macroscopic problems. However, it fails when the wave nature of the heat propagation or the time lags become dominant and the memory or/and spatial non-local effects significant—in ultrafast heating (pulsed laser heating and melting), rapid solidification of liquid metals, processes in glassy polymers near the glass transition temperature, in heat transfer at nanoscale, in heat transfer in a solid-state laser medium at the high pump density or under the ultrashort pulse duration, in granular and porous materials including polysilicon, at extremely high values of the heat flux, in heat transfer in biological tissues. In common materials the relaxation time ranges from 10−8 to 10−14 sec; however, it could be as high as 1 sec in the degenerate cores of aged stars, and its reported values in granular and biological objects vary up to 30 sec. The book considers numerous non-Fourier heat conduction models that incorporate the time non-locality for the materials with memory (hereditary materials, including fractional hereditary materials) or/and the spatial non-locality for the materials with non-homogeneous inner structure. Saint Petersburg, Russia November 2022

Alexander I. Zhmakin

Acknowledgements The author is grateful to Sergei Y. Karpov, Vladimir F. Mymrin, Georgii G. Prochorov, Alexandre A. Schmidt, Sergei L. Sobolev, Valentin S. Yuferev and Igor A. Zhmakin for useful discussions and to Marina N. Nemtseva for the help with the manuscript preparation.

v

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I

1 16

Classical Transport

2

Phase-Lag Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Maxwell–Cattaneo–Vernotte Equation . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 “Relativistic” Heat Conduction . . . . . . . . . . . . . . . . . . . . . 2.2 Dual-Phase-Lag Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Non-local Dual-Phase-Lag Model . . . . . . . . . . . . . . . . . . . 2.3 Triple-Phase-Lag Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Non-local Triple-Phase-Lag Model . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 43 61 63 73 75 75 76

3

Phonon Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Phonon Transport Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Guyer–Krumhansl (GK) Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Two-Fluid Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Ballistic–Diffusive Model . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Extended Ballistic–Diffusive Model . . . . . . . . . . . . . . . . . 3.3.3 Unified Non-diffusive-Diffusive Model . . . . . . . . . . . . . . 3.3.4 Enhanced Fourier Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Two-fluid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Generalized Fourier Law by Hua et al. . . . . . . . . . . . . . . . . . . . . . . . 3.5 Phonon Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Nonequilibrium Thermodynamics of Phonon Hydrodynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Flux-Limited Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Relaxon Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89 90 107 120 120 124 125 127 129 130 132 137 139 142 150

vii

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Contents

4

Thermomass Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Equation of State (EOS) of the Thermon Gas . . . . . . . . . . . . . . . . . 4.1.1 EOS of Thermon Gas in Ideal Gas . . . . . . . . . . . . . . . . . . 4.1.2 EOS of Thermon Gas in Dielectrics . . . . . . . . . . . . . . . . . 4.1.3 EOS of Thermon Gas in Metals . . . . . . . . . . . . . . . . . . . . . 4.2 Equations of Motion of Thermon Gas . . . . . . . . . . . . . . . . . . . . . . . 4.3 Heat Flow Choking Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Dispersion of Thermal Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 164 164 165 166 167 170 171 173

5

Mesoscopic Moment Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

6

Microtemperature and Micromorphic Temperature Models . . . . . . . 6.1 Microtemperature Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Micromorphic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

181 182 183 184

7

Thermodynamic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Jou and Cimmelli Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Heat Conduction in Thermoelectric Systems . . . . . . . . . . 7.2 Sellitto and Cimmelli Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Kovács and Ván Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Famá et al. Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Rogolino et al. Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Two-Temperature Model by Sellitto et al. . . . . . . . . . . . . . . . . . . . . 7.7 EIT Ballistic–Diffusive Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

187 188 190 191 193 195 196 199 201 202

8

Fractional Derivative Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Fractional Fourier Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Nonlinear Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Fractional Pennes Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Zingales’s Fractional-Order Model . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Fractional Cattaneo and SPL Models . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Fractional DPL Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Fractional TPL Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Non-local Fractional TPL Model . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205 205 209 211 212 213 216 217 219 220

9

Fractional Boltzmann and Fokker–Planck Equations . . . . . . . . . . . . . 9.1 Continuous-Time Random Walks . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Lévy (Khintchine–Lévy) Walks . . . . . . . . . . . . . . . . . . . . . 9.2 Kramers–Fokker–Planck Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Li and Cao Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

225 229 237 240 241 244

Contents

ix

10 Elasticity and Thermal Expansion Coupling . . . . . . . . . . . . . . . . . . . . . 10.1 Non-Fourier Thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Fractional Thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

247 250 251 254

11 Some Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Phase-Lag Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Phonon Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Fractional Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

257 258 265 267 273

Part II

Relativistic Transport

12 Relativistic Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 13 Relativistic Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 General Relativistic Boltzmann Equation . . . . . . . . . . . . . . . . . . . . 13.2 Particles in External Electromagnetic Fields . . . . . . . . . . . . . . . . . . 13.3 Relativistic Gas in Gravitational Field . . . . . . . . . . . . . . . . . . . . . . . 13.4 Grad’s Moment Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Chapman–Enskog Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.1 Anderson–Witting Transport Coefficients in General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

299 303 311 314 315 318

14 Variational Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Márkus and Gambár Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Multifluid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

325 325 328 332

320 321

15 Relativistic Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 Part III Quantum Transport 16 Landauer Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 17 Green–Kubo Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 18 Coherent Phonon Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 19 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 Appendix: An Introduction to Fractional Calculus . . . . . . . . . . . . . . . . . . . 375

Acronyms

BD BGK BTE CE CIT CTRW DODE DOM DPL DWE EBDM EIT EITBD FDBTE FFP FHCE FP FTFC FTPL GCCT GCE GENERIC GK GPU HBIM HF HHCE HIFU HPM HTP

Ballistic–Diffusive Bhatnagar–Gross–Krook Boltzmann Transport Equation Chapman–Enskog Classical Irreversible Thermodynamics Continuous-Time Random Walks Distributed-Order Differential Equations Discrete Ordinate Method Dual Phase Lag Damped Wave Extended Ballistic–Diffusive Model Extended Irreversible Thermodynamics EIT Ballistic–Diffusive Model Frequency-Dependent Boltzmann Transport Equation Fractional Fokker–Planck Fourier Heat Conduction Equation Fokker–Planck Fundamental Theorem of Fractional Calculus Fractional Tripple Phase Lag General-Covariant Continuum Thermodynamics Generalized Cattaneo Equation General Equation for Non-equilibrium Reversible–Irreversible Coupling Guyer–Krumhansl Graphics Processing Unit Heat-Balance Integral Method High Frequency Hyperbolic Heat Conduction Equation High Intensity-Focused Ultrasound Homotopy Perturbation Method Heat Transfer Paradox xi

xii

IID LA LBM LCM LDOS LF LHC LLNETT LO LRF LTNE MDA MDD MEPP MFP MOSFET NDPL NEMD NEMS NESS NET-IV NF NTPL NW PDF PETE PHC QBE QGP RHIC RTA SDE SL SMRTA SOI SPL TA TBR TC TDTR TMO TO TPL UND VIM

Acronyms

Independently Identically Distributed Longitudinal Acoustic Lattice Boltzmann Method Layered Correlated Materials Local Density of States Low Frequency Large Hadron Collider Local/Linear Non-equilibrium Thermodynamics Theory Longitudinal Optical Local Rest Frame Local Thermal Non-equilibrium Equations Modified Differential Approximation Memory-Dependent Derivative Maximum Entropy Production Principle Mean Free Path Metal-Oxide-Semiconductor Field-Effect Transistor Non-local Dual Phase Lag Non-equilibrium Molecular Dynamics Nanoelectromechanical System Non-equilibrium Steady State Thermodynamics with Internal Variables Nanofilm Non-local Triple Phase Lag Nanowire Probability Distribution Function Piezoelectric ThermoElasticity Paradox of Heat Conduction Quantum Boltzmann Equation Quark–Gluon Plasma Relativistic Heavy Ion Collider Relaxation Time Approximation Stochastic Differential Equation Superlattice Single-Mode Relaxation Time Approximation Silicon-on-Insulator Single Phase Lag Transverse Acoustic Thermal Boundary Resistance Temperonic Crystal Time-Domain Thermal Reflectance Transition Metal Oxide Transverse Optical Triple Phase Lag Unified Non-diffusive–Diffusive Variational Iteration Method

Chapter 1

Introduction

The heat conduction is probably the most important dissipative phenomenon; thus it is the starting point of any theories of dissipation, particularly for the non-equilibrium thermodynamics; the distinctive property of this process is that it does not have a reversible part [1]. The heat conduction model suggested in the beginning of the nineteenth century by Jean Baptiste Joseph Fourier [2] in his famous work named “Analytical Theory of Heat” is based on the constitutive relation q(r, t) = −λ∇T (r, t).

(1.1)

leading, after substitution it into the energy conservation equation for the solid body in rest ∂(cT ) =∇·q+Q (1.2) ∂t to the classical parabolic heat conduction equation (called sometimes the “FourierKirchhoff equation”, rarely “Maxwell-Fourier law” [3], or the “Fourier heat conduction equation” (FHCE) [4]) ∂(cT ) = ∇ · (λ∇T ) + Q ∂t

(1.3)

that for the case of constant thermophysical properties is written as ∂T = κT ∂t

(1.4)

where κ = λ/c is the thermal diffusivity. Specific heat capacity shows nonnegligible variations only in very wide temperature range [5]. However, Cimmelli et al. [6] analysed the effect of the temperature dependence of thermal conductivity and heat capacity [7, 8] on heat transfer. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. I. Zhmakin, Non-Fourier Heat Conduction, https://doi.org/10.1007/978-3-031-25973-9_1

1

2

1 Introduction

For bulk materials, the thermal conductivity λ is believed to be an intensive property that does not depend on the size and the geometry of the sample. This is valid when the phonon transport is diffusive; when the sample size along the transport direction is much smaller than the phonon mean free path, phonons propagate ballistically across the sample without experiencing an appreciable scattering [9–12]. This is, for example, the case in the thin films and nanowires where the thermal conductivity decreases as film becomes thinner or the nanowire radius decreases due to the increase of the phonon–boundary scattering [13]. The Fourier law can be directly derived from the classical irreversible thermodynamics based on the local equilibrium hypothesis [14, 15]. The entropy production is written as 1  = q · ∇ ≥ 0. T From the viewpoint of the irreversible thermodynamics, the Fourier law describes a linear relationship between the generalized force (the temperature gradient) and the generalized flux (the heat flux) [16]. For the isotropic materials the heat flux is q = λT ∇

λT 1 = − 2 ∇T. T T

Thus λ = λT /t 2 is the Fourier heat conduction coefficient with λT being a thermodynamic conduction coefficient [1]. The derivation of the Fourier equation is based on the principle of the local thermodynamic equilibrium and the continuum hypothesis, according to which each small element of the medium has a local equilibrium state and can be described by the local thermodynamic potentials that are dependent on the spatial variable and the time only through the thermodynamic parameters. The Fourier law is local with respect to time and space [17]. Accepting the local equilibrium principle is possible only if the rate of change in the system macroparameters due to the external influences is much less than the rate of system relaxation to the local equilibrium [17]. Based on the local equilibrium principle, it is assumed that the transport laws are valid not only for the entire system, but also for any arbitrarily small part of it. Thus it is possible to perform the limit transition (by tending the volume of integration to zero) in the integral conservation laws and obtain these laws in the form of the differential equations. If the characteristic microscale of the system and the characteristic time of its relaxation to equilibrium are significantly less than the characteristic macroscale and the total time of the process, then the differential equations derived based on the local equilibrium principle and the continuum hypothesis will be local both in space and in time. Thus we obtain the transport equations that do not contain the relaxation time τ and the characteristic scale of the microstructure l. The Fourier law is a prototype of the constitutive equations leading to a parabolic partial differential equation. The major drawback of the parabolic FHCE is the unphysical instantaneous propagation of the disturbances—it neglects the boundness

1 Introduction

3

of the speed of the thermal disturbances by the the internal structure of the media they travel on—[18] (“the heat transfer paradox” [1], “the paradox of propagation of the thermal signals” [19], “the paradox of heat conduction” (PHC) [1, 20–22], “the heat transfer paradox” (HTP) [23] or “the paradox of diffusion” [24]) called sometimes acausality [25]—it does not forecasts the propagation of the disturbances restricted by the characteristic causal light-cones [26]. Lars Onsager in 1931 [27] pointed out that “... Fourier law is only approximate description of the process of conduction neglecting the time needed for acceleration of the heat flow”. The solution of Eq. (1.4) in the one-dimensional case for the initial conditions T (x = 0, 0) = 0, T (0, 0) = 1 is   x2 1 . exp − T (x, t) = √ 4κt π κt Thus, for x as large as one wishes and for any t > 0 one gets T (x, t) > 0. Hence the heat has propagated from x = 0 to x in an interval of time no matter how small. The speed of the propagation of the heat is infinite, which is not acceptable. However, as noted Fichera [28], the Fourier law is a result of experimental observations; hence it is conditioned by the physical nature of the material under consideration and by the degree of refinement of the measurements. It results that the infinitely small heat signal that propagates at an infinite speed cannot be experimentally seen— the sensitivity of the experimental devices is finite, therefore they can measure only finite speeds [22, 29, 30]. Day [31] demonstrated that, if an infinitesimal small fraction of the heat signal does propagate at an infinite speed, the propagation of the bulk of the signal—which is physically significant—is at finite velocity. As stressed Márkus [32], one should distinguish the signal and action propagation. Auriault [33] claimed that the paradox is inconsistent using the underlying homogenization process to investigate the domains of validity of both the Fourier and the Maxwell “Cattaneo” Vernotte (Sect. 2.1) equations. The author used three different characteristic lengths l, L and L 1 associated with the atomic scale, the macroscopic scale where the Fourier law and the Maxwell “Cattaneo” Vernotte law are defined and an upper macroscopic scale: l/L = ε  1, L/L 1 = η  1. The smallness of the parameter of separation of scales η is the condition for the existence of the equivalent macroscopic models [33]. In the framework of the homogenization theory T = T (0) + εT (1) + ε2 T (2) + · · · and the Fourier law is ∂ T (0) = κT (0) + O(ε). ∂t Thus an infinite speed of a heat signal obtained from the classical Fourier equation cannot be demonstrated from the last equation since such a signal is very small—it is included in the term O(ε). Thus Auriault concluded that since the Fourier equation is an approximation and cannot forecast a heat signal with an infinite speed because such a signal would be infinitely small, it should not be considered paradoxical.

4

1 Introduction

The Fourier law is valid if [34, 35] • L  O(1), • τt  O(1), • T  0◦ K where L is the characteristic size of the system, is the mean free path of the heat carries, τ is the relaxation time. The ratio K n = /L is called the Knudsen number similar to the rarefied gas dynamics theory. The Fourier law is valid in the limit of small Knudsen number, i.e. when /L  1. The Knudsen number may increase either because of an increase of the mean free path of the heat carries as in rarefied gases or due to the reduction of L as in miniaturized systems. Sometimes the Fourier number is used Fo = λT0 /ρC L [36, 37]. Thus, the Fourier law is invalid if the “microscopic nature of heat” [38] could not be ignored. The Fourier law also fails in the case of the extremely high heat fluxes since the heat flux cannot reach the arbitrary high values but is limited by the quantity of the order of the energy density times the maximum speed of the signal propagation, thus the saturation of the fluxes occurs for the high enough values of the temperature gradient [39–44]. The heat waves in the form of the second sound [45]—the thermal transport regime where the heat is carried by the temperature waves similarly to the propagation of the sound wave in the ordinary gases—were predicted by Landau [46] and Tisza [47] (see also [48, 49]) who studied the behaviour of the quasi-particles in the superfluid liquid helium II using the two-fluid model (the normal fluid and the superfluid) and were observed in the helium II at 1.4 K by V. Peshkov in 1944 with velocity about 19 m/s that is one order of magnitude less than the speed of sound in the helium II [50–55]. As pointed out by Hardy [56, 57], the first sound (or simply “sound,” i.e. the mechanical lattice vibrations) and the second sound are described by a similar equation where the variables have a different physical meaning—pressure and temperature, respectively. The derivation of a dissipative extension of the Maxwell–Cattaneo–Vernotte equation from kinetic theory, the Guyer–Krumhansl equation (see Sect. 3.2) provided the “experimental window condition” , when dissipation is minimal. The second sound was observed at the cryogenic conditions in other materials [58–67]—solid helium-3 [7, 68–72], sodium fluoride (at ca. 10–20 K [73–77]), bismuth (at 1.2–4.0 K [78]), sapphire, strontium titanate SrTiO3 [79, 80], in the highly oriented pyrolytic graphite at the temperature above 100 K [81]. In some experiments second sound and ballistic propagation “heat pulses propagating with the speed of sound” were observed together [75]. At lower temperatures ballistic propagation appears without second sound [22]. At very low temperatures when both the normal and the Umklapp scattering MFP are longer than the sample length, the longitudinal and the transverse phonons travel ballistically. At higher temperatures, the normal scattering equilibrates phonons and the second sound forms. At even higher temperatures, the Umklapp scattering dominates and the heat propagation transitions to diffusion [82].

1 Introduction

5

Hydrodynamic effects of the phonon transport, which usually observed at the extremely low temperatures, become significant in the graphitic materials even at the moderate temperature due to both the high Debye temperature and the strong anharmonicity [83, 84]. The existence of the hydrodynamic effects in graphite—a three-dimensional material—above the liquid nitrogen temperature was predicted by ab initio computations by Ding et al. [85] who determined the phonon distribution function in an infinitely large graphite crystal under the constant temperature gradient. The authors demonstrated that at the temperature of 100 K heat transport in graphite is hydrodynamic in nature, as it is dominated by the drifting phonons. In order to investigate the robustness of the phonon hydrodynamics with respect to the sample quality, Ding et al. studied the effect of the vacancies that are treated as the mass disorder and find that the collective phonon drift motion is destroyed when the vacancy concentration is about 0.01%, but can still be observed when the vacancy concentration is about 0.001%. Ding et al. attribute the unusual behaviour of graphite to its strong intralayer sp2 hybrid bonding and to the weak van der Waals interlayer interactions. The authors also noted that the reflection symmetry associated with a single graphene layer is broken in the graphite, which opens up more momentum-conserving phonon–phonon scattering channels and results in the stronger hydrodynamic features in graphite than in graphene. The first-principles simulations also predicted the existence of the hydrodynamic phenomena at the non-cryogenic temperatures in the low-dimensional or the layered materials such as graphene [86–88], other 2D materials [86], carbon nanotubes [89]. In the all experimental observations of the second sound, the dominance of the momentum conserving phonon scattering (the Normal processes) with respect to the resistive phonon scattering (the Umklapp processes, the isotope or the impurity scattering) was found to be critical—the second sound was observed almost exclusively in the very low temperature regime, with the exception of the experiment by Huberman et al. in the graphite [81]. Thus the condition for the experimental detection of the second sound was found to be τ N < τexp < τ R , i.e. the typical experimental observation times τexp must be larger than the normal phonon scattering times τ N to allow the momentum redistribution but smaller than the resistive phonon scattering times τ R to avoid the decay of the phonon wave packet into the phonon equilibrium distribution [90]. Recently Beardo et al. [90] demonstrated the existence of the second sound in bulk Ge between 7 K and the room temperature by studying the phase lag of the thermal response under the harmonic high-frequency external thermal excitation (the pump laser with the wavelength λ = 405 nm modulated between 30 kHz and 200 MHz).

6

1 Introduction

The wave nature of the heat propagation or the time lags become dominant and the memory,1 nonlinear or spatial non-local2 effects significant [99, 100] • in the ultrafast heating – the pulsed laser heating and melting [101–110] (a short pulse laser results in a more localized heating than a continuous laser [111]; the ultrashort laser pulses (USLP) have various advantages for the precision micro-fabrication: the amount of energy deposited into the sample can be minimized and highly localized [112], the heat flux can exceed 1013 W/m2 [113]. The use of the Fourier equation for such problems results in the higher lattice temperature and the lower electron temperature), e.g. the femtosecond heating of the metal films [114–121] or the thin film of solid argon [122, 123]), – the welding and drilling of metals, – the surface annealing, – the sintering of ceramics [124], – the micro-machining [125], – the rapid solidification of liquid metals [126–129] (e.g. the crystallization front velocity in the undercooled melt can be as high as several tens of metres per second [130]—the solidification rates up to 70 m/s were observed for the pure Ni and Cu–Ni alloy [131]), – the glass transition of supercooled liquids that could involve the multiple steps of the energy relaxation [131, 132], – the glassy polymers near the glass transition temperature [133], – the heat pulse experiments at the room temperature, including “the book experiment” [22, 134–141], – the sudden contact of the two liquids such as the uranium oxide and sodium [142], • in the heat transfer at nanoscale [82, 143–147] – in the microelectronic and optoelectronic devices [148–156], e.g. the hot spots in the nanotransistors [157–164],3 1

The memory effects could be related to the inhomogeneity of the medium structure. For example, position-depending trapping of the carriers or the enhanced diffusion due to the disorder could occur [91]. 2 Non-local effects could originate from the material non-homogeneity since defects, fractures and lattice orientations interact with one another at a distance which can be relevant at a convenient scale [92]. Non-local effects become also important in nanoscale systems, such as various kinds of nanowires and thin layers, since even a small difference of temperature, or electrical potential, over a small-scale length may generate very high temperature gradients [93–95]. The non-local formulation of the thermal heat conduction is needed in other situations when the free paths of the energy carriers are large, e.g. in the electron heat conduction in the models of the flaring solar “loop” structures [96, 97] or in modelling of the tokamak power exhaust and scrape-off-layer thermal transport in the high-power fusion devices [98]. 3 The integrated schemes (IC) contain billions of the transistors (about 9 billions CPUs in 2015 [165]) that generate the huge heat fluxes in a very small area; these “hot spots” become the bottleneck of the future development in terms of both the performance [144] (many devices have the

1 Introduction

7

– the nanostructured devices for the solid-sate energy conversion, e.g. the thermoelectric and thermoionic refrigeration [170–178], – in the nanoelectromechanical systems (NEMS) [179], – in the two-dimensional materials [11], – in the heterostructures [11, 180], – in the layered strongly correlated materials (LCM) [181] such as, for example, the transition metal oxides (TMOs) [182] in which the strong short-ranged electronic interactions lead to the failure of the independent-electron approximation and which are characterized by the intertwining of the charge, spin and lattice degrees of freedom and exhibit the wavelike regime for the temperature propagation at the nanoscale [182, 183], – in the laser plasma in irradiating small targets [184]), – in the nanoengineered suspensions for the radiative cooling [185] and the volumetric solar thermal energy absorption [186], – in the heat transfer in the macromolecule of the deoxyribonucleic acid (DNA) during its denaturation (“melting”)—unravelling of the double-stranded structure into two single strands [187], – in the ultrahigh thermal isolation across the heterogeneously layered twodimensional materials [188], • in the heat transfer in a solid-state laser medium at the high pump density or under the ultrashort pulse duration [189], • in the heat transfer in the granular and porous materials [190–198] including porous silicon (pSi),4 in the fractured geological media [208–213], temperature-dependent figures of merit) and the reliability [166, 167] (overheating is the major source of the device failure). The hotspot within the transistor drain region where the energy transfer from electrons to the lattice is the most intense can be 10 nm thick [158] and lead to the increase of the drain series, the source injection electrical resistances and the thermal-induced breakdown [161]. Sverdrup et al. measured the ballistic phonon conduction near hotspot and the doped resistor thermometry in the suspended silicon membrane and found that the temperature exceeds by 60% a value predicted by the Fourier law [158]. Johnson et al. [10] by the transient thermal grating (TTG) experiments on nanofilms proved that the room-temperature thermal transport in silicon significantly deviates from the diffusion model already at micron distances. The Fourier law for heat conduction dramatically overpredicts the rate of the heat dissipation from sources with dimensions smaller than the dominant phonon free paths [168, 169] that is very important in the thermal management in microelectronics. However, the scaling problem for the thermal management in nanoelectronics is mitigated by the phenomenon discovered recently by Hoogeboom-Pot et al. [169]—“collective diffusion”: when the separation between nanoscale heat sources is small to compared the phonon mean free path, phonons can scatter with phonons originated from the neighbouring heat source increasing the heat transfer efficiency to near the diffusion limit. 4 Porous silicon was discovered in 1956 [199]. Now it is used in the light-emitting diodes (LEDs), sensors, the thermoelectric devices [200, 201], as the insulation for the microelectronic device; the thermal conductivity of pSi is two to five orders of magnitude smaller than that of the bulk Si [202–204]. The nanoscale porous materials are also called Cantor materials [205]. The silicon porous membranes show the thermal conductivity values lower than expected by porosity and the bulk phonon mean free path of silicon, and the small-pore membranes have smaller

8

1 Introduction

• at the extremely high values of the heat flux [214] (>≈ 107 W/cm2 [214])—for example, in the beam deep penetration welding with the heat flux greater than 108 W/cm2 [215] and • in the heat transfer in the biological tissues [216–228]. Note, however, that errors due to the use of the Fourier law at the nanoscale are not always important. For example, Wilson and Cahill [229] listed three reasons why the errors in the diamond thermal conductivity are not essential in the analysis of the thermal management of the microelectronic devices using diamond as the heat spreader: 1. The magnitude of the ballistic–diffusive effects in the polycrystalline diamond films grown by the chemical vapour deposition (CVD) [230] will be smaller than in the single crystal diamond due to the phonon scattering by the grain boundaries that reduces the amount of the heat carried by the long mean-free-path phonons [231]; 2. At least for the GaN devices such as the high electron mobility transistors (HEMTs) the substrate thermal conductivity is the major factor in the thermal performance only when the dimensions of the active region undergoing the selfheating exceed 1 µm; 3. For the sufficiently high areal density of devices there is no high in-plane temperature gradient. The non-Fourier effects are also reduced in the semitransparent media due to the medium radiation [102]. The extensions of the Fourier law allowing to avoid the paradox of heat conduction are important not only in the heat transfer itself but in the related physics areas such as the thermoelasticity [232–234] and the piezoelectric thermoelasticity (PETE) [235– 238]. The ultrafast lasers (with the pulse duration range from subnanoseconds down to femtoseconds [239] or even attoseconds [240]) are used in a wide spectrum of the biomedical technologies: the optical tomography [241] that provides both the physiological and morphological information about the of the living tissues and organs [242], the photodynamic therapy [111, 243], the hyperthermia [244–251], the laser thermokeratoplasty (LTK), the interstitial laser photocoagulation therapy (ILP) [221]. The control of the temperature in the treated tissues could be enhanced by the injection of the nanostructures [252, 253], for example, of the gold nanoparticles (GNPs) [254–256] . The relaxation time is the characteristic time taken by the system to return to the steady state (that could be or not be in the thermodynamic equilibrium) after it has been suddenly removed from it. The relaxation time is related to the mean collision time τc of the particles responsible for the heat transfer. However, there is no universal thermal conductivity compared to the large-pore ones despite that they have close porosity values [206]. The hydrodynamic-heat-transport models are needed to describe the experimental effective thermal conductivity of silicon thin films and periodic holey membranes for different sizes and temperatures [207].

1 Introduction

9

relation between τ and τc : in some cases τ may correspond to just a few τc while in others the difference could be huge (for, example, in the model of the early universe the relaxation time of the shear viscosity could be orders of magnitude larger than the collision time between photons and electrons due to the matter–radiation decoupling [26]). The origin of the time lag could be the existence of several energy carriers [257] (the well-known example in the solid-state physics is the relaxation between the electron and phonon subsystems [258, 259], e.g. the heat transfer from free electrons to the lattice [260, 261] in the metal heating by the ultrashort laser pulses [262–265]— the time necessary to established an electron temperature is under picosecond, for example, for gold is about 800 fs, while the electron-lattice relaxation time is on the order of a few picoseconds for metals [112]) or the heterogeneous inner structure. The biological tissues contain cells, membranes, organelles, superstructures, liquids, solid/soft elements (sometimes the tissues are referred to as to the mesoporous structure [266]). The heating or cooling of the living tissues induces a series of chemical, electrical and mechanical processes, e.g. diffusion, electrical potential change and osmosis across the cell membrane; the cell membranes could store energy [216]. Thus, the heat propagation involves the multimode energy conversion at the different cellular levels [216, 267, 268]. From a purely thermodynamic point of view, rocks do not differ significantly from the biological materials: they both porous, having various heterogeneities and irregularities, thus parallel time scales are present [269]. There are numerous experimental approaches to study the heat transfer at the micro/nanoscale [231]: • the 3 ω method (based on measuring the third harmonic in the voltage during heating of the sample with a sinusoidal wave of frequency ω) [270–274], including the differential 3 ω technique [150]; 5 • the 2 ω method [37] • the scanning thermal microscopy [272, 275–279]6 ; • the bimaterial cantilever; • the optical methods, including optical pump-probe method [280]7 ; • the coherent X-ray probing; • the thermal coherence tomography [281]; • the flash method [282, 283]; • the optothermal Raman method [11, 284];

3 ω method is based on measuring the third harmonic in voltage during heating of the sample with a sinusoidal wave of frequency ω. 6 A scanning thermal microscope (SThM) operates by placing a sharp temperature-sensing tip in close proximity to a solid surface of the sample and the local heat transfer changes the tip temperature [231]. 7 In the pump-probe technique the ultrashort laser pulse is split into two parts: one (“pump”) very intense excites the medium under investigation while the weaker second part (“probe”) to detect the physical effects induced by the “pump”. 5

10

1 Introduction

• the time-domain thermal reflectance (TDTR)8 and the frequency-domain thermal reflectance (FDTR) [288–290]; • the frequency modulated thermal wave imaging [291]9 ; • the laser flash method [4, 292]; • the resonant X-ray diffraction [182]; • the micro-bridge method [293, 294]; • the Raman thermometry [295–297]; • the electrical-resistance thermometry [298]; • the thermal conductivity spectroscopy [299]. The possibilities of the modern experimental techniques are impressive; for example, the recently reported ultrafast coherent sources operating at the extreme ultraviolet (EUV) and X-ray wavelengths [300] allow measurements of the thermal transport on the 10-nm scale. There are also several computational methods of the varying fidelity [301, 302]: • the first-principal (ab initio) computations such as the density functional method, frequently used to provide the required phonon dispersion dependencies and the scattering rates for the phonon Boltzmann equation [303];) • the lattice dynamics; • the non-equilibrium Green’s function [304]; • the molecular dynamics [218, 305–309]—either the equilibrium (Green-Kubo) method (EMD) based on the fluctuation-dissipation theorem [310] or the nonequilibrium (direct) method (NEMD) based on storing positions and velocities of all particles at each step of the simulation, the results of both approaches critically depend on the accuracy of the interatomic potential functions [311, 312] and on the nature of the thermal reservoirs on the contacts [313];10 • the Monte Carlo method [315–321]; • the solution of the Boltzmann transport equation11 [272, 326–330]; • the multiscale simulations [331–335]. 8

TDTR is an optical non-contact method that applicable to the materials with wide range of the thermal conductivity (from diamond to the ultralow thermal conductivity of disordered crystals) and different sample geometries [145, 272, 285, 286]. It is necessary in experiments to account for the reduction of the effective thermal conductivity of a semi-infinite semiconductor crystal at high frequencies of the oscillating heat source applied to its top surface. If the heating frequency is much smaller than the average dominant phonon scattering rate over all branches polarizations, the steady-state form of the Boltzmann equation can be used to describe the phonon transport process [287]. 9 This method is based on the thermal response of the sample surface to a heat stimulus—also known as the active thermography in contrast to the passive thermography without the external heat stimulus—and could be used in two variants: the pulsed-based thermography or the lock-in (single frequency) thermography. 10 Recent research [314] showed that both EMD and NEMD simulations underestimate the ballistic thermal conduction observed in carbon nanotubes. 11 It is not straightforward to solve the BTE in general. Even the linearized BTE is still very challenging to solve because of its high dimensionality. Three common methods have been used to solve the BTE:

1 Introduction

11

The relaxation time is the macroscopic parameter that integrates a series of microscopic interactions and is associated with the communication time between the particles such as photons, electrons, phonons [336]. In common materials the relaxation time ranges from 10−8 to 10−14 s [337–341] (e.g. it equals 3 ps–3.5 ps [151] for silicon, 4.5–6.4 ps for mercury, 5.1–7.3 ps for molten gallium [342]). Evidently, the non-Fourier effects in these case are rather small. For example, Saad and Didlake [343] studied the Stefan problem (melting of a semi-infinite slab subjected to a step change in temperature) using the Cattaneo heat conduction model and found for the aluminium (the estimated relaxation time 10−10 –10−12 s) the non-Fourier effects are significant only for times on the order of 10−9 –10−11 s and in the region within 10−4 –10−5 cm from the phase boundary. However, the thermal relaxation time could be as high as seconds in the degenerate cores of the aged stars [26] (e.g. about 4.5 s for a white dwarf [344]) and its reported value in the granular and biological objects varies up to 100 s [340]. For example, Kaminski [337] reported the relaxation time of 20 s for sand and 29 s for NaHCO3 , Mitra et al. [345] found the value of the relaxation time for the processed meat to be c.a. 15 s. However, Grassman and Peters [193] and Herwig and Beckert [346, 347] did not found the evidence for the hyperbolic heat conduction in materials with the non-homogeneous inner structure. These discrepancies were explained by Roetzel et al. [348] as an inconsistency in the early experiments with determination of the thermophysical properties independently from the relaxation time measurements. Roetzel et al. obtained all parameters simultaneously from the single experiment. They confirmed the non-Fourier character of heat transfer but the reported smaller values of the relaxation times (2.26 s for sand, 1.77 s for processed meat). The later experiments by Antaki [349] give a value of 2 s for the processed meat. Sudár et al. [269] analysed the experiments by Tang et al. [350] and by Jaunich et al. [351] and emphasized that one must separate the heat sources from the heat conduction effects as much as possible. Although volumetric heat generation is inevitable in many practical situations and also has a significant influence in medical diagnostics, it could dominate the time evolution of the temperature field. Thus, the experiments aiming to measure biological material’s thermal properties should be designed to let the heat conduction be the dominant heat transport mechanism. For observation of the temperature oscillation in the living tissue [352] see, e.g., [353] and the review [354]. The errors in the predicted temperature distribution in the case of the cryosurgery and the cryopreservation can manifest themselves in the thermal stress distribution [355–357] and estimation of the tissue fracture [358, 359] – the Monte Carlo method; – the lattice Boltzmann method [322, 323]; – various deterministic discretization-based methods. The Monte Carlo method is expensive for practical engineering application; the lattice Boltzmann method has only been used for simple 2D structures, the deterministic discretization-based methods [324, 325] require a large number of solid angles, which is computational challenging.

12

1 Introduction

due to the large the volumetric expansion [360]. The mechanical waves caused by the thermal expansion were also observed in the pure argon films that were suddenly heated [218]. The investigation of the thermal responses of the biological tissues is required to be ensure of the patients’ safety during the hyperthermia and cryotherapy. The study of skin biothermomechanics is also essential for military and space operations to provide astronauts and army personnel as well as firefighters with clothes for thermal protection. Yu et al. [361] used the dynamic low-frequency impedance (DLFI) method to register the biological tissue subjected to a minimally invasive therapy probe instantly switched between strong freezing and heating. To analyse the heat transfer in the biological tissue in vivo one should account for the heat transfer by the arterial and venous bloods [228, 362–365]. The continuum models of the microvascular heat transfer are derived with intention to average the effects of a large number of the blood vessels present in the region of interest [366–369]. The best-known and certainly the most important continuum model was suggested by H.H. Pennes in 1948 [370] and referred to as the Pennes equation or “bioheat” equation (sometimes this model is also called “heat and sink model” [371]) c

∂T = ∇ · λ∇T + Q p , ∂t

Q p = cb ωb (Ta − T ) + q˙met + Q ext ,

(1.5) (1.6)

where T , , c and λ are the temperature, the density, the specific heat and the thermal conductivity of the tissue as the homogeneous medium, ωb is the blood perfusion rate, cb is the blood specific heat, Ta is the temperature of the arterial blood, q˙met and Q ext are the heat sources due to the metabolic reactions (the metabolic reactions usually could be neglected in the cryobiology problems) and the external source of energy. An obvious extension leading to the nonlinear (“modified”) Pennes equation is to account for the temperature dependence of the blood perfusion rate ωb = ωb0 + ωb1 T [372]. The different models of the bioheat equations have been presented where the vascular structures of the tissues have been supposed to be uniformly distributed in order to consider the physical model as a uniform porous medium [373–378]. For example, Xuan and Roetzel [379] introduced a two-equation bioheat model that considers the heat transfer in the porous media called the local thermal non-equilibrium equations (LTNEs). They modelled the biological tissue by dividing it into two different regions: the tissue region (muscle, vascular tissues and other solid compounds—the extravascular region) and the blood region (the vascular region), without considering the local thermal equilibrium between the two media and introducing an equivalent effective thermal conductivity in the energy equations of blood and tissue. They also proposed an interfacial convective heat transfer term instead of the perfusion one.

1 Introduction

13

Yuan [375] found that the equivalent heat transfer coefficient between tissue and blood in a porous model is inversely related to the blood vessel diameter. The coupled differential equations for the conservation of energy in the tissue and blood are formulated as follows: • for the tissue phase (1 − ε)(ρC p )t

∂ Tt = (1 − ε)λt ∇ 2 Tt + ha(Tb − Tt ) + (1 − ε)Q t , ∂t

• for the blood phase  ε(ρC p )b

∂ Tb + ub · ∇Tb ∂t

 = ελt ∇ 2 Tb − ha(Tb − Tt ) + ε Q b

where Tt and Tb are temperatures averaged over the tissue and blood volumes, ε is the porosity, h is the heat transfer coefficient, u b is the blood velocity vector, a is the volumetric transfer area between tissue and blood, and Q is the absorbed power density. Later Roetzel and Xuan [380] developed a three-temperature model by dividing the biological tissue into the artery, vein and tissue regions and investigated the transient heat transfer process between the tissue, artery, and vein in a cylinder biological model. Nakayama and Kuwahara [374] developed a generalized three-equation bioheat models for vascular and extravascular space under the local thermal non-equilibrium condition and they incorporated the blood perfusion term within the two sub-volume equations. They considered the effect of heat transfer in the closely spaced countercurrent artery-vein pair. Bazett et al. [381] found that the axial temperature gradient in the limb artery of the human under a low ambient temperature is an order of magnitude higher than that of in the normal ambient condition due to the effect of countercurrent heat exchange in the bioheat transfer. The countercurrent heat exchange also reduces the heat loss from the extremity to the surroundings [382]. The three equations are derived for the arterial blood phase, the venous blood phase and the tissue phase with three different temperatures. If the two temperatures are eliminated, the equation for the single (tissue) temperature could be derived that contains the derivatives of the temperature up to the sixth order in space (eliminating of the tissue or the blood temperature from the two-temperature model leads to the bioheat transfer equation where the blood temperature or tissue temperature is a sole unknown variable that contains temperature derivatives up to the fourth order) [383, 384]. Thus, the non-local DPL bioheat transfer model and the two- and three-temperature bioheat transfer equation model have the same effect on the heat transport. One may obtain that the heat transfer processes with N energy carriers depend upon the high orders of the lagging times and the non-local characteristic length [384]. The deviations from the Fourier law are also observed for the low-dimensional objects (i.e. the spatially constrained systems [385]) such as the thin films, the carbon,

14

1 Introduction

boron-nitride, bismuth [386], silicon (both crystalline and with amorphous surface [387–391]), silicon-germanium (single crystalline [392] and core–shell [393]) and indium arsenide (both with wurtzite and zinc blende phases) nanotubes and nanowires [394–396], including InGaAs/GaA and GaN/a-Si core–shell structures [397], the graphene nanoribbons (GNRs, the flat monolayer of the sp 2 -hybridized carbon atoms tightly packed into the 2D honeycomb lattice [398–400]) [87, 307, 401–407] as well as the various graphene allotropes such as, for example, α-, β-, δ- and γ graphynes [408] and the derivatives of the graphene such as the graphane [409], the fluorographene [410] and the diamane [411], the trilayer graphene [412], the polymer chains [413, 414]; in the nanotubes the violation of the Fourier was observed even when the phonon mean free path was much shorter than the sample length as showed the experiments by Chang et al. [394] with the individual multiwalled carbon and boron-nitride nanotubes and by the non-equilibrium molecular dynamics simulations by Yang et al. [415]. The low-dimensional materials demonstrate the size effect [156] (“the Casimir size effect” [82])12 —the reducing the thermal conductivity with reducing the size of the sample. For example, the thermal conductivity of the crystalline nanowires (NWs) is significantly lower than the bulk value and decreases with the wire diameter [204]— the thermal conductivity of the SiNW (one of the most promising nanomaterials due to the ideal interface compatibility with the conventional Si-base technology) is about two orders of magnitude smaller that that of bulk Si [163]. The reason for such behaviour of the thermal conductivity is the increase of the roughening [420]. This model combines the incoherent surface scattering of the short-wavelength phonon with the nearly ballistic behaviour of the long-wavelength phonons. The thermal conductivity of the thinnest possible Si NWs reaches a superhigh level that is as large as more than an order of magnitude higher than its bulk counterpart and shows the non-monotonic diameter dependence [421]. The abnormality is explained in terms of the dominant normal process (energy and momentum conservation) of the low-frequency acoustic phonons that induces the hydrodynamic phonon flow in the Si NWs without being scattered. With diameter increasing, the downward shift of the optical phonons triggers the strong Umklapp scattering with the acoustic phonons. The two competing mechanisms result in the non-monotonic diameter dependence of the thermal conductivity with minima at the critical diameter of 2–3 nm. Since the mean free path (MFP) is inversely proportional to a certain power of the phonon frequency, no matter what the distance scale one has for the temperature variation, there are always the low-frequency phonons with the MFP greater than the linear scale, the non-local theory is actually needed in the analysis of all aspects of 12

The classical size effects also happen when the size of the heat source becomes comparable to or smaller than the MFP of the materials, even without the interface or the boundary scattering [416, 417]. This situation may arise, for example, near the drain of a MOSFET (metal-oxide semiconductor field effect transistor), where most of the heat is generated [82]. The size effect is usually less important in amorphous materials since the mean free path of phonons is short [418], and the opposite trend is observed in polyethylene chains because of the reduction of the chain–chain anharmonic scattering [419].

1 Introduction

15

the heat transport by phonons in the semiconductor and dielectric crystals, especially when the low-frequency phonons have a considerable contribution to the process [287]. For energy transport across a thin film or in a multilayer structure the effect of the thermal boundary resistance (TBR) becomes significant [422–424]; moreover, due to the wave–particle duality in some cases the electron wave or the phonon wave effects should be also considered [144]. The thermal conductivity of the superlattices (SLs) is significantly reduced compared to the bulk values of the corresponding alloys [425–427]and non-monotonously depends on the superlattice period thickness [180, 428, 429]. The reduced thermal conductivity of the superlattices is beneficial for applications such as the thermoelectrics [393, 430–432]. The coherent phonon heat conduction has been confirmed experimentally in the superlattice structures. Such travelling coherent phonon waves in the superlattices lead to increase of the thermal conductivity as the number of periods increases. For applications such as the thermal insulation or the thermoelectrics, minimization of the phonon coherent effect is desirable. It is found that either aperiodic SLs [433] or SLs with the rough interfaces can significantly disrupt the coherent heat conduction when the interface densities are high. As the size further shrinks, a nanowire becomes a molecular chain and a thin film becomes a molecular sheet. The famous experiment of Fermi, Pasta and Ulam started the study of the thermal conductivity in the long chains of the interacting particles that showed that thermal conductivity can diverge with the chain length (the thermal conductivity scales with a positive power of the system size in 1D and shows a logarithmic divergence in 2D [434]13 ) in the case of the so-called integrable systems (the FPU lattice, the disordered harmonic chain, the diatomic 1D gas, the diatomic Toda lattice).

13

The behaviour of one-dimensional chain is more complex, if both longitudinal and transverse motions are present, as, for example, in polymeric chain [435] that has flexibility—the divergence could be logarithmic, 1/3 power-law or 2/5 power-law, depending on the strength of the longitudinal and transverse motions coupling. The phononic thermal conduction in quasi-one-dimensional system might not be the same as that in 1D lattice models, because the atoms in the quasi-one-dimensional nanostructures can vibrate in the three-dimensional real space; however, anomalous thermal conduction behaviour has also been observed in numerical simulations for carbon nanotubes, nanowires and polymer chains [11]. It is interesting to note that electrical and thermal conductivities show very different behaviour [436]: for example, the nanotube can be metallic or semiconducting depending on its chirality while the thermal conductivity does not depend on the chirality; on the other hand, adding 14C isotope impurity decreases the thermal conductivity and has no effect on the electronic properties. The heat transport in chain also depends on the degree of its coupling to the environment—the reservoirs that supplies the heat; Velizhanin et al. [437] distinguish three regimes • weak coupling: energy input from the reservoir limits the heat flow; • strong coupling: the lattice dynamics are distorted by the presence of the reservoir; • intermediate coupling: thermal transport is determined by the intrinsic chain parameters. The discrete chain system with long-range interactions could be transformed into continuous medium models with fractional derivatives [438].

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Moreover, the heat transport through the chain depends on the boundary conditions and variation of these conditions could change the disordered chain from a heat superconductor to a heat insulator [439]. Casati and Mejia-Monasterio [440] showed that while the absence of the total momentum conservation is necessary for the emergence of the Fourier law in the dynamical system of the interacting particles, the exponential instability (Lyapunov chaos) is not needed for the establishment of the Fourier law. Hence, the Fourier law can be derived from the laws of quantum mechanics since the main feature of the quantum motion is the lack of the exponential dynamic instability [441]. These problems as well as their connection to the extremely high thermal conductivity of the carbon and boron nitride nanotubes [394, 395] and the graphene [284, 399, 442, 443], borophene [444], phosphorene [445, 446], silicene [447] sheets or hybrid graphene/silicene monolayers [448] and diamond-like bilayer graphene [449] as well as the strain effects on the heat transfer in nanostructures [449–453] are not considered in this book—these topics are examined in detail in the reviews • S. Lepri, R. Livi, A. Politi, Thermal conduction in classical low-dimensional lattices, Phys. Rep. 2003, v. 377, p. 1–80; • S. Liu, X. F. Xu, R. G. Xie, G. Zhang, B. W. Li, Anomalous heat conduction and anomalous diffusion in low dimensional nanoscale systems, Eur. Phys. J. B 85, 337 (2012) and • A monograph S. Lepri (ed.), Thermal Transport in Low Dimensions: from Statistical Physics to Nanoscale Heat Transfer, Lecture Notes in Physics 921, Springer, 2016. The book also does not consider the problems of the heat transport across interfaces (e.g. the diffuse mismatch model [37, 454]) that has an essential role in the nanodevices and cooling of the electronic circuits and such topics as the temperature jump boundary condition or the thermal boundary resistance [37, 424, 427, 455–458] or the heat transfer via convection or radiation [459].

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432. Qiu, B., Tian, Z., Chen, G.: Effects of aperiodicity and roughness on coherent heat conduction in superlattices. Nanosc. Microsc. Thermophys. Eng. 19, 272–278 (2015) 433. Hu, R., Iwamoto, S., Feng, L., Ju, S., Hu, S., Ohnishi, M., Nagai, N., Hirakawa, K., Shiomi, J.: Machine-learning-optimized aperiodic superlattice minimizes coherent phonon heat conduction. Phys. Rev. X 10, 021050 (2020) 434. Grassberger, P., Yang, L.: Heat conduction in low dimensions: from Fermi-Pasta-Ulam chains to single-walls nanotubes. arXiv: cond-mat/020424 (2002) 435. Wang, J.S., Li, B.: Intringiing heat conduction of polymer chain. arXiv: cond-mat/0308445 [cond.mat.stat-mech] (2003) 436. Zhang, G., Li, B.: Thermal conductivity of nanotubes revisited: effects of chirality, isotope impurity, tube length, and temperature. arXiv:cond-mat/0501194 [cond-mat.mtrl-sci] (2006) 437. Velizhanin, K.A., Sahu, S., Chien, S.S., Dubi, Y., Zwolak, M.: Crossover behavior of the thermal conductance and Kramer’s transition state theory. Sci. Rep. 5, 17506 (2015) 438. Tarasov, V.E.: Continuous limit of discrete chain system with long-range interaction. J. Phys. A 39, 14895–14910 (2006) 439. Lepri, S., Livi, R., Politi, A.: Anomalous heat transport. In: Klages, R., Radons, G., Sokolov, I.M. (eds.) Anomalous Transport. Foundations and Applications, pp. 293–325. Wiley-VCH (2008) 440. Casati, G., Mejia-Monasterio, C.: Classical and quantum chaos and control of heat flow. arXiv:cond-mat/0610269 (2006) 441. Casati, G., Chirikov, B.I., Guarneri, I., Shepelynski, D.L.: Dynamical stability of quantum “chaotic⣞ motion in a hydrogen atom. Phys. Rev. Lett. 56, 2437–2440 (1986) 442. Nika, D.L., Balandin, A.A.: Two-dimensional phonon transport in graphene. J. Phys.: Condens. Matter. 24, 233203 (2012) 443. Sadeghi, M.M., Pettes, M.T., Shi, L.: Thermal transport in graphene. Solid State Commun. 152, 1321–1330 (2012) 444. He, J., Ouyang, Y., Yu, C., Jiang, P., Pen, W., Chen, J.: Lattice thermal conductivity of β12 and χ3 borophene. Chin. Phys. B 29, 126503 (2020) 445. Qin, G., Yan, Q.B., Qin, Z., Yue, S.Y., Hu, M., Su, G.: Anisotropic intrinsic lattice thermal conductivity of phosphorene from first principles. Phys. Chem. Chem. Phys. 17, 4858 (2015) 446. Jain, A., McGaughey, A.J.: Strongly anisotropic in-plane thermal transport in single-layer black phosphorene. Sci. Rep. 5, 8501 (2015) 447. Fleurence, A., Friedlein, R., Ozaki, T., Wang, Y., Yamada-Takamura, Y.: Experimental evidence for epitaxial silicene on diboride thin films. Phys. Rev. Lett. 108, 245501 (2012) 448. Liu, B., Banimova, J.A., Reddy, C.D., Dmitriev, S.V., Law, W.K., Feng, X.Q., Zhou, K.: Interface thermal conductance and rectificatio in hybrid graphene/silicene monolayer. Carbon 79, 236–244 (2014) 449. Hu, X., Li, D., Yin, Y., Li, S., Ding, G., Zhou, H., Zhang, G.: The important role of strain on phonon hydrodynamics in diamond-like bi-layer graphene. Nanotechnology 31, 335711 (2020) 450. Abramson, A.R., Tien, C.L., Majumdar, A.: Interface and strain effects on the thermal conductivity of heterostructures: a molecular dynamics study. J. Heat Transf. 124, 963–970 (2002) 451. Li, X.B., Maute, K., Dunn, M.L., Yang, R.G.: Strain effects on the thermal conductivity of nanostructures. Phys. Rev. B 81, 245318 (2010) 452. Hu, M., Zhang, X.L., Poulikakos, D.: Anomalous thermal respose of silicene thermal to uniaxial stretching. Phys. Rev. B 87, 195417 (2013) 453. Zhang, G., Zhang, Y.W.: Strain effects on thermoelectric properties of two-dimensional materials. Mech. Mater. 91, 382–398 (2015) 454. Song, Q., Chen, G.: Evaluation of the diffuse mismatch model for phonon scattering at disordered interfaces. Phys. Rev. B 104, 085310 (2021) 455. Khater, A., Szeftel, J.: Theory of Kapitza resistance. Phys. Rev. B 35, 6749–6755 (1987) 456. Chen, W., Yang, J., Wei, Z., Liu, C., Bi, K., Xu, D., Li, D., Chen, Y.: Effects of interfacial roughness on phonon transport in bilayer silicon thin films. Phys. Rev. B 92, 134113 (2015)

References

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457. Li, D., McGaughey, A.J.H.: Phonon dynamics at surfaces and interfaces and its implications in energy transport in nanostructured materials. An opinion paper. Nanosc. Microsc. Thermophys. Eng. 19, 166–182 (2015) 458. Ran, X., Guo, Y., Wang, M.: Interfacial phonon transport with frequency-dependent transmissivity by Monte Carlo simulation. Int. J. Heat Mass Transf. 123, 616–628 (2018) 459. Zhmakin, A.I.: Modelling of mass and heat transfer in bulk single crystal growth. LAM Lambert Academic Publishing (2020)

Part I

Classical Transport

Chapter 2

Phase-Lag Models

A number of the non-Fourier models have been developed that are based on the modification of the constitutive relation between the heat flux and the temperature gradient1 for media with thermal memory or/and with a discrete structure [3]. Most of these models incorporate the time non-locality for materials with memory [4, 5] (the hereditary materials, including the fractional hereditary materials [6]), some are also considered as the space non-locality, i.e. materials with the non-homogeneous inner structure. Sobolev [7] suggested to distinguish the strong non-locality (the non-local effects in the transport processes are described in the integral form of the balance equations) and the weak non-locality (the parameters related to the internal characteristic time and space scales are introduced into the differential balance equations). Takahashi [8] noted that the space non-locality is related to the introduction of the scale intermediate between the microscale and the nanoscale—the mesoscale. Studying the non-local phenomena helps researchers to capture the microscopic effects at the macroscopic level with greater accuracy. The non-Fourier models must be compatible with the kinetic theory of gases—the rarefied gases have the best understood microscopic composition among continua (the macroscopic equations from the moment series expansion of the kinetic theory are instructive in this respect)—and with the second law of the thermodynamics

1

However, Razi-Naqvi and Waldenstrom [1] suggested a non-Fourier heat conduction equation ∂T = λ(1 − e−t/τ )∇ 2 T ∂t

termed “the new heat equation” (NHE) that is essentially based on the Fourier constitutive relation q = −λeff ∇T with the time-dependent thermal conductivity λeff = λ(1 − e−t/τ ). In the long-time limit the NHE reduces to the Fourier equation. It is stated that the NHE overcomes the artificial wavefront observed in some models [2]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. I. Zhmakin, Non-Fourier Heat Conduction, https://doi.org/10.1007/978-3-031-25973-9_2

39

40

2 Phase-Lag Models

and the spacetime requirements of the non-equilibrium thermodynamics that are universal, independent of the material composition and properties, and therefore their consequences are universal as well [9]. The thermophysical fields and the constitutive models should be frame-indifferent (sometimes this property is called objectivity) [10–13]. One should remember that strictly speaking, a temperature has a well-established meaning only in global equilibrium when the heat flux is zero [14]—a temperature at a point can be defined only under the local thermodynamic equilibrium that is valid only for a relatively weak deviation from equilibrium, thus a meaningful temperature can be defined only at points separated on an average by the mean free path of the energy carries [15, 16]. The meaning of temperature is related with the heat transport [14, 17]. Different degrees of freedom, far from equilibrium, may be associated with the different “temperatures”. Jou and Cimmelli [18] stated the question: how do the gradients of these different temperatures contribute to heat transport? The nonlinearity in the heat conduction equation is expressed in two ways—as the temperature dependence of the material thermophysical properties and as the presence of the nonlinear products of the temperature gradient or the heat flux [19, 20]. The constitutive relation (1.1) could be formulated in the integral form [21, 22] (“the integrated history of the temperature gradient” [2]) similar to the theory of the nonlinear materials (e.g. the viscoelastic materials [23]) in which the stress at the point depends on the deformation gradients at the previous instants [24–26]. A general expression for the heat flux in the nonlinear materials with memory as the convolution integral was written by Gurtin and Pipkin half a century ago in analogy with the stress–strain relationship of the viscoelastic materials as [27] (see also the paper by Coleman and Gurtin [28]) t q=−

Q(t − t  )∇T dt 

(2.1)

−∞

where Q(s) is a positive decreasing function (called the relaxation kernel of the Jeffreys’s type [29–32] after H. Jeffrey who used similar relations for the rheological description of the earth core [33], also called the memory kernel) tending to zero as s → ∞. This is an example of the strong time non-locality in terms of Sobolev [7]. Equation (2.1) is a particular case of the relation between the flux J(t) at a given time t and the conjugated thermodynamic force X(t  ) at the previous times t  [34] t J= −∞

K (t − t  ) · X(t  )dt  .

2 Phase-Lag Models

41

In the classical transport theory the fluxes J and the thermodynamics forces X are related by the linear law J (t) = Lˆ · X(t) where Lˆ denotes suitable phenomenological coefficients. Different choices of the memory kernel lead to different constitutive relations and thus different heat conduction equations. For example, if the memory kennel is chosen as the Mittag–Leffler function, a fractional-order constitutive equation will occur [35]. The condition that the relaxation kernel Q(t − t  ) is the decreasing function means the decrease of the relevance of the older temperature gradients in comparison with the newer ones, thus the term “fading memory” [36] is used. Herrera [37] has recently considered the heat conduction that contravenes the “fading memory” paradigm with the relaxation kernel of the form   t−t  Q(t − t  ) = c 1 − e− τ . Evidently, in this case the older temperature gradients have more influence on the heat flux than the present ones and the most impotent contribution to q is from the remote past (t  → −∞). Differentiation of Eq. (2.1) with this kernel that represents the extreme case of the violation of the fading memory paradigm gives the equation ∂q κ =− 2 ∂t τ

t

e−

t−t  τ

∇T (x, t  )dt 

(2.2)

∇T (x, t  )dt  .

(2.3)

−∞

that could be transformed to τ

∂q κ +q =− ∂t τ

t −∞

Combining Eq. (2.3) with the energy conservation law produces ∂2 T κ 1 ∂T = 2 + ∂t 2 τ ∂t τ cV

t

∇ 2 T dt 

−∞

and after differentiation the third-order equation 1 ∂2 T κ ∂3 T + = 2 ∇ 2 T. 3 2 ∂t τ ∂t τ cV

42

2 Phase-Lag Models

The author considered two problems where such kernel is relevant: 1. The thermohaline convection that is observed when a layer of warm salt water is above a layer of fresh cold water and due to the cooling of the warm salt water the “salt fingers” appear in the freshwater layer. Instabilities of this kind can also occur in stars [38]. 2. The secular stability of the nuclear burning when the hydrogen falling onto the surface of a neutron star in a close binary system undergoes nuclear fusion, independently on how low the temperature may be. The nuclear instability appears whenever the characteristic time for the increasing of the thermal energy generated by the nuclear burning is smaller than the time required for the removal of this energy. Note that selection of the integral domain (∞, t] instead of [0, t] means that the initial effects are implicitly neglected [37]. As Sobolev [39] noted when for the heat flux the constitutive relation in the integral form is used it is natural to apply the integral form of the energy conservation law for the non-local media    d ρcT dV = qdS + QdV. dt V

S

V

For Q(s) = λδ(s) where δ(s) is the one-sided Dirac delta function [29] 0+ δ(s)ds = 1 0

we get the Fourier law (1.1), i.e. the Fourier law corresponds to the “zero-memory” material. There are also a few other definitions related to the “memory” that are used in the literature [40]: • “Full memory” means that there is no fading, i.e. the relaxation kernel is constant and t q(t) = −λ ∇T (t  )dt  0

yielding the wave equation for the temperature ∂2 T = κT. ∂t 2 • “Short-tail memory” with the relaxation kernel

2.1 Maxwell–Cattaneo–Vernotte Equation

λ q(t) = − ζ

t 0

43

  t − t ∇T (t  )dt  exp − ζ

yielding the telegraph equation for the temperature ∂T ∂2 T + ζ 2 = κT. ∂t ∂t • “Long-tail memory”: such long-tail power kernel Q(t) ∝ t −α [41, 42] is realized via the fractional derivative and integral (see Appendix) q(t) = −λDaα+ ∇T (t), 0 < α ≤ 1, q(t) = −λJaα+ ∇T (t), 1 < α ≤ 2, and leads to the time-fractional heat conduction equation ∂αT = κT. ∂t α Different choices of the constitutive relation lead to numerous “lagging” models (see, for example, [43–46] and references therein).

2.1 Maxwell–Cattaneo–Vernotte Equation A well-known example of the constitutive heat flux equation of the Jeffreys type is [29, 47] ∂ ∂q + q = −λ∇T − τ λ1 ∇T. (2.4) τ ∂t ∂t For the case λ1 = 0, Eq. (2.4) reduces to the well-known Cattaneo (also called Maxwell2 –Cattaneo–Vernotte (MCV) [9, 49, 50] or Maxwell-Chester–Cattaneo– Vernotte [51]) constitutive relation that was independently formulated by Philip M. Morse and Hermann Feshbach in their famous monograph “Methods of Theoretical Physics” [52], by Harold Grad (1958), by Carlo Cattaneo (1958) [53] and by Pierre Vernotte (1958) [54, 55] (see also [56, 57]) τ 2

∂q + q = −λ∇T ∂t

(2.5)

As noted Müller [48], Maxwell indeed wrote the equation for the heat flux with a rate term derived in the kinetic theory of gases in his papers of 1867 and 1879, but he finally dismissed it as being small and uninteresting.

44

2 Phase-Lag Models

If additionally τ = 0 Eq. (2.4) reduces to the Fourier law. The term τ ∂q/∂t introduced in a reasonable (but arbitrary) manner is sometimes referred to as “thermal inertia” [58]. Coleman et al. [59, 60] extended the Cattaneo equation (2.5) to the anisotropic case ∂q + q = −λˆ ∇T τˆ ∂t where τˆ and λˆ are the non-singular second-order tensors that, as function of temperature, depend on the material under consideration. Sobolev [61] suggested to extend the Cattaneo equation with additional secondorder space derivatives ∂q τ + q = −λ∇T + le2 q ∂t where le is the space non-locality of the heat transfer process (correlation length), which in the case of metal is of the order of the mean free path of electrons; this equation is similar to the Guyer–Krumhansl equation. Note that similar relation is used to describe the viscoelastic effects [62] τ P˙ˆ ν + Pˆ ν = −2η Vˆ s where Pˆ ν is the viscous pressure tensor, Vˆ s is the symmetric part of the velocity gradient tensor ∇ V , and η is the shear viscosity. The Cattaneo constitutive relation (sometimes called “modification of the Fourier law” (MFL) [57]) is obtained if in the general Gurtin–Pipkin relation (2.1) the relaxation kernel is chosen as κ t−t  Q(s) = e− τ τ where κ = λ/(ρC) is the thermal diffusivity. Thus the Cattaneo equation for the temperature (also “the hyperbolic heat conduction equation” (HHCE) [57]) could be written (for constant material properties) as [63, 64] ∂T ∂2 T = κ∇ 2 T. (2.6) τ 2 + ∂t ∂t Earlier (in 1934) the similar equation for the diffusion ∂ 2n ∂n + τ 2 + = D∇ 2 n ∂t ∂t was derived by Boris Davydov [56] who suggested the phenomenological system for the particle density n(x, t) to account for the finite particle velocity during the molecular diffusion introducing an explicit time interval of the mean free path [65, 66]

2.1 Maxwell–Cattaneo–Vernotte Equation

∂q ∂n + = 0, ∂t ∂x

45

∂q q0 − q = ∂t τ

where q0 = −D(∂n/∂ x). Equation (2.6) in contrast to the parabolic Fourier equation is the hyperbolic differential equation and can be considered as a particular case of the telegrapher’s equation [67] (sometimes called the causal diffusion equation [68]). This equation describes the crossover between the ballistic motion and the onset of diffusion—the transition from the reversible to the irreversible behaviour—at the characteristic time τ: • for small times t  τ the first term in Eq. (2.6) is dominant and it reduces to the wave equation ∂2 T τ 2 = κ∇ 2 T ∂t that describes the reversible process since it is invariant with respect to the time inversion. Continuum mechanics and heat conduction are strongly related. Ballistic heat conduction is the benchmark where the continuum and kinetic theories can be compared and tested [9]. The ballistic propagation is predicted theoretically with kinetic theory for rarefied gases. It is attributed to the free propagation of the particles, when the average mean free path is larger than the characteristic length of the system. For the phonons the free propagation means propagation with the speed of sound. From a continuum point of view, it is the limit of the ideal elasticity; the ballistic propagation with the speed of sound is a thermoelastic effect in a continuum framework [9]. Based on phonon Boltzmann equation under the Callaway model, it was found that in the transient thermal grating (TTG) experiments of suspended silicon membrane, for short TTG periods compared to the phonon mean free path, the model yields a non-exponential or wavelike decay of grating amplitude with time than an exponential decay in the diffusive regime [69]. In contrast to the second-sound detection where the theoretical calculation of the experimental window condition [70–72] helped the authors to choose the optimal frequency for the excitation in order to find the lowest dissipation and making visible the wave phenomenon, such window condition for ballistic propagation does not exist. It is hard to find it experimentally, and it requires extremely pure crystals [73], mostly made from NaF. Moreover, the appearance of the ballistic signal is very sensitive for the temperature, probably due to the state dependence of material properties, including the thermal expansion coefficient [74, 75]. The ballistic and second-sound effects could appear both in rarefied and nanosystems under room temperature conditions [76]. In the kinetic theory the Knudsen number characterizes the rareness of the system. For instance, phonon hydrodynamics is applicable for processes with high Knudsen number K n > 0.01 [77]. In such a situation, the second sound and ballistic propagations become experimentally visible.

46

2 Phase-Lag Models

The proper coupling between the heat flux and thermal pressure requires to include the ballistic effects. In the case of rarefied gases the pressure is the complete mechanical pressure. Thus the generalization of the Navier–Stokes–Fourier system is needed with separation of the deviatoric and spherical parts of the pressure tensor [78–81]. Both have a time evolution equation with different relaxation times; the compressibility becomes significant, and hence the spherical part could possess a different timescale than the deviatoric one [79, 82, 83]. The ballistic contribution is visible at very low pressures, at the order of magnitude around 100 Pa, and observed as a change in the speed of sound respect to the mass density variation [84–86]. Therefore the mass density dependence of the material parameters (the thermal conductivity, shear and bulk viscosities, relaxation times etc.) must be accounted for. The observation of the ballistic phonon propagation at room temperature is possible in the nanostructures as was convincingly demonstrated in experiments reported by Siemens et al. [87], Hoogeboom-Pot et al. [88] and by Lee et al. [89]. The solution of Eq. (2.6) for the initial condition T (x, 0) = δ(x) is written as  T (x, t) = δ

 κ t−x , τ

√ i.e. the initial pulse propagates with the velocity κ/τ without changing the shape of the distribution [66]. • for large time t τ the first term could be neglected, and one gets the parabolic equation of the irreversible process—the heat diffusion. Thus, the hyperbolic nature of Eq. (2.6) is the most significant at short times τ ≈ t. From the physical point of view, this corresponds to an initial condition when all particles move in the same direction; after the characteristic time τ due to the effect of the randomization of the particle motion Eq. (2.6) reduces to the Fourier heat conduction equation [57]. The value of τ can be considered as the characteristic time for the crossover from the ballistic motion to diffusion [34] or as the Lyapunov time beyond which predictability is lost [90]. Jou et al. [34] suggested to describe the transition to the ballistic regime introducing the generalized conductivity λ(T, l/L) in such a way that for all regimes   l T q = λ T, L L with limiting values 

l λ T, L

 → λ(T ) f or

l → 0, L

2.1 Maxwell–Cattaneo–Vernotte Equation

47

  λ(T ) l l → ≡ (T ) f or λ T, L a L

l →∞ L

where a is constant depending on the system. The dynamic properties of Eq. (2.6) are analysed under assumption that the solution has the form T (x, t) = t0 exp[i(kx − ωt)] where k is a complex wavenumber and ω is a real frequency. The phase speed v ph and the attenuation factor α are given as √ 2κω 2κ v ph =  , α= . √ v ph τ ω + 1 + τ 2 ω2 In the high-frequency limit the phase velocity remains finite for the nonzero values of the relaxation time τ  κ v ph∞ = U = . τ The value of v ph∞ diverges when τ → 0 leading to the infinite speed of propagation. The velocity U is called the second sound and is a damped temperature wave, while the first sound is the pressure wave [91]. Cimmelli et al. [92] have shown that in the framework of the weakly non-local extended thermodynamics [93] a small heat pulse will travel with different velocity in the direction of the non-vanishing heat flow than in the opposite direction—the heat pulse propagating in a body conducting heat will travel more slowly in the direction of heat flow than in the opposite direction. Sellitto et al. [94] studied the second sound at the nanoscale using the model based on the dynamical non-equilibrium temperature that takes into account the effects of relaxation and non-locality. The authors examine the propagation of thermal waves in the heat conducting rigid solids and showed that non-locality alters the speed at which second sound propagates. The authors used as the reference state value of the speed of propagation of thermal signals in a rigid solid under the special case of the Cattaneo law  κ0 c0 = ρ0 cV0 τ0 and introduced the parameter λC λC =

1 L



κ0 τ0 τ0 cV0 , = ρ0 cV0 L

where L is the positive constant that denotes a length scale characteristic of the domain of propagation. The parameter λC plays the role of the Knudsen number, 0 < λC < 1. Evidently, λC ∝ O(1) (i.e. small L) corresponds to nanoscale and 0 < λC  1 (large L) corresponds to macroscale of the heat transport. Sellitto et al. estimated λC to be λC ≈ 6.2 nm/L in the case of gallium arsenide.

48

2 Phase-Lag Models

The authors also stressed that the Clausius version of the second law, which states that heat cannot spontaneously flow from cold to hot without external work being performed, does not strictly apply to small systems, such as the individual nano [95], and even micron-sized particles [96]. Chen et al. [97] using both theoretical analysis and numerical simulation based on the Cattaneo–Vernotte (CV) model demonstrated that the thermal conduction could be controlled in a one-dimensional periodical structure, named thermal wave crystal. The Cattaneo equation (2.6) can be derived in different ways: • from the Boltzmann transport equation under the relaxation time approximation; • in the framework of the extended irreversible thermodynamics (EIT). The basic assumption of the relaxation time approximation written as (in onedimensional case) ∂f f0 − f ∂f + vx = (2.7) ∂t ∂x τ is that the distribution function f is close to the equilibrium distribution function f 0 , i.e. one can assume that ∂ f0 ∂ f0 ∂ T ∂f ≈ = . ∂x ∂x ∂T ∂x Multiplying Eq. (2.7) by τ εvx (ε is the particle energy) and integrating over the momentum space give Eq. (2.6). In the EIT the dissipative fluxes such as the heat flux are considered as the basic independent variables [98, 99]. The classical irreversible thermodynamics, rational extended thermodynamics (RET) [100, 101], extended irreversible thermodynamics and thermodynamics with internal variables (NET-IV) [102, 103] are the special theories of the non-equilibrium thermodynamics [104, 105]. The most important difference between the rational extended thermodynamics and the extended irreversible thermodynamics is that the former assumes that the state space is strictly local, while the latter allows a non-local state space [106]. The kinetic theory-based RET builds its heat conduction models on phonon interactions, thus the transport coefficients can be calculated while the continuum formulation of NET-IV leaves the particular transport mechanism aside, and these coefficients can be determined through a fitting procedure [107]. Similarly to the CIT, NET-IV uses the balances of mass, linear momentum, angular momentum and internal energy as the constraints to evaluate the entropy production rate density from the balance of entropy, whose positive semidefiniteness is ensured via the Onsagerian force-flux relations. CIT is based on the hypothesis of local equilibrium. The NET-IV assumes that the state space is extended by one or more internal variables. In contrast to the EIT, they are not given any physical interpretation. Usually [108] it is assumed that entropy is shifted by a concave expression of the internal variables; the simplest choice is a quadratic function. If ξk , where k = 1, . . . , M are M internal variables with arbitrary tensorial orders and characters, the extended specific entropy is given as [109]

2.1 Maxwell–Cattaneo–Vernotte Equation

s˜ = s eq −

49 M

mk k

2

 · ξk · ξk

where m k are the coefficient functions of the appropriate tensorial orders and dot denotes the full contraction of tensors of an arbitrary order. In the framework of The NET-IV it is also possible to generalize the relationship between the heat current density q and the entropy current density J and instead of the usual expression J = 1/T q assume that  J=

 M 1 ˆ ˆ Ck · ξk I +C ·q + T k

where Cˆ and Ck are constitutive tensors of the appropriate orders called Nyíri multipliers. It should be mentioned that there is a close system—GENERIC—general equation for non-equilibrium reversible-irreversible coupling [109–115]. It is based on an abstract mesoscopic time evolution equation collecting the structure that guarantees its physical pertinence, also called a metriplectic time evolution equation in order to emphasize the presence of both the symplectic and the Riemannian (metric) geometry x˙ =

δ(x, x  ) δL . + δx δx  x  = δS δx

where x a mesoscopic state variable (e.g. the particle distribution function in kinetic theory or the hydrodynamic fields in fluid dynamics), x˙ is the time derivative of x, the first term on the right-hand side represents mechanics (symplectic geometry), and the second term is the thermodynamical (Riemannian geometry) contribution appearing due to the mesoscopic nature of the state variable x. The GENERIC equation provides a framework (a scaffold) for constructing mesoscopic dynamical models [110]. The basic idea of the GENERIC is the separation of the reversible and the irreversible contributions to dynamics. The object of the geometric mechanics is the description of dynamics without dissipation, and Hamiltonian mechanics gives a canonical description—the Hamiltonian plays the role of the total energy, i.e. the integral of the sum of the kinetic and internal energy densities. GENERIC is suitable for the multilevel description of processes [109, 116]. Heat current density is frequently identified with a vectorial internal variable; however, Szücs et al. [109] had shown that identification of heat current density with the conjugate of the vectorial internal variable is more advantageous. Physically new variables differ significantly from the classical ones that obey the conservation laws and changes relatively slowly in the system evolution—the fluxes generally do not obey the conservation laws and are the relatively “fast” variables [117].

50

2 Phase-Lag Models

The local or non-local models of the heat transport could be derived, depending on whether the gradients of the fluxes are included in the state space, or not [118]. Thus, an entropy in EIT depends on the internal energy and the heat flux s = s(u, q) and obeys the evolution equation ∂s + ∇ · J s = σ s ≥ 0, ∂t where J s is the entropy flux and σ s is the entropy production rate. The definition of the non-equilibrium temperature as T −1 = ∂s/∂u and an assumption that ∂s/∂q = −αT , where α is a material coefficient, lead (with the energy balance for the rigid conductor at rest—substantial time derivatives are equal with partial time derivatives in case of rigid heat conductors) to the equation   q ∂q ∂s −1 = −∇ · + q · ∇T − α , ∂t T ∂t and thus

  ∂q −1 . σ = q · ∇T − α ∂t s

The simplest way to provide the positiveness of the entropy production rate σ s is to assume a linear relation between the heat flux and the thermodynamic force in the parentheses ∂q = μq, ∇T −1 − α ∂t where μ is a positive coefficient. By introduction of the notation α = τ, μ

1 = λ, μT 2

one recovers the Cattaneo constitutive relation (2.5). Cimmelli et al. [20] derived the Cattaneo relation (2.5) using the dynamical semiempirical temperature β [92, 119]. The dynamical semiempirical temperature could be considered as the internal state variable [120] in the classical irreversible thermodynamics [121, 122] (the name “internal variables” denotes parameters which, when are in equilibrium, reduce to a function of the standard variables, such as, for example, temperature [92]). The dynamical temperature differs from the thermodynamic non-equilibrium temperature by a frictional term which is responsible of the finite speed of propagation of the thermal disturbances [123].

2.1 Maxwell–Cattaneo–Vernotte Equation

51

It is assumed that the heat flux is given by the relation qi = −λ

∂β ∂ xi

(2.8)

and the evolution equation for β is written as β˙ = f () where  denotes the state space. In the case it includes θ and β, the evolution equation for β is rewritten as F(θ, β) (2.9) β˙ = − τR where F is a suitable smooth function, θ is the non-equilibrium absolute temperature, and τ R is a temperature-dependent relaxation time related to the resistive processes of interaction among the heat carriers [106]. The function F must fulfil the conditions ∂F ≥ 0, ∂θ

∂F ≤0 ∂β

that ensure that [123] 1. In the thermodynamic equilibrium when θ equals to T [124] β is a regular function of T . 2. The solutions of Eq. (2.9) are stable. The simplest case these conditions are met if β˙ = −

1 (β − θ ). τR

(2.10)

If λ and τ R are constant, Eqs. (2.8) and (2.10) yield the Cattaneo relation (2.5). The dynamical temperature follows after θ with a certain delay, which is controlled by the relaxation time τ [125]. The dispersion relation of the heat waves along the core-shell nanowires showed no difference between the different definitions of nonequilibrium temperature—the absolute non-equilibrium temperature and a dynamical non-equilibrium temperature [126]. In the EIT the dissipative fluxes are considered as the independent thermodynamic variables, thus, accounting for the fact that the specific internal energy e is the function of the thermodynamic local equilibrium temperature T , the state space includes the heat flux  = {e, β, β,i }.

52

2 Phase-Lag Models

Carlomango et al. [123] postulated the following evolution equation for β β˙ =

e β A − + β,i β, j σ τ 2

where σ, τ and A are regular scalar function of e. For example, if τ (e) = τ R (θ ) and σ = CV τR 1 A β˙ = − (β − θ ) + β,i β, j . (2.11) τR 2 From Eq. (2.11) follows that even in the steady state θ and β are related by the partial differential equation [123]. When the temperature dependence of the thermal conductivity λ = λ(θ ), the heat capacity cV = cV (θ ) and the relaxation time τ R = τ R (θ ) are taken into account, the constitutive relation is expressed as  

τR λ τ R  ∂θ ˙ θ = −λ 1 − β˙ τ R q˙i + qi 1 − λ θ θ ∂ xi

(2.12)

instead of the Maxwell–Cattaneo relation. The combination of the last equation with the local balance of specific internal energy e ∂qi e˙i = − ∂ xi yields (neglecting the third-order terms) the hyperbolic equation for the nonequilibrium temperature [20] 



∂cV cV λ τR  ∂ 2θ − θ˙ 2 = λ 1 − β˙ (2.13) ∂θ λ θ θ ∂ xi2

  ∂θ τ R ∂λ ∂ θ˙ 2λ ∂τ R 2 ∂τ R ∂λ ∂θ − − + . qi − qi λ ∂θ ∂ xi τ R ∂θ ∂θ ∂ xi τ R ∂θ ∂ xi

τ R cV θ¨ + cV θ˙ + τ R

Cimmelli et al. studied the propagation of the longitudinal plane waves θ (x, t) = θ¯ (x) + θ0 exp[i(ωt − kx)] ¯ in one-dimensional case. Linearization around the non-equilibrium steady state θ(x) ¯ xi gives with the average heat flux qx0 = −λ∂ θ/∂ τ R cV θ¨ + cV θ˙ = λ

∂ 2θ ∂ ln λ ∂ ln λ ∂θ qx0 − τR qx0 . − 2 ∂x ∂θ ∂x ∂ θ˙

(2.14)

In Eq. (2.14) the nonlinear second-order terms such as θ˙ 2 and (∂θ/∂ x)02 were neglected.

2.1 Maxwell–Cattaneo–Vernotte Equation

53

In the case of the high-frequency waves (i.e. when θ˙  θ¨ and ∂θ/∂ x  ∂ 2 θ/∂ x 2 ) the dispersion relation is written as k2 +

τ R ∂ ln λ τR qx0 ωk − ω2 = 0 λ ∂θ χ

where χ is the thermal diffusivity. The phase velocity defined as U = |ω/Re(k)| is given as U± =



χ 1 √ τ R 1 + 2 ±

(2.15)

√ where 2 = qx0 ( τ R /λcV )∂ ln λ/∂θ . When the thermal conductivity λ does not depend on the temperature θ , = 0 √ and U + ≡ U − ≡ U0 = χ /τ R . Equation (2.15) shows that the temperature pulse travels with different velocities in the direction of the heat flow (U + ) than in the opposite direction (U − ). Similar results have been reported in the papers by Coleman, Casas-Vazquez, Cimmelli and others [92, 127, 128]. In the case  1 Eq. (2.15) becomes U ± = U0 (1 ∓ ) and the difference of the phase velocities of the temperature waves travelling in the different directions qx ∂ ln λ . U− − U+ = 0 cV ∂θ Cimmelli and Frischmuth [129] found this difference is rather small for waves in NaF at 15 K—about 2.1 × 10−4 cm (µs)−1 . Jou and Casas-Vazquez [17] have shown that it is possible to include the non-local term into the Cattaneo equation assuming that the generalized entropy, the entropy flux and the entropy production explicitly depend on the the flux of the heat flux, the ˆ to get tensor Q, ∂q + q = −λ∇T + 2 ∇ 2 q. τ ∂t This equation differs from the Guyer–Krumhansl equation (see Sect. 3.2) by an absence of the term of the form ∇∇ · q. The relaxation time τ is the time lag required to establish the steady heat conduction in a volume element once a temperature gradient has been imposed across it; this lag is the result of the “thermal inertia” of the material. The thermal disturbances in this model propagate with a finite speed  s=

λ . ρCτ

54

2 Phase-Lag Models

The estimates for the relaxation time in solids and rarefied gas could be written, respectively, as [130] 3λ τs = 2 c¯ and τg =

3ν c¯2

where c¯ is the phonon velocity in the solid or the mean molecular velocity in the gas  c¯ =

8 kB T π m

where ν is the gas kinematic viscosity and m is the mass of molecule. The relaxation time for metals at the low temperatures could be considered constant [131]. Sometimes the Cattaneo number is used [130] that is defined as Ca =

κτ L2

V e2 =

κτ . L2

or the Vernotte number [132, 133]

For example, Kundu and Lee [134] studied the effect of the Vernotte number on the heat conduction analysis in the absorber plates of a flat-plate solar collector and found a significant difference in the temperatures obtained from the Fourier and non-Fourier models for the higher values of the Vernotte number. The Cattaneo constitutive relation could be rewritten by the entropy and entropy flux distributions [135]. The entropic constitutive relations provide a perspective for heat conduction modelling. Analogous to hyperbolic heat conduction, an entropic relaxation is introduced between the entropy flux and entropy gradient, but this entropic relaxation can avoid non-positive absolute temperature for the well-posed problems because non-positive absolute temperature will generate a singularity. The Cattaneo constitutive relation (2.5) could be considered as Taylor’s series expansion of the relation with a time lag q(r, t + τ ) = −λ∇T (r, t)

(2.16)

which is called sometimes the “improved” Cattaneo model or the single-phase-lag (SPL) model [136]. Evidently, the SPL model (2.16) is Galilean invariant [137]. Chen et al. [138] have derived the SPL model equation from the Boltzmann equation using for the time derivative approximation [137]

2.1 Maxwell–Cattaneo–Vernotte Equation

55

f (t + t) f (t + τ ) ∂f ≈ = . ∂t t τ Recently Li and Cao [139] noted that the Cattaneo model should not be considered as a consequence of the SPL model since the accuracy of this approximation is uncertain—the remaining higher-order terms could be very large even if the relaxation time is very small and the predictions of these models could be different. The Cattaneo constitutive relation (2.5) could be written as an integral over the history of the temperature gradient λ q=− τ

t −∞

  t − t ∇T t  . exp − τ

(2.17)

Frankel et al. [140] noted that the alternative formulation—in terms of the heat flux (scalar equations for three components in the general case), could be useful for the problems involving the heat flux in the boundary conditions. The temperature distribution is recovered by integration of the general energy balance equation over time t 1 [−∇ · q(t  ) + Q]dt  T (t) = T (0) + (2.18) ρc t  =0

Nir and Cao [141] have compared the numerical simulations (using the alternating directions implicit, ADI, finite difference scheme) based on three representations: the temperature, the flux and the hybrid and found the last one to be preferable, and Kovács stated that q-representation is advantageous over the T-representation [142]. Sometimes the Cattaneo model is called a damped version of Fourier law (damped wave equation, DWE) [143–145]. In practice the Cattaneo model is useful for the low-temperature heat conduction problems [107, 146]. Jou et al. [147] extended in the framework of EIT generalized the Cattaneo relation to the case of graded materials with the composition gradient ∇ξ introducing into the constitutive relation the additional terms containing ∇ξ that account for the non-local and nonlinear effects. Cattaneo law removes the paradox of the infinite speed of propagation of disturbances but causes another one for the case of heat transfer in the moving body: the Cattaneo law is not Galilean invariant [148]: while the Fourier law is frameindifferent, but does not respect the causality principle, the Cattaneo’s model removes the causality problem, but its form depends on the choice of the observer [13]. The Cattaneo law is also not Lorentz-covariant [149], i.e. the form of the equation changes under the transformations ∂ =γ ∂x



 where γ = 1/ 1 − u 2 /c2 .

u ∂ ∂ + 2 ∂x c ∂t

 ,

∂ =γ ∂t 



∂ ∂ +u ∂t ∂t



56

2 Phase-Lag Models

The heat conduction in the moving medium is governed by the equation (in the dimensionless variables) [150] ∂2 T ∂(u · ∇T ) ∂ T + + + u · ∇T = T. ∂t 2 ∂t ∂t

(2.19)

In the simplest case of the one-dimensional problem and the constant velocity u(x, t) = U Eq. (2.19) reduces to ∂T ∂T ∂2 T ∂2 T + +U = T. + U 2 ∂t ∂ x∂t ∂t ∂x The speeds of disturbances in the coordinate system moving with a velocity U are the nonlinear functions of U [150]: c1,2 =

  1 U ± U2 + 4 . 2

Evidently, c1,2 = U ± 1, i.e. the sum or the difference of the dimensionless frame velocity and the thermal wave speed. This paradox is removed when instead of the partial time derivative a material derivative is used [150]. According to the Galilean principle of relativity the equation should be the same in the inertial moving frame, i.e. do not change under the Galilean transformation ξ = x − U t, τ = t.

as

In the general 3D case the Cattaneo relation in the material framework is written   ∂ + u · ∇ q = κ∇T. (2.20) q+τ ∂t

Equation (2.20) cannot be resolved with respect to the heat flux, and thus a single equation for the temperature cannot be derived [150]. Later Hristov [151] suggested to use the frame-indifferent upper-convected [152] Oldroyd derivative and to write the Cattaneo equation (2.5) as

∂q + v · ∇q + q · ∇v + (∇ · v)q + q = −κ∇T τ ∂t

that allows elimination of q to get for the temperature equation [151]

∂u ∂T ∂2 T + · ∇T + + 2u · ∇ τ ∂t 2 ∂t ∂t ∂T + + u · ∇T = ∇ · (κ∇T ). ∂t





∂T + u · ∇T (∇ · u) + u · ∇(u · ∇T ) ∂t (2.21)

2.1 Maxwell–Cattaneo–Vernotte Equation

57

Hristov showed that Eq. (2.21) retains its form under the change of variables corresponding to the frame that moves with a constant velocity. Müller and Ruggeri [48] formulated the constitutive equations for the heat and momentum transfer in fluids in the non-inertial frame using the extended thermodynamics and kinetic theory. They replaced the time derivative of the heat flux by ∂qi /∂t + qk (∂vi /∂ xk ) − 2qk Wik where v is the fluid velocity and W is the angular velocity matrix. The first additional term qk (∂vi /∂ xk ) follows from the requirement of objectivity (the so-called Jaumann derivative), and the second can be rewritten as −2c2 (ρh u h )k Wik and regarded as the Coriolis inertia term [153]. Al Nahas et al. [13] proposed to replace the notion of the frame-indifference with the covariance principle introduced by Einstein [154, 155] and derived a covariant spacetime heat conduction model using the spacetime formalism to ensure the indifference to changes of observer. Thermal convection with the Cattaneo–Hristov constitutive relation is studied by Straughan [156]. He observed that the thermal relaxation effect is prominent if the Cattaneo number is sufficiently high. Stability for the Cattaneo–Christov equation has been investigated by Ciarletta and Straughan [157]. Haddad [158] studied the thermal stability in the Brinkman permeable space in the existence of the improved heat conduction Cattaneo–Christov model. Fluid flow and heat transfer characteristics of the Maxwell material over a stretched sheet by employing this heat conduction model are addressed by Han et al. [159]. Hayat et al. [160] simulated the 3D flow of Prandtl fluid. Zhang et al. [161] studied the heat transfer of blood vessels subject to the transient laser irradiation, where the irradiation is extremely brief and has high power. Chu et al. [162] numerically studied the effect of the radiation on the propagation of the thermal waves. The authors found that the internal radiation in the medium significantly influences the wave nature. The thermal wave nature in the combined non-Fourier heat conduction with radiation is more obvious for large values of conduction-to-radiation parameter, small values of optical thickness and higher scattering medium. Joseph and Preziosi [29] suggested to use the relaxation kernel in the form R J P = λ1 δ(s) + (λ2 /τ )exp(−s/τ ) where λ1 is the effective thermal conductivity and λ2 is the elastic conductivity. In this case the heat flux equals (in the one-dimensional case) to   t λ2 ∂T t − s ∂T − ds. ex p q = −λ1 ∂x τ τ ∂x −∞

The Cattaneo (Cattaneo–Vernotte) model may give the unphysical predictions such as the negative temperature when the two cooling waves meet [163] or the jumps at the wavefront, indicating the discontinuity of the temperature distributions [164]. Kronberg et al. [165] derived a mixed-type boundary condition, and an accommodation coefficient was introduced. When the accommodation coefficient equalled to 1, no temperatures exceeding the boundary value exist. However, the discontinuity

58

2 Phase-Lag Models

was not avoided. In addition, it is found that for smaller accommodation coefficient, the results for the mixed boundary condition are same as that for Dirichlet boundary condition. Körner and Bergmann [166], Rubin [167], Criado-Sancho and Lebot [168], Bai and Lavine [169], Barletta and Zancini [170–172] analysed the compatibility of the Cattaneo law with the second law of thermodynamics. Barletta and Zancini also considered the Taitel’s paradox (a temperature exceeding the difference of the boundary values in the slab whose sides are kept at the different temperatures). In the frame of the classical irreversible thermodynamics (CIT) the entropy production could be written as [91] σs = λ

(∇T )2 τ ∂q + ·∇T. + 2 T2 T ∂t

Barletta and Zancini found that the production of the entropy could be negative in the regions where the heat flux decreases so steeply that |∂q/∂t| > |q|/τ . Note also that the second law does not say anything about fluctuations, and for a short time an instantaneous negative production could be observed [173]. Torii and Yang [174] studied the heat transfer in the thin film under the continuousoperated and pulse laser heat sources and found that CV model could lead to the overshooting phenomenon in the propagation of the thermal wave that depends on the frequency of the laser source and seemingly violates the second law of thermodynamics. However, these results do not contradict the second law since classical thermodynamics is based on the local equilibrium hypothesis [175] that is not fulfilled in this case [176] and the concept of “temperature” in the hyperbolic heat transfer equation cannot be interpreted in the conventional sense [91]. Cattaneo law is compatible with the second law in the extended irreversible thermodynamics that introduces the generalized entropy and entropy flux depending not only on the classical variables but also on the fluxes [34, 177, 178]. Li and Cao [179] studied the thermodynamics problems of the SPL model. Using the expression of the entropy production rate S=−

q · ∇T T2

from the classical irreversible thermodynamics (CIT) the authors got for the Fourier law q2 SF = − 2 ≥ 0 λT and for SPL SSCPI TL = −

q(t) · q(t + τ |) . λT 2

2.1 Maxwell–Cattaneo–Vernotte Equation

59

Evidently, the entropy production rate for the SPL model is not necessary positive or zero. The second law of thermodynamics is satisfied in the EIT [177] where the entropy production rate is expresses as EIT =− SSPL

q(t + τ ) · q(t + τ |) . λT 2

Still, the SPL model could violate the second law of thermodynamics by the breaking the equilibrium spontaneously in the special circumstances. Li and Cao [179] have constructed the simple example of such behaviour. They have considered the case 2τ n 2 π =1 ρC V 2 and obtained the solution T (x, t) = C1 sin

πt nπ x sin + C0 l 2τ

(2.22)

of the SPL model equations with the boundary conditions T (0, t) = C0 , T (l, 0) = C0 and the initial condition T (x, 0) = C0 . The coefficient C1 in the solution (3.51) is arbitrary, and the initial equilibrium T = C0 could be broken spontaneously. Note that both Fourier and Cattaneo models preserve the equilibrium [179]: dE F =− dt



dE C V 2 =− dt τ

λ (∇T )2 dV ≤ 0, ρCv  

∂T ∂t

2 dV ≤ 0.

Glass et al. [180] numerically studied the Cattaneo heat conduction in a semiinfinite slab with the temperature-dependent thermal conductivity λ = λ0 (1 + βT ) and found that the nonlinear thermal conductivity alters the speed of the thermal front. In biological problems the Cattaneo equation is called the thermal wave model or the thermal wave model bioheat transfer (TWMBT) [181–184]; sometimes the terms the “heat wave” or “temperature wave” are also used [185, 186]. Conejero et al. [176] studied the chaotic asymptotic behaviour of the solutions of the Cauchy problem for the hyperbolic heat transfer Eq. (2.6) 

T (0, x) = φ1 (x), x ∈ R ∂T (0, x) = φ2 (x), x ∈ R ∂t

where φ1 (x) and φ2 (x) are the initial temperature and the initial temperature variation.

60

2 Phase-Lag Models

The authors expressed the hyperbolic heat transfer equation as a first-order equation and represented the solutions as a C0 -semigroup3 on the product of the certain function space X with itself [187] and showed that this semigroup is hypercyclic and chaotic in the sense of Devaney [176]. Cattaneo equation can be derived using non-equilibrium statistical mechanics and the assumption of the near equilibrium state (thus accepting Maxwellian as the non-equilibrium distribution) [188, 189]. Podio-Guidugli [190] (see also [191]) derived the Maxwell–Cattaneo–Vernottetype equations using the splitting the entropy into the standard and dissipative parts. Compte and Metzler [35] suggested three possible generalization of the Cattaneo equations (GCE) using the generalization of the constitutive equation q(r, t) + τ γ and continuity equation

∂γ q = −λ∇T ∂t γ

∂γ T = −q(r, t). ∂t γ

The generalization is written as follows: • GCE I

• GCE II

• GCE III

∂γ T ∂ 2γ q + τ γ 2γ = −λT γ ∂t ∂t 2 ∂ 2−γ T γ ∂ q + τ = −λT ∂t 2−γ ∂t 2

∂2q ∂T ∂ 1−γ + τ 2 = −λ 1−γ T. ∂t ∂t ∂t

The equation GCE III is equivalent to ∂γ T ∂ 1+γ q + τ 1+γ = T. γ ∂t ∂t Compte and Metzler analysed the behaviour of solutions of GCE I–GCE III equations in the short-time and long-time limits and found that in the former GCE II describes the pure ballistic propagation, while the signal propagation velocity is infi-

3

A family {Tt }t≥0 of linear continuous operators on the Banach space X is called C0 -semigroup if

• Tt Ts = Tt+s for all t, s ≥ 0; • limt→s Tt = Ts for all x ∈ X and s ≥ 0. .

2.1 Maxwell–Cattaneo–Vernotte Equation

61

nite in the case of GCE I and GCE III. In the long-time limit GCE I and GCE III show that subdiffusion and GCE II correspond to the superdiffusive transport.

2.1.1 “Relativistic” Heat Conduction Ali and Zhang [51] introduced the HHCE-like “relativistic” heat conduction equation (RHE) using the weaker interpretation of the relativity theory (relativity in the strong sense deals with objects moving at speeds comparable with speed of light): the weak relativity is concerned with objects moving at speeds comparable with the limiting speed C of the signal propagation in the medium involved. Ali and Zhang used four-dimensional (Minkowski) space–time (τ, x, y, z) coordinate system where τ has a length dimension. The authors set τ = iCt with i being the imaginary unit constant; there are no restrictions on the numeric value of C The four-dimensional gradient  is given by =

∂ ∂ ∂ ∂ i ∂ ∂ o+ i+ j+ k= o+∇ = o+∇ ∂τ ∂x ∂y ∂z ∂τ C ∂t

where o, i, jand k are the unit vectors along the coordinate axes. Similarly, the four-dimensional Laplacian (the d’Alambertian) 2 is =

∂2 ∂2 ∂2 ∂2 1 ∂2 ∂2 2 + + + = + ∇ = − + ∇2. ∂τ 2 ∂x2 ∂ y2 ∂z 2 ∂τ 2 C 2 ∂t 2

The Fourier law takes the form q(x, t) = −λT (x, t) =

iλ ∂ T (x, t) o + λ∇T (x, t) C ∂t

(2.23)

and the heat conduction equation (“relativistic” heat conduction equation) ∂ κ ∂2 T (x, t) = κT (x, t) = − 2 2 T (x, t) + κ∇ 2 T (x, t) ∂t C ∂t

(2.24)

where C is the speed of heat propagation in the medium. Equation (2.24) is identical in form to the hyperbolic heat equation. Ali and Zhang stressed that the imaginary component in Eq. (2.23) is the manifestation of the wave nature of heat conduction and that no information on the microstructure of the heat conducting medium is needed. Taking the time derivative of Eq. (2.23) produces iλ ∂ 2 T (x, t) ∂q = o + λ∇ ∂t C ∂t 2



∂ T (x, t) ∂t



62

2 Phase-Lag Models

that allows the author to write “Cattaneo-like” relation

∂q ∂ T (x, t) τ0 + q = −λ∇ τ0 + T (x, t) ∂t ∂t that differs from the Cattaneo relation (2.5) by the term with the temperature time derivative. In contrast to the Cattaneo model the “relativistic” heat conduction equation does not contradict the second law even in the frame of the classical irreversible thermodynamics. The entropy production λ 1 σ = q · T = 2 T T



∂T ∂τ

2

 +

∂T ∂x

2

 +

∂T ∂y

2

 +

∂T ∂z

2 

is always non-negative since it can be rewritten using the relations ∂ T dt ∂T = , w = τ, x, y, z ∂w ∂t dw as



1 ∂T σ =λ T ∂t

2 

1 − 2+ C



dt dx

2

 +

dt dy

2

 +

dt dz

2  .

This equation is a statement of the fundamental property of the space–time since it is a direct consequence of the Minkowski metric where the interval is ds 2 = dx 2 + dy 2 + dz 2 − C 2 dt 2 . The alternative formulation of the “relativistic” heat conduction equation in terms of the heat flux alone could be useful if heat flux is present in boundary conditions ∂q = κ2 q. ∂t Introduction of the operator A[] as A[ f ] = τ0

∂f + f ∂τ

Ali and Zhang obtained the “relativistic” heat conduction equation in the form identical to the classical Fourier equation ∂ A[T ] = κ∇ 2 T. ∂t

2.2 Dual-Phase-Lag Model

63

2.2 Dual-Phase-Lag Model To include effects of both the relaxation and the microstructure, Tzou [192–194] introduced the dual-phase-lag (DPL) model q(r, t + τq ) = −λ∇T (r, t + τT )

(2.25)

where τq and τT are the phase lags for the heat flux vector and for the temperature gradient, respectively, arising from the “thermal inertia” and the “microstructural interaction” [195]; these parameters are the intrinsic properties of the material. The phase lag of the heat flux can be interpreted as the time delay due to the fast transient effect of the thermal inertia, while the phase lag of the temperature gradient represents the effect of the phonon–electron interactions and the phonon scattering during the ultrafast heat transfer [196]. Tang and Araki [197] (see also the paper by Shen and Zhang [198]) distinguish four heat propagation modes of the DPL model: • • • •

wave mode (τT = 0); wavelike mode (0 < τT < τq ); diffusion mode (τT = τq ); overdiffusion mode (τT > τq ).

Askarizadeh and Ahmadikia [199] suggested to distinguish the modes according to whether the heat flux is the cause of temperature gradient in the medium (over diffused or flux precedence heat flow) or the temperature gradient is the cause of heat flux (gradient precedence heat flow). Zhang et al. [200] considered the damping of the thermal waves measured as the relative decrease of the temperature and the heat flux for the Cattaneo and the DPL models (and for the thermomass √ model, see below ) and introduced the dimensionless “damping factor” ξ = L/ κτ . In the case of the heat conduction by the photon transport this factor is shown to be inversely proportional to the Knudsen √ number (K n = l/L, l is the mean free path of phonons) ξ = 3/K n—the decrease of the Knudsen number reflects stronger collisions between phonons and thus quicker damping of the energy transported by the thermal wave. In contrast to the SPL model that could lead to unphysical infinite value of the local heat [201] the DPL model smooths such wavefront and eliminates these infinite values [202]. The DPL model reduces to the SPL model when τT = 0 and to the Fourier law if τT = τq = 0. Both relaxation times are very small for common materials. For example, the phase lags τq and τT for gold are 8.5 ps and 90 ps, respectively [203]. The relaxation time τT was found by the molecular dynamics computations to be smaller than τq and both times being in the range from a few picoseconds to tens of picoseconds in the pure solid argon films and films with vacancy defects [204]. Goicochea et al. [205] found for the bulk silicon from the molecular dynamics computations that the relaxation time τq is inversely proportional to the cube of

64

2 Phase-Lag Models

the temperature τq ∝ T −3 and to the temperature τq ∝ T −1 when the temperature is higher than the Debye temperature. Thus, the thermal wave travels faster at the higher temperature [206]. There is a lot of controversies in the literature concerning the values of relaxation times for the biological tissues. For the processed meat the values of τq and τT are estimated as 14–16 s and 0.043–0.056 s[207], and experiments with the muscle tissue from cow give values 7.36–8.43 s and 14.54–21.03 s [208]. Zhang [209] estimated the relaxation times in terms of the blood and tissue properties and the interphase convective heat exchange coefficient and found them to be close to each other. Equation (2.25) could be rewritten as q(r, t) = −λ∇T (r, t + (τT − τq ))

(2.26)

Thus, the solution of the DPL model does not depend on the relaxation times τT and τq separately but only on their difference [210], and the SPL and the DPL models are mathematically equivalent [211–213]. Fabrizio et al. [212, 214] showed that there are mathematical conditions in addition to the physical ones to obtain an exponentially stable equilibrium solution for DPL equation. Such condition requires the negative time delay τd (called the retarded effect) between the heat flux and temperature gradient τd = τq − τT ≤ 0. The opposite case, i.e. τd > 0 is mathematically ill-posed which enlightens the validity of the MCV equation but excludes equations based on the arbitrary Taylor series expansion. The DPL model is closely related [215] to the hyperbolic models that describe the energy exchange in the systems composed of two subsystems, each with its own temperatures [34], for example, between the electrons and the phonons in the solids by a pair of the coupled nonlinear equations governing the effective temperatures of the electrons and the phonons [3, 136, 216–222] (the two-temperature models, TTM). Two-temperature models are valid if the time needed to establish the equilibrium within each subsystem is much less than the time needed to establish the equilibrium between them [3]. In this case it is possible to assign an individual temperature to each subsystem and reduce the thermal problem in the system to the determination of the space–time evolution of these temperatures with an account for the energy exchange between the subsystems [117]. For example, it is assumed that the electron and phonon subsystems in the solids are at their local equilibrium, the heat conduction by phonons is neglected, and the lattice temperature is near or above the Debye temperature (thus the electron–electron and electron-defect scatterings are insignificant compared with the electron–phonon scattering [186]). Electrons play a major role in the heat transport in metals; often it is assumed that electrons move with the velocity of the order of the Fermi velocity [3]. The equations of the TTM for the one-dimensional case could be written as ce

∂q ∂ Te =− − G(Te − Tl ) + Q, ∂t ∂x

(2.27)

2.2 Dual-Phase-Lag Model

65

∂ Tl = G(Te − Tl ), ∂t ∂q ∂ Te τ + q = −λ , ∂t ∂x cl

(2.28) (2.29)

where Te is the temperature of the electron gas, Tl is the temperature of the lattice, ce and cl are the heat capacity of the electron gas and the lattice, respectively, and G is the electron–phonon coupling factor that is estimated as [223] G=

π 2 m e n e cs2 τe (Te )Te

where m e is the mass of electron, n e is the density of the free electrons, √ τe is the electron relaxation time, cs is the speed of sound in bulk material cs = B/ρm , B is the bulk modulus, and ρm is the density. There are other physical two-temperature systems, for example, the protons and the electrons in the astrophysical models of the accretion flow or the ions and the electrons in the partly ionized gas [224]. Qiu and Tien [225, 226] suggested a similar model named a hyperbolic two-step (HTS) model that accounts for the lattice conductivity ∂qe ∂ Te =− − G(Te − Tl ) + Q, ∂t ∂x ∂qe ∂ Tl =− + G(Te − Tl ), cl ∂t ∂x ∂qe ∂ Te τe + qe = −λ , ∂t ∂x ∂qi ∂ Ti τi + qi = −λ . ∂t ∂x ce

For pure metals heat conduction in the lattice is small compared to that in electrons and can be neglected as well as the electron relaxation time, and thus the dualhyperbolic model reduces to the parabolic two-temperature model [227]. Tzou [193] had estimated the relaxation times of the DPL model in terms of the parameters G, ce , cl of this model and obtained for copper, silver, gold and lead the typical values of τT and τq about 10−11 s and 10−13 s, respectively. Zhang [209] proposed the explicit estimates of the relaxation times for the heat transfer in the tissues ε(1 − ε) ρb Cb ε(1 − ε) ρb Cb , τT = ε τq = ε + (1 − ε) G + (1 − ε) G Ctb K tb

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2 Phase-Lag Models

where Ctb = ρt Ct /(ρb Cb ) is the ratio of heat capacities of tissue and blood, K tb = λt /λb is the ratio of the thermal conductivities, ε is the porosity of the tissue, and G is “the lumped convection-perfusion parameter”. Gonzalez-Narvaez et al. [228] developed the two-temperature model for the heat conduction in solids with internal structure. The model is based on the fact that the total internal energy of the system u may be separated in two parts u = u M + u m , namely the part associated with the macroscopic length scale processes u M and the part related to processes taking place in the microscopic length scale u m . Each of the internal energies is related to a quasi-temperature referred to as the macrotemperature TM = u M /C M and the microtemperature Tm = u m /Cm where C is he heat capacity at constant volume. The total (measurable) temperature of the system is given by the average value T =

C M TM + Cm Tm . C M + Cm

The two-temperature model for heat transport in the rigid solid could be interpreted as the model in which one of the temperatures accounts for the deviation of the system from the Fourier behaviour [228]. There are three sources of dissipation: (1) processes on the macroscopic scale, (2) processes at the microscopic scale and (3) interaction processes—energy exchange— between the two subsystems. The heat fluxes could be written (in the one-dimensional case) as −q M = L 11

∂ TM ∂x TM2

+ L 12

∂ Tm ∂x Tm2

,

−qm = L 21

∂ TM ∂x TM2

+ L 22

∂ Tm ∂x Tm2

where L i j are phenomenological coefficients that obey the Onsager reciprocity relations L 12 = L 21 for the processes near the equilibrium [228]. Tzou and Dai [44] considered lagging in the multicarrier systems. The equations for the N-carrier system are written as ∂ T1 = λ1 ∇ 2 T1 − G 1i (T1 − Ti ), ∂t i=2 N

C1

Cm

m−1 N ∂ Tm G jm (T j − Tm ) − G mi (Tm − Ti ), = λm ∇ 2 Tm + ∂t j=1

m = 2, 3, ...(N − 1),

i=m+1

N −1

CN

∂ TN = λ N ∇ 2 TN + G i N (Ti − TN ). ∂t i=1

Deriving an equation for the single temperature in the three-carrier system (e.g. the composite with three constituents or the polar semiconductor where the heat

2.2 Dual-Phase-Lag Model

67

could be transported by electrons, holes and phonons) Tzou and Dai found that the nonlinear effects are related to the τq2 and τT2 . Depending on the value of phase lags, the alternative orders of the Taylor series expansion can be made to provide a series of DPL heat conduction models [229, 230]. Using the first-order expansion for both τq and τT in Eq. (2.25) gives the constitutive relation

∂q ∂∇T = −λ ∇T + τT . (2.30) q + τq ∂t ∂t Inserting this relation in the energy conservation law results in the so-called type I [213, 229, 231] (also linear [232] or first-order [230]) DPL model. The first-order DPL bioheat transfer equation is written as 





ρb cb wb ∂T ∂ + (T − Tb ) + τq = ∇ 2 T + τT (∇ 2 T ), λ ∂t ∂t (2.31) if the source terms the metabolic reactions and the external source of energy are omitted. The equations of this model could be rewritten in terms of the heat flux instead of the temperature [233] c λ

∂T ∂2 T + τq 2 ∂t ∂t

∇(∇q) + τT ∇(∇q) =

τq ∂ 2 q 1 ∂q + κ ∂t κ ∂t 2

and even in terms of the heat flux potential φ defined by the equation q = ∇φ [234]. Wang et al. [234] proved the well-posedness of the DPL model on a finite onedimensional region under the uniform Dirichlet, Neumann or Robin conditions. Later Wang and Xu [235] extended this result to the n-dimensional case. Type II DPL model [213] is obtained when the first-order and the second-order Taylor series expansions are used for the heat flux q and the temperature T , respectively,

τT ∂ 2 ∇T ∂q ∂∇T (2.32) = −λ ∇T + τT + q + τq ∂t ∂t 2 ∂t 2 and type III (second-order [230]) DPL model [213, 236])—when the second-order Taylor series expansions are used for both q and T

τq ∂ 2 q τT ∂ 2 ∇T ∂q ∂∇T . + + q + τq = −λ ∇T + τT ∂t 2 ∂t 2 ∂t 2 ∂t 2 The second-order DPL bioheat transfer equation is written as follows:

(2.33)

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2 Phase-Lag Models

c λ



τq2 ∂ 3 T ∂T ∂2 T + τq 2 + ∂t ∂t 2 ∂t 3

= ∇ 2 T + τT



  τq2 ∂ 2 T ∂T ρb cb wb + + (T − Tb ) + τq λ ∂t 2 ∂t 2

∂ τ 2 ∂2 (∇ 2 T ) + T 2 (∇ 2 T ). ∂t 2 ∂t

(2.34)

Sometimes the other notation is used to distinguish the DPL models, indicating the orders of the Taylor expansions, for example, DPL (2,1) [237] or just “1st–1st” order, “1st–2nd” order, etc. [238]. Rukolaine [239] found the solutions of the DPL model to be unstable. He investigated the initial value problem for the three-dimensional equation with the positive localized source of the short duration. The constitutive equation of the Jeffrey-type was used. Rukolaine obtained an analytical solution using the Fourier transform and the Laplace transform for DPL equation assuming the Gaussian source of finite duration   1 |x|2 pσ (x) = √ 3 exp − 2 . 2σ 2π σ The solution for τq > τT gives an unphysical behaviour of the temperature history, and it goes into the negative domain in the vicinity of origin at least for the small values of the parameter σ . Later Rukolaine confirmed his conclusion for the type III DPL model [240]. Quintanilla and Racke [241] (see also [242]) analysed the stability of the solution of the different DPL versions. Quintanilla introduced the ratio of the two relaxation times of the DPL model τT ξ= τq as a parameter that controls the stability of the DPL model. The author considered the characteristic polynomial associated to the Laplace operator for the case of the Dirichlet boundary conditions in a bounded domain. Stability or instability of the solution is determined by the real part of the eigenvalues. The results of the study could be summarized as • When approximation up to first order in τq and up to first or second order in τT is used, the system is always stable. • When approximation up to second order in τq and up to first order in τT is used, the system is stable if ξ > 1/2 and unstable if ξ < 1/2. • When approximation √ up to second order in both √ τq and τT is used, the system is stable if ξ > 2 − 3 and unstable if ξ < 2 − 3. • Whenever ξ > 1/2, the several models predict the same behaviour. Restrictions on the ratio ξ = τT /τq were derived from the second law of thermodynamics by Fabrizio and Lazzari [243]. The compatibility of the DPL with the second law of the extend irreversible thermodynamics was proved by Xu [244].

2.2 Dual-Phase-Lag Model

69

Both the TWMBT and DPL models are used widely to study the heat transfer in the biological objects. For example, interaction of the laser radiation with the tissue, including the laser interstitial thermal therapy (LITT) [9], has been considered by the numerous researches: Zhou et al. [245], Jaunich et al. [246], Liu [247], Afrin et al. [248], Ahmadikia et al. [249], Sahoo et al. [250], Liu and Wang [251], Poor et al. [252], Hooshmand et al. [253], Kumar and Srivastava [254], Mohajer et al. [255], Jasinsky et al. [256], Liu and Chen [257] and Zheng et al. [258]. Singh and Melnik [259] used the SPL and the DPL models incorporating both the tissue contraction and expansion during the procedure, considering DPL along with Helmholtz harmonic wave equation, the modified stress–strain and the thermoelastic wave equations to simulate the radiofrequency ablation (RFA) and microwave (MW) ablation. The authors reported that the effect of the non-Fourier based coupled thermoelectromechanical model is less pronounced in RFA as compared to the MWA. Recently Singh and Melnik published the review of the computational models and future directions of the thermal ablation [260]. The high intensity focused ultrasound (HIFU) ablation was investigated by Li et al. [261], Namakshenas and Mojra [262, 263] and Singh et al. [264]. As the power of the transducer increases, the deviation from the Fourier results also increases. Zhou et al. [265] solved the two-dimensional (axisymmetric) problem for the two cases: the surface heating and the body heating and found the multidimensional effect to be not negligible. Noroozi et al. used the thermal wave model to study the heat transfer in the slab heated by a laser radiation [266] and the DPL model for the case of a slab subjected to an internal heat source [267]. Liu et al. [182] computed the temperature elevation generated by the ultrasonic irradiation using the TWMBT model. Li et al. [261] used the DPL model to predict the ex vivo temperature response to the focused ultrasonic heating of a homogeneoustissue phantom and a heterogeneous liver tissue. Kumar et al. used the finite element wavelet Galerkin method to study the hyperthermia assuming the Gaussian external heat source [268]. Ho et al. exploited the lattice Boltzmann method (LBM) for solution of the DPL model studying the heat transfer in the two-layer system [269]. Kumar and Rai [270] used DPL model to investigate the heat transfer in the bilayer living tissues during the magnetic fluid hyperthermia treatment exploiting the finite element Legendre wavelet Galerkin method. Kumar et al. [271] used the non-local DPL model to compute the heat conduction approach in a bilayer tissue (tumour and normal tissue) during magnetic fluid hyperthermia. The effect of variability of magnetic heat source parameters (magnetic induction, frequency, diameter of magnetic nanoparticles, volume fractional of magnetic nanoparticles and ligand layer thickness) has been investigated. The effect of the phase-lagging times only appears in the tumour region: the temperature increases with the increment of the value of τq and decreases with the augmentation of the value of the τT in the tumour zone [9]. Liu and Chen [272, 273], Liu and Cheng [274], Liu and Yang [275], Raouf et al. [276] and Kumar et al. [271, 277] also used the DPL model to study the magnetic fluid hyperthermia.

70

2 Phase-Lag Models

Kumar et al. [278] numerically solved the nonlinear DPL model for the heat transfer within the skin tissue. The DPL of the bioheat transfer is used for the assessment of the protective clothing. Zhai and Li [279] have reviewed different burn prediction techniques used in clothing evaluation. These methods are founded on experiments performed on living bodies which are getting in touch with hot water or exposed to radiant source. Ye and De [280] published the review on the experimental and numerical studies of thermal injury of the skin and subdermal tissues. Moradi and Ahmadikia [281] used the DPL model to study the heat transfer during the extremely fast freezing of the biological tissues (the cooling rate about 1000◦ /s [282]) when the frozen region tends to form the amorphous ice [283]. Alghamdi and Yuossef [284] applied the DPL model to investigate the effect of the porosity, the evaporation rate and the ambient temperature on the heat transfer in the human eye subjected to the exponentially decaying laser radiation. Afrin et al. [285] used the DPL model to investigate the heating of a gas-saturated porous medium with short laser pulses. Chou and Yang [286] studied the two-dimensional problem of the heat transfer in the multilayered structures using the space–time conservative element and solution element (CESE) method [287]. This method was developed for solving the Euler and the Navier–Stokes equations of the fluid dynamics [288] and provides the local and global flux conservation due to using both the flow variables and their derivatives as unknowns. Chou and Yang found four modes of the heat propagation: hyperbolic, wavelike, diffusive and overdiffusive. Earlier these authors [289] used the CESE scheme for computations of the SPL and DPL thermal waves and found that this method is characterized by the low numerical dissipation and the dispersion errors. Li et al. [290] applied DPL to investigation of the thermoviscoelastic behaviour of the biological tissue subjected to the hyperthermia treatment. Liu et al. [291] applied the second-order Taylor expansion of dual-phase-lag model for analysis of the thermal behaviour in the laser-irradiated layered tissue, which was stratified into skin, fat and muscle. Jiang et al. [292] studied heat conduction in porous material heated by the microsecond laser pulse. The authors found that the both the HHC equation and the DPL model can predict the non-Fourier heat conduction phenomena observed in the experimental sample qualitatively. However, referring to the maximum temperature emerging in the experimental sample, the the DPL model is more agreeable to the experimental result. The maximum temperature predicted by the DPL model is still greater than but closer to that of the experimental result. Gandolpi et al. [293] used the first-order expansion of DPL (2.30) that results in the temperature equation

τ  ∂2 T ∂3 T ∂2 T 1 ∂T q − τT − + =0 2 2 κ ∂t ∂x κ ∂t ∂t∂ x 2

(2.35)

as the basis for the design of the temperonic crystal (TC)—a periodic structure with a unit cell made of two slabs sustaining the temperature wavelike oscillations on the

2.2 Dual-Phase-Lag Model

71

short timescales. The temperonic crystal is similar to the electronic, phononic and photonic superlattices. The complex-valued dispersion relation for the temperature scalar field in TCs discloses the frequency gaps, tunable upon varying the slabs thermal properties and dimensions, serving, for instance, as a frequency filter for a temperature pulse [293]. For analysis of Eq. (2.35) Gandolpi et al. [294] used the dimensionless variables (Teq is the equilibrium temperature) β=

x t , ξ=√ , τq κτq

Z=

τT T , θ= . τq Teq

Equation (2.35) is rewritten as ∂ 3θ ∂ 2θ 1 ∂θ ∂ 2θ − Z − + =0 ∂t 2 ∂x2 κ ∂t ∂t∂ x 2

(2.36)

Equation (2.35) is parabolic. Still, its solution may still bear the wavelike characteristics if [294] τT 1 Z= < τq 2 since the first two terms in Eq. (2.35) constitute the homogeneous wave equation,  while the last two are responsible for the damping effects; κ/τq is the speed of √ propagation, and κτq is the diffusion length [295]. The authors sought the solution of Eq. (2.36) in the form ˜

˜ θ (ξ, β) = θ0 ei(kξ +ωβ)

where the complex-valued dimensionless wave vector k˜ and the angular frequency √ ω˜ are linked to their dimensional counterparts k˜ = κτq k and ω˜ = τq ω. Equation (2.36) gives the dispersion relation   1 2 ˜k 2 (1 + i Z ω) . ˜ = ω˜ 1 − ω˜ In the case Z |ω| ˜  1 and 1/|ω| ˜  1 Eq. (2.36) reduces to the dispersion relation k˜ 2 = ω˜ 2 for a free-propagating wave. In the dimensional variables conditions can be written as 1  τq , τT  |ω| ˜ i.e. the period of the temperature oscillation in time 2π/|ω| ˜ must lay between the two relaxation times and that the temperature gradient must precede the onset of heat flux.

72

2 Phase-Lag Models

Gandolpi et al. [293] considered the system of two slabs, within each slab the thermal dynamics is described by Eq. (2.35), the continuity of temperature and heat flux is enforced at the boundary between slabs. The authors studied the propagation of a small temperature pulse that not substantially affect the TC’s initial temperature and can be considered as a linear superposition of the plane waves. Each single plane wave was assumed to have a real-valued k and a complex-valued ω˜ = ω1 + iω2 , where 2π/ω1 is the temperature oscillation period and 1/ω2 is the damping time. To quantify the damping of a given mode, the modal quality factor Q(k) = ω1 (k)/ω2 (k) is introduced which discriminates the overdamped (Q  1) from the underdamped regimes (Q 1) [295]. Dispersion curve ω1 (k) and dependence Q(k) are obtained numerically. However, Gandolphi et al. [295] obtain the analytic expressions of ω1 and ω2 as a function of k for the one-dimensional problem in which the spatial coordinate, z, is perpendicular to the sample surface. The authors showed that the nature of the solution strongly depends on the ratio τT /τq . The non-oscillatory solutions (i.e. ω1 = 0) are found for τT > τq , i.e. when the heat flux precedes the establishment of a thermal gradient, a wavelike behaviour of the temperature propagation may emerge in the case τT < τq , i.e. when the temperature gradient is established before the onset of the heat flux. The authors defined lower (klow ) and upper (kup ) bounds for the wave vectors that can sustain oscillatory solutions:     2 τq τT 1 τT klow = − 1− 1− , κτT τT 2 τq τq  kup =

2 τq κτT τT

   1 τT τT 1− . + 1− 2 τq τq

In the range klow < k < kup the allowed complex frequencies are:      

 κ 2 τT 2 4 κ 1 τT 1  2 k + −1 k + 2 ω1 = ± − 4 τq τq 2 τq 4τq and ω2 =

1 κ τT 2 + k . 2τq 2 τq

The most favourable condition to observe the underdamped oscillations corresponds to the maximum Q-factor:  Q max =

τq −1 τT

√ that is obtained at the wave vector k Qmax = 1/ κτT .

2.2 Dual-Phase-Lag Model

73

Numerous researches used the DPL model to study the heat transfer in the nanoelectronic devices, for example, the metal–oxide–semiconductor field-effect transistors (MOSFETs) [296–298], see the papers [299–304] and references therein.

2.2.1 Non-local Dual-Phase-Lag Model To accommodate the effect of thermomass (see Chap. 4), the distinctive mass of heat, of dielectric lattices in the heat conduction and the size dependency of the thermophysical properties, Tzou and Guo [305] and Tzou [306] have introduced the non-local (NL) behaviour of the heat transport in space in addition to the thermal lagging of the temperature gradient and heat flux, in time. The non-local dual-phaselag (NL DPL) heat conduction can be written as: q(r + λq , t + τq ) = −λ∇T (r, t + τT )

(2.37)

where λq is the correlating length of the non-local heat flux that has the same form of the length parameter in the thermomass model [196]. The natural extension of the NL DPL (2.37) is the introduction of another local space scale λT q(r + λq , t + τq ) = −λ∇T (r + λT , t + τT ). (2.38) Using the first-order Taylor series expansion of Eqs. (2.37 and 2.38) with respect to both the correlating lengths and phase lags, Tzou has developed the non-local DPL (NDPL) heat conduction models. The first-order expansion in the lagging time and the second-order effect in the non-local parameters of Eqs. (2.37 and 2.38) are given in the one-dimensional case [230] λq2 ∂ 2 q(x, t) ∂ T (x, t) ∂q(x, t) = −λ + τq q(x, t) − 2 ∂x2 ∂t ∂x and

λq2 ∂ 2 q(x, t) ∂ T (x, t) λ2T ∂ 3 q(x, t) ∂q(x, t) . =− λ − q(x, t) − + τq 2 ∂x2 ∂t ∂x 2 ∂x3 These expressions for the heat flux are used to derive the non-local DPL bioheat equations

3 ρb cb wb  ∂ 2 T ρb cb wb ∂ T 2 ρc ∂ T 1 + λq2 + λ = τq q 2λ ∂x2 2λ ∂t∂ x 2 λ ∂t

ρb cb wb ∂2 T ρc ∂ T + τq 2 + (T − Tb ) + λ ∂t ∂t λ

(2.39)

74

2 Phase-Lag Models

and ρb cb wb  ∂ 2 T ρc ∂ 3 T λ2 ∂ 4 T + λq2 − T 2 2 2λ ∂x 2λ ∂t∂ x 2 ∂x4



2 ∂ T ∂T ρc ∂ T ρb cb wb + τq 2 + = (T − Tb ) + τq . λ ∂t ∂t λ ∂t

1 + λq2

(2.40)

Li et al. [230] noted that the coefficient λq2 ρc/2λ at the term ∂ 3 T /∂t∂ x 2 in Eq. (2.39) is similar to the coefficient τt in the first-order DPL bioheat transfer Eq. (2.31) indicating that the lagging time terms with τT in Eq. (2.31) and the nonlocal items with λq in Eq. (2.39) have similar effect on the bioheat transfer. Thus it is possible to model the microstructural interaction effect in space using the lagging time τT in the temperature gradient instead of the non-local parameter λq in the heat flux [230]. Li et al. [230] considered the introduction of the non-local parameter λT into the one-dimensional non-local DPL model q(x, t + τq ) = −λ

∂ T (x + λT , t + τT ) ∂x

(2.41)

whose first-order expansion in the lagging time and second-order in the non-local parameter yields q(x, t) + τq



∂ T (x, t) λ2T ∂ 3 T (x, t) ∂q(x, t) ∂ 2 T (x, t) = −λ − + τ T ∂t ∂t 2 ∂t 3 ∂t∂ x

and the corresponding bioheat transfer equation c λ



ρb cb wb ∂T ∂2 T ∂T + τq 2 + (T − Tb ) + τq ∂t ∂t λ ∂t 2 4 2 3 ∂ T ∂ T λ ∂ T = + τT − T . ∂x2 ∂t∂ x 2 2 ∂x4



Li et al. [230] also investigated the fourth-order expansion λT and the second-order expansion in τq and τT in Eq. (2.41) τq2 ∂ 2 q ∂q + q + τq ∂t 2 ∂t 2

τT2 ∂ 3 T λ4T ∂ 5 T ∂2 T ∂T λ2T ∂ 3 T λ2T τT ∂ 4 T . + τT + + = −λ − − ∂t ∂t∂ x 2 ∂t 2 ∂ x 4! ∂ x 5 2 ∂x3 2 ∂t∂ x 3 The corresponding bioheat transfer equation could be found in the paper by Li et al. [230].

2.3 Triple-Phase-Lag Model

75

2.3 Triple-Phase-Lag Model The triple-phase-lag model is obtained by Choudhuri [307] (who extends the thermoelastic model suggested by Green and Naghdi [308]) by introduction to the DPL additionally to the relaxation times for the heat flux and the temperature gradient the relaxation time for the thermal displacement4 gradient [45, 307, 311, 312], sometimes simply the “phase lag of the thermal gradient” [313] or the “phase lag of the thermal displacement” [314]. q(r, t + τq ) = −[λ∇T (r, t + τT ) + λ ∇v(r, t + τv )],

(2.42)

λ is a positive material constant characteristics of the TPL theory (“the rate of thermal conductivity”). Falahatkar et al. [315] used the TPL model to study heat conduction in a laserirradiated tooth. The human tooth is composed of enamel, dentine and pulp with the unstructured shape and the uneven boundaries. Earlier Falahatkar et al. [316] studied this problem using the DPL model and found that the heat flux phase lag has the significant effect on the temperature profile at the early stages, while the temperature gradient phase lag is more important at the later stages. Akbarzadeh et al. [311] used the TPL model to investigate the heat transfer in the functionally graded hollow cylinder (earlier these authors exploited the DPL model to study the heat conduction in the one-dimensional functionally graded media [317]).

2.3.1 Non-local Triple-Phase-Lag Model Akbarzadeh et al. [196] derived the non-local three-phase-lag (NL TPL) constitutive equation for the non-Fourier heat conduction in the following form to include the effects of the thermal displacement (the scalar “history variable” used by Green and Naghdi [308, 310]) and its associated non-local length in the NL DPL thermal analysis: q(r + λq , t + τq ) = −[λ∇T (r + λT , t + τT ) + λ ∇v(r + λv , t + τv )]

(2.43)

4

Thermal displacement was introduced by H. von Helmholtz [196, 309]. It satisfies the definition v˙ = T . This quantity was used in models by Green and Naghdi [308, 310] as “a scalar history variable” t v = T (τ )dτ + v0 . t0

Bargmann and Favata [191] used the thermal displacement in their continuum thermomechanical analysis of the laser-pulsed heating in polycrystals including time derivative of the thermal displacement v˙ (i.e. the temperature), gradients of the thermal displacement ∇v and its time derivative ∇ v˙ as the state space variables for all quantities of the coupled equations—free energy, entropy, first Piola-Kirchhoff stress tensor, chemical potential, defect flux, heat flux.

76

2 Phase-Lag Models

where λq , λT and λv represent the correlating non-local lengths of heat flux, temperature gradient and thermal displacement gradient. The Taylor series expansion of the constitutive Eq. (2.43) with respect to either non-local lengths and/or phase lags leads to the alternative non-local phase-lag heat conduction models.

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Chapter 3

Phonon Models

Phonons are the quantized lattice vibrations (the elastic waves that can exist only at the discrete energies); phonons serve as the major heat carriers in the dielectric and semiconductor crystals (in contrast to metals [1–3]1 ) that undergo scattering in the course of propagation [8, 9]. In a simple monoatomic solid with only one atom per primitive cell one can have only three acoustic phonon branches corresponding to the three degrees of freedom of atomic motion, for monoatomic solids with two atoms per primitive cell such as diamond or diatomic compounds such as GaAs, one also has three optic phonon branches in addition to the three acoustic phonons. In compounds with a greater number of atoms and complex crystal structures, the number of optic phonons is more than three: if the crystal unit cell contains N atoms, then 3N degrees of freedom result in 3 acoustic phonons and 3N − 3 optical phonons [10]. The electrons and/or holes may also contribute to the heat transport [11, 12] (however, frequently the energy and entropy productions due to the presence of an electromagnetic field could be neglected). For example, the electronic thermal conductivity could be significant in the special cases, for example, in the channel layer of the AlGaN/GaN high electron mobility transistors (HEMTs) due to the formation of the high density electron two-dimensional gas [13, 14].2 1

In metals the relaxation of the heat and charge currents is closely related as they are carried by the same particles. The Wiedemann–Franz (WF) law states that the electronic contribution to a metal’s thermal conductivity is proportional to its electrical conductivity and temperature. However, violations of the WF law have been reported for in strongly interacting systems [4] such as Luttinger liquids [5], metallic ferromagnets [6], and underdoped cuprates [7]. 2 Electronic heat transport is also important in the layered quantum correlated materials (LCM). Moreover, LCM could be considered as the ideal platform to access the entire spectrum of the unconventional electronic heat transport regimes [15] The strong local Coulomb interaction can drive fast local thermalization processes leading to the rapid build up of a hot intralayer electronic temperature before relaxation via slower scattering © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. I. Zhmakin, Non-Fourier Heat Conduction, https://doi.org/10.1007/978-3-031-25973-9_3

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It is important to stress that both the phonons and the electrons are not in an equilibrium state during the heat transfer [16, 17].

3.1 Phonon Transport Regimes Phonons are the eigenstates of the atomic system and can propagate without dissipation in the purely harmonic solid that should have the infinite thermal conductivity [18, 19]. Peierls [20] stated that the origin of the thermal resistance is the combination of anharmonicity and the discrete nature of crystal lattice. However, the anharmonicity alone cannot induce resistance—an infinite thermal conductivity is expected if all the phonon scattering processes that conserve momentum [20]. Dissipation occurs due to the phonon scattering. One should consider the different mechanisms of the phonon scattering: the three-phonon inelastic scattering such as normal [21, 22] and Umklapp (both transverse and longitudinal [23]), the defect scattering (one should distinguish the scattering on the impurity and on the isotope atom [24]), scattering at the boundaries of the sample [25, 26] that in the general case should account for the root-mean-square roughness height [27] and the dependence of the phonon scattering probability on the phonon frequency, the incidence angle and the surface roughness [28, 29], the phonon–electron scattering [30]; the fourphonon processes could be important above the Debye temperature [31–34]. The Umklapp scattering, the isotope scattering and the impurity scattering are referred to as the momentum-destroying phonon scattering processes (R-scattering) [35]. Phonons, which are the main heat carriers in non-metallic systems, can undergo not only a diffusive regime that can be described by means of the Fourier law, but also a hydrodynamic regime (Poiseuille-like [36]) and a ballistic one [37]. Division of the phonon processes into the normal and Umklapp ones is also applicable to the amorphous materials [38], for example, polymers (except crystalline one) where the structural scattering can occur that reduces the phonon mean free path to a few monomer lengths [39]. Recent studies show that Umklapp scatterings are not necessarily resistive—no thermal resistance is induced if the projected momentum is conserved in the direction of heat flow [40]. This feature is especially important in anisotropic materials such as graphite where phonons dispersion along one direction (“soft” axis) is much softer than the other directions (“stiff” directions). Ding et al. proposed that a classification of N scattering and U scattering should be based on the projected phonon momentum in the heat flow direction and the condition of the quasi-momentum paths takes place. The interaction may thus act as a tuning parameter to control the relative interand intralayer energy exchange processes in LCM. As it increases, the two processes can effectively decouple, thus opening to novel electronic heat transport regimes. Mazza et al.[15] concluded that the subpicosecond thermal dynamics of electrons displays three regimes of heat transport: the ballistic propagation of energy; the hydrodynamic propagation of energy; the Fourier-like heat transport driving the recovery of thermal equilibrium.

3.1 Phonon Transport Regimes

91

conservation (3.57) should be modified to (k1 ) j + (k2 ) j = (k3 ) j + b j where j is the heat flux direction. Thus the scattering event is the normal scattering as long as b j = 0, which holds when b = 0 or b is a reciprocal lattice vector orthogonal to j [40]. By comparison with the exact solutions of the phonon BTE, Ding et al. demonstrated that the new classification of N-scattering and U-scattering processes leads to the more accurate predictions of the thermal conductivity using the Callaway model. Callaway [41] (see also [42]) obtained the following expression for the thermal conductivity     kB T 3 I22 kB I1 + λ= 2π 2 ν  I3 where θD /T

I1 =

−1

θD /T

τc ξ exp(ξ )[exp(ξ ) − 1] dξ, I2 = 4

0

0 θD /T

I3 = 0

τc 4 ξ exp(ξ )[exp(ξ ) − 1]−1 dξ, τN

1 − τc /τ N 4 ξ exp(ξ )[exp(ξ ) − 1]−1 dξ τN

where ξ = ω/(k B T ). Holland [21] tried to correct the Callaway approximation to improve the predictions of the thermal conductivity at the high temperatures. He accounted for the bifurcation of the phonon branches and separately considered the contributions of the LA and TA phonons. First-principles computations show that in the pure single crystals the dominant relaxation mechanism for the phonons near the Brillouin-zone centre is the normal phonon–phonon scattering while the dominant mechanism for phonons near the Brillouin-zone edge is the Umklapp phonon–phonon scattering [43]. The semiclassical Boltzmann transport equation (BTE) (also the Peierls– Boltzmann equation, BPTE) for the phonon distribution function [44] f is written as   ∂f ∂f ∂f +v·∇ f + F = (3.1) ∂t ∂P ∂t scatt where P is the momentum, F is the external force. In the disordered systems the phonon MFPs could be so short that the quasiparticle picture of the heat carriers is invalid and the BTE is not applicable [45]. To solve the Boltzmann one needs to know the phonon dispersion relation ω(k), the group velocity vg = ∂ω/∂ k and the rate of collisions.

92

3 Phonon Models

The phonon BTE combined with first-principles computations has achieved great success in exploring the thermal conductivity of various materials and understand the underlying phonon transport mechanism [46–53] provided the phonon branches remain well separated [54]. However, the convergence of the predicted value is a critical issue, leading to quite scattered results even for the same material [55]. The prediction accuracy depends on a number of input parameters [56–58]. Among them, the accuracy of interatomic force constants and the cutoff radius rc for the nearest neighbours’ interaction. Jiang et al. [55] studied, using the first-principles computations based on density functional theory, the origin for the convergence in the 2D materials (graphene and silicene) and in the bulk silicon. By iteratively solving the BTE and neglecting the four-phonon scattering process [59] in the framework of three-phonon scattering, the lattice thermal conductivity tensor κ αβ was obtained as [60] κ αβ =

 1 β f 0 ( f 0 + 1)(ωλ )2 vλα Fλ 2 kB T λ

where λ is the phonon mode index,  is the reduced Planck constant, ω is the phonon frequency, T is the temperature, is the system volume, f 0 is the equilibrium Bose– Einstein distribution and vλ phonon group velocity. Here α and β are the Cartesian β coordinate directions, and the term Fλ is the linear coefficient of the nonequilibrium phonon distribution function given by   β β β Fλ = τλ0 vλ + λ where τλ0 is the phonon lifetime obtained under single-mode relaxation time approxβ imation (RTA)—see below. The term λ is obtained by the fully iterative solution of the BTE, which is a correction term to reflect the phonon distribution deviation from the RTA scheme [55]. It was found that the thermal conductivity of these three materials exhibits diverse convergence behaviours with respect to the third-order interatomic force constants , which coincides with the strength of hydrodynamic phonon transport. In contrast to the fast convergence the thermal conductivity at very small rc in the 3D bulk silicon, the conductivity of the 2D materials exhibits a persistent dependence on rc in both graphene and silicene, which is more pronounced in graphene even up to the 15th nearest neighbour. By further analysing the phonon lifetime and scattering rates, the authors revealed that the dominance of the normal scattering process gives rise to the hydrodynamic phonon transport in both graphene and silicene, which results in long-range interaction and a large lifetime of low-frequency flexural acoustic phonons, while the same phenomenon is absent in the bulk silicon, thus proved the importance of long-

3.1 Phonon Transport Regimes

93

range interaction associated with hydrodynamic phonon transport in determining the thermal conductivity of the 2D materials. There are mainly two strategies to deal with the scattering term in the Boltzmann transport equation [61]: 1. the full integral scattering term; 2. relaxation time approximations. In the first strategy, the solution is obtained using the empirical phonon–boundary scattering term [46, 62–65] or via the Monte Carlo (MC) scheme developed to directly solve the BTE (3.1) under full scattering term [66]; however, this approach is very computationally expensive [67, 67]. The relaxation time approximations represent more feasible models for engineering applications. Frequently a number of assumptions are made: single mean relaxation time (grey approximation); linear phonon dispersion (Debye approximation) local nearto-equilibrium; local occupation number; ad hoc or fitted boundary scattering rates; neglecting the cross-mode correlations. The relaxation time approximation (RTA) [68] 

∂f ∂t

 =− scatt

f − f0 τ

where f 0 is the equilibrium distribution function. It is also called the single-mode relaxation time model approximation (SMRTA or just SMA) [44, 68–73]), the Callaway model, or the grey relaxation time model [74, 75] or the Bhatnagar–Gross–Krook (BGK) model [76, 77]. This approximation is known to underestimate the thermal conductivity [19, 78], including the conductivity of graphene, graphite, diamane and the ultrathin silicon nanowires [65, 79], especially when the strong normal scattering is present [40, 80]. The RTA, due to its simplicity, is useful as long as the size of the samples is quite large and the experiments are performed under the slow heating conditions. The new devices reduce the sample size down to few nanometres, and the heating times can be of the order of picoseconds. In these cases the RTA solution is often far from the real solution [81, 82]; thus the more accurate solutions are needed such as the direct solution, the iterative solution or the kinetic collective model (KCM) approach. Recently Torres et al. [83] investigated the transition metal dichalcogenides (TMDs, here MX2 (M = Mo, W ; X = S, Se) from the first principles (using the set of the interatomic force constants computed within the density functional theory (DFT)) by solving the Boltzmann Transport Equation and found that RTA can result in the underprediction of the thermal conductivity up to the 50%. In the local/linear non-equilibrium thermodynamics theory (LLNETT) the heat flux density current is a local and linear function of the temperature gradient ∇T that constitutes the only driving potential force. On the other hand, when the non-local and nonlinear effects are considered, other driving potential forces come to play with two kinds of terms ∇ n T and (∇T )n with n > 1; the former are non-local while the latter are nonlinear [84].

94

3 Phonon Models

Ezzahri et al. [84] studied the heat transfer in the semiconductors in one dimension and considered a semi-infinite geometry for the crystal that corresponds to TDTR and FDTR experiments using the modified Debye–Callaway model [85] in which both longitudinal and transverse phonon modes are included explicitly. The semiconductor crystal is assumed to have a cubic symmetry and is treated as a continuum, elastic and isotropic medium characterized by a linear phonon spectrum for each phonon branch polarization so that one considers heat transport due only to acoustic phonons and ignore any contribution from optical phonons. Callaway’s approximation of the collision operator in BPTE captures the effects of phonon– phonon normal scattering processes (N-processes). The contributions of the longitudinal and transverse acoustic branch polarizations are considered separately; furthermore, any conversion of the normal modes between both acoustic polarizations (inter-transitions) is neglected; only transitions within the same acoustic branch polarization (intratransitions) are considered. Ezzahri et al. wrote the steady-state Callaway’s form along the x-axis as vpm

m dUq, p

dx

=

m 0 Uq, p − Uq, p C τq, p

+

m gq, p C τq, p

,

here the deviational spectral energy density per phonon mode of the wave vector eq m m m q and the polarization p is introduced as Uq, p = ω(n q, p − n q, p ) where n q, p is the phonon distribution function at the absolute local thermal equilibrium temperature T , eq 0 m thus Uq, p = ω(n q, p − n q, p ) is the deviational equilibrium spectral energy density eq 0 per the phonon mode with n q, p and n q, p denoting the equilibrium Planck distribution functions at temperatures T and T0 , where T0 represents the reference temperature. m The term gq, p is given by m gq, p = −β

C τq, p

τq,N p

mv p

C −1 where (τq, = (τq,N p )−1 + (τq,R p )−1 is the “combined” relaxation time, β p is Callp) away parameter that has the dimension of a relaxation time. The authors found that the heat flux is formed under two different driving potential forces—the conventional Fourier’s temperature gradient dT /dx and the temperature Laplacian d2 T /dx 2 . The common formulas for the bulk thermal conductivity are based on the kinetic theory and can be derived from the Boltzmann transport equation in the relaxation time approximation by summation of the polarization dependence and integration of the frequency dependence [86]:



∞

κbulk =

s

0

Cvbulk dω

(3.2)

3.1 Phonon Transport Regimes

95

where C is the volumetric specific heat capacity per unit frequency, v is the group velocity, bulk is the bulk MFP, ω is the frequency, and s indexes the polarizations. It is assumed that the dispersion relation and the bulk MFPs are isotropic. This assumption is exact for gases (e.g. molecules, photons, free electrons) and for electrons and phonons in the amorphous materials. In the crystalline materials the dispersion relation of electrons and phonons depends on the direction and a more general form of the Eq. (3.2) is needed [49] (the thermal conductivity κbulk itself still can be is isotropic, for example, in crystals with cubic symmetries). The phonon dispersion relation could be determined experimentally or calculated using the elastic wave theory [23]. Yang and Dames [86] suggested to change the integration variable in the Eq. (3.2) from the frequency ω to the bulk MFP bulk to obtain    1 dbulk −1 Cvbulk =− dbulk . 3 dω s ∞

κbulk

(3.3)

0

The authors interpreted the change of variables as the change of the labelling scheme for the energy carriers from (q, s) to (bulk , s) and then applied Fubini’s theorem to exchange the orders of summation and integration to obtain ∞ κbulk =

K  dbulk 0

where K = −

1 s

3

 Cvbulk

dbulk dω

−1 (3.4)

is the thermal conductivity per MFP. This function is known as the MFP distribution or the MFP spectrum for the bulk thermal conductivity. Yang and Dames introduced the thermal conductivity accumulation function

α(α ) =

1 κbulk

α K  dbulk .

(3.5)

0

This function represents the fraction of the total thermal conductivity due to carriers with MFPs less than α . Eqs. (3.4) and (3.5) determine the range of MFPs that contribute to heat conduction flux. Often this range is described by a single “effective” (or grey) MFP

96

3 Phonon Models

gray

⎞−1 ⎛ ∞  1 Cvdω⎠ . = κbulk ⎝ 3 s

(3.6)

0

The Eq. (3.6) corresponds to the MFP distribution that given by the Dirac δ function with weight κbulk centred on gray that is a good approximation in systems where the real MFP distribution is narrow (e.g. including the ideal gases or the free electron gases). However, in the systems with the strongly frequency-dependent scattering (e.g. the semiconductor crystals), the distributions can be quite broad [86]. As noted by Yang and Dames [86], as long as the structure’s characteristic length L c is much larger than the thermal wavelengths of the energy carriers and there is no coherence effects, the group velocity and spectral heat capacity in the nanostructure are identical to those in bulk, so the only effect of the nanostructuring is to reduce the effective MFP by scattering at boundaries and interfaces. Thus, the nanostructure thermal conductivity could be written as 

∞

κnano,t =

s

Cvnano dω

(3.7)

0

where nano (ω, s) < bulk (ω, s) and the subscript t indicates the “type” of the geometry, e.g. “wire” or “film”. After change variables from ω to bulk one gets ∞ κnano,t =

K 0

nano dbulk . bulk

(3.8)

For a wide variety of the geometries (the wires of arbitrary cross section, the thin films both in-plane and cross-plane, the porous media with the arbitrary pore shapes and distributions) the ratio nano /bulk depends only on the Knudsen number K n = bulk /L c without any other explicit dependence on polarization, group velocity, frequency, etc. [86]. Often the so-called Callaway dual relaxation approximation [41] is exploited [87, 88]   ∂f f − fλ f − f0 =− − , ∂t scatt τN τR where f λ is the distribution function of the uniformly drifting phonon gas, τ N is the relaxation time for the normal scattering and τ R is the relaxation time for the Umklapp process. The Callaway dual relaxation approximation allows a simple separation of N-processes and U-processes. Callaway dual relaxation approximation provides better estimation for the 2D materials than the single-mode relaxation time approximation [89, 90].

3.1 Phonon Transport Regimes

97

Although the Callaway’s model has been widely used in modelling hydrodynamic phonon transport [35, 67, 88, 91–94], the direct numerical solution of the BTE using Callaway’s scattering term has been advanced only recently [91, 95–97]. Guo et al. [98] studied the thermal phonon vortices in the graphene ribbon using the deterministic discrete ordinate method (DOM) [95] for the solution of the phonon Boltzmann equation under the Callaway’s dual relaxation model. The phonon scattering rates of the normal and resistive processes are acquired from ab initio calculation without need of any empirical input parameters. In recent years there is the increasing interest in modelling hydrodynamic phonon transport in the graphene ribbon. The first works were mainly focused on the temperature profile and the thermal conductivity in-plane [67, 95, 99] and cross-plane [91, 92] in the relatively simple rectangular geometries and usually considered the steadystate phonon transport; the transient transport was studied only in few works [93, 96, 97]. The phonon transport falls into the hydrodynamic regime when the phonon scattering strengths of normal scattering (N), Umklapp scattering (U), isotope scattering (I) and extrinsic scattering rates (ε) satisfy this relationship N  U + I  ε [100]. The phonon transport in the hydrodynamic regime could be described by the the Guyer–Krumhansl Eq. (3.21) where the contribution of the heat flux may be neglected with respect to its spatial derivatives [37]—a signature of the hydrodynamic phonon transport is the collective drift motion of phonons that manifests itself in the phonon distribution function and is similar to the viscous fluid flow [101]. It is expected that the vortex phenomenon of phonons motion may take place in crystals, in particularity, in the graphene ribbons [102–105]. Yet it has attracted less attention than the electron hydrodynamic flow.3 There is a demonstration of heat vortex in a rectangular graphene ribbon based on the macroscopic phonon hydrodynamic equation in a recent paper [108]. Guo et al. [98, 109] wrote the phonon Boltzmann Eq. (3.1) under the Callaway’s dual relaxation model as eq

eq

f (Tl,N , u) − f f (Tl,R ) − f ∂f + vg · ∇ f = R + N ∂t τ R (ω, p, T ) τ N (ω, p, T )

(3.9)

where the phonon distribution function f = f (x, p, ω, s, t) = f (x, p, K , t), x is the spatial position, ω is the phonon angular frequency, p is the phonon polarization, K is the 2D wave vector that is assumed to be isotropic, i.e. K = |K |s where s is the 3

Strongly interacting electrons can move in a neatly coordinated way, reminiscent of the movement of viscous fluids, e.g. viscous flows interactions facilitate transport that can allow conductance to exceed the fundamental quantum-ballistic limit for nanoscale electron systems. This effect is particularly striking for the electron flow through a viscous point contact, a constriction exhibiting the quantum-mechanical ballistic transport at T = 0 but governed by electron hydrodynamics at a higher temperature [106]. Such flows have been observed in ultraclean GaAs, graphene and PdCoO2 [103]. For example, the experiments by Moll et al. [107] on the ultrapure PdCoO2 that show a large viscous contribution and yielded an estimate of the electronic viscosity compared to the water viscosity at the room temperature. The Dirac fluid—a strongly-interacting quasi-relativistic electron- hole plasma—in graphene demonstrates the hydrodynamic behaviour [4].

98

3 Phonon Models

unit directional vector in 2D coordinate system, t is the time. v = ∇ K ω is the group velocity calculated by the phonon dispersion. The relaxation times of the resistive and normal phonon scattering processes are denoted by τ R (ω, p, T ) and τ N (ω, p, T ), with their local equilibrium distribution functions, respectively, defined as: eq

f R (Tl,R ) =

and eq

f N (Tl,N , u) =

1  ω −1 exp k B Tl,R 

1  . (ωμ − k · u) −1 exp kB T 

Here with Tl , R and Tl , N are the local pseudo-temperatures and u is the local phonon drift velocity, which are determined by the energy and momentum conservation conditions of scattering processes. The two pseudo-temperatures are introduced as the mediate mathematical quantities to ensure the energy conservation of relaxationtype scattering term [110], the local temperature T defined from the local energy density represents the physical temperature of the phonon system, k is the phonon wave vector,  is the reduced Planck constant. The phonon BTE equation is written in terms of a deviational energy form as eq

eq

e (Tl,R ) − e e N (Tl,N , u) − e ∂e + vg · ∇e = R + ∂t τ R (ω, p, T ) τ N (ω, p, T )

(3.10)

where the deviational distribution functions of the energy density are [95, 96] eq

e=

ωD( f − f R (T0 )) , 2π eq

eq

eR =

eq

ωD( f R − f R (T0 )) , 2π

eq

ωD( f N − f R (T0 )) 2π

eq

eN =

(3.11)

(3.12)

eq

(3.13)

where D = |K |/(2π |v|) is the phonon density of states, T0 is the reference temperature. For the phonons D(ω) is proportional to ω2 if the dispersion is linear [111]. For a small temperature difference T  T0 and a small drift velocity K · u  ω the equations (3.12, 3.13) could be linearized eq

eR =

K ·u T − T0 T − T0 eq , eN = + CT 2π 2π 2π ω

3.1 Phonon Transport Regimes

99

where

eq

C(ω, p, T0 ) = 2π

∂e R |T =T0 ∂T

is the mode specific heat at T0 . The Eq. (3.10) in the stationary case is rewritten in the dimensionless form as v eq, eq, · ∇x  e = β R (e R − e ) + β N (e N − e ) |v| where x =

eq

eq

e x e e L L eq, eq, , e = , eR = R , eN = N , βN = , βN = , L C T0 C T0 C T0 |v|τ N |v|τ N

the parameters β N and β R represent the ratio of the system size L to the phonon mean free path of the normal (l N = |v|τ N ) and resistive (l R = |v|τ N ) scattering. Guo et al. distinguished three heat transfer regimes: diffusive: β N = 0 and β R → ∞, ballistic: β N = β R = 0, hydrodynamic: β N → ∞ and β R = 0. The authors used the grey model and the Debye approximation, i.e. ω = v|K | [92, 97] and compute the zeroth and the first-order moments of phonon BTE (3.10) to get ∂E ∂q q + ∇ · q = 0, + ∇ · Qˆ = − ∂t ∂t τR where E=

  p

ed dω, q =

 

ved dω,

p

Qˆ =

 

vved dω,

p

the integral is over the whole 2D solid angle space and the frequency space. In the diffusive limit the resistivity scattering dominates the phonon transport and causes the heat dissipation; at the steady state, the distribution function can be approximated as [98] eq eq e ≈ e R − τ R v · ∇e R and the thermal conductivity in the Fourier law q = −κd ∇T in the diffusive limit κd =

1 C|v|2 τ R . 2

The authors introduce the vorticity that is defined as the curl of heat flux  q · dr l

100

3 Phonon Models

where l is an arbitrary closed curve inside the thermal system, dr is the unit tangent vector of the curve l in a clockwise direction. In the diffusive regime 

 q · dr = −κd l

dT · dr = 0. dr

l

Thus, there are no heat vortices inside the system in the diffusive regime. In both hydrodynamic (the normal scattering conserves momentum and causes no thermal resistance [67]) and ballistic (rare phonon–phonon scattering) regimes, the heat flux is conserved in the interior domain [62, 63, 92] and there is no heat dissipation ∂q + ∇ · Qˆ = 0. (3.14) ∂t Although Eq. (3.14) is valid in the ballistic and hydrodynamic limits, these regimes are different. In the ballistic regime [67] the momentum conservation is satisfied due to rare phonon–phonon scattering and the phonon mean free path is much larger than the system characteristic length: at the steady state, the phonon distribution function does not vary with the spatial position until scattering with the boundaries and the appearance of heat vortices depends on the geometry settings or/and boundary conditions. However, it is hard to unmistakably distinguish the (quasi) ballistic or hydrodynamic phonon transport only by the wave like propagation of the heat, as noted by Zhang and Guo [112]: the wavelike propagation of heat can be observed in both the hydrodynamic [113–115] and (quasi) ballistic regimes [116, 117]. Up to now there is no general consent on the exact definition of the terms “hydrodynamic phonon transport” or “phonon hydrodynamic”, in particularity, whether the strong N scattering of phonons is required—see discussion in the paper by Zhang and Guo. These authors themselves assumed that the hydrodynamic phonon transport occurs when the N scattering is much stronger than the R scattering. Recently Zhang and Guo [112] suggested to use a transient heat conduction to distinguish the hydrodynamic and (quasi) ballistic phonon transport regimes. The authors studied the transient heat propagation in the homogeneous thermal system using the phonon Boltzmann transport equation (BTE) under the Callaway dual approximation. The authors considered the three-dimensional materials; the phonon dispersion was not accounted, and the Debye model was used that is acceptable for studies of the hydrodynamic phonon transport in the bulk materials [67, 92, 118, 119]. The temperature difference in the domain was assumed to be small compared to the reference temperature T0 , i.e. |T − T0 |  T0 . The phonon BTE for the deviational phonon distribution function of energy density e = e(x, s, t) which depends on spatial position x , unit directional vector s and time t is written as eq

eq

e − e eN − e ∂e + vg s · ∇ x e = R + ∂t τR τN

(3.15)

3.1 Phonon Transport Regimes eq

101

eq

where e R and e N are the equilibrium distribution function of the R scattering and N scattering linearized by the specific heat C eq

e R (T ) =

C(T − T0 ) C(T − T0 ) C T s · u eq , , e N (T ) = + 4π 4π 4π vg

τ R and τ N are the relaxation time of the R scattering and N scattering, respectively, vg is the value of group velocity, u is the drift velocity. To close the phonon BTE Wang and Guo used the energy conservation for both N scattering and R scattering 

eq

eN − e d = τN



eq

eR − e d = 0, τR

where the integral is over the whole solid angle space and used the momentum conservation for the N scattering 

eq

s

eN − e d = 0. τN

The macroscopic spatio-temporal distributions such as the local energy E(x, t) or the heat flux q(x, t) are obtained by taking the moments of the phonon distribution function [67, 95, 96, 112]   E(x, t) = ed , q(x, t) = vg sed . The drift velocity is given by [92] u = 3q/C T . To wrote the dimensionless version of the BTE Eq. (3.15) the authors related the spatial scale to L, the temporal scale to tr e f = L/vg , the distribution functions to er e f = CT /4π (T is the temperature difference in the domain) and introduced two Knudsen numbers which represent the ratio between the phonon mean free path of the R scattering and N scattering to the characteristic length K nR =

vg τ R , L

K nN =

vg τ N L

and mainly determine the transient heat propagation to get e N − e ∂e e R − e  e = + s · ∇ + . x ∂t  K nR K nN eq,

eq,

(3.16)

Evidently, in the diffusive limit K n R → 0, K n N = ∞ the Fourier law is valid; in the ballistic limit K n R = ∞, K n N = ∞ the BTE Eq. (3.15) simplifies to

102

3 Phonon Models

∂e + vg s · ∇ x e = 0. ∂t When the N scattering dominates heat conduction and the R scattering could be neglected, the phonon BTE Eq. (3.15) becomes eq

e −e ∂e + vg s · ∇ x e = N . ∂t τN

(3.17)

Taking the zeroth- and first-order moments of the Eq. (3.17) leads to [112] ∂E + ∇ x · q = 0, ∂t where Qˆ =

∂q + ∇ x · Qˆ = 0 ∂t

 vg ssed .

It is follows from the above equations ∂2 E = ∇ x · ∇ x · Qˆ ∂t 2 eq

that can be transformed into a wave equation for the temperature by assuming e = e N [112]. Zhang and Guo established that the transient temperature could be lower than the lowest value of the initial environment temperature in the hydrodynamic regime within a certain range of time and space. This phenomenon disappears in the ballistic or diffusive regime and in the quasi-one-dimensional simulations and thus could be used to distinguish the hydrodynamic phonon transport from the heat propagation in the ballistic regime. In the hydrodynamic regime, the phonon mean free path is much smaller than the local characteristic length due to the frequent normal scattering [62, 63] . At the steady state, the phonon distribution function can be approximated as [108, 120] eq

eq

e ≈ e N − τ N v · ∇e N . Thus in contrast to the ballistic regime the phonon distribution function varies with the spatial position. This conclusion was confirmed by the experiments exploiting the picosecond laser irradiation [121]. Zhang et al. [109] used the discrete ordinate method [95, 122] to solve the stationary phonon BTE. The authors studied the phonon dynamics in several situations: the heat vortices in the 2D ribbon and in the two-dimensional porous materials, including the porous graphene, using both the grey and non-grey models. Guo et al. [98] studied the phonon vortex dynamics in the rectangular and the Ttype graphene ribbon that influenced by the temperature, the ribbon size and carbon

3.1 Phonon Transport Regimes

103

isotope. The wide MFP distribution of the resistive phonon scattering processes is the crucial factor that destroys the vortex hydrodynamic effect. The phonon vortex configuration in the T-type graphene ribbon is found to depend on the height–width ratio, with the multihierarchy primary, secondary and ternary vortexes obtained. Shang and Lü [123] studied the hydrodynamic phonon heat transport in the twodimensional (2D) materials. The authors started from the Boltzmann equation with the Callaway model (3.9) and multiply each term by ωv and summing over all wave vectors and phonon indices to get κˆ q q1 ∂q + · ∇T = − − ∂t τR τR τN where q=

 s

dk ωsk vsk f sk (2π )2

is the heat current density, q 1 is the second term in the expansion in a small parameter ε = τ N /τ R (i.e. the case of the scattering rates of N-process being much larger than that of R-process is considered) q = q 0 + εq 1 + ε2 q 2 + · · · , and κˆ = τr q =

 s

∂ f sk dk ωsk vsk vsk (2π )2 ∂T

is the thermal conductivity tensor. The authors considered the case of small u when q 1 = 0 and finally derived a 2D Guyer–Krumhansl-like equation

κ0 q ∂q + ∇T + = η ∇ 2 q + 2∇(∇ · ∇q) − ζ ∇(∇ · q), ∂t τR τR

(3.18)

where the κ0 is the zeroth-order thermal conductivity, η and ζ are the first and second viscosity coefficients [123]. Shang and Lü considered the heat flow through the nanoribbon with the length L(x ∈ [0, L]) and the width w(y ∈ [0, w]) with the temperature difference applied along the ribbon (x direction). At the steady state neglecting q/τ R the Eq. (3.18) reduces to ∂ T κ0 d2 q = 2 dy ∂ x ητ R that gives the parabolic heat for the non-slip boundary condition q(0) = q(w) = 0 and the total flux that scales as w 3 —that is the signature of the Poiseuille flow (for the diffusive transport, the heat current scales linearly with the width) [123]. For computation of the pattern of vortices in the stationary 2D flow the authors introduced a streamfunction and used a simplified version of the Eq. (3.18)

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3 Phonon Models

η∇ 2 q −

q = C L vg2 ∇T. τR

The presence of the viscous terms in the Guyer–Krumhansl-like equation of the phonon hydrodynamics allowed Huberman [124] (similar to Cimmelli et al. [125, 126], see Sect .4.2) to introduce a “thermal Reynolds number” Rether m =

vss L |v|2 τ N

where vss is the speed of second sound. However, as noted by Zhang et al. [109], this definition is hardly fruitful since the “thermal Reynolds number” has only formal resemblance to the Reynolds number (and its importance) in fluid dynamics: there is no convective term in the Guyer–Krumhansl-like equation and this equation actually is similar to the Stokes equations—the low-Reynolds-number form of the Navier–Stokes equations. Recently the progress in studying the hydrodynamic phonon transport in 2D materials was reviewed by Yu et al. [127]. The alternative approach is based on the similarity of phonons and photons [128]. They are both bosons—their equilibrium distribution function is the Bose–Einstein distribution. Both phonons and photons are considered as non-local quasi-particles with zero mass carrying energy ω when the length scale exceeds the coherence length for phonons and the wavelength for photons. It is possible to define the phonon radiative intensity by analogy with the thermally emitted photons 1 D(ω)n ω (x, t, s)ωvω , Iω (x, t, s) = 4π where D(ω) is the number of modes per unit volume, vω = dω/dk is the group velocity and n ω is the average number of phonons with angular velocity ω moving in the direction of the unit vector s averaged over all branches of the dispersion curves of the phonons. The frequency-dependent BTE (FD-BTE) is useful for both transient and steadystate cases (see, e.g. [82, 129]). However, despite its high accuracy, the FD-BTE has two major limitations [130]: • It requires knowledge of the phonon dispersion curves and the phonon–phonon relaxation times, which are generally unknown for new materials; • FD-BTE requires the discretization of the dispersion curves and thus dense sampling close to zero group velocity zones. Furthermore, as all polarizations have to be included, it becomes computationally prohibitive for materials with complex materials, where phonon dispersion curves have several branches.

3.1 Phonon Transport Regimes

105

The internal energy e and the heat flux q could be presented as the spectral and the total quantities eω (x, t) =

1 vω



 f (x, t, ω, s)d ,

q ω (x, t) =

4π

and

f (x, t, ω, s)sd , 4π

ωmax 

ωmax 

eω (x, t)dω,

e(x, t) =

q(x, t) =

0

q ω (x, t)dω, 0

where ωmax is the cutoff frequency determined by the crystal lattice, d is an elementary solid angle around the direction s. The propagation of phonons is governed by equation dn ω n0 − nω ∂n ω + vω s · ∇n ω = |coll ≈ ω , ∂t dt τω or by the equation for the radiation intensity (phonon radiative transfer equation [128, 131]) 1 ∂ Iω 1 0 + s · ∇ Iω = (I − Iω ), (3.19) vω ∂t ω ω where 1/ω = κω is the analogue of the absorption coefficient in the photon transport and D(ω)n 0ω (T )vω Iω0 (T ) = 4π is the equilibrium radiation intensity. Integration of (3.19) over the whole spectrum and all directions leads to ∂e +∇ ·q = ∂t

ωmax 

⎛ ⎝4π Iω0 −

0



⎞ Iω d ⎠ dω.

4π

When the mean free path of phonons is small compared to the characteristic length of the sample L/ = κω L  1 (the diffusive regime) the radiation intensity can be expanded in a Taylor series Iω = I0 +

1 1 I1 + I2 + . . . κω L (κω L)2

Substitution of this expansion into the Eq. (3.19) gives to the first order Iω = Iω0 −

1 s · ∇ Iω0 . κω

106

3 Phonon Models

It is possible to define the so-called Rosseland thermal conductivity [128] 4π λR = 3

ωmax 

0

1 ∂ Iω dω κω ∂ T

and thus get q ≈ −λ R ∇T . The Rosseland approximation ignores the directional nature of the energy propagation. This feature is accounted for in the PN methods that are based on the expansion in terms of the spherical harmonics (that form the complete orthogonal basis) truncated at order N . The equation for the radiation intensity (3.19) in the stationary case is written as s · ∇ Iω +

1 1 0 Iω = I . ω ω ω

The decomposition of the intensity is written as Iω (x, s) =

∞  l 

Ilm (x)Ylm (s)

l=0 m=−l

where Ylm are the spherical harmonics—the angle-dependent azimuthally symmetric functions defined as

Ylm ( )

m + |m|  1 (l − |m|)! 2 imφ |m| 2 = (−1) e Pl (cos ψ) (1 + |m|)! |M|

where φ and ψ are the polar angles (zenith and azimuth), Pl are the associated Legendre polynomials. Experience shows that truncation at N = 1 (the P1 method) provides the sufficient accuracy in most cases Iω (x, s) = I00 (x)Y00 (s) + I1−1 (x)Y1−1 (s) + I10 (x)Y10 (s) + I11 (x)Y11 (s), the next approximation is P3 resulting in the increase of the number of unknown functions from 4 to 16 [128]. The presence of the phonons with a broad spectrum means that there is no the single value of the MFP that determines the heat transfer regime. The range of values of the effective mean free path of phonons is rather wide [86, 132], e.g. for silicon at the room temperature values from 40 to 260 nm are mentioned [133]. The first principal methods predict the different MFP distributions for different materials, e.g. in Si half of the heat is carried by phonons whose mean free paths is greater than 1μm [134] while in diamond 80% of heat is carried by phonons with

3.2 Guyer–Krumhansl (GK) Equation

107

the mean free paths between 0.3 and 2μm [43]; more than 95 % of heat in sapphire is carried by phonons with the MFP shorter than 1 μ m [135]. The formation of the nanometre-scale phonon hotspots in the semiconductor devices with the highly non-equilibrium distribution of phonons by the scattering of the high-energy electrons is important for the analysis of the conduction device cooling. This process involves the following nanoscale conduction phenomena [136]: the phonon–boundary scattering that reduces the thermal conductivity; the intense electron-phonon coupling in the transistor channel; the weak anharmonic coupling between the slow longitudinal optical (LO) phonons generated by the scattering with high energy electrons and faster acoustic phonons. Dong et al. [133] in the studies of the heat transfer in the nanostructures distinguishing the mean free paths of phonons in the normal (l N = vτ N ) and resistive √ (l R = vτ R ) scatterings and suggested to define an effective length scale as l = l N l R /5 and the Knudsen number as K n = l/L. The authors also considered different boundary conditions for the phonon gas motion: the Maxwell boundary [18]; the backscattering boundary; the MFP-proportional slip boundary.

3.2 Guyer–Krumhansl (GK) Equation Guyer and Krumhansl [120, 137] solved the linearized phonon Boltzmann equation assuming that the normal scattering rates are much larger than the resistive scattering rates that is valid at low temperatures. Earlier Krumhansl [138] solved the BTE in the presence of sole N processes. Nielsen and Shklovskii [139] noted that the intense N-processes lead to the phonon distribution function in the form of “Planck function with drift”; the drift velocity in the absence of the external influence relaxes to zero through the U-processes. Guyer and Krumhansl developed a phenomenological coupling between the phonons and elastic dilatational fields caused by the lattice anharmonicity. When the bulk phonon MFP is large compared to the sample size—the Knudsen number becomes comparable to (or higher than) 1, the heat transport is no longer diffusive, but ballistic. Moreover, when the MFP between successive collisions becomes large, there is a direct connection among non-adjacent regions of the system with different values of the temperature [140]. In this case the phonon-gas description is equivalent to the “Knudsen flow” [141] or “ballistic transport”. Such heat conduction can occur without energy dissipation [142] because phonons can ballistically travel for hundreds of nanometres. The extreme situation when boundary scattering dominates over the intrinsic scattering is often called the “Casimir limit” [143, 144] owing to the Casimir’s investigations using cylindrical rods [145]. However, the control over directionality of the ballistic transport and thus its practical use remains a challenge, as the directions of the individual phonons are chaotic. Anufriev et al. [146] demonstrated a method to control the directionality of the ballistic phonon transport using the silicon membranes with arrays of holes that

108

3 Phonon Models

form the fluxes of phonons oriented in the same direction (the silicon films with the controlled, periodic arrays of the cylindrical holes can be fabricated by patterning the silicon-on-insulator (SOI) device layer [144, 147]). To show the potential for practical applications, the authors introduced the thermal lens nanostructures and demonstrated the evidence of the nanoscale heat focusing. Guyer and Krumhansl established conditions when the Poiseuille flow can considerably contribute to the thermal conductivity. The Poiseuille regime emerges when energy exchange between phonons is frequent enough to keep the local temperature well defined, and the Umklapp collisions are very rare. Recently Martelli et al. [148] experimentally proved the existence of the Poiseuille flow of phonons in the single crystal of perovskite Sr1−x N bx T i O3 at low temperatures. The authors found that the thermal conductivity varies faster than cubic temperature dependence (that corresponds to the ballistic heat conduction regime) in a narrow (6 K < T < 13 K) temperature window. First-principles simulations seem to support the hydrodynamic transport claim in Sr T i O3 , although a much smaller grain size (4 μm) than the actual sample size (100 μm) had to be assumed to achieve agreement between simulation and experiment [149]. The thermal conductivity measurements in solid helium 4 in the size effect regime shows that thermal conductivity scales with the temperature with an exponent of 7–8, because the relaxation time for normal scattering has a strong temperature dependence [29, 150]. In the suspended graphene [63] and other 2D systems [62, 83] the Poiseuille phonon flow could be observed at higher temperatures than in the 3D systems, because in the 2D systems the normal momentum conserving phonon–phonon collisions are two orders of magnitude higher than those in the 3D systems [37]. Such phonon transport phenomena as the phonon interference [151–153] and the phonon localization [154, 155] are observed in the 2D materials. For example, Chen et al. [153] found in numerical simulations that the thermal conductivity of the Lorentz gas model oscillates with the degree of the boundary roughness due to the interference between the transporting particles and the periodic boundary conditions. In the 2D materials the phonon MFP can be affected by various factors [127], such as doping [156], defects [157], the edge chirality [158], stress [159], and the substrate coupling [160] . Sellitto et al. [37] studied the thermal transport in a 2D strip of width w assuming that the thickness of the layer is much smaller than its other two characteristic sizes. The authors used the simplified version of the Guyer–Krumhansl equation q = −λ∇T + a 2 (T )2 ∇ 2 q the general solution of which (for the case of the constant temperature gradient along the x axis) y  y T + A exp + B exp − . q(y) = λ L l l

3.2 Guyer–Krumhansl (GK) Equation

109

The slip boundary condition for this case is written as q and a symmetry condition

w 2 

 = C1l

∂q ∂y

∂q ∂y

 y=w/2

 =0 y=0

lead to the solution ⎡

⎛

⎢ ⎜ ⎜ q(y) = λ ⎢ ⎣1 − ⎝

1 1 + C1 tanh

y



1 2K n



⎞⎤

⎟⎥ T cosh l ⎥ ⎟  ⎠⎦ L 1 cosh 2K n

where K n = l/w is the Knudsen number. At the low values of K n (Fourier diffusive regime) the limit heat flux qdi f f,lim = λT /L; at the moderate values of K n the Poiseuille phonon flow is realized;  at the  large values of K n (ballistic regime) the heat flux qball,lim = λC1 tanh 2K1 n T /L. The Poiseuille flow is realized for the steady heat conduction in a cylinder described by the equation [161] 2 ∇ 2 q = λ∇T, λ∇T with the solution given by q(r ) = A(R 2 − r 2 ), A = − 2 .  Usually the Guyer–Krumhansl equation is written in the form τ

∂q + q = −λ∇T + β q + β

∇ · ∇q, ∂t

(3.20)

where β and β

are the Guyer–Krumhansl coefficients (in the rarefied gas they are related to the relaxation times of the Callaway collision integral [162]) or τ

∂q + q = −λ∇T + 2 (∇ 2 q + 2∇ · ∇q), ∂t

(3.21)

where  is the mean free path of phonons, λ is the Ziman limit for the bulk thermal conductivity λ = ρcV τ c¯2 /3 where ρ is the mass density, cV is the specific heat per unit mass at constant volume, c¯ is the average speed of phonons [163]. Evidently, when  = 0 the Eq. (3.21) reduces to the Cattaneo relation (2.5). The Guyer–Krumhansl equation is the macroscopic heat transport equation that provide the hydrodynamic description of the phonon system [164]. Similar equation is used to study the transport regimes and inhomogeneous superfluid turbulence in liquid helium II [165, 166].

110

3 Phonon Models

The Guyer–Krumhansl equation is successfully applied for the heat conduction in rocks, foams and biological material [167, 168]. Calvo-Schwarzwälder et al. investigated the one-dimensional growth of a solid into a liquid bath, starting from a small crystal and nanoscale solidification using the Guyer–Krumhansl and MaxwellCattaneo models of heat conduction [169, 170]. Sellitto and Alvarez [171, 172] used the Guyer–Krumhansl Eq. (3.21) to study the heat removal from the hot nanostructures characterized by the radius r0 and the thickness h 0 through the graphene layer with the outer radius r g and the thickness hg. Increase of the integrated circuits (IC) density and of the clock speed makes the heat removal one of the major problem of the chip design. The small temperature differences divided by the minute length yield the very high-temperature gradients [172–174]. Incorporation of the graphene could yield the devices that are faster and more reliable [175]. The heat removed from the hot spot per unit time through the graphene layer is higher than through the usual materials [171] since the mean free path of phonons in the graphene l is very long—of the order of hundreds nanometres at the room temperature [176–180]. 4 The authors considered the simplified case when the heat is carried away radially along the layer without being transversally transferred to the environment. The radial profile of the heat flux is given (in the first-order approximation in dr ) by an equation dq + q(r ) = 0 dr that results in q(r ) =

 r

(3.22)

(3.23)

where  = Q/(2π h g ) is a constant value, Q is the heat removed from the device per unit time. 4

Hu et al. [181] investigated the diffusive thermal conductivity of the diamond-like bilayer graphene combining the first-principles calculation and the phonon Boltzmann transport equation. This diamond-like nanoscale film formed by the two-layer graphene upon compression exhibits stiffness and hardness comparable to diamond. The diamond-like few-layer graphene sheets have attracted tremendous interest, because of their robust stability and excellent mechanical properties [182–188]. Diamane has high thermal conductivity due to its high phonon group velocity. Hu et al. computed the group velocity projected phonon dispersion of diamane. They found that the maximal phonon group velocity occurring in longitudinal acoustic (LA) branch is up to 18 Km/s, which is higher than most 2D materials and near to graphene (about 21 Km/s). The calculations also showed that the lowest two optical branches exhibit relative high phonon group velocity, which are close to 9 and 11 Km/s, respectively. Cocemasov et al. explored the effect of various rotation angles on phonon properties of twisted bilayer graphene [189]. In twisted bilayer graphene, phonon modes from different high-symmetry directions in the Brillouin zone can form hybrid folded phonons, which depend strongly on the rotation angle. Moreover, the lattice specific heat also depends strongly on the twist angle in bilayer graphene at low temperature [190]. Recently, the covalent-bonded bilayer graphene was synthesized experimentally [191].

3.2 Guyer–Krumhansl (GK) Equation

111

Using the relation (3.23) in the steady version of the Eq. (3.21) yields for r > r0   2 dT 1 l λ0 = 3 − dr r r and after integration (T (r ) ≡ T0 for r ≤ r0 )  T (r ) = T0 λ0



l2 2r02



r2 1 − 02 r

 + ln

 r  0

r

.

(3.24)

In the absence of the non-local effects the Eq. (3.24) is simplified to T (r ) = T0

  r0  . ln λ0 r

(3.25)

Assuming that the temperature of the outer edge of the graphene layer is Tg , it is possible to estimate the role of the non-local effects on the intensity of the heat removal from the device ⎡ ⎡ ⎤ ⎤ T T − T − T g 0 g 0   ⎦ .  Fourier = λ0 ⎣   ⎦ , nonlocal = λ0 ⎣ 2  r02 r0 l 1 − + ln ln rrg0 r2 rg 2r 2 0

g

Evidently, the greater the external radius of the graphene layer, the less important the role of the non-local effects. To account for the lateral heat transfer from the graphene layer, Sellitto and Alvarez [171] introduced the heat-exchange coefficient between graphene and environment σ and modified the Eq. (3.22) to q 2σ dq + = − [T (r ) − Tg ]. dr r hg Note that combination of the Guyer–Krumhansl equation with the energy balance produce the parabolic partial differential equation for the temperature and thus the paradox of the infinite velocity of the perturbations propagation [192, 193]. Jou and Cimmelli formulated the simplest heat conduction equation that accounts for the non-local effects written as [140] τ

∂q + q = −λ∇T + 2 ∇ 2 q. ∂t

(3.26)

Tzou reported on phenomena such as thermal wave resonance [194] and thermal shock waves generated by a moving heat source [195]. Both et al. [196] investigated the one-dimensional version of Eq. (3.26) and considered the so-called Fourier resonance condition

112

3 Phonon Models

2 = κ. τ The solutions of the energy conservation equation with the heat flux (3.26) show wavelike characteristics if 2 /τ > κ and are over-diffusive in the case 2 /τ < κ [197, 198]. In some cases in the nanosystems it is possible that 2 ∇ 2 q ≈ K n 2  q since the Knudsen number could be much greater than unity [140, 199]. Then the non-local term the Eq. (3.26) is more important than the heat flux q itself and the equation reduces to [163, 200, 201] ∇2q =

λ ∇T. 2

(3.27)

Equation (3.27) is similar to the Stokes equation of the classical hydrodynamics ∇2 =

1 ∇ η

that justifies the term “hydrodynamic regime” of the heat conduction and allows to define the “viscosity” of phonons in terms of the thermal conductivity and the mean free path. The hydrodynamics is a fundamental ingredient of the semiconductor heat transport in the nanoscale at the room temperature allowing to explain many experimental situations in terms of the hydrodynamic concepts, such as friction and vorticity [202]. Recently Machida et al. [203] showed that making the graphite samples thin expands the hydrodynamic regime from cryogenic to the room temperatures. The researchers measured an extremely high thermal conductivity in the very thin graphite samples. The Guyer–Krumhansl equation was later obtained in the framework of the ninemoment phonon hydrodynamics by Struchtrup et al. [204, 205]. Fryer and Struchtrup used the phonon Boltzmann equation and the Callaway model for the phonon–phonon interaction; for the phonon interaction with the crystal boundaries the condition similar to the Maxwell boundary conditions in the classical kinetic theory was exploited. The macroscopic transport equations for an arbitrary set of moments were developed and closed by means of the Grad’s moment method. As example, sets with 4, 9, 16 and 25 moments were considered and solved analytically for the one-dimensional heat transfer and the Poiseuille flow of phonons. The results showed the influence of Knudsen number on phonon drag at the solid boundaries that the Knudsen layers near the boundaries reduce the net heat conductivity of solids in the rarefied phonon regimes.

3.2 Guyer–Krumhansl (GK) Equation

113

Later Mohammadzadeh and Struchtrup [206] derive the macroscopic equations for the phonon transport at the room temperature from the phonon-Boltzmann equation. In these equations, the Callaway model with the frequency-dependent relaxation time is considered to describe both the resistive and normal processes in the phonon interactions. The Brillouin zone was considered to be a sphere with the diameter that depends on the temperature of the system. A model to describe the phonon interaction with the crystal boundary was employed to obtain the macroscopic boundary conditions, where the reflection kernel is the superposition of the diffusive reflection, specular reflection and isotropic scattering. The macroscopic moments were defined using a polynomial of the frequency and the wave vector of phonons. As an example, a system of the moment equations, consisting of three directional and seven frequency moments, i.e. 63 moments in total, was used to study the one-dimensional heat transfer and the Poiseuille flow of phonons. The Guyer–Krumhansl equation can be derived from the extended irreversible thermodynamics under an assumption that non-locality could be introduced into the entropy flux as [207] q J s = + γ (q · ∇q + 2q∇ · q) T where γ is a positive coefficient. Cimmelli [208] showed that the Guyer–Krumhansl heat transport equation could be also obtained within the frame of the weakly nonlocal rational thermodynamics [209]; they can also be derived using the extended irreversible thermodynamics [210]. Lebon and Dauby [211] used a variational principle that is a generalization of Prigogine’s minimum-entropy production criterion to derive the Guyer–Krumhansl equation from macroscopic arguments. Ramos et al. [212] studied the basic structure of the Guyer–Krumhansl equation proved the well-posedness of some initial and boundary value problems. The Guyer–Krumhansl equation could also be derived in the framework of the GENERIC approach (since the GK equation can be considered as a non-local extension of the MCV model, the reversible dynamics needs no modification, but it is only the dissipation potential has to be extended by a non-local term) [213]. Grmela et al. [214] suggested a unified notation for a number of models in the form ∂qi = Wi ∂t where Wi is as follows for • the Maxwell–Cattaneo–Vernotte model    1 1 ∂ −qi + λ , Wi = τ ∂ xi T • the Guyer–Krumhansl equations

114

3 Phonon Models

     2 1 1 ∂ ∂ ∂qi 2 ∂ qi + Wi = +2 , −qi + λ τ ∂ xi T ∂ xi ∂ x j ∂ x 2j • the nonlinear extension of the Guyer–Krumhansl model [125, 126]      2 1 ∂qi 1 ∂ ∂ ∂qi 2 ∂ qi + +2 Wi = Wi = . − μq j −qi + λ τ ∂ xi T ∂x j ∂ xi ∂ x j ∂ x 2j Cimmelli et al. derived the equation for the coupled heat transfer by phonon and electrons the restrictions that follow from the second law of thermodynamics using the generalized Coleman–Noll procedure [215–217]. The obtained equations account for the non-local effects, which may arise at the nanometric length-scale, when the mean free paths of phonons and electrons are higher than the physical dimensions of the system. These equations may be applied, e.g. to the simulation of the heat transport in metallic nanowires, in which both electrons and phonons contribute, and extended to heat heat transport in semiconductor nanowires, in which phonons, electrons and holes contribute. If the coupling can be neglected, the authors recovered two parabolic equations for the phonon and electron transport of the Guyer–Krumhansl type; the terms proportional to the mean free paths (i.e. the non-local effects) in these equations are negligible, the equations of the Maxwell-Cattaneo type are recovered. The EIT formulates the 13-fields theory and the heat propagation in rigid conductors whose state space is spanned by the thirteen thermodynamic variables: the ˆ temperature T , the heat flux q and flux of the heat flux Q: τ

∂ Qˆ ∂q + q = −λ∇T + 2 ∇ 2 q − τ2 λT 2 ∇ · , ∂t ∂t τ2

l2 ∂ Qˆ ˆ = (∇q − Q). ∂t λT 2

The non-local character of the heat transfer could leads to unusual results such as the heat flowing from the cold regions to the hotter ones. Cimmelli et al. [218] considered an axisymmetrical problem of the heat transfer in the circular thin layer of thickness h surrounding a steady source of heat—the hot nanodevice. The authors assumed that the total heat flux q arises from two contributions due to both phonons and electrons q = q p + qe . The evolution of the phonon and electron fluxes is governed by the Guyer–Krumhansl equations (3.21) τp

∂q p + q p = −λ∇T + 2 (∇ 2 q p + 2∇ · ∇q p ) ∂t

(3.28)

τe

∂qe + qe = −λ∇T + 2 (∇ 2 qe + 2∇ · ∇qe ). ∂t

(3.29)

and

3.2 Guyer–Krumhansl (GK) Equation

115

The temperatures of phonons and electrons are assumed to be equal (this restriction has been removed in the later authors’ papers [16, 219]). The radial dependence of the heat fluxes was obtained by considering two concentric circular areas at a radial from the source equal to r and r + dr to get in the first-order approximation in dr   dq dr 2π(r + dr )q(r + dr ) ≈ 2π(r + dr ) q(r ) + dr and thus r

dq + q(r ) = 0. dr

From the last equation follows q(r ) ≡ q p (r ) + qe (r ) =

 r

where  = (Q 0 /2π h) is a constant value, Q 0 is the heat produced by the hot source per unit time. Both fluxes q p and q e are divergence free and thus Eqs. (3.28, 3.29) reduce to τp

∂q p + q p = −λ∇T + 2 ∇ 2 q p ∂t

τe

∂qe + qe = −λ∇T + 2 ∇ 2 qe ∂t

and

and allow to determined the profiles of the phonon and electron fluxes. Finally, Cimmelli et al. obtained the first-order differential equation for the temperature dT = dr



l 2p λp



d2 q p 1 + dr 2 r



l 2p λp



dq p qp − . dr λp

Integration of the last equation showed that in the neighbourhood of the source the temperature increases with the radial distance. This anomalous temperature hump— the heat flux against the temperature gradient—is related to the non-local effects [220]. To prove the thermodynamic compatibility of this behaviour the authors considered the local balance of the entropy that in the steady case could be written as σ (s) = ∇ · J (s) where σ (s) is the entropy production per unit volume,

116

3 Phonon Models

J

(s)

q = + T





l 2p λpT 2

 ∇q p · q p +

le2 λe T 2

 ∇q e · q e

is the entropy flux. The computed entropy production was everywhere positive, thus the hump in the temperature distribution is physically possible since it agrees with the second law of thermodynamics. The non-classical terms    2  l 2p le ∇q p · q p , ∇q e · q e λpT 2 λe T 2 are directly related to a non-local contribution to a generalized transport equation for the heat flux, containing the Laplacians of the heat fluxes. Recently Calvo-Schwarzwälder et al. [169] used the Guyer–Krumhansl Eq. (3.21) to solve the one-dimensional Stefan problem. Calvo-Schwarzwälder [221] also investigated the heat transfer in the one-dimensional solid by considering a Fourier law with a size-dependent effective thermal conductivity proposed by Alvarez and Jou [222] ⎞ ⎛  2   2 L ⎝ l 1+ − 1⎠ κe f f (L) = 2κ l L and a Newton cooling condition at the interface between the solid and the cold environment. Calvo-Schwarzwälder found that non-local effects become less important as the Biot number Bi = hl/κ (h is the the heat transfer coefficient in the Newton cooling condition q(0, t) = h(Te − T (0, t))) decreases. The simplified version of Guyer–Krumhansl Eq. (3.27) was used by Jou et al. to model the heat transfer in the nanowires considered as the phonon flow [201, 223, 224]. The authors used the analogy with the flow of the viscous fluid along the cylindrical duct of the radius R under the pressure gradient p/L with the velocity profile p (R 2 − r 2 ) V (r ) = 4 Lη and the volume flow Q=

π R 4 p 8η η

to obtain for the total heat flow along a cylindrical conductor Qh =

π R 4 κ0 T . 82 L

(3.30)

3.2 Guyer–Krumhansl (GK) Equation

117

Thus an effective thermal conductivity if the nanowire that depends on the ratio l/R is written as Qh L κ0 R 2 (3.31) κe f f = = 2 2. 2 π R T 8 l If instead of Eq. (3.27) the full Guyer–Krumhansl equation is used, the effective thermal conductivity is [201] κ e f f = κ0

τ R c0 R

  2J1 (i z) 1− , i z J0 (i z)

where J0 and J1 are the cylindrical Bessel functions [225], c0 is the Debay velocity, √ l = c0 τr τ N is the phonon mean free path, τ N and τ R are the relaxation times of normal and resistive phonon scattering. The Eq. (3.31) gives a quadratic dependence of the effective thermal conductivity on the nanowire radius while experiments show the linear variation [201]. Alvarez et al. assumed that the heat flux, similar to the velocity in the rarefied gas dynamics, does not obey the no-slip boundary condition on the wall, i.e. takes no-zero value on the wall and as in microfluidics [226] is proportional to the phonon mean free path and to the flux gradient at the wall  qw = Cl

∂q ∂r

 r =R

.

(3.32)

This boundary condition is called sometimes “first-order slip condition” [163] or “Maxwell slip model” [227]. In the case of the general geometry the boundary condition (3.32) written as [37, 228]  qw = Cl

∂q ∂ξ

 γ

where ξ means the normal direction to the wall cross section (pointing towards the flow), and γ is the curve accounting for the outer surface of the transversal section of the system. Lebon et al. [229] investigated the heat slip flow along solid walls in the frame of EIT elevating the heat flux at the boundary to the status of the independent variable and formulated the boundary conditions obtained from the constraint imposed by the second law of thermodynamics expressing that the rate of the entropy production is non-negative. Xu [77] derived the slip boundary condition for the heat flux from the Boltzmann transport equation for phonon. Sellitto et al. [200, 224] suggested an extended version of this condition (“second-order slip condition” [163])  qw = Cl

∂q ∂r



 − α

2

r =R

∂ 2q ∂r 2

 r =R

.

(3.33)

118

3 Phonon Models

The Eq. (3.33) is analogous to the boundary condition for the fluid velocity in the gas dynamics that was suggested by Cercignani [230]. The coefficients C and α describe the effect of the interactions of phonons with walls: C describes the specular or diffusive reflections of the phonons while α accounts for the backscattering collisions [224]. Sellitto et al. [231] formulated the relations of the coefficients C and α to the characteristics of the wall and to the temperature    , C = C (T ) 1 − L

α = α (T )

 , L

the functions C (T ) and α (T ) were determined by the authors for both the smoothwalled and rough-walled silicon nanotubes. When the condition (3.33) is used with α = 2/9 the term “1.5-Order Slip-Flow Model” is also used [232]. The coefficient C describes the specular and diffusive collisions of the phonons with the wall while the coefficient α accounts for backscattering [163]. Both coefficients in the equations (3.32, 3.33) depend on the temperature and are related to the properties of the wall [200, 231]. Zhu et al. [233] studied the heat transport in a 2-D nanolayer in which the size along x-axis direction is much larger than the mean free path of the heat carriers using the Guyer–Krumhansl Eq. (3.21) with the second-order slip condition (3.33) where the coefficient C is written as suggested by Carlomango et al. [234] similar to the relation for gases and for electron collisions [143] C =2

1+ P 1− P

where P is the fraction of the heat carriers reflected back with a specular reflection from a solid surface. Since P ∈ [0, 1], the coefficient C ∈ [2, ∞]. Small values of C mean that the diffusive phonon–wall collisions are predominant over the specular ones. Conversely, the large values of C mean that the specular phonon–wall scattering dominates over the diffusive one. The boundary scattering is the main cause of the non-uniform heat-flux profile in the hydrodynamic regime [37]. Zhu et al. also used the developed for the rarefied gas dynamics by Bird [235] the direct simulation Monte Carlo (DSMC) to investigate the heat transfer in the argon gas nanolayer. The slip-flow contribution to the heat flux is essential in the so-called Knudsen layer with the thickness of the order of the phonon mean free path. However, in the nanosystems with dimension comparable or smaller than the phonon mean free path the influence of the slip boundary condition extends to the whole system.

3.2 Guyer–Krumhansl (GK) Equation

119

Alvarez et al. [163] stated that the slip boundary condition should be modified in the case of the high-frequency perturbations: the relaxation of the heat flux should be introduced similar to the Cattaneo law  2    ∂ q ∂qw ∂q + qw = Cl − α2 τw ∂t ∂r r =R ∂r 2 r =R The relaxation time τw should account for the specular, diffusive and backward reflection of the phonons from the wall and can be determined according to the Matthiessen’s rule 1 1 1 1 = + + . τw τspec τdi f f τback Alvarez et al. suggested a crude estimate of the relaxation time through the total frequency of the phonon–wall collisions assuming that the wall has the smooth regions of width D and the rough regions of width  and the relaxation time is expressed as D c¯  c¯ 1 + = τw d − D D+ d − D+ where the ratios D/(D + ) and /(D + ) are the probabilities of finding the smooth and rough regions, respectively, c¯ is the mean phonon speed. Use of the slip boundary condition for the heat flux increases the total heat flow along a cylindrical conductor: (3.30) is modified as Qh =

π R 4 κ0 T 8l 2 L

  l 1 + 4C R

and the effective thermal conductivity (3.31) as κe f f =

κ0 R 2 8 l2

  l 1 + 4C R

(3.34)

where C is a constant related to the wall properties. In terms of Knudsen number Eq. (3.34) is written as κe f f =

κ0 (1 + 4 C K n) . 8K n

For nanowires of small radius R  l the Eq. (3.34) simplifies to κe f f =

κ0 C R , 2 l

thus linear dependence of the effective thermal conductivity on the nanowire radius is obtained.

120

3 Phonon Models

It is also known [201] that quantum effects could lead to the cross-over from the linear dependence of the effective conductivity κe f f on the radius to the quadratic behaviour [236, 237]. For the thin film of thickness h Alvarez et al. [201] derived κeff

κ0 h 2 = 12 l 2

  l 1 + 6C h

when the simplified Guyer–Krumhansl equation is used and    2l h κe f f = κ0 1 − tanh h 2l in the case of the full equation.

3.3 Two-Fluid Models Such models are based on the division of the phonons into two populations. Usually it is assumed that the high-frequency part is in the quasi-thermal equilibrium with the well-defined local temperature [238] while the low-frequency modes are out of the equilibrium and do not interact with each other because of the small phase space for such scattering [143, 239–241] but can exchange energy with the high-frequency modes. The weakness of the two-fluid (two-channel) models is the arbitrariness of the choice of the cut-off between two channels [242].

3.3.1 Ballistic–Diffusive Model Ballistic–diffusive (BD) model was introduced by Chen [243, 244]. This model can be considered as an approximation to the grey unsteady BTE [32] and is close to the differential approximation used in the analysis of the radiative heat transfer [245] that involves replacing the integral equation for the heat transfer by the differential equation for the heat flux and its modification (MDA) suggested by Olfe [246]. BD model could be considered as the mixed macroscopic–mesoscopic method [164, 247]. BD model is based on the splitting of the distribution function (as well as the internal energy and the heat flux) into the two parts f = f b + f d , e = eb + ed , q = q b + q d reflecting the coexistence of two kinds of the heat carriers

3.3 Two-Fluid Models

121

• Ballistic phonons that experience mainly the collisions with the boundaries, • Diffusive phonons that undergo multiple collisions within the core of the system. The evolution of these two parts of the internal energy is governed by the classical balance equations ∂eb = −∇ · q b + rb , ∂t ∂ed = −∇ · q d + rd . ∂t

(3.35) (3.36)

The source terms rb and rd describe the energy exchange between the ballistic and the diffusive phonon populations and in the absence of the external sources rb = −rd . The evolution equations for the heat flux are ∂q b + q b = −λb ∇T + 2b (∇ 2 q b + 2∇∇ · q b , ) ∂t ∂q τd d + q d = −λd ∇T. ∂t τb

(3.37) (3.38)

The coupled equations (3.37, 3.38) lead to the Guyer–Krumhansl equation under the assumption that τd = τ, b = , λ = λb + λd . The relative contributions of these two components of the phonon distribution depend on the value of the Knudsen number K n and on the geometry of the considered system [248]. As noted Li and Cao in the just cited paper, the ballistic transport is responsible for the nonlinearity of the temperature distribution and this nonlinearity increases with increasing the Knudsen number. The advantage of the ballistic–diffusive model over the Boltzmann transport equation is the simplicity of the description—only the time and spatial coordinates are involved. Similar model called the “two-channel model” [81] is based on the assumption that there are two individual heat conduction channels and division of the phonon population into two parts: the long MFP and the short MFP phonons. Chen [243] used the Boltzmann equation under the relaxation time approximation assuming that the relaxation time depends on the phonon frequency ω (phonon energy) and does not depend on the on the wave vector that is linearly related to the phonon quasi-momentum p = k , i.e. the isotropic scattering is considered f − f0 ∂f +v·∇ f =− . ∂t τ (ω) where f 0 is the equilibrium phonon distribution function, v is the phonon group velocity v = ∇k ω. The single-mode relaxation time approximation leads to the following expression for the phonon thermal conductivity [179]

122

3 Phonon Models

λ=

1 2 v cτ, 3

where c is the specific heat capacity per unit volume. The equation for the ballistic part of the phonon distribution function f b is written as fb 1 ∂ fb +  · ∇ fb = − , (3.39) |v| ∂t |v| where  is the unit vector in the direction of the phonon propagation. The equation for the diffusive part of the phonon distribution function is ∂ fd fd − f0 + v · ∇ fd = − . ∂t τ

(3.40)

Chen [243] solved the BTE using the spherical harmonic expansion and keeping the first two terms f b = g0 + g 1 · , where g0 is the scalar average of f b over all directions, g 1 is a vector related to the heat flux. Using this solution, multiplication of the Eq. (3.40) by  and integration over the solid angle gives g 1 ∂ g1 + ∇g0 = − 1 , |v| ∂t   = |v|τ is the phonon mean free path. Thus the heat flux is  1 q= vω( f b + f d )d 3 v = q b + q d 4π and the internal energy 1 e= 4π

 ω( f b + f d )d 3 v = eb + ed .

Yang et al. [249] used the BTE in terms of the phonon intensity Iω = vω ω f D(ω)/4π where vω is the carrier group velocity, ω is the phonon circular frequency, D(ω) is the phonon density of states per unit volume, Sω is the phonon source term that could be determined, e.g. by the electron-phonon scattering Iω − I0ω ∂ Iω + Sω . + vω ∇ Iω = − ∂t τω

3.3 Two-Fluid Models

123

The equations for the ballistic and diffusive parts of the phonon distribution function were written, respectively, as Ibω ∂ Ibω + vω ∇ Ibω = − + Sω . ∂t τω and

Ibω − I0ω ∂ Idω . + vω ∇ Idω = − ∂t τω

Allen [250] performed an analysis of the cross-over from the ballistic to diffusive regime of the heat conduction using the computer simulations and a Fouriertransformed version of the phonon Boltzmann transport equation. The evolution equation of the average occupation in the reciprocal space of the phonon mode Q includes several terms (drift, scattering, external) ∂ NQ = ∂t 

d NQ dt



d NQ dt



 + dri f t

d NQ dt



 +

scatt



 = −v Q · ∇ N Q = −v Q ·

dri f t



d NQ dt

 =− scatt



d NQ dt

 , ext

 dn Q ∇T + ∇ Q , dT

S Q,Q  Q ,

Q

where n Q is the local equilibrium Bose–Einstein distribution,  Q = N Q − n Q and S Q,Q is the linearized scattering operator. Vazquez et al. [251] and Lebon et al. [207] have considered the two-temperature variant of the ballistic–diffusive model. Vazquez et al. used the Guyer–Krumhansl equation (see Sect. 3.2) to describe both the ballistic and diffusive heat fluxes. Lebon et al. assumed that the distributive function of the diffusive phonons is governed by the Cattaneo Eq. (2.5) while the ballistic phonons obeys the Guyer–Krumhansl equation. Direct nanoscale imaging of ballistic and diffusive thermal transport in graphene structures was reported by Pumarol et al. [252]. Siemens et al. [253] used the ultrafast coherent soft X-ray beams (the wavelength 29 nm) to study heat transfer from the nanoscale hotspot (the highly doped silicon resistor near a thin silicon membrane was investigated). The authors found a significant (as much as three times) decrease of the energy dissipation away from the heat source compared to the predictions by the Fourier law.

124

3 Phonon Models

3.3.2 Extended Ballistic–Diffusive Model Rezgui et al. [74] assumed that τ = τ R could be calculated according to the Matthiessen’s rule 1 1 1 1 = + + τR τU τi τb where τU is the relaxation time of Umklapp phonon–phonon collisions, τi is the relaxation time of phonon–impurity collisions and τb is the relaxation time of phonon– boundary collision and using the first-order Taylor expansion of Eq. (3.40) it is possible to get f d (r, ε, t + τ R ) − f d (r, ε, t) f d (r, ε, t) − f 0 (r, ε) + v∇ f d (r, ε, t) = τR τR

(3.41)

where ε is the kinetic energy; the diffusive and the ballistic fluxes could be defined as  qd (r, t) = v(r, T ) f d (r, ε, t)ε D(ε)dε, ε

 qb (r, t) =

v(r, T ) f b (r, ε, t)ε D(ε)dε, ε

D(ε) is the density of states, the total heat flux is clearly q = qd + qb . Rearranging the terms of Eq. (3.41) Rezgui et al. get f 0 (r, ε) = τ R v∇ f d (r, ε, t) + f d (r, ε, t + τ R ).

(3.42)

Multiplying of the Eq. (3.42) by ε D(ε)v and accounting for the relation  v f 0 ε D(ε)dε ε

yields

 qd (r, t + τr ) +

τ R v(r, T )∇ f d (r, ε, t)ε D(ε)vdε = 0 ε

or, assuming that ∇f =

df ∇T, λ dT

 τ R v2

d fd ε D(ε)dε dt

the equation τR

dqd + qd = −λ∇Td . dt

(3.43)

3.3 Two-Fluid Models

125

Using the conservation of the total internal energy is defined as u = u d + Ub in the form (q˙h is the volumetric heat generation) ∂u(r, t) ∂ T (r, t) =C = −∇q(r, t) + q˙h ∂t ∂t the authors finally wrote out the equation they called extended ballistic–diffusive model (EBDM) τR

1 τ R ∂ q˙h ∂ 2 Td (r, t) ∂ Td (r, t) 1 q˙h = ∇(λ∇Td (r, t)) − ∇qb (r, t) + + . + ∂t 2 ∂t C C C C ∂t

3.3.3 Unified Non-diffusive-Diffusive Model Ramu and Ma [254] suggested a two-fluid (two-channel) model similar to the BD model and called it the unified non-diffusive-diffusive (UND) model. The authors based this model on the Boltzmann transport equation, considered one-dimensional case and used different accuracy of the spherical harmonic expansions of the phonon distribution function for the low-frequency (LF) phonons with the mean free path (MFP) of the same order of magnitude as the length scale of interest and highfrequency (HF) phonons. The phonon spectrum is divided into two parts 1. High-heat-capacity high-frequency phonons that are in quasi-equilibrium with local temperature. 2. Low-heat low-frequency phonons that are farther out of equilibrium. The LF modes do not interact with each other due to the small phase space for for such scattering [239] but can exchange energy with the HF modes [254]. The distribution function for LF phonons g(x, k) is expanded in the spherical harmonic functions [255] that form a complete orthogonal set for expanding azimuthally-symmetric functions g(x, k) =

∞ 

gi (x, k)Pi (cos θ ),

(3.44)

i=0

where k is the phonon wave vector making an angle θ with the x-axis. The authors assumed that all LF modes have the same lifetime τ , the same group velocity magnitude v and that the phonon dispersion is isotropic (thus v = vk/k). The linearized steady-state Boltzmann equation for the LF part of spectrum is written as g − f0 ∂g =− . (3.45) v cos θ ∂t τ

126

3 Phonon Models

Substitution of the expansion (3.44) into the BTE (3.45), multiplying by Pi (cos θ ) sin θ (i = 0, 1, 2, ...) and integration over θ produces a hierarchy of coupled equations for gi . The first three of these equations are written as g0 − f 0 1 ∂gne1 v + = 0, 3 ∂x τ 2 ∂g2 ∂g0 g1 v +v + = 0, 5 ∂x ∂x τ 2 ∂g1 g2 3 ∂g3 v + v + =0 7 ∂x 3 ∂x τ

(3.46) (3.47) (3.48)

Ramu and Ma truncate the expansion at second order by requiring g3 = 0 that is the next approximation after the Fourier law which consists in setting g2 = 0. After elimination of g2 and g0 Ramu and Ma get the equation in terms of g1 ∂ 2 g1 3 ∂ f0 + g1 = 0. − (vτ )2 2 + vτ 5 ∂x ∂x

(3.49)

Since f (T ) depends on x only through T it is possible to use ∂ f 0 dT ∂ f0 = . ∂x ∂T dx Multiplying Eq. (3.49) by (4π/3)ωv and summing over all k gives 3 ∂ 2q L F 1 ∂T − (vτ )2 + qLF = 0 + C L F v2 τ 5 ∂x2 3 ∂x ⎛

where C L F = 4π

∂ ⎝ 1 ∂ T (2π )3



⎞ ω f 0 k 2 dk ⎠

k

is the volumetric heat capacity of the LF modes. The expression for the LF heat flux could be written defining the thermal conductivity of the LF modes as λ L F = 13 C L F v 2 τ and the MFP of the low-frequency phonons as  L F = vτ qLF =

∂T 3 L F 2 ∂ 2q L F − λL F ( ) . 2 5 ∂x ∂x

Similar analysis is performed for the HF phonons h(x, k) =

∞  i=0

h i Pi (cos θ )

3.3 Two-Fluid Models

127

with two distinctions 1. The spherical harmonic expansion is truncated at the first order, i.e. h(x, k) = h 0 + h 1 cos θ . 2. The Bose statistics is used for the symmetric part of the distribution, i.e. h 0 (x, k) = f0 . Thus the HF heat flux is written as q H F = −λ L F

∂T . ∂x

Using the definition λ = λ L F + λ H F the total heat flux is expressed as q=

∂T 3 L F 2 ∂ 2q L F ( ) . −λ 5 ∂x2 ∂x

3.3.4 Enhanced Fourier Law Ramu and Bowers [242, 256] derived an enhanced Fourier law from the BTE assuming the grey population of the quasi-ballistic phonon modes. Similar to the unified non-diffusive-diffusive model the authors propose to get the heat flux of the quasi-ballistic mode directly from BTE using the expansion of the phonon distribution function in the spherical harmonic functions (3.44) and truncating it at the second order in the angular momentum. The original version [254] exploited the two-channel model, later [242] it has been extended to incorporate the quasi-full spectral solution of the BTE. The authors derived the hierarchy of the coupled equations for harmonics gi similar to (3.46). The general ith equation of the hierarchy is written as i ∂gi−1 gi i + 1 ∂gi+1 v + v + = 0. 2i + 3 ∂ x 2i + 1 ∂ x τ Increasing the number of harmonics certainly improves the accuracy but introduces the difficulty with the specification of the boundary conditions [242]. Ramu and Bowers [257] use the BTE in the isotropic (thus the group velocity is ˆ where kˆ is the unit vector in the direction of k) relaxation time v(k) written as v(k) k, approximation f0 − f v(k) · ∇ f = τ (k) where f 0 (r, k) is the equilibrium Bose function.

128

3 Phonon Models

Introducing the model mean free path (k) = vτ produces f = f 0 −  kˆ · ∇ f. The authors “reiterate” the BTE to get f = f 0 −  kˆ · ∇ f + 2 ( kˆ · ∇)( kˆ · ∇) f.

(3.50)

Since cos θ is the first spherical harmonic, its orthogonality to all other harmonics allows to get the heat flux in the one dimension as [242] the sum over modes and the integral over angles. ˆ The authors multiply (3.50) by kvω(k) where ω(k) is the modal frequency, integrate over the polar angles of kˆ and use the identities [258] 

and



ˆ kˆ · ∇)n f 0 = 0, n = 0, 2, 4, . . . , d k(

ˆ kˆ · ∇)n f 0 = 4π ∇ n f 0 , n = 1, 3, 5, . . . , d k( n+2

to get the expression for the modal heat flux. This expression accounts for the orthogonality of the spherical harmonics and includes 81 angular integrals most of which evaluate to zero and is finally written as q(k, r) = −

∂ f0 3 4π 1 vω(k) ∇T + 2 (∇ · q(k, r)) − 2 ∇ × (∇ × q(k, r)). 3 ∂T 5 5

Introducing the thermal conductivity for the low-frequency modes κ L F = where  ∂ f0 CLF = 4π ω(k) ∂T k

1 LF  vC L F 3

is the heat capacity of the low-frequency modes, the modal heat flux is written as 3 1 q L F (r) = −κ L F ∇T + ( L F )2 (∇ · q L F (r)) − ( L F )2 ∇ × (∇ × q L F (r)). 5 5 (3.51) The total heat flux is the sum of the low-frequency and the high-frequency parts q H F (r) = −κ H F ∇T. The important feature of the Eq. (3.51) is the presence of the term 1 LF 2 ( ) ∇ × (∇ × q L F (r)) 5

3.3 Two-Fluid Models

129

that is the solenoidal (circulatory) quasi-ballistic component of the heat flux with zero divergence [257]. Ramu et al. [259] proposed to manipulate the patterns of the quasi-ballistic heat flow to reduce the effective thermal conductivity. There can be several the low-frequency channels, thus in the three-dimensional case the energy conservation equation is written as [242]    ∂T HF LF =∇· q + q . −Cv ∂t i

3.3.5 Two-fluid Model The two-fluid model [260–262] similar to the BD model divides the phonon population into two groups: • The reservoir group: longitudinal optical (LO), transverse optical (TO), transverse acoustic (TA) phonons; • The propagating group: longitudinal acoustic (LA) phonons that have a single group velocity. All phonons are assumed to have a single overall scattering time [136]. The Monte Carlo computations show that the high-energy electrons scatter preferentially with the LO phonons [263] thus the Boltzmann equation for the LO phonons includes the term Se−L O (that is absent in equations for other phonon modes) and could be written in the steady state as

v · ∇φ = −

φ φ − φ¯ − τimp (ω) τan (ω) 

+ ω ω

φ P(ω → ω )dω

τan (ω )

φ

P(ω → ω)dω + Se−L O τan (ω )

where v is the group velocity, ω is the angular frequency, φ is the number of phonons per unit volume and angular frequency, φ¯ is an average over all directions at a given point, indexes “imp” and “an” refer to the elastic impurity scattering and the anharmonic phonon scattering, respectively. Multiplication of equations by ω, integration over all frequencies for each branch and summing branches give energy balances for the reservoir and propagating groups [136] C R (TR − TL ) u L R 0= + q = − + q τ τ

130

3 Phonon Models

and ∇ · jP =

C p (TP − TL ) u P = τ τ

(3.52)

where u L R is the energy transferred out of the reservoir as it relaxes towards equilibrium with the lattice, q  is the power generated by hot electrons, C R is the heat capacity of the reservoir phonons, u P L is the energy gained by the propagating phonons from the phonon reservoir.

3.4 Generalized Fourier Law by Hua et al. Hua et al. [264, 265] developed a generalized Fourier model valid from ballistic to diffusive regimes. The authors start with the mode-dependent phonon Boltzmann equation under the relaxation time approximation (Bhatnagar–Gross–Krook (BGK) model [76, 77]) gμ − g0 (T, x, t) ∂gμ (x, t) + Q˙ μ , + vμ · gμ (x, t) = ∂t τμ

(3.53)

where gμ (x, t) = ωμ ( f μ (x, t) − f 0 (T )) is the deviational energy distribution function for the phonon state μ = (q, s); here q is the phonon wave vector, s is the phonon branch index, f 0 is the Bose–Einstein distribution, g0 (T ) = ωμ ( f 0 (T ) − f 0 (T0 )) ≈ Cμ T, Cμ is the mode-dependent specific heat, Q˙ μ is the heat input rate per mode. To close the problem, the energy conservation law is used ∂ E(x, t) + ∇ · q(x, t) = Q(x, t), ∂t where E(x, t) =

1  g V μ μ

is the total volumetric energy, q=

1  g vμ V μ μ

3.4 Generalized Fourier Law by Hua et al.

131

is the heat flux. The sum over μ denotes a sum over all phonon modes in the Brillouin zone. To get a generalized constitutive relation between the heat flux and the temperature gradient, the authors rearrange the Eq. (3.53) and performed the Fourier transformation in time to get μx

∂ g˜ μ ∂ g˜ μ ∂ g˜ μ + μy + μz + (1 + iητμ )g˜ μ = Cμ T˜ + Q˜ μ τμ , ∂x ∂t ∂z

(3.54)

η is the Fourier temporal frequency, μx , μy , μz are the directional mean free paths along x, y, z. The authors introduced the new independent variables ξ, ρ, ζ ξ = x, μy μx x− y ρ= μ μ μx μz x− z. ζ = μ μ  where μ = 2μx + 2μy + 2μz . The Eq. (3.54) after this transformation becomes μξ

∂ g˜ μ + αμ g˜ μ = Cμ T˜ + Q˜ μ τμ , ∂ξ

(3.55)

where αμ = 1 + iητμ . The authors, solving the Eq. (3.55) on the interval [L1, L2], finally related the temperature gradient to the mode-specific heat fluxes  q˜μξ = −

λμξ (ξ − ξ )



∂T

dξ + Bμ (ξ, ρ, ζ, η), ∂ξ

(3.56)

where Bμ (ξ, ρ, ζ, η) is determined by the boundary conditions and the volumetric heat input rate, [L1, ξ ) i f vμξ > 0, ∈ i f vμξ < 0, (ξ, L 2 ] and the model thermal conductivity along the direction ξ

λμξ = Cμ vμξ μξ

! ! ξ exp(−αmu !! 

μξ

αμ |μξ |

! ! !) !

.

132

3 Phonon Models

The first term in the Eq. (3.56) is a convolution a space- and time-dependent thermal conductivity and the temperature gradient along the the direction ξ and reflect the non-locality of the heat conduction.

3.5 Phonon Hydrodynamics Guo and Wang [164, 266] derived the macroscopic equations for the phonon gas motion from the phonon Boltzmann equation ∂f + vg = C( f ) ∂t where f = f (x, t, k) is the phonon distribution function, vg = ∇k ω is the phonon group velocity. The scattering term C( f ) includes the contribution of the two major processes 1. The normal scattering (N-process) and 2. The resistive scattering (R-process). The energy is conserved in both kinds of collision of phonons with the wave vectors k1 and k2 ω(k1 ) + ω(k2 ) = ω(k3 ) while the quasi-momentum of phonons k1 + k2 = k3 + b

(3.57)

is conserved only in the N-processes for which b is the reciprocal lattice vector or b = 0 [143, 267]. The simplification of the Boltzmann transport equation is based on the Callaway’s dual relaxation model that assumes that the N process and R process proceed separately. At the low temperatures the dominant process is N process while at the ordinary temperatures N process is negligible and the phonon Boltzmann equation becomes eq f − fR ∂f + vg = − (3.58) ∂t τR where the equilibrium distribution function for the R processes is the Planck distribution 1 eq   , (3.59) fR = ω −1 ex p kB T here  is the reduced Planck constant.

3.5 Phonon Hydrodynamics

133

The phonon relaxation time and phonon group velocity depend often on the phonon frequency that complicates the solution of the BTE and forces one to use numerical methods such as the discrete ordinate method (DOM) [50] or the Monte Carlo (MC) method. Guo and Wang made a few assumptions: • The phonon distribution is isotropic, i.e. the phonon properties in one crystalline direction are representative of those in the whole wave vector space; • Grey assumption—three acoustic phonon branches have an effective constant relaxation time, contribution from the optical phonon branches is neglected; • The phonon dispersion relation is linear ω = vg k (Debye model). The phonon hydrodynamic model is based on the macroscopic field variables that are defined as the statistical averages of the phonon distribution function: the phonon energy density  e=

ω f d k,



the heat flux q= the flux of the heat flux Qˆ =

vg ω f d k,

 ωvg vg f d k.

Multiplication of the Boltzmann Eq. (3.58) by the phonon energy quanta ω and integration over the wave vector space gives the balance equation for the energy density ∂e + ∇ · q = 0. (3.60) ∂t The right-hand side of this equation vanishes because of the energy conservation during phonon scattering. Using the similar procedure with the factor vg ω one obtains the balance equation for the heat flux q ∂q + ∇ · Qˆ = − . (3.61) ∂t τR These balance equations are the four-moment field equations of the phonon Boltzmann equation.5 To close the system of the phonon transport equations the flux of the heat flux Qˆ has to be specified in terms of the four basic field variables (the energy density e and the three components of the heat flux q). There are several approaches to deal with the closure problem in the kinetic theory: • The Hilbert method [268], 5

There is a misprint in [266]— Qˆ is printed as a vector in the equation for the heat flux.

134

• • • • •

3 Phonon Models

The Chapman–Enskog expansion [164, 269, 270], The Grad’s moment method [271, 272] The “regularized moment method” (the R13 moment method)6 Maximum entropy moment method7 [278–280], The invariant manifold method [281].

The authors close for phonon transport problem developing a perturbation equation to the Boltzmann equation around the four-moment non-equilibrium phonon distribution function obtained by the maximum entropy principle8 [217] (the idea of this principle is to resolve the distribution function through a maximization of entropy density under the constraints of field variables)—thus the derived phonon hydrodynamic equation should be able to describe the strong thermodynamic nonequilibrium effects in nanoscale heat transport. Thus the problem reduces to maximization of the following functional   = −k B

      [ f ln f − (1 + f ) ln(1 + f )]d k + β e − ω f d k + γi qi − vgi ω f d k ,

(3.62) where β and γi are the Lagrange multiplies. The extremum conditions of the functional Eq. (3.62) are as follows [266]: ∂ = ∂f

 

   1 k B ln 1 + − βω − γi gvgi ω dk = 0 f ∂ =e− ∂β

6

(3.63)

 ω f d k,

The R13 moment method consists in using the Chapman–Enskog expansion around the Grad’s thirteen-moment non-equilibrium distribution instead of the usual local Maxwell-Boltzmann equilibrium distribution. The R13 moment method differs from the classical Grad’s 13-moment system in the closure relations—the new closure process allows to add some terms of super-Burnett order to the classical 13-moment equations. Three new quantities, a third-order tensor m i jk , a second-order tensor Ri j , and a scalar , are introduced in order to represent the deviations of the effective values of the higher-order moments ρi jk , ρnni jk , and ρnnpp in 13-moment system, from the corresponding values determined through the usual Grad’s closure [208]. As a result, the R13 equations can be used to model the high-Knudsen-number gas flow. [273– 275]. 7 The maximum entropy method of the determination of the phonon distribution function is based on the maximization of the phonon entropy density under the constraints of the state variables with the Lagrange multiplier method (Liu procedure) [276]: the constraints may be removed by the use of the Lagrange multipliers [277]. Thus a closed set of balance equations for all the phonon state variables (energy, quasi-momentum, ... ) and the Lagrange multipliers [164]. 8 It is formulated by Jaynes in connection with the information theory [282, 283] as: The thermodynamic systems evolve naturally toward " states corresponding to a distribution function f (x, ξ , t) which maximizes the entropy h = −k B R3 f log f dξ where k B is the Boltzmann constant, f dxdξ is the number of particles in the volume dxdξ of the phase space centred at (x, ξ ) and included in R3 × R3 . The closely related notion is the maximization of the entropy production during nonequilibrium processes (the so-called maximum entropy production principle, ME PP) [284].

3.5 Phonon Hydrodynamics

135

∂ = qi − ∂γi

 vgi ω f d k.

Finally, from Eq. (3.63) follows the four-moment non-equilibrium phonon distribution function as 1   (3.64) f4 = vgi ω ω −1 exp β + γi kB kB where the subscript “4” in the phonon distribution function represents the dependence on the four basic field variables. At equilibrium state, the heat flux and its corresponding Lagrange multiplier vanish and Eq. (3.64) reduces to the Planck distribution Eq. (3.59). Thus the Lagrange multiplier for energy density is β = 1/T . Equation (3.64) could be linearized when heat transport is not too far away from the equilibrium state eq ∂f eq f 4 = f R − γi vgi T 2 R ∂T where under the grey and Debye assumptions the Lagrange multiplier for heat flux is [266] 3qi γi = − 2 2 . T vg cV Thus the four-moment non-equilibrium distribution close to the equilibrium is written as eq 3 ∂ fR eq qi vgi . f4 = f R − 2 vg cv ∂ T This approximation allows the authors to express the flux of heat flux in terms of the energy density: 1 (3.65) Qˆ = e Iˆ 3 where Iˆ the unit tensor. Inserting the Eq. (3.65) into the Eq. (3.61), one obtains the Cattaneo–Vernotte-type heat conduction equation. The higher-order approximation to the flux of heat flux Qˆ is derived from the balance equation   ∂ Mi jk ∂ Qi j 1 1 2 + v eδi j − Q i j , = ∂t ∂ xk τR 3 g the third-order tensor Mˆ is defined as  Mi jk = vgi vg j vgk ω f d k.

136

3 Phonon Models

Using the perturbation expansion in the Knudsen number as a small parameter ε = Kn (1) Q i j = Q i(0) j + ε Qi j + . . . and retaining the zeroth-order Q i(0) j =

1 2 v eδi j 3 g

and the first-order Q i(1) j

 = −τ R

∂ ∂ (Q i(0) (M (0) | f ) j | f4 ) + ∂t ∂ xk i jk 4



terms for the flux of the heat flux are written as ∂qk 1 2 1 δi j − τ R vg2 Q i j = vg2 ε Ri j + τ R vg2 3 15 ∂ xk 5



∂q j ∂qi + ∂x j ∂ xi

 .

Higher-order approximate terms for the flux of heat flux are thus derived in the form of the gradient of heat flux, which is crucial for modelling nanoscale heat transport. Finally the balance equation is written as   1 2 ∂q 1 2 + q = −λ∇T +  ∇ q + ∇(∇ · q) τR ∂t 5 3

(3.66)

where  = vg τ R is the mean free path. Since Eq. (3.66) uses the same field variables as the traditional Fourier’s description, this model avoids the complexity of classical moment methods that involve the governing equation of higher-order moments. In the diffusive limit where both relaxation and non-local effects are negligible, Eq. (3.66) reduces to the Fourier’s law. The Eq. (3.66) differs only by the numerical coefficient at the non-local term from the Guyer–Krumhansl Eq. (3.21) that was developed for the study of the heat transport in bulk dielectric crystals in low-temperature situations. The authors stressed that the mathematical structure of these equations is the same, but the underlying physical mechanisms of heat transport are different. The non-local terms in the Guyer–Krumhansl equation are originated from the non-resistive phonon normal scattering and this equation is suitable for the investigation of the heat transfer at low temperatures. The non-local terms in the phonon hydrodynamic by Guo and Wang represent the spatial non-equilibrium effects from the phonon–boundary scattering or from the large spatial thermal variation. The non-equilibrium phonon distribution function corresponding to the hydrodynamics Eq. (3.66) is written as [266]

3.5 Phonon Hydrodynamics

137 eq

f =

eq fR

eq

3τ R ∂qi eq 3 ∂ fR τ R ∂qi ∂ f R qi vgi + − . + vgi vg j C V vg2 ∂ T C V ∂ xi ∂ T C V vg2 ∂x j ∂T

Guo and Wang used the derived phonon hydrodynamics equations to solve a number of examples: • • • • • •

The in-plane phonon transport through a thin film.9 The cross-plane phonon transport through a thin film. The phonon transport through a nanowire. The one-dimensional transient phonon transport across a thin film. The high-frequency periodic heating of a semi-infinite surface. The heat conduction in a transient thermal grating.

3.5.1 Nonequilibrium Thermodynamics of Phonon Hydrodynamic Model As a phonon is a kind of boson, the kinetic (or statistical mechanical) definition of the entropy density is [204, 287, 288]  [ f ln f − ( f + 1) ln( f + 1)]dk

s = −k B

(3.67)

The temporal derivative of entropy density is ∂s = −k B ∂t

 #

$ ∂( f ln f ) ∂[(1 + f )ln(1 + f )] − dk ∂t ∂t

Using the phonon Boltzmann equation, the entropy balance equation for phonon transport is written as ∂s + ∇ · Js = σ s ∂t where entropy flux is  J s = −k B

9

vg [ f ln f − ( f + 1) ln( f + 1)]dk

This problem was studied in numerous work both experimentally and theoretically—see, for example, Hua and Gao [285] and references therein. Monte Carlo method is frequently used to solve the Boltzmann transport equation, intrinsic scattering processes such as phonon–phonon and phonon–impurity scatterings being accounted for in the relaxation time approximation. The lateral boundaries usually are assumed to be completely diffusive while other boundaries are treated as phonon blackbody, i.e. as absorbing phonons. One of conclusions drawn is that phonon–boundary scattering significantly suppresses the in-plane thermal transport [285, 286].

138

3 Phonon Models

and the entropy generation is  σ s = −k B

C( f )[ln f − ln( f + 1)]dk.

The authors considered the heat transport at ordinary temperatures, where the phonon resistive scattering is dominant over the normal scattering, i.e. the normal scattering and hydrodynamic phonon transport usually relevant at very low temperature [95, 137] are not taken into account. In the diffusive regime of heat conduction, the phonon distribution function is obtained through a Chapman–Enskog expansion within first order eq (3.68) f = fR + φ where the first-order perturbation quantity is eq

φ = −τ R vgβ

∂T ∂ fR . ∂ xβ ∂ T

Using the Eq. (3.68) in the kinetic definition of the heat flux produces the Fourier law q = −λ∇T with the bulk thermal conductivity λ = 1/3cV vg  where the phonon mean free path is defined as  = vg τ R . The authors used the approximations through Taylor’s expansion eq

eq

eq

f ln f ≈ f R ln f R + φ(1 + ln f R ) +

φ2 eq 2 fR

to get  s = −k B

[

eq fR

ln

eq fR

−(

eq fR

+ 1) ln(

eq fR

1 + 1)]dk − k B 2



φ2 eq dk. + fR )

eq f R (1

The second-order term in φ 2 is negligibly small in the near equilibrium regime where Fourier law is valid and the entropy density reduces to exactly the equilibrium value and thus is consistent with the local equilibrium hypothesis in the classical inverse thermodynamics. In the case of the phonon hydrodynamics Guo et al. derived the following expression for the entropy density s = seq −

τR τR q·q− 2 (∇q)s0 : (∇q)s0 . 2 2λT 5λT 2

where the symmetric traceless part of the second-order tensor is expressed as (∇q)s0 =

1 1 [∇q + (∇q)T ] − (∇ · q) Iˆ, 2 3

3.5 Phonon Hydrodynamics

139

here the superscript “T” denotes the transpose of a tensor and Iˆ is the unit tensor. The entropy flux is q 22 q · (∇q)s0 . Js = + T 5λT 2 With the use of the phonon hydrodynamic Eq. (3.66) the entropy generation is expressed as q·q 22 + (∇q)s0 : (∇q)s0 . σs = λT 2 5λT 2 The non-negativeness of entropy generation is explicitly ensured σ s ≥ 0 due to the quadratic form of its expression. Thus, the phonon hydrodynamic equation for nanoscale heat transport at ordinary temperature is consistent with the second law in the frame of the extended irreversible thermodynamics.

3.5.2 Flux-Limited Behaviour Usually three kinds of the non-Fourier behaviour are considered: relaxation, nonlocal, and nonlinear effects [125, 289]. At high-temperature gradients the nonlinear effects of the heat transport become important due to the limited velocity of the heat carriers and a saturation of the heat flux occurs: it cannot reach the arbitrary high values but is limited by the quantity of the order of the energy times the maximum speed of the signal propagation. This case may be met in nanostructures where a finite temperature difference is established over an extremely small-scale length. This kind of nonlinear behaviour may have important effect, for example, on the effective cooling of the microelectronic devices. Similar situations arise in the radiative heat transfer [290, 291] where the maximum heat flux is ∝ T 4 c (c is the speed of light), mass diffusion [292–294] and in the plasma physics in the study of the collapse of stars and in the laser–plasma interaction in the laser-induced nuclear system where the maximum heat flux is ∝ T 3/2 [295]. Guo et al. [289] studied the flux-limited behaviour in the phonon hydrodynamics and some other non-Fourier models: • The phonon hydrodynamic model derived from the phonon kinetic theory that the authors considered as a credible standard, since it is the direct result from the phonon Boltzmann equation. The zeroth-order solution to the BTE under the Callaway relaxation approximation by maximum entropy principle is [287, 296] τ

3vg qq ∂q  + q = −λ∇T − τ ∂t 2vg E + 4vg2 E 2 − 3q 2

(3.69)

140

3 Phonon Models

where vg is the average phonon group speed, E is the phonon energy density,  denotes the deviatoric part of tensor qq. The phonon " energy density E is integrated from the phonon distribution function f as E = ω f dk, ω is the energy quanta, k is the wave vector. • The Lagrange multiplier model, the heat flux is related to the Lagrange multiplier conjugate to the heat flux in information theory,which results in a nonlinear heat transport equation [297] ⎡ 3 q = ⎣1 − 2



q vg cV T



2 +

3 1+ 4



q vg cV T

2

⎤ ⎦ ∇T.

(3.70)

• The hierarchy moment model based on a continued-fraction technique [298] q=

1 2

+



λ∇T 1 4

+ 2 (∇ ln T )2

(3.71)

where l is the MFP of heat carriers. • The nonlinear phonon hydrodynamic model derived by a dynamical nonequilibrium temperature method [126, 299] that could be treated as a derivative of EIT leads to a non-local nonlinear heat conduction equation [126] τ

2 τ ∂q + q = −λ∇T + q · ∇q + 2 [∇ 2 q + 2∇(∇ · q)]. ∂t T cV

(3.72)

• The phenomenological tempered diffusion model [294, 300]  q = − 1 − (q/vg cV T )2 λ∇T.

(3.73)

• The thermomass model (Chap. 4) τT

 qq  ∂q + q + λ∇T = −τT ∇ · ∂t E

(3.74)

where τT = ρτ vg2 /6γ cV T is the relaxation time of thermon gas, γ is the Grüneisen constant of a dielectric material. • The generalized nonlinear heat transport equation [172] τT

∂q + q = −λ(1 + βq 2 )∇T + μq · ∇q + μ ∇q · q + 2 μq · ∇q ∂t

(3.75)

where β, μ and μ are the phenomenological coefficients. Guo et al. thus analysed three group of non-Fourier models: phonon hydrodynamic model from phonon kinetic theory (3.69), non-equilibrium thermodynamic

3.5 Phonon Hydrodynamics

141

models derived in the framework of the the irreversible thermodynamic theory. (3.70,3.71,3.72), the phenomenological models (3.73,3.74), and the generalized nonlinear heat transport equation that combines several models (3.75). To assess the models, the authors considered the one-dimensional steady-state heat conduction under a temperature gradient dT /d x. For example, the phonon hydrodynamic model (3.69) reduces to 

4

qx = − 5 − √ 1 − M2 where M =

 λ

dT dx

(3.76)

√ 3qx /2vg cV T . From the Eq. (3.76) evidently follows that M ≤ 1, i.e. 2 qx ≡ C ≤ q max = √ vg cV T. 3

The second law of thermodynamics requires a non-negative effective thermal conductivity, which gives a smaller upper bound and thus the upper limit for the heat flux is √ 2 3 vg cV T. C ≤ q max = 5 Guo et al. [289] listed the approximations for the heat flux qx derived from all other models. In the limit of small heat flux (small temperature gradient) all models reduce to the Fourier law qx = −λ|DT /dx. In contrast, in the limit of large heat flux the nonlinear terms are important and can no longer be neglected. Furthermore, there exists the upper bound for the heat flux in all the nonlinear heat conduction models except the nonlinear phonon hydrodynamic model (3.72). As Li and Cao [301] stated, Fourier heat conduction with the nonlinear thermal conductivity λ = λ(T ) can also predict flux-limited behaviour. In the “fast diffusion” model [302] with λ = λ0 /T α the heat flux always exists for arbitrary boundary temperatures and size in 1D steady-state problems and equals C=

λ0 (T2α (1−α)l λ0 ln TT21 l

− T2α ) α = 1 α = 1.

If α > 1, the limiting saturation heat flux is λ0 T1α−1 , (α − 1)l which will decrease with T1 increasing. In contrast, the saturation heat fluxes of the above non-Fourier models are increasing with T1 increasing (usually it is proportional to T ).

142

3 Phonon Models

Li and Cao [301] noted that some of the models considered above cannot be applied to the heat conduction problem with arbitrarily small size because the definitions of the local temperature and heat flux will be questionable or even undefinable for extremely small size. Therefore, besides the value of the heat flux varying along with the increasing temperature gradient, the applicable size of a heat conduction model should also be limited. The authors considered 1D steady-state heat conduction with the boundary conditions T (0) = T1 and T (l) = T2 , where flux-limited behaviours are usually discussed, and found that there exists a critical size determined by the boundary temperatures, and the heat flux will exist only when the size is larger than the critical one. The critical sizes of these models can be regarded as the limits of their applicable ranges. For the phonon hydrodynamic model (3.69) the authors found that in order to guarantee the existence of heat flux, the size should satisfy the following inequality √

    2  T2 3λ T2 5 −1 −4 −1 . l ≥l = 2vg cV T1 T12 c

If the ratio of the boundary temperatures is sufficiently small, the phonon hydrodynamic model could be applied to arbitrarily small-scale heat conduction problems. For 41/9 < T2 /T1 , the value of l c is positive and only when the size is larger than l c , the heat flux will exist. For the tempered diffusion model (3.73) Li and Cao obtained the following expression for the critical size    T22 λ T1 c − 1 − arccos l = vg cV T2 T12 that is always positive. Thus, for any problem with fixed boundary temperatures, there always exists a critical size l c for the existence of heat flux. The hierarchy moment model (3.71) and the nonlinear phonon hydrodynamic model (3.72) have no restrictions on the linear scale of the problem. The size and boundary effects for the existence of heat flux show different from Fourier heat conduction behaviour—even in the limit of small heat flux these nonFourier models with flux-limited behaviours will not reduce to Fourier law and thus nonlinear effects could not be negligible.

3.6 Relaxon Model Recently (2020) Simoncelli et al. [101] used the evolution of relaxons to derive two coupled equations for the temperature and for the drift velocity that describe the heat transfer in dielectric crystals.

3.6 Relaxon Model

143

The concept of relaxons was introduced several years ago by Cepellotti and Marzari [303] as the collective excitation of the out-of-equilibrium lattice vibrations consisting of a linear combination of the out-of-equilibrium phonon populations. Each relaxon is characterized by a well-defined relaxation time; each relaxon also has a well-defined drift velocity and the mean free path. The thermal conductivity can be interpreted as the relaxon gas motion. Simoncelli et al. restricted their analysis to the “simple” crystals, i.e. such crystals where the phonon interbranch spacings are much larger than their linewidths. The authors start with the linearized phonon Boltzmann transport equation (BTE) 1  ∂n μ (r, t) + vμ · ∇n μ (r, t) = −

μμ n μ (r, t) ∂t V μ

(3.77)

where the sum is over all possible phonon states μ (μ = (q, s) where q varies over the Brillouin zone and s over phonon branches), vμ is the phonon group velocity, V is the normalization volume (V = ν Nc , i.e. the unit cell volume ν times the number of unit cells that constitute the crystal Nc ), μμ is the linear phonon scattering operator (the phonon scattering matrix) [303]. The phonon Boltzmann transport Eq. (3.77) governs the evolution of the deviation of the phonon population n μ from the equilibrium, i.e. the Bose–Einstein distribution 1  , ωμ −1 exp kB T 

where ωμ is the phonon frequency, n μ (r, t) = Nμ (r, t) − N¯ μ . From the solution of the BTE the local lattice energy E(r, t) =

1  ωNμ (r, t) V μ

and the total crystal momentum P(r, t) =

1  q Nμ (r, t) V μ

could be derived [304]. The energy flux generated in response to the temperature gradient determines the thermal conductivity; correspondingly, the crystal-momentum flux generated in response to the perturbation of the drift velocity determines the thermal viscosity.

144

3 Phonon Models

The authors considered a crystal in the hydrodynamic regime of thermal transport; the local equilibrium phonon distribution, obtained maximizing the local entropy under the constraints of fixed local energy and momentum [305] is the phonon drifting distribution [306] 1   , (3.78) NμD = (ωμ − q · u) −1 exp kB T where u is the drift velocity. This distribution differs from the Bose–Einstein distribution due to the presence of the drift velocity u(r, t), a parameter expressing the amount of local momentum. The temperature T (r, t) and the velocity u(r, t) can be related to the Lagrange multipliers that enforce the constraints of fixed local energy and momentum, respectively [101]. The drifting distribution (3.78) depends on time and space implicitly through T (r, t) and u(r, t). Simoncelli et al. in order to study the small perturbation of the temperature and drift velocity split the deviation of the distribution function as n μ = n μT + n μD + n δμ 

where n μT

=

∂ NμD ∂T

(3.79)



 (T − T¯ ),

n μD

=

∂ NμD ∂u

eq

 u + n δμ eq

the subscript “eq” means that the derivatives are computed at equilibrium where T (r, t) = T¯ and u(r, t) = 0. Linearization of the Boltzmann equation around the constant temperature and drift velocity gradients at steady state gives ∂ N¯μ vμ · ∇T + vμ · ∂T



∂ NμD ∂u

 · ∇u = −

1 

μμ (n μT + n μD + n δμ ). V μ

(3.80)

Equation (3.80) can be recasted [101] into symmetric (hence, diagonalizable) form in terms of  N¯ μ ( N¯ μ + 1) ¯ μμ = μμ

N¯ μ ( N¯ μ + 1) and n˜ μ = 

nμ N¯ μ ( N¯ μ + 1)

.

3.6 Relaxon Model

145

¯ μμ is real symmetric and thus can be transformed into The scattering operator the diagonal form 1 1 

μμ θμα = θμα , V μ

τα where θμα is the relaxon (an eigenvector), α is the relaxon index, τα is the relaxon lifetime—the inverse eigenvalue. Any response n¯ μ can be represented as a linear combination of the eigenvectors θμα that are called relaxons.  n¯ μ = f α θμα . α

The BTE could be formulated in relaxon (collective excitation) that consists of a few phonons interacting through the scattering between themselves but uncoupled from the phonons belonging to other relaxons. Then BTE can be formulated in θ α basis as    ∂T ∂ fα  C fα + < 0|α > +∇T + v + Vαα · ∇ f α = − . α k B T 2 ∂t ∂t τ α α

¯ μμ have the well-defined parity, Since the eigenvectors of the scattering matrix n˜ δμ is splitted into even n˜ δμE and odd n˜ δμO components; Eq. (3.80) is decoupled into two parts, one for each parity [101]: • The odd part that describes the response to the temperature gradient 



vμ N¯ μ ( N¯ μ + 1)

·

∂ N¯μ ∇T ∂T

 =−

1 

μμ n˜ δμO

V μ

• And the even part that describes the response to the drift velocity gradient 



vμ N¯ μ ( N¯ μ + 1)

·

∂ N¯μD ∂u

 ∇u = −

1 

μμ n˜ δμE . V μ

The total crystal-momentum flux tensor is written as ij

tot =

1  i J q vμ Nμ V μ

or, accounting for the decomposition (3.79) ij

tot =

1  i J ¯ q vμ ( Nμ + n μT + n δμE ) V μ

146

3 Phonon Models

(only the even part of the phonon distribution function contributes to the crystalmomentum flux tensor). The asymmetric thermal viscosity tensor is formulated as ηi jkl =

√

Ai A k



J l wiα wkα

α>0

where Ai =

1  ¯ ¯ Nμ ( Nμ + 1)(q i )2 k B T¯ V μ

and the velocity tensor j

wiα =

1  i j α φ v θ V μ μ μ μ

for relaxon θμα and the eigenvector φμi . The symmetrized viscosity tensor is expressed as  w wl + wl w √ ηi jkl + ηilk j iα kα iα kα = Ai A k τα 2 2 α>0 j

μi jkl =

j

where the specific momentum in direction i  Ai =

∂ Pi ∂u i

 =

1  ¯ ¯ Nμ ( Nμ + 1)(q i )2 k B T¯ V μ

eq

is referred to as the specific momentum, τα is the relaxation time of relaxon α (i.e. the inverse eigenvalue associated with the eigenvector θμα of the symmetrized scattering j ¯ μμ ), and wiα matrix is the velocity tensor j

wiα =

1  i j α φ v θ V μ μ μ μ

for relaxon θμα and eigenvector φμi . The eigenvectors φμi are three special eigenvectors linked to the crystal momentum N ¯ μμ = ¯ μμ ¯U of the system: the scattering matrix could be decomposed as

+ μμ

N ¯ μμ ¯U where

and μμ contain only the momentum-conserving (normal) and the momentum dissipating (Umklapp) processes, respectively; since the normal part of the scattering matrix conserves crystal momentum, there exists a set of three eigenvectors φμi with zero eigenvalue which are associated to the conservation of crystal momentum in the three Cartesian directions.

3.6 Relaxon Model

147

The thermal conductivity and the viscosity are quantities describing the energy and crystal momentum transport due to the odd and even parts of the spectrum, respectively. The thermal conductivity in the harmonic approximation for the heat flux q=

1  ωμ vμ n μ V μ

is written as (λi j ) S M A =

1  Cμ vμi (μj ) S M A , V μ

j

where (μ ) S M A is the component of the phonon mean free path in the direction j. Thus, thermal conductivity is provided by phonons carrying a specific heat Cμ

1 n¯ μ (n¯ μ + 1)(ωμ )2 , kB T 2 j

travelling at velocity vμ and the mean free path (μ ) S M A before being thermalized by scattering. Cepellotti and Marzari stress that the definition of the phonon lifetime or the mean free path cannot be extended beyond the SMA since the off-diagonal terms of the scattering operator introduce the coupling between the phonons and the phonon thermalization cannot be described by the exponential relaxation. Finally, Cepellotti and Marzari derived the following relation for the thermal conductivity  1  ωμ vμj n μ = C Vαi αj . λi j = V ∇i T μ α The authors also showed that Matthiessen rule [143] for the relaxation time in systems with different scattering mechanisms leads to the overestimation of the thermal conductivity. Simoncelli et al. have derived two coupled equations for the temperature and drift-velocity fields %  ∂T ∂u j  i j ∂ 2 T = W0i j T¯ A j C i − λ =0 ∂t ∂r ∂r i ∂r j i, j i, j 3

C

3

(3.81)

and ∂u i + Ai ∂t



3 3 3   √ C Ai  j ∂ T ∂ 2uk ij Wi0 j − μi jkl j l = − − Ai A j DU u j , (3.82) ¯ ∂r ∂r ∂r T j j,k,l j

148

3 Phonon Models

where C=

 1 N¯ μ ( N¯ μ + 1)(ωμ ) k B T¯ 2 V μ

is the specific heat, W0i j =

 μ

φμ0 vμi φμj

ij is the the velocity tensor, T¯ is the reference temperature, DU is the momentum dissipation rate. The authors stressed that the derived viscous heat equations differ from the Stokes equations for fluids in two major ways:

• There is no analogy with the mass conservation satisfied by Stokes equations, since the total phonon number is not a constant of motion (e.g. a phonon coalescence event decreases the number of phonons in the system); • While collisions between molecules in the fluid conserve momentum, scattering among phonons does not conserve crystal momentum in the presence of Umklapp processes. The most important feature of the viscous heat equations is their capability to describe hydrodynamic thermal transport in terms of mesoscopic quantities, i.e. temperature and drift velocity, resulting in a much simpler and computationally less expensive approach than the solution of the microscopic BTE. The parameters entering these equations can be determined from first principles . Second sound is the coherent propagation of a temperature wave is an effect properly described by the viscous heat equations. From a phenomenological point of view, second sound appears when the temperature field satisfies the following damped wave equation (x as the second sound propagation direction): ∂ 2 T (x, t) 1 ∂ T (x, t) ∂ 2 T (x, t) − τ + =0 ss ∂t 2 τss ∂t ∂x2 where τss and τss are the second-sound relaxation time and propagation velocity, yet to be determined. Simonelli et al. derived the second-sound equation from the viscous heat equations following two different approaches (bottom up and top down). The temperature profile that solves the second-sound equation has the form of a damped wave: ¯ where the second-sound frequency ω(k) ¯ depends on T (x, t) = T¯ + δT ei(kx−ω(k)t)) the second-sound wave vector k. Simonelli et al. solved the one-dimensional problem for Eqs. (3.81, 3.82) analytij ically neglecting the dissipation of momentum by the Umklapp process (DU = 0) These equations can be rewritten (suppressing indices) as %

∂2 T ∂u −λ 2 =0 T¯ AC W ∂t ∂x

3.6 Relaxon Model

149

&

and

AC ∂ T ∂2 W − λ 2 = 0. ∂t ∂x T¯

For one-dimensional domain (with the length 2l) the no-slip boundary condition is used u(x = ±l) = 0 and Dirichlet conditions for the temperature T (x = ±l) = T¯ ± δT. The solution is written as  u(x) = δT and

λ μT¯





cosh(bx) − cosh(bl) sinh(bl)

sinh(bx) T (x) = T¯ + δT sinh(bl) 

where b=

AC W 2 . μλ

The factor 1/b is the length scale at which boundary scattering affects thermal transport. The authors calculated the normalized L2 distance between the temperature profile predicted by Fourier law and the viscous heat equations for a given sample length l T O T and reference temperature T¯ in the spatially homogeneous region G as " L = 2

G [TF (x, t) "

− Tv (x, t)]dxdy G dxdy

The authors found that the dependence of L2 on T¯ and T¯ is qualitatively unchanged for different sample shapes. The model of relaxons has also been used to predict the existence of a driftless second sound in addition to conventional drifting second sound, although this driftless second sound would be difficult to observe [29].

150

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Chapter 4

Thermomass Model

The thermomass model is based on the old ideas of Tolman [1] that, since according to the special relativity all forms of energy have inertia, the heat carriers own the mass-energy duality: these carriers exhibit both the energy-like characteristics (in the conversion processes) and the mass-like characteristics (in the transport processes) [2]. The thermomass theory applied to the heat conduction in the rigid solid bodies at rest formally could be classified as an example of the phonon model, and its equations are similar to that of the phonon hydrodynamics, but the underlying physical ideas are quite different. The mass of heat is determined by the mass-energy equivalence of Einstein [3–5] E = Mc2 = 

M0 c 2

1−

v2 c2

where E is the thermal energy, M0 is the rest mass, v is the velocity of the heat carrier, c is the speed of light in vacuum, and M is the relativistic mass. When v  c, this equation is simplified to E ≈ (M0 + Mk )c2 here Mk is the additive mass induced by the kinetic energy. The thermomass (TM) Mh is the relativistic mass of the internal energy U Mh =

U . c2

The thermomass is very small (10−16 kg for 1 J heat) [2, 6]. The density of the thermomass contained in the medium is [7]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. I. Zhmakin, Non-Fourier Heat Conduction, https://doi.org/10.1007/978-3-031-25973-9_4

163

164

4 Thermomass Model

ρh =

ρC V T c2

where ρC V T represents the thermal energy density. The thermon is defined as a unit quasi-particle carrying thermal energy. For fluids, the thermons are supposed to be attached to the particles of the medium; for solids, the thermon gas is the phonon gas that flows through the vibrating lattices or molecules [2]. The macroscopic drift velocity of the thermon gas that is considered as a continuum is given as q . vh = ρC T The total energy of thermon gas in a medium is the sum of the kinetic energy and the potential (pressure) energy [8]   ET =

(ρT u T du T + d pT )dV, V

where V is the total volume of the medium.

4.1 Equation of State (EOS) of the Thermon Gas A general form of the EOS of the thermomass gas can be the written as F( pt , ρT , T, ξ ) = 0, where pT is the thermomass pressure, ρT is the density of the thermon gas, and ξ the parameter related to the effects of the interaction between thermons; if the interaction between the thermons could be neglected, the EOS has an explicit form [2].

4.1.1 EOS of Thermon Gas in Ideal Gas Two assumptions are made for the thermon gas as an ideal gas: 1. The thermons are attached to the gas molecules and satisfy the Maxwell– Boltzmann distribution function. 2. The Newtonian mechanics is applicable to the thermon gas. The pressure for a system of n particles with mass m randomly moving in x direction with the velocity u x is P = nmu 2x

4.1 Equation of State (EOS) of the Thermon Gas

165

and accounting for the spatial symmetry in x, y, z directions (u 2x = u 2y = u 2z = 13 u¯ 2 ) [5],   1 2 1 m u¯ 2 1 1 nm u¯ 4 2 . P = nm h u¯ = n u¯ = 3 3 2 c2 6 c2 Using the classical Maxwell–Boltzmann distribution function  f M (u) = 4π u one gets

∞ u¯ = 4

2

m 2π k B T

3/2

  mu 2 exp − 2k B T 

kB T u f M (u)du = 15 m

2

4

0

and finally P=

5 ρC V RT 2 3 c2

where R is the gas constant. The last equation shows that the thermon gas pressure is proportional to the square of temperature. Dong et al. [7] formulated the isotropic thermomass pressure in the phonon Boltzmann method via the equilibrium Planck distribution function f Es as    P=

f Es (x, t, k)

ω 2 v dk x dk y dk z c2 x

which is similar to the classic expression in the kinetic theory of gases    px =

f (x, t, v)mvx2 dvx dv y dvz .

4.1.2 EOS of Thermon Gas in Dielectrics Phonons are thermons for dielectrics. The total energy of the lattice vibrations can be written as E D = E D0 + E h = (M0 + Mh )C V T where E h is the energy of TM. Thus the thermon gas pressure is P=

γG ρ (Cv T )2 c2

166

4 Thermomass Model

where γG is the Grüneisen parameter (also called “the Grüneisen anharmonicity constant” [9]) that describes the overall effect of the volume change of a crystal on vibrational properties, which is expressed as γG = V /C V (d P/dT )V , where V is the volume of a crystal, C V is the heat capacity at constant volume, and (d P/dT )V is the pressure change due to the temperature variation at constant volume [10]. The pressure of the phonon gas is proportional to the square of the temperature as in ideal gas. For silicon at the room temperature the thermon gas pressure is about 5 × 10−3 Pa [5].

4.1.3 EOS of Thermon Gas in Metals The thermons in metals are attached to electrons. The thermon gas pressure is given as 1 P = nm h u 2h 3 where m h = cε2 , ε√ is the internal energy that includes contributions of both electrons and lattice, u h = 2ε/m is the velocity of the randomly moving particles, and thus the pressure is 2 nε2 . P= 3 c2 m The general equation for the thermon pressure is written as 2 P= 3mc2

∞ ε2 f (ε, T )Z (ε)dε, 0

where f (ε, T ) =

1  ε − εF +1 exp kB T 

is the Fermi–Dirac distribution function and Z (ε) =

1 2π 2



2m 2

 23

1

ε2

is the Sommerfeld electron state density function. Wang [5] finally obtained the following expression for the thermon gas pressure P=

5 π 2 nk 2B 2 T . 12 c2 m

4.2 Equations of Motion of Thermon Gas

167

Wang et al. [2] noted that the electrical interaction between electrons in metals may be not be negligible and can influence the EOS below the Debye temperature.

4.2 Equations of Motion of Thermon Gas The one-dimensional conservation equations of mass and momentum are ∂ρh u h S ∂ρh + = 2 ∂t ∂x c where S is the internal heat source and ∂ ∂ ∂P (ρh u h ) + (u h · ρh u h ) + + fh = 0 ∂t ∂x ∂x where f h is the resistance. The continuity equation for the thermon gas is actually the energy conservation equation. The thermon flow in the solid can be seen as the flow of the compressible fluid in the porous medium, and the D’Arcy law (K is the permeability of the porous medium) dp u = −K dx can be used to estimate the TM resistance f h = βh u h , the proportionality constant [11, 12] 2γρ 2 C 3 T 2 . βh = c2 λ The effective resistance force f h is introduced instead of the viscous term (μh ∇ 2 uh ) to avoid [2] 1. determination of the viscosity μh for the complex materials; 2. the interaction effects between the thermomass gas and the lattice/solid molecules. The thermon gas flow in the solids (phonon flow) is driven by the pressure gradient, thus by the gradient of the square of the temperature [11]. The thermomass is too small to be observed under common conditions, but under the extreme conditions of the ultrafast heating or the ultrahigh heat flux the inertia of TM will represent itself and cause detectable influences to the heat conduction process. The conservation equation of the thermon momentum can be written as a heat conduction equation [5]

168

4 Thermomass Model

 τh

∂q ∂q ∂T + 2u h − u 2h ρC V ∂t ∂x ∂x

 +λ

∂T + q = 0. ∂x

The general heat conduction in the three dimensions could be written as [4] τh

∂q + 2(l · ∇)q − bλ∇T + λ∇T + q = 0, ∂t

where b=

(4.1)

q2 . 2γG ρh2 C V3 T 3

The coefficient b could be expressed through the thermal Mach number Mah [13, 14] as the ratio of the phonon gas velocity u h to the thermal wave speed in the phonon gas C h b = Mah2 . Equation (4.1) could be rewritten in the form similar to the Cattaneo model by introduction of the material derivative ∂ D = + 2(v h ∇) Dt ∂t as [4] τh

Dq + q = −λ(1 − b)∇T. Dt

Wang et al. [6] noted that relaxation times in the thermomass theory and in the Cattaneo model have different meaning: while in the thermomass theory the characteristic time means the lagging time from the temperature gradient to the heat flux, and in the Cattaneo equation it is relaxation time to the equilibrium state. Wang [5] developed a special two-step version of the thermomass theory for metals that are subjected to the ultrafast laser heating under the following assumptions 1. The electrons absorb the laser energy and then transfer it the lattice by the electron– phonon coupling. 2. The scattering at the defects and the grain boundaries is ignored. 3. The electron–phonon collisions are presented by an electron–phonon coupling factor. Similar to the porous flow hydrodynamics the Brinkman term μ∇ 2 q (μ is the effective viscosity that is defined as μ = 2τ/Cv T [15]) could be introduced into the equation of the thermon gas motion (4.1) that exhibits the additional drag by the walls in the system and is necessary only if the characteristic length of the system is comparable to the friction boundary layer of the thermomass, i.e. if the Knudsen number is large enough [7]

4.2 Equations of Motion of Thermon Gas

τh

169

∂q + 2(l · ∇)q − bλ∇T + λ∇T + q − μ∇ 2 q = 0. ∂t

(4.2)

The equation similar to Eq. (4.2) was suggested by Cimmelli et al. [15, 16] as the nonlinear extension of the Guyer–Krumhansl equation (3.20). The authors modified the evolution equation for the semiempirical dynamical temperature (2.10) to β˙ = −

1 l ∂ 2β (β − θ ) + 3 τR τR ∂ x 2

where l is the suitable coefficient accounting for the mean free path of phonon and finally get in the three-dimensional case τR

∂qxi ∂θ + qxi = −λ − μqxk qxi + 2 ∂t ∂ xi

with μ=



∂ 2 q xi ∂ 2 qxk + 2 ∂ xk2 ∂ xi2

 (4.3)

2τ R . Cv θ

In the one-dimensional case q = qx i where i is a unitary vector along the direction x, and Eq. (4.3) simplifies to τR

∂θ ∂qx ∂qx ∂ 2 qx + μqx = −λ + 3 2 2 . ∂t ∂x ∂x ∂x

(4.4)

The condition qx  2 ∂∂ xq2x necessary to neglect the heat flux qx in Eq. (4.3) is satisfied, for example, in the one-dimensional nanowires with very small characteristic 2 length L and thus ∂∂ xq2x ∝ qx /L 2 [16]. Formal analogy with the Navier–Stokes equations of the hydrodynamics (more exactly, their low Reynolds number version-Stokes equations) allowed Cimmelli et al. [15, 16] to introduce the thermal Reynolds number. As was stated above (Sect. 3.1) its usefulness is dubious. Sometimes the Brinkman number Br = l B /L is used [17] that compares √ the viscous friction with the D’Arcy friction; the characteristic length scale l B = μh /βρh : 2

• if Br  1, the boundary effect region is small compared to the channel width and the velocity profile is nearly uniform across the channel cross section. • if Br  1, the velocity profile is close to the Poiseuille flow. The entropy production during heat transport is due to the dissipation rate of the mechanical energy of the thermon gas similar to the viscous dissipation in hydrodynamics dE h + ∇ · J Eh = f h · u h , dt where E h is the mechanical energy of the thermon gas and J E h is the flux of E h .

170

4 Thermomass Model

Thus the entropy production in the thermomass theory can be written as [18] s =− σTM

1 1 q · q. F h · uh = T λT 2

Dong et al. [4, 18] formulated the total derivative of the entropy density as q ∇·q ds = −∇ · J s + σTM = · (q + λ∇T ) − 2 dt λT T where J s is the entropy flux. The non-equilibrium temperature is introduced by Dong et al. [4] as θ

−1

 =

∂s ∂e

 = V,q

1 1 ∂(τ/(λTeq2 )) q · q, − Teq 2 ∂e

where the subscript V, q means that the derivative is token at constant volume and heat flux. Evidently, the non-equilibrium temperature θ is lower than the local equilibrium temperature Teq . Under the assumption that τ does not depend on Teq θ=

1 τ + O(q 2 )]. ≈ Teq [1 − 1 τ λC V Teq2 q · q + Teq λC V Teq3 q · q

Dong et al. [19] exploited an analogy analysis between the non-Fourier heat conduction and the non-Newtonian momentum transport. Similar to the assumptions in the thermomass model, the authors derived a governing equation for momentum transport in the nanosystems, which accounts for the varying effective viscosity in steady flow. This shear thinning effect will be apparent in the nanochannel flow where the velocity gradient and the momentum transport flux are huge. The molecular dynamics simulation was performed in the Lennard–Jones fluid and the hard sphere gas and showed that the viscosity decreases with the shear rate.

4.3 Heat Flow Choking Phenomenon Wang et al. [20, 21] for the one-dimensional steady heat transfer without the internal heat source use the heat conduction equation in the form  dT  +q =0 κ 1 − Mah2 dt

4.4 Dispersion of Thermal Waves

171

where the thermal Mach number is introduced Mah =

uh , Ch

u is the drift velocity √ of the phonon gas, and the thermal sound speed is, e.g. for dielectrics, C h = 2γ C V T . Thermon gas is a compressible fluid; thus its flow demonstrates numerous features similar to those of the flow of the compressible gas such as air. One of such features is the flow behaviour in the convergent nozzle when the Mach number equals to unity. In the flow of compressible air driven by the pressure gradient in the converging nozzle the velocity increases and the pressure decreases in the flow direction. The flow choking occurs when the current Mach number equals unity and there is the pressure jump. The drift velocity of thermon gas driven by the temperature gradient increases as it flows in the opposite direction of temperature gradient. The heat flow choking that occurs at the thermal Mach number equalled to unity results in the temperature jump. The confirmation of the heat flow choking phenomenon was obtained in the experiments on the heat conduction in an individual single-walled carbon nanotube (CNT) suspended between two metal electrodes by Wang et al. [20]. The CNT was electrically heated by the internal Joule heat, the heat flowed from the middle to the two ends of the CNT. The heat flux was governed by the electrode temperature until the thermal Mach number reaches unity at the CNT ends; further decrease of the electrode temperature had no effect on the heat flux. However, this phenomenon was not confirmed by Zhang et al. [22] who studied using the molecular dynamics simulations of the energy transport in the open-ended single-walled carbon nanotube cantilevered out of the (001) surface of bulk silicon. The authors did not observed the heat flow choking as the flow increases but found that energy can be efficiently transported by the low-frequency mechanical wave. This energy can be calculated as PW =

1 2 2 μω A Vg 2

where μ is the mass of CNT per unit length, ω is the angular frequency, A is the wave amplitude, and Vg = ∂ω/∂k is the phonon group velocity.

4.4 Dispersion of Thermal Waves The speed of the thermal wave in the thermomass theory depends on whether the wave is moving towards the heat flux or in the opposite direction; the temperature dependence of the wave speed to the shrinking or extending of the heat pulses spatially [23].

172

4 Thermomass Model

Zhang et al. [24] investigated numerically using an implicit finite difference scheme (with the central differences for space discretization and the backward differences for time) the dispersion of the thermal waves. The authors considered the case of the cosine heat flux pulse boundary condition. As the wave moves forward, peaks appear in the rear of the thermal wave. The underlying mechanism for the dispersion is that thermal waves travel faster in the the regions with higher temperature. Computations were performed for the CV, DPL and TM models. Zhang et al. [24] started with the TM model written in the form q ∂q q ∂T q q − τTM + τTM ∇ · q − τTM · ∇T = −λ∇T ∂t T ∂t ρC V T ρC V T T (4.5) and in order to analyse the origin of the thermal wave dispersion considered the special versions of Eq. (4.5) q + τTM

∂q = −λ∇T, ∂t

(4.6)

∂q q ∂T − τTM = −λ∇T, ∂t T ∂t

(4.7)

∂q q + τTM ∇ · q = −λ∇T, ∂t ρC V T

(4.8)

q ∂q q − τTM · ∇T = −λ∇T. ∂t ρC V T T

(4.9)

q + τTM q + τTM0 q + τTM0 q + τTM0

Equation (4.6) is used to study the effects of the inertia term of heat flux to time on the dispersion of the TM-wave; this term is different from that of the CV model since the characteristic time τTM decreases with the temperature, while the relaxation time remains unchanged in the propagation of the CV-wave. The effects of the inertia term of temperature to time, the inertia term of heat flux to space and the inertia term of temperature to space could be investigated using Eqs. (4.7), (4.8) and (4.9), respectively [24]. It should be noted that according to the energy conservation equation ρCv

∂T +∇ ·q =0 ∂t

with the constant thermal properties and no heat sources Eq. (4.7) could be transformed into Eq. (4.8) by replacing ∂ T /∂t by −∇ · q/ρCv . Zhang et al. also showed that the increase of the amplitude of the heat flux pulse and the decrease of the initial temperature enhance the dispersion of TM-wave; the increase of the amplitude of the heat flux pulse and of the relaxation time τq enhance the dispersion of CV-wave and DPL-wave, while the increase of the relaxation time τT weaken the dispersion of the DPL-wave.

References

173

References 1. Tolman, R.C.: On the weight of heat and theral equilibrium in general relativity. Phys. Rev. 35, 904–924 (1930) 2. Wang, M., Yang, N., Guo, Z.Y.: Non-Foirier heat conductions in nanomaterials. J. Appl. Phys. 110, 064310 (2011) 3. Guo, Z.Y.: Motion and transfer of thermal mass—thermal mass and thermon gas. J. Eng. Thermophys. 27, 631–634 (2006) 4. Dong, Y., Cao, B.Y., Guo, Z.Y.: Temperature in nonequilibrium states and non-Fourier heat conduction. Phys. Rev. E 87, 032150 (2013) 5. Wang, H.D.: Theoretical and Experimental Studies on Non-Fourier Heat Conduction Based on Thermomass Theory. Springer (2014) 6. Wang, M., Guo, Z.Y.: Understanding of temperature and size dependences of thermal conductivity of nanotubes. Phys. Lett A 374, 4312–4315 (2010) 7. Dong, Y., Cao, B.Y., Guo, Z.Y.: Generalized heat conduction laws based on the thermomass theory and phonon hydrodynamics. J. Appl. Phys. 110, 063504 (2011) 8. Wang, M., Cao, B.Y., Guo, Z.Y.: General heat conduction equations based on the thermomass theory. Front. Heat Mass Transfer 1, 013004 (2010) 9. Liu, W., Balandin, A.A.: Thermal conduction in Al x Ga1−x N alloys and thin films. J. Appl. Phys. 97, 073710 (2005) 10. Gu, X., Wei, Y., Yin, X., Li, B., Yang, R.: Phononic thermal properties of two-dimensional materials. Rev. Mod. Phys. 90, 041002 (2018) 11. Cao, B.Y., Guo, Z.Y.: Equation of motion of phonon gas and non-Fourier heat conduction. J. Appl. Phys. 102, 053503 (2007) 12. Wu, J., Guo, Z., Song, B.: Application of Lagrange equations in heat conduction. Tsinghua Sci. Technol. 14, 12–16 (2009) 13. Tsou, D.Y.: Shock wave formation around a moving heat source in a solid with finite speed of heat propagation. Int. J. Heat Mass Transfer 32, 1979–1987 (1989) 14. Mandrusiak, G.D.: Analysis of non-Fourier conduction waves from a reciprocating heat source. J. Thermophys. Heat Transfer 11, 82–89 (1997) 15. Cimmelli, V.A., Sellitto, A., Jou, D.: Nonlinear evolution and stability of the heat flow in nanosystems: beyond linear phonon hydrodynamics. Phys. Rev. B 82, 184302 (2010) 16. Cimmelli, V.A., Sellitto, A., Jou, D.: Nonequilibrium temperatures, heat waves, and nonlinear heat transport equations. Phys. Rev. B 81, 054301 (2010) 17. Dong, Y., Cao, B.Y., Guo, Z.Y.: Size dependent thermal conductivity of Si nanosystems based on phonon gas dynamics. Physica E 56, 256–262 (2014) 18. Dong, Y., Cao, B.Y., Guo, Z.Y.: General expression for entropy production in transport processes based on the thermomass model. Phys. Rev. E 85, 061107 (2012) 19. Dong, R.Y., Dong, Y., Sellitto, A.: An analogy analysis between one-dimensional non-Fourier heat conduction and non-Newtonian flow in nanosystems. Int. J. Heat Mass Transfer 164, 120519 (2021) 20. Wang, H.D., Cao, B.Y., Guo, Z.Y.: Heat flow choking in carbon nano-tubes. Int. J. Heat Mass Transfer pp. 1796–1800 (2010) 21. Wang, H.D., Cao, B.Y., Guo, Z.Y.: Non-Fourier heat conduction in carbon nanotubes. J. Heat Transfer p. 051004 (2012) 22. Zhang, X., Hu, M., Poulikakos, D.: A low-frequency wave motion mechanism enables eficient energy transport in carbon nanotubes at high heat fluxes. Nano lett. 12, 3410–3416 (2012) 23. Sellitto, A., Rogolino, P., Carlomagno, I.: Heat-pulse propagation along nonequilibrium nanowires in thermomass theory. Commun. Appl. Industr. Math. 7, 39–55 (2016) 24. Zhang, M.K., Cao, B.Y., Guo, Y.C.: Numerical studies on dispersion of thermal waves. Int. J. Heat Mass Transfer 67, 1072–1082 (2013)

Chapter 5

Mesoscopic Moment Equations

In order to correctly describe the high-frequency and the short-wavelength processes Bergamasco et al. [1] developed in the frame of the kinetic theory a number of the mesoscopic moment systems (the two-moment and the three-moment systems) introducing the Knudsen number Kn as the ratio of the mean free path of the heat carriers to the characteristic size of the system and using the concept of ghost moment. The authors considered the following four-flow regimes [2]: continuum flow (K n ≤ 0.001), slip-flow regime (0.001 < K n ≤ 0.1), transition regime (0.1 < K n ≤ 10) and free molecular flow (K n > 10). The ghost moments are those that have a higher order with respect to those required to recover the hydrodynamic level (hydrodynamic moments). This approach provides a quantitative description of heat hydrodynamics at the mesoscopic level, i.e. in terms of the partial differential equations (PDEs) that are simpler than the microscopic integro-differential BTE (Simonelly et al. [3] used the term “mesoscopic model” to denote any description that requires more fields or PDEs than the Fourier’s law for the temperature field—probably the first example of such approach is the research by Sussmann and Thellung [4] who, starting from the linearized BTE in the absence of momentum-dissipating (Umklapp) phonon–phonon scattering events, derived mesoscopic equations in terms of the temperature and of the phonon drift velocity, i.e. the thermal counterparts of pressure and fluid velocity in liquids). The authors start from the Fourier equation rewritten as K n2

∂ ∂T + Kn ∂t ∂x

 −K nκ

∂T ∂x

 =0

and add a ghost moment φ = K nq, which has units of thermal flux, to increase the order of the physical description K n2

K n ∂φ ∂T + = 0, ∂t ρc p ∂ x

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. I. Zhmakin, Non-Fourier Heat Conduction, https://doi.org/10.1007/978-3-031-25973-9_5

(5.1)

175

176

5 Mesoscopic Moment Equations

∂T 1 K n 2 κ ∂φ + K nκ =− φ, ρc p c2 ∂ x ∂t ρc p

(5.2)

where an additional term involving the time derivative of the ghost moment responsible for the enriched mesoscopic description is introduced into Eq. (5.2) and c is the constant velocity of arbitrary value. Equations (5.1) and (5.2) are called the TwoMoment Hyperbolic Equation system from which the mesoscopic equation for the temperature follows ∂T ∂2T κ K n2 ∂ 2 T + (5.3) =κ 2. 2 2 c ∂t ∂t ∂t Equation (5.3) has the same form as the Cattaneo equation where the heat flux relaxation τ = κ K n 2 /c2 . Bergamasso et al. also suggested another two-moment system where the velocity appears in both equations: ∂T K n 2 ∂φ + K nρc p = 0, c2 ∂t ∂x K n2

K n ∂φ c2 ∂T + = − T. ∂t ρc p ∂ x κ

(5.4)

(5.5)

The system of Eqs. (5.4)–(5.5), called by the authors the Switched Two-Moment Hyperbolic system, leads, as the system (5.1)–(5.2), to the same equation for the temperature (5.3). The authors introduce another ghost moment e that has units of temperature and is defined such that e − T = 0 and two additional parameters γ and θ to get ThreeMoment Hyperbolic Equations system: K n2

K n ∂φ ∂T + = 0, ∂t ρc p ∂ x

K n 2 κ ∂φ ∂e 1 + κKn =− φ, ρc p c2 ∂t ∂x ρc p K n2

K n ∂φ 1 ∂e + = ∂t ρc p ∂ x γ



 θ −e . T

(5.6)

(5.7)

(5.8)

Three-moment system can be reduced to the single equation for the temperature      ∂2 ∂T κ ∂2T K n2 ∂ 3 T κ  ∂2T ∂T 2 . = + γκKn − 2 + 2 Kn γ + 2 + c ∂t 2 ∂t θ ∂x ∂x ∂t c ∂t 3 (5.9) Equation (5.9) reduces to Eq. (5.3) for γ = 0 and θ = 1. 2

5 Mesoscopic Moment Equations

177

To analyse the solution, Bergamasso et al. perform the Fourier transform ˆ t) = T (k,

∞

T (x, t)e−ikx dx

−∞

with the inverse Fourier transform 1 T (x, t) = 2π

∞ T (x, t)eikx dk. −∞

to get the Fourier equation in terms of the Fourier image d Tˆ = −κk 2 Tˆ dt that has the general solution ˆ t) = T (k, ˆ 0)e−κk 2 t . T (k, The authors introduce the complex temperature

ˆ 0)eikx−κk t = 0 ei(kx+ωt)

(k, x, t) = T (k, 2

where ω = iκk 2 . The authors assumed that the ghost moment has the same form as the complex temperature (k, x, t) = 0 ei(kx+ωt) Substituting of these solutions into the two-moment system (5.1)–(5.2) yields an eigenvalue problem with the characteristic polynomial κ K n2 2 ω − iω − κk 2 = 0 c2 that has roots ω1,2 =

ic2 ±

√ −c4 + 4κ 2 K n 2 k 2 c2 . 2κ K n 2

The solution can be written as

= 01 ei(kx+ω1 t) + 02 ei(kx+ω2 t)

(5.10)

178

5 Mesoscopic Moment Equations

and = 01 ei(kx+ω1 t) + 02 ei(kx+ω2 t) . The same procedure applied to the three-moment system (5.6)–(5.8) results in more complex characteristic polynomial 

κ K n2 2 κ K n2 ω − iω − (γ K n ω − i) c2 θ 2



  1 = κγ k ω 1 − . θ 2

Solution of the last equation requires a rather cumbersome algebra. Authors analyse in detail the two-moment systems. Equation (5.10) is rewritten as √ c2 ± c c2 − 4κ 2 K n 2 k 2 . ω1,2 = i 2κ K n 2 Two cases are κ K nk • < 21 c The argument of the square root is positive. The authors used the Taylor expansions for ω1 and ω2 and concluded that 1 and 2 depend on the time with the multiple scales   t 2 , t, K n t, . . .

1 = 1 K n2 and

2 = 2 (t, K n 2 t, . . . ). Thus the authors concluded that solution can have two modes: 1. a fast (advective) mode that goes to zero very quickly when K n is small; 2. a slow (diffusive) mode that does not depend on K n and recovers the diffusive behaviour of the macroscopic equation. •

κ K nk > 21 c The argument of the square root is negative; thus, the square root yields a complex number, and oscillations in the solution are expected.

References 1. Bergamasco, L., Alberhini, M., Fasano, M., Cardellini, A., Chiavazzo, E., Asinari, P.: Mesoscopic moment equations for heat conduction: characteristic features and slow-fast mode decomposition. Entropy 20, 126 (2018) 2. Beskok, A., Karniadakis, G.E.: A model for flow in channels, pipes and ducts at micro and nano scales. Microsc. Thermophys. Eng. 3, 43–77 (1999)

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3. Simoncelli, M., Marzari, N., Cepellotti, A.: Generalization of Fourier law into viscous heat equations. Phys. Rev. X 10, 011019 (2020) 4. Sussmann, J., Thellung, A.: Thermal conductivity of perfect dielectric crystalls in the absence of Umklapp processes. Proc. Phys. Soc. 81, 1122–1130 (1963)

Chapter 6

Microtemperature and Micromorphic Temperature Models

Liu et al. [1] suggested to divide the non-Fourier heat conduction models into four classes depending on the used theoretical framework: 1. Modifications of the classical Fourier models such as the Cattaneo and Vernotte, the Guyer–Krumhansl (GK), the dual-phase-lag model, the ballistic–diffusive heat-conduction equations. In fact, to obtain a finite velocity for the propagation of thermal signals it is not necessary to assume relaxation terms in heat equation. This result can be also obtained if one assumes a nonlinear equation with the thermal conductivity depending on the temperature [2–4]. 2. Inclusion the temperature gradient as an argument of the Helmholtz-free energy. There are three ways to comply with the second law of thermodynamics introducing the gradient of temperature or gradient of entropy: • By introducing an extra entropy flux: for example, Ireman and Nguyen [5] using the idea of Maugin’s [6, 7] work on gradients of internal variables introduced the gradient of temperature that leaded to the additional entropy production; • By modification using the extra energy production: Forest et al. [8] incorporated the temperature and temperature gradient into the second grade theory, with extra energy production (extra power) by extending the method of virtual power. Later Forest et al. [9] developed the entropy gradient theory and derived an enhanced heat equation which has the structure of the Cahn–Hilliard equation. • Modification with additional entropy production. Nguyen [10] proposed an additional entropy production and a new relationship between internal energy and free energy to introduce the gradient of temperature in the set of state variables. 3. Double temperature models such as models by Aifantis [11, 12] and by Sobolev [13, 14]. 4. The microtemperature or microentropy theory. The microtemperature that depends on the microcoordinates of the microelements is based on the work on kinematics and dynamics of a continuum with microstructure by Eringen and Suhubi [15, 16] (see also [17]). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. I. Zhmakin, Non-Fourier Heat Conduction, https://doi.org/10.1007/978-3-031-25973-9_6

181

182

6 Microtemperature and Micromorphic Temperature Models

6.1 Microtemperature Models Based on the theory by Grot [18] of the thermodynamics of elastic bodies with microstructure whose microelements possess a microtemperature vector, Iesan et al. [19–22] derived a linear theory of microstretch thermoelastic bodies with microtemperatures. The local forms of balance of energy and the balance of first moment of energy can be expressed as ρ e˙ = σˆ : ∇v + σˆ : vˆ + σˆm : ∇v + ∇ · q + ξ , and ˆ + ∇ · q + q − Q + ξ, ρ e˙ m = σˆ m : (∇v − v) where v is the velocity vector, σˆ is the microstress tensor, vˆ denotes the microgyration tensor, σˆ m is the first stress moment third-rank tensor, e˙ m represents the first moment of energy vector, q is the first heat flux moment second-rank tensor, Q is the microheat flux average, and ξ is the first heat supply moment tensor. The local form of the second law of thermodynamics is modified to include the microtemperature ρ s˙ − ∇ ·

q T

−

 1 q · T − (ξ + ξ · T) ≥ 0 T T

where T is the microtemperature vector. Finally the following field equations of temperature and microtemperature are obtained for the rigid body heat conduction (without body heat sources) [1] c T˙ = λT + k1 ∇ · T and

bT˙ = k6 T + (k4 + k5 )∇ (∇ · T) − k2 T − k3 ∇T

where ki , b and c are material constants. Considering the internal constraint that T = ∇T , the generalized heat equation becomes c T˙ = (λ + k1 − k2 − k3 )T − bT˙ + (k4 + k5 )2 T. If 2 T is small, the above heat equation is equivalent to the one derived from the hyper temperature model [1] ρcV T˙ + T0 A T T˙ = λT that has the same form as the heat equation of Cattaneo if τ = T0 A T /λ.

6.2 Micromorphic Approach

183

6.2 Micromorphic Approach Iesan and Nappa [23] also deduced the heat equation in micromorphic continua, while using the thermomechanical theory established by Green and Naghdi [24]. In particular, the authors showed that in the linearized theory according to this approach, heat can be transferred as thermal waves with finite speed. The heat equations of linear theory for temperature and microtemperature are given by: a T¨ + m∇ T˙ = λT and

b T¨ = d2 T + (d1 + d3 )∇ (∇ · T) − m∇ T˙

where a, m, k, b and di are material constants. For a hypothetical medium in which m = 0, these equations are uncoupled in the sense that the temperature is independent of microtemperatures. In this case the temperature satisfies the classical wave equation. Combining the above two heat equations and assuming that T = ∇T , the generalized heat equation becomes a T¨ + 2 mT˙ + bT¨ − (d1 + d2 + d3 )2 T = λT. Forest and Aifantis [12] proposed theories based on scalar microtemperature and microentropy model by applying the micromorphic approach by Forest [25] to the temperature and entropy. They showed that the gradient of entropy theory and gradient temperature theories can be regarded as a limit case of the microentropy and microtemperature theories. In the purely thermal case there exist additional independent power of internal and external generalized forces due to the introduced microentropy variable s˘ and ∇ s˘ . The virtual power of the generalized internal forces according to Germain [26] is enhanced Pint =

 

 ˙ ˙ σ˘ a s˘ + σ˘ b · ∇ s˘ dV

V

where σ˘ a and σ˘ b are generalized stresses or microforces [27]. In the quasi-static case the generalized principle of virtual power results in the following balance equation ∇ σ˘ b − σ˘ a = 0 with the Neumann-type boundary condition σ˘ b · n = F s˘ where F s˘ is the generalized contact forces.

184

6 Microtemperature and Micromorphic Temperature Models

The following generalized heat conduction equation is obtained 

ρ 2 As 4 ∇ s− ∇ s˘ ρT0 s˙ = λ 2β ρ



where β and As are material parameters. The micromorphic temperature T˘ is introduced, based on the micromorphic approach [25], into the constitutive equations as an additional degree of freedom (dof). It assumes that there exist contributions to the virtual power from the micromorphic temperature. The virtual power of the generalized inertia terms (acceleration forces) is according to the generalized principle of virtual power [28] is assumed to be    ˙ ˙ Pa = ρ u¨ · u˙ + ζ1 T˙˘ T˘ + ζ2 T¨˘ T˘ dV V

where ζi represents the generalized mass of micromorphic temperature. The first contribution one mimics the mechanical acceleration and introduces the second time derivative of the micromorphic temperature, it is similar to the inertia of displacement. The second one involves only the first time derivative of the micromorphic temperature and is based on justifications from kinetic theory of gas, where the kinetic energy is proportional to temperature [1]. If ζi = 0, the theory reduces to a micromorphic temperature model similar to the microentropy model derived by Forest and Aifantis [12].

References 1. Liu, W., Saanouni, K., Forest, F., Hu, P.: The micromorphic approach to generalized heat equations. J. Non-Equilib. Thermodyn. 42, 327–357 (2017) 2. Luikov, A.V., Bubnov, V.A., Soloviev, I.A.: On wave solutions of the heat-conduction equation. Int. J. Heat. Mass Transf. 19, 245–248 (1976) 3. Bubnov, V.A.: Wave concepts in the theory of heat. Int. J. Heat. Mass Transf. 19, 175–184 (1976) 4. Swenson, R.L.: Heat conduction–finite or infinite propagation. J. Non-Equilib. Thermodyn. 3, 39–48 (1978) 5. Ireman, P., Nguyen, Q.S.: Using the gradients of temperature and internal parameters in continuum thermodynamics. C. R. Mec. 332, 249–255 (2004) 6. Maugin, G.A., Muschik, W.: Thermodynamics with internal variables Part I. General concepts. J. Non-Equilib. Thermodyn. 19, 217–249 (1994) 7. Maugin, G.A., Muschik, W.: Thermodynamics with internal variables Part II. Applications. J. Non-Equilib. Thermodyn. 19, 250–289 (1994) 8. Forest, S., Cardona, J.M., Sievert, R.: Thermoelasticity of second-grade media. In: Maugin, G.A., Drouot, R., Sidoroff, F. (eds.) Continuum Thermomechanics, pp. 163–176. Springer, Netherlands, Dordrecht (2000) 9. Forest, S., Amestoy, M.: Hypertemperature in thermoelastic solids. C. R. Mec. 336, 347–353 (2008)

References

185

10. Nguyen, D.T., Colvin, M.E., Yeh, Y., Feeney, R.E., Fink, W.H.: Intermolecular interaction studies of winter flounder antifreeze protein reveal the existence of thermally accessible binding state. Biopolymers 75, 109–117 (2004) 11. Aifantis, E.C.: Further comments on the problem of heat extraction from hot dry rocks. Mech. Res. Commun. 7, 219–226 (1980) 12. Forest, S., Aifantis, E.S.: Some links between recent gradient thermo-elasto-plasticity theories and the thermomechanics of generalized continua. Int. J. Solids Struct. 47, 3367–3376 (2010) 13. Sobolev, S.L.: Two-temperature discrete model for nonlocal heat conduction. J. Phys. III 2261– 2269 (1993) 14. Sobolev, S.L.: Two-temperature Stefan problem. Phys. Lett. A 197, 243–246 (1995) 15. Eringen, A.C., Suhubi, E.S.: Nonlinear theory of simple micro-elastic solids - I. Int. J. Eng. Sci. 2, 189–203 (1964) 16. Suhubi, E.S., Eringen, A.C.: Nonlinear theory of simple micro-elastic solids–II. Int. J. Eng. Sci. 2, 389–404 (1964) 17. Eringen, A.C.: Microcontinuum Field Theories: I. Springer, Foundations and Solids (1999) 18. Grot, R.A.: Thermodynamics of a continuum with microstructure. Int. J. Eng. Sci. 801–814 (1969) 19. Iesan, D.: On the theory of heat conduction in micromorphic continua. Int. J. Eng. Sci. 40, 1859–1878 (2002) 20. Iesan, D.: Thermoelasticity of bodies with microstructure and microtemperatures. Int. J. Solids Struct. 44, 8648–8662 (2007) 21. Iesan, D., Quintanilla, R.: On thermoelastic bodies with inner structure and microtemperatures. J. Math. Anal. Appl. 354, 12?23 (2009) 22. Iesan, D., Scalia, A.: Plane deformation of elastic bodies with microtemperatures. Mech. Res. Commun. 37, 617–621 (2010) 23. Iesan, D., Nappa, L.: On the theory of heat for micromorphic bodies. Int. J. Eng. Sci. 43, 17–32 (2005) 24. Green, A., Naghdi, P.: Thermoelasticity without energy dissipation. J. Elast. 31, 189–209 (1993) 25. Forest, S.: Micromorphic approach for gradient elasticity, viscoplasticity, and damage. J. Eng. Mech. 135, 117–131 (2009) 26. Germain, P.: The method of virtual power in continuum mechanics. Part 2: microstructure. SIAM J. Appl. Math. 25, 556–575 (1973) 27. Gurtin, M.E.: Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Phys. D 92, 178–192 (1996) 28. Gerasimov, A.N.: Generalization of the linear deformation laws and applications to the problems of inner friction (in Russian). Appl. Math. Mech. 12, 529–539 (1948)

Chapter 7

Thermodynamic Models

Thermodynamic models are deduced from the thermodynamic constraints following from the second law of thermodynamics [1–15]. The maximum entropy method of the determination of the field variable is based on the maximization of the entropy density under the constraints of the state variables with the Lagrange multiplier method (Liu procedure) [16]: the constraints may be removed by the use of the Lagrange multipliers [17]. An alternative approach— Coleman–Noll procedure [18] and its generalization [19, 20] according to idea by Cimmelli [21] that one should consider as constraints for the entropy inequality both the governing equations of the wanted fields and their gradient extensions up to the order which appears in the state space. Triani et al. [22, 23] showed using the example of a rigid heat conductor with general entropy flux that the Coleman–Noll and the Liu procedures are equivalent, if in the Coleman–Noll procedure all relevant equations are taken into account as constraints. The wavelike propagation of heat such a second sound is due to the inertia of internal energy. The deviations from the Fourier law could be modelled by an additional non-equilibrium thermodynamic state variables. The internal variables are the powerful tool for modelling in other continuum theories, for example, rheology [24, 25], dislocations in semiconductor crystals and superlattices [26], porous nanocrystals filled by fluid flow [27, 28]. In the case of the heat transfer the simplest choice for the additional vectorial state variable is the heat flux as in the extended thermodynamics [29–32]. In such a way one obtains the Maxwell–Cattaneo–Vernotte and Guyer–Krumhansl equations. With the additional variables, including the tensorial variables, a more general theory can be derived that correctly describes, e.g. the ballistic propagation and the propagation of heat with the speed of sound [33]. The evolution equations for these new variables are direct consequences of the second law of the thermodynamics and could be obtained by solving the inequality of the entropy production. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. I. Zhmakin, Non-Fourier Heat Conduction, https://doi.org/10.1007/978-3-031-25973-9_7

187

188

7 Thermodynamic Models

7.1 Jou and Cimmelli Model Jou and Cimmelli [10] introduced an extra internal variable represented by a secondorder tensor Qˆ and wrote the balance of the heat flux as follows ˆ τ1 q˙ + q = −λ∇T + ∇ · Q,

(7.1)

where τ1 is the relaxation time. The tensor Qˆ is assumed to be symmetric and may be split Qˆ = Q Iˆ + Qˆ s , where ˆ Qˆ s is the symmetric traceless (deviatoric) the scalar Q is one-third of trace of Q, ˆ In the relaxation time approximation the evolution equations for the tensor part of Q. parts are written as τ0 Q˙ˆ + Qˆ = γ0 ∇ · q (7.2)   τ2 Q˙ˆ s + Qˆ s = 2γ2 (∇q)0 s

(7.3)

  where (∇q)0 s is the symmetric traceless part of ∇q. Under the assumption that τ0 and τ2 are small and only regular processes, for which the time derivatives at the left-hand side of Eqs. (7.2) and (7.3) do not diverge, are considered, the equation for the heat flux (7.1) could be rewritten as [10]   1 γ2 τ1 q˙ + q = −λ∇T + γ2 ∇ 2 q = γ0 + γ2 ∇∇ · q. 3

(7.4)

Evidently, the Eq. (7.4) reduces to the the Guyer–Krumhansl transport Eq. (3.21) if τ1 = τ R , γ0 =

5 2  , γ2 = 2 . 2

The thermodynamics underlying the Eq. (7.4) is derived by introduction constitutive equations [10] for the specific entropy s = seq −

τ1 τ2 τ0 q·q− Q2 Qˆ : Qˆ − 2λT 2 4λT 2 γ2 2λT 2 γ0

(7.5)

and for the entropy flux Js =

q T

  Q Qˆ s · q 1+ + λT λT 2

(7.6)

where seq is the local equilibrium entropy, “:” means the complete contraction of tensors producing a scalar, Qˆ s · q means the contraction over the last index of Qˆ s producing a vector.

7.1 Jou and Cimmelli Model

189

The absolute temperature is given by the reciprocal of the derivative of the entropy with respect to the internal energy at constant values of the other. When the extended entropy (7.5) is used instead of the local equilibrium entropy, the resulting absolute temperature is [34]     ∂seq τ2 τ0 1 ∂ ∂ ∂s ∂  τ1  Qˆ s : Qˆ s − q·q− Q2 = = − 2 2 2 T ∂e ∂e ∂e 2λT ∂e 4λT γ2 ∂e 2λT γ0

that differs from the local equilibrium temperature ∂seq 1 . = Teq ∂e The local equilibrium temperature loses its validity in situations where the deviation from equilibrium ensemble is significant [35, 36]. The dispersion relation of heat waves along nanowires which are not isolated from the environment allows to compare different definitions of non-equilibrium temperature, since thermal waves are predicted to propagate with different phase speed [10, 37]. The first-order flux as the sole independent variable is not sufficient for the correct description of the high-frequency processes when the frequency becomes comparable to the inverse of the relaxation time of the first-order flux, all the higher-order fluxes will also behave like independent variables. EIT allows one to introduce the higher-order fluxes Jˆ2 , . . . , Jˆn where Jˆk is the tensor of order k that serves as the flux of the preceding flux Jˆk−1 . Thus the specific entropy is written as [10, 38] α1 ˆ ˆ αn ˆ ˆ s = s(e, Jˆ1 , . . . Jˆn ) = seq + J1 : J1 + · · · + Jn : Jn 2 2 and entropy flux as Jˆ1 + β1 Jˆ2 · Jˆ1 + · · · + βn−1 Jˆn · Jˆn−1 Jˆs = T where α1 , . . . αn and β1 , . . . βn are functions that can depend on the internal energy, Jˆk · Jˆk−1 means the contraction over the last (k − 1) indices of the tensor Jˆk producing a vector. The evolution of fluxes is governed by the following equations that are compatible with a positive entropy production β1 τ1 τ1 J˙ˆ1 + Jˆ1 = −λ∇T + ∇ · Jˆ2 α1 ... βn τn βn−1 τn τn J˙ˆn + Jˆn = ∇ · Jˆn+1 + ∇ · Jˆn−1 . αn αn

190

7 Thermodynamic Models

If the relaxation time of the nth flux τn  τn+1 , the hierarchy can be stopped at the nth flux [1]. Consideration of this hierarchy of the equations for the higher-order fluxes in the Fourier space (ω, k) allows one to introduce a generalized thermal conductivity λ(ω, k) as continued-fraction expansion [39, 40] λ(ω, k) =

λ0 k 2 l12 1 + iωτ1 + iωτ2 + . . .

.

The authors extended the Guyer–Krumhansl equation to the nonlinear case as  τ R q˙ + q = −λ∇T +

2τ cV T

 + l 2p (∇ 2 q + 2∇∇ · q),

and finally derived a general constitutive equation (that includes the Cattaneo and Guyer–Krumhansl equations as a special cases) as τ R q˙ + q + (μ∇∇q + μ ∇q) = −λ(1 + ξ q · q)∇T + l 2p (∇ 2 q + 2∇∇ · q), where μ, μ , ξ are material coefficients. This equation accounts for non-local, nonlinear and memory effects.

7.1.1 Heat Conduction in Thermoelectric Systems The authors considered two-temperature model of the transport in the thermoelectric systems. Thermoelectric effects involve an interplay between the electric and thermal properties of the material. The two primary thermoelectric effects are the Seebeck effect and the Peltier one: the Seebeck effect describes how a temperature difference creates a charge flow; the Peltier effect describes how an electrical current creates a heat flow. Thermoelectric devices are attractive; they do not have moving parts, do not create pollution and do not make noise [41]. In practical applications, the thermodynamic efficiency of a thermoelectric device is determined by the dimensionless parameter Z T , where T is the operating temperature, and Z is the figure-of-merit Z =  2 σe /λ, where  is the Seebeck coefficient, σe is the electrical conductivity. Although the thermal conductivity λ of thermoelectric materials is usually dominated by that of the electrons, sometimes the lattice heat conductivity should be accounted for. The primary challenges in developing advanced thermoelectric materials are increasing the power factor  2 σe and reducing the thermal conductivity. Nanomaterials provide an interesting avenue to obtain more performing thermoelectric devices, for example, using the one-dimensional nanostructures [42, 43].

7.2 Sellitto and Cimmelli Model

191

Jou and Cimmelli [10] considered the case when the phonons and electrons have no the same temperature. Accounting for two different temperatures is necessary in several physical situations • The presence of hot electrons. When the electron mean free path corresponding to electron–phonon collisions is long, one may have a population of “hot electrons”, whose kinetic temperature is higher than that of the phonons [44]. • Non-equilibrium temperatures. As the electron mean free path is usually shorter than the phonon mean free path, when the l distance is bigger than the electron mean free path but smaller than the phonon mean free path, there will be a very high number of electron collisions, and only few phonon collisions. Thus, the electron temperature may reach its local equilibrium value, whereas the phonon temperature may be still far from it. • Fast laser pulses. When a laser pulse hits a system, initially the electrons capture the main amount of the incoming energy and subsequently, through electron–phonon collisions, they give a part of the energy to the phonons. The evolution equations for the internal energy of phonons per unit volume e p , the internal energy of electrons per unit volume ee and the electrical charge per unit volume of electrons ρe are e˙ p = −∇ · q p , e˙e = −∇ · qe + EI, ρ˙e = −∇ · I. The average temperature—a measurable quantity in practical applications—is p p defined as T = (cv T p + ceV Te )/cV where cV = cv + ceV the volumetric heat capacity at constant volume of the whole system [45].

7.2 Sellitto and Cimmelli Model Sellitto and Cimmelli [8] developed a continuum approach to the thermomass model. The authors introduced the absolute temperature θ and the vectorial variable c as the sate variables and postulate similar to the approach by Lebon et al. [46] that c is proportional to the heat flux. The authors assumed that the evolution of c is governed by the balance equation ˆ (c) + σ (c) c˙ = ∇ ·  where the flux and the production of c are given as ˆ (c) = 1 (θ ) Iˆ + 2 (θ ) 



1 2 c + cc , σ (c) = 0 (θ )c. 2

(7.7)

192

7 Thermodynamic Models

Here 0(1,2) are scalar functions, c = |c| the Eq. (7.7) is rewritten as  c˙ =

   ∂1 ∂2 1 ∂2 2 + c ∇θ + · c + 2 ∇ · c + 0 c + 2 (∇c · c + ∇c T · c). ∂θ 2 ∂θ ∂θ

(7.8)

In the absence of heat source the energy conservation is (u is the volumetric internal energy) u˙ = −∇ · q, (7.9) the constitutive equation is written as q = g(θ, c2 )c.

(7.10)

Sellitto and Cimmelli listed several reasons to use the variable c instead of the heat flux q (see also [47]): • This choice allows to get the constitutive equation in the form similar to that in phonon hydrodynamics. • c is proportional to τ , which is a small parameter, it may be very helpful in the determination of the closure of nonlinear series expansions. • Compliance with earlier models of the generalized Fourier law [48]. • The form of the function g could be chosen in accordance of the principle of the frame indifference. Jou et al. [47] consider c as the renormalized heat flux. Since c is collinear to q but not equal to q and c is not identified, c may be considered as an internal variable, instead of the heat flux. The internal variables constitute an efficient tool when dealing with non-equilibrium processes involving complex thermodynamical systems. These additional non-equilibrium parameters, whose nature depends on the phenomenon at hand, usually are introduced through ordinary differential equations, called kinetic equations. The Eq. (7.9) could be rewritten as  u˙ =

 ∂g ∂g ∇θ + 2 2 ∇c · c · c − g∇ · c. ∂θ ∂(c )

(7.11)

The Eqs. (7.8) and (7.11) form a closed set of equations. Solving these equations under the restrictions imposed by the second law of thermodynamics, the authors obtained the general equation for the heat flux that in one-dimensional case is written as [8] ln g qx θ˙ q˙ x − ∂ ∂θ =

g 1 − g2 ∂(1/g) q2 ∂(c2 ) x



∂1 3 + 2 ∂θ g



ln g 2 ∂ ∂θ 1 ∂2 ∂θ − qx2 2 ∂(1/g) 2 2 ∂θ ∂ x 1 − g ∂(c2 ) qx

7.3 Kovács and Ván Model

193

0 32 + qx + 2 g g



1 1−

2 ∂(1/g) 2 q g ∂(c2 ) x

∂qx qx . ∂x

This equation is greatly simplified in the case g(θ, c2 ) = g0 (θ ):  q˙ x − 0 qx =

∂1 3 g0 + ∂θ g0

+qx



1 ∂2 ∂ ln g 2 ∂θ − 2 qx 2 ∂θ ∂θ ∂x

∂ ln g0 32 ∂qx qx . θ˙ + ∂θ g0 ∂ x

This equation is compatible with the second law of thermodynamics. The entropy flux is written as   2γ C V 2 g0 1− (s) = c c θ 3ρθ and the net entropy production as σ (s) =

1 q·q λθ 2

where the following relations are used 1 = −θ, 2 = −

2γ Cv , 3ρ

∂g0 2γ Cv = , ∂θ ρ

∂g2 = 0. ∂θ

7.3 Kovács and Ván Model Kovács and Ván [7] introduced the heat flux and a second-order tensorial variable as additional internal field variables and assumed the following form of the entropy flux ˆ J = bˆ · q + Bˆ : Q, (7.12) where bˆ is second order and Bˆ is third-order tensorial functions called current multipliers (Nyiri multipliers [49]). The authors also assumed a quadratic dependence of the entropy density on the additional field variables s = seq (e) −

m2 ˆ ˆ m1 q·q− Q : Q, 2 2

where m 1 and m 2 are the constant positive material coefficients.

194

7 Thermodynamic Models

The entropy inequality could be written as [7] ∂s ∂ Qˆ ∂q 1 ˆ + Bˆ : ∇ Qˆ + Qˆ : (∇ · B) ˆ + ∇ · q = − ∇ · q − m1q · − m 2 Qˆ : + bˆ : ∇q + q · (∇ · b) ∂t T ∂t ∂t       ∂q ∂ Qˆ 1 · q + ∇ · Bˆ − m 2 : Qˆ + Bˆ : ∇ Qˆ ≥ 0, = bˆ − Iˆ : ∇q + ∇ · bˆ − m 1 T ∂t ∂t

where Iˆ is the unit tensor. The authors identified four generalized forces and four fluxes in this equations and assumed the linear relationships between them that are written as (in the onedimensional case) ∂b ∂q − = −l1 q, m1 ∂t ∂x m2

∂Q ∂ Qq ∂q − = −k1 Q + k12 , ∂t ∂x ∂x b−

1 ∂q = −k1 Q + k2 , T ∂x B=n

∂Q . ∂x

The material coefficients (l1 , k1 , k2 , k12 , k21 ) are subjected to the following restrictions from the second law of thermodynamics: l1 ≥ 0, k1 ≥ 0, k2 ≥ 0, k1 k2 − k12 k21 ≥ 0. Using the constraints from the second law of thermodynamics (the non-negative production of the entropy) and eliminating the internal variables, the authors get the general constitutive equation for the heat flux as m 1 m 2 ∂tt q + (m )l1 + m 1 k1 )∂t q − (m 1 n + m 2 k2 )∂x xt q + nk2 ∂x4 q − (l1 + K )∂x x q +k1l1 q = m 2 ∂xt

1 1 1 + k1 ∂x − ∂x3 . T T T

Choosing some of the materials coefficients to be equal to zero, one can get a number of known models such as • Fourier n = m 1 = m 2 = k2 = 0 and (k12 = 0 or k21 = 0) • Cattaneo n = m 2 = k2 = 0 and (k12 = 0 or k21 = 0) • Ballistic–diffusion n = k2 = 0

7.4 Famá et al. Model

195

• Jeffrey’s type n = m 1 = k2 = 0 and (k12 = 0 or k21 = 0) • Guyer–Krumhansl n = m2 = 0 • Cahn–Hilliard n = m1 = m2 = 0

7.4 Famá et al. Model Famá et al. [33] introduced the three-dimensional form of the equations of the heat ˆ transfer in the isotropic materials, with a second-order tensorial internal variable Q, including the possible Onsager reciprocity relations and second law requirements for the transport coefficients. Their model could be considered as the extension of the Kovács and Ván model (Sect. 7.3). The authors consider the balances of the internal energy e ρ

∂e +∇ ·q=0 ∂t

and of the entropy s in the rigid heat conductor ρ

∂s + ∇ · J = σ (s) ∂t

where σ (s) is the entropy production rate. The authors introduced an additional internal variable Qˆ (a second-order tensor) which incorporates higher-order effects in the heat transport. Qˆ may be interpreted as the flux of the heat flux in solids [12, 30] or as the pressure tensor in fluids [30, 50]. Similar to the Kovács and Ván model it is assumed that the entropy flux is zero ˆ the Eq. (7.12) is valid. if q and Q, ˆ up to second approximation around Expansion of the entropy function s(e, q, Q) the equilibrium state gives ˆ = seq (e) − 1 m i j qi q j − 1 Mi jkl Q i j qkl s(e, q, Q) 2ρ 2ρ where there are symmetry relations m i j = m ji ,

Mi jkl = Mkli j .

(7.13)

Note that relation (7.13) is valid for the anisotropic systems, too. For isotropic systems m i j and Mi jkl would reduce to a scalar and the three scalar components ˆ respectively [33]. Thermodyconjugate to the three scalar invariants of the tensor Q, namic stability requires that the inductivity tensors mˆ and Mˆ [51] are positive definite and the authors assumed that they are constant.

196

7 Thermodynamic Models

The entropy production rate must be non-negative [33] ρ

∂q j dseq ∂e 1 ∂qi ∂s 1 + Ji, j = ρ ρ − mi j q j − m i j qi ∂t de ∂t 2 ∂t 2 ∂t

∂ Qi j 1 ∂ Q kl 1 − Mi jkl Q kl − Mi jkl Q i j + bi j,i q j + bi j q j,i + Bi jk,i Q jk + Bi jk Q jk,i 2 ∂t 2 ∂t      δi j ∂q j ∂ Q kl b ji, j − m i j qi + Bk ji,k − Mi jkl Q i j Q i j + Bi jk Q jk, i ≥ 0. = bi j − T ∂t ∂t

The authors obtained the general 3D anisotropic linear relations between between the thermodynamic fluxes bi j − δi j/T, b ji, j − m i j ∂qi /∂t, Bi jk , Bki j,k − Mi jkl ∂ Q kl and forces qi , qi, j , Q i j , Q jk,i . Famá et al. obtained the Onsager coefficients taking into account that the body properties are invariant with respect to all rotations and inversion of the frame of axes and the rate equations for q and Qˆ accounting for the Onsager reciprocity relations. The authors investigated the one-dimensional case and showed that the list of the special case includes ballistic–conductive, Guyer–Krumhansl, Cahn–Hilliard, Jeffrey type, Maxwell–Cattaneo–Vernotte and Fourier equations. The authors stressed that the entropy density depends on the internal variables quadratically, in order to preserve the concavity, that is thermodynamic stability. The entropy flux depends on the internal variables linearly; therefore, it disappears when they are zero. As long as these two physical conditions and the entropy inequality are valid, the derived consequences are also valid.

7.5 Rogolino et al. Models Rogolino et al. [13] based their analysis on the work by Ván & Fülöp [5] that used two assumptions: 1. The deviation from the equilibrium state is described by the heat flux and the second-order tensorial internal variable; 2. The deviation from the classical form of the entropy current is described by two tensorial functions, called the current multipliers. Rogolino et al. developed two versions of the generalized heat conduction equations. The first one ignores the non-local effects, and the second equation is able to describe the heat conduction in the presence of the non-local effects. The authors used in the first case as the basic field variables the specific (per unitary volume) internal energy e, the heat flux q and the flux of heat flux q . In the second case they considered the differential consequences in the first case by the spatial and time derivatives of the equations and obtained the higher-order equation

7.5 Rogolino et al. Models

197

for the heat flux. Rogolino et al. proved that the entropy flux is non-local in both cases while the entropy is local in the first case and non-local in the second case. The authors assumed the following form of the corresponding balance laws (in the one-dimensional case) ∂e ∂q + =0 (7.14) ∂t ∂x ∂q ∂q + = rq ∂t ∂x

(7.15)

∂q ∂ + = r ∂t ∂x

(7.16)

where rq and r are the production rates of the heat flux q and flux of heat flux q ,  is the flux of q . For the closure of the system of Eqs. (7.14–7.16) it is necessary to use the constitutive equations for the flux  and the source terms rq and r . The system of Eqs. (7.14–7.16) is the one-dimensional version of the 13-moments system of extended irreversible thermodynamics [30, 31] that is related to the Grad’s 13-moments method. The entropy s and the entropy flux J depend on the system state  ∂q ∂q ∂e , q, , q , . Z = e, ∂x ∂x ∂x The authors using the entropy constraints and the Lagrange–Farkas multipliers1 λ, α and β calculate the entropy inequality and found that the Lagrange–Farkas multipliers are determined by the partial derivatives of the entropy with respect to the basic field variables λ=

∂s ∂s ∂s . , α= , β= ∂e ∂q ∂q

The general expression for the entropy flux is written as J = J0 (e, q, q ) +

∂s ∂q

and the entropy inequality is given by ∂ ∂x



∂s ∂q

 −

∂s ∂q ∂ J0 ∂s ∂s ∂q ∂s − + + rq + r ≥ 0. ∂e ∂ x ∂q ∂ x ∂x ∂q ∂q

(7.17)

1 The Lagrange multipliers are used to incorporate the constrains related to the entropy [16, 52]. Some of these multipliers have the classical counterparts such as, for example, the temperature, the chemical potential, while others related to the non-equilibrium constraints do not have analogous in the equilibrium theory.

198

7 Thermodynamic Models

The solution of the inequality (7.17) requires the additional assumptions. Rogolino et al. choose as the entropy a quadratic function of the non-equilibrium variables q and q q2 q2 . (7.18) s = s¯ (e) − m − M 2 2 where m and M are the constant positive coefficients and assumed the following compatibility condition to be valid ∂ J0 ∂s . = ∂q ∂q The chosen function (7.18) ensures the fulfilment of the principle of maximum entropy at the equilibrium. The authors also required that expression for the entropy flux reduces to the classical value J = ∂s/∂e at equilibrium and finally get the simplified form of entropy inequality (7.17) as ∂q  +q −M ∂x



∂ ∂x



∂s ∂e



 − mrq + q



∂ ∂x



∂s ∂q



 − Mr

≥ 0.

The authors introduce the phenomenological coefficients l1 , l2 , l3 and after the elimination of the production rates rq , r and the highest-order flux  get the system of the transport equations ∂e ∂q + = 0, (7.19) ∂t ∂x   ∂q 1 ∂q ∂ + q + τq = l2 , (7.20) τq ∂t ∂x ∂x T τ

∂q ∂ 2 q ∂q + q − τl1 M = −l3 m . ∂t ∂x2 ∂x

(7.21)

where τq = ml2 and τ = Ml3 . System of Eqs. (7.19–7.21) includes as special cases a number of known models: • If l3 = 0 then Eq. (7.21) yields q = 0 and Eq. (7.20) reduces to the MaxwellCattaneo-Vernotte Eq. (2.5) τ

∂q + q = −λ∇T. ∂t

• If τ is negligible than from Eq. (7.21) follows that q = 0 and inserting into Eq. (7.20) leads to one-dimensional Guyer–Krumhansl equation (provided τl3 = 32 )

7.6 Two-Temperature Model by Sellitto et al.

τ

199

∂T ∂ 2q ∂q +q +λ = 32 2 . ∂t ∂t ∂x

• If in the last equation q is also negligible, one gets the equation of Green–Naghdi type ∂T ∂q ∂ 2q +λ = 32 2 . τ ∂t ∂t ∂x Rogolino et al. derived two versions of the generalized heat equation 1. The second order in space and the first order in time equation neglecting the non-local effects. 2. The fourth order in space and second order in time equation incorporating the non-local effects.

7.6 Two-Temperature Model by Sellitto et al. Sellitto et al. [14] suggested the model that accounts for the non-local and nonlinear effects in the different regimes which electrons and phonons (that are considered as the gas-like collections, flowing through the crystal lattice [53–55]) can undergo in the heat conduction (when the equations for the electrons and the phonons are identical, the subscript “c” is used to denote the heat carrier, “e” or “p”) T˙ c +

q˙ic +

qic,i ccV

= 0,

2q cj q cj,i λc T,ic Q i j,ic qic + − − = 0, τ1c τ1c ccV T c τ1c Q˙ icj +

Q icj τ2c

−

lc2 qic, j τ2c

=0

where T c are the temperatures related to the internal energy as U c = ccV T c , ccV are the specific heats, Q icj are second-order tensors representing contribution to the heat flux, λc are thermal conductivities, τ c are the relaxation times, lc are the mean free paths. As the authors noted these equations do not follow from a rigorous microscopic derivation, thus it is necessary to check their thermodynamic compatibility. Following the EIT framework, the authors chosen the following state space:  = {T e , q e , Q e , T p , q p , Q p }.

200

7 Thermodynamic Models

The local balance of the specific entropy s˙ + Jis,i = σ s . The second law is formulated according with the classical Liu procedure [16] as  s˙ +

Jis,i

−

e

T˙ e +

qie,i

 −

ceV

ie

  2q ej q ej,i λe T,ie Q i j,ie qie e q˙i + e + e − e e − e − τ1 τ1 cV T τ1





p

− p T˙ p +

qi,i



 −

p

cV

p i

−

τ2e

τ2e p

p p p p 2q j q j,i λi T,i Q i j,i qi + p + − p − p p τ1 τ1 cV T p τ1

+

Qi j p

τ2

−





p

p

p Q˙ i j

−



lc2 qie, j

p

p q˙i

 p i j

Q iej

Q˙ iej +

−iej

lc2 qi, j

≥ 0.

p

τ2

(7.22)

The constitutive assumptions on s and Jis are given as s(T e , q e , Q e , T p , q p , Q p ) and Ji (T e , q e , Q e , T p , q p , Q p )s . Thus the following sets of necessary and sufficient conditions guarantee that second law of thermodynamics is fulfilled ∂s − e = 0, ∂T e

∂s − ie = 0, ∂qie

∂s − iej = 0, ∂ Q iej

∂s −  p = 0, ∂T p

∂s p p − i = 0, ∂qi

∂s p p − i j = 0. ∂ Qi j

and ∂ Jis e λe − e = 0, e ∂T τ1 ∂ Jis ∂T p

−

pλp p

τ1

= 0,

iej le2 ∂ Jis 2ie q j e δi j − + + = 0, ∂q ej ceV ceV T e τ2e ∂ Jis

 p δi j

∂q j

cV

p −

p

p

p

+

2i q j p

cV T p

+

i j l 2p p

τ2

= 0,

ej δi j ∂ Jis + = 0, ∂ Q ejk τ2e p

∂ Jis

 j δi j

∂ Q jk

τ2

p +

p

= 0.

together with the reduced entropy inequality p

p Q iej ∂s Q i j ∂s qie ∂s qi ∂s p + p p ≤ 0. e e + e e + p τ1 ∂qi τ2 ∂ Q i j τ1 ∂qi τ2 ∂ Q i j

The authors showed by direct calculations that the proposed model has well-posed theoretical basis. Sellitto et al. determined the speed of the heat waves for the cases when the wave propagates in the same direction of the average heat and in the opposite direction.

7.7 EIT Ballistic–Diffusive Model

201

The authors investigated the influence of both the non-local effects (represented in the model by terms le2 and l 2p ) and the nonlinear effects (represented in the model by p p terms 2q j qie, j /ceV T e and 2q j qi, j /cV T p ) on the the wave propagation. They found that the non-local effects enhance the speeds of propagation; the effect of the nonlinear terms depends on the direction of the wave propagation. The authors noted that applications of the model limited to the problems where the electron–phonon interactions could be neglected—in the proposed the twotemperature model the two different species of heat carriers have been completely decoupled. However, the electron–phonon interactions in graphene, e.g. are important for the describing the anomalies of photoemission spectra observed [56, 57].

7.7 EIT Ballistic–Diffusive Model The EIT ballistic–diffusive model [58] follows the purely macroscopic approach in contrast to the BD model introduced by Chen [59, 60] that is based on the mixture of the kinetic and macroscopic approaches. Similar to the Chen’s model EITBD by Lebon et al. relays on the coexistence of two kinds of heat carriers • The ballistic phonon that originate at the boundaries and collide mainly with the walls; • The diffusive phonons that undergo the multiple collisions within the core of the system. The internal energy and the heat flux are splitted into two parts e = eb + ed , q = qb + qd . The state variables of EIT are selected as 1. (eb , qb ) provide the description of the ballistic motion of phonons. 2. (ed , qd ) provide the description of the diffusive motion of phonons. The authors introduce the ballistic and the diffusive quasi-temperatures Tb = eb /cb and Td = ed /cd that do not represent the temperatures in the usual sense but serve as the measure of the internal energies [9]. Assuming that the heat capacities are equal, the total quasi temperature is introduced T = e/c = Tb + Td . Evolution of the internal energies is governed by the balance equations ∂eb = −∇ · qb + rb , ∂t ∂ed = −∇ · qd + rd , ∂t

(7.23) (7.24)

202

7 Thermodynamic Models

the total energy e = eb + ed satisfies the first law of thermodynamics ∂e = −∇ · q + r. ∂t To describe the evolution of the ballistic phonons the authors use Guyer– Krumhansl equation τb

∂qb + qb = −λb ∇T + lb2 (∇ 2 qb + 2∇ · ∇qb ), ∂t

while the evolution of diffusive phonons is governed by the Cattaneo equation τd

∂qd + qd = −λd ∇T. ∂t

References 1. Sobolev, S.L.: Local non-equilibrium transport models. Phys. Usp. 40, 1042–1053 (1997) 2. Ván: Weakly nonlocal irreversible thermodynamics—the Guyer-Krumhansl and the CahnHilliard equations. Phys. Lett. A 290, 88–92 (2001) 3. Serdyukov, S.I.: A new version of extended irreversible thermodynamics and dual-phase-lag model in heat transfer. Phys. Lett. A 281, 16–20 (2001) 4. Cimmelli, V.A.: Different thermodynamics theories and different heat conduction laws. J. NonEquilib. Thermodyn. 34, 299–333 (2009) 5. Ván, P., Fülöp, T.: Universality in heat conduction theory: weakly nonlocal thermodynamics. Ann. Phys. 524, 470–478 (2012) 6. Ván, P., Kovach, R., Fülöp, T.: Thermodynamic hierarchy of evolution equations. arXiv: 1412.4490 [cond-mat.stat-mech] (2014) 7. Kovács, R., Ván, P.: Generalzed heat conduction in heat pulse experiments. Int. J. Heat Mass Transf. 83, 613–620 (2015) 8. Sellitto, A., Cimmelli, V.A.: A continuum approah to thermomass theory. J. Heat Transfer 134 (2012) 9. Lebon, G.: Heat conduction at micro and nanoscales: a review through the prism of extended irreversible thermodynamics. J. Non-Equilib. Thermodyn. 39, 35–59 (2014) 10. Jou, D., Cimmelli, V.A.: Constitutive equations for heat conduction in nanosystems and nonequilibrium processes: an overview. Commun. Appl. Industr. Math. 7, 196–222 (2016) 11. Sellitto, A., Rogolino, P., Carlomagno, I.: Heat-pulse propagation along nonequilibrium nanowires in thermomass theory. Commun. Appl. Industr. Math. 7, 39–55 (2016) 12. Sellitto, A., Cimmelli, V.A., Jou, D.: Mesoscopic Theories of Heat Transfer in Nanosystems. Springer, Berlin (2016) 13. Rogolino, P., Kovács, R., Ván, P., Cimmelli, V.A.: Generalized heat conduction equations: parabolic and hyperbolic models. Cont. Mech. Thermodyn. 30, 1245–1258 (2018) 14. Sellitto, A., Carlomagno, I., Di Domenico, M.: Nonlocal and nonlinear effects in hyperbolic heat transfer in a two-temperature model. ZAMP 72, 7 (2021) 15. Cimmelli, V.A.: Local versus nonlocal constitutive theories of nonequilibrium thermodynamics: the Guyer-Krumhansl equation as an example. ZAMP 72, 195 (2021) 16. Liu, I.S.: Method of Lagrange multipliers for exploitation of the entropy principle. Arch. Ration. Mech. Anal. 46, 131–148 (1972)

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17. Müller, I.: Speed of propagation in classical and relativistic extended thermodynamics. Living Rev. Relativ. 2, 1–32 (1999) 18. Coleman, B.D., Noll, W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Ration. Mech. Anal. 13, 167–178 (1963) 19. Cimmelli, V.A., Sellitto, A., Triani, V.: A generalized Coleman-Noll procedure for the exploitation of the entropy principle. Proc. R. Soc. A 466, 911–925 (2010) 20. Cimmelli, V.A., Sellitto, A., Triani, V.: A new perspective on the form of the first and second laws in rational thermodynamics: korteweg fluids as an example. J. Non-Equilib. Thermodyn. 35, 251–265 (2010) 21. Cimmelli, V.A.: An extension of Liu procedure in weakly nonlocal thermodynamics. J. Math. Phys. 48, 113510 (2007) 22. Cimmelli, V.A., Sellitto, A., Triani, V.: Exploitation of the second law: Coleman-Noll and Liu procedure in comparison. J. Non-Equilib. Thermodyn. 33, 47–60 (2008) 23. Cimmelli, V.A., Muschik, W., Triani, V.: Non-equilibrium thermodynamics with higher order fluxes: Balance laws and exploitation of the entropy inequality. J. Non-Equilib. Thermodyn. 33, 47–60 (2008) 24. Verhás, J.: Thermodynamics and Rheology. Kluwer Academic Publisher (1997) 25. Öttinger, H.C.: Beyond Equilibrium Thermodynamics. Wiley-Interscience (2005) 26. Jou, D., Restuccia, L.: Non-equilibrium thermodynamics framework for dislocations in semiconductor crystals and superlattices. Ann. Acad. Rom. Sci. Ser. Math. Appl. 10 (2018) 27. Restuccia, L.: Non-equilibrium temperatures and heat transport in nanosystems with defects, described by a tensorial internal variable. Comm. Appl. Indust. Math. 7, 81–97 (2016) 28. Restuccia, L., Palese, L., Caccamo, M.T., Famá, A.: A description of anisotropic porous nanocrystals filled by a fluidflow, in the framework of extended thermodynamics with internal variables. Proc. Romanian Acad. Sci. Series A, Ser. Math. Appl. 2, 225–233 (2020) 29. Müller, I., Ruggeri, T.: Rational Extended Thermodynamics. Springer, N.Y.k (1998) 30. Jou, D., Casa-Vázquez, J., Lebon, G.: Extended irreversible thermodynamics. Rep. Prog. Phys. 51, 1105–1179 (1988) 31. Jou, D., Casa-Vázquez, J., Lebon, G.: Extended irreversible thermodynamics revisited (1988– 1998). Rep. Prog. Phys. 62, 1035–1142 (1999) 32. Machrafi, H.: Extended Non-Equilibrium Thermodynamics: From Principles to Applications in Nanosystems. CRC Press (2019) 33. Famá, A., Rstuccia, L., Ván, P.: Generalized ballistic-conductive heat transport laws inthreedimensional isotropic materialsa. arXiv:1902.10980 [cond-mat.stat-mech] (2020) 34. Casas-Vázquez, J., Jou, D.: Temperature in nonequilibrium states: a review of open problems and current proposals. Rep. Progr. Phys. 66, 1937–2023 (2003) 35. Luzzi, R., Vasconcellos, A.R., Casas-Vázquez, J., Jou, D.: Characterization and measurement of a nonequilibrium temperature-like variable in irreversible thermodynamics. Physica A 234, 699–714 (1997) 36. Criado-Sancho, J.M., Jou, D., Casas-Vázquez, J.: Nonequilibrium kinetic temperatures in flowing gases. Phys. Lett. A 350, 339–341 (2006) 37. Sellitto, A., Cimmelli, V.A., Jou, D.: Thermoelectric effects and size dependency of the figureof-merit in cylindrical nanowires. Int. J. Heat Mass Transfer 57, 109–116 (2013) 38. Cimmelli, V.A., Ván, P.: The effect of non-locality on the evolution of higher order fluxes in nonequilibrium thermodynamics. J. Math. Phys. 46, 112901 (2005) 39. Ferrer, M., Jou, D.: Higher-order fluxes and the speed of thermal waves. Int. J. Heat Mass Transfer 34, 3055–3060 (1991) 40. Jou, D., Casa-Vázquez, J., Lebon, G.: Extended irreversible thermodynamics of heat transport. A brief introduction. Proc. Eston. Acad. Sci. 57, 118–126 (2008) 41. Rowe, D.M.: Thermoelectrics Handbook: Macro to Nano. CRC Press, Boca Raton (2005) 42. Silicon nanowires as efficient thermoelectric materials: Boukai, A.I., Bunimovich, Y., TahirKheli, J., Goddard, W.A., III., Heath, J.R. Nature 451, 168–171 (2008) 43. Sellitto, A., Cimmelli, V.A., Jou, D.: Entropy flux and anomalous axial heat transport at the nanoscale. Phys. Rev. B 87, 054302 (2013)

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44. Balkan, N.: Hot Electrons in Semiconductors-Physics and Devices. OUP, New York (1998) 45. Benedict, L.X., Louie, S.G., Cohen, M.L.: Heat capacity of carbon nanotubes. Solid State Comm. 100, 177–180 (1996) 46. Lebon, G., Ruggieri, M., Valenti, A.: Extended thermodynamics revisited: renormalized flux variables and second sound in rigid solids. J. Phys.: Condens. Matter 20, 025223 (2008) 47. Jou, D., Cimmelli, V.A., Sellitto, A.: Dynamical temperature and renormalized flux variable in extended thermodynamics of rigid heat conductors. J. Non-Equil. Thermodyn. 36, 373–392 (2011) 48. Wang, L.Q.: Generalized Fourier law. Int. J. Heat Mass Transfer 37, 2627–2634 (1994) 49. Nyíri, B.: On the entropy current. J. Non-Equil. Thermodyn. 16, 179–186 (1991) 50. Kovács, R., Ván, P.: Second sound and ballistic heat conduction: NaF experiments revisited. Int. J. Heat Mass Transf. 117, 682–690 (2018) 51. Gyarmati, I.: The wave approach of thermodynamics and some problems of non-linear theories. J. Non-Equil. Thermodyn. 2, 233–260 (1977) 52. Casas-Vázquez, J., Jou, D.: Lagrange multipliers in extended irreversible thermodynamics and in informational statistical thermodynamics. Braz. J. Phys. 27, 547–559 (1997) 53. Chen, G.: Nanoscale Energy Transport and Conversion: A Parallel Treatment of Electrons, Molecules, Phonons, and Photons. OUP, Oxford (2005) 54. Jou, D., Sellitto, A., Cimmelli, V.A.: Phonon temperature and electron temperature in thermoelectric coupling. J. Non-Equil. Thermodyn. 38, 335–361 (2013) 55. Jou, D., Sellitto, A., Cimmelli, V.A.: Multi-temperature mixture of phonons and electrons and nonlocal thermoelectric transport in thin layers. Int. J. Heat Mass Transf. 71, 459–468 (2014) 56. Yan, J., Zhang, Y., Kim, P., Pinczuk, A.: Electric field effect tuning of electron-phonon coupling in graphene. Phys. Rev. Lett. 98, 166802 (2007) 57. Coco, M., Romano, V.: Simulation of electron-phonon coupling and heating dynamics in suspended monolayer graphene including all the phonon branches. J. Heat Trans. 140, 092404 (2018) 58. Lebon, J., Machraft, H., Grmela, M., Debois, C.: An extended thermodynamic model of transient heat conduction at sub-continuum scales. Proc. R. Soc. A 467, 3245–3256 (2011) 59. Chen, G.: Ballistic-diffusive heat conduction equations. Phys. Rev. Lett. 86, 2297–2300 (2001) 60. Chen, G.: Ballistic-diffusive equations for transient heat conduction from nano to microscale. J. Heat Transfer 124, 320–328 (2002)

Chapter 8

Fractional Derivative Models

8.1 Fractional Fourier Model Deng et al. [1] studied the steady heat transfer using the 2D fractional Helmholtz equation in the fractal media1 ∂ 2α T ∂ 2α T + + k 2 = f, ∂ x 2α ∂ y 2α where 0 < α ≤ 1, 0 < β ≤ 1. He and Liu [3] and Liu et al. [4] used the He’s fractional derivative dn dα 1 f = dx α (n − α) dx n

t [ f 0 (s) − f (s)]ds t0

where f 0 (s) is a known function and the fractional form of the Fourier law λ2α

dα T =q dxα

to study the heat transfer in a fractal medium arising in the silk cocoon hierarchy; the structure of the cocoon could be duplicated in the biometric fabric design. Similar approach was used by Beybalaev [5] to study the heat conduction in the fractal medium and Beybalaev et al. [6] to study of the ground freezing. He et al. [7, 8] used the stationary space-fractional equation

1

The fractal media cannot be described as the fractals since the main property of the fractal is the non-integer Hausdorff dimension that should be observed on all scales—the fractal structure of the real media cannot be observed on all scales [2].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. I. Zhmakin, Non-Fourier Heat Conduction, https://doi.org/10.1007/978-3-031-25973-9_8

205

206

8 Fractional Derivative Models

 α  ∂ T λ α =0 ∂x

∂α ∂xα

and Wang et al. [9] the non-stationary equation ∂T ∂α + α ∂t ∂x

 α  ∂ T λ α =0 ∂x

to study the heat transfer in the fractal medium of the polar bear hair. Meilanov and Shabanova [10] solved the one-dimensional problems for the space– time-fractional equation ∂αT ∂β T = λ (8.1) ∂t α ∂xβ where the time non-local effects are accounted by the Caputo fractional derivative and the spatial non-local effects—by the Riesz derivative ∂2 ∂β T 1  2 π = β ∂x (2 − β) ∂ x 2(2 − β) cos 2

∞ 0

T (x  , t) d x . |x − x |β−1

Earlier these authors considered Eq. (8.1) with the Riemann–Liouville fractional time derivative ∂ α T /∂t α and used the Fourier transform [11] in the spatial coordinate and the Laplace transform in time to get the solution [12]. Voller et al. [13] used the time-fractional and Meilanov et al. [14] the time–space fractional models to solve the Stefan problems. Voller et al. considered the cases of both the sharp and diffuse interfaces between the liquid and the solid phases. The authors neglected the surface tension and the kinetic effects and assumed that the heat flux, temperature and the latent heat are zero. Thus the heat balance is  t





q · ndt d A =

− l 0

(ρcT + ρ L)d V,

(8.2)

l

where L is the latent heat of fusion, l is the part of the liquid domain surface that coincides with the total domain surface, n is the outwards pointing normal. To account for the memory effects the heat flux is expressed as 1 ∂ q= (α) ∂t

t

(t − t  )α−1 ql dt  ,

0 < α ≤ 1,

(8.3)

0

where () is the gamma function, α is the measure of the memory retention (α = 0 corresponds to the instantaneous flux).

8.1 Fractional Fourier Model

207

Inserting the Eq. (8.3) into the Eq. (8.2) and taking the Caputo derivative the authors get    dα − ql · nd A = ρc α T d V + ρ Lv αf d A, dt l

l

l

here v αf is the normal “fractional speed” at a point at the phase interface pointing into the solid defined by the relationship  f

v αf d A

1 = (1 − α)

t

 −α



(t − t )

v · n f d Adt 

f

0

and after some transformations obtained    ∂αT l q · nd A = ρc α d V + (ρ Lv αf − ql · n)d A. − ∂t l + f

l

f

The above equations are derived for the case of the sharp phase boundary. In case of the diffuse interface Voller et al. use the enthalpy H = ρcT + ρ L f where the liquid fraction f changes smoothly from f = 1 to f = 0 across the temperature interval and get   α ∂ H − ql · nd A = dV. ∂t α 



To compare the sharp and the diffusive formulations the authors considered the one-dimensional problem where the melting is driven by imposing the fixed temperature at x = 0: • The sharp phase boundary: the governing equation is ρc

∂2 T ∂αT = λ ∂t α ∂t 2

with the boundary condition at the melt front x = s(t) ρL

dαs ∂T . = −λ α dt ∂x

• The diffuse phase boundary: the governing equation is ∂α H ∂2 T = λ . ∂t α ∂t 2 The memory of the latent heat accumulation is lumped on the interface through the fractional interphase speed d α s/dt α in the sharp interface model while in the

208

8 Fractional Derivative Models

diffusive model each point in the melt retains the memory of both the sensible and latent heat—the memory of the latent heat accumulation is distributed. Meilanov et al. [14] considered the non-local Stefan problem and solved the Eq. (8.1) written for the solid and the liquid phases coupled by the condition at the crystallization front ζ (τ ) (the dimensionless variables ξ = x/L and τ = t/τ0 are used) Ts (ζ, τ ) = Tl (ζ, τ ),   ∂ γ Ts ∂ γ Tl ∂ γ ζ (τ ) λs γ − λl γ =Q ∂ξ ∂ξ ∂τ γ ξ =ζ (τ ) where Q is the rate of heat release during the phase , γ = β/2 [14]. Sierociuk et al. [15] exploited the time-fractional Fourier equation to study the heat transfer in the non-uniform semi-infinite beam. Cheng and Pang [16] reviewed the existing definitions of the fractional Laplacian (−)s [17, 18] (when s = 2 the standard Laplacian is recovered) that is the spatial integro-differential operator describing the spatial non-locality and the power law behaviour in the heat conduction: • Definition A. The three-dimensional operator (x ∈ R 3 ) with an approximate finite difference  ly u(x) dy, l > s > 0, (−)s u(x) = c1 (s) ||y||3+s R3

where

ly u(x)

=

  l k u(x − ky) and || · || is the Euclidean norm. k=0 (−1)

l

k

• Definition B. Operator of Riesz–Marchaud type  (−)s u(x) = c2 (s) R3

u(y) − u(x) dy, ||x − y||3+s

0 3

• Superdiffusion (n > 1/2): – – – –

Nonlinearity only, β ∈ (0.5, 1] Memory + non-locality, 1 + α < 2γ Memory + nonlinearity, βγ > 1/2 Non-locality + nonlinearity, 3 > α + 1/β > 2

• Superdiffusion+ (n > 1): – Non-locality + nonlinearity, α + 1/β < 2

8.1.2 Fractional Pennes Model The time-fractional generalization (∂ α /∂t α is the Caputo fractional derivative of order α ∈ (0, 1] T, α is sometimes called the order of fractionality) of the Pennes bioheat Eq. (1.5) c

∂αT = ∇ · λ∇T + cb ωb (Ta − T ) + q˙met + Q ext , ∂t α

(8.5)

212

8 Fractional Derivative Models

was used by Damor et al. [22] to study the hyperthermia and the anomalous diffusion in the skin tissue with the constant and the sinusoidal heat flux at the boundary [23, 24] and by Ezzat et al. [25] to study the temperature transient in the skin exposed to the instantaneous surface heating. As noted by Ferras et al. [26], the Eq. (8.5) is not dimensionally consistent and one has to either redefine the coefficients of these equations or to introduce a factor τ 1−α to get a “new” thermal conductivity. Singh et al. [27] used the space–time-fractional bioheat equation ρC

∂α ∂β = λ + Q p, 0 < β ≤ 1 < α ≤ 2 ∂β ∂α

to study the heat transfer in the tissues during the thermal therapy.

8.2 Zingales’s Fractional-Order Model Zingales [28] (see also [18]) considered two components of the heat transfer in the rigid solid bodies at rest 1. The short-range heat flux governed by the conventional Fourier law. 2. The long-ranged heat transfer between the elementary volumes located at the points x and y that is proportional to • The product of the interacting masses; • The temperature difference T (x) − T (y); • The distance-decaying function g(||x − y||) . The long range correlations of the local thermodynamic variables in the nonequilibrium steady-state (NESS) systems are studied by Bertini et al. [29] using the Hamilton–Jacobi equation for the non-equilibrium free energy. Zingales assumed that the distance-decaying function g decays as the power law of the distance 1 1 g(||x − y||) = dn (α) ¯ ||x − y||n+α where dn (α) ¯ is the normalizing coefficient related to the decaying exponent α and to the dimension of the topological space of the body n. Finally the energy balance equation is written in the form ρC

∂T = −∇ · q + ρ 2 λα Dxα T ∂t

8.3 Fractional Cattaneo and SPL Models

213

where Dxα is the Marchaud fractional derivative of order α defined as Dxα T

1 = dn (α) ¯

 Vy

T (x) − T (y) d Vy . ||x − y||n+α

8.3 Fractional Cattaneo and SPL Models The hyperbolic models of the non-Fourier heat conduction suffer from the unrealistic singularity of the temperature gradient across the thermal wavefront; the fractional calculus for differentiation and integration can remove the thermal wave singularity [30]. Sometimes the fractional version of Cattaneo equation is called “non-local” Cattaneo–Vernotte equation, reflecting the basic properties of the fractional derivatives [31]. Liu et al. [32] used the modification of the Cattaneo model by Christov to develop the space-fractional equation with the Riesz derivative. A time-fractional SPL model for the bioheat transfer is formulated as [33]  c

∂ 1+α T ∂T + τ 1+α ∂t ∂t

 = ∇ · λ∇T + cb ωb (Ta − T ) + q˙met + Q ext .

(8.6)

Computations show that the fractional SPL equation give the same temperature distribution as the DPL model [33]. The space-fractional SPL model was formulated for the one-dimensional case by Kumar et al. [34] c

∂αq ∂T = − α + cb ωb (Ta − T ) + q˙met + Q ext , ∂t ∂

where q(x, t) + τ

∂q(x, t) ∂q(x, t) = −λ . ∂x ∂x

(8.7)

(8.8)

Fabrizio [35] formulated the fractional Cattaneo equation as γα (1 − α)

t −∞

q(x, t) − q(x, τ ) dτ = q(x, t) + λ∇T (x, t). (t − τ )1+α

Jiang and Qi [36] derive the fractional thermal wave model of the bioheat transfer changing the Cattaneo relation to

214

8 Fractional Derivative Models

τα α D q + q = −λ∇T, α! t where Dtα is a modified Riemann–Liouville derivative of order α. Qi et al. [37] studied the laser heating generalizing the Cattaneo relation as p

τ p Dt q + q = −λ∇T ; 0 < p < 1, p

where Dt is the Caputo derivative of order p; the factor τ p is introduced to keep the dimensionality in order. Povstenko [38, 39] and Jiang and Xu [40] proposed the time-fractional Fourier law as the constitutive relationship q = −κ D 1−α ∇T where D 1−α the Caputo fractional derivative. Povstenko [41] showed that many fractional generalizations of the Cattaneo relation could be obtained from the timenon-local genealization where the kernels are functions of the Mittag-Leffler type (see Sect. A.3.3.1). Xu et al. [42] formulated the fractional Cattaneo equation using two fractional derivatives (Caputo) of the different order. The authors started with the generalized constitutive equation ∂ α−1 q ∂ β−1 q + τ α−1 = −λ∇T, 0 < β ≤ α ≤ 2. β−1 ∂t ∂t and using the Laplace transform obtained the time-non-local constitutive relation λ q(t) = − τ

t

 α−2

(t − t ) 0

  (t − t  )α−β ∇T (t  )dt  E α−β,α−1 − τ

where the generalized Mittag-Leffler function is defined as E μ,ν (z) =

∞  k=0

zk . (μk + ν)

(8.9)

The Mittag-Leffler function E μ reduces to the exponential function for μ = 1. It provides the interpolation between the stretched exponential pattern and the inverse power behaviour [43]   ⎧ μ 1 ⎨exp − λn t t λμ μ (1 − μ) E μ (−λn t ) ∝ 1 ⎩ (λn t μ (1 − μ))−1 t  λ μ

8.3 Fractional Cattaneo and SPL Models

215

Xu et al. [42] wrote the generalized Cattaneo heat equation as ∂β T ∂αT + = DT, ∂t β ∂t α where the generalized thermal diffusivity D has dimension [D] = m 2 /s β . The fractional SPL model was used by Mishra and Rai [44, 45] to study the heat transfer in the thin films. Moroz et al. [46] used the space–time-fractional version of the SPL model to simulate the heat conduction in the ferroelectric material (triglycine sulphate). Hristov [47, 48] developed the space-fractional equation for the transient heat conduction with the damping term expressed via the Caputo–Fabrizio fractional derivative [49] that is the extension of the Caputo fractional derivative with the singular kernel Dtα

M(α) f (t) = 1−α

t 0

  α(t − s) d f (t) ex p − ds, 1−α dt

where M(α) is a normalization function such as M(0) = M(1) = 1. Numerical solution of this problem was considered by Alkahtani and Atangana [50]. Yang et al. [51] (see also [52]) for the study of fractional heat transfer introduced a new fractional derivative without the singular kernel that is the extension of the Riemann–Liouville fractional derivative with the singular kernel Dα(ν) +T

R(ν) d = 1 − ν dx

x a

  ν (x − x  ) T d x  , exp − 1−ν

where R(ν) is the normalization function. Ghazizadeh and Maerafat [53] used developed by Odibat and Shawagfeh [54] the generalized Taylor’s formula q(r, t + τ ) = q(r, t) +

τα , 0 < α ≤ 1. (1 + α)

Inserting this equation into the energy balance equations and eliminating the heat flux the authors get the “non-local” Cattaneo equation ρC

∂T ρCτ α ∂ α+1 T + = λ∇ 2 T. ∂t (1 + α) ∂t α+1

216

8 Fractional Derivative Models

8.4 Fractional DPL Model Ji et al. [55] derived the fractional DPL model starting with one-dimensional version of the original general definition (2.25) q(x, t + τq ) = −λTx (x, t + τT ) and using the fractional Taylor series expansion [37, 54] up to the first two terms as q(x, t + τq ) = q(x, t) + and Tx (x, t + τT ) = Tx (x, t) +

τqα (1 + α)

Dα q(x, t) + . . .

τTα D α Tx (x, t) + . . . . (1 + α) 

where 0 < α < 1 and the operator Dα denotes the Caputo fractional derivative of order α (Sect. A.1.3) to get q(x, t) +

τqα (1 + α)

Dα q(x, t)



τTα D α T (x, t) = −λ T (x, t) + (1 + α) 

 . x

and finally  ρC

τqα ∂T + Dα T ∂t (1 − α) 



 =λ T +

 τTα Dα T . (1 − α)

Ji et al. [56] used the time-fractional DPL and the developed by the authors earlier [55] the unconditionally stable finite difference scheme to study the heat transfer in the thin films. Xu et al. [57] studied the bioheat transfer using the following equation in the Caputo derivatives of orders α and β q + τqα

  β ∂αq β ∂ = −λ ∇T + τ ∇T . T ∂t α ∂t β

The authors have changed the relaxation times of DPL model τq and τT to τqα and β τT to maintain the dimensionality in order. Xu et al. used the nonlinear least-square method to estimate simultaneously two relaxation times and two or up to degrees of fractionality. Kumar and Rai [58] solved the time-fractional dual-phase-lag model to study the heat transfer within the skin tissue during the thermal therapy. To reduce the timefractional equation into the system of ordinary differential equations, the spatial descretization in space is applied. Then the time-fractional ODEs are converted into Sylvester matrix equation by using the finite element Legendre wavelet Galerkin

8.5 Fractional TPL Model

217

method with utilization of the block pulse function in terms of Caputo fractionalorder derivative. Famy [59] proposed a new hybrid algorithm based on integrating the local radial basis function collocation method and the general boundary element method for studying time fractional DPL problems in the functionally graded tissues. The author claims that the general solution, T , of the time-fractional DPL equation is the sum of the T f , the solution of the fractional-order governing equation without dual phase lags, and Td , the solution of the DPL governing equation without fractional derivative T = T f + Td . Liu et al. [60] used the convected derivative introduced by Christov [61] and the Caputo fractional derivative of order α to formulate a model  τq

   ∂αq ∂α + v · ∇q + q · ∇v + (∇ · v)q + q = −λ∇ 1 + τT α T. ∂t α ∂t

8.5 Fractional TPL Model Ezzat et al. [62, 63] obtained the Fractional TPL model (FTPL) taking the Taylor series expansion on the both sides of the constitutive relation of the TPL model (2.42) and retained the terms up to 2α F -order for the relaxation time τq and up to α F -order for τT and τv to get 

τqα F ∂ α F τq2α F ∂ 2α F 1+ + α F ! ∂t α F (2α F )! ∂t 2α F

 q=−

   τ αF ∂ αF  ∇T + λ ∇v τv + λ T α F ! ∂t α F (8.10)

where 0 < α ≤ 1, τv = λ + λ

τvαv ∂ α F −1 . α F ! ∂t α F −1

The fractional TPL model as well as ordinary TPL one was used to study the problems of the thermoelasticity [62, 63] and the piezoelectric thermoelasticity problems. Ezzat et al. [62] use the energy conservation equation for the homogeneous isotropic thermoelastic solid −∇ · q + ρ Q = ρC E T˙ + γ T0 e˙ii where Q is the heat source, C E is the specific heat at constant strain, eii are the components of the strain tensor, γ = (3ν + 2μ)αT , ν, μ are the Lame’s constants, αT is the coefficient of linear thermal expansion.

218

8 Fractional Derivative Models

Taking the divergence and the time derivative of the Eq. (8.10) Ezzat et al. get 

τqα F ∂ α F τq2α F ∂ 2α F 1+ + α F ! ∂t α F (2α F )! ∂t 2α F

 ∇ · q˙ = −

   τ α F ∂ α F +1 2 ˙  2 T + λ τv + λ T ∇ v ˙ . ∇ α F ! ∂t α F +1

and finally obtain the modified fractional heat transport equation 

τq2α F ∂ 2α F τqα F ∂ α F + 1+ α F ! ∂t α F (2α F )! ∂t 2α F

 ˙ = (ρC E T¨ + γ T0 e¨ − ρ Q) τv ∇ 2 T˙ + λ

τTα F ∂ α F +1 2 ∇ T + λ ∇ 2 T. α F ! ∂t α F +1

(8.11)

The authors consider several limiting cases: • When α F = 1 and the thermal conductivity λ is much smaller than λ , q = −λ ∇v, τv λ τv and (neglecting τq2 ) the Eq. (8.11) simplifies to     ∂ ∂  ˙ ¨ (ρC E T + γ T0 e¨ − ρ Q) = λ 1 + τv ∇ 2 T. 1 + τq ∂t ∂t This equation is an extension of the Green–Naghdi theory of type II. The heat conduction equation of the Green–Naghdi theory of type II [64] ρC E T¨ + γ T0 e¨ − ρ Q˙ = λ ∇ 2 T is obtained if τT = τv = τq = 0. • When α F = 1, τT = τv = τq = 0 (hence τv = λ) the Eq. (8.11) reduces to the heat conduction equation of the Green–Naghdi theory of type III [65, 66] ρC E T¨ + γ T0 e¨ − ρ Q˙ = λ ∇ 2 T + λ∇ 2 T. • When α F = 1 and λ = 0 the equation (8.11) reduces to the heat conduction equation of the DPL model [67, 68]. • When α F = 1 and λ = 0, τT = τv = 0, τq > 0, the Eq. (8.11) reduces (neglecting τq2 ) to the Lord-Shulman model [69]. • When α F = 1 the Eq. (8.11) reduces to the heat conduction equation of the TPL model. • When 0 < α F ≤ 1, τT = τv = 0 and neglecting τq2α F the fractional model due to Sherief et al. [70] is obtained.

8.5 Fractional TPL Model

219

8.5.1 Non-local Fractional TPL Model Akbarzadeh et al. [30] suggested the non-local fractional three-phase-lag (NL FTPL) heat conduction model. The constitutive equation of NL FTPL heat conduction is written as q(r + λq , t + τq )

= −[λJ α F −1 ∇T (r

+λT , t + τT ) + λ J α F −1 ∇v(r + λv , t + τv )]

(8.12) where 0 ≤ α F ≤ 2, J α is the Riemann–Liouville fractional integral. Alternative forms of the constitutive Eq. (8.12) can be derived by the Taylor series expansion of the correlation lengths λq , λT , λv in the space domain and of the relaxation times τq , τT , τv in the time domain. The heat conduction equation is obtained by the elimination of the heat flux from the constitutive equation and the energy conservation equation. In the case of the second-order Taylor expansion for both the correlation lengths and the relaxation times it is written (in the absence of the volumetric heat source) as [30] 

 1 + λq · ∇ +

λq · λq 2



  1 2 ∂2 T ∂ 1 2 ∂2 ∇ + τq + τq 2 τ ∂t 2 ∂t κ q ∂t 2   λ = ∇ · FT + Fv J α−1 ∇T, λ 2



where FT = 1 + λT · ∇ +



and Fv = 1 + λv · ∇ +

λT · λT 2 λv · λ T 2

 ∇ 2 + τT  ∇ 2 + τv

(8.13)

1 ∂2 ∂ + τT2 2 ∂t 2 ∂t

1 ∂2 ∂ + τv2 2 . ∂t 2 ∂t

Akbarzadeh et al. [30] also derived the wave-like NL FTPL for the homogenous medium by simplifying the NL FTPL heat conduction Eq. (8.13) using the first-order Taylor series expansion of λq in space and the second-order Taylor series expansion of τq and the first-order Taylor series expansion of τT and τv in time:  1 + λq · ∇ + τq

  1 2 ∂2 T 1 ∂2 ∂ + τq2 2 τq 2 = ∂t 2 ∂t κ ∂t    λ ∂ ∂ 1 + τT + 1 + τv J α−1 ∇ 2 T. ∂t λ ∂t

The authors noted that τq , τT , τv could present the delayed thermal responses due to the collisions of electrons and phonons, the temporary momentum loss, the normal relaxation in the phonon scattering, and the internal energy relaxation [71,

220

8 Fractional Derivative Models

72] and the time-fractional derivative removes the thermal wave singularity across the thermal wavefront and is used to present the subdiffusion (0 ≤ α < 1), normal diffusion (α = 1) and superdiffusion phenomena (1 < α < 2) [73]. The NL FTPL heat conduction shows that the non-local length of heat flux in the absence of heat generation only contributes in the mixed derivative term implying the negligible effect of heat flux non-locality on the thermal responses for long-term and steady-state heat transport analysis and the reduction of thermal behaviour of NL FTPL to transient Fourier heat conduction [72, 74].

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69. Lord, H.W., Shulman, Y.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299–309 (1967) 70. Fractional order theory of thermoelasicity: Sherief, H.H., El-Sayed, A.M.A., Abd El-Latief, A.M. Int. J. Solid Struct. 47, 269–273 (2010) 71. Tzou, D., Guo, Z.Y.: Nonlocal behavior in thermal lagging. Int. J. Therm. Sci. 49, 1133–1137 (2010) 72. Akbarzadeh, A.H., Pasini, D.: Phase-lag heat conduction in multilayered cellular media with imperfect bonds. Int. J. Heat Mass Transf. 75, 656–667 (2014) 73. Youssef, H.M.: Theory of fractional order generalized thermoelasticity. J. Heat Transf. 132, 1–7 (2010) 74. Akbarzadeh, A.H., Chen, Z.: Heat conduction in onedimensional functionally graded media based on the dualphaselag theory. Proc. Inst. Mech. Eng. Part C 227, 744–759 (2013)

Chapter 9

Fractional Boltzmann and Fokker–Planck Equations

Li and Cao [1] studied the entropic functionals including the entropy density, the entropy flux and the entropy production rate for the standard and fractional phonon Boltzmann transport equations (BTEs) using the relaxation time approximation f − f0 ∂f + vg ∇ f = ∂t τ

(9.1)

where v g is the phonon group velocity, f (x, t, k) is the distribution function, k is the wave vector, 1   f0 = ω exp k B T − 1 is the equilibrium distribution function,  is the reduced Planck constant, ω is the angular frequency, and k B is the Boltzmann constant. ˆ The phonon energy density e(x, t), heat flux q(x, t) and flux of heat flux Q(x, t) are defined as    ˆ e(x, t) = f ωdk, q(x, t) = f ωv g dk, Q(x, t) = f ωv g v g dk. Multiplying Eq. (9.1) by v g and integrating it over the wave vector space give (using the definition of the temperature as the measure of the local energy density de = cV dT [2]) the energy conservation equation ∂e = −∇ · q. ∂t

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. I. Zhmakin, Non-Fourier Heat Conduction, https://doi.org/10.1007/978-3-031-25973-9_9

225

226

9 Fractional Boltzmann and Fokker–Planck Equations

Similarly, multiplying Eq. (9.1) by v g v g and integrating yield q+τ

∂q ˆ = −τ ∇ · Q. ∂t

The “Boltzmann–Gibbs” entropy density for the phonon distribution function (3.67) has the time derivative  ∂s f ∂f = kB [ln( f + 1) − ln f ]dk ∂t ∂t or, using Eq. (9.1)   ∂s f = −∇ · v g k B [( f + 1) ln( f + 1) − f ln f ]dk ∂t  f +1 f − f0 ln dk. +k B τ f Thus the entropy flux J f (x, t) is  Jf =

v g k B [( f + 1) ln( f + 1) − f ln f ]dk,

and the entropy production rate σ f (x, t) is 

f − f0 f +1 ln dk. τ f

σ f = kB

 If f = f 0 , the entropy density s = T −1 cV dT , the entropy flux JC = T −1 q, and one gets the classical entropy production rate σ f = f0 ≡ σC = q · ∇

 1 . T

In the case | f − f 0 |  f 0 the authors used the expansion and finally get the entropy balance equation σ f ≈ σC +

2 ∂σ f = σC + ∇ · K τ ∂t

where the entropy density extra flux is expressed by τ K = − Qˆ · 2

 1 + O( f − f 0 ). T

(9.2)

9 Fractional Boltzmann and Fokker–Planck Equations

227

Thus 2 σ f (x, t) = σ f (x, 0) + τ

t 0

 2ξ dξ. (σC + ∇ · K)(x, t − ξ ) exp − τ

The authors stressed that the last equation does not require small temporal or spatial derivatives and therefore is valid in the supertransient and large-gradient heat conduction problems in contrast to Eq. (9.2) that requires small temporal and spatial derivatives of ( f − f 0 )2 . The authors investigated the fractional BTE in the framework of models of the GCE class (see Sect. 2.1). The fractional BTE in the RTA is written as follows: γ

τ γ −1 Dt f + v g · ∇ f =

f − f0 , 0 0 and sequence of the IID random jumps X 1 , X 2 , . . . , ∈ R with the PDF ψ(x − x  , t − t  ). Frequently these probability distribution functions are temporal and spatial invariant—ψ(t) and w(x)—that enables the use of the Laplace–Fourier transform [17]. An essential assumption is that the waiting time distribution and the jump length distribution are independent of each other. The stochastic process is Markovian if and only if the wating time PDF is of the form ψ(t) = m exp(−mt) with positive m [16]. The CTRW model is a generalization of the compound Poisson model in which the underlying Poisson process is replaced by the renewal process [18]. In the Poisson process the waiting times are the exponentially distributed IID random variables. In the renewal process the waiting times have the general distribution. The only “memoryless” distribution is the exponential one, and thus the only Markov renewal process is the Poisson one. The transition from the microscopic level to the macroscopic level by scaling produces four classes of the motion: the Brownian motion, the Lévy motion, the subordinated Brownian motion and the subordinated Lévy motion [18]. These classes are similar to the random motions derived by Eliazar and Schlesinger [18] from the Langevin motion (two Markov motions—the Brownian motion and the Lévy motion, two non-Markov motions—the fractional Brownian motion, the fractional Lévy motion). Eliazar and Schlesinger noted that these classes of motion are the self-similar processes, but have the different statistical behaviour: the increments of the subordinated Brownian/Lévy motions are not Gauss/Lévy distributed and are non-stationary, while the increments of the fractional Brownian/Lévy motions are Gauss/Lévy distributed and stationary. The evolution equation for the probability p(x, t) of finding the random walker at the position x at the time instant t (the Montrol–Weiss equation [19]) can be written (for the initial condition p(x, 0), i.e. the walker is initially at the origin) as t p(x, t) = δ(x)(t) +

⎡ ψ(t − t  ) ⎣

⎤ w(x − x  ) p(x  , t  )dx  ⎦ dt  ,

(9.7)

−∞

0

where

∞

∞ (t) =





t

ψ(t )dt = 1 − t

ψ(t  )dt  .

0

The function (t) is the survival function—the probability that at the instant t the walker is still sitting in the starting position x = 0. The special choice w(x) = δ(x − 1) gives the “pure renewal process” with the position x(t) = N (t) denoting the counting function with all jumps of the length 1 in the positive direction at the renewal instants [20].

9.1 Continuous-Time Random Walks

231

The resulting equation can be interpreted as the evolution equation of the generalized Fokker–Planck–Kolmogorov type [21] t 0

∂ p(x, t  )  (t − t ) dt = − p(x, t) + ∂t  

∞

w(x − x  ) p(x  , t  )dx 

−∞

where the auxillary function (t) is such as t (t) =

(t − t  )ψ(t  )dt  .

0

If the CTRW is Markovian, by the choice of unit of time (t) = δ(t) [21] we get the exponential waiting time φ(t) = e−t and the most general master equation for the Markovian CTRW (also called the Kolmogorov–Feller equation [20]) ∞

∂ p(x, t) = − p(x, t) + ∂t

w(x − x  ) p(x  , t  )dx  .

−∞

The choice (t) =

t −β , 0 0

t−td

where the kernel function K (t, t  ) is used to measure the degree of memory effect from the past to the present in the delayed interval [t, t − td ]. The kernel function can be  b t − t  K (t, t ) = + 1 , b ≥ 0. td The generalized piezoelectric thermoelasticity model formulated by Guo et al. [29] is written as ci jkl εkl, j + ei jk φ,k j − βi j θ, j = ρ u¨ i ,

254

10 Elasticity and Thermal Expansion Coupling

ei jk ε jk,i − di j φ,i j + pi θ,i = 0, (1 + τq Dtd )(βi j T0 ε˙ i j − pi T0 φ˙ ,i + ρcV θ˙,i ) = [κ  (θ )θ,i ],i . To deal with nonlinear thermal conductivity, Kirchhoff transformation method based the integral relation is used [37–39]; it permits the conversion of the nonlinear problem into linear one  1 θ κ(θ )dθ ϑ= κ 0 that leads to the relations κϑ,i = κ  (θ )θ,i , κϑ,ii = [κ  (θ )θ,i ],i , κ ϑ˙ = κ  (θ )θ˙ . The authors investigated the dynamic responses of a transient heated thin piezoelectric plate composed of a material with the variable thermal conductivity κ  (θ ) = κ(1 + κ1 ) where κ1 is the small quantity to measure the influence of temperature on thermal conductivity.

References 1. Fülöp, T., Kovács, R., Lovas, A., Rieth, A., Fodor, T., Szücs, M., Ván, P., Grof, G.: Emergence of non-Fourier hierarchies. Entropy 20, 832 (2018) 2. Bargmann, S., Favata, A.: Continuum mechanical modeling of laser-pulsed heating in polycrystals: a multi-physics problem of coupling diffusion, mechanics, and thermal waves. ZAMM 94, 487–498 (2014) 3. Sellitto, A., Cimmelli, V.A., Jou, D.: Nonlinear propagation of coupled first- and second-sound waves in thermoelastic solids. J. Elast. 138, 93–109 (2020) 4. Biot, M.A.: Thermoelasticity and irreversible thermodynamics. J. Appl. Phys. 27, 240–253 (1956) 5. Chandrasekharaiah, D.S.: Thermoelasticity with second sound. Appl. Mech. Rev. 39, 355–376 (1986) 6. Chandrasekharaiah, D.S.: A generalized linear thermoelasticity theory of piezoelectric media. Acta. Mech. 71, 39–49 (1988) 7. Chandrasekharaiah, D.S.: Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. pp. 705–729 (1998) 8. Hetnarski, R.B., Ignaczak, J.: Generalized thermoelasticity. J. Therm. Stress. 22, 451–476 (1999) 9. Lord, H.W., Shulman, Y.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299–309 (1967) 10. Green, A.E., Lindsay, K.A.: Thermoelasticity. J. Elast. 2, 1–7 (1972) 11. Arai, M., Yamazaki, I.: Numerical simulation of thermoelastic wave coupled with non-Fourier heat conduction equation. AIP Conf. Proc. 2309, 020019 (2020) 12. Dhaliwal, R., Sherief, H.: Generalized thermoelasticity for anisotropic media. Quart. Appl. Math. 33, 1–8 (1980) 13. Green, A., Naghdi, P.: A re-examnation of the basic postulates of thermomechanics. Proc. Roy. Soc. London 357, 171–194 (1991)

References

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14. Green, A., Naghdi, P.: On undamped heat waves in an elastic solid. J. Thermal Stresses 15, 253–264 (1992) 15. Green, A., Naghdi, P.: Thermoelasticity without energy dissipation. J. Elast. 31, 189–209 (1993) 16. Taheri, H., Fariborz, S.J., Eslami, M.R.: Thermoelastic analysis of an annulus using the GreenNaghdi model. J. Therm. Stresses 28, 911–927 (2005) 17. Green, A.E., Laws, N.: On the entropy production inequality. Arch. Rat. Mech. Anal. 45, 47–53 (1972) 18. Yu, Y.J., Hu, W., Tian, X.G.: A generalized thermoelasicity model based on memory-dependent derivative. Int. J. Eng. Sci. 81, 123–134 (2014) 19. Choudhuri, S.: On a thermoelastic three-phase-lag model. J. Therm. Sci. 30, 231–238 (2007) 20. Kumar, R., Gupta, V.: Plane wave propagation and domain of influence in fractional order thermoelastic materials with three-phase-lag heat transfer. Mech. Adv. Mater. Struct. 23, 896– 908 (2016) 21. Tiwari, R., Kumar, R., Abouelregal, A.E.: Analysis of a magneto-thermoelastic problem in a piezoelastic medium using the non-local memory-dependent heat conduction theory involving three phase lags. Mech. Time-Depend. Mater. 1–17 (2021) 22. Borjalilou, V., Asghari, M., Bagheri, E.: Small-scale thermoelastic damping in micro-beams utilizing the modified couple stress theory and dual-phase-lag heat conduction model. J. Therm. Stresses 42, 1–14 (2019) 23. Luo, P., Li, X., Tian, X.: Nonlocal thermoelasticity and its application in thermoelastic problem with temperature-dependent thermal conductivity. Europ. J. Mech. / A Solids 87, 104204 (2021) 24. Sheoran, S.S., Kundu, P.: Fractional order generalized thermoelasticity theories: a review. Int. J. Adv. Appl. Math. Mech. 3, 76–81 (2016) 25. Fractional order theory of thermoelasicity: Sherief, H.H., El-Sayed, A.M.A., Abd El-Latief, A.M. Int. J. Solid Struct. 47, 269–273 (2010) 26. Youssef, H.M.: Theory of fractional order generalized thermoelasticity. J. Heat Transf. 132, 1–7 (2010) 27. Ezzat, M.A.: Thermoelectric MHD non-Newtonian fluid with fractional derivative heat transfer. Phys. B 405, 4188–4194 (2010) 28. Abbas, I.A.: Fractional order GN model on thermoelastic interaction in an infinite fibrereinforced anosothropic plate conatining a circular hole. J. Comput. Theor. Nanosci. 11, 380– 384 (2014) 29. Guo, H., Li, C., Tian, X.: A modified fractional-order generalized piezoelectric thermoelasticity model with variable thermal conductivity. J. Therm. Stresses 41, 1538–1557 (2018) 30. Abouelregal, A.E.: Fractional order generalized thermo-piezoelectric semi-infinite medium with temperature-dependent properties subjected to a ramp-type heating. J. Therm. Stress. 34, 1139–1155 (2011) 31. Islam, M., Kanoria, M.: One-dimensional problem of a fractional order two-temperature generalized thermo-piezoelasticity. Math. Mech. Solids 19, 672–693 (2014) 32. Ma, Y.B., He, T.H.: Dynamic response of a generalized piezoelectric-thermoelastic problem under fractional order theory thermoelasticity. Mech. Adv. Mater. Struc. 23, 1173–1180 (2016) 33. Wang, L., Li, H.F.: Surpassing the fractional derivative: concept of the memory-dependent derivative. Comput. Math. Appl. 62, 1562–1567 (2011) 34. Ezzat, M.A., El Karamany, A.S., Bary, A.A.E.: Generalized thermo-viscoelasticity with memory dependent derivatives. Int. J. Mech. Sci. 89, 470–475 (2014) 35. Li, C.E., Guo, H.L., Tian, X.G.: Shock-induced thermal wave propagation and response analysis of a viscoelastic thin plate under transient heating loads. Waves Random Complex Media 28, 270–286 (2018) 36. Ezzat, M.A., El Karamany, A.S., Bary, A.A.E.: Electro-thermoelasticity theory with memory dependent derivative heat transfer. Int. J. Eng. Sci. 99, 22–38 (2016) 37. Ezzat, M.A., El-Bary, A.A.: Effects of variable thermal conductivity and fractional rod of heat transfer on a perfect conducting infinitely long hollow cylinder. Int. J. Therm. Sci. 108, 62–69 (2016)

256

10 Elasticity and Thermal Expansion Coupling

38. Sherief, H.H., Hamza, F.A.: Modeling of variable thermal conductivity in a generalized thermoelastic infinitely long hollow cylinder. Meccanica 51, 551–558 (2016) 39. Li, C.L., Guo, H.L., Tian, X., Tian, X.G.: Transient response for a half-space with variable thermal conductivity and diffusivity under thermal and chemical shock. J. Therm. Stress. 40, 389–401 (2017)

Chapter 11

Some Exact Solutions

The exact analytical solutions assist in the analysis of the physical phenomena and serve as the benchmarks for the numerical methods. Fan and Wang published an extensive review of the analytical solutions of the bioheat problems that includes the Pennes, the thermal wave and the DPL models [1]. Kulish and Lage [2] developed the relation between the temperature and the heat flux at the same point, thus avoiding the need to solve the three-dimensional timedependent heat conduction equation ∂ T (x1 , x2 , x3 , t) − κ∇ 2 T (x1 , x2 , x3 , t) = 0. ∂t The authors considered the unidirectional problems within a semi-infinite domain assuming the constant and uniform properties. Using the change of variables θ = T − T0 and (ξ1 , ξ2 , ξ3 ) = κ −1/2 (x1 , x2 , x3 ) the authors get ∂ T (ξ1 , ξ2 , ξ3 , t) − ∇ 2 T (ξ1 , ξ2 , ξ3 , t) = 0 ∂t and after performing the Laplace transform [3] accounting for the initial condition θ (ξ1 , ξ2 , ξ3 , 0) = 0 obtained sθ  (ξ1 , ξ2 , ξ3 , s) − ∇ 2 θ  (ξ1 , ξ2 , ξ3 , s) = 0.

(11.1)

The separation of variables θ  (ξ1 , ξ2 , ξ3 , s) = X 1 (ξ1 )X 2 (ξ2 )X 3 (ξ3 ) reduces Eq. (11.1) to the system of equations ⎡ ⎤   h p hq ∂ X j ⎦ 1 ⎣ 1 ∂ = λ2j s,  Xj ∂ξ h ∂ξ j j hj j

p = q = j = 1, 2, 3.

(11.2)

j

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. I. Zhmakin, Non-Fourier Heat Conduction, https://doi.org/10.1007/978-3-031-25973-9_11

257

258

11 Some Exact Solutions

where h 1 , h 2 , h 3 are multipliers of the orthogonal coordinate system. The solution of Eq. (11.2) in the Cartesian coordinates is X j (ξ j , s) =

∞

[A jm j (s)es

1/2

λ jm j ξ j

+ B jm j (s)e−s

1/2

λ jm j ξ j

].

m j =−∞

Thus the Laplace image of the temperature is θ =

m1 m1 ∞ m 1 =−∞ m 3 =0 m 2 =0

⎡ ⎣



⎤ s 1/2 λ jm j ξ j

(A jm j (s)e

+ B jm j (s)e

−s 1/2 λ jm j ξ j

)⎦ . (11.3)

j=1,2,3

It follows from Eq. (11.3) that the equality ∂θ  (ξ, s) = −s 1/2 θ  (ξ, s) ∂ξ

(11.4)

holds when, depending on the sign of λ jm j , A jm j = 0 or B jm j = 0 and when λ jm j is unity. Sarkar et al. [4] considered heat transfer in the skin tissue using the Pennes equation for the cases of the temperature or the heat flux prescribed on surface.

11.1 Phase-Lag Models Tang and Araki [5] used the Cattaneo equation to study the 1D heat transfer in the finite medium under the laser-pulsed heating and in the semi-infinite medium under the periodic heating. The authors exploited the dimensionless variables x η= √ , 2 κτ

ξ=

t , 2τ

θ (η, ξ ) =

√ λ τ √ . q0 κ

The following boundary and initial conditions are used: ∂ T (0, t) ∂ T (L , t) ∂ T (x, 0) = = 0, T (x, 0) = = 0. ∂x ∂x ∂t The solution is obtained in the form of infinite series. The boundary and initial conditions for the periodic heating case are −λ

∂ T (0, t) ∂ T (x, 0) ∂ T (∞, t) = q0 eiωt , T (∞, t) = = 0, T (x, 0) = = 0. ∂x ∂x ∂x

11.1 Phase-Lag Models

259

The solution is written with the use of the Heviside unit step function and the modified Bessel function of zero order. Ordonez-Miranda and Alvarado-Gil [6] solved the Cattaneo–Vernotte equation for the system of the finite and semi-infinite layers in the perfect thermal contact 1 ∂ T (x, t) τ ∂ 2 T (x, t) 1 ∂ 2 T (x, t) − =− − 2 2 ∂x κ ∂x κ ∂t λ



 ∂ S(x, t) S(x, t) + τ , (11.5) ∂t

where S(x, t) = F(x)(1 + cos(ωt)) = Re[F(x)(1 + exp(iωt))] is the rate per unit volume of heat generation. The temperature in the sample is written as T (x, t) = T0 + Tdc (x) + Tac (x, t), T0 is the ambient temperature, and Tdc (x) and Tac (x, t) = Re[θ (x)eiωt ] are the stationary and periodic components of the temperature. Thus the general solution of Equation (11.5) for the spatial part of the oscillatory temperature θ is given by θ (x) = Beq x + Ce−q x , where the constants B and C depend on the boundary conditions and q is [6]

q=

iω √ χ + iχ −1 iωτ = , κ μ

χ=



1+

(ωτ )2

+ ωτ ,

μ=

2κ . ω

The perfect contact means the continuity of both the temperature and the heat flux at the interface x = L between the layers θ (x = L − ) = θ (x = L + ), dθ (x = L − ) λ dθ (x = L + ) λ = , 1 + iωτ dx 1 + iωτ dx where L − and L + denote the limit x → L from the left and from the right side. Mikic [7], Baumeister and Hamil [8] and Amos and Chen [9] derived the temperature distribution in a semi-infinite medium due to a step change in the temperature at its surface. Glass et al. [10] and Maurer and Thompson [11] considered the heat conduction in a semi-infinite slab with the periodic temperature at the surface. Taifel [12] and Lewandowska and Malinowski [13] obtained the solution for the heat transport in a thin layer due to a step change of the temperature on the surfaces. Özisik and Vick [14] derived an analytical solution for the temperature distribution in a finite layer with insulated boundaries due to a volumetric energy source. AbdalHamid [15] and Tang and Araki [16] studied the heat conduction in a finite slab under periodic surface thermal disturbance and Gembarovich and Majernik [17] under the heat pulse flux boundary conditions.

260

11 Some Exact Solutions

Gembarovich and Gembarovich [18] considered the temperature distrubution in an isotropic homogeneous medium (0 ≤ x ≤ L) with zero initial temperature and adiabatically insulated boundaries when one surface is heated by the stepwise heat pulse of duration t1 using the Cattaneo model. The solution is written as 1 V (x, t) = t1

t1  0

F(x, t − t  ) − τ

 ∂ F(x, t − t  ) dt  ∂t 

where      ∞    t 1 1 L t 2 − γ12 H (t − γ1 ) F = √ e(− 2τ ) t 2 − γ 2 H (t − γ ) + I0 I0 2τ 2τ κτ k=0

√ √ where γ = (2k L + x) τ/κ, γ1 = (2k L + 2L + x) τ/κ and H (t) is the Heaviside unit step function, I0 (z) is the modified Bessel function of the first kind of zero order. Tzou [19] obtained an analytical solution of the propagation of the thermal disturbance produced by the moving crack. Jiang [20] considered the radial heat conduction in the hollow sphere with constant properties, where ri is the inner radius and ro is the outer radius. The initial temperature is uniform and equals to T0 . The heat transfer is caused by the sudden change of the inner surface temperature to Twi and the outer surface temperature to Two . The steady-state equation using dimensionless variables η=

r κt T − To , ξ = 2, θ = ro ro Two − To

is written as [20] ∂ 2 θ (η, ξ ) 2 ∂θ (η, ξ ) ∂ 2 θ (η, ξ ) 2 ∂θ (η, ξ ) =ε + + 2 ∂η η ∂η ∂ξ 2 η ∂ξ

(11.6)

where ε = κτ/eo2 , the relaxation time τ = 3κ/v 2 , and v is the phonon velocity. Initial and boundary conditions are θ (η, 0) = 0,

∂θ |ξ =0 = 0, (rγ ≤ η ≤ 1), θ (rγ , ξ ) = Tγ , θ (1, ξ ) = 1 (ξ > 0) ∂ξ

where rγ =

ri Twi − T0 , Tγ = . ro Two − T0

11.1 Phase-Lag Models

261

Applying the Laplace transform to Eq. (11.6) with respect to the variable ξ and accounting for the initial conditions, Jiang got ¯ s) 2 ∂ θ¯ (η, s) ∂ 2 θ(η, ¯ s) = 0 + − (s + εs 2 )θ(η, ∂η2 η ∂η

(11.7)

with the boundary conditions θ¯ (rγ , s) = Tγ /s, θ¯ (1, s) = 1/s. The solution of Eq. (11.7) was transformed with the use of the expansion 1

= √ 1 − exp[−2 s + εs 2 (1 − rγ )]

∞

exp[−2 s + εs 2 (1 − rγ )].

0

and a table of inverse Laplace transform to get a cumbersome expression for ηθ (η, ξ ) that reduces to the solution for the solid sphere when rγ → 0 (i.e. when ri → 0 or ro → ∞) and to the solution of the corresponding parabolic problem for ε → 0. Barletta and Zancini [21] analysed the non-stationary heat transfer in an infinitely wide slab with the prescribed boundary heat flux using the Cattaneo equation. Similar problem in the cylindrical coordinates was considered by Saerdodin et al. [22]. Ahmadikia et al. [23, 24] and Kundu and Dewanjee [25] used the TWMBT model (as well as the Pennes equation) to simulate the action of the constant and the pulse train heat flux on the surface of the skin tissue. Al-Khairy et al. [26] studied the effect of the laser heat source (e.g. constant or exponential) on the moving semi-infinite medium using the Cattaneo equation. Choi et al. [27] solved 1D CV equation with boundary conditions T (0, t) = Th and T (l, t) = Ti seeking the solution in the form T (x, t) = Aeax ebt to get A = Th ,

b1 =

− τ1 +



1 τ2

+ 4a 2 4C T2

2

1 a= l



, b2 =

Ti Th

 ,

− τ1 −



1 τ2

+ 4a 2 4CT2

2

,

√ where the propagation speed CT = κ/τ , thus T (x, t) = Aeax (eb1 t + eb2 t ). The DPL model was used by Askarizadeh and Ahmadikia [28, 29] and by Lin [30] to investigate the transient heating of the skin tissue. Askarizadeh and Ahmadikia considered the cases of constant, pulse train and periodic heat flux. Biomechanics of skin under the thermal treatment was studied also by Xu et al. [31–33]. Dai and Nassar [34] used the initial form of the DPL model. Kulish and Novozhilov [35] derived using the Laplace transform the integral equation that relates the temperature and its gradient that holds everywhere inside the domain. The authors considered the 1D version of the DPL model (2.26) ∂ 2 θ (x, t − τ ) ∂θ (x, t) =α , −∞ < x < ∞, −∞ < t < ∞ ∂x ∂x2

(11.8)

262

11 Some Exact Solutions

with the following initial and boundary conditions θ (x, 0) = 0,

1 ∂θ (0, t − τ ) = − q n (0, t), θ (κ, t) = 0. ∂x k

Taking Laplace transform of Equation (11.8) produces ODE d 2 s − exp(τ s) = 0. dx 2 α with the general solution   



s s (x, s) = C1 (s) exp − exp(τ s)x + C2 (s) exp exp(τ s)x . α α The solution must be bounded at t → ∞ thus C2 = 0. Taking the time derivative of the solution and combining with the solution gives

  τ d(x, s) α exp − s . (x, s) = − s 2 dx The inverse Laplace transform involving the table transform produces the relation between the temperature and its gradient [35] (t − t  = ζ )

T (x, t) = T0 −

α π

t− τ 2

0

∂ T (x,ζ )

∂x  dζ.  t − τ − ζ 2

Zhang et al. [36] used elements of the method of the “manufactured solutions” (the authors called it “the method of trial and error”) that is based on guessing of the possible exact solution and obtaining the initial and boundary conditions by substituting the “guess” solution into the studied equation. The method of manufactured solutions [37–39] is a simple process, although it involves the cumbersome algebra: a chosen analytical “exact” solution φex (t, x, y, z)   , φex ) is substituted into the equations to produce the source terms f ex = L(φex , φex and the boundary conditions. This approach is attractive since it could be applied to the multidimensional domains while allowing the assessment of the solution accuracy. However, its usefulness depends on the similarity of the guessed exact solution to the typical solution of the real problem in question. An availability of the analytical exact solution greatly reduces the CPU time needed for testing of the non-stationary codes: one can use an exact solution for the specification of the initial and, if necessary, the (non-steady) boundary conditions and avoid the need of using large time integration intervals. Zhang et al. [36] considered the three-dimensional DPL equation written as

11.1 Phase-Lag Models

∂θ ∂ 2θ + τq 2 = κ ∂t ∂t

263



∂ 2θ ∂ 2θ ∂ 2θ + + ∂x2 ∂ y2 ∂z 2



∂ + κτT ∂t



∂ 2θ ∂ 2θ ∂ 2θ + + ∂x2 ∂ y2 ∂z 2



where θ is the excess temperature, κ is the thermal diffusivity, τq is the relaxation time in establishing the heat flux, and τq is the relaxation time in establishing the temperature gradient. Zhang et al. [36] derived two exact solutions 1. It is assumed that u = C1 x + C2 y + C3 z, θ = f (t) exp(u) + g(u). Separation of variables produces two equations for f (t) and g(u) τq f  (t) + f  − κ f (t)(C12 + C22 + C32 ) − κτT (C12 + C22 + C32 = C4 , κ(C12 + C22 + C32 )g  (u) = C4 exp(u). Solution of these equations  f (t) = C5 exp  C6 exp

κτT CC − 1 +

κτT CC − 1 −

g(u) =



 [κτT CC − 1]2 + 4κτq CC t + 2τq

 [κτT CC − 1]2 + 4κτq CC C4 t − , 2τq CC

C4 exp(u) + C7 u + C8 , κCC

where CC = C12 + C22 + C32 . 2. It is assumed that θ = f (t)[exp(C1 x) + exp(C1 y) + exp(C1 z)] + g(x, y, z). As in the previous case separation of variables is used: τq f  (t) + (1 − κτT C12 ) f  (t) − κC12 f (t) = C2 ,  κ

∂2g ∂2 g ∂2g + + ∂x2 ∂ y2 ∂z 2

 = C2 (eC1 x + eC1 y + eC1 z ).

Thus f (t) and g(u) are determined as follows: ⎡ f (t) = C3 exp ⎣

κτT C12 − 1 +



(κτT C12 − 1)2 + 4κτq C12 2τq

⎤ t⎦ +

264

11 Some Exact Solutions

⎡ C4 exp ⎣

g=

κτT C12 − 1 −

 ⎤ (κτT C12 − 1)2 + 4κτq C12 C2 t⎦ − , 2τq κC12

C2 (exp(C1 x) + exp(C1 y) + exp(C1 z)) + C5 x + C6 y + C7 z + C8 . κC12

Molina et al. [40] investigated one of the energy-based ablative techniques—the radiofrequency thermal ablation—using the model based on the electrode of the radius r0 embedded into the biological tissue. The authors considered the Fourier, hyperbolic and “relativistic” heat conduction equations. The latter is written as  −κ

∂ 2 T (r, t) 2 ∂ T (r, t) + ∂r 2 r ∂r

 +

∂ T (r, t) κ ∂ 2 T (r, t) + 2 = Q(r ) ∂t C ∂t 2

where Q(r ) = Pr0 /4πr 4 and r ≥ r0 is the Joule heat produced per unit volume of the tissue with the initial and boundary conditions T (r, 0) = t0 ,

∂T (x, t) = 0, lim T (r, t) = 0. r →∞ ∂t

The boundary condition at the interface between the electrode and the tissue (r = r0 ) was obtained under assumption that it is mainly governed by the thermal inertia of the electrode and the heat increment in the mass of electrode equal to ρ0 c0

4πr03 ∂ T (r0 , t) 3 ∂t

where ρ0 and c0 are the density and the specific heat of the electrode, thus  4π κr02

−

1 C2



∂T (r0 , t) ∂t

2

 +

∂T (r0 , t) ∂r

2 = ρ0 c0

4πr03 ∂ T (r0 , t). 3 ∂t

The authors used the dimensionless variables ρ=

κt r , ξ = 2, μ = r0 r0



κ r0 C

2 , V =

4π κr0 (T − T0 ) P

to solve the equation  −

∂2 V 2 ∂V + 2 ∂ρ ρ ∂ρ

 +

∂V ∂2 V 1 +μ 2 = . ∂ξ ∂ξ ρ

Applying the Laplace transform L(ρ, s) the authors obtained

11.2 Phonon Models

265



∂ 2L 2 ∂L − + ∂ρ 2 ρ ∂ρ

 + (s + μs 2 ) =

1 ρ4 s

with the general solution ⎞ ⎞ ⎛ ρ ⎛ ρ

−ν

ν −νρ eνρ ⎝ 1 1 e u e u e ⎝ L(ρ, s) = du + M1 (s)⎠ + du + M1 (s)⎠ 2ρν s u3 2ρν s u3 1

where ν = [40].



1

s + μs 2 and M1 (s) and M2 (s) determined by the boundary conditions

11.2 Phonon Models Dong et al. [41] considered the heat transport in the nanofilm (NF) and in the nanowire (NW) using the GK-type (phonon hydrodynamics) equation describing the D’Arcy– Brinkman phonons flow in the porous medium −λ∇T = q −

μh τR 2 ∇ q = q − τB2 ∇ 2 q. ρh

The heat flow is in the x direction. If the heat flow is zero at the boundaries, the flux profile in NF is  ⎤ ⎡ r cosh ⎥ ⎢ l  B ⎥ q = −λ∇T ⎢ ⎣1 − L ⎦ cosh lB where r ∈ [0, L/2] is the distance from the centreline of the nanofilm of thickness L. The heat flux profile in NW is ⎤ ir J0 ⎢ l ⎥  B⎥ q = −λ∇T ⎢ 1 − ⎣ R ⎦ J0 lB ⎡



where R is the radius of NW and J0 is the Bessel function.  The effective thermal conductivity (an average over the structure thickness) λeff = qdy/(−∇T ) [41]

266

NF λeff

11 Some Exact Solutions

   1 , = λ0 1 − 2Br tanh 2Br



⎞ i ⎜ 2Br ⎟  ⎟ = λ0 ⎜ ⎠. ⎝1 − 4Br i i J0 2Br ⎛

J1

NW λeff

Zhukovsky [42] (see also Zhukovski et al. [43]) derived the exact solution to the one-dimensional Guyer–Krumhansl equation (krum) for the temperature written as 

∂2 ∂3 ∂ − δ + ε ∂t 2 ∂t ∂t x 2



  ∂2 2 T (x, t) = α 2 + λ T (x, t). ∂x

Zhukovsky rewrote this equations using notation τ = 1/ε, μ = κ 2 /ε and λT = α/ε—Fourier-type thermal conductivity, λb = δ/ε—ballistic-type conductivity as  2    ∂ ∂3 ∂3 ∂ τ 2+ T (x, t) = λb + λ + μ T (x, t) T ∂t ∂t ∂t x 2 ∂x3 and noted that this equation is a special case of more general equation 

∂2 ∂ +ε 2 ∂t ∂t



ˆ T (x, t) = D(x)T (x, t),

where Dˆ is the differential operator acting on the coordinate x. To solve this equation Laplace transform is used, and the particular solution is written as √ tε 1 2 ˆ T (x, t) = Ce− 2 e− 2 ε +4 D . Zhukovsky used the operational approach to solve the problem and claimed that in some regimes the solution could be negative and that the maximum principle could be violated. However, this conclusion was not confirmed in the recent paper by Kovács [44]. The initial and boundary conditions were chosen as in the laser flash experiment. The solution was obtained for two intervals 0 < t < τ and t > τ (τ is the duration of the laser impulse) in the form of an infinite sum. No negative temperature domains were detected. Kovács used the GK constitutive equation written in the terms of the heat flux τq

∂ 2q ∂q ∂T + q + τ −λ 2 =0 ∂t ∂x ∂x

τq

∂ 2q ∂ 2q ∂ 3q ∂q = + + λ2 . 2 2 ∂x ∂t ∂x ∂t x 2

and the energy equation

11.3 Fractional Models

267

The heat flux boundary condition at the front side of the sample is formulated as   ⎧ ⎨ 1 − cos 2π t τ q(0, t) = q0 (t) = ⎩ 0 if t > τ

if

0 < t ≤ τ ,

and the condition on the rear side q(L , t) = qL (t) = 0; the initial condition is T (x, 0) = 0.

11.3 Fractional Models Kulish and Lage [2] developed the fractional equation applying the inverse Laplace transformation to Eq. (11.4) and finally obtained the local relation between the heat flux and the temperature  1/2  ∂ T (x, t) T0 λ − q(x, t) = 1/2 κ ∂t 1/2 (π t)1/2 where ∂ 1/2 /∂t 1/2 is the Riemann–Liouville derivative. Oane et al. [45] used the Zhukovsky approach (named by the authors the “Fourier– Zhukovsky model”) to study the heat transfer in the Au nanoparticles—the onedimensional and two-dimensional lattices—under the intense nanoseconds laser irradiation pulses and remarked that the Zhukovsky approach is closely related to the integral transform technique [46]. Hristov [47] used the heat-balance integral method (HBIM) to solve the fractional (“half-time”) heat conduction equation √ ∂ T (x, t) ∂ 1/2 T (x, t) =− κ ∂t 1/2 ∂x also called the Dirac’s equation [47]. The method is based on the following energy balance (δ is the thermal penetration depth): 

δ  1/2 √ ∂ T (x, t) dx = − κ(T |x=0 − T |x=δ ). 1/2 ∂t 0

Li et al. [48] considered the two-dimensional time-fractional homogeneous heat conduction equation   2 ∂ T ∂2 T ∂αT , (11.9) =D + ∂t α ∂x2 ∂ y2 t > 0, 0 < x < a, 0 < y < b, 0 < α ≤ 1.

268

11 Some Exact Solutions

The authors applied to Eq. (11.9) the fractional complex transform ξ=

qt α + px + ky (1 + α)

to obtain D( p 2 + k 2 )

∂2 T ∂T = 0. −q ∂ξ 2 ∂ξ

(11.10)

Solution of Eq. (11.10) is written as [48]  T (ξ ) = c1 + c2 exp

qξ D( p 2 + k 2 )



or  T (x, y, t) = c1 + c2 exp

qky q 2t α qpx + + D( p 2 + k 2 ) D( p 2 + k 2 ) D( p 2 + k 2 )(1 + α)



where c1 and c2 are constants. Zhang et al. [49] derived the solution of the one-dimensional heat condition governed by the equation ∂ 2α T ∂αT = κ , 0