Generalization of Fourier’s Law into Viscous Heat Equations

Abstract

Heat conduction in dielectric crystals originates from the propagation of atomic vibrational waves, whose microscopic dynamics is well described by linearized or generalized phonon Boltzmann transport. Recently, it was shown that the thermal conductivity can be resolved exactly and in a closed form as a sum over relaxons, i.e. the collective phonon excitations that are eigenvectors of Boltzmann equation's scattering matrix [Cepellotti and Marzari, Phys. Rev. X 6, 041013 (2016)]. Relaxons have a well-defined parity and only odd relaxons contribute to the thermal conductivity. Here, we show that the complementary set of even relaxons determines another quantity -the thermal viscosity -that enters into the description of heat transport in the hydrodynamic regime, where dissipation of crystal momentum by Umklapp scattering phases out. We also show how the thermal viscosity and conductivity parametrize two novel viscous heat equations -two coupled equations for the local temperature and drift velocity fields -which represent the thermal counterpart of the Navier-Stokes equations of hydrodynamics in the linear, laminar regime. These viscous heat equations are derived from a coarse-graining of the linearized Boltzmann transport equation for phonons, and encompass both limits of Fourier's law or second sound for strong or weak Umklapp dissipation, respectively. Last, we introduce the Fourier deviation number, a dimensionless parameter that quantifies the steady-state deviations from Fourier's law due to hydrodynamic effects. We showcase these findings in a test case of a complex-shaped device made of silicon or diamond. This formulation generalizes rigorously Fourier's heat equation, and extends the reach of microscopic computational techniques to characterize the fundamental parameters governing heat conduction. * michele.simoncelli@epfl.ch ics. Sussmann and Thellung [12], starting from the linearized BTE (LBTE) in absence of momentumdissipating (Umklapp) phonon-phonon scattering events, derived mesoscopic equations in terms of temperature and phonon drift velocity, the thermal counterpart of pressure and fluid velocity in liquids. Further advances came from Gurzhi [13,14] and Guyer & Krumhansl [15,16] who, including the effect of weak crystal momentum dissipation, obtained equations for damped second sound and Poiseuille heat flow. Among early works, we also mention the discussions on phonon hydrodynamics using approaches different from the LBTE of Refs. [17,18]. While correctly capturing the qualitative features of phonon hydrodynamics, all the theoretical investigations mentioned above are heuristic, e.g. they assume simplified phonon dispersion relations (either power-law [13,14] or linear isotropic [12,15,16]), or neglect momentum dissipation. A more rigorous and general formulation -albeit valid only in the hydrodynamic regime of weak Umklapp scattering -was introduced by Hardy, who extended the discussion of second sound [19] and, together with Albers, of Poiseuille flow in terms of mesoscopic transport equations...

Observation of second sound in a rapidly varying temperature field in Ge

“…Second sound in solids was first experimentally observed in solid He ( 6 ); later in NaF ( 7 ), Bi ( 8 ), and SrTiO 3 ( 9 ); and most recently in highly oriented pyrolytic graphite ( 10 ). Several theoretical works have also recently addressed its occurrence in low-dimensional systems ( 11 – 13 ). In all these experimental observations of second sound, the dominance of momentum conserving phonon scattering (Normal processes) with respect to resistive phonon scattering (Umklapp processes) was found to be the key mechanism leading to its observation.…”

Heat vortex in hydrodynamic phonon transport of two-dimensional materials

“…It is well known that the transport of the heat carriers including phonon, electron and gas, can be described by the Boltzmann transport equation with similar mathematical form, although their physical meanings are different [1,2]. In addition, as normal scattering dominates the heat conduction, the corresponding macroscopic Guyer-Krumhansl (G-K) equation [14,22,30,[45][46][47] is similar to the Navier-Stokes equation in fluid dynamics [36,38]. Hence, will phonon transport like the vortexes in fluid dynamics [36][37][38] or viscous electron flow in hall bar geometries [48][49][50][51]?…”

Ultrafast Thermo-Optical Dynamics of a Single Metal Nano-Object

“…[7][8][9][10] Second, Fourier law of heat conduction, which assumes a diffusive motion of heat carriers, ceases to be applicable to describe heat propagation over distances smaller than their characteristic mean free paths (ballistic regime) 11,12 and in situations where the causality relation between the temperature gradient and heat flux becomes manifest (hydrodynamic regime). 13,14 A lot of the current understanding of nanoscale heat transfer has been obtained using noncontact optical pump-probe techniques. The main idea of this approach consists in investigating the dissipation of the energy injected by a pump light beam in a sample by monitoring the induced modifications of the sample optical properties, e.g.…”