• Open Access

Generalization of Fourier’s Law into Viscous Heat Equations

Michele Simoncelli, Nicola Marzari, and Andrea Cepellotti
Phys. Rev. X 10, 011019 – Published 28 January 2020

Abstract

Heat conduction in dielectric crystals originates from the dynamics of atomic vibrations, whose evolution is well described by the linearized Boltzmann transport equation for the phonon populations. Recently, it was shown that thermal conductivity can be resolved exactly and in a closed form as a sum over relaxons, i.e., collective phonon excitations that are the eigenvectors of Boltzmann equation’s scattering matrix [A. Cepellotti and N. 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, and is especially relevant in the hydrodynamic regime, where dissipation of crystal momentum by umklapp scattering phases out. We also show how the thermal conductivity and viscosity parametrize two novel viscous heat equations—two coupled equations for the 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 the limit of Fourier’s law and that of second sound, taking place, respectively, in the regime of strong or weak momentum dissipation. Last, we introduce the Fourier deviation number as a descriptor that captures the deviations from Fourier’s law due to hydrodynamic effects. We showcase these findings in a test case of a complex-shaped device made of graphite, obtaining a remarkable agreement with the recent experimental demonstration of hydrodynamic transport in this material, and also suggesting that hydrodynamic behavior can appear at room temperature in micrometer-sized diamond crystals. The present formulation rigorously generalizes Fourier’s heat equation, extending the reach of physical and computational models for heat conduction also to the hydrodynamic regime.

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  • Received 26 June 2019
  • Revised 16 October 2019

DOI:https://doi.org/10.1103/PhysRevX.10.011019

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Michele Simoncelli1,*, Nicola Marzari1, and Andrea Cepellotti2

  • 1Theory and Simulation of Materials (THEOS) and National Centre for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne, Lausanne 1015, Switzerland
  • 2Department of Physics, University of California at Berkeley and Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

  • *michele.simoncelli@epfl.ch

Popular Summary

The celebrated Fourier’s equation for heat conduction was developed in 1822 and nowadays is still used to model heat transfer, such as in heating or refrigeration systems. Despite being accurate at scales larger than a millimeter and at room temperatures or higher, Fourier’s law fails when heat transfer is hydrodynamic (that is, when the heat flow is akin to the flow of a fluid). Here, we overcome this failure by deriving two viscous heat equations that explain hydrodynamic thermal transport. The new insight from these equations deepens our understanding of heat transport and sets the stage for the development of more efficient electronic devices.

The study of the inadequacies of Fourier’s law has an illustrious history, beginning in the 1960s to explain the thermal hydrodynamic phenomena observed in several materials at cryogenic conditions. Recently, it has received renewed interest because of the observation of these phenomena at higher-than-cryogenic temperatures in graphite. We exploit the recently developed microscopic thermal transport theory of relaxons (collective excitations of phonon populations) to define the thermal viscosity and derive a set of mesoscopic viscous heat equations that describe heat conduction in the hydrodynamic, mesoscopic, and diffusive (Fourier) heat transport regimes.

Our formulation dramatically improves the mesoscopic description of heat transport and paves the way for the design of heat management in next-generation electronic and phononic devices.

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Vol. 10, Iss. 1 — January - March 2020

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