Heat Conduction Law in Dielectric Nanosystems Based on the Thermomass Theory

Abstract

In this paper a general heat conduction law has been proposed based on the thermomass theory, which can be derived from the Boltzmann equation for phonons in dielectrics. The Boltzmann equation for phonons gives a balance between the drift and friction parts of the distribution function. When the normal scattering term is omitted in the friction part, the Fourier’s law and Cattaneo-Vernotte thermal wave equation can be obtained by the zeroth order and first order approximations of the drift term, respectively. A second order approximation of the drift part lead to the thermomass theory based general heat conduction law, which is a nonlocal damped heat wave equation and consists of driving, inertial and resistant forces for phonon gas motion. In nanosystems, the normal scattering term of the friction part reflecting the boundary effect is required and induces a Laplacian term in governing equations by solving the phonon Boltzmann equation. A general law containing the viscosity of the thermomass fluid is obtained in analogy with the Brinkman extension in porous hydrodynamics. The general law is then applied to investigate the effective thermal conductivity of a nanosystem, which covers the effect of the ultra-high heat flux and the boundary confinement and scattering. The present research not only presents a powerful heat conduction law for heat transport in nanosystems, but also bridges the microscopic Boltzmann transport equation and the macroscopic gas dynamics of phonons in terms of the thermomass theory.