On the Equations and Boundary Conditions Governing Phonon-Mediated Heat Transfer in the Small Mean Free Path Limit: An Asymptotic Solution of the Boltzmann Equation

Abstract

Using an asymptotic solution procedure, we construct solutions of the Boltzmann transport equation in the relaxation-time approximation in the limit of small Knudsen number, Kn << 1, to obtain continuum equations and boundary conditions governing phonon-mediated heat transfer in this limit. Our results show that, in the bulk, heat transfer is governed by the Fourier law of heat conduction, as expected. However, this description does not hold within distances on the order of a few mean free paths from the boundary; fortunately, this deviation from Fourier behavior can be captured by a universal boundary-layer solution of the Boltzmann equation that depends only on the material model and the phonon-boundary interaction model (Boltzmann boundary condition). Boundary conditions for the Fourier description follow from matching this inner solution to the outer (Fourier) solution. This procedure shows that the traditional no-jump boundary conditions are appropriate only to zeroth order in Kn. Solution to first order in Kn shows that the Fourier law needs to be complemented by jump boundary conditions with jump coefficients that depend on the material model and the phonon-boundary interaction model. In this work, we calculate these coefficients and the form of the jump conditions for an adiabatic-diffuse and a prescribed-temperature boundary in contact with a constant-relaxation-time material. Extension of this work to variable relaxation-time models is straightforward and will be discussed elsewhere. Our results are validated via comparisons with low-variance deviational Monte Carlo simulations.