A Simple Boltzmann Transport Equation for Ballistic to Diffusive Transient Heat Transport

TL;DR

This paper introduces a simplified Boltzmann transport equation approach that accurately models heat transfer from ballistic to diffusive regimes, capturing transient effects and phonon dynamics with ease.

Contribution

It presents a new simplified model for transient heat transport that naturally incorporates ballistic phonon effects and finite-velocity propagation, aligning well with detailed phonon BTE results.

Findings

Fundamental temperature jumps depend on ballistic thermal resistance.

Phonon transport approaches ballistic limit at early times.

Ballistic effects cause apparent reductions in heat conduction due to lower temperature gradients.

Abstract

Developing simplified, but accurate, theoretical approaches to treat heat transport on all length and time scales is needed to further enable scientific insight and technology innovation. Using a simplified form of the Boltzmann transport equation (BTE), originally developed for electron transport, we demonstrate how ballistic phonon effects and finite-velocity propagation are easily and naturally captured. We show how this approach compares well to the phonon BTE, and readily handles a full phonon dispersion and energy-dependent mean-free-path. This study of transient heat transport shows i) how fundamental temperature jumps at the contacts depend simply on the ballistic thermal resistance, ii) that phonon transport at early times approach the ballistic limit in samples of any length, and iii) perceived reductions in heat conduction, when ballistic effects are present, originate from reductions in temperature gradient. Importantly, this framework can be recast exactly as the Cattaneo and hyperbolic heat equations, and we discuss how the key to capturing ballistic heat effects is to use the correct physical boundary conditions.