Modeling ballistic effects in frequency-dependent transient thermal transport using diffusion equations

TL;DR

This paper demonstrates that classical diffusion equations, with proper boundary conditions, can accurately model ballistic phonon effects in frequency-dependent thermal transport, aligning well with detailed phonon Boltzmann results.

Contribution

It shows how hyperbolic heat and Cattaneo equations can be used to model ballistic phonon effects, providing a simple, transparent approach that matches rigorous numerical solutions.

Findings

Diffusion equations can model ballistic effects accurately with correct boundary conditions.

Analytical solutions agree well with phonon Boltzmann transport simulations.

The approach can incorporate detailed material properties like phonon dispersion.

Abstract

Understanding ballistic phonon transport effects in transient thermoreflectance experiments and explaining the observed deviations from classical theory remains a challenge. Diffusion equations are simple and computationally efficient but are widely believed to break down when the characteristic length scale is similar or less than the phonon mean-free-path. Building on our prior work, we demonstrate how well-known diffusion equations, namely the hyperbolic heat equation and the Cattaneo equation, can be used to model ballistic phonon effects in frequency-dependent periodic steady-state thermal transport. Our analytical solutions are found to compare excellently to rigorous numerical results of the phonon Boltzmann transport equation. The correct physical boundary conditions can be different from those traditionally used and are paramount for accurately capturing ballistic effects. To illustrate the technique, we consider a simple model problem using two different, commonly-used heating conditions. We demonstrate how this framework can easily handle detailed material properties, by considering the case of bulk silicon using a full phonon dispersion and mean-free-path distribution. This physically transparent approach provides clear insights into the nonequilibrium physics of quasi-ballistic phonon transport and its impact on thermal transport properties.