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

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.

9 pages, 4 figures

References in corpus (6)

• Direct Measurement of Room Temperature Non-diffusive Thermal Transport Over Micron Distances in a Silicon Membrane
• Superdiffusive heat conduction in semiconductor alloys -- I. Theoretical foundations
• Superdiffusive heat conduction in semiconductor alloys -- II. Truncated Lévy formalism for experimental analysis
• Steady-State Heat Transport: Ballistic-to-Diffusive with Fourier's Law
• Radial Quasiballistic Transport in Time-Domain Thermoreflectance Studied Using Monte Carlo Simulations
• A Simple Boltzmann Transport Equation for Ballistic to Diffusive Transient Heat Transport

Cited by in corpus (6)

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• Spectral analysis of non-equilibrium molecular dynamics: spectral phonon temperature and phonon local non-equilibrium in thin films and across interfaces
• Thermal Transport at the Nanoscale - A Fourier's Law vs. Phonon Boltzmann Equation Study
• Modeling quasi-ballistic transient thermal transport with spatially sinusoidal heating: a McKelvey-Shockley flux approach
• Compact models for multidimensional quasiballistic thermal transport
• Relativistic stable processes in quasi-ballistic heat conduction in thin film semiconductors