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Ballistic phonon effects, arising on length scales comparable to themean-free-path, result in non-diffusive heat flow and alter the thermalproperties of materials. Simple theoretical models that accurately capturenon-diffusive transport physics are valuable for experimental analysis,technology design, and providing physical insight. In this work, we utilize andextend the McKelvey-Shockley (McK-S) flux method, a simple and accurateframework, to investigate ballistic effects in transient phonon transportsubmitted to a spatially sinusoidal heating profile, simulating a transientthermal grating. We begin by extending a previous McK-S formulation to includeinelastic scattering, then obtain an analytical solution in the single phononenergy case (gray approximation), and after show how this approach can readilysupport a full phonon dispersion and mean-free-path distribution. The resultsagree with experimental data and compare very well to solutions of the phononBoltzmann transport equation in the diffusive and weakly quasi-ballistictransport regimes. We discuss the role of ballistic and non-equilibriumphysics, and show that inelastic scattering is key to retrieving the heatequation solution in the diffusive limit. Overall the McK-S flux method, whichtakes the form of a diffusion-like equation, proves to be a simple and accurateframework that is applicable from the ballistic to diffusive transport regime.

Modeling quasi-ballistic transient thermal transport with spatially sinusoidal heating: a McKelvey-Shockley flux approach