Transport regimes in quasiballistic heat conduction

Transient grating (TG) spectroscopy is an important experimental technique to measure mean-free-path (MFP) spectra using observations of quasiballistic heat conduction. To obtain MFP spectra, the measurements must be interpreted within the framework of the frequency-dependent Boltzmann transport equation (BTE), but previous solutions have restricted validity due to simplifying assumptions. Here we analyze heat conduction in TG using a new analytical solution of the frequency-dependent BTE that accurately describes thermal transport from the diffusive to ballistic regimes. We demonstrate that our result enables a more accurate measurement of MFP spectra and thus will lead to an improved understanding of heat conduction in solids.

Figure 1

Temperature decay curves $\Delta \tilde{T}$ in the (a) diffusive limit, (b) ballistic limit, (c) weakly quasiballistic regime, and (d) strongly quasiballistic regime. The BTE solutions are given by the solid lines, the Fourier solution by the dashed lines, the ballistic conduction solution by the dotted line, and the modified Fourier solution by the dash-dotted lines.

Figure 2

Spectral thermal conductance ${\sigma }_{\omega }$ in (a) the weakly quasiballistic regime and (b) the strongly quasiballistic regime. The BTE solutions are given by the solid lines, the Fourier solution by the dashed lines, and the weak BTE solution by the dash-dotted lines.

Figure 3

Example MFP reconstructions for silicon at (a) 100 K and (b) 500 K and (c) PbSe at 300 K using numerically simulated data. Plotted are the analytical MFP distribution (solid line), the numerical apparent thermal conductivities (squares), and the reconstructed MFP distributions by the general suppression function (triangles) and by the weak suppression function (circles). The $x$ axis corresponds to the MFP for the distributions and to the grating wavelength for the thermal conductivity data.