On the dual-phase-lag thermal response in the pulsed photoacoustic effect: A theoretical and experimental 1D-approach

In a recent work, assuming a Beer–Lambert optical absorption and a Gaussian laser time profile, it was shown that the exact solutions for a 1D photoacoustic (PA) boundary-value-problem predict a null pressure for optically strong absorbent materials. In order to overcome this inconsistency, a heuristic correction was introduced by assuming that heat flux travels a characteristic length during the duration of the laser pulse [M. Ruiz-Veloz et al., J. Appl. Phys. 130, 025104 (2021)] τp. In this work, we obtained exact analytical solutions in the frequency domain for a 1D boundary-value-problem for the Dual-Phase-Lag (DPL) heat equation coupled with a 1D PA-boundary-value-problem via the acoustic wave equation. Temperature and pressure solutions were studied by assuming that the sample and its surroundings have a similar characteristic thermal lag response time τT; therefore, the whole system is assumed to have a similar thermal relaxation. A second assumption for τT is that it is considered as a free parameter that can be adjusted to reproduce experimental results. Solutions for temperature and pressure were obtained for a one-layer 1D system. It was found that for τT<τp, the DPL temperature has a similar thermal profile of the Fourier heat equation; however, when τT≥τp, this profile is very different from the Fourier case. Additionally, via a numerical Fourier transform, the wave-like behavior of DPL temperature is explored, and it was found that as τT increases, thermal wave amplitude is increasingly attenuated. Exact solutions for pressure were compared with experimental PA signals, showing a close resemblance between both data sets, particularly in time domain, for an appropriated value of τT; the transference function was also calculated, which allowed us to find the maximum response in frequency for the considered experimental setup.