Direct simulation of second sound in graphene by solving the phonon Boltzmann equation via a multiscale scheme

The direct simulation of the dynamics of second sound in graphitic materials remains a challenging task due to lack of methodology for solving the phonon Boltzmann equation in such a stiff hydrodynamic regime. In this work we aim to tackle this challenge by developing a multiscale numerical scheme for the transient phonon Boltzmann equation under Callaway's dual relaxation model which captures well the collective phonon kinetics. Compared to traditional numerical methods, the present multiscale scheme is efficient, accurate, and stable in all transport regimes attributed to avoiding the use of time and spatial steps smaller than the relaxation time and mean free path of phonons. The formation, propagation, and composition of ballistic pulses and second sound in graphene ribbon in two classical paradigms for experimental detection are investigated via the multiscale scheme. The second sound is declared to be mainly contributed by ZA phonon modes, whereas the ballistic pulses are mainly contributed by LA and TA phonon modes. The influence of temperature, isotope abundance, ribbon size, and structure asymmetry on the second sound propagation is also explored. The speed of second sound in the observation window is found to be at most 20% smaller than the theoretical value in the hydrodynamic limit due to the finite umklapp, isotope, and edge resistive scattering. The present study will contribute to not only the solution methodology of the phonon Boltzmann equation, but also the physics of transient hydrodynamic phonon transport as guidance for future experimental detection.