Lattice-Boltzmann modeling of phonon hydrodynamics

Abstract

Based on the phonon Boltzmann equation, a lattice-Boltzmann model for phonon hydrodynamics is developed. Both transverse and longitudinal polarized phonons that interact through normal and umklapp processes are considered in the model. The collision term is approximated by the relaxation time model where normal and umklapp processes tend to relax distributions of phonons to their corresponding equilibrium distribution functions-the displaced Planck distribution and the Planck distribution, respectively. A macroscopic phonon thermal wave equation (PTWE), valid for the second-sound mode, is derived through the technique of Chapman-Enskog expansion. Compared to the dual-phase-lag (DPL) -based thermal wave equation, the PTWE has an additional fourth-ordered spatial derivative term. The fundamental difference between the two models is discussed through examining a propagating thermal pulse in a single-phased medium and the transient and steady-state transport phenomena on a two-layered structure subjected to different temperatures at boundaries. Results show that transport phenomena are significantly different between the two models. The behavior exhibited by the DPL model, as thermal wave behavior goes over to diffusive behavior, tau_{T}-->tau_{q} is incompatible with any microscopic phonon propagating mode. Unlike the DPL model, in which tau_{T} only has an effect on the transient phenomena, in the PTWE model tau_{T} shows effects on phenomena at both transient and steady state. With the intrinsic compatibility to the microscopic state, discontinuous quantities, such as a jump of temperature at a boundary or at an interface, can be calculated naturally and straightforwardly with the present lattice-Boltzmann method.