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

Abstract

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. Comparing 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 and ribbon size on the second sound propagation is also explored. The speed of second sound in the observation window is found to be at most 20 percentages smaller than the theoretical value in hydrodynamic limit due to the finite umklapp, isotope and edge resistive scattering. The present study will contribute to not only the solution methodology of phonon Boltzmann equation, but also the physics of transient hydrodynamic phonon transport as guidance for future experimental detection.

Heat vortex in hydrodynamic phonon transport of two-dimensional materials

“…In order to predict the thermal transport in two-dimensional materials correctly [7,23,24,28], the phonon Boltzmann transport equation (BTE) under the Callaway's dual relaxation model [34,[52][53][54] is used, i.e., ∂f ∂t…”

“…where D = |K|/ (2π|v|) is the phonon density of states [53,54]. Assuming a small temperature difference ∆T and a small drift velocity, i.e., ∆T /T 0 1, K • u ω, where T 0 is the reference temperature, then the equilibrium distribution functions (Eqs.…”

“…In addition, boundary conditions also play an indispensable role in numerical simulations [54,61]. For the solution of the phonon BTE, the diffusely reflecting boundary condition, periodic boundary condition and isothermal boundary condition are presented as follows, 1.…”

Size effect on phonon hydrodynamics in graphite microstructures and nanostructures

“…Callaway's dual relaxation model [21] represents a good approximation to the full scattering term in the phonon Boltzmann equation [10,22] and has been widely adopted in analyzing heat transport in the hydrodynamic regime by analytical or semi-analytical methods [5,[23][24][25][26][27][28][29]. The direct solution of the phonon Boltzmann equation under Callaway's model has been advanced recently by a few numerical schemes including both deterministic methods [30,31] and stochastic ones [32,33]. However, the deterministic numerical method [30,31] was designed for heat transport in 2D graphene ribbons with empirical isotropic phonon properties.…”

“…The direct solution of the phonon Boltzmann equation under Callaway's model has been advanced recently by a few numerical schemes including both deterministic methods [30,31] and stochastic ones [32,33]. However, the deterministic numerical method [30,31] was designed for heat transport in 2D graphene ribbons with empirical isotropic phonon properties. The gray Monte Carlo simulation [32,33] was conducted in hypothetical graphitic materials due to the lack of knowledge of normal and Umklapp scattering rates and the pending development of the methodology.…”