The lattice Boltzmann Peierls Callaway equation for mesoscopic thermal transport modeling

Abstract

The lattice Boltzmann Peierls Callaway (LBPC) method is a recent development of the versatile lattice Boltzmann formalism aimed at a numerical experiment on mesoscale thermal transport in a multiphase phonon gas. Two aspects of mesoscopic thermal transport are discussed: the finite phonon mean free path and the interface thermal resistance. Based on the phonon momentum screening length measured in the LBPC computational apparatus, the validity of the Umklapp collision relaxation time in the Callaway collision operator is examined quantitatively. The discrete nature of the spatio-temporal domain in the LBPC method, along with the linear approximation of the exponential screening mechanism in the Callaway operator, reveals a large discrepancy between the effective phonon mean free path and the analytic phonon mean free path when the relaxation time is small. The link bounce back interface phonon collision rule is used to realize the interface thermal resistance between phonon gases with dissimilar dispersion relations. Consistent with the Callaway collision operator for the bulk phonon dynamics, the interface phonon collision process is regarded as a linear relaxation mechanism toward the local pseudo-equilibrium phonon distribution uniquely defined by the energy conservation principle. The interface thermal resistance is linearly proportional to the relaxation time of the proposed phonon interface collision rule.