Sub-micron Lattice Boltzmann Study of Phonon Transport

Heat transport at micro-nanoscales departs substantially from the well established classical laws. The Fourier Law of heat conduction cannot be applied at sub-continuum level due to its inability in modeling non-equilibrium energy transport. Therefore one must resort to a rigorous solution to the Boltzmann Transport Equation (BTE) in the realm of nanoscale transport regime. Some recent studies show that a relatively inexpensive and accurate way to predict the behavior of sub continuum energy transport in solids is via the discrete representation of the BTE referred to as the Lattice Boltzmann method (LBM). Although quite a few numerical simulations involving LBM have been exercised in the literature, there exists an ambiguity over employing the right lattice configurations describing phonon transport. The objective of the present investigation is to evaluate a general lattice structure to be employed in different problems of energy transport involving multiple length and timescales. In order to reduce the modeling complexity, gray model assumption based on Debye approximation is adopted throughout the analysis. The performance of various D2Q9 lattice configurations has been tested involving different weights and further a novel weight distribution is suggested on the basis of their comparison. It has been observed that the choice of proper weight distribution of a particular lattice structure play a crucial role in estimating the local temperature, especially at sub-micron regime.