Bridging scales in modeling and simulation of thermal transport processes

Abstract

The vastly disparate length and time scales existing in new devices and materials born out of nanotechnology have made thermal modeling and simulation more important and more difficult. The experimental thermal characterization of such systems, e.g. modern computer processors, can be prohibitively difficult or expensive making numerical simulation the only route to effective technology design. However, obtaining solutions that account for small scales, but are still computationally feasible, requires innovative modeling approaches. The research contained herein represents three independent contributions to the understanding of the modeling of thermal transport processes in systems with nano-sized features. At their common core, all contributions in this thesis are rooted in transport theory--the solution or approximation of the Boltzmann equation (BE)--to statistically describe a system made up of a great many energy-carrying particles. The work roughly divides into the three modes of heat transfer--convection, conduction, and radiation. First, a framework for the discretization of the BE (in its many forms) based on lattices is presented. The widely-used lattice Boltzmann method for the simulation of fluid flow is shown to be a sub-case. The framework gives a new rigorous foundation to the use of lattice methods which have emerged in recent years with applications ranging from Brownian motion to astrophysical radiation. Second, we give a thorough presentation of recently proposed models of heat conduction derived from the phonon BE which provides rigor and insight into the different approaches. Most notably, the "new heat equation" is derived directly from the phonon BE for the first time along with a novel boundary condition. The result is shown to give excellent agreement with the more detailed description provided by the equation of phonon radiative transport. Last, we provide the radiative characterization of a nano-porous material using Maxwell's equations in order to recover coefficients to the linear BE governing thermal radiative transfer.