Efficient Multiscale Methods for Micro/Nanoscale Solid State Heat Transfer
In this thesis, we develop methods for solving the linearized Boltzmann transport equation (BTE) in the relaxation-time approximation for describing small-scale solidstate heat transfer. We first discuss a Monte Carlo (MC) solution method that builds upon the deviational energy-based Monte Carlo method presented in [J.-P. Péraud and N.G. Hadjiconstantinou, Physical Review B, 84(20), p. 205331, 2011]. By linearizing the deviational Boltzmann equation we formulate a kinetic-type algorithm in which each computational particle is treated independently; this feature is shown to be consequence of the energy-based formulation and the linearity of the governing equation and results in an “event-driven” algorithm that requires no time discretization. In addition to a much simpler and more accurate algorithm (no time discretization error), this formulation leads to considerable speedup and memory savings, as well as the ability to efficiently treat materials with wide ranges of phonon relaxation times, such as silicon. A second, complementary, simulation method developed in this thesis is based on the adjoint formulation of the linearized BTE, also derived here. The adjoint formulation describes the dynamics of phonons travelling backward in time, that is, being emitted from the “detectors” and detected by the “sources” of the original problem. By switching the detector with the source in cases where the former is small, that is when high accuracy is needed in small regions of phase-space, the adjoint formulation provides significant computational savings and in some cases makes previously intractable problems possible. We also develop an asymptotic theory for solving the BTE at small Knudsen numbers, namely at scales where Monte Carlo methods or other existing computational methods become inefficient. The asymptotic approach, which is based on a Hilbert expansion of the distribution function, shows that the macroscopic equation governing heat transport for non-zero but small Knudsen numbers is the heat equation, albeit supplemented with jump-type boundary conditions. Specifically, we show that the traditional no-jump boundary condition is only applicable in the macroscopic limit where the Knudsen number approaches zero. Kinetic effects, always present at the boundaries, become increasingly important as the Knudsen number increases, and manifest themselves in the form of temperature jumps that enter as boundary conditions to the heat equation, as well as local corrections in the form of kinetic boundary layers that need to be superposed to the heat equation solution. We present techniques for efficiently calculating the associated jump coefficients and boundary layers for different material models when analytical results are not available. All results are validated using deviational Monte Carlo methods primarily developed in this thesis. We finally demonstrate that the asymptotic solution method developed here can be used for calculating the Kapitza conductance (and temperature jump) associated with the interface between materials.