Fully Anisotropic Solution of the Three Dimensional Boltzmann Transport Equation

The development of accurate modeling techniques for nanoscale thermal transport is an active area of research. Modern day nanoscale devices have length scales of tens of nanometers and are prone to overheating, which reduces device performance and lifetime. Therefore, accurate temperature profiles are needed to predict the reliability of nanoscale devices. The majority of models that appear in the literature obtain temperature profiles through the solution of the Boltzmann transport equation (BTE). These models often make simplifying assumptions about the nature of the quantized energy carriers (phonons). Additionally, most previous work has focused on simulation of planar two dimensional structures. This thesis presents a method which captures the full anisotropy of the Brillouin zone within a three dimensional solution to the BTE. The anisotropy of the Brillouin zone is captured by solving the BTE for all vibrational modes allowed by the Born Von-Karman boundary conditions.

POSITIVE FILTERED P_N METHOD FOR LINEAR TRANSPORT EQUATIONS AND THE ASSOCIATED OPTIMIZATION ALGORITHM

We propose a positive, accurate moment closure for linear kinetic transport equations based on a filtered spherical harmonic (FP_N) expansion in the angular variable. The FP_N moment equations are accurate approximations to linear kinetic equations, but they are known to suffer from the occurrence of unphysical, negative particle concentrations. The new positive filtered P_N (FP_N+) closure is developed to address this issue. The FP_N+ closure approximates the kinetic distribution by a spherical harmonic expansion that is non-negative on a finite, predetermined set of quadrature points. With an appropriate numerical PDE solver, the FP_N+ closure generates particle concentrations that are guaranteed to be non-negative. Under an additional, mild regularity assumption, we prove that as the moment order tends to infinity, the FP_N+ approximation converges, in the L2 sense, at the same rate as the FP_N approximation; numerical tests suggest that this assumption may not be necessary. By numerical experiments on the challenging line source benchmark problem, we confirm that the FP_N+ method indeed produces accurate and non-negative solutions. To apply the FP_N+ closure on problems at large temporal-spatial scales, we develop a positive asymptotic preserving (AP) numerical PDE solver. We prove that the propose AP scheme maintains stability and accuracy with standard mesh sizes at large temporal-spatial scales, while, for generic numerical schemes, excessive refinements on temporal-spatial meshes are required. We also show that the proposed scheme preserves positivity of the particle concentration, under some time step restriction. Numerical results confirm that the proposed AP scheme is capable for solving linear transport equations at large temporal-spatial scales, for which a generic scheme could fail. Constrained optimization problems are involved in the formulation of the FP_N+ closure to enforce non-negativity of the FP_N+ approximation on the set of quadrature points. These optimization problems can be written as strictly convex quadratic programs (CQPs) with a large number of inequality constraints. To efficiently solve the CQPs, we propose a constraint-reduced variant of a Mehrotra-predictor-corrector algorithm, with a novel constraint selection rule. We prove that, under appropriate assumptions, the proposed optimization algorithm converges globally to the solution at a locally q-quadratic rate. We test the algorithm on randomly generated problems, and the numerical results indicate that the combination of the proposed algorithm and the constraint selection rule outperforms other compared constraint-reduced algorithms, especially for problems with many more inequality constraints than variables.

Nano-Engineering of Densely Packed Electrochemical Energy Storage Architectures by Atomic Layer Deposition

Nanostructures are highly attractive for future electrical energy storage devices because they enable large surface area and short ion transport time through thin electrode layers for high power devices. Significant enhancement in power density of batteries has been achieved by nano-engineered structures, particularly anode and cathode nanostructures spatially separated far apart by a porous membrane and/or a defined electrolyte region. A self-aligned nanostructured battery fully confined within a single nanopore presents a powerful platform to determine the rate performance and cyclability limits of nanostructured storage devices. Atomic layer deposition (ALD) has enabled us to create and evaluate such structures, comprised of nanotubular electrodes and electrolyte confined within anodic aluminum oxide (AAO) nanopores. The V2O5- V2O5 symmetric nanopore battery displays exceptional power-energy performance and cyclability when tested as a massively parallel device (~2billion/cm2), each with ~1m3 volume (~1fL). Cycled between 0.2V and 1.8V, this full cell has capacity retention of 95% at 5C rate and 46% at 150C, with more than 1000 charge/discharge cycles. These results demonstrate the promise of ultrasmall, self-aligned/regular, densely packed nanobattery structures as a testbed to study ionics and electrodics at the nanoscale with various geometrical modifications and as a building block for high performance energy storage systems[1, 2]. Further increase of full cell output potential is also demonstrated in asymmetric full cell configurations with various low voltage anode materials. The asymmetric full cell nanopore batteries, comprised of V2O5 as cathode and prelithiated SnO2 or anatase phase TiO2 as anode, with integrated nanotubular metal current collectors underneath each nanotubular storage electrode, also enabled by ALD. By controlling the amount of lithium ion prelithiated into SnO2 anode, we can tune full cell output voltage in the range of 0.3V and 3V. This asymmetric nanopore battery array displays exceptional rate performance and cyclability. When cycled between 1V and 3V, it has capacity retention of approximately 73% at 200C rate compared to 1C, with only 2% capacity loss after more than 500 charge/discharge cycles. With increased full cell output potential, the asymmetric V2O5-SnO2 nanopore battery shows significantly improved energy and power density. This configuration presents a more realistic test - through its asymmetric (vs symmetric) configuration – of performance and cyclability in nanoconfined environment. This dissertation covers (1) Ultra small electrochemical storage platform design and fabrication, (2) Electron and ion transport in nanostructured electrodes inside a half cell configuration, (3) Ion transport between anode and cathode in confined nanochannels in symmetric full cells, (4) Scale up energy and power density with geometry optimization and low voltage anode materials in asymmetric full cell configurations. As a supplement, selective growth of ALD to improve graphene conductance will also be discussed[3]. References: 1. Liu, C., et al., (Invited) A Rational Design for Batteries at Nanoscale by Atomic Layer Deposition. ECS Transactions, 2015. 69(7): p. 23-30. 2. Liu, C.Y., et al., An all-in-one nanopore battery array. Nature Nanotechnology, 2014. 9(12): p. 1031-1039. 3. Liu, C., et al., Improving Graphene Conductivity through Selective Atomic Layer Deposition. ECS Transactions, 2015. 69(7): p. 133-138.