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Multi-Scale Multi-Species Modeling for Plasma Devices

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Multi-Scale Multi-Species Modeling for Plasma Devices University of California Los Angeles Multi-Scale Multi-Species Modeling for Plasma Devices A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Aerospace Engineering by Samuel Jun Araki 2014 c Copyright by Samuel Jun Araki 2014 Abstract of the Dissertation Multi-Scale Multi-Species Modeling for Plasma Devices by Samuel Jun Araki Doctor of Philosophy in Aerospace Engineering University of California, Los Angeles, 2014 Professor Richard E. Wirz, Chair This dissertation describes three computational models developed to simulate important as- pects of low-temperature plasma devices, most notably ring-cusp ion discharges and thrusters. The main findings of this dissertation are related to (1) the mechanisms of cusp confinement for micro-scale plasmas, (2) the implementation and merits of magnetic field aligned meshes, and (3) an improved method for describing heavy species interactions. The Single Cusp (SC) model focuses on the near-cusp region of the discharge chamber to investigate the near surface cusp confinement of a micro-scale plasma. The model employs the multi-species iterative Monte Carlo method and uses various advanced methods such as electric field calculation and particle weighting algorithm that are compatible with a non- uniform mesh in cylindrical coordinates. Three different plasma conditions are simulated with the SC model, including an electron plasma, a sparse plasma, and a weakly ionized plasma. It is found that the scaling of plasma loss to the cusp for a sparse plasma can be similar to that for a weakly ionized plasma, while the loss mechanism is significantly different; the primary electrons strongly influence the loss structure of the sparse plasma. The model is also used, along with experimental results, to describe the importance of the local magnetic field on the primary electron loss behavior at the cusp. Many components of the 2D/3D hybrid fluid/particle model (DC-ION) are improved from the original version. The DC-ION code looks at the macroscopic structure of the discharge ii plasma and can be used to address the design and optimization challenges of miniature to micro discharges on the order of 3 cm to 1 cm in diameter. Among the work done for DC-ION, detailed steps for the magnetic field aligned (MFA) mesh are provided. Solving the plasma diffusion equation in the ring-cusp configuration, the benefit of the MFA mesh has been fully investigated by comparing the solution with a uniform mesh. It is found that the MFA mesh can still produce a relatively large error due to the misalignment at the domain boundary but still provides a significant improvement in the bulk plasma region. The mesh generation routine can be further improved by enhancing the smoothness of the near-boundary grid elements. The Ion Beam (IB) model is similar to the SC model but primarily focuses on heavy species. The model implements detailed calculation of the heavy species collisions, solving the classical scattering equation with higher order interaction potentials for different collision pairs. A parametric study has been conducted, and the simulation results have shown very good agreement with the experimental results when using an appropriate value for the secondary electron yield. Further study on the elastic collision has shown that the initial atom velocity should not be neglected in order to accurately compute the post-collision CEX ion velocity. The knowledge gained has led to an improvement on the current method by defining an effective elastic collision cross-section and significantly reducing the frequency of the collision calculation. The same method can be applied to collisions between fast and slow atoms. iii The dissertation of Samuel Jun Araki is approved. Troy A. Carter Jeff D. Eldredge Xiaolin Zhong Richard E. Wirz, Committee Chair University of California, Los Angeles 2014 iv To my mother, for her inspiration. v Table of Contents List of Figures :::::::::::::::::::::::::::::::::::::: xi List of Tables ::::::::::::::::::::::::::::::::::::::: xxiii 1 Introduction :::::::::::::::::::::::::::::::::::::: 1 1.1 Electric Propulsion . .1 1.2 Ion Thruster . .3 1.2.1 Historical Background . .3 1.2.2 Types of Ion Thrusters . .4 1.2.3 Discharge Chamber . .5 1.2.4 Extraction Grids . .6 1.2.5 Hollow Cathode . .7 1.3 Survey of Numerical Models for a DC Cusped Ion Thruster . .8 1.3.1 Discharge Chamber Models . .8 1.3.2 Ion Optics Models . 11 1.3.3 Plume Models . 15 1.4 Organization of Dissertation . 18 2 Numerical Investigation of Near-Surface Cusp Confinement of Micro-Scale Plasma ::::::::::::::::::::::::::::::::::::::::::: 20 2.1 Introduction . 21 2.2 Survey of Literature . 22 2.2.1 Experimental and Theoretical Studies on Leak Width . 22 2.2.2 Computational Models for Cusp Confinement Devices . 26 vi 2.3 Description of Single Cusp Model . 28 2.3.1 Overview . 28 2.3.2 Analytical Equations for Magnetic Fields . 29 2.3.3 Particle Tracking Method . 31 2.3.4 Particle Weighting Algorithm . 34 2.3.5 Collisions . 41 2.3.6 Potential Solver . 50 2.3.7 Electric Field Solver for Non-Uniform Mesh . 56 2.3.8 Mixing and Smoothing . 62 2.3.9 Convergence . 64 2.4 Component Level Validations . 65 2.4.1 Particle Weighting in Cylindrical Coordinates . 65 2.4.2 Potential Calculation . 73 2.4.3 Electric Field Calculation . 76 2.5 Simulation Results . 78 2.5.1 Primary Electron Confinement . 79 2.5.2 Sparse Plasma . 81 2.5.3 Weakly Ionized Plasma . 97 2.6 Chapter Summary . 105 3 Techniques for a Ring-Cusp Discharge Chamber Modeling for a Micro- Discharge Design ::::::::::::::::::::::::::::::::::::: 108 3.1 General Description of a Ring-Cusp Discharge Model . 109 3.2 Magnetic Field Aligned Mesh . 110 3.2.1 Introduction . 110 vii 3.2.2 Process of MFA Mesh Generation . 112 3.2.3 MFA Meshes for Various Field Configurations . 125 3.3 Numerical Method for Solving Plasma Diffusion Equation in a MFA Mesh . 127 3.3.1 Introduction . 127 3.3.2 Plasma Diffusion Equation . 127 3.3.3 Numerical Method . 129 3.3.4 Validation in a Simple Configuration . 139 3.3.5 Comparison of Errors in MFA and Uniform Mesh . 141 3.4 Improved View Factor Method for Neutral Atom Density . 152 3.5 Chapter Summary . 156 4 Heavy Species Collision Model for Ion Beam Experiment and Electric Propulsion Devices ::::::::::::::::::::::::::::::::::: 158 4.1 Introduction . 158 4.2 Previous Models with Ion-Atom Elastic Collisions . 160 4.3 Computational Model Description . 161 4.3.1 Overview . 161 4.3.2 Collision Dynamics . 163 4.3.3 Xe+ + Xe Interaction Potential . 165 4.3.4 Deflection Function . 167 4.3.5 Charge-Exchange Probability . 171 4.3.6 Differential Cross-Section . 174 4.3.7 Xe + Xe Collisions . 176 4.3.8 Secondary Electron Emission . 180 4.4 Analytical Model for Exit Orifice Current . 188 viii 4.5 Results and Discussion . 189 4.5.1 Electrode Currents . 189 4.5.2 Current Density Profiles along Electrodes . 194 4.5.3 Importance of Initial Target Particle Velocity . 198 4.5.4 Improved Collision Calculation Method . 201 4.6 Chapter Summary . 206 5 Conclusions and Future Work ::::::::::::::::::::::::::: 208 5.1 Single Cusp Model . 208 5.2 Discharge Model . 210 5.3 Ion Beam Model . 211 A Useful Geometric Algorithms ::::::::::::::::::::::::::: 213 A.1 Point inside a Convex Polygon . 213 A.2 Intersection of Two Lines . 214 A.3 Intersection of a Line with a Surface . 216 A.3.1 Annulus . 216 A.3.2 Shell . 217 A.3.3 Truncated Cone . 218 B Data Structure for Computational Mesh :::::::::::::::::::: 219 B.1 Data Structure . 219 B.2 Numbering System . 222 B.3 Connectivity . 222 C Magnetic Field Equations ::::::::::::::::::::::::::::: 224 C.1 Magnetic Dipole . 224 ix C.2 Axially Magnetized Cylindrical Magnet . 225 C.3 Block Magnet . 226 C.4 Axially Magnetized Ring Magnet . 227 C.5 Radially Magnetized Ring Magnet . 228 C.6 Current Loop . 229 C.7 Evaluation of Elliptic Integrals . 229 C.7.1 Fukushima's Fast Computation Method . 230 C.7.2 Carlson's Method . 232 C.8 Transformation between Inertial and Magnet Reference Frame . 232 D Supplemental Details for Particle Simulations :::::::::::::::: 234 D.1 Particle Loading and Injection . 234 D.2 Determination of Post-Collision Velocity . 236 D.3 Primary Electron Equilibration Rate Constant . 238 References ::::::::::::::::::::::::::::::::::::::::: 239 x List of Figures 1.1 Schematic of DC electron bombardment ring-cusp ion thruster (Redrawn from Ref. [13]). .5 2.1 Simplified flowchart of the SC model. 28 2.2 Mapping of a triangular element in natural coordinates. 36 2.3 Mapping of a quadrilateral element in natural coordinates..
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