Fully Kinetic Numerical Modeling of a Plasma Thruster by James Joseph Szabo, Jr. B.S., Mechanical Engineering, Cornell University (1991) M.S., Aeronautics and Astronautics, Stanford University (1992) Submitted to the Department of Aeronautics and Astronautics in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Aeronautics and Astronautics at the Massachusetts Institute of Technology February 2001 @ Massachusetts Institute of Technology 2001. All rights reserved. Author ..... A4partn}4q of Aeonautics and IYronautics Jan 29, 2001 Certified by.. ....................... Manuel Martinez-Sanchez Professor, Department of Aeronautics and Astronautics Thesis Supervisor Certified by.. ..... ~-~. ....I ...... leg Batishchev Researeb i_ .t' t Depa/tn t of clear Engineering Certified by.. ....................... .......... ~Iin Molvig Associate 4 rofessor, Department ofi Nucleat nguAering Certified by.. I Jhime Pc-aire Professor, Department of Aeronautics and Astronautics Certified by.. Saul Rappaport . Professor, DApartrlgent of Phygics / Accepted by........................ Wallace Vander Velde MASSACHUSETTS INSTITUTE Chairman, Department Graduate Committee OF TECHNOLOGY SEP 11 2001 AERO LIBRARIES Fully Kinetic Numerical Modeling of a Plasma Thruster by James Joseph Szabo, Jr. Submitted to the Department of Aeronautics and Astronautics on Jan 29, 2001, in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Aeronautics and Astronautics Abstract A Hall effect plasma thruster with conductive acceleration channel walls was numerically modeled using 2D3V Particle-in-Cell (PIC) and Monte-Carlo Collision (MCC) methodolo- gies. Electron, ion, and neutral dynamics were treated kinetically on the electron time scale to study transport, instabilities, and the electron energy distribution function. Axisymmet- ric R-Z coordinates were used with a non-orthogonal variable mesh to account for important small-scale plasma structures and a complex physical geometry. Electric field and sheath structures were treated self-consistently. Conductive channel walls were allowed to float electrically. The simulation included, via MCC, elastic and inelastic electron-neutral colli- sions, ion-neutral scattering and charge exchange collisions, and Coulomb collisions. The latter were also treated through a Langevin (stochastic) differential equation for the particle trajectories in velocity space. Ion-electron recombination was modeled at the boundaries, and neutrals were recycled into the flow. The cathode was modeled indirectly by inject- ing electrons at a rate which preserved quasineutrality. Anomalous diffusion was included through an equivalent scattering frequency. Free space permittivity was increased to allow a coarser grid and longer time-step. A method for changing the ion to electron mass ratio and retrieving physical results was developed and used throughout. Results were compared with theory, experiments. Gradients and anisotropy in electron temperature were observed. Non-Maxwellian electron energy distribution functions were observed. The thruster was numerically redesigned; substantial performance benefits were predicted. Thesis Supervisor: Manuel Martinez-Sanchez Title: Professor, Department of Aeronautics and Astronautics Biographical Note James Szabo graduated from Louis E. Dieruff High School in Allentown, Pennsylvania in June, 1987. That fall, he entered Cornell University with the aid of an Air Force Reserve Officer Training Corps (ROTC) scholarship. While there, he was elected to the Mechanical Engineering honor society, Pi Tau Sigma, and to the oldest Greek letter social organization, The Kappa Alpha Society. In 1991, he was named a Bachelor of Science with Distinction by the Sibley School of Mechanical and Aerospace Engineering. Szabo next studied at Stanford University, from which he was conferred a Master of Science degree in Aeronautics and Astronautics in 1992. That same year, he entered active duty with the United States Air Force. He served four years at Space and Missile Systems Center, Los Angeles Air Force Base, before separating with the rank of Captain. He arrived at MIT in 1996. He is a member of Sigma Xi and the American Institute of Aeronautics and Astronautics (AIAA). He has authored many papers on electric propulsion. Acknowledgments I would first like to thank my advisor, Manuel Martinez Sanchez, for making my tenure at MIT possible. I would next like to thank Oleg Batishchev, whose contributions to the numerical model described herein were invaluable. I would also like to thank Saul Rappaport, Kim Molvig, and Jaime Peraire for their advice and for serving on my thesis committee. I owe special thanks to AFOSR and to Busek Co. for sponsoring my research. I would especially like to thank Vlad Hruby and Jeff Monheiser. I would also like to thank the scientists at The Aerospace Corporation who introduced me to the field of Space Propulsion, Siegfried Janson, Ron Cohen, and Jim Pollard. On a more personal note, I would like to thank Debie, Monster, and Belle for putting up with my incessant ramblings about "the code." Finally, I would like to thank the graduate students with whom I shared an office for several years, Tatsuo Onishi, Paulo Lozano and Darrel Robertson. I wish them all a window and a working air conditioner. This thesis is dedicated to my late grandfather, Frank Szabo, who said a few sunflower seeds would change the surface of Mars. We shall see. "We ought then to regard the present state of the universe as the effect of its anterior state and as the cause of the one which is to follow. Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it - an in- telligence sufficiently vast to submit these data to analysis - it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes. The human mind offers, in the perfection which it has been able to give to astronomy, a feeble idea of this intelligence. Its discoveries in mechanics and geometry, added to that of universal gravity, have enabled it to comprehend in the same analytical expressions the past and future states of the system of the world. Applying the same method to some other objects of its knowledge, it has succeeded in referring to general laws observed phenomena and in foreseeing those which given circumstances ought to produce. All these efforts in the search for truth tend to lead it back continually to the vast intelligence which we have just mentioned, but from which it will always remain infinitely removed." -From "A Philosophical Essay on Probabilities" by Pierre Simon, Marquis de Laplace [27] Contents 1 Introduction 34 1.1 Electric Propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.2 Hall Thrusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.2.1 General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.2.2 Types of Hall Thrusters . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.3 Statement of Technical Problem . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.4 Literature Survey/Brief Summary of Previous Work . . . . . . . . . . . . . 38 1.4.1 1-D Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.4.2 Measurements of the EEDF . . . . . . . . . . . . . . . . . . . . . . . 40 1.4.3 The mini-TAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.5 Thesis Topic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 1.6 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 1.7 Summary of Theory (Chapter 2) . . . . . . . . . . . . . . . . . . . . . . . . 44 1.7.1 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 1.7.2 Equations and Units . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 1.7.3 Mean Free Path Analysis . . . . . . . . . . . . . . . . . . . . . . . . 45 1.7.4 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.7.5 Speeding up Heavy Particles . . . . . . . . . . . . . . . . . . . . . . 48 1.8 Summary of Numerical Method (Chapter 3) . . . . . . . . . . . . . . . . . . 49 1.8.1 New Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 1.8.2 Running the Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.8.3 Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.8.4 Time-Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 1.8.5 Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7 1.8.6 Initial Distributions and Particle Injection . . . . . . . . . . . . . 54 1.8.7 Interpolation and Computational Coordinates . . . . . . . . . . . 54 1.8.8 Calculating the Electric Potential and Field . . . . . . . . . . . . 54 1.8.9 Moving the Particles . . . . . . . . . . . . . . . . . . . . . . . . . 55 1.8.10 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 55 1.8.11 Inter-particle Collisions . . . . . . . . . . . . . . . . . . . . . . . 57 1.8.12 Particle Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 1.9 Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 1.9.1 Future Requirements . . . . . . . . . . . . . . . . . . . . . . . . . 59 1.10 Summary of Code Validation (Chapter 4) . . . . . . . . . . . . . . . . . 59 1.11 Summary of Results and Conclusions (Chapters 5 and 6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2 Theory 61 2.1 Dimensions of Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.2 Simulation Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.3 Current Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.3.1 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . 67 2.4 Performance Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.5 Maxwell's Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.5.1 Maxwell's Equations in CGS Units . . . . . . . . . . . . . . . . . 71 2.5.2 Normalized Unit System . . . . . . . . . . . . . . . . . . . . . . . 72 2.6 Simulation Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2.6.1 Basic Length and Time Scales . . . . . . . . . . . . . . . . . . . 77 2.6.2 Simplifying and Accelerating the Simulation . . . . . . . . . . . . 79 2.6.3 Modified Estimates of Plasma Parameters . . . . . . . . . . . . . 82 2.6.4 Recovery of Physical Solution . . . . . . . . . . ... ....... 86 2.6.5 Limits of Artificial Mass Approximation . . . . . . . . . . . . . . 88 2.7 Characteristic Times for Convergence . . . . . . . . . . . . . . . . . . . 90 2.8 Mean Free Path Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 92 2.8.1 Collisions and Mean Free Paths . . . . . . . . . ... ....... 93 2.8.2 Electron-Neutral Scattering . . . . . . . . . . . ... ....... 94 8 2.8.3 Ion-Neutral Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2.8.4 Neutral-Neutral Scattering . . . . . . . . . . . . . . . . . . . . . . . 102 2.8.5 Coulomb Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 2.8.6 Bulk Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 2.8.7 Summary of Mean Free Paths . . . . . . . . . . . . . . . . . . . . . . 106 2.9 Simple Orbit Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 2.9.1 Hall Thruster Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 108 2.9.2 Motion of a Charged Particle . . . . . . . . . . . . . . . . . . . . . . 108 2.10 Electron Transport in a Hall Thruster . . . . . . . . . . . . . . . . . . . . . 110 2.10.1 Secondary Emission and Wall Effects . . . . . . . . . . . . . . . . . . 110 2.10.2 Classical Diffusion and Guiding Center Drifts . . . . . . . . . . . . . 110 2.10.3 Perpendicular Transport due to Guiding Center Drifts . . . . . . . . 113 2.10.4 The Meaning of Mobility and Diffusion . . . . .. . . . 114 2.10.5 Anomalous Diffusion . . . . . . . . . . . . . . . .. . . . . 115 2.10.6 Coulomb Collisions and Electron Transport . . . . . . . 117 2.11 The Extent of the Ionization Layer . . . . . . . . . . . .. . . . 117 2.12 Boundary Conditions for Particle Impact . . . . . . . .. . . . . 118 2.12.1 Specular Reflection . . . . . . . . . . . . . . . . . . . . . 119 2.12.2 Diffuse Reflection . . . . . . . . . . . . . . . . . . . . . . 119 2.12.3 Application to mini-TAL . . . . . . . . . . . . .. . . . 120 2.12.4 Energy Loss to Walls . . . . . . . . . . . . . . . . . . . 121 2.12.5 Secondary Electron Emission . . . . . . . . . . .. . . . 122 2.13 Wall Potential and Sheath Formation . . . . . . . . . . .. . . . 124 2.13.1 Insulators, Conductors, and Capacitance . . . . .. . . . 124 2.13.2 Wall Potential . . . . . . . . . . . . . . . . . . .. . . . 125 3 The Numerical Method 127 3.1 The Particle-In-Cell Methodology . . . . . . . . . . . . .. . . . 127 3.2 The Monte Carlo Methodology . . . . . . . . . . . . . .. . . . 128 3.3 The Langevin Equation . . . . . . . . . . . . . . . . . . . . . . 130 3.4 Code and Data Structure . . . . . . . . . . . . . . . . . . . . . 130 3.5 Initializing the Simulation . . . . . . . . . . . . . . . . .. . . . 134 9 3.5.1 Important Note on Units . . . . . . . . . . . . . . . . 134 3.6 T im e-Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 3.6.1 Leapfrog Stability Criterion . . . . . . . . . . . . . . . . . . . . . . . 135 3.6.2 Gyro Frequency Criterion . . . . . . . . . . . . . . . . . . . . . . . . 135 3.6.3 Plasma Frequency Criterion . . . . . . . . . . . . . . . . . . . . . . . 136 3.6.4 Numerical Heating Criterion . . . . . . . . . . . . . . . . . . . . . . 136 3.7 Grid ..... ..... .......... ........ ..... . . . . ... ... 137 3.7.1 Elliptic Grid Generator . . . . . . . . . . . . . . . . . . . . . . . . . 137 3.7.2 Grids Used in Simulation . . . . . . . . . . . . . . . . . . . . . . . . 138 3.7.3 Node Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 3.7.4 Geometry of a Grid Cell.. . . . . . . . . . . . . . . . . . . . . . . . . 142 3.8 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 3.8.1 The Interpolation Concept . . . . . . . . . . . . . . . . . . . . . . . 144 3.8.2 Square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 3.8.3 Arbitrary Quadrilateral . . . . . . . . . . . . . . . . . . . . . . . . . 145 3.8.4 Computational Implementation . . . . . . . . . . . . . . . . . . . . . 147 3.8.5 Accuracy of the Interpolation Scheme . . . . . . . . . . . . . . . . . 147 3.9 Creating Particle Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 147 3.9.1 Creating a Distribution in Space . . . . . . . . . . . . . . . . . . . . 147 3.9.2 Maxwellian Velocity Distribution . . . . . . . . . . . . . . . . . . . . 148 3.9.3 Creating a Maxwellian Distribution in Velocity . . . . . . . . . . . . 149 3.10 Moments of a Particle Distribution . . . . . . . . . . . . . . . . . . . . . . . 150 3.10.1 Mass Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 3.10.2 Velocity and Momentum . . . . . . . . . . . . . . . . . . . . . . . . . 152 3.10.3 E nergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 3.10.4 Numerical Implementation of Particle Moments . . . . . . . . . . . . 155 3.11 The Numerical Electron Energy Distribution Function . . . . . . . . . . . . 155 3.11.1 Theoretical Distributions . . . . . . . . . . . . . . . . . . . . . . . . 156 3.11.2 Testing the Algorithm on a Numerical Distribution . . . . . . . . . 158 3.12 Calculating the Electric Potential and Field . . . . . . . . . . . . . . . . . . 158 3.12.1 Method 1: Poisson's Equation . . . . . . . . . . . . . . . . . . . . . . 164 3.12.2 Method 2: Gauss's Law . . . . . . . . . . . . . . . . . . . . . . . . . 165 10