Plasma Simulations by Example

Lubos Brieda

• Publisher: CRC Press
• Year: 2019
• ISBN:
  9780429801051
• Edition: 1
• Topic: Scienze fisiche
• Subtopic: Astronomia e astrofisica

The study of plasmas is crucial in improving our understanding of the universe, and they are being increasingly utilised in key technologies such as spacecraft thrusters, plasma medicine, and fusion energy. Providing readers with an easy to follow set of examples that clearly illustrate how simulation codes are written, this book guides readers through how to develop C++ computer codes for simulating plasmas primarily with the kinetic Particle in Cell (PIC) method. This text will be invaluable to advanced undergraduates and graduate students in physics and engineering looking to learn how to put the theory to the test.

Features:

• 
  
  

• Provides a step-by-step introduction to plasma simulations with easy to follow examples

• 
  
  

• Discusses the electrostatic and electromagnetic Particle in Cell (PIC) method on structured and unstructured meshes, magnetohydrodynamics (MHD), and Vlasov solvers

• 
  
  

• Covered topics include Direct Simulation Monte Carlo (DSMC) collisions, surface interactions, axisymmetry, and parallelization strategies.

Lubos Brieda has over 15 years of experience developing plasma and gas simulation codes for electric propulsion, contamination transport, and plasma-surface interactions. As part of his master's research work, he developed a 3D ES-PIC electric propulsion plume code, Draco, which is to this date utilized by government labs and private aerospace firms to study plasma thruster plumes. His Ph.D, obtained in 2012 from George Washington University, USA, focused on a multi-scale model for Hall thrusters utilizing fluid-kinetic hybrid PIC codes. He has since then been involved in numerous projects involving development and the use of plasma simulation tools. Since 2014 he has been teaching online courses on plasma simulations through his website: particleincell.com.

CHAPTER 1

Fundamentals

1.1 INTRODUCTION

THIS CHAPTER starts with a review of kinetic and fluid approaches for simulating plasma flows. The governing equations of the Electrostatic Particle in Cell (ES-PIC) method are then introduced. We next discuss the Finite Difference Method for discretizing differential equations. We use the method to simulate an electron trapped in a potential well.

1.2 GAS SIMULATION APPROACHES

Plasma, on the microscopic scale, is simply a gas containing charged particles: ions and electrons. That is at least the simplistic view we take in this book. More complex flows can be found in nature. Plasma may contain negative ions or large dust particulates that collect charge through tribolectric charging or photoemission. There may be multiphase flows involving gaseous components and liquid droplets. We do not consider such cases here, and instead focus solely on the basic combination of neutral atoms, positive ions, and enough electrons to neutralize the space charge. The objective of flow simulations is to predict the evolution of particle velocities and positions given some initial conditions and governing laws. Within the realms of Newtonian physics, we assume that all neutrals, ions, and electrons are rigid bodies governed by the equations of motions,

$\begin{array}{ll}\frac{d\overrightarrow{x}}{dt}=\overrightarrow{v}\hfill & \frac{d\overrightarrow{v}}{dt}=\overrightarrow{a}\hfill \end{array}$

(1.1)

where $\overrightarrow{x}$, $\overrightarrow{v}$, and $\overrightarrow{a}$ are the position, velocity, and acceleration of a single particle. If the particle mass m remains constant, as is assumed throughout this book, the acceleration can be written in terms of total force, $\overrightarrow{a}={\displaystyle \sum \overrightarrow{F}}/m$ per Newton’s Second Law. Some forces, such as gravity, originate from the external environment. Other forces may be intrinsic to the system. Charged particles interact with each other through the Coulomb force,

${\overrightarrow{F}}_{c,ij}=\frac{1}{4\pi {ϵ}_0}\frac{q_i{q}_j}{{\left|{\overrightarrow{x}}_i-{\overrightarrow{x}}_j\right|}^3}\left({\overrightarrow{x}}_i-{\overrightarrow{x}}_j\right)$

(1.2)

where the i and j are indexes of two unique particles with charges q_i and q_j. ϵ₀ ≈ 8.8542 × 10⁻¹² C/(Vm) is a physical constant known as permittivity of free space. This formulation ignores magnetic field effects found with moving charges. The total force acting on a single particle i is

${\overrightarrow{F}}_{c,i}={\displaystyle \sum _j^N{\overrightarrow{F}}_{c,ij}}i\ne j$

(1.3)

where the sum is over all particles in the system. Particle velocity can also change through collisions. We thus have

$\frac{d\overrightarrow{v}}{dt}=\frac{1}{m}\left({\overrightarrow{F}}_g+{\overrightarrow{F}}_c+\dots \right)+{\left(\frac{d\overrightarrow{v}}{dt}\right)}_{col}$

(1.4)

Although gravity is included in the above formulation, it is customary to ignore it. A quick “back of the envelope” calculation demonstrates why. Large vacuum chambers used in plasma processing operate at pressures p around 10⁻⁶ Torr (1 Torr is 1/760th of 1 atmosphere). Using the ideal gas law, p = nk_BT, where k_B ≈ 1.381 × 10⁻²³ J/K is the Boltzmann constant and T is the room temperature, the average number density n is around 3.3 × 10¹⁶ particles per cubic meter. The mean inter-particle spacing can be estimated from the volume needed to be assigned to each ion to fill a unit cube. Modeling each particle as a cube with sides a, the particles are a = (1 / n)^1/3 or approximately 3 μm apart. At this distance, the Coulomb force is around 10⁻¹⁷ N. Even for the heavy xenon used in plasma propulsion, the force of gravity, F = mg, is only 10⁻²⁴ N. The gravitational force is thus seven orders of magnitude smaller than the electrostatic Coulomb force. In the Low Earth Orbit (LEO), the ambient plasma density drops to 10¹² m⁻³, but the Coulomb force still dominates by four orders of magnitude. Therefore, unless we are dealing with pro...