Plasma-material boundary conditions for discontinuous Galerkin continuum-kinetic simulations, with a focus on secondary electron emission

Abstract

Continuum kinetic simulations of plasmas, where particle distribution functions are directly discretized in phase-space, permit fully kinetic simulations without the statistical noise of particle-in-cell methods. Recent advances in numerical algorithms have made continuum kinetic simulations computationally competitive. This work presents a continuum kinetic description of high-fidelity wall boundary conditions that utilize the readily available particle distribution function without coupling to additional physical models. The boundary condition is realized through a reflection function that can capture a wide range of cases from simple specular reflection to more involved first principles models. While the framework is usable for various numerical methods and boundary conditions, this work focuses on the discontinuous Galerkin implementation of electron emission using a first-principles quantum-mechanical model. Presented results demonstrate effects of electron emission from a dielectric material on formation of a classical plasma sheath.

Introduction

Kinetic models of plasmas are necessary to capture processes that occur at small spatial and temporal scales and depend on the shape of a particle distribution. An example of such a process is collisionless Landau damping where an electromagnetic wave is damped in a plasma resulting in a flattening of the particle distribution around the phase velocity of the wave. Kinetic simulations are most commonly performed using particle-in-cell (PIC) methods [2]. PIC models have a significant advantage of vast libraries of various physical processes, however, they suffer from particle noise. That is not the case for continuum kinetic methods which directly discretize particle distributions in phase space and evolve them in time using Vlasov (collision-less) [27], [16], [47], [38], [22] or Boltzmann (collisional) [14], [51], [26], [24], [18] equations. With advances in numerical algorithms, the continuum kinetic methods are becoming computationally competitive.

This work is motivated by simulations of electron emission (SEE) in wall-bounded plasmas using continuum kinetic methods. The method described here is intended to be more general, with details of the SEE implementation in the second part of the paper. In wall-bounded plasmas, the plasma dynamics is influenced by formation of plasma sheaths [41]. These are narrow regions of net space charge that form where electrons and ions come into contact with a solid surface. The process of sheath formation results from significant differences in electron and ion masses and, consequently, their thermal flows. The faster outflow of electrons gives rise to a potential barrier, which equalizes electron and ion fluxes to the wall. This behavior can be reproduced in the simplest case by setting the particle distribution function to zero for both species, ions and electrons, at the edge of the domain [8], [11] representing an ideal sink.

Despite the small spatial scales associated with plasma sheaths, they play an important role in particle momentum, energy, and heat transfer and on surface erosion, which can have global effects on the plasma. This can have significant implications for plasma thrusters [21], fusion devices [52], dielectric barrier discharges, and RF discharges [34]. Devaux and Manfredi [19], [20] performed Vlasov simulations of plasma-wall interactions including effects of magnetic fields and studied theoretical ion sputtering based on modeled distribution functions at the wall. One way to dynamically include SEE in a computational model is by using a constant gain function [13]; however, this technique does not account for the dynamic role of the incoming distribution on the SEE. Self-consistent implementations have been performed using a plasma model coupled with a surface model like TRIDYN [29], [30]. This work presents an alternative approach of a generalized boundary condition framework for continuum kinetic methods by directly utilizing the particle distribution functions and enabling a straightforward implementation of various boundary models. This is demonstrated in detail on a first-principles quantum-mechanics-based model for electron emission from dielectrics [4]. The boundary conditions and infrastructure to incorporate electron emission can be extended to other general boundary conditions allowing for computationally efficient solutions of physics-relevant surface models. The SEE boundary conditions and results described in this work are presented using a discontinuous Galerkin (DG) scheme that is extendable to arbitrarily high order, however, the boundary condition descriptions are independent of the numerical method.

The paper is organized as follows. Following the introduction, a brief review of plasma sheath physics and electron emission are provided in Sec. 2 and Sec. 3 respectively. We would like to emphasize that the Sec. 2 is included for the benefit of readers not familiar with plasma sheaths to provide context for the last part of this work and does not add anything on top of classical plasma physics textbooks [15]. Familiar readers are encouraged to skip this section. Section 4 presents the description of a continuum kinetic model and the general boundary conditions. Specific examples showing applications of phenomenological and first-principles models are in Sec. 5 with implementation details for the discontinuous Galerkin continuum kinetic model in the Gkeyll framework (https://gkeyll.rtfd.org/). Results are presented for the first continuum-kinetic plasma sheath simulations using high-fidelity first-principles SEE boundary conditions.