Feasibility Study of Hall Thruster's Wall Erosion Modelling Using Multiphysics Software

Transcription

1 Feasibility Study of Hall Thruster's Wall Erosion Modelling Using Multiphysics Software Amin Mirzai Space Engineering, masters level 2016 Luleå University of Technology Department of Computer Science, Electrical and Space Engineering

2 Amin Mirzai Feasibility Study of Hall Thruster s Wall Erosion Modelling Using Multiphysics Software School of Electrical Engineering Department of Radio Science and Engineering Master s Thesis in Space Science and Technology Espoo, August 25, 2016 Supervisor: Instructor: Examiner: Assistant Professor Jaan Praks, Aalto University Antti Kestilä, Aalto University Prof. Javier Martin-Torres, Lulea University of Technology

3 Aalto University School of Electrical Engineering ABSTRACT OF MASTER S THESIS Author: Amin Mirzai Title: Feasibility Study of Hall Thruster s Wall Erosion Modelling Using Multiphysics Software Date: August 25, 2016 Pages: 60 Major: Supervisor: Instructor: Examiner: Space Science and Technology Assistant Professor Jaan Praks, Aalto University Antti Kestilä, Aalto University Prof. Javier Martin-Torres, Lulea University of Technology The most common type of electric propulsion in space exploration is the Hall Effect Thruster (HET), mainly due to its high specific impulse and high thrust to power ratio. However, uncertainties about the thruster s lifetime prediction have prevented widespread integration of HETs. Among these limitations, wall erosion of acceleration channel is of greatest concern. The experimental methods of erosion are time consuming and costly, and they are often limited to one single configuration. Hence, developing a computational model not only decreases the costs but also shortens the design time of a HET. This thesis investigates the feasibility of a fluid erosion modelling with a multiphysics software (COMSOL) to further decrease the time and the development cost. First of all, this thesis provides an overview of available plasma modelling techniques and the physics behind the erosion phenomenon. Moreover, the effective parameters and available modules in the multiphysics software as well as their theoretical background were studied and discussed in detail. The Electron Anomalous phenomenon and pressure instability are determined as the main limiting factors for such a model. A non-magnetized model is included to find an optimal value for pressure and to reduce the probability of pressure instability occurrence in magnetized model. To fulfill this task, several simulations for various pressure values (0.005 Torr, 0.05 Torr, and 0.5 Torr) were conducted. Next, the simulation of magnetized/full model has been carried out with addition of magnetic coils in non-magnetized model. To avoid the Electron Anomalous phenomenon, the Bohm diffusion approach was implemented. In addition, a full Particle-In-Cell (PIC) simulation of a typical HET (SPT-100) with the similar input parameters as in fluid model was conducted, and the results were compared and validated using experimental data. The PIC model was intended to be utilized to investigate the accuracy of erosion model in multiphysics software. The results of this thesis indicate that current application of erosion model in COMSOL is not possible whilst high accuracy of the erosion model based on PIC approach can be achieved. Finally, the application of semi-empirical method through direct input of magnetic field data can allow short time simulation of a HET in COMSOL to gain insight about the preliminary behaviour of plasma, however, the simulation of an erosion model requires either a built-in PIC algorithm in COMSOL or a PIC based code. i Keywords: Hall Effect Thruster, SPT-100, Erosion modelling, Fluid method, Electron Anomalous, Bohm diffusion, PIC Language: English

4 Acknowledgements I would like to express my sincere gratitude to my advisor Antti Kestilä and supervisor D.Sc. Jaan Praks for providing constructive comments and guidance in writing of this thesis. I am also grateful for the useful comments and suggestions of Laura Mendoza. Finally, I would like to thank the European Commission for providing financial support under the Erasmus Mundus framework throughout this master program. Otaniemi, August 25, 2016 Amin Mirzai ii

5 Symbols and Abbreviations Symbols A A D B B B r B z B φ D e D k,m Dk T D ɛ E e int F F L g 0 I sp j J e j k k B k k L M M M k M n m e Area Vector field Bohm diffusion Magnetic field Radial magnetic field Axial magnetic field Azimuthal magnetic field Electron diffusion The mixture-averaged diffusion coefficient Thermal diffusion coefficient Electron energy diffusivity Electric field Internal energy Body force Lorentz force The acceleration due to gravity at the Earth s surface Specific impulse Current density Externally generated current density Diffusive flux vector Boltzmann constant Rate coefficient Heat flux Number of reactions Magnetic moment Molar mass for species k Mean molar mass Electron mass iii

6 ṁ Mass flow rate N Number of super particles N n The total neutral number density n Number density n e Electron number density n ɛ Electron energy density p Pressure of plasma p e Electron pressure tensor Q E Heat transfer due to collision Q p Source term for transport of momentum Q ρ Source term for mass production and annihilation q Electron charge R e The sum of sink and sources of electrons R k Rate expression for species k R ɛ The electron energy loss or gain due to inelastic collisions S Source term T e Electron temperature t Time u Mass average fluid velocity vector u Velocity vector u e Electron drift velocity V k Multi-component diffusion velocity v Velocity v exit Exhaust velocity x k Mole fraction z k Charge number for species k Γ Incomplete Gamma function Γ e Electron flux vector Γ ɛ Electron energy flux vector γ Specific heat ratio ζ Constant ɛ 0 Permittivity of free space ɛ Electron energy ɛ Mean electron energy λ D Debye length µ dc Electron mobility in absence of a magnetic field µ e Electron mobility µ e Parallel electron mobility µ e Perpendicular electron mobility µ k,m Mixture-averaged mobility iv

7 µ ɛ Electron energy mobility µ 0 Permeability of the conductor ν Collision frequency ν m Momentum transfer frequency ρ Density ρ surf Surface charge density σ Collision cross section σ k Reaction cross section Φ Electric potential φ Mean electron energy ω k Mass fraction of k th species Electron plasma frequency ω pe Abbreviations BE BN CFD DC DD DSMC EEDF FDTD HET h-bn MCC ODE PIC PDE r-bn SPT TAL Boltzmann Equation Boron Nitride Computational Fluid Dynamics Direct Current Drift Diffusion Direct Simulation Monte Carlo Electron Energy Distribution Function Finite-Difference-Time-Domain Hall Effect Thruster Hexagonal Boron Nitride Monte Carlo Collision Ordinary Differential Equations Particle-In-Cell Partial Differential Equations Rhombohedral Boron Nitride Stationary Plasma Thruster Thruster with Anode Layer v

8 Contents Abbreviations and Acronyms iii 1 Introduction 1 2 Electric Propulsion and Plasma Modelling Electric Propulsion Hall Effect Thruster HET Channel Wall Erosion Plasma Modelling Methods Fluid Method Kinetic Method Hybrid method Sputtering Incident Energy Incident Angle Fluid Model of Plasma and Erosion Model Fluid Governing Equations Electron Energy Distribution Function Maxwellian Description Druyvesteyn Description The Boltzmann Equation Comparison of Electron Energy Distribution Function Approaches Drift Diffusion Electron Impact Reactions Plasma-Sheath Region Diffusion and Mobility in Magnetic Field Heavy Species Transport Source Terms vi

9 4.6 Neutral Atoms Magnetic Field Erosion Model Remarks On Fluid Model Particle-In-Cell Plasma Model and Erosion Model Particle-In-Cell Methodology Numerical Approach Monte Carlo Collisions Erosion Model Particle-In-Cell Stability Limits Grid Cell Effect Time Step Effect Density of Computational Particles Implementation of Models Fluid Model Implementation Non-magnetized Model Magnetized Model Implementation of Particle-In-Cell Model Results and Discussion Implementation Remarks on Fluid Model Non-magnetized Model Magnetized Model Results of Particle-In-Cell Model Conclusions and Recommendations 53 vii

10 Chapter 1 Introduction An effective, efficient and reliable propulsion system is one of the key requirements for space exploration. The classical means of propulsion, chemical propulsion, has high cost and is not efficient enough to be applied specifically for deep space (e.g. asteroid mining) missions. One way to overcome such issue, is application of Electric Propulsion (EP) thrusters that offer high specific impulse. One of the most common types of EPs for both commercial and scientific missions is the Hall Effect Thruster (HET). This is mainly due to its high specific impulse (1600 s s) and high thrust to power ratio[1]. Hence, the HETs are of particular interest among the spacecraft developers. Despite many application of the HETs in past and present missions, there still exist some constraints requiring further research to understand limiting factors of such thrusters[2]. The erosion of the acceleration of channel due to ion bombardment is the primary limiting factor for HETs[3]. As mission requirement and determination of spacecraft lifetime involving HETs increases, the need for prediction and validation of thruster lifetime increases as well. The experimental techniques of lifetime prediction are time consuming and costly, requiring up to tens of thousands of hours, and costs hundreds and thousands of dollars for equipments and fuel[4]. Therefore, a computational tool that can deliver a quick and accurate prediction of erosion is in great demand. The goal of this thesis is to develop a computational model that is inexpensive in terms of both computational time and development cost. Hence, the model is carried out in a multiphysics software (COMSOL). The COM- SOL is a commercial multiphysics software that recently has added the plasma package into its library. This way, the design iteration is quick due to built-in modules and requires less time to implement a new geometry configuration. Furthermore, the computational time reduces dramatically since the plasma model in COMSOL is based on fluid method[4]. The physics 1

11 CHAPTER 1. INTRODUCTION 2 modules in COMSOL must be combined and customized according to application. To validate the model, the results should be compared with a Full Particle-In-Cell (PIC) code. The first chapter of the thesis motivates the need for erosion modelling. The Chapter 2 provides a short description of EP technologies and available methods for plasma modelling. The energy transfer mechanism and the effective properties on sputter yield process are described in Chapter 3. The Chapter 4 elaborates the modelling of plasma in a HET based on fluid method as well as available module to capture the sputtering effect. Chapter 5 illustrates the PIC/Monte Carlo Collisions (MCC) algorithm. The implementation plan to model a SPT-100 based on proposed models are illustrated in Chapter 6. The discussion of implementation and results are included in Chapter 7. Finally, a summary of findings and future recommendation are added in Chapter 8.

12 Chapter 2 Electric Propulsion and Plasma Modelling 2.1 Electric Propulsion The operation principle of all the thrusters is based upon Newton s third law of motion, where the action of expelled gas causes a reaction force in the opposite direction. The utilization of different materials as expelled gas, and the means of acceleration of the propellant, define the type of propulsion technology such as, chemical propulsion, cold gas propulsion, and EP[1]. The general thrust equation based on Newton s law for an ideal rocket is T = ṁ propellant v exit = ṁ propellant I sp g 0, (2.1) where T, ṁ propellant, I sp, and g 0 stand for thrust, propellant mass flow rate, specific impulse and the acceleration due to gravity measured at the surface of the Earth, respectively[5]. The I sp is the measurement term for achievable thrust per specific amount of propellant mass[1], and it can be obtained by rearranging Equation 2.1 as: I sp = T ṁ propellant g 0. (2.2) Since the beginning of the space era, chemical rockets have dominated the space missions due to their high thrust to mass ratio and available technology. However, the high cost and the short term operation of chemical rockets have limited the space missions. In order to overcome this issue, the EP technology was introduced. In general, any type of propulsion system that uses electricity to accelerate the propellant is called EP[1]. This general definition of EP leads to a wide range of thrusters. Based on the mechanism 3

13 CHAPTER 2. ELECTRIC PROPULSION AND PLASMA MODELLING4 through which electric power is utilized to accelerate the exhaust flow they can be categorized into three different groups as it is shown in Figure 2.1[1]. As EPs achieve high specific impulse (Figure 2.2), the overall mass of spacecrafts reduces, which leads to reduction of the space mission costs. Besides, the operation time of EP thrusters are usually in order of thousands of hours, which make them suitable for long term missions such as interplanetary missions. Furthermore, due to the long operational lifetime and lower thrust compared to chemical or nuclear propulsion systems, the EPs are often used in secondary duties such as satellite station-keeping[5]. Figure 2.1: Electric propulsion technologies 2.2 Hall Effect Thruster The HET is an electrostatic thruster (Figure 2.1) that accelerates ions by means of an electric field. A HET consists of a relatively simple configuration: a cylindrical channel with an interior anode, a magnetic circuit generates radial magnetic field throughout the channel, where the pick of the magnetic field is designed to take place at the exit, and a hollow cathode external to the channel that provides electrons to initiate ionization and to neutralize the ions at plume region[1]. As electrons move toward the anode due to difference in electric potential between anode and cathode, they get trapped by applying magnetic field at the channel exit. Furthermore, due to acting electric and magnetic fields, the electrons begins to move in azimuthal direction. The azimuthal movement of the electrons forms Hall current, from which the name of this type of thruster is taken[1]. Later on, when neutral

14 CHAPTER 2. ELECTRIC PROPULSION AND PLASMA MODELLING5 Figure 2.2: Thrust & I sp ranges of propulsion systems [4] propellant particles fed from anode collide with these energetic electrons ionization occurs. Finally, the ions generated due to ionization process, are accelerated axially toward the exit under influence of an electric field[5][1]. There are two types of HETs, Stationary Plasma Thruster (SPT) and Thruster with Anode Layer (TAL)[1]. Although the basic structure for both thruster types remains unchanged, use of different material in acceleration channel is the primary reason behind the division of these two types[2][1]. The acceleration channel in SPT type is made of dielectric material such as Boron Nitride or Silicon Carbide that allows charge building along the axial length of the acceleration channel[5]. In TAL type thruster, since the channel wall material is made of metallic materials such as stainless steel or molybdenum a constant potential exist along the entire wall. Furthermore, the length of acceleration channel in TAL type is shortened (a few mm) and higher electron temperature has observed[5]. In this thesis, the selected HET type for acceleration channel erosion modelling is SPT-100. Although both types of HETs are commercially available, the application of SPT type in commercial space application is much more common, hence, the study and research on this type of HET is in higher desire[5]. A basic schematic view of the SPT HET is shown in Figure 2.3.

15 CHAPTER 2. ELECTRIC PROPULSION AND PLASMA MODELLING6 Figure 2.3: Schematic view of SPT type HET[2] 2.3 HET Channel Wall Erosion There are many factors affecting lifetime of a HET, such as near wall electric field, degradation of insulating and structural materials, erosion of hollow cathode due to ion bombardment, evaporation of thermo-emitter, and deformation and cracking from thermal shocks[6]. Nonetheless, the erosion of the acceleration channel s wall due to ion bombardment known as sputtering is the main lifetime limiting factor of HETs[3][7]. The erosion in the channel wall occurs whilst accelerated ions with enough energy collide with the wall and remove part of wall material due to sputtering. The channel wall material protects the magnetic circuit from hazardous particles. Once the magnetic circuitry is exposed to plasma due to gradual erosion of the channel wall s material, the thruster s end of life is declared[3][8]. The lifetime of a thruster is important since often the limiting factor for operation time of commercial communication satellites are fuel and the lifetime of the thruster. Moreover, the increasing number of satellites, which use HET for station-keeping purposes, makes the lifetime estimation of HETs crucial[4]. The most straightforward method to identify the lifetime of a HET, is to conduct experiment. However, to avoid the long operation time of lifetime prediction that could be in range of hundreds up to thousands of hours, the numerical models are often applied[4]. Apart from operation time, empirical methods are very costly due to fuel requirement and experimental setup

16 CHAPTER 2. ELECTRIC PROPULSION AND PLASMA MODELLING7 such as vacuum chamber[4]. Therefore, a numerical modeling technique to reduce the cost and operation time is highly desirable. Furthermore, numerical methods are often configurable; meaning the condition and the type of geometry of the HET can be altered whereas empirical methods are often applicable to a single condition for a specific thruster[7][5]. 2.4 Plasma Modelling Methods In order to model the wall erosion of a HET, one must model accurately the plasma environment in the channel and interaction with the wall material. Fundamentally, there are two types of numerical approaches: continuum or fluid methods, and kinetic method. In addition, there is also Hybrid method, which has emerged from the combination of both methods. In the following sections, the different numerical methods of plasma modelling will be discussed, advantages and disadvantages will be compared Fluid Method In this method, all plasma species (electrons, ions, neutral atoms) are treated as fluid, and a set of governing equations are solved in a way similar to the standard computational fluid dynamics (CFD) to simulate plasma. This method has been under development by CFD community since the 1940 s. As a result, the fluid method is well understood and it is the dominant method to model electron dynamics, since it shows a fair representation with reasonable computational time[4]. Fluid-based method can be adapted for 1-D, 2-D, or axisymmetric geometries, as well as both steady and unsteady time-dependent solvers. The main advantage of the fluid method is the computational speed. Therefore, it is a better choice when a quick set of results is required to initiate the HET design. Some of the early fluid modelling codes are based on the works of Yim[4], Roy & Pandey [9], Ahedo et al[10], Barral et al[11], Keider et al[12] Kinetic Method In a fully-kinetic method, all the species (electrons, ions, neutral atoms) in the plasma are treated as particles[13]. Furthermore, the particles are a discrete representation of the velocity distribution function. This approach solves for the distribution function for the ions and electrons in a plasma by solving either the Boltzmann equation or an approximation, like the Fokker- Planck equation[14].

17 CHAPTER 2. ELECTRIC PROPULSION AND PLASMA MODELLING8 Although this technique yields highly accurate results, it is an extremely time consuming process mainly due to the fact that electrons are several orders of magnitude lighter than the ions, thus, they move on a much smaller timescale. As a result, the time steps for electrons must be around 500 times smaller in comparison with ion dynamics to fulfill the requirement[5]. Despite all difficulties, several different fully-kinetic modeling codes have been developed with some successes in application for HETs. For example, Hirkawa [15], Szabo [16], and Smirnov [17] Hybrid method In hybrid modelling, the heavy species (ions and neutral atoms) and electrons are modeled different from each other. In a typical hybrid method, the heavy species are modeled as particles and the electrons as fluid[13]. In this way, the method can keep the accuracy of heavy species, and capture non-maxwellian features while it maintains the computational cost as low as possible[13][5]. The hybrid method is currently the most popular approach for HET simulation, and the most common code based on this approach is HPHall by Fife[18]. Since then further research and efforts have carried out based on the layout of Fife s work.

18 Chapter 3 Sputtering In the section 2.3, the erosion of acceleration channel due to sputtering introduced as the main limiting factor of the HETs. Therefore, it is important to have a clear understanding of sputtering before modelling the erosion. Hence, this section is dedicated to study of sputtering and the mechanisms that drives such phenomenon. The study of the sputtering process is already conducted in a comprehensive manner by Smentkowski[19]. In this thesis, a brief description of sputtering phenomenon is presented based on the work of Cheng[7]. Sputtering is the process of the material removal from a solid surface due to ion bombardment, and it occurs when and only the striking ion has the sufficient energy. Due to the collisions, the energy stored in striking ions are spent to break the bonds between target atoms (Figure 3.1) leading to the removal of atoms. Figure 3.1: Ion bombardment of surface atoms[7] There are three fundamental mechanisms of energy transfer in which the sputtering happens (Figure 3.2); (a) direct removal of the target atoms (b) the removal of the neighboring atom after colliding with the target atom 9

19 CHAPTER 3. SPUTTERING 10 (a) Direct knock-on regime (b) Single recoil regime (c) Linear cascade regime Figure 3.2: Mechanisms of energy transfer during sputtering[7] (c) the removal of atom due to collision cascade initiated by the energetic particles[7]. The main properties that determine the sputtering process from a solid surface depend on incident species, incident energy, incident angle, surface lattice structure, surface binding energy and surface morphology, and all the sputtering mechanisms can happen simultaneously. Here the two main properties (incident energy and incident angle) of sputtering are discussed in greater details. In the mean time, as Boron Nitride (BN) is the typical material to cover the magnetic circuitry of the HETs, the experiment results shown in here are associated to BN[7]. 3.1 Incident Energy The average energy of singly charged ions in HETs is directly related to discharge voltage. This means for the 300 V discharge voltage in SPT-100 the single charged ion energy is about 300 ev and double-charged ions (not considered here) have an average of 600 ev. Thus, the sputter yield of BN in HETs are listed as low energy since the determined threshold for this category is 600 ev[7]. The Figure 3.3 compares the experimental results of BN sputter yield conducted by Abgaryan et al. [8], and Elovikov et al. for hexagonal boron nitride (h-bn) and rhombohedral boron nitride (r-bn) [20],

20 CHAPTER 3. SPUTTERING 11 Figure 3.3: The sputter yield of BN bombarded by Xe + at normal incidence[7] Garnier et al. [21], and Yalin et al. [22]. The Figure 3.3 clearly exhibits that the sputtering increases as the incident energy of ions increase. Cheng states that this increase with incident energy, continues up to 10 kev kev. The ions with higher energies, penetrate deeper into the material and less energy deposits in surface material, thus, the sputtering decreases[7]. 3.2 Incident Angle The Figure 3.4 is comparing the experimental results of Garnier and Yalin for variable oblique angles. The magnitude of Garnier s results are higher, however, both show the same pattern. This means the sputtering increases as the angle increases from 0 up to 60 70, then the sputtering magnitude starts to decrease. Apart from the properties mentioned earlier, the role of temperature is determinative as well. Unfortunately, the temperature dependence of BN sputter yield process has not been measured. However, the temperature dependence for Borosil ceramic that is also used in construction of SPT type HETs, is available. As seen in Figure 3.5 there is a sharp increase in sputtering after 600 C that is of great concern in HETs. This is due to the fact that the sudden increase in sputtering can not be explained with the physical sputtering mechanism, and the sublimation temperature of the both

21 CHAPTER 3. SPUTTERING 12 Figure 3.4: Sputter yield of BN by Xe + at oblique angles[7] BN and Borosil is at 3000 K, which is way above the critical temperature (around 900 K)[7]. Nevertheless, the thermal effect of sputtering should only be treated carefully when the energy transfer due to ion collisions with walls are included in the sputtering code. Otherwise, the thermal effect can simply be ignored. Figure 3.5: Temperature dependence of Borosil due to sputtering of Xe + [19]

22 Chapter 4 Fluid Model of Plasma and Erosion Model The fluid model of plasma is based on a set of macroscopic quantities. These macroscopic quantities are the electron number density, the mean electron momentum and the mean electron energy, wherein the value can be achieved by solving the continuity, the flux and the energy equation for the constituent species[23]. These governing equations can be derived by taking velocity moments of the Boltzmann equation. Finally, the fluid governing equations are coupled with either Maxwell s equations or Poisson s equation (in case of electrostatic modelling) to constitute a self-consistent electric and/or magnetic fields[24]. The resulting derived equations are a set of Partial Differential equations(pdes) that are usually solved by applying finite elements or finite volume discretization techniques[23][25]. In this thesis, the fluid modelling of SPT-100 is investigated in COMSOL. A HET essentially is a DC discharge aided by a static magnetic field. Here, the plasma is modeled based on following sub-modules: DC discharge (drift diffusion, heavy species transport and electrostatic force), magnetic field, laminar flow, and charged particle tracing. 4.1 Fluid Governing Equations The conservation equations of mass, momentum and the energy are the fundamental governing equations to characterize the flow of the plasma as fluid. The conservation of mass, shows that the mass of input and output species passing through a fixed test volume over time remains constant. So, the change of density ρ over time in volume V during inflow, is equal to change of density in outflow whilst it passes through a surface A, on a normal di- 13

23 CHAPTER 4. FLUID MODEL OF PLASMA AND EROSION MODEL14 rection n, with velocity v. Equation 4.1 is the integral representation of conservation of mass. In the meantime, S constant represents any other source terms[4][5][1]. δ ρdv = ρ v. nda + S constant (4.1) δt V A The conservation of momentum as given in Equation 4.2 explains that, the change of momentum in a system with volume V is equal to change of momentum flowing in or out, plus the change of forces acting on the species. The p and F are pressure of the plasma and body forces respectively, the viscous forces are ignored, and S mom represents any other sources of momentum[4][5][1]. δ δt V ρ vdv = A ρ v v. vda A p vda + V ρ F m dv + S mom (4.2) Finally, Equation 4.3 explains that the conversion of energy from one form to another is possible, however, the total internal energy e int in a system with volume V remains constant. S energy represents any other sources of energy[4][5][1]. δ δt V ρe int dv = A ρe int v nda + S energy (4.3) 4.2 Electron Energy Distribution Function Electron transport properties and source coefficients are required to calculate the electron energy and density. The Electron Energy Distribution Function (EEDF) and collision cross-section data are used to calculate these properties and coefficients of electron transport[26]. The study of complete theory behind EEDF is not in the scope of this thesis. Therefore, only the available options and their influences on a plasma model have discussed. For further discussion regarding the EEDF topic, please refer to [26]. In COMSOL, there are three explicit different approaches of the EEDF available: Maxwellian, Druyvesteyn, Two-Term Approximation. Each of these approaches will be discussed briefly Maxwellian Description The Maxwellian description is based on constant collision frequency. The distribution function can be addressed as Maxwellian when ionization degree is

24 CHAPTER 4. FLUID MODEL OF PLASMA AND EROSION MODEL15 high, thus, electrons are in thermodynamic equilibrium. In this approach, the electron-electron collisions push the distribution toward Maxwellian shape. At higher electron energies inelastic collisions (Excitation and/or ionization) between electrons and the heavy species decrease the EEDF. The mathematical description of this approach is f(ɛ) = φ 3/2 β 1 exp( ɛβ 2 ), (4.4) φ β 1 = Γ(5/2) 3/2 Γ(3/2) 5/2, β 2 = Γ(5/4)Γ(3/2) 1, where φ is the mean electron energy, ɛ is electron energy, and Γ is the incomplete Gamma function[27] Druyvesteyn Description The Druyvesteyn description is based on constant cross section. In the Druyvesteyn description, the distribution function assumes that the elastic collisions dominate the plasma. Therefore, the effects of inelastic collisions on distribution function are insignificant. In such a case, the distribution function becomes spherically symmetric. In elastic collisions happening between electrons and neutral atoms only the electrons direction of motion changes but not their energies. The Druyvesteyn distribution function (Equation 4.5) often gives more accurate results for a lower ionization degree[27]. f(ɛ) = 2φ 3/2 β 1 exp( (ɛβ 2 /φ) 2 ), (4.5) β 1 = Γ(5/4) 3/2 Γ(3/4) 5/2, β = Γ(5/4)Γ(3/4) The Boltzmann Equation Another approach to calculate the EEDF is to solve the Boltzmann Equation (BE) explicitly. Equation 4.6, BE, describes evolution of the distribution function f, in a six-dimensional phase space. δf δt + v f q m (E vf) = C[f] (4.6) High number of simplification is required to solve the BE. One common approach is to assume the distribution function is spherically symmetric. In this case, the series used for solving BE can be shortened to two terms instead of solving six or more terms in case of high accuracy. This is the reason such technique is often called as two-term approximation[27].

25 CHAPTER 4. FLUID MODEL OF PLASMA AND EROSION MODEL Comparison of Electron Energy Distribution Function Approaches The electron collision reactions in a plasma model is determined with rate coefficient, k k. The EEDF is used to calculate the rate coefficient as k k = γ 0 ɛσ k (ɛ)f(ɛ)dɛ, (4.7) where γ is equal to 2q/m e, ɛ is electron energy, and in here σ k is the cross section for reaction. The rate coefficients for excitation and ionization highly depend on the shape of the EEDF, which makes it dependant on the approach we choose to solve EEDF[27]. Figure 4.1: Plot of rate coefficients for argon ionization computed with different EEDF approaches are shown in different colors [27] As seen in Figure 4.1, the Maxwellian approach leads to overestimation of the ionization rate. However, the transport properties has less dependency on the type of EEDF as it has shown in Figure 4.2. Hence, the selection of two-term approximation will lead to more accurate results. This is due to the fact that the plasma model in HETs is not dominated by elastic collisions. Furthermore, this way the overestimation of the ionization rate that occurs in Maxwellian approach is avoided.

26 CHAPTER 4. FLUID MODEL OF PLASMA AND EROSION MODEL17 Figure 4.2: Plot of experimental and computed drift velocity for different EEDFs approaches are shown in different colors[27] 4.3 Drift Diffusion The Drift Diffusion (DD) sub-module is required to describe the electron transport in the model. It also determines the mathematical description of the electron diffusivity, electron mobility, electron energy diffusivity and electron energy mobility[27]. The DD approach is best when it is used for the gas pressure above 1 mtorr and a moderate electric field. Furthermore, the discharge plasma should be weakly ionized, meaning the number density of charged species should be much less than the number density of background gas[23]. Lastly, the mean free path between electrons and neutral atoms must be less than the characteristic dimension of the system, such condition is called collisional[25]. Usually, the BE expresses the electron transport. It is a non-local continuity equation in phase space (r, u), which is a highly complicated integrodifferential equation. As it was mentioned earlier, in fluid approach BE is approximated by taking its velocity moment[23]. The zeroth moment of BE gives the equation that describes the rate of change of particle density δ Qρ (n) + (n u) = δt m, (4.8) manipulation of the Equation 4.8 gives the rate of change of electron δ δt (n e) + Γ e = R e, (4.9)

27 CHAPTER 4. FLUID MODEL OF PLASMA AND EROSION MODEL18 where n e is the electron density, u is the average velocity vector, Q ρ is the source term for mass production and annihilation, Γ e is the electron flux vector, n is number density, R e is the sum of sink and source of electrons[25][23]. The mathematical representation of R e is R e = M x j k j N n n e, (4.10) j=1 where M is the number of reactions which contribute to the growth or decay of electron density. x j is the mole fraction of the species, N n is the total neutral number density and k j is the rate coefficient of the reaction j[27]. The first moment of BE gives the general momentum transport equations δ(n nu) m + p + m (n u u) nq( E δt + u B) = uq ρ + Q p, (4.11) mn δ u δt + mn( u ) u + p nq( E + u B) = Q p. (4.12) Manipulation of Equations 4.11 and 4.12 gives the electron momentum as δ δt (n em e u e ) + n e m e u e u T e = ( p e ) + qn e E n e m e u e v m, (4.13) where Q p is the source term for transport of momentum, m e is the mass of electron, p e is the electron pressure tensor, ν m is the momentum transfer frequency, q is the electron charge, E is the electric field and u e is the electron drift velocity[23][27]. Moreover, the second velocity moment of BE gives the heat transport or energy equation. The general form for all species is: 1 γ 1 (δp δt + (p u)) + ( P ) u + L = Q E, (4.14) and the rate of change of the electron energy density can be found from manipulation of Equation 4.14 as δ δt (n ɛ) + Γ ɛ + E Γ e = R ɛ, (4.15) where n ɛ (the subscript ɛ refers to electron energy) is the electron energy density, Q E is heat transfer due to collision, L defines heat flux, γ is the specific heat ratio, and R ɛ is the electron energy loss or gain due to inelastic

28 CHAPTER 4. FLUID MODEL OF PLASMA AND EROSION MODEL19 collisions[23][27]. It can be obtained by summing the collisional energy loss over all reactions as shown in Equation R ɛ = P x j k j N n n e ɛ j, (4.16) j=1 where ɛ j is the energy loss from reaction. The first term on the left hand side of Equation 4.13 can be ignored if we assume that the ionization, attachment, angular frequencies are much less than the momentum transfer frequency. The second term on the left hand side of the same equation can also be ignored based on the assumption that the electron drift velocity is less than the thermal velocity[27]. Furthermore, under the assumption of Maxwellian distribution the pressure term p e can be substitute with the equation of state as p e = n e k B T e. (4.17) Based on these assumptions, which were mentioned earlier the electron drift velocity can obtained as u e = k B m e ν m T e k BT e n e m e ν m n e + Next, the electron flux Γ e is derived from Equation 4.13 as q m e ν m E. (4.18) Γ e = n e u e = (µ e E)n e (D e n e ), (4.19) the two new terms on the right hand side of the Equation 4.19, denote electron mobility µ e and the electron diffusivity D e [25][27]. The electron mobility can be divided to two component as parallel electron mobility and the perpendicular electron mobility µ e = q m e ν, (4.20) µ e = qν/m e. (4.21) ν 2 + ωc 2 Moreover, the mathematical description of electron diffusivity is D e = k BT e m e ν m. (4.22) Similarly, the electron energy flux in Equation 4.15 can be written as Γ ɛ = (µ ɛ E)n ɛ (D ɛ n ɛ ), (4.23)

29 CHAPTER 4. FLUID MODEL OF PLASMA AND EROSION MODEL20 where D ɛ and µ ɛ are electron energy diffusivity and mobility respectively[25][27]. Finally, since the expressions of both fluxes are introduced the mean electron energy can be written as ɛ = n ɛ n e. (4.24) Electron Impact Reactions The charged particles in plasma mediums interact with each other, primarily by coulomb collisions. Moreover, they can also collide with neutral atoms present in the plasma. These collisions are very important when describing diffusion, mobility, and resistivity in the plasma[1]. In COMSOL Multiphysics, the collisions as a result of electron impact on target species are categorized in four groups of elastic, ionization, excitation, and attachment[27]. The cross section data and energy loss associated with the inelastic collisions of Xenon are imported to the COMSOL. The collision types are divided based on the energy of colliding electron and the cross section σ of the target species[27]. The following list categorize the electron impact collisions with respect to energy. During Elastic collision, no energy loss takes place. Excitation reaction is an inelastic collision in which collided particles loose their energy but the energy loss is below the ionization threshold (the threshold depend on the cross section of neutral atoms). When ionization occurs the energy loss of impacted electron pass the threshold. Attachment includes collisions which result in the electron attaching to the target species, forming a negative ion Plasma-Sheath Region The plasma sheath is a phenomenon happening naturally in HET s acceleration channel, or any plasma medium that is bounded by walls[4]. In the region close to the walls, since the electron mobility is higher than ions they get depleted quicker due to absorption of the walls. Accumulation of these electrons near the walls arises a potential difference that repel electrons and consequently absorbs ions. The electric field generated in this region (plasma-sheath region) due to potential difference causes a substantial increase on electron mean energy. Therefore, the electron number density spans by 10 orders of magnitude over the plasma-sheath region[28]. The best

30 CHAPTER 4. FLUID MODEL OF PLASMA AND EROSION MODEL21 way to avoid this, is by taking the logarithm of electron number density and energy density[25]. This avoids the division of Equation 4.24 by zero during evaluation. Thus, we should substitute N e = lnn e and E n = lnn ɛ into Equations 4.9 and 4.15 respectively, to stabilize the original equations for the rate of change of electron density and energy[27]. In addition, the electron temperature can be simplified to T e = 2 3 ɛ. (4.25) Equation 4.25 is a very important analysis tool to analyze the ionization rate within a plasma medium Diffusion and Mobility in Magnetic Field Recalling the Equation 4.19 of the electron flux, the first term on the right hand side of the equation describes the electron motions in the presence of the electric field by means of the electron mobility µ e. While the diffusion is related to the Einstein s relation, it is defined by electron diffusion coefficient D e in the second term of the Equation 4.19 [25]. Due to presence of a magnetic field the mobility becomes a tensor[25]. The inverse mobility tensor can be written as µ 1 e = 1 µ dc B z B φ 1 B z µ dc B r, (4.26) 1 B φ B r µ dc where B r, B z, and B φ are the radial, axial, and azimuthal component of the magnetic field. The actual expression for the electron mobility in presence of DC magnetic field can not be compacted instead the quantity µ dc is introduced to compact the mobility tensor. The quantity is the mobility in absence of magnetic field. The mobility tensor can be multiplied with electric field vector to replace the E B drift[29]. Moreover, the electron diffusivity, energy diffusivity and energy mobility can be computed based on the electron mobility tensor as: D e = µ e T e, (4.27) D ɛ = µ ɛ T e, (4.28) µ ɛ = 5 3 µ e. (4.29)

31 CHAPTER 4. FLUID MODEL OF PLASMA AND EROSION MODEL Heavy Species Transport In COMSOL Multiphysics, the transport of heavy species is specified through modified version of the Maxwell-Stefan equation ρ δ δt (ω k) + ρ(u )ω k = j k + R k, (4.30) which is the only expression that conserves the total mass in the system and also satisfies a number of auxiliary constraints[27]. In Equation 4.30, j k is the diffusive flux vector, R k is the rate expression for species k (there are two types of heavy species here: Xe and Xe + ), u is the mass average fluid velocity vector, ρ is the density of the mixture, and ω k is the mass fraction of the kth species. The expression of diffusive flux vector is j k = ρω k V k, (4.31) where V k is the multi-component diffusion velocity for species k (Xe or Xe + )[27]. The expression of V k depends on type of diffusion model that are defined as: Mixture-averaged and Fick s law. The Mixture-averaged is the more accurate one and it fulfills all the criteria for mass conservation (the author has chosen the Mixture-averaged option for modelling of SPT- 100 HET)[27]. The expression for V k is V k = D k,m ω k ω k + D k,m M n M n + DT k ρω k T T z kµ k,m E, (4.32) where D k,m is the Mixture-averaged diffusion coefficient, M n is the mean molar mass of the mixture, T is the gas temperature, Dk T is the thermal diffusion coefficient for species k, z k is the charge number for species k, µ k,m is the Mixture-averaged mobility for species k and E is the electric field. The Mixture-averaged diffusion coefficient D k,m for species k can be specified as: D k,m = 1 ω k j the mean molar mass M n 1 M n = the mole fraction for species k Q k=1 Q x j /D kj, (4.33) j k ω k M k, (4.34) x k = ω k M k M n, (4.35)

32 CHAPTER 4. FLUID MODEL OF PLASMA AND EROSION MODEL23 and the mobility tensor for Xe + is calculated from Einstein s relation µ Xe + = qd k,m k B T. (4.36) Moreover, the electric field is derived from the electric potential Φ. The electric potential is calculated using the electron and ion densities through Poisson s equation[27][1] 2 Φ = q ɛ 0 (n Xe + n e ), (4.37) where ɛ 0 is permittivity of vacuum, and n Xe + is the Xenon ion density that can be obtained using DD Equation 4.9 for ions as: δ δt (n Xe +) + (n Xe +µ Xe +E D Xe + n Xe +) = R Xe +. (4.38) 4.5 Source Terms The creation and loss of particles must be captured to model plasma discharge. The inelastic reactions are responsible for the creation and the loss of particles[30]. The calculation of the source coefficients are carried either using experimental data or cross section data. In case of using cross section data the term is called the rate coefficient[27], and the rate coefficient of a particular reaction j is obtained as R i = j σ j x j Nn e ν j. (4.39) However, when Townsend coefficients, which are the measure of the number of collisions an electron will experience as it travels through 1 cm of the plasma[27], are used the source term equation is modified to R i = j α j x j N Γ e. (4.40) The fundamental difference between these two source terms is that the rate coefficient is based on the particle density and the cross section data whilst the Townsend coefficients are derived from the total incoming flux of particle. Generally, the Townsend coefficients lead to more accurate results when the ionization is driven by an electric field[30][27]. Here, the rate coefficient approach is chosen due to simplicity.

33 CHAPTER 4. FLUID MODEL OF PLASMA AND EROSION MODEL Neutral Atoms A one-dimensional axial flow of neutral Xenon particles is included to simulate the constant mass flow rate of 5 mg/s. The gas flow is assumed laminar, adiabatic, and inviscid. The neutral gas is not driven by any pressure or velocity differences on the boundary condition. Hence, the mass flow of neutral particles is driven only by diffusion. The governing equations of the flow is based on compressible Naiver-Stokes equations[27]. 4.7 Magnetic Field As it was mentioned before, the main application of magnetic field in HET is to trap electrons and to prevent their motion toward anode. Only the electrons get magnetized, and the only external force acting on ions is the electrostatic force due to the fact that the Debye length λ D (determines the distance over which Coulomb forces of individual particles are effective) of ions are larger the physical length of the acceleration channel[1]. Additionally, the magnetic field modelling has direct impact on the structure of the plasma that can affect the accumulation location of the electrons consequently ionization. Nonetheless, the most notable effect is the location of erosion in the acceleration channel[4]. In this thesis, the modelling of the magnetic field is carried out through application of DC magnetic field interface in COMSOL. The magnetic field formulation is derived from Ampere s law. The modified version of Ampere s law for a magneto-static field can be written as: (µ 1 0 A M) σv ( A) + σ φ = J e, (4.41) where J e is an externally generated current density, v is the velocity of the conductor, A is the vector field, φ is the electric potential, µ 0 is the permeability of the conductor, and M is the magnetic moment. It should be noted that the magnetic field is only applied on radial direction while the magnitude on other directions are set to zero[31]. 4.8 Erosion Model The sputter yield will be modeled using the charged particle tracing module of the COMSOL. This module is a particle based model and trace every single sputtered atoms.this module is a custom tool to trace the charged particles trajectories in the presence of the external fields[32]. It provides a

34 CHAPTER 4. FLUID MODEL OF PLASMA AND EROSION MODEL25 Lagrangian approach of a problem by solving ordinary differential equations (ODEs) using Newton s law of motions. The ODEs are solved for every single particle (Xenon ion in this case) and every component of the position vector to determine the trajectories[32]. The equation of motion for particles is based on Lorentz formulation after manipulation of the Lagrangian equation. The Equation 4.42 shows the Lorentz force formulation. The effects of the ion-surface material interaction can be captured through available sub-nodes such as particle-matter interaction and particle counter[32]. This way, the impact of ions on dielectric walls as well as the number of impacted ions that can be removed from the solid surface atoms are determined. F L = qe + q(v B) (4.42) 4.9 Remarks On Fluid Model Developing a HET model using the DD technique requires many assumptions to simplify the process. As a result, many details of electron velocity distribution and microscopic electron-field interaction are ignored during the process, and only the macroscopic motion and interactions are conserved. However, macroscopic details are not sufficient to describe the dynamics of magnetized electrons[33]. One of the important problem associated with ignorance of the microscopic dynamic, is the lack of knowledge on cross-field electron mobility. The experimental data shows that the magnetized electrons are more mobile and vibrant than their description in Equation 4.21[34]. The shortcoming could be due to microscopic turbulent electron-field interactions that are not illustrated well enough in classical cross field diffusion (Equation 4.21)[33]. One of the proposal to overcome the problem is introduction of Bohm diffusion D B = 1 k B T e 16 qb. (4.43) Although Bohm diffusion can be considered as an improvement over the classical diffusion, it still does not accurately describe wide range of plasma discharge[30]. This phenomenon, high flow of the electrons across magnetic field lines is called anomalous electron transport and it has also been observed in particle simulations. Currently, the only way to improve the results even further than Bohm approach is to establish a relation between the empirical transport parameters and the given discharge configuration of HET[33]. Apart from the anomalous electron transport, the DD is only valid if the mean free path is small with respect to plasma dimensions and gradient

35 CHAPTER 4. FLUID MODEL OF PLASMA AND EROSION MODEL26 length since it is derived from the local balance between forces and collisional momentum loss. It should be added that electron fluid equations are not comprehensive enough to describe the ionization process in an accurate way. The ionization often occurs due to the motion of high speed electrons, whose their behavior different from the average electrons speed described in fluid equations. Moreover, their properties such as temperature are not well captured using fluid equations[4][33]. Additionally, the DD approach is not particularly well enough to describe the low-pressure plasma discharge whereas the typical pressure condition of HETs is in order of 0.1 mtorr - 20 mtorr. Therefore, the DD approach of fluid model can not deliver very accurate results. Thus, it is customary to model the plasma below 1 Torr with PIC codes[33]. Nonetheless, the fluid model could provide initial knowledge of plasma behaviour and computational cost advantage with respect to other numerical methods.

36 Chapter 5 Particle-In-Cell Plasma Model and Erosion Model The fluid modeling of plasma is often incomplete and gives inaccurate results in low pressure or density plasma such as rarefied gas and partially ionized gas, since the plasma can no longer be treated as continuous medium[4]. In these situations, Kinetic modelling can be addressed to the problem. However, the direct application of the integro-partial differential BE for solving the distribution function requires very complex boundary conditions and usually is not practical[35]. One way of getting around this problem is to follow the trajectory of finite size discrete particles over the finite computational grid. Such technique is called Particle-In-Cell (PIC) method. The PIC method is linearly dependant on the number of particles N existing in simulation, and the electromagnetic fields acting on charged particles are computed on the finite number of grids[35][36]. 5.1 Particle-In-Cell Methodology The PIC method is the most frequently used approach to study numerically the plasma dynamics on the kinetic level[35]. The introduction of finite size particles, that has mainly applied to reduce the number of particles in a simulation in comparison with the physical system, is the key property for popularity of this method[37]. In the PIC method, the distribution function computed using a Lagrangian approach, where ordinary differential equations are solved for the electron position and velocity[38]. The finite size particles are also known as superparticle. Each superparticle is in fact, represents a collective cloud of physical particles[37]. Such assumption avoids the singularity at zero separation when two point parti- 27

37 CHAPTER 5. PARTICLE-IN-CELL PLASMA MODEL AND EROSION MODEL28 cles reach together due to increase of acting forces on them such as Coulomb force. As Figure 5.1 shows when the distance between two superparticles are getting smaller, and finally overlap each other, the force on the (overlap) area becomes zero. Similarly at zero separation, the force is entirely zero. As a result, a system defined with the finite size particles never reach singularity at zero separation[37]. In addition, the application of finite size particle technique in PIC method reduces the potential energy for the same kinetic energy, which means less interaction between particles and finally accurate plasma particles by using fewer number of particles than the physical system[37], which can be interpreted as less complexity and computational cost. Furthermore, in a fully PIC method the Coulomb interaction between particles becomes collisionless so that the collisions should be re-introduced as sub-grid phenomenon using the Monte Carlo methodology, which is going to be discussed in section 5.2. Figure 5.1: Finite size particle[37] The concept of operation for PIC based codes are as follows: first of all, charge density for each cell is determined by weighting particle onto a computational grid at a certain time t. Then, calculated electric and magnetic fields through solving Maxwell s equations are integrated back to the particle at t + 1 timestep as leapfrog scheme (Figure 5.2), to move particle into its new location[16]. The cycle repeats for convergence. The Figures 5.3 and 5.4 explain the cycle in several steps Numerical Approach The most common PIC codes for HET simulation are often conducted in 2D axisymmetric, some of these codes are HPHall[18], StarFish[41], and the VORPAL[42]. VORPAL is a commercially available full PIC code that is originally developed by the University of Colorado and later it has been incorporated into the Vsim plasma package of the Tech-X company[43]. In PIC method, a specific functional form is assigned to each superparticle for its distribution. The phase space functional form has a number of free parameters whose time evolution will determine the numerical solution of

38 CHAPTER 5. PARTICLE-IN-CELL PLASMA MODEL AND EROSION MODEL29 Figure 5.2: Visual representation of leap-frog algorithm[39] Figure 5.3: The Simple Flowchart of PIC-MCC Method[40] Vlasov equation[37]. In other words, the number density per unit element of phase space determines the phase space distribution function which is the definition of Vlasov equation 1 in an ideal plasma δf δt + v xf + q m (E + 1 c v B) vf = 0, (5.1) where c is the speed of the sound. On the scale of simplicity the PIC codes can be divided into electrostatic and electromagnetic cases[44]. In electrostatic 1 The equation is written for 1D, since expansion to 3D is not difficult but here it avoids complicated notations

39 CHAPTER 5. PARTICLE-IN-CELL PLASMA MODEL AND EROSION MODEL30 Figure 5.4: Illustration of PIC procedure for a single particle per cell[7] case, the field is calculated using Poisson s equation ρ = ɛ 0 δ 2 φ δx 2, (5.2) and the fields in electromagnetic case are obtained by solving Maxwell s equations[38] E = δb δt, (5.3) B = µ 0 ɛ 0 δe δt + µ 0J, (5.4) B = 0, (5.5)

40 CHAPTER 5. PARTICLE-IN-CELL PLASMA MODEL AND EROSION MODEL31 E = ρ ɛ 0. (5.6) 5.2 Monte Carlo Collisions The two most commonly used methods to simulate stochastic (a system with a time dependent random variable, such as HET) collisions are, direct simulation Monte Carlo (DSMC)[44] [45], and the Monte Carlo collisions (MCC)[46]. Here, the focus of the study is on the MCC method since the MCC method is the most common method in HET simulation models due to introduction of less complexity compare to DSMC method[35]. Moreover, in Vsim the interaction between different species are randomly modeled using a built-in block known as MonteCarloInteractions that is based on the MCC approach[47]. The main principles of MCC is listed as follows, to provide initial insight regarding this methodology as full study of MCC is not in the scope of this thesis (see Szabo[16]). The probability of collisions between each particle of species a with species b is calculated as P a,b = fn(n b, v a,b, Q(v a,b )). To specify whether an event takes place or not, the probability P a,b is compared to a random number. A collision cross section σ, is used to determine the occurrence of collision. The momentum of the initial particle varies momentarily with some magnitude/vector, if an event occurs. The magnitude of the change must be determined randomly. The application of MCC is most efficient where it represents the collisions between species with different masses and velocities m 1 m 2 << 1. Due to the fact that there is no clear particle pairing in MCC, the calculation of the relative velocity is conducted through average quantities of one species in a given cell[35]. Therefore, electron-neutral collisions and arguably electronions are the best cases of MCC application. In addition, since there is no need for particles to be individually paired together to specify the occurrence of the collision, the computational steps are reduced[44].

41 CHAPTER 5. PARTICLE-IN-CELL PLASMA MODEL AND EROSION MODEL Erosion Model In Vsim, the impact of ion bombardment on dielectric walls is modeled using a built-in block known as Sputter. The sputter can simulate the emission of neutral atoms from the solid surface by considering their incident angles and deposited energy. The physics of the sputter yield process is based on Yamamura[48] sputtering model. Only ions are capable of removal of solid surface atoms. Moreover, the underlying block is responsible for determining the number of sputtered atoms[49]. 5.4 Particle-In-Cell Stability Limits In application of PIC method, there are three important parameters that restrain the accuracy of numerical solution. These parameters are listed as: the grid-cell width r, the time step t, and the superparticle density n o. Therefore, the selection of these parameters must be done very carefully to ensure the simulation results are only affected by the physics, not the numerical characteristics[37][50]. Furthermore unlike the fluid method, the PIC method is a very well suited choice for inspecting and analyzing sheath phenomenon[28]. Thus, the parameters should be chosen as small as possible to increase the accuracy. Unfortunately, this would increase the computational cost of simulation as well. Therefore, a trade-off study is applicable to determine the particular parameter s scales Grid Cell Effect The grid-cell width r, is chosen such so that it will be able to resolve the Debye length of the electron[50]. Such condition is required to avoid finite grid instability originating from coupling of the Fourier modes[37]. This phenomenon that is known as aliasing, occurs due to attempt in solving a continuous function using finite sampling points. The oscillation due to aliasing varies with the factor of 2π/ r. More detailed analysis and derivations are available in [38]. The condition to avoid such phenomenon is r < ζλ D, (5.7) where r is the grid cell size, ζ is a constant of order 1 whose exact value depends on the choice of interpolation and assignment functions, and λ D is Debye length[50].

42 CHAPTER 5. PARTICLE-IN-CELL PLASMA MODEL AND EROSION MODEL Time Step Effect The time step must be chosen so that it could resolve both Langmuir wave and light wave[37]. In a general case, the electron plasma period is the shortest time scale. This means adapting the electron plasma period will resolve the ion motion as well. Therefore, the time step of simulations are chosen based on the electron plasma frequency as ω pe t < 2, (5.8) where ω pe is the electron plasma frequency, and t is the time step[50] Density of Computational Particles Due to graininess characteristic of plasma as particles, the density fluctuate once the particles passes over the boundaries of the cells. The generated noise as a result of fluctuation, affects potential and electric fields. As, electrons are sensitive to such oscillation in the field, they would respond with non-physical increase of energy[50]. Therefore, the density of superparticle should be kept at a number to avoid the fluctuation of the field. The required condition to be held is χ = (ω pe) 2 << 1, (5.9) 2N where N is the number of superparticles, and χ denotes the condition that must be hold[51].

43 Chapter 6 Implementation of Models 6.1 Fluid Model Implementation To model SPT-100 (Figure 6.1) in COMSOL, an axisymmetric, time-dependent simulation was conducted. In this way not only the total simulation time decreases but also it allows us to ignore the azimuthal direction in the model. This helps to avoid the application of 3D model. As it was discussed before, the fluid model is not capable of delivering accurate results in low pressure, this is the reason the low-pressure discharges below 1 Torr is usually modeled by the PIC model[33]. Hence, to investigate the effect of pressure that was described in section 4.9, first a non-magnetized model is performed, then the magnetized model is simulated Non-magnetized Model The main input parameters and physical dimensions of the model are listed in Table 6.1. The selected internal boundary material is h-bn, and its thickness is 5 mm while the anode is modelled as metallic boundary. The hollow cathode that inject the required electrons to initiate and maintain ionization is located at the end of exit region and its boundary modeled as dielectric with zero charge accumulation. The external boundary conditions are set to ground potential to simulate the effect of the neutralizing hollow cathode. Furthermore, the exit boundaries are meant to represent the plasma plume region and the outer space so they are set to 0 V potential. As a result, all the ions and electrons in contacts with these boundaries are considered to be lost. The Schematic view of this model is shown in Figure 6.2. The model has started with uniform plasma comprised of singly charged ions (the double charged-ions are ignored), electrons, background gas, and excited Xenon atoms. These species are the most dominant at the electron 34

44 CHAPTER 6. IMPLEMENTATION OF MODELS 35 Figure 6.1: The view of a SPT-100 thruster Table 6.1: Input Parameters of the simulation SPT-100 Operation Condition & physical characteristics Variable Magnitude Unit Initial pressure Torr Initial plasma density 1E16 1/m 3 Initial electron energy 25 V Nominal discharge voltage 300 V Discharge current 4.5 A Mass flow rate(anode) 5 mg Channel length 25 mm Radial magnetic field 0 G Channel r max 50 mm channel r min 35 mm energies of interest, and they have much greater cross section than other excited species so that the other species were neglected. Hence, the dominant electron impact reactions in the plasma model can be listed as:

45 CHAPTER 6. IMPLEMENTATION OF MODELS 36 Figure 6.2: Schematic view of the Non-magnetized model in COMSOL Xe + e Xe + e Elastic, (6.1) Xe + e Xes + e Excitation, (6.2) Xe + e Xe + + e Ionization, (6.3) Xes + e Xe + + e two wise Ionization. (6.4) Finally, the effect of ion bombardment of dielectric walls are modeled through adding a surface reaction node. The node assures the release of a BN atom, as an ion with sufficient energy collides with the dielectric wall. The equation for such surface reaction is Xe + Xe + BN. (6.5) Here, collisions are modelled based on their cross section data that were taken from LXCAT databases[27]. From the cross section data, the two-term

46 CHAPTER 6. IMPLEMENTATION OF MODELS 37 approximation of the BE (section 4.2) is solved to create swarm coefficients for ionization reactions. These coefficients were specifically used since a more accurate description of the ionization in the direct discharge plasma model can be achieved. The effect of thermodynamic and convection are ignored, and the reduced electron transport is chosen to model the electron. This means the electron and ion transport tensor calculation is cancelled to observe the evolution of plasma without the magnetic field. The tested values of plasma pressures are 0.5 Torr, 0.05 Torr and Torr. The electric and chemical reactions of the plasma model with physical boundaries are defined as follows. All excited species in contact with the walls lose their excitation energy and revert back to neutral gas atoms. All ions in contact with the walls lose their ionization energy and revert back to neutral atoms. The temperature increase of the wall due to collision of the ions and excited atoms is not considered. The emission of secondary electrons due to collision is ignored. The accumulation of electrons on metallic boundaries can be ignored due to the magnitude of applied voltage. However, the surface charge accumulation on dielectric surface is accounted by δ δt (ρ surf) = n J e + n J Xe +, (6.6) where the J e and J Xe + are the electron and ion currents normal to the surface, respectively. And n is the plasma density Magnetized Model At this section the input parameters listed in Table 6.1 is applied here as well. The exception is the magnitude of magnetic field that has changed to 120 Gauss since it is an optimal value for SPT-100, and the same magnitude is applied in the PIC model. The magnetic field is applied through external coils with operation current of 4.5 A. The boundary description and collisions are kept similar to the non-magnetized model. To capture the effect of the magnetic field, the computation of electron and ion tensors are included. To exclude the pressure instability of fluid method and finite element approach, the optimal pressure found in the nonmagnetized model will be applied. The schematic view of magnetized model is shown in Figure 6.3.

47 CHAPTER 6. IMPLEMENTATION OF MODELS 38 Figure 6.3: The schematic view of the magnetized model in COMSOL 6.2 Implementation of Particle-In-Cell Model The kinetic modelling of SPT-100 in this thesis has been carried out in Vsim code. The Vsim is an electromagnetic fully PIC code, which solve the motion of particles and the kinetic equations of plasma based on Finite-Difference- Time-Domain (FDTD) numerical algorithm. The Vsim runs on Vorpal engine that is an electromagnetic computational framework. It was first developed by University of Coloroda, and later integrated into the Vsim after acquisition by Tech-X company[42]. The Figure 6.4 shows the setup of SPT-100 simulation in the Vsim. The model is designed for a 2D axisymmetric cylindrical geometry. The length of acceleration is set to 25 mm, and an extra 10 mm of length is added to accommodate the cathode (electron source) in the model. Furthermore, it allows sufficient space for expansion of the plasma after exiting the channel. The anode is set to 300 discharge and the maximum magnitude of the radial magnetic field is 120 Gauss. The inner thickness of the dielectric material (h-bn) is 10 mm, and the thickness of dielectric material at the outer wall is

48 CHAPTER 6. IMPLEMENTATION OF MODELS 39 5 mm[43]. The input parameters are similar to the conditions used in fluid model to allow the comparison. Figure 6.4: Schematic view of SPT-100 setup in Vsim [43] An electron source is located right after the channel exit to model the neutralizer cathode and the cathode s potential is set to zero. The exit boundaries are also set to 0 V to simulate the outer space. The neutral gas considered as an one dimensional static fluid background[43]. Furthermore, the initial electron energy is 25 ev, and the simulation ran for time steps with step time of The model initiated with a uniform plasma containing both electrons and Xenon ions, where its number density is equal to Finally, the sputter yield process is modeled using a built-in block in Vsim known as sputter.

49 Chapter 7 Results and Discussion 7.1 Implementation Remarks on Fluid Model As it was discussed in section 6.1, to find an optimal pressure for the magnetized model, the non-magnetized model is set. The increase of pressure comes with the cost of getting further away from the actual model. Additionally since COMSOL utilizes the DD approximation, the accuracy of results degrade at low pressures. Thus, a trade-off is required to capture the optimal pressure. This way one of the limiting factors introduced by the DD approximation can be ignored for the rest of simulation attempts. Additionally, it helps to focus on reaching initial results by the magnetized model Non-magnetized Model During this stage, the time and other input parameters are kept constant and the pressure was changed from Torr to 0.5 Torr. In the first attempt, the simulation time set to 10 ns and the pressure was set to Torr. After a long operation time (order of hours), the simulation was interrupted (Figure 7.1), knowing that it would not converge to provide results. Next, the simulation time kept constant while the pressure increased to 0.05 Torr. However, the simulation was not conclusive for such pressure as well (Figure 7.2). The curves in these figures, indicate the elapsed number of time steps carried out by the solver to reach the steady state. The interpretation of the cyclic path (specially in Figure 7.1) marks the crucial instability of DD approach in low-pressure plasma. The reason for such instability can be referred to the fact that, at very low pressure the medium inside acceleration channel is a rarefied flow rather than a fluid. In the rarefied flow, the Knudsen 40

50 CHAPTER 7. RESULTS AND DISCUSSION 41 number, which is defined as the ratio of the mean free path and characteristic length of flow is above To clarify, the DD approximation fails under the rarefied flow condition since there is no sufficient rate of collisions to maintain the velocity distribution function[52]. Furthermore, the simulation attempts indicate that implementation of low-pressure plasma would require high computational time, which is utterly opposite to the main goal of this thesis (low computational time). At the last attempt, the pressure was set to 0.5 Torr the simulation while the simulation time was kept constant. As seen in Figure 7.3, the simulation was conclusive after 90 time steps. The comparison of time steps between the 3 attempts highlights the dramatic reduction in last attempts. As a result, observing such instability in non-magnetized model provides an insight regarding the computational challenge that would be generated during the magnetized model simulation. Figure 7.1: The evolution of plasma for p = Torr The electric potential distribution after 10 ns of simulation is shown in Figure 7.4. The peak value of electric potential is seen near the anode (bottom end), and it begins to drop along the channel. The main potential drop occurs within the acceleration channel noticeably at the exit plane, moreover, the potential drop continues at the plume region. The behaviour of the electric potential inside and outside of the channel agrees with PIC result in Figure 7.8. The potential drops can also be seen along the channel walls, as the walls are covered with dielectric materials and their corresponded poten-

51 CHAPTER 7. RESULTS AND DISCUSSION 42 Figure 7.2: The evolution of plasma for p = 0.05 Torr Figure 7.3: The evolution of plasma for p = 0.5 Torr tials are set to ground. Finally, the electric potential reaches the minimum value at the exit wall (top end), where its exterior electric potential is also set to ground. The electron density provides an excellent knowledge about the location

52 CHAPTER 7. RESULTS AND DISCUSSION 43 Figure 7.4: Electric potential distribution in the non-magnetized model after 10 ns of ionization and erosion. In a full HET model, it is expected to have the peak of electron density at the exit region as the applied radial magnetic field lines traps the electrons. However, in a non-magnetized model due to existence of strong electric field and absence of magnetic field the bulk of electrons migrate toward the source of electric potential (anode) as seen in Figure 7.5. The peak value of electron density is near the anode and it begins to degrade in a small amount toward the channel exit. However, a sudden decrease happens at the exit plane. Furthermore, the electron density drops along all the walls due to exclusion of secondary electron emissions. It means collisions of electrons with channel walls do not produce secondary electrons. Since the external walls model the outer space, all the electrons in contact with these boundaries are lost, hence, the lowest electron density is visible at the exit boundary (top end). The electron temperature determines the mean kinetic energy of electrons that directly affects the ionization process. Particularly, higher electron temperature leads to higher ionization. Hence, it is a definitive parameter on eroded depth as high ionization can be interpreted as a high number of ion bombardment. Additionally, if the electron temperature increases too much, the number of doubly charged Ions increases which makes them as decisive species for erosion. However, in this model only the impact of singly charged

53 CHAPTER 7. RESULTS AND DISCUSSION 44 Figure 7.5: Electron density distribution in the non-magnetized model after 10 ns Ions are included. In a typical HET, the electron temperature reaches the peak value at the exit region, where bulk of electron density exist. However in Figure 7.6, there is small change of electron temperature from the anode toward the exit plane. This can be referred to Equations 4.25 and 4.24, where the electron temperature depends on electron energy density and electron number density. Since the variation of these two parameters are proportional to each other, the variation of electron temperature behaves accordingly. At the exit plane, there is a sudden drop of electron temperature and reaches to near zero at the exit boundary as both electron energy and electron number densities decrease dramatically. The overall results of unmagnified model matches with physics of plasma discharge. Moreover, it has provided an optimal pressure value to be used in full model Magnetized Model The full model of a typical HET is completed by adding the magnetic coils around the acceleration channel as it is described in Figure 6.3. The modelling of a low-pressure and magnetized plasma comes with many issues that were described in section 4.9. Particularly, the electron anomalous is of great-

54 CHAPTER 7. RESULTS AND DISCUSSION 45 Figure 7.6: Electron temperature distribution in the non-magnetized model after 10 ns est concern. In order to overcome this issue, the electron mobility is modeled according to the Bohm diffusion approach as described in Equation Such technique has already been used in hybrid code where electrons were modeled as fluid and ions are modeled as particles[53]. As seen in Figure 7.7, the time-dependent solver was reached singularity after time steps (order of days). The long simulation time indicate the computational challenge caused by magnetization. This computational challenge must be aroused from a high ratio of cross-field electron mobility to electron mobility along the magnetic field lines that can reach up to in COMSOL[27]. Due to such high ratio, the solver can not converge even with the application of Bohm diffusion approach, thus, singularity occurs. Further attempts were carried out by increasing the pressure and the initial plasma density while the simulation time has kept constant. Next, the simulation time has also decreased to check the effect of time. At least, more than 30 different attempts were tested to obtain a conclusive results. However, each time singularity prevented from obtaining the result. Despite the fact that obtaining a meaningful erosion profile requires hours of operations. Hence, even if the simulation for 1s would have been successful the simulation time must have been increased in order to obtain acceptable results for lifetime prediction. This means due to the current shortcomings (the electron

55 CHAPTER 7. RESULTS AND DISCUSSION 46 Figure 7.7: Visualization of time-dependant solver of magnetized model anomalous and low-pressure instability) using such model is not advisable for lifetime prediction. 7.2 Results of Particle-In-Cell Model The contour plots of the electric potential generated by anode discharge voltage is displayed in Figure 7.8. The Z axis represents the axial direction and the vertical axis represents the R direction. As seen in the figure, a uniform electric potential throughout the acceleration channel is maintained. The steady state reaches after time steps, when the bulk of plasma inside the channel has almost 10% higher (328 V) voltage than anode voltage. The excessive voltage is due to electron energy in HETs. As the peak of electron energy in HETs reaches to 10% of discharge potential[54]. Finally, the plasma sheath near the inner and outer cylinder walls can be seen since the plasma is confined. As it was mentioned before, the plasma sheath occurs naturally when the plasma is bounded by the walls. The strength and distribution of the magnetic field is crucial in HETs, since the interaction of electric and magnetic fields ensure the confinement of electrons and acceleration of ions. Moreover, the erosion location is directly

56 CHAPTER 7. RESULTS AND DISCUSSION 47 Figure 7.8: The exhibition of uniform electric field along the channel after time steps related with distribution of the magnetic field. It means the dislocation of the magnetic field causes misinterpretation of erosion profile location within the channel[4]. The optimization of the magnetic field is based on power consumption and the electron confinement. While a stronger field increases the chance of confinement it increases the power consumption as well. On the other hand, the weak confinement leads to migration of electron field side-wise toward the anode and outside of the channel. Outside of the channel the neutralized ion beam can gain a radial component, which results in the reduction of thrust and thruster s efficiency. Moreover, the migration of electrons towards the anode reduces the biased potential of the thruster, which again results in the reduction of thrust. This is the reason the magnetic field is not stretched near the anode, as it is shown in Figure 7.9. In this simulation, an optimal magnetic field strength for SPT-100 of 120 G is chosen[42]. The color distribution in Figure 7.9 illustrates the magnitude of the magnetic field, the lighter the color the more intensified magnetic field. As it is visible from the figure, the lightest area is attached to the bottom wall. This shows the magnitude of the magnetic field is highest near the inner radius of the acceleration channel, and it diminishes to near zero close to the outer boundary. Lastly, the strength of the field is only available in radial direction R since the interaction of electric and magnetic fields are exerting the azimuthal Lorentz force on electrons and the axial electrostatic force on ions.

57 CHAPTER 7. RESULTS AND DISCUSSION 48 Figure 7.9: Intensity & location of the magnetic field after time steps In Figure 7.10, the bulk of electron population is concentrated near exit region, this proves the confinement of the electrons by the magnetic field lines. The electrons injected from right end (channel exit) are accelerated toward the biased anode wall on left end, where the hollow cathode is located (Figure 6.4). Additionally, a less denser electron region still exists close to the anode, which proves the cross-field electron motion in PIC code. The electron accumulation near the exit region (Figure 7.10) of the acceleration channel increases the probability of electrons-neutral collisions. In HETs, the neutral propellant atoms are fed from left end, where the anode exists. To simulate the neutral flow, a continuous axial flow of neutral atoms are included in the model as shown in Figure The flow is linearly varying with a peak density (as color distribution implies) near the anode, and diminishing density towards the channel exit. The flow of background gas has a fixed mass flow rate of 5 mg/s, as it is the optimal value for most SPT-100 thrusters. Near the exit region, the neutral gas atoms meet the highest density of energetic electrons. However, due to existence of electrons throughout the channel the ion particles can also be seen along the channel with a less denser region near the anode as seen in Figure The color distribution (the brighter it is, the denser it is) in Figure 7.12 shows that the peak of ionization begins at 5 mm - 6 mm from the channel exit. As the electron s density increases toward the exit, the number of ionized neutral atoms increases as

58 CHAPTER 7. RESULTS AND DISCUSSION 49 Figure 7.10: Phase-space distribution of electrons after time steps Figure 7.11: The linear flow of background gas in the channel after time steps well. The higher number of Xe + ions at this region should be interpreted as high risk of ion bombardment of insulation material. In addition, due to a

59 CHAPTER 7. RESULTS AND DISCUSSION 50 high number of collisions and high electron mean energy at this region the generation of doubly charged ions increases. The double-charged ions Xe 2+, with twice as energy as normal ions can dramatically increase the sputter yield process. At the plume region, the ions start to diverge. The divergent occurs due to the fact that there is no physical wall to confine the ions. Additionally, the electrostatic force acting on ions are reduced drastically to near zero, allowing the ions to gain radial component. Figure 7.12: Phase-space distribution of Xe + after time steps particles along the channel The result of erosion modelling of a SPT-100 after time steps is shown in Figure This figure shows the beginning lifetime erosion of the insulation material. Although time steps are not sufficient to draw a full erosion profile of the acceleration channel, it provides an insight on erosion profile. The figure indicates the high number of sputter yields at the channel exit, which is an expected result due to high density of energetic ions at this region. In addition, the sputtering curve shows a stretched profile of erosion along the channel, while the experimental data exhibits a shorter erosion profile. This contradiction must be referred to short simulation time that suggests a longer period of simulation for more accurate results. Therefore, a simulated erosion profile with a longer period of time is included as shown in Figure The figure compares the simulated erosion profile with experimental data over 600 operating hours. The dashed lines mark the initial location of the channel wall while the blue and the square curved lines