42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit 9-12 July 2006, Sacramento, California AIAA 2006-4656 Wall Erosion in 2D Hall Thruster Simulations Emmanuelle Sommier *, Michelle K. Allis, Nicolas Gascon, and Mark A. Cappelli Stanford University, Stanford, CA, 94305 Hall Thruster degradation due to wall sputtering by heavy particle bombardment is assessed by developing and adding wall erosion process to a 2D radial-axial hybrid PIC model of the discharge. The simulation includes ion-neutral charge-exchange and momentum-exchange collisions as the former is found to have a significant effect on the instantaneous wall erosion rate. The erosion of the Stanford Hall Thruster channel walls was simulated for a time of 4300 hours leading to wall erosion of 2 mm. The potential impact the progression of eroded walls may have on the discharge characteristics in the Stanford Hall Thruster is presented. The erosion process is found to decrease the thruster performances due to a gradual decrease in the plasma density. Finally, in order to verify the accuracy of the erosion model, the discharge geometry is adapted to an SPT-100 Hall thruster. A comparison is made between the simulated erosion rate and experimental erosion rate data for the SPT-100. The results suggest that the computed ion flux to the channel walls is too low, probably due to an oversimplified wall sheath and presheath model. Nomenclature A = area e = electron charge E = electric field g = relative velocity h(z,ϕ,ε) = incident angle and energy distribution function k = Boltzmann constant K = thermal diffusivity M = mass ˆn = direction normal to the magnetic field contours n e,i = plasma/ion density n n = neutral density r = radial position S = sputter yield Ŝ = normalized sputter yield t = time T e = electron temperature u = electron velocity enˆ z = axial position v = velocity β = random number between 0 and 1 χ = scattering angle Φ = ion production cost γ = secondary electron emission coefficient Γ = wall flux ω = cyclotron frequency σ Τ = total cross section * Visiting Researcher, Mechanical Engineering Department, Building 520. Member, AIAA, Research Assistant, Mechanical Engineering Department, Building 520. Member, AIAA, Research Associate, Mechanical Engineering Department, Building 520. Member, AIAA, Professor, Mechanical Engineering Department, Building 520. Member, AIAA. 1 Copyright 2006 by the, Inc. All rights reserved.
I. Introduction ue to their high specific impulse, Hall effect thrusters are ideally suited for long duration space missions such Das satellite station keeping and deep space exploration. One motivation for developing a Hall thruster simulation is to construct an accurate model of the time-evolution of the channel walls due to erosion by energetic particles. The erosion of the Hall thruster channel walls is caused by sputtering due to ions and neutrals, and is a significant factor in determining operational lifetime. Since future missions will require thruster lifetimes on the order of several years, determining this lifetime experimentally in ground test facilities is both time-consuming and expensive, especially when attempting to optimize operating conditions or geometry for maximum life. A robust and accurate numerical simulation is a necessary alternative for evaluating performance and lifetime. In the present work, a sputtering model has been introduced into a two-dimensional (2D) hybrid Particle-In-Cell (PIC) simulation in order to determine the erosion behavior of the channel walls of a laboratory Hall discharge. Originally corresponding to the Stanford Hall Thruster, the computational domain extends from the anode, through the interior channel, and into the near field plume region. In order to assess thruster degradation due to sputtering, a model of the wall erosion process has been developed and added to the simulation. The model calculates the erosion rate by using the method described by Manzella et al. 10 The effect of charge-exchange collisions and momentum-exchange collisions on the sputtering process has been added to the model. Successive simulations that use the computed local erosion rate to update the geometry of the channel walls, are used to predict the erosion evolution of the channel of the Stanford Hall Thruster over a 4000 hour period. The potential impact the progression of eroded walls may have on the discharge characteristics in the Stanford Hall Thruster has been investigated. Finally, in order to verify the accuracy of the erosion model, the thruster geometry has been adapted to an SPT-100 Hall thruster. A comparison is made between numerical erosion results and experimental erosion rate data for an SPT-100. II. Numerical Model A. Main Model The 2D (radial-axial, or r-z) hybrid particle-in-cell (PIC) model used here was originally developed by Fernandez et al. 1 This model was inspired by the simulation of Fife 2 and the basic elements of this model are described in more detail by Allis et al. 3 In this hybrid model, the electrons are treated as a quasi-one-dimensional fluid governed by a generalized Ohm s law, and the ions and neutrals are described as discrete particles advanced in space using a PIC approach. An important element in the model is the ability to account for finite-rate ionization, for which the rate (per unit volume), n&, depends critically on the computed electron temperature, T e e. The transient spatially varying electron temperature profile is calculated at each timestep by solving the electron energy equation: Here, n e is the plasma density, E i is the ionization energy, 3 5 Te nekte + neuen ˆkTe K = ne Φ ( Te ) Ei γω + jen ˆEnˆ t 2 nˆ 2 nˆ & (1) ue n ˆ is the electron velocity in the direction normal to the magnetic field contours ( nˆ ), is the component of the computed electric field je n ˆ is the electron current density, En ˆ along nˆ, γω accounts for the wall damping, and K is the thermal diffusivity, which incorporates an anomalous electron transport coefficient 4. The ion production cost, Φ(T e ), is determined from the expression given by Dugan 5, and is evolved in the original nonlinear form. The terms on the right hand side of Eq. (1) represent the ionization and wall damping sink terms and the Joule heating source term. In this model it is assumed that all ions created are singly charged, and the effect of the azimuthal current on the magnetic field within the channel is neglected. In order to reduce the computational time, ions and neutrals are regrouped into macroparticles of varying sizes, which are spread over the computational grid cells. When striking the walls, neutrals are diffusely scattered back into the channel using a Maxwellian distribution, whereas ions are assumed to recombine into neutrals and scatter diffusely. The ion and neutral timestep is fixed at 25 ns whereas the electron timestep is on the order of 0.1 ns. The total amount of simulated time is 625 µs which takes one day to run on a single processor (AMD Athlon 64 3500 MHz) desktop computer. 2
In order to model the Stanford Hall Thruster, the magnetic field used as an input into the code has been generated by a magnetic model created with the Finite Element Method Magnetics (FEMM) software package 14 and accounts for the induced magnetic field generated by the azimuthal current. An experimentally based effective mobility is imposed to account for the anomalously high electron transport across magnetic field lines 6. The corresponding computational geometry has an 8 cm long and 1.2 cm wide channel, within a 9 cm (outer diameter) annulus. The nominal mass flow rate is 2 mg/s, the peak radial magnetic field is 16 mt (axial width of the distribution is 15 mm), and the discharge voltage and current are 200 V and 2.5 A respectively. The computational grid consists of 101 grid points in the axial direction and 13 grid points in the radial direction. The simulation is initialized with about 500,000 neutral macroparticles, and about 350,000 ion macroparticles. Originally corresponding to the Stanford Hall thruster, the simulation has been also adapted to the SPT-100. In the latter case, the magnetic field used was constructed from the experimental results by Dorval et al. 7 and the magnetic field configuration given by Garrigues et al. 8. The computational geometry has a 25 mm long and 15.5 mm wide channel, within a 10 cm (outer diameter) annulus. The nominal mass flow rate is 5 mg/s, the peak radial magnetic field is 16 mtesla on the center line located at the exit 7, and the discharge voltage and current are 300 V and 4.5 A respectively. The computational grid consists of 40 grid points in the axial direction and 13 grid points in the radial direction. The simulation is initialized with approximately 150,000 neutral macroparticles and 200,000 ion macroparticles. The main problem encountered in simulating the SPT-100 was that the effective mobility of this thruster was not known. Therefore, in order to solve this problem, an effective mobility was first constructed from the SPT-70 mobility given by Fife 2. Then, different configurations of the Hall parameter were tested in order to approach the discharge characteristics of a SPT-100 given by Dorval et al. 7. Anode & Injector Channel Exit Figure 1. Computed ion density and ion velocities in the Stanford Hall thruster. Figure 2. Computed ion density and ion velocities in the SPT-100. Representative results of ion density distribution (color map) and ion velocity (superimposed vectors) averaged over time are illustrated in Figs. 1 and 2. An examination of those figures indicates that the plasma density is the highest at approximately 3 cm and 1.5 cm upstream of the exit of the channel for the Stanford Hall thruster and the SPT-100 respectively. However, as shown below, in both cases, erosion takes place mainly in the last centimeter of the channel, due to the potential drop required to accelerate the ions to energies that are above the erosion threshold for the particular wall material, as discussed in the section below. 3
B. Sputtering Model Sputtering is a physical phenomenon in which atoms of a solid target are ejected due to the bombardment of this target by heavy particles of enough kinetic energy to overcome the inter-atomic binding energy of the atoms bound within the vicinity of the target s surface. In the model, the target is the ceramic insulating material which constitutes the walls of the thruster. Sputtering is driven primarily by momentum exchange between the energetic particles and the atoms of the wall. The number of atoms ejected from the surface per incident particle is called the sputter yield, and is a representation of the efficiency of the sputtering process. The sputter yield depends on the incidence angle and energy of the particle, the mass of incident and target particles, and the binding energy of atoms in the wall. The definition of the sputter yield used in the model is the one given by Manzella et al. 10 : ) ) S( z) = S( ϕ) h( z, ϕ, ε) S( ε) (2) ) Here S(z) is the sputter yield at one position z on the wall. S( ϕ) and S( ε) are, respectively, the normalized sputter yield angular dependence (shown in Fig. 3) and the sputter yield energy dependence as discussed in Ref. 10. For ) each position z, incidence angle ϕ, and energy ε, h( z, ϕε, ) represents the distribution function of the incident particles playing a role in the sputtering process, and it has been normalized at each position z on the wall by the number of the impacting particles with the wall at this position. Only ion particles with energy greater than a threshold energy of 50 ev and neutral particles with energies greater than 60 ev (the energy threshold of ions plus the ionization energy of the atom) are considered to participate to the sputtering process, since particles with energy less than the threshold energy are unable to break the bonds between atoms in the ceramic walls. To determine the volumetric erosion rate at a given position, z, along the channel wall, the calculated sputter yield is multiplied with the flux, Γ(z), of energetic particles scattering with the walls. dr ( z ) = S ( z ) Γ ( z ) (3) dt The wall flux is defined as: Γ h( z,, ) ( z) = ϕε ϕε, δ A( z) δt (4) Here, δa(z) is the annular wall area at one position z on the wall, h( z, ϕε, ) is the distribution function of the incident energetic particles, and δt is the time interval in the simulation for which the distribution function is evaluated. Figure 3. Normalized sputter yield angular dependence for xenon on boron nitride [from Ref. 10]. Figure 4. Average ion energy on the channel walls of the Stanford Hall thruster. 4
The sputtering model has been integrated into the thruster simulation as follows. Particle positions and velocities are updated for each time-step using a leap-frog method. The model then checks the particle location. In the case that a particle is found outside of the boundaries of the computational grid, the position where the particle would have collided with the wall is determined if such a position exists (i.e., some particles may exit the computational boundaries in the plume without having collided with the wall). The energy and incidence angle of the impacting particles are then calculated. If the particle is found to be energetic enough to contribute to the sputtering process, its contribution is accounted for in the distribution function that leads to the determination of the sputter yield. Finally, the overall instantaneous (local) erosion rate is calculated from the rate at which these particles collide with the surface (wall flux). It is found from the simulations that nearly all of the erosion, due mainly to high energy ions, occurs in a region very near the thruster exit. This is confirmed by examining the ion energy distribution along the walls (Fig. 4). The peak in ion energy for both inner and outer insulator walls is observed in a region close to the channel exit. In our simulations, the 1 cm long region where most sputtering occurs was divided into 14 segments. At each segment position, the model calculates the distribution function of the high energy particles impacting the walls and deduces the wall flux, sputter yields, and the local wall erosion rate. The walls are subsequently advanced (with a corresponding computational grid adjustment), and the plasma and thruster properties are then recomputed for the modified geometry. III. Results and Discussion Figures 5 and 6 present a computational prediction of the erosion history of the inner and outer Stanford Hall thruster channel walls. The simulations are carried out to an effective thruster operating time of 4300 hours. It is observed that the erosion rate decreases substantially over time, with more than half of the erosion occurring in the first 1000 hours of life. This result is in qualitative agreement with similar calculations using 1-D plasma flow models although on a considerably different thruster 8. This drop over time in the instantaneous erosion rate is believed to be mainly due to the angular dependence in the sputter yield (Fig. 3), which implies that the majority of high energy ions contributing to sputtering have incident angles greater than 60 o relative to the surface normal. It is noteworthy that by 4300 hours, the surface is predicted to erode to an angle of about 11 o relative to the thruster axis. Figure 5. Simulation of the erosion on the inner wall of the Stanford Hall Thruster. Figure 6. Simulation of the erosion on the outer wall of the Stanford Hall Thruster. 5
Figure 7. Evolution of the discharge current and thrust over time for the Stanford Hall Thruster. Figure 8. Evolution of the ion density over time for the Stanford Hall Thruster The erosion process is expected to affect the discharge operation and performance of the thruster. The effect of the erosion of the channel walls has been studied by examining the evolution of the discharge current and thrust during the simulation of the thruster for an operating time of 4300 hours. The results presented in Fig. 7 show that, as the wall is eroded, the performance of the thruster decreases, primarily as a result of diminished discharge current, likely due to the gradual decrease in the plasma density observed in Fig. 8. Figure 9. Examples of different configurations of Hall parameters used in the simulation. 6
The 2D hybrid PIC model used to simulate the behavior of the Stanford Hall thruster has been adapted to another Hall thruster, the SPT-100. The main difficulty was that the effective mobility of the SPT-100 was unknown. A first Hall parameter used in the simulation was built from the SPT-70 mobility given by Fife 2. Then, in order to get closer to the experimental data given by Dorval et al. 7, this first Hall parameter was successively altered by moving the peak position of the Hall parameter downstream or upstream, and/or increasing the Hall parameter value. Fig. 9 shows some of the different Hall parameters configurations that were tried in the simulation: a Hall parameter with a peak located a few millimeters before the exit plane (case 1), a few millimeters after the exit plane (case 2), and on the exit plane (case 3). Also considered were a Hall parameter with a peak located on the exit plane and with a higher value than case 3 at the exit plane and at the anode (case 4) and a higher value at the anode only (case 5). It was observed that an increased value of the Hall parameter at the anode by a factor of 2 or 3 leads to a decrease in the ion density at the anode that agrees more with expected results. It was also observed that a slight increase in the width of the Hall parameter peak leads to a decrease in the plasma properties (plasma density, temperature, ion velocity). Fig. 10 compares the computed results obtained for the different configurations of Hall parameter of Fig. 9, with the experimental axial ion velocity and axial electric field given by Dorval et al. 7. It can be seen that computed results match well with experimental data in cases 3 to 5. It appears that the position of the peak in electric field and the start of the ion acceleration zone is directly related to the position of the peak in the Hall parameter. Given the results from Fig. 10, the peak must be located at the exit plane. Figure 10. Computed results of axial ion velocity and axial electric field (obtained with cases 1 to 5 of Fig. 9) vs. experimental data The plots in Fig. 11 map the plasma density, the axial velocity, the electron temperature, and the plasma potential obtained with the model for the case 5 described above. Figures 12 and 13 present a normalized view of the different main plasma properties obtained from the simulations of the Stanford Hall Thruster and the SPT-100 (with the case 5 configuration). Plasma properties for the SPT-100 and the Stanford Hall Thruster are qualitatively similar and share several distinctive features. Both exhibit a local maximum in the ion velocity profile and a sharp electric field profile. Those features are expected, and are representative of experimental observations. Furthermore, the SPT-100 simulation shares greater agreement with experimental data than the Stanford Hall Thruster simulation. For example, in the case of the Stanford Hall Thruster, the electric potential is simulated to first increase from the anode inside the channel, whereas it is only decreasing in the case of the SPT-100. 7
Figure 11. Ion density, plasma potential, electron temperature and ion velocity computed for the SPT-100 Figure 12. Discharge characteristics (Hall parameter, Magnetic field, Ion density, Ion axial velocity, Neutral density, Electron Temperature, Electric potential) normalized for the Stanford Hall thruster. Figure 13. Discharge characteristics (Hall parameter, Magnetic field, Ion density, Ion axial velocity, Neutral density, Electron Temperature, Electric potential) normalized for the SPT-100. 8
Fig. 14 shows the impact of the Hall parameter on the channel wall erosion rate. First, it can be seen that the erosion zone begins where the ion acceleration starts (see Fig. 10). In general, if the maximum in Hall parameter is located further toward the exit (or outside the channel), the erosion is reduced. This is expected since fewer high energy ions can strike the channel walls. If the peak of the Hall parameter is increased (case 4) or moved slightly further inside the channel (case 5), the erosion is increased. This is observed in all examples, except in case 1 whose different pattern is thought to be more certainly due to a reduced plasma density than a change in the ion energy distribution function. Figure 14. Influence of the Hall parameter on the erosion rate (outer wall values). Figure 15. Erosion rate on the Stanford Hall thruster and the SPT-100 channel walls with and without collision. Fig. 15 illustrates the effect of collisions on the erosion rates obtained for the Stanford Hall Thruster and the SPT-100 on both channel walls. In general, similarities in erosion rate trends exist between the two thruster types. First, for both thrusters, the maximum amplitude of erosion rate is approximately the same for both walls. Also, the calculated erosion rate is higher without collisions than with collisions and is higher on the outer wall than on the inner wall. However, the simulated erosion zone is two times longer for the Stanford Hall thruster than for the SPT-100. As was already observed in Fig. 14, this is likely due to the fact that the Stanford Hall thruster has a Hall parameter peak (Fig. 12) located further upstream from the channel exit than the SPT-100 (Fig. 13). Experimental measurements 15,16 show that our simulated erosion rate for the SPT-100 is under-estimated. Absalamov et al. 15 and Garnier et al. 16 measured an erosion rate of about 8 µm/hr at the exit plane for non-eroded walls. This discrepancy is believed to be due to a simulated ion flux to the wall which is too low, despite the plasma density being approximately correct. This implies that the incidence energy distribution function does not contain a large enough population on the walls of high energy particles. The present sheath model used in the simulation is oversimplified 12, and a more appropriate sheath model in which the presheath is directly simulated would increase the ion flux to the wall. This would likely improve the calculated erosion rate and give better agreement with experimental results. IV. Summary and Future Work We present the results of numerical studies aimed at understanding the evolution of the erosion process in a Hall thruster channel. We have incorporated a model of ion and energetic neutral-induced sputtering into a 2D hybrid PIC simulation. The hybrid PIC simulation includes ion-neutral charge-exchange and momentum-exchange collisions as the former is found to have a significant effect on the instantaneous wall erosion rate. The erosion of the Stanford Hall Thruster channel walls was simulated for a time of 4300 hours leading to wall erosion of 2 mm. The erosion process was found to decrease the thruster performances due to a gradual decrease in the plasma density. Then, in order to verify the accuracy of the erosion model, the simulation was adapted to a SPT-100. The results suggest that the evaluated erosion rate was too low compared to experiments. Future work will involve an implementation of a more accurate sheath model in the simulation in order to better estimate the wall ion flux and obtain the expected erosion rate. 9
Acknowledgments This work was supported in part by the Air Force Office of Scientific Research. Support for E. Sommier was provided by the SAFRAN company. References 1 Fernandez, E., Cappelli, M. A., Mahesh, K., 2D Simulations of Hall Thrusters, CTR Annual Research Briefs, 1998, pp. 81. 2 Fife, J. M., Nonlinear Hybrid-PIC Modeling and Electrostatic Probe Survey of Hall Thrusters, Ph.D. Dissertation, Aeronautics and Astronautics Dept., Massachusetts Institute of Technology, Cambridge, MA, 1998. 3 Allis, M. K., Gascon, N., Vialard-Goudou, C., A Comparison of 2-D Hybrid Hall Thruster Model to Experimental Measurements, 40th AIAA/ASME/ASEE Joint Propulsion Conference, Ft. Lauderdale, FL, July 11-14, 2004. 4 Meezan, N., Hargus, W., Cappelli, M., Physical Review E 63, 026420 (2001). 5 Dugan, J. V. and Sovie, R. J., Volume Ion Production Costs in Tenuous Plasmas: A General Atom Theory and Detailed Results for Helium, Argon and Cesium," NASA TN D-4150. 6 Fernandez, E., Borelli, N., Cappelli, M. A., Gascon, N., Investigation of Fluctuation-Induced Electron Transport in Hall Thrusters with a 2D Hybrid Code in the Azimuthal and Axial Coordinates, 45th Annual Meeting of the Division of Plasma Physics, Albuquerque, New Mexico, October 27-31, 2003. 7 Dorval, N., Bonnet, J., Marque, J.P., Rosencher, E., Chable, S., Rogier, F., Lasgorceix, P., «Determination of the ionization and acceleration zones in a stationary plasma thruster by optical spectroscopy study: Experiments and model, Journal of Applied Physics, Vol.91, Nb. 8, April 2002. 8 Garrigues, L., Hagelaar, G.J.M., Bareilles, J., Boniface, C., Bœuf, J.P., Model study of the influence of the magnetic field configuration on the performance and lifetime of a Hall thruster, Physics of plasmas, Vol.10,Nb.12, Dec. 2003. 9 Thomas, C. A., Gascon, N., Cappelli, M. A., The non-intrusive characterization of the azimuthal drift current in a coaxial ExB discharge plasma, submitted to Physical Review E, 2006. 10 Manzella, D., NASA Glenn research Center, Cleveland, OH 44135, Yim, J. and Boyd, I.,University of Michigan, Ann arbor, MI 48109, Predicting Hall Thruster Operational Lifetime, 40th AIAA/ASME/ASEE Joint Propulsion Conference, Ft. Lauderdale, FL, July 11-14, 2004. 11 Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas Flow, Oxford Science Publication, Ch.2&11, 1994. 12 Pullins, S., Chiu, Y., Levandier, D. and Dressler, R., Ion Dynamics in Hall Effect and Ion Thrusters: Xe + + Xe Symmetric Charge Transfer, AIAA Paper 00-0603, Jan. 2000. 13 Boyd, I. D., Dressler, R. A., Far Field Modeling of the plasma Plume of a Hall Thruster, Journal of Applied Physics,Vol.92, nb 4, pp.1764-1774, 2002 August 15. 14 Sommier, E., Allis, K.M., Cappelli, M.A., Wall Erosion in 2D Hall Thruster Simulations, IEPC-2005-189, Proceedings of the International Electric Propulsion Conference, Princeton University, NJ, Oct. 31-Nov. 4, 2005. 15 FEMM, Finite Element Method Magnetics, Software Package, Ver. 4.0, Foster-Miller, Inc., Boston, MA, 2004 16 Abasalamov, S.K. et al., Measurement of Plasma Parameters in the Stationary Plasma Thruster (SPT-100) Plume and Its Effect on Spacecraft Components, 28th AIAA/SAE/ASME/ASEE Joint Propulsion Conference and Exhibit, 1992, AIAA92-3156 17 Garner, C. E., Brophy, J. R., Polk, J. E., and Pless, L. C., performance Evaluation and Life Testing of the SPT- 100, 30th AIAA/SAE/ASME/ASEE Joint Propulsion Conference and Exhibit, 1994, AIAA 94-2856 10