Physics of a Vacuum Arc for a Plasma Thruster Application

Transcription

1 To IEPC-2013 Physics of a Vacuum Arc for a Plasma Thruster Application IEPC Presented at the 33rd International Electric Propulsion Conference, The George Washington University Washington, D.C. USA Isak I. Beilis Electrical Discharge and Plasma Laboratory, Depart of Electrical Engineering - Physical Electronics, Faculty of Engineering, Tel Aviv University, P.O.B , Tel Aviv 69978, Israel. Abstract: The cathode phenomena (local cathode heating, vaporization, electron emission, cathode sheath etc.) were modeled considering the kinetics of cathode mass flow. A self-consistent simulation was conducted for given spot current I to obtain the cathode erosion rate G and jet velocity V determined the force on the cathode. The time dependent cathode and plasma characteristics were calculated for Cu, Ag, Al and W cathodes and for various primary plasma parameters defined by different spot ignition time at which the spot can be developed. The model shows that the time dependent range of G agrees well with the measured data and the jet velocity V calculated taking into account the experimental cathode force also close to that obtained in the literature. The larger trust performance for W cathode was obtained for the primary plasmas with times with respect to those for Cu and Ag cathodes. Nomenclature CPD = cathode potential drop, V t = time step, s F = cathode force, din/a G = Cathode erosion rate in (g/s) or in (g/c) q = the heat flux density to the cathode from the plasma = time primary plasma initiating, s m = atom or ion mass N0 = equilibrium heavy particle density na,i = atom or ion density at the characteristic boundaries v3 = atom or ion velocity at the external boundary of Knudsen layer j = current density, A/cm 2. Ken = performance of cathode plasma thrust Ker = Cathode evaporation fraction with respect to Langmuir evaporation into vacuum s = electron current fraction Te = electron temperature T(t) = time dependent cathode surface temperature in the spot V = plasma jet velocity vth = heavy particle thermal velocity, (kt3/m) 0.5 Ua = arc voltage, V Up = plasma jet potential drop, V Depart of Electrical Engineering - Physical Electronics, Tel Aviv University: beilis@eng.tau.ac.il 1

2 I. Introduction It is well known that a vacuum arc is a unique source of supersonic, highly ionized and quasineutral metallic plasma jet. The jet was ejected from cathode spots with a velocity V [cm/s] and supported by the cathode mass loss with a rate of G [g/s]. As result the supersonic jet reaction produced a driving force on the cathode surface provoking the cathode motion in direction opposite to the jet expansion direction. The mentioned advantageous and the simple construction of arc plasma source has potential to use metal as a propellant for an arc thruster in order to microsatellite orbit correction 1,2. A different developed designs and the advantages of this arc application were widely discussed 3,4 comparing to other micro thrusters used a gas as propellant 5. An important characteristic is the performance of the vacuum arc thruster that determined by the plasma jet parameters. Polk et al 6 showed that the performance can be improved when the arc plasma was subsequently accelerated electrostatically. It was found also that the application of a magnetic field leads to ion acceleration by a factor of ~2 in the free expanding plasma plume 7. An important question is how the arc plasma parameters were changed in time during the cathode spot development and what is the influence of the conditions of spot initiation, pulse duration, cathode material etc.? This work describes the time dependent cathode phenomena in a vacuum arc produced the force by the cathode spot development on a bulk cathode in frame of a model considered the kinetics of cathode vaporization into the adjacent to the surface plasma and the hydrodynamic plasma flow of the erosion material. II. General problem Firstly the Cu cathode plasma jet force was measured by Tanberg 8 for arc current in the range of A, and the vacuum chamber pressure was 10 4 torr. The average jet velocity from the cathode of the order of 1.6 x10 6 cm/s and the average reaction force from the freely swinging cathode of 17 dyn/a were obtained. Robertson 9 measured the reaction force of a copper cathode for a range of pressures from 1 to 10 torr. The force increased when the pressure in the chamber decreased, and the largest force measured was 15 dyn/a at a pressure of 1 torr with arc currents of 7 20 A. Kobel 10 measured the force on a mercury cathode indicating the force in range of dyn/a for currents of 30-37A and pressures of torr. Our measurements 11 using pendulum deflection due to cathode plasma jet action showed about 40 din/a for Cu and about 20 din/a for Al cathodes. According to the measurements the force F(dyn) is determined by the cathode erosion rate in G(g/s) and plasma velocity V (cm/s) as 2

3 F=GV... (1) The plasma velocity V and the erosion rate G measurements indicated a relatively narrow range of V from 6x10 5 to 3x10 6 cm/s and (20-60) g/c respectively for different cathode materials 6. For copper cathode V~1x10 6 cm/s and G~35 g/c the force is agree with that measured in ref. 11. The measured arc parameters G, V, and F are averaged over some time of arc duration. At present, the time dependent data of the mentioned quantities as well as the thrust performance are absent for transition arc operation. To understand the physical mechanism of time dependent cathode mass loss and plasma jet formation below a physical model for transition spot was developed and the calculation method and calculated results are described. III. Physical model A. General description The heated cathode evaporates and the mass flow from the surface is produced. The cathode vapor was ionized and several characteristic regions were originated. The model 12,13 considers a highly ionized cathode vapor plasma, whose structure consists of the regions partially overlapping and starting from the cathode surface (boundary 1 in Fig. 1): a ballistic zone comprising a space charge sheath, from the cathode surface to boundary 2, a non-equilibrium Knudsen layer (of ion-atom mean free path size), from the cathode surface to boundary 3, an electron relaxation zone (where the plasma electrons are heated) from the cathode surface to boundary 4, and a plasma acceleration region beyond boundary 4. Fig.1. Schematic diagram of the experimental setup In the Knudsen layer (1-3), the evaporated and returned heavy particle fluxes are formed; their difference is the net cathode erosion rate G. The flux of ions and electrons flowing towards the cathode surface is formed here. Also here the difference between the electron emission and the returning plasma electron flux determine the resulting electron flux flowing away from the cathode. However the returning plasma electron flux depends on the CPD in the ballistic zone. The electron momentum equation in ballistic zone (1-2) determines the CPD. The mathematical details of the electron kinetics were presented previously 12. According to the model 12,13 the atoms are ionized by electrons emitted from the cathode, as well as by plasma electrons with temperature Te in the electron relaxation zone 1-4. The length of 3

4 this zone is by factor of ~10 2 larger than the heavy particle mean free path, and the plasma therein is quasi-neutral, dense (~ cm -3 ) and collisional. In the electron relaxation zone, the plasma flows and carries current, and energy is dissipated. The plasma is accelerated by a gasdynamic mechanism, forming a jet with velocity V. The cathode heat balance includes Ohmic heating and incident fluxes from the plasma of ions and electrons. Cooling is from heat conduction into the body of the cathode, radiation, evaporation and electron emission. The incoming heat flux to the cathode needed for selfconsistent spot operation changes in time and depends on the current and plasma parameters during the spot evolution. The system of equations includes the equations of electron emission in general form 12, equations of total spot current, ion and electron back fluxes, electric field at the cathode surface, the kinetics of emitted and back flowing heavy particle fluxes and plasma energy 12. A system of Saha equations in the quasi-neutral plasma determines the fraction of atoms/ions in various charge states. In the Knudsen layer (1-3), the difference between evaporated and returned heavy particle fluxes produces the net cathode mass loss flux G(g/s)=m(na3+ni3)v3,... (2) that determined the cathode erosion rate G(g/C)=G(g/s)/I, where m=mi=ma and I is the electrical current. The relation of b3=v3/vth defined the velocity fraction relatively to the thermal velocity vth=(kt3/m) 0.5 at the external boundary of the Knudsen layer. The plasma is accelerated mainly by the ion and electron pressures and also by electron-ion friction 12. In general, the jet velocity V is determined by the equations of momentum and energy conservation in the plasma jet. Previously the distribution of Cu cathode plasma flow parameters along the jet was calculated using the equations of momentum and energy conservation in differential form 12,14. It was shown that the plasma accelerated by the jet expansion in conical configuration from subsonic in the spot region to supsonic velocity 10 6 cm/s at distance of about two spot radii from the cathode. While the result obtained from equations in differential approximation was in good agreement with the measurements 6,15, the velocity calculated from an integral momentum equation using a spot plasma pressure can exceed the measured value. This is because the plasma expands with a frozen electron temperature of about one ev indicating that only part of internal momentum was converted to the direct jet velocity. This point is critical when jet force is calculated where the higher accuracy for V is requested. Therefore below the plasma jet velocity will be determined from equation (1) using the measured force F and the cathode erosion obtained from the above model and cathode spot system of equation 12,13. 4

5 The calculated parameters are CPD, electron temperature T e, heavy particle density n, degree of ionization, cathode temperature T, erosion rate G, cathode electric field E c, current density j, and electron current fraction s for transient spot on bulk cathode. One of important calculated parameters is the fraction of cathode material evaporation K er, which is the ratio of the net atom evaporation rate into the adjacent dense plasma, to the Langmuir evaporation rate, i.e. into vacuum. The given parameters are the cathode material properties, and a characteristic small life time during which the system of equation was solved to determine the primary plasma parameters that initiates the spot. The physical phenomena, assumptions and the system of equations for the cathode plasma regions have been described in detail previously 12,13. B. Calculation method The cathode spot in a vacuum arc can be initiated when an adjacent to the surface primary plasma can be originated. The life time of the primary (initiating) plasma depends on the kind of arc triggering or on the plasma of a previously extinguished spot. The initiating plasma parameters (density, temperature) determine the initial heat flux to the cathode, which support the future spot development. Therefore the behaviour of spot parameters during the spot development depends on. The heat model takes in account that the plasma plume and consequently the heating are localized on the cathode surface by a concentrated heat source. The solution of the time-dependent heat conduction equation for a solid body heated by a local heat source is presented in the following general form 16 : t 0... (3) 0 Tt () qtt (, ') ftt (' ) dt' T in where T(t) is the time dependent cathode temperature, q(t,t') is the time dependent heat flux to the cathode surface and f(t',t0) is a time dependent function resulting from integrating the heat conduction equation in differential form, and t0 is a parameter describing heat flux concentration. The cathode thermal model takes in account the heat conduction equation (3) and the cathode energy balance to determine the surface temperature in the spot. According to Eq. (3) the temperature is implicitly depends on time due to non-linear q(t,t') dependence. Therefore the system of equations was solved numerically using an iteration method. The initial condition (initial temperature Tin) is determined by the heat flux from the primary plasma. The primary plasma density and temperature should be produced at a level that an initiated spot can be developed. Therefore the initiating parameters of the primary plasma were calculated self-consistently from the cathode system of equations assuming that the heat flux q is constant during a small. Using the initial parameters the future solution of the cathode system of equations will determine the dependences qn(t,t), fn(t) at each time step t and consequently 5

6 the temperature T(t). A relatively small t was chosen so that qn( t) can be assumed constant during each t. The solution continues up to time t=n t for which the T(t) as well as the cathode plasma parameters reached steady state. The calculations were conducted for different cathode materials such as Cu, Ag and Al which have intermediate thermophysical properties, and also for W, a refractory material. Cathode potential drop, in V IV. Solution and results A. Cathodes with intermediate thermophysical properties. The cathode plasma jet formation and the plasma jet force are determined by the energy dissipation in Ag, =10ns the cathode body and in the cathode spot dense Ag, =50ns plasma and by returned plasma flux. The energy 40 depends on the CPD, electron temperatures, cathode Cu, 50ns Cu, Ag, =100ns temperature, current density, and electron current Cu, =10ns 10 fraction and etc. The dependencies of these Time, ns 10 3 parameters on spot time development are presented below. Fig.2. CPD dependencies on spot time t with τ as parameter for Cu and Ag cathodes. For Cu, Al and Ag the spot initiation time was chosen in range =6-100 ns and the spot current was 10A. t was chosen sufficiently small so that the results of calculation for all CS parameters were independent of t. The calculated time dependent CPD is presented in Fig. 2. The calculation shows that CPD dependence on time is very sensitive near the value of τ. The larger CPD~70 V at first step with τ=10 ns. For Al cathode (not shown in Fig. 2) the CPD is relatively lower and increase from 20 to 50 V when τ decreased from 100 to 6 ns. Fig.3 shows the cathode evaporation fraction that is comparatively low with respect to Langmuir evaporation flux into a vacuum. Ker decreases from about 0.25 to about 0.04 with the spot time in range shown in Fig.3. For Al: Ker= ; Fig.3. Cathode evaporation fractions as dependencies on spot time t with τ as parameter for Cu and Ag cathodes. When the time of spot development increases from a few ns to about 100ns the other spot parameters are changed as it is shown in Table 1: the heavy particle density N0=(0.4-2)x10 20 cm -3, 6

7 electron current fraction s= , current density j= MA/cm 2, electron temperature Te=8-2 ev, total degree of ionization I0=1-0.7 for both Cu and Ag cathodes. Cathode temperature T= K (Cu) and K (Ag); erosion rate G=15-30 µg/c (Cu) and G=30-60 µg/c (Ag). The characteristic parameters for Al were: G= (28-experiment 6 ) g/c; V~2x10-6 (1.5x10-6 -experiment 6 ) cm/s; s= ; j~(4-2)x10 6 A/cm 2 ; Te=6-1.5eV; a b Fig.4. The jet velocity as dependence on spot time t with τ as parameter for Cu (a) and Ag (b) cathodes. The jet velocity dependencies are obtained using the calculated parameters of G(g/C) for Cu (fig.4a) and for Ag (Fig.4b). It was used the measured force for Cu cathode about 40din/A that assumed also for Ag. It can be seen that the jet velocity depends on primary time of spot ignition and sharply decreased with spot time near τ. While the value of V is around of the measured 10 6 cm/s the velocity for Cu is larger than for Ag by factor 2 for and spot time t>10ns. The cathode thrust performance Ken was defined as K en 2 mv... (4) 2IUa Where m is the heavy particle mass and Ua is the arc voltage which is the sum of CPD and the plasma jet voltage. Fig. 5 illustrates the time dependencies of Ken for Cu and Ag cathodes with as parameter. A minimum of Ken is calculated in the dependencies of Fig. 5 and the range of Ken for Cu ( ) is larger than that for Ag (0.1-2). For Al cathode Ken= ; Fig.5. Cathode thrust performance as dependencies on spot time t with τ as parameter for Cu and Ag cathodes. 7

8 Table 1. Cathode characteristics for Cu, Ag and W cathodes. Copper and Silver-f(t=6-100ns) Tungsten-f( =6-1000ns) N0x10 2 j T K Te I0 G s N0x10 2 j G 0 MA/cm 2 ev µg/c 0 MA/cm 2 µg/c cm -3 cm (Cu) 0.63 (Cu) K (Ag); (Ag) B. Refractory W cathode The main problem for applying the spot theory to refractory materials is that there is a large electron emission to atom evaporation rate ratio over a wide range of temperatures 13. A virtual plasma cathode model was developed taking into account the large space charge of emitted electrons to solve this problem 12,13,17. A minimum of Um=kmT appears in the potential distribution near the cathode surface where the low energy electron were repealed decreasing the electron current density emitted from the cathode. Cathode potential drop, V j CPD j Fig.6 W T CPD , ns Spot current density, j, MA/cm 2 Cathode temperature, T, kk Electron temperature, T e, ev W T e , ns Fig.7 V Jet velocity, V.10 6, cm/s Fig.6. CPD, cathode temperature T K, and spot current density j (MA/cm 2 ) as function on τ for W. Fig.7. Plasma jet velocity and electron temperature in the spot as function on τ for W. The primary plasma parameters for W cathode was calculated for initial times in range from =2 ns to 10 s. The spot current was 10 A. The calculated parameters: CPD, cathode 8

9 temperature T and current density in the spot are decreased as it is shown in Fig.6 and electron temperature Te decreasing with is shown in Fig.7. The plasma jet velocity slightly increases with (Fig.7) and it is around 1.6x10 6 (1.4x10 6 -experiment 6 ) cm/s. Fig.8 illustrates the dependencies of cathode plasma thrust performance Ken, the evaporation fraction Ker and velocity fraction in Knudsen layer b3. It can be seen that Ken is relatively large and increases from 0.5 to 0.85 with while relatively small values of b3 (0.02) and Ker (0.08) slightly changed with (Fig.8). The calculations also shown (see Table 1) where the heavy particle density significantly Performance, evaporation & velocity fractions K en K er , ns W b 3 decreases (from 7x10 20 to 1x10 19 cm -3 ), the low level of erosion rate is mostly constant (25-24 g/c), electron current fraction slightly increases from 0.7 to 0.8 when increase in the above indicated range. The plasma in the spot is fully ionized and mainly consists of ions with charge fraction of 1 + (0.09), 2 + (0.40), 3 + (0.44) and 4 + (0.06) for =2ns and 1 + (0.03) 2 + (0.39), 3 + (0.56) and 4 + (0.02) for 10 s. Fig.8. Cathode K en,, K en and velocity b 3 at the external boundary of Knudsen layer as dependencies on primary time τ for W. V. Discussion The CPD sharply decreases with spot time t around the time τ. In this time range the cathode was not enough heated and therefore a relatively large CPD is formed to increase the energy flux from the plasma to reach sufficiently cathode temperature and vaporization rate in order to support the self-sustained spot development. After ignition the cathode temperature and vaporization increased with t and the requested CPD decreased to some steady state level. The initial CPD also decreased with τ because the large time of primary plasma action leads to larger initial cathode temperature and intensity of cathode vaporization. As result the heavy particle and plasma density significantly rise with spot time t. This fact promotes larger back particle (atoms and ions) flux to the cathode which explains the calculated significant decrease of the cathode evaporation fraction (4-7% at steady state, Fig.3) relatively to Langmuir vaporization flux into vacuum. This calculated result indicate that the cathode evaporation rate in an arc is far from a metal evaporation rate into vacuum due to relatively large density of the cathode plasma supported the request current density. 9

10 The difference between direct and back heavy particle fluxes in the Knudsen layer formed the erosion rate that according to the calculations changed by factor two during the considered ranges of spot time. This result explains the change of plasma jet velocity presented in Fig's 4a and 4b by assuming the measured cathode plasma force 40 din/a. The difference between calculate velocities for Cu and Ag cathodes is due to difference of G (g/c) for these cathodes. The small minimum in the dependencies of V on time t for both cathodes is caused by those dependencies of G (g/c) on t which is determined by a time dependent combination of heavy particle and plasma velocity at the external boundary of Knudsen layer b3. The minimum in V(t) dependence produces similar minimum in the time dependence of cathode thrust performance which is adequately different for Cu and Ag cathodes and reaches about 0.5 for Cu and 0.25 for Ag. It should be noted that Kev is lower for Ag although the larger atom mass (Ai=108) than for Cu (Ai=64). It can be understood considering the expression for jet kinetic energy 2 F 2 F 2 Kev mv m( ) m( ) G mn v... (5) 3 3 Taking into account that the calculated heavy particle density n3 and plasma velocity fraction v3 at the external boundary of the Knudsen layer are close for both Cu and Ag cathodes the following dependence is derived (with weak change of F and IUa) from (5): K ev 1 m... (6) The relation (6) shows that Kev inverse proportional to the atom mass although the plasma force increases with m. The erosion rate (~m) agrees approximately with that measured lower for Cu (0.3-0,4 g/c) than for Ag (0.8 g/c). The larger Ken is obtained for W cathode at primary time of spot development due to lower Ua and lower rate of calculated cathode erosion that determined by lower heavy particle density with respect to the Cu and Ag cathodes. In frame of proposed transition model the calculation show that G and V are changed during the spot development. This change is relatively weak (factor 2) and indicate the resulting relatively weak change of cathode jet force with spot time. Using the complicated solution for jet expansion in differential form the force and the jet velocity can be obtained independently. The present work illustrates the mechanism of cathode plasma flow in the kinetic region studied selfconsistently with the plasma flow in hydrodynamic regions during the spot initiation and 10

11 development. This result allows understand and determine the contribution of cathode mass loss and plasma jet acceleration in the reactive force formation at the cathode of a vacuum arc. VI. Conclusions 1. The cathode evaporation fraction Ker decrease from about 0.25 to about when the spot time decreased from few ns to ~1 s. According to the kinetics of plasma flow the cathode vaporization rate in a vacuum arc is far from that into vacuum due to relatively large density of the cathode plasma supported the request current density. 2. The calculated cathode erosion rates in form as plasma for Cu & Ag are in good agreement with those obtained experimentally. 3. The calculated time dependent jet velocity is in range of measured value. 4. The observed cathode force can be understood studying the kinetic cathode loss mass flow self-consistently with the complicated cathode and plasma physical phenomena during the spot initiation and development. Acknowledgments This research was supported by a grant from the Israel Science Foundation, No. 912/11. 11

12 References 1 M. Keidar, I.I. Beilis, Plasma Engineering: Applications from Aerospace to Bio and Nanotechnology, (Acad Press, Elsevier, London-NY, 2013). 2 J. Schein, N. Qi, R. Binder, M. Krishnan, J. K. Ziemer, J. E. Polk and A. Anders, Inductive energy storage driven vacuum arc thruster, Rev. Sce. Instr., vol. 73, no2, , M. Keidar, J. Schein, K. Wilson, A. Gerhan, M. Au, B. Tang, L. Idzkowski, M. Krishnan, and I.I. Beilis, Magnetically enhanced vacuum arc thruster, Plasma Sources Sci. Technol., vol. 14, no 4, , M.A. Kemp, and S.D. Kovaleski, Thrust Measurements of the Ferroelectric Plasma Thruster, IEEE TPS, Vol. 36, No.2, 356, R. Wirz, R. Sullivan, J. Przybylowski, and M. Silva, Hollow Cathode and Low-Thrust Extraction Grid Analysis for a Miniature Ion Thruster, International Journal of Plasma Science and Engineering, Vol.1, No1, Article ID , 11 pages, J.E. Polk, M.J. Sekerak, J.K. Ziemer, J. Schein, N. Qi, and A. Anders, A Theoretical Analysis of Vacuum Arc Thruster and Vacuum Arc Ion Thruster Performance, IEEE TPS, Vol. 36, No.5, 2167, T. Zhuang, A Shashurin, I.I. Beilis, M. Keidar, Ion velocities in a micro-cathode arc thruster, Phys. Plasmas, Vol.19, No6, pp , R. Tanberg, On the cathode of an arc drawn in a vacuum, Phys. Rev., vol. 35, no. 9, pp , E. Kobel, Pressure and high velocity vapour jets at cathodes of a mercury vacuum arc, Phys. Rev., vol. 36, no. 11, pp , R. M. Robertson, The force on the cathode of a copper arc, Phys. Rev., vol. 53, no. 7, pp , H.S. Marks, I.I. Beilis, and R.L. Boxman, Measurement of the Vacuum Arc Plasma Force, IEEE TPS, Vol.37, No7, 1332, I.I. Beilis, Theoretical modeling of cathode spot phenomena, In Handbook of vacuum arc science and technology, ed. by R.L. Boxman, P.J. Martin, D.M. Sanders, Noyes Publ. Park Ridge, N.J, 1995, p I.I. Beilis, "State of the theory of vacuum arcs", IEEE Trans. Plasma Sci., vol.29, N5, p.621, I.I. Beilis, M.P. Zektser, Calculation of the parameters of the cathode stream an arc discharge, High Temp., 29, N4, , W.D. Davis, H.C.Miller, Analysis of the electrode products emitted by DC arcs in a vacuum ambient, J. Appl. Phys., 40, N5, , H.S. Carslaw and J.C. Jaeger, Conduction of heat in solids, Clarendon Press, Oxford, I.I. Beilis, On the mechanism function of the vacuum arc spot operation on refractory cathodes. High Temp., vol.26, No.6, ,