PHYSICS OF PLASMAS 13, 063504 共2006兲
Plasma processes inside dispenser hollow cathodes Ioannis G. Mikellides,a兲 Ira Katz, Dan M. Goebel, James E. Polk, and Kristina K. Jameson Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109
共Received 16 November 2005; accepted 4 May 2006; published online 12 June 2006兲 A two-dimensional fluid model of the plasma and neutral gas inside dispenser orificed hollow cathodes has been developed to quantify plasma processes that ultimately determine the life of the porous emitters inserted in these devices. The model self-consistently accounts for electron emission from the insert as well as for electron and ion flux losses from the plasma. Two cathodes, which are distinctively different in size and operating conditions, have been simulated numerically. It is found that the larger cathode, with outer tube diameter of 1.5 cm and orifice diameter of 0.3 cm, establishes an effective emission zone that spans approximately the full length of the emitter when operated at a discharge current of 25 A and a flow rate of 5.5 sccm. The net heating of the emitter is caused by ions that are produced by ionization of the neutral gas inside the tube and are then accelerated by the sheath along the emitter. The smaller cathode, with an outer diameter of 0.635 cm and an orifice diameter of 0.1 cm, does not exhibit the same operational characteristics. At a flow rate of 4.25 sccm and discharge current of 12 A, the smaller cathode requires 4.5 times the current density near the orifice and operates with more than 6 times the neutral particle density compared to the large cathode. As a result, the plasma particle density is almost one order of magnitude higher compared to the large cathode. The plasma density in this small cathode is high enough such that the Debye length is sufficiently small to allow “sheath funneling” into the pores of the emitter. By accessing areas deeper into the insert material, it is postulated that the overall emission of electrons is significantly enhanced. The maximum emission current density is found to be about 1 A / mm2 in the small cathode, which is about one order of magnitude higher than attained in the large cathode. The effective emission zone in the small cathode extends to about 15% of the emitter length only, and the power deposited at the emitter surface by returning electrons is found to be twice that deposited by ions. A previous study suggested that the computed particle flux and energy of ions to the emitter of the 1.5 cm cathode were not high enough to change the barium evaporation rate compared to thermally induced evaporation. The same suggestion is made here for the 0.635 cm cathode. The peak ion flux to the emitter is found to be 1.2 A / cm2 共7.6⫻ 1018 / s cm2兲, and the corresponding peak sheath drop is 2.9 V. Consequently, once the emitter operating temperature is known it is possible to determine directly the barium depletion-limited life of these cathodes using existing vacuum-cathode data. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2208292兴 I. INTRODUCTION
Orificed hollow cathode 共OHC兲 failure is one of the most difficult challenges the electric propulsion 共EP兲 community faces in its efforts to establish ion and/or Hall as the propulsion technologies of choice for the National Aeronautics and Space Administration’s long-duration missions. Many missions to the outer planets of our solar system and beyond would require more than ten years of continuous thruster operation.1 Cathodes made of porous tungsten inserts that are impregnated with barium-calcium aluminates are potential candidates to meet the lifetime requirements for many of these missions. Two main life-limiting mechanisms in OHCs are 共1兲 depletion of the barium in the emitter and 共2兲 erosion of the keeper electrode. Cathode orifice erosion and pore blockage by tungstates and tungsten crystals may also limit the life of these devices. The longest operation of an impregnated OHC in an EP application was achieved during the extended life test 共ELT兲 of the NASA Solar Electric Propulsion Technology Applicaa兲
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tions Readiness 共NSTAR兲 engine—a 2.3 kWe ion engine—in which the 0.635 cm diameter discharge OHC continued to operate after 30 352 h.2 However, the keeper electrode had completely eroded and post-ELT analyses suggested that cathode failure would be the most likely near-term cause for the failure of the engine.3 The complexity associated with the physics of the multicomponent gas inside these devices, and the difficulty of accessing empirically this region, have limited our ability to design cathodes for EP that perform better and last longer. The most prominent experimental efforts to measure the plasma properties inside an OHC have been reported by Siegfried and Wilbur4 in 1978 in a mercury OHC and more recently by Goebel et al. in a xenon OHC.5,6 Both studies showed that plasma particle densities reach high values near the orifice. In a 0.635 cm cathode operating with xenon the plasma particle density in this region has been measured to exceed 1021 m−3.6 In a 1.5 cm diameter xenon OHC, designed for a 25 kWe ion engine,7 the peak plasma particle density has been found to be in the order of 1020 m−3.5 These empirical studies provided valuable plasma measurements by
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probing the region inside the cathode along the axis of symmetry, but provided no data of ion fluxes and sheath potentials off-axis. In the absence of this information the life of the emitter may not be assessed accurately. Several zero-dimensional phenomenological and onedimensional theoretical models of the OHC have been reported in the literature.8–13 However, the inherent twodimensionality of the plasma both inside and outside the device allows such models to provide only a global picture of some of the driving physics. Only two-dimensional 共2-D兲 models supported by detailed measurements may provide the level of detail needed to accurately quantify the mechanisms that determine cathode life. Recent experiments for example have shown that if a flux of 25– 30 eV ions bombards the impregnated emitter the effective barium evaporation rate from the porous tungsten surface increases by an order of magnitude14 compared to that expected by thermal evaporation only 共as is the case in most vacuum cathode technologies兲. If this occurs then the life of the emitter would be significantly reduced. Thus, it is imperative that the 2-D plasma environment inside the cathode be quantified to determine the particle and energy fluxes along the emitter. A 2-D numerical model of the OHC that follows a fluid approach was developed by the authors of this article and reported in Ref. 15. The numerical model was used to simulate one specific set of operating conditions and geometry associated with the 1.5 cm xenon OHC mentioned above. A coupled system of eight governing laws for the plasma and neutral gas was solved numerically on a 2-D axisymmetric rectilinear computational mesh. All dominant elastic collisions were incorporated including electron-neutral 共e-n兲, electron-ion 共e-i兲, and ion-neutral resonant charge exchange 共i-n兲. The fluid approach was justified by the high collisionality attained inside that cathode. The numerical results and comparisons with measurements in the 1.5 cm cathode were reported in Ref. 15. In the present article we apply the same numerical model to simulate a smaller cathode; namely, a 0.635 cm OHC, operating at 12 A and 4.25 sccm. Although the work described here has led to two new augmentations of the model 共associated with the boundary conditions兲, the numerical model itself is not the focus of this work. Emphasis is given here on the different operating characteristics of the plasma inside these two cathodes as computed by the model. The results produced by the model for the 0.635 cm cathode, and the new understanding generated by the comparison of the results between both cathodes are discussed in detail in Secs. III and IV. Prior to the 2-D numerical work described in Ref. 15, only a few attempts to model the ionized and neutral gases, in 2-D, inside OHCs were reported. Salhi and Turchi16,17 developed a first-principles fluid model that included a 2-D variation of the plasma properties inside a 0.635 cm cathode similar to the NSTAR cathode geometry studied here. Ohmic heating of the plasma was determined using the classical 共Spitzer兲 electrical conductivity for fully ionized plasma. The work neglected e-n and i-n collisions. Nonetheless, comparisons with measurements obtained in a mercury-OHC showed a maximum discrepancy of less than 35% between theory and experiment.17 To obtain a numerical solution of the neu-
tral gas dynamics throughout a 0.2 cm diameter orificed microhollow cathode, in the absence of the plasma and i-n collisions, Crawford18 used the direct simulation Monte Carlo19 approach. A preliminary simulation of the electrons has been attempted using the electrostatic particle-in-cell20 method, as reported by the same author, but a steady-state solution was not attained.21
II. THEORETICAL MODEL
The main motivation for developing the 2-D computer code in Ref. 15 has been the identification and quantification of the mechanisms that affect the life of the emitter. The objective has been to develop a theoretical model that predicts the steady-state, 2-D distributions of all pertinent plasma properties, including electron/ion fluxes and the sheath potential drop along the emitter. By simulating only the geometrically simple emitter region, the numerical complexity and computational times are reduced, but the calculation depends on measurements at z = 0, which is defined as the orifice boundary 共as will be explained in greater detail below兲. Only the steady-state solution is sought, so that the time-dependent terms do not appear in the plasma conservation equations. The self- and applied magnetic fields have been neglected since their effect is negligible inside the cathode. The naming convention followed in this article uses the outer tube diameter dc to distinguish the two cathodes. It is noted that the diameter of the computational region is smaller than dc; it equals the inner diameter of the emitter.
A. Governing laws
The governing conservation laws have been presented in detail in Ref. 15 and will not be repeated here. In summary, the coupled system consists of eight governing equations for the plasma and neutral gas and yields the steady-state profiles of the following 共main兲 variables: plasma particle density n, ion and electron current densities ji and je, respectively, electron temperature Te, electric field E, plasma potential , neutral particle density nn, and heavy species temperature T. The system includes two energy equations and nonequilibrium ionization. The inertia terms are neglected in both the electron and ion momentum equations. The functional forms of the collision frequencies used here have also been provided in Ref. 15. It is assumed that ions and neutrals are in thermal equilibrium, so that a single equation is derived for the conservation of energy of the heavy species. The neutral gas dynamics are neglected. Although previous work has suggested that the classical resistivity may be enhanced anomalously by streaming instabilities downstream of the orifice entrance,22 only classical resistivity has been used in the present simulations. The comparisons with the measurements in the 0.635 cm cathode, to be shown later, suggest that classical resistivity is sufficient to characterize the plasma inside the emitter region. The total pressure P is assumed to be uniform throughout the cathode channel and the neutral particle density is determined using the ideal gas law.
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B. Boundary conditions
The boundary conditions have also been described in detail in Ref. 15 and will only be briefly summarized here. Two new major boundary condition improvements related to emission enhancement and emission turn-off have been introduced as a result of this work. The 0.635 cm cathode simulations 共as will be shown later兲 have shown that emission enhancement is critical in cathodes with porous emitters when the Debye length D becomes smaller than the average pore diameter. Emission turn-off is implemented when the sheath drop at the wall becomes negative enough so that thermionically emitted electrons do not have sufficient energy to overcome the sheath barrier and enter the plasma region. Both emission-related mechanisms have been found negligible in the 1.5 cm cathode. A minor boundary condition improvement related to the effect of the presheath on the Bohm velocity was also implemented in the 0.635 cm cathode simulations. The improved condition, which leads to the multiplication of the Bohm velocity by the factor exp共− 21 兲, was not included in the simulations reported in Ref. 15. Thus, the results presented here include this effect for the 0.635 cm cathode but not for the 1.5 cm cathode. The improvement makes only small quantitative differences in the results of the 1.5 cm cathode, in the order of 10% or less, and causes no qualitative changes. Both ions and electrons are allowed to penetrate the sheath and be absorbed by the insert walls. Ions at wall boundaries are assumed to have attained the Bohm velocity. As mentioned above, the latter is effectively reduced by a factor of exp共− 21 兲 as a result of the presheath, but the effect is found to have a negligible impact on the overall behavior of the plasma. The absorbed electron current density follows the one-sided thermal flux assuming Boltzmann electrons. The emitted electron current density from the insert is modeled after the Richardson-Dushman equation for thermionic emission,23 with ␣ = 1.202⫻ 106 A / m2 / K, and includes the effect of the Schottky potential SH, where eff ⬅ WF − SH. For all simulations of the 0.635 cm cathode, the work function WF 共in volts兲 as a function of Tw 共in kelvin兲 is given by Eq. 共1兲 according to Cronin:24
WF = 1.67 + 2.82 ⫻ 10−4Tw .
共1兲
The 1.5 cm cathode simulations used a slightly different functional form15 as available at the time, given by WF = 1.41+ 5 ⫻ 10−4Tw. For the temperature boundary conditions, the model requires two inputs: the peak emitter temperature Tw,max 共in kelvin兲 and a nondimensional polynomial that expresses the variation of this value as a function of position along the emitter in the form of 2 ¯ − 1兲 + c2Lins ¯2 − 1兲 共z Tw ⬅ Tw,max + c1Lins共z 3 4 ¯3 − 1兲 + c4Lins ¯4 − 1兲, ¯z ⬅ z/Lins , + c3Lins 共z 共z
共2兲
where the length of the insert is Lins, and c1 , c2 , c3 , c4 are constants. Tw has been measured25 in the 0.635 cm cathode and has been prescribed in the model using appropriate values of the constants “c.” The maximum and minimum temperatures along the insert were 1474 and 1300 K, respec-
FIG. 1. Schematic of a simplified porous emitter geometry and sheath funneling.
tively. Emitter temperature measurements have not yet been conducted in the 1.5 cm cathode. Therefore, Tw,max and the functional form 共through the constants “c”兲 must be provided differently. This is done as follows. First, a nondimensional functional form very similar to that used in the 0.635 cm cathode simulations is assumed in the 1.5 cm cathode simulations. The two functional forms follow the same monotonic trend with distance from the emitter edge. The differences between the two forms do not impact any of the conclusions made in this work. Second, to determine Tw,max, a different iterative scheme is employed that makes use of the measured plasma potential to self-consistently calculate Tw,max. The details of this iterative process are described in Sec. C. The heavy species temperature T is set equal to the surface temperature at all wall boundaries. An improvement to the original cathode model15 that has proven by the 0.635 cm cathode simulations to be critical is emission enhancement under high plasma density conditions. The mechanism may be best characterized as “sheath funneling” and is depicted by the schematic in Fig. 1. In cathodes with porous emitters when the plasma density is high enough so that the Debye length becomes smaller than the mean pore radius r p, it is postulated here that the sheath can be funneled into the pores thereby enhancing the effective emission area. In light of the uncertainties associated with the microstructure of the pores the increase in emission area is modeled here very simply. It is assumed that the pores are cylinders with radius r p and that the sheath penetrates the cylinder a distance h from the pore entrance. The penetration height h is approximated by assuming that the 共collisionless兲 ions come from a single source and enter the pore along straight-line trajectories with a velocity component normal to the emitter surface that equals the Bohm speed 共kTe / m兲1/2 and a parallel component that equals the ion thermal speed 共kT / m兲1/2, where m is the heavy species atomic mass. The ion particle density drop from ni共0兲 to ni共h兲, assuming spherical expansion from a single point source, may be then expressed by the ratio
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冉 冊
ni共0兲 ␦r = 1+ ni共h兲 rp
−2
= 共1 + ¯x兲−2,
¯x ⬅
ui,x1/ui,x2 冑T/Te = r p/h r p/h
TABLE I. Comparison of the two cathodes simulated with the 2-D orificed hollow cathode model.
共3兲 since ␦r = h共ui,x1 / ui,x2兲. Since the density is inversely proportional to the square of the Debye length, we write
冉 冑 冊
ni共0兲 = 1+ ni共h兲
T h Te r p
2
D共h兲2 = . D共0兲2
共4兲
Taking D共h兲 = r p, the height h is given by h = rp
冑
Te 共a − 1兲, T
a⬅
rp . D共0兲
共5兲
To account for the enhancement, the emitted electron density jem e is multiplied by an emission enhancement factor f when a at any location along the emitter exceeds 1. Based on the assumed geometry of the pores, the factor f is easily found to be f=
冦
1,
a艋1
1 + 2b
h = 1 + 2b rp
冑
Te 共a − 1兲, T
a⬎1
冧
,
共6兲
where b = A p / Aem is the ratio of open area 共i.e., the sum of all pore entrance areas兲 A p over the total emitter surface area Aem. It is approximated that for the porous emitters used in the 0.635 and 1.5 cm cathodes, b may vary between 0.2 and 0.5 assuming a pore diameter range of 1 – 7 m.24 C. Numerical approach
The conservation equations are discretized using finite volumes. The flux vectors are edge-centered and the scalar variables are cell-centered. The fluxes are determined using second-order accurate finite differences. The system of equations is solved in a time-split manner using explicit timemarching for the plasma particle density and electron temperature. Initial estimates of the electron current density vector field, plasma particle density and electron temperature are used to compute all required fluxes, transport coefficients and related quantities. The ion continuity and electron energy equations are then time-marched to yield new values of n and Te. The evolution of these equations at fixed current density is repeated for N iterations. When N reaches a specified value conservation of total current is solved implicitly to determine a new value of the plasma potential , which is in turn used to compute the new electric field 共E = −兲. The electron current density vector field is then updated using the electron momentum equation. Concurrently, the heavy species energy equation is solved implicitly to determine the new heavy species temperature T. The procedure is repeated until the solution for all quantities has reached steady state. There are five main inputs to the numerical model: 共1兲 the total discharge current, 共2兲 the internal cathode pressure 共P兲, 共3兲 the plasma density at the orifice boundary, 共4兲 the plasma potential at the orifice boundary, and 共5兲 the emitter temperature as a function of position. There are two numerical simulation scenarios: the emitter temperature is 共1兲 known and 共2兲 unknown. The total discharge current and
Designed for Discharge current Id Mass flow rate Cathode tube diameter dc Orifice diametera do
0.635 cm cathode
1.5 cm cathode
2.3 kWe ion engine 12 A
25 kWe ion engine 25 A
4.25 sccm 0.635 cm
5.5 sccm 1.5 cm
0.1 cm 0.3 cm 2.54 cm 2.54 cm Emitter length Le Internal pressure P 7.88 Torr 1.07 Torr 2.5⫻ 1020 m−3 Peak plasma particle density n 1.7⫻ 1021 m−3 Characteristic lengths in dense-plasma region 0.4 mm 2.5 mm e-n mfp, en e-i mfp, ei i-n mfp, in Debye length D
0.03 mm 0.06 mm 2.3⫻ 10−4 mm
0.2 mm 0.4 mm 5.9⫻ 10−4 mm
The actual orifice diameter of the larger cathode 共2.79 mm兲 is slightly smaller than the dimension used in the numerical simulations.
a
internal pressure must always be specified. Only two of the three remaining inputs are required for any given simulation scenario. For all cathode simulations performed thus far the orifice plasma density has been specified based on measurements. For the first simulation scenario, the plasma potential at the orifice boundary is varied by iteration until the total discharge current is satisfied. In the second simulation scenario, the plasma potential at the boundary 共or any other single point along the axis of symmetry兲 is held fixed and the peak emitter temperature is varied until the total discharge current is satisfied. In both cases, the functional form of temperature as a function of position along the emitter is assumed or specified by measurement. III. NUMERICAL SIMULATION RESULTS
The hollow cathode model predictions were found to be in good agreement with measurements of the plasma particle density, plasma potential, and electron temperature along the axis of symmetry of the 1.5 cm cathode.15 This larger cathode was designed for a higher power ion engine than the 0.635 cm cathode and thus operates nominally at much different conditions. The two cathodes are compared in Table I. It is noted that one of the significant differences between the two cathodes operating at nominal conditions is that the peak plasma density measured in the 0.635 cm cathode is almost one order of magnitude higher than the value measured in the 1.5 cm cathode. The computed Debye length along the emitter wall is compared in Fig. 2. It is seen that emission enhancement is expected to be non-negligible in the 0.635 cm cathode since the Debye length is as much as 2.5 times smaller than the minimum average pore radius 共assumed to be 1 m in this study兲 and as much as 7.5 times the maximum pore radius 共assumed to be 3 m兲. By contrast, the same effect is expected to be small in the 1.5 cm cathode. For all simulations reported herein the average pore radius has been taken to be 2 m. A comparison between the computed and measured plasma particle densities along the axis of symmetry of the
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FIG. 4. Electron current density 共unit兲 vectors in the 1.5 cm cathode. Top half: without emission enhancement 共b = 0兲. Bottom half: with emission enhancement 共b = 0.5兲.
FIG. 2. Debye length along the emitter compared to limiting values of the mean pore radius.
0.635 cm cathode is shown in Fig. 3. The measured internal pressure during operation of the cathode was 7.9 Torr. The comparison suggests poor agreement without emission enhancement, but the agreement is excellent if enhancement is included using b = 0.5. By contrast, the effect of emission enhancement is found to be negligible in the 1.5 cm cathode, as suggested by the computed 2-D profiles of the electron current density unit vectors compared in Fig. 4 with and without emission enhancement. In view of the complexity associated with the micromorphology of the emitter pores, it is noted that the agreement at b = 0.5 for the 0.635 cm cathode and the simplicity of the formulation presented in Sec. II B only suggest that a likely mechanism for the sharp drop in plasma density 共within a few millimeters of the orifice entrance兲 has been identified. The agreement at b = 0.5 is not meant to imply that the formulation is exact nor that the pores are perfect cylinders. The important deduction here is that the emission enhancement has a dependence on the inverse of the Debye length and thus is proportional to 冑n. Therefore, cathodes that employ the same type of porous emitters as the 0.635 cm cathode studied here and operate at high enough plasma densities, emission enhancement is likely, which can lead to the utili-
FIG. 3. Comparison between theory at various values of b and measurements of the plasma particle density along the axis of symmetry of the 0.635 cm cathode. The experimental error is approximately ±40%.
zation of only a small portion of the emitter while operating with much higher emission current density in that region. The mechanism may in fact be used favorably in the design of longer-life cathodes since, by accessing emission areas deeper into the emitter, it may be possible to operate the insert at lower temperatures. Similar design approaches to enhance emission by artificially increasing the emission area, as opposed to naturally, as is the case described here, have been reported in the past.26 A closer look at the plasma particle density comparison for b = 0.5 in the 0.635 cm cathode on logarithmic scale 共Fig. 5兲 shows that the computed density is in fact much higher than the measured density beyond the emission-enhanced region; the deviation begins at about 6 mm upstream of the orifice, which is approximately where the plasma potential sign changes from positive to negative. The higher plasma densities computed by the model point to an over prediction of electron emission in this region of negative plasma potentials. Specifically, when the plasma potential along the wall becomes negative enough thermionically emitted electrons do not have sufficient energy to overcome the sheath barrier and enter the plasma, but this has not been accounted for in the simulations up to this point. The result with emission turn-off when the sheath potential becomes negative is shown in Fig. 5 共for b = 0.5兲. The agreement is now found to be much better for z ⬍ 1.4 cm predicting well the measured trend of an exponentially decreasing plasma density. As ex-
FIG. 5. Comparison between theory and measurements of the plasma particle density along the axis of symmetry of the 0.635 cm cathode. The computed results are for b = 0.5 and are shown for the cases with and without emission turn-off. The experimental error is approximately ±40%.
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FIG. 8. Computed 2-D profiles of the plasma potential 共in volts兲 in the 0.635 cm cathode for the case of b = 0.5 and emission turn-off. FIG. 6. Comparison between theory and measurements of the electron temperature along the axis of symmetry of the 0.635 cm cathode. The computed results are for b = 0.5 and are shown for the cases with and without emission turn-off. The experimental error is ±0.5 eV.
pected the agreement achieved for z ⬎ 1.4 cm is only slightly affected. With emission turn-off the computed electron temperature is also found to decrease monotonically 共Fig. 6兲 and does not exhibit the minimum seen in the case with no emission turn-off, which is in better agreement with the measured trend. It is noted that, as presently implemented, the “emission turn-off” boundary condition turns off the emission exponentially as the sheath potential transitions from positive to negative, which is quite abrupt and ad hoc. In reality, it is likely that a smoother transition of the emission flux exists as the sheath potential changes sign. Modeling the details of the transition is much more complex, considering issues such as the effect of the emitter morphology on the sheath, and is not attempted here. A closer look at the particle density comparison in Fig. 5 shows that the slope of the computed result beyond 0.4 cm is somewhat steeper than the slope of the data based on a fitted 共dashed兲 line aimed to extrapolate the data beyond the noisy portion 共z ⬃ 0.75 cm兲. If the emission cutoff was not as abrupt along the emitter, the computed density and electron temperature would both be higher. The overall distribution of the je streamlines has also changed dramatically as a result of the boundary condition modifications. The streamlines with both emission enhancement and turn-off exhibit the same general trend observed in the 1.5 cm cathode 共Fig. 4兲 but the saddle point for the electron flow 共electrons moving away from this point兲 now oc-
FIG. 7. Computed electron current density 共je = −enue兲 unit vectors in the 0.635 cm cathode for the case of b = 0.5 and emission turn-off.
curs at only ⬃4 mm upstream of the orifice entrance as shown in Fig. 7. By contrast, without emission enhancement the saddle point in the 0.635 cm cathode is found to occur more than 11 mm upstream. In the 1.5 cm cathode, the saddle point extends almost all the way up to the emitter upstream edge. The importance of the saddle point location is that it defines the effective emission region. The influence of the boundary condition modifications on the plasma potential is shown in Figs. 8 and 9. It is seen that with both emission enhancement and turn-off the plasma potential falls steeply with distance from the orifice entrance, reaching negative values at about z = 0.8 cm. Without emission enhancement, positive plasma potentials persist well beyond z = 1 cm. This is a direct corollary of ⬇ −je + 共1 / n兲 共nTe兲: the computed plasma potential is driven mostly by the electron pressure gradient since the resistive contribution to the electric field is relatively small. For example, around the saddle point, je共z = 0.4 cm兲 = −5 V / m, but 共nTe兲 / n = 1140 V / m. Written as ⬇ −je + 共1 / n兲 共nTe兲, the equation implies that ions play a negligible role, which is a good approximation in this region. IV. PLASMA FLUXES TO THE EMITTER
As explained earlier, one of the motivations for seeking the solution of the hollow cathode plasma in two dimensions is to quantify the profiles of the particle and energy fluxes along the cathode surfaces. The strong dependence of the
FIG. 9. Computed 2-D profiles of the plasma potential 共in volts兲 in the 0.635 cm cathode without emission enhancement 共b = 0兲.
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FIG. 10. Computed particle fluxes along the emitter boundary of the 0.635 cm cathode.
FIG. 12. Computed particle fluxes along the emitter boundary of the 1.5 cm cathode.
emitter life on the operating temperature makes its accurate prediction crucial in any assessments of cathode life. As described above, the present theoretical model does not couple the emitter temperature with the plasma solution in a single self-consistent calculation. Instead, the emitter temperature is specified 共if a direct measurement exists兲 or it is iteratively solved for by fixing the plasma potential and particle density at the orifice entrance boundary. The latter must be provided by direct measurements 共or estimated兲. However, the computed fluxes may in turn be implemented as boundary conditions in a thermal model of the OHC components to predict the emitter temperature. A plasma-thermal model is beyond the scope of this article. Here, we present the computed fluxes along the emitter and identify unique trends in the two cathodes. The main heat sources for the emitter are ions that are accelerated through the sheath, and plasma electrons with energies high enough to traverse the sheath and bombard the emitter. Metastable atoms and photons from the plasma are assumed negligible in this model. The emitter cools by electron emission, by thermal conduction with adjoining surfaces and by radiation. The kinetic energy flux at the emitter surface associated with the acceleration of ions through the sheath of height s is ji · nˆ 共 21 Te + s兲. The 21 Te appears because
the ions are assumed to enter the sheath with the Bohm velocity in order to satisfy the Bohm criterion. For ions to acquire this velocity, the presheath imposes a potential at the sheath edge that equals 21 Te. The power released at the emitter surface 共per unit area兲 by recombination of ions is ji · nˆ 共-eff兲 with being the first ionization potential of xenon 共12.13 eV兲. By assuming that the absorbed electrons are characterized by a Maxwellian distribution function, it has been shown that the average energy of electrons leaving the plasma is 共2Te + s兲.15 Upon arrival at the emitter, the Maxwellian electrons traversing the 共collisionless兲 sheath retain their average thermal energy, 2Te, and are absorbed by the conductor at a potential energy that equals the 共effective兲 work function—the minimum energy needed to remove an electron from the conductor. Thus, the energy flux of the ˆ 共2Te + eff兲. Cooling absorbed electrons at the surface is jab e ·n by electron emission is expressed by the energy flux ˆ eff. jem e ·n The computed particle and energy fluxes versus z 共cm兲 for the 0.635 cm cathode are shown in Figs. 10 and 11, respectively. The corresponding values for the 1.5 cm cathode are plotted in Figs. 12 and 13. The net electron flux along the emitter boundary is found to be almost one order of magnitude higher in the 0.635 cm cathode compared to the 1.5 cm
FIG. 11. Computed energy fluxes along the emitter boundary of the 0.635 cm cathode.
FIG. 13. Computed energy fluxes along the emitter boundary of the 1.5 cm cathode.
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Mikellides et al.
TABLE II. Integrated fluxes 共absolute values兲 along the emitter in the 0.635 and 1.5 cm hollow cathodes. 0.635 cm cathode
FIG. 14. Computed sheath potentials along the emitter wall boundary in the 0.635 cm and 1.5 cm cathodes.
cathode. This is largely due to the higher plasma densities and lower sheath potentials 共see Fig. 14兲 established in the smaller cathode. The higher densities yield Debye lengths that are small enough to allow sheath funneling into the emitter pores of the 0.635 cm cathode. The enhanced emission in turn reduces the size of the effective emission zone 共e.g., compare Fig. 4 with Fig. 7兲. Only ⬃15% of the emitter is utilized in the 0.635 cm cathode, as opposed to almost 100% in the 1.5 cm cathode. The order-of-magnitude increase of the plasma density in the small cathode is largely a result of operating a small orifice size cathode with a relatively high circuit current. While for the two cases studied here the circuit current in the 0.635 cm cathode is only about 1 2 that in the 1.5 cm cathode, the orifice area is about 9 times smaller. Thus, the smaller cathode is forced to operate with higher electron current densities near the orifice. The numerical simulations presented here, which exclude all regions downstream of the orifice entrance but fix the orifice plasma density using direct measurement, predict that the electron current density near the orifice entrance is indeed about 4.5 times higher in the small cathode. In addition, it is noted that even in the absence of sheath funneling 共e.g., if the emitter was not porous兲 Fig. 3 共see curve for b = 0兲 suggests that the effective emission zone would still only extend to less than 50% of the emitter length, which is far less than the effective emission zone calculated in the larger cathode 共Fig. 4兲. Except for a very small region of the emitter around the location of the peak emission enhancement, where heating by ions and cooling by electrons are comparable, the majority of the emitter inner surface in the 0.635 cm cathode is heated by electrons, not by ions, as shown in Fig. 11. Table II compares the emitter current and power in the two cathodes. The total power deposition by electrons in the smaller cathode is 73 W and the net 共heating兲 power by electrons is 9.5 W. By comparison, only 4.1 W are due to ion heating. The current of emitted electrons in the 0.635 cm cathode totals more than 2.5 times the discharge current, which means that this cathode is not operating in an emission-limited mode. Thus, to attain the required current of 12 A, a high current of electrons back to the emitter is
1.5 cm cathode
Emitted electrons
I 共A兲 31.7
P 共W兲 63.5
I 共A兲 33.5
P 共W兲 67
Absorbed ions
0.31
4.1
4.78
88.8
Absorbed electrons
17.2
73
11.2
60.2
generated 共17.2 A兲. To accomplish this high return current, the cathode operates at high plasma particle densities 共Fig. 3兲 and low plasma potentials along the emitter 共Fig. 14兲. In the 1.5 cm cathode the trends are noticeably different. Although the emission current is also higher than the circuit current 共25 A兲, it is so by only 25% as opposed to more than 60% in the smaller cathode. It is found that electrons in fact cause net cooling 共6.8 W兲 in the 1.5 cm cathode. The main heat source is ions, totaling 88.8 W of power deposited to the emitter. In the dense-plasma region, the neutral gas particle density is found to be 6 times higher in the small cathode 共⬃2 ⫻ 1022 m−3兲 with a heavy species temperature maximum of 2523 K. Finally, with regard to possible enhancements of the effective barium evaporation rate from the porous emitter surface by high-energy ions, the numerical results predict peak ion fluxes to the emitter of 1.2 A / cm2 共7.6⫻ 1018 / s cm2兲 and 1.0 A / cm2 共6.2⫻ 1018 / s cm2兲 for the 0.635 and 1.5 cm cathodes, respectively. The corresponding peak sheath drops are 2.9 V 共0.635 cm cathode兲 and 8.3 V 共1.5 cm cathode兲. As noted in Sec. II B, the effect of the presheath was not included in the simulations of the large cathode. A simulation performed specifically to quantify this effect shows only minor quantitative differences; both the peak sheath drop and the ion flux increase by about 10% and 7%, respectively, when the presheath factor is included. In both cathodes, such fluxes and energies of xenon ions are not expected to change the evaporation rate compared to thermally induced evaporation. This conclusion is based on the studies of Doerner et al., who used measurements made at about a hundred times lower ion fluxes but at energies as high as 40 V, and a validated model, to extrapolate the results to the 1.5 cm cathode conditions. It was concluded that in the range of emitter temperatures at which these cathodes typically operate 共⬎1000 °C兲, no enhancement of the evaporation rate is expected for ion energies below 10 V.14,27 Consequently, it may be assumed that the barium evaporation rate of the emitter surface inside both cathodes is driven by thermal evaporation, and is therefore the same as for a similar cathode operating in vacuum. In the absence of enhanced evaporation the evaporation rate depends only on the surface temperature, so that it is possible to calculate the barium depletionlimited life of these hollow cathodes based on vacuumcathode results once the emitter operating temperature has been determined 共by measurement or calculation兲.
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Phys. Plasmas 13, 063504 共2006兲
Plasma processes inside dispenser hollow cathodes
V. CONCLUSIONS
It has been customary to perceive orificed hollow cathodes as devices that operate with the following few characteristic features. The insert runs in an emission-limited mode emitting electron current that is comparable to the circuit current 共when in self-heating mode兲. The emission occurs from the majority of a porous insert that is being heated primarily by ion bombardment, while the electrons cause net cooling of the emitter. A 2-D fluid model of the plasma and neutral gas inside the OHC, that incorporates all the dominant driving physics believed to exist in these devices, was developed recently by the authors and used here for the first time to quantify the plasma fluxes to the emitter inside two OHCs. The two simulated cathodes are significantly different in size and operating conditions. This work has been motivated by the need of a life-predictive capability for these devices that avoids costly and time consuming life experiments. It is found that the larger of the two cathodes 共dc = 1.5 cm and do = 0.3 cm兲 operates closely to the conventional OHC mode. The emitted electron current is only ⬃25% higher than the circuit current 共25 A兲, so that the electron current returning back to the emitter is only a fraction of the circuit current 共45%兲. To accomplish this the plasma is forced to establish a relatively high sheath potential, which peaks at about 8 V, allowing only a small fraction of the electron thermal flux back to the emitter. The main source of heat for the emitter is ions 共⬃90 W兲, while the electrons cool it at a rate of about 7 W. The effective emission zone extends to approximately 100% of the emitter length. By contrast, it is found that the smaller cathode 共dc = 0.635 cm and do = 0.1 cm兲 operates in a completely different mode. Both the emitted 共31.7 A兲 and absorbed 共17.2 A兲 electron currents exceed the circuit current 共12 A兲. The net energy flux of electrons heats the emitter at a rate of 9.5 W, and results in twice the heating power deposited by ions. The effective emission zone extends only a few millimeters deep into the cathode tube. The distinctively different operational features of the two cathodes are due to two main reasons. The smaller cathode is forced to operate at much higher current densities near the orifice due to its 9 times smaller orifice area. The overall smaller size of the cathode 共both the tube and orifice areas兲 also lead to about 6 times higher neutral gas density in the smaller cathode when it is operated with a similar mass flow rate as the large cathode, which in turn increases the local ionization rate near the orifice. The two effects lead to almost an order of magnitude increase of the plasma density inside the small cathode compared to the large cathode. The Debye length thereby becomes small enough to allow sheath penetration into the pores and in turn most likely enhance the emission. The maximum emission current density is found to be about an order of magnitude higher in the small cathode. Despite the aforementioned differences, both cathodes generate low enough fluxes and energies of xenon ions to the emitter such that no enhancement of the effective barium evaporation rate is expected. Thus, the computed fluxes may
be implemented as boundary conditions into a thermal model to predict the emitter temperature, which can then be used to obtain the barium depletion-limited life based on the large volume of existing vacuum-cathode data. ACKNOWLEDGMENTS
The research described in this article was carried out by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration for the Prometheus Advanced Systems and Technology Office. 1
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