Fully Kinetic Particle-in-Cell Simulation of a Hall Thruster

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1 Fully Kintic Particl-in-Cll Simulation of a Hall Thrustr Francsco Taccogna 1, Savino Longo 1,2, Mario Capitlli 1,2, and Ralf Schnidr 3 1 Dipartimnto di Chimica dll Univrsitài di Bari, via Orabona 4, Bari, Italy cscpft61@ara.ba.cnr.it 2 IMIP-CR, sct. Bari, via Orabona 4, Bari, Italy 3 Max Planck Institut für Plasmaphysik, Wndlstinstr. 1, D Grifswald, Grmany Abstract. A 2D axisymmtric fully kintic Particl-in-Cll (PIC) modl of th atom (X), ion (X + ) and lctron dynamics of a stationary plasma thrustr (SPT) is dvlopd. Elctron-nutral lastic scattring, xcitation and ionization procsss ar modlld by Mont Carlo collision mthodology. Th intraction of th plasma discharg with th cramic walls lads to plasma rcombination, nrgy loss and scondary lctron. Ths phnomna ar includd into th modl by diffrnt approachs. Th lctric fild is slfconsistntly solvd from th Poisson quation, whil th magntostatic fild is prcomputd. Th cod is applid to a scald SPT thrustr gomtry whr fundamntal physics paramtrs ar kpt constant. Th modl rproducs th discharg ignition dynamics. Th numrical rsults will provid a bttr undrstanding of th xprimntally obsrvd nhancd axial lctron currnt and high frquncy oscillations. 1 Introduction Th modlling of lctric thrustr is a vry important issu in viw of th incrasing importanc of such propulsion in all spac applications whn spcific impuls, and not just powr, is important, i.. for satllit guidanc, orbit transfr and dp spac xploration projcts. Fig. 1. Schmatic rprsntation of th discharg chambr in th SPT-100 thrustr M. Bubak t al. (Eds.): ICCS 2004, LCS 3039, pp , Springr-Vrlag Brlin Hidlbrg 2004

2 Fully Kintic Particl-in-Cll Simulation of a Hall Thrustr 589 A SPT can b schmatically dscribd (s Fig. 1) as an anod-cathod systm, with a dilctric annular chambr whr th propllant ionization and acclration procss occurs [1]. This thrustr works using a prpndicular lctric and magntic filds configuration. A magntic circuit gnrats an axisymmtric and quasi-radial magntic fild btwn th innr and outr pols. In opration, an lctrical discharg is stablishd btwn an anod (dp insid th channl), which is acting also as a gas distributor, and an xtrnal cathod, which is usd also as an lctron mittr. In this configuration, cathod lctrons ar drawn to th positivly chargd anod, but th radial magntic fild crats a strong impdanc, trapping th lctrons in cyclotron motion which follows a closd drift path insid th annular chambr. Th trappd lctrons act as a volumtric zon of ionization for nutral propllant atoms and as a virtual cathod to acclrat th ions which ar not significantly affctd by th magntic fild du to thir largr Larmor radii. Gnrally, xnon is usd as propllant. Th quasi-radial magntic fild and th channl lngth L ar chosn so that: r ω L, c,i << L << r τ i L,i << 1 << ω c, τ (1) whr r L is th Larmor radius, ω c is th angular cyclotron frquncy, τ is th man tim btwn collisions and th subscripts and i mans lctrons and ions. Elctrons ar strongly magntizd, whras ions ar non-magntizd. Th rsulting xtrnal jt composd by th high spd ion bam is subsquntly nutralizd by part of lctrons coming from th xtrnal cathod-compnsator. Th prsnc of an insulator as wall matrial has a profound ffct on th plasma within a Hall typ thrustr. Aftr an impact with dilctric walls, high nrgy lctrons ar absorbd and rlas lss nrgtic scondary lctrons that ar mor firmly confind by th magntic fild. Th rsult is that th dilctric wall limits th tmpratur of th lctrons confind into th channl. By limiting th lctron tmpratur, a smooth continuous variation of th plasma potntial rsults. Ths ffcts will b spcially addrssd within this work. Modls of Hall thrustrs hav bn dvlopd using hybrid fluid-particl approachs [2-7] to aid in th optimization of th prformanc of th thrustr. In all ths modls, lctrons ar dscribd as a fluid by th first thr momnts of th Boltzmann quation. Thy ar 1D (axial or radial), quasi-1d (considring wall losss) or 2D (in th plan (r,z) or (r,θ)), and stady-stat or transint (on th ion tim scal). Most of th modls ar basd on th quasi-nutrality (Q) assumption. Thrfor, Poisson s quation is not solvd and th constraints rlatd to th xplicit tim intgration of th transport quations and xplicit spac intgration of th Poisson s quation wr thrfor liminatd. This assumption considrably simplifid th numrical aspct of th simulation. Usually, in th lctron momntum transport quation, Bohm and/or nar-wall conductivity wr includd by mans of mpirical fitting paramtrs. A qustion which cannot b rsolvd by ths modls, and which in fact strongly limits th rliability of thir rsults, is th lctron transport in SPTs, and in particular th important rol that th lctron intractions with th channl walls play togthr with volum procsss. As a rsult,

3 590 F. Taccogna t al. SPT prformanc is affctd by both th stat of th wall surfac and th proprtis of plasma structurs on Dby and lctron-larmor scals. Thrfor, th construction of a slf-consistnt thory of SPT procsss rquirs a kintic dscription of not only havy particls (atoms and ions), but of lctrons as wll. To accomplish this, w prsnt a two-dimnsional axisymmtric Particl-In-Cll (PIC) [8,9] modl using Mont Carlo Collision (MCC) mthod to tak into account lctron-nutral intractions. Scondary lctron mission from th wall is simulatd by a probabilistic modl [10]. Th lctric potntial is calculatd solving Poisson quation solution without assuming quasi-nutrality. W first dscrib in Sc. 2 th numrical modl. Th rsults ar prsntd and discussd in Sc umrical Mthod 2.1 Gomtrical Scaling To captur lctron dynamics, w nd a tim-stp on th ordr of th plasma oscillation. Unfortunatly, using th ral mass ratio, nutral particls will rquir millions of such tim-stps to cross th simulation rgion. In ordr to spd-up th cod w hav rducd th dimnsion of th discharg whil prsrving th valus of th rlvant paramtrs that govrn th physics [11]. All th basic plasma charactristics in gas-phas ionization dvics rly havily on th ionization procss, whrby th nutrals ar ionizd in collisions with th lctrons. As th siz of th dvic is rducd, vrything ls rmaining constant, th numbr of collisions that th lctrons and th nutrals xprinc with ach othr as thy travrs th ffctiv lngth of th dvic is rducd. Thus, in ordr to maintain th ffctiv collision probability, it is ncssary to incras th numbr dnsitis of all th spcis in proportion, that is th mass flow rat of propllant should scal as: m = M X n c s A L (2) whr M X, n and c s ar th nutral mass, dnsity and thrmal vlocity rspctivly, whil A is th ara of th anod. Morovr, to prsrv th ffctivnss of th lctron confinmnt schm (th ratio of th Larmor radius to th thrustr dimnsion) undr scaling, th strngth of th magntic fild must vary invrsly with lngth. Th discharg currnt I d, dtrmind as th product of th currnt dnsity tims th ara of th dvic, scals proportionally with lngth. Consquntly th thrust T, that is th total forc undrgon by th SPT in raction to th acclration of th ions, scals linarly with th lngth whil th spcific impuls I sp = T / mg rmains invariant undr gomtrical scaling. Hr w hav assumd that th lctron tmpratur is constant and indpndnt of scal, as was shown in Rf. [11]. Howvr, th bnfits of this trick ar limitd. First, th plasma frquncy must rmain shortr than th lctron gyrofrquncy. Also, th Dby lngth must rmain a small quantity with rspct to ovrall thrustr dimnsions. If th shaths bcom too larg, thy can intrfr with th discharg. Undr ths constraints th gomtrical rducing factor was ζ=0.02.

4 Fully Kintic Particl-in-Cll Simulation of a Hall Thrustr Magntic Fild Bcaus th slf-inducd magntic fild is ngligibl compard to th applid fild from th Hall thrustr s coils, th simulation uss only a constant magntic fild. Undr this condition w can dfin th magntic potntial φ B and us th corrsponding Laplac quation to solv for th magntic fild. Boundary conditions ar th sam usd by Fif [3]: at th right and lft boundaris, on has zro flux, whil on th outr magntic pol, on has φ B = 1, and on th innr pol on has φ B = 0. W thn solvd for φ B across th domain, and took th gradint to arriv at th magntic fild B. Finally, w spcifid a control point and a fild strngth at that control point, and usd that to normaliz th fild. 2.3 utral Gas Particl Kintics Th nutral dnsity in this tst cas is two ordr of magnitud highr than th plasma dnsity. Thrfor w lt nutral supr-particls b som intgral numbr s =100 tims largr than plasma supr-particls. As a nutral undrgos ionization vnts, s dcrass by on unit until s =0 and th particl disappars. W initializ th simulation by xpanding a plum of nutrals from th anod with a long tim-stp until w approach a stady stat. Th numbr of nutrals macroparticls cratd at th anod lin (z=0) pr tim-stp t, is a function of th scald mass flow rat according to: m = t (3) Mw whr w is th nutral statistical wight. Thy ar injctd with a radial position sampld from an uniform cylindrical dnsity distribution and thir initial vlocity distribution is takn to b half-maxwllian with a tmpratur of typically 500 K, by using th polar form of th Box-Mullr transformation for vlocity. utrals can disappar whn thy rach th right boundary of th gomtry. utrals which hit th anod and walls ar r-mittd according to an half-maxwllian at a wall tmpratur (900 K) basd on xprimntal data. On must considr th probability for a particl crossing th surfac to hav a givn dirction. This probability dnsity actually follows a cosin law, du to th fact that particls with a larg vlocity componnt along th normal to th surfac scap mor frquntly than othrs. Th probability dnsitis for axial and azimuthal vlocity componnts ar Gaussian distribution (th polar form of th Box-Mullr transformation was usd), whil th probability dnsity for th radial componnt is a Rayligh distribution. utralnutral collisions may b ignord, assuming a nutral fr molcular rgim bcaus th man fr path is much longr than thrustr dimnsion. 2.4 Plasma Phas Whn nutral particls hav filld th simulation rgion, lctrons ar introducd ach tim stp t from th xit plan (th cathod is not includd in th simulation rgion) with a stady stat currnt control mthod of lctron injction [12]:

5 592 F. Taccogna t al. I t d c c n = n + n i (4) qw whr w is th statistical wight of lctron supr-particls, q is th lmntary c c charg, n i and n ar th numbr of ions and lctrons passing th fr spac boundary ach itration. Elctron initial vlocity distribution is takn to b half- Maxwllian with a tmpratur of typically about 15 V. Thy can los nrgy and chang momntum by collisions (lastic scattring, xcitation and ionization) with nutrals. Ths intractions ar modlld by MCC tchniqu and scattring cross sctions ar gathrd from th litratur [12]. For ach lctron w calculat th probability of an - scattring in a tim t coll short with rspct to th man fr flight tim: P tot = n σ tot (v )v t = 3 k= 1 n σ k (v )v t whr n is th nutral dnsity, σ tot is th total lctron-nutral cross sction, v is th lctron vlocity and P 1, P 2 and P 3 ar th probability for th occurrnc of collisional vnt 1, 2, and 3, rspctivly. P tot is compard with a random numbr r d sampld from an uniform distribution in th rang [0,1] in ordr to dcid if a collision vnt happns (in our cas t coll is chosn so that P tot <10-2 ). If P tot >r d, w compar anothr random numbr to th cross sctions for lastic scattring, xcitation, and ionization to dtrmin which typ of vnt occurs. W choos th collisional vnt j if j 1 j Pk Pk < rd (6) P P k= 1 tot In all cass, th lctron is scattrd isotropically. If th collision is inlastic, nrgy (8.32 V for th first xcitation and 12.1 V for th first ionization) is subtractd from th lctrons. In th cas of ionization, ions and scondary lctrons ar cratd at th primary lctron's location. Th nrgy of primary and scondary lctrons is dividd randomly. Whn an lctron striks th dilctric wall (Boron itrid B), w dcid th numbr of lctrons mittd on th basis of th nrgy and th angl of impact of th incidnt lctron implmnting th probabilistic modl of Furman and Pivi [10]. Th nrgy spctrum of th mittd lctrons is abl to rproduc th thr diffrnt typ of scondary lctrons, that is backscattrd (high nrgy rgion), rdiffusd (middl nrgy rgion) and tru scondary lctrons (low nrgy rgion). For simplicity, w assum th sam mission-angl distribution ( cosθ and uniform azimuthal angl) for all lctrons, rgardlss of th physical mchanism by which thy wr gnratd. Vrboncour intrpolation mthod [13] which guarants charg dnsity consrvation on a radial mtrics is usd to wight particls to th grid nods, whr k= 1 tot = 3 k= 1 P k (5)

6 Fully Kintic Particl-in-Cll Simulation of a Hall Thrustr 593 th fild quations ar solvd, and to wight th filds back to th particls. Th lctric potntial is r-calculatd ach tim-stp using finit diffrncs mthod for Poisson quation which is solvd itrativly using succssiv ovr-rlaxation (SOR) with Chbyscv acclration tchniqu [8]. As rgards th boundary conditions, w kp th lctric potntial constant at th anod and at th xit plan and w also assum that th lctric fild changs its sign at th dilctric surfac and its magnitud rmains th sam [14]. Thn, th boundary condition at th walls can b writtn as: ρ E w = (7) 2ε whr ρ is th surfac charg dnsity and ε 0 is th fr spac prmittivity constant; th possibl surfac conductivity of th dilctric is nglctd. Th potntial at th wall surfacs is assumd to b zro. Elctrons ar movd discrtizing th quations of motion by th lapfrog mthod of Boris [9]. All lctrons which hit th anod boundary and th fr spac boundary ar dltd. At th nd of ach lctron loop, ions ar cratd at th ionization locations. Th initial ion vlocity is st qual to th local nutral background vlocity, but ions and nutrals ar movd onc ach 100 lctron loop. An ion which impacts th anod and th walls disappars and a nutral is cratd with on half of th ions initial kintic nrgy, but with a random vlocity dirction. Th particl is thus partially accommodatd in nrgy, and fully accommodatd in momntum. At th fr spac boundaris, all particls ar dltd and a count of lctron, ion and nutral fluxs is maintaind. 0 3 Rsults and Discussions In this sction, w prsnt th simulation rsults (scald valus) at th following oprating conditions: channl lngth L=2.5 cm, innr radius R in =3.5 cm, outr radius R out =5 cm, discharg voltag φ d =300 V, mass flow rat m = 3 mg/s, discharg prssur P=2x10-5 mbar, discharg currnt I d =3.2 A, maximum radial magntic fild B r,max =150 Gauss. Figur 2(a) shows long tim history of lctron, ion and nutral total numbr macroparticls. W can s that chargd particls continu to oscillat (th simulatd discharg is not stabl), whil th numbr of nutrals is not strongly changd du to th fact that on nutral transit tim corrsponding to th simulation tim is not sufficint to s th low frquncy phnomna rlatd to th nutral scals. Th colour plots of Figs. 2(b)-3 dmonstrat th main faturs of th discharg and show, rspctivly, th spac variation of th lctric potntial, plasma dnsity and lctron tmpratur. As it can s in Fig. 4(b), most of th potntial drop occurs in th xhaust rgion, whr th magntic fild is larg. This dcras compnsats th low lctron conductivity in this rgion and nsurs currnt continuity. It is customary to allocat th acclration rgion hr. Th larg axial lctric fild rsulting from this voltag drop is rsponsibl for acclrating th ions from th ionization rgion to th xit plan and th lctron from th outlt to th

7 594 F. Taccogna t al. anod. From th figur, also th littl anod drop ( 20 V) and th latral wall shaths, whos voltag drop dcrass from th anod to th outlt, ar vidnt. Howvr, th computd potntial vanishs at th channl xit, whil obsrvations [15] indicat that only on-half to on-third of th potntial drop taks plac downstram of th thrustr xit. This diffrnc is du imposing th zro potntial boundary condition at th xit plan in th numrical simulation, i.., th full potntial drop is forcd to occur insid th channl , E2 2.1E2 1.8E2 2.5E2 macroparticls lctrons ions nutrals r (mm) E E , , t (s) 1.4E z (mm) Fig. 2. (a) Long tim history of lctron, ion and nutral macroparticls. (b) Profil of th plasma potntial Th spac variation of th plasma dnsity (Fig. 3.a) shows that th plasma rachs its maximum in th cntr of th channl and in th ionization rgion, whil it dcrass in th acclration rgion du to th incrasing ion vlocity. Th lctron tmpratur (Fig. 3.b), calculatd in ach cll from th usual formula rachs a maximum (18 V) clos to th xhaust and dcrass to about 6.7 V in th low lctric fild rgion. Th pak can b attributd to Ohmic hating du to th maximum azimuthal drift vlocity in this rgion. Furthrmor, th drop nar th walls is a consqunc of th mission of cold scondary lctrons from th insulators. Ths numrical rsults ar consistnt with our global undrstanding of th stationary plasma thrustr and rproduc quit wll th xprimntal obsrvations [15] E19 1.4E19 2.8E19 4.2E r (mm) E E18 4.9E19 5.6E z (mm) r (mm) z (mm) Fig. 3. Profil of th lctron (a) dnsity and tmpratur (b)

8 Fully Kintic Particl-in-Cll Simulation of a Hall Thrustr Conclusions A 2D(r,z)-3V numrical modl was dvlopd to assss th ffct of dilctric walls in stationary plasma thrustrs. Th modl consists of a fully kintic PIC-MCC for th plasma phas. Th mission of scondary lctron by lctron impact from th walls is takn into account by a probabilistic modl simulating th diffrnt kind of lctrons cratd at th wall, backscattrd, rdiffusd and tru scondaris. In ordr to mak th simulation possibl and modl th lctron dynamics, a nw kind of scaling-law is applid for th PIC modl. Rfrncs 1. Zhurin, V. V., Kaufman, H. R., Robinson, R. S.: Plasma Sourcs Sci. Tchnol. 8 (1999) R1. 2. Komurasaki, K., Arakawa, Y.: J. Prop. Pow. 11(6) (1995) Fif, J. M.: PhD thsis, Massachustts Institut of Tchnologis (1998). 4. Haglaar, G. J. M., Barills, J., Garrigus, L., Bœuf, J.-P.: J. Appl. Phys. 91(9) (2002) Koo, J. W., Boyd, I. D.: AIAA papr (2003). 6. Garrigus L. : PhD thsis, Univrsité Paul Sabatir, Toulous (1998). 7. Lvchnko I., Kidar M.: IEPC (2003). 8. Eastwood, J. W., Hockny, R. W.: Computr Simulation using particls, McGraw- Hill, w York (1981). 9. Birdsall, C. K., Langdon, A. B.: Plasma Physics via Computr Simulation, Mc-Graw- Hill, w York (1985). 10. Furman, M. A., Pivi, M. T. F.: Phys. Rv. Spcial Topics Accl. and Bams 5(12) (2002) Khayms, V.: PhD thsis, Massachustts Institut of Tchnology (2000). 12. Szabo, J. J. Jr.: PhD thsis, Massachustts Institut of Tchnology (2002). 13. Vrboncour, J. P. : J. Comp. Phys. 174 (2001) Morozov, A. I., Savl v, V. V.: Plasma Phys. Rp. 28(12) (2002) Bishav, A. M., and Kim, V. P.: Sov. Phys. Tch. Phis. 23 (1978) 1055.