Numerical study of the normal current density behaviour in a narrow -gap glow discharge

Transcription

1 Numrical study of th normal currnt dnsity bhaviour in a narrow -gap glow discharg S.A. Starostin 1, P.J.M. Ptrs 2,, E.Kindl 1, A.V. Azarov 2, S.V. Mitko 2, and K.J.Bollr 2 1 Istitut für Nidrtmpratur Plasmaphysik, Fridrich-Ludwig-Jahn-Str Grifswald, Grmany 2 Univrsity of Twnt, Lasr and Non-linar Optics Group, Faculty of Tchnical Scincs, PO Box 217, 7500 AE, Enschd, Th Nthrlands Abstract A numrical study of normal glow discharg proprtis was prformd in th cas of small lctrods sparations ( cm) and modrat gas prssurs (10-46 Torr). A rcntly obsrvd nw xprimntal ffct of a considrabl rduction in th normal currnt dnsity for smallr discharg lngths was analyzd both by mans of 2D fluid modl and by a minimal 1D drift modl of gas discharg. A good agrmnt btwn thortical and xprimntal bhaviour was dmonstratd. An influnc of th lctrods sparation and of th gas hating on th valu of th normal currnt dnsity is discussd. starostin@inp-grifswald.d, p.j.m.ptrs@utwnt.nl 1

2 Introduction Th normal glow discharg invstigation history spans mor than a cntury thus on could xpct that this is on of th bst-studid subjcts in modrn physics. Still, in a varity of conditions and applications whr normal dischargs ar usd (.g. lighting, plasma procssing and lasr pumping) th physics of th normal glow discharg rmains an intnsivly discussd issu. According to th gnrally adoptd viw [1,2], a diffusiv, slf-sustaind discharg btwn two paralll plan lctrods can b classifid as follows: at vry small currnts (a fw µa or lss) whn th spac charg dos not affct th lctric fild distribution, th discharg occupis th whol lctrod surfac. Such discharg is calld a Townsnd discharg. Th sustaining voltag is qual to th brakdown voltag and dos not chang with th currnt. As th currnt gradually incrass th influnc of spac charg bcoms mor pronouncd and th discharg bgins to contract. Th slop of th voltag-currnt charactristic in this rgim is ngativ (causing instabilitis). This discharg is calld subnormal. Whn th currnt dnsity rachs a crtain valu, with a furthr currnt incras th discharg bgins to sprad again. Whil th total currnt incrass, th currnt dnsity and voltag rmain constant. This is known as th normal discharg and is charactrisd by th so-calld normal currnt dnsity, th normal voltag drop and th normal cathod layr thicknss. As soon as th currnt spot on th cathod will covr th whol surfac of th lctrod, th voltag will start to ris again with a furthr incras in currnt. This is known as th abnormal" rgim. For small valus of th pd paramtr (pd < pd c, wr pd c is usually in ordr of 1 Torr.cm) a normal phas dos not xist and a dirct transition from Townsnd to abnormal discharg, with incrasing currnt, occurs. In this cas th discharg will always occupy th whol lctrod surfac. Th dvlopmnt of micro-dischargs for UV light sourcs, plasma display lmnts or as activ mdium for gas lasrs rcntly ld to a rnwd intrst in dischargs occurring in short gaps at lvatd gas prssurs. It appars that thr is a lack of information on glow dischargs in discharg gaps with th pd valu in th ordr of pd c and with th discharg in a transint stat btwn subnormal and normal. Morovr in rlation to th prsnt intrst for bifurcation phnomna in gas dischargs, th rlaxation to a normal glow at low pd valus can b considrd as an important mchanism for structurd discharg formation [3]. Th procss of normal cathod spot apparanc is important not only for 2

3 continuous wav (dirct currnt) opration, but for pulsd or altrnating currnt (AC) and barrir dischargs as wll. Th transition from Townsnd to normal discharg was alrady studid in a numbr of thortical and xprimntal articls [4,5,6,7,8]. Howvr ths paprs ar not focusd on th proprtis of th normal dischargs itslf. Th most important scaling paramtr charactrizing a normal discharg is th rducd normal currnt dnsity J n /p 2, whr J n is th normal currnt dnsity and p th prssur. It is blivd (s for instanc [1]) that th valu of J n /p 2 is solly dpndnt on both th gas and lctrod matrial and is indpndnt of th discharg lngth. It should b notd that th thortical vrification is basd on th classical, on-dimnsional thory of th cathod layr dvlopd in th first half of 20 th cntury by von Engl and Stnbck [9]. Howvr, as w found in rcnt xprimnts [10], J n /p 2 bcoms dpndnt on th discharg lngth, whn th distanc btwn th lctrods is in ordr of th ngativ glow lngth. Th classical discharg modl was furthr dvlopd by Ward in th lat 50s - arly 60s [11,12]. Ward prformd on of th first numrical invstigations of th gas discharg, making slf-consistnt calculations of th lctric fild and drift currnt. Howvr this work was by far complt and no spcial invstigation was don to study th bhaviour of th J n /p 2 valu vrsus th pd paramtr. Th rcnt comprhnsiv analytical and numrical analysis of th 1-dimnsional discharg modl don in [3] rvals nw typs of voltag-currnt charactristics at low pd valus. Howvr, th normal currnt bhaviour was not studid in this work. It is important to not that within th 1-dimnsional modl th normal currnt dnsity can not b calculatd dirctly. It is gnrally accptd to follow th assumption, mad by von Engl and Stnbck, that th normal currnt corrsponds to th abscissa of th minimum on th calculatd voltag-currnt charactristic. Howvr this assumption rmains a constantly argud issu (s [1]) and is not provn yt. Strictly spaking th formation of th normal currnt spot is at last a two-dimnsional ffct so its dscription rquirs adquat modls. On of th arlist two-dimnsional numrical study of th normal discharg in nitrogn was don by Raizr and Surzhikov in [13]. At that tim 2D calculations wr vry tim consuming but vn this somwhat incomplt two-dimnsional modlling contributd a lot to a dpr insight on th procss of th normal currnt spot formation [1]. Mor rcnt numrical invstigations of th normal dischargs and of th transition from Townsnd to normal rgims for a pd rang of 1-10 Torr cm wr prsntd in [4,5]. Th numrical study don in [4] with a 2D fluid modl both with a local and non-local 3

4 ionization trm was mainly focusd on th low currnt Townsnd and subnormal rgims. In th sam articl a comprhnsiv smi-analytical modl of th currnt spot was proposd. Particularly it was prdictd that xistnc rgion of th normal currnt spot can b xtndd to lowr pd valus if scondary mission is a growing function of th lctric fild [4]. In [5] th applid numrical modl of th discharg is similar to th on dscribd in th prsnt work, but without accounting for gas hating. It will b shown blow that gas hating can significantly influnc th proprtis of normal discharg spcially at modrat gas prssurs. Th modlling in [5] was prformd for a discharg in Ar at a fixd intr-lctrod distanc of 1 cm and for two diffrnt valus of th gas prssur (3 and 6 Torr). By incrasing th discharg currnt a transition from Townsnd to subnormal, to normal and to an abnormal discharg was followd. Additionally subnormal oscillations wr invstigatd. Nvrthlss from th rsults prsntd in [4,5] it is difficult to mak a conclusion about th bhaviour of th normal currnt dnsity in th vicinity of th valu pd c, thrfor an additional invstigation is ndd. Dspit th imprssiv dvlopmnt of gas discharg modls, spcially during last yars, such a fundamntal paramtr as th normal currnt dnsity was not studid in dtail. So, th knowldg on th normal currnt dnsity bhaviour and its scaling proprtis hardly changd sinc th classical discharg modl was proposd by von Engl and Stnbck svnty yars ago. This situation is also causd by th lack of nw xprimntal information on th subjct. In th prsnt work a two-dimnsional fluid modl of th discharg was applid to analys th rcnt xprimntal rsults [10]. At th sam tim w mad calculations using th classical discharg modl [11] in ordr to s if a minimal approach to discharg modlling can rflct th nwly obsrvd bhaviour of th normal currnt dnsity vrsus pd. It should b additionally noticd that although in this work w ar considring th sam rang of pd paramtr as in [4,5], hr it corrsponds to th cas of small lctrod sparations and modrat gas prssurs. Ths conditions ar intrsting for many nowadays applications of glow discharg. Modlling Th primary goal of th modl dvlopd hr was to invstigat if th xprimntally obsrvd dcras of th normal currnt dnsity with th gap lngth at short lctrod 4

5 distancs [10] can b undrstood by mans of on of th standard approachs that hav so far bn usd to simulat gas dischargs. It is known that dischargs in rar gass ar influncd in thir qualitativ bhaviour by impuritis. Th xprimnts dscribd in [10] wr carrid out without gas flow thus a contamination of th gass usd in ths xprimnts is not xcludd. That is why w dcidd to modl a discharg in air for th sak of quantitativ comparison with xprimntal data. Additionally, w mad calculations of th normal currnt dnsitis at low pd valus, utilizing Ward s 1D minimal modl of th gas discharg [11]. A fully spatial dpndnt simulation of th discharg gomtry with rctangular-shapd lctrods usd in [10] would rquir a 3 dimnsional modl in Cartsian co-ordinats. Howvr, as it follows from th masurmnts [10], th valus of th normal currnt dnsity ar indpndnt on th lctrod width. At th sam tim it is important to show by mans of modlling that th dpndncy of th normal currnt dnsity on th discharg lngth is not xclusivly a proprty of th discharg gomtry applid in [10]. From this considration for our modlling w hav chosn a mor gnral 2D cylindrical gomtry, kping gas prssur and lctrod sparation corrsponding to [10] According to th numrical rsults of kintic simulations [4] th assumption of a local dpndncy of th ionisation cofficint on th lctric fild is a good approximation for subnormal dischargs, whil for normal dischargs at pd rangs of 1-10 Torr.cm th nonlocal natur of th ionisation cofficint bcoms mor pronouncd. In our modlling w ar using a drift-diffusion approximation whr th ionisation cofficint, lctron mobility and lctron diffusion cofficint ar functions of th man lctron nrgy. This approach still allows accounting for small non-local ffcts. Th possibl influnc of run-away lctrons on th discharg charactristics rmains out of considration. A 3-momnt drift-diffusion fluid quation st (s Appndix A and [14]) was applid for modlling of th normal discharg in air in a two-dimnsional cylindrical gomtry. Th corrsponding st of quations includs th continuity quations for lctrons and positiv ions, th quation for th man lctron nrgy, th Poisson quation for th lctric potntial, and th gas thrmal conductivity quation. Though in gnral th lctron nrgy distribution function (EEDF) hardly rsmbls a Maxwllian on, according to rf. [14], it is possibl to introduc th man lctron nrgy as th main charactristic of th distribution function and to considr all of th cofficints as bing dpndnt only on this man nrgy. To driv th man lctron nrgy, a corrsponding quation that taks into account th nrgy loss, th gain and th lctron 5

6 hat conduction has to b solvd (A4). Th rat and transport cofficints as functions of th man lctron nrgy ar dtrmind bforhand by numrical solving th 0D Boltzmann quation for th EEDF. Thn ths dpndncis, in th form of tabulatd functions, ar usd for rsolving quation st A1-A12. This approach rquirs th assumption that ths rats and cofficints dpnd on th man lctron nrgy in th sam way as thy do in th quilibrium condition, whn th local lctron nrgy losss ar balancd by th local nrgy gain [14]. For a discharg in air as wll as for dischargs in atomic gass, at diffrnt prssurs a similar bhaviour of th normal currnt dnsity vrsus lctrod sparation was obsrvd [10]. On th basis of ths xprimntal obsrvations it can b assumd that this bhaviour is not dirctly rlatd to volumtric procsss as rcombination or lctron attachmnt. Thrfor in our modl w ar limiting ourslvs to a simpl sourc trm in th continuity quations including only on ionization procss. Crtainly, an in dpth numrical study of a discharg in air would rquir accounting also for ngativ ions and for various plasmachmical ractions but this is not th aim of th prsnt work. According to our calculations in th studid rang of tmpraturs, th tmpratur dpndncy of th thrmal conductivity cofficint λ can b approximatd by a linar function. (S Appndix A7). Firstly w calculatd th tabulatd dpndncy of th thrmal conductivity cofficint on th gas tmpratur and thn dducd th cofficints A and B for th linar function (A7). Furthr, it was assumd that thr is no convctiv gas flow in th discharg volum and that th local dnsity can b dtrmind from th idal gas quation. Our numrical mthod is basd on an xponntial discrtisation schm [15] of th driftdiffusion quations on a non-quidistant msh. Th transport and Poisson quations wr intgratd succssivly in tim. Discrtisation of th Poisson and continuity quations on a two dimnsional grid, rsults in 5-points linar systms that wr solvd with th SIP (strongly implicit procdur) mthod [16]. Starting from initial conditions th volution of th discharg paramtrs wr advancd in tim till a stationary solution was rachd. A smi-implicit rprsntation of th sourc trms in th Poisson and lctron nrgy quation [17] was usd to allow largr tim stps. Th discharg was modlld in a cylindrical numrical domain with a radius of 0.7 cm. (Th latral siz of this domain is not important as long as th cathod spot is smallr than th domain). Th distanc btwn th lctrods was chosn in th rang of 0.5 to 4 mm. 6

7 Various non-quidistant mshs wr usd to nsur that th grid siz dos not affct th valu of calculatd normal currnt dnsity (lik it was obsrvd in [13]). Th kintic cofficints for lctrons wr calculatd with th frwar BOLSIG program [18]. For th discharg in air w assumd a gas mixtur of N 2 : O 2 : Ar = 78 : 21 : 1; th gas prssur in our calculations was varid from 10 to 46 Torr. Th mobility of th positiv ions was takn as 1.5 cm 2 /(V sc) which corrsponds to a gas dnsity of cm -3. A valu of 0.1 was takn for th scondary ion lctron mission cofficint γ. Th dpndncy of th gas thrmal conductivity cofficint λ [W/cm K] on th tmpratur (s Appndix A7) was approximatd with A air = , B air = It should b noticd that th dpndncy of th ffctiv scondary lctron mission cofficint γ on th lctric fild valu at th cathod is now an intnsivly discussd issu [5,19,20] and can significantly influnc th rsults of th fluid modl [5,17]. According to a rcnt comprhnsiv analysis [19] various mchanisms such as ion or lctron mission, photo-mission by rsonanc photons, mission causd by mta-stabls and fast atoms contribut to th total ffctiv lctron mission from th cathod. Th influnc of ach procss is dpndnt on th valu of th E/N paramtr on th cathod as wll as on th stat ( clan or dirty ) of th cathod surfac. If on considrs only th procss of scondary ion-lctron mission thn γ is a fast growing function of th E/N paramtr at low E/N valus but it rachs a constant valu at high E/N. Th natur of this functionality has its origin in th ffct of lctron back-scattring towards th cathod. This dpndncy can considrably affct th sustaining voltag of th Townsnd discharg and th proprtis of th subnormal discharg. It also shifts th minimal pd at which th normal discharg can xist to th lowr valus. Yt bcaus, th γ dpndncy on th lctric fild is unknown for a discharg in air, w hav dcidd to us, for a first modlling, a constant γ and thus do not tak into account back-scattring ffcts. A study of th influnc of th variation of γ with th lctric fild will b a subjct for furthr invstigation. Th quation st for th 1-dimnsional classical drift modl is listd in Appndix B. Taking into account th continuity quation for th drift currnt and th lctrostatic quation for th lctric fild rsults in two first ordr diffrntial quations. Th numrical solution of systm (Appndix B: B1-B3) is prformd by a shooting mthod whr an initially chosn lctrical fild valu at th cathod (or anod) is varid until th boundary conditions (B3) ar mt. For th calculation of th normal currnt dnsity th 7

8 gnrally accptd assumption that normal currnt corrsponds to th minimum on th calculatd voltag currnt charactristic was usd. Th input paramtrs for th 1D drift modl ar as follows: th mobility of positiv ions and th scondary lctron mission cofficint wr takn th sam as for th 2D fluid modl. Th mobility of th lctrons was chosn as 434 cm 2 /(V sc) at a gas dnsity of cm -3 (according to th rsults of th BOLSIG cod [18] for a N 2 - O 2 - Ar gas mixtur and 4 V lctron nrgy). Two cass wr considrd for th Townsnd s ionization cofficint. In th first cas th ionization cofficint was calculatd as a function of th rducd lctric fild E/p and thn usd as a tabulatd function. In th scond cas according Raizr s txtbook [1] an xponntial approximation was usd (s Appndix B4; with s = 1, C= 15, D = 365 for air ). Th currnt dnsity profils calculatd with th 2D fluid modl at th cathod for a discharg lngth of 1 mm ar prsntd in Fig. 1 for thr diffrnt valus of th total discharg currnt I = 7, 12 and 17 ma. On can s that a charactristic normal cathod spot is formd with a uniform currnt dnsity in th cntr and rathr stp dgs. Also can b sn that an incras of th discharg currnt rsults in a largr cathod spot, whil th currnt dnsity (and voltag drop) rmains th sam. Th xprimntal [10] and diffrnt thortical dpndncis of th rducd normal currnt dnsity J n /p 2 vrsus pd paramtr ar prsntd in Fig. 2. On can s that th rducd normal currnt dnsity dcrass for smallr discharg gaps in all thortical curvs. Th calculation is don for diffrnt gas prssurs and th rsults ar prsntd in this figur for 10, 20 and 46 Torr. In ordr to illustrat th influnc of gas hating som calculations wr don without th ffct of gas hating, for a constant gas tmpratur of 290 K (curvs 4 and 5). Th rsults obtaind using th 1D drift modl ar plottd for two diffrnt ionisation cofficint approximations. An analysis of ths rsults can b found in discussion sction. Th contour maps of th qui-potntial lins for a normal discharg in air at 20 Torr and for lctrod sparations of 0.5, 1, 2 and 4 mm calculatd with th 2D modl ar prsntd in Fig.3 (a-d). It can b sn that with th rduction of th intr-lctrod gap, whil th normal currnt dnsity dcrass (s also Fig.2, curv 2) th lngth of th cathod fall rgion incrass. In Fig.4 (a-d) th calculatd contour plots for constant ionization rats (sourc trm in th continuity quation A1: k i (ε)n N [cm -3 / sc]) ar shown for diffrnt discharg lngths, 8

9 corrsponding to potntial distribution at Fig.3 (a-d). It can b sn that for th smallst lctrod sparation of 0.5 mm (pd = 1 Torr cm) whn th normal currnt spot still xist th cathod fall lngth (Fig.3(a)) as wll as th high ionization rgion (Fig.4(a)) bcom of th sam ordr as discharg gap. It should b notd though that alrady for an intrlctrod distanc of 1 mm (pd = 2 Torr cm) both th cathod voltag fall lngth and lngth of th rgion whr th main ionization occurs ar significantly smallr than total discharg lngth, yt th currnt rduction is still apparnt for this lctrod sparation (s Fig.2, curv 2). Thrfor th xistnc of th normal discharg is limitd by a pd c valu which is in ordr of cathod fall dimnsion pd n. At th sam tim th dpndncy of th normal currnt dnsity on th lctrod sparation can b sn for a rang of pd valus which significantly xcds pd n. Discussion In [10] it was obsrvd that th normal currnt dnsity dpnds on th distanc btwn th lctrods in th pd rang of a fw Torr.cm: th J n dcrass with rduction of th discharg lngth. For incrasing discharg lngth th normal currnt dnsity attains a constant valu. It sms that this dpndncy on discharg lngth is of gnral natur bcaus it was obsrvd for diffrnt gass, prssurs and lctrod matrials. So far it was commonly assumd that th normal currnt dnsity dos not dpnd on th discharg lngth. This assumption was mainly basd on a classical, smi-analytical modl of th normal cathod layr proposd by von Engl and Stnbck [9] (som aspcts of this thory ar discussd in [1]). In thir modl th fild distribution insid th discharg was approximatd by a linar function. Th absolut valu of th lctric fild linarly rducs from E = E 0 at th cathod surfac (x = 0) to E = 0 at x = d s, whr d s is th shath thicknss. For x > d s th lctric fild is assumd to b zro. Th ionisation cofficint was dpndnt on th local valu of th lctric fild. From such approximation immdiatly follows that th shath proprtis ar indpndnt on th discharg lngth if th distanc btwn th lctrods is largr than th shath thicknss. In th drift modl dvlopd by Ward [11] no a priori givn lctric fild profil was implid. Nvrthlss th calculatd fild insid th cathod fall rgion was wll approximatd by a linar function which was in agrmnt with th xprimntal obsrvations and confirming th prvious assumption mad by von Engl and Stnbck [9]. At th sam tim th rsulting lctric fild was at no plac across th discharg gap 9

10 zro, bcaus this is a strict rquirmnt for th drift currnt continuity. As it was mntiond in th introduction Ward s modl for a long tim rmaind insufficintly analysd. Particularly this is valid for th dpndncy of th J n /p 2 paramtr on th discharg lngth. Our calculations show that th currnt dnsity, which corrsponds to th minimum on th voltag-currnt curv (attributd to th normal rgim), shifts toward lowr valus for shortr discharg lngths (s curvs (6) and (7) in Fig 5). This dpndncy coincids with th on found in our xprimnt. It should b notd hr that a on dimnsional modl can not dirctly dscrib th normal currnt spot. Howvr, vn such a simpl discharg simulation approach givs alrady a valuabl indication about th functionality btwn J n /p 2 valu and th intr-lctrod distanc. In comparison with th classical on dimnsional approach th 2-dimnsional fluid modl applid hr is accounting for th drift-diffusion transport of th chargd particls in a slf-consistnt 2D lctrical fild and also is charactrisd by th non-local ionization and transport cofficints which dpnd on th man lctron nrgy valu. A two-dimnsional analysis is ssntial for th dscription of th radial bhaviour of th cathod spot. According to [4] th shap of currnt spot is govrnd by th ion drift in th radial lctric fild for normal dischargs and th radial lctron diffusion for subnormal dischargs. Th last rgim is charactrizd by considrably lowr currnt dnsitis in comparison with th normal discharg. Th rduction in lctrod sparation for a normal discharg lads to a dcras in th radial lctric fild componnt at th currnt spot boundary (s Fig.3). Th shap of currnt spot for a shortr discharg lngth is to a gratr xtnt govrnd by lctron diffusion which is a fast procss. If th voltag currnt charactristic of such short discharg still has a minimum, thn a normal rgim may xist, but it will hav a lowr currnt dnsity. Two argumnts should b considrd whn analyzing th rsults of th prsnt numrical modlling concrning th normal currnt dnsity bhaviour. Firstly, alrady th simpl 1D drift modl givs a dpndncy of th J n /p 2 paramtr on th discharg lngth which is a consqunc of th slf-consistnt solution for th transvrs fild and drift currnt. Scondly, for diffrnt lctrod sparations th balanc conditions at th currnt spot boundary ar changing, rsulting, in turn, to a chang in th sustaind currnt dnsity. Th influnc of gas hating on th normal currnt dnsity can b sn by comparing curvs (2) and (4) in Fig 5, calculatd rspctivly with and without taking into account th ffct of gas hating. Gas hating lads to a dcras of th J/p 2 paramtr. Th rduction bcoms significant at highr currnts (and consquntly highr hating rats). 10

11 This ffct can b undrstood by th fact that th currnt dnsity J scals not as J/p 2 but, mor prcisly, is proportional to th squar valu of th gas concntration (N 2 ). With a constant gas prssur a highr gas tmpratur rsults in a lowr gas dnsity and thrfor lowr currnt dnsity. Th sam rason is valid for th dviation from th J/p 2 scaling for th curvs calculatd for diffrnt gas prssurs (s curvs (2) and (3) in Fig.2). Whn gas hating is not takn into account th J/p 2 scaling holds (s curvs (4) and (5) in Fig.2). It can b also sn from Fig.2 that th calculatd curvs (2) and (3) ar rproducing th xprimntal bhaviour (curv (1)) vry wll, although having a lowr absolut valu of th J/p 2 paramtr. This diffrnc btwn curvs (1) and (3) is approximatly of factor of 2. This can b attributd to th simplicity of th applid modl which did not tak into account attachmnt and rcombination procsss. It should b also notd that th calculations and xprimnts wr don for diffrnt discharg gomtris. It can b xpctd, according to Fig.2, that diffrnt rgims of hat rmoval can caus a considrabl variation in absolut valus of currnt dnsity. It can b argud that fast non-local lctrons cratd in th cathod fall ar producing a significant amount of ion-lctron pairs in th ngativ glow which can xtnd considrably byond th cathod fall rgion. In this cas th rduction in discharg lngth will diminish th lctron multiplication whil th lctrod sparation still gratly xcds th siz of cathod voltag fall. A dtaild study of th influnc of fast lctrons on th formation of th discharg with a crtain valu of th normal currnt dnsity would rquir a pur kintic or hybrid modl (s for instanc [ 21 ]). This will b a subjct for furthr invstigation. Howvr th non-local fluid modl applid in th prsnt work alrady givs a sufficintly good dscription of th xprimntal data. It is important to not that th fact, that th normal currnt dnsity dpnds on th distanc btwn th lctrods, imposs crtain rquirmnts on th xprimntal mthodologis and st-ups for th masurmnt of handbook valus of th J n /p 2 paramtr. Thrfor th rsults of prvious masurmnts nd to b analysd from this point of viw. Summary Th normal glow discharg was invstigatd numrically in a narrow gap gomtry for a pd rang of 1-10 Torr.cm at modrat gas prssurs. A considrabl rduction in currnt dnsity for small discharg lngths was rcntly xprimntally obsrvd in [10] for 11

12 diffrnt gass and lctrod matrials. It was found that such fundamntal discharg similarity paramtr as rducd normal currnt dnsity J n /p 2 bcoms a function of th lctrod sparation at low pd valus. Whil it was traditionally assumd that J n /p 2 is dfind only by gas and lctrod matrial. In prsnt work this nw ffct was analyzd with a 3-momnts 2D fluid modl. Th valu of J n /p 2 was found to b strongly dpndnt on discharg lngth in th cas whn gas hating is nglctd. An influnc of th gas hating on th J n /p 2 valu and scaling proprtis wr discussd. A good qualitativ agrmnt was shown btwn modlling and xprimntal rsults. It was also shown that alrady a minimal 1D drift discharg modl can giv an indication of th J n /p 2 functionality on th discharg lngth. W bliv that mor insight on th ffct can b gaind by analysing th influnc of th non-local ionization with a hybrid (kintic-fluid) modlling which can b a subjct for furthr invstigation. 12

13 APPENDIX A Equation st for th two-dimnsional fluid modl of a DC discharg 1 Continuity quations for positiv ions and lctrons: n p, t F p F + div( F ) p, = k ( ε) n N i (A1) = n µ E D grad( n ) (A2) p p p p = n µ ( ε ) E D ( ε ) grad ( n ) (A3) In ths quations is n p, th dnsity of positiv ions and lctrons rspctivly, k i (ε) th ionization cofficint, F p, th chargd particl flux, N th nutral gas dnsity, µ p, is th mobility, D p, th diffusion cofficints and E th lctric fild. 2 Equation for th man lctron nrgy ε: n ε + t 5 3 div { n εµ ( ε) E D ( ε) grad( n ε) } = F E n Nk ( ε) whr k l (ε) is th lctron nrgy loss cofficint. l (A4) 3 Poisson quation for th lctric potntial φ: ϕ = 4π( n p n ) (A5) 4 Gas thrmal conductivity quation: whr T gas { ( )} + w T gas = div λ T gas )grad(t t gas (A6) is gas tmpratur, λ th cofficint for thrmal conductivity and w is th dnsity of hat sourcs. For th thrmal conductivity cofficint a linar approximation was usd: λ ( T ) = AT B (A7) gas gas + It was assumd that thr is no convctiv gas flow in th discharg volum and th local gas dnsity was dtrmind according th idal gas quation: P = Nk b T gas (A8) Th voltag ovr th discharg was calculatd according to: 13

14 U = U mf - R b I d (A9) Whr U mf is th voltag of th gnrator, R b th xtrnal ballast rsistanc, and I d is discharg currnt. Boundary conditions: At th cathod z = 0: F F p = γ ; = 0 z F p At th anod z = d: ( ) = ; p = 0 ; F 1 4 n ν, th And at th latral walls r = R: F = ( ) p ; = 0 r F 1 4 n ν, th ; ε 0 ε = ; ϕ = 0 (A10) { } n Q 4 n ν ( 2k T ) = 1, th b ; = U ; Q 4 n ν ( 2k T ) ϕ (A11) ϕ = 1 {, th b }; = 4πσ (A12) r A symmtry condition was applid at th discharg axis at r = 0. Hr γ is th scondary ion-lctron mission cofficint, ε 0 nrgy of th scondary lctrons (assumd to b 1 V), v,th th thrmal vlocity of th lctrons, T th lctron tmpratur, Q th lctron nrgy flux and σ th surfac charg dnsity. For th thrmal conductivity quation th tmpratur was assumd to b known (Dirichlt conditions) and qual to 290 K. A dtaild dscription of th solution procdur for th continuity quations coupld with th Poisson quation in two dimnsions can b found lswhr (s for instanc [14, 17] and rfrncs thrin). 14

15 APPENDIX B 1D drift modl This st of quations is analogous to th on usd by Ward [11] and can b drivd from th continuity condition for th drift currnt and th lctrostatic quation for th lctric fild. This rsults in a systm of two first ordr diffrntial quations: dj dz de dz = α( E) J (B1) 4π = { J J (1 + µ p / µ )} (B2) µ p E With boundary conditions for th currnt: γ J J (0) = ; J ( d) = J (B3) 1 + γ Hr J and E ar th lctron currnt dnsity and lctric fild rspctivly; J is th total currnt dnsity; µ p, µ ar th mobilitis of ions and lctrons; γ is th scondary lctron mission cofficint and α is th Townsnd s ionization cofficint; z = 0 corrsponds to th cathod and z = d to th anod location. For α usually an xponntial (Townsnd) approximation is usd: s p α / p = C xp D (B4) E whr th indx s is typically qual to 1 or to 1/2. Following th original work [11] th solution of th systm abov is found by intgrating (B1 and B2) ovr th discharg gap whil varying th assumd lctric fild valu at cathod (or anod) till th boundary conditions (B3) ar mt. Th valu of th total currnt dnsity J is supposd to b known. Standard library routins can b utilizd for intgration, for instanc, in th rcnt articl [3] a rfrnc is givn to th ODE-PACK FORTRAN library from th frwar sit: ntlib.org. It should b notd that th solution of (B1-3) utilizing th dscribd procdur is rstrictd to low valus of currnt dnsitis. In gnral th validity of th local modl for highr currnts in th abnormal branch of th voltag currnt charactristic is qustionabl. 15

16 FIGURES: 0,025 currnt dnsity J [A/cm 2 ] 0,020 0,015 0,010 0,005 0,000 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 Spot radius [cm] Fig. 1. Calculatd profils of th currnt dnsity vs th spot radius at th cathod for thr diffrnt valus of th total discharg currnt: I = 7, 12 and 17 ma. (Normal discharg in air at a prssur of 20 Torr and with lctrod sparation of 1 mm) 16

17 3,0x10-4 J/p 2 [A/cm Torr 2 ] 2,0x10-4 1,0x , pd [Torr cm] Fig. 2. Dpndncy of th rducd normal currnt dnsity J n /p 2 on th pd paramtr for discharg in air: xprimntal data for 46 Torr (1)[10]; modlling 20 Torr (2); modlling 46 Torr (3); modlling 20 Torr, no gas hating (4), modlling 10 Torr, no gas hating (5), rsults of 1D drift modl with ionization cofficint from Boltzmann cod (6), rsults of 1D drift modl, ionization cofficint approximatd according [1] (7). 17

18 0,05 0,10 Axial position z [cm] 0,04 0,03 0,02 0,01 a 0,08 0,06 0,04 0,02 b Axial position z [cm] 0,00 0,00 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,20 0,18 0,16 0,14 0,12 0,10 0,08 0,06 0,04 0,02 c 0,00 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,40 0,35 0,30 0,25 0,20 0,15 0,10 0,05 d 0,00 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 Radial position r [cm] Radial position r [cm] Fig. 3. Calculatd quipotntial contour maps for a normal discharg in air at a prssur 20 Torr. Graphs a, b, c, and d ar rspctivly for lctrod sparations of 0.5, 1, 2 and 4 mm. z = 0 corrsponds to th cathod plan. 18

19 0,05 0,10 Axial position z [cm] 0,04 0,03 0,02 0,01 a 0,08 0,06 0,04 0,02 b 0,00 0,20 0,00 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,40 0,1 0,2 0,3 0,4 0,5 0,6 0,7 Axial position z [cm] 0,18 0,16 0,14 0,12 0,10 0,08 0,06 0,04 0,02 c 0,35 0,30 0,25 0,20 0,15 0,10 0,05 d 0,00 0,00 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,1 0,2 0,3 0,4 0,5 0,6 0,7 Radial position r [cm] Radial position r [cm] Fig. 4. Contours of constant ionization rats (k i (ε)n N [cm -3 / sc]) corrsponding to th potntial distributions in Fig.3. Graphs a, b, c, and d ar rspctivly for lctrod sparations of 0.5, 1, 2 and 4 mm. (Normal discharg in air at a prssur of 20 Torr) 19

20 Rfrncs: 1 Yu.P.Raizr Gas Discharg Physics Springr (1997) 2 B.N. Klyarfld, L.G.Gusva, A.S.Pokrovskaya-Sobolva, Sov. Phys. Tch. Phys., 11, 520 (1966) 3 D.D.Šijačić, U.Ebrt, Phys. Rv. E, 66, (2002) 4 V.I.Kolobov, A.Fiala, Phys.Rv. E, 50, (1994) 5 R.Arslanbkov and V.Kolobov, J.Phys.D Appl. Phys., 36, 2986 (2003) 6 Z.Lj.Ptrovic and A.V.Phlps, Phys. Rv.E, 47, 2806 (1993) 7 B.G.Salamov, S.Ellialtioglu, B.G.Akinoglu t al., J. Phys. D: Appl. Phys., 29, 628 (1996) 8 M.A.Fdotov, I.D. Kaganovich, L.D. Tsndin, Tch. Phys., 39, 241 (1994) 9 A.von Engl, M.Stnbck., Elctrisch Gasntladungn: Ihr Physik und Tchnik, Vol II (Springr, Brlin 1934) 10 A.V.Azarov, S.V.Mitko, V.N.Ochkin, Bulltin of th Lbdv Physics Institut, 4, 14-23, (2002) 11 A.L.Ward, Phys. Rv., 112, 1852 (1958) 12 A.L.Ward, J.Appl. Phys., 33, 2789 (1962) 13 Yu.P.Raizr, S.T.Surzhikov, Tplofiz. Vysokikh Tmpratur, 26, 428 (1988) 14 J.P.Bouf and L.C.Pitchford, Phys. Rv. E, 51, (1995) 15 D.L.Sharfttr and H.K.Gumml, IEEE Trans. Elctron Dvics, ED-16, 64 (1969) 16 H.L.Ston, SIAM J. on Num. Anal., 21, 530 (1968) 17 G.J.M.Haglaar, PhD thsis, Tchnical Univrsity, Eindhovn, (2000) 18 Kinma softwar & CPAT 19. A.V.Phlps, Z.Lj.Ptrović, Plasma Sourcs Sci., Tchnology, 8, R21 (1999) 20 A.A.Kudriavtsv, L.D.Tsndin, Tch. Phys. Ltt., 28, 1 (2002) 21 A.Bogarts, Plasma Sourcs Sci. Tchnology, 8, 210 (1999) 20