Problem of Radio-Frequency Discharge at. Atmospheric Pressure in Local Statement

Transcription

1 Applid Mathatical Scics, Vol. 10, 016, o. 1, HIKARI Ltd, Probl of Radio-Frqucy Discharg at Atosphric Prssur i Local Statt V. Ju. Chbakova Kaza Fdral Uivrsity 18 Krlyovskaya Strt 40008, Kaza, Russia Fdratio V. S. Zhltukhi Kaza Natioal Rsarch Tchological Uivrsity 68 Karl Marx Strt 40015, Kaza, Russia Fdratio V. T. Dubrovi Kaza Fdral Uivrsity 18 Krlyovskaya Strt 40008, Kaza, Russia Fdratio Copyright 015 V. Ju. Chbakova, V. S. Zhltukhi ad V. T. Dubrovi. This articl is distributd udr th Crativ Coos Attributio Lics, which prits urstrictd us, distributio, ad rproductio i ay diu, providd th origial work is proprly citd. Abstract Noliar probl of fidig of paratrs of atosphric prssur capacitiv coupld radio-frqucy discharg (CCRFD) btw sytric lctrods is cosidrd i local approach. Elctros, uclar ad olcular ios, tastabl atos ad dir argo, ad also atos i th ai stat ar cosidrd i th odl. Rsults of urical calculatios for th odl probl ar giv. Kywords: Mathatical siulatio, low-tpratur plasa, radio-frqucy capacitiv discharg, diffrc sch, itrativ thod, local statt

2 1014 V. Ju. Chbakova t al. Itroductio Plasa is ffctivly usd i th fild of aotchology, for cratig ad odify i aostructurs (.g., for th tratt of atrials i ordr to iprov durability ad rliability of achiry, cratig a light ad strog polyr coposits, polythyl plastics, aopowdrs of tals ad thir coposits tc.) [1,,,, 8, 9]. This papr is furthr work [7] whr th odl of itractio btw CCRFD ad a sapl allowig to calculat charactristics of th discharg i a itr-lctrod is giv. Th odl [7] icluds th covctio diffusios quatios for lctroic ad ioic gas, Poisso quatio for lctric fild pottial, th quatio of balac of tastabl atos ad utral atos, ad also statioary quatio of hat coductivity of uclar ad io gas. Dpdc of charactristics of th discharg o th boudary coditios takig ito cosidratio proprtis of a sapl lctrod is show. Th coclusio is draw o d of iclusio i th kitic sch of olcular ios ad dir of argo. A o-disioal slf-cosistt odl of a atosphric prssur CCRFD i argo cosidrig is prstd i this papr. To urical ralizatio of this probl th approxiat thod is offrd. Th rsults of calculatios for th odl probl ar giv illustratig th procss of discharg Not that i th paprs [-14] wr also cosidrd th statts of so oliar probls of athatical physics ad approxiat thods of solvig th. 1 Probl statt O-disioal slf-cosistt odl of atosphric prssur CCRFD at 1.56 MHz i argo is prstd. Th odl of CCRFD btw two paralll plats, o of which is groudd, ad aothr is coctd to RF grator is showd. Elctric fild i such dischargs is clos to pottial ad th discharg is uifor alog lctrods. That allows costructig o-disioal odl. Estiats of lctro rgy dissipatio ti ad lctro rgy rlaxatio lgth at atosphric prssur show that local approach is possibl to us i ths coditios [1]. Th followig faturs of th CCRFD as xistc of aras whr th coditio of quasi utrality is't carrid out, ad chagig th closd tsio, ad also procsss with participatio of tastabl atos, olcular ios ad dir ar cosidrd i th odl. Th procsss i CCRFD dscrib by as of th followig iitial ad boudary-valu probls ad Cauchy's probls. 1. Covctio diffusios quatio for lctroic gas: t D x R 5 E R1 N R x R 10 R 1 R 16 R * R 17 R * 4

3 Probl of radio-frqucy discharg 1015 with th boudary coditios at x = 0, x = b, if thfild is dirctd to a lctrod, / x 0, if thfild is dirctd fro a lctrod. Hr,, ar th dsity of lctros, atoic ad olcular positiv chargd ios, rspctivly, * is th dsity of argo dirs, γ is th cofficit of scodary lctroic issio, D / x E, D / x E ar lctroic ad ioic fluxs,, ar lctro ad atoic io obility, D, D ar lctro ad io diffusio cofficits, E is lctric fild ts, is th dsity of tastabl argo atos, 4 N is th dsity of atos i groud stat. Th iitial valu is ( x,0) 10 c, that corrspods to particl dsity at oral coditios. It is assud that approxiatio of a idal gas is satisfy ad P knta, whr k is th Boltza costat, T a is th atoic tpratur. Hriaftr Ri, i=1 19, ar rat cofficits of th followig plasa chical ractios i th followig ordr: Ar+ Ar + +, Ar * + Ar * Ar + Ar + +, Ar * + Ar + +, Ar + + Ar+hv, Ar + + Ar+, Ar + Ar * +, Ar * Ar+hv, Ar * + Ar Ar, Ar * + Ar+, Ar * + Ar * Ar + +, Ar+ Ar + Ar + +Ar, Ar + + Ar + Ar + +, Ar + + Ar + Ar, Ar * Ar+hv, Ar * + Ar Ar * +Ar, Ar * + Ar * +Ar+ Ar +, Ar * + Ar * +Ar+ Ar +, + Ar * Ar * +, Ar * + Ar Ar * +Ar Ar+hv, Ar + + Ar Ar + Ar +.. Covctio diffusio quatio for th ioic gas t D x x R 5 E R1 N R R 1 R 11 N R R 4 with th boudary coditios at x = 0, x = b / x 0, if thfild is dirctd to a lctrod, 0, if thfild is dirctd fro a lctrod, ad with iitial coditio ( x,0) 510 см. It is assud that atoic ad olcular dsitis ar qual to ach othr.

4 1016 V. Ju. Chbakova t al.. Kitic quatios for olcular ios Ar + t D x x R1 R16 R17* with boudary coditios at x=0, x=b / x 0, if thfild is dirctd to a lctrod, 0, if thfild is dirctd fro a lctrod, ad with iitial coditio ( x,0) 5 10 c. E R * 10 R 11 N R 1 q 4. Poisso's quatio for lctric fild pottial ( ) x 0 with boudary coditios V a si( t), o th load lctrod ( x l), 0, o th groudd lctrod ( x 0). Hr q is th lctro charg, 0 is th lctrical costat, ω is circular frqucy, Va is a voltag agitud. 5. Equatio of tastabl atos balac: D R6 N R18 t x x * R R R R9 R10 R15N R17* R19N Hr D is a diffusio cofficit of tastabl ato of argo. Th boudary coditio =0 is spcifid at x=0, x=b. 6. Kitic quatios for argo dir Ar * * R19N R15N R14* R t with iitial coditio *( x,0) 0 7 R N 16* R17 N * R18* R19 7. Equatio of hat coductivity of atoic-ioic gas Th quatio of atoic tpratur balac is avragd o th priod of Ta fluctuatio of lctric fild a QупN, whr a is a hat x x coductivity cofficit of atoic-ioic gas, Q уп is rgy gttig by atos at lastic collisios. O lctrods th coditio of hat xchag of plasa with th watr coolig a lctrod is spcifid at a / x k full ( T a Tв ). Hr k full is a of sapl full hat issio cofficit, T в is coolig watr tpratur. 8

5 Probl of radio-frqucy discharg 1017 Approxiatio of trasport cofficits Diffusio cofficit D, lctro obility, rats of dirct ioizatio R 1 ad tastabl xcitatio R 6, cotributio of lastic collisio ito gas hatig Qс ad avrag rgy kt / wr approxiatd by BOLSIG+ softwar [19] takig ito accout dpdc fro lctrical ts ad lctro-lctro collisios. Hr T is th lctro tpratur. Th diffusio cofficit of olcular ios D+ is st by Eisti ratio D. Othr cofficits wr st i accordac with [15-18, 0, 6]. Nurical thod Th oliar syst of boudary ad iitial-boudary probls is solvd by fiit diffrc approxiat ad itratio procss. Dsitis of particls (such as olcular ad atoic ios, lctros, dirs, tastabl atos), lctric fild pottial, ad gas tpratur ar dfid i grid poits whil lctric fild tsio ad fluxs ar dfid i half grid poits. A iplicit fiit-diffrc approxiatio is costructd by itgral-itrpolatio thod [4] usig th thod of dirctioal diffrcs [5]. Th liarizatio of th syst is carrid out by th thod of Sidl typ, ad th urical algorith is basd o th lowrig th oliarity of icoig cofficits at th lowr layr. Noliar quadratic trs i th right sid wr liarizd by usig Nwto sch [4]. Th solutio of th quatio was carrid out oc durig th priod of oscillatio of th lctric fild by th itrativ thod of Jacobi typ. Th fiit diffrc schs for th covctio diffusio quatio of chargd particls (lctros ad ios), for th Poisso quatio ad for th balac quatio of tastabl atos ad gas tpratur ar giv i [7]. Chags ar ad i th itrprtatio of th boudary coditio for th balac of particls quatio i th cas wh th chargd particls ov toward of th lctrod: z / x 0, whr z=,+,+. Thus th boudary coditio at x=h, b-h ar rgardd as t t t R R R N R 1 R 1 R R 16 R * 4 R 17 R 5 * R 1 N R R R4 R5 R1 R11 N 10 R 11 N R 1 R 1 R 16 * 10 R whr h is th spatial stp. For solvig th abov probls ad th probl for th argo dir w us th Rug-Kutta thod. 17 *

6 1018 V. Ju. Chbakova t al. 4 Nurical rsults Th rsults of solvig of th odl probl showd chags i gas copositio i th procss of dvlopt of th discharg. If at first ar doiatd th olcular ios at low gas tpratur ad th coctratio of xcitd atos ad dirs ar approxiatly qual th durig hatig th gas th dcras of coctratios of olcular ios ad dirs with icrasig of coctratio of atoic ios ad dcrasig of coctratio of dirs taks plac (s Figur 1). Th coctratio of olcular ios i th lctrod layrs is gratr whr th gas tpratur is lss (s Figur ). At th sa ti i th discharg a doai of cosrvatio of quasi-utrality xists. Figur 1. Th spatial distributio of th avrag for th priod of coctratios of chargd particls Figur. Th spatial distributio of th atoic tpratur, calculatd by th avrag paratrs

7 Probl of radio-frqucy discharg Coclusios Th papr prsts a odl of radio-frqucy capacitiv discharg, icludig th covctio-diffusio quatio for th lctro ad io gass, Poisso quatio for th pottial of th lctric fild, th balac quatio of tastabl atos ad utral atos. I additio, th odl icluds a statioary hat quatio of atoic io gas, thе covctio-diffusio quatio for th olcular io ad kitic quatio for argo dir. W dscrib a odificatio of th calculatio thod proposd i th papr [1]. Th rsults of calculatios for th odl probl ar giv illustratig th procss of discharg. Ackowldgts. Th work was supportd by th Russia Foudatio for Basic Rsarch (projcts os , ) ad th basic part of stat task projct No. 196 of gratd by Miistry of Educatio ad Scic of th Russia Fdratio. Rfrcs [1] I. S. Abdulli, V. S. Zhltukhi, I. A. Borodav, E. V. Strbkov, A. A. Khubatkhuzi, Th ifluc of RF plasa tratt at low prssur o th prability of a polyurtha aocoposit, Moscow Uivrsity Physics Bullti, 69 (014), o., [] I. Sh. Abdulli, V. S. Zhltukhi, M. F. Shakhov, Mathatical odl of itractio of low-prssur RF discharg plasa with solid surfacs, Bullti of th Russia Acady of Scics: Physics, 70 (006), o. 8, [] I. B. Badriv, O th solvig of variatioal iqualitis of statioary probls of two-phas flow i porous dia, Applid Mchaics ad Matrials, 9 (01), [4] I. B. Badriv, M. M. Karchvskii, Covrgc of th itrativ Uzawa thod for th solutio of th statioary probl of spag thory with a liit gradit, Joural of Sovit Mathatics, 45 (1989), o. 4, [5] I. B. Badriv, V. V. Badrov, Itrativ thods for solvig variatioal iqualitis of th thory of soft shlls, Lobachvskii Joural of Mathatics, 5 (014), o. 4, [6] I. B. Badriv, V. V. Badrov, Nurical thod for solvig variatio probls i athatical physics, Applid Mchaics ad Matrials, (014),

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