DYNAMIC ION BEHAVIOR IN PLASMA SOURCE ION IMPLANTATION BİLGE BOZKURT

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1 B. BOZKURT METU 6 DYNAMIC ION BEHAVIOR IN PLASMA SOURCE ION IMPLANTATION BİLGE BOZKURT JANUARY 6

2 DYNAMIC ION BEHAVIOR IN PLASMA SOURCE ION IMPLANTATION A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY BY BİLGE BOZKURT IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN PHYSICS JANUARY 6

3 Appoval of he Gaduae School of Naual and Applied Sciences Pof. D. Canan Özgen Dieco I ceify ha his hesis saisfies all he equieens as a hesis fo he degee of Mase of Science. Pof. D. Sinan Biliken Head of Depaen This is o ceify ha we have ead his hesis and ha in ou opinion i is fully adequae, in scope and qualiy, as a hesis fo he degee of Mase of Science. Pof. D. Sinan Biliken Supeviso Exaining Coiee Mebes Pof. D. Aif Dei (Kocaeli Unv, PHYS Pof. D. Sinan Biliken (METU, PHYS Assis. Pof. D. İsail Rafaov (METU, PHYS Assoc. Pof. D. Akif Esendei (METU, PHYS Assoc. Pof. D. Seha Çakõ (METU, PHYS

4 I heeby declae ha all infoaion in his docuen has been obained and pesened in accodance wih acadeic ules and ehical conduc. I also declae ha, as equied by hese ules and conduc, I have fully cied and efeenced all aeial and esuls ha ae no oiginal o his wok. Nae, Las nae: Bilge Bozku Signaue : iii

5 ABSTRACT DYNAMIC ION BEHAVIOR IN PLASMA SOURCE ION IMPLANTATION Bozku, Bilge M. S., Depaen of Physics Supeviso: Pof. D. Sinan Biliken Januay 6, 7 pages The ai of his wok is o analyically ea he dynaic ion behavio duing he evoluion of he ion aix sheah, consideing he indusial applicaion plasa souce ion iplanaion fo boh plana and cylindical ages, and hen o develop a code ha siulaes his dynaic ion behavio nueically. If he sepaaion beween he elecodes in a dischage ube is sall, upon he applicaion of a lage poenial beween he elecodes, an ion aix sheah is foed, which fills he whole ine-elecode space. Afe a sho ie, he ion aix sheah sas oving owads he cahode and disappeas hee. Two egions ae foed as he aix sheah evolves. The poenial pofiles of hese wo egions ae deived and he ion flux on he cahode is esiaed. Then, by using he finie-diffeences ehod, he poble is siulaed nueically. I has been seen ha he esuls of boh analyical calculaions and nueical siulaions ae in a good ageeen. Keywods: Ion iplanaion, Plasa Souce Ion Iplanaion, Plasa Iesion Ion Iplanaion, Dynaic Ion Behavio, Modeling iv

6 ÖZ PLAZMA KAYNAKLI İYON EKİMİNDE DİNAMİK İYON HAREKETİ Bozku, Bilge Yüksek Lisans, Fizik Bölüü Tez Yöneicisi: Pof. D. Sinan Biliken Ocak 6, 7 sayfa Bu çalõşanõn aacõ, plaza kaynaklõ iyon ekii ile düzlesel ve silindiik hedefle için endüsiyel uygulaa düşünüleek iyon-ais plakalaõnõn evii esnasõndaki dinaik iyon haekeini analiik olaak çalõşak ve sona bu dinaik iyon haekeini sayõsal olaak siüle eden bi kod gelişieki. Eğe bi deşa üpünde elekola aasõ esafe az ise, elekola aasõna yüksek bi poansiyel uygulandõğõnda, ü elekola aasõ esafeyi dolduan bi iyonais plakasõ oluşu. Kõsa bi süe sona iyon-ais plakasõ kaoda doğu haeke eeye başla ve oada kaybolu. Mais plakasõnõn evii esnasõnda iki bölge oluşu. Bu iki bölgenin poansiyel pofili üeiliş ve kaoaki iyon akõsõ hesap edilişi. Sona, sonlu fakla eodu kullanõlaak poble sayõsal olaak siüle edilişi. Analiik hesaplaalaõn sonuçlaõ ile sayõsal siülasyonlaõn uyulu olduğu gözlelenişi. Anaha Keliele: İyon Ekii, Plaza Kaynaklõ İyon Ekii, Dinaik İyon Haekei, Sayõsal Çözülee, Modellee v

7 FOR PROF. DR. ORDAL DEMOKAN vi

8 ACKNOWLEDGEMENTS I would like o expess y deepes gaiude o dea Pof. D. Odal Deokan fo suggesing his eseach and his guidance, advice, suppo and insigh in his sudy unil his soowful deise. I would like o expess y gaiude o y supeviso Pof. D. Sinan Biliken fo his guidance, suggesions, coens, conibuion and suppo. I would also like o expess y gaeful hanks o Assis. Pof. D. İsail Rafaov and Pof. D. Ayşe Kaasu fo hei guidance, conibuions and suggesions. I would like o hank y faily and y fiancé fo hei oal suppo and encouageen. I would also like o hank İnanç Kanõk and Yasein Filiz fo hei invaluable discussions, suggesions, coens and suppo. vii

9 TABLE OF CONTENTS PLAGIARISM...iii ABSTRACT...iv ÖZ...v DEDICATION...vi ACKNOWLEDGEMENTS...vii TABLE OF CONTENTS...viii LIST OF FIGURES...xii CHAPTER. INTRODUCTION.... Plasa Pocessing.. Plasa Souce Ion Iplanaion...3 Basic Plasa Sheah Physics.. viii

10 .3. Phase..3. Phase Phase The Obecives and Conen of he Thesis 8. THEORY AND CALCULATIONS.. 9. Plana Configuaion. 9.. The Model The Deeinaion of Poenials..3 The Deeinaion of Ion Velociy and Flux 7.. Gaphical Repesenaion.. 9. Cylindical Configuaion..... The Model The Deeinaion of Poenials The Deeinaion of Ion Velociy and Flux 35 ix

11 3. SIMULATION The Basics of Siulaion Nueical Analysis The Finie-Diffeences Mehod Plana Configuaion The Model Equaions Bounday and Iniial Condiions Nueical Pocedue Poisson s Equaion Coninuiy Equaion Moenu Balance Equaion Nueical Resuls Cylindical Configuaion The Model 6 x

12 3.3.. Equaions Bounday and Iniial Condiions Nueical Pocedue Poisson s Equaion Coninuiy Equaion Moenu Balance Equaion Nueical Resuls. 6. CONCLUSIONS.. 67 REFERENCES 7 xi

13 LIST OF FIGURES FIGURES Figue. Block diaga of a ypical PSII syse Figue. When a lage negaive sep poenial o a ain of high volage negaive pulses,, is applied o he cahode, elecons ae quickly epelled fo he cahode on he ie scale of elecon plasa fequency 5 Figue.3 An ion-aix sheah is lef behind 5 Figue. Afe a ie scale of he ion plasa fequency, ions sa esponding and he ion aix sheah sas o expand.. 6 Figue.5 The seady sae poenial pofile 7 Figue. When he applied poenial,, is lage (~ kv and he elecode sepaaion is sall (~ -, he aix can fill he enie ineelecode space ( / ω pe < < / ω pi, whee ω pe and ω pi ae elecon and ion plasa fequencies, especively 9 Figue. Afe a sho ie, he aix sas oving owads he cahode. The noaion s( epesens he ie-dependen locaion of he sheah edge... Figue.3 The aix disappeas as all ions ae iplaned. Figue. Ion velociy as a funcion of x a ie = ax xii

14 Figue.5 Ion velociy as a funcion of x a ie =. ax Figue.6 Ion velociy as a funcion of x a ie = 3 ax Figue.7 Ion velociy as a funcion of x a ie = ax.. Figue.8 Ion flux as a funcion of a x = L Figue.9 When he applied poenial, elecode sepaaion is sall (~ -, is lage (~ kv and he, he aix can fill he enie ineelecode space ( / ω pe < < / ω pi, whee ω pe and ω pi ae elecon and ion plasa fequencies, especively 3 Figue. Afe a sho ie, he aix sas oving owads he cahode. s( is he ie-dependen locaion of he sheah edge 3 Figue. The aix disappeas as all ions ae iplaned Figue. The gaph of i π Ef ( i ln x whee a Figue.3 The gaph of i π Ef ( i / x = /. 37 = ln( wih aken as = 38 Figue. A copaison of he gaphs of equaion (..6 and (..67 fo = ax = and 39 Figue.5 The gaph of ( Figue.6 The egions whee he oos of coespond. 3 xiii

15 Figue.7 The ion velociy as a funcion of a he insans = ax, = ax, 3 = ax and = ax obained as a esul of nueic calculaions Figue.8: The ion flux as a funcion of a he age suface obained as a esul of nueic calculaions 5 Figue 3. Copuaional doain 5 Figue 3. Ion densiy and ion velociy as a funcion of = ax, = ax, x a he insans 3 = ax and = ax 57 Figue 3.3 Ion velociy vesus posiion gaphs ploed using analyical esuls 59 Figue 3. The analyically calculaed ion velociy as a funcion of posiion and ie 59 Figue 3.5 Copuaional doain 6 Figue 3.6 Ion densiy and ion velociy as a funcion of a he insans = ax, = ax, 3 = ax and = ax 65 Figue 3.7 Ion velociy as a funcion of a fou paicula insans = ax, = ax, 3 = ax and = ax 66 xiv

16 CHAPTER INTRODUCTION.. Plasa Pocessing Plasa pocessing is a plasa-based aeial pocessing echnology, he ai of which is o odify he popeies of he sufaces of aeials such as eals, plasics and ceaics by vaious pocesses like deposiion, iplanaion, and eching ec [] [3]. The pincipal coponen of plasa pocessing is he plasa ediu, in which he aeial o be pocessed is iesed. To ainain he desied plasa ediu, geneally DC and adio fequency (RF dischages ae used. RF dischages ae used fo a boad ange of pocesses such as deposiion, eching and eaen while DC dischages ae pefeed in ion iplanaion pocesses as DC dischages involve lage poenials []. The applicaions of plasa pocessing include: he anufacuing of inegaed cicuis; he deconainaion of wase aeials and sufaces; plasa souce ion iplanaion and he eching of silicon wafes fo seiconduco fabicaion [, ]... Plasa Souce Ion Iplanaion Plasa souce ion iplanaion (PSII, which was invened by Conad in 987 [3], has becoe well-known as one of he well-esablished plasa-based anufacuing echnique, fequenly used in he suface odificaion of aeials fo indusial applicaions [5] [8].

17 PSII is a oo epeaue suface enhanceen echnique ha uses a plasa ediu suounding a age and high-negaive-volage, high-cuen pulses o acceleae ions ino a age suface fo all diecions. PSII odifies he age suface in beneficial ways, aking i hade, ipoving wea popeies, educing he coefficien of ficion, enhancing is esisance o coosion and daaically ipoving he wea-life of anufacuing ools in acual indusial applicaions. The block diaga of a ypical PSII syse is given in figue. [9] []. Figue.: Block diaga of a ypical PSII syse. PSII depas adically fo convenional ion iplanaion echnology in he following aspecs [7,, ]:

18 . In PSII, ages ae placed diecly ino a plasa ediu and ions ae acceleaed ono he age hough a plasa sheah ha suounds he age. Hence, PSII is a non-line-of-sigh echnique which eans a pah fo a single ion souce o he suface of he age is no equied. This enables PSII o ea he enie age suface siulaneously wihou he need of he ie-consuing oaion of he age and o ea uliple age sufaces siulaneously wihou he need fo in-vacuu anipulaion of he age assebly.. In PSII, he ions ae acceleaed noal o he suface of he age so hee is no eained dose poble. Even wih age oaion, he convenional ion iplanaion echnique equies age asking in ode o iniize he gazing incidence, which poduces excessive spueing, and heefoe liis he eained dose. 3. PSII is no liied by ion opics chaaceisics o by he Child-Langui space chage liied flow popeies of convenional ion iplanaion echniques. Due o his fac, he aveage ion flux o he age suface can be oe han an ode of agniude lage han using convenional echniques a low enegies, which significanly educes he equied eaen ie fo lage, coplex age asseblies and enables PSII o iplan ions efficienly o concenaions and dephs equied fo suface odificaion.. PSII is a siple and low-cos echnique because a he ion acceleao sage, he age oaion appaaus is eliinaed in PSII. 5. Since he age oaion sage does no exis in PSII, lage and heavy ages can also be eaed by PSII. The age oaion in convenional ion iplanaion echnique adds coplexiy o he pocess and educes he size of he age. 6. PSII is a oo epeaue pocess. The oaion poble in he convenional ion iplanaion echnique is aggavaed by he need fo adequae hea sinks a he ages o lii he epeaue ise duing iplanaion. 7. I has high dose aes, ininsic chage neualizaion capabiliy, and adapabiliy o ohe suface odificaion pocesses. 3

19 .3. Basic Plasa Sheah Physics In he PSII echnique, he aeial, whose suface is o be odified, acs as he cahode in a gounded vacuu chabe, in which a flow of he gas equied fo iplanaion is ainained. The gas inside he vacuu chabe is ionized by one of he vaious dischage ehods [5, 6, 7, ]..3.. Phase When a lage negaive sep poenial o a ain of high volage negaive pulses, a ie, ha has a agniude beween kv and kv is applied o he cahode =, he elecons ae quickly epelled fo he cahode on he ie scale of elecon plasa fequency ( as shown in figue. [5] [7]. On /ω pe his ie scale, as elecons ecede, he ions oion is negligible so ha he epelled elecons leave behind an elecon-fee egion, naely he ion aix sheah []. The ion aix sheah consiss of unifoly disibued, saionay ions, confoally apping he cahode suface, which is shown in figue.3 [5] [7]. An ion aix sheah is a egion beween a quasi-neual plasa and a negaive elecode o which a lage negaive sep poenial,, is applied [].

20 Figue.: When a lage negaive sep poenial o a ain of high volage negaive pulses,, is applied o he cahode, elecons ae quickly epelled fo he cahode on he ie scale of elecon plasa fequency. Figue.3: An ion-aix sheah is lef behind. 5

21 .3.. Phase Afe a ie scale of he ion plasa fequency ( / ω pi, he ions sa esponding o he sheah elecic field and sa o acceleae owads he cahode [5] [7], which is usually he age in PSII. The deceasing ion densiy in he sheah egion causes a decease in he elecon densiy [] and consequenly he sheah sas expanding a appoxiaely he ion acousic velociy owads he Child Langui lii, uncoveing ions which ae hen acceleaed owads he cahode by he field in he sheah [5] [7], which is shown in figue.. Figue.: Afe a ie scale of he ion plasa fequency, ions sa esponding and he ion aix sheah sas o expand. These ions evenually bobad he age cahode pependiculaly and unifoly, wih sufficien enegies o poduce iplanaion. The expeienal and nueical 6

22 [] [5] invesigaions show ha he ion-aix phase plays a vial ole in deeining he consequen ion flux o he age suface [5, 6, ] Phase 3 Afe he sheah edge eaches he Child-Langui lii, i sops expanding and launches an ion-acousic wave ino he plasa, which poduces ion aeficaion as i popagaes. This aeficaion ceaes a quasi-neual pesheah egion beween he plasa and he sheah, which acceleaes ions up o Boh s speed befoe eneing he sheah. The poenial pofile in his sae is given in figue.5. Figue.5: The seady sae poenial pofile. Sheah volages ae ofen vey lage copaed o he elecon epeaue,. The poenial in ion aix sheahs is highly negaive wih espec o he / T plasa-sheah edge; hence n ~ n e e whee n is he elecon densiy and e s e T e n s is he densiy of elecons a he sheah edge, which eans ha hee ae only ions in he aix sheah. Theefoe, fo a high-volage sheah, he cuen a he cahode is alos all ion cuen [6]. 7

23 .. The Obecives and Conen of he Thesis Theoeical invesigaions of ion-aix sheahs fo plana, cylindical, and spheical ages have been successfully caied ou by Conad [], whee he plana and cylindical ages wee assued o have infinie aeas and lenghs, especively, educing he analysis o one-diensional cases [5]. Sheidan [7] and Zeng [8] have applied he sae foulaion o sudy he ion-aix sheahs in cylindical boes, wih and wihou axial elecodes, especively. Deokan [7] and Filiz [] pesened he fis analyic eaens of ion-aix sheahs in wo diensions, by consideing fis a ecangula and hen a sei-cylindical goove of infinie lenghs in an infinie, plana age. Finally, Deokan [5] epesened he poble of ion aix sheahs inside cylindical boes wih sall adii and longiudinal gooves analyically, which eaed he poble oe ealisically o pooe he applicaion of plasa souce ion iplanaion echniques o a boad ange of poducs, especially hose concening defense indusies. Howeve, fo he ages consideed in his wok, iplanaion on he side walls of he gooves is elaively poo, since he ions ae acceleaed pedoinanly in he adial diecion befoe eneing he gooves [5]. In his pape Deokan suggesed axiizing he aio of he ion velociy coponens inside he gooves by using a new echnique, inceasing he aio n n (whee is iniial ion densiy and is he age poenial and leing he ine-elecode space appoxiaely equal he age adius ies he age poenial. In his hesis, a new analyical echnique o sudy he evoluion of ion aix sheahs will be exained fo plana and cylindical configuaions, a copue poga will be developed o siulae he sheah evoluion nueically and a basis fo fuue sudies on he poble of cylindical boes wih sall adii and gooves will be esablished. 8

24 CHAPTER THEORY AND CALCULATIONS.. PLANAR CONFIGURATION... The Model The poble, subec o ou wok is illusaed in figues.,. and.3, consideing he iniial phase and he poceeding evoluion. Figue.: When he applied poenial,, is lage (~ kv and he elecode sepaaion is sall (~ -, he aix can fill he enie ine-elecode space ( / ω pe < / ω < pi, whee pe ω and ω pi ae elecon and ion plasa fequencies, especively. 9

25 Figue.: Afe a sho ie, he aix sas oving owads he cahode. The noaion s( epesens he ie-dependen locaion of he sheah edge. Figue.3: The aix disappeas as all ions ae iplaned. The odel pesened hee is a one-diensional odel, wih a plana anode of infinie aea placed a is placed a x = L x = and a plana cahode whee he age o be iplaned. The age consideed is also a plana plae ha has an infinie

26 - aea. The agniude of L is ~. The negaive poenial applied o he cahode is ~ kv, wheeas he anode is kep a zeo poenial. Poenial fo < x < s( is labeled as > and he poenial fo s ( < x < L is labeled as <. The following assupions have been ade: The ion densiy n, he poenials < and >, and he ion velociy have a fo, which is a zeoh ode e ha does no ake ino accoun he conibuions fo he ion densiy plus a fis ode e due o he conibuions fo he ion densiy; Since he conibuion of n is aleady a fis ode e, any flucuaion in his e is a second ode e, and can be negleced in Poisson s Equaion; 3 Howeve, boh < and > will be ie dependen due o s(. u i Theefoe, ( ( < = < < ( ( > = > > ( n n = i, (.., (.. = n, (..3 u u u =. (.. ( i ( i... The Deeinaion of Poenials To deeine he poenials in boh egions < x < s( and s ( < x < L, he Poisson s equaions in hese egions have o be solved and he bounday condiions have o be iposed.

27 ( > ( < Since he zeoh ode poenials, and, neglec he effec of n, which is a fis ode e, hey ae equal, ( x = < ( ( > ( x, and he fis ode poenials ( > and ( < will depend on ie. ( > ( < The Poisson s Equaions fo and ae ( > x ( < x =, (..5 =. (..6 ( > ( < The bounday condiions o be iposed fo and ae ( x =, ( =, (..7 > ( ( x = L, ( L = ( =. (..8 > < L ( > ( < Then, he soluions fo and ae found as ( x = L ( ( ( x = < x >. (..9 ( > ( < The Poisson s equaions fo and ae ( > x ( < x =, (.. en = ε ( en = ε. (..

28 ( > ( < The bounday condiions o be iposed fo and ae ( x =, (, =, (.. > x = s(, ( x,, (..3 ( ( > ( x, = s( < s( x = s(, ( ( > < x s( = x s(, (.. ( x = L, ( L, = (..5 < ( > ( < Then, he soluions fo and ae found as ( en > ( x, = ( L s( x, (..6 ε L en x ε en [( s( L ] x ( s( ( < (, =. (..7 x en ε L ε Theefoe, he poenials in he egions < x < s( and s ( < x < L, i.e. and <, ae found as > en > ( x, = x ( L s( x, (..8 L ε L L en x ε en [( s( L ] x ( s( < (, =. (..9 x x en ε L ε These poenials do no depend on ie explicily. To find he explici ie dependence of he poenials, he explici fo of s( us be deeined. 3

29 To do his, he equaion of oion a x = s( is used, which is d s d > = e. (.. x x= s( Subsiuing > gives d s d e = L enl ε s(. (.. L ( ( To be consisen, s ( is also assued o have a fo s ( = s ( s (. d s d d s = d ( d s d ( e = L enl s ε ( s L ( ( L ( (.. To find he zeoh and fis ode es in his diffeenial equaion, le us exaine he poenial of oion is <. Since ( ( < ( x = ( x = x, he zeoh ode equaion L > s d e = (..3 L ( d which gives ds d ( ( e = ui =. (.. L

30 Reaanging is es, equaion (..7 becoes en x ( x, = L ε ( s( ( s( ( < x L x. (..6 The es in backes have a axiu value of ( s( ( s( L x = L L x ax. Then, ( < is on he ode of enxl ε, which gives ( < enl ( < ε en L. So, ε is a fis ode e. Then, he zeoh ode equaion of oion is given by equaion (..3 and he fis ode equaion of oion is given by d s d ( e = L ( enl s ( ε L (..6 Theefoe, fo he diffeenial equaions (..3 and (..6, s ( ( and s ( ( ae found as given in equaions (..7 and (..8. s s ( ( e ( L = (..7 e A e ( A e ( A ( L L L = ( whee A enl = ε, which is a fis ode e. 5

31 To copae hese es, axiu ie,, involved in his pocess us be esiaed. Then, ax ax ax ( ( ( L e L s gives / ax e L. (..9 So, he es in, in equaion (..8, coespond o, /3, /5, especively. Then, he fis wo es can be consideed of equal ode and is found as given ( ( s s( 6 ( L A e A L e s =. (..3 Neglecing he second ode es, which coe fo he squae of A, he explici fos of he poenials > and < ae given by ( ( 3 8 (, ( x A L n e x A L n e x A L n e x L en x L x ε ε ε ε = >, (..3 ( ( ( ( 8 ( 8 (, ( A L n e x A L n e A L n e x A L n e x L en x en x L x ε ε ε ε ε ε = <. (..3 6

32 ..3. The Deeinaion of Ion Velociy and Flux To exaine he ion behavio, he ion velociy and he flux us be deeined. To do his, he equaion of oion is consideed. dui d ui ui e ui = x x = <. (..33 u i ( x, Subsiuing equaions (.. and (..3, i.e. and ( x,, he zeoh and fis ode equaions of he ion velociy can be found as given by equaions (..3 and (..35. < u ui ui x e L ( ( i ( ui = ( u ( i ui x ( u ( i ui x ( 5 3 e no e ( n A = x 5 ε ε L e ( n ( A 3 3 8ε L 6 e nol ε (..3 (..35 Since he igh hand side of equaion (..3 is a consan, his diffeenial equaion can be wien as ( u i ( x, = ( du i d, i has a soluion in he fo e =, and iposing he iniial condiion L ( e u i ( L =. (..36 Subsiuing his soluion, equaion (..36, ino equaion (..35, he diffeenial equaion becoes 7

33 L n e L A n e L A n e x n e x u L e u o o i i ( ( 8 ( ( ( ε ε ε ε =. (..37 The soluion of a quasi-linea equaion of fis ode ha has a geneal fo of,, (, (, (, (, ( x u R x u x Q x x u x P = is found by he ehod of chaaceisics, ha is by siulaneously solving he equaions R du Q d P dx = =. Applying his ehod o ou diffeenial equaion, equaion (..37, and iposing he iniial condiion,, he soluion is found o be, ( ( = x u i L n e L A n e L A n e L e n e x n e x u o o o i ( 5 8 ( ( 7 ( 3, ( ε ε ε ε ε =. (..38 Using he esiaed value of axiu ie,, involved in his pocess, which is given by equaion (..9, he es of ae copaed wih he e of and he coefficiens of and ae found o be negligible. Then, he fis ode ion velociy becoes ax, ( ax ( x u i ( ax ( u i 5 7 L n e L e n e x n e x u o o o i 3 ( 3, ( ε ε ε =. (..39 8

34 Cobining equaions (..36 and (..39, he ion velociy has he fo e ui ( x, = L e n L o e no e no e x ε ε ε L 3 3. (.. Theefoe, he ion flux has he fo Γ( x, = n u ( x, en = L i e ( n L e ( no ε ε e ( no x ε e L 3 3. (..... Gaphical Repesenaion The ion velociy is given by equaion (... A a paicula ie, his ion velociy has a fo u ( x = Cx D whee C and D ae consans. In figues., i.5,.6, and.7, his alos linea dependence is shown gaphically a = ax, 3 = ax, and = ax, especively. = ax, In he calculaions, he ions ae assued o be ianiu ions and he ion ass is aken o be kg. Also in he calculaions, he agniude of he disance beween he elecodes, L, is aken o be ~ -, he negaive poenial applied o he cahode,, is aken o be ~ kv, poenial applied o he anode is aken o be. kv, he iniial ion velociy is aken o be. /s and he iniial ion densiy, n, is aken o be ~ 5 / 3. 9

35 Figue.: Ion velociy as a funcion of x a ie = ax. Figue.5: Ion velociy as a funcion of x a ie = ax.

36 Figue.6: Ion velociy as a funcion of x a ie =. 3 ax Figue.7: Ion velociy as a funcion of x a ie = ax.

37 Also, he ion flux is given by equaion (... Figue.8 shows he ion flux as a funcion of ie a x = L. Figue.8: Ion flux as a funcion of a x = L... CYLINDRICAL CONFIGURATION... The Model The poble subec o ou wok is illusaed in figues.9,. and., consideing he iniial phase and he poceeding evoluion.

38 Figue.9: When he applied poenial,, is lage (~ kv and he elecode sepaaion is sall (~ -, he aix can fill he enie ine-elecode space ( / ω pe < / ω < pi, whee pe ω and ω pi ae elecon and ion plasa fequencies, especively. Figue.: Afe a sho ie, he aix sas oving owads he cahode. s( is he ie-dependen locaion of he sheah edge. 3

39 Figue.: The aix disappeas as all ions ae iplaned. The odel pesened hee is a one-diensional odel, wih a cylindical anode of infinie lengh and adius. The age consideed is he inenal suface of an infiniely long cylinde whose inne adius is. The anode is axially placed inside he age. The agniude of is ~ -, is ~ 3 -, he negaive poenial applied o he age,, is ~ kv, wheeas he anode is kep a zeo poenial. Poenial fo < < s( is labeled as > and he poenial fo s ( < < < is labeled as. The following assupions have been ade: The ion densiy n, he poenials < and >, and he ion velociy have a fo, which is a zeoh ode e ha does no ake ino accoun he conibuions fo he ion densiy plus a fis ode e due o he conibuions fo he ion densiy; u i

40 Since he conibuion of n is aleady a fis ode e, any flucuaion in his e is a second ode e, and can be negleced in Poisson s Equaion; 3 Howeve, boh < and > will be ie dependen due o s(. Theefoe, equaions (.., (.., (..3 and (.. ae also assued o be ue fo he cylindical case.... The Deeinaion of Poenials To deeine he poenials in boh egions < < s( and s ( < <, he Poisson s Equaions in hese egions have o be solved and he bounday condiions have o be iposed. ( ( ( ( Since and neglec he effec of n, which is fis ode, ( = ( > < ( > ( < and he fis ode poenials and will depend on ie. > < ( > ( < The Poisson s equaions fo and ae > = ( ( ( > > < ( =, (.. =. (.. ( > ( < The bounday condiions o be iposed fo and ae ( =, ( =, (..3 > ( ( =, ( = ( =. (.. > < 5

41 ( > ( < Then, he soluions fo and ae found as ( ( = ( ln ( ( ( < > =. (..5 ln ( > ( < The Poisson s equaions fo and ae ( > =, (..6 ( < en en = = ε ε. (..7 ( > ( < The bounday condiions o be iposed fo and ae ( =, =, (..8 > (, = s(, (,, (..9 ( ( > (, = s( < s( = s(, ( ( > < s( = s(, (.. ( =, =. (.. < (, ( > ( < Then, he soluions fo and ae found as ( > ln = ln ( ( en ε ( [ s s ln( s ] ( ( ( en ln en < = s s ln ε ln ε [, (.. ( s ]. (..3 6

42 Theefoe, he poenials in he egions < < s( and s ( < <, i.e. > and <, ae found as > < ( ( ( ( [ s s ( s ] ln ln en = ln ln ln ε = ln ln ln( ln( ( ( en ε en ε ( [ s s ln( s ], (... (..5 These poenials do no depend on ie explicily. To find he explici ie dependence of he poenials, he explici fo of s( us be deeined. To do his, he equaion of oion a = s( is used, which is d s d > = e. (..6 = s( Subsiuing > gives d s d e en = [ s s ln( s ]. (..7 ln( s ε ( ( To be consisen, s ( is also assued o have a fo s ( = s ( s (. Then, d s d ( ( d s d s = d d e en = ln( s ε [ s s ln( s ]. (..8 7

43 In s(, he fis ode e is uch salle han he zeoh ode e. Then, and s can be expanded as given by equaions (..9 and (... s = s s s = s s s = s ( ( ( ( ( ( s s ( (, (..9 = ( ( ( ( ( s ( = ( s s ( s s. (.. s s Then, he equaion of oion a = s( becoes d s d ( d s d ( = e ln ( s ( s ε s s s ( ( ( ( ( s en ( s s ( s ( ( ( s ln ( s s ( ( (.. ( en ( s Noe ha en = ε ε ax, which is a fis ode e. Then, oiing he second ode es in equaion (.. equaion (.. is obained. d s d ( d s d ( = ln e s s ( ( ( s en ( s ( ( ε s ( ln s ( (.. 8

44 Since ae fis ode es, he diffeenial equaion of s s ( ( s ( ( and is ( en s en = ε ε ax d s d ( e ln =. (..3 ( ( s The diffeenial equaion of s ( ( is d s d ( s s ( ( s s ( e en = ( ( ln ( ( ( s ln ε s. (.. Theefoe, fo he diffeenial equaions (..3, o find s ( ( le ds d ( = p, hen d s d ( d ( p e = = (..5 ( ( ds ln( s which can be sepaaed as d ( p e ln ( = (..6 ( ( s ds Hence, inegaing boh sides equaion (..7 is obained. ( ds e ( p = = s C d ln ln(. (..7 9

45 The iniial condiions o be iposed fo ( s and ds ( ae d ( ds =, ( = = = d p (..8 = ( =, s ( = =. (..9 Then, he following esuls ae found. C e ln ln = (..3 ( ds d ( e ( ( s = ln ln( / (..3 To siplify he equaion, he following definiions ae ade / e 7 =.696 ln( = s a (..3 ( ( s x =. (..33 Then, he diffeenial equaion (..3 akes he fo ( x d d [ ln( x ] / = a (..3 which is given in equaion (..35 in inegal fo. = x dx' a x [ ln '] / (..35 3

46 To find he soluion of his inegal, assupion / / [ ln x' ] [ x' ].5[ x' ] fo x 3, (..36 is ade wih a axiu eo of %. Then, he inegal equaion becoes x a =.5[ x' ] dx' / fo x 3. (..37 [ x' ] [ ] / u = x' Defining, du =, he soluion is found as [ x' ] dx' / u a = u.3 (..38 which can be wien as u 6.6u 3.3a =. (..39 Noe ha [] u ax. and [ u ] ax (. = 3. 8 which is vey sall copaed o [ 6.6u] ax 37.. So i can be ignoed. Then, a u. (.. 3

47 To ge a bee esul u is defined as u = u ( (.. δ whee a u (.. δ <<. (..3 Subsiuing hese values ino equaion (..39 and using he fac ha a a a u ax and << 6.6 []. ax ( a, δ is found as 3 δ << ( Then, u is found as 3 ( a u = u( δ = ( a ( ( / u = x Since, x is found as ( a ( 3 5 a x = ( a = ( a (

48 a 5 << 3.3 Again using he fac ha (, x a (. (..7 ( Reebe ha ( s ( s ( x =. Theefoe s is found as [ ( ] ( a (..8 whee a is defined by he equaion (..3. ( ( To find s, s is subsiued ino equaion (... d s d ( = e e n ε ln( [ ( a ] [ ( ] a ( ln( [ ( a ] s ( ln [ ] ( ln ( a (..9 To siplify his equaion, he change of vaiables = ( a ω is ade. d s dω ( s ( [ ω ] 3 en = ε [ ω ] ( [ ω ] ln ln[ ω ] (..5 This equaion can be solved using vaiaion of paaees bu his is no necessay. Dividing expessions fo and by gives fis ode es on he lef hand side. ( > ( < ( 33

49 ( < ( en ε ( > ( ( ln ln ( ( ens = ε ln = ln ( ( [ ( s ln( s ] ens ε [ ( s ln( s ] (..5 Since ( ( s en en = ε ε ax is a fis ode e, fo he es on he igh ( hand side no o be second ode es, only ( s is needed as ε ( ( ( s s en will be of second ode. Also fo his eason, he appoxiaion of ( s given by equaion (..8 can be ade. Then, he explici fos of he poenials > and < ae given by > < ln = ln ln ln = ( ( ( ( ln ln en ε ( ( ln ln en ( ( en ( a [ ( ] ( ε a ( a [ ( ] ( ε a ( ln a ln [ ] [ ] a ln (..5 (..53 3

50 ..3. The Deeinaion of Ion Velociy and Flux To exaine he ion behavio, he ion velociy and he flux us be deeined. To do his, he equaion of oion is consideed. e u u u d du i i i i = = <. (..5 Subsiuing <, equaion (..5 akes he fo ( ( [ ] ( ( [ ] n e a en e u u u d du a a i i i i ln ln ε ε = = (..55 Using he assupion given by equaion (.., wo diffeenial equaions fo he zeoh ode ion velociy and fo he fis ode ion velociy ae obained. ( e u u u i i i ln ( ( ( = (..56 ( [ ] ( ( ( [ ] n e a n e u u u u u a a i i i i i ( ( ( ( ( ln ln ε ε = (..57 Equaion (..56 is a quasi-linea paial diffeenial equaion and can be solved by he ehod of chaaceisics, i.e. by solving he odinay diffeenial equaions given by equaion (

51 d ( d dui = = (..58 ( u e i ln( The diffeenial equaion d u ( dui = (..59 e ln( ( i has a soluion in he fo / [ ln( / ( u i = a ] (..6 whee denoes he iniial posiion a =, which can be any poin in he ineval, and a is given by equaion (..3. Using ( u i given by equaion (..6, he diffeenial equaion d d = ( (..6 ui can be wien in inegal fo as d = a / d( / [ ln( / ] /. (..6 Wih he siplificaion, he inegal in equaion (..6 becoes x = / 36

52 a = x dx' ln x'. (..63 The soluion of equaion (..63 is found o be a i π Ef ( i ln x =. (..6 Figue. shows he gaph of i π Ef ( i ln x gaph of he soluion given by equaion (..6. and figue.3 shows he Figue.: The gaph of i π Ef ( i ln x x = / whee. 37

53 a Figue.3: The gaph of i π Ef ( i / = wih aken as ln( =. The soluion given by (..6 canno be used in his fo since i is iaginay. When i π Ef ( i / ln( is wien in he infinie seies fo, equaion (..65, i is obseved ha he soluion is eal. ( i ln( / n= [ ln( / ] n i π Ef = (..65 n!(n Using he fac ha can ake values in he ineval / 3 /, ln( / can ake values in he ineval ln( /. 8. The axiu value of ln( / is used o copae he fis five es ( n in he infinie seies 38

54 given by equaion (..65. These five es coespond o,.37,.,.3 and.7, especively. Then, he fis hee es ( n =,, can be consideed of equal odes and he ohe es can be excluded. Howeve, i is no possible o solve equaion (..65 analyically afe his appoxiaion unless he hid e is also excluded. Then, equaion (..65 can be wien in he fo given by equaion (..66 whee only he fis wo es in he expansion ae aken wih an eo of 9 %. The soluion of equaion (..6 is given by (..67. ( i ln( / = ln( / [ ln( / ] 3 i π Ef ( a [ ln( / ] 3 ln( / 3 =. (..67 When he gaphs of equaion (..6 and (..67 ae ploed fo =, he diffeence in soluions can be seen in figue.. = ax and Figue.: A copaison of he gaphs of equaion (..6 and (..67 fo = ax and =. 39

55 Equaion (..67 is solved fo ( by using he aheaical packe poga Maheaica and hus hee oos ae obained. The only eal oo is given in equaion (..68. ( = e f ( (..68 whee f ( = / 3 / 3 (..69 When equaion (..68 is aken as he soluion fo ( and his soluion is subsiued ino he equaion (..6, equaion (..7 is obained. ( u i = a [ f (] / (..7 I can be easily seen ha his soluion ehod has lead o a soluion fo he zeoh ode ion velociy wih he fo u ( i = u ( i (. Howeve, siulaions have shown ha he zeoh ode ion velociy has he fo ( u i (,. Since solving ( (, (, equaion (..67 fo, which ay lead o a soluion of he fo, is no analyically possible, i can be concluded ha expessing of he iaginay eo funcion in infinie seies fo will no give an accepable soluion. u i

56 To find an accepable soluion fo he zeoh ode ion velociy, he fac ha equaion (..63 has he sae fo wih equaion (..35 is aken ino accoun. Then, doing siila calculaions, ( / is found as a ( =, ( 3 (..7 whee a is given by equaion (..3. Equaion (..7 can be wien in quadaic fo as a = (..7 This quadaic equaion has wo oos in he following fo (, ± a = (..73 ( To find he oo ha is valid, he gaph of, which is shown in figue.5, is ploed and exained. Noe ha he gaph in figue.5 acually belongs o a. ( =

57 Figue.5: The gaph of. ( Fo he gaph i is easily seen ha boh oos of ae valid soluions. The negaive oo coesponds o he egion and he posiive oo coesponds o he egion. In ou odel = and he egions whee oos coespond ae shown in figue.6.

58 Figue.6: The egions whee he oos of coespond. Subsiuing he oos of ino equaion (..6 and using he appoxiaion given by equaion (..36, ( u i is found a = a a z [ z ].3[ z ] z ( u i (..7 a z, [ ] [ ] z.3 z z, whee = a and a is defined by he equaion (..3. Howeve, equaion z (..7, which is obained analyically unde he appoxiaion (..36, needs o be sudied fuhe because i does no fully saisfy equaion (

59 Solving equaion (..56 and (..57 by using nueic ehods, he ion velociy, u = u u i ( i ( i, was obained as shown in figue.7. In he nueical calculaions, he ions ae assued o be ianiu ions and he ion ass is aken o be kg, he adius of anode,, is aken o be ~ -, he inne adius of cahode,, is aken o be ~3 -, he agniude of he disance beween he elecodes, L, is aken o be ~ -, he negaive poenial applied o he cahode,, is aken o be ~ kv, poenial applied o he anode is aken o be. kv, he iniial ion velociy is aken o be. /s and he iniial ion densiy, n, is aken o be ~ 5 / 3. Figue.7: The ion velociy as a funcion of a he insans = ax, 3 = ax and = ax obained as a esul of nueic calculaions. = ax, Then, he ion flux on he suface of he age as a funcion of ie is found nueically. Figue.8 shows he gaph of he ion flux obained as a esul of hese nueic calculaions.

60 Figue.8: The ion flux as a funcion of a he age suface obained as a esul of nueic calculaions. 5

61 CHAPTER 3 SIMULATION 3.. THE BASICS OF SIMULATION 3... Nueical Analysis Many pobles in aheaics do no have a soluion in closed fo. In hese siuaions, an appoxiae soluion using asypoic analysis o a nueical soluion is sough. The nueical analysis is he sudy of algoihs fo he pobles of coninuous aheaics using basic aiheical opeaions [9]. In nueical analysis, he algoihs ha can solve a poble exacly ae called diec ehods; he algoihs ha solve he poble by successive appoxiaions ae called ieaive ehods. The pocess of eplacing a coninuous poble wih a discee poble whose soluion is appoxiaely equal o ha of he coninuous poble is called disceizaion. Nueical analysis is concened wih copuing he soluion of boh odinay diffeenial equaions and paial diffeenial equaions in is applicaion aeas, which include copuaional physics, copuaional fluid dynaics, weahe foecasing, cliae odels, he analysis and design of olecules (copuaional cheisy and econoy. Paial diffeenial equaions ae solved by fis disceizing he equaion. Thee coon ehods of disceizaion of paial diffeenial equaions ae he finie eleen ehod, he finie volue ehod and he finie-diffeences ehod. 6

62 3... The Finie-Diffeences Mehod The finie-diffeences ehod is a nueical ehod fo solving paial and odinay diffeenial equaions, which ais o eplace he coninuous deivaives wih diffeence equaions ha involve only he discee values of he unknown funcion elaed o he posiion in he doain of he poble []. The finie-diffeences ehod disceizes he doain of he poble ino a egula gid (esh defined by a ceain nube of nodes which ae sepaaed in he diecion of coodinaes by a ceain spaial and ie ineval. Fo a poble solved wih he explici finie-diffeences ehod o convege o he exac soluion, he ie sep us be salle han a ceain axial value []. Fo exaple, he sable explici diffeencing schees fo solving he advecion equaion ae subec o he Couan-Fiedichs-Levy (CFL condiion, which deeines he axiu allowable ie sep []. The CFL condiion, which gives he condiion of he convegence of a diffeence appoxiaion in es of he concep of doain dependence, saes ha fo a convegen algoih, he doain of dependence of he paial diffeenial equaion us lie wihin he doain of dependence of he nueical algoih. This, in un iplies a condiion on he ie sep, given by equaion (3.. [3]. The CFL condiion on he ie sep: δ x δ (3.. v whee v is a phase speed o an advecion velociy and v >. On he ohe hand, fo a poble solved wih he iplici finie-diffeences ehod, hee is no sabiliy equieen on he ie sep; he value of he ie sep is dicaed by accuacy only. 7

63 When he finie-diffeences ehod is applied ove he esh, he unknown funcion is appoxiaed a each one of he nodes, giving a diffeence equaion ha elaes he value of he funcion a a paicula node wih is value a he neighboing nodes []. This pocedue is epeaed a each node coposing he gid esuling in a syse of diffeence equaions which ay be nueically solved eihe wih ieaive appoxiae ehods o wih diec decoposiion ehods. The disceizaion of one-diensional paial diffeenial equaions usually leads o sandad hee-poin pobles (also known as idiagonal equaion syses [5 6] which ay be wien as. (3.. = = = =.,,..,,, n n n n n d b U a U n d U c b U a U d cu bu In hee-poin pobles, he equaion nube involves only he es wih nubes, and so ha he aix of he syse has non-zeo eleens only on he diagonal and in he posiions iediaely o he lef and igh of he diagonal as shown in equaion (3..3 [6]. (3..3 = n n n n n d d d U U U b a c b a c b a c b To solve hee-poin pobles, idiagonal aix algoihs such as he Thoas algoih ae used. The Thoas algoih, which is a diec ehod, equies he known coefficiens, and o saisfy he condiions given by equaions (3.. and (3..5, a b c 8

64 which ensue ha he aix fo of he hee-poin poble is diagonally doinan [6]. a >, b >, > (3.. c b > a c (3..5 The Thoas algoih solves he hee-poin poble by educing he syse of equaions o uppe iangula fo, by eliinaing he e U in each equaion uning he ino a new fo given by equaion (3..6 [6]. U = e U f, =,,..., n (3..6 e whee and saisfy he ecuence elaions given by equaions (3..7 and f (3..8 and have iniial values given by equaion (3..9 [6]. c e =, =,,..., n (3..7 b a e d a f f =, =,,..., n (3..8 b a e c e = and b d f =. (3..9 b Afe finding he coefficiens by using he ecuence elaions given by equaions (3..7 and (3..8, he values of U ae easily calculaed using equaion (3..6. Fo sabiliy, he condiion each e < in equaion (3..6 us hold, which is aleady guaaneed by he equaions (3.. and (

65 3.. PLANAR CONFIGURATION 3... The Model The odel pesened in he plana pa of ou poble is a one-diensional odel, wih a plana anode of infinie aea placed a x = and a plana cahode a x = L whee he age o be iplaned is placed. The age consideed is also a plana plae ha has an infinie aea. The agniude of is ~ -, and he negaive poenial applied o he cahode,, is ~ kv, wheeas he anode is kep a zeo poenial, he ion velociy is iniially zeo and he iniial ion densiy, n, has a value of ~ 5 / 3. L 3... Equaions The equaions ha have o be solved o siulae he plana pa of ou poble ae given by equaions (3.. (3..3. Poisson s Equaion: x = en ε. (3.. Coninuiy Equaion: n ( un =. (3.. Moenu Balance Equaion: u u u x = e i E. (3..3 5

66 In equaions (3.. (3..3, is he poenial, e is he elecic chage of an ion, u is he ion velociy, n is he ion densiy, ε is he elecic peiiviy of fee space, E is he elecic field beween he elecodes. i is he ass of an ion, and 3... Bounday and Iniial Condiions The bounday condiions o be iposed ae given by equaions (3.. (3..7. x =, (, =, (3.. x = L, ( L, =, (3..5 x =, u (, =, (3..6 x =, n (, = n. (3..7 The iniial condiions o be iposed ae given by equaions (3..8 and (3..9. =, ( x, = u, (3..8 { [ ] } n 3 =, x, = acan 5 ( x π n. (3..9 π ( 3... Nueical Pocedue x = x = L Equaions (3.. (3..3, which ae subec o he bounday and iniial condiions given by equaions (3.. (3..9, have been solved nueically using a finie-diffeences echnique. The copuaional doain is esiaed o be beween and, which is illusaed by figue 3.. 5

67 Figue 3.: Copuaional doain A unifo veex-ceneed gid x x, x = L N, =,,.., N = (3.. is used whee is he spaial index. The ie level is given by =, >, =,,.. (3.. = whee subscip denoes he ie level wih a sep size. Densiy n and elecic poenial ae evaluaed a he nodes of he gids, while he ion velociy u and he elecic field E ae a he cenes of he copuaional cells. 5

68 3... Poisson s Equaion To obain a finie-diffeences appoxiaion of Poisson s equaion (3.., which has he bounday condiions given by equaions (3.. and (3..5, a sandad second ode discee schee of he fo given by equaion (3.. is used. ( x, x O ( x. (3.. ( x As a esul, Poisson s equaion uns o he sandad hee-poin poble given by equaion (3..3. en = ( x ε (3..3 Equaion (3..3 is solved by he Thoas algoih, using bounday condiions descibed in equaions (3.. and (3..5 in discee fo, which is given by equaions (3.. and (3..5. = (3.. N = ( Coninuiy Equaion x x x To obain a finie-diffeences epesenaion of equaion (3.., i is fis inegaed ove he cell volue / /. As a esul, we have dn d ( un = ( un / / x (

69 whee u is he ion velociy and n is ion densiy. A choice of n ±/ a he cenes of he copuaional cells deeines he concee disceizaion ehod fo he convecive es of equaion (3... We used he hid-ode upwind-biased schee [7], which has he fo ( un [ u ( n 5n n u ( n n n ] / = / / 5 6 (3..7 whee u ( u, and ( u / = ax, / u. (3..8 / = ax, / Fo he nueical ie inegaion, he exapolaed second ode backwad diffeeniaion foula (BDF is used [8]: 3 n n n = (, F n n,. (3..9 Hee, F conains he disceized convecive es. Noe, ha spaial indices in equaion (3..9 have been dopped. Since wo-sep ehod needs n and n as saing values, he explici Eule ehod given by equaion (3.. is used on he fis sep. n = n F(, n (3.. Because of he explici ie inegaion, we ae esiced by he sandad CFL condiion fo sabiliy. 5

70 Moenu Balance Equaion Le us conside he oenu equaion, which is given by equaion (3..3. This equaion can be ewien in he fo of u u x = e i E. (3.. To have a consevaive diffeence schee fo (3..3, i is consuced by he inego-inepolaed ehod; inegaing equaion (3..3 ove he copuaional cell x x x equaion (3.. is obained. x e ( u u dx ( u u d = x i x x Edxd (3.. Diffeen appoxiaions of he inegals in equaion (3.. lead o diffeen diffeence schees. One of he is a schee iplici in ie and of he following fo: u u ( u ( u x = f (3..3 wih e f = E(, x. (3.. i I is obvious ha equaion (3..3 epesens a quadaic algebaic equaion wih espec o u. Using he ion velociy known fo he pevious ie level u 55

71 ( =,,..., N, a nueical pocedue can be oganized by a successive calcula- ion of saing fo he lef bounday ( x = u u = (3..5 o he igh by he equaion u x x x = u ( u xf (3..6 = =,,3,... whee,3,..., N, Nueical Resuls Nueical pocedue is oganized as follows. On evey new h ie sep, fis, Poisson s equaion is solved, using he known ion densiy in is souce e; hence, elecic field in he new ie sep is deeined. Then, he oenu balance equaion is solved, using he known elecic field, n E, and he ion velociy in he new ie sep is deeined. Finally, he coninuiy equaion is solved, using he known ion velociy and heefoe he ion densiy in he new ie sep, n, is deeined. The nueical convegence is checked by pefoing seveal calculaions using efineen of he space gid and diffeen ie sepping paaees. The nube of gid nodes used in he calculaions was. In he siulaion, he ions ae assued o be ianiu ions and he ion ass is aken o be kg, he agniude of he disance beween he elecodes, L, is aken o be ~. -, he negaive poenial applied o he cahode,, is aken o be ~. kv, poenial applied o he anode is aken o 56

72 be. kv, he iniial ion velociy is aken o be. /s and he iniial ion densiy, n, is aken o be ~. 5 / 3 The daa obained fo he siulaion is used o plo he ion densiy vesus posiion gaph and ion velociy vesus posiion gaph a a paicula ie. Figue 3. shows hese gaphs a he insans = ax, = ax, 3 = ax and = ax. Figue 3.: Ion densiy and ion velociy as a funcion of x a he insans = ax, = ax, 3 = ax and = ax. I can be easily seen fo figue 3. ha he ions a a paicula posiion x ove wih alos he sae velociy as hey ae acceleaed owads he age. The oion of he aix sheah and he posiion of he sheah edge can be seen fo he densiy vesus posiion gaph in figue 3.. The iplanaion of ions ends, i.e. he ion aix sheah disappeas, a a ie ax, which is found as s in he analyical calculaions and s in he siulaion. The final ion velociy a x = L and = ax is obained as.8 5 /s fo he siulaion daa. The analyically calculaed final ion velociy, which 57

73 is.78 5 /s, is appoxiaely equal o he siulaion esul wih an eo of.7 %. I is also obseved fo figue 3. ha he ion velociy inside he ion aix a a paicula insan is alos linea. Howeve, he ion velociy vesus posiion gaphs ploed by using he analyical daa, figue 3.3, and he nueical daa, figue 3., indicae diffeen elaions beween he ion velociy and posiion. The gaphs of he analyical daa indicae a linea elaionship beween ion velociy and posiion wheeas he gaphs of he nueical daa indicae a cuvilinea elaionship beween ion velociy and posiion. This lineaizaion of analyically calculaed ion velociy is due o he appoxiaions ade in he analyical calculaions. This causes he analyically calculaed ion velociy o incease oe slowly han he siulaed velociy in he ie ineval siulaed values in he ie ineval 3 ax < <. ax 3 < < ax and o cach he The gaph in figue 3. gives he analyically calculaed ion velociy as a funcion of posiion and ie. Alhough he analyically calculaed ion velociy is slighly diffeen fo he siulaed ion velociy egading he dependence on posiion, i can be concluded ha hee is a good ageeen beween he analyical and he nueical soluions. 58

74 Figue 3.3: Ion velociy vesus posiion gaphs ploed using analyical esuls. Figue 3.: The analyically calculaed ion velociy as a funcion of posiion and ie. 59

75 3.3. CYLINDRICAL CONFIGURATION The Model The odel pesened hee is a one-diensional odel, wih a cylindical anode of infinie lengh and adius and a cylindical cahode whee he age o be iplaned is placed. The age consideed is he inenal suface of an infiniely long cylinde whose inne adius is. The anode is axially placed inside he age. The agniude of is ~ -, is ~ 3 -, he negaive poenial applied o he cahode,, is ~ kv, wheeas he anode is kep a zeo poenial. The ion velociy is iniially zeo and he iniial ion densiy, n, has a value of ~ 5 / Equaions The equaions ha have o be solved o siulae he cylindical pa of ou poble ae given by equaions (3.3. ( Poisson s Equaion: = en ε. (3.3. Coninuiy Equaion: n ( un =. (3.3. Moenu Balance Equaion: u u u = e i E. (

76 In equaions (3.3. (3.3.3, is he poenial, e is he elecic chage of an ion, u is he ion velociy, n is he ion densiy, ε is he elecic peiiviy of fee space, E is he elecic field beween he elecodes. i is he ass of an ion, and Bounday and Iniial Condiions The bounday condiions o be iposed ae given by equaions (3.3. ( =, (, =, (3.3. =, (, =, (3.3.5 =, u (, =, (3.3.6, =, n ( = n. (3.3.7 The iniial condiions o be iposed ae given by equaions (3.3.8 and ( =, u (, =, (3.3.8 n 3 π =, (, = { acan[ 5 ( ] } n. (3.3.9 π Nueical Pocedue Equaions (3.3. (3.3.3, which ae subec o he bounday and iniial condiions given by equaions (3.3. (3.3.9, have been solved nueically using a finie-diffeences echnique. The copuaional doain is esiaed o be beween he adius of anode,, and he inne adius of cahode,, whee he disance beween he elecodes is figue 3.5. = L. The copuaional doain is given by 6

77 Figue 3.5: Copuaional doain A unifo veex-ceneed gid, = L N, =,.., N = (3.3. is used whee is he spaial index. The ie level is given by equaion (3... Densiy n and elecic poenial ae evaluaed a he nodes of he gids, while he ion velociy u and he elecic field E ae a he cenes of he copuaional cells Poisson s Equaion To obain a finie-diffeences appoxiaion of Poisson s equaion (3.3., which has he bounday condiions given by equaions (3.3. and (3.3.5, he inegoinepolaed ehod is used; inegaing equaion (3.3. ove he copuaional cell equaion (3.3. is obained. 6

78 (, / / ( O (3.3. ( As a esul, his poble uns o he sandad hee-poin poble of he fo (3.. which is again solved by he Thoas algoih using he foulaion given by equaions (3..3 ( Coninuiy Equaion To obain a finie-diffeences epesenaion of equaion (3.3., i is inegaed i ove he cell volue / / and equaion (3.3. is obained. dn d = / ( un / / ( un / (3.3. The hid-ode upwind-biased schee [7], which has a fo given by equaions (3..7 and (3..8, and he exapolaed second ode BDF [8] given by equaion (3.3.3 ae used. 3 n n n = F(,n n,, (3.3.3 Since he wo-sep ehod equies n and n as saing values, he explici Eule ehod, given by equaion (3.., is used again on he fis sep wih he sandad CFL condiion fo sabiliy Moenu Balance Equaion Equaion (3.3.3 can be ewien in he fo given by equaion (3... The inego-inepolaed ehod ove he copuaional cell and he 63

79 u schee iplici in ie given by equaions (3..3 and (3.. ae again used in he nueical copuaion of oenu balance equaion by a successive cal- culaion of saing fo he lef bounday ( = o he igh by he equaion ( Nueical Resuls The nueical pocedue is oganized in he sae fo as he nueical pocedue given in secion In he siulaion, he ions ae assued o be ianiu ions and he ion ass is aken o be kg, he adius of he anode,, is aken o be ~. -, he inne adius of he cahode,, is aken o be ~3. -, he - agniude of he disance beween he elecodes, L, is aken o be ~., he negaive poenial applied o he cahode,, is aken o be ~. kv, poenial applied o he anode is aken o be. kv, he iniial ion velociy is 5 aken o be. /s and he iniial ion densiy, n, is aken o be ~. / 3. The daa obained fo he siulaion is used o plo he ion densiy vesus posiion gaph and ion velociy vesus posiion gaph a a paicula ie. Figue 3.6 shows hese gaphs a he insans = ax, = ax, 3 = ax and = ax. 6