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SolvingtheBoltzmannequationtoobtainelectrontransportcoefficientsandratecoefficientsforfluidmodelsThisarticlehasbeendownloadedfromIOPscience.Pleasescrolldowntoseethefulltextarticle.2005PlasmaSourcesSci.Technol.14722()Downloaddetails:IPAddress:61.53.181.202Thearticlewasdownloadedon15/09/2011at03:16Pleasenotethattermsandconditionsapply.Viewthetableofcontentsforthisissue,orgotothejournalhomepageformoreHomeSearchCollectionsJournalsAboutContactusMyIOPscienceINSTITUTEOFPHYSICSPUBLISHINGPLASMASOURCESSCIENCEANDTECHNOLOGYPlasmaSourcesSci.Technol.14(2005)722–733doi:10.1088/0963-0252/14/4/011SolvingtheBoltzmannequationtoobtainelectrontransportcoefficientsandratecoefficientsforfluidmodelsGJMHagelaarandLCPitchfordCentredePhysiquedesPlasmasetdeleursApplicationsdeToulouse,Universit´ePaulSabatier,118routedeNarbonne,31062ToulouseCedex9,FranceReceived13June2005Published5October2005Onlineatstacks.iop.org/PSST/14/722AbstractFluidmodelsofgasdischargesrequiretheinputoftransportcoefficientsandratecoefficientsthatdependontheelectronenergydistributionfunction.Suchcoefficientsareusuallycalculatedfromcollisioncross-sectiondatabysolvingtheelectronBoltzmannequation(BE).Inthispaperwepresentanewuser-friendlyBEsolverdevelopedespeciallyforthispurpose,freelyavailableunderthenameBOLSIG+,whichismoregeneralandeasiertousethanmostotherBEsolversavailable.Thesolverprovidessteady-statesolutionsoftheBEforelectronsinauniformelectricfield,usingtheclassicaltwo-termexpansion,andisabletoaccountfordifferentgrowthmodels,quasi-stationaryandoscillatingfields,electron–neutralcollisionsandelectron–electroncollisions.Weshowthatfortheapproximationsweuse,theBEtakestheformofaconvection-diffusioncontinuity-equationwithanon-localsourceterminenergyspace.Tosolvethisequationweuseanexponentialschemecommonlyusedforconvection-diffusionproblems.Thecalculatedelectrontransportcoefficientsandratecoefficientsaredefinedsoastoensuremaximumconsistencywiththefluidequations.Wediscusshowthesecoefficientsarebestusedinfluidmodelsandillustratetheinfluenceofsomeessentialparametersandapproximations.1.IntroductionFluidmodelsofgasdischargesdescribethetransportofelectrons,ionsandpossiblyotherreactiveparticlespeciesbythefirstfewmomentsoftheBoltzmannequation(BE):(1)thecontinuityequation,(2)themomentumequation,usuallyapproximatedbythedrift-diffusionequationand(3)theenergyequation,usuallyonlyforelectrons.Eachoftheseequationscontainstransportcoefficientsorratecoefficientswhichrepresenttheeffectofcollisionsandwhichareinputdataforthefluidmodel[1–4](seealsoreferencestherein).Transportcoefficientsandratecoefficientsmayberatherspecificforthedischargeconditions.Inparticular,coefficientsconcerningelectronsdependontheelectronenergydistributionfunction(EEDF),whichingeneralisnotMaxwellianbutvariesconsiderablydependingontheconditions.Forsimpleconditions(swarmexperiments)andcommongases,theelectrontransportcoefficientsandratecoefficientshavebeenmeasuredandtabulatedasfunctionsofthereducedelectricfieldE/N(ratiooftheelectricfieldstrengthtothegasparticlenumberdensity)[5].Ingeneral,theEEDFandtheelectroncoefficientsforthegivendischargeconditionscanbecalculatedfromthefundamentalcollisioncross-sectiondatabysolvingtheelectronBE[6].AcommonapproachistosolvesomeapproximationoftheBEforaseriesofreducedelectric-fieldvaluesandtoputtheresultingcoefficientsintablesversusthereducedfieldorversusthemeanelectronenergy(disregardingthefieldvalues),whicharethenusedinthefluidmodeltofindthetransportcoefficientsandratecoefficientsbyinterpolation.Fluidmodelswithoutelectronenergyequationtreattheelectroncoefficientsasfunctionsofthelocalreducedfield;modelswithanelectronenergyequationtreatthemasfunctionsofthelocalmeanelectronenergy.TheBEsolversusedtogeneratetheelectron-relatedinputdataforfluidmodelsareusuallybasedontechniquesdevelopedduringthe1970sand1980s,whenmuchworkwasdoneonthesolutionoftheBEforthepurposeofcheckingtheconsistency0963-0252/05/040722+12$30.00©2005IOPPublishingLtdPrintedintheUK722ElectronBoltzmannequationsolverforfluidmodelsbetweencross-sectiondataandtransportcoefficientsorratecoefficientsmeasuredindifferentexperiments[7–16].Thesesolutiontechniquesoriginallyaimedatsimulatingspecificexperimentsandcalculatingtheexactphysicalquantitiesmeasuredintheseexperimentswithhighnumericalprecision.Forfluiddischargemodelling,however,onehassomewhatdifferentobjectives:(1)theBEsolvershouldworkoveralargerangeofdischargeconditions(reducedelectricfield,ionizationdegree,gascomposition,fieldfrequency)ratherthansimulateaspecificexperiment;(2)thecalculatedtransportcoefficientsandratecoefficientsshouldcorrespondformallytothesamecoefficientsappearinginthefluidequations(momentsoftheBE)ratherthantoquantitiesmeasuredinexperiments;notethattheliteraturegivesdifferentdefinitionsofthetransportcoefficients,someofwhicharenotcompletelyconsistentwiththefluidequations;(3)theerrorsinthecalculatedtransportcoefficientsandratecoefficientsshouldnotlimittheaccuracyofthefluidmodel;thisisalessstrictrequirementthantheextremeprecision(e.g.0.1%inthedriftvelocity)neededforthecross-section