comsol等离子体放电二维模型.docx

comsol等离子体放电二维模型.docx

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comsol等离子体放电二维模型

SolvedwithCOMSOLMultiphysics5.2

DCGlowDischarge

Introduction

DCglowdischargesinthelowpressureregimehavelongbeenusedforgaslasersand

fluorescentlamps.DCdischargesareattractivetostudybecausethesolutionistime

independent.ThismodelshowshowtousetheDCDischargeinterfacetosetupan

analysisofapositivecolumn.Thedischargeissustainedbyemissionofsecondary

electronsatthecathode.

ModelDefinition

TheDCdischargeconsistsoftwoelectrodes,onepowered（theanode）andone

grounded（thecathode）.Thepositivecolumniscoupledtoanexternalcircuit:

1000Ω

CathodePlasmaAnode

1pF

V

Figure1:

SchematicoftheDCdischargeandexternalcircuit.

DOMAINEQUATIONS

Theelectrondensityandmeanelectronenergyarecomputedbysolvingapairof

drift-diffusionequationsfortheelectrondensityandmeanelectronenergy.

Convectionofelectronsduetofluidmotionisneglected.Fordetailedinformationon

electrontransportseeTheoryfortheDriftDiffusionInterfaceinthePlasmaModule

User’sGuide.

∂n

（）+∇⋅[–ne（μe•E）–De•∇ne]=Re

∂t

e

∂n

（）+∇⋅[–nε（με•E）–Dε•∇nε]+E⋅Γe=Rε

ε

∂t

where:

1|DCGLOWDISCHARGE

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Γe=–（μe•E）ne–De•∇ne

TheelectronsourceReandtheenergylossduetoinelasticcollisionsRεaredefinedlater.Theelectrondiffusivity,energymobilityandenergydiffusivityarecomputedfromtheelectronmobilityusing:

De=μeTe,με

5

μ

=--,Dε=μεTe

e

3

Thesourcecoefficientsintheaboveequationsaredeterminedbytheplasmachemistry

usingratecoefficients.SupposethatthereareMreactionswhichcontributetothe

growthordecayofelectrondensityandPinelasticelectron-neutralcollisions.In

generalP>>M.Inthecaseofratecoefficients,theelectronsourcetermisgivenby:

Re=xjkjNnne

j=1

wherexjisthemolefractionofthetargetspeciesforreactionj,kjistheratecoefficient

forreactionj（m3/s）,andNnisthetotalneutralnumberdensity（1/m3）.ForDCdischargesitisbetterpracticetouseTownsendcoefficientsinsteadofratecoefficientstodefinereactionrates.Townsendcoefficientsprovideabetterdescriptionofwhat

happensinthecathodefallregionRef1.WhenTownsendcoefficientsareused,the

electronsourcetermisgivenby:

Re=xjαjNnΓe

j=1

whereαjistheTownsendcoefficientforreactionj（m2）andΓeistheelectronfluxas

definedabove（1/（m2·s））.TownsendcoefficientscanincreasethestabilityofthenumericalschemewhentheelectronfluxisfielddrivenasisthecasewithDC

discharges.Theelectronenergylossisobtainedbysummingthecollisionalenergyloss

overallreactions:

Rε=xjkjNnneΔεj

j=1

whereΔεjistheenergylossfromreactionj（V）.Theratecoefficientsmaybecomputed

fromcrosssectiondatabythefollowingintegral:

2|DCGLOWDISCHARGE

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∞

kkγεσk（ε）f（ε）dε

=

0

whereγ=（2q/me）1/2（C1/2/kg1/2）,meistheelectronmass（kg）,εisenergy（V）,σk

isthecollisioncrosssection（m2）andfistheelectronenergydistributionfunction.InthiscaseaMaxwellianEEDFisassumed.WhenTownsendcoefficientsareused,the

electronenergylossistakenas:

Rε=xjαjNnΓeΔεj

j=1

Fornon-electronspecies,thefollowingequationissolvedforthemassfractionofeach

species.Fordetailedinformationonthetransportofthenon-electronspeciessee

TheoryfortheHeavySpeciesTransportInterfaceinthePlasmaModuleUser’s

Guide.

∂wρ

（）+ρ（u⋅∇）wk=∇⋅jk+Rk

∂t

k

Theelectrostaticfieldiscomputedusingthefollowingequation:

–∇⋅ε0εr∇V=ρ

Thespacechargedensityρisautomaticallycomputedbasedontheplasmachemistry

specifiedinthemodelusingtheformula:

nρ=qZknk

–

e

k=1

FordetailedinformationaboutelectrostaticsseeTheoryfortheElectrostaticsInterfaceinthePlasmaModuleUser’sGuide.

BOUNDARYCONDITIONS

UnlikeRFdischarges,themechanismforsustainingthedischargeisemissionof

secondaryelectronsfromthecathode.Anelectronisemittedfromthecathodesurface

withaspecifiedprobabilitywhenstruckbyanion.Theseelectronsarethenaccelerated

bythestrongelectricfieldclosetothecathodewheretheyacquireenoughenergyto

initiateionization.Thenetresultisarapidincreaseintheelectrondensityclosetothe

cathodeinaregionoftenknownasthecathodefallorCrookesdarkspace.

3|DCGLOWDISCHARGE

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Electronsarelosttothewallduetorandommotionwithinafewmeanfreepathsof

thewallandgainedduetosecondaryemissioneffects,resultinginthefollowing

boundaryconditionfortheelectronflux:

–n⋅Γe

=

1

γ

-p（Γp⋅n）

-νe,thne

–

2

p

（1）

andtheelectronenergyflux:

–n⋅Γε

=

5

ε

--νe,thnε–pγp（Γp⋅n）

6

p

（2）

Thesecondtermontheright-handsideofEquation1isthegainofelectronsduetosecondaryemissioneffects,γpbeingthesecondaryemissioncoefficient.ThesecondterminEquation2isthesecondaryemissionenergyflux,εpbeingthemeanenergyofthesecondaryelectrons.Fortheheavyspecies,ionsarelosttothewallduetosurfacereactionsandthefactthattheelectricfieldisdirectedtowardsthewall:

–⋅jk=MwRk+MwckZμk（E⋅n）[Zkμk（E⋅n）>0]

n

PLASMACHEMISTRY

Argonisoneofthesimplestmechanismstoimplementatlowpressures.The

electronicallyexcitedstatescanbelumpedintoasinglespecieswhichresultsina

chemicalmechanismconsistingofonly3speciesand7reactions:

TABLE1:

TABLEOFCOLLISIONSANDREACTIONSMODELED

REACTIONFORMULATYPE

Δε（eV）

1e+Ar=>e+ArElastic0

2e+Ar=>e+ArsExcitation11.5

3e+Ars=>e+ArSuperelastic-11.5

4e+Ar=>2e+Ar+Ionization15.8

5e+Ars=>2e+Ar+Ionization4.24

6Ars+Ars=>e+Ar+Ar+Penningionization-

7Ars+Ar=>Ar+ArMetastablequenching-

4|DCGLOWDISCHARGE

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InaCCPreactortheelectrondensityanddensityofexcitedspeciesisrelativelylowso

stepwiseionizationisnotasimportantasinhighdensitydischarges.Inadditionto

volumetricreactions,thefollowingsurfacereactionsareimplemented:

TABLE2:

TABLEOFSURFACEREACTIONS

REACTIONFORMULASTICKINGCOEFFICIENT

1Ars=>Ar1

2Ar+=>Ar1

Whenametastableargonatommakescontactwiththewall,itrevertstotheground

stateargonatomwithsomeprobability（thestickingcoefficient）.

ResultsandDiscussion

Theelectricpotential,electrondensityandmeanelectronenergyareallquantitiesof

interest.Mostofthevariationineachofthesequantitiesoccursalongtheaxiallength

ofthecolumn.Figure2plotstheelectrondensityinthecolumn.Theelectrondensity

peaksintheregionbetweenthecathodefallandpositivecolumn.Thisregionis

sometimesreferredtoasFaradaydarkspace.Theelectrondensityalsodecreases

rapidlyintheradialdirection.Theiscausedbydiffusivelossofelectronstotheouter

5|DCGLOWDISCHARGE

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wallsofthecolumnwheretheyaccumulateasurfacecharge.Thebuildupofnegative

chargeleadstoapositivepotentialinthecenterofthecolumnwithrespecttothewalls.

Figure2:

Surfaceplotofelectrondensityinsidethecolumn.

InFigure4theelectricpotentialisplottedalongtheaxiallengthofthecolumn.Notice

thatthepotentialprofileismarkedlydifferentfromthelineardropinpotentialwhich

resultsintheabsenceoftheplasma.Thestrongelectricfieldinthecathoderegioncan

leadtohighenergyionbombardmentofthecathode.Heatingofthecathodesurface

6|DCGLOWDISCHARGE

SolvedwithCOMSOLMultiphysics5.2

occurswhichmayinturnleadtothermalelectronemissionwhereadditionalelectrons

areemittedfromthecathodesurface.

Figure3:

Plotofelectron“temperature”alongtheaxiallengthofthepositivecolumn.

7|DCGLOWDISCHARGE

SolvedwithCOMSOLMultiphysics5.1

Figure4:

Plotoftheelectricpotentialalongtheaxiallengthofthepositivecolumn.

8|DCGLOWDISCHARGE

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Figure5:

Plotofelectricpotentialalongtheaxiallengthofthepositivecolumn.

Figure6:

Plotoftheelectrontemperaturealongtheaxiallengthofthepositivecolumn.

9|DCGLOWDISCHARGE

SolvedwithCOMSOLMultiphysics5.1

Figure7:

Plotoftheelectrondensityalongtheaxiallengthofthepositivecolumn.

Figure8:

Plotofthenumberdensityofexcitedargonatoms.

10|DCGLOWDISCHARGE

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Figure9:

Plotofthenumberdensityofargonions.

Reference

1.M.A.LiebermanandA.J.Lichtenberg,PrinciplesofPlasmaDischargesand

MaterialsProcessing,JohnWiley&Sons,2005.

ApplicationLibrarypath:

Plasma_Module/Direct_Current_Discharges/

positive_column_2d

ModelingInstructions

FromtheFilemenu,chooseNew.

NEW

1IntheNewwindow,clickModelWizard.

11|DCGLOWDISCHARGE

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MODELWIZARD

1IntheModelWizardwindow,click2DAxisymmetric.

2IntheSelectphysicstree,selectPlasma>DCDischarge（dc）.

3ClickAdd.

4ClickStudy.

5IntheSelectstudytree,selectPresetStudies>TimeDependent.

6ClickDone.

GEOMETRY1

Rectangle1（r1）

1OntheGeometrytoolbar,clickPrimitivesandchooseRectangle.

2IntheSettingswindowforRectangle,locatetheSizeandShapesection.

3IntheWidthtextfield,type0.05.

4IntheHeighttextfield,type0.4.

Rectangle2（r2）

1OntheGeometrytoolba