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To cite this version:

Monrolin, Nicolas and Plouraboué, Franck Multi-scale two-domain numerical modeling of stationary positive DC corona discharge/drift-region coupling. (2021) Journal of Computational Physics, 443. 110517. ISSN 0021-9991 .

Official URL:

https://doi.org/10.1016/j.jcp.2021.110517

Open Archive Toulouse Archive Ouverte

Multi-scale two-domain numerical modeling of stationary positive DC corona discharge/drift-region coupling

Nicolas Monrolin

^b

, Franck Plouraboué

^a,∗

aInstituteofFluidMechanicsofToulouse(IMFT),ToulouseUniversity,CNRS,INPT,UPS,Toulouse,France bÉcoleNationaledel’AviationCivile(ENAC),Toulouse,France

a b s t ra c t

Keywords:

Coronadischarge Multi-scalemodeling Multi-domaincoupling Lagrangemultipliers Asymptoticanalysis Ionicwind

Coronadischargemodelingmostlyreliesontwo,mostlydistinct,approaches:high-fidelity, numerically challenging, unsteady simulations having high-computational cost or low- fidelitysimulationsbasedonempiricalassumptionssuchasconstantelectricfieldatthe emitter electrode.For the purpose of steady discharge currentpredictions, high-fidelity models are very costly to use whilst empirical models have limited range of validity owing the subtle use of tuned parameters. We propose an intermediate approach: an asymptoticmulti-scale/two-domainnumericalmodelingbasedupongeneralizingprevious asymptotic axi-symmetrical analysis [1,2]. We show how the initial elliptic (electric potential), hyperbolic (charge transport), non-local (photo-ionization) problem can be formulated intotwolocalproblemscoupledbymatchingconditions.Theapproachrelies onamultipoleexpansionoftheradiativephoto-ionizationsourceterm(intwodimensions for cylindrical emitters). The analytical asymptotic matching conditions derived in [2]

result influxcontinuityconditionsattheboundaryofthe twodomains.Thesecoupling conditionsareenforcedbyLagrangemultipliers,withinavariationalformulation,leading toahierarchyofnon-linearcoupledproblems.Theproposedapproachisbothmonolithic andtwo-domains:twoasymptoticregions,aninner-oneassociatedwithcoronadischarge, and an outer-one, the ion drift region. Numerical convergence and validations of the finite element implementation is provided. A comparison with various experimental results convincingly demonstrate the applicability of the method, which avoids tuning parameters dedicated to each specific configuration, but, on the contrary, exclusively reliesonknownand measurablephysicalquantities(e.g.,ionmobilities,photo-ionization coefficient,ionizationelectricfield,Townsenddischargecoefficient,etc...).

1. Introductionandcontext

DC-coronadischargeisacomplexphenomenonarisingwithinagaswhentheelectricfieldreachesathresholdforwhich electroncollisions cascadeandproducepositiveandnegative ionchargedmoleculesinsome confinedregions.Thesecon- finedregionsarecalled‘corona’or‘glowingregions’whereacoldplasmaisset-upandejectsunipolarchargesinasecond region calledthe‘driftregion’ inthegaswhere electronsdie-away. Since theseunipolarchargescanfurther collidewith neutralgasmoleculesinthe‘driftregion’,undertheactionofanappliedelectricfield,theycanthengeneratenetmomen-

*

Correspondingauthor.

E-mailaddresses:[email protected](N. Monrolin),[email protected](F. Plouraboué).

tumandproduceionicwindthere.Aback-couplingbetweenthesetworegionscomesfromtheactionofphoto-ionization.

Lightisindeedemittedfromthe‘glowingregion’intothe‘driftregion’andproducesasmallamountofsecondaryelectrons inathinzoneofthe‘driftregion’,ofcrucialimportancetosustainthecoldplasma creation.Thisverybriefandsynthetic descriptionofDC-corona dischargedepictsits complexity,so thatits modeling raiseschallenges.Ifone addsthe factthat the time-scalesassociatedwithcharge creation andelectro-driftcan be very different, onerealizes that thedynamicsof corona, (e.g. associatedwithso-calledstreamers),isevenmorechallenging[3–9].Furthermorethedetailedphysics ofthe modeling associatedwiththevariousnon-stationaryaspectsofcoronarenderitscomparisonwithexperimentalresults(e.g.

the so-calledTrichelpulses) delicate,eitherusing commercialcodes [10] ormore elaborated ones[11], albeitfeasible in 2D [12].Nevertheless, atintermediate voltages, above theinception voltage,a steady-state can be sustained, the model- ing of which isstill difficult when coupled withdrift-region. Here,we focusour interest on thenumerical computation ofsteady-state DC-coronadischarge whichisalreadya difficultissue,as,forexample,studied in[13] for thedrift region orin[14–17] for thecorona region.Fromthe applicativeview-point, themodelingofsteadycorona discharge isrelevant inmanyapplicationssuch aselectrostaticprecipitors[18], EHD(Electro-Hydro-Dynamic)gas pump[19], particleanalyzer [20], miniaturized heat cooler [21,22] andxerography, i.e. electrophotography. In theseapplications, manyconfigurations involvecoronadischargesgeneratedfromwiresintoacavity,thewallofwhichareplacedatreferencepotential.Inthese casestheKaptzov assumption(which iscorrectforawireina infinitedomain,orcentered intoanaxi-symmetriccavity) mightoversimplifytherealelectricfieldatemitters,sothatamoreelaboratedapproachtakingcareofthecoronadischarge physicsisnecessary.

Historically,manyapproacheshavetried to avoidthemodelingofthe completecouplingbetweenglowingregion and drift region. Mostof theseapproaches reliedon experimental measurements, providing some approximate expression of theelectricfieldandthechargedensityattheedgeoftheglowingregion.Moreprecisely, theseapproachesaregenerally calibratedforairatatmosphericpressure,andprovidethe current-potentiallaw I−φ neededtoset thecharge distribu- tion andtheelectricfield atthefrontier betweenglowinganddrift regions. Forasingle cylindricalelectrode (calledthe emitter), insidea finiteco-cylindricalgeometryTownsend’s lawhasbeensuccessfullyused[23–25]. Consideringnonaxi- symmetricdriftregionproblemswhilstusingaxi-symmetricchargeinjectionsand/orelectricpotential(suchasPeek’slaw) hasalsobeenused(e.g. inpoint/planeconfiguration[26],cylinder/cylinderconfigurations[27],etc...)whichmightbeafair approximationissomecases.Nevertheless,ingeneralnonaxi-symmetricconfigurationsnotonlytheparametersofcurrent- potential law(and/or charge injection-electric field law) have to be adapted, but alsothe hypothesis ofaxi-symmetrical emitted chargeshas to be reconsidered. Forexample, based upon experimental measurements [28,29] have shownthat the current-potentiallawis modified inthepresence ofexternal airflow inthedrift region ina tip/planeconfiguration.

Morerecently,themodificationofthechargeinjectionboundaryconditionshasalsopermittedtoreproduceexperimental measurementsinapoint-to-ringconfiguration[30].Otherexperimentalevidencescallsfornonaxi-symmetricchargesinjec- tions,suchastheobservationsoflightintensityvariations(inDielectricBarrierDischarge,i.eDBD,configurations)resulting from thegas flow effects [31], asrecently confirmedby [32]. Inthis context [33] has recentlyproposed to use a Robin boundaryconditionforthechargedensityninjectionatthedriftregionedge,,n(x)|=β(E(x)|−^Ep).

The boundary condition associated withthedrift region is clearly resulting fromtheinteraction betweenthe various fields (electric potential, ions,electrons) betweenthe glowing anddrift regions. This is whymany modeling approaches haveconsideredacoupledmulti-domainor‘hybrid’approachesinordertomodelthephysicsofDCcorona[34–37].Amajor issueinthisareaistoforeseearelevantmodelingusingphysicalparametersonly, (kineticallybasedparameters available fromopendata-bases)butavoidingtheneedofdedicatedphenomenologicalparameters.Effortstowardthisdirectionhave been addressedusingmulti-domain approacheswithin a partitionedstrategy, iteratively seekingforthe solution ineach sub-domain withafixedpoint method.Nevertheless,inmanyproblemsasimilarpartitionedstrategy isknowntobe less stablethanamonolithicone.Monolithicfully-coupledapproacheshavealsoindeedbeenpursuedtonumericallycompute the non-linear elliptic/hyperbolic problemassociated withelectricpotential, electrons, ions charge creations, electro-drift andsecondaryphoto-ionization (Cf[38,39] amongothers).Thesemonolithicfully-coupledapproachesmightbeinteresting in order to get physicallydetailed, chemical composition of corona [39]. Theyhave been mainly applied to very simple coronageometries,sincethenumericalcomplexityofthecompletephysicsisdifficulttoaddressincomplexdomains.

Inthispaperweproposean alternativemethodbothmonolithicandtwo-domain,derived fromtheasymptoticanalysis of the fully-coupled problem, producing two asymptotic regions, an inner-one associatedwith corona discharge, andan outer-one, thedrift region. Thismethod generalizes theanalytical axi-symmetrical analysisperformedin [2] to domains having any regular shapesfor which no analytical solution is available. The approach combines the advantage of being stableandefficientsoastobeabletoaddresspotentiallycomplexdomains inthedriftregion.Thenumericalapproachis alsoinspiredbydomaindecompositiontechniques[40,41] usingLagrangemultipliersdefinedattheinterfacebetweentwo domains to matchsuitable boundary conditionsbetweenthe variousfields involved.The ideabehind ourapproach isto gain understandingonthecorona dischargemechanisms soasto set-upanasymptotichierarchyofmaincoupledeffects, whilstretrievingirrelevantones.

The paperisorganizedasfollows.Section 2.1describestheconstitutivemodel,its underlyingphysics,thegeometrical settingandcontext aswell asits dimensionlessformulation. Section 3discussesits asymptoticformulationanddevelops ontheresultingmulti-scale/two-domainstrategy.Section4providesthenumericaldetailsoftheimplementation,thecon- vergencestudy,andvalidationtest-casescombiningpreviousanalytic,numericalandexperimentalresults.Finallysection 5 showcasessomeillustrationsandcomparisonwithpreviouslypublishedexperimentalresults.

Fig. 1.(a)Schematicrepresentationofthepositivecoronadischargeproblem:collectorsizeLismuchlargerthanemitterdiametera(inblack).Theorigin ofthepositionvectorrin(xy)planeistheemittercenter.(b)Thetwo-domainapproachofsection3:=1∪2,boundaryistheinterfacebetween 1and2.∂^cand∂^e arethesurfaceofthecollectorandtheemitterrespectively.

2. Coronadischargemodel 2.1. Constitutiveequations

We considera positiveDC-corona discharge arising intoan infinitetwo-dimensional configurationsketchedinFig. 1a.

Eventhough,thegeneralideasandmethodproposedinthispapermightbegeneralizedto3D,thereareherebydistinctly derivedin2Dfornotationsandmethodologicalsimplifications.

As mentionedintheintroduction,theeffectivefluid modelofthepositiveDCcorona isconsidered.The productionof positiveions,electronsandnegativeions(respectivedensitynp,ne andnn)isgovernedbytheimpactionizationcoefficient

α

andtheattachmentcoefficient

η

.TheionizationcoefficientdependencywithelectricfieldfollowsthestandardTownsend form

α = β

exp

(−

Ei

/

E

),

(1)

whereβ andEi aretwophysicalparameterswhichdependsonthegascomposition, thermodynamicconditionsandthey aresupposedtobeknown.Ei istheionizationelectricfield,i.e., thefieldbeyondwhichthecoronadischargelightens.The impactionizationcoefficient

α

isassumedtovanishatlowelectricfieldintensityE= ∇

ϕ

^{.Thecompletesetofequations} describingtheelectricpotential

ϕ

,electrondensityne,positiveandnegativeionchargesdensitiesnp andnn is

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎩

∇

²

ϕ =

^e

0

(

ne

+

nn

−

np

),

∇ ·

jp

= α

je

+

S

,

∇ ·

^je

= ( α − η )

j_e

+

^S

,

∇ ·

^jn

= η

j_e

,

(2)

where e is the elementary charge, je=

μ

ene∇

ϕ

, jp= −

μ

pnp∇

ϕ

,and jn=

μ

nnn∇

ϕ

are the local fluxesof the electron, positive and negative ion charges, je= |^je|^{, associated with their respective mobility (i.e.}

μ

e forthe electrons,

μ

p,

μ

n for the positive andnegative ion charges).

α

is theimpact ionization Townsendcoefficient (1) and

η

is theattachment coefficient.Inthefollowingweintroducenotation

α

_{e f}

₌ α ₋ η ,

(3)

andconsider that

α

e f isa known smooth functionof E. Furthermore,Appendix C showsthat bothcoefficients

α

and

η

(and thus

α

e f) havean exponentialdependencewiththe inverseofthelocalelectricfield, similarto (1) thatwe willbe subsequentlyused.Finally,Sisthesourcetermassociatedwithsecondaryionizationwhichisonecomplexaspectofcorona dischargemodeling.Notethat,inthisformulation,photo-ionizationprovidesanequallybalancedsourcetermforelectrons andpositive charges,sinceit both generatesan electronanda positive chargeout ofa neutralmolecule. Such abalance isnotalways takenintoaccount,butthispointwillbediscussedfurtherinthenextsection.Evenifsecondaryionization isvery smallcomparedtothe impactionization,it isnecessarytoexplain theonsetandtosustain thedischarge. Photo- ionizationisthesourceofsecondaryelectronandresultsfromanon-localcreationcomingfromaconvolutionofthecharge fluxwitharadiativekernel.In3D,usingpositionvectorR=^r+^zez builtfromhorizontalpositionrandverticaldistance along z,

S

(

R

) = γ

g

(

R

,

R

) ( α (

R

) − η (

R

))

j_e

(

R

)

d³R

,

(4) where,again, je(r)= |^je(r)| ^{and S}(R)is thenumberof photo-ionizingevents atpositionR per unit time andvolume.

Thecoefficient

γ

isthesecondaryelectronefficiency,identicaltotheoneintroducedbyZheng [42],adimensionlesssmall quantity,i.e.

γ

1, toaccount for thephoto-ionization cross-section andprobability as in[43,44]. Thephoton radiative kernel mayhave differentforms [45,1,46,47]. We hereby derive a generaltheory which can be adapted to any(regular) formofkernel.Inthispaper,werestrictourattentionto2Dproblemsbeingtranslationalyinvariantalongz.Inthiscontext, we derive in Appendix B a specific 2D kernel froma well-established3D one [43,44]. Hence, inthe hereby considered contextcylindricalcoordinatesareusedand(4) reducesto

S

(

r

) = γ

G

(

r

,

r

) ( α (

r

) − η (

r

))

j_e

(

r

)

d²r

,

(5) wherethephoto-ionizationsourceS(r)alsobeinginvariantalongz,becauseitonlydependsonr.Forthesakeofsimplicity, in the following,each time we willspecify the 2D domain ofintegration, we will omitthe differential incrementin all integrals,i.e. (5) willbedenoted

S

(

r

) = γ

G

(

r

,

r

) ( α (

r

) − η (

r

))

je

(

r

).

(6)

The boundaryconditionsassociatedwithproblem(2) arebased uponnotationsofFig.1a. Theelectricpotential

ϕ

fulfills Dirichletboundaryconditionsontheelectrodes,withahightension

ϕ

a appliedatemitterandareferencezeropotentialat collector,i.e.

ϕ |

∂^e

= ϕ

a

, ϕ |

∂^c

=

⁰

.

(7)

Bothn_p andn_e fulfill apurelyhyperbolicproblemsothatoneupstream boundarycondition foreach fieldisneeded.Ina positive coronadischarge, forpositivechargestravelingalongtheelectricfield linesfromtheemittersurface ∂^e toward thecollectorone∂^c,zeropositivechargesfluxissetattheemitter

jp

·

n

|

∂^e

=

0

.

(8)

Symmetrically,fortheelectronsandnegativechargestravelingagainsttheelectricfield,azerofluxinletboundarycondition issetatthecollector

j_e

·

ⁿ

|

∂^c

=

⁰

,

(9)

sothatitisassumedthatnoelectronsareinjectedatcollector,whichmightresultinnoelectronsatall.Butafewsecondary electronsarecreatedbyphoto-ionizationnearbytheemitterthatwillfeedthecoronadischarge.Thissimplifiedframework ismeaningfullsinceweassume thattheelectricfield attheemitterismuchlargerthantheone nearbycollectorsothat thegeneratedphoto-ionization sourcetermismuchsmallerthere,thusofnegligibleeffect.

2.2. Dimensionlessformulation

Thefirstmainphysicalparametersassociatedwiththecoronadischargearetheappliedelectricpotentialdifference

ϕ

a, betweentheemitterandthecollector,beingatdistanceLapart,witharesultingappliedelectricfieldmagnitudeof

ϕ

a/L.It isinterestingtocomparethisappliedfieldtothe“internal”onedefinedbytheelectricionizationfieldE_iusedinTownsend relation(1).Fromthiscomparisonasmallasymptoticparameter

ε

isdefinedasin[1,2]

ε ₌ ϕ

_a

L E_i

.

(10)

This ratiobeingsmall indicates that theapplied electric field issmall comparedto the ionization field ofthe discharge.

Dimensionlessvariablesarechosenfromtheexternal(outerordriftregion)lengthreferenceLby

ˆ

r

=

^r

L

, ϕ ˆ ₌ ϕ ϕ

a

,

n

ˆ

_k

=

^nk

n_k

,

a

ˆ =

^a

L

,

(11)

withk≡^e,p,n forelectrons, positiveionsandnegative ionsrespectivelyanda the emitterradius.Thereferencenumber densityn_kis

n_k

=

0

ϕ

a

eL²

μ

p

μ

k

.

(12)

NotethatcontrarytoDurbin&Turyn[1] wedifferentiatetheadimensionalizationforionsandelectronssothatnˆe∼^O(1) inthecoronaregion1 andⁿˆp∼^O(1)inthedriftregion2.Thisiswhythesmallparameterδμ=ⁿe/n_p=

μ

p/

μ

e later- onappears in(25).δ_μ typicallytakesvaluessmallerthan10⁻² inair.Inthefollowing,we alsouseinner(coronaregion) variablescaling

R

ˆ =

^r

L

≡

¹

^ˆ

^r

^,

⁽¹³⁾

so that rˆ=

R.ˆ Definingouter non-dimensional gradient ˆ∇ ≡∂_rˆ, andinner onesas ˆ∇Rˆ ≡∂_Rˆ also leadto ˆ∇ = ¹ˆ∇Rˆ. It is interesting to mentionthat, here,the chosen referencenumberdensityn_k differs fromprevious contributions [1,2] since it doesnot containtheelectriccurrent I.Thischoiceisjustified becausethetotalcurrent I isapriori unknown,butwas takenascontrol parameter foreasiertheoretical derivations in[1,2]. Sincethe purposeofthiscontribution isto provide a numericalformulationbasedonknownimposed parameters, thecurrentbeingoneresultofthecomputation, we built n_k on known parameters. Doing so, the non-dimensional equation for the electric potential, will not contain unknown parameter(suchasdimensionlesscurrentdenoted J in[1,2]).Usingdimensionlesselectricfieldin(1),asin[1],thereaction coefficientsscaleasfollows

ˆ α

ε ⁼

^L

α = β ˆ ε

^exp

−

¹

ε

E

^ˆ

,

(14)

ˆ η

ε ⁼

^L

η ,

(15)

ˆ α

e f

ε ⁼

^L

α

e f

≡

L

( α − η ),

(16)

withβˆ=βL

ε

,and Eˆ = | ˆ∇ ˆ

ϕ

|^{. Both}

α

and

η

dimensionbeingthe inverseofareferencelength-scale, (14)-(16) statethat, thislength-scaleistheinnerone L.Forthesakeofbrevity,inthefollowingweuse

α

ˆe f = ˆ

α

_{− ˆ}

η

astheeffectiveionization coefficient.

Letusnowconsiderthenon-dimensionalizationofthephoto-ionizationterm(5).First,itisimportanttomentionthat, since the convolution integral arises over =1∪2, it can be decomposed into two distinct contributions fromthe corona dischargedomain1 andthedriftdomain2.Inthesecontributions,sincethereferencelength-scaleis Lin1 (resp.Lin2),theelectricfieldrespectivelyscalesasE=^ϕa_LEˆ in1 (resp.E=^ϕ_L^aEˆ in2).Thus,usingpreviouslydefined non-dimensionalizationandparticularly(14)-(16) in(5) leadsto

S

(

r

) =

μ

en_e

ϕ

a

L²

γ

²

⎛

⎜ ⎝

1

G

(

r

,

r

) α ˆ

_{e f}

(

r

) ˆ

j_e

(

r

)

d²r

+

2

G

(

r

,

r

) α ˆ

_{e f}

(

r

)ˆ

j_e

(

r

)

d²r

⎞

⎟ ⎠ .

(17)

Then, one needs toconsider thenon-dimensionalization ofthehereby considered 2D photo-ionizationkernel G.In most contributions, photo-ionization kernels g(R) are discussed anddefinedin 3D, withR²= |^r−^r|²+^{z2 the3D Cartesian} distance,z beingthedirectionorthogonaltotheherebyconsideredplane.AsdetailedinAppendixB,therelationbetween g(R)andG(r,r)≡^G(|^r−^r|)being

G

( |

^r

−

^r

| ) =

R g

(

R

)

4

π

R^2dz

.

(18)

Then,non-dimensionalizationofkernelg(R)^{leadsto g}(R)= ˆ^g(R)/^{L(CfAppendixBformoredetails),andfrom(18)} G

( |

^r

−

^r

| ) =

¹

L²

R

ˆ

g

4

π

_R

ˆ

^2d

ˆ

z

=

¹

L²G

ˆ ( |ˆ

^r

− ˆ

^r

| ).

(19)

Fromusing(19) in(17) leadsto

S

(

r

) =

μ

en_e

ϕ

a

L²

γ

⎛

⎜ ⎝

ˆ1

G

ˆ (

r

, ε

R

^ˆ

) α ˆ

e f

(

R

) ˆ

_je

(

R

)

d²_R

ˆ

+

¹

ˆ2

G

ˆ (

r

,

r

) α ˆ

e f

(

r

) ˆ

_je

(

r

)

d²r

ˆ

⎞

⎟ ⎠ ,

(20)

wherewehavenowre-scaledcoordinatesinthecoronausinginnervariable Rˆ (13),anddefiningˆ1beingadimensionless (order O(1))domain1.Now,realizingthatthesecondtermof(20)’sr.h.s. issmallbecauseboththeTownsendcoefficient

ˆ

α

andtheattachmentterm

η

ˆ decayasexp(−¹/

)inregionˆ2,sodoes

α

ˆe f from(3) and(14)-(16),dominatingoverany algebraicpowerin ,onegets,

S

(

r

) =

μ

en_e

ϕ

a

L²

γ

⎡

⎢ ⎣

ˆ1

G

ˆ (ˆ

r

, ε

R

^ˆ

) α ˆ

e f

(

R

ˆ

) ˆ

j_e

(

R

ˆ

)

d²R

ˆ

+

^O

exp

(−

1

/ )

⎤

⎥ ⎦ .

(21)

Sothat,onecanthendefinethenon-dimensionalphoto-ionizationkernel ˆSfromS=^μe_Lⁿ2^eϕa

γ

^ˆS,i.e.

S

ˆ (ˆ

r

) =

ˆ1

G

ˆ (ˆ

r

, ε

R

^ˆ

) α ˆ

e f

(

R

ˆ

) ˆ

j_e

(

R

ˆ

) ≡

ˆ1

G

(ˆ

r

, ε

R

^ˆ

) α ˆ

e f

ˆ

j_e

ˆ

R

.

(22)

Then,amultipoleasymptoticexpansionof(22),togetherwiththeformof(19) reads, S

ˆ (ˆ

r

) =

^G

(ˆ

r

)

ˆ1

α ˆ

_{e f}

^ˆ

je

Rˆ

+ ε _∇

G

(ˆ

r

) ·

ˆ1

α ˆ

_{e f}

^ˆ

je

RˆR

ˆ

+

^O

(

²

),

(23)

S

ˆ (ˆ

r

) =

S

ˆ

⁰

(ˆ

r

) + ε ˆ

S¹

(ˆ

r

) +

^O

(

²

),

(24)

neglecting quadrupolar O(

²)corrections. Using referencecharge density(12),outer dimensionless variable^rˆ ^{(13) in(2)} whilstusingnon-dimensionalization(21),leadstothefollowingdimensionlessdriftregionformulation

ˆ∇

²

ϕ ˆ = −(

n

ˆ

_p

− δ

μn

ˆ

_e

− ˆ

ⁿn

),

(25)

ˆ∇ · ˆ

jp

= α ˆ

^ˆ

^je

⁺ γ

S

^ˆ (ˆ

r

),

(26)

ˆ∇ · ˆ

^je

= α ˆ − ˆ η

^ˆ

^je

⁺ γ

S

ˆ (ˆ

r

),

(27)

ˆ∇ · ˆ

jn

= η ˆ

^ˆ

^je

^,

⁽²⁸⁾

where ˆj_e= |ˆ^je|= ˆⁿeEˆ,ˆj_p= |ˆ^jp|= ˆⁿpE,ˆ andˆj_n= |ˆ^jn|= ˆⁿnEˆ.

It is interesting to note that the non-dimensionalization leading to (25) produces a smaller contribution of electron density compared to positive charge in the drift region. The main reason is based on flux considerations: the electron currentdensityattheemittershouldbalancetheioncurrentdensityatcollector.Theratiobetweenthemaximumnumber densityofunipolarpositiveionsn_p andthemaximumnumberdensityofelectronsn_e isthengivenbythemobilityratio δμ.Onemightquestionthishierarchyinthecoronaregion1 sincetheionnumberdensitydecreasesdrasticallynearthe emitter surface:nˆp∼^O(1)indrift regionbutnˆp=^{0 at theemitterwhilst} nˆe∼^O(1)inthe coronaregion andnˆe=^{0 at} the collector.Inpractice thisisnot aconcern sincein thecorona region,a re-scaling ofthecoordinates produces O(

²) smallterminfrontof(25)’sr.h.s., leadingtonegligiblecharge effectatleading orderintheelectrostaticproblem(34). In anutshell,thespacechargeplaysan importantroleonlyinthedriftregionandisstronglydominatedbythepositiveions charge,there.Lastbutnotleast,itisimportanttorealizethatthenegativechargesconcentrationdonotplayanactiverole in theproblem. First, in thecorona region, negative charges doesnot contribute to thepotential (asanyother charges), fortheaforementionedreasonofhavinganegligibleimpactonelectrostatic problem(34).Builtintothecoronaregionby attachmentcoefficient

η

fromelectronflux,negativechargesonlymigrate totheemittersoastoproduce,togetherwiththe electrons, thenecessary(negative)charge flux balancetothepositive chargesdriftingaway fromit.Secondly,inthe drift region,theonlysource termfornegative chargesin(28) istheproduct ofattachment coefficient

η

withelectronflux.As discussedjustafter(20),

η

decayasexp(−¹/

)inthedriftregion,leadingtonegligibleproductionofnegativechargesflux, thusleadingtonegligiblenegativeionnumberdensitythere.Thisiswhy,inthesequel,negativechargesarenotconsidered.

Dimensionlessproblem(25)-(28) iscomplementedwithdimensionlessboundaryconditions

ˆ

ϕ |

_∂ˆe

=

¹

, ϕ |

_∂ˆc

=

⁰

,

(29)

ˆ

j_p

·

ⁿ

|

_∂ˆe

=

⁰

,

(30)

and

ˆ

j_e

·

ⁿ

|

_∂ˆc

=

⁰

.

(31)

Thus (25)-(28) associatedwithboundaryconditions (29)-(31) andsource term(23) representsa couplednon-linearnon- local system of equations.In the following we show how a multi-scale approach can be used to transform it into two coupledlocalproblems,withnotationsprovidedinFig.1b.

3. Multi-scaleasymptotic expansion

We now seek for a regular asymptotic expansion with respect to parameter

of the problem, neglecting O(δμ), O(^exp(−1/))aswellasO(

²),butkeeping O(

γ

)andO(

)terms,i.e.

( ϕ ˆ ,

n

ˆ

_p

,

n

ˆ

_e

, α ˆ , α ˆ

e f

) = ( ϕ ˆ

⁰

,

n

ˆ

⁰_p

,

n

ˆ

⁰_e

, α ˆ

⁰

, α ˆ

_{e f}⁰

) + ε ( ϕ ˆ

¹

,

n

ˆ

¹_p

,

n

ˆ

¹_e

, α ˆ

¹

, α ˆ

_{e f}¹

) +

^O

²

, δ

μ

,

^exp

(−

¹

/ )

.

(32)

Wealsosubsequentlydefine

α

ˆ⁰_{e f}≡ ˆ

α

e f(ˆE⁰),

α

ˆ⁰≡ ˆ

α

(Eˆ⁰),whilst,obviously,Eˆⁿ= | ˆ∇ ˆ

ϕ

ⁿ|^forn=⁰,1.Furthermore,fromTaylor expandingtheelectricfieldexpansionEˆ= ˆ^E0+

Eˆ¹+^O(

²)in(14),leadsto

α

ˆ = ˆ

α

⁰+

α

ˆ¹+^O(

²)with

ˆ

α

¹

= α ˆ

⁰E

ˆ

¹

(

E

ˆ

⁰

)

²

, α ˆ

_{e f}¹

= ∂ α ˆ

_{e f}⁰

(

E

ˆ

⁰

)

∂

E

ˆ = ˆ α

¹

− ∂ η ˆ

⁰

∂

E

ˆ .

(33)

Since from(3),

α

ˆe f = ˆ

α

− ˆ

η

, whilst alsousingnotation

η

ˆ⁰≡ ˆ

η

(Eˆ⁰).Some explicit relationfor

η

(E) anditsderivative are given in (C.2) and (C.3). In the following, we will index the fields

ϕ

ˆ,Eˆ,nˆe,nˆp by j, j=¹,2 for specifying into which domaintheyfallunder.

3.1. Coronadomain1problem

Atleadingorder,thecoronaproblemreads

ˆ∇

_R²_ˆ

ϕ ˆ

⁰

1

=

⁰

,

(34)

ˆ∇

_Rˆ

· (

n

ˆ

⁰_p1

ˆ∇

_Rˆ

ϕ ˆ

⁰₁

) = − ˆ α

⁰n

ˆ

⁰_e1E

ˆ

⁰₁

,

(35)

ˆ∇

_Rˆ

· (

n

ˆ

⁰_e1

ˆ∇

_Rˆ

ϕ ˆ

⁰₁

) = ˆ α

⁰_{e f}n

ˆ

⁰_e1E

ˆ

⁰

1

.

(36)

Notethat,surprisingly,thereisnomoresourcetermontheright-hand-sideof(34),asopposedtomanyothertwo-region modeling forcorona models alreadyproposed intheliterature, (e.g. [34,35]),some of themnot derived fromasymptotic considerations [36]. This issueis muchmore benignthan what could be though atfirst sight.As a matter offact, since (34) is expressed in internal variable Rˆ whichis stretched upon theexternal one, r,ˆ Rˆ= ˆ^r/

, theresulting re-scaling of theLaplacianappliedonthe right-hand-sideof(25) multiplies itbyan O(

²)term. Thismeans thatthecharge effecton thecoronaregiononlyaddsa verysmallcorrectiontothepotential.Furthermore,takingintoaccountthiscorrectionwhen discarding other O(

²) termsassociatedwiththecouplingbetween1 and2 isnot asymptoticallyconsistent.At order O(

),wehave

ˆ∇

_R²_ˆ

ϕ ˆ

¹₁

=

0

,

(37)

ˆ∇

_Rˆ

· (

n

ˆ

⁰_p1

ˆ∇

_Rˆ

ϕ ˆ

¹₁

+ ˆ

n¹_p1

ˆ∇

_Rˆ

ϕ ˆ

⁰₁

) = − ˆ α

⁰n

ˆ

_e⁰_1E

ˆ

¹

1

− ˆ α

⁰n

ˆ

¹_e1E

ˆ

⁰

1

− ˆ α

¹n

ˆ

⁰_e1E

ˆ

⁰

1

,

(38)

ˆ∇

_Rˆ

· (

ⁿ

ˆ

⁰_e1

ˆ∇

_Rˆ

ϕ ˆ

¹₁

+ ˆ

ⁿ¹_e1

ˆ∇

_Rˆ

ϕ ˆ

⁰₁

) = ˆ α

_{e f}⁰ⁿ

ˆ

⁰_e1E

ˆ

¹

1

+ ˆ α

_{e f}⁰ⁿ

ˆ

_e¹₁E

ˆ

⁰

1

+ ˆ α

¹_{e f}ⁿ

ˆ

⁰_e1E

ˆ

⁰

1

.

(39)

3.2. Driftdomain2problem

Forthepotential andpositive chargesinthedrift domain,atleading order,theelectrostatic (25) and positive charges conservationproblem(26) reads

ˆ∇

²

ϕ ˆ

⁰

2

= −ˆ

ⁿ⁰_p2

,

(40)

ˆ∇ · (

n

ˆ

⁰_p

2

ˆ∇ ˆ ϕ

⁰

2

) = γ

S

ˆ

⁰

,

(41)

becausethe

α

ˆ termisO(^exp(−1/))in2.Atorder O(

),wehave

ˆ∇

²

ϕ ˆ

¹₂

= −ˆ

ⁿ¹_p2

,

(42)

ˆ∇ · (

n

ˆ

⁰_p

2

ˆ∇ ˆ ϕ

¹

2

+ ˆ

ⁿ¹p2

ˆ∇ ˆ ϕ

⁰

2

) = γ

S

ˆ

¹

, .

(43)

Finally,inthefollowing,wewillnotsolvetheelectronprobleminthedriftdomain2,but,fornow,weleaveitasin(27), butforneglectingthecontributionofthe

α

ˆ_{e f} termwhichis O(^exp(−1/)),withoutexpandingitin ,i.e.

ˆ∇ · (

n

ˆ

e2

ˆ∇ ˆ ϕ

2

) = γ

S

^ˆ (ˆ

r

).

(44)

Sincethephoto-ionizationtermˆS(ˆ^r)isevanescent,i.e. exponentiallydecayingalong^rˆ^{from(23),sodoestheelectrondensity} in the drift region. Hence, except for a smallevanescent region ofwidth λ, i.e., a very thinlayer λ/L in dimensionless

Inthefollowing,we willuse∂n≡ ∇ ·^{nfortheprojection ofgradientoperatorto theoutwardnormalofaboundary.}

Thisleadsto

ˆ

n⁰_p

∂

n

ϕ ˆ

⁰

|

∂^e₁

=

⁰

,

(51)

and,

ˆ

n¹_p

∂

n

ϕ ˆ

⁰

_|

_∂^e

1

+ ˆ

ⁿ⁰p

∂

n

ϕ ˆ

¹

_|

_∂^e

1

=

⁰

.

(52)

In 1,nofurther conditionis needed.Toenforceionflux continuity,2 mustbe fedwiththeionflux comingfrom 1.Thisleadstoa“one-way”coupling,i.e.nˆp2 directlydependsonnˆp1 butnotreciprocally

ˆ

jp2

·

ⁿ2

|

= −ˆ

^jp1

·

ⁿ1

|

,

(53)

withagainaminussignbecauseofthenormal.Sotheinletcondition ofboundaryconditionof2 isgivenby1 and nooutletconditionisrequired.

•Symmetrically,fortheelectronsandnegativechargestravelingagainsttheelectricfield,theinletboundaryconditionis setatthecollector:

ˆ

j_e

·

ⁿ2

|

∂^c₂

=

⁰

,

(54)

leadingto,

ˆ

n⁰_e

∂

n

ϕ ˆ

⁰

|

∂^c₂

=

⁰

,

(55)

and

ˆ

n¹_e

∂

n

ϕ ˆ

⁰

_|

_∂^c

2

+ ˆ

ⁿ⁰e

∂

n

ϕ ˆ

¹

_|

_∂^c

2

=

⁰

.

(56)

We assumethat noelectrons are injectedin atthecollector,whichshould resultinnoelectrons atall.But afew secondaryelectronsarecreatedbyphoto-ionizationin2 thatwillfeed1throughtheinterface

ˆ

je1

·

ⁿ1

|

= −ˆ

^je2

·

ⁿ2

|

.

(57)

Thisisagaina“one-way”coupling,sinceˆje1directlydependsonˆje2 andnotvice-versa.Furthermore,photo-ionization in drift domain 2 dependson the ionization rate in 1, in a rather complex way.Thus, given(23) in (44) in 2 domain,leadsto

ˆ∇ · ˆ

^je

= γ

G

(ˆ

^r

)

M₀

+ ε ∇

^G

(ˆ

^r

) ·

^M1

+

^O

(

²

)

,

(58)

withthemulti-polarexpansionassociatedwithherebydefinedmono-polarscalarM0 anddipolarvectorM1

M₀

=

1

α ˆ

_{e f}

ˆ

j_e

R

,

(59)

M₁

=

1

α ˆ

e f

ˆ

j_e

RR

,

(60)

whilst,again,omittingthedifferentialincrementintheintegrals.Insertingexpansion(32) in(59),onefinds

M0

=

^M0⁰

+ ε

M₀¹

+

^O

(

²

),

with (61)

M⁰₀

=

1

α

_{e f}⁰

ˆ

j⁰_e

R

,

(62)

M¹₀

=

1

α ˆ

_{e f}¹n

ˆ

⁰_e

1E

ˆ

⁰

1

+ ˆ α

_{e f}⁰n

ˆ

¹_e

1E

ˆ

⁰

1

+ ˆ α

⁰_{e f}n

ˆ

⁰_e

1E

ˆ

¹

1

R

.

(63)

And,similarly,insertingexpansion(32) in(60) keepingonlytheleadingordercontributiontothedipolarcorrection, M⁰₁

=

1

α ˆ

_{e f}⁰

^ˆ

j⁰_e

RR

.

(64)

Then,(58) reads,

ˆ∇ · ˆ

^je

= γ

G

(ˆ

^r

)

M⁰₀

+ ε

G

(ˆ

^r

)

M₀¹

+ ∂

_ˆrG

(ˆ

^r

)

M⁰₁

·

^er

+

^O

(

²

)

.

(65)

Realizingfrom(32) thattheelectronfluxˆjefollowsthesameregularasymptoticexpansion

ˆ

j_e

= ˆ

^j0e

+ ε ^ˆ

j¹_e

+

^O

( ε

²

).

(66)

Weseektosolve,ateachorder,theelectronfluxcomingfromphoto-ionizationonly.Atleading orderin ,theforcing termdisplaysan axi-symmetricalradialdependence,

ˆ∇ · ˆ

^j0e

= γ

G

(ˆ

r

)

M⁰₀

.

(67)

The solution of (67) ˆj⁰_e can be decomposed into a general(conservative, i.e. divergence-free) contributionˆj⁰_eG anda particularsolutionˆ_j⁰_{e P whosedivergenceequalsthe}right-hand-sidephoto-ionizationtermof(67),i.e.ˆ_j⁰_e= ˆj⁰_eG+ˆj⁰_{e P},and

ˆ∇ · ˆ

j⁰_eG

=

0

,

(68)

ˆ∇ · ˆ

^j0_{e P}

= γ

G

(ˆ

r

)

M⁰₀

.

(69)

Sinceweconsiderno-incomingelectronfromanyother source,thegeneralconservativecontribution,beingunique,is zero,ˆj⁰_eG=^{0.Hence, we are left withfinding the particularsolution} ˆj⁰_{e P}. From theaxi-symmetry ofboth thesource termandthe boundary,we can assume thatˆj⁰_{e P} = ^j0e(r)er andthusdevelop the divergenceoperator incylindrical coordinates, only keeping the radial part. Integratingbetween ˆr (dimensionless radius of boundary ) andinfinity leadsto

j_e⁰

(ˆ

^r

) =

^M₀⁰

γ

¹

ˆ

r

∞ ˆ r

G

(ˆ

^r

)ˆ

^rdr

ˆ =

^M₀⁰

γ

0

(ˆ

^r

),

(70)

with,

γ

0

(ˆ

r

) = γ

¹

ˆ

r

∞ ˆ r

G

(ˆ

r

)ˆ

rdr

ˆ .

(71)

AppendixBprovidesdetailsconcerningGandtheexplicitcomputationof

γ

0.NowconsideringtheorderO(

),inserting (66) in(65),and(32) in(59) onefinds

ˆ∇ · ˆ

^j1e

= γ

G

(ˆ

r

)

M¹₀

+ ∂

_ˆrG

(ˆ

r

)

M⁰₁

·

^er

.

(72)

Thesameconsiderationapplies,atthisorderandthegeneralconservativecontributiontoˆj¹_e isthuszero.Theelectron flux(72) thusresultsfromtwocontributions.Anaxi-symetricone,providedbytheM¹₀term,andadipolaroneresulting from M⁰₁. The first one, is similar to the leading-order in

,having an amplitude M¹₀ instead of M₀⁰. Seeking fora particularsolution,fromtheaxi-symmetryoftheboundary,aswellastheradialdependenceofthephoto-ionization termontheright-hand-sideof(72),onefindsthat,

ˆ

j¹_e

(ˆ

r

) =

^M1₀

γ

0

+

^M0₁

·

^er

γ

1

,

(73)

γ

1

(ˆ

r

) = γ ˆ

r

∞ ˆ r

∂

_ˆrG

(ˆ

r

)ˆ

rdr

ˆ .

(74)

Again,onecanfindanexplicitexpressionfor

γ

1(ˆ^r)using

γ

1

(ˆ

r

) = γ

G

(

r

ˆ

) − γ ˆ

r

∞ ˆ r

G

(ˆ

r

)

d

ˆ

r

.

(75)

Hence,providedfluxatinterface(70) and(73) wefoundtheelectronfluxindomain2 tobe

ˆ

je2

·

ⁿ2

|

= ˆ

^j0e

|

+ ε ^ˆ

j¹_e

|

(θ ) +

^O

( ε

²

),

(76) with,