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Mechanistic model of multi-frequency complex conductivity of porous media containing water-wet nonconductive and conductive particles at various water
saturations
Yuteng Jin, Siddharth Misra, Dean Homan, John Rasmus, André Revil
To cite this version:
Yuteng Jin, Siddharth Misra, Dean Homan, John Rasmus, André Revil. Mechanistic model of multi- frequency complex conductivity of porous media containing water-wet nonconductive and conductive particles at various water saturations. Advances in Water Resources, Elsevier, 2019, 130, pp.244-257.
�10.1016/j.advwatres.2019.06.015�. �hal-02324310�
ContentslistsavailableatScienceDirect
Advances in Water Resources
journalhomepage:www.elsevier.com/locate/advwatres
Mechanistic model of multi-frequency complex conductivity of porous media containing water-wet nonconductive and conductive particles at various water saturations
Yuteng Jin
a, Siddharth Misra
a,∗, Dean Homan
b, John Rasmus
b, André Revil
caMewbourne College of Earth and Energy, University of Oklahoma, Norman, OK, USA
bSchlumberger Technology Corporation, Sugarland, TX, USA
cCNRS-ISTERRE, France
a b s t r a c t
Electricallyconductiveparticles,suchaspyrites,andsurface-charge-bearingnonconductiveparticles,suchasclays,arecommonlypresentinwater-bearingsubsurface formations.Underanexternalelectricfieldgeneratedbyelectromagneticmeasurementtool,theseparticlesgiverisetointerfacialpolarization(IFP)effects,which causesfrequencydispersionofeffectiveconductivityandeffectivepermittivityofthemixturecontainingsuchparticles.TheneglectofIFPeffectscanleadtoinaccurate estimationofpetrophysicalpropertiesofformations,especiallyinclay-andpyrite-richformations.Inthispaper,wedevelopedamechanisticmodelthatcouples surface-conductance-assistedinterfacialpolarization(SCAIP)modelwithperfectlypolarizedinterfacialpolarization(PPIP)modeltoestimateeffectiveconductivity andeffectivepermittivityofhomogeneousformationscontainingbothnonconductiveandconductiveparticlesatvariousfluidssaturations.Themodelisdeveloped basedonthePoisson-Nernst-Planck(PNP)equationsforadilutesolutioninaweakelectricalfieldregimetocalculatethedipolarizabilityoftherepresentative volumecomprisingasingleisolatedsphericalparticleinanelectrolytehost.Thentheeffectivemediumtheoryisusedtodetermineeffectivecomplexconductivity ofthewholemixture.TheresultshowsthattheconductiveparticlesdominatethefrequencydispersionofcomplexconductivityduetoIFPeffectscomparedto nonconductiveparticles.
1. Introduction
Interfacial polarization phenomena (Dukhin et al., 1974; Wong, 1979; Schmuck and Bazant, 2015) influences the migration, accu- mulation, depletion, and diffusion of charge carriers. If neglected, interfacial polarization (IFP) effects will lead to inaccuracy when estimating petrophysicalproperties of formations usingconventional resistivity/conductivity/permittivity interpretation methods (Clavier etal.,1976;Misraetal.,2016a;Zhaoetal.,2016).Someoftheinter- pretationtechniquesforthesubsurfacegalvanicresistivity(laterolog), electromagnetic(EM)induction andEM dielectricdispersionlogs do notconsidertheIFPeffects(Andersonetal.,2007;Corleyetal.,2010), which cause inaccurate estimates for pyrite-rich sedimentary rocks (Altmanetal.,2008)andpyrite-andgraphite-richorganicsourcerocks (Altmanetal.,2008).Althoughinthelastdecade,somepapersincluded IFPeffectinEMinductionlogs(MacLennanetal.,2013),orindielectric modelwhichconsiderscationexchangecapacity(Revil, 2013),there isstillaneedtoinvestigatetheIFPeffect.Recently,forhydrocarbon volumeestimation,Dengetal.(2018)appliedspectralinducedpolar- ization methodto estimate oilsaturation in oil-contaminatedclayey soils.Freedetal.(2018)alsodevelopedaphysics-basedmodelforthe dielectricresponsethat accountsfortheIFP effectdue tothecation exchangecapacityinlow-salinityshalysandsformations.
∗Correspondingauthor.
E-mailaddress:[email protected](S.Misra).
Mechanistic model of the IFP phenomena can improve resistiv- ity/conductivity/permittivity interpretation in clay- and conductive- mineral-rich formations. Tomodel theIFP effect of electrically con- ductiveinclusions,Misraetal.(2016b)appliedPoisson-Nernst-Planck (PNP)equation.Theirmodelpredictionshaveagoodmatchwithlab- oratorymeasurementsonconductive-mineral-bearingmixtures.More- over,severalmathematicalmodelshavebeendevelopedinthefieldsof petrology(Reviletal.,2017),geophysics(Revil,2012;Placencia-Gómez andSlater,2014),biology(GrosseandSchwan,1992;ZhengandWei, 2011),electrochemistry(ChuandBazant,2006)andcolloidalscience (GrosseandBarchini,1992;Grosseetal.,1998),allofwhichfacilitate thestudyofinterfacialpolarizationeffectsarisingfromvariousmecha- nisms.
Inthispaper,wedevelopamodelthatcouplestheinterfacialpo- larizationofuniformlydistributedwater-wetnonconductivespherical grainspossessingsurfaceconductancewithinterfacialpolarizationof uniformlydistributedconductivesphericalinclusionsinredox-inactive conditionsatvariouswatersaturations.Theproposedmodelcanbeap- pliedtoestimateeffectiveconductivityandeffectivepermittivityofho- mogeneousformationscontainingbothconductiveandnonconductive particlesatvariousfluidssaturations.
https://doi.org/10.1016/j.advwatres.2019.06.015
Received4February2019;Receivedinrevisedform25June2019;Accepted27June2019 Availableonline28June2019
0309-1708/PublishedbyElsevierLtd.
Acronyms
EM electromagnetic IFP interfacialpolarization PDE partialdifferentialequations PNP Poisson–Nernst–Planck
PPIP perfectlypolarizedinterfacialpolarization PS PPIP-SCAIP
SCAIP surface-conductance-assistedinterfacialpolarization Symbols
a characteristiclengthofinclusionphase(m) c chargedensityvariation(1/m3)
d netchargedensityvariation(1/m3)
D diffusioncoefficientofchargecarriers(m2/s) Δ(∇2) Laplace’soperator
e Euler’snumber e electricfieldvector
E0 amplitudeoftheelectricfield(V) 𝜀 dielectricpermittivity(F/m)
𝜀0 vacuumpermittivity(8.854×10−12F/m)
𝜀eff effectivedielectricpermittivityofthemixture(F/m) 𝜀r relativepermittivity
f frequency(Hz)
f(𝜔) dipolarizability(dipolarfieldcoefficient) i squarerootof−1
in modifiedsphericalBesselfunctionofthefirstkindofnth order
In modifiedBesselfunctionofthefirstkindofnthorder j currentdensity(A/m3)
kB Boltzmann’sconstant
kn modifiedsphericalBesselfunctionofthesecondkindof nthorder
Kn modifiedBesselfunctionofthesecondkindofnthorder 𝜆 surfaceconductance(S)
𝜆D Debyescreeninglength(m) 𝜇 electricalmobility[m2/(V·s)]
n anintegerreferringtotheorderofthestandingwave solution
N chargecarrierdensity(1/m3)
𝜔 angularfrequencyoftheelectricfield(rad/s) Pf netfreechargedensity(C/m3)
𝑃𝑛0 associatedLegendrefunctionsofthefirstkindofnthor- der
𝜑 electricalpotential(V) 𝜙 volumefraction(%)
q elementarycharge(1.6×10−19C)
𝑄0𝑛 associatedLegendrefunctionsofthesecondkindofnth order
r radialdistancealongthenormaltotheinterface(m) 𝜌 surfacechargedensity(C/m2)
s totaliondensityvariation(1/m3) 𝜎 electricalconductivity(S/m) 𝜎∗ complexelectricalconductivity(S/m)
𝜎eff effectiveelectricalconductivityofthemixture(S/m) 𝜎𝑒𝑓𝑓∗ effectivecomplexelectricalconductivityofthemixture
(S/m)
t time(s)
T absolutetemperature(K)
𝜃 anglebetweenthenormaltotheinterfaceandtheinci- dentexternalelectricfield(°)
Z chargenumber Subscripts
0 attimeequalto0s
c clay
cond conductiveparticles
eff effective
h hostmedium
i inclusionphase j typeofmedium/phase n anintegerreferringtotheorder
̂𝑛 unitvector
ncond nonconductiveparticles
o oil
p pyrite
r relative
s sand
Superscripts
+ positivelychargedcarrier
− negativelychargedcarrier
1.1. Interfacialpolarizationaroundsurface-charge-bearingspherical nonconductiveparticles
Variousmixingmodelshavebeendevelopedtoquantifytheeffects ofvariousinterfacialpolarizationphenomena.Themodelproposedby Schwarz (1962)considers interfacialpolarization(IFP)effect around chargednonconductiveparticles.Itassumesadiffusionofcounterion layermovingalongthesurfaceofthechargedparticlebycalculatingthe potentialoutsidethecounterionlayerasasolutionofLaplace’sequation ratherthanPoisson’sequation.However,thismodelfailstoaccountfor allthebulkdiffusioneffects.Incontrast,Dukhinetal.(1974)concluded thatthemechanismbehindinterfacialpolarizationisthediffusionof ionsinthebulkelectrolytearoundtheparticle.Theywereunabletopro- videanalyticalexpressionsforIFPeffectsintermsofvariousrelaxation parametersduetomathematicalcomplexitycausedbynon-linearityof Dukhinetal.(1974)equation.Thismodel,calledthestandardmodel incolloidalchemistry,doesnotconsidertheexistenceofaSternlayer withmobileions.GrosseandFoster(1987)developedananalyticalso- lutionofIFPeffectbydevelopingasimplifiedmodelofchargednoncon- ductivesphericalparticlesinbulkelectrolyte.Intheirmodel,positive ionsfromthebulkelectrolytecanfreelyexchangewiththepositively chargedcounterionlayerwhilethenegativeionsareexcludedfromthe counterionlayer.ThismodelwasgeneralizedinGrosse(1988)byal- lowingarbitrarychargeinnonsymmetricelectrolytes,assumingfinite surfaceconductivityandconsideringtheentirefrequencyspectrum.
1.2. Interfacialpolarizationaroundsphericalconductiveparticles
Garciaetal.(1985)developedamodelforconductivesphericalpar- ticles with insulating shells(fore.g. oxidizedsurfaceof pyrite)in a conductivemediumwherethediffusiveeffectsplayanimportantrole.
GrosseandBarchini(1992)improvedtheprevioustheoryforinfinitely conductivesphericalparticlesinbulkelectrolytebyconsideringionflow acrosstheinterface.Moreover,incomparisontodielectricmixturefor- mulas,Tunceretal.(2001)appliedafiniteelementmethodoncylinder- likeconductiveinclusionphasetoinvestigatethedielectricrelaxation phenomena.Theirresultshowsthetwomethodsmatchwellatlowin- clusionconcentrations.However,astheconcentrationofinclusionin- creases,mutualinteractionoftheinclusionsbecomessignificant.
2. Methodology 2.1. Assumptions
BoththeSCAIPmodelandPPIPmodelarebasedonthePoisson- Nernst-Planck(PNP)equationsforadilutesolutioninaweakelectrical
Fig.1. Cross-sectionof a nonconductivesphericalinclusionpossessing sur- facechargesurroundedbyanionichostmedium.Theinclusionisnegatively charged,surroundedbyapositivechargedcounterion layer,whichformsa Gouy–Chapmanmodel.Chargecarriersintheionichostmediumarecations, identifiedby“+” symbol,andanions,identifiedby“−” symbol.Thedirection oftheexternallyappliedelectricalfield,e,isidentifiedwithaboldarrownext tothesymbol“e”.Thedirectionofmovementofthechargecarriersintheionic hostmediumisrepresentedbythearrownexttothesymbolofthechargecar- rier.
fieldregime.ByapplyingthePNP equations,we analyzetheEM re- sponseofarepresentativevolumecomprisingasingle,isolatednoncon- ductiveinclusionpossessingsurfacechargeorelectricallyconductivein- clusionsurroundedbyanelectrolyte-saturatedhostmedium(Zhengand Wei,2011). Tosimplifythe model,we assume only sphericalparti- clesare presentin the porous media.Also, the host,inclusion, and pore-fillingfluidareassumedtohavehomogeneous,isotropic,andnon- dispersiveelectricalproperties.Therefore,thefrequencydispersionand dielectricenhancementpredictedbytheSCAIPmodelorPPIPmodel solelystemsfromtheSCAIPorPPIPphenomenaaroundthenegatively chargednonconductiveorelectrically conductiveinclusions.Wealso assumeallthechargecarriersbearunitarychargeandbothhostand inclusionphasesbearbinary,symmetricchargecarriers.
2.2. Mechanisticmodelofinterfacialpolarizationdueto surface-charge-bearingsphericalnonconductiveparticle
Thesurfaceofanonmetallic(nonconductive)mineral,suchasclay, acquireschargesifthemineralissurroundedbyelectrolytesduetoionic adsorption,protonation/deprotonationofthehydroxylgroups,anddis- sociationofotherpotentiallyactivesurfacegroups,alsocombinedlyre- ferredassurfacecomplexation reactions(LeroyandRevil, 2004).In thispaper,surface-conductance-assistedinterfacialpolarization(SCAIP) modelisdevelopedtoinvestigatetheinterfacialpolarizationphenom- enaaroundsurface-charge-bearingsphericalnonconductive particles.
Fig.1showsSCAIPphenomenainarepresentativevolumeofadilute mixtureofuniformlydistributedsurface-charge-bearingnonconductive sphericalinclusionsinanelectrolyte-saturatedhostmedium,wherein- terfacialpolarizationisindependentofthedirectionoftheexternally appliedelectricfieldduetosphericalsymmetry.
The phenomenological basis of interfacial polarization consid- eredin ourwork buildsonthe mechanisticdescriptionsoutlinedby Grosse(1988).Thenegativelychargedinclusion,togetherwithitspos- itivecounterionlayer,essentiallybehaves asaconductor ofpositive chargecarriers,whichallowsthepositiveionsinthehostmediumto freelyexchange withtheionsin thecounterion layer,andasanon- conductorofnegativecharges,whichexcludesthenegativeionsfrom thecounterionlayer.
In the absence of an externally applied electric field, a Gouy–
Chapman doublelayeris assumedaroundthesurface-charge-bearing nonconductiveinclusions,wherethepositivecounterionlayerischar- acterizedbyafinitesurfaceconductivity.Weassumethethicknessof counterionlayerisnegligible,whichisvalidwhena≫ 𝜆D,where𝜆D
istheDebyescreeninglengthandaisthecharacteristiclengthofthe inclusionphase.
2.2.1. DevelopmentofSCAIPmodel
ThePoisson–Nernst–Planck(PNP)equationhasbeenusedtomodel electromigrationanddiffusionofionicchargecarriersin electrolytes (ZhengandWei,2011)andthatduetoholesandelectronsinsemicon- ductors(SchmuckandBazant,2015).Itis basedonamean-fieldap- proximationofchargecarrierinteractionsandcontinuumdescriptions ofchargeconcentrationandelectrostaticpotential.WeapplythePNP equationstomodelchargedynamicsandrelaxationintherepresenta- tivevolumecontainingonlytwophases:thehostmedium,denotedby subscripth,andtheconductive(tobediscussedinthefollowingsection) ornonconductiveparticles(inclusions),denotedbysubscripti.Inour formulation,thehostmediumcanbeassumedasahomogeneousmix- tureofelectrolyteandnonconductivematrixorasapureelectrolyte.At timet<0,itisassumedthatthereisnoexternalelectricfieldexciting therepresentativevolume.Initialchargecarrierdensitiesatequilibrium conditionsinboththehostandinclusionphasesaredenotedas𝑁0±,𝑗, wheresubscriptjtakestheformofifortheinclusionphaseandhforthe hostphase.Startingattimet=0,therepresentativevolumeexperiences auniformexternallyappliedelectricfieldE=E0ei𝜔t,whereE0istheam- plitudeoftheexternallyappliedelectricfield,iissquarerootof−1,𝜔is theangularfrequency(rad/s)oftheexternallyappliedelectricfield,and eisEuler’snumber.Note𝜔=2𝜋f,wherefisfrequency(Hz).Weassume thenegativelychargedsphericalnonconductiveparticleissurrounded byalayerofpositivelycharged,conductingcounterionlayer,whichhas asurfaceconductance𝜆andbearsafield-inducedsurfacechargeden- sity𝜌ei𝜔tcos𝜃,where𝜃istheanglebetweenthenormaltotheinterface andtheincidentexternalelectricfield.Underaweakfieldapproxima- tion,chargecarrierdensitiesinhostandinclusionphasesareperturbed from their equilibrium conditions near the host-inclusion interfaces, resulting in a new linearly approximated charge distribution, given by
𝑁𝑗±(𝑟,𝑡,𝜃)=𝑁0±,𝑗+𝑐±𝑗(𝑟)𝑒𝑖𝜔𝑡cos𝜃 (1) suchthat|𝑐±𝑗|≤𝑁0±,𝑗,𝑐𝑗±isthechargedensityvariationnearthehost- inclusioninterfacein mediumjduetotheexternallyappliedelectric fieldandristheradialdistancealongthenormaltotheinterface.Note thatinthissection,fornonconductiveinclusion,𝑐𝑖±(𝑟)=0.Inaddition, oneassumptionistheabsenceofchargecarriersinthenonconductive inclusionphase,𝑁0±,𝑖=0. Further,thesymbol“+” identifiespositive- chargecarrierssuchasholesandcations,whilethesymbol“−” identifies negative-chargecarrierssuchaselectronsandanions.
Weassumethatthecharacteristiclengthaoftheinclusionsphaseis fargreaterthantheDebyescreeninglength𝜆D.Notethat𝜆Disameasure ofinducedchargedistributionthatformsaroundaninclusionparticle duetosurfacechargesthatexistontheinclusionparticleintheabsence ofanexternallyappliedelectricfield.Inotherwords,𝜆Drepresentsa volumeoutside ofwhichsurfacechargesonaninclusionparticleare electricallyscreened.Thecharacteristiclengthaisequaltotheradiusof sphericalinclusion. Mathematically,𝜆𝐷=
√𝜀ℎ𝑘𝐵𝑇∕(2𝑍ℎ+𝑍ℎ−𝑞2𝑁0,ℎ),
where𝜀hisdielectricpermittivityofthehost,kBisBoltzmann’sconstant, Tisabsolutetemperature,𝑍ℎ±ischargenumberofpositiveandnegative chargecarriersinthehost,andqistheelementarycharge.Thevolume fractionofconductive(fore.g.,pyrite)andnonconductiveparticles(for e.g.clays)isassumedtobeintherangeof5−15%.Anothersimplifying assumptionisthatallthechargecarriersbearunitarychargeandthat bothhostandinclusionphasesbearbinary,symmetricchargecarriers.
Inotherwords,
𝑍𝑗±=1,𝜇ℎ+=𝜇ℎ−=𝜇ℎ, 𝜇+𝑖 =𝜇−𝑖 =𝜇𝑖, 𝑁0+,𝑖=𝑁0−,𝑖=𝑁0,𝑖,
𝑁0+,ℎ =𝑁0−,ℎ=𝑁0,ℎ (2)
where𝜇±𝑗 istheelectricalmobilityofpositiveandnegativechargecar- riersinmediumj,and𝑍𝑗±ischargenumberofpositiveandnegative chargecarriersinmediumj.
Thecurrentdensityofeachchargecarriertypeinthehostandinclu- sionphasesisthesumofcurrentdensityduetodriftcurrentanddiffu- sioncurrent.Intheabsenceofgeneration/recombinationreactions,the transportequationrepresentingconservationlawsforcharge-carrying speciescanbewrittenas
𝒋±𝑗 =𝒋±𝑗,𝑑𝑟𝑖𝑓𝑡+𝒋±𝑗,𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛=𝑞𝑁𝑗±𝜇𝑗𝒆𝑗∓𝑞𝐷±𝑗∇𝑁𝑗± (3) where𝒋±𝑗 isthecurrentdensityofpositiveandnegativechargecarriers, respectively,inmediumj,ejisthenetelectricfieldvectorinmedium j,and𝐷±𝑗 isdiffusioncoefficientofpositiveandnegativechargecarri- ers,respectively,inmediumj.Whenusingthesimplifyingassumption forelectricalmobilityofchargecarriers,asmentionedinEq.(2),and Einstein’srelationshipofdiffusioncoefficientwithelectricalmobility, namelyDj=(𝜇jkBT)/q,weobtain
𝐷+ℎ =𝐷−ℎ=𝐷ℎ; 𝐷𝑖+=𝐷−𝑖 =𝐷𝑖 (4) Bysubstitutingej=−∇𝜑jintothelow-frequencylimitofMaxwell’s equations(inductionneglected)andsubstitutingEq.(4)intoEq.(3),we expressthechargespeciesconservationconditionas
𝒋±𝑗 =−𝑞𝑁𝑗±𝜇𝑗∇𝜑𝑗∓𝑞𝐷𝑗∇𝑁𝑗± (5) where𝜑j is theelectrical potential in medium j. Eq.(5) is Nernst–
Planck’sequationthatdescribestherelationshipofthefluxofcharge- carryingspeciestoitsconcentrationgradientandthattotheapplied electricalpotentialgradientinagivenmedium.Nernst-Planck’sequa- tioncanalternativelybeexpressedas
𝒋±𝑗 =−𝐷±𝑗𝑁𝑗±∇𝜑±𝑐𝑗 (6)
where 𝜑±𝑐𝑗=𝑘𝐵𝑇ln𝑁𝑗±±𝑞𝑍𝑗±𝜑𝑗 is the electrochemical potential of chargecarriers.Thecontinuityequationforchargecarrierdensitybased onmassconservationforeachchargecarriertypeinanincompressible mediumwithoutanyconvectiveflowcanbewrittenas
∓𝑞𝜕𝑁𝑗±
𝜕𝑡 =∇⋅𝒋±𝑗 (7)
ByapplyingEqs.(5)–(7),weobtain
𝜕𝑁𝑗+
𝜕𝑡 =∇⋅(
𝐷𝑗∇𝑁𝑗++𝜇𝑗𝑁𝑗+∇𝜑𝑗)
(8)
and
𝜕𝑁𝑗−
𝜕𝑡 =∇⋅(
𝐷𝑗∇𝑁𝑗−−𝜇𝑗𝑁𝑗−∇𝜑𝑗)
(9)
ThetimederivativeofEq.(1)assumingaxialsymmetryis
𝜕𝑁𝑗±
𝜕𝑡 =𝑖𝜔𝑐𝑗± (10)
where𝑐𝑗±=𝑐𝑗±(𝑟,𝑡,𝜃)=𝑐±𝑗(𝑟)𝑒𝑖𝜔𝑡cos𝜃.WeapplyEq.(10)toEqs.(8)and 9,thenweaddandsubtractEqs.(8)and(9)toobtainEqs.(11)and (12)expressedas:
−𝑖𝑞𝜔𝑑𝑗=−2𝑞𝑁0,𝑗𝜇𝑗Δ𝜑𝑗−𝑞𝐷𝑗Δ𝑑𝑗 (11) and
−𝑖𝑞𝜔𝑠𝑗=−𝑞𝐷𝑗Δ𝑠𝑗 (12)
where𝑑𝑗=𝑐+𝑗 −𝑐−𝑗 representsnetchargedensityvariation,𝑠𝑗=𝑐𝑗++𝑐𝑗− representstotaliondensityvariation,andΔ(∇2)isLaplace’soperator.
Notedjandsjarefiniteeverywhereintherepresentativevolume,andfor
nonconductiveparticles,di=si=0.WeobtainedEqs.(11)and(12)by assumingdj𝜇j≪1andsj𝜇j≪1as|𝑐±𝑗|≤𝑁0±,𝑗.
UndertheinfluenceofanexternallyappliedEMfield,thedistribu- tionofchargecarriersinbothmedialeadstoatime-varyingelectricpo- tentialthatisexpressedas𝜑j(r,t,𝜃)=𝜑j(r)ei𝜔tcos𝜃.UsingGauss’slaw andEq.(1),weobtain
∇⋅( 𝜀𝑗𝒆𝑗)
=𝑃𝑓,𝑗=𝑞(
𝑁𝑗+−𝑁𝑗−)
=𝑞( 𝑐𝑗+−𝑐𝑗−)
=𝑞𝑑𝑗 (13)
wherePf,jisthenetfreechargedensityinmediumjduetochargere- distributioninthepresenceofanexternallyappliedEMfield,ej,and 𝜀j= 𝜀r,j𝜀0isthedielectricpermittivityofmediumj,𝜀r,jistherelative permittivityofmediumj,and𝜀0=8.854×10−12F/misthevacuumper- mittivity.Eq.(13)relatesthespatialdistributionofelectricchargetothe time-varyingelectricfield.Assumingbothmediaarelinear,isotropic, andhomogeneous,andthattheelectricfieldcanbedefinedbyascalar electricalpotentialfield,𝜑j,weobtain
∇⋅( 𝜀𝑗𝒆𝑗)
=−∇⋅( 𝜀𝑗∇𝜑𝑗)
=−𝜀𝑗Δ𝜑𝑗 (14)
BycombiningEqs.(13)and(14),weobtainanalternateexpression ofPoisson’sequation,expressedas
Δ𝜑𝑗=−𝑞𝑑𝑗
𝜀𝑗 (15)
Poisson’sequationisappliedtodescribetheelectricfieldintermsof theelectricalpotential,thegradientofwhichgovernselectromigration in bothmedia.BysubstitutingEq.(15) intoEq.(11),we obtainthe Poisson-Nernst-Planck(PNP)equation,givenby
−𝑖𝑞𝜔𝑑𝑗=2𝑞2𝑁0,𝑗𝜇𝑗𝑑𝑗∕𝜀𝑗−𝑞𝐷𝑗Δ𝑑𝑗 (16) whichcanbere-writtenas
Δ𝑑𝑗= (𝑖𝜔
𝐷𝑗 + 𝜎𝑗 𝜀𝑗𝐷𝑗
)
𝑑𝑗 (17)
where𝜎j=2N0,j𝜇jqistheelectricalconductivityofmediumj.Werewrite Eqs.(17)and(12)as
Δ𝑑𝑗=𝛾𝑗2𝑑𝑗 (18)
where 𝛾𝑗2=
(𝑖𝜔 𝐷𝑗 + 𝜎𝑗
𝜀𝑗𝐷𝑗 )
(19)
and
Δ𝑠𝑗=𝜉𝑗2𝑠𝑗 (20)
where 𝜉𝑗2= 𝑖𝜔
𝐷𝑗 (21)
respectively.Eqs.(18)and(20)areHelmholtzpartialdifferentialequa- tions(PDE)whichcanbesolvedtoobtaindistinctanalyticalexpressions ofdj andsj forthehostandinclusionphases,respectively.Eq.(18)is insertedintoEq.(15)toobtainthefollowingLaplacePDEthatcanbe solvedfortheelectricpotentialfieldintherepresentativevolume:
Δ𝜗𝑗=0 (22)
where 𝜗𝑗=𝜑𝑗+(
𝑞𝑑𝑗)
∕ (𝛾𝑗2𝜀𝑗
)
(23)
2.2.2. SolutionofHelmholtzPDE
Asmentioned before,fornonconductiveinclusions, di=si=0.So, we’resolvingtheHelmholtzPDEstoobtainthedistinctanalyticalex- pressionsofdhandshforthehostphase.Asphereofradiusequaltoa exhibitsdipolarizability(dipolemoment)intheradialdirection.Such aninclusionidentifiesagrainorvug.Inordertocomputethedipolar- izabilityoftherepresentativevolumecomprisingasphericalinclusion
inanelectrolytichost,Eq.(18) canbeexpressedinsphericalcoordi- nates,assumingazimuthalsymmetry,axialsymmetry,andaseparable solution(Young,2009)fordh(r,𝜃)=Rh(r)Th(𝜃),as
1 𝑅ℎ 𝜕
𝜕𝑟 (
𝑟2𝜕𝑅ℎ
𝜕𝑟 )
−𝛾ℎ2𝑟2+ 1 𝑇ℎsin𝜃 𝜕
𝜕𝜃
(sin𝜃𝜕𝑇ℎ
𝜕𝜃 )
=0 (24)
and 1 sin𝜃 𝜕
𝜕𝜃
(sin𝜃𝜕𝑇ℎ
𝜕𝜃 )
=−𝑛(𝑛+1)𝑇ℎ (25)
wherenisanintegerreferringtotheorderofthestandingwavesolu- tion.Astandingwavesolution(Young,2009)totheabovedifferential equationis
𝑇ℎ=
∑∞ 𝑛=1
[𝐴𝑛,ℎ𝑃𝑛0(cos𝜃)+𝐵𝑛,ℎ𝑄0𝑛(cos𝜃)]
(26)
where𝑃𝑛0and𝑄0𝑛areassociatedLegendrefunctionsofthefirstandsec- ondkind(Weisstein,2018a),respectively,ofnthorderandAn,handBn,h areunknowncomplex-valuedcoefficientsofthegeneralsolutionofthe partialdifferentialEq.(25).SubstitutingEq.(25)inEq.(24),weobtain
𝜕𝑟𝜕 (
𝑟2𝜕𝑅ℎ
𝜕𝑟 )
−[
𝛾ℎ2𝑟2+𝑛(𝑛+1)]
𝑅ℎ=0 (27)
Astandingwavesolutiontotheabovedifferentialequationis 𝑅ℎ=
∑∞ 𝑛=1
[𝐶𝑛,ℎ𝑖𝑛( 𝑟𝛾ℎ)
+𝐷𝑛,ℎ𝑘𝑛( 𝑟𝛾ℎ)]
(28)
wherenis anintegerforthestandingwavesolution(Young,2009), in andkn arethemodified sphericalBesselfunction of thefirstand secondkind(Weisstein,2018b),respectively,ofn−thorder.Cn,hand Dn,hareunknowncomplex-valuedcoefficientsofthegeneralsolution ofthepartialdifferentialEq.(27).inandkncanbeexpressedinterms ofmodifiedBesselfunctionofthefirstandsecondkind,respectively, as𝑖𝑛(𝑟𝛾ℎ)=√ 𝜋
2𝑟𝛾ℎ𝐼𝑛+1
2
(𝑟𝛾ℎ)and𝑘𝑛(𝑟𝛾ℎ)=√
2 𝜋𝑟𝛾ℎ𝐾𝑛+1
2
(𝑟𝛾ℎ),where𝐼𝑛+1 and𝐾𝑛+1 2
2
arethemodifiedBesselfunctionofthefirstandsecondkind, respectively, of (𝑛+1
2)−thorder. Intheanalytical expressionof our model,theseriesisreducedtoasingletermforn=1andBn,h=0by consideringthefollowingsymmetries ofthecharge density:(1)axial symmetry,(2)anti-symmetrywithrespectto𝜃,(3)uniformityofthe externalappliedfield,and(4)dipolarnatureoftheexternallyapplied field.Thissimplificationisalignedwithboundaryconditionsthatcan- notbesatisfiedbyothervaluesofn.ThisreducesEqs.(28)and(26)to 𝑅ℎ=𝐶ℎ𝑖1
(𝑟𝛾ℎ) +𝐷ℎ𝑘1
(𝑟𝛾ℎ)
(29)
and
𝑇ℎ=𝐴ℎcos𝜃 (30)
respectively,whereCh, Dh,andAh andareunknowncomplex-valued coefficientsoftheparticularsolutionobtainedfromEqs.(26)and(28). Thegeneralrepresentationofdh(r,𝜃)cannowbewritten,bycombining Eqs.(29)and(30),as
𝑑ℎ(𝑟,𝜃)=𝐴ℎ[ 𝐶ℎ𝑖1
(𝑟𝛾ℎ) +𝐷ℎ𝑘1
(𝑟𝛾ℎ)]
cos𝜃 (31)
Usingtheconditionthatdh(r,𝜃)shouldbefiniteatr→∞,weobtain aparticularsolutionofdhforthehostphasethatcanberepresentedas 𝑑ℎ(𝑟,𝜃)=𝐵ℎ1𝑘1
(𝑟𝛾ℎ)
cos𝜃 (32a)
or
𝑑ℎ(𝑟,𝜃)=𝐵ℎ1
[ 𝑒−𝑟𝛾ℎ
( 1 𝑟𝛾ℎ+ 1
(𝑟𝛾ℎ)2
)]
cos𝜃 (32b)
whereBh1isunknowncomplex-valuedcoefficientoftheparticularso- lutioninthehostmediumobtainedfromEq.(31).Notewhenr→∞,
dh(r,𝜃)=0.Repeattheaboveprocedure,wecanobtainaparticularso- lutionofshforthehostphasefromEq.(20)thatcanberepresentedas 𝑠ℎ(𝑟,𝜃)=𝐵ℎ2𝑘1
(𝑟𝜉ℎ)
cos𝜃 (33a)
or
𝑠ℎ(𝑟,𝜃)=𝐵ℎ2
[ 𝑒−𝑟𝜉ℎ
( 1 𝑟𝜉ℎ+ 1
(𝑟𝜉ℎ)2
)]
cos𝜃 (33b)
whereBh2isunknowncomplex-valuedcoefficientoftheparticularso- lutioninthehostmedium.
2.2.3. SolutionofLaplacePDE
The Laplacianpartial differentialequation (PDE) mustbe solved to obtain the electric potential field in the representative volume.
Assuming azimuthal symmetry and a separable solution for ϑj(r,𝜃, 𝜑)=R𝜀j(r)T𝜀j(𝜃),Eq.(22)canbeexpressedinsphericalcoordinatesas Δ𝜗𝑗= 1
𝑅𝜀𝑗 𝜕
𝜕𝑟 (
𝑟2𝜕𝑅𝜀𝑗
𝜕𝑟 )
+ 1
𝑇𝜀𝑗sin𝜃 𝜕
𝜕𝜃
(sin𝜃𝜕𝑇𝜀𝑗
𝜕𝜃 )
=0 (34)
Assumingaxial symmetry,ageneralsolution(Hogg,2001)tothe abovePDEcanbeexpressedas
𝜗𝑗(𝑟,𝜃)=
∑∞ 𝑛=0
[𝐴𝑛,𝑗𝑟𝑛+𝐶𝑛,𝑗𝑟−(𝑛+1)][
𝐸𝑛,𝑗𝑃𝑛0(cos(𝑛𝜃))+𝐹𝑛,𝑗𝑄0𝑛(sin(𝑛𝜃))] (35) wherenisanintegerandAn,j,Cn,j,En,j,andFn,jareunknowncomplex- valued coefficients of the general solution of the PDE expressed in Eq.(34).Fortheanalyticalmodelingpurposes,weassumen=1,Fn,j=0, A0,j=0 andC0,j=0, which ensures that the remaining termssatisfy thepolarangledependenceofthemodel.Simplifiedrepresentationof Eq.(35)isexpressedas
𝜗𝑗(𝑟,𝜃)=(
𝐴1,𝑗𝑟+𝐶1,𝑗𝑟−2)
𝐸1,𝑗cos𝜃 (36a)
whichcanberewrittenusingEq.(23)as 𝜑𝑗(𝑟,𝜃)=(
𝐴𝑗𝑟+𝐶𝑗𝑟−2)
cos𝜃−𝑞𝑑𝑗(𝑟,𝜃) 𝛾𝑗2𝜀𝑗
(36b)
Usingtheconditionthatdi=0and𝜑ishouldbefinitewhenr→0, wecanobtainCi=0.So,astandingwaverepresentationofEq.(36b)for thenonconductiveinclusionphaseis
𝜑𝑖(𝑟,𝜃)=𝐴𝑖𝑟cos𝜃 (37)
whereAiisunknowncomplex-valuedcoefficientoftheparticularsolu- tioninthenonconductiveinclusionphaseobtainedfromEq.(36b).Us- ingtheconditionwhenr→∞,dh=0,wecanobtainAh=−E0.Astanding waverepresentationofEq.(36b)forthehostphase,usingEq.(32b),is
𝜑ℎ(𝑟,𝜃)=(
−𝐸0𝑟+𝐶ℎ𝑟−2)
cos𝜃−𝑞𝐵ℎ1
𝛾ℎ2𝜀ℎ [
𝑒−𝑟𝛾ℎ (
1 𝑟𝛾ℎ+ 1
(𝑟𝛾ℎ)2
)] cos𝜃
(38) whereChisunknowncomplex-valuedcoefficientoftheparticularsolu- tioninthehostobtainedfromEq.(36b)andE0istheamplitudeofthe externallyappliedelectricfield.
2.2.4. Boundaryconditions
Toobtainanexpressionforthedipolarizability(dipolemoment),we needfirsttoidentifytheboundaryconditions(Grosse,1988):
(a) Continuityoftheelectricpotentialattheinterface.
𝜑𝑖(𝑟=𝑎)=𝜑ℎ(𝑟=𝑎) (39a)
(b) Discontinuityofthenormalcomponentofthedisplacementcurrent attheinterfacebecauseofthesurfacechargedistributiononthein- clusionphase.ThisboundaryconditionisderivedfromGauss’Law.
𝜀𝑖𝜕𝜑𝑖
𝜕𝑟||
||𝑟=𝑎−𝜀ℎ𝜕𝜑ℎ
𝜕𝑟 ||
||𝑟=𝑎=𝜌cos𝜃 (39b)
(c) Continuityofthesurfacechargedensityatthehost-inclusioninter- facequalitativelyexpressedas:Rateofchangeofsurfacechargeden- sitynormaldrift/conductioncurrentattheinterfaceduetopoten- tialgradientarisingfromtheexternalelectromagneticfield+normal diffusioncurrentdue toconcentrationgradientinthehostmedia attheinterface+tangentialconductioncurrentduetothepotential gradientarisingfromthesurface-charge-bearinginclusionphase.In otherwords,thisboundaryconditionshowsthatthetimederivative ofsurfacechargedensityinthecounterionlayerisequaltothesum ofthenormalconductionanddiffusioncurrentduetopotentialand concentrationdifference,separately,fromthehostmediumandthe tangentialconductioncurrentdue topotential fromtheinclusion phase.
−𝑖𝜔𝜌cos𝜃=−𝜎ℎ 2
𝜕𝜑ℎ
𝜕𝑟 ||
||𝑟=𝑎−𝑞𝐷ℎ𝜕𝑐+ℎ
𝜕𝑟 ||
|||𝑟=𝑎
+2𝜆
𝑎𝐴𝑖cos𝜃 (39c) (d) Thenormalcomponentof thecurrentdensityof negativeionsin thehostmediummustvanishattheinterfaceduetotheassumption thatthenegativeionsareexcludedfromthecounterionlayer.
𝒋−ℎ=−𝜎ℎ 2
𝜕𝜑ℎ
𝜕𝑟 ||
||𝑟=𝑎+𝑞𝐷ℎ𝜕𝑐−ℎ
𝜕𝑟||
|||𝑟=𝑎
=0 (39d)
(e) Duetotheapplicationoftheexternalelectricfield,weuseasimpli- fyingassumptionthattherelativechangeofthepositiveiondensity inthecounterionlayer(whichisassumedtobenegligiblythin)and thatinthehostmediummustbetheequalbecausethepositiveions inthehostmediumcanfreelyexchangewiththeionsinthecounte- rionlayer.
𝑐ℎ+(𝑟=𝑎,𝜃) 𝑁0,ℎ =𝜌cos𝜃
𝜌0
(39e)
where𝜌0istheinitialequilibriumsurfacechangedensityinthecounte- rionlayerand𝜌isthenetresultantsurfacechargedensityinthecoun- terionlayeraftertheapplicationofelectricfield.
2.2.5. SolutionfortheDipolarizability
Usingboundarycondition(39a),Eqs.(37)and(38)canbeequated onthesurfaceofthesphereofradiusequaltoa.Theresultingequation canbeabbreviatedas
−𝐸0𝑎+𝐶ℎ
𝑎2 −𝐸ℎ𝐵ℎ1=𝐴𝑖𝑎 (40a)
where 𝐸ℎ= 𝑞
𝛾ℎ2𝜀ℎ𝑒−𝑎𝛾ℎ [
1 𝑎𝛾ℎ + 1
(𝑎𝛾ℎ)2
]
(40b)
Theequationobtainedusingboundarycondition(39b)atthesurface ofthespherecanbeabbreviatedas
𝜀ℎ (
𝐸0+2𝐶ℎ 𝑎3 −𝐺ℎ𝐵ℎ1
)
+𝜀𝑖𝐴𝑖=𝜌 (41a)
where 𝐺ℎ= 𝑞
𝛾ℎ𝜀ℎ𝑒−𝑎𝛾ℎ [
1 𝑎𝛾ℎ + 2
(𝑎𝛾ℎ)2 + 2 (𝑎𝛾ℎ)3
]
(41b)
Boundarycondition(39c)givesusthefollowingabbreviatedequa- tion:
𝑖𝜔𝜌=−𝜎ℎ
2𝐸0−𝜎ℎ𝐶ℎ 𝑎3 +𝜎ℎ
2𝐺ℎ𝐵ℎ1
−𝐷ℎ
2 𝛾ℎ2𝐺ℎ𝐵ℎ1𝜀ℎ−𝐷ℎ
2 𝜉ℎ2𝐿ℎ𝐵ℎ2𝜀ℎ−2𝜆
𝑎𝐴𝑖 (42a)
where 𝐿ℎ= 𝑞
𝜉ℎ𝜀ℎ𝑒−𝑎𝜉ℎ [
1 𝑎𝜉ℎ+ 2
(𝑎𝜉ℎ)2 + 2 (𝑎𝜉ℎ)3
]
(42b)
Similarly,theequationobtainedusingboundarycondition(39d)can beabbreviatedas
𝜎ℎ
2𝐸0+𝜎ℎ𝐶ℎ 𝑎3 −𝜎ℎ
2𝐺ℎ𝐵ℎ1=𝐷ℎ
2 𝜉ℎ2𝐿ℎ𝐵ℎ2𝜀ℎ−𝐷ℎ
2 𝛾ℎ2𝐺ℎ𝐵ℎ1𝜀ℎ (43) Forboundarycondition(39e),weassumetheelectricalmobilitiesin thetworegionsarethesametore-writethisboundaryconditionas 2𝑞𝑐ℎ+(𝑟=𝑎)=𝜌𝜎ℎ
𝜆 (44)
AftersolvingEqs.(40a),(41a),(42a),(43)and(44),weobtainthe dipolarizability(dipolarfieldcoefficient)oftherepresentativevolume comprisingasphericalnonconductiveinclusioninanelectrolytichost as
𝑓𝑛𝑐𝑜𝑛𝑑(𝜔)= 𝐶ℎ
𝐸0𝑎3 = 𝑄(𝑅+𝐴)−𝑃
𝑄(𝑅−2𝐴)+2𝑃 (45)
where 𝐴= 1
𝑎2 (45a)
𝑃 =𝛾ℎ2+𝜉ℎ2𝐺 𝐻 + 2𝐺
𝑎2𝐿 (45b)
𝑄= 1 𝑖𝐹+1
[ 2−𝑎2𝜉ℎ2
𝐻 (𝐿
𝑖𝐹 +𝐸)
−2𝐸 𝐿
]
(45c) 𝑅=𝑃
𝑄
(𝑖𝐹𝐸+𝐿 𝑖𝐹+1
)
(45d)
𝐻= 𝑎𝐿ℎ
𝐹ℎ ,𝐺=𝑎𝐺ℎ
𝐸ℎ, 𝐿= 2𝜆 𝑎𝜎ℎ, 𝐸= 𝜀𝑖
𝜀ℎ, 𝐹= 𝜔𝜀ℎ
𝜎ℎ (45e)
𝐹ℎ= 𝑞 𝜉ℎ2𝜀ℎ𝑒−𝑎𝜉ℎ
[ 1 𝑎𝜉ℎ + 1
(𝑎𝜉ℎ)2
]
(45f)
2.3. Mechanisticmodelofinterfacialpolarizationduetospherical conductiveparticle
In this paper, perfectly polarized interfacial polarization (PPIP) model is applied to investigate interfacial polarization phenomena aroundconductiveparticle.Fig.2showsPPIPphenomenainarepre- sentativevolumeofadilutemixtureofuniformlydistributedelectrically conductivesphericalinclusionsinanelectrolyte-saturatedhostmedium, whereinterfacialpolarizationisindependentofthedirectionoftheex- ternallyappliedelectricfieldduetosphericalsymmetry.
The phenomenological basis of interfacial polarization consid- ered inour workbuildson themechanisticdescriptionsoutlinedby Reviletal.(2015).Chargecarriersinconductivemineralshavehigher mobilitycomparedtoionsinporousgeomaterials.Inthepresenceofan externallyappliedEMfield,chargecarriersinthedisseminatedelectri- callyconductiveinclusionsmigratefasterandaccumulateatimperme- ableinterfaces.Consequently,electricallyconductiveinclusionsbehave asdipolesinthepresenceofanexternallyappliedelectricfield.Subse- quently,chargecarriersinthehostmediummigrateandaccumulateon host-inclusioninterfacesundertheinfluenceoftheexternallyapplied electricfieldandthatoftheinducedchargesinconductiveinclusions.
Intheabsenceofanexternallyappliedelectricfield,anegligible initialsurfacechargeisassumedonelectricallyconductiveinclusions.
Thus,thereistypicallyanegligibledoublelayeraroundthesurfaceof electricallyconductiveinclusions,wherebythesurfaceconductanceof aconductiveinclusionisnegligible. Similarassumptionsaremadein electrochemistryandcolloidsciencewithrespecttoelectrochemicalre- laxationaroundmetallicsurfaces(ChuandBazant,2006).Also,weas- sumeabsenceofredox-activespeciesandneglecttheinfluenceofpHof
Fig.2. Cross-sectionofaperfectlypolarizedconductivesphericalinclusionsur- roundedbyanionichostmedium.Chargecarriersintheionichostmediumare cations,identifiedby“+” symbol,andanions,identifiedby“−” symbol.Charge carriersintheconductivesphericalinclusionaren-andp-chargecarriers,identi- fiedbysymbol“n” and“p”,respectively.Thedirectionoftheexternallyapplied electricalfield,e,isidentifiedwithaboldarrownexttothesymbol“e”.Thedi- rectionofmovementofthefourdifferenttypesofchargecarriersisrepresented bythearrownexttothesymbolsofthechargecarriers.
porewater(Reviletal.,2015).Thehostandinclusionphasescanbe modeledasanelectricallyconductive,insulating,ordielectricmaterial.
Also,pore-fillingfluidcanbemodeledaselectricallyconductive(e.g.
brine)ornon-conductivematerial(e.g.oil).
2.3.1. DevelopmentofPPIPmodel
Thedevelopmentof thePPIPmodel(Misraetal.,2016b)is very similartothatoftheSCAIPmodel.ForPPIPmodeldevelopment,spon- taneousinitialaccumulationofchargesisassumedtobeabsentonthe host-inclusioninterfaces.Attimet<0,electro-neutralityis assumed throughoutthesystem.
2.3.2. SolutionofHelmholtzPDE
Theabove-mentionedEq.(18)mustbesolvedtoobtainananalytical expressionfordjinthehostandinclusionphasesaroundtheperfectly polarizedhost-inclusioninterfaceofconductivesphericalinclusion.Re- callthat𝑑𝑗=𝑐+𝑗 −𝑐−𝑗 representsnetchargedensityvariation,where𝑐±𝑗 is thechargedensityvariationnearthehost-inclusioninterfaceinmedium jduetotheexternallyappliedelectricfield.Expressionfordh(r,𝜃)for themixturecontainingconductivesphericalinclusionisthesameasthat forthemixturecontainingnonconductivesphericalinclusion.Usingthe conditionthatdi(r,𝜃)shouldbefiniteatr→0,weobtainaparticular solutionfordiforthemixturecontainingconductivesphericalinclusion thatcanberepresentedas
𝑑𝑖(𝑟,𝜃)=𝐵𝑖𝑖1
(𝑟𝛾𝑖)
cos𝜃 (46a)
or 𝑑𝑖(𝑟,𝜃)=𝐵𝑖
[cosℎ( 𝑟𝛾𝑖) 𝑟𝛾𝑖 −sinℎ(
𝑟𝛾𝑖) (𝑟𝛾𝑖)2
]
cos𝜃 (46b)
whereBi isunknowncomplex-valuedcoefficientof theparticularso- lutionintheinclusionphaseobtainedfromEq.(31),substitutingthe subscripthwithi.Notewhenr→0,itisassumedthatdi(r,𝜃)=0.
2.3.3. SolutionofLaplacePDE
Theabove-mentionedEq.(22)mustbesolvedtoobtaintheelectric potentialfieldintherepresentativevolume.Theexpressionfor𝜑h(r,𝜃) forthemixturecontainingconductivesphericalinclusionisthesameas thatforthemixturecontainingnonconductivesphericalinclusion.Using
theconditionwhenr→0,di=0and𝜑ishouldbefinite,wecanobtain Ci=0.AstandingwaverepresentationofEq.(36b)fortheconductive inclusionphase,usingEq.(46b),is
𝜑𝑖(𝑟,𝜃)=𝐴𝑖𝑟cos𝜃− 𝑞𝐵𝑖 𝛾2𝑖𝜀𝑖
[cosℎ( 𝑟𝛾𝑖) 𝑟𝛾𝑖 −sinℎ(
𝑟𝛾𝑖) (𝑟𝛾𝑖)2
]
cos𝜃 (47)
whereAiisunknowncomplex-valuedcoefficientoftheparticularsolu- tionintheconductiveinclusionphaseobtainedfromEq.(36b).
2.3.4. Boundaryconditions
Toobtainanexpressionforthedipolarizability,weneedfirsttoiden- tifytheboundaryconditions(GrosseandFoster,1987):
(a) Assumingazero-intrinsiccapacitanceofthehost-inclusioninterface, theelectricpotentialmustbecontinuousattheinterface.
𝜑𝑖(𝑟=𝑎)=𝜑ℎ(𝑟=𝑎) (48a)
(b) Thenormalcomponentofthedisplacementcurrentmustbecontinu- ousattheinterface.Thisconditioncorrespondstothefactthatthere isnonetsurface-chargedistributiononanelectricallyconductivein- clusionphase.
𝜀𝑖𝜕𝜑𝑖
𝜕𝑟||
||𝑟=𝑎=𝜀ℎ𝜕𝜑ℎ
𝜕𝑟 ||
||𝑟=𝑎 (48b)
(c)Thenormalcomponentofthecurrentdensitymustvanishatthein- terfaceforbothmedia.Thisconditionexpressesthefactthatinthe absenceoftransportofchargecarriersandexchangeofchargesalong theinterface,thediffusiveandelectro-migrativecurrentsmustcan- celeachotherattheinterface.Ourfocusisperfectlypolarizableor completelyblockinginterfaceswithoutFaradicprocesses,wherein fluxesofchargecarriersmustvanishonbothsidesoftheinterface.
Notethatthisboundaryconditionisusedtoobtaintwoequations:
onefortheoutervolumeofthesphereinthehostmedium,andthe otherfortheinnervolumeofthesphereintheinclusionmedium.
𝒋+𝑗 +𝒋−𝑗 =−2𝑁0,𝑗𝑞𝜇𝑗 𝜕𝜑𝑗
𝜕𝑟||
|||𝑟=𝑎
−𝑞𝐷𝑗 𝜕𝑑𝑗
𝜕𝑟||
|||𝑟=𝑎
=0(𝑗=ℎ𝑜𝑟𝑖) (48c)
2.3.5. Solutionforthedipolarizability
Usingboundarycondition(48a),Eqs.(47)and(38)canbeequated onthesurfaceofthesphereofradiusequaltoa.Theresultingequation canbeabbreviatedas
−𝐸0𝑎+𝐶ℎ
𝑎2 −𝐸ℎ𝐵ℎ=𝐴𝑖𝑎−𝐹𝑖𝐵𝑖 (49a)
where 𝐹𝑖= 𝑞
𝛾𝑖2𝜀𝑖 [cosℎ(
𝑎𝛾𝑖) 𝑎𝛾𝑖 −sinℎ(
𝑎𝛾𝑖) (𝑎𝛾𝑖)2
]
(49b)
Theequationobtainedusingboundarycondition(48b)atthesurface ofthespherecanbeabbreviatedas
𝜀ℎ
(
−𝐸0−2𝐶ℎ 𝑎3 +𝐺ℎ𝐵ℎ
)
=𝜀𝑖(
𝐴𝑖+𝐻𝑖𝐵𝑖)
(50a) where
𝐻𝑖= 𝑞 𝛾𝑖𝜀𝑖
[2cosℎ( 𝑎𝛾𝑖) (𝑎𝛾𝑖)2 −sinℎ(
𝑎𝛾𝑖)
𝑎𝛾𝑖 −2sinℎ( 𝑎𝛾𝑖) (𝑎𝛾𝑖)3
]
(50b)
Similarly,theequationobtainedusingboundarycondition(48c)at theoutersurfaceofthesphereinthehostmediumcanbeabbreviated as
𝐶ℎ=−𝑎3 (𝐸0
2 +𝑖𝜔𝜀ℎ𝐺ℎ𝐵ℎ 2𝜎ℎ
)
(51)
Ontheotherhand,theequationobtainedusingboundarycondition (48c)attheinnersurfaceofthesphereintheinclusionmediumcanbe abbreviatedas
𝐴𝑖=𝑖𝜔𝜀𝑖𝐻𝑖𝐵𝑖
𝜎𝑖 (52)