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1、61Chapter6:TheCurrentDistribution*1.Significance2.WhatControlstheCurrentDistribution?3.DeterminationoftheCurrentDistribution4.ControllingMode3sandApproximations5.PrimaryDistribution6.SecondaryDistribution*Thefollowingtextisbased,inpart,onmaterialthatisinpressforpublicationin“AdvancesinElectrochemist
2、ryandElectrochemicalEngineering”,Vol.45,Elsevier.621.INTRODUCTIONANDOVERVIEWThetopicofcurrentdistributionmodelingiscentraltotheanalysisofelectrochemicalsystemsandhasbeenaddressedintextbooks(1),reviews(e.g.,24)andnumerousjournalpublications.Newmanstextbook(1)providesameticulousandcomprehensivetreatme
3、ntofthesubject.PrenticeandTobias(2)presentareviewoftheearly(upto1980)publicationsinthearea.Dukovicsmorerecentreview(3)isverycomprehensive,providingcriticalanalysisofboththeelectrochemicalandthenumericalaspectsofthetopic.ArecentreviewbySchlesinger(4)focuses primarily on the numerical techniques.The p
4、resent monograph introduces thefundamentalprocessesandequationsunderlyingthemodelingofthecurrentdistribution,andcritically analyzes common assumptions and approximations.Focus is placed on discussingscaling parameters for the characterization of the current distribution.Commonly usedalgorithmsfornum
5、ericaldeterminationofthecurrentdistributionarecomparedandafewnumericalimplementationsarediscussed.Lastly,themodelingofthecurrentdistributioninsomespecialconfigurationsandapplicationsisintroduced,emphasizingrecentpublications.2.SIGNIFICANCEOFMODELINGTHECURRENTDISTRIBUTIONThecurrentdistributionisamong
6、themostsignificantparameterscharacterizingtheoperationoftheelectrochemicalcell.Thecurrentdensityontheelectrodesisdirectlyproportionaltothereactionrateanditsdistributioncriticallyaffectstheelectrochemicalprocess.Inelectroplating,thedepositthicknessdistribution,andpropertiessuchasthedepositsurfacetext
7、ureanditsmorphology are directly linked to the current distribution.When multiple simultaneouselectrodereactionsarepresent,suchasinalloydepositionorinhydrogencoevolution,thealloycompositionintheformercase,andthecurrentefficiencyinthelatter,arecontrolledbyImportance of Modeling Identify critical proc
8、ess parameters and their impact Relate global measurable parameters to local variables in which we are interested Predictive and rationale design&scale-up eliminate trial&error save time and costs Optimize&control processes Interpretation of experiments63the overpotential distribution,which,as discu
9、ssed below,is directlyrelated to the currentdistribution.Electrolyticprocesseswhichdonotinvolvedepositionarealsostronglyaffectedbythecurrentdistribution.Examplesincludeoptimizedutilizationofcatalyticelectrodesandtheneedtopreventthecurrentdensityfromsurgingonelectrodesections,onseparatorsandonmembran
10、es.Thepowerrequiredforoperatinganelectrochemicalcell,andparticularlytheohmiclossarealsodependentonthecurrentdistribution.Lastly,thecorrectinterpretationofexperimentaldatahingesonunderstandingtherangeofcurrentdensitiestowhichthetestedelectrodehasbeensubjected.Thecurrentdistributioncanbeanalyzedondiff
11、erentscales.Themacroscopiccurrentdistributionofthe,wherethedistributionisresolvedonlengthscaleoftheorderofcm,isimportantincharacterizingthedepositthicknessuniformityonaplatedpart,orinselectiveplating,whereanonuniformcurrentdistributionissought.Themicroscaledistribution,ontheotherhand,wherethecurrent
12、densityisresolvedonsubmmlengthscalesaffectsprimarilyparameters such as the deposit texture and roughness,nucleation and deposition withinmicronandnanoscalefeatures.For many applications,numerical simulation capability which provides the currentdistribution in a given configuration and plating condit
13、ions,or for a given set of suchparameters,issufficient.However,forpredictiveprocessdesignandforscaleupofcellsandprocesses(andscaledownofindustrialprocessesforlaboratorytesting)analyticalmodelsthatelucidate the dependence of the current distribution on the process parameters are morebeneficial.643.EX
14、PERIMENTALDETERMINATIONOFTHECURRENTDISTRIBUTIONElectroplating processes,where a solid deposit is formed and its thickness can be directlymeasured,providearelativelyconvenientmeansfordeterminationofthecurrentdistribution.Thedepositthicknesscanbemeasuredbyanumberofcommerciallyavailabledevices,basedone
15、.g.,xrayfluorescence,betabackscatter,magneticproperties,orcontrolleddissolution.Adirectprobebasedontheinducedfieldassociatedwiththecurrentflowhasbeenrecentlyintroduced.Opticalandelectronmicroscopyofcrosssectioneddepositsprovideacommonmeansformeasuringthedepositthickness.Oncethedepositthickness,d,ism
16、easured,itcanberelatedtothecurrentdensity,i,throughFaradayslaw:FM tdiF n1Here,tistheplatingtime.FisFaradaysconstant,M,andnaretheplatedmetalatomicweight,its density,and the number of electrons transferred in the deposition reaction,respectively.F is the Faradaic efficiency,accounting for side(parasit
17、ic)reactions.MetalsnobletohydrogentypicallyplatefromaqueoussolutionsatF1correspondingtocloseto100%Faradaicefficiency(unlesstheyaredriventothelimitingcurrent).WhentheFaradaic65efficiencyislessthan100%,eq.1canbeusedtodeterminethecurrentefficiencyoncethecurrentdensityisevaluated.Forthecaseofredoxorgase
18、volvingreactions,wherenosoliddepositisformed,thecurrent distribution on the electrodes can be determined using segmented electrodes,orinsulatedprobeelectrodes.Here,theelectrodeonwhichthecurrentdistributionissoughtissectionedintomultiple,electricallyisolatedsegments,towhichthecurrentmaybeindividually
19、fedandmeasured.Ifmultiple,narrow,segmentsareprovided,theaveragesegmentalcurrentdensities,obtainedbydividingthesegmentalcurrentsbythecorrespondingsegmentalareas,provideanapproximationtothecurrentdistribution.Forthesegmentedelectrodetoresembleacontinuouselectrode,allsegmentsmustbecoplanarandessentiall
20、yequipotential.ToassurethatthepotentialofallsegmentsiswithinafewmV,multichannelpotentiostatsmustbeused.Alesscostlyapproachistoconnecteachsegmenttothecommonbusviaverylow(typicallym)shuntresistor,whichenablesthemeasurementofthecurrentyetintroducesinsignificantvoltagevariation.Modeling Analytical Appro
21、ximations Numerical Analogies Electrostatics Magnetic Electrolytic trough Experimental Cross section back scatter X-ray fluorescence Cutting and weighing Striping with coulometry Isotope deposition Magnetic flux Sound reflection Direct probe measurement(Hall effect)Sectioned electrode Potential fiel
22、d mappingDetermination of the current distribution66 Chemistry Kinetics(additives)Conductivity Complexing agents Flow Agitation Ultrasound Temperature Cell GeometryShields Thieves Shape of the anode Part shape Voltage(current)Non DC waveforms(pulsing)How can we control the current distribution?What
23、controls the current distribution?Why is the current distribution not uniform?i1,L1i2,L2i2,L2i1,L167What controls the current distribution?Why is the current distribution not uniform?i1,L1i2,L2I=-?V/R (ohms law)IViAR ARi1Assume ohmic control(primary):VVViLRAALALi11221LLiiIn general:V1=E0+a(i1)+C(i1)
24、+O(i1)V2=E0+a(i2)+C(i2)+O(i2)V1=V2E0+a(i1)+C(i1)+O(i1)=E0+a(i2)+C(i2)+O(i2)The current density will adjust to maintain the voltage balanceNote:1 i=i(x,y,z)2 system is non-linear 3 L is a-priory unknownAssumption:Electrodes are equi-potential,i.e.no terminal effect 1Overpotentials The Driving ForceAp
25、plied Voltage=Standard Potential+Overpotential?V =E0+?T(i)?T =?a(i)+?C(i)+?O(i)li0lniiFRTaLCiiFnRT1lnLength scale(cm)Time scale(sec)10-610-610-31110-13Activation:Concentration:Ohmic:VXE0?a?C?O?V?C?ORbLtCDFni1Limiting current:?a684.ANALYTICALDERIVATIONOFTHECURRENTDISTRIBUTIONThistopiciscoveredinsigni
26、ficantdetailinNewmanstextbooks(1).Asummary,relevanttotheensuingdiscussionisprovidedhere.TheCurrentDensityThecurrentdensityisdirectlyrelatedtotheionicflux,Nj,intheelectrochemicalcell.Thefluxistypically described in terms of three major components:diffusion of ions across aconcentrationgradient,migrat
27、ionofchargedionsdowntheelectricfield,andtransportofionsduetobulkelectrolyteconvection.Consequently,thefluxofanionicspeciesjisgivenby:jjjjjjJNDcu z Fcc v 2ThecurrentdensityisdeterminedbyassigningthechargeFzjtothefluxofeachspeciesjandsummingoverallionicspecies:jjjiFz N 3SubstitutingEq.2intoEq.3,22jjjj
28、jjjJjjjiFz DcFu z cFz cv 4Electroneutrality,expressedas:0jjjz c 5ispresentthroughoutthecell(exceptforthevanishinglythindoublelayer)andrendersthelasttermontherightofEq.4tozero,providingforthetotalcurrentdensity:22jjjjjjjjiFz DcFu z c 6Wealsorecognize(1)thattheelectrolyteconductivity,isgivenby:22jjjjF
29、u z c7Hencewecanrewriteeq.6as:69jjjjiFz Dc 8Eq.8 indicates that the current density is determined by both the potential andconcentration gradients.The explicit velocity term is absent from eq.8(due toelectroneutrality),however,convection still affects the current density by controlling theconcentrat
30、ionfield.ItshouldalsobenotedthattheelectrodekineticswhichdonotappearexplicitlyinEq.8establishtheboundaryconditionsrequiredforitssolution.Assubsequentlyshown,theelectrodekineticsmayinfluencequitesignificantlythecurrentdistribution.Whilerepresentingthecurrentdensityasafunctionofthepotentialandconcentr
31、ationdistributions,eq.8 does not provide the necessary relationships required for solving thedistribution.Thisisderivedfromtheconstitutiveequationsdescribedbelow.Ionic flux is due to:Diffusion +Electric Migration +Convection+.Diffusion=-DjCjMigration=-UjZjFCjConvection=CjvIonic TransportBoundary lay
32、ers+-Ionic Flux:Nj=-DjCj-UjZjFCj+Cjv +.moles/sec cm2Current density is due to flux of all charge carrying species:i=ZjFNj=-FZjDjCj-F2UjZj2Cj+F ZjCjv Electroneutrality:Current density:i=-F ZjDj Cj-Compare with(the differential form of)Ohms law:i=-I=-?V/R0jjjZCVNjZj?610MaterialbalanceThegoverningequat
33、ionsforacellwithdiffusion,migrationandconvectionarederivedbyperformingamaterialbalanceonavolumeelement,foreachoftheionicspecies:jjc=NtjR9Rj is the rate of species j generation due to a homogeneous reaction within the volumeelement.Such reactions are uncommon in electrochemical cells(may be encounter
34、ed e.g.,whenacomplexdissociates,releasingtheionicspeciesj),andthereforewesetidentically,Rj=0.When the flux expression,eq.2 is substituted into eq.9 the general equation(NernstPlanck)fortheconcentrationandpotentialfieldsisobtained.jjjjjjjc+vc =F(z u c)+(Dc)t10Foramulticomponentelectrolytewithjionicsp
35、ecies,equation10representsasystemofjequations,oneforeachionicspecies.Sincethepotentialispresentineachequation,thereareatotalofj+2unknowns(jspeciesconcentrations,cj+theelectrostaticpotential,thefluidvelocity,v).The extra equations required for solving the system are the electroneutralitycondition5,an
36、dthemomentumequationwhichdescribesthefluidvelocityatalllocationswithin thecell.Themomentum equation is typicallyrepresentedin terms of the NavierStokesapproximation(5):221()jiiTijTjijjjiVVpVVVxxxxxx 11TheNavierStokesequationiswrittenhereforaCartesiantwodimensionalcoordinatesystemwhereiandjrepresentt
37、hetwoaxes.Accordingly,viandvjarethevelocitycomponentsinthedirections i and j.P is the hydrostatic pressure,and andT are the molecular and theturbulentkinematicviscosity,respectively(5).Forsystemsinvolvingforcedconvection,thefluid flow equations are typically decoupled from the electrochemical proces
38、s,and can besolvedseparately.The set of j+2 equations(j eqs.10+electroneutrality(eq.5)+the momentumequation11)fullydescribethecurrent,potentialandconcentrationdistributionsinthecellas611a function of time.This set of equations must be solved subject to the electrochemicalboundaryconditions.Material
39、BalanceRate of accumulation =net flux in +generationjjjRNtCCbThin boundary layer:?CjjjjjjjCDCZ FU CVCt Nernst-Plank Eq.This is the fundamental equation to solveUnknowns:UnknownSymbolNumberConcentrations CjjPotentialF1VelocityV1 (vector)totalj+2Equations:NameNumberNernst-Plank jElectroneutrality1Flow
40、1 (vector)totalj+2jjjjjjjNDCUZF CC V BoundaryconditionsTwotypesofboundariesarepresentinelectrochemicalcells:(i)electrodesand(ii)insulatingboundaries.(i)Insulator:Here,nocurrentmayflowintotheboundary,andaccordingly,thecurrentdensitynormaltotheinsulatingboundaryisspecifiedaszero,i =0n12(ii)Electrodes:
41、On an electrode,an expression for the reaction kinetics that relates thepotentialtothenormalcurrentmustbeprovided:612neji=f(c,)e13ElectrodekineticsandoverpotentialsTypically,thecurrentdensityisrelatedtotheoverpotential,which is the driving force for the electrochemical reaction.The overalloverpotent
42、ialattheelectrodeisgivenby:VE14Vistheelectrodepotential.Eisthethermodynamicequilibriumpotentialcorrespondingtotheconditionofnocurrentflow,andistheelectrostaticpotentialwithinthesolutionnexttheelectrode,measuredattheouteredgeofthemasstransport(=concentration)boundarylayer.Thetotaloverpotentialattheel
43、ectrodecanbefurtherresolvedintotwooverpotentialcomponents,sandc,Thefirst,s,isthesurface(oractivation)overpotential,whichrelatesdirectlytothekineticsoftheelectrodeprocesses.Thesecondoverpotentialcomponent,c,istheconcentrationoverpotential,accountingforthevoltagedissipationassociatedwithtransportlimit
44、ations.Thesurfaceoverpotential,S,istypicallyrelatedtothecurrentdensitythroughtheButlerVolmerequation(6).0,eni =iexp()exp()ACssFFRTRT15Eq.15incorporatesthreeempiricallymeasuredparameters:theexchangecurrentdensity,i0,(givenhereintermsofitsvalueontheelectrode,i0,e)andtheanodicandcathodictransfercoeffic
45、ients,A andC,respectively.These parameters are obtained from polarizationmeasurements(7).Often,butnotalways,A+C=n.ItshouldbenotedthatwhiletheButlerVolmer equation correlates well many electrode reactions,there are numerous others,particularlywhencarriedinthepresenceofplatingadditiveswhichdonotfollow
46、it.TheButlerVolmerequationintheformpresentedbyEq.15relatestopureelectrodekinetics and does not consider transport limitations,which cause the concentration at theelectrode,ce,tovaryfromitsbulkvalue,cb.Thisconcentrationvariationaffectsmostlytwoparameters:theexchangecurrentdensity,i0,whichisafunctiono
47、ftheconcentration,andtheoverpotential,whichnowalsoincludesthecomponentassociatedwithtransportlimitations,C,.Wecanwrite=SC16613Wheretheconcentrationoverpotential,C,isgivenby:=lnnFeCbRTcc17Itshouldbenotedthatthedivisionofthetotaloverpotentialintoapurekineticscomponent,(S)andamasstransportcomponent(C)a
48、spresentedbyeq.16issomewhatarbitraryandisusedmainlytocharacterizethetwotypesofdissipativeprocesses.Bothtermsarestronglycoupledanditisverydifficulttodirectlymeasureeithercomponentseparately.Whilechemicalengineersoftendiscussthetwooverpotentialcomponentsseparately(1),chemists(e.g.,6,7)tendtocombinebot
49、htermstogetherandcharacterizetheelectrochemicalsystemintermsofthetotaloverpotential,.Toaccountfortheconcentrationvariationswhicharealwayspresentatelectrodesincurrentcarryingcells,acorrectionmustbeintroducedintoeq.15.Eitheroftwoapproachesistypicallytaken:thefirst,morecharacteristictoengineeringpublic
50、ations(1),presentseq.15as:,cAssbFFoReeRTRTo coRbbCCiieeCC 18Writtenhereforthegeneralfirstorderreaction:O+ne R19andareparametersadjustingthevalueofi0,fromitsbulkvaluetoitsvalueattheelectrodeandaregenerallydeterminedempirically.Ifthereducedspeciesdoesnotdissolvewithintheelectrolyte(as in most plating