Influence of normal and radial contributions of local current density on local electrochemical impedance spectroscopy.

To link to this article: DOI:10.1016/j.electacta.2011.11.053

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To cite this version:

Ferrari, J.V. and De Melo, Hercílio Gomes and Keddam, M. and Orazem,

Mark E. and Pébère, Nadine and Tribollet, Bernard and Vivier , Vincent

Influence of normal and radial contributions of local current density on

local electrochemical impedance spectroscopy. (2012) Electrochimica

Acta, vol. 60 . pp. 244-252. ISSN 0013-4686

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Influence

of

normal

and

radial

contributions

of

local

current

density

on

local

electrochemical

impedance

spectroscopy

J.V.

Ferrari

a,b,c

_,

_H.G.

_De

_Melo

c

_,

_M.

_Keddam

a,b

_,

_M.E.

_Orazem

d

_,

_N.

_Pébère

e

_,

_B.

_Tribollet

a,b

_,

_V.

_Vivier

a,b,∗

a_{CNRS,UPR15,LaboratoireInterfacesetSystèmesElectrochimiques,F-75005Paris,France} b_{UPMCUnivParis06,UPR15,LISE,4placeJussieu,F-75005Paris,France}

c_{ChemicalEngineeringDepartmentofthePolytechnicSchooloftheSãoPauloUniversity,P.O.Box61548,CEP05424-970,SãoPaulo,SP,Brazil} d_{DepartmentofChemicalEngineering,UniversityofFlorida,Gainesville,FL32611,USA}

e_{UniversitédeToulouse,CIRIMAT,UPS/INPT/CNRS,ENSIACET,4,alléeEmileMonso,BP44362,31030Toulousecedex4,France}

Keywords: LEIS Currentdistribution Microprobes Blockingelectrode

a

b

s

t

r

a

c

t

Anewtri-electrodeprobeispresentedandappliedtolocalelectrochemicalimpedancespectroscopy (LEIS)measurements.Asopposedtotwo-probesystems,thethree-probeoneallowsmeasurementnot onlyofnormal,butalsoofradialcontributionsoflocalcurrentdensitiestothelocalimpedancevalues. Theresultsconcerningthecasesoftheblockingelectrodeandtheelectrodewithfaradaicreactionare discussedfromthetheoreticalpointofviewforadiskelectrode.Numericalsimulationsandexperimental resultsarecomparedforthecaseoftheferri/ferrocyanideelectrodereactionatthePtworkingelectrode disk.Atthecentreofthedisk,theimpedancetakingintoaccountbothnormalandradialcontributions wasingoodagreementwiththelocalimpedancemeasuredintermsofonlythenormalcontribution.At theperipheryoftheelectrode,theimpedancetakingintoaccountbothnormalandradialcontributions differedsignificantlyfromthelocalimpedancemeasuredintermsofonlythenormalcontribution.The radialimpedanceresultsattheperipheryoftheelectrodeareingoodagreementwiththeusual expla-nationthattheassociatedlargercurrentdensityisattributedtothegeometryoftheelectrode,which exhibitsagreateraccessibilityattheelectrodeedge.

1. Introduction

Local electrochemical techniques such as scanning elec-trochemical microscopy (SECM), scanning vibrating electrode technique(SVET)orlocalelectrochemicalimpedancespectroscopy (LEIS)arenowwidelyusedforthecharacterizationofsurface reac-tivity[1–6],thedeterminationofelectrontransferkinetics[7–9], as wellasstudyingcomplex electrochemicalreactions[10–12]. Amongthesetechniques,localelectrochemicalimpedance spec-troscopy(LEIS),pioneeredbyIsaacsetal.[13,14],takesadvantage ofelectrochemicalimpedancespectroscopytoprobethelocal elec-trochemicalreactivityofaninterface[15–17].Fromthisseminal work,differentapproaches havebeenenvisionedfor measuring LEIS.Themostwidespreadprocedureemploysabi-electrodeto sensesimultaneouslythelocalpotentialattwolocationsabovethe substrate[18,19].Thiscanbeachievedusingalargebi-electrode(Pt electrodesinthemillimetrerangewithaninter-electrodedistance offewmillimetresarecommerciallyavailable)[20–23]orsmaller

∗ Correspondingauthorat:CNRS,UPR15,LaboratoireInterfacesetSystèmes Elec-trochimiques,F-75005Paris,France.

E-mailaddress:vincent.vivier@upmc.fr(V.Vivier).

probesthatconsistoftwomicro-electrodesembeddedinaglass capillary[14,19,24,25].Inthelattercase,Ag/AgClmicro-reference electrodescanbeused[19].Thelocalcurrentdensityisthus calcu-latedfromthislocalpotentialdifferenceandtheOhm’slawforthe electrolyte.

AnadaptationofSVETtoACpolarizationwasalsodevisedand tookadvantageoftheuseofasingleprobethatisvibrated,allowing afineandeasycontroloftheinter-electrodedistancebycontrolling theamplitudeofthevibration[26,27].Thistechnique,however, suffersfromlocalconvectioninducedbytheprobevibrationand fromthecontributionoftheredoxpotentialatametallicelectrode [26].SincethedepositofAg/AgClattheapexofthevibratingtip resultsinafragileprobe,nosignificantimprovementinthe mea-surementofthelocalpotentialcanbereached.

Acommonfeatureoftheaboveconfigurationsisthatonlythe normalcomponentoftheAC-currentinsolutionismonitored,i.e., curvature ofequipotentialsurfacesin solutionis nottakeninto account.Thereforetheresultingestimateforlocalcurrentdensity isvalidonlywhenthesensingprobeislocatedabovethecentreof thedomainofinterest;otherwisethelocalcurrentinsolutionisthe vectorsumofboththenormalandtheradialcomponents.In addi-tion,inspiteofthecapabilityofSVETtomeasuresimultaneously thenormalandradialdc-currentcomponents[28–32],tothebest

ofourknowledge,onlythenormalcomponentofthecurrenthas beenconsideredforthemeasurementofLEIS.

Morerecently,somedevelopmentsofthemicrocelltechnique havebeenreported[33,34],whereonlyasmallareaofanelectrode isisolatedbyaglasscapillaryandplacedindirectcontactwiththe electrolyteforperformingLEISmeasurements.Itshouldbenoted thatusingamicrocapillary,equipotentialsurfacesareconstrained bythecapillarygeometry,thusequipotentialsurfacesareparallel totheelectrodesurfacealongthecylinderbetweentheworking electrodeandthecounterelectrode.Withsuchadevice,local mea-surementcanbeperformed,buttheelectrochemicalresponsedoes notaccountforthesurroundingenvironmentofthelocaldomain analyzed.

The objective of the present work is to report on the use of a tri-electrode system which is used for the first time to performLEIS measurements.A theoretical framework is devel-oped for a classical blocking electrode and for an electrode withfaradaicreaction,allowingadirectcomparisonwith previ-ousworks[24,25,35–37].Then,experimentalinvestigationfor a modelsystem(ferri/ferrocyanideredoxcoupleatPtelectrode)was performedtoillustratetherelevanceofthesimultaneous measure-mentofbothnormalandradialcurrentcomponents.

2. Experimental

Theinstrumentation,thenewtri-electrodecurrentsensor,and chemicalsusedintheexperimentalworkaredescribedbelow.

2.1. Instrumentation

The LEIS apparatus consisted of an in-house made device witha3-axispositioningsystem(UTM25,Newport)drivenbya motionencoder(MM4005,Newport)allowingaspatialresolution of0.2mminthethreedirections.Thepotentialswerecontrolledby ahome-madepotentiostat.Theexperimentalsetupwascomputer controlledby a single software developed under theLabview®

environment.Useofafour-channelfrequencyresponseanalyzer (Solartron–FRA1254)allowedbothglobalandlocalimpedances toberecordedsimultaneously[24,36,38].Home-madelow-noise analogdifferentialamplifierswithbothvariablegainandhighinput impedanceweredevelopedforrecordingboththelocalpotentials andcurrentvariations.

TheLEISexperimentswereperformedusing50mV peak-to-peaksinewaveperturbation,50acquisitioncyclesoverafrequency rangeof65kHzto100mHzwith7pointsperdecades.

2.2. Tri-electrodecurrentsensor

Thetri-electrodeconsistedofthreesilvermicrowiresof100mm indiameter,eachofthemlaterallyinsulatedusingacataphoretic paint(fewmicrometersthick),andthensealedinacapillaryglass withanepoxyresin.Theelectrodearrangementwasoptimizedfor themeasurementofbothnormalandradiallocalcurrentdensities (Fig.1).Suchasetupisequivalenttothevibratingprobedeveloped forSVETexperiments,exceptthattheuseofamultisensoravoided localforcedconvection.Theapexoftheelectrodewaspolishedwith SiCemerypaperupto1200grade,andanelectrochemicaldeposit ofAgClwasperformedbypotentiostaticoxidationofAginaKCl 0.5Msolutiononeachelectrode.Thissetofreference microelec-trodesallowedsimultaneousmeasurementofthreelocalpotentials intheclosevicinityofthesubstrate.Itshouldbementionedthat, aspreviouslydemonstrated,therelevantparametersforthelocal measurementsarethesizeofeachprobe,thedistancebetweentwo probes,andthetip-to-sampledistance[15].

Fig.1.Opticalimageofasilvertri-electrodeusedforLEISmeasurement.Eachwire is100mmindiameter.

2.3. Chemicalsandsamples

All the experiments were performed using analytical grade chemicals as received. Electrolytic solutions were prepared in twice-distilledwater(18Mcm).A10mMferri/ferrocyanide solu-tionwaspreparedina0.5MKClelectrolyte.APtworkingelectrode waslaterallyinsulated witha cathaphoreticpaint,heat treated for 1h at 150◦_{C, and then molded into an epoxy resin}

(Buh-ler,EpoxycureTM_{).Thiselectrodewassecuredatthebottomofa}

TefloncellwithanO-ringlargerthanthenominalPtdiameter.The counterelectrodewasalargeplatinumgauzesurroundingallthe electrochemicalcellinordertominimizecurrentandpotential dis-tributionsduetothecellgeometry.Asaturatedcalomelelectrode (SCE)wasusedasreferenceelectrode.Allpotentialsarereported withrespecttothisreferenceunlessotherwisestated.

3. Mathematicalmodels

Themathematicalmodelsdevelopedforthisworkandthe defi-nitionsemployedinthedescriptionoftheimpedanceresponseare presentedbelow.First,thedefinitionsemployedin the descrip-tionoftheimpedanceresponsearepresented.Then,simulations weredeveloped for ablocking electrodeand an electrodewith Faradaicreaction.Allsimulationswereperformedusingfinite ele-mentmethod,implementedinComsolMultiphysicsonaPC. 3.1. Definitions

Previousworksdefinedglobal,local,localinterfacial,andlocal ohmic impedances using a bi-electrode for probing the solu-tionpotentialanda multi-channelfrequency responseanalyzer [35–37].ThelocalAC-currentdensityiloc(ω)wasobtainedthrough

theOhm’slawusing[13]: i_loc(ω)= 1Vprobe(ω)

d (1)

where  is the electrolyte conductivity, 1Vprobe(ω) is the

AC-potentialdifferencebetweenthetwoprobes,anddisthedistance betweenthetwoprobes.AsshowninFig.2,forthediskgeometry, thelocalcurrentdensityinsolutionisthevectorsumofradialand normalcontributions: in(ω)=1V n_(ω) dn (2) ir_(ω)=1Vr(ω) dr (3)

Fig.2. Currentandpotentialdistributionsinthevicinityofaninlaiddiskelectrode.

where1Vn_{isthepotentialdifferencemeasuredbytheprobeina}

directionnormaltotheelectrodesurface,and1Vr_{isthepotential}

differencemeasuredbytheprobeinaradialdirection,parallelto theelectrodesurface.

Thetwolocalimpedances,zn_(ω)andzr_{(ω),involvetheelectrode}

potentialmeasuredwithrespecttoareferenceelectrodelocatedfar fromtheelectrodesurface:

zn_(ω)= V (ω)˜ −˚ref in_(ω) = ˜ V (ω) 1Vn_(ω) dn  (4) zr_(ω)= V (ω)˜ −˚ref ir_(ω) = ˜ V (ω) 1Vr_(ω) dr  (5)

where ˜V (ω)−˚refrepresentstheAC-potentialdifferencebetween

theworkingelectrodeandthereferenceelectrodeinthebulk solu-tion.

Thetwolocalinterfacialimpedances,zn

0(ω)andz0r(ω),involve

thepotentialof theelectrode referencedtothepotentialofthe electrolytemeasuredattheinnerlimitofthediffuselayer. zn₀(ω)= V (ω)˜ − ˜˚0(ω) in_(ω) = ˜ V (ω)− ˜˚0(ω) 1Vn_(ω) dn  (6) zr 0(ω)= ˜ V (ω)− ˜˚0(ω) ir_(ω) = ˜ V (ω)− ˜˚0(ω) 1Vr_(ω) dr  (7)

Thus, thelocal Ohmic impedances, zn

e(ω) and zer(ω), can be

deducedbycalculatingthedifferencebetweenthelocalimpedance andlocalinterfacialimpedanceforbothnormalandradial contri-butions.

zne(ω)=zn(ω)−zn0(ω) (8)

zr

e(ω)=zr(ω)−zr0(ω) (9)

Following previous developments in this area [34–37], the Ohmic impedance can be a complex number. Throughout this paper,theresultsareexpressedintermsofadimensionless fre-quency,K,whichisdefinedby:

K= C0ωr0

 (10)

whereC0istheinterfacialcapacitanceandr0istheelectroderadius.

3.2. Blockingelectrode

Thepotential˚insolutioninthevicinityofaninlaiddisk elec-trodeisgovernedbytheLaplace’sequation.

∇

2˚=0 (11)

Using cylindrical coordinates and taking into account the cylindricalsymmetrycondition,thepotentialdistributioncanbe expressedas: ∂2_˚ ∂r2 + 1 r



∂˚ ∂r



+∂ 2_˚ ∂y2 =0 (12)

whereyisthenormaldistancetotheelectrodesurface,andristhe radialcoordinate.Asablockingelectrodecanbedescribedbyapure capacitivebehaviour,thefluxboundaryconditionattheelectrode surfacewaswrittenas:

C0 ∂(V−˚0) ∂t =− ∂˚ ∂y









_y=0 (13)

whereistheelectrolyteconductivity,Vistheelectrodepotential, and˚0isthepotentialjustoutsidethedoublelayer.Onthe

sur-roundinginsulatorandfarfromtheelectrodesurface,theboundary conditionsweregivenby:

∂˚ ∂y









_y=0 =0 for r>r0 (14) and: ˚→0 when r2+y2→∞ (15)

3.3. ElectrodewithFaradaicreaction

The second model describes an electrochemical reaction of redoxmoietyinsolutionthatalsotakesintoaccountcurrentand potentialdistributions.Aone-stepelectrochemicalreaction involv-ingasingleelectronexchangewasassumedtooccuratthedisk interface,e.g.,

ox+e−

⇄red (16)

Thereactionrate(

v

)followstheButler–Volmerrelationship:

v

=k0



_C redexp

h

˛F RT(V−E 0₎

i

_−C oxexp

h

−(1−˛)F RT(V−E 0₎

i

(17) where Cred and Cox are the interfacial concentrations of

elec-troactivespecies,k0 _{isthestandardrateconstant,˛thetransfer}

coefficient,E0_{thestandardpotential.Iftheelectrochemicalcellis}

assumedtobeconvectionfreeduringthedurationofthe exper-iment,themasstransportofelectroactivespeciesisgovernedby thesecondFick’slaw:

∂Ci ∂t =Di



∂2_C i ∂r2 + 1 r



∂Ci ∂r



+∂ 2_C i ∂y2



(18)

wherethesubscriptiaccountsforoxorred,andDiisthe

diffu-sioncoefficientofthespeciesi.Fortheelectrochemicalimpedance simulations,thisproblemwassplitintwocontributions:a steady-state contributionand a harmonic contribution.The latter was obtained from the linearization of Butler–Volmer equation as alreadydescribedelsewhere[39,40].Thus,thesystemtobesolved consistedinfivecoupleddifferentialequations(twoforeachredox moietydescribingthesteady-stateandtheharmoniccontributions, andoneforthepotential).Thefaradaicadmittancewasobtained byintegratingthefluxalongtheradialdirectionoftheelectrode foreachangularfrequency(ω=2f).

Fig.3.Localnormalandlocalradialimpedances(a),andlocalnormalandlocal radialinterfacialimpedances(b)calculatedabovetheelectrodesurface(ata dis-tancey=100mm)closetothecentreofthedisk(r=100mm)withC0=10mFand =0.01S/cm.

4. Resultsanddiscussion

Simulations were performed for a blocking electrode, and bothsimulationsandexperimentsweredoneforamodel ferri-cyanide/ferrocyanideredoxsystem.

4.1. Blockingelectrode

In thefollowing, thecase of a blockingelectrode is investi-gated.Itconsistsofadiskelectrodeof0.25cminradius,embedded in an infinite insulator and which behaves as a purecapacitor (C0=10mF).Theelectrolyteconductivityis=0.01S/cm.

Calculatednormalandradiallocalelectrochemicalimpedances arepresentedinFig.3aforaprobepositionedabovetheelectrode surface(ata distancey=100mm)and closetothecentreofthe disk(r=100mm).Thehigh-frequencylimitofthespectrumisabout 0.5forthenormallocalimpedance,andismuchlarger(about 320)inthecaseoftheradiallocalimpedance.Similarly,inthe caseofthelocalinterfacialimpedanceshowninFig.3b,thehigh fre-quencylimitisabout12mforthenormalcontribution;whereas,

Fig.4.Localnormalandlocaltotalimpedances(a);andlocalnormalandlocal totalOhmic impedances(b)calculatedabove theelectrodesurface(ata dis-tancey=100mm)closetothecentreofthedisk(r=100mm)withC0=10mFand =0.01S/cm.

itisabout8fortheradialcontribution.Itshouldbenotedthat, inbothcases,theratioofthehighfrequencylimitisthesame.

Thelocalimpedancecalculatedfrombothnormal andradial contributionsundertheconditionsdescribedforFig.3iscompared inFig.4atothelocalimpedancecalculatedonlywiththenormal contributionofthecurrentdensity.Diagramsareverysimilarin bothshapeandintensity.Theerrormadebyusingonlythenormal contributionisabout0.2%inthehighfrequencies,andabout1.5%in themediumfrequencies.ThelocalOhmicimpedancediagrams cal-culatedfrombothnormalandradialcontributionarecomparedin Fig.4btothelocalOhmicimpedancecalculatedonlywiththe nor-malcontributionofthecurrentdensity.Asalreadyobservedinthe caseofthenormalcomponentonly[35–37],thediagramsexhibita singleinductiveloopinthehigh-frequencydomainwiththesame timeconstantthantheoneobservedforthelocalimpedance.Thus, abovetheelectrodecentre,theuseofonlythenormalcurrent den-sityforthedeterminationofthelocalimpedancesresultsinminor error,usuallyintherangeoftheexperimentalerror.

The local impedance calculated close to the electrode edge (r=0.24cm) is presented in Fig. 5a. In that case, both local impedancescalculatedfromthenormalorfromtheradialcurrent

Fig.5.Localnormal,localradial,andlocaltotalimpedances(a),andlocalnormal, localradial,andlocaltotalinterfacialimpedances(b)calculatedabovetheelectrode surface(atadistancey=100mm)closetotheedgeofthedisk(r=0.24cm)with C0=10mFand=0.01S/cm.

densityareofthesameorderofmagnitude.Asaresult,thelocal impedanceclosetotheelectrodeedgeissmallerthantheone cal-culatedwithonlyonecontribution.Forinstance,ifonlythenormal currentdensityisconsidered,theerrorislargerthan50%inthe wholefrequencyrange.Suchabehaviourisattributedtothe geom-etryoftheelectrode,whichexhibitsagreateraccessibilityatthe electrodeedge.ThesametrendsareshowninFig.5bforthelocal interfacialimpedance.However,thedifferencebetweentheradial andthenormallocalinterfacialimpedancesislargerbecause,close totheelectrodesurface,thecurvatureofthecurrentlinesisless significant.

ThelocalOhmicimpedancecalculated closetotheelectrode edgeisshowninFig.6a.ThelocalOhmicimpedancecalculated withthenormalcurrentdensityismainlycapacitivewithasmall inductive feature in thehigh frequencies. Conversely, the con-tribution calculated with the radial current density only is an inductiveloopinahigherfrequencyrange.Thus,thelocalOhmic impedanceis characterizedbytwo time constantsasshown in Fig.6b.

Thus in the case of a blocking electrode, the geometry of the cell cannot be neglected. The radial component of the

Fig.6.Localnormal,localradial,andlocaltotalOhmicimpedancescalculated abovetheelectrodesurface(atadistancey=100mm)closetotheedgeofthe disk(r=0.24cm)withC0=10mFand=0.01S/cm;zoomonthelocaltotalOhmic impedance.

current is to be considered when the probe is close to the boundarybetweentheconductingpartoftheelectrodeandthe surrounding insulator. It is also interesting to note that both contributionsofthelocalcurrentdensityresultintwodifferent time-constantsthatcanbeobservedinthelocalOhmicimpedance spectra.

4.2. Ferri/ferrocyanideataPtelectrode

Experimentsandsimulationswerealsoperformedonamodel system (Fe(CN)64−/Fe(CN)63−) in order to investigate the

con-tribution of normal and radial current densities on the most commoncontributionssuchaschargetransferresistanceand dif-fusionimpedances.Threesuccessiveglobalimpedancediagrams, performed with a 0.25cm in radius Pt electrode in a 10mM ferri/ferrocyanideand0.5MKClsolutionattheequilibrium poten-tial,areshowninFig.7.ThePtdiskelectrode wasnotpolished norelectrochemicallyactivatedinordertodecreasetherateofthe electronexchangereaction(i.e.,causinga largerchargetransfer resistance).Thediagramswereobtainedinabsenceorinpresence ofthetri-electrodeintheclosevicinityofthesubstrate.Itcanbe

Fig.7.Globalimpedancemeasurements(successiveexperiments)onaPt elec-trodeimmersedinaferri/ferrocyanidesolutionwiththetri-electrodeclosetothe electrodecentreorclosetotheelectrodeedgeandwithoutthetri-electrode.

seenthatevenifthepresenceoftheprobeundoubtedlymodifies thecurrentandpotentialdistributionsatthedisk,nosignificant effectisseen.

ExperimentallocalimpedancediagramsareshowninFig.8a fora probe positionclosetotheedge orclosetothecentre of theelectrode. Thesediagramshavethesameshape asdoesthe globalmeasurement(Fig.7)andhavesimilartimeconstants. How-ever,theimpedanceclosetotheelectrodeedgeislarger,which correspondstosmallercurrentdensityattheelectrodeedgethan atthecentre.Thisbehaviourwaspreviouslyobservedby perform-ingsystematiclocalimpedancemeasurementsalonganelectrode radius of AZ91Mg alloy[41]. Due tothe edge effect, one can imaginethatthelocalcurrentdensityshouldbelargeratthe elec-trodeedgethanatthecentre[42,43],thustheradialcontribution cannotbe neglected. Thiswasconfirmedbythe useofthe tri-electrode(Fig.8b),whichallowstheradiallocalimpedancetobe measuredsimultaneouslywiththenormalcontribution.The low-frequencylimit(0.1Hz)showninFig.8bisabout8timeslargerat theelectrodecentrethanattheedge,butthegeneralshapeofthe signalcorrespondstoahigh-frequencycapacitiveloopattributed tothechargetransferresistanceinparallelwiththedoublelayer capacitance,and theWarburg impedance inthe low frequency domainfor thediffusion ofelectroactivespecies. In addition, a highfrequencyinductivefeatureappearswhentheprobeisabove theelectrodecentre.Suchabehaviourwaspreviouslyattributed tothediskgeometryofthesubstrate[35–37].Simulationswere performedtoaccountforsuch abehaviourusingfiniteelement methods to describe simultaneously diffusion of electro-active speciesandcurrentandpotentialdistributionsinthe electrochemi-calcell.Allcalculationswereperformedwithadiffusioncoefficient of10−5_cm2_{/s,aheterogeneousrateconstantof10}−3_cm/s,a

trans-fercoefficientof0.5,anequimolarconcentrationofredoxmoieties (10−2_{mol/l)atip-to-substratedistanceof100mm,an}

interelec-trodedistanceof100mm,C0=10mFand=0.01S/cm;andresults forbothradialandnormallocalimpedanceareshowninFig.9.The simulationresultsareinagreementwiththeexperimentalones, showingthatwhenthemeasurementisperformedabovethe elec-trodecentre,thenormalcomponentofthelocalcurrentdensity (thus,thenormallocalimpedance)describeswithagood approx-imationthelocalbehaviour.However,closetotheelectrodeedge, theradialcurrentdensityincreasesandcannolongerbeneglected. Itshouldbenotedthatsuchbehaviourdoesnotapplyonlyforadisk electrode.Itcanbeeasilyextendedtomorecomplicatedsystem, e.g.,formultiphaseelectrodesforwhichdiscontinuityexistsateach boundarybetweengrains.Infact,theinfluenceoftheradial com-ponentofthecurrentwaspreviouslyarguedinthecaseofgalvanic couplingbetweentwometalsinordertoexplainthevariationsin thehighfrequencydomainoflocalimpedancemeasuredwitha bi-electrode[44,45].

Fig.8. Localnormal(a)andlocalradialimpedances(b)experimentallyobtainedat thecentreandclosetotheedgeoftheelectrodeinferri/ferrocyanidesolutionatthe equilibriumpotential.

However,somedifferencesbetweenexperimentaland simu-lateddiagramscanbenoticed.Inparticular,theamplitudeofthe diagramsisdifferentandthereisa smallshiftofthe character-isticfrequencies.Thisisduetolimitationsofthesimulation for whichwecannotextendthenumberofmeshesthroughoutthe wholedomain.Tocircumventthisproblem,wehaveused geomet-ricparameters(probe-to-substratedistance,distancebetweenthe twoprobes)forthecalculationsslightlydifferentfromthe exper-imentalconditions.Accordingly,thesimulationsaccountenough fortheobservedexperimentalvariationsandvalidatedthe contri-butionsofthenormalandradialcontributionsonlocalimpedance measurements.Theobjectivewasnottofitaccuratelythe experi-mentalresults.

Fig. 10a shows the experimental local normal Ohmic impedances measured at the centre and at the edge of the electrode.Inbothcase,twotimeconstantsareobservedat650Hz andat1Hz.Itshouldbementionedthatthisimpedancechanges frominductivetocapacitivebehaviourfromthecentretotheedge,

Fig.9. Simulationoflocalnormal(a)andlocalradial(b)impedancesatthe cen-treandclosetotheedgeforafaradaicsysteminvolvingdiffusion;(c)zoomof thelocalradialimpedanceclosetotheedge.Calculationswereperformedwith D=10−5_cm2_/s,k

0=10−3cm/s,˛=0.5,Cox=Cred=10−2mol/l,atip-to-substrate

dis-tanceof100mm,aninterelectrodedistanceof100mm,C0=10mFand=0.01S/cm.

respectively. The HFtime constant should beattributed tothe relaxationofchargetransferresistanceinparallelwiththedouble layercapacitance,andthelowfrequencyloopistobelinkedtothe diffusionprocess.

Fig.10. Experimentallocal normalohmic impedance (a)andlocal radial (b) impedancesatthecentreandclosetotheedgeinferri/ferrocyanidesolutionat theequilibriumpotential.

Theradialcontributionswerealsomeasured(Fig.10b).Close to the electrode edge, this local Ohmicimpedance exhibits an additional high-frequencytime constant, which isattributedto the geometry of the electrode, and a shift towards smaller value of the time constant is also observed. Above the elec-trode centre, no reliable measurement could be obtained (see below).

Thenumericalsimulationsofthetwocomponentsofthelocal OhmicimpedancearepresentedinFig.11a.Agoodagreementwith theexperimentalresultspresentedinFig.10forboththeshapeand thefrequencyshiftisobserved.AsshowninFig.11b,theradiallocal Ohmicimpedanceisverylarge(inthemegaOhmrange),which cor-respondstoverysmallcurrentdensity.Thisexplainswhytheradial contributionstolocalOhmicimpedancecouldnotbemeasuredat thecentreoftheelectrode,andwhyonlythecontributionatthe electrodeedgeisreportedinFig.10b.

Fig.11.Simulationof localnormal ohmicimpedance(a)and localradial (b) impedancesatthecentreandclosetotheedgeforafaradaicsysteminvolving diffusion.SameconditionsthanforFig.9.

5. Conclusions

Thenumericalsimulationsshowthatthecontributionofthe radialcurrenttotheimpedanceresponseissignificantnearthe electrode peripheryfor electrodesexhibiting bothblocking and faradaicbehaviour.Thus,neartheelectrodeperiphery,thelocal impedancecalculatedabovetheelectrodesurfaceusingonlythe normalcurrentcomponentislargerthanwouldbeobtainedusing thecorrectcurrentdensitycontainingbothradialandnormal com-ponents.Atthecentreoftheelectrode,theradialcomponentof currentdensitymaybeneglected,andthelocalimpedance calcu-latedusingonlythenormalcurrentcomponent isequaltothat whichwouldbeobtainedusingboth radialand normal compo-nents.

Thedevelopmentofatri-electrodeprobehasallowed measure-mentofaxialandradialcontributionstolocalcurrentdensityandto localimpedancevalues.Atthecentreofadiskelectrode,theradial componentcontributedlittletotheoverallimpedanceresponse, and theimpedancetakinginto accountboth normaland radial contributionsisingoodagreementwiththelocalimpedance mea-suredintheusualwayintermsofonlythenormalcontribution.At theperipheryoftheelectrode,theradialcomponentofthecurrent densityissignificant,andtheimpedancetakingintoaccountboth normalandradialcontributionsdifferssignificantlyfromthelocal impedancemeasuredintheusualwayintermsofonlythe nor-malcontribution.Thus,abi-microelectrodemaybeusedtoobtain quantitativedatawhentheprobeislocatedabovethecentreofthe electrode;whereas,onlyqualitativeinformationcanbeobtained abovetheedge.Theradialimpedanceresultsattheperipheryof theelectrodeareingoodagreementwiththeusualexplanation thattheassociatedlargercurrentdensityisattributedtothe geom-etryoftheelectrode,whichexhibitsagreateraccessibilityatthe electrodeedge.Inaddition,theinductiveandcapacitivebehaviour observedonthelocalOhmicimpedancefortheblockingelectrode arelinkedtoradialandnormallocalcurrentdensity,respectively. Acknowledgments

M.E.OrazemacknowledgesfinancialsupportfromtheCharles A.Stokesprofessorship,andJ.FerrariacknowledgeshisPh.D.grant supportedbyCAPES.

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