Local electrochemical impedance spectroscopy: A review and some recent developments

To link to this article: DOI: 10.1016/j.electacta.2011.03.018

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Huang , Vicky Mei-Wen and Wu, Shao-Ling and Orazem, Mark E. and

Pébère, Nadine and Tribollet, Bernard and Vivier , Vincent Local

electrochemical impedance spectroscopy: A review and some recent

developments. (2011) Electrochimica Acta, vol. 56 (n° 23). pp. 8048-8057.

ISSN 0013-4686

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Local

electrochemical

impedance

spectroscopy:

A

review

and

some

recent

developments

Vicky

MeiWen

Huang

a,1

_,

_Shao-Ling

_Wu

a

_,

_Mark

_E.

_Orazem

a

_,

_Nadine

_Pébère

b

_,

Bernard

Tribollet

c

_,

_Vincent

_Vivier

a,∗

a_{DepartmentofChemicalEngineering,UniversityofFlorida,Gainesville,FL32611,USA}

b_{UniversitédeToulouse,CIRIMAT,UPS/INPT/CNRS,ENSIACET–4AlléeÉmileMonso,BP44362,31030ToulouseCedex04,France} c_{LISE,UPR15duCNRS,UniversitéP.etM.Curie,CP133,4PlaceJussieu,75252ParisCedex05,France}

Keywords: LEIS Mathematicalmodel Electrodeheterogeneities Electrodekinetics Ohmicimpedance

a

b

s

t

r

a

c

t

Localelectrochemicalimpedancespectroscopy(LEIS),whichprovidesapowerfultoolforexploration ofelectrodeheterogeneity,hasitsrootsinthedevelopmentofelectrochemicaltechniquesemploying scanningofmicroelectrodes.Thehistoricaldevelopmentoflocalimpedancespectroscopymeasurements isreviewed,andguidelinesarepresentedforimplementationofLEIS.Thefactorswhichcontrolthe limitingspatialresolutionofthetechniqueareidentified.Themathematicalfoundationforthetechnique isreviewed,includingdefinitionsofinterfacialandlocalOhmicimpedancesonbothlocalandglobal scales.Experimentalresultsforthereductionofferricyanideshowthecorrespondencebetweenlocal andglobalimpedances.SimulationsforasingleFaradaicreactiononadiskelectrodeembeddedinan insulatorareusedtoshowthattheOhmiccontribution,traditionallyconsideredtobearealvalue,can havecomplexcharacterincertainfrequencyranges.

Contents

1. Introduction... 8048

2. TheLEISsetup... 8050

2.1. Electrochemicaldevice... 8050

2.2. Probepreparation... 8050

3. Definitionsoflocalimpedances... 8050

4. Mathematicalfoundation... 8052

5. Potentialdifferencemeasurementandspatialresolution... 8052

6. Asimpleexample:theferri/ferrocyanidesystem... 8053

7. Influenceofthecellgeometryonlocalimpedanceresponse... 8053

8. Conclusions... 8056

AppendixA. Notation... 8056

References... 8056

1. Introduction

Electrochemicaltechniquessuchascyclicvoltammetryor elec-trochemical impedance spectroscopy (EIS) are widely used to improvetheunderstandingofmulti-stepreactions[1,2],allowing the kinetics of heterogeneous electron-transfer reactions,

cou-∗ Correspondingauthor.

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

1_{Presentaddress:IndustrialTechnologyResearchInstitute,Materialand}

Chem-icalResearchLab,Hsinchu30011,Taiwan.

pled chemical reactions, or adsorptionprocesses tobe studied. Insuchconventionalelectrochemicalexperiments,theelectrode responsetoaperturbationsignalcorrespondstoasurface-averaged measurementascribabletothebehaviourofthewholeelectrode surface.However,electrochemicalsystemsrarelyshowanideal behaviour,andthiscanleadtodifficultieswithdatainterpretation. Forinstance,inthecaseoflocalizedcorrosion,surface-averaged techniquescannotidentifythetimeofinitiationorthelocationof asingleattackamongmany.

To overcome these difficulties, several scanning techniques usingelectrodesofsmalldimensionsuchasmetalmicroelectrodes [3,4] have beendeveloped to probe insitu theelectrochemical

interface[5].Thescanningreferenceelectrodetechnique(SRET)is aratheroldtechniquewhichwasfirstintroducedbyEvansand Thornhill[6–8]in 1938.Itconsistedof measuringthepotential distributioninsolutionwithaLuggincapillaryatdifferent posi-tionsalongtheelectrodesurface,anditwassuccessfullyapplied overthepassingyearsformappingheterogeneitiesduring corro-sionprocesses[9–11].Thescanning-vibrating-electrodetechnique (SVET),firstdevelopedforinsitumonitoringofthesteady-state cur-rentdensitynearindividuallivingcellsbyJaffeandNuccitelli[12], wasalsowell-suitedforstudyingcorrodinginterfaces[13–17].Itis basedontheuseofasinglemicroelectrodewhichisvibratedinone ortwodirectionsacrossthesamplesurface,allowingthe poten-tialgradienttobemeasuredaccuratelywitha lock-inamplifier referencedtotheprobe-vibrationfrequency[5,12].Furthermore, theintense developmentofboth microelectrodes and scanning electrochemicalmicroscopy[18–22]allowedvariouskindsof inter-facetobeimagedwitharesolutioninthemicronrangeorlower, dependingonboththeelectrochemicalprobedimensionsandthe probe-to-substratedistance.Inaddition,variouskindsofprobes, suchasamperometricorpotentiometricsensors,canbeusedto selectivelydetectalargevarietyofspecies[23,24].

Theuseoflocaldc-current-densitymeasurementssuchasSVET allowssurfaceheterogeneitiestobeidentifiedbutcannotexplain thelocalreactivity.Inanattempttoevaluatethelocalimpedance ofrestricted activeareas, Isaacsand Kendig[25] pioneeredthe developmentofthescanning-probeimpedancetechnique.Inthis technique,asmallprobecontainingboththecounterandthe ref-erenceelectrodeswasrastered ataheightof about30mmover theworking electrode surface forming a thin-layer cell config-uration. These measurements allowed qualitative resultsto be acquiredonstainlesssteelweldsandoncoatedgalvanizedsteel plates.However,thecurrentmeasuredcannotbeuniquely ascrib-abletotheareaunderstudy,and,thus,noquantitativeresultscould beobtained.Moreover,suchanelectrochemical-cellconfiguration withasmallcounterelectrodefacingalargeworkingelectrodeis notsuitedforimpedancemeasurements.

Afewyearslater,theIsaacsgroupintroducedanovelmethod forgeneratingquantitativelocalelectrochemicalimpedance spec-troscopy[26,27].Thistechniqueisbasedonthehypothesisthat thelocalimpedancecanbegeneratedbymeasuringthe ac-local-currentdensityinthevicinityoftheworkingelectrodeinausual three-electrode cellconfiguration [26].This was achievedwith theuseofadualmicroelectrodeforsensingthelocalac-potential gradient,thelocalcurrentbeingobtainedfromthedirect appli-cationoftheOhm’slaw.Itwasalsoshownthat10mmdiameter platinummicroelectrodesspaced170mmapartalloweda resolu-tionof30–40mm,whichcomparedfavorablywiththecalculated resultsdescribedpreviouslyforSVETexperiments[15,27]. How-ever,theseauthorsmentionedasignificantdiscrepancybetween theoryandexperimentswhentheprobewasclosetotheelectrode underconditionswhenthebestresolutionwasexpected.Using thisexperimentalsetup,impedancediagramsoveranactivepit wereobtainedallowingthedirectcomparisonoflocalandglobal impedances[28].Avariantofthetechniqueisthemappingmodeat asinglefrequency,whichpermitsidentificationofelectrochemical activeareas[26–28]ortime-resolvedimagingforthe investiga-tionofthedynamicsoftheinterfacemodifications.Forinstance, Wittmannetal.[29]showedthatdefectsinorganiccoatingcanbe identifiedbythistechniquepriortodetectionbyvisualobservation. Bayetetal.[30–32]developedLEISmeasurementsbasedonthe useofaSVETsetup.Bythisway,theauthorstookadvantageof thesmallvibratingprobesize,allowingreductionofthescreening effectoftheprobe.Theyalsoproposedtomeasurethelocal poten-tialincombinationwiththelocalcurrenttodefinelocalimpedance [30].Suchanapproachtakesadvantageofusingauniqueprobefor sensingthepotentialattwolocations.However,oneofthemain

drawbacksistheforcedconvectioninducedbythevibrationofthe probeintheclosevicinityoftheanalyzedsurfacearea.

SomeLEISinvestigationswerealsoreportedusinglargerprobes. For instance, Baril et al. [33] investigated the electrochemical behaviouroftheAZ91magnesiumalloyinNa2SO4solution.Some

localimpedancediagramswereobtainedoveranactivepitafter4 daysimmersioninNa2SO4electrolyte.However,theresolutionof

theLEISsetupwasnotsufficienttodescribethesurfacereactivity ascribabletothealloystructureofthesample.Usingahome-made setupwithprobesoffewtensofmicrometers,Galiciaetal.[34] wereabletoobservetheinfluenceofthemicrostructureonthe cor-rosionrateoftheAZ91alloy.Thesameexperimentalset-upwith largeprobeswasusedbyJorcinetal.[35]toinvestigatethe initi-ationandpropagationofdelaminationatthesteel/organiccoating interface and also by Lima-Neto [36] to determine the exten-sionofsensitizedzonesinweldedAISI304stainlesssteel.Itwas shownthatthelengthofthesensitizedzoneintheheat-affected zonewasdetectedby theLEIStechnique.For thesetwo exam-ples,industrialsampleswereinvestigated.Thispointisimportant becauseitemphasizesthat,dependingontheprobesize, informa-tioncanbeobtainedatdifferentscales.Usingasimilartechnique, Philippeetal.[37]investigatedpolymer-coatedgalvanizedsteel. Theyshowedthattheglobalelectrochemicalimpedance measure-mentsprovidedsurface-averagedresponsescorrespondingtoboth thepolymerpropertiesanditsdefects,whereasLEISallowed coat-ingdefectstobeisolated.

Pilaskietal.[38]developedaLEISsystembasedonthe micro-capillarycelltechnique[39].Themainadvantageofthistechnique isthepossibilitytoinvestigateasmallsurfaceareaindependentlyof thesurroundingmaterial.However,thiscanbeseenasadrawback sincetheeffectofthesurroundingmaterial(forinstancegalvanic coupling)ishindered.

It shouldalsobementioned thatthedevelopmentof a spe-cificapparatuscanalsoberequiredforspecialapplicationssuchas fuelcellinvestigations[40]orforstudyingmetalhydridebattery electrodes[41].Forinstance,anexperimentalsetupallowingthe simultaneousmeasurementof10-localelectrochemicalimpedance responsestoberecordedinparallelhasbeendevelopedforthe studyofapolymerelectrolytefuelcell[42,43].Auniquefeature ofthissetupistheuseofazeroresistanceammeterbasedonthe utilizationofHalleffectcurrentsensorsforthelocalcurrent mea-surements[42].Thelocalimpactofwateranddryingeffectscould beobservedbyuseofthelocalmeasurements.

Ina recent seriesofpapers [44–46],ourgrouprevisitedthe basis of theLEIStechnique. Akey contributionwas the defini-tionofthreelocalimpedances.Thelocalinterfacialimpedance(z0)

wasdefinedtoinvolvebothalocalcurrentdensityandthelocal potentialdropacrossthe diffusedouble layer.ThelocalOhmic impedance(ze)wasdefinedtoinvolvealocalcurrentdensityand

potentialdropfromtheouterregionofthediffusedoublelayerto thedistantreferenceelectrode.Thelocalimpedance(z)wasthus thesumof thelocalinterfacialimpedanceandthelocalOhmic impedance.Theinfluenceofthecellgeometryandofprobeheight overadiskelectrodewasexploredtheoretically[47,48].The cal-culated and experimentally observed frequency dispersion and imaginarycontributionstoOhmicimpedancewereattributedto thecurrentandpotentialdistributionsassociatedwiththe geom-etry of a diskelectrode embeddedin an insulating plane.This worksuggestedthatthefrequencydispersioneffectsshouldnot beapparentforgeometries,suchasarecessedelectrode,forwhich theprimarycurrentdistributionisuniform.Itwasalsoshownthat low-frequencydispersioncanbeseenwhenadsorbedspeciesare involved[49,50].

Theobjectiveofthepresentworkistoprovideguidelinesfor implementationofLEISandreviewexperimentalandsimulation resultswhicharecharacteristicofthetechnique.

2. TheLEISsetup

A fundamental requirement for the LEISdevice is the mea-surementofthelocalcurrentdensityintheclosevicinityofthe interfaceunderinvestigation.Amongthedifferentpossibilities pre-sentedintheintroductionpart,thedualprobesystemappearstobe wellsuitedsincethefabricationofsturdymetallicmicroelectrodes (UME)isnowwellestablished.Furthermore,theresolutionofthe probeshoulddepend,infirstapproximation,onbothUME dimen-sionandtheinter-microelectrodedistance.Thesetwoparameters canbecontrolledduringthefabricationprocess.

Probesusedwithcommercialdevicesaregenerallylargeand willhavealimitedspatialresolution(somemm2_).Allthe

exper-imentalresultspresentedherewereobtainedwithahome-made system.However,asstatedbefore,thesizeoftheprobeshastobe choseninaccordancewiththesampledimension.

2.1. Electrochemicaldevice

Theexperimentalsetup(Fig.1)forperformingLEIS measure-ments consisted of a home-made potentiostat coupled with a Solartron1254four-channelfrequencyresponseanalyzer(FRA), allowingbothglobalandlocalimpedancestoberecorded simul-taneously. Two home-made analog differential amplifiers with bothvariablegainandhighinputimpedancewereusedtorecord simultaneouslythelocalpotentialandcurrentvariations.Thelocal currentwasobtainedbymeasuringthelocalpotentialdifference sensedbythetwomicroprobes;whereas,thelocalpotentialwas measured as the potential differencebetween the potential of theelectrodeand theclosestmicro referenceelectrode.The bi-electrodewasmovedwitha3-axispositioningsystem(UTM25, Newport)drivenbyamotionencoder(MM4005,Newport) allow-ing a spatial resolution of 0.2mm in the three directions. All measurementsweregenerallyperformedunderpotentiostatic reg-ulation,withtheamplitudeoftheappliedsinusoidalvoltagesetto beaslargeaspossibletoimprovetheratiosignal/noisebut suffi-cientlylowthatthelinearapproximationofthepotential/current curvecanbeused.Forinstance,someexperimentshavebeen per-formedusinga100mVpeak-to-peaksignal,50acquisitioncycles, and7pointsperdecadeoffrequency[45].Home-madesoftware developedforscanningelectrochemicalmicroscopywasusedfor dataacquisition.Oneofthegreatadvantagesofusingahome-made apparatusisthatitprovidesthecapabilityformeasuring simulta-neouslylocalandglobalimpedances,whichisnot,toourbestof knowledge,possiblewithcommercialLEISdevices.

2.2. Probepreparation

Thebi-electrodeconsistedoftwometallicwires,thedimension ofwhichcanusuallyvaryfromfewmicrometerstotensof microm-etersindiameter.InthecaseofaPtbi-electrode,wiresweresealed intothebi-capillarybymeltingtheglassusingaresistanceheater withacontrollablecurrentthroughacoilednichromewire.The tipwasthen polishedwithSiCpaperanda deposit ofPtblack fromhydrogenhexachloroplatinate(IV)wasperformeddailyon eachmicrodisktoreducetheinterfacialimpedanceofthese elec-trodes[12].Ag/AgClbi-microelectrodescanbeconstructedfrom silverwiressealedintoadualcapillaryusinganepoxyresin.The AgCllayerwasformedbyanodizingtheAgmicroelectrodeinaKCl solutionaccordingtothefollowingelectrochemicalreaction: Ag+Cl−

⇄AgCl+e− ₍₁₎

Thesilvermicroelectrodewasfirstcycledbetween−0.3 and 0.2V/SCEatarateof100mVs−1 _{ina2MKClsolutionfor}

clean-ing,andthenapotentiostaticoxidationofAgwasperformedinthe

sameelectrolyteat0.4V/SCEduring5–10min.Itshouldbe men-tionedthatthespatialresolutioncanbeadaptedtothesizeofthe areatobeinvestigatedbychangingthesizeoftheAgwiresused formeasuringthelocalcurrentandpotential.

Theelectrochemicalmeasurementsweregenerallycarriedout withaclassicalthree-electrodecellatroomtemperature.The coun-terelectrodewasalargeplatinumgrid,andthepotentialswere measuredwithrespecttoareferenceelectrodelocatedfarfrom thesampleinthesolutionbulkasshownintheschematic repre-sentationoftheexperimentalcellgiveninFig.1a.

3. Definitionsoflocalimpedances

Usinga bi-electrodefor probingthesolutionpotentialand a 4-channel frequency response analyzer, global, local, and local interfacialimpedancescanbemeasuredsimultaneously[44].In thefollowing,theuseofanupper-caselettersignifiesthatZisa globalvalue;whereas,theuseofalower-caselettermeansthatzis alocalvalue,followingthenotationproposedbyHuangetal.[44] andsummarizedintheNotationsection.Forlocalelectrochemical measurements,thelocalAC-currentdensityiloc(ω)canbeobtained

throughtheOhm’slawusing iloc(ω)=

1Vprobe(ω)

d (2)

whereistheelectrolyteconductivity,1Vprobe(ω)istheAC

poten-tial difference between the two probes, and d is the distance betweenthetwoprobes(seeFig.1b).

Thelocalimpedance(z)involvestheelectrodepotential mea-suredwithrespecttoareferenceelectrodelocatedfarfromthe electrodesurface(Fig.2).

z(ω)= V (ω)˜ −˚ref iloc(ω) = V (ω)˜ 1Vprobe(ω) d  (3)

where ˜V (ω)−˚refrepresentstheACpotentialdifferencebetween

theelectrodesurfaceandthereferenceelectrodeinthebulk solu-tion.

Thelocalinterfacialimpedance(z0)involvesthepotentialofthe

electrodereferencedtothepotentialoftheelectrolytemeasuredat theinnerlimitofthediffusionlayer.

z0(ω)= ˜ V (ω)− ˜_˚0(ω) iloc(ω) = ˜ V (ω)− ˜_˚0(ω) 1Vprobe(ω) d  (4)

Thus the local Ohmic impedance (ze) can be deduced by

calculatingthedifferencebetweenthelocalimpedanceandlocal interfacialimpedance.

ze(ω)=z(ω)−z0(ω) (5)

Fromapracticalpointofview,itisnotpossibletoperforma potentialmeasurementjustoutsidethedoublelayer(Fig.2).Thus thedistancebetweentheprobeandthesubstrate,h,mustbetaken intoaccount.Usingthedefinitionspresentedabove,thelocal inter-facialimpedance,zh(ω),estimatedaty=h,canbeobtainedusing

zh(ω)= ˜ V (ω)− ˜˚h iloc(ω) = ˜ V (ω)− ˜˚h 1Vprobe(ω) d  (6)

where ˜V (ω)− ˜_˚hrepresentstheACpotentialdifferencebetween theelectrodesurfaceandtheclosestofthetwoprobesofthe bi-electrode, locatedata distancey=hfromtheelectrode surface (Figs. 1and 2). HoweverEq. (6) is validunder theassumption thatthespreadingofthecurrentinsolutioncanbeneglected,as previouslyshown byZouandco-workers[27].ThelocalOhmic impedanceze,hcanthusbededucedbycalculatingthedifference

betweenthelocalimpedanceandlocalinterfacialimpedance,i.e. ze,h(ω)=z(ω)−zh(ω) (7)

Fig.1.(a)BlockdiagramoftheLEISsetup;(b)zoomonthemicroprobeclosetothesubstrate.Theprobe-to-sampledistanceishandthedistancebetweenthetwo microreferenceelectrodesisd.

Experimental measurements can therefore be employed to verifytheappearanceofalocalOhmicimpedancepredictedby sim-ulations.Onecanstressagainthattheuseofhome-madedevicefor performingLEISallowedsimultaneousmeasurementoftheglobal impedanceandallthelocalimpedances.

4. Mathematicalfoundation

ThemathematicaltreatmentpresentedbyHuangetal.[44] fol-lowsthat presentedby Newmanfor asimple Faradaicreaction onaplanardiskelectrodeembeddedinacoplanarinsulator[51], butitcanbeextendedtoothercellgeometries[47,48]inorderto investigatetheedgeeffectoftheelectrodeontheelectrochemical impedanceresponse,andtodifferentmechanismsinvolving inter-mediates[49,50].Thegeometryofthesystemwasdefinedbythe radiusoftheelectroder0.Thepotential˚insolutionsurrounding

thiselectrodeisgovernedbyLaplace’sequation.

∇

2˚=0 (8)

Incylindricalcoordinates(r,,y),Eq.(8)canbeexpressedas 1 r ∂ ∂r



r∂˚ ∂r



+ 1 r2 ∂2_˚ ∂2 + ∂2_˚ ∂y2 =0 (9)

whereyisthenormaldistancetotheelectrodesurface,risthe radialcoordinate,andistheazimuth.Thecylindricalsymmetry conditionrequiresthatthegeometryisinvariantunderrotation abouttheyaxis(i.e.thesymmetryaxis),thus

∂˚

∂ =0 (10)

Onthesurroundinginsulatorandfarfromtheelectrodesurface, theboundaryconditionsweregivenby

∂˚ ∂y









_y=0 =0atr>r0 (11) and ˚→0asr2_+y2_{→∞ (12)}

Forapurecapacitivebehaviour,thefluxboundaryconditionat theelectrodesurfacewaswrittenas

C0∂(V −˚0) ∂t =− ∂˚ ∂y









_y=0 (13) whereC0 istheinterfacialcapacitance,is theelectrolyte

con-ductivity,Vistheelectrodepotential,and˚0isthepotentialjust

outsidethedoublelayer.InthecaseofaCPEbehaviour,the gov-erningequationsweresimilar;theonlychangetobedonewasthe substitutionofthecapacitanceC0onthefluxboundarycondition

oftheelectrodesurfacebyQ(jω)˛_{.Formorecomplexsituations,}

applicableboundaryconditionsaredetailedinRefs.[46,49,50].The currentdensitywasthuscalculatedbyintegratingthelocal admit-tanceofthesystemovertheelectrodedisksurface.Aspreviously observedforaplanarembeddeddiskelectrode[44–46],theresults couldbeexpressedintermsofadimensionlessfrequency,K,which isdefinedby

K= Qω

˛_r 0

 (14)

inwhich˛istakentounityandQ=C0inthecaseofapurecapacitor.

Fordataanalysis,theoriginofthenormalaxisy=0isalways definedbythediskelectrodesurface,thatisallthedistancewere measuredfromthisorigin.Allcalculationswereperformedusing finiteelementmethodsoftware(COMSOL®_{)withtheconductive}

mediaDCmoduleina2Daxialsymmetry.Thegeometryandthe positionofthereferenceelectrodewereshowntoplayasignificant

10-2 10-1 100 101 102 103 104 105 106 10-12 10-10 10-8 10-6 10-4 10-2 pure capacitance _RC Δ E / V f / Hz detection limit

Fig. 3.Local potential differencemeasured at h=100mmwith a bi-electrode (d=50mm) over a planar-disk electrode. Calculations were performed for =0.01Scm−1_;r

0=0.25cm;C0=10mF;R=infinity(purecapacitance—circles)and R=1k(squares);1V=30mVpp.

roleonthenumericalresultofthecalculatedimpedance.Thus,a sphericalgeometry,andadistance2000timeslargerthanthedisk electrodedimensionwereusedtoperformnumericalcalculation withinanerrorrangesmallerthan0.2%.

5. Potentialdifferencemeasurementandspatialresolution Independent of the mechanism under investigation, LEIS is basedonthemeasurementofalocalpotentialdifferenceinthe close vicinity of the electrochemical interface. The size of the probe and the distance between the probe and the substrate aretherelevantparameters.Fig.3showsthelocalpotential dif-ference measuredover a planar-diskelectrode (r0=0.25cm)for

a probe locatedh=100mmof theinterface witha bi-electrode (distance between thetwo probes, d=50mm).The calculations were performed for =0.01Scm−1 _{and a double layer}

capaci-tanceC0=10mF.Twocaseswereconsidered:ablockingelectrode (R=infinity—circles)andR=1k,whichrepresentsa kinetically-controlledelectron-transferreaction(squaresonFig.3).Inthecase ofablockingelectrode,thepotentialdifferencedecreaseslinearly withthefrequencyandreaches1nVfor0.5Hz.As1nVisalower limitforacommercialapparatus,thisisastronglimitationofthe technique.However,whenthesysteminvolvesasingleelectron exchange,thecurveisS-shapedandtendstowards500nV.Itshould benotedthatthis lattervaluedependsonboth electrolyte con-ductivityasshownbyEq.(14)andcharge-transferresistance.As aguideline,adecreaseoftheelectrolyteconductivityallows mea-surementatlowerfrequencies.However,ifthesysteminvolves a smallertimeconstant, associated,forexample, withdiffusion or relaxation of adsorbed species, preliminaryexperiments are requiredforthedeterminationofthelowestmeasurablefrequency. Oneapproachforincreasingthespatialresolutionwouldbeto decreasetheprobesize,butthismakessenseonlyifthedistance betweenthetwoprobesalsodiminishessincethespatialresolution shoulddependonthetotalsizeoftheprobe.Fig.4showsthe influ-enceofthedistancebetweenthetwomicroreferenceelectrodes whentheprobeislocatedataposition50mmfromtheinterface. Thepotentialdifferencebetweenthetwoelectrodesdecreaseswith theinter-electrodedistanceandreachesavaluesmallerthan1nV whentheprobeseparationissmallerthan1mm.Fig.5showsthat whenaprobewiththeinter-electrodedistanceof1mmisused, thedistancebetweentheprobeandthesubstratedoesnotplaya significantroleifh>d.

10-2 10-1 100 101 102 103 104 105 106 10-12 10-10 10-8 10-6 10-4 10-2 h = 50 µm d = 50 µm d = 10 µm d = 1 µm d = 50 nm Δ E / V f / Hz

Fig.4. Localpotentialdifferencemeasuredath=50mmoveraplanar-diskelectrode withthedistancebetweenthetwosensingelectrode(d)asaparameter.Calculations wereperformedfor=0.01Scm−1_;r

0=0.25cm;C0=10mF;R=1k;1V=30mVpp.

Fromthis series of simulations,it is shown thatto increase thespatialresolution,theprobesizemustdecrease.However,a dimensioninthemicrometerrangeorlowerseemstobetheactual limit.Thecommercialprobesizes(electrodedimensionand inter-electrodedistance)aresignificantlylarger,whichhastheeffectof

10-2 10-1 100 101 102 103 104 105 106 10-10 10-8 10-6 10-4

a

d = 1 µm h = 50 µm h = 10 µm h = 1 µm f / Hz 10-2 10-1 100 101 102 103 104 105 106 9x10-9 10-8 1.1x10-8 1.2x10-8

b

d = 1 µm h = 50 µm h = 10 µm h = 1 µm Δ E / V Δ E / V f / Hz

Fig.5.(a)Localpotentialdifferencemeasuredwithabi-electrode(d=1mm)over aplanar-diskelectrodewiththeprobe-to-substratedistance(h)asaparameter. Calculationswereperformedfor=0.01Scm−1_;r

0=0.25cm;C0=10mF;R=1k; 1V=30mVpp.(b)Zoomonthepotentialdifferenceinthelowfrequencyrange.

Fig.6.Global(a)andlocal(b)impedancemeasurementsperformedsimultaneously ona0.5cmindiametercarbonelectrodeinferri-ferrocyanidesolution(10mM)+KCl (0.5M)solutionattheequilibriumpotential.Thelocalprobeconsistedoftwo40mm indiameterAg/AgClwiresat100mmoverthecenterofthecarbonelectrode.

decreasingthespatialresolution.However,useof largerprobes allowsmeasurementoveralargerfrequency domain(especially thelowerfrequencyrange).

6. Asimpleexample:theferri/ferrocyanidesystem

Fig.6showsexperimentalresultsforglobalandlocalimpedance measurementsperformedina ferri/ferrocyanidesolutionatthe equilibriumpotential.Thebi-microelectrodewhich consistedof two silver/silver chloride microelectrodes of 40mm in diame-tereach, waspositionedat100mmabovethecarbon electrode (0.5cmindiameter).Fig.6ashowstheglobalimpedance.Fromthe high-frequencyloop,theelectrolyteresistance,thechargetransfer resistanceandthedoublelayercapacitancecanbeobtained.The low-frequencybehaviourcorrespondstotheWarburgimpedance (diffusion)asexpectedforasimpleredoxsystematsteady-state. Fig.6bshows thelocalimpedancediagram performedoverthe center of the carbon electrode, simultaneously withthe global one. It is noteworthyto seethat the two diagramsare identi-cal,thusvalidatingthelocalimpedancemeasurement.Inaddition, thecharacteristic frequencies of the time constantsare similar foreachprocess.Thissimpleexperimentallowscalibrationofthe experiment.Italsoshowsthatthesameprocessescanbe charac-terizedwithbothglobalandlocalimpedancespectroscopy,taking advantageofthespatialresolutionoftheLEIStechnique.Froman experimentalpointofview,alltherequirementsforperforminga globalimpedancemeasurement(i.e.linearity,stability,and causal-ity)remainthesameforLEIS,andthevalidityofthemeasurements canbecheckedusingtheKramers-Kronigrelationshipdirectlyor throughuseofthemeasurementmodel[52–54].

7. Influenceofthecellgeometryonlocalimpedance response

Huangetal.[44–46]haveshownthattheOhmiccontributionto theimpedanceresponseofadiskelectrodeembeddedinan

insu-10-5 10-4 10-3 10-2 10-1 100 101 102 0.1 1 5 10 1 Z' κ / r0 π K J = 0.1

a

b

10-5 10-4 10-3 10-2 10-1 100 101 102 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 10 1 J = 0.1 -Z " κ / r0 π K

Fig.7. DimensionlessimpedanceresponseforadiskelectrodewithasingleFaradaicreactionunderTafelkineticswithJasaparameter:(a)realpart;and(b)imaginarypart.

latingplanetakestheformofanimpedance.Theappearanceofthe complexOhmicimpedancewasattributedtothenonuniform cur-rentandpotentialdistributionsinducedbyelectrodegeometry.For thediskembeddedinaninsulatingplane,theradialcurrentdensity isafunctionofradialpositionandhasanassociateddecreasein

cur-1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 _{J = 0.1} J = 1 J = 10 Z '' κ / r0 π /( Z "ma x κ / r0 π -0 .2 5 ) (Z'κ / r₀π-0.25)/(Z'_maxκ / r₀π-0.25) K=0.1 10 1

Fig.8.ScaledimpedanceresponseinNyquistformatfortheimpedanceresults presentedinFig.7. 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0 0.1 0.2 K=100 K=10 K=1 -z '' κ / r0 π z'κ / r 0π r/r=0 r/r=0.5 r/r=0.8 r/r=0.96 K=0.1 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0 0.1 0.2 K=100 10 K=1 -z '' κ / r0 π z'κ / r₀π r/r=0 r/r =0.5 r/r =0.8 r/r=0.96

b

a

Fig.9.LocalimpedanceresponsefortheimpedanceresultspresentedinFig.7.(a) J=1;and(b)J=10.

rentdensitywithaxialposition.Inthiscase,theOhmicimpedance mustberepresentedbyacomplexnumber.Fortherecesseddisk electrode geometry,there is no radialcurrent and the current densityisindependentofaxialposition.In thiscase,theOhmic impedancecanberepresentedbyarealnumber.Thecomplex char-acteroftheOhmicimpedanceisthereforenotonlyapropertyof electrolyteconductivity,butalsoapropertyofelectrodegeometry andinterfacialimpedance.

ThesimulationspresentedbyHuang etal.[46]areextended hereforasingleFaradaicreactionunderTafelkinetics.The electro-chemicalreactionischaracterizedbytheparameter

J(r)=˛F





¯i(r)





r0 RT = 4  Re Rt. (15)

TheglobalimpedanceresponseispresentedinFig.7withJasa parameter.WhenJislarge, thekineticsarefastandRt issmall

ascomparedtotheOhmicresistanceRe.Forallcases,theslope

ofthelinesgiveninFig.7bforlowfrequenciesisequaltounity,

0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1 0.0 0.1 -z "e κ / r0 π z'_eκ / r₀π r/r₀=0 r/r₀=0.5 r/r₀=0.8 r/r₀=0.96

a

0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1 0.0 0.1 -z "e κ / r0 π z'_eκ / r₀π r/r₀=0 r/r₀=0.5 r/r₀=0.8 r/r 0=0.96

b

Fig.10.LocalOhmicimpedanceresponsefortheimpedanceresultspresentedin Fig.7.(a)J=1;and(b)J=10.

buttheslopeisgreaterthan−1atlargefrequencies.Theresults presented in Fig.7 arecharacterizedby two frequencies:K=1, abovewhichthegeometryinfluencestheimpedanceresponseand inducespseudo-CPEbehaviour,andK/J=1,whichcorrespondsto thefrequencyω=1/RtC0.

ThescaledglobalimpedanceresultspresentedinFig.8shows thattheinfluenceofalargevalueofJistoincreasetheappearance ofadepressedsemicircleusuallyassociatedwithCPEbehaviour. Thiseffectismadedramaticbecausethecharacteristicfrequency atwhichK/J=1isgreaterthanthefrequencyK=1.

Theinfluenceofgeometryontheimpedanceresponsecanbe understoodbyexaminingthelocalimpedanceresponseshownin Fig.9aforJ=1.Thelocalimpedanceislargestatthecenterofthe electrode,consistentwiththesmallercurrentdensityseenatthe electrodecenter.Thelocalimpedanceissmallestattheperipheryof theelectrode.ThesizeoftheloopsisdecreasedinFig.9bforJ=10, consistentwiththesmallervalueofRt relativetoRe.The

high-frequencylimitforthelocalimpedance,however,isunchanged. Thehigh-frequencyinductiveand/orcapacitiveloopsseeninFig.9 areaconsequenceoftheelectrodegeometryandhave,infact,been experimentallyconfirmed.Theseloopscanbeexpressedinterms ofthelocalOhmicimpedancediscussedabove.

ThelocalOhmicimpedanceforJ=1andJ=10ispresentedin Fig.10aandb,respectively.Theloopsareinductiveatthecenterof theelectrodeandcapacitiveattheelectrodeperiphery.Again,these resultshavebeenconfirmedbyuseoflocalimpedance measure-ments.TheloopsassociatedwiththelocalOhmicimpedanceare smallerforthecasewhereJ=10,whichmayseemtobeinconflict withtheresultshowninFig.8showingthatthesemicircle

depres-10-5 10-4 10-3 10-2 10-1 100 101 102 0.1 0.2 0.3 0.4 0.5 0.6 z'e κ / r0 π K

a

b

r/r₀=0 r/r₀=0.5 r/r₀=0.8 r/r₀=0.96 10-5 10-4 10-3 10-2 10-1 100 101 102 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 -z "e κ / r0 π K r/r₀=0 r/r 0=0.5 r/r 0=0.8 r/r₀=0.96

Fig.11.LocalOhmicimpedanceresponsefortheimpedanceresultspresentedin Fig.10aforJ=1:(a)realpart;and(b)imaginarypart.

sionincreaseswithincreasingJ.Thereasonisthat,whiletheOhmic impedanceloopsaresmall,theyrepresentalargercontributionas comparedtothecontributionassociatedwiththekinetics.

Thefrequency dependence of thelocal Ohmicimpedance is showninFigs.11and12forJ=1andJ=10,respectively.At fre-quencies below K=1, the imaginary contribution, as shown in Figs.11band12b,tendstowardzero.Atfrequencieswellabove K=1,theimaginarycontribution, again,tendstowardzero.In a rangeoffrequenciesnearK=1,thelocalOhmiccontributionhas a complexcharacter.While not shown here,a similarcomplex characterisseenfortheglobalOhmiccontribution.

Thetime-constantdispersionoftheglobalimpedanceresponse diskelectrodesexhibitinggeometry-inducedcurrentandpotential distributionsisseenfordimensionlessfrequencyK=ωC0r0/>1,

whichcanbewithintheexperimentallyaccessiblerange[2].The originofthetime-constantdispersionisconsideredinthepresent worktobeacomplexOhmicimpedance.Blancetal.[55] demon-stratedthattheconceptofanOhmicimpedanceisfullyconsistent withthepioneeringcalculationoffrequencydispersionpresented by Newman in 1970. To show that this approach is in agree-mentwithNewman’sresults,theyconsideredablockinginterface witha frequency-independentcapacitanceC0 andaninterfacial

impedanceZ0=1/jωC0.Thisinterfacialimpedanceisindependent

oftheelectrodegeometry.Theoverallimpedance,whichincludes theOhmiccontribution,isZ=Ze+1/jωC0,wherethecapacitanceC0

isindependentoffrequencyandZeistermedtheOhmicimpedance.

Newman,incontrast,representedtheoverallimpedanceasthe sum of a frequency-dependent resistance Reff in series with a

10-5 10-4 10-3 10-2 10-1 100 101 102 0.1 0.2 0.3 0.4 0.5 0.6 z'e κ / r0 π K

a

b

r/r 0=0 r/r₀=0.5 r/r₀=0.8 r/r₀=0.96 10-5 10-4 10-3 10-2 10-1 100 101 102 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 -z ''e κ / r0 π K r/r₀=0 r/r₀=0.5 r/r₀=0.8 r/r₀=0.96

Fig.12.LocalOhmicimpedanceresponsefortheimpedanceresultspresentedin Fig.10bforJ=10:(a)realpart;and(b)imaginarypart.

frequency-dependentcapacitanceCeff.Thetwodescriptionsofthe

samephenomenagives Ze+ 1 jωC0 =Reff+ 1 jωCeff (16) whichyields Ze=Reff− j ω



_C0−_Ceff C0Ceff



(17) ThefactthatZeisfrequencydependentisinperfectagreementwith

Newman’sresult.Whenthefrequencytendstowardsinfinity,the currentdistributioncorrespondstotheprimarycurrent distribu-tionand lim

ω→∞Ze=1/4r0,inagreementwithNewman’sformula.

Simulationsandexperimentswereusedtoshowthatcomplex Ohmicimpedancescouldbeobservedatfrequenciessubstantially lowerthan K=1in caseswheretheFaradaicreactionsinvolved adsorbedintermediates[49,50].

8. Conclusions

The origin of local electrochemical impedance spectroscopy measurementscanbetracedbacktotheoriginaldevelopmentof scanningreferenceelectrodesin1938.ThedevelopmentofLEIS wasmotivatedbytheneedtoprobethelocalreactivityof hetero-geneoussurfaces.Thepowerofthetechniquecanbedemonstrated bytheagreementreportedbetweenexperimentalresultsand sim-ulations.Localimpedancemeasurementswereused,forexample, toconfirmtheappearanceofalocalOhmicimpedancethatwas predictedbysimulations.

LEIScanbeperformedeasilybymeasuringthelocalpotential differenceinsolution,allowingthelocalcurrenttobecalculated. Useofamultichannelfrequencyresponseanalyzerallows simulta-neousmeasurementofglobalandlocalbehaviours.Theresolution of thetechniquedepends onthesize oftheelectrodes usedin thebi-electrodeprobetosensethelocalpotentialandthe spac-ingbetweentheelectrodes.Theultimateresolutionachievableis constrainedbythesensitivityofthepotentialmeasuringcircuitry. Measurementsensitivityontheorderof1nVlimitstheresolution tothemicrometerrange.

Calculatedand measuredlocaland Ohmic impedanceshave beenshowntoprovideinsightintothefrequencydispersion asso-ciatedwiththegeometryofdiskelectrodes.Theglobalimpedance associated with a simple Faradaicreaction on a disk electrode ispurelycapacitive,butthelocalimpedancehashigh-frequency inductiveloops.The localimpedance is influenced bythelocal Ohmic impedance, which has complex behaviour near dimen-sionless frequency K=1. The imaginary part of both the local andglobalOhmicimpedancesisequaltozeroatbothhighand lowfrequencieswheretheOhmicimpedancehaspurelyresistive character.

AppendixA. Notation SymbolMeaningunits

C0 interfacialcapacitance(Fcm−2)

ddistancebetweenthetwoprobes(cm)

hdistancebetweentheprobeandthesample(cm) iloclocalaccurrentdensity(Acm2)

Kdimensionlessfrequency(K=ωC0r0/)

rradialcoordinate(cm) r0 electroderadius(cm)

Velectrodepotential(V)

1Vprobe acpotentialdifferencebetweenthetwoprobes(V)

y axialcoordinate(cm)

Zglobalimpedance(orcm2₎

Z′ _{realpartoftheglobalimpedance(orcm}2₎

Z′ _{imaginarypartoftheglobalimpedance(orcm}2₎

zlocalimpedance(cm2₎

z′ _{realpartofthelocalimpedance(cm}2₎

z′′ _{imaginarypartofthelocalimpedance(cm}2₎

ze localOhmicimpedance(cm2)

z′

e realpartofthelocalOhmicimpedance(cm2)

z′′

e imaginarypartofthelocalOhmicimpedance(cm2)

z0 localinterfacialimpedance(cm2)

z′

0 realpartofthelocalinterfacialimpedance(cm2)

z′′

0 imaginarypartofthelocalinterfacialimpedance(cm2)

zh localinterfacialimpedanceestimatedaty=h(cm2)

z′

h realpartofthelocalinterfacialimpedanceestimatedat

y=h(cm2₎

z′′

h imaginary part of the local interfacial impedance

esti-matedaty=h(cm2₎

ze,h localOhmicimpedanceestimatedaty=h(cm2)

electrolyteconductivity(−1_cm−1₎

˚ref potentialofthereferenceelectrode(V)

˚0 potentialattheinnerlimitofthediffusionlayer(V)

ω angularfrequency,ω=2fs−1

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