HAL Id: hal-00726407
https://hal.archives-ouvertes.fr/hal-00726407
Submitted on 30 Aug 2012
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Local electrochemical impedance spectroscopy: A review
and some recent developments
Vicky Mei-Wen Huang, Shao-Ling Wu, Mark E. Orazem, Nadine Pébère,
Bernard Tribollet, Vincent Vivier
To cite this version:
Vicky Mei-Wen Huang, Shao-Ling Wu, Mark E. Orazem, Nadine Pébère, Bernard Tribollet, et al..
Lo-cal electrochemiLo-cal impedance spectroscopy: A review and some recent developments. Electrochimica
Acta, Elsevier, 2011, 56, pp.8048-8057. �10.1016/j.electacta.2011.03.018�. �hal-00726407�
To link to this article: DOI: 10.1016/j.electacta.2011.03.018
URL:
http://dx.doi.org/10.1016/j.electacta.2011.03.018
This is an author-deposited version published in:
http://oatao.univ-toulouse.fr/
Eprints ID: 5557
To cite this version:
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
O
pen
A
rchive
T
oulouse
A
rchive
O
uverte (
OATAO
)
OATAO is an open access repository that collects the work of Toulouse researchers
and makes it freely available over the web where possible.
Any correspondence concerning this service should be sent to the repository
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,∗aDepartmentofChemicalEngineering,UniversityofFlorida,Gainesville,FL32611,USA
bUniversitédeToulouse,CIRIMAT,UPS/INPT/CNRS,ENSIACET–4AlléeÉmileMonso,BP44362,31030ToulouseCedex04,France cLISE,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).
1Presentaddress: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.
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 isdefinedbyK= 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 1.2x10-8b
d = 1 µm h = 50 µm h = 10 µm h = 1 µm Δ E / V Δ E / V f / HzFig.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 KFig.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=0.96b
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=0.96Fig.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=0.96Fig.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 limω→∞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(orcm2) Z′ imaginarypartoftheglobalimpedance(orcm2)
zlocalimpedance(cm2)
z′ realpartofthelocalimpedance(cm2) z′′ imaginarypartofthelocalimpedance(cm2) 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(−1cm−1)
˚ref potentialofthereferenceelectrode(V)
˚0 potentialattheinnerlimitofthediffusionlayer(V) ω angularfrequency,ω=2fs−1
References
[1]A.J.Bard,L.R.Faulkner,ElectrochemicalMethods:Fundamentalsand Applica-tions,2nded.,Wiley-VCH,NewYork,2001.
[2] C.Gabrielli,in:I.Rubinstein(Ed.),PhysicalElectrochemistry.Principles, Meth-ods,andApplications,MarcelDekker,NewYork,1995(Ch.6).
[3]R.M.Wightman,Anal.Chem.53(1981)1125A.
[4] C.Amatore,in:I.Rubinstein(Ed.),PhysicalElectrochemistry.Principles, Meth-ods,andApplications,MarcelDekker,Inc.,NewYork,1995(Ch.4).
[5]R.S.Lillard,in:P.Marcus,F.Mansfeld(Eds.),AnalyticalMethodsinCorrosion ScienceandEngineering,CRCPress,BocaRaton,2006.
[6] U.R.Evans,Nature(London,UnitedKingdom)142(1938)160. [7] R.S.Thornhill,U.R.Evans,J.Chem.Soc.(1938)2109. [8]R.S.Thornhill,U.R.Evans,J.Chem.Soc.(1938)614. [9]H.S.Isaacs,G.Kissel,J.Electrochem.Soc.119(1972)1628. [10] J.W.Campbell,R.D.Braun,Instrum.Sci.Technol.24(1996)195. [11] N.Cui,H.Y.Ma,J.L.Luo,S.Chiovelli,Electrochem.Commun.3(2001)716. [12]L.F.Jaffe,R.Nuccitelli,J.Cell.Biol.63(1974)614.
[13]M.J.Franklin,D.C.White,H.S.Isaacs,Corros.Sci.33(1992)251. [14] H.S.Isaacs,Corros.Sci.28(1988)547.
[15] H.S.Isaacs,J.Electrochem.Soc.138(1991)722. [16]R.Oltra,V.Vignal,Corros.Sci.49(2007)158.
[17]B.Vuillemin,X.Philippe,R.Oltra,V.Vignal,L.Coudreuse,L.C.Dufour,E.Finot, Corros.Sci.45(2003)1143.
[18]A.J.Bard,F.-R.F.Fan,J.Kwak,O.Lev,Anal.Chem.61(1989)132. [19]P.Sun,F.O.Laforge,M.V.Mirkin,Phys.Chem.Chem.Phys.9(2007)802. [20]S.E.Pust,D.Scharnweber,C.N.Kirchner,G.Wittstock,AdvancedMaterials,19,
Weinheim,Germany,2007,p.878.
[21]C.Gabrielli,F.Huet,M.Keddam,P.Rousseau,V.Vivier,J.Phys.Chem.B108 (2004)11620.
[22]C.Gabrielli,S.Joiret,M.Keddam,H.Perrot,N.Portail,P.Rousseau,V.Vivier,J. Electrochem.Soc.153(2006)B68.
[23]G.Denuault,G.Nagy,K.Toth,in:A.J.Bard,M.V.Mirkin(Eds.),Scanning Elec-trochemicalMicroscopy,MarcelDekker,Inc.,NewYork,2001(Ch.10). [24]M.Etienne,A.Schulte,S.Mann,G.Jordan,I.D.Dietzel,W.Schuhmann,Anal.
Chem.76(2004)3682.
[25]H.S.Isaacs,M.W.Kendig,Corrosion(Houston,TX,UnitedStates)36(1980) 269.
[26]R.S.Lillard,P.J.Moran,H.S.Isaacs,J.Electrochem.Soc.139(1992)1007. [27]F.Zou,D.Thierry,H.S.Isaacs,J.Electrochem.Soc.144(1997)1957. [28]I.Annergren,D.Thierry,F.Zou,J.Electrochem.Soc.144(1997)1208. [29]M.W.Wittmann, R.B.Leggat, S.R.Taylor, J. Electrochem.Soc.146 (1999)
4071.
[30]E.Bayet,F.Huet,M.Keddam,K.Ogle,H.Takenouti,J.Electrochem.Soc.144 (1997)L87.
[31]E.Bayet,F.Huet,M.Keddam,K.Ogle,H.Takenouti,Mater.Sci.Forum289–292 (1998)57.
[32]E.Bayet,F.Huet,M.Keddam,K.Ogle,H.Takenouti,Electrochim.Acta44(1999) 4117.
[33]G.Baril,C.Blanc,M.Keddam,N.Pebere,J. Electrochem.Soc.150(2003) B488.
[34] G.Galicia,N.Pebere,B.Tribollet,V.Vivier,Corros.Sci.51(2009)1789. [35] J.B.Jorcin,E.Aragon,C.Merlatti,N.Pebere,Corros.Sci.48(2006)1779. [36]P.Lima-Neto,J.P.Farias,L.F.Herculano,H.C.deMiranda,W.S.Araujo,J.B.Jorcin,
[37]L.V.S. Philippe, G.W. Walter, S.B. Lyon, J. Electrochem. Soc. 150 (2003) B111.
[38]M.Pilaski,T.Hamelmann,A.Moehring,M.M.Lohrengel,Electrochim.Acta47 (2002)2127.
[39]H.Krawiec,V.Vignal,R.Oltra,Electrochem.Commun.6(2004)655. [40]D.J.L.Brett,S.Atkins,N.P.Brandon,V.Vesovic,N.Vasileiadis,A.Kucernak,
Elec-trochem.SolidStateLett.6(2003)A63.
[41] P.Georen,A.K.Hjelm,G.Lindbergh,A.Lundqvist,J.Electrochem.Soc.150(2003) A234.
[42] I.A.Schneider,H.Kuhn,A.Wokaun,G.G.Scherer,J.Electrochem.Soc.152(2005) A2092.
[43] I.A.Schneider,D.Kramer,A.Wokaun,G.G.Scherer,Electrochem.Commun.7 (2005)1393.
[44] V.M.-W.Huang,V.Vivier,M.E.Orazem,N.Pebere,B.Tribollet,J.Electrochem. Soc.154(2007)C81.
[45] V.M.-W.Huang,V.Vivier,I.Frateur,M.E.Orazem,B.Tribollet,J.Electrochem. Soc.154(2007)C89.
[46] V.M.-W.Huang,V.Vivier,M.E.Orazem,N.Pebere,B.Tribollet,J.Electrochem. Soc.154(2007)C99.
[47]I.Frateur,V.M.Huang,M.E.Orazem,B.Tribollet,V.Vivier,J.Electrochem.Soc. 154(2007)C719.
[48]I.Frateur,V.M.-W.Huang,M.E.Orazem,N.Pebere,B.Tribollet,V.Vivier, Elec-trochim.Acta53(2008)7386.
[49]S.L.Wu,M.E.Orazem,B.Tribollet,V.Vivier,J.Electrochem.Soc.159(2009) C28.
[50]S.L.Wu,M.E.Orazem,B.Tribollet,V.Vivier,J.Electrochem.Soc.156(2009) C214.
[51]J.Newman,J.Electrochem.Soc.117(1970)198.
[52] P.Agarwal,M.E.Orazem,L.H.Garci-Rubio,J.Electrochem.Soc.139(1992) 1917.
[53] P.Agarwal,M.E.Orazem,L.H.Garcia-Rubio,J.Electrochem.Soc.142(1995) 4159.
[54] P.Agarwal,O.D.Crisalle,M.E.Orazem,L.H.Garcia-Rubio,J.Electrochem.Soc. 142(1995)4149.
[55] C.Blanc,M.E.Orazem,N.Pebere,B.Tribollet,V.Vivier,S.Wu,Electrochim.Acta 55(2010)6313.