Citation
Influence of Electrode Geometry on Local and Global Impedance Response

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Title:
Influence of Electrode Geometry on Local and Global Impedance Response
Creator:
Wu, Shao-Ling
Place of Publication:
[Gainesville, Fla.]
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (176 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Chemical Engineering
Committee Chair:
Orazem, Mark E.
Committee Members:
Chauhan, Anuj
Martin, Charles R.
Vivier, Vincent
Graduation Date:
8/7/2010

Subjects

Subjects / Keywords:
Capacitance ( jstor )
Current density ( jstor )
Electric current ( jstor )
Electrodes ( jstor )
Electrolytes ( jstor )
Geometry ( jstor )
Kinetics ( jstor )
Mass transfer ( jstor )
Rotating disks ( jstor )
Silver ( jstor )
Chemical Engineering -- Dissertations, Academic -- UF
adsorbed, disk, double, electrode, impedance, intermediate, layer
Genre:
Electronic Thesis or Dissertation
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
Chemical Engineering thesis, Ph.D.

Notes

Abstract:
DISSERTATION TITLE: INFLUENCE OF ELECTRODE GEOMETRY ON LOCAL AND GLOBAL IMPEDANCE RESPONSE ABSTRACT: Distributions of current and potential associated with the electrode geometry are essential issues in studying the electrochemical systems. The nonuniform distributions that cause time-constant dispersion along the electrode surface can obscure results from the electrochemical measurements and lead to an incorrect interpretation of experimental data. The electrode configuration of interest is a disk electrode embedded in an insulator, which is one of the most popular geometries used in the electrochemical measurements. The geometry effect can be observed at high frequencies for a blocking disk electrode and for a disk electrode subject to a single Faradaic reaction. The present study involves more complicated electrode processes that include, first, a coupled Faradaic reactions by an adsorbed intermediate, and then incorporate the nonuniform mass transfer on a rotating disk electrode (RDE) for a general redox reaction. On a stationary disk electrode, while the frequency or time-constant dispersion due to the dependence of the radial distribution of interfacial potential was shown to have an effect at high frequencies, the time-constant dispersion was also found to influence the impedance response at low frequencies due to the potential dependence of the fractional surface coverage of the adsorbed intermediate. The geometry effects were reflected in values for the local Ohmic impedance, which had complex behavior at both high and low frequencies. The dispersion of time constant was described in terms of a local constant-phase element (CPE) that represented the impedance response at low frequencies as well as at high frequencies. The geometry effect can be eliminated by use of a recessed electrode on which the current and potential distributions are uniform. Experimental verification was obtained by applying local electrochemical impedance spectroscopy (LEIS) on an iron disk electrode immersed in a 1 M sulfuric acid solution. The nonuniform mass transfer distribution together with the effect of electrode geometry was investigated on a RDE below the mass-transfer-limited current. A two-dimensional impedance model was proposed to study the influence of nonuniform current and potential distributions associated with both mass-transfer and Ohmic effects on the global and local impedance response. The concentration and potential distributions were calculated simultaneously throughout the system domain by taking into account the transport of species from diffusion, electric migration, and convection. Under the assumption that the Faradaic reaction and the charging of the double layer cannot be separated a priori, part of the flux of reacting species contributes to the charging of the interface as well as to Faradaic reaction. A double-layer model following the Gouy-Chapman type of double layer was used to assess the charge density on electrode without specific adsorption, and to evaluate interfacial properties, such as the double-layer capacitance and the change of charge associated with the variation of ionic concentrations. The local interfacial impedance showed a depressed semicircle that cannot be attributed to the geometry-induced current and potential distributions. The appearance of CPE behavior was attributed to the frequency dependent effective double-layer capacitance that accounted for the contribution of flux in charging the double layer. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2010.
Local:
Adviser: Orazem, Mark E.
Statement of Responsibility:
by Shao-Ling Wu.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Embargo Date:
10/9/2010
Resource Identifier:
004979745 ( ALEPH )
705931189 ( OCLC )
Classification:
LD1780 2010 ( lcc )

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INFLUENCEOFELECTRODEGEOMETRYONLOCALANDGLOBALIMPEDANCE RESPONSE By SHAO-LINGWU ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2010

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c 2010Shao-LingWu 2

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Tomyparents. 3

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ACKNOWLEDGMENTS Isincerelythankmyadvisor,Prof.MarkOrazem,whohasalwaysbeenpatient andencouragingduringmydoctoralresearchattheUniversityofFlorida.Notonlyhis enthusiasmandwideknowledgeshowedmethebeautyofresearchinelectrochemistry, butalsohisgreatpersonalitiestaughtmetobecomeabetterperson.TheyearsIspent intheresearchgroupwilldenitelybenettherestofmylife. IwouldliketothankDr.BernardTribolletandDr.VincentVivierfromtheCNRS LaboratoryinParisfortheirconstanthelpandadvicesduringvariousstagesofthis researchwork.IalsothankProf.AnujChauhanandProf.CharlesMartinfortheir suggestionsonmyproposalanddissertation.Ithankmylovelygroupmembers,Vicky Huang,SunilRoy,PatrickMcKinney,BryanHirschorn,andErinPatrickwhogaveme lotsofmemoriesduringmydoctoralyearsintheresearchgroup.Ireallyappreciatetheir companyandfriendship. Mydeepestgratitudegoestomyfamilyfortheircontinuousloveandsupport throughoutmylife.Thisdissertationisdedicatedtothem. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS..................................4 LISTOFTABLES......................................8 LISTOFFIGURES.....................................9 LISTOFSYMBOLS....................................17 ABSTRACT.........................................21 CHAPTER 1INTRODUCTION...................................23 2ELECTROCHEMICALIMPEDANCESPECTROSCOPY.............27 2.1Constant-PhaseElement...........................29 2.2CurrentandPotentialDistributions......................30 2.2.1PrimaryDistribution..........................31 2.2.2SecondaryDistribution.........................33 2.2.3TertiaryDistribution...........................35 2.3LocalElectrochemicalImpedanceSpectroscopy..............35 2.3.1ExperimentalConguration......................35 2.3.2DenitionofTerms...........................37 3LITERATUREREVIEW...............................40 3.1Geometry-InducedCurrentandPotentialDistributions...........40 3.2NonuniformMassTransferonaRotatingDiskElectrode..........42 4ELECTRODEREACTIONWITHADSORBEDINTERMEDIATES:MODEL...45 4.1MathematicalDevelopment..........................46 4.1.1LinearKineticsneartheEquilibriumPotential............46 4.1.2TafelKineticsforAnodicReactions..................48 4.1.3PotentialDistribution..........................50 4.2CalculatedImpedanceResults........................54 4.2.1GlobalImpedance...........................56 4.2.2LocalInterfacialImpedance......................60 4.2.3LocalOhmicImpedance........................62 4.2.4LocalImpedance............................65 4.3ValidationofCalculations...........................68 4.4EvaluationofCPEExponent.........................70 5

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5ELECTRODEREACTIONWITHADSORBEDINTERMEDIATES:EXPERIMENTAL.......................................74 5.1DissolutionofIron...............................74 5.2ExperimentalSetup..............................76 5.3ExperimentalResults.............................77 5.4SimulationResults...............................83 5.5Discussion...................................86 6INFLUENCEOFMASSTRANSFERONSTEADY-STATECURRENTAND POTENTIALDISTRIBUTIONS...........................91 6.1MathematicalDevelopment..........................91 6.1.1MassTransportinDiluteSolutions..................91 6.1.2FluidFlowonaRotatingDiskElectrode...............92 6.1.3FluxandCurrentatElectrodeBoundary...............95 6.2NumericalSimulation.............................97 6.3SimulationResults...............................100 6.3.1PotentialandConcentrationProlesnearElectrodeSurface....100 6.3.2CurrentDistributiononElectrodeSurface..............103 7MODELOFELECTRICALDOUBLELAYER...................109 7.1TheGouy-Chapman-SternTheory......................109 7.2NumericalApproach..............................113 7.3SurfaceChargeintheDiffusePartoftheDoubleLayer...........113 7.3.1VariationofSurfaceChargewithPotential:Double-LayerCapacitance...................................117 7.3.2VariationofSurfaceChargewithConcentration...........122 7.3.3VariationofExcessConcentrationofIndividualSpecieswithConcentration................................126 7.4CouplingofDouble-LayerChargingWithMassTransfer..........128 8INFLUENCEOFMASSTRANSFERONIMPEDANCERESPONSE......133 8.1MathematicalDevelopmentofImpedanceModel..............133 8.1.1No APriori SeparationofFaradaicandChargingCurrents.....134 8.1.2 APriori SeparationofFaradaicandChargingCurrents.......135 8.2NumericalSimulation.............................136 8.3CalculatedImpedanceResponsesforReductionofFerricyanide.....136 8.3.1GlobalImpedance...........................137 8.3.2LocalImpedance............................138 8.3.3LocalInterfacialImpedance......................138 8.3.4LocalOhmicImpedance........................141 8.4CalculatedImpedanceResponsesforDepositionofSilver.........141 8.4.1GlobalImpedance...........................144 8.4.2LocalInterfacialImpedance......................148 6

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8.4.3LocalOhmicImpedance........................153 8.4.4LocalImpedance............................153 8.5EffectiveDouble-LayerCapacitance.....................158 8.6Discussion...................................162 9CONCLUSIONS...................................164 9.1InuenceofAdsorbedIntermediates.....................164 9.2InuenceofNonuniformMassTransfer....................165 10SUGGESTIONSFORFUTUREWORK......................168 REFERENCES.......................................170 BIOGRAPHICALSKETCH................................176 7

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LISTOFTABLES Table page 4-1Thevaluesofkineticparametersusedforthesimulations.............53 4-2ValuesofCPEexponent calculatedusingequations4and4for thehigh-frequencyandlow-frequencyimpedanceloopsinFigure4-5......72 5-1Parametersusedinthesimulationsofirondissolutioninsulfuricacidsolution..83 6-1Parametersusedforcalculatingthesteady-statecurrentandpotentialdistributionsonaRDEatroomtemperature.......................98 8-1Effectivedouble-layercapacitanceforthecaseNAPSatdifferentpositions onelectrodeandthesurface-averageddouble-layercapacitanceforthecase APS.TheeffectivecapacitanceforNAPSwerecalculatedatthecharacteristicfrequencyassociatedwiththeFaradaicreactionandthechargingofthe interface.Theunitforthecapacitanceis F/cm 2 ..................160 8-2Comparisonofresultsunderdifferentsimulationconditions............162 8

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LISTOFFIGURES Figure page 1-1Onlinerecordsfortheuseofdiskelectrodesintheareaofelectrochemistry providedbyEngineeringVillagedatabasebeforeJune,2010...........24 2-1Perturbationofanelectrochemicalsystematsteadystatewithasmallsinusoidalsignal,where 4 V and 4 I representthepotentialandcurrentoscillatingatthesamefrequency .Thephasedifferencebetweenpotentialand currentis .......................................28 2-2NyquistplotofimpedancedatacorrespondingtoaRCcircuitof R t =100 cm 2 and C 0 =10 F = cm 2 .Theresistancefromtheelectrolyteisnotshownin thisgure........................................29 2-3Schematicrepresentationofatime-constantdistributionAattheelectrodeelectrolyteinterface,andBintheoxidelayer,where R t isthecharge-transfer resistance, C 0 isthedouble-layercapacitance,and R f and C f aretheresistanceandcapacitanceofanoxidelm.......................31 2-4Primarycurrentandpotentialdistributionsonadiskelectrode. 1 .........32 2-5SecondarycurrentdistributiononadiskelectrodeforAlinearkineticsandB Tafelkinetics. 1 ....................................34 2-6Schematicrepresentationoftheelectrochemicalcellusedtoperformlocal electrochemicalimpedancemeasurements. 2 ...................36 2-7Localequivalentcircuitscorrespondtolocalimpedancesthatvarywiththe radialpositionsontheelectrodesurface.......................37 4-1Impedanceplotsinresponsetodifferentsignsof A ................50 4-2Thedomainusedforthenite-elementsimulations.Thesolidlinesrepresent steady-stateiso-potentialplanes,anddashedlinesrepresentsteady-state trajectoriesforowofcurrent.............................54 4-3Radialdistributionofthenormalizedsteady-statefractionalsurfacecoverageonadiskelectrode:Aforpositivesurface-averagedvaluesof h A i ;B fornegativesurface-averagedvaluesof h A i ;Ctherelationshipbetweenthe surface-averagedvaluesof h A i reportedinpartsAandBandtheapplied steady-stateelectrodepotential;andDthecorrespondingsurface-averaged valuesof h J i totheappliedsteady-stateelectrodepotential............55 9

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4-4ThevariationofAsteady-statesurfacecoveragedensity,andBsteady-state currentdensitywiththeinterfacialpotential.Dashedsquaresareusedtoidentifytherangeofcurrentandsurfacecoveragecorrespondingtothesimulationsperformedat m = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 15 V h A i =0 : 011 S/cm 2 s, m = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 1 V h A i =0 S/cm 2 s,and m =0 : 1 V h A i = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 83 S/cm 2 s.Theposition r =0 correspondstothelower-leftcornerofeachbox.....................57 4-5CalculatedNyquistrepresentationoftheglobalimpedanceresponsefora diskelectrodeconsideringtheinuenceofelectrodegeometrysolidlines andintheabsenceofgeometryeffectdashedlines:A h A i > 0 0 : 011 S/cm 2 s; B h A i =0 ;andC h A i < 0 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 83 S/cm 2 s.....................59 4-6Thevalueof R t ; e =R t and R ; e =R evaluatedfromtheglobalimpedanceasa functionof h J i .....................................60 4-7CalculatedNyquistrepresentationofthelocalinterfacialimpedanceresponse ofadiskelectrodewithnormalizedradialposition r=r 0 asaparameter:A h A i > 0 0 : 011 S/cm 2 s;B h A i =0 ;andc h A i < 0 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 83 S/cm 2 s........61 4-8CalculatedNyquistrepresentationofthelocalOhmicimpedanceresponseof adiskelectrodewithnormalizedradialposition r=r 0 asaparameter:A h A i > 0 0 : 011 S/cm 2 s;B h A i =0 ;andC h A i < 0 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 83 S/cm 2 s...........63 4-9CalculatedrealandimaginarypartsofthelocalOhmicimpedanceresponse ofadiskelectrodeasafunctionofdimensionlessfrequency K :Arealpart for h A i =0 : 011 S/cm 2 s;Bimaginarypartfor h A i =0 : 011 S/cm 2 s;Creal partfor h A i =0 ;Dimaginarypartfor h A i =0 ;Erealpartfor h A i = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 83 S/cm 2 s;andFimaginarypartfor h A i = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 83 S/cm 2 s...............64 4-10CalculatedNyquistrepresentationofthelocalimpedanceresponseofadisk electrodewithnormalizedradialposition r=r 0 asaparameter:A h A i > 0 0 : 011 S/cm 2 s;B h A i =0 ;andC h A i < 0 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 83 S/cm 2 s............66 4-11Calculatedrepresentationoftheimaginarycomponentofthelocalimpedance responseonadiskelectrodeasafunctionofdimensionlessfrequency K :A h A i > 0 0 : 011 S/cm 2 s;B h A i =0 ;andC h A i < 0 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 83 S/cm 2 s........67 4-12Theimpedanceresponseforarecessedelectrode:A h A i > 0 ;andB h A i < 0 .69 4-13Agraphicalrepresentationofasingleimpedanceloopseparatedintoahigherfrequencyhfhalfandalower-frequencylfhalf..................71 5-1Thepolarizationcurveforastationaryironelectrodein0.5MH 2 SO 4 :Aa scanincludingboththeactiveandpassivatedregions;Bzoomedportionof theboxinpartAshowingthepotentialatwhichtheimpedancemeasurementswereperformed................................77 10

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5-2Globalelectrochemicalresponseforapureironelectrodein0.5MH 2 SO 4 :A globalimpedanceresponsemeasuredatpotentialIasindicatedinFigure51B;andBglobalimpedanceresponsemeasuredatpotentialII.........78 5-3Experimentallocalimpedanceforapureironelectrodein0.5MH 2 SO 4 measuredatApotentialI;andBpotentialIIinFigure5-1B..............79 5-4LocalOhmicimpedanceforapureironelectrodein0.5MH 2 SO 4 atpotentialIinFigure5-1B.ALocalOhmicimpedanceatthecenterandtheedge oftheelectrode;Btheenlargementforthelow-frequencyendofthelocal Ohmicimpedanceatelectrodecenter;andCasimilarenlargementforthe electrodeedge.....................................81 5-5TheimaginarypartofthelocalOhmicimpedanceasafunctionofthedimensionlessfrequencyforapureironelectrodein0.5MH 2 SO 4 measuredatpotentialIinFigure5-1B.................................82 5-6LocalOhmicimpedanceforapureironelectrodein0.5MH 2 SO 4 atpotential IIinFigure5-1B.ANyquistplotforthelocalOhmicimpedancesatcenterof theelectrode;andBimaginarypartofthelocalOhmicimpedanceasafunctionofdimensionlessfrequency...........................82 5-7ThecalculatedglobalimpedancecorrespondingtoApolarizationpointIand BpointIIinFigure5-1B.TheseresultsaretobecomparedtotheexperimentalresultspresentedinFigure5-2Aand5-2B.................84 5-8ThecalculatedlocalimpedancecorrespondingtoApolarizationpointIand BpointIIinFigure5-1B.TheseresultsaretobecomparedtotheexperimentalresultspresentedinFigure5-3........................85 5-9ThelocalOhmicimpedancecalculatedatpotentialIinFigure5-1B.ALocalOhmicimpedanceatthecenterandtheedgeoftheelectrode;Btheenlargementforthelow-frequencyendofthelocalOhmicimpedanceattheelectrodecenter;andCtheelectrodeedge.Theseresultsaretobecomparedto theexperimentalresultspresentedinFigure5-4..................87 5-10TheimaginarypartofthelocalOhmicimpedanceasafunctionofthedimensionlessfrequencycalculatedatpotentialIinFigure5-1B.Thisplotistobe comparedtotheexperimentalresultpresentedinFigure5-5...........88 5-11TheimaginarypartofthelocalOhmicimpedanceasafunctionofthedimensionlessfrequencycalculatedatpotentialIIinFigure5-1B.Thisplotistobe comparedtotheexperimentalresultpresentedinFigure6-1B..........88 11

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5-12CalculatedlocalOhmicimpedanceforthecase h A i =0 presentedbyWu et al. 3 ANyquistplotforthelocalOhmicimpedancesatthecenteroftheelectrode;andBimaginarypartofthelocalOhmicimpedanceasafunctionof thedimensionlessfrequency.TheseresultsaretobecomparedtotheexperimentalresultspresentedinFigure5-6.......................89 6-1Velocityprolesonarotatingdiskelectrode:AradialandBaxialcomponentsofthedimensionlessvelocity.........................94 6-2Meshedmodelusedtocalculatethepotentialandconcentrationdistributions inthewholedomainandneartheelectrode-insulatorinterface..........99 6-3Steady-statepotentialandconcentrationdistributionsforAreductionofferricyanideandBdepositionofsilveratone-fourthoflimitingcurrentonarotatingdiskelectroderotatingat120rpm.Thesurfacecolorsrepresentthe concentrationvariationsofAferricyanideandBsilverion.Thecontourlines representthepotentialdistributions.Theinsertsshowthepotentialandconcentrationprolesnearelectrodesurface......................101 6-4ProlesofApotentialandBconcentrationatelectrodecenterforreduction offerricyanideonarotatingdiskelectrodeofrotationspeed120rpmandat one-fourthofthelimitingcurrent...........................102 6-5ProlesofApotentialandBconcentrationatelectrodecenterfordeposition ofsilveronarotatingdiskelectrodeofrotationspeed120rpmandatonefourthofthelimitingcurrent..............................102 6-6Polarizationbehaviorforthereductionofferricyanideandoxidationofferrocyanideonarotatingdiskelectrode.........................103 6-7Calculatedcurrentdistributionsforthereductionofferricyanideonarotating diskelectroderotatingatA120,B600,andC2400rpm............104 6-8Polarizationbehaviorforthedissolutionanddepositionofsilveronarotatingdiskelectrodeofrotationspeed120rpm.Thesolutionconsistsof0.1M AgNO 3 and1M,0.1M,and0.01MofsupportingelectrolyteKNO 3 .......105 6-9Calculatedcurrentdistributionsforsilverdepositiononarotatingdiskelectroderotatingat120rpm.Thesolutioncontains0.1MAgNO 3 andA1M,B 0.1M,andC0.01MKNO 3 .............................107 7-1Thestructureofelectricaldoublelayer.Thesketchisnottoscale........110 7-2Calculatedtotalchargedensityinthediffusepartofthedoublelayerandthe contributionofeachionicspecies.Theelectrolyticsolutioncontains0.01M K 3 FeCN 6 ,0.01MK 4 FeCN 6 and1MKCl.....................114 12

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7-3Calculatedaveragedionicconcentrationsattheouterlimitofthediffuse.The electrolyticsolutioncontains0.01MK 3 FeCN 6 ,0.01MK 4 FeCN 6 and1M KCl...........................................114 7-4Calculatedtotalchargedensityinthediffusepartofthedoublelayerandthe contributionofeachionicspecies.Theelectrolyticsolutioncontains0.1M AgNO 3 andA1M,andB0.01MKNO 3 ......................115 7-5Calculatedaveragedionicconcentrationsattheouterlimitofthediffuse.The electrolyticsolutioncontains0.1MAgNO 3 andA1M,andB0.01MKNO 3 ..116 7-6Potentialdependenceofdouble-layercapacitanceforsystemofferro/ferricyanide redoxcoupleinthepresenceofanexcesssupportingelectrolyte.........118 7-7Radiallydependentdouble-layercapacitancecalculatedatdifferentfractions oflimitingcurrentforreductionofferricyanideinthepresenceofanexcess supportingelectrolyte.................................119 7-8Potentialdependenceofdouble-layercapacitanceforasilverelectrodeinsolutioncontaining0.1MAgNO 3 anddifferentconcentrationsofsupportingelectrolytewhenthesystemisAatequilibriumandBnotatequilibrium......120 7-9Radiallydependentdouble-layercapacitancecalculatedatdifferentfractions oflimitingcurrentforsilverdepositioninsolutioncontaining0.1MAgNO 3 and A1M,B0.1M,andC0.01MKNO 3 .......................121 7-10Potentialdependenceof @q m =@c i; 0 inAlinearscaleandBlogarithmscale forreductionofferricyanideandoxidationofferrocyanide.............122 7-11Potentialdependenceof @q m =@c i; 0 inAlinearscaleandBlogarithmscale fordepositionanddissolutionofsilver........................123 7-12Radialdistributionsof @q m =@c i; 0 forreductionofferricyanideatdifferentfractionsoflimitingcurrentforAferrocyanide,Bferricyanide,Cchloride,and Dpotassiumions...................................124 7-13Radialdistributionsof @q m =@c i; 0 fordepositionofsilveratdifferentfractionsof limitingcurrentforA,C,Epotassiumandsilverions,andB,D,Fnitrate. Thesolutioncontains0.1MAgNO 3 andA,B1M,C,D0.1M,andE,F 0.01MKNO 3 ......................................125 7-14Potentialdependenceof @ )]TJ/F27 7.9701 Tf 7.314 -1.793 Td [(i =@c i; 0 forreductionofferricyanideandoxidation offerrocyanide.....................................127 7-15Potentialdependenceof @ )]TJ/F27 7.9701 Tf 7.314 -1.793 Td [(i =@c i; 0 forApotassiumandsilverions,andBnitrateionfordepositionanddissolutionofsilver...................127 13

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7-16Radialdistributionsof @ )]TJ/F27 7.9701 Tf 7.314 -1.793 Td [(i =@c i; 0 forreductionofferricyanideatdifferentfractionsoflimitingcurrentforAferrocyanide,Bferricyanide,Cchloride,and Dpotassiumions...................................129 7-17Radialdistributionsof @ )]TJ/F27 7.9701 Tf 7.314 -1.793 Td [(i =@c i; 0 fordepositionofsilveratdifferentfractionsof limitingcurrentforA,C,Epotassiumandsilverions,andB,D,Fnitrate. Thesolutioncontains0.1MAgNO 3 andA,B1M,C,D0.1M,andE,F 0.01MKNO 3 ......................................130 8-1NyquistrepresentationoftheglobalimpedanceresponseforreductionofferricyanideonarotatingdiskelectroderotatingatA120rpmandB600rpm..137 8-2Realandimaginarycomponentsoftheglobalimpedanceresponseforreductionofferricyanideatthree-fourthsoflimitingcurrentonarotatingdiskelectroderotatingat120rpm...............................138 8-3Nyquistrepresentationofthelocalimpedanceforreductionofferricyanideon arotatingdiskelectroderotatingat120rpmcalculatedatA,Bone-fourth, C,Done-half,andE,Fthree-fourthsoflimitingcurrent.Theenlargements ofthehigh-frequencyinductiveloopsareshowninB,D,andE........139 8-4Nyquistrepresentationofthelocalinterfacialimpedanceforreductionofferricyanideonarotatingdiskelectroderotatingat120rpmcalculatedatA,B one-fourth,C,Done-half,andE,Fthree-fourthsoflimitingcurrent.The enlargementsofthehigh-frequencyinductiveloopsareshowninB,D,and E...........................................140 8-5NyquistrepresentationofthelocalOhmicimpedanceforreductionofferricyanideonarotatingdiskelectroderotatingat120rpmandatAone-fourth, Bone-half,andCthree-fourthsoflimitingcurrent.................142 8-6ImaginarycomponentsofthelocalOhmicimpedanceforreductionofferricyanideonarotatingdiskelectroderotatingatA,B,C120rpm,D,E,F 600rpm,andG,H,I2,400rpm..........................143 8-7Nyquistrepresentationoftheglobalimpedanceresponsefordepositionof silveratone-fourthoflimitingcurrentonarotatingdiskelectroderotatingat 120rpm.Thesolutionconsistsof0.1MAgNO 3 andA1MandB0.01M KNO 3 asasupportingelectrolyte..........................145 8-8Calculatedrealandimaginarycomponentsoftheglobalimpedanceresponse asafunctionoffrequencyfordepositionofsilveratone-fourthoflimitingcurrentonarotatingdiskelectroderotatingat120rpm:Arealpartfor1MKNO 3 Brealpartfor0.01MKNO 3 ,Cimaginarypartfor1MKNO 3 ,andDimaginarypartfor0.01MKNO 3 ..............................146 8-9NormalizedOhmicresistanceofelectrolyticsolutionconsistingofAgNO 3 and KNO 3 asafunctionofsilverdepositionrate.....................147 14

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8-10Axialdistributionofsolutionconductivityatelectrodecenterfordepositionof silver.Theelectrolyticsolutionconsistsof0.1MAgNO 3 andA1MandB 0.01MKNO 3 ......................................148 8-11Normalizedeffectiveconductivityattheinterfaceasafunctionofsilverdepositionrate........................................149 8-12Nyquistrepresentationofthelocalinterfacialimpedancefordepositionofsilveronarotatingdiskelectroderotatingat120rpmcalculatedatAone-fourth, Bone-half,andCthree-fourthsoflimitingcurrent.Theelectrolyticsolution consistsof0.1MAgNO 3 and1MKNO 3 ......................150 8-13Nyquistrepresentationofthelocalinterfacialimpedancefordepositionofsilveronarotatingdiskelectroderotatingat120rpmcalculatedatAone-fourth, Bone-half,andCthree-fourthsoflimitingcurrent.Theelectrolyticsolution consistsof0.1MAgNO 3 and0.01MKNO 3 .....................151 8-14Nyquistrepresentationoftheinterfacialimpedanceatelectrodecenterfordepositionofsilveratone-fourthoflimitingcurrentonarotatingdiskelectrode rotatingat120rpm.Thesolutionconsistsof0.1MAgNO 3 andA1MandB 0.01MKNO 3 asasupportingelectrolyte......................152 8-15NyquistrepresentationofthelocalOhmicimpedancefordepositionofsilver onarotatingdiskelectroderotatingat120rpmcalculatedatAone-fourth, Bone-half,andCthree-fourthsoflimitingcurrent.Theelectrolyticsolution consistsof0.1MAgNO 3 and1MKNO 3 ......................154 8-16NyquistrepresentationofthelocalOhmicimpedancefordepositionofsilver onarotatingdiskelectroderotatingat120rpmcalculatedatAone-fourth, Bone-half,andCthree-fourthsoflimitingcurrent.Theelectrolyticsolution consistsof0.1MAgNO 3 and0.01MKNO 3 .....................155 8-17ImaginarycomponentsofthelocalOhmicimpedancefordepositionofsilveronarotatingdiskelectroderotatingat120rpmcalculatedatA,Bonefourth,C,Done-half,andE,Fthree-fourthsoflimitingcurrent.Thesolutionconsistsof0.1MAgNO 3 andA,C,E1MandB,D,F0.01MKNO 3 ..156 8-18Nyquistrepresentationofthelocalimpedancefordepositionofsilveronarotatingdiskelectroderotatingat120rpmcalculatedatAone-fourth,Bonehalf,andCthree-fourthsoflimitingcurrent.Theelectrolyticsolutionconsists of0.1MAgNO 3 and1MKNO 3 ...........................157 8-19Nyquistrepresentationofthelocalimpedancefordepositionofsilveronarotatingdiskelectroderotatingat120rpmcalculatedatAone-fourth,Bonehalf,andCthree-fourthsoflimitingcurrent.Theelectrolyticsolutionconsisted of0.1MAgNO 3 and0.01MKNO 3 ..........................159 15

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8-20Effectivedouble-layercapacitancefordepositionofsilveronarotatingdisk electroderotatingat120rpmandatone-fourthoflimitingcurrent.Thesolutionconsistsof0.1MAgNO 3 andA0.01MandB1MKNO 3 ..........161 8-21Effectivedouble-layercapacitanceforreductionofferricyanideonarotating diskelectroderotatingat120rpmandatone-fourthoflimitingcurrent.The solution........................................161 16

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LISTOFSYMBOLS Roman A lumpedparameterinexpressionforimpedanceresponseofreactionswith adsorbedintermediate,S/cm 2 s,seeequation4 A constantusedforthevelocityexpansionforarotatingdiskinequations612 and6, A = 0.934 a constantusedforthevelocityexpansionforarotatingdiskinequations6 and6, a = 0.51023 B lumpedparameterinexpressionforimpedanceresponseofreactionswith adsorbedintermediate,1/s,seeequation4 B constantusedforthevelocityexpansionforarotatingdiskinequations6 and6, B = 1.208 b constantusedforthevelocityexpansionforarotatingdiskinequation6, b = 0.616 b kineticparameter, F=RT or )]TJ/F26 11.9552 Tf 11.955 0 Td [( F=RT ,V )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 C 0 double-layercapacitance, F/cm 2 c constantusedforthevelocityexpansionforarotatingdiskinequations612 and6, c = 0.88447 c i concentrationofspecies i ,mol/cm 3 D i diffusioncoefcientofspecies i ,cm 2 /s F dimensionlessuidvelocityintheradialdirection F Faraday'sconstant,96,487C/equiv f frequency,Hz H dimensionlessuidvelocityintheaxialdirection I totalcurrent,A i currentdensity,A/cm 2 i 0 exchangecurrentdensity,A/cm 2 17

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i C double-layerchargingcurrentdensity,A/cm 2 i F Faradaiccurrentdensity,A/cm 2 i lim mass-transfer-limitedcurrentdensity,A/cm 2 J dimensionlessexchangecurrentdensity j complexnumber, p )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 K dimensionlessfrequency k reactionrateconstant,A/cm 2 orAcm/mol n numberofelectronstransferredinelectrodereaction N i uxofspecies i ,mol/cm 2 s Q CPEcoefcient,s / cm 2 q surfacechargedensity,C/cm 2 R universalgasconstant,8.314J/molK r radialdistance,cm r 0 radiusofdiskelectrode,cm R e electrolyteorOhmicresistance, or cm 2 R t charge-transferresistance, or cm 2 s i stoichiometriccoefcientofspecies i T temperature,K t time,s u i mobilityofspecies i ,cm 2 mol/Js v uidvelocity,cm/s V interfacialpotential,V V 0 equilibriumpotential,V r uidvelocityintheradialdirection,cm/s y uidvelocityintheaxialdirection,cm/s y axialdistance,cm Z globalimpedance, or cm 2 18

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Z 0 globalinterfacialimpedance, or cm 2 Z e globalOhimcimpedance, or cm 2 Z F globalFaradaicimpedance, or cm 2 z localimpedance, cm 2 z 0 localinterfacialimpedance, cm 2 z e localOhmicimpedance, cm 2 z F localFaradaicimpedance, cm 2 z i chargeassociatedwithspecies i Greek CPEexponent constantusedintheinterpolationfunction6toweighthevelocityexpansionsforarotatingdisk symmetryfactor thickness,cm dielectricconstant 0 permittivityofvacuum,8.8542 10 )]TJ/F24 7.9701 Tf 6.586 0 Td [(14 F/cm totaloverpotential,V c concentrationoverpotential,V s surfaceoverpotential,V )]TJ0 g 0 G/F20 11.9552 Tf 41.853 0 Td [(maximumsurfacecoverage,mol/cm 2 )]TJ/F27 7.9701 Tf 7.314 -1.793 Td [(i surfaceexcessconcentrationofspecies i ,mol/cm 2 fractionalsurfacecoverage electrolyteconductivity,S/cm electrolyteviscosity,cm 2 /s rotationspeedofdisk,rad/s angularfrequency,rad/s electricpotential,V 19

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phaseangle,radordegree dimensionlessdistancefromthedisk GeneralNotation h X i averagedvalueofXoverelectrodesurface e X oscillatingpartof X t Re f X g realpartofX X steady-statepartof X t Subscripts 0 attheinnerlimitofdiffusionlayer a anodichalfreaction c cathodichalfreaction d inthediffuselayer ihp atinnerHelmholtzplane j imaginarypartofimpedance m atmetalsurface ohp atouterHelmholtzplane r realpartofimpedance 1 inthebulksolutionorfarawayfromelectrodesurface 20

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy INFLUENCEOFELECTRODEGEOMETRYONLOCALANDGLOBALIMPEDANCE RESPONSE By Shao-LingWu August2010 Chair:MarkE.Orazem Major:ChemicalEngineering Distributionsofcurrentandpotentialassociatedwiththeelectrodegeometryare essentialissuesinstudyingtheelectrochemicalsystems.Thenonuniformdistributions thatcausetime-constantdispersionalongtheelectrodesurfacecanobscureresults fromtheelectrochemicalmeasurementsandleadtoanincorrectinterpretationof experimentaldata.Theelectrodecongurationofinterestisadiskelectrodeembedded inaninsulator,whichisoneofthemostpopulargeometriesusedintheelectrochemical measurements.Thegeometryeffectcanbeobservedathighfrequenciesforablocking diskelectrodeandforadiskelectrodesubjecttoasingleFaradaicreaction.Thepresent studyinvolvesmorecomplicatedelectrodeprocessesthatinclude,rst,acoupled Faradaicreactionsbyanadsorbedintermediate,andthenincorporatethenonuniform masstransferonarotatingdiskelectrodeRDEforageneralredoxreaction. Onastationarydiskelectrode,whilethefrequencyortime-constantdispersiondue tothedependenceoftheradialdistributionofinterfacialpotentialwasshowntohave aneffectathighfrequencies,thetime-constantdispersionwasalsofoundtoinuence theimpedanceresponseatlowfrequenciesduetothepotentialdependenceofthe fractionalsurfacecoverageoftheadsorbedintermediate.Thegeometryeffectswere reectedinvaluesforthelocalOhmicimpedance,whichhadcomplexbehavioratboth highandlowfrequencies.Thedispersionoftimeconstantwasdescribedintermsofa localconstant-phaseelementCPEthatrepresentedtheimpedanceresponseatlow 21

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frequenciesaswellasathighfrequencies.Thegeometryeffectcanbeeliminatedby useofarecessedelectrodeonwhichthecurrentandpotentialdistributionsareuniform. Experimentalvericationwasobtainedbyapplyinglocalelectrochemicalimpedance spectroscopyLEISonanirondiskelectrodeimmersedina1Msulfuricacidsolution. Thenonuniformmasstransferdistributiontogetherwiththeeffectofelectrode geometrywasinvestigatedonaRDEbelowthemass-transfer-limitedcurrent.Atwodimensionalimpedancemodelwasproposedtostudytheinuenceofnonuniform currentandpotentialdistributionsassociatedwithbothmass-transferandOhmic effectsontheglobalandlocalimpedanceresponse.Theconcentrationandpotential distributionswerecalculatedsimultaneouslythroughoutthesystemdomainbytaking intoaccountthetransportofspeciesfromdiffusion,electricmigration,andconvection. UndertheassumptionthattheFaradaicreactionandthechargingofthedouble layercannotbeseparated apriori ,partoftheuxofreactingspeciescontributestothe chargingoftheinterfaceaswellastoFaradaicreaction.Adouble-layermodelfollowing theGouy-Chapmantypeofdoublelayerwasusedtoassessthechargedensityon electrodewithoutspecicadsorption,andtoevaluateinterfacialproperties,suchas thedouble-layercapacitanceandthechangeofchargeassociatedwiththevariationof ionicconcentrations.Thelocalinterfacialimpedanceshowedadepressedsemicircle thatcannotbeattributedtothegeometry-inducedcurrentandpotentialdistributions. TheappearanceofCPEbehaviorwasattributedtothefrequency-dependenteffective double-layercapacitancethataccountedforthecontributionofuxinchargingthe doublelayer. 22

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CHAPTER1 INTRODUCTION ElectrochemicalimpedancespectroscopyEISisatransienttechniquethatis widelyusedtocharacterizeelectrodeprocess.Impedancemeasurementisperformed byapplyingasmallperturbationofpotentialorcurrenttoanelectrochemicalsystem atsteadystateandmeasuringtheoutputresponsecurrentorpotentialatdifferent perturbationfrequencies.ThefeaturesofEISincludecapturingthereal-timechanges ofelectrodeprocesswithoutdestroyingthespecimensurfaceandalsocharacterizing differentelectrodeprocessesbytheirassociatedtimeconstantswithinjustonesingle measurementoverasufcientbroadrangeoffrequency. WhileEISisparticularlyusefultodistinguishamongtransportandkineticphenomena,geometry-inducedcurrentandpotentialdistributionsgiverisetofrequency dispersionthatdistortstheimpedanceresponseandcouldleadtoanincorrectestimationofkineticandtransportparameters. 4 Thefrequencyortime-constantdispersion wasoriginallyattributedtothedispersionofdouble-layercapacitanceorthedependenceofcapacitanceonfrequency.Itisnowgenerallyattributedtothenonuniform currentandpotentialdistributionsassociatedwithfactorssuchaselectrodemorphology andheterogeneity. Amongallkindsofelectrode,diskelectrodeisoneofthepopulargeometriesused inelectrochemicalmeasurements.Thegeometryofadiskelectrodeembeddedinan insulatorissimpletoconstructandeasytoperformpostprocessingontheelectrode surface.Thewell-denedgeometryalsomakespossibletheanalyticalsolutionsfor currentandpotentialdistributions,frequencydispersion,anduidmechanicsonadisk electrode. 1,5,6 ThecontributionsrelatedtotheuseofdiskelectrodesintheareaofelectrochemistryareshowninFigure1-1.Theamountofstudiesincreasesdramatically sincethe1980'sandkeepincreasingafter2000.Theapplicationofimpedancetechniqueaccounts10%ofthesework.Therefore,itisimportanttounderstandtheeffectof 23

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Figure1-1.Onlinerecordsfortheuseofdiskelectrodesintheareaofelectrochemistry providedbyEngineeringVillagedatabasebeforeJune,2010. diskgeometryonimpedanceresponseandgiveaproperinterpretationtoexperimental results. Numerousapproacheshavebeenapplytostudythetransientbehaviorofadisk electrodeinthepresenceofatime-constantdispersion.Analyticalsolutionswere obtainedforthetransientpotentialandcurrentresponsesundergalvanostaticand potentiostaticcontrol. 79 Numericalsimulationsfacilitatedthecalculationofimpedance responseofadiskelectrodethatwasideallypolarized 10,11 orwassubjecttoasingle Faradaicreaction. 12 LocalelectrochemicalimpedancespectroscopyLEIScouldprovidelocalinformationofelectrodesurfaceandconrmthevariationoflocalimpedance resultingfromthenonuniformcurrentandpotentialdistributions. 2,13,14 Thepresenceof geometry-inducedcurrentandpotentialdistributionscausesadepressionofimpedance plotthatdeviatesfromthestandardsemicircle.Themeasuredglobalimpedance,or theconventionalimpedance,cannotbeproperlyinterpretedbyknownelectricalcircuit elementssuchasresistor,capacitors,andinductors,andisgenerallyexpressedin termsofaconstant-phaseelementCPEinequivalentcircuits. 24

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Inthisdissertation,fundamentalconceptsforthetransienttechnique,EISandLEIS, andtherelevantissuessuchasgeometry-inducedcurrentandpotentialdistributions, andtheoriginofCPEbehaviorarepresentedinChapter2.Theoreticaldevelopments andsimulationandexperimentalstudiesontheimpedanceresponseassociatedwith nonuniformcurrentandpotentialdistributionsonadiskelectrodeandmass-transfer distributiononarotatingdiskelectrodearereviewedinChapter3. InaseriesofpaperspresentedbyHuangandcoworkers, 1012 thenonuniform currentandpotentialdistributionsonadiskelectrodewereshowntoinuencethe impedanceresponseonlyathighfrequencies.Theworksuggestedexperimentalists couldavoidtheassociatedcomplicationsbysimplyperformingimpedancemeasurementsbelowacriticalfrequencygivenasafunctionofdiskradius,diskcapacitance, andsolutionconductivity.Asanextensiontothisseriesofwork,theobjectiveofthe presentworkistoinvestigatewhetherthegeometryeffectmayalsoplayaroleatlower frequencies.Themathematicaldevelopment,thecalculatedlocalandglobalimpedance results,andthediscussiononCPEbehaviorandcomplexOhmicimpedancearepresentedinChapter4.Theimpedancemodelisexpectedtopredictexperimentalresults andtoprovideguidelinesofinspectionandinterpretationofthemeasuredimpedance data.Anexperimentalvericationperformedonanirondiskelectrodeinacidicsolution ispresentedinChapter5. Forreactionsassociatedwithmasstransferofreactingspecies,thecurrentand potentialdistributionsareaffectedbythenonuniformmasstransferonelectrode.The rotatingdiskelectrodeRDEisusedextensivelyinelectrochemistryandtheuid mechanicsiswellunderstood.Thecurrentdistributionisuniformonlyatthemasstransfer-limitedcurrentwheretheconcentrationofreactingspeciesisequaltozeroover theentiredisk.One-dimensionalmodelscannotaccountfortheradialdistributionof convectivediffusionatcurrentsbelowthelimitingcurrent. 15 Newmanconsideredboth theradialandaxialdistributionofmasstransferinathindiffusionlayeraboveelectrode 25

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surface.Two-termvelocityexpansionswereusedforuidowneartheelectrode surface.Forexploringthenonuniformmasstransferinadditiontothegeometryeffect onaRDE,atwo-dimensionalmodelispresentedinChapter6toaccountfortheux ofeachspeciesinelectrolyteassociatedwithdiffusion,migration,andconvection. Thesystemistreatedasasingleintegraldomaingovernedbytheconvectivediffusion equation.Ananalyticvelocityexpressionvalidfortheentiredomainisdevelopedusing anappropriateinterpolationfunctiontoweightinnerandoutervelocityexpansions. Thisapproachispreferabletodividingthedomainofinterestintoadiffusionregion neartheelectrodesurface,andanouterregionwithuniformconcentrationaswas donebyNewman. 1 Thesteady-stateanalysisofthenonuniformcurrentdistribution belowthemass-transfer-limitedcurrentcanbeusedtoexplorethecontributionofthe radiallydependentconvectivediffusiontotheimpedanceresponsewhichispresentedin Chapter8. Mostimpedancemodelsassumenegligiblecontributionofmassuxtothecharging ofelectricdoublelayerattheelectrode-electrolyteinterface, i.e., themassuxonly contributestothecharge-transferreactions.Theassumptionofthe apriori separation ofFaradaicanddouble-layerchargingcurrentsisrelaxedinthepresentstudy.Asimple double-layermodelfollowingtheGouy-ChapmantheoryisgiveninChapter7.The nonuniformmassandpotentialdistributionsresultinanonuniformchargedistribution, andthereforeanonuniformdistributionofdouble-layercapacitanceintheinterfacial region.Thecurrentandtheuxofeachspeciesatelectrodeboundaryarecorrected bythepresenceofdouble-layercharging.Theimprovedtwo-dimensionalmodelis expectedtoprovideamoregeneralapproachinassessingtheimpedanceresponse onaRDEassociatedwiththeelectrodegeometryandmass-transfereffects.The impedanceresponsewiththecorrectionofdouble-layereffectisalsopresentedin Chapter8. 26

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CHAPTER2 ELECTROCHEMICALIMPEDANCESPECTROSCOPY ElectrochemicalimpedancespectroscopyEISisapowerfultechniquethathas beenwidelyusedinstudyingtheinterfacialelectrochemistry.Thechangeofelectrode propertiesarerevealedduringimpedancemeasurements.Theelectrodeprocess associatedwithreactionkineticsandsurfacemorphologycanthereforebediagnosed. Theimpedancemeasurementinvolvestheperturbationofanelectrochemical systemwithasmallsinusoidalsignalandrecordingtheoutputresponse.Asshown inFigure2-1,theoutputsignalrespondstotheinputsignalwiththesamefrequency andaphaseshift.Theconversionofthetime-dependentsignalsintofrequencydomain givesthetransferfunctionacomplexfeature.Impedanceisthereforeacomplexquantity denedbytheratioofpotentialandcurrent Z = j4 V j j4 I j e j = Z r + jZ j where isthephasedifferencebetweenthepotentialandcurrent, j isacomplex numberequalto p )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 ,and Z r and Z j aretherealandimaginarycomponentsofthe impedance,respectively.Ifthepotentialandcurrentareinphase,theimpedanceisa realnumberandisactuallyaresistance Z resistor = R Ifthecurrentleadsorlagstheappliedpotentialby90degrees,theimpedanceisapure imaginarynumber,whichrelatestocapacitanceorinductanceby Z capacitor = 1 j!C and Z inductor = j!L 27

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Figure2-1.Perturbationofanelectrochemicalsystematsteadystatewithasmall sinusoidalsignal,where 4 V and 4 I representthepotentialandcurrent oscillatingatthesamefrequency .Thephasedifferencebetweenpotential andcurrentis Inanelectrochemicalsystem,themeasuredimpedanceisusuallyacomplexnumber havingrealandimaginarycomponents,whichmeansthatthesystemcanbedescribed byacombinationofresisters,capacitors,andinductors. Impedancedataareoftenrepresentedinacomplexplane,whichisalsoknown astheNyquistplot.Figure2-2showsatypicalimpedanceplotcorrespondingtoaRC circuitthecombinationofadouble-layercapacitanceinparallelwithapolarization resistance.Thevalueofthepolarizationresistancecanbereadfromthelow-frequency limit.Thecharacteristicfrequencyortimeconstantforthesystem f = 1 2 = 1 2 RC canbeobtainedfromthepeakatwhichthenegativevalueofimaginarypartofthe impedanceismaximum.OneoftheattractivefeaturesofEISisthecharacterization ofdifferentelectrodeprocessesbytheirassociatedtimeconstantswithinonesingle measurementoverasufcientbroadrangeoffrequency.Thetransientresponseof 28

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Figure2-2.NyquistplotofimpedancedatacorrespondingtoaRCcircuitof R t =100 cm 2 and C 0 =10 F = cm 2 .Theresistancefromtheelectrolyteis notshowninthisgure. currentandpotentialabouttheirsteady-statevaluesgivestheinformationofelectron andmasstransferattheelectrode-electrolyteinterface.Thechargingoftheelectric doublelayercanbeobservedathighfrequencies,andthediffusionofionicspeciesto theinterfacialregionismoresignicantatlowerfrequencies.Therefore,EISisusefulto distinguishamongtransportandkineticphenomena. WhiletheimpedancedatagiveninFigure2-2tracesaperfectsemicircle,experimentaldatararelyshowidealbehaviorinrealsystems.Thenonidealbehaviorforan electrodeprocessisobservedintheimpedanceplaneofadepressedsemicircle,which isdifculttobeexplainedbysimplecircuitelements.Theimpedancedatareectinga nonuniformdistributionofreactivityonelectrodesurfacecabbedescribedbyusinga constant-phaseelementCPEinequivalentcircuits. 2.1Constant-PhaseElement Aconstant-phaseelementliterallymeansacircuitelementthatdisplaysaconstant phaseangle,suchastheresistor,capacitor,andinductor.Theterm,however,hasbeen 29

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specicallyusedtodescribethenonidealbehavioroftheinterfacialcapacitanceshowingafrequency-dependentphaseangledifferentfrom90degrees. 16 Theimpedance expressionoftheCPEisgivenby Z CPE = 1 j! Q where and Q areconstant.When =1 Q hasunitsofacapacitance, i.e., F/cm 2 andresemblesasanidealcapacitor.When 6 =1 Q hasunitsof s = -cm 2 .Usually,the electrochemicalinterfaceofarealcellisnotidealandbehaveslikeaCPEinwhichthe exponent isbetween0.5and1. ThenonidealbehaviorleadingtoaCPEcanbeattributedtothefrequencyortimeconstantdistributionalongtheareaoftheelectrodeoralongthedirectionnormaltothe electrodesurface.Thesurfacedistributionmayarisefromthesurfaceheterogeneities suchasdifferentcrystallinesizeorfaceswithdifferentelectrochemicalcharacteristics. 17 Anormaldistributionmaybeattributedtothechangeofcompositionofoxidelayers 18 or tothesurfaceroughnessandporosity. 19,20 Theschematicrepresentationsforsurface andnormaldistributionsatelectrodearegiveninFigure2-3byusingequivalent circuits.Thevariationsofreactionreactivityanddouble-layercapacitanceatthe electrode-electrolyteinterface,andthevariationoflmpropertiesintheoxidelayer causeafrequencyortime-constantdistributionatelectrodesurface.Thisdistributions areobservedduringtheimpedancemeasurementsintheformofaCPE. ThepresenceofCPEbehavior,however,isverycommonevenforahomogenous andrelativelysmoothsurface.Theaccessibilitytotheelectrodesurfacecouldbe constrainedbythecongurationofelectrodewiththesurroundinginsulator,andcausea geometry-inducedtimeconstantdispersionleadingtoCPEbehavior. 2.2CurrentandPotentialDistributions Currentandpotentialdistributionsareessentialpropertiesofanelectrode. Impedanceresponsecanbestronglyinuencedbythenonuniformdistributionsof 30

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A B Figure2-3.Schematicrepresentationofatime-constantdistributionAatthe electrode-electrolyteinterface,andBintheoxidelayer,where R t isthe charge-transferresistance, C 0 isthedouble-layercapacitance,and R f and C f aretheresistanceandcapacitanceofanoxidelm. currentandpotentialassociatedwiththegeometryofelectrodeunderstudy.Oneofthe commongeometriesusedinelectrochemicalmeasurementsisadiskelectrodeembeddedinaninsulator.Thegeometryofthediskconstrainsthecurrentowandpotential distributioninsuchamannerthatbothcannotsimultaneouslybeuniform.Undertheassumptionofnegligibleconcentrationgradient,Newman 1,5 solvedtheLaplace'sequation forpotentialusingrotationalellipticcoordinate.Thedistributionsareconsideredtobe primaryorsecondarydependingonthepresenceofelectrodepolarization. 2.2.1PrimaryDistribution WhenthepolarizationresistanceatelectrodeissmallcomparedtotheOhmic resistanceinelectrolyte,thepotentialinsolutionadjacenttotheelectrodecanbe depictedbyanequipotentialsurface.ThecurrentowstoelectrodefollowingtheOhm's law.Thisconditionistakenasaprimarydistribution.Theprimarycurrentdistributionon 31

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Figure2-4.Primarycurrentandpotentialdistributionsonadiskelectrode. 1 adiskelectrodeembeddedinaninniteinsulatingplaneisgivenby i i avg = 1 2 p 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( r=r 0 2 where r 0 istheradiusofthedisk,and i avg istheaveragecurrentdensityontheelectrode.Thetotalcurrenttothediskis I =4 1 r 0 0 where 1 istheconductivityofthebulksolution,and 0 isthepotentialissolution adjacenttotheelectrodesurface.TheanalyticalresultshowninFigure2-4illustrates aninnitecurrentdensityattheedgeandhalfthevalueofaveragecurrentdensityat thecenterofthediskwhentheelectrodeissubjecttoauniformpotentialdistribution. TheprimarydistributionrepresentsanextremecasewheretheOhmicresistance dominatesandthecurrentdistributionisthemostnonuniform. 32

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Ontheotherhand,theprimarypotentialdistributionwascalculatedunderthe conditionwhenthecurrentislimitedbythemasstransferofreactingspeciestothe disk. 1 ThepotentialcurveinFigure2-4isnormalizedbythetotalcurrentgivenin equation2soastoeasilycomparewiththeprimarycurrentdistribution.Alarger Ohmicpotentialdropispresentatelectrodecenter,andasmallerpotentialdropis observedatelectrodeperiphery.Thepotentialdistributesinsuchawaythatmore currentisforcedtoowfromtheperipherytothecenterofthediskinordertomaintain auniformcurrentdensityontheelectrodesurfaceascomparedtotheprimarycurrent distribution. AclassicsolutionfortheOhmicresistanceonadiskelectrodewasprovidedby Newman 5 as R e = 1 4 1 r 0 Thissimpleformwasobtainedassumingthatahemisphericalcounterelectrodeis placedatinnity.TheOhmicresistanceisonlyassociatedwiththegeometryfactorof thedisk, i.e., theelectroderadius,andthevalueofOhmicresistancevarieswhenthe probepositionchanges. 2.2.2SecondaryDistribution Whenelectrodekineticsistakenintoaccount,thepotentialadjacenttoelectrode isaffectedbythecharge-transferreactionstakingplaceintheinterfacialregion,and cannolongerbeconsideredasanequipotentialsurface.Thecurrentiscontrolledby theOhmicpotentialdropandtheinterfacialpotential,andthedistributionistakenasa secondarydistribution. Thesecondarycurrentdistributionwerediscussedintwocaseswheretheelectrodereactionsfollowthelinearkineticsatsmallcurrentdensities,andtheTafelkinetics atlargercurrent.Inbothcases,thecurrentdistributionsaredeterminedbytherelative contributionfromthecharge-transferresistanceandtheOhmicresistance.Adimensionlessparameter J wasgiventoweighthecontributionfromthetworesistances 33

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A B Figure2-5.SecondarycurrentdistributiononadiskelectrodeforAlinearkineticsand BTafelkinetics. 1 as J = 4 R e R t Largervaluesof J areseenwhentheOhmicresistanceisimportant,andsmallervalues areseenwhenthesystemisdominatedbyslowelectrodekinetics.Thecalculated secondarycurrentdistributionsforlinearandTafelpolarizationsareshowninFigure 2-5asafunctionof J .When J approachesinnity,fastkineticsappliesandtheOhmic potentialdropinsolutiondominates.Thecurrentunderthisconditionisconsideredto betheprimarydistribution.When J approacheszero,auniformcurrentdistributionis observedimplyingthatthecurrentisonlydependentonthenatureofthepolarization reactionandnottheelectrodegeometry. 34

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2.2.3TertiaryDistribution Tertiarycurrentandpotentialdistributionsapplywhentheassumptionofuniform concentrationisrelaxed.Theconcentrationsofchargedspeciesarenowtakeninto accountinadditiontothepotential.Newman 1 solvedtheconvectivediffusionequation forconcentrationwithinthediffusionlayer,andtheLaplace'sequationforpotential inthebulksolution.Theseequationsweresolvedsimultaneouslyandwerecoupled throughtheuxandcurrentdensityatelectrodeboundary.Adimensionlessparameter N ,similarto J inthesecondarydistribution,wasusedinrepresentingtherelative importanceoftheOhmicresistanceandthemass-transferresistance.Morediscussions ontheeffectofconcentrationpolarizationonthecurrentandpotentialdistributionsare givenelsewhere. 1,21 Theassumptionofconcentrationvariationonlyinthediffusionlayerisrelaxed inthepresentstudy.TheapproachmadebyNewmanismodiedbycouplingthe concentrationsandpotentialthroughthemassconservationandchargeconservation equations.Themathematicaldevelopmentandthecalculationresultsforthecurrent andpotentialdistributionsonarotatingdiskelectrodearepresentedinChapter6. 2.3LocalElectrochemicalImpedanceSpectroscopy TheresultsofconventionalEISrepresentanaverageresponseovertheentire surface.Localimpedancemeasurementsprovidelocalinformationofspecimensurface andareusefulinstudyingthetime-constantdistributionleadingtoaCPEbehavior.The inuenceofelectrodegeometryontheimpedanceresponsewerestudiednumerically andexperimentallybyOrazemandcoworkers 2,3,1014,22,23 inwhichtheLEIStechnique wasusedtoconrmtheresultingnonuniformbehavior. 2.3.1ExperimentalConguration TheuseoflocalelectrochemicalimpedancespectroscopyLEISwaspioneered byIsaacs etal. 2426 forthedeterminationofsurfaceheterogeneities.Thetechnique 35

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Figure2-6.Schematicrepresentationoftheelectrochemicalcellusedtoperformlocal electrochemicalimpedancemeasurements. 2 isperformedwithabielectrodeconsistingoftwoplatinumwires.TheschematicrepresentationofLEISmeasurementsisgiveninFigure2-6.Aperturbationofpotential isappliedtotheelectrode.Thepotentialdifferencebetweentheprobes 4 V probe aremeasuredatdifferentperturbationfrequencies.Thelocalcurrentdensity i loc oscillatedatthesamefrequencycanbeobtainedthroughtheOhm'slawfollowing i loc = 4 V probe d where istheelectrolyteconductivity, d isthedistancebetweentheprobes,and isthe oscillationfrequency.Thelocalimpedance z isthendenedby z = e V )]TJ/F15 11.9552 Tf 11.955 0 Td [( ref ; 1 i loc = e V 4 V probe d where e V istheperturbationofelectrodepotentialand ref ; 1 isthepotentialof referenceelectrodeplacedinthebulksolution. 36

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Figure2-7.Localequivalentcircuitscorrespondtolocalimpedancesthatvarywiththe radialpositionsontheelectrodesurface. Theuseofabielectrodeenablesthemeasurementoflocalcurrentdensity.The CommercialLEISinstrumentationyieldsaresolutionwithadimensionofabout1mm. Custom-builtinstrumentationcanyieldresolutionsontheorderof100 m. 14,27 With afour-channelfrequencyresponseanalyzer,theglobalandlocalimpedancescanbe measuredsimultaneously. 11 2.3.2DenitionofTerms ThenotationsoflocalimpedancepresentedinthisstudyfollowthedenitionsproposedbyHuang etal. 10 wherethelower-caseletter z wasusedtorepresentthelocal impedance,andtheupper-caseletter Z wasusedtorepresenttheglobalimpedance obtainedfromtheconventionalEISmeasurements.Tohelpenvisionthelocalandglobal properties,equivalentcircuitsareusedtodescribetheinterfacialelectrochemistry.As seeninFigure2-7aseriesofequivalentcircuitsarepresentattheelectrodesurface. EachequivalentcircuitcorrespondstoalocalelectrodeprocessconsistedofadoublelayercapacitanceinparallelwithaFaradaicimpedance.Blocksareusedtostandfor undeterminedreactionmechanismsattheelectrodesurface.Theelectrolyteproperties arealsodepictedbyblocksinordertoreectthecomplexfeatureoftheOhmiccontribution.Theperturbationofelectrodepotentialisdenotedby e m ,andtheoscillationin 37

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theadjacentsolutioninresponsetotheelectrodeperturbationisdenotedby e 0 .The positionof e 0 islocatedattheouterlimitofthediffusedoublelayer.Thesepotentials arereferencedtoadistantelectrodewhichvalueiszero.Thedenitionsoflocaland globalimpedancesaregivenasfollowusingtheconventionsillustratedinFigure2-7. Thelocalimpedancevariablesaredependentonradialpositionalongtheelectrode surfaceandarecalculatedfromlocalcurrentdensitiesandpotentialsdenedatdifferent locations.Thelocalimpedanceinvolvestheelectrodepotentialwithrespecttoadistant electrodeandisexpressedby z r = e m e i r where e i istheoscillationofthelocalcurrentdensity.Thelocalinterfacialimpedance involvesthepotentialdifferenceacrosstheelectrode-electrolyteinterfaceandisdened by z 0 r = e m )]TJ/F31 11.9552 Tf 12.862 3.022 Td [(e 0 r e i r ThelocalOhmicimpedanceinvolvestheOhmicpotentialdropinthesolutionandis givenby z e r = e 0 r e i r Fromthedenitionsofthelocalimpedancevariablesgivenabove,thelocalimpedance z r = z 0 r + z e r canberepresentedbythesumoflocalinterfacialandlocalOhmicimpedances. Theglobalimpedancevariablesrepresentaveragedpropertiesoftheelectrode surfaceandarenotdependentonradialpositions.Theglobalimpedance,similartothe localimpedance,involvestheelectrodepotentialwithrespecttothereferenceelectrode placedfarawayfromthedisk,andisdenedby Z = e m )]TJ/F31 11.9552 Tf 12.862 3.022 Td [(e 0 I 38

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wheretheoscillationoftotalcurrentisrelatedtothelocalcurrentdensityby I = Z r 0 0 e i r r d r Theglobalimpedancecanalsobeobtainedfromthelocalimpedancevaluesby 1 Z = Z r 0 0 1 z r 2 r d r whichisactuallytheglobaladmittancerepresentbytheintegrationoflocaladmittance overthedisksurface.Followingthesamestrategy,theglobalinterfacialimpedancecan beobtainedbyintegratingthelocalinterfacialadmittance Z 0 = Z r 0 0 1 z 0 r 2 r d r )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 TheglobalOhmicimpedanceisthereforedenedtobe Z e = Z )]TJ/F26 11.9552 Tf 11.955 0 Td [(Z 0 whichisthedifferencebetweentheglobalandglobalinterfacialimpedances.The high-frequencylimitoftheglobalimpedance,whichcorrespondstotheglobalOhmic impedance,shouldreachadimensionlessvalueof0.25whichisobtainedfromthe analyticalsolutionbyNewman. 5 39

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CHAPTER3 LITERATUREREVIEW Impedancemodelsareusuallybasedontheassumptionofuniformlyactive electrodesurface.Electrochemicalsystems,however,rarelyshowanidealbehavior becausethecurrentandpotentialadjacenttotheelectrodeareconstrainedbyelectrode geometry.Numerousstudiesweremadetoinvestigatethecurrentandpotential distributionsassociatedwithelectrodegeometry.Aliteraturereviewispresentedinthis Chapterforthegeometry-inducedcurrentandpotentialdistributionsandthenonuniform masstransferonarotatingdiskelectrode. 3.1Geometry-InducedCurrentandPotentialDistributions Thecurrentandpotentialdistributionsintheelectrolyteadjacenttoadiskelectrode embeddedinaninsulatingplaneareconstrainedbytheelectrodegeometry.Newman hasshownthattheOhmicpotentialdropinsolutioncausesanonuniformcurrentdistributionattheelectrodesurface. 5 Healsocalculatedthesecondarycurrentdistribution, whichaccountsfortheadditionalinuenceofthecharge-transferresistance. 1 The presenceofpotentialdropacrosstheelectrode-electrolyteinterfacereducesthecontributionfromtheOhmicpotentialdropinsolutionandthereforemakesthedistributionsof currentandpotentialmoreuniform.NisanciogluandNewman 7,8 haveinvestigatedthe transientresponseofadiskelectrodewithasingleFaradaicreactionsubjecttoastep changeinappliedcurrentandastepchangeinappliedpotential.Themodeldidnotaccountformasstransfereffects,andtheanalyticalsolutiontotheLaplace'sequationwas obtainedusingatransformationtorotationalellipticcoordinateswithaseriesexpansion intermsofLegendrepolynomials. Geometry-inducedcurrentandpotentialdistributionscauseafrequencydispersion thatdistortstheimpedanceresponseandreectsadistributionofelectrodereactivity. 6 Thefrequencyortime-constantdispersionwasoriginallyattributedtothedispersion ofdouble-layercapacitanceorthedependenceofcapacitanceonfrequency.Itisnow 40

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generallyattributedtothenonuniformcurrentandpotentialdistributionsassociatedwith factorssuchaselectrodemorphology, 20 heterogeneity, 17 orcompositionvariationin oxidelayers, 18 andisgenerallyexpressedintermsofaconstantphaseelementCPE inequivalentcircuits. 16 Thefrequencydispersionresultingfromelectrodegeometryis apparentinsystemswheretheOhmicresistancedominates,leadingtoaCPEbehavior. Thenonuniformcurrentandpotentialdistributionscouldleadtoanerrorinestimationofboththecharge-transferresistanceandtheinterfacialcapacitance. 4 Huang andcoworkers 2,1012 havepresentedaseriesofpapersdescribingtheinuenceof electrodegeometryontheimpedanceresponse.Theydenedthreelocalimpedances inadditiontotheglobalimpedanceobtainedfromtheusualimpedancemeasurements. SimulationsshowedthatthelocalimpedanceandlocalOhmicimpedanceexhibited time-constantdispersionassociatedwiththediskgeometry,andthecalculatedglobal impedancehadquasi-CPEbehaviorathighfrequenciesforblockingelectrode 10,11 andelectrodessubjecttoasingleFaradaicreaction. 12 ThegeometryeffectwasreectedinthelocalOhmicimpedanceinwhichnonzeroimaginarycomponentswere observed.Theseeffectscanbeeliminatedbyrecordingtheimpedancedatabelowthe characteristicfrequency.PredictionsmadebyHuang etal. 1012 wereinagreementwith observationsofFrateur etal. 2 forthelocalelectrochemicalimpedancespectroscopy LEISmeasurementsonastainlesssteelelectrode.Jorcin etal. 13 alsoobserveda CPEbehavioronadiskelectrodemadeofmagnesiumalloythatmaybeassociatedwith aradialdistributionoflocalresistance. TheoriginofOhmicimpedancewasdiscussedbyBlanc etal. 23 Thelocalvariations ofaxialandradialcurrentdensitiescausetheOhmiccontributiontoberepresentedbya complexnumber.ThecomplexcharacteroftheOhmicimpedanceisnotonlyaproperty ofelectrolyteconductivity,butalsoapropertyofelectrodegeometryandinterfacial impedance. 41

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3.2NonuniformMassTransferonaRotatingDiskElectrode Thewell-developedhydrodynamicsandconvectivetransportcharacteristicsmake rotatingdiskelectrodeRDEapopulartooltostudythereactionkineticsandmass transferinthediffusionlayer.Atthemass-transfer-limitedcurrent,theconcentrationof reactingspeciesiszeroovertheentiresurfaceofthediskandtheconvectiveuid bringsfreshreactanttotheelectrodesurface.Undersuchcondition,thecurrent distributionmaybeuniformontheelectrodesurfaceandtheuidvelocityintheradial directioncanbeneglected.Levich 28 calculatedthemass-transfer-limitedcurrentby usingthersttermofvelocityexpansionintheaxialdirection, 29 whichsatisesonly whentheSchmidtScnumberisinnitelylarge.Newman 30 providedacorrectionfor aScnumberof1,000whichreduced3%ofthevalueoflimitingcurrentfromLevich's results. TheassumptionofuniformcurrentdistributionwasrelaxedbyNewman 1 bytaking intoaccounttheconcentrationdistributioninbothradialandaxialdirectionsinthe diffusionlayer.Theconvectivediffusionequationisgivenby r @c @r + y @c @y = D @ 2 c @y 2 where c istheconcentrationand D isthediffusioncoefcientofthereactingspecies, and r and y aretheradialandaxialcomponentsofthevelocitywhichcanbeexpressedbyCochran's 29 two-termexpansionforuidvelocityneartheelectrodesurface. Outsidethediffusionlayerassumingauniformconcentrationinbulksolution,the electrolyticpotentialcanbeobtainedfromLaplace'sequationfollowing r 2 =0 Themigrationofspecieswasmodiedbytransferencenumber.Thenonuniformcurrent distributiononelectrodesurfacewasthencalculatedforelectrodereactionofmetal 42

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deposition.Thisapproachwasextendedforgeneralelectrodereactionsdependingon bothreactantandproductconcentrations. 31 Thesteady-statesolutionfortheuniformdistributionsofconcentrationandcurrentcannotbeappliedtothedevelopmentofimpedancemodelduetothefactthat impedanceisatransienttechniquethatreliesontheperturbationofpotentialorcurrentbelowthemass-transfer-limitedcurrent.Deslouis etal. 32,33 developedanalytical solutionsfortheresponsesofcurrentandpotentialtoasinusoidalperturbationofrotationspeed,alsoknownaselectrohydrodynamicEHDimpedancespectroscopy,at themass-transfer-limitedplateau.Theyusedonlythersttermoftheaxialvelocity expansion.Thetheoreticalsolutionswereinagreementwiththeexperimentalresults obtainedfromredoxreactionswithfastkineticsandlargevaluesofScnumberSc= 3,400and8,600.Intheunsteady-statecalculations,neglectinghighordertermsinthe velocityexpansioncouldresultinalargererrorwhenassessingtheScnumber.For aScnumberof1,000,anerrorof24.4%wasfound, 34 whichismoresignicantthan thatderivedfromsteady-statecalculations.TribolletandNewman 15 usedtwotermsof thevelocityexpansioninthederivationofEHDimpedanceandtabulatedtheWarburg impedanceasafunctionofScnumber.Theuseofthislook-uptablereducesthetime forregressiontoexperimentalresults.Theapplicationoftheseone-dimensionalmodels toimpedancemeasurements,however,arenotvalidbelowthemass-transfer-limited currentandoftenleadtoanomalouslylargeScnumbers. ThenonuniformmasstransferonaRDEandthenonuniformOhmicpotential dropduetoelectrodegeometryresulttoadistributionofcurrentonelectrodesurface thatrequiresatwo-dimensionalanalysisatbothsteadyandunsteadystates.Appel andNewman 35 providedamathematicalmodelthatconsideredtheradialconvective diffusion.Thediskelectrodewassubjecttoastepchangeinconcentrationandthe oscillatingconcentrationdistributionwascalculated.Thispreliminarydevelopment, validforinniteScnumber,couldbeusedaspartofamodelfortheinuenceofradially 43

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dependentconvectivediffusionontheimpedanceresponse.DurbhaandOrazem 36 extendedtheworktoasteady-statetreatmentofcurrentandpotentialdistributionsthat accountedforanitevalueofScnumberandforadistributionofchargeinthediffuse partofdoublelayer.SimilartoNewman'scalculation, 1 threetermsoftheradialandaxial velocityexpansionswereused.Inthesubsequentwork, 37 atwo-dimensionalimpedance modelwasdeveloped.Discrepancieswereseenbetweenthetwo-dimensionaland one-dimensionalmodels.Althoughtheone-dimensionalmodelprovidedagoodtto experimentaldata,theregressedScnumbercouldbeasmuchas22%higherthanthe expectedvalueincaseswherethediscrepancybetweenmodelswassignicant. 38 44

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CHAPTER4 ELECTRODEREACTIONWITHADSORBEDINTERMEDIATES:MODEL ThebehaviorofadsorbedintermediatesinFaradaicprocesseshasbeenstudied inthe1950s. 39,40 TheproceduresinvolvingelectrosorptionstepsinH 2 evolutionorO 2 evolutionreactionhaveanadsorptioncapacitancewhichisrelatedtothecoverageby intermediatesdependentonelectrodepotential. 41 EpelbinandLoric 42 analyzedsystems formetaldissolutionandthemeasuredimpedancesshowedlow-frequencyinductive loopscausedbyadsorbedintermediates.Thekineticsofreactionswasdiscussedby Armstrong etal. 43 withconsideringthediffusionofspeciesinsolutionandbyEpelboin et al. 44 withmorethanoneadsorbedspecies. SystematicclassicationsoftheimpedanceresponseformultireactionsinvolvinganadsorbedintermediateunderpotentiostaticcontrolwasgivenbyArmstrong andEdmondson 45 andCao. 46 Thehigh-frequencyimpedanceloopisattributedtothe chargingofdoublelayerandcharge-transferreactionsatelectrodesurface,andthe low-frequencyloopisrelatedtotherelaxationofsurfacecoveragebyadsorbedintermediates. 44 BaiandConway 47 discussedthedependenceofimpedanceresponseon electrodepotentialandcharacterizedthevaluesofsurfacecoverageatsteadystate.For areversiblereactionwithtwoelectron-transfersteps,therelativevalueofrateconstant foreachreactionstephadaneffectontheshapeofimpedance.Low-frequencyinductiveloopswereobservedinanarrowpotentialrangenearthemaximumorminimum valueofthesurfacecoverage. Inthischapter,thegeometry-inducedcurrentandpotentialdistributionsareapplied toelectrochemicalsystemswithadsorbedintermediates.Ageneraldiscussionforthe inuenceofelectrodegeometryontheglobalandlocalimpedanceresponseispresented.Thisworkisanextensionofthepreviousstudiesforablockingelectrode, 10,11 andforelectrodewithsingleFaradaicreaction. 12 Theobjectiveofthepresentstudyisto 45

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investigatewhetherthegeometryeffectsmayplayaroleatlowfrequencieswherethe relaxationofsurfacecoverageisobserved. 4.1MathematicalDevelopment Theapproachpresentedheretoaccountforlow-frequencyimpedanceloops associatedwithadsorbedintermediateswaspioneeredbyEpelboin etal. 44,48 Therst partofthissectiondemonstratesthatsuchlow-frequencyloopscannotbeobserved forreactionsrepresentedbylinearkineticexpressions.Thesubsequentpartsprovides themodeldevelopmentundertheassumptionofTafelkineticsandpotentialdistribution constrainedbyelectrodegeometry. 4.1.1LinearKineticsneartheEquilibriumPotential Thegeneralreactionsfortwosuccessivecharge-transferstepswithanadsorbed intermediatecanbedescribedby M X + ads +e )]TJ/F20 11.9552 Tf 174.015 -4.936 Td [( and X + ads P 2+ +e )]TJ/F20 11.9552 Tf 169.737 -4.936 Td [( Reactionsnearequilibriumpotentialcanbeexpressedaslinearfunctionsofsurface overpotential.Thus,thecurrentdensityattheelectrodesurfaceforreactions4and 4canbeexpressedby i M = K M )]TJ/F26 11.9552 Tf 11.955 0 Td [( b Ma + b Mc V and i X = K X b Xa + b Xc V where K M and K X aretheeffectiverateconstants, isthesurfacecoveragebythe adsorbedintermediate X + ads V istheinterfacialpotentialdenedbythedifference betweentheelectrodepotential m andthesolutionpotentialadjacenttotheelectrode 46

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0 ,and b M and b X arekineticparametersforreactions4and4wherethe subscriptsaandc,respectively,representtheanodicandcathodichalfreactions.The summationofthecurrentdensityofeachreactionyieldstheFaradaiccurrentdensityas i F = i M + i X Thevariationofthesurfacecoverageisrelatedtothereactionratesas d d t = i M )]TJ/F26 11.9552 Tf 11.955 0 Td [(i X )]TJ/F26 11.9552 Tf 7.315 0 Td [(F where )]TJ/F20 11.9552 Tf 10.638 0 Td [(isthemaximumsurfaceconcentrationbyintermediate,and F istheFaraday constant.Atsteadystate,thereactionratesreachconstantvalues,andthesurface coveragedoesnotchangewithtime, i.e., d = d t =0 .Thesteady-statevalueof can thereforebecalculatedfromequations4and4as = K M b Ma + b Mc K M b Ma + b Mc + K X b Xa + b Xc wherethebarnotationrepresentsthesteady-statecondition.Equation4indicates that,inthelinearregime,thesteady-statesurfacecoverageisnotdependentuponthe surfaceoverpotential.Thesurfacecoverageremainsuniformalongtheelectrodeandis afunctionofkineticparametersonly.Thecurrentdensityforeachreaction,however,is afunctionofsurfaceoverpotential,andthereforetheFaradaicimpedanceofthesystem isthecharge-transferresistancegivenby R t = @ V @ i F = 1 K M )]TJETq1 0 0 1 277.778 216.546 cm[]0 d 0 J 0.478 w 0 0 m 6.722 0 l SQBT/F26 11.9552 Tf 277.778 209.725 Td [( b Ma + b Mc + K X b Xa + b Xc AstheFaradaicimpedanceisinparallelwiththedouble-layercapacitance,the impedancediagramshowsasinglecapacitiveloopintheNyquistplane.Inductive andcapacitiveloopsarenotevidentatlowfrequenciesundertheassumptionoflinearkinetics.Inordertoinvestigatetheinuenceofadsorbedintermediatesonthe impedanceresponse,theelectrodekineticsisassumedtobeintheanodicTafelregime. 47

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4.1.2TafelKineticsforAnodicReactions Whenthesurfaceoverpotentialislarge,theelectrodebehaviorobeysTafelkinetics. Reactions4and4arenowtakentobeirreversibleas M X ads + + e )]TJ/F20 11.9552 Tf 170.361 -4.936 Td [( and X ads + P 2+ + e )]TJ/F20 11.9552 Tf 159.436 -4.936 Td [( Thereactantcouldbeametal M whichdissolvesthroughanadsorbedintermediate X ads + ,andthenfurtherreactstoformthenalproduct P 2+ .Similarmechanismswere proposedbyEpelboinandKeddam 48 forcalculatingtheimpedanceofirondissolution throughtwostepsinvolvinganadsorbedFeOHintermediate,andbyPeter etal. 49 for theimpedancemodelofthedissolutionofaluminuminthreeconsecutivestepswith twoadsorbedintermediates.UndertheassumptionofTafelkineticsandnegligible diffusionprocesses,thesteady-statecurrentdensitiesforreactions4and4are expressedby i M = K M )]TJETq1 0 0 1 311.08 349.145 cm[]0 d 0 J 0.478 w 0 0 m 6.722 0 l SQBT/F26 11.9552 Tf 311.08 342.324 Td [( exp[ b M V ] and i X = K X exp[ b X V ] Theexpressionforsteady-statesurfacecoveragecanbecalculatedfromequations 4and4as = K M exp[ b M V ] K M exp[ b M V ]+ K X exp[ b X V ] Incontrasttothesurfacecoveragegiveninequation4forlinearkinetics,equation 4showsthatthesteady-statesurfacecoverageisdependentonthesurface overpotentialundertheassumptionofTafelkinetics. 48

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FollowingthemethoddevelopedbyEpelboin etal. 44 theFaradaicimpedance Z F at agivenpotentialisgivenby 1 Z F = 1 R t + A j! + B wherethecharge-transferresistance R t isdenedby 1 R t = 1 R t ; M + 1 R t; X = b M j i M j + b X j i X j andparameters A and B arepotentialdependentvariablesgivenby A = @ i F @ @ @ V d d t = R )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 t; M )]TJ/F26 11.9552 Tf 11.955 0 Td [(R )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 t; X [ K X exp b X V )]TJ/F26 11.9552 Tf 11.955 0 Td [(K M exp b M V ] )]TJ/F26 11.9552 Tf 7.314 0 Td [(F and B = )]TJ/F26 11.9552 Tf 13.855 8.088 Td [(@ @ d d t = K X exp b X V + K M exp b M V )]TJ/F26 11.9552 Tf 7.314 0 Td [(F While B isalwayspositive,thesignof A varieswiththepotentialacrosstheelectrodeelectrolyteinterface,andthefeatureoftheimpedanceplanechangesaccordingtothe signof A AsshowninFigure4-1fortheFaradaicimpedance Z F inparallelwiththedouble layercapacitance C 0 atelectrodesurface,andinserieswiththeOhmicresistance R e inelectrolyte,theoverallimpedancehasdifferentfeaturesatlowfrequencieswith differentsignsof A .Athighfrequencies,theFaradaicimpedanceisobservedthrough thecharge-transferresistance R t ,whichisthersttermofequation4,andthe impedanceshowsahigh-frequencycapacitiveloopcorrespondingto R t inparallelwith C 0 R t jj C 0 .Atlowfrequencies,theFaradaicimpedanceisobservedthroughthesecond termofequation4.Thelow-frequencyloopexhibitsinductivebehaviorwhen A ispositive,andcapacitivebehaviorwhen A isnegative.When A isequaltozero,the twotermsofthenumeratorinequation4cancel, i.e., @ i F =@ =0 .Inthiscase, thereactioncurrentdensityisnotdependentonthesurfacecoverage,and,therefore, theimpedanceplotshowsasinglecapacitiveloopcorrespondingto R t jj C 0 withno low-frequencyloop. 49

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Figure4-1.Impedanceplotsinresponsetodifferentsignsof A 4.1.3PotentialDistribution ThepotentialdistributionimpliedbyFigure2-7resultsfromthegeometryofadisk electrodewithradius r 0 embeddedinaninsulatingplane.Thecounterelectrodewas assumedtobeplacedinnitelyfarfromthediskelectrode.Thepotentialinthesolution canbesolvedbyusingLaplace'sequationincylindricalcoordinates, i.e., r 2 =0 Thesystemisassumedtohavecylindricalsymmetrysuchthatthepotentialinsolution isdependentonlyontheradialposition r alongtheelectrodesurfaceandthenormal distance y .Inresponsetoanalternatingcurrentwithaparticularangularfrequency =2 f ,thepotentialcanbeseparatedintosteadyandtime-dependentpartsas = + Re f e e j!t g 50

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where isthesteady-statesolutionforpotentialand e isthecomplexoscillating component,whichisafunctionofpositiononly.Therefore,Laplace'sequationbecomes 1 r @ @r r @ e @r + @ 2 e @y 2 =0 Theboundaryconditionsatinsulatorsandfarfromtheelectrodesurfacearegivenby @ e @y y =0 =0at r>r 0 and e =0as r 2 + y 2 !1 Thecurrentdensityattheelectrodesurfacecanbeexpressedas i = C 0 @V @t + i M + i X = )]TJ/F26 11.9552 Tf 9.299 0 Td [( @ @y y =0 where C 0 istheinterfacialcapacitanceand istheelectrolyteconductivity. Thecurrentattheelectrodesurfacecanbewrittenbyuseofthereactionkinetics developedinequations4and4,andexpressedinfrequencydomainby Kj e V + J 1 e V + J 2 e V = )]TJ/F26 11.9552 Tf 9.299 0 Td [(r 0 @ e @y y =0 where e V istheoscillationofsurfaceoverpotentialdenedby e V = e m )]TJ/F31 11.9552 Tf 12.861 3.022 Td [(e 0 where e m istheimposedperturbationinelectrodepotentialand e 0 isthecorresponding oscillationinthesolutionpotentialadjacenttoelectrodesurface,and K isthedimensionlessfrequencydenedby K = !C 0 r 0 51

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Theparameters J 1 and J 2 aredimensionlessfunctionsdependentonangularfrequency andradialpositionontheelectrodesurface, i.e., J 1 r;! = J M r )]TJ 18.568 17.473 Td [(j i M r j 1 )]TJETq1 0 0 1 313.02 653.596 cm[]0 d 0 J 0.359 w 0 0 m 4.832 0 l SQBT/F27 7.9701 Tf 313.02 648.909 Td [( r [ J M r )]TJ/F26 11.9552 Tf 11.955 0 Td [(J X r ] )]TJ/F26 11.9552 Tf 7.314 0 Td [(Fj! + j i M r j 1 )]TJETq1 0 0 1 352.968 628.472 cm[]0 d 0 J 0.359 w 0 0 m 4.832 0 l SQBT/F27 7.9701 Tf 352.968 623.785 Td [( r + j i X r j r and J 2 r;! = J X r )]TJ 19.694 17.472 Td [(j i X r j r [ J M r )]TJ/F26 11.9552 Tf 11.955 0 Td [(J X r ] )]TJ/F26 11.9552 Tf 7.314 0 Td [(Fj! + j i M r j 1 )]TJETq1 0 0 1 351.717 558.79 cm[]0 d 0 J 0.359 w 0 0 m 4.832 0 l SQBT/F27 7.9701 Tf 351.717 554.103 Td [( r + j i X r j r Theparameters J M and J X aredenedtobethedimensionlesscurrentdensitiesfor reactions4and4,respectively,andarefunctionsofradialpositiononthe electrodesurface,asgivenby J M r = b M j i M r j r 0 and J X r = b X j i X r j r 0 respectively.Thesumof J M and J X representsthedimensionlesscurrentdensitywhich owsthroughthecharge-transfersteps J r = J M r + J X r = r 0 [ K M b M )]TJETq1 0 0 1 304.148 309.122 cm[]0 d 0 J 0.478 w 0 0 m 6.722 0 l SQBT/F26 11.9552 Tf 304.148 302.301 Td [( exp b M V + K X b X exp b X V ] Therelationshipbetweentheparameter J andthecharge-transferandOhmicresistancescanbeestablishedusingthehigh-frequencylimitfortheOhmicresistancetoa diskelectrodeobtainedbyNewman 5 R e = r 0 4 where R e hasunitsof cm 2 .Theparameter J canthereforebeexpressedintermsof theOhmicresistance R e andcharge-transferresistance R t as 12 J = 4 R e R t 52

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Table4-1.Thevaluesofkineticparametersusedforthesimulations. SymbolMeaningValueUnits b M M F=RT 40V )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 b X X F=RT 10V )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 K M Effectivereactionconstantforreaction477.2A/cm 2 K X Effectivereactionconstantforreaction40.19A/cm 2 m Steady-stateelectrodepotential For h A i > 0 0 : 011 S/cm 2 s-0.15V For h A i =0 )]TJ/F15 11.9552 Tf 9.299 0 Td [(6 : 6 10 )]TJ/F24 7.9701 Tf 6.587 0 Td [(4 S/cm 2 s-0.10V For h A i < 0 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 83 S/cm 2 s0.10V e m Perturbationofelectrodepotential0.01V Largevaluesof J areseenwhentheOhmicresistanceismuchlargerthanthechargetransferresistance,andsmallvaluesof J areseenwhenthecharge-transferresistance dominates.Thedenitionofparameter J inequation4isthereciprocalofthe Wagnernumber, 50 whichisadimensionlessquantitythatmeasurestheuniformityofthe currentdistributioninanelectrolyticcell. Simulationswereperformedtoinvestigatetheelectrochemicalimpedancebehavior fordifferentpotentiostaticsituationswhenthesurface-averagevalueof h A i ispositive, negative,andzero.Thevaluesofkineticparametersusedforthesimulationsare giveninTable4-1.ThevaluesinTable4-1for b M and b X areclosetothevaluesof 38.4V )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 and7V )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 reportedbyKeddam etal. 51 forthedissolutionofironinacidic media.Thecalculatedresultforeachsimulationcorrespondstoaparticularvalueof J representingdifferentcontributionsfromtheOhmicandthecharge-transferresistances. Theequationsweresolvedbyusingthenite-elementpackage COMSOLMultiphysics r withtheconductivemediaDCmoduleina2Daxialsymmetriccoordinatesystem.A quarter-circlewasconstructedwithanaxisofsymmetryat r =0 andtheelectrode positionedat y =0 .ThedomainsizeshowninFigure4-2was2,000timeslargerthan thediskelectrodedimensioninordertomeettheassumptionthatthecounterelectrode waslocatedinnitelyfarfromtheelectrodesurface. 53

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Figure4-2.Thedomainusedforthenite-elementsimulations.Thesolidlinesrepresent steady-stateiso-potentialplanes,anddashedlinesrepresentsteady-state trajectoriesforowofcurrent. 4.2CalculatedImpedanceResults Asindicatedbyequation4,thesteady-statefractionofsurfacecoverage varieswiththeinterfacialpotential V = m )]TJETq1 0 0 1 316.434 372.674 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 316.434 362.831 Td [( 0 .While m isassumedtobeuniform onaconductiveelectrode,thepotentialoutsidethediffusedoublelayer 0 isafunction ofradialposition.Thus, m )]TJETq1 0 0 1 230.649 324.858 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 230.649 315.015 Td [( 0 and arefunctionsofradialposition.Thedistribution ofthenormalizedsteady-statefractionalsurfacecoverageispresentedinFigures4-3A and4-3Bforpositiveandnegativeaveragevaluesof h A i ,respectively.Thevalueof averagesurfacecoverage h i is0.069forcurve1,and0.98forcurve9.Theparameter A isitselfafunctionofradialpositionduetoitsdependenceoninterfacialpotential. Therangeofthevalueof A fromelectrodecentertoelectrodeperipheryis0.0109to 0.00509S/cm 2 sforcurve4,0.0106to-0.0665S/cm 2 sforcurve5,and-0.109to-7.60 S/cm 2 sforcurve6.Thevariationforlocalvalue A alongtheelectrodesurfaceislarger astheappliedpotentialisincreased. Therelationshipbetweenthesurface-averagedvaluesof h A i reportedinFigures 4-3Aand4-3Bandtheappliedsteady-stateelectrodepotentialispresentedinFigure 54

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A B C D Figure4-3.Radialdistributionofthenormalizedsteady-statefractionalsurfacecoverage onadiskelectrode:Aforpositivesurface-averagedvaluesof h A i ;Bfor negativesurface-averagedvaluesof h A i ;Ctherelationshipbetweenthe surface-averagedvaluesof h A i reportedinpartsAandBandtheapplied steady-stateelectrodepotential;andDthecorrespondingsurface-averaged valuesof h J i totheappliedsteady-stateelectrodepotential. 55

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4-3C.Thecoveragebyintermediateismostnonuniformat m = -0.1V,wherethe correspondingaveragevalueof h A i isequaltozero.Itbecomesmoreuniformatlarger orsmallervaluesof m .Thecorrespondingsurface-averagedvalueof h J i totheapplied potentialatsteadystatewascalculatedfromequation4andisgiveninFigure 4-3D.As m increases,thevalueof h J i increasesandthesignof h A i changesfrom positivetonegative.Thesignof h A i determinestheshapeoflow-frequencyfeatures oftheimpedanceplot.Inordertounderstandtheimpedancebehaviorunderdifferent potentiostaticconditions,threecasesfor h A i > 0 h A i =0 ,and h A i < 0 points4,5,and 6inFigure4-3Carediscussed. ThemaximumvariabilityofsurfacecoveragewasshowninFigures4-3Aand4-3B tooccurfor h A i =0 .Interestingly,thepotentialyieldingthemaximumvariabilityof surfacecoveragedoesnotcoincidewiththepotentialyieldingthemostnonuniform distributionofcurrentorpotential.TheadsorptionisothermgiveninFigure4-4Ashows aninectionat h A i =0 ,representingthestrongerdependenceofthesurfacecoverage ontheinterfacialpotential;whereas,at h A i = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 83 S/cm 2 s,theisothermcrossesthe largestpotentialinterval,indicatingthemostnonuniformpotentialdistributiononthe electrodesurface.ThedistributionofcurrentcanalsobeobservedinFigure4-4B,which showsamorenonuniformdistributionofcurrentwhentheappliedpotentialisincreased. For h A i = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 83 S/cm 2 s,theparameter J hasthelargestvalue,meaning,asshown inequation4,thattheOhmicresistanceismuchlargerthanthecharge-transfer resistance.Hencethecurrentdistributionismorenonuniform. 21 Thediscussionpresentedbelowfollowstheinuenceofkineticparametersonthe global,localinterfacial,localOhmic,andlocalimpedances. 4.2.1GlobalImpedance Theglobalimpedancerepresentsanaveragedresponseoftheelectrode.The calculatedresultsofglobalimpedanceforthecaseswith h A i > 0 0 : 011 S/cm 2 s, h A i =0 ,and h A i < 0 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 83 S/cm 2 sarepresentedinNyquistformatinFigures4-5A, 56

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A B Figure4-4.ThevariationofAsteady-statesurfacecoveragedensity,andB steady-statecurrentdensitywiththeinterfacialpotential.Dashedsquares areusedtoidentifytherangeofcurrentandsurfacecoveragecorresponding tothesimulationsperformedat m = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 15 V h A i =0 : 011 S/cm 2 s, m = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 1 V h A i =0 S/cm 2 s,and m =0 : 1 V h A i = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 83 S/cm 2 s.The position r =0 correspondstothelower-leftcornerofeachbox. 57

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4-5B,and4-5C,respectively.ThesolidlinesinFigure4-5representthesimulation resultsbysolvingtheLaplace'sequationcouplingtheboundaryconditionsthataccount forthetime-constantdispersionassociatedwiththeelectrodegeometry.Thedashed curvesrepresenttheglobalimpedancescalculatedbyuseofamathematicalexpression Z = R e + 1 1 =Z F + j!C 0 associatedwiththeOhmicresistanceinsolutionandacombinationofFaradaicreaction andelectricdoublelayeratelectrode.TheOhmicresistancedevelopedbyNewman 5 isgiveninequation4.TheFaradaicimpedanceiscalculatedfromequation 4intermsofthesurface-averageparametersgivenfromequation4to 4andthereforedidnotaccountfortheinuenceofelectrodegeometry.The geometryofthediskelectrodeisshowntodistorttheglobalimpedanceresponse.The geometry-induceddistortionofimpedanceresponseandcorrespondingdepressionsof semicirclesathighandlowfrequenciesaremoreobviousinFigure4-5Cwhere h A i < 0 Thecharge-transferresistance R t forthecoupledreactionscanbeevaluatedfrom thediameterofthehigh-frequencyloopoftheglobalimpedance,andthelow-frequency loopyieldstheresistance R associatedwiththeconcentrationofadsorbedspecies. Acomparisonbetweentheeffectivekineticparameters R t ; e and R ; e ,whichaccount forelectrodegeometry,totherespectivevaluesthatassumeanuniformelectrode ispresentedinFigure4-6asafunctionof h J i .Therelationshipbetween h J i and theelectrodepotential m isgiveninFigure4-3D.Theratios R t ; e =R t and R ; e =R approachunityas h J i! 0 .Thevalueof R t ; e =R t increaseswhen h J i increases,which isinagreementwiththeresultpresentedbyHuang etal. 12 thattheinuenceoftimeconstantdispersionisgreaterwhen h J i islarge.Thevalueof R ; e =R islargestfor h J i closeto3.09where h A i =0 .Thesurfacecoveragebythereactionintermediatehas thegreatestnonuniformityat h A i =0 ,asshowninFigure4-3Aand4-3B,andhencea signicanterrorin R ; e isseenwhen h A i approacheszero. 58

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A B C Figure4-5.CalculatedNyquistrepresentationoftheglobalimpedanceresponsefora diskelectrodeconsideringtheinuenceofelectrodegeometrysolidlines andintheabsenceofgeometryeffectdashedlines:A h A i > 0 0 : 011 S/cm 2 s;B h A i =0 ;andC h A i < 0 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 83 S/cm 2 s. 59

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Figure4-6.Thevalueof R t ; e =R t and R ; e =R evaluatedfromtheglobalimpedanceas afunctionof h J i Theresultsforglobalimpedancecanbeunderstoodthroughtheexaminationof thelocalimpedancedistributions.Inthefollowingsections,thecalculatedresultsfor local,localinterfacialandlocalOhmicimpedancearepresentedandcomparedforthree differentpotentiostaticconditions: m = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 15 V h A i =0 : 011 S/cm 2 s, m = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 1 V h A i =0 S/cm 2 s,and m =0 : 1 V h A i = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 83 S/cm 2 s. 4.2.2LocalInterfacialImpedance NyquistplotsforthecalculatedlocalinterfacialimpedancearepresentedinFigures 4-7A,4-7B,and4-7Cfor h A i > 0 h A i =0 ,and h A i > 0 ,respectively,withnormalized radialposition r=r 0 asaparameter.Theimpedancediagramsaresuperposedathigh frequencies,showingthatcurrentowsmainlythroughthedouble-layercapacitance whichwasassumedtobeuniformattheelectrodesurface.TheshapeofthelowfrequencyFaradaicloopisdependentonthesignofparameter h A i .Thelocalinterfacial impedancehasalow-frequencyinductiveloopatallpositionsontheelectrodewhen h A i > 0 ,andshowslow-frequencycapacitivefeatureswhen h A i < 0 .For h A i =0 althoughonlyasinglecapacitiveloopisobservedintheglobalimpedanceFigure 60

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A B C Figure4-7.CalculatedNyquistrepresentationofthelocalinterfacialimpedance responseofadiskelectrodewithnormalizedradialposition r=r 0 asa parameter:A h A i > 0 0 : 011 S/cm 2 s;B h A i =0 ;andc h A i < 0 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 83 S/cm 2 s. 61

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4-5B,twotimeconstantsareseeninthelocalinterfacialimpedance.Attheperiphery oftheelectrode,alow-frequencycapacitiveloopisseen,representingalocalpositive valueof A .Neartheelectrodecenter,alow-frequencyinductiveloopisobserved, indicatingthelocalnegativevalueof A .Thedifferentlow-frequencyfeaturesseen atdifferentradialpositionsdemonstratesthattheglobalimpedanceisanaverage representationoftheelectrodesurface. Forthethreepotentiostaticconditions,thevaluesofthelocalinterfacialimpedance arelargerattheelectrodecenterandsmallerattheperiphery,indicatingagreater accessibilityneartheedgesoftheelectrode.Theappliedelectrodepotentialisthe largestinthecase h A i < 0 ,drivinglargercurrentdensitiesthroughtheelectrodeelectrolyteinterface,thustheinterfacialimpedancehasthesmallestvaluewhen h A i is negative. 4.2.3LocalOhmicImpedance ThecalculatedlocalOhmicimpedancesfor h A i > 0 h A i =0 ,and h A i < 0 inNyquist formatareshowninFigures4-8A,4-8B,and4-8C,respectively,withnormalizedradial positionasaparameter.AsdiscussedbyHuang etal. 1012 theresistanceinthe electrolyteisnotapureresistance,butactsasanimpedancewithcomplexfeatures. Thehigh-frequencyloopsat K> 10 )]TJ/F24 7.9701 Tf 6.586 0 Td [(2 areinagreementwiththepreviousstudies forablockingelectrodeandadiskelectrodesubjecttoasingleFaradaicreaction,in whichthegeometry-inducedcurrentandpotentialdistributionsareobserved. 12 Inthe presentcase,however,low-frequencyloopscanalsobeobserved.Thesizeofthese low-frequencyloopsincreaseswithappliedpotential, i.e., for h A i < 0 ThedependenceofthelocalOhmicimpedancewithfrequencyisshownmore clearlyintherepresentationoftherealandimaginarycomponentsasgiveninFigure 4-9.Fortherealcomponent,thevaluesathigh-frequencylimitareindependentof h A i ; whereas,atthelow-frequencylimit,thedifferenceofrealvaluesbetweentheelectrode centerandtheperipheryislargerfor h A i < 0 .Thenonzerovaluesintheimaginary 62

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A B C Figure4-8.CalculatedNyquistrepresentationofthelocalOhmicimpedanceresponse ofadiskelectrodewithnormalizedradialposition r=r 0 asaparameter:A h A i > 0 0 : 011 S/cm 2 s;B h A i =0 ;andC h A i < 0 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 83 S/cm 2 s. 63

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A B C D E F Figure4-9.CalculatedrealandimaginarypartsofthelocalOhmicimpedanceresponse ofadiskelectrodeasafunctionofdimensionlessfrequency K :Arealpart for h A i =0 : 011 S/cm 2 s;Bimaginarypartfor h A i =0 : 011 S/cm 2 s;Creal partfor h A i =0 ;Dimaginarypartfor h A i =0 ;Erealpartfor h A i = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 83 S/cm 2 s;andFimaginarypartfor h A i = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 83 S/cm 2 s. 64

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componentoftheimpedanceat 10 )]TJ/F24 7.9701 Tf 6.587 0 Td [(7
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A B C Figure4-10.CalculatedNyquistrepresentationofthelocalimpedanceresponseofa diskelectrodewithnormalizedradialposition r=r 0 asaparameter:A h A i > 0 0 : 011 S/cm 2 s;B h A i =0 ;andC h A i < 0 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 83 S/cm 2 s. 66

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A B C Figure4-11.Calculatedrepresentationoftheimaginarycomponentofthelocal impedanceresponseonadiskelectrodeasafunctionofdimensionless frequency K :A h A i > 0 0 : 011 S/cm 2 s;B h A i =0 ;andC h A i < 0 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 83 S/cm 2 s. 67

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ispresentedasafunctionofthedimensionlessfrequency K .Changesinsignare evidentinthefrequencyrange 1
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A B Figure4-12.Theimpedanceresponseforarecessedelectrode:A h A i > 0 ;andB h A i < 0 69

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electrodewith P =4 .Thelocalimpedanceresponseatdifferentradialpositionsshows nodispersionalongtheelectrodesurface.Inaddition,idealfeatureswereobservedin theglobalimpedanceplots.Theeliminationofthegeometryeffectbyuseofarecessed electrodedemonstratesthatthecomplexbehaviorforthelocalOhmicimpedanceat bothlowandhighfrequenciescannotbeattributedtocalculationartifacts. 4.4EvaluationofCPEExponent Geometry-inducedcurrentandpotentialdistributionswereshowntohaveinuence ontheimpedanceresponseonlyatdimensionlessfrequency K> 1 forablockingelectrode 6,10 andadiskelectrodesubjecttoasingleFaradaicreaction. 11,12 Theresultsof thepresentworkshowedthat,foramorecomplicatedsystemwithreactionsassociated withanadsorbedintermediate,theimpedanceresponseisaffectedbythegeometryof thediskelectrodeatlowfrequenciesaswellasathighfrequencies. Theconceptthatthenon-idealimpedancecausedbygeometry-inducedcurrent andpotentialdistributionscouldbeexpressedintermsofCPEbehaviorathighfrequencieshasbeendiscussedbyHuang etal. 11,12 Theparameter intheCPEexpression wasobtainedintheirworkfromtheslopeofthemagnitudeoftheimaginarypartof theglobalimpedanceplottedasafunctionoffrequencyinlogarithmicscales.This approachworkedwellforthehigh-frequencybehavior.Inthepresentstudy,however, theslopeforthelow-frequencyinductiveorcapacitiveloopscouldnotbeclearlyresolvedbecausetherangeoffrequencywastooshort.Theimpedancedatawithinthis frequencyrangewasinuencedbythehigh-frequencyloopwhichobscuredtheslope. AnotherapproachforgraphicalquanticationforCPEbehaviorcanbedeveloped byexploringhowtheshapeofasingleimpedanceloopdeviatesfromthatofaperfect semicircle.Themaximummagnitudeoftheimaginarypartofimpedance Z j and thedifferencebetweenhighandlow-frequencyasymptotesfortheofrealpartofthe impedance Z r showninFigure4-13forasingleloopcorrespondtotheradiusandthe diameterforasemicircle,respectively.Theabsoluteratioof Z j and Z r isrelatedto 70

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Figure4-13.Agraphicalrepresentationofasingleimpedanceloopseparatedintoa higher-frequencyhfhalfandalower-frequencylfhalf. theCPEparameter by Z j Z r = 1 2 tan 4 If =1 j Z j = Z r j =0 : 5 ,whichrepresentsaperfectsemicircleintheimpedance plane.IfanelectrodeexhibitsalocalCPEbehavior,thevalueof j Z j = Z r j islessthan 0.5andadepressedsemicircleintheimpedanceplotisobserved.Thedistortionof theimpedanceresponsecanbefoundbydividingtheimpedanceloopintoahigherfrequencyhalfandalower-frequencyhalfasshowninFigure4-13.Thedependenceof j Z j = Z r; hf j and j Z j = Z r; lf j totheparameter canbefoundfromequation4to be Z j Z r; hf or Z j Z r; lf =tan 4 Ifthevaluesofthetworatiosaredifferent,theimpedanceloopisnotsymmetricand doesnotcorrespondtoatrueCPEforwhich mustbeconstant. TheshapeoftheglobalimpedanceloopscanbeobservedinFigure4-5.The calculatedvaluesof fromequations4and4fortheimpedanceloops arepresentedinTable4-2.Forthethreepotentiostaticconditions, m = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 15 V 71

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Table4-2.ValuesofCPEexponent calculatedusingequations4and4for thehigh-frequencyandlow-frequencyimpedanceloopsinFigure4-5. Higher-frequencyhalfLower-frequencyhalfAverage h A i > 0 0 : 011 S/cm 2 s High-frequencyloop0.9280.9950.960 Low-frequencyloop0.9980.9970.997 h A i =0 S/cm 2 s High-frequencyloop0.9190.9910.954 h A i < 0 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 83 S/cm 2 s High-frequencyloop0.9050.9750.939 Low-frequencyloop0.8810.9260.903 h A i =0 : 011 S/cm 2 s, m = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 1 V h A i =0 S/cm 2 s,and m =0 : 1 V h A i = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 83 S/cm 2 s,allhigh-frequencyloopsshowdepressedsemicirclesandthecorresponding valuesaresmallerthanunity.IncontrasttotheresultsgivenbyHuang etal. 12 fora singleFaradaicreaction,thelower-frequencyhalfshowsCPEbehaviorinsteadofan idealimpedanceresponsebecausethecharacteristicfrequencyofthehigh-frequency loopshiftstoahighervalueastheappliedelectrodepotentialincreasesand,thus,the CPEfeaturesinducedbytheelectrodegeometrycanbeseeninthelower-frequency half. Forthepresentstudy,thelow-frequencyloopsfor h A i6 =0 arealsodepressed.For h A i =0 thereisnolow-frequencyloopobserved.Thedeviationof fromunitybecomes largerwhen A isnegative,representingthemorenonuniformcurrentandpotential distributionsas V increases.Thetime-constantdispersionassociatedwiththegeometry ofthediskelectroderesultsinaCPEbehavioratlowfrequencyduetotherelationship betweentheradialdistributionofadsorbedintermediateandtheradialdistributionof interfacialpotential.AsdescribedbyHuang etal. 12 CPEbehaviorisseenaswellat highfrequenciesduetotheradialdistributionofinterfacialpotential. The valuescalculatedfromtheglobalimpedanceresponseofarecessed electrodewereequaltounityforbothhigh-frequencyandlow-frequencyloops.This 72

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idealbehaviorshowsthatthepotentialandcurrentwereuniformfortherecessed electrodegeometry. Theevidenceofafrequencyortime-constantdispersionhasrelevancetotheuse ofmicroelectrodestostudythekineticsoffastelectrochemicalreactions.Forsimple reactions,itissufcienttodesigntheelectrodesuchthatthecharacteristicfrequency thatyields K =1 issufcientlylargerthanthecharacteristicfrequencyforthereaction. Thiscanbeachievedbymakingtheelectrodesmaller.Forreactionsinvolvingadsorbed intermediates,however,frequencyortime-constantdispersionwillbeobservedatlow frequenciesaswell,thuscomplicatingtheinterpretationoftheexperimentalresults. 73

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CHAPTER5 ELECTRODEREACTIONWITHADSORBEDINTERMEDIATES:EXPERIMENTAL Thegeometry-inducedcurrentandpotentialdistributionswereinvestigatedina Faradaicsysteminvolvinganadsorbedintermediate.Thedispersionofimpedance responsewaspredictedtobeobservednotonlyathighfrequencies,butalsoatlow frequenciesinwhichthecharacteristicfrequencyfortheadsorptionreactionwas involved.Anexperimentalvericationofthetheoreticalresultsisprovidedinthischapter usingtheirondissolutionasanexample. 5.1DissolutionofIron Asmostoftheelectrochemicalsystemsdonotfollowthesimplecaseoflinear kinetics,theimpedancemodelpresentedhereisassumedtobeintheTafelregime. Thesystemofinterestisthecorrosionofapureironelectrodein0.5Msulfuricacid. Bockrisandcoworkers 52 proposedareactionmodelinwhichtwoconsecutivestepsare coupledbyanadsorbedintermediate.Theanodicdissolutionofironcanbedescribedin simpliedformas Fe K 1 )167(! Fe + ads +e )]TJ/F20 11.9552 Tf 168.247 -4.936 Td [( and Fe + ads K 2 )167(! Fe 2+ +e )]TJ/F20 11.9552 Tf 162.587 -4.936 Td [( Theironrstoxidizesandformsamonovalentintermediateadsorbedontheelectrode surface.Thisreactionisfollowedbytheoxidationoftheintermediate.Theferrousion issolubleanddiffusesawayfromtheelectrode.Forthiswork,thesystemwasata potentialsuchthatthesystemremainedintheactivedissolutiondomain. UndertheassumptionofTafelkinetics,thereactionratesforreactions5and 5canbeexpressedby i 1 = K 1 )]TJ/F26 11.9552 Tf 11.955 0 Td [( exp[ b 1 m )]TJ/F15 11.9552 Tf 11.955 0 Td [( 0 ] 74

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and i 2 = K 2 exp[ b 2 m )]TJ/F15 11.9552 Tf 11.955 0 Td [( 0 ] where K 1 and K 2 aretheeffectiverateconstants,and b 1 and b 2 aretheTafelslopesfor eachreaction. Thevariationofsurfacecoverageisassociatedwiththetwoanodicreactionsby @ @t = i 1 )]TJ/F26 11.9552 Tf 11.956 0 Td [(i 2 )]TJ/F26 11.9552 Tf 7.314 0 Td [(F Whenthesteadystateisachieved,thesurfacecoverageatsteadystatecanbecalculatedfromequations5and5as = K 1 exp[ b 1 m )]TJETq1 0 0 1 347.714 492.372 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 347.714 482.529 Td [( 0 ] K 1 exp[ b 1 m )]TJETq1 0 0 1 288.659 475.039 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 288.659 465.196 Td [( 0 ]+ K 2 exp[ b 2 m )]TJETq1 0 0 1 406.77 475.039 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 406.77 465.196 Td [( 0 ] Incontrasttothecasepresentedinequation4forlinearkinetics,equation5 showsthatthesteady-statesurfacecoverageisdependentonthesurfaceoverpotential undertheassumptionofTafelkinetics. TheFaradaicimpedanceassociatedwiththeTafelkineticsisexpressedbyequation 4withthecharge-transferresistance R t = 1 K 1 )]TJETq1 0 0 1 217.438 304.66 cm[]0 d 0 J 0.478 w 0 0 m 6.722 0 l SQBT/F26 11.9552 Tf 217.438 297.839 Td [( b 1 exp[ b 1 m )]TJETq1 0 0 1 306.433 307.682 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 306.433 297.839 Td [( 0 ]+ K 2 b 2 exp[ b 2 m )]TJETq1 0 0 1 440.976 307.682 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 440.976 297.839 Td [( 0 ] andtheparameters A and B aregivenby A = f K 2 b 2 exp[ b 2 m )]TJETq1 0 0 1 219.582 248.717 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 219.582 238.874 Td [( 0 ] )]TJ/F26 11.9552 Tf 11.955 0 Td [(K 1 )]TJETq1 0 0 1 153.044 215.889 cm[]0 d 0 J 0.478 w 0 0 m 6.722 0 l SQBT/F26 11.9552 Tf 153.044 209.068 Td [( b 1 exp[ b 1 m )]TJETq1 0 0 1 242.039 218.911 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 242.039 209.068 Td [( 0 ] g K 1 exp[ b 1 m )]TJETq1 0 0 1 352.905 226.999 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 352.905 217.156 Td [( 0 ] )]TJ/F26 11.9552 Tf 11.955 0 Td [(K 2 exp[ b 2 m )]TJETq1 0 0 1 471.211 226.999 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 471.211 217.156 Td [( 0 ] )]TJ/F26 11.9552 Tf 7.314 0 Td [(F and B = K 1 exp[ b 1 m )]TJETq1 0 0 1 290.046 157.456 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 290.046 147.613 Td [( 0 ]+ K 2 exp[ b 2 m )]TJETq1 0 0 1 408.157 157.456 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 408.157 147.613 Td [( 0 ] )]TJ/F26 11.9552 Tf 7.314 0 Td [(F respectively.Accordingtothevaluesofthekineticparameters, A canbepositiveor negative,but B isalwayspositive.Substitutionofequation5intoequations5 75

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and5yieldsthesteady-statepolarizationcurveexpressedas i = 2 K 1 exp[ b 1 m )]TJETq1 0 0 1 297.43 690.113 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 297.43 680.27 Td [( 0 ] K 2 exp[ b 2 m )]TJETq1 0 0 1 401.123 690.113 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 401.123 680.27 Td [( 0 ] K 1 exp[ b 1 m )]TJETq1 0 0 1 287.294 672.78 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 287.294 662.937 Td [( 0 ]+ K 2 exp[ b 2 m )]TJETq1 0 0 1 405.405 672.78 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 405.405 662.937 Td [( 0 ] TheFaradaicimpedancewasthencalculatedfromequation4. InthecaseoflinearkineticsdevelopedSection4.1.1,itiseasytoverifythat A =0 TheFaradaicimpedanceisonlyacharge-transferresistancewhichisinagreementwith equation4. ThedispersionoflocalimpedanceswasdiscussedinChapter4forelectrochemical reactionsinvolvingoneadsorbedintermediate.Whiletheglobalimpedanceatagiven potentiostaticconditionmayhaveaspeciclow-frequencyfeature, e.g., capacitive orinductive,thelocalimpedancesshowavariationoflow-frequencyfeaturesalong theelectrodesurface,whichresultsfromthedistributionofpotentialandcorresponds todifferentvaluesofthelocalparameters A and B .Theglobalimpedancecouldbe consideredtoprovideanaveragerepresentationoftheelectrodesurface,andthelowfrequencyfeatureisdeterminedbythesurfaceaveragevaluesof h A i and h B i .Asthe polarizationconditionchangesalongthecurrent-voltage I V curve,thelow-frequency loopintheglobalimpedancediagramcouldchangefrominductivetocapacitive,and viceversa. 5.2ExperimentalSetup Thediskelectrodeusedintheexperimentwasa0.5cmdiameterpureironelectrodeembeddedinaninsulatingepoxyresin.Theelectrolytesolutionwasa0.5M H 2 SO 4 solutionpreparedwithdistilledwater. TheexperimentalmeasurementsofLEISwereperformedwithabielectrodethat consistedoftwoplatinumwiressealedinaborosilicatebicapillary.Theconguration andpreparationofthebielectrodeweredescribedinearlierpapers. 2,14 Theuseofa four-channelfrequencyresponseanalyzerallowedtheglobal,local,andlocal-interfacial impedancestobemeasuredsimultaneously.Allimpedancemeasurementswere 76

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A B Figure5-1.Thepolarizationcurveforastationaryironelectrodein0.5MH 2 SO 4 :Aa scanincludingboththeactiveandpassivatedregions;Bzoomedportionof theboxinpartAshowingthepotentialatwhichtheimpedance measurementswereperformed. performedatroomtemperatureabout19 C.Thepotentialappliedtotheironelectrode wasselectedtoachievetheactivedissolutionofirononthepolarizationcurve. Theimpedancemeasurementwasperformedusinga50mVpeak-to-peaksinusoidalperturbation,afrequencyrangeof65,000Hzto0.1Hz,andsevenpointsper decadeoffrequency.The50mVperturbationamplitudewasselectedtomaximizethe signal-to-noiseratiowhilemaintaininglinearity. 53,54 Thelinearresponsewasconrmed bymeasuringtheimpedanceusingdifferentamplitudesanddeterminingthemaximum amplitudethatdidnotchangetheimpedanceresponse.Thedenitionsforallthelocal impedanceswerepresentedbyHuang etal. 10 theoreticallyandbyFrateur etal. 2 in anexperimentalperspective.Thesamenotationwillbeusedthroughoutthefollowing resultsanddiscussion. 5.3ExperimentalResults Thecurrent-potentialcurveobtainedonastationaryironelectrodein0.5Msulfuric acidsolutionispresentedinFigure5-1A.Thescanratewas2mV/sec.Thepotential wasappliedtotheironelectrodefromtheopen-circuitpotential,activatingtheiron 77

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A B Figure5-2.Globalelectrochemicalresponseforapureironelectrodein0.5MH 2 SO 4 : AglobalimpedanceresponsemeasuredatpotentialIasindicatedinFigure 5-1B;andBglobalimpedanceresponsemeasuredatpotentialII. electrode,toahigherpotential,passivatingtheelectrodesurface.ThepotentialsIand IIinFigure5-1Brepresentthetwosteady-stateconditionsforwhichtheimpedance measurementswereperformed.TheglobalimpedanceresponsesgiveninFigure 5-2Aand5-2Bshowlow-frequencyloopswhichareassociatedwiththeformationof FeIspeciesadsorbedontheelectrodesurface,andcorrespondtodifferentvaluesof surface-averagevalue h A i .Tofacilitatethecomparisonofresults,allfrequencieswere madedimensionlessaccordingto K = !C 0 r 0 =4 !C 0 r 2 0 R e where r 0 istheelectroderadius, C 0 isthedouble-layercapacitance,whichwasassumed tobeuniform, istheelectrolyteconductivity,and R e istheelectrolyteresistance.The parameters C 0 and R e weredeterminedfromtheglobalimpedancemeasurements.The electrolyteresistance R e wasevaluatedfromthelimitingvalueoftherealpartofthe impedanceinthehigh-frequencyrange;whereas, C 0 canbeeasilydeterminedfrom 78

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A B Figure5-3.Experimentallocalimpedanceforapureironelectrodein0.5MH 2 SO 4 measuredatApotentialI;andBpotentialIIinFigure5-1B. thecharacteristicfrequencyoftherstRCtimeconstant,whichrepresentsthehighfrequencycapacitiveloopcorrespondingto R t inparallelwith C 0 .Whentheelectrode potentialincreases,thefeatureoftheimpedancediagramatlowfrequencieschanges fromaninductivetoacapacitivebehavior,implyingthat h A i changessign. Theexperimentallocalimpedancediagramsobtainedwithabielectrodelocated atdifferentpositionsonthediskelectrodearedepictedinFigure5-3.WhenmeasurementswereperformedatthepolarizationpointI,thevariationsofthelocalimpedance showdifferentreactivityasafunctionoftheprobelocationovertheironelectrode.Such abehaviorisattributedtothegeometry-inducedcurrentandpotentialdistributions. 1012 InFigure5-3A,thelow-frequencyinductiveloopislargeratcenteroftheelectrodeand smallerattheperiphery,whichisinagreementwithapreviousstudy 3 andindicates agreateraccessibilityandmorereactivedomainneartheedgesoftheelectrode.In Figure5-3B,onlythelocalimpedanceatthecenterofthediskispresented.Themeasurementisdifculttoperformatamoreanodicpotential i.e., atpotentialIIorforlarger 79

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potentialsbecauseofanincreaseinthereactionrateleadingtoanincreaseintherate ofirondissolution,especiallyatelectrodeedges.Theremayalsobesomeaccumulation ofcorrosionproductspecies.Atlowfrequencies,thelocalelectrochemicalimpedance showsaninductivefeaturewhichisdifferentfromthecapacitivebehaviorshowninthe globalimpedanceplotinFigure5-2B,representingalocallypositivevalueof A atthe electrodecenter. ThecontributionofthelocalOhmicimpedancecanbecalculatedfromthedifferenceofthelocalandlocal-interfacialimpedancemeasurements.Forthesteady-state conditionI,thelocalOhmicimpedanceispresentedinFigure5-4Awiththeradial positionoftheprobeasaparameter.Asalreadyobservedforblockingelectrodesorfor simpleelectrontransferreactions, 1012 thiscontributionactsasanimpedancewithcomplexfeaturesandisstronglydependentontheradialpositionalongtheelectroderadius. However,whenanadsorbedintermediateisinvolved,complexbehaviorsareobserved atlowfrequenciesaswellasathighfrequencies.Expandedlow-frequencyregionsare presentedinFigures5-4Band5-4CforthelocalOhmicimpedanceatthecenterand edgeoftheelectrode,respectively.AsdiscussedinChapter4,suchageometryeffectis atypicalfeatureatlowfrequenciesforelectrochemicalsysteminvolvinganadsorbedintermediate.FromtherepresentationoftheimaginarypartsoflocalOhmicimpedanceas afunctionofthedimensionlessfrequencyinFigure5-5,nonzerovaluesareobserved atlowfrequencies,butthesevariationsremainsmallerthanthoseobservedathigh frequencies. Astheelectrodepotentialisbiasedtoamoreanodicpotential,low-frequency complexfeaturesinthelocalOhmicimpedancebecomemoresignicantasshownin Figure5-6,whichisinagreementwiththepreviousstudy. 3 Thecharacteristicfrequency atlowfrequenciesisassociatedwiththepotentialdependenceofthesurfacecoverage bytheadsorbedintermediate.Thecharacteristicfrequencydependsontheeffective rateconstantsandmaynotbethesameforadifferentreactionsystem. 80

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A B C Figure5-4.LocalOhmicimpedanceforapureironelectrodein0.5MH 2 SO 4 atpotential IinFigure5-1B.ALocalOhmicimpedanceatthecenterandtheedgeof theelectrode;Btheenlargementforthelow-frequencyendofthelocal Ohmicimpedanceatelectrodecenter;andCasimilarenlargementforthe electrodeedge. 81

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Figure5-5.TheimaginarypartofthelocalOhmicimpedanceasafunctionofthe dimensionlessfrequencyforapureironelectrodein0.5MH 2 SO 4 measured atpotentialIinFigure5-1B. A B Figure5-6.LocalOhmicimpedanceforapureironelectrodein0.5MH 2 SO 4 atpotential IIinFigure5-1B.ANyquistplotforthelocalOhmicimpedancesatcenterof theelectrode;andBimaginarypartofthelocalOhmicimpedanceasa functionofdimensionlessfrequency. 82

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Table5-1.Parametersusedinthesimulationsofirondissolutioninsulfuricacidsolution. SymbolMeaningValueUnits b 1 1 F=RT 38.2V )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 b 2 2 F=RT 8.05V )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 K 1 effectiverateconstantforreaction5 3 10 13 A/cm 2 K 2 effectiverateconstantforreaction530A/cm 2 C 0 double-layercapacitance55 F/cm 2 electrolyteconductivity0.2S/cm )]TJ/F20 11.9552 Tf 51.813 0 Td [(maximumsurfacecoveragebyintermediate 1 10 )]TJ/F24 7.9701 Tf 6.587 0 Td [(8 mole/cm 2 5.4SimulationResults Themathematicaldevelopmentforthecalculationofimpedancewasdescribed byinChapter4.Attheelectrodesurface,thepotentialperturbationwascoupledby thecurrentassociatedwiththechargingoftheelectricaldoublelayerandtheFaradic reactions.ThepotentialinthebulksolutionsatisesLaplace'sequationandwassolved incylindricalcoordinates. Theexperimentalpolarizationcurvewasrstttedbyequation5allowing thedeterminationofthekineticparameters K 1 K 2 b 1 ,and b 2 .TheLEISresultswere stronglydependentontheseparameters.Then,thekineticvariableswerevaried toobtainsimulationsinagreementwithboththepolarizationcurveandtheglobal impedancesatthetwosteady-stateconditions.Thevaluesofkineticvariablesand otherparametersusedinthesimulationsaregiveninTable5-1.Itshouldbementioned thatthesolepurposeofthecomparisonofexperimentandsimulationsistoverifythe predictionmadeinChapter4thattheOhmicimpedanceforsystemsinvolvingadsorbed intermediatesmayhavecomplexcharactersatbothlowandhighfrequencies.Aswill bedescribedintheDiscussionsection,thisworkwasnotintendedtodeterminethe kineticconstantsforthereactionsystem.Thus,calculatedresultspresentedinthis sectioncanbeusedtoshowaqualitativecomparisonwiththeexperimentaldata. Theexperimentalcurvescanpotentiallybettedbyothersetsofkineticparameters. TheparameterspresentedinTable5-1areinagreementwiththosepresentedby 83

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A B Figure5-7.ThecalculatedglobalimpedancecorrespondingtoApolarizationpointI andBpointIIinFigure5-1B.Theseresultsaretobecomparedtothe experimentalresultspresentedinFigure5-2Aand5-2B. EpelboinandKeddam. 48 Thevaluesof 1 and 2 obtainedfromthevaluesof b 1 and b 2 respectively,differfrom0.5,consistentwithnon-elementarystepreactions. 21 ThesimulationresultsfortheglobalimpedancescorrespondingtopotentialI andIIintheexperimentalpolarizationcurvearepresentedinFigure5-7Aand57B,respectively,andshouldbecomparedtoexperimentalEISdiagramspresented inFigures5-2Aand5-2B.Theimpedancevaluesforthesimulationcurvesarein agreementwiththeexperimentalresults,withtheexceptionthatthecharacteristic frequencyhasasmallervalueforsteady-stateconditionII.Thiscanbeattributedto theuncertaintiesintheestimationofthekineticparameters.Thecalculatedglobal impedancehasalow-frequencyinductiveloopatpotentialI,correspondingtoapositive valueof h A i .Thelow-frequencyfeatureatpotentialIIisnotveryobvious,butaslight capacitivebehaviorcanstillbeobservedintheenlargedregionatthelowfrequency limit,correspondingtoasmallnegativevalueof h A i 84

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A B Figure5-8.ThecalculatedlocalimpedancecorrespondingtoApolarizationpointIand BpointIIinFigure5-1B.Theseresultsaretobecomparedtothe experimentalresultspresentedinFigure5-3. ThecalculatedlocalimpedancevaluesareshowninFigure5-8fortwodifferent steady-stateconditions.InFigure5-8A,thelocalimpedancecalculatedatpotentialI showslittledifferencesbetweenelectrodecenterandelectrodeedge,incontrasttothe experimentalresultsinFigure5-3A,especiallytheinductiveloopsatlowfrequencies. Thereasonforthelargerdifferencesintheexperimentaldatacouldresultfromthe dissolutionofironelectrodeduringtheimpedancemeasurementsatlowfrequencies. Theironelectrodeisrecessedmoreattheedgeoftheelectrode,and,therefore,the impedancedataweremeasuredatdifferentheights.Thediscrepancyofthelocal impedanceamongdifferentpositionsislargeratahigherpotentiostaticcondition,which isalsoingoodagreementwiththeaboveexplanation.AtpotentialII,thecalculated impedanceatthecenteroftheelectrodehasainductiveloopatlowfrequenciesas shownintheenlargedregion,whichisinagreementwiththeexperimentalresult inFigure5-3B.Itcorrespondstoalocalpositivevalueof A attheelectrodecenter. Thecalculatedimpedanceattheelectrodeedgehasacapacitivefeature,which 85

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correspondstoalocalnegativevalueof A .Suchabehaviorisinaccordancewiththe resultspresentedinChapter4thatpredictedalocalvariationof A withtheelectrode radius. ThecalculatedlocalOhmicimpedancesforsteady-stateconditionIarepresented inFigure5-9.AscomparedtoFigure5-4,thelocalOhmicimpedancesatthecenter andtheedgeoftheelectrodehavesimilarbehaviors.Complexfeaturesareevidentat lowfrequenciesinbothexperimentalFigures5-4Band5-4Candsimulationresults Figures5-9Band5-9C.However,fromthedependenceoftheimaginarypartofthe localOhmicimpedanceonfrequencyshowninFigure5-10,thecomplexvaluesatlow frequenciesareverysmallascomparedtothatathighfrequenciesandaretherefore barelyevident. ThevariationoftheimaginarypartofthelocalOhmicimpedancewithfrequencyat potentialIIaregiveninFigure5-11.Fordimensionlessfrequencies 10 )]TJ/F24 7.9701 Tf 6.586 0 Td [(5
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A B C Figure5-9.ThelocalOhmicimpedancecalculatedatpotentialIinFigure5-1B.ALocal Ohmicimpedanceatthecenterandtheedgeoftheelectrode;Bthe enlargementforthelow-frequencyendofthelocalOhmicimpedanceatthe electrodecenter;andCtheelectrodeedge.Theseresultsaretobe comparedtotheexperimentalresultspresentedinFigure5-4. 87

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Figure5-10.TheimaginarypartofthelocalOhmicimpedanceasafunctionofthe dimensionlessfrequencycalculatedatpotentialIinFigure5-1B.Thisplot istobecomparedtotheexperimentalresultpresentedinFigure5-5. Figure5-11.TheimaginarypartofthelocalOhmicimpedanceasafunctionofthe dimensionlessfrequencycalculatedatpotentialIIinFigure5-1B.Thisplot istobecomparedtotheexperimentalresultpresentedinFigure6-1B. 88

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A B Figure5-12.CalculatedlocalOhmicimpedanceforthecase h A i =0 presentedbyWu etal. 3 ANyquistplotforthelocalOhmicimpedancesatthecenterofthe electrode;andBimaginarypartofthelocalOhmicimpedanceasa functionofthedimensionlessfrequency.Theseresultsaretobecompared totheexperimentalresultspresentedinFigure5-6. aremadeasmalldistanceabovetheelectrodesurface.Inaddition,therapiddissolution oftheironelectrodecausedthedistancebetweentheprobeandtheelectrodeto changewithtime.Duringthecourseofacyclicvoltammetrymeasurement,forexample, theelectroderecededby2mm.Impedancemeasurementsatfrequencieslowerthan 0.1Hz, i.e., K< 4 10 )]TJ/F24 7.9701 Tf 6.586 0 Td [(5 ,couldnotbemadebecausethetimerequiredledto signicantelectroderecession.Inthesimulation,theelectrodesurfaceisassumedtobe smooth,withapositionthatisindependentoftime,andwithanactiveareaunaffected bydepositionofcorrosionproducts.Thereforethesimulationcannotgiveaquantitative comparisontothemeasuredresults. Nevertheless,thequalitativecomparisonbetweentheexperimentalandthe calculatedimpedanceresponseprovidessignicantsupportsforthesimulationresults. TheexperimentallocalOhmicimpedanceexhibitscomplexvaluesatlowfrequencies. Thesecomplexbehaviorsaremoreevidentatahigherpotential,inagreementwiththe theoreticalcalculations.Moreover,theradialdependenceofthelow-frequencyresponse 89

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demonstratesthatthegeometry-inducedcurrentandpotentialdistributionsinuence theimpedanceresponseatlowfrequencieswhenadsorbedintermediatesareinvolved inthereactions.Thecalculationsareinbetteragreementwiththeexperimentalresults atsteady-stateconditionI.Atsteady-stateconditionII,theappliedpotentialismore anodic,andthecongurationoftheironelectrodechangedduringthecourseofthe experiment.However,theseresultsshowadistributionoflocalimpedancecausedby thenonuniformdistributionsofcurrentandpotentialinagreementwiththeprediction presentedinChapter4. 90

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CHAPTER6 INFLUENCEOFMASSTRANSFERONSTEADY-STATECURRENTANDPOTENTIAL DISTRIBUTIONS Inthischapter,wearetoexplorethecurrentandpotentialdistributionsassociated withthenonuniformmasstransferandtheelectrodegeometry.Thetheoreticaldevelopmentincorporatingthepotentialdistributionwiththemasstransferandelectrode kineticsisprovided.Thecoupledsolutionsforthepotentialandconcentrationatsteady statearediscussedintworedoxsystems:reductionofferricyanideandoxidationof ferrocyanide,andelectrochemicaldepositionanddissolutionofsilver. 6.1MathematicalDevelopment InsteadofusingNewman's 1 approachofdividingtheproblemintoseparatebut coupleddomainsconsistingofainnerdiffusionregionneartheelectrodesurfaceand anouterregionwithuniformconcentration,theuxofeachspeciesisnowconsidered inaintegraldomaingovernedbytheconvectivediffusionequation.Amathematical developmentforthesteady-statepotentialandconcentrationdistributionsonaRDEis presentedinthissection. 6.1.1MassTransportinDiluteSolutions Themasstransportofspeciesinadilutesolutioncanbedescribedbythreetypes ofmotion:convectionifthespeciesismovingwiththeuidofbulkvelocity v ,diffusionif thereisaconcentrationgradient r c i ,andmigrationforspeciesofcharge z i subjectedto anelectriceld r .Theuxofaspecies i canbeexpressedby N i = )]TJ/F26 11.9552 Tf 9.298 0 Td [(z i u i Fc i r )]TJ/F26 11.9552 Tf 11.955 0 Td [(D i r c i + c i v where u i isthemobilityofspecies i andisrelatedtothediffusioncoefcient D i by u i = D i RT 91

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inadilutesolution.Theelectriccurrentisrelatedtotheuxofthechargedspeciesby i = F X i z i N i Themassofeachspeciesisconservedandcanbeexpressedbythemassbalance equationfollowing @c i @t = r N i + R i where R i representsthesourceorsinkofmaterialduetohomogeneousreactionsinthe bulksolution.Assumingthatthereisnohomogeneousreactionintheelectrolyticsystem andthediffusioncoefcientofeachspeciesisconstant,substitutionofequation6 intoequation6yieldsthemassbalanceequationforeachionicspeciesas v r c i = z i u i F r c i r + D i r 2 c i Theaboveequationappliesatsteadystateandusestherelationforincompressible uid, i.e., r v =0 .Foranaqueoussystemwith n species,thereare n massbalance expressionsintheformofequation6.Theelectricalpotential canbeobtainedby addingtheconservationofcharge r i = r F X i z i )]TJ/F26 11.9552 Tf 9.299 0 Td [(z i u i Fc i r )]TJ/F26 11.9552 Tf 11.955 0 Td [(D i r c i # =0 astheadditionalequation. Theconcentrationofeachspeciesandthesolutionpotentialaregovernedbythe massandchargeconservationequations.Beforeapplyingtheboundaryconditionsto getthecoupledsolutionforconcentrationandpotential,theuidvelocityinthemass conservationequation6needstobedened. 6.1.2FluidFlowonaRotatingDiskElectrode TheuidmechanicsforowonaRDEiswell-understood. 29,21,55 Therotationof diskwithanangularvelocity dragstheuidatitssurfacewiththesamevelocity. Becauseofthecentrifugalforce,theuidnearthedisksurfaceisdrivenoutwardfrom 92

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thecentertothediskperiphery,resultingavelocitydistributionintheradialdirection. Theuidatelectrodesurfaceisrelledbytheowthatbringsuidfromthedistanceto thedisk,resultingavelocitydistributionintheaxialdirection. VonK arm an 55 andCochran 29 solvedtheequationsofcontinuityandmotion forasteadymotionofincompressibleuid,andderivedthevelocityprolebyusing cylindricalcoordinates.Thesolutionwasobtainedbyusingseparationofvariableswith adimensionlessdistancefromthedisk = y r andthedimensionlessvelocitiesinradialandaxialdirections r = r F and y = p H where isthesolutionviscosity.Forsmalldistancesabovethedisksurface,the dimensionlessradialvelocity F andaxialvelocity H areexpressedinpowerseriesof as F = a )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 2 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 3 3 + ::: and H = )]TJ/F26 11.9552 Tf 9.299 0 Td [(a 2 + 1 3 3 + b 6 4 + ::: withcoefcients a =0 : 51023 and b = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 616 .Forlargerdistancesfromthedisk,the dimensionlessvelocityareexpressedinpowersof e )]TJ/F27 7.9701 Tf 6.586 0 Td [(c as F = Ae )]TJ/F27 7.9701 Tf 6.587 0 Td [(c )]TJ/F26 11.9552 Tf 13.151 8.088 Td [(A 2 + B 2 2 c 2 e )]TJ/F24 7.9701 Tf 6.586 0 Td [(2 c + A A 2 + B 2 4 c 4 e )]TJ/F24 7.9701 Tf 6.586 0 Td [(3 + ::: and H = )]TJ/F26 11.9552 Tf 9.298 0 Td [(c + 2 A c e )]TJ/F27 7.9701 Tf 6.586 0 Td [(c + A 2 + B 2 2 c 2 e )]TJ/F24 7.9701 Tf 6.586 0 Td [(2 c + ::: 93

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A B Figure6-1.Velocityprolesonarotatingdiskelectrode:AradialandBaxial componentsofthedimensionlessvelocity. withcoefcients c =0 : 88447 A =0 : 934 ,and B =1 : 208 Tosolvethemassandchargeconservationequations,avelocityproleisrequired todescribetheuidowinthewholedomain, i.e., = )]TJ/F26 11.9552 Tf 11.955 0 Td [(f 0 + f !1 where f isaninterpolationfunctionthatisusedtoweighthevelocityexpansionsinthe innerandouterregionsofdiffusionlayerfollowing f = 1 1+ e )]TJ/F27 7.9701 Tf 6.587 0 Td [( )]TJ/F27 7.9701 Tf 6.586 0 Td [( 0 where and 0 areconstant.ThisinterpolationfunctionissimilartotheFermi-Dirac functionappliedinquantummechanicsfordescribingthedistributionoffermions.Inthe presentcalculation,weused =25 and 0 =1 toestimatetheuidvelocityprole. Figure6-1showstheradialandaxialcomponentsofthedimensionlessvelocityas afunctionofdimensionlessdistancefromtheelectrodesurface.Thevelocityexpression applyingtheinterpolationfunctionsatisesthevelocitiesforsmallandlargevaluesof 94

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andshowsasmoothtransitioninthemediumdistancefromthedisksurface.Therefore, theuseofthevelocityproleinequation6onaRDEisjustied. 6.1.3FluxandCurrentatElectrodeBoundary Atelectrodeboundary,theuxofeachspeciesisdeterminedbythepotentialand concentrationgradientsatelectrodesurfaceby N i = )]TJ/F26 11.9552 Tf 9.299 0 Td [(z i u i Fc i @ @y y =0 )]TJ/F26 11.9552 Tf 11.955 0 Td [(D i @c i @y y =0 Theuxisalsorelatedtothecurrentdensityofanelectrodereactionby N i = )]TJ/F26 11.9552 Tf 14.142 8.088 Td [(s i nF i where s i isthestoichiometriccoefcientofspecies i ,and n isthenumberofelectrons transferredinthatelectrodereaction.Foranelectrochemicalreactioninuencedby thetransportofreactants,thereactionrateisdependentontheconcentrationsofthe reactantsinadditiontothepotentialatelectrodesurface.Forageneralredoxreaction R O+ n e )]TJ/F20 11.9552 Tf 171.154 -4.936 Td [( thereactionrateisgivenby i = k a c R ; 0 exp )]TJ/F26 11.9552 Tf 11.955 0 Td [( nF RT V )]TJ/F26 11.9552 Tf 11.955 0 Td [(k c c O ; 0 exp )]TJ/F26 11.9552 Tf 9.299 0 Td [(nF RT V where k a and k c aretherateconstantsfortheanodicandcathodicreactions,respectively, c R ; 0 and c O ; 0 aretheconcentrationsforthereductant R andtheoxidant O ,respectively,measuredattheinnerlimitofdiffusionlayer,and V istheinterfacialpotential denedbythedifferencebetweentheelectrodepotential m andthesolutionpotential adjacenttotheelectrode 0 .Thesymmetryfactor ,ortransfercoefcient,represents thefractionofenergyorpotentialthatisusedtopromotethecathodicreaction.Therate expressioninequation6assumesthatboththeanodicandcathodicreactionsare rst-order. 95

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Whenthenetcurrentforthereactioniszero,thesystemisatequilibrium.The equilibriumpotentialcanbeobtainedbyrearrangingequation6andisexpressed by V 0 = RT nF ln k c k a )]TJ/F15 11.9552 Tf 11.955 0 Td [(ln c O ; 0 c R ; 0 Thedepartureof V fromitsequilibriumvalue V 0 isdenedasthesurfaceoverpotential, i.e., s = V )]TJ/F26 11.9552 Tf 11.955 0 Td [(V 0 Substitutionofthesurfaceoverpotentialintoequation6yieldstheButler-Volmer equationas i = i 0 exp )]TJ/F26 11.9552 Tf 11.955 0 Td [( nF RT s )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F26 11.9552 Tf 9.299 0 Td [(nF RT s wheretheexchangecurrentdensity i 0 = k a k 1 )]TJ/F27 7.9701 Tf 6.587 0 Td [( c c R ; 0 c 1 )]TJ/F27 7.9701 Tf 6.587 0 Td [( O ; 0 isafunctionofionicconcentrationsattheelectrodesurface,and,therefore,isafunction ofpotential. AnotherexpressionfortheButler-Volmerequationcanbeobtainedbyusingthe totaloverpotential intherateequation i = i 1 0 c R ; 0 c R ; 1 exp )]TJ/F26 11.9552 Tf 11.956 0 Td [( nF RT )]TJ/F26 11.9552 Tf 15.267 8.088 Td [(c O ; 0 c O ; 1 exp )]TJ/F26 11.9552 Tf 9.298 0 Td [(nF RT where c R ; 1 and c O ; 1 arethebulkconcentrationsforthereactants R and O ,respectively.Thetotaloverpotentialisthesummationofthesurfaceoverpotentialandthe concentrationoverpotential, = s + c wheretheconcentrationoverpotentialisdenedby c = RT nF ln c R ; 1 c R ; 0 )]TJ/F15 11.9552 Tf 11.956 0 Td [(ln c O ; 1 c O ; 0 96

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Theexchangecurrentdensityinreaction6 i 1 0 = k a k 1 )]TJ/F27 7.9701 Tf 6.586 0 Td [( c c R ; 1 c 1 )]TJ/F27 7.9701 Tf 6.586 0 Td [( O ; 1 isnowdependentonthecompositionofthebulksolutionandisnotafunctionof electrodepotential.Withtheuseofappropriateexpressionsforthesurfaceuxand currentdensityatelectrode,theconcentrationsofeachspeciesandthepotentialinthe solutioncanbeobtainedbysolvingtheconservationequations6and6. 6.2NumericalSimulation Simulationswereappliedtotwoelectrochemicalsystemsincludingtheferro/ferricyanide redoxcouple FeCN 4 )]TJ/F24 7.9701 Tf -4.234 -7.892 Td [(6 FeCN 3 )]TJ/F24 7.9701 Tf -4.235 -7.892 Td [(6 +e )]TJ/F20 11.9552 Tf 133.988 -4.936 Td [( andthesilverredoxreaction Ag Ag + +e )]TJ/F20 11.9552 Tf 165.174 -4.937 Td [( Thereductionofferricyanideandoxidationofferrocyanidehavefastkineticsandare oftenusedtostudymasstransferinelectrolyticsystems. 36,37,56 Thesolutionunder studywasconsistedofK 3 FeCN 6 andK 4 FeCN 6 ofequalconcentration0.01M, andsupportingelectrolyteKClofconcentration1M.Therateconstantswerechosen tomaketheequilibriumpotentialbe0.23Vandtheexchangecurrentdensitybe 0.3A/cm 2 38,57 Forthesilverredoxreaction,thediffusionofsilverionwasassumedto havenegligibleeffectontheanodicdissolutionreaction.Thesolutionwasconsidered tobe0.1MAgNO 3 inthepresenceofdifferentconcentrationsofsupportingelectrolyte KNO 3 .Therateconstantswerechosentomaketheequilibriumpotentialbe0.799V, 21 andtheexchangecurrentdensitybe1A/cm 2 58,59 Thesolutioncompositionsandkinetic parametersusedinbothsystemsarelistedinTable6-1. Theequationsweresolvedbyusingthenite-elementpackage COMSOLMultiphysics r withtheNernst-Planckmoduleina2Daxialsymmetriccoordinatesystem.A quarter-circlewasconstructedwithanaxisofsymmetryat r =0 andtheelectrodeof 97

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Table6-1.Parametersusedforcalculatingthesteady-statecurrentandpotential distributionsonaRDEatroomtemperature. Redoxofferricyanide/ferrocyanideElectrodepositionofsilver c K 3 FeCN 6 ; 1 0.01 10 )]TJ/F24 7.9701 Tf 6.587 0 Td [(3 mol/cm 3 c AgNO 3 ; 1 0.01 10 )]TJ/F24 7.9701 Tf 6.587 0 Td [(3 mol/cm 3 c K 4 FeCN 6 ; 1 0.01 10 )]TJ/F24 7.9701 Tf 6.587 0 Td [(3 mol/cm 3 c KNO 3 ; 1 1 10 )]TJ/F24 7.9701 Tf 6.586 0 Td [(3 mol/cm 3 c KCl ; 1 1 10 )]TJ/F24 7.9701 Tf 6.586 0 Td [(3 mol/cm 3 0.1 10 )]TJ/F24 7.9701 Tf 6.586 0 Td [(3 mol/cm 3 D FeCN )]TJ/F25 5.9776 Tf 5.756 0 Td [(3 6 0.896 10 )]TJ/F24 7.9701 Tf 6.586 0 Td [(5 cm 2 /s0.01 10 )]TJ/F24 7.9701 Tf 6.586 0 Td [(3 mol/cm 3 D FeCN )]TJ/F25 5.9776 Tf 5.756 0 Td [(4 6 0.739 10 )]TJ/F24 7.9701 Tf 6.586 0 Td [(5 cm 2 /s D Ag + 1.648 10 )]TJ/F24 7.9701 Tf 6.586 0 Td [(5 mol/cm 3 D Cl )]TJ/F20 11.9552 Tf 52.632 -0.299 Td [(2.032 10 )]TJ/F24 7.9701 Tf 6.586 0 Td [(5 cm 2 /s D NO )]TJ/F25 5.9776 Tf 0 -6.117 Td [(3 1.902 10 )]TJ/F24 7.9701 Tf 6.586 0 Td [(5 mol/cm 3 D K + 1.957 10 )]TJ/F24 7.9701 Tf 6.586 0 Td [(5 cm 2 /s D K + 1.957 10 )]TJ/F24 7.9701 Tf 6.586 0 Td [(5 mol/cm 3 0.951 10 )]TJ/F24 7.9701 Tf 6.586 0 Td [(2 cm 2 /s 1 10 )]TJ/F24 7.9701 Tf 6.586 0 Td [(2 cm 2 /s 0.5 0.5 k a 340A-cm/mol k a 1.75 10 )]TJ/F24 7.9701 Tf 6.586 0 Td [(7 A/cm 2 k c 2.64 10 6 A-cm/mol k c 5.72 10 10 A-cm/mol radius0.25cmwaspositionedat y =0 .Thedomainsizewas2,000timeslargerthan thediskradiusinordertomeettheassumptionthatthecounterelectrodewaslocated innitelyfarfromtheelectrodesurface. Themesheddomainusedtocalculatethecoupledsolutionforpotentialand concentrationsareshowninFigure6-2.Acoarsemeshwasusedinthemostofthe domaindistantfromtheelectrodetoreducephysicalmemoryusageandcalculation time.Anermeshwasappliedintheregionthatis20timeslargerthanthediskradius tocapturethevariationofpotentialinthevicinityofelectrode.Sincetheconcentration ofionicspeciesvariesonlyinasmalldistanceabovetheelectrodesurface,amuch nermeshwasconstructedintheregionthatistentimeslargerthanthecharacteristic thicknessofthediffusionlayer = 3 D i a 1 = 3 1 = 2 tocapturetheconcentrationvariationinthethinregion.Foratypicalelectrolyticsystem withionicdiffusivity D i = 1 10 )]TJ/F24 7.9701 Tf 6.587 0 Td [(5 cm 2 /sandviscosity = 1 10 )]TJ/F24 7.9701 Tf 6.587 0 Td [(2 cm 2 /s,andforthedisk rotationspeedof120rpm,thediffusionlayerthicknesshasavalueof0.005cm.The numberofelementattheelectrode-insulatorboundary.25 0.001cmis100. 98

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Figure6-2.Meshedmodelusedtocalculatethepotentialandconcentrationdistributionsinthewholedomainandnearthe electrode-insulatorinterface. 99

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6.3SimulationResults Thecoupledsteady-statesolutionforpotentialandconcentrationinthesystem containingferro/ferricyanideredoxcoupleandthesilverredoxreactionarepresented inthissection.Theinuenceofnonuniformmasstransferonthepotentialandcurrent distributionsarediscussedfordifferentfractionsoflimitingcurrentandatdifferent rotationspeeds. 6.3.1PotentialandConcentrationProlesnearElectrodeSurface Themotionofionicspeciesinanelectrolyticsolutionisdeterminedbytheelectrode reactionandthecontributionsofdiffusion,migration,andconvectiontothemass ux.The2Dsurfaceplotsofthepotentialandconcentrationdistributionsinsystems ofreductionofferricyanideanddepositionofsilverionareshowninFigures6-3A and6-3B,respectively.Thecontourlinesindicatethepotentialdistributionsandthe surfacecolorsrepresenttheconcentrationvariationsofferricyanideFigure6-3A andsilverFigure6-3B.Inbothgures,theconcentrationsvaryonlyinathinregion abovetheelectrodesurfaceshowingthepresenceofaconcentrationdiffusionlayer. Theequipotentiallinesaresmoothinthebulksolutionandshowatransitionnearthe diffusionlayerboundary.Thetransitionismoresignicantinthesystemofreduction offerricyanideduetothelargerchargenumbersassociatedwithferricyanideand ferrocyanide. ThepotentialandconcentrationprolesnormaltotheelectrodesurfaceforreductionofferricyanideanddepositionofsilveraregiveninFigures6-4and6-5, respectively.Thepotentialgradientforthesystemofreductionofferricyanideshowsa clearchangeatthepositionwheretheconcentrationofeachspeciesapproachesthe bulkvalue.Atelectrodecenter,thenetcurrentintheradialdirectioniszero.Theux ofsupportingspeciesisalsozeroatelectrodesurface.Theuxofpotassiumionat 100

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A B Figure6-3.Steady-statepotentialandconcentrationdistributionsforAreductionof ferricyanideandBdepositionofsilveratone-fourthoflimitingcurrentona rotatingdiskelectroderotatingat120rpm.Thesurfacecolorsrepresentthe concentrationvariationsofAferricyanideandBsilverion.Thecontour linesrepresentthepotentialdistributions.Theinsertsshowthepotentialand concentrationprolesnearelectrodesurface. 101

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A B Figure6-4.ProlesofApotentialandBconcentrationatelectrodecenterforreduction offerricyanideonarotatingdiskelectrodeofrotationspeed120rpmandat one-fourthofthelimitingcurrent. A B Figure6-5.ProlesofApotentialandBconcentrationatelectrodecenterfor depositionofsilveronarotatingdiskelectrodeofrotationspeed120rpm andatone-fourthofthelimitingcurrent. 102

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Figure6-6.Polarizationbehaviorforthereductionofferricyanideandoxidationof ferrocyanideonarotatingdiskelectrode. electrodeisgivenby N K + = )]TJ/F26 11.9552 Tf 9.299 0 Td [(z K + u K + Fc K + @ @y y =0 )]TJ/F26 11.9552 Tf 11.955 0 Td [(D K + @c K + @y y =0 =0 Sincetheconcentrationgradientofpotassiumionisnegativeinbothsystems,the potentialgradientatelectrodemustbepositivetobalancethediffusionofpotassiumion bythemigrationofitself,andgivesazerouxatelectrodesurface.Thesignofpotential gradientcanalsobedeterminedfromtheconcentrationgradientsofothersupporting speciesintheelectrolyte.Thepotentialgradientinsolutionisaffectedbythecurrent densityandtheconcentrationsofallionicspecies,andthereforethepotentialgradient hasdifferentvalueswithinthediffusionlayerandinthebulksolution. 6.3.2CurrentDistributiononElectrodeSurface ThepolarizationbehaviorforthereductionofferricyanideandoxidationofferrocyanideisshowninFigure6-6withrotationspeedasasparameter.Athigherpotentials,theelectrodereactionislimitedbythemasstransferofreactingspeciestothe electrodesurface.Thecathodicandanodiclimitingcurrentshavedifferentvaluesdueto 103

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A B C Figure6-7.Calculatedcurrentdistributionsforthereductionofferricyanideonarotating diskelectroderotatingatA120,B600,andC2400rpm. thefactthatthediffusioncoefcientsforferricyanideandferrocyanidearedifferent.The valueoflimitingcurrentisproportionaltothesquarerootoftherotationspeed. 28 Thecurrentdistributionsonelectrodesurfacearecalculatedatone-fourth,one-half, andthree-fourthsofthecathodiclimitingcurrentasshowninFigure6-7.Thesurface currentdensityisnormalizedbytheaveragecurrentdensityobtainedateachfractionof limitingcurrent.Thesurfacecurrentexhibitsnonuniformdistributionsinspiteoftheuse ofexcesssupportingelectrolyte.Theseresultsareinagreementwiththecalculationsof DurbhaandOrazem. 36 Atalargerfractionoflimitingcurrent,thesurfacecurrentdensity 104

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Figure6-8.Polarizationbehaviorforthedissolutionanddepositionofsilveronarotating diskelectrodeofrotationspeed120rpm.Thesolutionconsistsof0.1M AgNO 3 and1M,0.1M,and0.01MofsupportingelectrolyteKNO 3 becomesmoreuniformbecausethemasstransferresistanceismoreimportantthanthe Ohmicresistanceofelectrolyte.Forthesamereason,theoverallcurrentdistributionis moreuniformathigherrotationspeed. Thereactionkineticsofreductionofferricyanideandoxidationofferrocyanide arehighlyrelatedtotheconcentrationofcation, i.e., theconcentrationofsupporting electrolyte. 60 Toignoretheeffectofreactionkinetics,theredoxreactionofsilveris appliedtoinvestigatetheroleofsupportingelectrolyteonthecurrentdistribution.The polarizationcurvesforthedepositionanddissolutionofsilverindifferentconcentrations ofsupportingelectrolytearegiveninFigure6-8.Inthecaseofmetaldeposition,the absenceofsupportingelectrolyteincreasestheOhmicpotentialdropinthebulksolution andalsostrengthentheelectriceldinthediffusionlayer.Therefore,themigration ofreactingspeciesinthediffusionlayerenhancesthelimitingcurrentdensity.Inthe presenceofanexcesssupportingelectrolyte,thelimitingcurrentonarotatingdisk 105

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electrodeisexpressedby 61 i lim ; D = )]TJ/F26 11.9552 Tf 9.298 0 Td [(s i nFD i c 1 ;i 1 \0504 = 3 whichisalsocalledthediffusionlimitingcurrent.Theeffectofmigrationinanelectrolytic systemcanbeestimatedbytheratiooftheobservedlimitingcurrenttothediffusion limitingcurrent i lim =i lim ; D .Themigrationenhancesthelimitingcurrentwhentheratiois greaterthanone,andreducesthelimitingcurrentwhentheratioissmallerthanone. Themigrationofreactantmakesnocontributionwhentheratioisunity.Forthesystem understudy,theratioof i lim =i lim ; D hasavalueof1.48whenthesolutioncontains0.1M AgNO 3 and0.01MKNO 3 ,meaningthatthemigrationofsilverioninthediffusionlayer enhancesthelimitingcurrentbyafactorof1.48. Figure6-9showsthecurrentdistributionsonthedisksurfaceindifferentconcentrationsofsupportingelectrolyte.ThereducedOhmicpotentialdropinthepresence ofexcesssupportingelectrolytemakesthedistributionofcurrentmoreuniform.The currentdensityincreaseswiththeincreasingoverpotentialtowardstheedgeofthe disk.Whentheoverpotentialissufcientlylargeandtheconcentrationofreactantis sufcientlysmallatelectrodeedge,theelectrodereactionislimitedbythemasstransfer ofreactant.Therefore,thecurrentdensitystartstodecreaseatelectrodeedgeasseen inFigure6-9Cforthecaseoflargerfractionoflimitingcurrent. Inthesystemoffastelectrodekinetics,thecurrentandpotentialdistributionsona rotatingdiskelectrodeareaffectedbythenonuniformmasstransferandtheelectrode geometry.Theeffectofmasstransfercanbereducedwithhigherrotationspeedor eliminatedbyperformingthereactionatthemass-transfer-limitedcurrent.However,the distributionsofcurrentandpotentialcanneverbeuniformevenwiththeuseofexcess supportingelectrolyte.ThereasonisbecauseanonuniformOhmicpotentialdropis presentinthebulksolutionassociatedwiththeelectrodegeometry.Theopposing effectsoftheOhmicresistanceandthemass-transferresistancecanbeobserved 106

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A B C Figure6-9.Calculatedcurrentdistributionsforsilverdepositiononarotatingdisk electroderotatingat120rpm.Thesolutioncontains0.1MAgNO 3 andA 1M,B0.1M,andC0.01MKNO 3 107

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whenbothvaluesarelargeinsolutionsofsmallconductivitiesandapproachingthe mass-transfer-limitedconditions. 108

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CHAPTER7 MODELOFELECTRICALDOUBLELAYER Thenatureoftheelectrode-electrolyteinterfaceinvolvesthedistributionofelectrons,ions,andmolecules.Achargedelectrodeholdsexcessionsintheadjacent solution,andadiffuselayerofchargeisformed.Theelectricbehavioroftheinterfaceis thereforeverydifferentfromthatofthebulksolution,andisrelatedtotheexcessordecientconcentrationofspeciesandthepotentialdropinthethininterfacialregion.The termelectricaldoublelayercamefromtheimageoftheinterfaceasconsistingoftwo layersofcharge,oneattheelectrodesurfaceandtheotherintheadjacentelectrolyte. Therealstructureofinterfaceissurelymorecomplicatedwhentakingintoaccountthe orientationofdipolesandthespecicadsorptionofionicspecies. Numerousmodels 62,63 havebeenappliedtodescribethethermalbehaviorof thechargedionsintheelectricaldoublelayer.Inthepresentstudy,weusetheGouyChapman-Sternmodeltodescribetheelectricalbehaviorinthediffusedoublelayer. Thechangeofsurfacechargedensityresultingfromthevariationsofpotentialand concentrationsattheinterfacearediscussedinthischapter.Thesepropertiesofthe interfacewillbeusedtomodifytheelectrodeprocessassociatedwiththedouble-layer chargingandtheFaradaicchargingcurrents,whichareimportantinstudyingthe impedanceresponseofsystemswithmasstransfer. 7.1TheGouy-Chapman-SternTheory TheclassicalGouy-Chapmantheoryshowsanexponentialdecayofpotentialina diffuseregionofchargeextendingfromtheelectrodesurface.Stern'smodicationtothe Gouy-Chapmanmodelassumesacompactlayerbetweentheelectrodeandthediffuse layer.TheouterlimitofthecompactlayerisalsoreferredtoastheouterHelmholtz planeOHPwhichisthelocusofcentersofmobilespeciesintheirpositionofcloset approachtotheelectrodesurface.Whenthereisspecicadsorptiononelectrode surface,thelocusofcentersofadsorbedspeciesistakentobetheinnerHelmholtz 109

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Figure7-1.Thestructureofelectricaldoublelayer.Thesketchisnottoscale. planeIHP.ThestructureoftheelectricaldoublelayerisillustratedinFigure7-1.The thicknessofthedoublelayerisverythin.Hencethedoublelayerisoftenconsidered tobeapartoftheelectrode-electrolyteinterface.Theinterfacialregionasawhole obeyselectricalneutrality, i.e., theexcesssurfacechargedensityatelectrodemustbe balancedbythesurfacechargesattheIHPandinthediffusepartofthedoublelayer suchthat q m + q ihp + q d =0 Thesurfacechargedensityisrelatedtothesurfaceexcessconcentrationofcharged speciesby q m = )]TJ/F15 11.9552 Tf 11.291 0 Td [( q ihp + q d = )]TJ/F26 11.9552 Tf 9.298 0 Td [(F X i z i )]TJ/F27 7.9701 Tf 7.314 -1.794 Td [(i Themeanelectrostaticpotentialsatthemetalsurface,theIHP,andtheOHParedenotedby m ihp ,and ohp ,respectively.Thesurfaceconcentrations c i; 0 andpotential 0 usedintherateexpressionsareusuallyevaluatedattheouterlimitofthediffuse layer,ortheinnerlimitofthediffusionlayer. BeyondtheOHP,thetotalchargeinthediffusepartofthedoublelayerisnot electricallyneutral.Theionicconcentrationsinthediffuselayerareassumedtohavea 110

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Boltzmanndistribution c i = c i; 1 exp )]TJ/F26 11.9552 Tf 10.494 8.088 Td [(z i F RT andthePoissonequationgivesthecorrelationbetweentheconcentrationsandpotential by d 2 d y 2 = )]TJ/F26 11.9552 Tf 15.586 8.088 Td [(F d 0 X i z i c i = )]TJ/F26 11.9552 Tf 15.587 8.088 Td [(F d 0 X i z i c i; 1 exp )]TJ/F26 11.9552 Tf 10.494 8.088 Td [(z i F RT where y isthedistancefromtheelectrode, d isthedielectricconstantinthediffuse layer,and 0 isthepermittivityofvacuum 0 = 8.8542 10 )]TJ/F24 7.9701 Tf 6.586 0 Td [(14 F/cm.Integrationofthe Poissonequationgivestherelationofpotentialgradienttothesurfacechargedensityin thediffuselayeras d d y = q d d 0 at y = where isattheinnerlimitofthediffuselayer.Atequilibrium,thepotentialapproaches zeroinalargedistancefromtheelectrodesurface, i.e., 0as y !1 ThePoissonequationcanthenbesolvedbyapplyingtheboundaryconditions,andthe chargedensityinthediffuselayerisfoundtobe q d = 2 RT d 0 X i c i; 1 exp )]TJ/F26 11.9552 Tf 9.298 0 Td [(z i F ohp RT )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 1 2 Theuppersignisusedifthepotentialispositiveand,conversely,thelowersignisused ifthepotentialisnegative.Thechargedensityassociatedwithindividualspeciescanbe furtherobtainedbyanintegraloverthepotentialdropinthediffuseregionfollowing q d ;i = Z ohp 0 z i Fc i; 1 exp )]TJ/F29 7.9701 Tf 6.675 -4.763 Td [()]TJ/F27 7.9701 Tf 6.587 0 Td [(z i F RT )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 n 2 RT d 0 P k c k; 1 exp )]TJ/F29 7.9701 Tf 6.675 -4.572 Td [()]TJ/F27 7.9701 Tf 6.586 0 Td [(z k F RT )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 o 1 2 d 111

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IfthereisnospecicadsorptionofspeciesontheIHP, i.e., q ihp =0 ,theindividual chargedensityisrelatedtothesurfaceconcentrationby q d ;i = z i F )]TJ/F27 7.9701 Tf 7.314 -1.793 Td [(i Theaboveexpressionsforchargedensitiesarederivedbyassumingatrueequilibrium ofthesysteminwhichtheionicconcentrationsattheouterlimitofthediffuselayer arethesameasthebulkvalues, c i; 0 = c i; 1 ,andthepotentialattheouterlimitofthe diffuselayerisequaltothatofareferenceelectrodeplacedatadistance;thus, 0 =0 Whenthenetcurrentowingtotheelectrodeisnotequaltozero,thesystemisnotat equilibrium, i.e., 0 6 =0 .Equations7and7become q d = 2 RT d 0 X i c i; 0 exp )]TJ/F26 11.9552 Tf 9.299 0 Td [(z i F ohp )]TJ/F15 11.9552 Tf 11.955 0 Td [( 0 RT )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 1 2 and q d ;i = Z ohp 0 z i Fc i; 0 exp )]TJ/F29 7.9701 Tf 6.675 -4.763 Td [()]TJ/F27 7.9701 Tf 6.586 0 Td [(z i F RT )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 n 2 RT d 0 P k c k; 0 exp )]TJ/F29 7.9701 Tf 6.675 -4.572 Td [()]TJ/F27 7.9701 Tf 6.586 0 Td [(z k F RT )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 o 1 2 d respectively.Thechargedensitiesinthediffuselayerarenowrelatedtotheconcentrationsandpotentialattheouterlimitofthediffuselayer,whichcanbeobtainedbysolving themassandchargeconservationequationsoutsidethediffuseregionofcharge. Theevaluationofthesurfacechargedensityinequation7or7requires additionalinformationintheelectricaldoublelayer.Gauss'slawrelatesthesurface chargedensitytotheelectriceldwithintheOHPby q m = )]TJ/F26 11.9552 Tf 9.298 0 Td [(q d = 0 m )]TJ/F15 11.9552 Tf 11.956 0 Td [( ohp where isthedielectricconstantbetweenthemetalsurfaceandtheOHP.Equation 7canbeusedasasecondequationtosolve q m and ohp intheelectricaldouble layer. 112

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7.2NumericalApproach Thenonlinearequationsrelatingtothechargeandpotentialdistributionsinthe electricaldoublelayerweresolvedbyusingtheNewton-Raphsonmethod.Fornonequilibriumsystems,alocalequilibriumwasassumedinwhichtheconcentrationsand potentialattheouterlimitofthediffuselayerweregivenbythesteady-statesolutions calculatedinChapter6.Theradiallydependentsurfaceconcentrationsandpotential gavearadialdistributionofchargeovertheelectrodesurface.Thesurface-averaged valueofchargewasobtainedfromthesurface-averagedvaluesofconcentrationandpotential.Thechargeinthediffuselayerassociatedwithindividualspecieswasobtained bysubsequentnumericalintegrationofequation7or7,usingtheMATLAB integrationfunction,theadaptiveGauss-Kronrodquadraturemethod.Thethicknessof thecompactlayerbetweenthemetalsurfaceandtheOHPwasassumedtobe3 A.The dielectricconstantinthecompactregionwithintheOHPwasapproximately6according toBockris 63 forafullyorientedwaterlayernexttotheelectrodesurface.Thedielectric constantinthediffuselayerwasassumedtobe78,whichisthevalueofwateratroom temperature. Thecalculationofsurfacechargedensityintheelectricaldoublelayerwasapplied tosystemsofferro/ferricyanideredoxcoupleandelectrolyticdepositionanddissolution ofsilver. 7.3SurfaceChargeintheDiffusePartoftheDoubleLayer Thecalculatedchargedensityandthechargeassociatedwithindividualspeciesin thediffusepartofthedoublelayerareshowninFigure7-2forthesystemcontaining ferro/ferricyanide.Thetotalchargedensityisequaltozerowhentheelectrodeisat thepotentialofzerocharge pzc ,whichisassumedtobezerointhissystem.The contributionofchargefromeachspeciesisaffectedbytheredoxreactiontakingplace attheinterface.Theconcentrationsofeachspeciescalculatedattheouterlimitofthe diffuselayerareshowninFigure7-3.Whentheelectrodepotentialisnegative,the 113

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Figure7-2.Calculatedtotalchargedensityinthediffusepartofthedoublelayerandthe contributionofeachionicspecies.Theelectrolyticsolutioncontains0.01M K 3 FeCN 6 ,0.01MK 4 FeCN 6 and1MKCl. Figure7-3.Calculatedaveragedionicconcentrationsattheouterlimitofthediffuse. Theelectrolyticsolutioncontains0.01MK 3 FeCN 6 ,0.01MK 4 FeCN 6 and 1MKCl. 114

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A B Figure7-4.Calculatedtotalchargedensityinthediffusepartofthedoublelayerandthe contributionofeachionicspecies.Theelectrolyticsolutioncontains0.1M AgNO 3 andA1M,andB0.01MKNO 3 reactionisinthecathodiclimitedcasewheretheconcentrationofferricyanideiszero. Thechargedensityassociatedwithferricyanideisalsozeroatnegativepotentialsas seeninFigure7-2.Thechargecontributionfromferrocyanideispositive,representing adepletionofferrocyanidespeciesinthediffuseregion.Whentheelectrodepotential becomespositive,thechargedensityassociatedwithferricyanideincreasesdueto theproductionofferricyanidefromtheoxidationofferrocyanide.Thechargedensityof ferricyanidekeepsincreasingatevenhigherpotentialswhentheanodiclimitedcondition isattainedbecausethepositivechargeontheelectrodetendstoholdmorenegative chargedspeciesinthediffuselayer.Thechargedensityassociatedwithferrocyanide alsoincreaseswhentheelectrodepotentialturnspositive.However,whenapproaching theequilibriumpotential V 0 = 0.23V,theconcentrationofferrocyanidestartsto decreaseduetotheoxidationreaction,resultingadecreaseofchargecontributionfrom theferrocyanidespecies. Forthereactionofsilverdepositionanddissolution,thesilverionistheonlyreactingspecies.Thecalculatedchargedensityinthediffuselayerandtheconcentrationsof eachspeciesattheouterlimitofthediffuselayerarepresentedinFigure7-4andFig115

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A B Figure7-5.Calculatedaveragedionicconcentrationsattheouterlimitofthediffuse. Theelectrolyticsolutioncontains0.1MAgNO 3 andA1M,andB0.01M KNO 3 ure7-5,respectively.Thepotentialofzerochargeforthesilverelectrodeisassumedto be pzc = -0.44V. 63 Thesilverionstartstohaveacontributiontothetotalchargewhen thedepositionofsilverisnotlimitedbythemasstransferofsilverion.Thenegative chargecontributionfromthesilverionatpositivepotentialsrepresentsthedepletionof silverioninthediffuselayer.Thedependenceofchargedensityonelectrodepotential isassociatedwiththemasstransferofsilverion.AsseeninFigure7-4,theslopefor q d changesatthepotentialcorrespondingtotheincreaseofsilverconcentrationobserved inFigure7-5. Undertheassumptionthatthereisnospecicadsorption,thesurfacechargeon theelectrodesideisbalancedbythechargedensityinthediffusepartofthedouble layertomaintainelectroneutralityintheinterfacialregion.Sincethesurfacecharge densityisdependentontheinterfacialpotentialandconcentrationsofeachspecies, thevariationofthesurfacechargedensityisassociatedwiththevariationofinterfacial potentialandthevariationofconcentrationofindividualspeciesby d q m = @q m @V c i; 0 d V + X i @q m @c i; 0 V;c j; 0 ;j 6 = i d c i; 0 116

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Thetwoderivatives, @q m =@V c i; 0 and @q m =@c i; 0 V;c j; 0 ;j 6 = i ,areessentialpropertiesofthe electricaldoublelayer.Grahame 64 andDelahay 62,6567 discussedthemethodtoevaluate theseproperties.Inthissection,thecalculationforthechangeofsurfacecharge densityresultingfromthevariationoftheinterfacialpotentialandtheconcentrationare presented. 7.3.1VariationofSurfaceChargewithPotential:Double-LayerCapacitance Thederivativeofthesurfacechargedensitywithrespecttotheinterfacialpotential atconstantcompositionisknownasthedouble-layercapacitance C 0 = @q m @V c i; 0 Sincetheinterfacialpotentialisdenedby V = m )]TJ/F15 11.9552 Tf 11.956 0 Td [( 0 = m )]TJ/F15 11.9552 Tf 11.955 0 Td [( ohp + ohp )]TJ/F15 11.9552 Tf 11.955 0 Td [( 0 thederivativeofpotentialwithrespecttothesurfacechargedensityyields @V @q m c i; 0 = @ m )]TJ/F15 11.9552 Tf 11.955 0 Td [( ohp @q m c i; 0 + @ ohp )]TJ/F15 11.9552 Tf 11.955 0 Td [( 0 @q m c i; 0 or 1 C 0 = 1 C m )]TJ/F24 7.9701 Tf 6.586 0 Td [(ohp + 1 C d where C m )]TJ/F24 7.9701 Tf 6.587 0 Td [(ohp isthecapacityofthecompactlayernexttotheelectrodesurfacewithin OHP,and C d isthecapacityofthediffusepartofthedoublelayer.Fromtheintegrated formofGauss'slaw, C m )]TJ/F24 7.9701 Tf 6.587 0 Td [(ohp canbecalculatedbyassumingauniformdielectricconstant betweenthemetalandtheOHPexpressedby C m )]TJ/F24 7.9701 Tf 6.587 0 Td [(ohp = 0 Theanalyticalexpressionforthecapacityinthediffuselayerisobtainedfromthe derivativeofsurfacechargeinequation7withrespecttothepotentialdropacross 117

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Figure7-6.Potentialdependenceofdouble-layercapacitanceforsystemof ferro/ferricyanideredoxcoupleinthepresenceofanexcesssupporting electrolyte. thediffuselayer C d = )]TJ/F26 11.9552 Tf 9.299 0 Td [(F P i z i c i; 0 exp )]TJ/F27 7.9701 Tf 10.494 7.691 Td [(z i F ohp )]TJ/F24 7.9701 Tf 6.587 0 Td [( 0 RT n 2 RT d 0 P i c i; 0 exp )]TJ/F27 7.9701 Tf 10.494 4.921 Td [(z i F RT ohp )]TJ/F15 11.9552 Tf 11.955 0 Td [( 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 o 1 2 Withtheabovedenitions,thecapacitanceofthedoublelayercanbeevaluatedwiththe useofappropriateparametersinsteadofassumingacertainvalueinthesubsequent calculationofimpedanceresponse. Thesurface-averageddouble-layercapacitanceforaninertelectrodeincontact withferro/ferricyanideredoxcoupleispresentedinFigure7-6.Thedouble-layer capacitancewascalculatedundertwoconditions.Therstassumedtheequilibrium condition 0 =0 ,inwhichcasetheconcentrationsandpotentialattheouterlimit ofthedoublelayerwereequaltothevaluesfarawayfromtheelectrode.Thesecond condition 0 6 =0 usedthethesteady-statesolutionsfortheconcentrationsand potentialcalculatedattheouterlimitofthediffuseregionofcharge.Whenthesystem 118

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Figure7-7.Radiallydependentdouble-layercapacitancecalculatedatdifferentfractions oflimitingcurrentforreductionofferricyanideinthepresenceofanexcess supportingelectrolyte. isatequilibrium,thecalculatedcapacitanceattheelectrodeinterfaceshouldfollow thetypicalbehaviorpredictedbytheGouy-Chapmanmodelofdiffuselayer,thatis, asymmetriccapacitanceoverthepotentialofzerocharge pzc = 0.Sincethe electrolyticsolutioncontainsasymmetricionicspecies,thecapacitance-potentialcurve shiftstoapotentialnegativethan pzc .Whenthecompositionofelectrolyteadjacent totheelectrodeisdifferentfromthebulksolution,thedouble-layercapacitanceshows ahumpneartheequilibriumpotential,whichshouldberelatedtotherelativecharge contributionsfromthereactingspeciesinthediffuseregion. Thevariationsofsurfaceconcentrationsandpotentialalongelectrodesurfaceresult inadistributionofsurfacechargedensity,and,therefore,adistributionofdouble-layer capacitanceassociatedwiththeelectrodegeometryandkinetics.AsshowninFigure 7-7,thevariationofdouble-layercapacitancewithradialpositionisnotsignicantand thecapacitanceisalmostuniform.Thepresenceofsupportingelectrolytesuppresses thepotentialgradientandleadstoasmallervariationofoverpotentialovertheelectrode 119

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A B Figure7-8.Potentialdependenceofdouble-layercapacitanceforasilverelectrodein solutioncontaining0.1MAgNO 3 anddifferentconcentrationsofsupporting electrolytewhenthesystemisAatequilibriumandBnotatequilibrium. surface.Therefore,theassumptionofusingauniformdouble-layercapacitancein systemswithexcesssupportingelectrolyteisjustied. Toinvestigatetheeffectofsupportingelectrolyte,thereactionsofsilverdeposition anddissolutionwereconsidered.Forelectrolyticsolutioncontainingsymmetricelectrolyte,thecapacitance-potentialcurvesaresymmetricover pzc whenthesystemisat equilibriumasshowninFigure7-8A.WhenthesystemisnotatequilibriumFigure78B,theminimumvalueofthedouble-layercapacitanceisobservedatapotentialmore negativeto pzc = -0.44V.Thedouble-layercapacitanceshowslargerdependencyon electrodepotentialwhenthesolutionconcentrationissmaller,indicatingtheincreasing potentialgradientintheinterfacialregionwiththedecreasingsolutionconductivity. Figure7-9showsthedistributionsofdouble-layercapacitanceonasilverelectrode incontactwithAgNO 3 anddifferentconcentrationsofsupportingelectrolyteKNO 3 Whenasmallamountofsupportingelectrolyteispresent,thedistributionofthedoublelayercapacitanceismorenonuniform.Althoughtheoverallvariationsarestillnot signicant,theabsenceofsupportingelectrolyteincreasestheinterfacialpotentialand leadstoamorenonuniformdistributionofdouble-layercapacitance. 120

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A B C Figure7-9.Radiallydependentdouble-layercapacitancecalculatedatdifferentfractions oflimitingcurrentforsilverdepositioninsolutioncontaining0.1MAgNO 3 andA1M,B0.1M,andC0.01MKNO 3 121

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A B Figure7-10.Potentialdependenceof @q m =@c i; 0 inAlinearscaleandBlogarithmscale forreductionofferricyanideandoxidationofferrocyanide. 7.3.2VariationofSurfaceChargewithConcentration Thevariationofthesurfacechargedensitywiththeconcentrationofeachspecies atconstantpotentialshownin7isexpressedby @q m @c i; 0 V = exp )]TJ/F27 7.9701 Tf 6.586 0 Td [(z i F ohp )]TJ/F24 7.9701 Tf 6.587 0 Td [( 0 RT )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 RT d 0 P i c i; 0 exp )]TJ/F27 7.9701 Tf 6.587 0 Td [(z i F ohp )]TJ/F24 7.9701 Tf 6.586 0 Td [( 0 RT )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 1 2 Theuppersignisusedwhenthepotentialacrossthediffusepartofthedoublelayeris largerthanthepotentialofzerocharge, i.e., ohp )]TJ/F15 11.9552 Tf 11.491 0 Td [( 0 > pzc ,and,conversely,thelower signisusedwhen ohp )]TJ/F15 11.9552 Tf 12.255 0 Td [( 0 < pzc .Fromequation7,thevariationofthesurface chargedensitywithionicconcentrationsataxedpotentialisonlydependentonthe chargeofeachspecies. Thepotentialdependenceof @q m =@c i; 0 forthesystemofreductionofferricyanide andoxidationofferrocyanideisshowninFigures7-10.Sincethechargeassociated withferrocyanideisthelargestamongallionicspecies,thevariationofchargeonmetal surfaceishighlydependentontheconcentrationofferrocyanide.Inthesystemofsilver depositionanddissolution,silverionandpotassiumionhavethesamechargenumber. 122

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A B Figure7-11.Potentialdependenceof @q m =@c i; 0 inAlinearscaleandBlogarithmscale fordepositionanddissolutionofsilver. Hencethevariationsofsurfacechargewiththeconcentrationsofthesetwospecies arethesameasshowninFigure7-11.Thesurfacechargeishighlydependentonthe concentrationofnitrate,andthevalueof @q m =@c NO )]TJ/F25 5.9776 Tf 0 -6.117 Td [(3 ; 0 increaseswithdecreasingsolution concentration.Thepeakassociatedwith @q m =@c NO )]TJ/F25 5.9776 Tf 0 -6.117 Td [(3 ; 0 occursatelectrodepotential wherethedepositionofsilverisnolongerlimitedbythemasstransferofsilverionas seeninFigure7-5. Theradialdistributionsof @q m =@c i; 0 forreductionofferricyanideandelectrolytic depositionofsilverarepresentedinFigures7-12and7-13,respectively.Similar tothedouble-layercapacitance,thedependenceof @q m =@c i; 0 onradialpositionis moresignicantatlargerfractionoflimitingcurrentandinsolutioncontainingsmall concentrationofsupportingelectrolyte,inwhichconditionstheinterfacialpotentialand theconcentrationofeachspeciesarelessuniform. Thechargeonthemetalsurfaceisshowntohavecorrelationwiththepotential andthecompositionofsolutionadjacenttotheelectrodesurface.Thevariationof thesurfacechargedensitywiththeconcentrationofeachspecies,likethedoublelayercapacitance,isalsoapropertyoftheelectrode-electrolyteinterface.Thecharge 123

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A B C D Figure7-12.Radialdistributionsof @q m =@c i; 0 forreductionofferricyanideatdifferent fractionsoflimitingcurrentforAferrocyanide,Bferricyanide,Cchloride, andDpotassiumions. 124

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A B C D E F Figure7-13.Radialdistributionsof @q m =@c i; 0 fordepositionofsilveratdifferentfractions oflimitingcurrentforA,C,Epotassiumandsilverions,andB,D,F nitrate.Thesolutioncontains0.1MAgNO 3 andA,B1M,C,D0.1M, andE,F0.01MKNO 3 125

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associatedwiththemobilespeciescouldalsobeattributedtothedouble-layercharging andcannotbeconsideredasanegligiblequantity.Thecharingcurrentundertransient conditionsmustaccountforthecontributionsofbothproperties. 7.3.3VariationofExcessConcentrationofIndividualSpecieswithConcentration Anotherimportantpropertyoftheelectricaldoublelayeristhechangeofexcess concentrationinthediffuseregionofagivenspeciesresultingfromthevariationofits concentration.Thispropertyisrelatedtothechargedensityby @ )]TJ/F27 7.9701 Tf 7.315 -1.793 Td [(i @c i; 0 = 1 z i F @q d ;i @c i; 0 whichcanbeusedtocalculatethecontributionofmassuxassociatedwitheach speciestothechargingprocess.Thechargecontributionofagivenspeciescanbe evaluatedbynumericalintegrationusingequation7.Thederivative @q d ;i =@c i; 0 can furtherbeobtainedbynumericaldifferentiation.Inthepresentcalculation,thevalueof @q d ;i =@c i; 0 wasobtainedbyusingve-pointcentraldifference. Thepotentialdependenceof @ )]TJ/F27 7.9701 Tf 7.314 -1.793 Td [(i =@c i; 0 ofeachspeciesforthesystemofreduction offerricyanideandoxidationofferrocyanideisshowninFigure7-14.Thevaluesof allderivativesarezeroatthepotentialofzerocharge pzc =0 .Whentheelectrode ispositivelycharged,thederivativeforferrocyanidehasthelargestdependencyon potential,representingthechangeoftheexcessconcentrationofferrocyanideinthe diffuseregionisstronglyaffectedbyitsconcentrationattheouterlimitofthediffuse region.ForthesystemofsilverdepositionanddissolutionshowninFigure7-15,the variationsoftheconcentrationsofpotassiumandsilverionshavethesameeffecton theirexcessconcentrationsinthediffuselayer.Whenthesystemcontainsasmaller amountofsupportingelectrolyte,thepotentialatwhichthederivativeisequaltozero shiftstoavaluemorenegativeto pzc = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 44 V .Thevaluesofderivativesofall speciesbecomelargerwhenthesystemislessconductive,representingalarger variationoftheinterfacialpropertywithionicconcentrations.Thepotentialatwhichthe 126

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Figure7-14.Potentialdependenceof @ )]TJ/F27 7.9701 Tf 7.314 -1.793 Td [(i =@c i; 0 forreductionofferricyanideandoxidation offerrocyanide. A B Figure7-15.Potentialdependenceof @ )]TJ/F27 7.9701 Tf 7.314 -1.793 Td [(i =@c i; 0 forApotassiumandsilverions,andB nitrateionfordepositionanddissolutionofsilver. 127

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peakofthederivativeofnitrateisobservedisassociatedwiththepotentialwherethe depositionofsilverislimitedbythemasstransferofsilverion. Thedistributionsof @ )]TJ/F27 7.9701 Tf 7.314 -1.793 Td [(i =@c i; 0 onelectrodesurfaceforsystemsofreductionof ferricyanideandelectrolyticdepositionofsilverarepresentedinFigures7-16and7-17, respectively.Atalargerfractionoflimitingcurrent,theradialdistributionsof @ )]TJ/F27 7.9701 Tf 7.314 -1.794 Td [(i =@c i; 0 for allspeciesaremorenonuniformduetothemorenonuniformdistributionsofsurfaceand concentrationoverpotentials.Thedecreasingconcentrationofsupportingelectrolyte resultsinmorenonuniformdistributionsofconcentrationsandpotential,andleadstoa morenonuniformdistributionoftheinterfacialproperty. Theinterfacialpropertiesdiscussedinthissectionareimportantindescribing thethermalbehaviorofthechargedspeciesattheelectrode-electrolyteinterface. Thechargingprocessisrelatedtothechangesofpotentialandconcentrationsat theinterface.Theuxofeachspecies,whichisassociatedwiththepotentialand concentrationgradients,thereforecontributestothechargingprocessatelectrodes.The inuenceofmasstransferinthechargingoftheinterfaceisdiscussedinthefollowing section. 7.4CouplingofDouble-LayerChargingWithMassTransfer Inelectrochemicalsystems,thepassageofcurrentthroughanelectrodecan beattributedtotwoprocesses,Faradaicreactionsanddouble-layercharging.The twoprocessesareusuallyconsideredseparatelyforsimulationsofunsteady-state systems.Theuxofreactantsandproductsaregivenbytherateoftheelectrode electrochemicalreaction.TheFaradaiccurrentisusuallyderivedbyneglectingthe double-layerchargingcurrent,whichisconsideredtobeanindependentprocesson anidealpolarizedelectrode.Thetotalcurrentissubsequentlyobtainedbyaddingthe double-layerchargingcurrenttotheFaradaiccurrent.Thisassumptionwascriticized byDelahay 6567 forthereasonthatpartoftheuxofreactingspeciescontributesto thechargingoftheinterfaceaswellastotheFaradaicreaction.Faradiccurrentisnot 128

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A B C D Figure7-16.Radialdistributionsof @ )]TJ/F27 7.9701 Tf 7.314 -1.794 Td [(i =@c i; 0 forreductionofferricyanideatdifferent fractionsoflimitingcurrentforAferrocyanide,Bferricyanide,Cchloride, andDpotassiumions. 129

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A B C D E F Figure7-17.Radialdistributionsof @ )]TJ/F27 7.9701 Tf 7.314 -1.793 Td [(i =@c i; 0 fordepositionofsilveratdifferentfractions oflimitingcurrentforA,C,Epotassiumandsilverions,andB,D,F nitrate.Thesolutioncontains0.1MAgNO 3 andA,B1M,C,D0.1M, andE,F0.01MKNO 3 130

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aseparatemeasurablequantity.Therefore,theFaradaiccurrentandthedouble-layer chargingcurrentcannotbeseparated apriori Themeasurabletotalcurrentshouldincludethevariationoftheexcessconcentrationofeachspecies )]TJ/F27 7.9701 Tf 7.314 -1.793 Td [(i inadditiontothecontributionsoftheusualdouble-layer capacitanceattheinterface.Theassessmentof )]TJ/F27 7.9701 Tf 7.314 -1.793 Td [(i requirestheknowledgeofthe chargedistributioninthediffusedoublelayer.Grahame 64 andDelahay 62 reviewedthe theoryandthestructureofdoublelayeranditsrelationtotheelectrodekinetics. 62 Inthis dissertation,weusetheGouy-Chapman-Sternmodeltoevaluatethechargedensityin thediffusepartofthedoublelayer.Thedouble-layertheorywasappliedtomodifythe boundaryequationsforuxandcurrentdensity. Attheelectrodeboundary,thesurfaceuxisusuallyconsideredtobecontributed onlyfromthecharge-transferreactions.Thesurfaceuxofeachspeciesisrelatedto theFaradaiccurrentby N i; 0 = )]TJ/F26 11.9552 Tf 14.142 8.088 Td [(s i nF i F Thisexpressionisnotquitecorrectbecausethecontributionfromthedouble-layer chargingisneglected.FollowingthederivationsbyDelahay 6567 andNisancioglu, 68,69 thesurfaceuxiscorrectedwithno apriori separationNAPSofFaradaicanddoublelayerchargingcurrentsby N i; 0 = )]TJ/F26 11.9552 Tf 10.494 8.088 Td [(@ )]TJ/F27 7.9701 Tf 7.314 -1.793 Td [(i @t )]TJ/F26 11.9552 Tf 16.799 8.088 Td [(s i nF i F Byapplyingasimplechainrule,thechangeofsurfaceexcessconcentrationofspecies i withtimecanbeexpressedintermsofthetotalchargedensityonmetalsurfaceandthe concentrationofthecorrespondingspeciesattheouterlimitofthediffuselayerby @ )]TJ/F27 7.9701 Tf 7.314 -1.793 Td [(i @t = )]TJ/F26 11.9552 Tf 12.649 8.088 Td [(@ )]TJ/F27 7.9701 Tf 7.314 -1.793 Td [(i @c i; 0 @c i; 0 @q m @q m @t ThetotalcurrentowstotheelectrodesurfaceisalsocorrectedinthecaseofNAPSby i = @q m @t + i F 131

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Sincethesurfacechargedensityisafunctionofpotentialandconcentrationsofeach species,thetimederivativeof q m inequations7and7canbeexpressedin termsofthecorrelatingvariablesfollowing @q m @t = @q m @V c i; 0 @V @t + X i @q m @c i; 0 V;c j; 0 ;j 6 = i @c i; 0 @t Thebehaviorofthethermodynamicpropertieswerediscussedintheprevioussection. Withtheuseoftheseproperties,thechargingprocessiscoupledwithmasstransferat electrodes. ThemassuxofeachspeciescontributesnotonlytotheFaradaicreactionsbut alsotothechargingthedoublelayer.Mostimpedancemodels 7073 neglectthecontributionofmassuxtothechargingoftheinterface,andassumeauniformdistribution ofdouble-layercapacitance.Inthepresentstudy,wetakeintoaccountthecontribution ofmasstransfertothedouble-layercharging,andalsotheradialdistributionofcharge onelectrodesurface.Thelocalandglobalimpedanceresponsesassociatedwiththe mass-transferandgeometryeffectsareinvestigatedundertheassumptionthatthereis no apriori separationofchargingandFaradaiccurrentsatelectrodes. 132

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CHAPTER8 INFLUENCEOFMASSTRANSFERONIMPEDANCERESPONSE Atwo-dimensionalimpedancemodelispresentedinthischaptertostudytheeffect ofnonuniformmasstransferinadditiontotheelectrodegeometryontheimpedance response.Themodelalsotakesintoaccountthecontributionofmasstransfertothe double-layerchargingcurrent.Thecalculationofthetransientresponserequiresthe steady-statesolutionsofconcentrationsandpotentialobtainedinChapter6.Themodel ofelectricdoublelayerpresentedinChapter7isappliedtomodifythechargingprocess attheelectrode-electrolyteinterface. 8.1MathematicalDevelopmentofImpedanceModel Electrochemicalimpedancemeasurestheoutputsignalofcurrentorpotential inresponsetoasinusoidalchangeofpotentialorcurrent.Forelectrochemicalreactionsdependentonmasstransfer,theconcentrationsofreactingspeciesalsoexhibit sinusoidalresponsescorrespondingtotheinputsignal.Inthefrequencydomain,the potentialandconcentrationsofeachspeciescanbedescribedintermsofsteadyand time-dependentpartsby = +Re f e e j!t g and c i = c i +Re f e c i e j!t g respectively,wherethebarnotationrepresentsthesteady-statecomponent,andthe tildenotationrepresentstheoscillatingcomponentwhichisafunctionofonlyposition. FollowingthedevelopmentdescribedinChapter6fornonuniformmasstransferona RDE,andusingtheaboveconventionsforpotentialandconcentrations,themassand chargeconservationequations6and6become j! e c i + v r e c i = r D i r e c i + z i u i F c i r e + z i u i F e c i r 133

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and r F X i z i )]TJ/F26 11.9552 Tf 9.299 0 Td [(D i r e c i )]TJ/F26 11.9552 Tf 11.955 0 Td [(z i u i F c i r e )]TJ/F26 11.9552 Tf 11.955 0 Td [(z i u i F e c i r # =0 wherethehigherordertermssuchas e c i r e areneglected.Attheelectrodeboundary, theuxofeachspeciesexpressedinequation6inresponsetoasmallperturbationofcurrentorpotentialbecomes e N i = )]TJ/F26 11.9552 Tf 9.298 0 Td [(D i @ e c i @y y =0 + z i u i F c i @ e @y y =0 + z i u i F e c i @ @y y =0 Thecorrelationbetweentheuxandthecurrentoscillationsatelectrodeboundary isdiscussedintwocaseswheretheFaradaiccurrentandthedouble-layercharging currentareconsideredwith apriori separationAPSandwithout apriori separation NAPS. 8.1.1No APriori SeparationofFaradaicandChargingCurrents Inthefrequencydomain,thesurfacechargedensityandthecurrentdensity, followingthesameconventionsforpotentialandconcentrationinequations8and 8,canbeexpressedby q = q +Re f e qe j!t g and i = i +Re f e ie j!t g respectively.Whenapplyingasmallperturbationtothesystem,thecurrentatelectrode surfacecorrespondingtoequation7becomes e i = j! e q m + e i F wheretheoscillationsofthesurfacechargedensityandtheFaradaiccurrentdensityare approximatedbyTaylorseriesexpansionsabouttheirsteadyvaluesas e q m = @ q m @ V c i; 0 e V + X i @ q m @ c i; 0 V; c j; 0 ;j 6 = i e c i; 0 134

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and e i F = @ i F @ V c i; 0 e V + X i @ i F @ c i; 0 V; c j; 0 ;j 6 = i e c i; 0 respectively.Thesurfaceuxcorrespondingtoequation7canalsobeinterpreted inthefrequencydomainas e N i; 0 = )]TJ/F26 11.9552 Tf 12.649 8.087 Td [(@ )]TJ/F27 7.9701 Tf 7.314 -1.793 Td [(i @c i; 0 @c i; 0 @q m j! e q m )]TJ/F26 11.9552 Tf 16.799 8.087 Td [(s i nF e i F Equations8and8canbeappliedastheboundaryconditionstoevaluatethe impedanceresponsewithouttheassumptionof apriori separationofFaradaicand double-layerchargingcurrents. 8.1.2 APriori SeparationofFaradaicandChargingCurrents Theconventionalapproachintreatingtheboundaryconditionsistoconsiderthe double-layerchargingandtheFaradaiccurrentsasseparablequantities.Thetotal currentdensityisusuallyexpressedby i = i C + i F = C 0 d V d t + i F Fromtheoscillationofthedouble-layerchargingcurrent e i C = j!C 0 e V andapplyingequation7,theoscillationofthetotalcurrentdensityisexpressedby e i = j!C 0 e V = j! @q m @V c i; 0 e V + e i F Thecomparisonofequation8withequations8and8showsthatthe secondterminequation8isneglected.Thisimpliesthatthecontributionofmass uxinchargingthedoublelayerisneglectedandtheFaradaiccurrentandthedoublelayerchargingcurrentareconsideredseparately.Sincethechangeofsurfacecharge densityresultingfromtheconcentrationvariationisneglected,theoscillationofux 135

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becomes e N i; 0 = )]TJ/F26 11.9552 Tf 14.142 8.088 Td [(s i nF e i F Undertheassumptionthatthereis apriori separationofFaradaicanddouble-layer chargingcurrents,equations8and8areusedastheboundaryconditionsto evaluatethetransientresponsesofpotentialandconcentrations. 8.2NumericalSimulation Thenonlinearequationsweresolvedbyusingthenite-elementpackage COMSOL Multiphysics r withtheNernst-Planckmoduleina2Daxialsymmetriccoordinate system.Thedomainsizeandthemeshcriteriaarethesameasthoseusedinthe steady-statemodelandweredescribedinChapter6.Thesteady-statesolutions ofpotentialandconcentrationsofallspecieswereusedasinputfunctionsinthe impedancemodelintheformoflookuptablesfromwhichappropriatevaluescanbe obtainedbyinterpolation.Theuseofthesamemesheddomaininthesteady-state andtheimpedancemodelsreducestheerrorfrominterpolationbetweennodalpoints. Thedistributionsofthedouble-layercapacitanceandotherthermodynamicparameters atelectrodeboundarywerealsoappliedintheimpedancemodeltoevaluatethe impedanceresponseundertheassumptionofno apriori separationofFaradaiccurrent anddouble-layerchargingcurrentcaseNAPS.When apriori separationofFaradaic andchargingcurrentappliescaseAPS,asurfaceaverageddouble-layercapacitance h C 0 i = 1 r 2 0 Z r 0 0 C 0 r r d r wasusedtodeterminethechargingcurrentwhichdidnotincludethecontributionof masstransferinthediffuseregionofcharge. 8.3CalculatedImpedanceResponsesforReductionofFerricyanide Theimpedanceresponseforreductionofferricyanidewascalculatedinsolution withexcesssupportingelectrolyte.Althoughthedistributionsofsurfacechargeand double-layercapacitancearealmostuniformwhenthesystemisinthepresenceof 136

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A B Figure8-1.Nyquistrepresentationoftheglobalimpedanceresponseforreductionof ferricyanideonarotatingdiskelectroderotatingatA120rpmandB 600rpm. asupportingelectrolyte,thecalculationsstillappliedthemodiedelectrodeprocess thatdidnotaccountforthe apriori separationofFaradaicanddouble-layercharging currentscaseNAPS.Thepresentedimpedancevaluesarenormalizedbyelectrolyte conductivityandelectroderadius.Thedimensionlessfrequencyisdenedas K = h C 0 i r 0 inbothcases. 8.3.1GlobalImpedance Theglobalimpedancecalculatedatdifferentrotationspeeds120rpmand600rpm arepresentedinNyquistformatinFigure8-1Aand8-1B,respectively.Thehighfrequencyloopassociatedwiththecharge-transferreactionissmallcomparedtothe low-frequencyloopassociatedwithmasstransfer,representingareactionoffastkineticsandcontrolledbymasstransferofreactingspeciestoelectrode.Differentrotation speedsshouldonlyhaveaneffectatlowfrequencies.Thesmallerlow-frequencyloop andlargercharacteristicfrequencyinFigure8-1Bindicateasmallermass-transferresistanceatahigherrotationspeed.Thecomparisonoftheglobalimpedancebyassuming 137

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A B Figure8-2.Realandimaginarycomponentsoftheglobalimpedanceresponsefor reductionofferricyanideatthree-fourthsoflimitingcurrentonarotatingdisk electroderotatingat120rpm. apriori andno apriori separationofFaradaicanddouble-layerchargingcurrentsis showninFigure8-2fortherealandimaginarycomponentsasafunctionoffrequency. Inthepresenceofanexcesssupportingelectrolyte,thecontributionofmassuxtothe chargingcurrentdoesnotaffecttheimpedanceresponsesignicantly. Theglobalimpedanceisanaveragerepresentationforthelocalimpedances.The localimpedances,however,donotalwaysbehavethesameastheglobalresponse. 8.3.2LocalImpedance TheNyquistrepresentationofthelocalimpedancesareshowninFigure8-3with radialpositionasaparameter.Thelocalimpedancehasalargervalueatelectrode centerandasmallervalueneartheedgeofthedisk,representingagreateraccessibility neartheelectrodeperiphery.Inductiveloopsareobservedathighfrequencies,whichis duetothenonuniformcurrentandpotentialdistributionsassociatedwiththeelectrode geometry. 8.3.3LocalInterfacialImpedance TheNyquistrepresentationofthelocalinterfacialimpedancesareshownin Figure8-4withradialpositionasaparameter.Interfacialimpedancerepresentsthe 138

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A B C D E F Figure8-3.Nyquistrepresentationofthelocalimpedanceforreductionofferricyanide onarotatingdiskelectroderotatingat120rpmcalculatedatA,B one-fourth,C,Done-half,andE,Fthree-fourthsoflimitingcurrent.The enlargementsofthehigh-frequencyinductiveloopsareshowninB,D, andE. 139

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A B C D E F Figure8-4.Nyquistrepresentationofthelocalinterfacialimpedanceforreductionof ferricyanideonarotatingdiskelectroderotatingat120rpmcalculatedatA, Bone-fourth,C,Done-half,andE,Fthree-fourthsoflimitingcurrent. Theenlargementsofthehigh-frequencyinductiveloopsareshownin B,D,andE. 140

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transientresponseofelectricpropertiesintheinterfacialregion.Thehigh-frequency loopisassociatedwiththecharge-transferreactionandthechargingoftheinterface. Sincetheradialdistributionsofdouble-layercapacitanceandotherthermodynamic parametersarealmostuniformonelectrodesurfaceinthepresencesofanexcess supportingelectrolyte,thedispersionoflocalinterfacialimpedanceisnotsignicantat highfrequencies. 8.3.4LocalOhmicImpedance TheNyquistplotsforthelocalOhmicimpedancearepresentedinFigure8-5as radialpositionasaparameter.Theresistanceofelectrolyteexhibitingcomplexfeatures isinagreementwiththepreviousstudies. 2,3,1012,14,22 ThelocalOhmicimpedance showsonlyinductivebehaviorneartheelectrodecenter,andshowsbothinductiveand capacitivebehavioratelectrodeperiphery.Thedependenceofthecomplexvaluesof thelocalOhmicimpedancewithfrequencyisgiveninFigure8-6Complexvaluesare observedinthewholefrequencyrange.Atthesamerotationspeed,thehigh-frequency complexvaluesbecomelargerwhenelectrodereactionrateislarger.Thelow-frequency complexvalues,however,donotaffectedbytheelectrodereactionandremainthe samewhenthediskrotationspeedisxed.Withincreasingrotationspeed,thecomplex valuesatlowfrequenciesarediminishedandbecomealmostzerowhentherotation speedisupto2,400rpm.Ontheotherhand,thecomplexvaluesathighfrequencies donovarywithrotationspeed.Theseresultsdemonstratethatthehigh-frequency complexbehaviorofthelocalOhmicimpedanceisassociatedwiththenonuniform Ohmicpotentialdropinsolution,whichismoresignicantwhentheelectrodekinetics arefast.Thelow-frequencycomplexbehaviorisassociatedwiththenonuniformmass transferofreactingspeciestoelectrodewhichcorrelateswiththediskrotationspeed. 8.4CalculatedImpedanceResponsesforDepositionofSilver TheglobalandlocalimpedanceresponsesofthedepositionofsilverwerecalculatedwithdifferenttreatmentsoftheboundaryconditionscaseAPSandcaseNAPS 141

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A B C Figure8-5.NyquistrepresentationofthelocalOhmicimpedanceforreductionof ferricyanideonarotatingdiskelectroderotatingat120rpmandatA one-fourth,Bone-half,andCthree-fourthsoflimitingcurrent. 142

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A B C D E F G H I Figure8-6.ImaginarycomponentsofthelocalOhmicimpedanceforreductionof ferricyanideonarotatingdiskelectroderotatingatA,B,C120rpm,D, E,F600rpm,andG,H,I2,400rpm. 143

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andindifferentconcentrationsofsupportingelectrolyte.Thepresentedimpedance valuesarenormalizedbyelectrolyteconductivityandelectroderadius.Thefrequencyis normalizedbyelectrolyteconductivity,electroderadius,anddouble-layercapacitanceas denedinequation8. 8.4.1GlobalImpedance Thecalculatedglobalimpedancesforsilverdepositionindifferentconcentrations ofsupportingelectrolyteareshowninFigure8-7.Theinuenceofelectrodeprocess withandwithout apriori separationofFaradaicanddouble-layerchargingcurrentson theglobalimpedanceisobservedathighfrequencieswherethechargingprocessis important.ThedepressedsemicirclesobservedathighfrequenciesincasesAPSand NAPSisduetothegeometryinducedcurrentandpotentialdistributions,andcouldalso beduetothedistributionsofinterfacialpropertiessuchasdouble-layercapacitance andotherthermodynamicpropertiesdiscussedinChapter7.InFigure8-7Bforasmall amountofsupportingelectrolytepresentinthesystem,thehigh-frequencyloopshows adistortioninthecaseofNAPS.Thediscrepanciesofimpedancevaluesbetween casesAPSandNAPScanbeobservedmoreclearlyintherepresentationofthe realandimaginarycomponentsasafunctionoffrequencygiveinFigure8-8.The characteristicfrequencyofagivenelectrodeprocesscorrespondstothefrequency wheretheimaginarycomponentoftheimpedancehasamaximumvalue.InFigure88D,alargercharacteristicfrequencycorrespondingtothechargingprocessisobserved incaseNAPS.Thediscrepanciesinthevalueofcharacteristicfrequencycouldbedue totheextracontributionofuxinchargingthedoublelayer.Thevaluesofeffective capacitancefordifferentelectrolyticsystemsareestimatedandwillbediscussedatthe endofthischapter. TheOhmicresistancecanbeobtainedfromthehigh-frequencylimitoftheglobal impedance.ThedimensionlessOhmicresistanceobtainedintheimpedancemodel hasavalueof0.249atzerocurrent.Thisvalueisdifferentfrom0.25,whichwasgiven 144

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A B Figure8-7.Nyquistrepresentationoftheglobalimpedanceresponsefordepositionof silveratone-fourthoflimitingcurrentonarotatingdiskelectroderotatingat 120rpm.Thesolutionconsistsof0.1MAgNO 3 andA1MandB0.01M KNO 3 asasupportingelectrolyte. 145

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A B C D Figure8-8.Calculatedrealandimaginarycomponentsoftheglobalimpedance responseasafunctionoffrequencyfordepositionofsilveratone-fourthof limitingcurrentonarotatingdiskelectroderotatingat120rpm:Arealpart for1MKNO 3 ,Brealpartfor0.01MKNO 3 ,Cimaginarypartfor1MKNO 3 andDimaginarypartfor0.01MKNO 3 146

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Figure8-9.NormalizedOhmicresistanceofelectrolyticsolutionconsistingofAgNO 3 andKNO 3 asafunctionofsilverdepositionrate. byNewman 5 fortheOhmicresistanceonadiskelectrodewiththecounterelectrode placedatinnity.TheerrorinassessingthevaluesofOhmicresistancebyusinganite domainwasconsideredtobethesamewhenthereiscurrentowinginthesystem. Figure8-9showstheOhmicresistancesatdifferentfractionsoflimitingcurrent.The Ohmicresistancesobtainedatdifferentfractionsoflimitingcurrentwerecorrectedby addingagivenvaluethatmakestheOhmicresistancetobe0.25atzerocurrent.The valuesofOhmicresistancearenotaffectedbytheassumptionthatthereis apriori or no apriori separationofFaradaicanddouble-layerchargingcurrents,butarerelated tothesolutioncompositionandelectrodereactionrate.Whentheionicconcentration islarger,theconductivityofsolutionislarger,andthereforetheOhmicresistancehas asmallervalue.TheincreasingOhmicresistancewithincreasingreactionrateisa consequenceofchangeofsolutioncompositionattheelectrode-electrolyteinterface. Figure8-10showstheaxialdistributionofsolutionconductivityinthediffusionlayer forsilverdepositionattheelectrodecenter.Atalargerfractionoflimitingcurrent,silver 147

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A B Figure8-10.Axialdistributionofsolutionconductivityatelectrodecenterfordeposition ofsilver.Theelectrolyticsolutionconsistsof0.1MAgNO 3 andA1Mand B0.01MKNO 3 ionsarekeptconsuminganddepletingattheelectrodesurface,resultingasmaller conductivityandlargerresistanceattheinterface. TheeffectiveconductivityintheinterfacialregioncanbeestimatedfromtheOhmic impedancethatisobtainedfromthehigh-frequencylimitoftheglobalimpedanceby e = 4 r 0 Z r !1 Figure8-11.showstheeffectiveconductivityattheinterfacenormalizedbytheconductivityinthebulksolution.Theconductivityattheinterfaceisequaltothevalueof bulksolutionatzerocurrent.Thepresenceofanexcesssupportingelectrolytereducestheelectriceldinthediffusionlayer,andtheOhmicresistanceandtheeffective conductivityshowaweakerdependencyonreactionrate. 8.4.2LocalInterfacialImpedance ThecalculatedlocalinterfacialimpedancesforthecaseNAPSarepresentedin Figures8-12and8-13for1Mand0.01MKNO 3 ,respectively.Atone-fourthoflimiting current,theinterfacialimpedanceshavelargervaluesatelectrodecenterandsmaller 148

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Figure8-11.Normalizedeffectiveconductivityattheinterfaceasafunctionofsilver depositionrate. valuesatelectrodeperiphery,showingalargerreactionrateandagreateraccessibilityattheedgeofelectrodeduetothediskgeometry.Theincreasingoverpotential enhancesthereactionrateandreducestheconcentrationofreactanttowardsthe electrodeperiphery.Thelocalinterfacialimpedanceattheelectrodeperipherycan beincreasedduetotheincreasingresistanceofthemasstransferofreactingspecies totheelectrodesurface.Thedistributionofthelocalinterfacialimpedanceatlarge electrodepotentialsisthereforeinuencedbythecoupledeffectofelectrodegeometry andmasstransferasseeninFigures8-12and8-13atone-halfandthree-fourthsof limitingcurrent,respectively.Theanomalouslylargeinterfacialimpedanceobservedat theelectrodeperipheryshowninFigure8-13Cisneededtobeconrmedthatitisnota numericalartifact. Theinterfacialimpedancesathighfrequenciesandatthecenteroftheelectrode aregiveninFigure8-14.AdepressedsemicircleisobservedincaseNAPSeven inthepresenceofanexcesssupportingelectrolyteasshowninFigure8-14A.The appearanceofthedepressedsemicircleinthelocalinterfacialimpedancecannot 149

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A B C Figure8-12.Nyquistrepresentationofthelocalinterfacialimpedancefordepositionof silveronarotatingdiskelectroderotatingat120rpmcalculatedatA one-fourth,Bone-half,andCthree-fourthsoflimitingcurrent.The electrolyticsolutionconsistsof0.1MAgNO 3 and1MKNO 3 150

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A B C Figure8-13.Nyquistrepresentationofthelocalinterfacialimpedancefordepositionof silveronarotatingdiskelectroderotatingat120rpmcalculatedatA one-fourth,Bone-half,andCthree-fourthsoflimitingcurrent.The electrolyticsolutionconsistsof0.1MAgNO 3 and0.01MKNO 3 151

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A B Figure8-14.Nyquistrepresentationoftheinterfacialimpedanceatelectrodecenterfor depositionofsilveratone-fourthoflimitingcurrentonarotatingdisk electroderotatingat120rpm.Thesolutionconsistsof0.1MAgNO 3 andA 1MandB0.01MKNO 3 asasupportingelectrolyte. 152

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beattributedtothegeometry-inducedcurrentandpotentialdistributionsandthe distributionsofinterfacialproperty.Whentheconcentrationofsupportingelectrolyteis small,thecharacteristicfrequencyislargerincaseNAPSasshowninFigure8-14B. Thedepressedsemicircleandthelargervalueofcharacteristicfrequencyobservedin caseNAPScouldbeduetothecontributionofuxinthechargingoftheinterface,and willbediscussedlaterinthischapter. 8.4.3LocalOhmicImpedance NyquistrepresentationsofthelocalOhmicimpedanceforthecaseofNAPSare presentedinFigures8-15and8-16for1Mand0.01MKNO 3 ,respectively.Inductive loopsareobservedatallpositionsontheelectrodeandcapacitiveloopisobserved nearthediskperiphery.TheimaginarycomponentofthelocalOhmicimpedanceis givenFigure8-17.Whilethehigh-frequencycomplexbehaviorisassociatedwiththe nonuniformOhmicpotentialdropintheelectrolyte,thecomplexbehaviorobservedat lowfrequenciesisattributedtothenonuniformmasstransfertoelectrodesurface.Inthe presenceofanexcesssupportingelectrolyte,thelocalOhmicimpedanceinthesystem ofsilverdepositionissimilartothatobservedinthesystemofreductionofferricyanide thatthelocal-frequencycomplexvaluesdonotaffectedbytheelectrodereaction rate.Insolutioncontainingsmallamountofsupportingelectrolyte,thelow-frequency complexvalues,however,arestronglydependentontheelectrodereactionrate.The anomalouslylargeOhmicimpedanceattheperipheryisneededtobeconrmedthatit isnotanumericalartifact. 8.4.4LocalImpedance NyquistrepresentationsofthelocalimpedanceforthecaseofNAPSarepresented inFigure8-18for1MKNO 3 .Thecalculatedlocalimpedancesshowinductivebehavior athighfrequencies.Thehigh-frequencyinductiveloopswerenotobservedinthelocal interfacialimpedancesandmustbeattributedtothelocalOhmicimpedances.Forthe localimpedancesinthesysteminabsenceofsupportingelectrolyteasgiveninFigure 153

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A B C Figure8-15.NyquistrepresentationofthelocalOhmicimpedancefordepositionof silveronarotatingdiskelectroderotatingat120rpmcalculatedatA one-fourth,Bone-half,andCthree-fourthsoflimitingcurrent.The electrolyticsolutionconsistsof0.1MAgNO 3 and1MKNO 3 154

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A B C Figure8-16.NyquistrepresentationofthelocalOhmicimpedancefordepositionof silveronarotatingdiskelectroderotatingat120rpmcalculatedatA one-fourth,Bone-half,andCthree-fourthsoflimitingcurrent.The electrolyticsolutionconsistsof0.1MAgNO 3 and0.01MKNO 3 155

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A B C D E F Figure8-17.ImaginarycomponentsofthelocalOhmicimpedancefordepositionof silveronarotatingdiskelectroderotatingat120rpmcalculatedatA,B one-fourth,C,Done-half,andE,Fthree-fourthsoflimitingcurrent.The solutionconsistsof0.1MAgNO 3 andA,C,E1MandB,D,F0.01M KNO 3 156

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A B C Figure8-18.Nyquistrepresentationofthelocalimpedancefordepositionofsilverona rotatingdiskelectroderotatingat120rpmcalculatedatAone-fourth,B one-half,andCthree-fourthsoflimitingcurrent.Theelectrolyticsolution consistsof0.1MAgNO 3 and1MKNO 3 157

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8-19,thelocalimpedancesarecompletelydifferentfromtheglobalimpedance.The localimpedancerepresentsthesummationofthelocalinterfacialandlocalOhmic impedances.SincethevaluesofthelocalinterfacialimpedancesshowninFigure8-13 areoneorderofmagnitudesmallerthanthevaluesofthelocalOhmicimpedancesin Figure8-16,thelocalimpedancesarestronglyaffectedbythelocalOhmicimpedance, showingonlyinductivefeaturesnearthecenterofelectrodeandbothinductiveand capacitiveloopsatelectrodeperiphery.Theextremelylargerlow-frequencyloopatthe diskperipheryisneededtobeconrmedthatitisnotanumericalartifact. 8.5EffectiveDouble-LayerCapacitance UndertheassumptionthattheFaradaiccurrentandthechargingcurrentcanbe separated apriori ,theuxofreactantsandproductsaregivenbytherateoftheelectrochemicalreaction.Thedouble-layercapacitance C 0 isassociatedwiththechange ofchargeonelectrodewiththevariationofinterfacialpotential.Forelectrochemical reactionsdependentonmasstransfer,partoftheuxofreactingspeciescontributesto thechargingoftheinterface. Thevariationofchargeonelectrodeisalsorelatedtothevariationsofionic concentrationinthesolutionadjacenttotheelectrodesurface.Theeffectivedoublelayercapacitancethataccountsforthecontributionoftheuxtothechargingprocessis denedby C e = C 0 + X i @ q m @ c i; 0 V; c j; 0 ;j 6 = i e c i; 0 e V Thersttermontheright-handsideisthedouble-layercapacitanceusedinthecase APS,andthesecondtermisthecontributionofuxofeachspeciesinthechargingof thedoublelayer.Theeffectivedouble-layercapacitanceisafunctionofradialpotential andfrequency.Thevaluesofdouble-layercapacitanceusedinthecaseAPSandthe effectivecapacitanceobtainedinthecaseNAPSatvariousradialpositionsarelisted inTable8-1.Theradialdistributionoftheeffectivedouble-layercapacitanceisalmost uniforminsolutionswithanexcesssupportingelectrolyte.Forthesystemofreduction 158

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A B C Figure8-19.Nyquistrepresentationofthelocalimpedancefordepositionofsilverona rotatingdiskelectroderotatingat120rpmcalculatedatAone-fourth,B one-half,andCthree-fourthsoflimitingcurrent.Theelectrolyticsolution consistedof0.1MAgNO 3 and0.01MKNO 3 159

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Table8-1.Effectivedouble-layercapacitanceforthecaseNAPSatdifferentpositionson electrodeandthesurface-averageddouble-layercapacitanceforthecase APS.TheeffectivecapacitanceforNAPSwerecalculatedatthe characteristicfrequencyassociatedwiththeFaradaicreactionandthe chargingoftheinterface.Theunitforthecapacitanceis F/cm 2 ElectrolyticNAPS: C e APS system i=i lim r=r 0 =0 r=r 0 =0 : 5 r=r 0 =0 : 8 r=r 0 =0 : 96 h C 0 i [AgNO 3 ]=0 : 1 M1/40.06160.06060.05750.050116.9 [KNO 3 ]=0 : 01 M1/20.05080.04870.04280.030516.9 3/40.04060.03730.02810.016816.9 [AgNO 3 ]=0 : 1 M1/412.912.912.912.917.1 [KNO 3 ]=1 M1/213.313.313.313.317.1 3/414.114.114.114.117.1 [K 3 FeCN 6 ]=0 : 01 M1/415.815.815.815.816.7 [K 4 FeCN 6 ]=0 : 01 M1/216.016.016.016.016.7 [KCl]=1 M3/414.116.216.216.216.7 offerricyanide,thevaluesoftheeffectivecapacitanceincaseNAPSareclosertothe valuesofthedouble-layercapacitanceincaseAPS.Forthesystemofsilverdeposition andinthepresenceofasmallamountofsupportingelectrolyte,thevaluesofthe effectivecapacitanceareabouttwoorthreeordersofmagnitudesmallerthanthatof thedouble-layercapacitance.Asmallereffectivecapacitanceresultsinasmallertime constantandalargercharacteristicfrequencyasobservedinthehigh-frequencyloopof theglobalandlocalinterfacialimpedances. Figure8-20showstheeffectivedouble-layercapacitanceasafunctionoffrequencyforthedepositionofsilver.Inthepresenceofasmallconcentrationofsupportingelectrolyte,theeffectivedouble-layercapacitanceisstronglydependenton frequency.Thevaluesof C e athighfrequenciesbecomesthreeordersofmagnitude smallerthan C 0 .Thefrequency-dependenteffectivecapacitanceleadstotheappearanceofCPEbehaviorintheglobalandlocalinterfacialimpedancesevenwhenthe solutioncontainsanexcessamountofsupportingelectrolyte.Forthereductionof ferricyanidegiveninFigure8-21,thedependenceoftheeffectivecapacitanceonthe frequencyislesssignicantthanthatinthesilverdepositionsystem.Thisresultcould 160

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A B Figure8-20.Effectivedouble-layercapacitancefordepositionofsilveronarotatingdisk electroderotatingat120rpmandatone-fourthoflimitingcurrent.The solutionconsistsof0.1MAgNO 3 andA0.01MandB1MKNO 3 Figure8-21.Effectivedouble-layercapacitanceforreductionofferricyanideonarotating diskelectroderotatingat120rpmandatone-fourthoflimitingcurrent.The solution 161

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Table8-2.Comparisonofresultsunderdifferentsimulationconditions. ElectrochemicalElectrode i 0 r y i=i lim NAPS reactiongeometryA/cm 2 rpm=APS ReductionofRDE0.30.931201/4,1/2,3/4Yes ferricyanide0.30.936001/4,1/2,3/4Yes 0.30.932,4001/4,1/2,3/4Yes DepositionofRDE10.911201/4,1/2,3/4No silver10.501201/4,1/2,3/4No 10.0911201/4,1/2,3/4No 0.010.911201/2Yes 0.010.0911201/2No Recessed10.9101/4No electrode10.09103/4No y r=[Cl )]TJ/F15 11.9552 Tf 7.084 -5.147 Td [(] = [K + ] forreductionofferricyanide, r=[K + ] = [NO )]TJ/F24 7.9701 Tf 0 -7.97 Td [(3 ] fordepositionofsilver explaintheimpedanceresponsesforthereductionofferricyanidethatdidnotshow signicantdifferencebetweenthecasesAPSandNAPS. 8.6Discussion Calculationswereperformedindifferentelectrochemicalsystemsanddifferent kineticandmasstransferconditionstoexaminetheimpedanceresponsesunderthe assumptionsthatthereis apriori separationandno apriori separationofFaradaic reactionandchargingtheinterface.Thecomparisonofthesimulationresultsisgiven inTable8-2.Inthesystemofreductionofferricyanide,theimpedanceresponses werealmostthesameincasesNAPSandAPS.Therefore,itisappropriatetoassume thattheFaradaicreactionandthedouble-layerchargingprocesscanbeseparated a priori .Inthesystemofdepositionofsilver,theimpedanceresultsshowingnosignicant differenceinthetwocaseswereobservedonlywhenthereactionrateisslowand whentheconcentrationofsupportingelectrolyteislarge.Thechargingprocessat electrodesseemstoberelatedtothenatureofthesystem,thereactionrate,andthe amountofsupportingelectrolyte.Adiscrepancybetweentheimpedanceresponsesin thecasesNAPSandAPSwasstillobservedonarecessedelectrode,demonstrating thattheCPEbehaviorobservedinthelocalinterfacialimpedancewasnotduetothe 162

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nonuniformcurrentandpotentialdistributions.TheappearanceofCPEwasattributed tothefrequencydependenteffectivedouble-layercapacitancethataccountsforthe contributionofuxinchargingthedoublelayer. Whiletheelectrodereactionrateincreaseswithincreasingsurfaceoverpotential towardstheelectrodeperiphery,thereactionratecouldbelimitedbythemasstransfer ofreactingspecieswhenthesurfaceoverpotentialissufcientlylarge.Thecoupled effectofmasstransferandelectrodegeometryismoresignicantatalargerfraction oflimitingcurrent.Inthesolutioncontainingasmallamountofsupportingelectrolyte, anumericalartifactmightbepresentneartheelectrodeperipheryduetotheapproach towardzerooftheionconcentrationsandisneededtobeconrmed. 163

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CHAPTER9 CONCLUSIONS Theinuenceofcurrentandpotentialdistributionsonthelocalandglobal impedanceresponseisdiscussedinthepresentstudy.Thecurrentandpotential distributionscanbeattributedtotheelectrodegeometryandthenonuniformmasstransferonarotatingdiskelectrodebelowthemass-transfer-limitedcurrent.Animpedance modelforelectrochemicalsystemswithadsorbedintermediatesisdevelopedtostudy theroleofgeometryeffectatlowerfrequencies.Anothermodelisdevelopedtoinvestigatethemass-transfereffectinadditiontothegeometryeffectontheimpedance response. 9.1InuenceofAdsorbedIntermediates Thecalculatedimpedanceresponseforthesysteminvolvinganadsorbedintermediateshowstheinuenceofgeometry-inducedcurrentandpotentialdistributionsat bothhighandlowfrequencies.Thecharacterofthelow-frequencyloopisdependenton thesignofthekineticparameter A ,whichisitselfafunctionoftheinterfacialpotential. When A> 0 ,thelow-frequencyloopisinductive,andwhen A< 0 ,thelow-frequency loopiscapacitive.Whilehigh-frequencyloopsappearinginthelocalimpedanceareattributedtothedistributionofcurrentassociatedwithelectrodegeometry,low-frequency loopsassociatedwithintermediatesalsoshowgeometry-induceddispersionofthe impedanceresponse.ThegeometryeffectsarereectedinthelocalOhmicimpedance, whichhascomplexbehavioratbothhighandlowfrequencies.Thesurfacecoverage byintermediatesisdependentontheinterfacialpotentialwhich,inturn,isrelatedtothe radialpositionoftheelectrode.Ithasthemostnonuniformdistributionontheelectrode when h A i =0 becausethedependenceontheinterfacialpotentialisstrongerinthis case. Theresultsofthepresentworkshowedthatthedispersionoftimeconstant attributedtothediskelectrodewouldalsoleadtoaCPEbehavioratlowfrequencies. 164

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Fordifferentpotentiostaticconditions,theextentofinuencebyelectrodegeometry canbeobservedbyexaminingthevalueof j Z j = Z r j foraparticularimpedance loop.Whentheappliedpotentialtotheelectrodeincreases,theCPEbehaviorismore signicantasthevalueof deviatesmorefromunity.Thedeviationofthelow-frequency loopfromthestandardsemicirclerepresentsadispersionoftimeconstantexhibitingon theelectrodesurfacewhichisaconsequenceofthegeometryeffect. Thedistortionoftheimpedanceresponseassociatedwiththeelectrodegeometry canhaveasignicantimpactontheinterpretationoftheresultingspectra.Depending onthevalueof h J i ,errorof10%canbefoundforthehigh-frequencyloop,anderroras largeas30%maybeseeninidenticationofthelow-frequencyloopassociatedwiththe reactionintermediate. Thereactionmechanismforcorrosionofironisconsideredinthepresentstudyto comprisetwosuccessivecharge-transferstepsinvolvingoneadsorbedintermediate. ThelocalimpedancemeasuredbyuseoftheLEIStechniqueshowsfrequencydispersioninducedbythediskgeometry.Thelow-frequencyeffectwasmoresignicant atthemoreanodicpotential.Usingsmallbielectrodeforperforminglocalimpedance measurements,itwaspossibletoobservetheradialdependanceoftheironelectrode reactivity.Thecomparisonbetweenmodelandexperimentwasconstrainedtobequalitativeonlyduetochangesoftheelectrodesurfaceduringthecourseoftheexperiment. Nevertheless,thequalitativeagreementconrmsthepredictedinuencesofelectrode geometryonthelow-frequencyimpedanceresponseforreactionsystemsinvolving adsorbedintermediates. 9.2InuenceofNonuniformMassTransfer Thecurrentdensityincreaseswithincreasingsurfaceoverpotentialtowardsthe peripheryofthediskelectrode,resultingasmallerlocalimpedancethatindicatesa greateraccessibilityneartheelectrodeperiphery.Whentheconcentrationofreacting speciesontheelectrodeissufcientlysmall,thereactionratestartstobelimitedby 165

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themasstransferofreactant.Theincreasingoverpotentialnolongerhasaneffect onthereactionrate,andthelocalimpedancebecomeslargerattheperipheryofthe diskelectrode.Themasstransfereffectismoresignicantinthesystemofsmall concentrationofsupportingelectrolyte.Inthepresenceofasmallamountofsupporting electrolyte,anextremelylargevalueofthelocalimpedanceatelectrodeperipheryis observed.Anumericalartifactmightbepresentduetotheapproachtowardzeroofthe ionconcentrationsneartheelectrodeperipheryandisneededtobeconrmed. Theeffectsofelectrodegeometryandmasstransferresultsintheappearanceof complexfeaturesinthelocalOhmicimpedanceresponse.Thehigh-frequencycomplex OhmicbehaviorofthelocalOhmicimpedanceisassociatedwiththenonuniformOhmic potentialdropinsolution,whichismoresignicantwhentheelectrodekineticsarefast. Thelow-frequencycomplexbehaviorisassociatedwiththenonuniformmasstransferof reactingspeciestoelectrode,whichcorrelateswiththediskrotationspeed. ElectrodeprocessassociatedwithFaradaicreactionandchargingtheinterface canbeobservedinthelocalandglobalimpedanceresponsesathighfrequencies. Therearetwocontributionstocurrent, i.e., aFaradaicreactionandthechargingof theinterface.UndertheassumptionthattheFaradaicreactionandthechargingofthe interfacecanbeseparated apriori ,theuxofreactantsandproductsaregivenbythe rateoftheelectrochemicalreaction.Thisassumptionisnotquitecorrectbecausepart oftheuxofreactingspeciescontributestothechargingoftheinterfaceaswellasto theFaradaicreaction.Theapplicationoftheassumptionthatthereis apriori separation ofFaradaicandchargingcurrentsisjustiedforthereactionsystemofreductionof ferricyanide.Inthesystemofdepositionofsilver,theassumptionofNAPSisnotvalid unlessthereactionrateisslowandtheconcentrationofsupportingelectrolyteis large.TheappearanceofCPEobservedinthelocalinterfacialimpedancecannotbe attributedtothegeometry-inducedcurrentandpotentialdistributions,butwasattributed 166

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tothefrequencydependenteffectivedouble-layercapacitancethataccountedforthe contributionofuxinchargingthedoublelayer. 167

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CHAPTER10 SUGGESTIONSFORFUTUREWORK Theproposedimpedancemodelincorporatesthenonuniformmass-transfer andOhmicpotentialdroponarotatingdiskelectrode.Theassumptionsthatthe concentrationisuniforminthebulksolutionandvariesonlyinthediffusionlayerare relaxed.Theradialdistributionofsurfacechargedensitythatdependsonthepotential andconcentrationattheouterlimitofdiffuselayerwasalsodiscussedinthepresent study.Theassumptionsofuniformdouble-layercapacitanceonelectrodesurfaceand negligiblecontributionofmassuxtothecharingoftheinterfacearealsorelaxed.The electrodeprocesswasmodiedwithout apriori separationofdouble-layerchargingand Faradaicchargingcurrents. Thedouble-layermodelinthepresentstudyfollowedtheGouy-Chapmantheory thatshowsanexponentialdecayofchargeinthediffuseregion.Stern'spostulation ofacombinationofacompactionlayerincontactwithmetalandanouterregionwith looselyscatteredionswasappliedtoestimatethedistributionofchargedensityon themetalsurface.Thecapacityofthecompactlayerwasassumedtobeaconstant. Grahame 64 reportedthatthecapacityofthecompactlayerisdependentonthecharge ontheelectrode.Thedouble-layermodelcanbemodiedbyrelaxingtheassumption ofaconstantcapacity C m )]TJ/F24 7.9701 Tf 6.587 0 Td [(ohp .Thevariationof C m )]TJ/F24 7.9701 Tf 6.586 0 Td [(ohp withthesurfacechargedensity andtheconsiderationofspecicadsorptionofspeciesontothemetalsurfacecan bethenextsteptoimprovethedouble-layermodel.Also,insteadofconsideringa continuousmotionofspeciesinthediffuseregionofcharge,adouble-layermodelwith molecular-scalesimulations 74 couldbeusedtoexplorethechargedistributionatthe interface. Furthermore,thethicknessofthediffusedoublelayerisassociatedwiththeionic concentration.Thesmallerionicconcentrationgivesalargervalueofdouble-layer thicknesswhichcouldleadtoanerrorinassessingthechargedistributioninthe 168

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interfacialregion.Theelectricaldouble-layermodelshouldbeimprovedtotakesinto accounttheionicconcentration. Undertheassumptionthatthereisno apriori separationofFaradaicanddoublelayerchargingcurrents,theimpedanceresponsesshowedadepressedsemicircleat highfrequencieseveninthepresenceofanexcesssupportingelectrolyte.Experimental vericationoftheimpedanceresultsisrequired. 169

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BIOGRAPHICALSKETCH ShaolingWu'sinterestintheeldofelectrochemistrystartedfromherjunior yearattheNationalTaiwanUniversity.SincethenshejoinedtheElectrochemical EngineeringLabandworkedonanundergraduateresearchprojectinvolvingthe characterizationofnano-particlesusedintheslurryofchemicalmechanicalpolishing process.ShereceivedtheBachelorofSciencedegreeinChemicalEngineeringfrom theNationalTaiwanUniversityinJune,2004.Shaolingpursuedherstudyinthesame groupandworkedonthedevelopmentofmetallizationbyintegratingelectrolessplating andink-jetprintingtechnology.AftershereceivedtheMasterofSciencedegreein Juneof2006,sheenteredthegraduateschoolattheUniversityofFlorida.Shaoling joinedtheresearchgroupunderthedirectionofProfessorMarkOrazeminJanuary, 2007.ShehasbeenworkingontheresearchprojectincollaborationwiththeCNRS LaboratoryinParisforthenumericalsimulationoftheinuenceofelectrodegeometry onimpedanceresponse.Shaolinggraduatedinthesummerof2010withaPh.D.in ChemicalEngineering. 176