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Distributed Time-Constant Impedance Responses Interpreted in Terms of Physically Meaningful Properties

Permanent Link: http://ufdc.ufl.edu/UFE0041984/00001

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Title: Distributed Time-Constant Impedance Responses Interpreted in Terms of Physically Meaningful Properties
Physical Description: 1 online resource (188 p.)
Language: english
Creator: Hirschorn, Bryan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: constant, element, impedance, kramers, kronig, nonlinear, oxide, phase, spectroscopy
Chemical Engineering -- Dissertations, Academic -- UF
Genre: Chemical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Models invoking Constant-Phase Elements (CPE) are often used to fit impedance data arising from a broad range of experimental systems. The physical origins of the CPE remain controversial. CPE parameters are considered to arise from a distribution of time-constants that may be distributed along the surface of an electrode or in the direction normal to the electrode. The capacitance of electrochemical systems is used to calculate properties, such as permittivity, layer thickness, and active surface area. The determination of capacitance from CPE data is often inadequate, leading to erroneous prediction of physical properties. In the present work, two different mathematical formulas for estimating effective capacitance from CPE parameters, taken from the literature, are associated unambiguously with either surface or normal time-constant distributions. However, these equations were not developed from a physical model and do not properly account for characteristic frequencies outside the measured frequency range. For a broad class of systems, these formulations for capacitance are insufficient, which illustrates the need to develop mechanisms to account for the CPE. CPE behavior may be attributed to the distribution of physical properties in films, in the direction normal to the electrode surface. Numerical simulations were used to show that, under assumption that the dielectric constant is independent of position, a normal power-law distribution of local resistivity is consistent with the CPE. An analytic expression, based on the power-law resistivity distribution, was found that relates CPE parameters to the physical properties of a film. This expression yielded physical properties, such as film thickness and resistivity, that were in good agreement with expected or independently measured values for such diverse systems as aluminum oxides, oxides on stainless steel, and human skin. The agreement obtained using the power-law model can be explained by the fact that it is based on formal solution for the impedance associated with a specified resistivity distribution, rather than using formulations for capacitance that do not take any physical model into account. The power-law model yields a CPE impedance behavior in an appropriate frequency range, defined by two characteristic frequencies. Ideal capacitive behavior is seen above the upper characteristic frequency and below the lower characteristic frequency. A symmetric CPE response at both high and low frequencies can be obtained by adding a parallel resistive pathway. CPE behavior may also be attributed to the distribution of physical properties along the surface of an electrode. Numerical simulations were used to show that a power-law distribution of Ohmic resistance along a blocking surface with uniform capacitance yielded an impedance response that was consistent with the CPE. The broad distribution necessary suggested that observed CPE behavior cannot be considered to arise from a distribution of Ohmic resistance alone. Nevertheless, the developed relationship between capacitance and CPE parameters for a surface distribution was shown to be different than the relationship developed for a normal distribution indicating that the physical origin of the CPE needs to be considered when assessing capacitance from impedance spectra. Analysis of systems exhibiting the CPE requires accurate estimates of model parameters. In support of the mechanistic development of the CPE, a generalized method was developed for identifying and minimizing nonlinear distortions in impedance spectra for increased confidence in model development and parameter estimation. A characteristic transition frequency was defined that can be used to tailor a frequency-dependent input signal to optimize signal-to-noise levels while maintaining a linear response. The Kramers-Kronig relations, which provide an essential tool for assessing the internal consistency of impedance data, are understood to be sensitive to failures of causality, but insensitive to failures of linearity. Numerical simulations showed that the Kramers-Kronig relations are not satisfied for measurements which include the characteristic transition frequency. However, the relations were satisfied for measurements taken below the characteristic frequency, even for very nonlinear systems.
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.
Statement of Responsibility: by Bryan Hirschorn.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Orazem, Mark E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041984:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041984/00001

Material Information

Title: Distributed Time-Constant Impedance Responses Interpreted in Terms of Physically Meaningful Properties
Physical Description: 1 online resource (188 p.)
Language: english
Creator: Hirschorn, Bryan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: constant, element, impedance, kramers, kronig, nonlinear, oxide, phase, spectroscopy
Chemical Engineering -- Dissertations, Academic -- UF
Genre: Chemical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Models invoking Constant-Phase Elements (CPE) are often used to fit impedance data arising from a broad range of experimental systems. The physical origins of the CPE remain controversial. CPE parameters are considered to arise from a distribution of time-constants that may be distributed along the surface of an electrode or in the direction normal to the electrode. The capacitance of electrochemical systems is used to calculate properties, such as permittivity, layer thickness, and active surface area. The determination of capacitance from CPE data is often inadequate, leading to erroneous prediction of physical properties. In the present work, two different mathematical formulas for estimating effective capacitance from CPE parameters, taken from the literature, are associated unambiguously with either surface or normal time-constant distributions. However, these equations were not developed from a physical model and do not properly account for characteristic frequencies outside the measured frequency range. For a broad class of systems, these formulations for capacitance are insufficient, which illustrates the need to develop mechanisms to account for the CPE. CPE behavior may be attributed to the distribution of physical properties in films, in the direction normal to the electrode surface. Numerical simulations were used to show that, under assumption that the dielectric constant is independent of position, a normal power-law distribution of local resistivity is consistent with the CPE. An analytic expression, based on the power-law resistivity distribution, was found that relates CPE parameters to the physical properties of a film. This expression yielded physical properties, such as film thickness and resistivity, that were in good agreement with expected or independently measured values for such diverse systems as aluminum oxides, oxides on stainless steel, and human skin. The agreement obtained using the power-law model can be explained by the fact that it is based on formal solution for the impedance associated with a specified resistivity distribution, rather than using formulations for capacitance that do not take any physical model into account. The power-law model yields a CPE impedance behavior in an appropriate frequency range, defined by two characteristic frequencies. Ideal capacitive behavior is seen above the upper characteristic frequency and below the lower characteristic frequency. A symmetric CPE response at both high and low frequencies can be obtained by adding a parallel resistive pathway. CPE behavior may also be attributed to the distribution of physical properties along the surface of an electrode. Numerical simulations were used to show that a power-law distribution of Ohmic resistance along a blocking surface with uniform capacitance yielded an impedance response that was consistent with the CPE. The broad distribution necessary suggested that observed CPE behavior cannot be considered to arise from a distribution of Ohmic resistance alone. Nevertheless, the developed relationship between capacitance and CPE parameters for a surface distribution was shown to be different than the relationship developed for a normal distribution indicating that the physical origin of the CPE needs to be considered when assessing capacitance from impedance spectra. Analysis of systems exhibiting the CPE requires accurate estimates of model parameters. In support of the mechanistic development of the CPE, a generalized method was developed for identifying and minimizing nonlinear distortions in impedance spectra for increased confidence in model development and parameter estimation. A characteristic transition frequency was defined that can be used to tailor a frequency-dependent input signal to optimize signal-to-noise levels while maintaining a linear response. The Kramers-Kronig relations, which provide an essential tool for assessing the internal consistency of impedance data, are understood to be sensitive to failures of causality, but insensitive to failures of linearity. Numerical simulations showed that the Kramers-Kronig relations are not satisfied for measurements which include the characteristic transition frequency. However, the relations were satisfied for measurements taken below the characteristic frequency, even for very nonlinear systems.
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.
Statement of Responsibility: by Bryan Hirschorn.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Orazem, Mark E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041984:00001


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DISTRIBUTEDTIME-CONSTANTIMPEDANCERESPONSESINTERPRETEDIN TERMSOFPHYSICALLYMEANINGFULPROPERTIES By BRYAND.HIRSCHORN ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2010

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c 2010BryanD.Hirschorn 2

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Tomywife,ErinHirschorn,andmydaughter,SiennaHirschorn 3

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ACKNOWLEDGMENTS Ithankmyadvisor,ProfessorMarkE.Orazem,forhisexpertise,insight,and supportinthiswork.Hispatiencewithstudentsisunmatchedandhealwaystakesthe timetolistenandofferhisassistance.Hisinterest,thoroughness,andattentiontodetail hassignicantlyimprovedmystudyandhashelpedpushmetorealizemycapabilities andtoimprovemyweaknesses. IthankDr.BernardTribollet,Dr.IsabelleFrateur,Dr.MarcoMusiani,andDr. VincentVivierfortheirhelpdevelopingandimprovingthisstudy.Theextensionof theirareasofresearchbroadenedthescopeofthiswork.Ialsothankmycommittee membersProfessorJasonWeaver,ProfessorAnujChauhan,andProfessorJuanNino. Ithankmywife,mymother,andmyfatherfortheirloveandencouragementduring mystudies. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS..................................4 LISTOFTABLES......................................8 LISTOFFIGURES.....................................9 LISTOFSYMBOLS....................................17 ABSTRACT.........................................20 CHAPTER 1INTRODUCTION...................................23 2LITERATUREREVIEW...............................28 2.1TheConstant-PhaseElement.........................28 2.1.1SurfaceDistributions..........................28 2.1.2NormalDistributions..........................30 2.2DeterminationofCapacitancefromtheCPE.................31 2.3ErrorsAssociatedwithNonlinearity......................32 2.4LinearityandtheKramers-KronigRelations.................35 3OPTIMIZATIONOFSIGNAL-TO-NOISERATIOUNDERALINEARRESPONSE37 3.1CircuitModelsIncorporatingFaradaicReactions..............37 3.2NumericalSolutionofNonlinearCircuitModels...............39 3.3SimulationResults...............................41 3.3.1ErrorsinAssessmentofCharge-TransferResistance........41 3.3.2OptimalPerturbationAmplitude....................42 3.3.3ExperimentalAssessmentofLinearity................46 3.3.4FrequencyDependenceoftheInterfacialPotential.........47 3.3.5OptimizationoftheInputSignal....................52 3.3.6Potential-DependentCapacitance...................54 3.4Conclusions...................................55 4THESENSITIVITYOFTHEKRAMERS-KRONIGRELATIONSTONONLINEARRESPONSES..................................56 4.1ApplicationoftheKramers-KronigRelations.................57 4.2SimulationResults...............................58 4.3TheApplicabilityoftheKramers-KronigRelationstoDetectingNonlinearity......................................64 4.3.1InuenceofTransitionFrequency...................64 4.3.2ApplicationtoExperimentalSystems.................69 4.4Conclusions...................................72 5

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5CHARACTERISTICSOFTHECONSTANT-PHASEELEMENT.........74 6THECAPACITIVERESPONSEOFELECTROCHEMICALSYSTEMS.....79 6.1CapacitanceoftheDiffuseLayer.......................79 6.2CapacitanceofaDielectricLayer.......................81 6.3CalculationofCapacitancefromImpedanceSpectra............83 6.3.1SingleTime-ConstantResponses...................83 6.3.2SurfaceDistributions..........................84 6.3.3NormalDistributions..........................87 6.4Conclusions...................................89 7ASSESSMENTOFCAPACITANCE-CPERELATIONSTAKENFROMTHE LITERATURE.....................................91 7.1SurfaceDistributions..............................91 7.2NormalDistributions..............................95 7.2.1Niobium.................................95 7.2.2HumanSkin...............................99 7.2.3FilmswithanExponentialDecayofResistivity............102 7.3ApplicationoftheYoungModeltoNiobiumandSkin............106 7.4Conclusions...................................110 8CPEBEHAVIORCAUSEDBYRESISTIVITYDISTRIBUTIONSINFILMS...112 8.1ResistivityDistribution.............................112 8.2ImpedanceExpression.............................118 8.3Discussion...................................122 8.3.1ExtractionofPhysicalParameters...................122 8.3.2ComparisontoYoungModel......................123 8.3.3VariableDielectricConstant......................125 8.4Conclusions...................................126 9APPLICATIONOFTHEPOWER-LAWMODELTOEXPERIMENTALSYSTEMS.........................................127 9.1Method.....................................127 9.2ResultsandDiscussion............................128 9.2.1AluminumOxideLarge 0 andSmall ...............130 9.2.2StainlessSteelFinite 0 andSmall ................133 9.2.3HumanSkinPowerLawwithParallelPath.............136 9.3Discussion...................................139 9.4Conclusions...................................141 10CPEBEHAVIORCAUSEDBYSURFACEDISTRIBUTIONSOFOHMICRESISTANCE......................................142 10.1MathematicalDevelopment..........................142 6

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10.2DiskElectrodes.................................148 10.2.1Increaseofresistancewithincreasingradius............148 10.2.2Decreaseofresistancewithincreasingradius............148 10.3Conclusions...................................152 11OVERVIEWOFCAPACITANCE-CPERELATIONS................153 12CONCLUSIONS...................................156 13SUGGESTIONSFORFUTUREWORK......................158 13.1CPEBehaviorCausedbySurfaceDistributionsofReactivity........158 13.1.1MathematicalDevelopment......................159 13.1.2Interpretation..............................161 13.2CPEBehaviorCausedbyNormalDistributionsofProperties.......164 APPENDIX APROGRAMCODEFORLARGEAMPLITUDEPERTURBATIONS.......165 BPROGRAMCODEFORNORMALDISTRIBUTIONS..............170 CPROGRAMCODEFORSURFACEDISTRIBUTIONS..............173 DPROGRAMCODEFORDISKELECTRODEDISTRIBUTION..........175 REFERENCES.......................................179 BIOGRAPHICALSKETCH................................188 7

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LISTOFTABLES Table page 4-1SimulationresultsusedtoexploretheroleoftheKramers-Kronigrelationsfor nonlinearsystemswithparameters: U =100 mV, C dl =20 F/cm 2 K a = K c =1 mA/cm 2 b a = b c =19 V )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ,and V =0 V...................58 7-1CPEparameters,resistance,effectivecapacitance,andthicknessofoxide lmsformedonaNbdiskelectrodein0.1MNH 4 FsolutionpH2asafunctionoftheanodizationpotential...........................97 7-2ThicknessofoxidelmsdevelopedonaNbelectrode,asafunctionofthe anodizationpotential.Comparisonofvaluesdeducedfromimpedancedata withthosefromtheliterature.............................98 7-3CPEparameters,resistance,effectivecapacitance,andthicknessforheatstrippedhumanstratumcorneumin50mMbufferedCaCl 2 electrolyteasa functionofimmersiontime.DatatakenfromMembrino. 1 ............100 7-4Physicalpropertiesobtainedbymatchingthehigh-frequencyportionofthe impedanceresponsegiveninFigure7-5forheat-strippedstratumcorneumin 50mMbufferedCaCl 2 electrolyteasafunctionofimmersiontime........109 8

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LISTOFFIGURES Figure page 3-1CircuitmodelswithFaradaicreaction:anon-OhmicFaradaicsystem;bOhmic Faradaicsystem;andcOhmicconstantcharge-transferresistancesystem...38 3-2Calculatedimpedanceresponsewithappliedperturbationamplitudeasaparameter.Thesystemparameterswere R e =1 cm 2 K a = K c =1 mA/cm 2 b a = b c =19 V )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 C dl =20 F/cm 2 ,and V =0 V,givingrisetoalinearchargetransferresistance R t ; 0 =26 cm 2 ..........................42 3-3Theerrorinthelow-frequencyimpedanceasymptoteassociatedwithuseof alargeamplitudepotentialperturbation.......................46 3-4Lissajousplotswithperturbationamplitudeandfrequencyasparameters. Thesystemparameterswere R e =0 cm 2 K a = K c =1 mA/cm 2 b a = b c = 19 V )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 C dl =20 F/cm 2 ,and V =0 V,givingrisetoalinearcharge-transfer resistance R t ; 0 =26 cm 2 ..............................47 3-5Maximumvariationoftheinterfacialpotentialsignalasafunctionoffrequency forparameters U =100 mV, R e =1 cm 2 C dl =20 F/cm 2 ,and R t ; 0 = 26 cm 2 ........................................48 3-6Maximumvariationoftheinterfacialpotentialsignalasafunctionoffrequency forparameters U =100 mV, R e =1 cm 2 C dl =20 F/cm 2 R t ; 0 =26 cm 2 Thesolidcurveis V max resultingfromthenumericsimulation.Thedashed curveis V predictedfromequation3using R t ; obs =19 cm 2 ,whichdecreasesfromthelinearvalue, R t ; 0 =26 cm 2 ,duetothelargeinputperturbation..........................................49 3-7Calculatedresultsforparameters U =100 mV, C dl =20 F/cm 2 ,and R t ; 0 = 26 cm 2 withOhmicresistanceasaparameter.aMaximumvariationofthe interfacialpotentialsignalasafunctionoffrequency;andbthecorrespondingLissajousplotsatafrequencyof0.016Hz...................50 3-8Inectionpointof V max islocatedatthetransitionalfrequencydenedby equation3 U =100 mV, R e =1 cm 2 C dl =20 F/cm 2 ,and R t ; 0 = 26 cm 2 ........................................51 3-9Calculatedresultsforparameters R e =1 cm 2 C dl =20 F/cm 2 ,and R t ; 0 = 26 cm 2 withappliedperturbationamplitudeasaparameter:aMaximum variationoftheinterfacialpotentialsignalasafunctionoffrequency;andb Theeffectivecharge-transferresistanceasafunctionoffrequency........52 3-10Effectivecharge-transferresistanceasafunctionoffrequency:atheeffectivecharge-transferresistancefordifferentOhmicresistancesandinputamplitudes;andbthedimensionlessformoftheeffectivecharge-transferresistanceversusdimensionlessfrequency.......................53 9

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3-11Systemwith R e =2 R t andbaselinenoisethatisconstantat20percentof lowfrequencycurrentsignal.a U =10 mVforall .b U =30 mVfor !< 10 t and U =300 mVfor !> 10 t ......................54 4-1Residualserrorsresultingfromameasurementmodelt Z m tosimulateddata Z s forthesystemwith R e =0 cm 2 :arealpart;andbimaginarypart.The linescorrespondtothe95.4% condenceintervalfortheregression.The systemparameterspresentedinTable4-1giveriseto R t ; obs / R t ; 0 =0.658and R e / R t ; obs =0......................................60 4-2Normalizedresidualerrorsresultingfromameasurementmodelt Z m tosimulatedimpedancedata Z s forthesystemwith R e = : 01 cm 2 :arealpart; andbimaginarypart.Thelinescorrespondtothe95.4%condenceinterval fortheregression.ThesystemparameterspresentedinTable4-1giveriseto R t ; obs / R t ; 0 =0.658and R e / R t ; obs = 5 : 8 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(4 ....................61 4-3Normalizedresidualerrorsresultingfromameasurementmodelt Z m tosimulatedimpedancedata Z s forthesystemwith R e =1 cm 2 :arealpart;and bimaginarypart.Thelinescorrespondtothe95.4%condenceintervalfor theregression.ThesystemparameterspresentedinTable4-1giveriseto R t ; obs / R t ; 0 =0.684and R e / R t ; obs = 5 : 6 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 ....................62 4-4Normalizedresidualerrorsresultingfromameasurementmodelt Z m tosimulatedimpedancedata Z s forthesystemwith R e =100 cm 2 :arealpart; andbimaginarypart.Thelinescorrespondtothe95.4%condenceinterval fortheregression.ThesystemparameterspresentedinTable4-1giveriseto R t ; obs / R t ; 0 =0.981and R e / R t ; obs =3.9........................62 4-5Acomparisonofsimulationresultstotherealcomponentofimpedancepredictedusingequation4forthesystemswith R e =0 cm 2 and R e = 1 cm 2 .ThesystemparameterspresentedinTable4-1giveriseto R t ; obs / R t ; 0 =0.658and R t ; obs / R t ; 0 =0.684,respectively.IntheabsenceofOhmicresistance,thesimulateddataandthepredictedvaluesareequal...........63 4-6Acomparisonofsimulationresultstotherealcomponentofimpedancepredictedusingequation4forthesystemswith R e =1 cm 2 and R e = 100 cm 2 .ThesystemparameterspresentedinTable4-1giveriseto R t ; obs / R t ; 0 =0.684and R t ; obs / R t ; 0 =0.981,respectively....................64 4-7Interfacialparametersasfunctionsoffrequencyforthesimulationspresented inTable4-1:aMaximumvariationoftheinterfacialpotential;andbtheeffectivecharge-transferresistance.Verticallinescorrespondtothetransition frequencygivenbyequation3.........................65 4-8Thenormalizedrealpartoftheimpedanceasafunctionofnormalizedfrequencyforthesystemwith R e =1 cm 2 solidline.Thedashedlinesrepresenttheideallinearresponsesforsystemswith R t ; 0 =26 : 3 cm 2 andwith R t ; obs =18 : 0 cm 2 ...................................66 10

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4-9Thenormalizedimpedanceresponseasfunctionsofnormalizedfrequency forthesystemswith R e =0 : 01 1 ,and 100 cm 2 :arealpart;andbimaginarypart.Thesolidcurveistheideallinearresponseandthedashedcurves arethenonlinearimpedanceresponsesarisingfromalargeinputamplitude of U =100 mVforsystemparameterspresentedinTable4-1..........67 4-10Thenormalizedimpedanceresponseasfunctionsofnormalizedfrequency forthesystemwith R e =0 :arealpart;andbimaginarypart.Boththeideal linearresponseandthenonlinearimpedanceresponsearesuperposed....68 4-11Thetransitionfrequencygivenbyequation3asafunctionof RC time constantwith R e =R t asaparameter.........................70 4-12Normalizedresidualerrorsresultingfromameasurementmodelt Z m tosimulatedimpedancedata Z s withnormallydistributedadditivestochasticerrors withstandarddeviationof 0 : 1 percentofthemodulusforthesystemwith R e = 1 cm 2 :arealpart;andbimaginarypart.Thelinescorrespondtothe95.4% condenceintervalfortheregression.Theinputpotentialperturbationamplitudewas U =1 mV.................................71 4-13Normalizedresidualerrorsresultingfromameasurementmodelt Z m tosimulatedimpedancedata Z s withnormallydistributedadditivestochasticerrors withstandarddeviationof 0 : 1 percentofthemodulusforthesystemwith R e = 1 cm 2 :arealpart;andbimaginarypart.Thelinescorrespondtothe95.4% condenceintervalfortheregression.Theinputpotentialperturbationamplitudewas U =100 mV................................72 5-1ImpedanceplanerepresentationoftheCPE,equation5,with asaparameterand Q =1 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(6 Fs )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 = cm 2 .......................75 5-2Thephase-angleassociatedwiththeCPE,equation5,with asaparameter.........................................75 5-3ImpedanceresponseoftheCPE,equation5,with asaparameterand Q =1 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(6 Fs )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 = cm 2 ;atherealcomponent;andbtheimaginarycomponent.........................................76 5-4Impedanceplanerepresentationofequation1with asaparameterand Q =1 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(6 Fs )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 = cm 2 and R =10kcm 2 ....................77 5-5Impedanceresponseofequation1with asaparameterand R =10kcm 2 and Q =1 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(6 Fs )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 = cm 2 ;atherealcomponent;andbtheimaginary component.......................................77 11

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6-1Schematicrepresentationofasurfacedistributionoftimeconstants:adistributionoftimeconstantsinthepresenceofanOhmicresistanceresultingina distributedtime-constantbehaviorthat,foranappropriatetime-constantdistribution,maybeexpressedasaCPE;andbdistributionoftimeconstants intheabsenceofanOhmicresistanceresultinginaneffectiveRCbehavior. Theadmittance Y i showninaincludesthelocalinterfacialandOhmiccontributions........................................85 6-2Schematicrepresentationofanormaldistributionoftimeconstantsresulting inadistributedtime-constantbehaviorthat,foranappropriatetime-constant distribution,maybeexpressedasaCPE......................88 7-1EffectiveCPEcoefcientscaledbytheinterfacialcapacitanceasafunction ofdimensionlessfrequency K with J asaparameter.TakenfromHuang et al. 2 ..........................................92 7-2Normalizedeffectivecapacitancecalculatedfromrelationshipspresentedby Brug etal. 3 foradiskelectrodeasafunctionofdimensionlessfrequency K with J asaparameter:awithcorrectionforOhmicresistance R e equation 6;andbwithcorrectionforbothOhmicresistance R e andcharge-transfer resistance R t equation6.TakenfromHuang etal. 2 ...........94 7-3Effectivecapacitancecalculatedfromequation6andnormalizedbythe inputinterfacialcapacitanceforadiskelectrodeasafunctionofdimensionlessfrequency K with J asaparameter.TakenfromHuang etal. 2 .......95 7-4ExperimentalimpedancedataobtainedwithaNbrotatingdiskelectrode rpmin0.1MNH 4 FsolutionpH2,at6VSCE:aComplexplaneplot;and btheimaginarypartoftheimpedanceasafunctionoffrequency.Datataken fromCattarin etal. 4 .................................96 7-5Experimentalimpedancedataobtainedforheat-separatedexcisedhuman stratumcorneumin50mMbufferedCaCl 2 electrolytewithimmersiontimeas aparameter:aComplexplaneplot;andbtheimaginarypartoftheimpedance asafunctionoffrequency.DatatakenfromMembrino. 1 .............101 7-6Nyquistplotsforsimulationoftheimpedanceassociatedwithanexponential decayofresistivitywith = asaparameter.Thecharacteristicfrequencyindicatedisindimensionlessformfollowing !"" 0 0 .................104 7-7Thederivativeofthelogarithmofthemagnitudeoftheimaginarypartofthe impedancewithrespecttothelogarithmoffrequencyasafunctionofdimensionlessfrequencyforthesimulationspresentedinFigure7-6..........105 7-8TheeffectivelmthicknessobtainedforthesimulationspresentedinFigure 7-6usingequations6and7:anormalizedbytheknownlmthickness ;andbnormalizedbythecharacteristiclength .............105 12

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7-9Circuitrepresentationofanormaldistributionofresistivityinwhichsomecapacitanceelementsarenotobservedoveranexperimentallyaccessiblefrequencyrangeduetolocalvariationofresisitivity..................107 7-10ComparisonoftheYoungmodeltothehigh-frequencypartoftheexperimentalimaginarypartoftheimpedanceasafunctionoffrequency:aNiobium oxideatapotentialof6VSCEseeFigure7-4;andbhumanstratumcorneum withimmersiontimeasaparameterseeFigure7-5.Thelinesrepresentthe model,andsymbolsrepresentthedata.......................108 7-11Resistivityprolesassociatedwiththesimulationoftheimpedanceresponse forNb 2 O 5 at 6 VSCEandskinusingauniformdielectricconstantandanexponentiallydecayingresistivity............................110 8-1Adistributionof RC elementsthatcorrespondstotheimpedanceresponseof alm..........................................113 8-2Syntheticdatasymbolsfollowingequation1with Q =1 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(6 s = cm 2 with asaparameterandthecorresponding RC measurementmodelts linesforatherealcomponentoftheimpedance;btheimaginarycomponentoftheimpedance;andcthephaseangle.Theregressedelementsare showninFigures8-3aand8-3b.........................114 8-3Theregressedmeasurementmodelparametersasafunctionoftime-constant forthesyntheticCPEdatashowninFigures8-2aand8-2b:aresistance; andbcapacitance.Thecircledvalueswerenotusedinthesubsequentanalysis...........................................115 8-4Resistivityasafunctionofdimensionlessposition.Thesymbolsarethediscreteresistivityvaluescalculatedfromequations8and8usingthe regressedvaluesofresistancesandcapacitancesgiveninFigures8-3aand 8-3band =10 .Thelinesrepresentequation8withparameter determinedaccordingtoequation8.......................117 8-5Acomparisonoftheimpedanceresponsegeneratedbynumericalintegration ofequation8symbolsandtheanalyticexpressionprovidedbyequation8lineswith 0 =1 10 16 cm =100cm =10 =100 nm, and asaparameter:atherealcomponentofimpedance;andbtheimaginarycomponentofimpedance............................120 8-6Thenumericalevaluationof g asafunctionof 1 = wherethesymbolsrepresentresultsobtainedfromequation8.Thelinerepresentstheinterpolationformulagivenasequation8........................121 8-7NyquistrepresentationoftheimpedancegiveninFigure8-5for =6 : 67 .The markedimpedanceatafrequencyof 2 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(5 Hzisclosetothecharacteristic frequency f 0 =1 : 8 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(5 Hz.............................122 13

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8-8Normalizedimpedanceresponseassociatedwithnormaldistributionsofresistivitywithaxeddielectricconstant =10 andathickness =100 nm. Thedashedlineprovidestheresultsforaresistivitygivenasequation8 with 0 =1 10 12 cm =2 10 7 cm ,and =6 : 67 .Thesolidlineprovides theresultforaYoungmodelwitharesistivityprolefollowingequation830 withthesamevaluesof and 0 ,yielding =9 : 24 nm.aNyquistplot;b realpartoftheimpedance;andcimaginarypartoftheimpedance.......124 8-9Resistivityprolesandestimatedvaluesof forthesimulationsreportedin Figure8-8:aresistivityversusposition;andbthevalueof d log j Z j j =d log f obtainedfromtheslopesgiveninFigure8-8c..................125 9-1Representationof ZQ where Z isgeneratedbynumericalintegrationofequation8and Q isobtainedfromequation8for =4 =0 : 75 and =10 with 0 and asparameters:atherealcomponentofimpedance; andbtheimaginarycomponentofimpedance.Thelinerepresents j! )]TJ/F22 7.9701 Tf 6.587 0 Td [(0 : 75 inagreementwithequation1.Thesymbolsrepresentcalculationsperformedfor 4 0 =10 18 cmand =10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 cm; 0 =10 14 cm; = 10 2 cm;and 0 =10 10 cm; =10 5 cm...................128 9-2NyquistplotofthedatapresentedinFigure9-1for 0 =10 10 cmand = 10 5 cm:aplotshowingthecharacteristicfrequency f 0 = 0 0 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 = 18 Hz;andbzoomedregionshowing f = 0 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 =1 : 8 10 6 Hz.....129 9-3Impedanceresponseassociatedwithafrequencyrangewhichexcludesthe characteristicfrequencies f 0 and f :asimulationsobtainedfor =10 and with asaparameter;andbexperimentalNyquistplotforpassiveAluminum ina0.1MNa 2 SO 4 electrolytedatatakenfromJorcin etal. 5 .Thedashed linerepresentsaCPEttothedataaccordingtoequation1........130 9-4Thevalueof ; max obtainedfromequation9withdielectricconstantasa parameter.......................................132 9-5Thecalculatedvalueof C e ; f asafunctionofthecut-offfrequency f with asaparameter. C 0 isthecapacitanceatthemaximumfrequencyexperimentallymeasured.....................................132 9-6ImpedancediagramofoxideonaFe17Crstainlesssteeldisksymbols:a experimentalfrequencyrange.Thesolidlineisthepower-lawmodelfollowingequation8withparameters 0 =4 : 5 10 13 cm =450cm =3 nm, =12 ,and =9 : 1 ,andthedashedlineistheCPEimpedance with =0 : 89 and Q =3 : 7 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(5 Fcm )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 s )]TJ/F22 7.9701 Tf 6.587 0 Td [(0 : 11 ;andbextrapolationtozero frequencywherethedashedlinerepresentsthetofaVoigtmeasurement modelandthesolidlinerepresentsthetofthepower-lawmodel........134 14

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9-7ImpedanceresponseofoxideonaFe17Crstainlesssteeldisksymbolsand thetheoreticalmodellinewithparametersreportedinFigure9-6a:athe realcomponent;andbtheimaginarycomponent.Theelectrolyteresistance valuewas 23cm 2 ...................................135 9-8Theimpedanceresponsesymbolsofhumanstratumcorneumimmersed in50mMbufferedCaCl 2 electrolytefor1.9hours.Thesolidlineisobtained followingequation8withalargevalueof 0 =49 =6 : 02 ,and = 48cm andaparallelresistance R p =56kcm 2 .Thedashedlineisobtained usingequation8with 0 =2 : 2 10 8 cm =49 =6 : 02 ,and = 48cm :aNyquistplot;brealpartoftheimpedance;andcimaginarypart oftheimpedance...................................137 10-1Asurfacedistributionofblockingelementswithauniformdistributionoflocal capacitance......................................143 10-2Acomparisonoftheimpedanceresponsegeneratedbynumericalintegration ofequation10symbolsandtheanalyticexpressionprovidedbyequation10lineswith R b =1 10 7 cm 2 R s =1 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 cm 2 C 0 = 10 F= cm 2 ,and asaparameter:atherealcomponentofimpedance;and btheimaginarycomponentofimpedance.....................146 10-3Thenumericalevaluationof g asafunctionof 1 )]TJ/F24 11.9552 Tf 11.836 0 Td [( wherethesymbolsrepresenttheresultsfromequation10.Thelinerepresentstheinterpolation formulagivenasequation8..........................147 10-4Thesimulationresultsfollowingequation10with R b =1 10 10 cm 2 R s =1cm 2 C 0 =10 F= cm 2 ,and asaparameter:atherealcomponent ofimpedance;btheimaginarycomponentofimpedance;cthegraphically determinedvalueof ;anddtheresistivitydistributionsfollowingequation 10.........................................149 10-5Thesimulationresultsfollowingequation10with R b =1 10 10 cm 2 R s =1cm 2 C 0 =10 F= cm 2 ,and asaparameter:atherealcomponent ofimpedance;btheimaginarycomponentofimpedance;cthegraphically determinedvalueof ;anddthenumericallydeterminedresistivitydistributions R r .......................................151 11-1Changeincapacitanceasafunctionofachangein R with Q heldconstant followingequation11;andchangeincapacitanceasafunctionofachange in Q with R heldconstantfollowingequation11.Unitsarearbitrary......154 15

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13-1Acomparisonoftheimpedanceresponsegeneratedbynumericalintegration ofequation13symbolsandtheanalyticalexpressionprovidedbyequation13lineswith R t ; m =1 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(4 cm 2 C 0 =10 F= cm 2 R e =10cm 2 and asaparameter:atherealcomponentofimpedance;btheimaginary componentofimpedance;andcthegraphicallydeterminedvalueof .The symbolsrepresentcalculationsperformedfor =3 =4 ,and 4 = 6 : 67 ...........................................162 13-2Thegraphicallydeterminedvalueof foranimpedanceresponsegenerated bynumericalintegrationofequation13with R t ; m =1 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(4 cm 2 C 0 = 10 F= cm 2 =4 ,and R e asaparameter.Thesymbolsrepresentcalculationsperformedfor R e =1cm 2 R e =10cm 2 4 R e =100cm 2 .............................................163 16

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LISTOFSYMBOLS A ^ normalizedarea b a anodiccoefcient, V )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 b c cathodiccoefcient, V )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 C capacitance, F= cm 2 C 0 uniformcapacitanceassociatedwithasurfacedistribution, F= cm 2 C B capacitance-CPErelationderivedbyBrug etal. 3 F= cm 2 C d diffuselayercapacitance, F= cm 2 C di dielectriclayercapacitance, F= cm 2 C dl doublelayercapacitance, F= cm 2 C e effectivecapacitanceofasystem, F= cm 2 C e ; f effectivecapacitanceofadielectriclm, F= cm 2 C e ; n effectivecapacitanceassociatedwithanormaldistribution, F= cm 2 C e ; s effectivecapacitanceassociatedwithasurfacedistribution, F= cm 2 C HM capacitance-CPErelationderivedbyHsuandMansfeld, 6 F= cm 2 C RC capacitanceassociatedwithasingletime-constant, F= cm 2 F Faraday'sconstant,96,487 C= equiv f 0 characteristicfrequencyassociatedwithresistivity 0 f 0 = 0 0 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ,Hz f frequency, f = != 2 ,Hz f characteristicfrequencyassociatedwithresistivity f = 0 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 ,Hz f max frequencycorrespondingtolargestmeasuredfrequency,Hz f peak frequencycorrespondingtothepeakoftheimaginaryimpedance,Hz f t characteristictransitionalfrequency,Hz g tabulatedfunction,seeequation8 i 0 exchangecurrentdensity, A= cm 2 i C capacitivecurrentdensity, A= cm 2 i f faradaiccurrentdensity, A= cm 2 17

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j imaginarynumber, j = p )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 Q CPEcoefcient,s / cm 2 R b largestvalueofresistanceinasurfacedistribution, cm 2 R e Ohmicresistance, cm 2 R f resistanceofalm, cm 2 R p polarizationresistance, cm 2 R s smallestvalueofresistanceinasurfacedistribution, cm 2 R t charge-transferresistance, cm 2 R t ; 0 linearvalueofcharge-transferresistance, cm 2 R t ; obs observedvalueofcharge-transferresistance, cm 2 U appliedcellpotential, V V interfacialpotential, V Y admittance, )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 cm )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 Z impedance, cm 2 Z CPE impedanceresponsethatexhibitsconstant-phaseelementbehavior, cm 2 Z CPE ;R impedanceresponsethatexhibitssymmetricconstant-phaseelementbehavior, cm 2 Z j pertainingtotheimaginarypartoftheimpedance, cm 2 Z r pertainingtotherealpartoftheimpedance, cm 2 CPEexponent lmthickness, cm dielectricconstant 0 permittivityofvacuum, 8 : 8541 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(14 F= cm power-lawexponent Debyelengthorcharacteristiclengthofanexponentialfunction resistivity, cm 0 interfacialresistivityatposition 0 cm 18

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interfacialresistivityatposition cm time-constant,s angularfrequency,s )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 normalizeddistance 19

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy DISTRIBUTEDTIME-CONSTANTIMPEDANCERESPONSESINTERPRETEDIN TERMSOFPHYSICALLYMEANINGFULPROPERTIES By BryanD.Hirschorn August2010 Chair:MarkE.Orazem Major:ChemicalEngineering ModelsinvokingConstant-PhaseElementsCPEareoftenusedtotimpedance dataarisingfromabroadrangeofexperimentalsystems.Thephysicaloriginsofthe CPEremaincontroversial.CPEparametersareconsideredtoarisefromadistribution oftime-constantsthatmaybedistributedalongthesurfaceofanelectrodeorinthe directionnormaltotheelectrode.Thecapacitanceofelectrochemicalsystemsis usedtocalculateproperties,suchaspermittivity,layerthickness,andactivesurface area.ThedeterminationofcapacitancefromCPEdataisofteninadequate,leadingto erroneouspredictionofphysicalproperties. Inthepresentwork,twodifferentmathematicalformulasforestimatingeffective capacitancefromCPEparameters,takenfromtheliterature,areassociatedunambiguouslywitheithersurfaceornormaltime-constantdistributions.However,these equationswerenotdevelopedfromaphysicalmodelanddonotproperlyaccountfor characteristicfrequenciesoutsidethemeasuredfrequencyrange.Forabroadclassof systems,theseformulationsforcapacitanceareinsufcient,whichillustratestheneedto developmechanismstoaccountfortheCPE. CPEbehaviormaybeattributedtothedistributionofphysicalpropertiesinlms, inthedirectionnormaltotheelectrodesurface.Numericalsimulationswereusedto showthat,underassumptionthatthedielectricconstantisindependentofposition,a normalpower-lawdistributionoflocalresistivityisconsistentwiththeCPE.Ananalytic 20

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expression,basedonthepower-lawresistivitydistribution,wasfoundthatrelates CPEparameterstothephysicalpropertiesofalm.Thisexpressionyieldedphysical properties,suchaslmthicknessandresistivity,thatwereingoodagreementwith expectedorindependentlymeasuredvaluesforsuchdiversesystemsasaluminum oxides,oxidesonstainlesssteel,andhumanskin. Theagreementobtainedusingthepower-lawmodelcanbeexplainedbythe factthatitisbasedonformalsolutionfortheimpedanceassociatedwithaspecied resistivitydistribution,ratherthanusingformulationsforcapacitancethatdonottake anyphysicalmodelintoaccount.Thepower-lawmodelyieldsaCPEimpedance behaviorinanappropriatefrequencyrange,denedbytwocharacteristicfrequencies. Idealcapacitivebehaviorisseenabovetheuppercharacteristicfrequencyandbelow thelowercharacteristicfrequency.AsymmetricCPEresponseatbothhighandlow frequenciescanbeobtainedbyaddingaparallelresistivepathway. CPEbehaviormayalsobeattributedtothedistributionofphysicalpropertiesalong thesurfaceofanelectrode.Numericalsimulationswereusedtoshowthatapower-law distributionofOhmicresistancealongablockingsurfacewithuniformcapacitance yieldedanimpedanceresponsethatwasconsistentwiththeCPE.ThebroaddistributionnecessarysuggestedthatobservedCPEbehaviorcannotbeconsideredtoarise fromadistributionofOhmicresistancealone.Nevertheless,thedevelopedrelationship betweencapacitanceandCPEparametersforasurfacedistributionwasshowntobe differentthantherelationshipdevelopedforanormaldistributionindicatingthatthe physicaloriginoftheCPEneedstobeconsideredwhenassessingcapacitancefrom impedancespectra. AnalysisofsystemsexhibitingtheCPErequiresaccurateestimatesofmodelparameters.InsupportofthemechanisticdevelopmentoftheCPE,ageneralizedmethod wasdevelopedforidentifyingandminimizingnonlineardistortionsinimpedancespectra 21

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forincreasedcondenceinmodeldevelopmentandparameterestimation.Acharacteristictransitionfrequencywasdenedthatcanbeusedtotailorafrequency-dependent inputsignaltooptimizesignal-to-noiselevelswhilemaintainingalinearresponse.The Kramers-Kronigrelations,whichprovideanessentialtoolforassessingtheinternalconsistencyofimpedancedata,areunderstoodtobesensitivetofailuresofcausality,but insensitivetofailuresoflinearity.NumericalsimulationsshowedthattheKramers-Kronig relationsarenotsatisedformeasurementswhichincludethecharacteristictransition frequency.However,therelationsweresatisedformeasurementstakenbelowthe characteristicfrequency,evenforverynonlinearsystems. 22

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CHAPTER1 INTRODUCTION ElectrochemicalimpedancespectroscopyEISisanin-situ,non-invasivetechniquewidelyutilizedforcharacterizingelectrochemicalsystems.EIShasbeenusedto investigateabroadrangeofexperimentalsystemswithverydifferentelectrochemical propertiesandisusedtoadvancemanyareasofscienceandengineeringincluding productdevelopment,diagnostictesting,materialsanalysis,andmechanisticstudies.ThescopeofEISisbroad,includingoptics,wetanddrychemistry,solid-state applications,andbiochemicalprocesses. Whileitcanbeconsideredageneralizedtransferfunctionapproach,EISusually involvesameasuredcurrentresponsetoapotentialinput,wheretheimpedancespectra aregeneratedbychangingthefrequencyoftheinputsignal.Advancesinelectrical equipmentanddigitaltechnologyhaveallowedimpedancespectratobecollected quicklyandaccurately.Itistheinterpretationofresultsthatposesthechallengeand thefocusofabroadrangeofresearchintheeld.Ingeneral,ifimpedancetechniques arenotproperlyimplementedorassessmentofthedataareunsound,thenconclusions drawnfromtheanalysismaybeerroneous. EISisanappealingtechniqueforelectrochemicalstudiesbecauseitallowsfor theseparationofsystemcomponents,whichcannotbeachievedthroughsteadystatemeasurements.Forinstance,impedancespectrayieldsinformationonsolution resistance,charge-transferresistance,andsystemcapacitance.Thecharacteristicsof theimpedanceresponseprovidesinformationondiffusion,convection,kinetics,and reactionmechanisms.Importantparametersandphysicalproperties,suchasdiffusion coefcients,exchangecurrentdensities,anodicandcathodictransfercoefcients, permittivity,activesurfacearea,andlmandcoatingthicknessescanbeobtainedfrom impedanceanalysis. Thedeterminationofsystemcapacitanceisimportantformanyscienceand engineeringapplicationsbecauseitsvalueprovidesinformationregardingactive 23

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surfacearea,layerthickness,andmaterialpermittivity.Thearea-scaledcapacitanceof electrode/electrolyteinterfacesdoesnotvarysignicantly,and,therefore,theunscaled capacitancevalueextractedfromimpedancemeasurementscanbeusedtoestimate activesurfaceareausingtypicaldouble-layercapacitancevalues.Thedetermination ofreactiveareaiscriticalforchemicalsynthesisapplicationsandenergytechnologies suchasbatteriesandfuelcells. Fordielectricmaterialsthecapacitanceisusedtoobtainpermittivityandlayer thickness.Thedeterminationofcapacitanceisimportant,forinstance,forthecharacterizationofoxidelms.Oxidespassivatemetalsbyprovidingaresistiveboundaryto corrosionallowingthemtobeusedasbuildingmaterials.Inasimilarmanner,organic coatingsareoftenusedtopreventcorrosion.Thedielectricpropertiesofoxidesare usedinthedesignandfabricationofsemiconductorsandintegratedcircuits.Thegrowth oflmsoncatalyticsurfaces,suchastheelectrodesofbatteriesandfuelcells,canact tobothpromoteandinhibitmasstransferandthereforesignicantlyinuenceperformance.Thecharacterizationofhumanskinisimportantforthedesignofelectriceld drivendrugdeliverysystems.EISiswidelyusedforthethestudyofoxides,organic coatings,biologicalmembranes,andevenhumanskin. Analysisofimpedancespectrarequiresdevelopingmodelsthataccountforthe physicalprocessesofasystemsuchthatthedesiredinformationcanbeobtained.Itis insufcienttosimplytimpedancespectratoamathematicalmodelortoacollection ofpassivecircuitelements,assuchanapproachprovideslittleinsightintothephysical processesthatareoccurring.Impedancespectracannotbeanalyzedfromexamination ofrawdataalone.TheintegratedapproachprovidedbyOrazemandTribollet 7 isto proposeaphysicalmodeltoaccountfortheimpedanceresponseofanelectrochemical system.Conrmationofamodelrequiressupportingexperimentalevidence. Modelsinvokingconstant-phaseelementsCPEareoftenusedtotimpedance dataarisingfromabroadrangeofexperimentalsystems.TheCPEisexpressedin 24

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termsofmodelparameters and Q as Z CPE = 1 j! Q where istheangularfrequencyoftheinputsignaland j = p )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 .Equation1is representativeofablockingsystemwithaninnitelow-frequencyimpedance.When =1 thesystemisdescribedbyasingletime-constantandtheparameter Q hasunits ofcapacitance,otherwise Q hasunitsofs / cm 2 orFs )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 /cm 2 7 Generally, ranges between0.5and1.Forreactivesystems Z CPE ;R = R 1+ j! QR where R isanitelow-frequencyimpedance.TheCPEmaybeincludedinimpedance modelsincorporatingmass-transporteffectsand/orcomplicatedreactionmechanisms. Surprisingly,theCPE,whichrequiresonlytwoadjustableparameters,accurately tstheimpedanceresponsesofabroadrangeofexperimentalsystems.Thephysical originsoftheCPEarecontroversial.Generally,theCPEisconsideredtoarisefroma distributionofcapacitance.AhistoricalreviewoftheCPEisprovidedinChapter2.In spiteofsomeexperimentalandtheoreticalsuccess,theproposedphysicalmodelsfrom theliteratureyieldpseudo-CPEbehavior,inwhich and Q arefrequencydependent andareroughlyconstantonlyinasmallfrequencyrange.Incontrast,theCPEbehavior forexperimentalsystemsgenerallyappliesoveralargerangeoffrequencyinwhich and Q areindependentoffrequency. TheCPE,whichispurelyamathematicaldescription,mayaccuratelyrepresent impedancedata,butitgivesnoinsightintothephysicalprocessesthatyieldsucha response.Nevertheless,capacitanceisoftenextractedfromCPEdatausingexpressionsprovidedbyBrug etal. 3 orbyHsuandMansfeld 6 thatarederivedindependentof physicalmodels.Applicationoftheseexpressionstoexperimentalsystems,presented 25

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inChapter7,oftenleadstoassessmentofcapacitancethatdoesnotagreewithindependentmeasurements.Inmanycases,capacitanceisobtainedusingsingle-frequency measurements,andtherefore,thepresenceoftheCPEisnoteventakenintoconsideration.Singlefrequencyapproachescanleadtomisinterpretationofresults.Following theapproachtakenbyOrazemandTribollet, 7 theinterpretationofimpedancespectra, andthereforethedeterminationofsystemcapacitance,requiresthedevelopmentof physicalmodels. Themotivationforthepresentworkarisesfromthefactthat,ingeneral,thephysical originsofCPEbehaviorarenotwellunderstood.Withouttheaidofphysicalmodels thedeterminationofphysicalparametersfromimpedancespectraisambiguous. TheobjectiveistodevelopmechanismsthataccountfortheCPEandtoprovide relationshipsbetweenthemeasuredCPEparametersandthephysicalpropertiesofa system. ThedevelopmentofmechanisticmodelsrequiresproperimplementationofEIS. Modeldevelopmentisenhancedwhenexperimentaltechniquesareoptimizedand dataisveriedforconsistency.Althoughtherequirementoflinearityandtheerrorsthat resultwhenlinearityisviolatedarewellestablished,ageneralizedsystem-dependent procedureforoptimizingexperimentaltechniquesislacking.InChapters3and4,a generalizedmethodisdevelopedforidentifyingandminimizingnonlineardistortions inimpedancespectraforincreasedcondenceinmodeldevelopmentandparameter estimation. ThemathematicalcharacteristicsoftheCPEareprovidedinChapter5.InChapter 6,differentcapacitance-CPErelations,originallyderivedbyBrug etal. andHsuand Mansfeld,areassociatedunambiguouslywitheithersurfaceornormaltime-constant distributions.Theformulasforcapacitanceareappliedtodifferentexperimentalsystems inChapter7andthelimitationsarediscussed.ThedevelopmentofresistivitydistributionsinlmsthataccountfortheCPEispresentedinChapter8.Ananalyticexpression 26

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isdeveloped,basedontheresistivitydistributions,thatrelatesCPEparameterstothe physicalproperties.InChapter9,thisexpressionisappliedtosuchdiversesystemsas aluminumoxides,oxidesonstainlesssteel,andhumanskinyieldingphysicalproperties, suchaslmthicknessandresistivity,thatwereingoodagreementwithexpectedor independentlymeasuredvalues.SurfacedistributionsofOhmicresistancethatresult inCPEbehaviorarederivedinChapter10.Anoverviewoftherelationshipbetween capacitanceandCPEparametersisprovidedinChapter11.InChapter13,preliminary workforCPEbehaviorcausedbysurfacedistributionsofreactivityisprovided. 27

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CHAPTER2 LITERATUREREVIEW ThepresentchapterprovidestheoriginsoftheempiricalConstant-PhaseElement model,proposedphysicalexplanationsoftheCPE,andderivedcapacitance-CPE relationships.Ahistoricalperspectiveisalsoprovidedontheinuenceofnonlinearity onimpedancespectraandtheutilityoftheKramers-Kronigrelationsforvalidating impedancedata. 2.1TheConstant-PhaseElement TheCPEhasbeenconsideredtoarisefromeitheradistributionofpropertiesalong thesurfaceofanelectrodeorinthedirectionnormaltotheelectrode. 2.1.1SurfaceDistributions In1941,ColeandColeintroducedanempiricalformula,nowknownasaConstantPhaseElement,thataccountedforthedielectricresponseofabroadrangeofliquids. 8 Thedielectricresponsewascharacterizedasadepressedsemicircleinacomplex admittanceplaneplot.ColeandColenotedthat,althoughthedispersionandabsorptionofthedielectricsdiffered,thegeneralizedbehaviorcouldbeattributedtoasingle parameter seeequation1.ColeandColeattributedthecauseofthisbehavior toadistributionofrelaxationtimes,ortime-constants,andwereabletocalculatethe necessarydistributionsoftime-constantsfollowingthemethodsofFuossandKirkwood. 9 ColeandColewereunabletoprovidephysicalsignicancetothedistributionof time-constantsandconsideredthedistributionfunctionassimplyamathematicalmeans ofrepresentingtheexperimentalresults.Notingthatthesamecharacteristicformula couldaccountforotherwisedissimilardielectrics,ColeandColesuggestedthatamore fundamentalmechanismmustbeinvolved. Sincethe1940s,abroadrangeofresearchershaveinvestigatedthepossible physicalandgeometricoriginsofCPEbehavior.Intheliterature,theoriginofCPE behaviorhasbeenattributedtoporosity,surfaceroughness,fractalgeometry,nonuniformcurrentdistributions,andthepresenceofgrainboundaries.Theamountof 28

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workontheoriginsofCPEbehaviorisextensiveandathoroughreviewisnotintended. Rather,ageneraloutlinehighlightingsomeoftheresearchisintendedtoillustratethe ambiguityassociatedwiththeCPE. Brug etal. havedevelopedmathematicaldistributionsoftime-constantsthat resultinimpedanceresponsesthatcanbeexpressedintermsofCPEmodels.In theirwork,thetime-constantswereconsideredtobedistributedradiallyalongthe electrodesurfaceDandtheimpedancewasobtainedfromasumoftheadmittance oftheindividualelements. 3 Themathematicaldevelopmentofthedistributionfunctions usingthemethodsofFuossandKirkwoodwaspossibleonlywhentheOhmicand kineticresistanceswereheldconstantandcapacitiveelementswereallowedtobe distributed. 3,9 Thetime-constantdistributionsthatleadtoCPEbehaviorforsucha modelrequiredthatthecapacitancevaryovermanyordersofmagnitude.Therefore,the models,althoughmathematicallysound,werephysicallyunreasonableascapacitance isnotknowntohavesuchabroadrangeofvalues. Theimpedanceofporouselectrodeshasbeenanalyticallycalculatedbyde Levie. 10,11 Theimpedanceofasingleporewasderivedandtheoverallimpedanceof theporouselectrodewasobtainedbyaccountingforanensembleofindividualpores. Theimpedanceresponseoftheoverallsystemledtodistortionoftheimpedanceinthe high-frequencyregionsuchthatan parameteroflessthanunitywasobserved. 12,13 However,thecalculatedvalueof fortheporouselectrodemodelwasfrequencydependentandthereforenotcharacteristicoftrueCPEbehavior. CPEbehaviorhasbeenattributedtoheterogeneityofelectrodesurfaces.TheexperimentalworkofdeLevieshowedarelationshipbetweenelectrodesurfaceroughness andthephase-angleassociatedwithCPEbehavior. 14 Scheiderusedabranched-ladder networkofresistorsandcapacitorstoaccountforCPEbehaviorofroughorunevensurfaces. 15 Theladder-networkwasintendedtorelatemicroscopicheterogeneitieswiththe macroscopicresponse.LeMehauteandCrepyconnectedfractalgeometryofelectrode 29

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surfaceswithCPEbehavior. 16 NyikosandPajkossyshowedthattheCPEparameter couldbeexpressedasafunctionoffractaldimension,andinterpreted asameasure ofsurfaceirregularityregardlessoftheshapeandstructureoftheirregularities. 17 Huang etal. haveshownthatpseudo-CPEbehavior,where and Q arefrequencydependent, canarisefromgeometricaleffectsduetocurrentandpotentialdistributionsonelectrode surfaces. 18,19,2 Inspiteofsomeexperimentalandtheoreticalsuccess,thephysicalphenomena thatcauseCPEbehaviorremaincontroversial.KeddamandTakenouti 20 andWang 21 havequestionedthevalidityoftherelationshipbetweentheCPEparameter and fractaldimension.Bates etal. experimentallyshowednocorrelationbetween and fractaldimension. 22 ArgumentsthatthesourceofCPEbehaviorispurelyaninterfacial phenomenonorduetointerfacialandbulkpropertycouplingweresummarizedby Pajkossy. 23 Anothercomprehensivereviewoffractalsandroughelectrodesasthey pertaintoimpedancemeasurementswasprovidedbydeLevie. 24 2.1.2NormalDistributions Jorcin etal. 5 haveusedLocalElectrochemicalImpedanceSpectroscopyLEIS toattributeCPEbehaviorseenintheglobalmeasurementstoeithersurfaceornormal time-constantdistributions.Normaldistributionsoftime-constantscanbeexpected insystemssuchasoxidelms,organiccoatings,andhumanskin.Suchnormaltimeconstantdistributionsmaybecausedbydistributionsofresistivityand/ordielectric constant.Therangeofvaluesexpectedforadielectricconstant,however,shouldbe muchnarrowerthanthatexpectedforresistivity. YamamotoandYamamoto 25 haveusedarectangularprobabilityfunctiontomodel resistivitydistributions.TheYoungmodel,developedforniobiumoxide,assumesan exponentialdistributionofresistivitywithinamaterial. 26 YamamotoandYamamoto 27 andPoonandChoy 28 usedexponentialresistivityprolestomodeltheimpedanceof humanstratumcorneum.Bojinov etal. 29 andSchillerandStrunz 30 usedtheYoung 30

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modeltotelectrochemicalimpedancedataandpredictphysicalproperties,including lmthickness.Bojinov etal. summarizedthejusticationofanexponentialdecay ofresistivitybyusingpoint-defectconductiontheoryinpassivelms. 31 Schillerand Strunz 30 derivedanapproximaterelationshipbetweentheYoungmodelparameter, andtheCPEparameter, Anexponentialdecayofresistivityisagoodrstapproximationofaphysical modelthatresultsinanimpedanceresponsethatcanbeexpressedintermsofa CPE.However,themodelisinsufcientinthesensethatitresultsinpseudo-CPE behavior;specically,the and Q valuesthatareextractedfromtheEISresponseare functionsoffrequency.Inaddition,onlyalimitedrangeof valuesarepossiblefrom suchamodel,andthusabroadclassofsystemscannotbeattributedtoanexponential resistivitydecay.ThecharacteristicsoftheYoungmodelareprovidedinChapter8. 2.2DeterminationofCapacitancefromtheCPE InChapter1,theimportanceofdeterminingthecapacitanceofelectrochemical systemswasdiscussed.Asshownbyequations1and1,whenanelectrochemicalsystemisdescribedbyasingletime-constant =1 and Q hasunitsof capacitance.When < 1 ,therelationshipbetweentheimpedanceresponseandthe valueoftheinterfacialcapacitanceisambiguous.ItisclearthattheCPEparameter Q cannotrepresentthecapacitancewhen < 1 .Anumberofresearchershaveexplored therelationshipbetweenCPEparametersandtheinterfacialcapacitance.Bytreating asurfacedistributionoftimeconstants,Brug etal. 3 developedarelationshipbetween interfacialcapacitanceandCPEparametersforbothblockingandFaradaicsystems. HsuandMansfeld 6 proposedadifferentrelationshipforcapacitanceintermsoftheCPE parameters.TheseexpressionsarepresentedwithderivationinChapter6. Theformulasyielddifferentresultsfortheeffectivecapacitance.Usingnumerical simulationsfortheinuenceofgeometry-inducedcurrentdistributions,Huang etal. 18 31

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haveshownthatcurrentandpotentialdistributionsinduceahigh-frequencypseudoCPEbehaviorintheglobalimpedanceresponseofadiskelectrodewithaFaradaic reaction. 2 TheirworkdemonstratedthattheBrugformulaforeffectivecapacitance yieldedamoreaccurateestimatethandidtheHsuandMansfeldequation. BoththeBrugformulasandtheHsuandMansfeldformulahavebeenwidelyused toextracteffectivecapacitancevaluesfromCPEparameters.TheBrugformulashave beenusedtoextractcapacitancevaluesfromCPEparametersforstudiesondouble layers, 32 hydrogensorptioninmetals, 37,38 hydrogenevolution, 39 oxygenevolution, 45 porouselectrodes, 46 self-assembledmonolayers, 47,48 polymerlms, 49 andpassive lms. 50,51 Similarly,theHsuandMansfeldformulahasbeenusedtoextractcapacitance valuesfromCPEparametersforstudiesonpassivelms, 50 protectivecoatings, 53 andcorrosioninhibitors. 57 ForagivensetofCPEparameters,theBrugformulasand theHsuandMansfeldformulayielddifferentvalues;yet,insomecases,bothsetsof equationshavebeenappliedtosimilarsystems. 2.3ErrorsAssociatedwithNonlinearity ModeldevelopmentandparameterestimationrequiresthatEISisproperlyimplemented.Whileitcanbeconsideredageneralizedtransferfunctionapproach, electrochemicalimpedancespectroscopyusuallyinvolvesameasuredcurrentresponse toapotentialinput.Initscommonapplication,thetechniquereliesonuseofasmall inputsignalamplitudetoensurealinearresponsewhichcanbeinterpretedusingtheoriesoflineartransferfunctions.Ingeneral,thereisatrade-offbetweenimplementing asignalthatislargeenoughtoachieveanadequatesignal-to-noiseratio,whileatthe sametimeensuringthatthesignalisnottoolargesuchthatlinearityisviolated.Most experimentalistsemploya10mVinputsignalamplitude,butthereisreasontoexpect, giventhewiderangeofelectrochemicalpropertiesinvestigatedwiththistechnique,that thisamplitudemaynotbeoptimalformanyexperimentalsystems. 32

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Orazemandco-authorshaveinvestigatedtheerrorstructureofimpedancemeasurements,usingameasurementmodelapproachtoquantifybothstochasticandbias errorsinreplicatedspectra. 58 Minimizationofstochasticerrorsservestoimprove theregressionanalysisforinterpretationofspectra.Alargeinputamplitudegenerallyreducesthestochasticerrors,butanamplitudethatistoolargeresultsinerrors associatedwiththenonlinearresponse. Theselectionofappropriateinputamplitudeshasdrawninterestintheliterature. Useofaninputperturbationthatistoolargeyieldsanincorrectvalueforthechargetransferresistance.Darowickiinvestigatedtheeffectoftheinputamplitudeontheerror ofcharge-transferresistanceobtainedfromimpedancemeasurements. 62 Heshowed thattheimpedancespectrumofanonlinearelectricalsystemdependsonboththe frequencyandamplitudeoftheinputsignal.Hedemonstratedthatthepolarization resistanceuncorruptedbynonlineareffectscanbedeterminedbyextrapolatingtothe zerovalueoftheamplitudeoftheinputsignal.Diard etal. studiedthedependenceof impedancemeasurementerrorontheelectrodepotentialandthesinusoidalvoltage amplitudeforanernstianredoxsystem. 63 Heshowedthatforhisgivensystemthe impedancemeasurementerrorwasindependentoffrequencyinthelowfrequency range. Inaseparatework,Darowickishowedthat,forsystemswithanon-negligible Ohmicresistance,theinterfacialpotentialdiffersfromtheappliedpotentialsignal. 64 He derivedanexpressionfortheinterfacialpotentialusingaseriesexpansionapproachthat relatestheinterfacialpotentialtotheamplitudeoftheinputsignal,theinputfrequency, theelectrolyteresistance,thedoublelayercapacitance,andthekineticparameters. Darowickifoundthat,forallinputamplitudes,theeffectiveinterfacialpotentialchanges withfrequencyduetothefrequencydependenceofthechargingcurrent,havinga maximumamplitudeatlowfrequencyandtendingtowardzeroathighfrequency.As aresultofthiseffect,theinuenceofalargeinputamplitudechangeswithfrequency. 33

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Darowickiprovidedamethodfordeterminingthefrequencyforwhichimpedance measurementswillbelinearincharacter. 64 Darowicki'sobservationsweresupported bythemodelingworkofPopkirovandSchindlerwhodevelopedsyntheticdatafora charge-transferresistorobeyingButler-Volmerkineticsinparallelwithadoublelayer capacitance. 65 Theirresultsshowedthattheperturbationamplitudehadnoeffectonthe impedancevaluesinthehigh-frequencyrangewherethechargingcurrentdominates. Alternatively,inthelow-frequencyrange,adecreaseoftheimpedancevalueswas observedwithincreasinginputsignalamplitude. Therehasbeensignicantefforttodeterminethelinearimpedancevalueswhen nonlinearerrorsarenotnegligible.Diard etal. quantiedthedeviationofthemeasured polarizationresistanceduetononlinearityusingasuccessivederivativeapproach. 66 Diard etal. developedexpressionsfortheelectrochemicalresponseofatwo-step reactiontoasinusoidalperturbationthatresultsinnonlinearimpedance. 69 Heused numericalmethodstoshowthatdeviationfromthelinearizedsystemdependedon thekineticparameters,theelectrodepotential,theinputamplitude,andthefrequency. MiloccousedaTaylorseriesmethodtodeterminethelinearimpedanceresponsewhen theperturbationcausedanonlinearresponse. 70 Fromanexperimentalperspective,VanGheem etal. 71 andBlajiev etal. 72 used multisinebroadbandsignalstodetectnonlinearitiesinelectrochemicalsystems.These groupswereabletodistinguishmeasurementerrorscausedbystochasticnoiseand errorscausedbynonlineardistortions. Asmentionedpreviously,theeffectoffrequency,kineticparameters,andOhmic resistanceonlinearresponsesinEISiswelldocumented.However,kineticcoefcients areoftennotknownornoteasilyobtained,therefore,itwouldbebenecialtorelate theconditionsnecessaryforlinearitytoglobalsystemparametersthatcanbeobtained directlyfromEIS.Inaddition,theeffectoffrequencyisgenerallydiscussedqualitatively, specically,thatatlimitinglow-frequenciesthegreatestdegreeofnonlinearityis 34

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observedandatlimitinghigh-frequenciestheresponseislinear.Itwouldbebenecial toquantitativelyrelatethemeasurementfrequencytothedegreeofnonlinearityusing globalsystemparameterssuchthatanoptimizedsignal-to-noiseratiocanbeachieved duringEIS.Basedonnumericalsimulations,ageneralizedmethodforoptimizing signal-to-noiselevelswhilemaintainingalinearresponseisprovidedinChapter3. 2.4LinearityandtheKramers-KronigRelations WhendevelopingphysicalmodelsfromEISitiscriticalthattheexperimentaldata isreliableanduseful.TheKramers-Kronigrelations,derivedforsystemsthatcanbe assumedtobelinear,stable,andcausal,haveprovenusefulforconrmingtheselfconsistencyofelectrochemicalimpedancedata.Failureofimpedancedatatosatisfythe Kramers-Kronigrelationsathighfrequenciescangenerallybeattributedtoinstrumental artifacts,andlow-frequencydeviationscanbeattributedtononstationarybehavior. Instrumentalartifactsandnonstationarybehaviorrepresentviolationsofcausality. WhileassumptionoflinearityisessentialforthederivationoftheKramers-Kronig relations,theKramers-Kronigrelationsaregenerallyconsideredtobeinsensitiveto nonlinearbehaviorinelectrochemicalsystems. 73 Urquidi-Macdonald etal. 74 used experimentaldatatoshowthattheKramers-Kronigtransformsarehighlysensitivetothe conditionofcausalityandareinsensitivetotheconditionoflinearity.Theirevaluationof theeffectofthelinearityconditionontheKramers-Kronigtransformswasaccomplished byvaryingtheamplitudeoftheinputpotentialperturbationsignalduringsubsequent impedancescansforthecorrosionofironina1M H 2 SO 4 solution.Forthelargest amplitudes,themagnitudeoftheimpedancedecreasedsignicantlyfromthesmall amplitudecase,indicatingviolationofthelinearityconditionfortheirsystem.Thedata wereneverthelessshowntoremainconsistentwithKramers-Kronigtransformsforall inputamplitudestested.TheresultshowedthattheKramers-Kronigrelationswere insensitivetotheconditionoflinearitywhichwasclearlyviolatedforlargeperturbation inputs.Urquidi-Macdonald etal. attributedthecauseofthisinsensitivitytoanequal 35

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decreaseintherealandimaginarycomponentsoftheimpedancewhentheperturbation amplitudewasincreasedandtotheabilityofthefrequencyresponseanalyzertoreject harmonics. Theissueofnonlinearityinimpedancemeasurementsisimportant.Whilethe Kramers-Kronigrelationshavenotbeenfoundusefulforassessingtheappearanceof nonlinearity,experimentalmethods,suchasexaminationoflow-frequencyLissajous plots,canbeusedtoidentifynonlinearresponses. 7,75 Applicationofarandomphase multisineinputcanbeusedtoresolvenonlinearcontributionstotheerrorstructureof impedancemeasurements. 76 Insupportofmodeldevelopment,numericalsimulationswereusedtoidentify theconditionsunderwhichtheKramers-Kronigrelationsaresensitivetononlinear behavior.TheutilityoftheKramers-Kronigrelationsforidentifyingnonlineardistortions andanexplanationforthelackofsensitivityoftheKramers-Kronigrelationstononlinear behaviorreportedbyUrquidi-Macdonald etal. 74 isprovidedinChapter4. 36

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CHAPTER3 OPTIMIZATIONOFSIGNAL-TO-NOISERATIOUNDERALINEARRESPONSE Developingphysicalmodelsthataccountforimpedancespectrarstrequiresthat experimentaltechniquesareproperlyimplementedandfullyoptimized.Inthepresent chapter,numericalsimulationsofelectrochemicalsystemswereusedtoexplorethe inuenceoflarge-amplitudepotentialperturbationsonthemeasuredimpedanceresponse.Theamplitudeoftheinputpotentialperturbationusedforimpedancemeasurements,normallyxedatavalueof10mVforallsystems,shouldinsteadbeadjusted foreachexperimentalsystem.Guidelinesaredevelopedforselectionofappropriate perturbationamplitudes.Acharacteristictransitionfrequencyisdenedthatcanbe usedtotailorafrequency-dependentinputsignaltooptimizesignal-to-noiselevelswhile maintainingalinearresponse. 3.1CircuitModelsIncorporatingFaradaicReactions Thenonlinearresponseinelectrochemicalsystemstypicallyresultsfromthe potentialdependenceofFaradaicreactions.Forexample,bothTafelandButler-Volmer reactionkineticsdisplayanexponentialdependenceontheinterfacialpotential.The totalcurrentpassedthroughtheelectrodecontributestochargingtheinterfaceand totheFaradiacreaction.Thesecontributionsarepresentedinparallelinthecircuit presentedinFigure3-1a,wheretheuseofaboxfortheFaradaicreactionisintended toemphasizethecomplicatedandnonlinearpotentialdependence.Additionofan Ohmiccharacteroftheelectrolytecausestheinterfacialpotential V todifferfromthe appliedpotential U .ThiseffectisillustratedinFigure3-1b. Theappliedpotential U canbeexpressedasasinusoidalperturbationabouta steadyvalue U as U = U + U cos !t where U istheinputamplitude, istheinputangularfrequency,and t istime.Inthe absenceofanOhmicresistance,asshowninFigure3-1a,theappliedcellpotential U 37

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a b c Figure3-1.CircuitmodelswithFaradaicreaction:anon-OhmicFaradaicsystem;b OhmicFaradaicsystem;andcOhmicconstantcharge-transferresistance system. andtheinterfacialpotential V areequal.InthepresenceofanOhmicresistance R e the appliedcellpotentialisrelatedtotheinterfacialpotentialby U = V + i f + i C R e TheFaradaiccurrentdensitycanbeexpressedas i f = i 0 [exp b a V )]TJ/F24 11.9552 Tf 11.955 0 Td [(V 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F24 11.9552 Tf 9.299 0 Td [(b c V )]TJ/F24 11.9552 Tf 11.955 0 Td [(V 0 ] orequivalently, i f = K a exp b a V )]TJ/F24 11.9552 Tf 11.955 0 Td [(K c exp )]TJ/F24 11.9552 Tf 9.299 0 Td [(b c V where b a and b c aretheanodicandcathodiccoefcientswithunitsofinversepotential and K includesthetheexchangecurrent i 0 andtheequilibriumpotentialdifference V 0 as K a = i 0 exp )]TJ/F24 11.9552 Tf 9.298 0 Td [(b a V 0 and K c = i 0 exp b c V 0 .When b a and b c arerelatedthroughthe symmetryfactor,thegeneralformofEquation3forindependentreactionssimplies tothatofButler-Volmerkinetics.Thecapacitivecurrentisexpressedas i C = C dl dV dt 38

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where C dl isthedoublelayercapacitance.Thetotalcurrentpassingthroughthecellis thesumoftheFaradaicandcapacitivecontributions, i.e., i = i f + i C IntheabsenceofanOhmicresistance, i.e., asshowninFigure3-1a,equations3 1-3canyieldananalyticexpressionforcurrentdensityasafunctionofapplied potential U = V ; i = )]TJ/F24 11.9552 Tf 9.299 0 Td [(!C dl V sin !t + K a exp b a V + V cos !t )]TJ/F24 11.9552 Tf 11.955 0 Td [(K c exp )]TJ/F24 11.9552 Tf 9.299 0 Td [(b c V + V cos !t Thecurrentandpotentialtermscannotbeseparatedinthemoregeneralcasegivenin Figure3-1b,andanumericalmethodmustbeemployed. 3.2NumericalSolutionofNonlinearCircuitModels Anumericalmethodwasusedtoestimatethetime-dependentcurrentresponse toasinusoidalpotentialinputusingtheelectricalcircuitpresentedasFigure3-1b forwhichthecharge-transferresistance R t isanonlinearfunctionofpotential.The relationshipbetweencurrentandpotentialcanbeexpressedintheformofasingle differentialequation, dV dt C dl R e + V 1+ R e R t t = U + U cos !t inwhich R t t isafunctionofpotentialand,therefore,afunctionoftime.Equation 3canbesolvedanalyticallyforxed R t usingtheintegratingfactorapproach.The equivalentcircuitofsuchasystemisshowninFigure3-1c.Thesolutionofequation 3forxed R t canbeexpressedas V t = A cos !t + !C dl R e R t R t + R e sin !t + V where A = U R t + R e R t C dl R e 2 R t + R e 2 R t C dl R e 2 + 2 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 39

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and V isaconstantofintegration.AsimilarapproachwastakenbyXiaoandLalvani, whosolvedalinearizedformoftheTafelequationtodevelopexpressionsforpotential andcurrentinacorrosionsystem. 79 Thevalueofthecharge-transferresistanceatagivenpotential V t canbecalculatedfromtheslopeoftheinterfacialpolarizationcurve, i.e., R t t = K a b a exp b a V t + K c b c exp )]TJ/F24 11.9552 Tf 9.299 0 Td [(b c V t )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 Undertheassumptionthat,forshorttimeperiods, i.e., smallmovementsonthe polarizationcurve,thecharge-transferresistanceisconstant,aniterativeprocedure usingequations3and3wasusedtocalculatethedevelopmentof V and i as functionsoftime.Thisprocedureallowedforthecompletedeterminationofthesystem describedbyapotential-dependentcharge-transferresistance.Theanalyticequations derivedforaxedcharge-transferresistancecanbeusedtoapproximatethesolution toFigure3-1bforwhichthechargetransferresistancevarieswithinterfacialpotential. TherationaleforthisapproximationisdevelopedinSection3.3.4. TheimpedanceresponsewascalculateddirectlyforeachfrequencyusingFourier integralanalysis. 80 Thefundamentaloftherealandimaginarycomponentsofthe currentsignal,forexample,canbeexpressedas I r = 1 T Z T 0 I t cos !t dt and I j = )]TJ/F15 11.9552 Tf 11.811 8.088 Td [(1 T Z T 0 I t sin !t dt respectively,where I t isthecurrentsignal, istheinputfrequency,and T istheperiodofoscillation.Similarexpressionscanbefoundforrealandimaginarycomponents ofthepotentialsignal.Therealandimaginarycomponentsoftheimpedancecanbe foundfrom Z r =Re U r + jU j I r + jI j 40

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and Z j =Im U r + jU j I r + jI j respectively,where j representstheimaginarynumber.Theadvantageofthenumerical approachemployedherewasthatitcouldbeappliedtogeneralformsofnonlinear behavior,includingconsiderationofapotential-dependentcapacitance.Detailsof algorithmusedforthenumericalmethodisprovidedinAppendixA. 3.3SimulationResults Theobjectiveofthispresentationistomaketheanalysisofsystemnonlinearity usefultotheexperimentalist.Tothatend,guidelinesareprovidedtoassessappropriate perturbationamplitudesasfunctionsofkineticandOhmicparameters,andexperimentalmethodsarediscussedforassessingtheconditionoflinearity.Thefrequency dependenceoftheinterfacialpotentialcanbeexploitedtotailorinputsignals. 3.3.1ErrorsinAssessmentofCharge-TransferResistance Inthelimitthattheperturbationamplitudetendstowardzero,thepolarization resistancecanbeexpressedas R p; 0 =lim U 0 @U @i f c i ; k where U isthecellpotential, U istheamplitudeoftheinputcellpotentialsignal, i f istheFaradaiccurrentdensity, c i istheconcentrationofspecies i evaluatedatthe electrodesurface,and k isthefractionalsurfacecoverageofadsorbedspecies k Equation3canbeexpressedintermsofaneffectivecharge-transferresistanceas R t ; 0 =lim U 0 @V @i f c i ; k where V istheinterfacialpotential.Forthekineticsdescribedintheprevioussections, thelinearvalueofthecharge-transferresistanceisgivenas R t ; 0 = K a b a exp b a V + K c b c exp )]TJ/F24 11.9552 Tf 9.298 0 Td [(b c V )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 41

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Figure3-2.Calculatedimpedanceresponsewithappliedperturbationamplitudeasa parameter.Thesystemparameterswere R e =1 cm 2 K a = K c =1 mA/cm 2 b a = b c =19 V )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 C dl =20 F/cm 2 ,and V =0 V,giving risetoalinearcharge-transferresistance R t ; 0 =26 cm 2 where V representsthepotentialatwhichtheimpedancemeasurementismade. ThecalculatedimpedanceresponseisgiveninFigure3-2withappliedperturbationamplitudeasaparameter.Thesystemparameterswere R e =1 cm 2 K a = K c =1 mA/cm 2 b a = b c =19 V )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 C dl =20 F/cm 2 ,and V =0 V,givingriseto alinearcharge-transferresistance R t ; 0 =26 cm 2 .TheresultspresentedinFigure3-2 areconsistentwiththeobservationofDarowickithatthemeasuredcharge-transferresistancedecreaseswithincreasedamplitudeoftheperturbationsignal. 62 Assuggested byequation3,thedecreaseinthemeasuredcharge-transferresistancewith increasedamplitudeisnotageneralresultanddependsonthepolarizationbehavior. 66 3.3.2OptimalPerturbationAmplitude Aguidelineforselectionoftheperturbationamplitudeneededtomaintainlinearity underpotentiostaticregulationcanbeobtainedbyusingaseriesexpansionforthe currentdensity.Similarseries-expansionapproachesthatexpressdeviationsfrom linearityinelectrochemicalsystemshavebeenprovidedbyDiard etal. ,Kooyman etal. ,andGabrielli etal. 66,67,69,81,82 ForasystemthatfollowsaFaradaicreaction,thecurrent 42

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densityresponsetoaninterfacialpotentialperturbation V t = V + V cos !t isgivenby i f t = K a exp b a V )]TJ/F24 11.9552 Tf 11.955 0 Td [(K c exp )]TJ/F24 11.9552 Tf 9.299 0 Td [(b c V Thus, i f t = K a exp )]TJ/F24 11.9552 Tf 5.479 -9.684 Td [(b a V + V cos !t )]TJ/F24 11.9552 Tf 11.955 0 Td [(K c exp )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [()]TJ/F24 11.9552 Tf 9.298 0 Td [(b c V + V cos !t or i f t = K a exp b a V cos !t )]TJETq1 0 0 1 328.369 496.237 cm[]0 d 0 J 0.478 w 0 0 m 10.818 0 l SQBT/F24 11.9552 Tf 328.369 486.394 Td [(K c exp )]TJ/F24 11.9552 Tf 9.298 0 Td [(b c V cos !t where K a = K a exp b a V and K c = K c exp )]TJ/F24 11.9552 Tf 9.299 0 Td [(b c V ATaylorseriesexpansionyields i f t = K a + b a V cos !t + b 2 a V 2 cos 2 !t 2! + b 3 a V 3 cos 3 !t 3! + ::::: + b n a V n cos n !t n + ::: )]TJETq1 0 0 1 207.688 264.702 cm[]0 d 0 J 0.478 w 0 0 m 10.818 0 l SQBT/F24 11.9552 Tf 207.688 254.859 Td [(K c )]TJ/F24 11.9552 Tf 11.955 0 Td [(b c V cos !t + b 2 c V 2 cos 2 !t 2! )]TJ/F24 11.9552 Tf 13.151 8.088 Td [(b 3 c V 3 cos 3 !t 3! + ::: + b n c V n cos n !t n + ::: Themeanvalueofthecurrent i f t is,for T equaltoanintegernumberofcycles, i f = 1 T T Z 0 i f t dt Bytakingintoaccounttheformula Z cos n xdx = 1 n cos n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 x sin x + n )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 n Z cos n )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 xdx 43

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andobservingthat sin T =0 T Z 0 cos n xdx = n )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 n T Z 0 cos n )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 xdx If n isanevennumber, T Z 0 cos n xdx = n )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 n n )]TJ/F15 11.9552 Tf 11.956 0 Td [(3 n )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 ::: 1 2 T andif n isanoddnumber,thevalueoftheintegralisequaltozero.Thus,themean valueof i f t is i f = K a 1+ 1 X n =1 b 2 n a V 2 n n n 2 )]TJETq1 0 0 1 325.937 512.572 cm[]0 d 0 J 0.478 w 0 0 m 10.818 0 l SQBT/F24 11.9552 Tf 325.937 502.729 Td [(K c 1+ 1 X n =1 b 2 n c V 2 n n n 2 Evaluationoftheharmonicsofthenonlinearcurrentresponsecanbeachievedby introductionofthetrigonometricexpressions cos2 x =2cos 2 x )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 and cos3 x =4cos 3 x )]TJ/F15 11.9552 Tf 11.955 0 Td [(3cos x ByconsideringonlythethreersttermsoftheTaylorseries, i f t becomes i f t = K a + b 2 a V 2 4 + b a V + 3 b 3 a V 3 24 cos !t + b 2 a V 2 4 cos !t + b 3 a V 3 24 cos !t )]TJETq1 0 0 1 206.768 233.43 cm[]0 d 0 J 0.478 w 0 0 m 10.818 0 l SQBT/F24 11.9552 Tf 206.768 223.587 Td [(K c + b 2 c V 2 4 )]TJ/F15 11.9552 Tf 11.955 0 Td [( b c V + 3 b 3 c V 3 24 cos !t + b 2 c V 2 4 cos !t )]TJ/F24 11.9552 Tf 13.15 8.088 Td [(b 3 c V 3 24 cos !t ThelimitationtotherstthreetermsoftheTaylorseriesgivesforthemeanvalueonly thersttermoftheseriesseeequation3. Equation3canbewrittenas i f t = i f + i f; 1 cos !t + i f; 2 cos !t + i f; 3 cos !t ::: 44

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wherethedccurrentisgivenby i f = K a 1+ b 2 a V 2 4 )]TJETq1 0 0 1 326.069 682.025 cm[]0 d 0 J 0.478 w 0 0 m 10.818 0 l SQBT/F24 11.9552 Tf 326.069 672.182 Td [(K c 1+ b 2 c V 2 4 andtherstharmonicorfundamentalisgivenby i f; 1 = K a b a V + 3 b 3 a V 3 24 + K c b c V + 3 b 3 c V 3 24 For V smallerthan 0 : 2 q K a )]TJETq1 0 0 1 238.728 573.287 cm[]0 d 0 J 0.359 w 0 0 m 7.705 0 l SQBT/F25 7.9701 Tf 238.728 566.585 Td [(K c K a b 2 a )]TJETq1 0 0 1 238.919 563.611 cm[]0 d 0 J 0.359 w 0 0 m 7.705 0 l SQBT/F25 7.9701 Tf 238.919 556.91 Td [(K c b 2 c ,thevariationofthedccurrentissmallerthan1 percent.For V smallerthan 0 : 2 q K a b a + K c b c K a b 3 a + K c b 3 c ,thevariationofthefundamentalissmaller than0.5percent. Applicationofalarge-amplitudepotentialperturbationtoanonlinearsystem resultsinharmonicsthatappearatfrequenciescorrespondingtomultiplesofthe fundamentalorappliedfrequency.Observationthatapplicationofalarge-amplitude potentialperturbationtoanonlinearsystemchangesboththesteady-statecurrent densityandthefundamentalcurrentresponse.Theimplicationofthisresultisthatthe impedanceresponsewillalsobedistortedbyapplicationofalarge-amplitudepotential perturbation. InthepresenceofasignicantOhmicresistance,theguidelineforthelowfrequencyperturbationamplitudeis U =0 : 2 s K a b a + K c b c K a b 3 a + K c b 3 c 1+ R e R t ; obs where R t ; obs istheeffectivecharge-transferresistancemeasuredatthegivenperturbationamplitude.Thus,alargerperturbationamplitudeshouldbeappliedforsystems where + R e =R t ; obs ismuchlargerthanunity.Therationaleforequation3is developedinSection3.3.4. Thepercenterrorinthelow-frequencyimpedanceasymptoteassociatedwithuse ofalarge-amplitudepotentialperturbationisgiveninFigure3-3undertheassumption ofTafelkineticswith b V asaparameter.Atavalueof b V =0 : 2 ,theerrorinthe 45

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Figure3-3.Theerrorinthelow-frequencyimpedanceasymptoteassociatedwithuseof alargeamplitudepotentialperturbation. low-frequencyimpedanceasymptoteis0.5percent.Thecorrespondingperturbation amplitudeis10.4mVforaTafelslopeof120mV/decade,and5.2mVforaTafelslopeof 60mV/decade. 3.3.3ExperimentalAssessmentofLinearity AsindicatedbyUrquidi-Macdonald etal. 74 theKramers-Kronigrelationsdo notprovideausefultoolforidentifyingerrorsassociatedwithanonlinearresponse toalargeperturbationamplitude.ThegeneralutilityofKramers-Kronigrelationsfor identifyingerrorsassociatedwithnonlinearityisprovidedinChapter4.Sequential impedancemeasurementsconductedwithdifferentperturbationamplitudescanbe usedtondtheoptimalinputperturbation,butthisprocessistimeconsuming. Amorerapidassessmentofanonlinearsystemresponsecanbeobtainedby observingdistortionsinLissajousplotsatlowfrequency.Lissajousplotsarepresented inFigure3-4withperturbationamplitudeandfrequencyasparameters.Thesystem parameterswere R e =0 cm 2 K a = K c =1 mA/cm 2 b a = b c =19 V )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 C dl =20 F/cm 2 and V =0 V,givingrisetoalinearcharge-transferresistance R t ; 0 =26 cm 2 .Atthe 46

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Figure3-4.Lissajousplotswithperturbationamplitudeandfrequencyasparameters. Thesystemparameterswere R e =0 cm 2 K a = K c =1 mA/cm 2 b a = b c =19 V )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 C dl =20 F/cm 2 ,and V =0 V,givingrisetoalinear charge-transferresistance R t ; 0 =26 cm 2 lowfrequencyof0.016Hz,astraightlineisobservedforaperturbationamplitudeof 10mV;whereas,asigmoidalshapeisevidentforaperturbationamplitudeof100mV. Thesigmoidalshapeisseenbecausethecalculationswereperformedat V =0 V. Adeviationfromastraightlinewillbeseenforlargeamplitudesatlargerorsmaller appliedpotentials,buttheshapewillbealtered.Atthelargerfrequencyof160Hz, thedifferencesbetweenthesmallerandlargerperturbationamplitudesbecomesless apparent,andthetwocurvessuperimposeasaperfectcircleatlargefrequenciesdue tothedominationofthecapacitivecurrent.Similarresultsareseenforthecasewhere R e 6 =0 ,withtheexceptionthattheLissajousplotappearsasastraightlineatbothlow andhighfrequencies.Theinuenceofnonlinearitiesisseenatlowfrequency. 3.3.4FrequencyDependenceoftheInterfacialPotential IntheabsenceofOhmicresistanceorwhenalinearapproximationissufcient, theinterfacialpotentialisasinusoidalquantityand V representstheamplitudeof 47

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Figure3-5.Maximumvariationoftheinterfacialpotentialsignalasafunctionof frequencyforparameters U =100 mV, R e =1 cm 2 C dl =20 F/cm 2 ,and R t ; 0 =26 cm 2 theinterfacialpotential.Forlargeperturbationstheinterfacialpotentialsignalcontains nonlineardistortions,thus,inthefollowingdiscussion V max representsthemaximum variationoftheinterfacialpotentialsignal.Thecalculated V max ispresentedinFigure 3-5asafunctionoffrequencyforasystemwithparameters U =100 mV, R e = 1 cm 2 C dl =20 F/cm 2 ,and R t ; 0 =26 cm 2 .Athighfrequencies V max isdampedand tendstowardzero. Equation3,althoughderivedforaconstantcharge-transferresistance,can beusedtoapproximatetheinterfacialpotentialofanonlinearsystemifthechargetransferobservedatlowfrequency, R t ; obs ,isusedintheequation.AsshowninFigure 3-6, V max resultingfromthenumericcalculationiscomparedto V calculatedfrom equation3.Theagreementshownbetweenthenumericalsolutionandthesolution obtainedusingequation3conrmsthatequation3isusefulforapproximating thebehavioroftheinterfacialpotential,eventhoughitisderivedfromaconstantchargetransferresistance.Itshouldbenotedthatthesinusoidaltime-domainapproximationwill notcontainthenonlineardistortionsandwillhavemaximumerroratlowfrequency. 48

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Figure3-6.Maximumvariationoftheinterfacialpotentialsignalasafunctionof frequencyforparameters U =100 mV, R e =1 cm 2 C dl =20 F/cm 2 R t ; 0 =26 cm 2 .Thesolidcurveis V max resultingfromthenumeric simulation.Thedashedcurveis V predictedfromequation3using R t ; obs =19 cm 2 ,whichdecreasesfromthelinearvalue, R t ; 0 =26 cm 2 duetothelargeinputperturbation. Inspectionofthelow-frequencyandhigh-frequencylimitsofequation3provides insightintotheconditionsatwhichideallinearityareapproached, i.e., lim 0 V = UR t ; obs R t ; obs + R e and lim !1 V = U !C dl R e respectively,where R t ; obs istheobservedcharge-transferresistanceatthegiven perturbationamplitude.Althoughequations3and3arederivedforthelinear system,theresultsshowninFigure3-6conrmsthattheseequationsareusefulfor approximating V max ,aslongasthecharge-transferresistance R t isreplacedbythe charge-transferresistanceinuencedbyanonlinearresponse R t ; obs .Asshownin equation3, V max decreasesinthelow-frequencyrangewithincreasingOhmic resistance.Asshowninequation3, V max decreasesinthehigh-frequencyrange 49

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a b Figure3-7.Calculatedresultsforparameters U =100 mV, C dl =20 F/cm 2 ,and R t ; 0 =26 cm 2 withOhmicresistanceasaparameter.aMaximumvariation oftheinterfacialpotentialsignalasafunctionoffrequency;andbthe correspondingLissajousplotsatafrequencyof0.016Hz. withincreasingfrequency.BoththelimitsofhighOhmicresistanceandhighfrequency approachtheconditionofideallinearity. Thefrequencydependenceof V max andthecorrespondingLissajousplotsare showninFigures3-7aand3-7b,respectively.AsshownintheFigures3-7aand 3-7b,alinearresponseisobtainedfora100mVinputamplitudewhentheOhmic resistanceislarge;whereas,anonlinearresponseisseenforthesameperturbation amplitudewhentheOhmicresistanceissmall.Thisresultisconsistentwithequation 3.Thelinearityofthesystemresponseisgovernedbythevalueof V max Acharacteristicfrequencyforthetransitionfromthelow-frequencybehaviortothe high-frequencybehaviorwasobtainedas f t = 1 2 R t ; obs C dl + 1 2 R e C dl = 1 2 R t ; obs C dl 1+ R t ; obs R e where f t istheinectionpointof V max versusfrequency,asshowninFigure3-8.This frequencymarksthetransitionfromlow-frequencynonlinearbehaviortohigh-frequency linearbehavior. 50

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Figure3-8.Inectionpointof V max islocatedatthetransitionalfrequencydenedby equation3 U =100 mV, R e =1 cm 2 C dl =20 F/cm 2 ,and R t ; 0 =26 cm 2 Thelow-frequencylimitgivenbyequation3isequivalenttothatderivedby Darowickisee e.g., equationinDarowicki 83 .Theadvantageofusingequation3 9isthatitapproximatestheinterfacialpotentialacrossallfrequencieswhileprovidinga muchsimplerexpressionthanthosederivedbytheseriesexpansionapproachusedby Darowicki. Thecharge-transferresistancewascalculatedusingequation3foreach time-dependentvalueof V generatedduringthedevelopmentofsyntheticdata.Ateach frequency,thecharge-transferresistancewasaveragedoveracompletesinusoidal cycleyieldingtheeffectivecharge-transferresistance,whichatlowfrequencyisapproximatelytheobservedcharge-transferresistance R t ; obs .Theconsequenceofthechange ininterfacialpotentialwithfrequencyisillustratedinFigures3-9aand3-9bforparameters R e =1 cm 2 C dl =20 F/cm 2 ,and R t ; 0 =26 cm 2 withappliedperturbation amplitudeasaparameter.Atlowfrequencies,theeffectivecharge-transferresistance decreaseswithincreasedinputamplitudeasexpected.Athigherfrequencies,however, theeffectivecharge-transferresistanceapproachestheexpectedvalue.Asdescribedby 51

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a b Figure3-9.Calculatedresultsforparameters R e =1 cm 2 C dl =20 F/cm 2 ,and R t ; 0 =26 cm 2 withappliedperturbationamplitudeasaparameter:a Maximumvariationoftheinterfacialpotentialsignalasafunctionof frequency;andbTheeffectivecharge-transferresistanceasafunctionof frequency. equation3, V max changesvalueatthetransitionfrequency.Correspondingly,the effectivecharge-transferresistancechangesvalueatthistransitionalfrequency.Forthe 100mVperturbationthevariationinthecharge-transferresistanceissignicant.Forthe 10mVperturbationthevariationisnegligible.InthepresenceofanOhmicresistance V max isdampedinthelimitofhighfrequencyandthevaluesforthecharge-transfer resistancewillbesuperimposed. Theeffectivecharge-transferresistanceisgiveninFigure3-10aasafunctionof frequencyfordifferentvaluesofOhmicresistanceandinputamplitudes.Thevalidity ofequation3isconrmedbythesuperpositionofthecurvespresentedinFigure 3-10bwherethenormalizedeffectivecharge-transferresistanceispresentedas functionsofnormalizedfrequency. 3.3.5OptimizationoftheInputSignal TheresultspresentedinFigure3-10bsuggestthatanoptimizedprotocolcanbe establishedforsystemswithOhmicresistance.Asmallerperturbationamplitudecanbe 52

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a b Figure3-10.Effectivecharge-transferresistanceasafunctionoffrequency:athe effectivecharge-transferresistancefordifferentOhmicresistancesand inputamplitudes;andbthedimensionlessformoftheeffective charge-transferresistanceversusdimensionlessfrequency. employedatfrequenciesbelowthetransitionfrequencydenedbyequation340,and alargeramplitudecanbeemployedatfrequenciesabovethetransitionfrequency. AsshowninFigure3-10a,atmoderatetolargevaluesofOhmicresistance thetransitionfrequencydenedbyequation3iswellwithintheexperimentally assessablerange.Largeamplitudeinputscanbeemployedatfrequenciesabove thetransitionfrequencyduetothedampeningoftheinterfacialpotential.Forlarge valuesofOhmicresistancethesecondterminequation3becomessignicantand inuencesselectionoftheappropriateinputpotentialamplitude. Toillustratetheconcept,anelectrochemicalsystemwasmodeledforwhichthe electrolyticresistancewastwicethevalueofthecharge-transferresistance.Aconstant baselinenoiseof20percentofthelow-frequencycurrentsignalwasaddedtothe currentsignal.Theresultingimpedanceresponsetothe10mVinputperturbation employedincommonpracticeispresentedinFigure3-11a.Substantialscatteris observedatallfrequencies. 53

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a b Figure3-11.Systemwith R e =2 R t andbaselinenoisethatisconstantat20percentof lowfrequencycurrentsignal.a U =10 mVforall .b U =30 mVfor !< 10 t and U =300 mVfor !> 10 t Theinputsignalcanbemodiedintwoways.Inthelow-frequencylimitequation 3,when R e =2 R t ,theexperimentalistcanusethreetimestheinputamplitude signalandstillachieveanadequatelinearresponse.Inthehigh-frequencylimitequation3, V max isdampedtozeroand,accordingly,amuchhigherinputamplitude signalcanbeused.TheimpedanceresponsegiveninFigure3-11bwasobtained whena30mVvoltageperturbationwasintroducedintothesystemforfrequenciesless thantentimesthetransitionalfrequencyanda300mVperturbationwasintroducedfor frequenciesgreaterthan10timesthetransitionalfrequency.Thedashedlineshows theimpedanceresponsethatwouldhaveresultedifthe300mVperturbationamplitude wasemployedforallfrequencies.Thescatterwassignicantlyreducedusingtheinput signalemployedforFigure3-11b.Thevariable-amplitudemethodyieldsmoreaccurate resultsandprovidesahighercondencefortheextractionofsystemparameters. 3.3.6Potential-DependentCapacitance Aconstantdoublelayercapacitancewasusedforthepurposesofthiswork.In general,thecapacitanceisafunctionofpotential.InthepresenceofasignicantOhmic resistance,thedampeningoftheinterfacialpotentialabovethetransitionfrequency denedbyequation3allowsforadequatelinearizationofcapacitancewhenthe 54

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capacitivecurrentdominates.ForsufcientlylowOhmicresistance,theinterfacial potentialwillnotbedampedandextractionofcapacitancevaluesmaybecompromised duetononlineareffects. 3.4Conclusions Theamplitudeoftheinputpotentialperturbationusedforimpedancemeasurements,normallyxedatavalueof10mVforallsystems,shouldinsteadbeadjustedfor eachexperimentalsystem.IfsystemparameterssuchasTafelslope,chargetransfer resistance,andOhmicresistanceareknown,equation3providesausefulguide forselectionofperturbationamplitudeatlowfrequencies.Thetransitionfrequency denedbyequation3canbeusedtotailorafrequency-dependentinputsignal. Whentheseparametersareunknown,distortionsoflow-frequencyLissajousplotsare associatedwithperturbationamplitudesthataretoolargetoensurealinearresponse. AssessmentofdatathatmaybeinuencedbynonlineareffectsisdiscussedinChapter 4. 55

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CHAPTER4 THESENSITIVITYOFTHEKRAMERS-KRONIGRELATIONSTONONLINEAR RESPONSES Priortointerpretingimpedancedataformodeldevelopmentitiscriticalthatthe obtaineddataisrepresentativeofthesystemthatwasintendedforinvestigation.Due tothetransientandnonlinearnatureofelectrochemicalsystemsimpedancedatamust beveriedforconsistency.Asystemthatischangingwithinthetimeframeofanexperiment,forexampleduetothegrowthofasurfacelm,willyieldinconsistentresults.For suchasystem,interpretationoftheresultswillbemisleadingandconclusionsdrawn fromthedatamaybeerroneous. Inadditiontotransientbehavior,theinherentnonlinearnatureofelectrochemical systemsmayalsoleadtoerroneousevaluationofimpedancespectra.Thecurrentand potentialsignalsobtainedfromanEISexperimentareinterpretedintermsoflinear transfer-functiontheory.Therefore,aresponsedistortedduetotheinherentnonlinear natureofasystemresultsinanerrorinducedimpedancespectrawhenlineartransferfunctiontheoryisapplied.InChapter3,theoptimizationofexperimentaldesignfor maximizingsignal-to-noiseratiowhilemaintainingalinearresponsewasdiscussed. However,impedancedatastilldemandsvericationforconformitywiththelinearity requirementpriortomodeldevelopment.Inthepresentchapter,anevaluationofthe toolsavailabletotheexperimentalistforassessingthelinearityrequirementisprovided. TheKramers-Kronigrelations,whichapplystrictlyforsystemsthatarelinear, stable,andcausal,provideanessentialtoolforassessingtheinternalconsistencyof impedancedata.TheKramers-Kronigrelationsareunderstoodtobesensitivetofailures ofcausality,butinsensitivetofailuresoflinearity.Numericalsimulationswereperformed toexploretheconditionsunderwhichtheKramers-Kronigrelationsaresensitiveto nonlinearbehaviorofelectrochemicalsystems.Thecharacteristictransitionfrequency, identiedinSection3.3.4asausefulguidefortailoringafrequency-dependentinput signal,alsohasutilityfordeterminingthesensitivityoftheKramers-Kronigrelations 56

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tononlinearbehavior.BelowthetransitionfrequencytheKramers-Kronigrelations aresatised,evenforverynonlinearsystems.Thisresultisobservedforsystems withasmallOhmicresistance.TheKramers-Kronigrelationsarenotsatisedfor measurementswhichincludethetransitionfrequency.ForsystemswithalargeOhmic resistance,theKramers-Kronigrelationsmayprovideabettertoolforassessingthe presenceofnonlinearbehaviorascomparedtoanalysisoflow-frequencyLissajous plots. 4.1ApplicationoftheKramers-KronigRelations Thedevelopmentoftheimpedanceresponsesfromthenonlinearsystemsshown inFigures3-1aand3-1bisdescribedinSection3.1.Thesimulatedimpedance datawastestedforcompliancewiththeKramers-Kronigrelationsusingmeasurement modelanalysis.TheprocedurefordeterminationofKramers-Kronigconsistency recommendedbyAgarwal etal. istottheimaginarycomponentofimpedancedatato ameasurementmodelofsequentialVoigtelementsandthenpredicttherealcomponent ofimpedancefromtheextractedparameters. 58,84 Inthepresentwork,thefrequencydependentcharge-transferresistancethatresultsfromtheinuenceofnonlinearity preventedaccurateregressionofthedatatotheimaginary-onlycomponent.Asaresult ofthislimitation,theanalysisinSection4.2isbasedonbest-tcomplexregressionof simulatedimpedancedatatoameasurementmodel.SequentialVoigtelementsare addedtothemeasurementmodeluntiltheadditionofanelementdoesnotresultinan improvementofthetwithina95percentcondence.Sincethemeasurementmodel isinherentlyconsistentwiththeKramers-Kronigtransforms,datathatfallwithinthe condenceintervalofaregressedmodelhavetransformedsuccessfully.Nonconformity withthemeasurementmodelindicatesnoncomplianceand,therefore,violationof linearity. Inadditiontothemeasurementmodelanalysis,thesimulatedimpedancedatawas testedforcompliancewiththeKramers-Kronigtransformationsdirectly.Theformofthe 57

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Table4-1.SimulationresultsusedtoexploretheroleoftheKramers-Kronigrelationsfor nonlinearsystemswithparameters: U =100 mV, C dl =20 F/cm 2 K a = K c =1 mA/cm 2 b a = b c =19 V )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ,and V =0 V. R e cm 2 R t ; 0 cm 2 U R t ; obs cm 2 R t ; obs / R t ; 0 R e / R t ; obs 026.39.5017.30.6580 .0126.39.5017.30.658 5 : 8 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(4 126.39.1518.00.684 5 : 6 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 10026.31.9825.80.9813.9 Kramers-Kronigintegralsusedisgivenby Z r = Z r; 1 )]TJ/F15 11.9552 Tf 13.759 8.088 Td [(2 Z 1 0 xZ j x )]TJ/F24 11.9552 Tf 11.955 0 Td [(!Z j x 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(! 2 dx where Z j isananalyticfunctionoftheimaginarycomponentofimpedanceand Z r; 1 isanadjustableparameterrepresentingthevalueoftheOhmicresistance.Theutility ofequation4isthattherealcomponentofimpedance Z r canbepredictedfrom ananalyticalfunctionoftheimaginarycomponentiftheconditionsoflinearity,stability, andcausalityarenotviolated.Theintegralexpressedinequation4wasevaluated byinsertingtheimaginarycomponent Z j x ofthesimulatednonlinearimpedancedata intotheintegrandandthenperforminganumericalintegrationateachfrequency.This allowedforthepredictionoftherealcomponent Z r .Thetestfordatacompliancewas achievedbycomparingthepredictedvaluefromequation4totherealcomponentof thesimulateddata. 4.2SimulationResults ThesimulationresultsusedtoexploretheroleoftheKramers-Kronigrelations fornonlinearsystemsaresummarizedinTable4-1.Foreachsimulation,thesystem parametersgaverisetoalinearcharge-transferresistance R t ; 0 =26 : 3 cm 2 .TheOhmic resistance R e wasvariedfrom 0 to 100cm 2 .Equation3canbeusedtodenea scaledpotentialperturbationas U = U 0 : 2 q K a b a + K c b c K a b 3 a + K c b 3 c + R e =R t ; obs 58

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whereavalue U =1 yieldsanalmostlinearresponse,resultinginanerrorof lessthan0.5percentinthemeasuredchargetransferresistance,and R t ; obs isthe observedcharge-transferresistancemeasuredatlowfrequency.Thevaluesofthe scaledpotentialperturbationgiveninTable4-1reecttheinuenceofOhmicresistance ontheinterfacialpotential V max resultingfromanappliedpotential U .Asdiscussed extensivelyintheliterature, 75,65,62,64,66 thelargepotentialperturbationcausesan errorintheobservedcharge-transferresistance.Themagnitudeoftheeffectcanbe assessedbyusingthedimensionlessratio R t ; obs / R t ; 0 .Themagnitudeoftheinduced errorsdependsontheOhmicresistance, 75 i.e., theratio R t ; obs / R t ; 0 approachesunityas R e / R t ; obs increases.Theerrorintheimpedanceresponsecausedbyalargeinputsignal isshowninFigure3-2forthesimulationswith R e =1 cm 2 SimulateddatageneratedfromthesystemspresentedinTable4-1wereanalyzed forconsistencywiththeKramers-Kronigrelations.Undertheseconditions,theintroducedinputamplitudeof U =100 mVcausedsignicanterrorsintheimpedance response.ThesimulateddatawereanalyzedusingboththemeasurementmodelapproachanddirectevaluationoftheKramers-Kronigintegrals,asdescribedinSection 4.1. Theresidualerrorsresultingfromameasurementmodelttosimulatedimpedance datageneratedfromthesystemwithnoOhmicresistance R e =0 areshownin Figures4-1aand4-1bfortherealandimaginaryparts,respectively.Thedashed linerepresentsthecondenceintervalfortheregressedmodel.Alloftheresidual errorsfellwithinthecondenceinterval,suggestingthattheKramers-Kronigrelations weresatised.Themagnitudeoftheresiduals,ontheorderof 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(13 ,showthatthe measurementmodelcouldtthedatatowithin12signicantdigits.Thedatawere showntosatisfytheKramers-Kronigrelations,eventhoughtheerrorsduetoanonlinear responsewereverylarge, i.e., R t ; obs / R t ; 0 =0.658. 59

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a b Figure4-1.Residualserrorsresultingfromameasurementmodelt Z m tosimulated data Z s forthesystemwith R e =0 cm 2 :arealpart;andbimaginarypart. Thelinescorrespondtothe95.4% condenceintervalforthe regression.ThesystemparameterspresentedinTable4-1giveriseto R t ; obs / R t ; 0 =0.658and R e / R t ; obs =0. Therealandimaginarypartsofthenormalizedresidualerrorsresultingfroma measurementmodelttosimulatedimpedancedatageneratedfromthesystemwith R e = : 01 cm 2 areshowninFigures4-2aand4-2b,respectively.Theresidual errorsfelloutsidethecondenceintervalatfrequenciesgreaterthan 10 5 Hz.Thescaled valuesof 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(4 athighfrequenciesshowsthatthedeviationsfromKramers-Kronig relationsareinthefourthsignicantdigit,whichmaynotbevisibleforexperimental data.Nevertheless,thesimulationresultsdonotconformtotheKramers-Kronig transforms. Therealandimaginarypartsofthenormalizedresidualerrorsresultingfroma measurementmodelttosimulatedimpedancedatageneratedfromthesystemwith R e =1 cm 2 areshowninFigures4-3aand4-3b,respectively.Thedeviation fromconsistencywiththeKramers-Kronigrelations,markedbyresidualerrorsthat falloutsidethecondentintervalfortheregressedmodel,areevidentforfrequencies greaterthan 10 3 Hz.Themagnitudeofthescaledresidualerrorsislargerthanseenin 60

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a b Figure4-2.Normalizedresidualerrorsresultingfromameasurementmodelt Z m to simulatedimpedancedata Z s forthesystemwith R e = : 01 cm 2 :areal part;andbimaginarypart.Thelinescorrespondtothe95.4%condence intervalfortheregression.ThesystemparameterspresentedinTable4-1 giveriseto R t ; obs / R t ; 0 =0.658and R e / R t ; obs = 5 : 8 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(4 Figure4-2.Inaddition,theresidualerrorsfalloutsidethecondenceintervalatalower frequencyascomparedtoFigure4-2. Theresidualerrorsresultingfromameasurementmodelttosimulatedimpedance datageneratedfromthesystemwith R e =100 cm 2 areshowninFigures4-4aand 4-4b.DuetothelargeOhmicresistance,theerrorduetononlinearitywassmall, i.e., R t ; obs / R t ; 0 =0.981.Nevertheless,thenormalizedresidualerrorsfelloutsidethe condenceintervalforallfrequencies. ThemeasurementmodelanalysisofconsistencywiththeKramers-Kronigrelations presentedinFigures4-1,4-2,4-3,and4-4wascomplementedbyanindependent analysisusingdirectevaluationoftheKramers-Kronigintegralequation4.Asshown inFigure4-5,therealpartoftheimpedancepredictedfromequation4wasin perfectagreementwiththesimulationvalueforthesystemwithnoOhmicresistance R e =0 cm 2 .ThisresultisinagreementwiththeresultpresentedinFigure4-1, showingthatevenaverynonlinearimpedanceresponseyielding R t ; obs / R t ; 0 =0.658 isconsistentwiththeKramers-KronigrelationsforanOhmicresistanceequaltozero. 61

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a b Figure4-3.Normalizedresidualerrorsresultingfromameasurementmodelt Z m to simulatedimpedancedata Z s forthesystemwith R e =1 cm 2 :arealpart; andbimaginarypart.Thelinescorrespondtothe95.4%condence intervalfortheregression.ThesystemparameterspresentedinTable4-1 giveriseto R t ; obs / R t ; 0 =0.684and R e / R t ; obs = 5 : 6 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 a b Figure4-4.Normalizedresidualerrorsresultingfromameasurementmodelt Z m to simulatedimpedancedata Z s forthesystemwith R e =100 cm 2 :areal part;andbimaginarypart.Thelinescorrespondtothe95.4%condence intervalfortheregression.ThesystemparameterspresentedinTable4-1 giveriseto R t ; obs / R t ; 0 =0.981and R e / R t ; obs =3.9. 62

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Figure4-5.Acomparisonofsimulationresultstotherealcomponentofimpedance predictedusingequation4forthesystemswith R e =0 cm 2 and R e =1 cm 2 .ThesystemparameterspresentedinTable4-1giveriseto R t ; obs / R t ; 0 =0.658and R t ; obs / R t ; 0 =0.684,respectively.Intheabsenceof Ohmicresistance,thesimulateddataandthepredictedvaluesareequal. Forthesystemwith R e =1 cm 2 ,therealcomponentofimpedancepredictedfrom equation4deviatedfromtherealcomponentofthesimulateddata,indicating noncompliancewiththeKramers-Kronigtransformsduetoviolationoflinearity.This resultisinagreementwiththeresultspresentedinFigure4-3. Thedirectintegrationofequation4wasalsoabletorevealaninconsistency withtheKramers-Kronigrelationsforthesystemwith R e =100 cm 2 system,as isshowninFigure4-6.Thepercenterrorinthelowfrequencyregion,correctedfor Ohmicresistance,is4timesgreaterforthe R e =100 cm 2 systemascomparedto the R e =1 cm 2 systemshowninFigure4-5.Forthesystemwith R e = : 01 cm 2 however,directintegrationofequation4didnotrevealtheinconsistencieswiththe Kramers-KronigrelationsshownbythemeasurementmodelanalysisinFigure4-2.This discrepancymayberegardedtobeatestimonytothesensitivityofthemeasurement modelanalysisforfailuresofconsistencywiththeKramers-Kronigrelations. 63

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Figure4-6.Acomparisonofsimulationresultstotherealcomponentofimpedance predictedusingequation4forthesystemswith R e =1 cm 2 and R e =100 cm 2 .ThesystemparameterspresentedinTable4-1giveriseto R t ; obs / R t ; 0 =0.684and R t ; obs / R t ; 0 =0.981,respectively. 4.3TheApplicabilityoftheKramers-KronigRelationstoDetectingNonlinearity ThesensitivityoftheKramers-Kronigtransformstononlinearityclearlydependson boththemagnitudeoftheerrors R t ; obs / R t ; 0 andontheOhmicresistance.Theobjective ofthefollowingsectionistoidentifytheconditionsunderwhichtheKramers-Kronig relationsmaydetecterrorscausedbyanonlinearimpedanceresponse. 4.3.1InuenceofTransitionFrequency Themaximumvariationofinterfacialpotential V max correspondingtoaninput perturbation U =0 : 1 VisgiveninFigure4-7aforthesimulationspresentedinTable 4-1.OneinuenceoftheOhmicresistanceisseenatlowfrequencies,where lim 0 V max = U + R e =R t ; obs ThepresenceoftheOhmicresistancefurtherdecreasestheperturbationamplitudeat higherfrequencieswheretheinterfacialimpedancebecomessmallandtheroleofthe Faradaiccurrentisdiminished. 64

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a b Figure4-7.Interfacialparametersasfunctionsoffrequencyforthesimulations presentedinTable4-1:aMaximumvariationoftheinterfacialpotential;and btheeffectivecharge-transferresistance.Verticallinescorrespondtothe transitionfrequencygivenbyequation3. AsdiscussedinSection3.3.4,thefrequencydependenceoftheinterfacialpotential causesacorrespondingchangeintheapparentchargetransferresistance,asshown inFigure4-7b.ThefrequencycharacteristicofthetransformationshowninFigure4-7 fromlow-frequencybehaviortohigh-frequencybehaviorisgivenbyequation30. Thetransitionfrequencydependsonthedimensionlessratio R t ; obs / R e andisgivenin unitsofHz. TheinuenceofthetransitionfrequencycanbeseeninFigure4-8wherethenormalizedrealpartoftheimpedanceispresentedasafunctionofnormalizedfrequency forthesystemwith R e =1 cm 2 .Thefrequencyisscaledbythefrequencycharacteristicofthe R t ; 0 C timeconstant,andtherealpartoftheimpedanceiscorrectedfor theOhmicresistanceandscaledbythe R t ; obs seenatlowfrequency.Atthetransition frequencygivenbyequation3,theeffectivecharge-transferresistancechanges fromthelow-frequencyvalue R t ; obs ,whichisaffectedbythenonlinearresponse,tothe linearvalue R t ; 0 65

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Figure4-8.Thenormalizedrealpartoftheimpedanceasafunctionofnormalized frequencyforthesystemwith R e =1 cm 2 solidline.Thedashedlines representtheideallinearresponsesforsystemswith R t ; 0 =26 : 3 cm 2 and with R t ; obs =18 : 0 cm 2 Thechangeinapparentcharge-transferresistancehasaninuenceaswellinthe imaginarypartoftheimpedance.Thenormalizedimpedanceresponsesarepresented asfunctionsofnormalizedfrequencyforthesystemswith R e =0 : 01 1 ,and 100 cm 2 inFigures4-9aand4-9bfortherealandimaginaryparts,respectively.Thesolid curveistheideallinearresponseandthedashedcurvesarethenonlinearimpedance responsesarisingfromalargeinputamplitudeof U =100 mVforsystemparameters presentedinTable4-1.AsshowninFigure4-9a,therealcomponentofimpedance isdistortedfromtheideallinearresponseforthe R e =1 cm 2 and R e = : 01 cm 2 systems.Distortionisalsopresentforthe R e =100 cm 2 system,however,itisnot visuallyevidentinFigure4-9aduetothesmalldeviationoftheobservedchargetransferresistancefromthelinearvalue, i.e., R t ; obs / R t ; 0 =0.981.Thedistortionfromthe linearresponseoccursatthetransitionfrequencydescribedbyequation30.As showninFigure4-9b,theimaginarycomponentofimpedanceisdistortedfromthe ideallinearresponseforthe R e =1 cm 2 and R e =100 cm 2 systems.Distortionisnot 66

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a b Figure4-9.Thenormalizedimpedanceresponseasfunctionsofnormalizedfrequency forthesystemswith R e =0 : 01 1 ,and 100 cm 2 :arealpart;andb imaginarypart.Thesolidcurveistheideallinearresponseandthedashed curvesarethenonlinearimpedanceresponsesarisingfromalargeinput amplitudeof U =100 mVforsystemparameterspresentedinTable4-1. evidentforthe R e = : 01 cm 2 system.Forthecasewith R e = : 01 cm 2 thetransition frequencywas f t =8 10 5 Hzwhichwasinthecalculatedrangeoffrequencies,butthe transitionto R t ; 0 takesplaceinafrequencyrangewherethecurrentispredominately chargingandthevalueofthecharge-transferresistanceisinconsequential. Incontrastnodistortionoftheimpedanceresponseisseenforthecaseinwhich R e =0 .Thenormalizedimpedanceresponseforthesystemwith R e =0 ispresented asafunctionofnormalizedfrequencyinFigures4-10aand4-10bforthereal andimaginaryparts,respectively.Boththeideallinearresponseandthenonlinear impedanceresponsearesuperposedinspiteofthelargepotentialamplitudeapplied, yieldingavalue R t ; obs =R t ; 0 =0 : 658 TheworkpresentedheredemonstratesthatthesensitivityoftheKramers-Kronig relationsonthenonlinearityofanelectrochemicalsystemdependsonboththemagnitudeofthepotentialperturbationandthevalueofthetransitionfrequencygivenby equation3.Thecriticalparametersare U ,givenbyequation4, R t ; obs C dl 67

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a b Figure4-10.Thenormalizedimpedanceresponseasfunctionsofnormalizedfrequency forthesystemwith R e =0 :arealpart;andbimaginarypart.Boththe ideallinearresponseandthenonlinearimpedanceresponseare superposed. and R e =R t ; obs .When R e =0 ,thetransitionfrequencygivenbyequation3isequal toinnity,andtheimpedanceresponseisgivenby Z = R t ; obs 1+ j!R t ; obs C where R t ; obs differsfrom R t ; 0 butisindependentoffrequency.Inthiscase,theKramersKronigrelationsaresatised.WhentheOhmicresistanceissmallandthetransition frequencyisoutsidetheexperimentallyassessablerange,theeffectivecharge-transfer resistanceisapproximatelyindependentoffrequency,asshowninFigure4-7b,and theKramers-Kronigrelationswillbesatised.TheKramers-Kronigrelationswillalsobe satisedifthethedeparturefromlinearbehaviorissufcientlysmallthat R t ; obs =R t ; 0 1 TheKramers-Kronigrelationswillbeviolatedforconditionswhere R t ; obs =R t ; 0 6 =1 and thetransitionfrequencygivenbyequation3fallswithintheexperimentalfrequency range.Insuchacase,thevaryingeffectivecharge-transferresistanceisthemechanism thatcausestheKramers-Kronigrelationstofail,justasatime-dependent R t would causetheKramers-Kronigrelationstofailduetoviolationoftheconditionofcausality. 68

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ThereexistsaninterestingbalanceofeffectsforthesystemswithalargeOhmic resistance.ThepresenceofalargeOhmicresistancereducestheportionoftheapplied potentialperturbationthatcontributestotheinterfacialpotential,andthereforereduces thedepartureof R t ; obs =R t ; 0 fromunity.Atthesametime,thetransitionfrequency approaches 1 =R t ; obs C dl ,thusmakinganydeparturefromlinearbehaviordetectableby useoftheKramers-Kronigrelations.Inthesecases,theuseofLissajousguresatlow frequenciesmaybelesssensitivetononlinearbehaviorascomparedtotheuseofthe Kramers-Kronigrelations.Forexample,theLissajousanalysisofthe R e =100 cm 2 system,showninFigure7oftheworkbyHirschorn etal. 75 didnotdetectthepresence ofnonlinearity,whiletheKramers-KroniganalysisshowninFigure4-6diddetect nonlinearbehavior.ItshouldbenotedthatfurtherincreaseintheOhmicresistance willeventuallyleadtoanapproximatelylinearresponsethatisincompliancewiththe Kramers-Kronigrelations. ForsmallvaluesofOhmicresistance,thetransitionfrequenciesaresignicantly greaterthan 1 =R t ; obs C dl andthetransitionto R t ; 0 takesplaceinafrequencyrange wherethevalueofthecharge-transferresistancehasanegligibleinuenceonthe imaginaryimpedance.Therefore,itwillbedifculttodetectdiscrepanciesatlow frequenciesbetweenthedataandthepredictedvaluesusingtheKramers-Kronig transformexpressedbyequation4.Thisisinagreementwiththeresultspresented inFigure4-6,wherethe R e =100 cm 2 systemwith f t =4 10 2 Hzismoresensitiveto nonlinearbehaviorthanisthe R e =1 cm 2 systemwith f t =8 10 3 Hz. 4.3.2ApplicationtoExperimentalSystems Thetransitionfrequencygivenbyequation3ispresentedinFigure4-11asa functionof RC timeconstantwith R e =R t asaparameter.Thetimeconstantforfastreactions,suchasthereductionofferricyanideonaplatinumelectrodeatanappreciable fractionofthemass-transfer-limitedcurrentdensity,canbeontheorderof 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(5 s.For thesesystems,thetransitionfrequencymayfalloutsidetheexperimentallyaccessible 69

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Figure4-11.Thetransitionfrequencygivenbyequation3asafunctionof RC time constantwith R e =R t asaparameter. frequencyrange.The RC timeconstantforreactionsneartheequilibriumpotentialmay, however,besignicantlylarger.Forthesesystems,thetransitionfrequencymayfall withintheexperimentalrange,evenforsmallvaluesof R e =R t Urquidi-Macdonald etal. havereported,basedonexperimentalobservations, thattheKramers-Kronigrelationsarenotsensitivetoanonlinearsystemresponse. 74 Theirconclusionswerebasedonexperimentsperformedwithdifferentperturbationamplitudesonanironelectrodeina1M H 2 SO 4 electrolyte.Theyfoundthatthe Kramers-Kronigrelationsweresatisedevenforpotentialperturbationamplitudes sufcientlylargetocausemeasurabledistortionsintheimpedanceresponse.Their resultscanbeplacedintothecontextofFigure4-11.Systemparameters R e =2 cm 2 C dl =10 F/cm 2 ,and R t ; obs =14 cm 2 wereestimatedfromthesmall-amplitude impedancedatafromthepublishedexperimentalresultsshowninFigure4oftheir work. 74 Thecorrespondingtransitionfrequencywasapproximately9000Hz.Their experimentalfrequencyrangeextendedonlyto5000Hz,asshowninFigure5oftheir work.Therefore,thetransitionfrequencywasnotinthemeasuredfrequencyrangeand 70

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a b Figure4-12.Normalizedresidualerrorsresultingfromameasurementmodelt Z m to simulatedimpedancedata Z s withnormallydistributedadditivestochastic errorswithstandarddeviationof 0 : 1 percentofthemodulusforthesystem with R e =1 cm 2 :arealpart;andbimaginarypart.Thelinescorrespond tothe95.4%condenceintervalfortheregression.Theinputpotential perturbationamplitudewas U =1 mV. themeasuredcharge-transferresistance,althoughinerrorduetononlinearityassociatedwithlargeperturbationamplitudes,wasapproximatelyfrequencyindependent.As aresult,thedatacompliedwithKramers-Kronigrelations. Itisworthasking,forsystemsforwhichthemeasuredfrequencyrangeincludes thetransitionfrequency,whetherthedistortionsassociatedwithnonlinearbehavior aresufcientlylargetobediscernableinexperimentalmeasurements.Normally distributedadditivestochasticerrorswithameanvalueofzeroandastandarddeviation of 0 : 1 percentofthemodulusofthecalculatedimpedanceresponsewereappliedto thesystemwith R e =1 cm 2 .Thislevelofnoisehasbeenreportedtobetypicalof impedancemeasurements. 85,86 Themeasurementmodelanalysisforthissystemwith aninputpotentialperturbationamplitudeof U =100 mVwaspresentedinFigure 4-3intheabsenceofaddednoise.Themeasurementmodelanalysisispresented inFigure4-12forsimulateddatausinganinputpotentialperturbationamplitudeof U =1 mV.Thenormalizedresidualerrorsforbothrealandimaginarypartsofthe 71

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a b Figure4-13.Normalizedresidualerrorsresultingfromameasurementmodelt Z m to simulatedimpedancedata Z s withnormallydistributedadditivestochastic errorswithstandarddeviationof 0 : 1 percentofthemodulusforthesystem with R e =1 cm 2 :arealpart;andbimaginarypart.Thelinescorrespond tothe95.4%condenceintervalfortheregression.Theinputpotential perturbationamplitudewas U =100 mV. impedance,showninFigures4-12aand4-12b,respectively,aredistributedaround zero,indicatingthattheKramers-Kronigrelationsaresatised. Thecorrespondingmeasurementmodelanalysisusinganinputpotentialperturbationamplitudeof U =100 mVispresentedinFigure4-13.Thenormalizedresidual errorsforbothrealandimaginarypartsoftheimpedance,showninFigures4-13aand 4-13b,respectively,arenotdistributedaroundzero,indicatingthattheKramers-Kronig relationsarenotsatised.Thus,aKramers-Kroniganalysisbasedonthemeasurementmodelwilldetectnonlinearityofsystemsforwhichthemeasuredfrequencyrange includesthetransitionfrequency,evenwhenreasonableexperimentalerrorispresent. 4.4Conclusions Whiletheresultspresentedhereareconsistentwiththeobservationsreportedby Urquidi-Macdonald etal. 74 thattheKramers-Kronigrelationswereinsensitivetofailures oflinearity,thisworkalsoshowsthat,underappropriateconditions,theKramersKronigrelationsprovideausefultoolfordetectionofnonlinearsystemresponses.The 72

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sensitivityoftheKramers-Kronigrelationsonthenonlinearityofanelectrochemical systemdependsonboththemagnitudeofthepotentialperturbationandonwhether thetransitionfrequencygivenbyequation3fallswithintheexperimentalfrequency range.Thevalueofthetransitionfrequencydependson R t ; obs C dl and R e =R t ; obs .The Kramers-Kronigrelationswillbeviolatedforconditionswhere R t ; obs =R t ; 0 6 =1 andthe transitionfrequencygivenbyequation3fallswithintheexperimentalfrequency range. Forsmallvaluesof R e =R t ; obs ,theKramers-Kronigrelationsmaybeoflimitedutility fordetectingerrorsassociatedwithanonlinearresponse.Inthiscase,itwillbemore appropriatetouseexperimentaltestsinvolvingeitherrepeatedmeasurementswith differentperturbationamplitudesorobservationofnonlinearresponsesinlow-frequency Lissajousplots. 7,75 InspectionofLissajousplotsmaybelessusefulforsystemswith largevaluesof R e =R t ; obs .Inthiscase,theKramers-Kronigrelationsmayprovidea moreusefultoolfordetectionofnonlinearresponsestolargepotentialperturbations. InChapter3,ageneralizedmethodforoptimizingtheimplementationofimpedance experimentswasdeveloped.Conrmationofcollecteddatawaspresentedinthecurrent Chapter.Bothexperimentaltechniquesandproperdataanalysisarecriticalfortting impedancedataandestimatingmodelparameters.TheworkpresentedinChapter3 andthepresentchapterprovidesthefoundationforaccuratelyinterpretingdatainterms ofphysicallymeaningfulparameters. 73

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CHAPTER5 CHARACTERISTICSOFTHECONSTANT-PHASEELEMENT InChapters1and2,abackgroundontheuseandinterpretationoftheconstantphaseelementwasprovided.InChapters3and4,ageneralizedmethodwasdevelopedforoptimizingexperimentalanddataanalysistechniquesforenhancingparameter estimation.Inthepresentchapter,themathematicsoftheCPEarereviewedand graphicalmethodsareprovidedforextractingCPEparameters. TheimpedanceoftheCPE,expressedbyequation1,iswrittenintermsitsreal Z r andimaginary Z j componentsas Z cpe = f )]TJ/F25 7.9701 Tf 6.586 0 Td [( Q cos )]TJ/F24 11.9552 Tf 9.298 0 Td [(= 2+ j f )]TJ/F25 7.9701 Tf 6.586 0 Td [( Q sin )]TJ/F24 11.9552 Tf 9.299 0 Td [(= 2 Thephase-angleisexpressedas cpe =arctan Z j Z r = )]TJ/F24 11.9552 Tf 10.494 8.088 Td [( 2 Asshownbyequation5,thephase-angledoesnotdependonfrequency,whichis theoriginoftheterm constant-phaseelement Forexperimentalsystemsthevalueof generallyrangesbetween0.5and1.As showninFigure5-1,for =1 theCPEdisplaysaverticallineintheimpedanceplane, whichisrepresentativeoftheimpedanceresponseofanidealcapacitor.When < 1 theimpedanceresponseisinclinedfromthevertical.Thesmallerthevalueof the greaterthedeviationfromtheverticalresponse.Thephase-anglewith asaparameter isshowninFigure5-2. TherealandimaginarycomponentsoftheCPEasafunctionoffrequencyare showninFigures5-3aand5-3b,respectively.AsdiscussedbyOrazem etal. 87 and shownbyequation5, d log j Z j j =d log f = )]TJ/F24 11.9552 Tf 9.299 0 Td [( .Therefore,theCPEparameter can beobtainedfromexperimentaldataas = d log j Z j j d log f 74

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Figure5-1.ImpedanceplanerepresentationoftheCPE,equation5,with asa parameterand Q =1 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(6 Fs )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 = cm 2 Figure5-2.Thephase-angleassociatedwiththeCPE,equation5,with asa parameter. 75

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a b Figure5-3.ImpedanceresponseoftheCPE,equation5,with asaparameterand Q =1 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(6 Fs )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 = cm 2 ;atherealcomponent;andbtheimaginary component. and Q canbeobtainedfrom Q =sin 2 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 Z j f TheimpedanceofareactivesystemexhibitingtheCPEisexpressedbyequation 1.AsshowninFigure5-4,for =1 theimpedancedisplaysasemi-circleinthe impedanceplane,whichisrepresentativeoftheimpedanceresponseofasingletimeconstantRC.When < 1 ,theimpedanceresponseisadepressedsemi-circle.The smallerthevalueof thegreaterthedepressionfromtheidealsemi-circle.Thepeakof theimaginarycomponentoccursatthecharacteristicfrequency f c = 1 2 RQ 1 = Therealandimaginarycomponentsoftheimpedanceasafunctionoffrequencyare showninFigures5-5aand5-5b,respectively.Theimpedanceexpressedbyequation1ischaracterizedbyasymmetricresponseaboutthecharacteristicfrequency expressedbyequation5.Thevalueof canbeobtainedfromexperimentaldata usingequation5. 76

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Figure5-4.Impedanceplanerepresentationofequation1with asaparameter and Q =1 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(6 Fs )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 = cm 2 and R =10kcm 2 a b Figure5-5.Impedanceresponseofequation1with asaparameterand R =10kcm 2 and Q =1 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(6 Fs )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 = cm 2 ;atherealcomponent;andb theimaginarycomponent. 77

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TheCPEisamathematicalmodelwidelyusedtotexperimentalimpedance data.AsdiscussedinChapters1and2,theinterpretationoftheCPEintermsof physicallymeaningfulparameters,suchascapacitance,iscontroversial.InChapter6, thecapacitanceofelectrochemicalsystemsisreviewedandrelationshipsbetweenCPE parametersandcapacitance,takenfromtheliterature,aredeveloped. 78

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CHAPTER6 THECAPACITIVERESPONSEOFELECTROCHEMICALSYSTEMS Thepresentchapterprovidesthetheoreticalframeworknecessaryforinterpreting thecapacitiveresponseofsystemsexhibitingtheCPE.Thediffuselayercapacitance andthecapacitanceofadielectriclayerisreviewed.Relationshipsrelatingcapacitance andCPEparametersprovidedbyBrug etal. 3 andHsuandMansfeld 6 yielddifferent results.Inmanycases,bothsetsofequationshavebeenappliedtosimilarsystems andareusedwithoutregardingthesourceoftheCPE.Thetwodifferentmathematical formulasforestimatingeffectivecapacitancefromCPEparametersareassociatedunambiguouslywitheithersurfaceornormaltime-constantdistributions.Thebackground presentedinthischaptersuppliesthecontextneededforanalysisofsimulatedand experimentaldatapresentedinChapters7-10. 6.1CapacitanceoftheDiffuseLayer Anyinterfacebetweendissimilarphases,suchastheelectrode/electrolyteinterface inanelectrochemicalcell,promoteschargeseparationthatcanbeinterpretedin termsofacapacitance.Theelectrode/electrolyteinterfaceischaracterizedbydistinct regionsadjacenttothemetalelectrode.TheinnerHelmholtzplaneIHPmarksthe distanceofclosestapproachofspecicallyadsorbedionsandtheouterHelmholtzplane OHPmarksthedistanceofclosestapproachofsolvatedadsorbedions.Followingthe notationofNewman, 88 theIHPisatadistance y 1 fromtheelectrodeandtheOHPisat adistance y 2 .Thedistancebetweenthemetalboundary y m andtheOHPboundary y 2 isonthesub-nanometerscaleandcontainsthecharge q 1 .ExtendingbeyondtheOHP isthediffuseregionwhichisontheorderofnanometersandcontainsthecharge q 2 Adjacenttothediffuseregionisthediffusionregionwhichiselectricallyneutralbutmay containconcentrationgradients.Beyondthediffusionregionisbulkelectrolyte.The doublelayerreferstotheHelmholtzregionandthediffuseregion. 79

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Thedoublelayercapacitanceisdenedas C dl = @q @U ;T where q isthechargeassociatedwiththedoublelayerand U isthepotential.The chargeonthemetalinterfaceisrepresentedby q suchthat, q + q 1 + q 2 =0 Thediffuselayercapacitanceisdenedas C d = )]TJ/F29 11.9552 Tf 11.291 16.857 Td [( @q 2 @ 2 ;T wherethesubscript 2 representsthechargeandpotentialassociatedwiththediffuse layer.Derivinganexpressionforthediffuselayercapacitancebeginswiththestatement ofPoisson'sequation r 2 = )]TJ/F24 11.9552 Tf 12.355 8.087 Td [( e 0 = )]TJ/F24 11.9552 Tf 12.985 8.087 Td [(F 0 X i z i c i whichrelatesthespacialdistributionofthepotentialeldwiththechargedistributionthat gaverisetotheeld.Equation6relatesthenetchargedensityinthediffuselayer withtheconcentrationofchargedspecies.Arelationbetweenconcentrationandthe potentialeldforthediffuselayercanbegeneratedbyapplyingthegeneralspeciesux equation N i = )]TJ/F24 11.9552 Tf 9.298 0 Td [(z i u i Fc i r )]TJ/F24 11.9552 Tf 11.955 0 Td [(D i r c i + c i v where u i isthemobilityand D i isthediffusioncoefcient.Inthediffuselayerthereisno netuxandnoconvection.Therefore,withuseoftheNerst-Einsteinrelation D i = RTu i equation6becomes r c i = )]TJ/F24 11.9552 Tf 10.494 8.088 Td [(z i F RT c i r Thesolutionofequation6is c i = c i 1 exp )]TJ/F24 11.9552 Tf 10.494 8.088 Td [(z i F RT 80

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whichistheBoltzmanndistributionofionicconcentrations.Equation67canbe substitutedintoequation6yieldingadifferentialequationrelatingthepotentialeld anddistancewithinthediffuseregion. Thesolutionofthedifferentialequationdescribedbyequations6and6 forthediffuselayerrequirestheboundaryconditionsatthemetalinterfaceandinthe solutionbulk.Theelectricpotentialiszerofarfromtheelectrodesurface 0 ;y !1 ThemetalinterfaceboundaryconditionisastatementofGauss'slaw d dy = q 2 0 ;y = y 2 statingthattheelectriceldatthediffuselayerboundaryisequivalenttothecharge encloseddividedbythepermittivity. Newman 88 hasshownthedetailsofsolvingequations6-6wherean expressionrelatingchargeintermsofpotentialisusedtoevaluatethediffuselayer capacitanceaccordingtoequation6 C d = 0 cosh zF 2 2 RT where istheDebyelength, = r 0 RT 2 z 2 F 2 c 1 Forthespecialcasewheretheexponentialtermofequation6canbelinearly approximated, i.e., theDebye-Huckelapproximation,equation6becomes C d = 0 6.2CapacitanceofaDielectricLayer Thecapacitanceassociatedwithadielectricmaterialisderivedfollowingthe assumptionthatnofreechargesarepresentinthelayer.Laplace'sequation,ratherthan 81

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equation6,applies r 2 =0 Theequivalentchargetothemetalcharge q isassumedtobelocatedcompletelyatthe dielectricmaterialboundary .Themetalchargeiscalled q andthematerialboundary chargeiscalled q 2 .Incontrast,inthecaseofthediffuselayertheequivalentchargeis distributedwithinthelayer.Integratingequation6inone-dimensionyields d dy = k where k isaconstantofintegration.Therstboundaryconditionisequivalenttothe boundaryconditionexpressedbyequation6, d dy = q 2 0 ;y =0 Equation6isastatementofGauss'slawthattheelectriceldisproportionaltothe chargeenclosed.Foradielectricmaterialthereisnofreechargesandthusnochange intheamountofchargeenclosedasyoumoveawayfromtheinterface.Therefore,the electriceldisconstantandequation6becomes d dy = q 2 0 for y lessthan .Integrationofequation6yields = q 2 0 y + C where C isaconstantofintegration.Thesecondboundaryconditionis =0 ;y = Equation6isastatementofGauss'slawthatthechargeenclosedasyouextend pastthesecondboundarybecomeszeroandthustheelectriceldandelectricpotential iszero.Thisisanequivalentstatementtoequation6.Usingtheboundarycondition 82

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equation6inequation6yields = q 2 0 y )]TJ/F24 11.9552 Tf 15.282 8.088 Td [(q 2 0 AswasthecasepresentedinSection6.1,thecapacitanceisdeterminedviaevaluation ofthepotentialatthemetalinterfaceasexpressedbyequation6.Thepotentialat themetalinterfaceoccursat y =0 and 2 = )]TJ/F24 11.9552 Tf 12.626 8.088 Td [(q 2 0 Rearrangingequation6andapplyingequation6yields C di = 0 where C di isthecapacitanceassociatedwithadielectricmaterial.Equation6 impliesthatthecapacitanceofthediffuselayerismathematicallyequivalenttothe capacitanceofadielectricwithallofthechargelocatedattheDebyelength.Itshould benotedthattheDebyelengthistypicallymuchsmallerthanthethicknessofadielectriclayer, i.e., anoxidelmonthesurfaceofelectrode,andtherefore,theeffective capacitanceassociatedwithadielectricinserieswiththediffuselayercapacitanceis dominatedbythecapacitanceofthedielectric. 6.3CalculationofCapacitancefromImpedanceSpectra AsdiscussedinChapter1,theCPEisconsideredtoarisefromadistribution oftime-constantsinasystem.Calculatingcapacitanceusingasingletime-constant modeldoesnotapplyforsystemsexhibitingtheCPE.TheformulasprovidedbyBrug etal. 3 andHsuandMansfeld 6 yielddifferentresultsforcapacitanceandareassociated unambiguouslywitheithersurfaceornormaltime-constantdistributions. 6.3.1SingleTime-ConstantResponses Whentheimpedanceresponseofasystemcanbedescribedbyasingletimeconstantthecalculationofthesystem'seffectivecapacitanceisstraight-forward.The 83

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impedanceresponseofasingletime-constantcorrectedforOhmicresistanceis Z = R 1+ j!RC andinthehigh-frequencyregionequation6reducesto Z = 1 j!C Therefore,thecapacitanceofasystemthatcanbemodeledbyasingleRCiscalculatedfromthemeasuredimaginaryimpedanceinthehigh-frequencyregionas C RC = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 !Z j Alternatively,thecapacitanceofasingletime-constantmodelcanbedeterminedby identifyingthefrequencycorrespondingtothepeakoftheimaginaryimpedance C RC = 1 2 Rf peak where R isdeterminedatthelimitoflow-frequency. 6.3.2SurfaceDistributions Inthecaseofasurfacetime-constantdistribution,theglobaladmittanceresponse oftheelectrodemustincludeadditivecontributionsfromeachpartoftheelectrode surface.ThesituationisdemonstratedinFigure6-1a,whereasurfacedistribution oftimeconstantsinthepresenceofanOhmicresistanceresultsinadistributedtimeconstantbehaviorexpressedasasummationofadmittances.Foranappropriate time-constantdistribution,theimpedanceresponsemaybeexpressedintermsofa CPE.Interestingly,intheabsenceofanOhmicresistance,showninFigure6-1b,the surfacedistributionoftimeconstantsresultsinaneffective RC behaviorinwhich 1 R e ; s = X 1 R i 84

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a b Figure6-1.Schematicrepresentationofasurfacedistributionoftimeconstants:a distributionoftimeconstantsinthepresenceofanOhmicresistance resultinginadistributedtime-constantbehaviorthat,foranappropriate time-constantdistribution,maybeexpressedasaCPE;andbdistributionof timeconstantsintheabsenceofanOhmicresistanceresultinginan effectiveRCbehavior.Theadmittance Y i showninaincludesthelocal interfacialandOhmiccontributions. 85

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and C e ; s = X C i Thus,theappearanceofaCPEbehaviorassociatedwithasurfacedistributionof timeconstantsrequiresthecontributionofanOhmicresistance. 89,90 WhileanOhmic resistanceinphysicalsystemscannotbeavoided,theexampleillustratedinFigure 6-1billustratesthecrucialroleplayedbytheOhmicresistanceinCPEbehavior associatedwithsurfacedistributions. FollowingthedevelopmentofBrug etal. 3 therelationshipbetweenCPEparametersandcapacitancerequiresanassessmentofthecharacteristictimeconstant correspondingtotheadmittanceoftheelectrode.Thus, Y = X i Y i = X i R e ;i + R i 1+ j!R i C i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 where Y i isthelocaladmittancerepresentedinFigure6-1a, R e ;i isthelocalOhmic resistanceand R i and C i representthelocalsurfaceproperties.Ontheotherhand,the totaladmittanceoftheelectrodecanalsobeexpressedintermsofthesymmetricCPE representedbyequation1as Y = 1 R e 1 )]TJ/F24 11.9552 Tf 26.946 8.088 Td [(R t R e + R t 1+ R e R t R e + R t Q j! )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 # where R e istheglobalOhmicresistanceand R t Q ,and representglobalproperties. Equation6canbeexpressedintermsofacharacteristictimeconstantassociated withtheadmittancespectra Y as Y = 1 R e 1 )]TJ/F24 11.9552 Tf 26.946 8.087 Td [(R t R e + R t + j! Y )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 where Y = R e R t R e + R t C Thecharacteristictime-constant Y correspondstothefrequencyatwhichtheimaginary componentoftheadmittancespectraofthesymmetricCPE,equation6,reaches 86

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itspeakvalue.Comparisonofequations6and6yields Y = Q R e R t R e + R t = Q 1 R e + 1 R t )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ThecapacitanceassociatedwiththeCPEcanthereforebeexpressedas C B = Q 1 = )]TJ/F24 11.9552 Tf 5.479 -9.684 Td [(R )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 e + R )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 t )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 = or C B = Q 1 = R e R t R e + R t )]TJ/F25 7.9701 Tf 6.586 0 Td [( = wherethesubscript B referstotheauthorBrug.Equations6and6are equivalenttoequationderivedbyBrug etal. 3 forasurfacedistributionwitha differentdenitionofCPEparameters.Inthelimitthat R t becomesinnitelylarge, equation6becomes C B = Q 1 = R )]TJ/F25 7.9701 Tf 6.587 0 Td [( = e whichisequivalenttoequationpresentedbyBrug etal. 3 forablockingelectrode. 6.3.3NormalDistributions Inthecaseofanormaltime-constantdistributionthroughasurfacelayer,theglobal impedanceresponseoftheelectrodemustincludeadditivecontributionsfromeachpart ofthelayer.ThesituationisdemonstratedinFigure6-2,whereanormaldistributionof timeconstantsresultsinadistributedtime-constantbehaviorexpressedasasummation ofimpedances.Foranappropriatetime-constantdistribution,theimpedanceresponse maybeexpressedintermsofaCPE.Inthiscase,theappearanceofaCPEbehavior doesnotrequirethecontributionofanOhmicresistance.TheappearanceofaCPE fromaseriesofVoigt RC elementsdoes,however,requirecontributionsfromboth resistiveandcapacitiveelements. TherelationshipbetweenCPEparametersandcapacitancerequiresanassessmentofthecharacteristictimeconstantcorrespondingtotheimpedanceofthelayer Z 87

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Figure6-2.Schematicrepresentationofanormaldistributionoftimeconstantsresulting inadistributedtime-constantbehaviorthat,foranappropriatetime-constant distribution,maybeexpressedasaCPE. Thus, Z = R e + X i Z i = R e + X i R i 1+ j!R i C i where R e istheOhmicresistanceand R i and C i representthelocalpropertiesofthe layer.ForaseriesofRCelementstheeffectivecapacitanceisexpressedas C e ; n = 1 P 1 C i SincetheOhmicresistancedoesnotcontributetothetime-constantdispersionsof alm,thedevelopmentcanbeperformedintermsofanOhmicresistance-corrected impedance Z )]TJ/F24 11.9552 Tf 11.955 0 Td [(R e TheOhmicresistance-correctedimpedanceofalmcanbeexpressedintermsofa CPEas Z )]TJ/F24 11.9552 Tf 11.955 0 Td [(R e = R f 1+ j! QR f where R f = X i R i representsthelmresistance.Alternatively, Z )]TJ/F24 11.9552 Tf 11.955 0 Td [(R e = R f 1+ j! Z 88

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Comparisonofequations6and6yields Z = R f C = QR f Thecharacteristictime-constant Z correspondstothefrequencyatwhichtheimaginary componentoftheimpedancespectraofthesymmetricCPE,equation69,reaches itspeakvalue.ThecapacitanceassociatedwiththeCPEcanthereforebeexpressedas C HM = Q 1 = R )]TJ/F25 7.9701 Tf 6.586 0 Td [( = f Equation6isequivalenttoequationpresentedwithoutderivationbyHsuand Mansfeld 6 intermsof max 6.4Conclusions Thecapacitanceofadielectric,asshownbyequation6,isinverselyrelated tolayerthickness.Theeffectivecapacitance,equations6and6,wasdened asthecompositecapacitanceoftheindividualcapacitancesofasystem.Calculation ofcapacitanceusingequation6isonlyvalidforsystemsdescribedbyasingle time-constant, i.e., =1 .ThederivationsoftheBrugandHsu-Mansfeldformulas forcalculatingcapacitanceofCPEsystemsarebasedonthepremisethatthereisa characteristictime-constantcorrespondingtothepeakoftheimaginaryadmittance andimaginaryimpedance,respectively.Equations6,6,and62haveall thesameform,buttheresistanceusedinthecalculationsofcapacitanceisdifferentin thethreecases,beingrespectivelytheparallelcombinationof R t and R e forequation 6, R e forequation6and R f forequation6. WithouttheaidofaphysicalmodeltoaccountfortheCPEitisgenerallyassumed thattheBrugandHsu-Mansfeldformulasyieldtheeffectivecapacitanceofasystem. TheseformulasyielddifferentresultsforthesameCPEparameters.Adistinctionwas madebetweensystemswhereasurfacedistributionofpropertiesisexpectedtobethe sourceoftheCPEfromsystemswhereanormaldistributionofpropertiesisexpectedto 89

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bethesourceoftheCPE.ForsurfacedistributionstheCPEisenvisionedtoarisefrom thesumofadmittancesandtheOhmicresistanceplaysacrucialroleinthepresenceof theCPE.Therefore,equation6isassumedtoapplyforsurfacedistributions.For normaldistributionstheCPEisenvisionedtoarisefromthesumofimpedancesandthe OhmicresistanceplaysnoroleinthepresenceoftheCPE.Therefore,equation6 isassumedtoapplyfornormaldistributions.TheBrugandHsu-Mansfeldequationsfor capacitanceareevaluatedusingsimulatedandexperimentalsystemsinChapter7. 90

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CHAPTER7 ASSESSMENTOFCAPACITANCE-CPERELATIONSTAKENFROMTHE LITERATURE Twodifferentmathematicalformulasforestimatingeffectivecapacitancefrom CPEparameters,presentedinChapter6,wereassociatedunambiguouslywitheither surfaceornormaltime-constantdistributions.Inmanycases,theseformulasare usedwithoutregardforthetypeofdistributionthatisthesourceoftheCPE.The objectiveofthepresentchapteristoexploretheconditionsofvalidityformodels whichrelatecapacitancetoCPEparameters.Applicationtodifferentexperimentaland simulatedsystemsareusedtoillustratetheimportanceofusingthecorrectformulathat correspondstoagiventypeofdistribution.Whenthelocalresistivityvariesconsiderably overthethicknessofalm,theexperimentalfrequencyrangemayprecludeobservation ofthecapacitancecontributionofaportionofthelm,resultinginunderpredictionof thelmthickness.Inmanycases,calculatingcapacitancefromCPEparameterswithout theaidofaphysicalmodelisunreliable,whichprovidesthemotivationfordeveloping mechanismstoaccountfortheCPEpresentedinChapter8. 7.1SurfaceDistributions Huang etal. 18 haveshownthatcurrentandpotentialdistributionsinduceahighfrequencypseudo-CPEbehaviorintheglobalimpedanceresponseofanideallypolarizedblockingelectrodewithalocalideallycapacitivebehavior.Inarelatedwork,Huang etal. 19 exploredtheroleofcurrentandpotentialdistributionsontheglobalandlocal impedanceresponsesofablockingelectrodeexhibitingalocalCPEbehavior.They wereabletorelatetheglobalimpedanceresponsetolocalimpedance,anddistinctive featuresofthecalculatedglobalandlocalimpedanceresponsewereveriedexperimentally.AsimilardevelopmentwaspresentedforadiskelectrodewithaFaradaic reaction. 2 Thisworkwasusedtoexploretheapplicabilityofequations6,6,and 6fordeterminationofeffectivecapacitance.Theapproachallowedcomparison 91

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Figure7-1.EffectiveCPEcoefcientscaledbytheinterfacialcapacitanceasafunction ofdimensionlessfrequency K with J asaparameter.TakenfromHuang et al. 2 betweentheestimatedcapacitanceandthevalueassumedforthesimulations.The graphicalmethodspresentedbyOrazem etal. 87 wereusedtoobtainCPEparameters and Q .Theparameters and Q wereobtainedfromequations5and5, respectively.Theparameters and Q obtainedbygraphicalevaluationarethesame aswouldbeobtainedbyregressionanalysis.AsdiscussedbyHuang etal. 2 the frequenciesusedfortheanalysiswerelimitedtothosethatwereonedecadelargerthan thecharacteristicpeakfrequencybecause,inthisfrequencyrange,thevalueof was well-dened.Theanalysistookintoaccounttheobservationthatthevalueof was dependentonthefrequencyatwhichtheslopewasevaluated. ThevalueofeffectiveCPEcoefcient, Q ,scaledbytheinterfacialcapacitance C 0 usedforthesimulations,ispresentedinFigure7-1asafunctionofdimensionless frequency K= !C 0 r 0 = ,where r 0 isthediskradius,and istheconductivityofthe electrolyte.TheresultsgiveninFigure7-1arepresentedasafunctionoftheparameter J ,incorporatedaspartoftheboundaryconditionforFaradaicreactionsattheelectrode surface.Undertheassumptionoflinearkinetics,validforsteady-statecurrentdensities i muchsmallerthanthetheexchangecurrentdensity i 0 ,theparameter J wasdenedto 92

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be J = a + c Fi 0 r 0 RT where a and c are,respectively,anodicandcathodicapparenttransfercoefcients. ForTafelkinetics,validfor i>>i 0 ,theparameter J wasdenedtobeafunctionofradial positionontheelectrodesurfaceas J r = c F j i r j r 0 RT where i r wasobtainedfromthesteady-statesolutionas i r = )]TJ/F24 11.9552 Tf 9.298 0 Td [(i 0 exp )]TJ/F24 11.9552 Tf 10.494 8.088 Td [( c F RT )]TJ/F15 11.9552 Tf 7.266 -6.662 Td [( V )]TJ/F15 11.9552 Tf 13.256 3.022 Td [( 0 r where V )]TJ/F15 11.9552 Tf 13.256 3.022 Td [( 0 r representsthelocalinterfacialpotentialdrivingforceforthereaction. Theparameter J canbeexpressedintermsoftheOhmicresistance R e andcharge transferresistance R t as J = 4 R e R t Largevaluesof J areseenwhentheOhmicresistanceismuchlargerthanthechargetransferresistance,andsmallvaluesof J areseenwhenthecharge-transferresistance dominates.Athigh-frequencies,wherefrequencydispersionplaysasignicantrole, theeffectiveCPEcoefcient Q providesaninaccurateestimatefortheinterfacial capacitanceusedasaninputforthesimulations,evenforsmallvaluesof J where,as shownbyHuang etal. 2 isclosetounity.AsshowninFigure7-1,assumptionthat Q representstheinterfacialcapacitanceresultsinerrorsontheorderof500percentat K=100 Equations6and6, i.e., C B ,arecomparedtotheinputvalueofinterfacial capacitanceinFigures7-2aand7-2b,respectively.Followingtheobservationby Huang etal. 2 thatthegeometry-inducedpotentialandcurrentdistributionsyielda pseudo-CPEbehaviorinwhichthecoefcient isaweakfunctionoffrequency,Figures 7-2aand7-2bweredevelopedusingfrequency-dependentvaluesof and Q .Thus, 93

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a b Figure7-2.Normalizedeffectivecapacitancecalculatedfromrelationshipspresentedby Brug etal. 3 foradiskelectrodeasafunctionofdimensionlessfrequency K with J asaparameter:awithcorrectionforOhmicresistance R e equation 6;andbwithcorrectionforbothOhmicresistance R e and charge-transferresistance R t equation6.TakenfromHuang etal. 2 thevalueof Q reportedisthatcorrespondingtothevalueof atagivenfrequency K Theerrorinequation6isafunctionofbothfrequency K and J .Whileequation 6appliesstrictlyforablockingelectrode,itgivesthecorrectanswerforFaradaic systemsifonechoosestocalculate atfrequencies K < 5 ,butfailsfor K > 5 .The dependenceon J isreducedsignicantlywhenboththeOhmicresistance R e and charge-transferresistance R t aretakenintoaccount,andtheerrorsinestimating interfacialcapacitancearelessthan20percent.Thecorrectionfor R t inequation6 isimportantforfrequencies K > 5 .Oftherelationshipstested,equation6provides thebestmeansforestimatinginterfacialcapacitancewhenfrequencydispersionis signicant. Equation6, i.e., C HM ,wastestedagainsttheinputvalueofinterfacialcapacitanceinFigure7-3where C 0 istheknowninterfacialcapacitance.Whileequation 6representsanimprovementascomparedtodirectuseoftheCPEcoefcient Q ,theerrorsinestimatingtheinterfacialcapacitancedependonboth J and K and 94

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Figure7-3.Effectivecapacitancecalculatedfromequation6andnormalizedbythe inputinterfacialcapacitanceforadiskelectrodeasafunctionof dimensionlessfrequency K with J asaparameter.TakenfromHuang etal. 2 rangebetween )]TJ/F15 11.9552 Tf 9.298 0 Td [(70 to +100 percent.Equation6,developedforanormaltimeconstantdistribution,isnotappropriateforinterpretationofresultsaffectedbyasurface time-constantdistribution. 7.2NormalDistributions Equation6,developedfornormaltime-constantdistributions,wasappliedfor determinationofeffectivecapacitanceintwosystemsinwhichanormalvariationof resistivitymaybeexpected. 7.2.1Niobium Theanodicdissolutionofa0.25cm 2 Nb.9%,Goodfellowrotatingdiskelectrode wasstudiedinapH2solutioncontaining0.1MNH 4 Fandsodiumsulfateassupporting electrolyte. 4 Theexperimentalimpedancedatacorrespondingtoananodization potentialof6VSCEareshowninFigure7-4a.Accordingtothesurfacecharge approachdevelopedbyBojinov, 91,92 thedielectricpropertiesoftheoxidelmdominate thehigh-frequencyresponse.Themedium-frequencyinductiveloopandlow-frequency capacitivelinearealsodescribedbyCattarin etal. 4 andFrateur. 93 Figure7-4bshows thataplotofthelogarithmoftheimaginarypartoftheimpedanceasafunctionofthe logarithmofthefrequencyyieldsastraightlinewithaslopeof )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 90 forfrequencies 95

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a b Figure7-4.ExperimentalimpedancedataobtainedwithaNbrotatingdiskelectrode rpmin0.1MNH 4 FsolutionpH2,at6VSCE:aComplexplane plot;andbtheimaginarypartoftheimpedanceasafunctionoffrequency. DatatakenfromCattarin etal. 4 higherthan300Hz,whichindicatesaCPEbehaviorwithaCPEexponentof 0 : 90 87 ratherthantheresponseofatrue R f C f parallelcombination,where R f istheoxide lmresistanceand C f istheoxidelmcapacitance.Furthermore,inFigure7-4b,the surfacedistributioncausedbynon-uniformcurrentandpotentialdistributionscannotbe observedinthehigh-frequencyrangesince K=1 wouldcorrespondto65kHzandthe maximumfrequencythatwasusedintheexperimentswas63.1kHz. 19 TheCPEparameters and Q forthehigh-frequencyloopwereobtainedusing thegraphicalmethodspresentedbyOrazem etal. 87 equations5and5, respectively.Theresultingvaluesfordifferentanodizationpotentialsarepresentedin Table7-1.Thevaluesoftheoxidelmresistancecorrespondingtothediameterofthe high-frequencylooparealsoreportedinTable7-1.FromtheCPEparametersandthe lmresistance,thecapacitanceoftheoxidelmwascalculatedviaequation62, i.e., C HM ,andappliedasthesystem'seffectivecapacitanceseeTable7-1. Fornormaltime-constantdistributionsforwhichthedielectricconstantmaybe assumedtobeindependentofposition,thecapacitanceshouldberelatedtolm 96

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Table7-1.CPEparameters,resistance,effectivecapacitance,andthicknessofoxide lmsformedonaNbdiskelectrodein0.1MNH 4 FsolutionpH2asa functionoftheanodizationpotential. Potential/VSCE Q / )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 cm )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 s R f /k cm 2 C HM / Fcm )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 d e /nm 20 : 955 : 9 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(6 1 : 304 : 68 60 : 903 : 5 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(6 2 : 012 : 018 100 : 882 : 5 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(6 3 : 651 : 329 thicknessaccordingtoequation6.If C HM istakenasthesystem'seffective capacitancethen C HM = "" 0 d e where isthedielectricconstantand 0 =8 : 8542 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(14 F/cmisthepermittivity ofvacuum.Inequation7 hasbeenreplacedwith d e torepresentthefactthat thecalculatedcapacitanceaccordingtotheHsu-Mansfeldformula, i.e., equation6 42,isconsideredtorepresenttheeffectivecapacitanceofthesystem.Inthecaseof anodicdissolutionofNbinaciduoridemedium,theoxideisassumedtobeNb 2 O 5 and =42 94,95 Thevaluesof d e arepresentedinTable7-1fordifferentanodization potentials. Thecalculatedvaluesof d e canbecomparedtothosegivenintheliterature. InLohrengel'sreviewofmetaloxides, 96 differentvaluesforthethicknessofNb 2 O 5 lmsformedonNbelectrodesat E =0 VSHEaregiventhatvarybetween3and 6.7nm.Moreover,theformationratio i.e., thethicknessincreasecausedbyaunit increaseofthepolarizationpotentialisreportedtobe2.6or2.8nm/V.Therefore, accordingtoLohrengel, 96 thethickness canbeestimatedapproximatelytobe = 5 : 0+2 : 70 E where hasunitsofnmand E isexpressedinVSHE.Themethods usedtodeterminetheoxidelmthicknessesandtheelectrolytecompositionarenot mentionedinReference[96].Arsova etal. 97 measuredthethicknessofNb 2 O 5 lms formedin1MH 2 SO 4 byellipsometry.Theformationratiodeterminedbytheseauthors is2.26nm/V.Extrapolationoftheir to E =0 Vyields =5 nm.Therefore,the 97

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Table7-2.ThicknessofoxidelmsdevelopedonaNbelectrode,asafunctionofthe anodizationpotential.Comparisonofvaluesdeducedfromimpedancedata withthosefromtheliterature. PresentWorkValuesfromLiterature Potential/VSCE d e /nm /nm[96] /nm[97] /nm[98] 2811107 618221915 1029332824 thicknessfoundbyArsova etal. 97 canbeestimatedtobe =5 : 0+2 : 26 E where has unitsofnmand E isexpressedinVSHE.Habazaki etal. 98 measuredthethickness oftheoxidebyTEMofanultramicrotomedsectionandbyimpedance.Fromtheirdata, thethicknesscanbeestimatedtobe =2 : 4+2 : 08 E where hasunitsofnmand E isreferencedtoPtin0.1MH 3 PO 4 .ThevaluesofNb 2 O 5 lmthicknesscalculatedby usingequations6and7fordifferentanodizationpotentialsarecomparedwith thosecalculatedfromthedataofLohrengel, 96 Arsova etal. 97 andHabazaki etal. 98 in Table7-2.Thevaluesof d e calculatedbyapplicationofequations6and7to impedancedataareinverygoodagreementwiththeliteraturevalues,inparticularwith thoseobtainedfromnon-electrochemicalmeasurements.Ourresultsagreealsowith thoseofHeidelberg etal. 99 whoreportedontheoxidationof10nmthickNblayersin micro-andnano-cells. Toshowtheconsequenceofthemisuseoftheresistanceterminthecalculation oftheeffectivecapacitance,thevaluesof C B werecalculatedusingequation6 inwhich R t wasreplacedby R f .Equation6yieldedthesamevaluesfor C B as equation6since R f >>R e .Asbefore,equation7wasusedtoestimate d e with C HM replacedby C B .Useofequation6yielded:for2VSCE, C B = 3 : 2 F/cm 2 and d e =12 nm,for6VSCE, C B =0 : 9 F/cm 2 and d e =41 nm,andfor 10VSCE, C B =0 : 5 F/cm 2 and d e =74 nm.Comparisontothevaluespresentedin Table7-1showsthattheeffectivelmthicknessobtainedusingtheeffectivecapacitance 98

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obtainedfromequation6canbesignicantlylargerthantheactuallmthickness, especiallyathighpotential. 7.2.2HumanSkin Impedancedatawerecollectedforheat-separatedexcisedhumanstratumcorneum obtainedfromtheabdomenortheback. 1 Theseparationprocedureinvolvedphysical andmechanicalmanipulationstoseparatethestratumcorneumfromtheunderlying dermis.Deionizedwaterwastheonlysolventaddedduringtheprocess.Theskin samplesweremountedbetweenglassdiffusioncellspriortotheimpedancestudy.The skinandthesolutionweremaintainedatconstanttemperatureof32 Cwithawaterjacketeddiffusioncell.Magneticstirbarswereusedforeachchamberofthediffusion celltokeepthesolutionswellmixed.Theelectrochemicalimpedancemeasurements wereconductedwithaSolartron1286potentiostatandaSolartron1250frequencyresponseanalyzer.Afour-electrodecongurationwasusedforallofthestudies.The Ag/AgClcounterandworkingelectrodeswereproducedbyInVivoMetric.TheAg/AgCl referenceelectrodeswerefabricatedbyMicroElectrodes,Inc. Theslightlymoistenedepidermiswasstoredinbetweentwosheetsofpolymer lminarefrigerator.Atthestartofatypicalexperimenttheskinwasremovedfrom therefrigeratorandimmersedina32 C50mMCaCl 2 /20mMHEPESsolutionpHof 6.95whichprovidedapproximatelythesamepHandionicstrengthastheelectrolytic uidwithinthebody. 100 Replicateelectrochemicalimpedancespectrawerecollected periodicallyusingVariable-AmplitudeGalvanostaticVAGmodulation. 101,102 The sinusoidalcurrentperturbationwassuperimposedabouta0 ADCcurrentbias,and theamplitudeofthepotentialresponseacrosstheskinwasmaintainedat 10mV. Theindividualscanstookapproximatelyveminutesandwereshowntosatisfythe Kramers-Kronigrelations,indicatingthatthesystemwasstationaryonthetimescaleof theexperiments.Uponcompletionofanimpedancescantheskinwasallowedtorelax forthreeminutesbeforethereplicatedspectrumwascollected. 99

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Table7-3.CPEparameters,resistance,effectivecapacitance,andthicknessfor heat-strippedhumanstratumcorneumin50mMbufferedCaCl 2 electrolyteas afunctionofimmersiontime.DatatakenfromMembrino. 1 Time/hr Q / )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 cm )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 s R f /k cm 2 C HM / Fcm )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 HM / m 0 : 00 : 8246 : 13 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(8 601 : 86 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 2 : 3 1 : 90 : 8345 : 36 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(8 511 : 66 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 2 : 6 5 : 10 : 8385 : 40 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(8 421 : 66 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 2 : 6 ImpedanceresultsarepresentedinFigure7-5awithimmersiontimeasaparameter.ThestraightlinesevidentathighfrequenciesinFigure7-5bshowahighfrequencyconstant-phasebehavior.TheCPEparameters and Q wereobtainedusing thegraphicalmethodspresentedbyOrazem etal. 87 Theresultingvaluesfordifferent immersiontimesarepresentedinTable7-3.Thevalueofthethicknessoftheskin dependsonitsdielectricconstant .Theestimatedthicknesses d e reportedinTable 7-3wereobtainedfromequations6and7undertheassumptionthat =49 Thevalueofdielectricconstantusedinthepresentworkwasobtainedbythecomparison,showninasubsequentsection,oftheYoungmodeltotheimpedancedata.The resultingvaluesof d e ofaround 2 maresubstantiallysmallerthanthethicknessofthe stratumcorneum,whichisacceptedtohaveavaluebetween10and40 m. 103 Equation6wasalsousedtocalculatetheeffectivecapacitance.Asforthe caseofNb 2 O 5 ,equation6yieldedsimilarvaluesfor C B asequation6since R f >>R e .Equation7with C HM replacedby C B wasusedtoestimate d e .Useof equation6yielded:for 0 : 0 h, C B =4 : 5 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 F/cm 2 and d e =9 : 6 m,for 1 : 9 h, C B =3 : 7 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 F/cm 2 and d e =12 m,andfor 5 : 1 h, C B =3 : 1 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 F/cm 2 and d e =14 m.Interestingly,theeffectivelmthicknesscalculatedusingtheeffective capacitanceobtainedfromequation6wasclosertotheexpectedvaluethanwas thethicknessestimatedusingequation6.Theapparentbetteragreementisfound inspiteofthefactthattheOhmicresistance R e hasnorelationshiptothedielectric propertyoftheskin.Thisworkillustratesaneedforabetterunderstandingofthe inuenceofstrongvariationsofresistivityontheimpedanceresponse. 100

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a b Figure7-5.Experimentalimpedancedataobtainedforheat-separatedexcisedhuman stratumcorneumin50mMbufferedCaCl 2 electrolytewithimmersiontime asaparameter:aComplexplaneplot;andbtheimaginarypartofthe impedanceasafunctionoffrequency.DatatakenfromMembrino. 1 101

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7.2.3FilmswithanExponentialDecayofResistivity Thevaluesof d e calculatedusingequations6and7wereinverygood agreementwiththeliteraturevaluesforNb 2 O 5 lms,butthevaluesobtainedforhuman stratumcorneumweresubstantiallysmallerthantheexpectedvalues.BoththeNiobium oxide 93,26,104 andtheskinsystems 27,105 havebeendescribedashavingaresistivitythat decaysexponentiallywithposition.Thecaseofalmwithauniformdielectricconstant andanexponentialdecayoflocalresistivityismathematicallyequivalenttotheYoung model,inwhichanexponentialincreaseinconductivityisassumed. 26,104 Thelocal resistance R x canbeexpressedas R x = 0 e )]TJ/F25 7.9701 Tf 6.586 0 Td [(x= d x where 0 isthemaximumvalueofresistivityfoundat x =0 ,whichcorrespondsto theoxideelectrolyteinterface,and representsacharacteristiclength.Theeffective resistanceofthelmcanbeobtainedbyintegrationoverthelmthickness i.e., R e = Z 0 0 e )]TJ/F25 7.9701 Tf 6.586 0 Td [(x= d x toyield R e = 0 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(e )]TJ/F25 7.9701 Tf 6.586 0 Td [(= Thelocalcapacitancecanbeexpressedas C x = "" 0 d x wherethedielectricconstant wasassumedtohaveauniformvalue.Theeffective capacitance,obtainedbyintegrationoverthelmthickness,following 1 C e = Z 0 1 "" 0 d x isfoundtobe C e = "" 0 102

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Equation7isidenticaltoequation7andequation6,astheintegrationis validforalldielectrics. Theimpedanceofthelmisobtainedfromintegrationacrossthelmthickness following Z = Z 0 0 e )]TJ/F25 7.9701 Tf 6.587 0 Td [(x= 1+ j! 0 e )]TJ/F25 7.9701 Tf 6.587 0 Td [(x= "" 0 d x Theresultis Z = )]TJ/F24 11.9552 Tf 21.541 8.088 Td [( j!"" 0 ln 1+ j!"" 0 0 e )]TJ/F25 7.9701 Tf 6.586 0 Td [(= 1+ j!"" 0 0 aswasalreadycalculatedbyG ohr etal. 106,107 Equation7isreferredtoasthe Youngimpedance.Inthelow-frequencylimit,applicationofL'H opitalsruleyields lim 0 Z = 0 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(e )]TJ/F25 7.9701 Tf 6.586 0 Td [(= Thisresultisinagreementwiththedirectintegrationofresistivitywhichyieldedequation 7.Inthehigh-frequencylimit, lim !1 Z = )]TJ/F24 11.9552 Tf 9.299 0 Td [(j !"" 0 Thisresultisalsoinagreementwithdirectintegrationofcapacitancewhichyielded equation7. Theimpedanceresponseassociatedwithequation7ispresentedinFigure 7-6indimensionlessformwith = asaparameter.Theimpedancewasscaledby thezero-frequencyasymptotegivenbyequation7.Thecharacteristicfrequency indicatedinthegureisindimensionlessformfollowing !"" 0 0 .For = =1 ,thepeakin theimaginaryimpedanceisslightlysmallerthan0.5andtheNyquistplotisonlyslightly depressedfromperfect RC behavior.Forlargervaluesof = ,distortionisevidentat higherfrequencies.Theshapeoftheplotremainsunchangedfor => 5 .Theabsence oftime-constantdispersionatlowfrequenciesindicatesthattheYoungmodelcannot accountforthelow-frequencybehaviorforskinseeninFigure7-5a. 103

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Figure7-6.Nyquistplotsforsimulationoftheimpedanceassociatedwithanexponential decayofresistivitywith = asaparameter.Thecharacteristicfrequency indicatedisindimensionlessformfollowing !"" 0 0 TounderstandtherelationshipbetweenYoungimpedanceandCPEbehavior, impedancevaluescalculatedaccordingtoequation7wereanalyzedbyan R f -CPE parallelcombination,inserieswiththeelectrolyteresistance.TheCPEparameters Q and canbeobtainedfollowingthegraphicalmethodsoutlinedbyOrazem et al. 87 Theparameter canbeobtainedfromtheslopeoftheimaginarypartofthe impedanceplottedasafunctionoffrequencyinalogarithmicscale.Theslopeis presentedinFigure7-7asafunctionofdimensionlessfrequency !"" 0 0 .Atlow frequency, dlog j Z j j = dlog =1 ,showingthattheexponentialdecayofresistivitydoes notresultinlow-frequencytime-constantdispersion.Athighfrequencyandfor = =1 dlog j Z j j = dlog = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 ,againshowingthattheexponentialdecayofresistivitydoesnot resultinhigh-frequencytime-constantdispersion.Forlargervaluesof = ,asignicant frequencyrangeabove !"" 0 0 =1 isseenforwhich dlog j Z j j = dlog differsfrom )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 TheCPEparameters Q and weredeterminedgraphicallyforeachfrequency above !"" 0 0 =1 .Theeffectivecapacitancewascalculatedusingequation6, andtheeffectivelmthicknesswascalculatedusingequation7.Theresultsare presentedinFigure7-8aasafunctionofdimensionlessfrequencyandwith = as aparameter.Ifthefrequencyissufcientlylarge,theeffectivelmthickness d e is 104

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Figure7-7.Thederivativeofthelogarithmofthemagnitudeoftheimaginarypartofthe impedancewithrespecttothelogarithmoffrequencyasafunctionof dimensionlessfrequencyforthesimulationspresentedinFigure7-6. a b Figure7-8.TheeffectivelmthicknessobtainedforthesimulationspresentedinFigure 7-6usingequations6and7:anormalizedbytheknownlm thickness ;andbnormalizedbythecharacteristiclength 105

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equaltotheactuallmthickness foranyvalueof = .Forabroadfrequencyrange andfor => 1 ,theeffectivelmthicknessobtainedfromthecapacitancecanclearly besignicantlysmallerthantheactuallmthickness.For = =400 ,theeffective lmthicknessistwoordersofmagnitudesmallerthantheactuallmthicknessfor frequenciesashighas !"" 0 0 =10 5 Thesimulationsshowthat,for = =5 ,theeffectivelmthicknessabruptly approachestheactuallmthicknessat !"" 0 0 =10 2 .Asimilarabruptchangeisseenfor = =10 at !"" 0 0 =10 4 .Thesimulationsindicatethat,whilethecapacitanceobtained fromtheimpedanceresponseforalmwithanexponentialdecayofresistivityshould yield,inthelimitofinnitefrequency,thecorrectthicknessofthelm,measurement overanitefrequencyrangewillyieldalmthicknessthatissubstantiallysmaller.As showninFigure7-8b,theeffectivelmthicknessislargerthanthecharacteristiclength .Thus,thelmthicknessobtainedfromthecapacitancecanliebetweentheactuallm thickness andthecharacteristiclength 7.3ApplicationoftheYoungModeltoNiobiumandSkin Theresultspresentedaboveshowthat,whileequation7willbevalidinthe limitofinnitefrequency,aniteexperimentallyaccessiblefrequencyrangemayrender undetectablethecapacitancecontributionsfromaportionofthelm.Thiseffectmay besignicantforcaseswherethelocalresistivityvariessignicantlywithposition.The effectisillustratedinFigure7-9.Foraconstantcapacitance C andlocalresistance R 1 ,measurementatfrequenciesmuchbelow =1 =R 1 C willyieldonlytheresistance R 1 becausethecapacitoractsasanopencircuitatthesefrequencies.Inthiscase, representedbytheuppertwo RC elementsinFigure7-9,theeffectivecapacitance obtainedfromequation6willunderpredictthethicknessofthelm.Onlythemost resistivepartofthelmisprobedbyimpedance.Thecapacitanceelementsthatarenot observedbyimpedancecanbeeitheratthemetal/lminterfaceoratthelm/electrolyte interface. 106

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Figure7-9.Circuitrepresentationofanormaldistributionofresistivityinwhichsome capacitanceelementsarenotobservedoveranexperimentallyaccessible frequencyrangeduetolocalvariationofresisitivity. Theconsequenceisexploredhereforthespeciccaseofalmwithauniform dielectricconstantandanexponentialdecayofresistivity.Thedielectricresponseof bothNiobiumoxide 93,26,104 andskin 27,105 havebeendescribedintermsofexponential decaysofresistivity.Regressionofsuchamodeltothedataobtainedforanodic dissolutionofNiobiumatapotentialof6VSCEinanaciduoridemediumyielded,for =42 ,valuesof 0 =2 : 66 10 9 cm, =30 nm,and =8 nm.Undertheseconditions, thesimulationofanexponentialdecayofresistivityindicatesthat,forfrequenciesabove 5 kHz, 1 and d e = 1 .AcomparisonoftheYoungmodeltotheimpedancedata forNiobiumoxideatapotentialof6VSCE,presentedinFigure7-10a,showsthat theslopeoftheYoungmodelisequaltounityforfrequenciesgreaterthan 5 kHz.The disagreementbetweenthevalueof =0 : 90 obtainedfromexperimentandthevalue of =1 obtainedfromthemodelindicatesthattheexponentialdecayofresistivity providesonlyanapproximatedescriptionofthehigh-frequencybehavioroftheNb 2 O 5 Nevertheless,thesimulationvalueof d e = =1 isconsistentwiththegoodagreement 107

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a b Figure7-10.ComparisonoftheYoungmodeltothehigh-frequencypartofthe experimentalimaginarypartoftheimpedanceasafunctionoffrequency: aNiobiumoxideatapotentialof6VSCEseeFigure7-4;andbhuman stratumcorneumwithimmersiontimeasaparameterseeFigure7-5. Thelinesrepresentthemodel,andsymbolsrepresentthedata. foundbetweenthethicknessestimatedfromtheimpedancemeasurementusing equations6and7andvaluesobtainedbyindependentmethods. In-vivoimpedanceexperimentsobtainedbytape-strippingsuccessivelayersof skinfromhumansubjectsdemonstratedthattheresistivityofhumanstratumcorneum decaysexponentiallywithposition. 27,105 AYoungmodelanalysiswasthereforeperformedforthein-vitrodatapresentedinFigure7-5forheat-strippedhumanstratum corneum.Modelparameterswereobtainedbymatchingthehigh-frequencyportionof theimpedanceresponsegiveninFigure7-5.Themodelparameters 0 ,and were selectedtomatchthezero-frequencyasymptotefortherealpartoftheimpedance, matchthecharacteristicfrequencyatwhichtheimaginarypartoftheimpedancehad amaximummagnitude,andyieldadielectricconstantbetweenthedielectricconstant ofwater =80 andlipid =2 108 Thevalueofskinthicknesswasassumedtobe either 20 mor 40 m,inkeepingwithreportedvalues. 27,105 Thecomparisonbetween theYoungmodelandthedataisgiveninFigure7-10b,andtheresultingparameters 108

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Table7-4.Physicalpropertiesobtainedbymatchingthehigh-frequencyportionofthe impedanceresponsegiveninFigure7-5forheat-strippedstratumcorneumin 50mMbufferedCaCl 2 electrolyteasafunctionofimmersiontime. Time/h 0 / cm / m / m d e = 0 6 10 8 49 2010 : 8480 : 152 0 6 10 8 49 4010 : 8580 : 078 1.9 5 10 8 49 2010 : 8440 : 143 1.9 5 10 8 49 4010 : 8540 : 076 5.1 4 10 8 49 2010 : 8390 : 139 5.1 4 10 8 49 4010 : 8500 : 074 arepresentedinTable7-4.Thecharacteristiclength =1 misinagreementwith thedatapresentedbyKalia etal. 105 butissmallerthanthevalue =5 mreported byYamamotoandYamamoto. 27 Valuesof > 1 : 5 yielded,forthepresentexperimental data,dielectricconstantsthatweregreaterthanthatofwater. Thevaluesof and d e = reportedinTable7-4wereestimatedfromthesimulation atafrequencyof 50 kHz.Thisfrequencywaschosenforthisanalysisbecauseitis attheupperlimitoftheexperimentalfrequencyrange.AsshowninFigure7-7,the Youngmodelprovidesonlyapseudo-CPEbehavioroverabroadhigh-frequencyrange inwhichtheCPEparametersareweakfunctionsoffrequency.Thegoodagreement betweenthevalueof obtainedfromexperimentandfromthemodelsuggeststhat theexponentialdecayofresistivityprovidesagooddescriptionforthehigh-frequency behavioroftheskin.Inaddition,thesimulationvaluesfor d e = areconsistentwith theobservationthatthethicknessesestimatedfromimpedancemeasurementsusing equations6and7weresubstantiallysmallerthanthosereportedinthe literature.TheresultspresentedheresupporttheobservationbyOh etal. thatthe capacitanceoftheskincouldnotbemeasuredafterrepeatedremovalofskinlayers reducedtheimpedancetoavalueindistinguishablefromthatofthebathingmedium. 109 Theresistivityprolescorrespondingtothesimulationspresentedinthissection aregiveninFigure7-11.ThechangeinresistivityfortheNb 2 O 5 lmissmallenough thattheentiredielectricresponseofthelmcanbeseenintheexperimentalfrequency 109

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Figure7-11.Resistivityprolesassociatedwiththesimulationoftheimpedance responseforNb 2 O 5 at 6 VSCEandskinusingauniformdielectric constantandanexponentiallydecayingresistivity. range.Incontrast,thechangeinlocalresistivityoftheskinismuchlarger,andthe capacitanceassociatedwiththeregionofsmallerresistivityvaluesisnotseenin theexperimentalfrequencyrange.Accordingly,thethicknessestimatedfromthe effectivecapacitanceismuchsmallerthantheactualthicknessoftheskin.Inthis case,thethicknessobtainedfromimpedancemeasurementsisthethicknessofthe higherresistivityregion.AsillustratedinFigure7-8fortheYoungmodel, = isthe keyparameterfordeterminingthemeaningoftheeffectivelmthickness.When = issmall,theeffectivethicknessdeterminedfromthecapacitanceistheactuallm thickness.When = islarge,theeffectivethicknessdeterminedfromthecapacitanceis thethicknessofonlytheresistiveportionofthelm. 7.4Conclusions MethodsfordeterminationofeffectivecapacitancefromCPEparametershavebeen employedextensivelyintheimpedanceliterature.Itisnotobviousthatalltheauthors whousedtherelationshipsderivedbyBrug etal. 3 andpresentedbyHsuandMansfeld 6 werefullyawarethattheformerareappropriateonlyforasurfacedistributionoftime constantsandthelatterappliesonlytoanormaldistribution,asisdemonstratedin 110

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Chapter6.TheresultspresentedinthecurrentChapterillustratetheimportanceof usingthecorrectformulathatcorrespondstoagiventypeofdistribution.Misuseof theformulas,forexamplebyusinganincorrecteffectiveresistanceinthecalculationof capacitance,mayleadtomacroscopicerrors,sincethevaluesofelectrolyteresistance, chargetransferresistanceandlmresistancemaybequitedifferent.Theselection oftherightformulashouldrestontheknowledgeofthesystemunderinvestigation, obtainedbydifferentmethods.Forinstance,localimpedancemayprovideevidence forsurface/normalinhomogeneity;whereas,spectroscopicmethodsmayshowthe presenceoflmsforwhichpropertiesmightbedistributedinthenormaldirection. TheresultspresentedinthecurrentChapterillustratethattheformulasprovided forcalculatingcapacitancedonotnecessarilyprovidethecorrectvalueevenifthe appropriateformulaforagiventypeofdistributionisused.Forinstance,theYoung modelprovidedanadequaterepresentationoftheimpedanceforthehumanskin system,however,theapplicationoftheHsu-Mansfeldformulayieldedaninaccurate estimateofcapacitance.Conversely,theHsu-Mansfeldformulayieldedanaccurate estimateofcapacitancefortheNiobiumoxidesystem,however,regressiontotheYoung modeldidnotprovideanadequaterepresentationoftheimpedanceathigh-frequency. Inaddition,theHsu-Mansfeldformuladoesnotapplytoblockingsystemswhere thelow-frequencyresistanceisundened.Thisworkdemonstratestheimportance ofdevelopingphysicallyreasonablemodelsthataccountfortheCPE.Aphysical modelthataccountsforCPEbehaviorinlmsisdevelopedinChapter8.Ananalytic expressionforcapacitance,basedontheproposedmodel,isdevelopedinChapter9 andappliedtoexperimentalsystems. 111

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CHAPTER8 CPEBEHAVIORCAUSEDBYRESISTIVITYDISTRIBUTIONSINFILMS AsshowninChapter7,theexponentialresistivitydistributionproposedbyYoung doesnotgiverisetoCPEbehavior.TheHsu-Mansfeldformulawasdevelopedfor normaldistributions,butthelmthicknessobtainedusingthisformulacanbetoosmall. TheresultspresentedinChapter7illustratedtheneedtodevelopmechanismsto accountfortheCPE.Inthepresentchapter,physicalmodelsaredevelopedthatcan accountfortheappearanceoftheCPEinsystemswherethevariationofpropertiesis expectedinthedirectionnormaltotheelectrode. 8.1ResistivityDistribution TheapproachtakenbyBrug etal. 3 tomodelCPEbehaviorwastoassumethatthe CPEoriginatedfromasurfacedistributionoftimeconstantswithuniformOhmicand kineticresistances.Thetime-constantdispersionthereforewasassumedtooriginate fromadistributionofcapacitances.Themathematicaldevelopmentofthedistribution functionsusingthemethodsofFuossandKirkwood 9 yieldedanormalizableprobability distribution.ThemethodofFuossandKirkwoodcouldnot,however,beappliedtoa normaldistributionoftime-constantswithequalvaluesofcapacitancebecausethe resultingprobabilitydistributioncouldnotbenormalized. Anoriginalapproachwasemployedinthepresentworkinwhichregressionofa measurementmodel 58,59 tosyntheticdatayieldedadistributionoftimeconstants.The syntheticdataweregeneratedfollowingequation1andwereregressedtotheVoigt measurementmodel,expressedas Z RC = X i R i 1+ j! i yieldingadiscretenumberoftime-constants i = R i C i andresistancevalues R i thatt thesyntheticdata.FollowingtheproceduredescribedbyAgarwal etal. 58,59 sequential Voigtelementswereaddedtothemodeluntiltheadditionofanelementdidnotimprove 112

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Figure8-1.Adistributionof RC elementsthatcorrespondstotheimpedanceresponse ofalm. thet.Modelparameterswererejectedthatincludedzerowithintheir95.4percent condenceinterval. AccordingtothemodelpresentedinFigure8-1,the RC time-constantsare assumedtobeassociatedwithdifferentiallayersofthelm.Thedifferentialcapacitance C i wasobtainedfromtheregressedparameters R i and i by C i = i =R i Thetsobtainedbyregressionofthe RC measurementmodeltosyntheticCPEdata areshowninFigures8-2a,8-2b,and8-2cfortherealpartoftheimpedance,the imaginarypartsoftheimpedance,andthephaseangle,respectively.ThecorrespondingvaluesforresistanceandcapacitancearepresentedinFigures8-3aand8-3bas functionsofthetime-constant .Thevaluesof R i and C i forthelargesttimeconstants donotconformtothepatternseenforthevaluesatothertimeconstants.Thiscanbe attributedtothedifcultyinttingblockingsystemswithaVoigtmodel,whichhasanite impedanceattheDClimit.Forthisreason,thesepointswerenotconsideredinthe subsequentanalysis. 113

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a b c Figure8-2.Syntheticdatasymbolsfollowingequation1with Q =1 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(6 s = cm 2 with asaparameterandthecorresponding RC measurementmodelts linesforatherealcomponentoftheimpedance;btheimaginary componentoftheimpedance;andcthephaseangle.Theregressed elementsareshowninFigures8-3aand8-3b. 114

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a b Figure8-3.Theregressedmeasurementmodelparametersasafunctionof time-constantforthesyntheticCPEdatashowninFigures8-2aand8-2b: aresistance;andbcapacitance.Thecircledvalueswerenotusedinthe subsequentanalysis. AsimilarapproachwastakenbyOrazem etal. 110 wherethedistributionofresistancevaluesrepresentedtheweightingappliedtoadistributionoftimeconstants. Orazem etal. 110 however,didnotexplorethevariationofcapacitancerequiredtosimulateCPEbehaviorwithaVoigtmodel.AsshowninFigure8-3b,thecapacitance valuesrequiredtottheCPEdatavariedasmuchastwoordersofmagnitude.For manyexperimentalsystemsthevariationinlocaldielectricconstantisnotexpectedto encompasssuchabroadrange. Thelocalcapacitanceisrelatedtolocaldielectricconstant i by C i = i 0 d i where 0 isthepermittivityofvacuumand d i isthethicknessassociatedwithelement i Thelocalresistancecanbeexpressedintermsofalocalresistivity i as R i = i d i 115

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Thetimeconstant i = i i 0 isindependentoftheelementthickness.Thevariabilityofcapacitancecouldbeinterpreted,forauniformdielectricconstant,asbeingaconsequenceofachangingelement thickness.Thisinterpretationhasconsequencesforlocalresistivity.Thus,if d i = 0 C i thecorrespondingresistivityisgivenby i = R i d i = i 0 Aresistivitydistributionmodelcan,therefore,beinferredfromtheregressedvaluesfor i and C i Thelocalresistivitywascalculatedaccordingtoequation8.Thediscreteresistivityvalueswerearrangedmonotonicallywiththeircorrespondingelementthickness interpretedintermsofalocalpositionsuchthat x k = k X i =0 d i TheresultsarepresentedinFigure8-4,wherethesymbolsrepresentthediscrete valuesofresistivitycalculatedfromtheregressedparameters, = x= representsthe dimensionlessposition,and isthethicknessofthelayer = n X i =0 d i AsshowninFigure8-4,theresistivityfollowsanearlylinearproleonalogarithmic scalewhichcanbeexpressedaccordingto = )]TJ/F25 7.9701 Tf 6.587 0 Td [( 116

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Figure8-4.Resistivityasafunctionofdimensionlessposition.Thesymbolsarethe discreteresistivityvaluescalculatedfromequations8and8using theregressedvaluesofresistancesandcapacitancesgiveninFigures 8-3aand8-3band =10 .Thelinesrepresentequation8with parameter determinedaccordingtoequation8. where istheresistivityat =1 and isaconstantindicatinghowsharplytheresistivityvaries.Adistributionofresistivitywhichprovidesaboundedvalueforresistivityis proposedtobe = 0 + 1 )]TJ/F24 11.9552 Tf 13.244 8.087 Td [( 0 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 where 0 and aretheboundaryvaluesofresistivityattheinterfaces. Itisworthnotingthat,whileAgarwal etal. 58,59 haveinsistedthatthemeasurement model,asusedforanalysisoferrorstructure,hasnophysicalmeaning,inthepresent application,theVoigtmeasurementmodelhasmeaningintermsofthenormaldistributionofresistivityanddielectricconstant.Underassumptionofauniformdielectric constant,animpedanceresponseshowingblockingCPEbehaviorcanbeexplained intermsofapower-lawdistributionofresistivity.Inthesubsequentsection,equation 8isusedtodevelopmodelsfortheimpedanceresponseofsurfacelms. 117

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8.2ImpedanceExpression Underassumptionthatthedielectricconstantisuniform,theimpedanceofthelm canbewrittenforanarbitraryresistivitydistribution x as Z f = Z 0 x 1+ j! 0 x dx Equation8canbewrittenintermsofdimensionlessposition = x= Z f = Z 1 0 1 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 + j! 0 d Whenthefrequencytendstowardzero, Z f = Z 1 0 d whichcanbeexpressedintermsoftheimpedanceofthecircuitshowninFigure8-1as Z f = n X 1 R i Whenthefrequencytendstowardinnity Z f 1 = j! 0 = 1 j! n X 1 1 C i = 1 j!C Introductionoftheresistivityprolegiveninequation8yields Z f = Z 1 0 1 a + b d where a = )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 0 + j! 0 and b = )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 )]TJ/F24 11.9552 Tf 11.956 0 Td [( )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 0 118

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Ananalyticsolutiontoequation8ispossibleforsomeintegervaluesof .For example,when =3 Z f = k 3 a 1 2 log k +1 3 1+ k 3 + p 3arctan 2 )]TJ/F24 11.9552 Tf 11.956 0 Td [(k k p 3 + p 3 6 # where k = a =b 1 = 3 Undertheconditionthat 0 >> k islessthan1for !< 0 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 ,andequation 8reducesto Z f = 2 3 p 3 b 1 = 3 a 2 = 3 = 2 3 p 3 1 = 3 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 0 + j! 0 2 = 3 Equation8isderivedforthespecialcaseof =3 .Ageneralexpressionofthe impedancecanbeproposedinthesameformas Z f = g 1 = )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 0 + j! 0 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 = where g isafunctionof and,inthecaseof =3 g =2 = 3 p 3 .Thecomparison ofequation8tothenumericalintegrationofequation8showsthatthis expressionisgeneralandcanbeappliedforall > 2 overabroadrangeoffrequencies. Therealandimaginarypartsoftheimpedanceobtainedbynumericalintegration ofequation8arepresentedinFigures8-5aand8-5b,respectively,with asa parameter.Thelinesrepresenttheevaluationofequation8wherethenumerical valueof g wasobtainedatthezerofrequencylimitofequation8accordingtothe expression g = Z f )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 = 0 1 = andwhere Z f isobtainedfromnumericalintegrationofequation8at =0 Equation8providesagoodagreementwithnumericalintegrationofequation8 17forfrequenciesbelowacharacteristicfrequencygivenas f = 0 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 .Detailsof algorithmusedforthenumericalintegrationofequation8isprovidedinAppendix B. 119

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a b Figure8-5.Acomparisonoftheimpedanceresponsegeneratedbynumerical integrationofequation8symbolsandtheanalyticexpression providedbyequation8lineswith 0 =1 10 16 cm =100cm =10 =100 nm,and asaparameter:atherealcomponentof impedance;andbtheimaginarycomponentofimpedance. Anumericalevaluationwasusedtoconrmthat g canbeexpressedasafunction ofonly andisindependentofothersystemparameters.AsshownbyFigure8-6,the valueof g rangesbetween1and1.6for 0 1 = 0 : 5 .Aninterpolationformula g =1+2 : 88 )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 : 375 couldbeobtainedthatadequatelyrepresentsthefunctionintherange 0 1 = 0 : 5 AsshowninFigures8-5aand8-5b,theanalyticexpressionprovidedbyequation 8isinagreementwiththenumericalsolutionofequation8for !< 0 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 Inthelow-frequencyrange,for !< 0 0 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 or f 0 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 Z j isequalto 1 =j!C forallvaluesof ,in agreementwithequation8. 120

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Figure8-6.Thenumericalevaluationof g asafunctionof 1 = wherethesymbols representresultsobtainedfromequation8.Thelinerepresentsthe interpolationformulagivenasequation8. Equation8isintheformoftheCPEfor !> 0 0 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 i.e., Z f = g 1 = j! 0 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 = = 1 j! Q Therefore,equation8yieldstheimpedancegivenbyequation1for 0 0 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 < !< 0 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 .Inspectionofequation8suggeststhat = )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 or 1 = =1 )]TJ/F24 11.9552 Tf 11.955 0 Td [( where 2 for 0 : 5 1 .Thus, g =1+2 : 88 )]TJ/F24 11.9552 Tf 11.955 0 Td [( 2 : 375 and Q = 0 g 1 )]TJ/F25 7.9701 Tf 6.587 0 Td [( Thus,equation8providestheanalyticexpressionfortherelationshipbetween and suggestedbytheresultspresentedinFigure8-4. Theimpedancecorrespondingtoequation8ispresentedinFigure8-7for 121

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Figure8-7.NyquistrepresentationoftheimpedancegiveninFigure8-5for =6 : 67 Themarkedimpedanceatafrequencyof 2 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(5 Hzisclosetothe characteristicfrequency f 0 =1 : 8 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(5 Hz. =6 : 67 =0 : 85 inNyquistcoordinates.Thehighandlow-frequencybehaviorsofthe impedancearenotsymmetric.Inhighfrequency,aCPEresponseisevident;whereas, thelowfrequencybehaviorcorrespondstoapurecapacitiveloop. 8.3Discussion Ananalyticexpressionfortheimpedanceresponseassociatedwithapower-law distributionsofresistivitywasdevelopedintheprevioussection.Itisusefultoexplore theconditionsunderwhichthismodelcanbeusedtoextractphysicalparameters fromexperimentaldata.Itisalsousefultocomparethisimpedanceresponsetothe impedanceobtainedfromotherresistivitydistributionsreportedintheliterature. 8.3.1ExtractionofPhysicalParameters Thefrequencyrangeforwhichtheimpedanceresponseprovidedbyequation 8isconsistentwiththeCPEispresentedinFigure8-5.Generally, isknownfrom independentmeasurements.Operationinthefrequencyrange f 0 f arerequiredtoobtainseparatelythelmthickness andtheinterfacialresistivity 122

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Measurementsfor f
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a b c Figure8-8.Normalizedimpedanceresponseassociatedwithnormaldistributionsof resistivitywithaxeddielectricconstant =10 andathickness =100 nm. Thedashedlineprovidestheresultsforaresistivitygivenasequation 8with 0 =1 10 12 cm =2 10 7 cm ,and =6 : 67 .Thesolidline providestheresultforaYoungmodelwitharesistivityprolefollowing equation8withthesamevaluesof and 0 ,yielding =9 : 24 nm.a Nyquistplot;brealpartoftheimpedance;andcimaginarypartofthe impedance. 124

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a b Figure8-9.Resistivityprolesandestimatedvaluesof forthesimulationsreportedin Figure8-8:aresistivityversusposition;andbthevalueof d log j Z j j =d log f obtainedfromtheslopesgiveninFigure8-8c. andthecorrespondingvaluesof d log j Z j j =d log f arepresentedinFigure8-9b.As discussedbyOrazem etal. 87 theslope d log j Z j j =d log f isequalto )]TJ/F24 11.9552 Tf 9.299 0 Td [( forasystemthat showsCPEbehavior.Thevalueof forthepower-lawmodelathighfrequenciesis independentoffrequencyintherangeof 1 Hzto 9 kHz,butchangesabruptlytoavalue ofunityforfrequencieshigherthan 9 kHz.Apseudo-CPEbehaviorisseenfortheYoung modelatfrequenciesbetween 1 Hzand 9 kHz,but,asisseenforthepower-lawmodel, changesabruptlytoavalueofunityforfrequencieshigherthan 9 kHz.Forboth distributions,thelow-frequencyresponseyields =1 ,aswasmentionedpreviously forthepower-lawmodel.Underconditionswhere f iswithinthemeasuredfrequency range,CPEbehavior,or,fortheYoungmodel,pseudo-CPEbehavior,isnotseenatthe high-frequencylimit.Forsuchacase, C e canbedirectlydeterminedat f>f 8.3.3VariableDielectricConstant Iftheresistivitydistributionresultsfromaninhomogeneouslayercomposition,then aproleforthedielectricconstantmayalsobeexpected.Thevariationofdielectric constant,however,shouldbesmallandgenerallylimitedtolessthanafactorof2or 3.Thepresentworkshowsthatadistributionof RC timeconstantovermanyordersof 125

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magnitudeisrequiredtoyieldCPEbehavioroverabroadfrequencyrange;thus,this distributioncannotbeexplainedbyvariationofdielectricconstantalone.Thepresent limitingcaseofadistributedresistivityandauniformdielectricconstantaccountsforthe dominanteffects,sinceavariabledielectricconstantrepresentsasecondordereffectas comparedtothevariableresistivity. 8.4Conclusions Thepresentworkshowsthat,underassumptionthatthedielectricconstantis independentofposition,anormalpower-lawdistributionoflocalresistivityisconsistent withtheCPE.Thepower-lawresistivitydistributionprovidesaphysicallyreasonable modelthatoffersaninterpretationoftheCPEforabroadclassofsystemswherea variationinpropertiesisexpectedinthedirectionnormaltotheelectrode.Theanalytic expressionfortheresultingimpedanceprovidesausefulrelationshipbetweensystem propertiesandCPEparameters.Applicationofthismodeltoexperimentalsystems ispresentedinChapter9,wherephysicalpropertiesareestimatedfromtheCPE response. 126

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CHAPTER9 APPLICATIONOFTHEPOWER-LAWMODELTOEXPERIMENTALSYSTEMS Theobjectiveofthepresentworkistodevelopamethodtoextractphysically meaningfulparametersfromimpedancedatayieldingCPEbehaviorcorresponding tosystemsforwhichavariationofpropertiesisexpectedinthedirectionnormalto theelectrode.Theresultingapproachisappliedtoexperimentaldata.Thedatareexaminedherewerealreadypublished,andtheexperimentaldetailsmaybefoundin thereferences. 9.1Method Inmanycases,theimpedanceresponsefollowsCPEbehavior,expressedfor blockingsystemsbyequation1andforreactivesystemsbyequation1.For 0 0 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 f arerequiredtoobtainseparatelythelmthickness and 127

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a b Figure9-1.Representationof ZQ where Z isgeneratedbynumericalintegrationof equation8and Q isobtainedfromequation8for =4 =0 : 75 and =10 with 0 and asparameters:atherealcomponentof impedance;andbtheimaginarycomponentofimpedance.Theline represents j! )]TJ/F22 7.9701 Tf 6.587 0 Td [(0 : 75 inagreementwithequation1.Thesymbols representcalculationsperformedfor 4 0 =10 18 cmand =10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 cm; 0 =10 14 cm; =10 2 cm;and 0 =10 10 cm; =10 5 cm. theinterfacialresistivity .Measurementsfor f
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a b Figure9-2.NyquistplotofthedatapresentedinFigure9-1for 0 =10 10 cmand =10 5 cm:aplotshowingthecharacteristicfrequency f 0 = 0 0 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 =18 Hz;andbzoomedregionshowing f = 0 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 =1 : 8 10 6 Hz. 129

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a b Figure9-3.Impedanceresponseassociatedwithafrequencyrangewhichexcludesthe characteristicfrequencies f 0 and f :asimulationsobtainedfor =10 and with asaparameter;andbexperimentalNyquistplotforpassive Aluminumina0.1MNa 2 SO 4 electrolytedatatakenfromJorcin etal. 5 .The dashedlinerepresentsaCPEttothedataaccordingtoequation1. 9.2.1AluminumOxideLarge 0 andSmall Whentheresistivitylimit issmall,thehigh-frequencydomaincorresponds toaCPE.Whentheresistivitylimit 0 isverylarge,itsinuenceontheimpedance responsecanbeoutsidetheexperimentallyaccessiblefrequencyrange.Inthiscase, theresultingimpedanceresponseisthatofablockingelectrode,asisseeninFigure 9-3aforafrequencyrangethatexcludes f 0 and f .Suchabehaviorcanbeobserved experimentally,asisshowninFigure9-3bforpassiveAluminumina0.1MNa 2 SO 4 electrolyte. 5 ThedashedlineinFigure9-3brepresentsaCPEttothedataaccording toequation1. Theeffectivecapacitanceofalmcanbeexpressedas C e ; f = 0 130

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Introductionofequation9intoequation8yields C e ; f = Q 0 1 )]TJ/F25 7.9701 Tf 6.586 0 Td [( g When =1 g =1 and,asshownbyequation9,theCPEparameter Q represents thecapacitanceofthelayer.Theparameters Q and inequation9canbeobtained fromgraphicalanalysisofimpedancedata, 87 and isoftenknownfromindependent measurements.Theparameter cannotbeknownexactlyfordatashowinghighfrequencyCPEbehavior.Nevertheless,equation9maybeattractiveforanalysis ofCPEdataforwhichapolarizationresistancecannotbeestimatedduetoblocking behaviorinthemeasuredfrequencyrangeorforwhichthepolarizationresistanceis inuencedbyphenomenanotassociatedwiththedielectricresponseofthelm. Althoughthevalueof isunknownfordatashowinghigh-frequencyCPEbehavior, anupperboundonitsvaluecanbedenedbecausethecharacteristicfrequency f mustbelargerthanthelargestmeasuredfrequency f max .Thus,amaximumvalueof canbeobtained ; max = 1 2 0 f max Therelationshipbetween ; max and f max fordifferentvaluesofdielectricconstantis showninFigure9-4.Sinceequation9canbewritten C e ; f = Q f )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 g asimilarboundontheeffectivecapacitancecanbefoundtobe C max = Q f max )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 g Therefore, C e ; f C max = f f max )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 = ; max )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 Theuncertaintyincalculatingeffectivecapacitanceduetouncertaintyinthevalueof canbeascertainedfromtheresultspresentedinFigure9-5.When iscloseto 131

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Figure9-4.Thevalueof ; max obtainedfromequation9withdielectricconstantasa parameter. Figure9-5.Thecalculatedvalueof C e ; f asafunctionofthecut-offfrequency f with asaparameter. C 0 isthecapacitanceatthemaximumfrequency experimentallymeasured. 132

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unity,theestimationof C e ; f fromequation9isrelativelyinsensitivetothevalue of ;whereas,for =0 : 5 ,anuncertaintyof2ordersofmagnitudein resultsinan uncertaintyof1orderofmagnitudein C e ; f InFigure9-3b,theimpedanceofapassivealuminumelectrodewasgiven wherethehighestmeasuredfrequency f max was30kHz.ThecorrespondingCPE parameterswere =0 : 77 and Q =1 : 7 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(5 Fcm )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 s )]TJ/F22 7.9701 Tf 6.586 0 Td [(0 : 23 .Followingequation9, C max =1 : 1 F/cm 2 .Underassumptionofadielectricconstantof11.5,aminimumlm thicknesscanbeestimatedtobe 9 nm.Followingequation9, ; max =5 : 2 10 6 cm Inafashionsimilartothedevelopmentof ; max ,aminimumvalue 0 ; min canbeobtained fromthelowestmeasuredfrequency f min =0 : 1 Hztobe 1 : 6 10 12 cm .Therefore, theminimumamplitudeofresistivityvariationwithinthealuminalayeris 5 : 2 10 6 to 1 : 6 10 12 cm .Thesevaluesfallwithintherangestypicalofsemiconductorsand insulators,respectively. TheapplicationoftheHsu-Mansfeldformula,givenasequation6,isnot possiblebecauseacapacitiveloopisnotapparentintheexperimentalresults,anda valuefor R f cannotbeestimated.Thisexampleshowsthatapplicationofapower-law distributionofresistivityallowsestimationoflmthicknessofablockinglm,forwhich theexistingformulasdonotapply.Inthepresentcase,thelowerlimitforthethickness andtheminimumrangeofresistivityvaluesaredetermined. 9.2.2StainlessSteelFinite 0 andSmall When 0 issufcientlysmall,thecharacteristicfrequency f 0 fallswithintheexperimentalfrequencyrange.Undertheseconditions,anitevalueisobtainedforthe impedanceatlowfrequencies. Anexperimentalexamplecanbegivenwiththeimpedanceofoxidesdeveloped onstainlesssteel.ExperimentaldataareshowninFigure9-6afortheimpedance responseofaFe17Crdiskpolarizedinthepassivedomainfor 1 hat-0.1Vmeasured withrespecttoamercury/mercuroussulfateelectrodeinsaturatedK 2 SO 4 indeaerated 133

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a b Figure9-6.ImpedancediagramofoxideonaFe17Crstainlesssteeldisksymbols:a experimentalfrequencyrange.Thesolidlineisthepower-lawmodel followingequation8withparameters 0 =4 : 5 10 13 cm =450cm =3 nm, =12 ,and =9 : 1 ,andthedashedlineistheCPE impedancewith =0 : 89 and Q =3 : 7 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(5 Fcm )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 s )]TJ/F22 7.9701 Tf 6.586 0 Td [(0 : 11 ;andb extrapolationtozerofrequencywherethedashedlinerepresentsthetofa Voigtmeasurementmodelandthesolidlinerepresentsthetofthe power-lawmodel. pH4,0.05MNa 2 SO 4 electrolyte.TheimpedancediagramresemblesblockingCPE behavior;however,theresponseisnotapureCPEaswasthecaseforthealuminum datapresentedabove. FromXPSanalysis,Frateur etal. 112 showedthatthepassivelmdevelopedon Fe17CrconsistedofaninnerlayerofFe 2 O 3 andCr 2 O 3 coveredbyanouterlayerof CrOH 3 andthatthethicknesswasabout 3 nm.Inthefollowing,avalueof 12 was assumedfor ,whichcorrespondstothedielectricconstantforFe 2 O 3 andCr 2 O 3 Graphicalanalysisoftheimpedanceyielded =0 : 89 and Q =3 : 7 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(5 Fcm )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 s )]TJ/F22 7.9701 Tf 6.587 0 Td [(0 : 11 Thelmthickness beingknown,equation8wasusedtoobtain =450cm whichisatypicalvalueforasemiconductor.AsshowninFigure9-6a,theimpedance 134

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a b Figure9-7.ImpedanceresponseofoxideonaFe17Crstainlesssteeldisksymbols andthetheoreticalmodellinewithparametersreportedinFigure9-6a:a therealcomponent;andbtheimaginarycomponent.Theelectrolyte resistancevaluewas 23cm 2 responsedoesnotfollowastraightline,butitsshapesuggestsanitelow-frequency impedance.AsshowninFigures9-6and9-7,introductionof 0 =4 : 5 10 13 cm intoequation8providedacalculatedresponsethatagreeswiththeexperimental results.Thisvalueof 0 correspondstoaninsulatorand,with =12 ,themaximumof theimaginaryimpedanceisexpectedtooccurat f 0 =3 : 3 mHz,whichissmallerthanthe lowestmeasuredfrequency. Ifex-situmeasurementsarenotavailablefortheevaluationoflayerthickness,only anestimateof ispossiblethroughuseofequation8duetouncertaintyinthe parameter .Equation9providesanupperboundon .Forthisexperiment,witha maximummeasurementfrequencyof100kHzandwith =12 ; max =1 : 5 10 6 cm Alowerboundfor mayalsobeestimatedonphysicalgrounds.Foranoxide,for example, isnotexpectedtobesmallerthanminimumresistivityvalueexpectedfor semiconductors, i.e., 1 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(3 cm .Usingequation8,thisconservativerangeof yieldsanestimatedlayerthicknessof =1 : 2 to12.6nm,whichencompassesthevalue of3nmobtainedfromXPS.For =0 : 89 ,anuncertaintyin of9ordersofmagnitude 135

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yieldedanuncertaintyin ofonlyoneorderofmagnitude.Inmanycases,atighter boundontheestimated maybepossible,asisthecaseforhumanstratumcorneum presentedinasubsequentsection. Incontrasttothedatapresentedforaluminumoxide,acapacitiveloopisevident intheimpedanceresponseforoxidesonsteel.Thezero-frequencyimpedancewas estimatedusinganextensionofthemeasurementmodelapproach 113 tobe R f = 0 : 756 0 : 09 M cm 2 .TheextrapolationisshowninFigure9-6b.Thecorresponding estimateoflmthicknessfromtheHsu-Mansfeldformula,givenasequation642, is =0 : 190 nm.Extrapolationtozerofrequencyofthepower-lawmodel,alsoshown inFigure9-6b,yields R f =0 : 85 M cm 2 and =0 : 187 nm.Thelmthicknessvalue obtainedfromapplicationoftheHsu-Mansfeldformulaissubstantiallysmallerthan theexperimentallymeasuredvalueof 3 nm.Incontrast,applicationofapower-law distributionofresistivityprovidesanestimationoflmthicknessthatencompassesthe experimentallydeterminedvalue. 9.2.3HumanSkinPowerLawwithParallelPath ImpedancedatafromMembrino 1 arepresentedinFigure9-8forheat-separated excisedhumanstratumcorneumimmersedin50mMbufferedCaCl 2 electrolyte for1.9hours.Ananalysisbasedonapplicationofequation6waspresented byHirschorn etal. 111 who,usinggraphicalmethods, 87 foundthat =0 : 834 and Q =5 : 36 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(8 Fcm )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 s )]TJ/F22 7.9701 Tf 6.587 0 Td [(0 : 166 .Theyreportedthattheskinthicknessestimatedusingthe capacitancefromequation6andadielectricconstantof 49 was 2 : 6 m,aboutone orderofmagnitudesmallerthantheexpectedvalue. Applicationofequation8forestimationoflmthicknessrequiresanestimate for .AsseenbythestraightlineinFigure9-8c,CPEbehaviorisevidentatthe largestmeasuredfrequencies.Thus,anupperboundfor canbeestablishedfromthe maximummeasuredfrequencyof 21 kHztobe 1 : 7 10 6 cm.Theresistivityofbody 136

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a b c Figure9-8.Theimpedanceresponsesymbolsofhumanstratumcorneumimmersedin 50mMbufferedCaCl 2 electrolytefor1.9hours.Thesolidlineisobtained followingequation8withalargevalueof 0 =49 =6 : 02 ,and =48cm andaparallelresistance R p =56kcm 2 .Thedashedlineis obtainedusingequation8with 0 =2 : 2 10 8 cm =49 =6 : 02 and =48cm :aNyquistplot;brealpartoftheimpedance;andc imaginarypartoftheimpedance. 137

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uidsis 48 cm.Ifthisvalueisconsideredtobealowerbound,therangeofpossible valuesis 1 : 7 10 6 to 48 cm.For =0 : 834 g =1 : 04 fromequation8. Thecorrespondingestimatedthicknessofskinisbetween 6 and 31 m,ingood agreementwiththeexpectedvalueof 10 )]TJ/F15 11.9552 Tf 13.02 0 Td [(40 m.Thethicknessestimatedusing equation9isinbetteragreementwithexpectedvaluesthanarethevaluesobtained byuseofequation6.Hirschorn etal. 111 explainedthatthecapacitanceobtained usingequation6doesnotaccountproperlyforthelowresistivityregionsofskin thathavecharacteristicfrequenciesoutsidethemeasuredfrequencyrange.Equation 6isbasedoncalculationofthecharacteristic RC timeconstantanddoesnot takeanyspecicdistributionofresistivityordielectricconstantintoaccount.Thebetter agreementobtainedusingequation9canbeexplainedbythefactthatitisbasedon formalsolutionfortheimpedanceassociatedwithaspeciedresistivitydistributionand requiresonlythehigh-frequencyportionofthemeasurement. Thepower-lawimpedancemodelmayalsobeappliedtoexplorethelow-frequency impedanceresponseforskin.Thelow-frequencylimitfortheimpedanceresponse expressedasequation8is Z f = g 1 )]TJ/F25 7.9701 Tf 6.587 0 Td [( 0 Thefunction g canbeeliminatedbyintroductionofequation8toyield Z f = 0 0 Q Thevalueof 0 =2 : 2 10 8 cmcanbeobtainedfromthecharacteristicfrequency f 0 =170 Hzbyusingtherelationship 0 = 1 2 0 f 0 andthemaximumvalueoftheimpedanceisobtainedfromequation9tobe Z f = 56 k cm 2 .Undertheassumptionthat =48 cmor =31 m,equation8 138

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yieldstheimpedancesimulationshownasadashedlineinFigure9-8.Asdiscussed inChapter8,theimpedanceresponseisasymmetricinaNyquistplot,yieldingCPE behaviorathighfrequencywith =0 : 834 andidealcapacitivebehavioratlowfrequency with =1 Analternativeextensiontolowfrequencyisobtainedbyconsidering 0 tobe innitelylargeandincludingaparallelpathforcurrentowwithresistance R p .Thevalue for R p wasobtainedfromthecharacteristicfrequency f 0 =170 Hzusing R p = 2 0 f 0 where,aswasusedabove, =31 m.Thevalueof 56 k cm 2 obtainedfor R p is ingoodagreementwiththevalueof Z f obtainedfortheimpedanceexpressedin termsof 0 .TheresultingimpedanceresponseisshownasasolidlineinFigure9-8. TheimpedanceresponseinthiscaseissymmetricinaNyquistplot,yieldingCPE behaviorwith =0 : 834 atbothhighandlowfrequency.Theparallelpathforcurrent owmaybeconsideredtoarisefromtransportthroughskinpores.Itisevidentthat, whilethemodelwithaparallelcurrentpathshowsbetteragreementwithexperimental data,neithermodelaccountsfullyforthecomplexityofskinbehavior.Thislackof agreement,however,doesnotinuencetheapplicationofequation9forassessing skinthickness,sincethisinterpretationrequiresonlythehighfrequencyvalues. 9.3Discussion CPEbehavioriscommonlyseenintheimpedanceresponseofelectrochemicalsystems,andthedeterminationofphysicalpropertiesfrominterpretationofthe impedanceresponseremainsachallengingproblem.Thetwoprevailingapproachesin theliteraturearethoseofBrug etal. 3 andHsuandMansfeld. 6 TheBrugformulawas developedforsurfacedistributionsofcapacitanceanddoesnotapplytothedielectric responseoflms.TheHsu-Mansfeldformulawasdevelopedfornormaldistributions,but 139

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Hirschorn etal. 111 showedthatthelmthicknessobtainedusingthisformulacanbetoo small. TheHsu-Mansfeldformulawasderivedsolelyonthepremisethat,independent oftheoriginofthetime-constantdistribution,thetimeconstantcorrespondingtothe frequencyforwhichtheimaginarypartoftheimpedancehasamaximummagnitude canbeexpressedas 0 = R f C HM .Incontrast,themodeldevelopmentinChapter8 identiedaspecicnormalresistivitydistributionthatexhibitsCPEbehavior. Theresistivityvaluesattheextremitiesofthelm, 0 and representkeyparametersinthepower-lawmodel.CPEbehaviorisseenforfrequenciesthatliebetween thecorrespondingcharacteristicfrequencies f 0 and f .Forsuchdata,neither 0 nor canbedeterminedunambiguously.Thelow-frequencybehavioratfrequenciesbelow f 0 reectsidealcapacitivebehaviorforwhich =1 .Theparameter 0 canbedetermined unambiguouslyinthiscase.Thehigh-frequencybehaviorforfrequenciesgreaterthan f alsoreectsidealcapacitivebehaviorforwhich =1 .Theparameter canbe determinedunambiguouslyinthiscase. Whilethepower-lawimpedancemaybeappliedoveracompleterangeoffrequency,aconsequenceofusingaspecicdistributionisthatphysicalpropertiescan beinferredfromthehigh-frequencyportionofthespectrum,evenfordatathatshow CPEbehaviorovertheentirehigh-frequencyrange.Graphicalmethodsdetailedby Orazem etal. 87 canbeusedtoobtaintheCPEparameters Q and .Often,thedielectricconstantisknownforspeciclmcompositions.Whileinsertionof Q ,and inequation8providesonlytheproduct 1 )]TJ/F25 7.9701 Tf 6.587 0 Td [( ,thisquantityisweaklydependent on for closetounity.Thus,anestimateforlmthicknesscanbeobtained.The examplespresentedhereshowthatthepower-lawmodelforresistivitydistributionyields estimatedvaluesthatareingoodagreementwitheithermeasuredorexpectedvalues forlmthickness.Theexamplespresentedherealsoshowsituationsforwhichthe 140

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Hsu-Mansfeldformulaeithercannotbeusedaluminumoxideoryieldsincorrectvalues forlmthicknessstainlesssteelandhumanskin. Thepresentworkshouldnotbeconsideredasubstitutefordevelopmentofsystemspecicprocessmodels.TheapproachislimitedtosystemsforwhichtheCPEbehavior canbeattributedtodistributionsoflmpropertiesinthedirectionnormaltotheelectrodesurface.Suchsystems,however,arecommonlyencountered.Thisworkmay applytothestudyofsystemsthatexhibitadistributeddielectricresponsesuchas oxides,organiccoatings,biologicalmembranes. 9.4Conclusions TheworkpresentedinChapter8showedthat,underassumptionthatthedielectric constantisindependentofposition,anormalpower-lawdistributionoflocalresistivityis consistentwiththeCPE.Ananalyticexpressionwasdeveloped,basedonthepowerlawresistivitydistribution,thatrelatesCPEparameterstothephysicalpropertiesof alm.Inthepresentwork,thisexpressionyieldedphysicalproperties,suchaslm thicknessandresistivity,thatwereingoodagreementwithexpectedorindependently measuredvaluesforsuchdiversesystemsasaluminumoxides,oxidesonstainless steel,andhumanskin. Forthepower-lawresistivitydistribution,CPEbehaviorwasseenformeasurements madebetweenthecharacteristicfrequencies f 0 and f .Thepower-lawdistributionof localresistivitywasshowntoyieldidealcapacitivebehavioratfrequenciesthatare sufcientlylargeandsufcientlysmall,ascanbeexpectedforanymodelthataccounts foradistributedlmresistivity.AsymmetricCPEresponsewasobtainedbyaddinga parallelresistivepathway.InChapter10,surfacedistributionsofresistancethatcan accountfortheCPEareconsidered. 141

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CHAPTER10 CPEBEHAVIORCAUSEDBYSURFACEDISTRIBUTIONSOFOHMICRESISTANCE Apower-lawdistributionofresistivityinthedirectionnormaltotheelectrodewas showninChapter8toprovideaphysicaloriginoftheCPE.TheCPEparameterswere relateddirectlytophysicalpropertiesofthesystem.However,inmanycases,theCPE parametersareconsideredtoarisefromadistributionoftime-constantsalongthe surfaceofanelectrode.Generally,theCPEisconsideredtoarisefromadispersionof capacitance.TheapproachtakenbyBrug etal. wastoassumethattheCPEoriginated fromasurfacedistributionoftimeconstantswithuniformOhmicandcharge-transfer resistances. 3 Thetime-constantdispersionthereforewasassumedtooriginatefrom adistributionofcapacitance.However,theresultingdistributionrequiredarangeof capacitanceovermanyordersofmagnitudethatisnotconsideredreasonablefor experimentalsystems. Theresistivepropertiesofasurfacemayinfactvarysignicantlyduetoelectrode geometricaleffects.Huang etal. haveshownthatpseudo-CPEbehavior,where and Q arefrequencydependent,canarisefromgeometricaleffectsduetocurrent andpotentialdistributionsonideallypolarizableelectrodesurfaces. 18 Thecurrent distributionscanbeinterpretedasadistributionofOhmicresistancealongthesurface. However,thecurrentdistributionspresentedbyHuang etal. arenotrepresentative oftheCPEduetothefrequencydependenceof and Q andthelimitedrangeof frequencyforwhichthepseudo-CPEbehaviorapplies. Inthepresentchapter,surfacedistributionsofresistancewithuniformcapacitance arederivedthatresultinimpedanceresponsesthatareconsistentwiththeCPE. Themathematicsaredevelopedforageneralelectrodegeometryandthenapplied specicallyforadiskelectrode. 10.1MathematicalDevelopment Asurfacedistributionofblockingelementswithauniformsurfacecapacitanceis showninFigure10-1.Thegeneralexpressionfortheadmittanceofablockingsurface 142

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Figure10-1.Asurfacedistributionofblockingelementswithauniformdistributionof localcapacitance. withuniformcapacitanceoveranincrementalareaiswritten Y i = dA R A + j!C 0 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 where R A isafunctionofareawithunitsof cm 2 and C 0 isuniformwithunitsof F= cm 2 ,respectively.Therefore, Y i hasunitsof )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 .Thesumoftheadmittanceoverthe entiresurfaceisexpressedbytheintegral Y s = Z A T 0 dA R A + j!C 0 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 where A T isthetotalsurfacearea.Thesum Y s istheunscaledadmittanceanddividing bythesurfacearea A T isrequiredtoobtainedscaledunitsofadmittance.Equation 10canbewrittenintermsofadimensionlessparameteras Y = Z 1 0 d ^ A R ^ A + j!C 0 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 where ^ A = A=A T and Y = Y s =A T suchthattheadmittanceisscaled.Asshownby equation10,asfrequencytendstowardinnitytheobservedOhmicresistanceis obtainedas Y 1 = R )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 e = Z 1 0 R ^ A )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 d ^ A 143

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andasthefrequencytendstowardzerothecapacitanceisobtainedas Y j! = C 0 InChapter8,thedevelopmentofanormaltime-constantdistributionthatisconsistentwiththeCPEwasaidedbyregressionofsyntheticdatatosequential RC elements. Forauniformcapacitanceitwasshownthatapower-lawdistributionofresistancewas necessarytoachievetheCPEresponse.Theaidofregressionwasnotrequiredforthe followingdevelopmentofasurfacedistributionthatisconsistentwiththeCPE. Theimpedanceresponseofanindividual RC yieldsanidealsemi-circlewhen representedintheimpedanceplane.Foraseriescontributionofindividualelements thesummationyieldsthetotalimpedance.Theadmittanceresponseofanindividual blockingelementyieldsanidealsemi-circlewhenrepresentedintheadmittance plane.Foraparallelcontributionofindividualelementsthesummationyieldsthetotal admittance.Forthesetwocases,asimilardistributionoftime-constantsshouldyield similarresponsesintheimpedanceandadmittanceplanes,respectively.Therefore,itis inferredthatapower-lawdistributionofsurfaceresistancewillyieldaCPE. Adistributionofresistancewhichprovidesaboundedvalueforresistanceis proposedtobe R = R s + R b )]TJ/F24 11.9552 Tf 11.955 0 Td [(R s ^ A where R b and R s areboundaryvaluesofresistance.Introductionofequation10into equation10yields Y = Z 1 0 1 a + b ^ A d ^ A where a = R s + j!C 0 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 and b = R b )]TJ/F24 11.9552 Tf 11.955 0 Td [(R s 144

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Ananalyticsolutiontoequation10ispossibleforsomeintegervaluesof .For example,when =3 Y = k 3 a 1 2 log k +1 3 1+ k 3 + p 3arctan 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(k k p 3 + p 3 6 # where k = a =b 1 = 3 .Undertheconditionthat R b >>R s k islessthan1for !> R b C 0 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 andequation10reducesto Y = 2 3 p 3 R )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 = 3 b R s + j!C 0 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 = 3 Theimpedanceofthesystem Z = Y )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 is Z = 3 p 3 2 R 1 = 3 b R s + j!C 0 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 = 3 Equation10isderivedforthespecialcase =3 .Ageneralexpressionofthe impedancecanbeproposedinthesameformas Z = 1 g R 1 = b R s + j!C 0 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 )]TJ/F26 5.9776 Tf 7.782 3.693 Td [( )]TJ/F23 5.9776 Tf 5.756 0 Td [(1 where g isafunctionof and,inthecaseof =3 g = 2 3 p 3 .Thecomparisonofequation 10tothenumericalintegrationofequation10showsthatthisexpressionis generalandcanbeappliedforall > 2 overbroadrangeoffrequencies.Detailsofthe algorithmusedforthenumericalintegrationareprovidedinAppendixC. Therealandimaginarypartsoftheimpedanceobtainedbynumericalintegrationof equation10arepresentedinFigures10-2aand10-2b,respectively,with asa parameter.Thelinesrepresenttheevaluationofequation10wherethenumerical valueof g wasobtainedatthehigh-frequencylimitofequation10accordingtothe expression g = R 1 = b R )]TJ/F23 5.9776 Tf 5.756 0 Td [(1 s Z 1 andwhere Z 1 isobtainedfromnumericalintegrationofequation10at = 1 Equation10providesgoodagreementwithnumericalintegrationofequation 145

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a b Figure10-2.Acomparisonoftheimpedanceresponsegeneratedbynumerical integrationofequation10symbolsandtheanalyticexpression providedbyequation10lineswith R b =1 10 7 cm 2 R s =1 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 cm 2 C 0 =10 F= cm 2 ,and asaparameter:athereal componentofimpedance;andbtheimaginarycomponentofimpedance. 10forfrequenciesaboveacharacteristicfrequency f b = 1 2 R b C 0 AsshownbyFigure10-3,thevalueof g isidenticaltothevalueof g obtainedfromthe normalpower-lawdistributionpresentedinChapter8,exceptthatinthiscase g appears inthedenominatoroftheimpedanceexpression. Forfrequenciesbelowacharacteristicfrequency f s < R s C 0 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 equation10isintheformoftheCPE, i.e., Z = R 1 = b g j!C 0 )]TJ/F23 5.9776 Tf 5.756 0 Td [(1 = 1 j! Q 146

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Figure10-3.Thenumericalevaluationof g asafunctionof 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [( wherethesymbols representtheresultsfromequation10.Thelinerepresentsthe interpolationformulagivenasequation8. Therefore,equation10yieldstheimpedancegivenbyequation1for R b C 0 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 f s whichis thefrequencyrangeatwhichequation10applies.AsshownbyFigure10-2b,the capacitanceisobtainedat f
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10.2DiskElectrodes Intheprevioussectiontheimpedancewasderivedintermsofageneralarea A Inthefollowingsectiontheobjectiveistoderivetheimpedanceresponseintermsofa diskelectrode.Twocasesareconsidered.First,thecasewherethemagnitudeofthe resistanceincreaseswithincreasingradius;andsecond,thecasewherethemagnitude oftheimpedancedecreaseswithincreasingradius. 10.2.1Increaseofresistancewithincreasingradius Foradisk ^ A = r 2 =r 2 0 andequation10andequation10yield Y = 2 r 2 0 Z r 0 0 rdr R s + R b )]TJ/F24 11.9552 Tf 11.955 0 Td [(R s r r 0 2 + j!C 0 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 AsshowninFigure10-4,asimulatedimpedanceresponsefollowingequation10 isconsistentwiththeCPE.AsshowninFigure10-4c,thegraphicallydeterminedvalue of isfrequencyindependentfor f
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a b c d Figure10-4.Thesimulationresultsfollowingequation10with R b =1 10 10 cm 2 R s =1cm 2 C 0 =10 F= cm 2 ,and asaparameter:athereal componentofimpedance;btheimaginarycomponentofimpedance;c thegraphicallydeterminedvalueof ;anddtheresistivitydistributions followingequation10. 149

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Thenumericalmethodrequiresdividingthediskintoincrementalrings.The admittanceoftheouterincrementalareacalculatedfromequation106issubstituted asthevalueofadmittanceintheinnerincrementalareaofthedisk.Thisprocedureis repeatedforeveryincrementalarea.Therefore, ^ A n )]TJ/F25 7.9701 Tf 6.587 0 Td [(k )]TJ/F15 11.9552 Tf 15.043 3.022 Td [(^ A n )]TJ/F22 7.9701 Tf 6.587 0 Td [( k +1 = r 2 k +1 )]TJ/F24 11.9552 Tf 11.956 0 Td [(r 2 k and Y k = Z ^ A n )]TJ/F26 5.9776 Tf 5.756 0 Td [(k ^ A n )]TJ/F23 5.9776 Tf 5.756 0 Td [( k +1 d ^ A R s + R b )]TJ/F24 11.9552 Tf 11.955 0 Td [(Rs ^ A =2 Z r k +1 r k y k r rdr where n isthetotalnumberofincrementalboundariesincludingthediskcenterand edgeand k representstheindividualboundaries.Thevalueof k isanintegerthat rangesbetween0and n )]TJ/F15 11.9552 Tf 12.76 0 Td [(1 .Alinearapproximationisassumedfortheadmittance between r k and r k +1 as y k r = m k r + c k where m k and c k areconstants.Thevalueof c k istheinnerboundaryadmittance,for instance, c 0 = R )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 b .Substitutionofequation10intoequation10yieldsan expressionwithasingleunknown m k .Thenumericalprocedureallowsforthecalculationofanadmittancevalueateveryringboundary k .Thecorrespondingresistance values, i.e., R k = y k r k )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ,yieldadistributionthatdecreaseswithincreasingradius. Thenumericallydeterminedresistancedistribution R r canbeusedtoobtaintheglobal admittanceofthediskas Y = 2 r 2 0 Z r 0 0 rdr R r + j!C 0 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 DetailsofthealgorithmisprovidedinAppendixD. Thesimulatedresultsfollowingequation10arepresentedinFigure10-5. TheimpedanceresponseshowninFigures10-5aand10-5bareidenticaltoFigures 10-4aand10-5b,respectively.However,inthiscase,theresistancedistribution thataccountsfortheimpedancedecreaseswithincreasingradius,asshowninFigure 150

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a b c d Figure10-5.Thesimulationresultsfollowingequation10with R b =1 10 10 cm 2 R s =1cm 2 C 0 =10 F= cm 2 ,and asaparameter:athereal componentofimpedance;btheimaginarycomponentofimpedance;c thegraphicallydeterminedvalueof ;anddthenumericallydetermined resistivitydistributions R r 151

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10-5d.Theresistanceapproachesaniteresistance R s attheouterareaofthedisk. AbroaddistributionofOhmicresistanceisnecessarytoaccountfortheCPEoverthe frequencyrangeof1mHzto100kHz. 10.3Conclusions ThesurfacedistributionsofOhmicresistancenecessarytoaccountforCPE behaviorwerederivedforcaseswherethesurfacecapacitancecanassumedtobe uniform.AbroaddistributionisrequiredtoaccountfortheCPEoveranextended frequencyrange.ThedistributionsofOhmicresistanceduetogeometricaleffectsofa diskelectrodederivedbyHuang etal. arenarrowincomparison. 18 Forexperimental systems,abroaddistributionofOhmicresistanceisnotexpected.Therefore,itis unlikelythattheCPEcanbeattributedtoadistributionofOhmicresistancealone. Nevertheless,theexpressionfor Q givenbyequation10isfoundtobeconsistent withthegeneralformofcapacitance-CPEparameterrelations,asisdiscussedin Chapter11. 152

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CHAPTER11 OVERVIEWOFCAPACITANCE-CPERELATIONS InChapter6,anexpressionforcalculatingcapacitancefromCPEparameters followingtheworkofBrug etal. 3 wasderivedas C B = Q 1 = R e R t R e + R t )]TJ/F25 7.9701 Tf 6.586 0 Td [( = where R t isthecharge-transferresistance, R e istheOhmicresistance,and Q and are theCPEparameters.AdifferentexpressionfollowingtheworkofHsuandMansfeld 6 wasderivedas C HM = Q 1 = R )]TJ/F25 7.9701 Tf 6.586 0 Td [( = f where R f representstheresistanceofalm.Asdiscussedearlier,theseexpressions werederivedwithoutconsiderationofaphysicalmodel.InChapter9,foralmwitha power-lawdistributionofresistivityanduniformdielectricconstant,thecapacitancewas relatedtophysicalpropertiesbyequation9.Equation9canbewritten C e ; f = Q 1 = R )]TJ/F25 7.9701 Tf 6.587 0 Td [( = g 1 = where R = and g isafunctiondependentonlyon thatwasexpressedbyequation 8.InChapter10,theCPEparameter Q wasrelatedtomodelparametersfora surfacedistributionofOhmicresistancebyequation10.Equation10canbe written C 0 = Q 1 = R )]TJ/F25 7.9701 Tf 6.587 0 Td [( = b g )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 = where R b isthelargestvalueofresistanceinthesurfacedistributionofblockingelements. Equations11-11allhavethesameform,buttheresistancevaluesusedin theexpressionshavedifferentmeanings.Clearly,considerationofthephysicaloriginof theCPEisrequiredfordeterminingcapacitancefromimpedancespectra.Byanalog,it 153

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Figure11-1.Changeincapacitanceasafunctionofachangein R with Q heldconstant followingequation11;andchangeincapacitanceasafunctionofa changein Q with R heldconstantfollowingequation11.Unitsare arbitrary. maybereasonabletosuspectthatthegeneralformofthecapacitance-CPErelationis C = Q 1 = R )]TJ/F25 7.9701 Tf 6.587 0 Td [( = g 1 = wherethemeaningof R dependsonthesystemunderconsiderationand g isasystemdependentfunctionthatdependsonlyon .For valuesclosetounity,the R )]TJ/F25 7.9701 Tf 6.586 0 Td [( = termisinsensitivetochangesin R .AsshowninFigure11-1,alargechangein R has littleeffectonthecalculatedcapacitance,whereas,achangein Q hasasignicant effectonthecalculatedcapacitance.Therefore,for closetounity,changesincapacitanceofasystemcanbedeterminedapproximatelyfromchangesofthemeasuredCPE parametersas C Q 1 = Equation11maybeusefulinassessingchangesinactivesurfaceareaorlayer thicknessforchangingsystems.Forinstance,considerasystemwhereequation11 appliesand =0 : 9 .Ifadielectriclayerdoublesinthicknessthen R doublesand C e ; f 154

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ishalved.Withoutaprioriknowledgeofthesystem,usingequation11willresultin anerroroflessthan8percentincalculatingthechangeincapacitance.Thisresultis duetotheinsensitivityofthe R )]TJ/F25 7.9701 Tf 6.587 0 Td [( = term.Achangeinsystemcapacitanceduetolm growthmanifestsitselfprimarilyinachangeinthevalueof Q 155

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CHAPTER12 CONCLUSIONS Thepresentworkshowedthattheempiricalconstant-phaseelement,generally usedfordata-tting,canbeinterpretedintermsofphysicalproperties.Underassumptionofauniformdielectricconstant,anormalpower-lawdistributionoflocalresistivity isconsistentwiththeCPE.AnalyticexpressionsthatrelatetheCPEttingparameters, and Q ,tophysicalpropertiesofalmweredeveloped.Forsuchdiversesystemsas aluminumoxides,oxidesonstainlesssteel,andhumanskin,theseexpressionsyielded physicalproperties,suchaslmthicknessandresistivity,thatwereingoodagreement withexpectedorindependentlymeasuredvalues.Thepower-lawimpedancemodelmay haveextensiveutilityforcharacterizingabroadrangeofsystemswhereavariationin propertiesisexpectedinthedirectionnormaltotheelectrode. TheanalyticexpressiondevelopedinthepresentworkrelatingCPEparameters tophysicalpropertiesmayalsobeusedtopredictdielectricconstantsofthinlms.In thepresentwork,layerthicknesswaspredictedbyusingliteraturevaluesofdielectric constant.Conversely,forsystemswithunknownvaluesofpermittivity,thedielectric constantmaybecalculatedfromindependentlymeasuredvaluesoflmthicknessusing thedevelopedmodel. Inthepresentwork,apower-lawdistributionofOhmicresistancealongablocking surfacewithuniformcapacitancewasshowntobeconsistentwiththeCPE.Thebroad distributionthatisnecessarytoaccountfortheCPEisnotexpectedexperimentally. Therefore,observationofCPEbehaviorcannotbeconsideredtoarisefromadistributionofOhmicresistancealone.Nevertheless,thedevelopedrelationshipbetween capacitanceandCPEparameterswasshowntobeinageneralformthatwasconsistentwithothercapacitance-CPErelations.However,thevalueofresistancethatshould beusedintherelationsdependsonthesourceoftheCPE.Therefore,considerationof thephysicaloriginoftheCPEisrequiredfordeterminingcapacitancefromimpedance spectra. 156

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Interpretingimpedancespectraintermsofphysicalpropertiesrequiresthatexperimentalanddataanalysistechniquesareproperlyimplementedandfullyoptimized. InsupportofthemechanisticdevelopmentoftheCPE,anintegratedapproachwas developedforidentifyingandminimizingnonlineardistortionsinimpedancespectrafor increasedcondenceinmodeldevelopmentandparameterestimation.Acharacteristic transitionfrequencywasdenedthatcanbeusedtotailorafrequency-dependent inputsignaltooptimizesignal-to-noiselevelswhilemaintainingalinearresponse.The Kramers-Kronigrelations,usedfordetectingnonlinearresponses,arenotsatisedfor measurementswhichincludethecharacteristictransitionfrequency.Therefore,the transitionfrequencycanbeusedasatoolforbothoptimizingexperimentaldesignand understandingtheutilityoftheKramers-Kronigrelationswhenverifyingimpedancedata forconsistency. 157

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CHAPTER13 SUGGESTIONSFORFUTUREWORK Distributionsofphysicalpropertiesweredevelopedthatcouldaccountforthe CPEbehaviorobservedintheimpedanceresponsesofelectrochemicalsystems.The distributionsofresistivityandOhmicresistance,presentedinthiswork,represents onlyasmallportionofthemanypossiblepropertydistributionsthatmaybeconsistent withtheCPE.Possibilitiesforadditionalinvestigationarepresentedbelow,including preliminaryworkforCPEbehaviorcausedbysurfacedistributionsofreactivity. 13.1CPEBehaviorCausedbySurfaceDistributionsofReactivity Brug etal. 3 developedsurfacedistributionsoftime-constantsthataccountedfor theCPEwherecapacitancewasdistributedandOhmicandkineticresistanceswere uniform.Inthepresentwork,surfacedistributionsofblockingelementsthataccounted fortheCPEweredevelopedwheretheOhmicresistancewasdistributedandthe surfacecapacitancewasuniform.LargedistributionsofsurfacecapacitanceorOhmic resistancearenotexpectedinrealsystems.However,forareactivesystemalarge distributionofkineticresistancesmaybereasonable.Adistributionofkineticresistances thatcanaccountfortheCPEwasnotconsideredinthepresentwork.Developingsuch adistributionwouldbeasignicantcontributionasitwouldprovideimportantinsightinto theactiveregionsofsurfaces. AsshowninChapter6,intheabsenceofanOhmicresistanceadistributionof time-constantsalongasurfacereducestoasingleeffectivetime-constant.Therefore, theOhmicresistancemustplayaroleintheobservationoftheCPE.However,the conditionsunderwhichtheCPEwillbeobservableforsuchsystemswerenotexamined andthedistributionsofsurfacepropertiesnecessarytoaccountfortheCPEwere notdeveloped.Preliminaryworkisprovidedherefordevelopingadistributionof reactivitythatcanaccountfortheappearanceoftheCPEandfordevelopingthesystem conditionsunderwhichtheCPEmaybeobservable. 158

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13.1.1MathematicalDevelopment Asurfacedistributionoftime-constantsisshowninFigure6-1a.Thegeneral expressionfortheadmittanceofareactivesurfaceoveranincrementalareacanbe written Y i = dA R e )]TJ/F24 11.9552 Tf 63.8 8.088 Td [(dA R e + R 2 e R )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 t + j!R 2 e C 0 where R t A isafunctionofpositionwithunitsof cm 2 ,and R e and C 0 areuniform withunitsof cm 2 and F= cm 2 ,respectively.Thesumoftheadmittanceovertheentire surfaceisexpressedbytheintegral Y s = Z A T 0 dA R e )]TJ/F29 11.9552 Tf 11.956 16.272 Td [(Z A T 0 dA R e + R 2 e R )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 t + j!R 2 e C 0 where A T isthetotalsurfacearea.Thesum Y s istheunscaledadmittanceanddividing bythesurfacearea A T isrequiredtoobtainscaledunitsofadmittance.Equation13 canbewrittenintermsofadimensionlessparameteras Y = Z 1 0 d ^ A R e )]TJ/F29 11.9552 Tf 11.955 16.273 Td [(Z 1 0 d ^ A R e + R 2 e R )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 t + j!R 2 e C 0 where ^ A = A=A T and Y = Y s =A T suchthattheadmittanceisscaled.Adistributionof resistanceisproposedtobe R t = R t ; m ^ A )]TJ/F25 7.9701 Tf 6.587 0 Td [( where R t ; m representstheminimumvalueofcharge-transferresistancealongthe surface.Equation13isunboundedbecauseitisenvisionedthattherecouldbe portionsofthesurfacethatareinactiveandthereforehaveaninnitecharge-transfer resistance.Introductionofequation13intoequation13yields Y = 1 R e )]TJ/F29 11.9552 Tf 11.955 16.272 Td [(Z 1 0 1 a + b ^ A d ^ A where a = R e + j!R 2 e C 0 159

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and b = R 2 e R t ; m Ananalyticsolutiontoequation13ispossibleforsomeintegervaluesof .For example,when =3 Y = 1 R e )]TJ/F24 11.9552 Tf 15.904 8.087 Td [(k 3 a 1 2 log k +1 3 1+ k 3 + p 3arctan 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(k k p 3 + p 3 6 # where k = a =b 1 = 3 .Undertheconditionthat R t ; m =R e issmall, k islessthan1for !< R t ; m C 0 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 andequation13reducesto Y = 1 R e )]TJ/F15 11.9552 Tf 44.697 8.088 Td [(2 3 p 3 b 1 = 3 a 2 = 3 = 1 R e )]TJ/F15 11.9552 Tf 17.524 8.088 Td [(2 3 p 3 R 2 e =R t ; m )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 = 3 R )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 = 3 e + j!R e C 0 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 = 3 Forfrequencies !> R e C 0 )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 ,theimpedanceofthesystem Z = Y )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 is Z = 1 1 R e )]TJ/F22 7.9701 Tf 27.909 4.707 Td [(1 T j! 2 = 3 where T = 3 p 3 R 2 e C 2 = 3 0 2 R 1 = 3 t ; m .Equation13canbewritten Z = 1 c )]TJ/F24 11.9552 Tf 11.955 0 Td [(jd where c = 1 R e )]TJ/F25 7.9701 Tf 13.186 4.707 Td [(! )]TJ/F23 5.9776 Tf 5.756 0 Td [(2 = 3 T cos )]TJ/F22 7.9701 Tf 10.494 4.707 Td [(2 3 2 and d = )]TJ/F23 5.9776 Tf 5.756 0 Td [(2 = 3 T sin )]TJ/F22 7.9701 Tf 10.494 4.707 Td [(2 3 2 .Multiplicationofequation13 byitscomplexconjugateyields Z = 1 R e D )]TJ/F24 11.9552 Tf 13.15 8.088 Td [(! )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 = 3 TD cos )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(2 3 2 + j )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 = 3 TD sin )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(2 3 2 where D = 1 R 2 e )]TJ/F15 11.9552 Tf 13.15 8.087 Td [(2 )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 = 3 R e T cos )]TJ/F15 11.9552 Tf 10.494 8.087 Td [(2 3 2 + )]TJ/F22 7.9701 Tf 6.586 0 Td [(4 = 3 T 2 Forlargefrequencies, i.e., intherange !> R e C 0 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ,thesecondandthirdtermsin equation13arenegligibleandtheimaginarycomponentofequation13 reducesto Z j = )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 = 3 Q s sin )]TJ/F15 11.9552 Tf 10.494 8.087 Td [(2 3 2 160

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where Q s = 3 p 3 C 2 = 3 0 2 R 1 = 3 t ; m .Equation13isintheformoftheimaginarycomponentofthe CPEwith =2 = 3 .Therealcomponentofequation13doesnotappeartobeinthe sameformastherealcomponentoftheCPE.However,theCPEisconsistentwiththe Kramers-Kronigrelations,therefore,thefactthat Z j = Z j; CPE requiresthat Z r = Z r; CPE Equation13isderivedforthespecialcase =3 .Ageneralexpressionofthe impedance,inthefrequencyrange R e C 0 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1
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a b c Figure13-1.Acomparisonoftheimpedanceresponsegeneratedbynumerical integrationofequation13symbolsandtheanalyticalexpression providedbyequation13lineswith R t ; m =1 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(4 cm 2 C 0 =10 F= cm 2 R e =10cm 2 ,and asaparameter:athereal componentofimpedance;btheimaginarycomponentofimpedance;and cthegraphicallydeterminedvalueof .Thesymbolsrepresent calculationsperformedfor =3 =4 ,and 4 =6 : 67 162

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Figure13-2.Thegraphicallydeterminedvalueof foranimpedanceresponse generatedbynumericalintegrationofequation13with R t ; m =1 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(4 cm 2 C 0 =10 F= cm 2 =4 ,and R e asaparameter.The symbolsrepresentcalculationsperformedfor R e =1cm 2 R e =10cm 2 4 R e =100cm 2 TheeffectofOhmicresistanceonthefrequencyrangewheretheCPEisobservable isshowninFigure13-2.AsshowninFigure13-2,theCPEbecomesobservableat approximatelyonefrequencydecadegreaterthan f R e C 0 .Singletime-constantbehavior isobservedatfrequencieslessthan f R e C 0 Typically,theexperimentallyassessablefrequencyrangeislessthan100kHz. Therefore, f R e C 0 musthaveavalueoflessthan10kHzforCPEcausedbyadistribution ofreactivitytobeobservableintheassessablefrequencyrange.Foratypicalvalueof thedoublelayer, i.e., C 0 =10 F= cm 2 ,theOhmicresistanceofthesystemmustbeat least R e =1 : 6cm 2 toobservetheCPE.ObservationoftheCPEatfrequenciesless than f R e C 0 mustbeattributedtoaphysicalsourceotherthanadistributionofreactivity. OnesuchsourceisadistributionofresistivitywithinalmprovidedinChapter8. 163

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13.2CPEBehaviorCausedbyNormalDistributionsofProperties Inthepresentwork,apower-lawdistributionoflocalresistivitywithinalmwas showntobeconsistentwiththeCPEundertheconditionthatthedielectricconstantis independentofposition.However,foraninhomogeneouslayer,itmayalsobeexpected thatthedielectricconstantvarieswithposition.Forinstance,forhumanskin,itmaybe expectedthatthattheouterfattylayerhasadielectricconstantconsistentwithlipids, i.e., 2 ,andthelayeradjacenttobodyuidshasadielectricconstantconsistentwith salinesolution, i.e., 50 .Anon-uniformdistributionofdielectricconstantwillcause theresistivitydistributionnecessarytoaccountfortheCPEtodifferfromthatofthe power-law.Anon-uniformdielectricconstantwasnotconsideredinthepresentwork andtheinuenceofaspecieddistributionofpermittivityontheinterpretationofthe CPEoffersanareaforadditionalinvestigation. 164

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APPENDIXA PROGRAMCODEFORLARGEAMPLITUDEPERTURBATIONS TheMATLABalgorithmusedforthenumericalsolutionofthenonlinearcircuit modelsdiscussedinSection3.2ispresentedbelow. functionlargeperturbation ZR=0; ZJ=0; wM=0; Vamp=0; Rtmean=0; forx=-1:.2:5 deltaU=.1;[inputpotentialamplitude: V ] Cd=100e-6;[capacitance: F= cm 2 ] Re=1;[Ohmicresistance: cm 2 ] ba=19;[anodiccoefcient: V )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 ] bc=19;[cathodiccoefcient: V )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 ] Vbar=0;[steady-stateinterfacialpotential: V ] io=1e-3;[exchangecurrentdensity: A= cm 2 ] w=10 x ; f=w/2/pi;[frequency:Hz] cyclet=1/f; scale=2000; unittime=cyclet/scale; segfac=1; numdatapercycle=cyclet/unittime; Rt1=1/io*ba*expba*Vbar+bc*exp-bc*Vbar; t=[0:unittime:segfac*unittime]';[time:s] Rt1nal=1/io*ba*expba*Vbar+bc*exp-bc*Vbar; 165

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ifardiffnal=2; whileabsifardiffnale-14 Anal=deltaU Rt1nal+Re = Rt1nal Cd Re 2 = Rt1nal+Re 2 = Rt1nal Cd Re 2 +w 2 ; Vnal=Anal*sinw*t+w*Cd*Re*Rt1nal/Rt1nal+Re*cosw*t; ifar1nal=Vnal/Rt1nal; Vnal=Vnal,1; ifar1nal=ifar1nal,1; ifarexpnal=io*expba*Vnal-exp-bc*Vnal; Rt2nal=Vnal/ifarexpnal; Rt1nal=Rt2nal; ifardiffnal=ifar1nal-ifarexpnal; end tnal=0; factor=roundnumdatapercycle/segfac; forloop1=1:2*factor ifardiff=2; whileabsifardiffe-14 A=deltaU Rt1+Re = Rt1 Cd Re 2 = Rt1+Re 2 = Rt1 Cd Re 2 +w 2 ; V=A*sinw*t+w*Cd*Re*Rt1/Rt1+Re*cosw*t; ifar1=V/Rt1; V=VlengthV,1; ifar1=ifar1lengthifar1,1; ifarexp=io*expba*V-exp-bc*V; Rt2=V/ifarexp; Rt1=Rt2; ifardiff=ifar1-ifarexp; 166

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end t,:=[]; tnal=[tnal;t]; tnext=[tlengtht,1:unittime:tlengtht,1+segfac*unittime]'; Vbar=VlengthV,1; Rt1=1/io*ba*expba*Vbar+bc*exp-bc*Vbar; t=tnext; Vnext=V; Vnal=[Vnal;Vnext]; ifar1next=ifar1; ifarexpnext=ifarexp; ifar1nal=[ifar1nal;ifar1next]; ifarexpnal=[ifarexpnal;ifarexpnext]; end assignin'base','tnal',tnal Rtmatrix=Vnal./ifar1nal; assignin'base','Rtmatrix',Rtmatrix RtM=meanRtmatrix; Rtmean=[Rtmean;RtM]; V=deltaU*sinw*tnal; ic=V-Vnal-ifar1nal*Re/Re; iT=ifar1nal+ic; ifarcheck=io*expba*Vnal-exp-bc*Vnal; assignin'base','iT',iT assignin'base','V',V iTmax=maxiT; Vmax=maxV; 167

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LissM=[iT/iTmax,V/Vmax]; assignin'base','LissM',LissM MiT=[tnal,iT]; lengthMiT; l=roundnumdatapercycle; go=20; MV=[tnal,V]; [CV1 ; IV1]=maxMV:roundnumdatapercycle ; 2 ; [CV2 ; IV2]=maxMVIV1+go:2 roundnumdatapercycle ; 2 ; index11=IV1; index22=IV1+IV2+go-1; sine=sinw*tnal; cose=cosw*tnal; integrandVr=Vindex11:index22.*sineindex11:index22; integrandVj=Vindex11:index22.*coseindex11:index22; integrandIr=iTindex11:index22.*sineindex11:index22; integrandIj=iTindex11:index22.*coseindex11:index22; AVr=cumtrapztnalindex11:index22,integrandVr; Vr=AVrlengthAVr; AVj=cumtrapztnalindex11:index22,integrandVj; Vj=AVjlengthAVj; AIr=cumtrapztnalindex11:index22,integrandIr; Ir=AIrlengthAIr; AIj=cumtrapztnalindex11:index22,integrandIj; Ij=AIjlengthAIj; assignin'base','Ij',Ij; assignin'base','Ir',Ir; 168

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j=sqrt-1; Zr=realVr+j*Vj/Ir+j*Ij; Zj=imagVr+j*Vj/Ir+j*Ij; ZR=[ZR;Zr];[impedance: cm 2 ] ZJ=[ZJ;Zj];[impedance: cm 2 ] wM=[wM;w]; Vmax=maxVnal; Vamp=[Vamp;Vmax]; end Vamp,:=[]; ZR,:=[]; ZJ,:=[]; wM,:=[]; Rtmean,:=[] w=wM; fdet=w/2/pi; assignin'base','w',w assignin'base','ZR',ZR; assignin'base','ZJ',ZJ; assignin'base','fdet',fdet assignin'base','Vamp',Vamp assignin'base','Rtmean',Rtmean gure;plotZR,ZJ,'o';axisequal; gure;loglogfdet,ZJ,'o'; gure;loglogfdet,ZR,'o'; end 169

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APPENDIXB PROGRAMCODEFORNORMALDISTRIBUTIONS TheMATLABcodeusedforsimulatingtheimpedanceresultingfromanormal power-lawdistributionofresistivityasdiscussedinChapter8ispresentedbelow.The programnumericallyintegratesequation8andcomparestheresultwithequation 8. functionpowerlaw e=10;[dielectricconstant] e0=8.8542e-14;[permittivityofvacuum:F/cm] delta=100e-7;[lmthickness:cm] alphai=0.667;[CPEparameter] gamma=1/-alphai;[power-lawexponent] rhodelta=1e5;[interfacialresistivity: cm ] rho0=1e18;[interfacialresistivity: cm ] start=1e-12; nish=1; startlog10=log10start; nishlog10=log10nish; inc=.025; dlog10=[startlog10:inc:nishlog10]'; xxN=10 : dlog10 ;[normalizedposition:cm] xx=xxN*delta;[position:cm] res=rhodelta xxN : )]TJ/F22 7.9701 Tf 6.586 0 Td [(gamma ; resMMM=res/rhodelta; res=1.//rho0+/rhodelta-1/rho0*1./resMMM;[resistivity: cm ] fdet=0; ZM=0; forx=-3:.1:5 170

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f=10 x ;[frequency:Hz] w=2*pi*f; integrand=1././res+j*w*e*e0; AT=cumtrapzxx,integrand; Aint=ATlengthAT; Zi=Aint; ZM=[ZM;Zi]; fdet=[fdet;f]; end fdet,:=[]; ZM,:=[];[impedance: cm 2 ] wdet=2*pi*fdet; Zr=realZM; Zj=imagZM; alphaM=0; QM=0; forx=1:1:lengthfdet-1 alpha1=log10-Zjx+1,1-log10-Zjx,1/log10fdetx+1,1-log10fdetx,1; alphaM=[alphaM;alpha1]; Q1=wdetx ; 1 alpha1 = Zjx ; 1 sinalpha1 pi = 2 ; QM=[QM;Q1]; end g=1+2 : 88261 )]TJ/F15 11.9552 Tf 11.955 0 Td [(alphai 2 : 37476 ; Zcalc=g delta rhodelta = gamma := rho0 )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 +j wdet e e0 : gamma )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 = gamma ; Zcalcr=realZcalc; Zcalcj=imagZcalc; gure;semilogxfdet,alphaM,'o' 171

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gure;semilogxfdet,QM,'o' gure;plotZr,-Zj,'o',Zcalcr,-Zcalcj;axisequal gure;loglogfdet,Zr,'o',fdet,Zcalcr gure;loglogfdet,-Zj,'o',fdet,-Zcalcj gure;loglogxx,res,'o' end 172

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APPENDIXC PROGRAMCODEFORSURFACEDISTRIBUTIONS TheMATLABcodeusedforsimulatingtheimpedanceresultingfromasurface distributionofOhmicresistanceasdiscussedinSection10.1ispresentedbelow.The programnumericallyintegratesequation10. functionsurfacedistributiongeneralareaRb=1e7;[boundaryresistance:[ cm 2 ] Rs=1e-3;[boundaryresistance:[ cm 2 ] alpha=0.85;[CPEparameter] gamma=1/-alpha;[power-lawexponent] C0=10e-6;[surfacecapacitance: F= cm 2 ] inc=.01; start=-7; logA=[start:inc:0]'; A=10 : l ogA ; Ahat=[0;A];[normalizedarea: cm 2 ] R=Rs+Rb )]TJ/F15 11.9552 Tf 11.956 0 Td [(Rs Ahat : g amma ;[localresistance: cm 2 ] fdet=0; YM=0; forx=-3:.1:8 f=10 x ;[frequency:Hz] w=2*pi*f; integrand=1 := R+j w C0 )]TJ/F15 11.9552 Tf 7.085 -4.339 Td [(1 ; AT=cumtrapzAhat,integrand; Aint=ATlengthAT; Yi=Aint; YM=[YM;Yi]; fdet=[fdet;f]; end 173

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fdet,:=[]; YM,:=[]; wdet=2*pi*fdet; ZM=1./YM;[impedance: cm 2 ] Zr=realZM; Zj=imagZM; alphaM=0; QM=0; forx=1:1:lengthfdet-1 alpha1=log10-Zjx+1,1-log10-Zjx,1/log10fdetx+1,1-log10fdetx,1; alphaM=[alphaM;alpha1]; Q1=wdetx ; 1 a lpha1 = Zjx ; 1 sinalpha1 pi = 2 ; QM=[QM;Q1]; end gure;semilogxfdet,alphaM,'o' gure;loglogAhat,R,'o' fb=1/*pi*Rb*C0; fs=1/*pi*Rs*C0; gs=ZrlengthZr ; 1 = Rb 1 = gamma Rs gamma )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 = gamma ; Zmod=gs Rb 1 = gamma := Rs+j wdet C0 : )]TJ/F15 11.9552 Tf 7.085 -4.339 Td [(1 : )]TJ/F15 11.9552 Tf 11.955 0 Td [(gamma )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 = gamma ; Zmodr=realZmod; Zmodj=imagZmod; gure;loglogfdet,Zr,'o',fdet,Zmodr; gure;loglogfdet,-Zj,'o',fdet,-Zmodj; g=1/gs; 2*pi/3/sqrt; end 174

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APPENDIXD PROGRAMCODEFORDISKELECTRODEDISTRIBUTION TheMATLABcodeusedforthenumericalmethodoutlinedinSection10.2.2is presentedbelow. functiondiskdistributionRb=1e10;[ cm 2 ] Rs=1e0;[ cm 2 ] alpha=0.85;[CPEparameter] gamma=1/-alpha;[power-lawexponent] C0=10e-6;[surfacecapacitance: F= cm 2 ] inc=.01; start=-3; logA=[start:inc:0]'; A=10 : l ogA ; Ahat=[0;A];[normalizedarea: cm 2 ] R=Rs+Rb )]TJ/F15 11.9552 Tf 11.956 0 Td [(Rs Ahat : g amma ;[localresistance: cm 2 ] gure;loglogAhat,R,'o' integrand=1./R; YM=cumtrapzAhat,integrand; integrandA=oneslengthAhat,1; AM=cumtrapzAhat,integrandA; YunitM=0; AunitM=0; forx=1:1:lengthYM-1 Yunit=YMx+1,1-YMx,1; YunitM=[YunitM;Yunit]; A=AMx+1,1-AMx,1; AunitM=[AunitM;A]; end 175

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YunitM,:=[]; AunitM,:=[]; sumAunitM sumYunitM M=[AunitM,YunitM]; Aunitip=ipudAunitM; Yunitip=ipudYunitM; r=0; rnext=0; forx=1:1:lengthAunitip rnext=sqrtAunitipx ; 1+pi rnext 2 = pi ; r=[r;rnext]; end r;[radius: cm ] lengthr; lengthYunitip; Yb=1/Rb; bnext=Yb; YMM=Yb; forx=1:1:lengthYunitip b=bnext; YT=Yunitipx,1; ri=rx,1; rf=rx+1,1; m=YT )]TJ/F15 11.9552 Tf 11.955 0 Td [(pi b rf 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(ri 2 = = 3 pi rf 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [(ri 3 ; Yf=m*rf+b; bnext=Yf; 176

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YMM=[YMM;Yf]; end RM=1./YMM; L=[r,RM]; gure;semilogyr,RM,'o' r0=rlengthr,1; Ahat2=r : 2 = r0 2 ; trapzAhat,YM trapzAhat2,YMM R=RM; fdet=0; YM=0; forx=-3:.1:5 f=10 x ;[frequency:Hz] w=2*pi*f; integrand=r := R+j w C0 )]TJ/F15 11.9552 Tf 7.084 -4.339 Td [(1 AT=cumtrapzr,integrand; Aint=ATlengthAT; Yi=Aint; YM=[YM;Yi]; fdet=[fdet;f]; end fdet,:=[]; YM,:=[]; wdet=2*pi*fdet; YM=YM 2 = r0 2 ; ZM=1./YM;[impedance: cm 2 ] 177

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Zr=realZM; Zj=imagZM; alphaM=0; QM=0; forx=1:1:lengthfdet-1 alpha1=log10-Zjx+1,1-log10-Zjx,1/log10fdetx+1,1-log10fdetx,1; alphaM=[alphaM;alpha1]; Q1=wdetx ; 1 a lpha1 = Zjx ; 1 sinalpha1 pi = 2 ; QM=[QM;Q1]; end rhat=r/r0; gure;semilogxfdet,alphaM,'o' gure;semilogyr,R,'o' gure;loglogfdet,Zr,'o'; gure;loglogfdet,-Zj,'o'; end 178

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BIOGRAPHICALSKETCH BryanHirschornwasbornin1981inRockville,Connecticut.HegrewupinStorrs, ConnecticutandgraduatedfromE.O.HighSchoolin1999.HeearnedhisB.S.in chemicalengineeringfromtheUniversityofConnecticutin2003.Uponhisgraduation, heworkedasaresearchanddevelopmentengineerforSaint-GobainCorporationin Northboro,Massachusetts.In2006,BryanenrolledattheUniversityofFloridatopursue hisdoctorateinchemicalengineering.DuringhistimeattheUniversityofFlorida,Bryan wasmarriedtoErinWeedenin2007,whomhemetatSquamLake.In2008,they welcomedtheirrstchild,SiennaCatherineHirschorn,intotheworld.Bryan'shobbies includebasketball,soccer,golf,boating,andwater-sports. 188