Citation
Impedance Spectroscopy

Material Information

Title:
Impedance Spectroscopy The Influence of Surface Heterogeneity and Application to Corrosion Monitoring of Bridge Tendons
Creator:
Alexander, Christopher L
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (177 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Chemical Engineering
Committee Chair:
ORAZEM,MARK E
Committee Co-Chair:
JIANG,PENG
Committee Members:
ZIEGLER,KIRK JEREMY
BLOOMQUIST,DAVID G
VIVIER,VINCENT

Subjects

Subjects / Keywords:
corrosion -- electrochemistry -- impedance
Chemical Engineering -- Dissertations, Academic -- UF
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Chemical Engineering thesis, Ph.D.

Notes

Abstract:
External post-tensioned tendons are used in bridge construction to link precast concrete segments together. The tendons consist of 7-wire high-strength steel strands contained within a HDPE duct, which is filled with an alkaline grout to protect against corrosion. Despite the use of grout, corrosion has occurred in as little as 7 years compromising the integrity and safety of the bridge. Non-destructive techniques are needed to determine the integrity of the steel without having to break open the tendon. An indirect impedance technique is proposed as a way to detect corrosion within the tendons before failure of the steel occurs. By inserting 4 electrodes into the duct of the tendon to establish electric contact to the grout, the impedance of the grout and the grout and steel interface may be measured. Bench-top experiments were performed on fabricated tendons with and without induced corrosion to show that the indirect impedance was sensitive to the properties of the steel and grout interface. The biggest obstacle in the application of this technique is the lack of a reliable way of interpreting the data. Due to the geometry of the tendon and the configuration of the indirect impedance, frequency dispersion occurs at low frequencies which obscures the actual impedance of the steel. Finite-element models were used to simulate the indirect impedance and determine how the impedance of the grout contributed to the indirect impedance. Frequency dispersion is sometimes an unavoidable aspect of impedance measurements and has multiple contributing factors such as electrode geometry and surface heterogeneity. In terms of electrode geometry, any electrochemical cell setup which does not produce a uniform current distribution along the working electrode surface at all frequencies will yield an impedance with frequency dispersion. For example, a blocking disk electrode embedded within an insulating plane produces frequency dispersion at high frequencies. The frequency at which this occurs is inversely related to the radius of the disk such that frequency dispersion may be avoided with the use of smaller electrodes. Frequency dispersion may also be a result of surface heterogeneity of the electrode. The influence of surface heterogeneity on impedance measurements was explored to determine if this could be a physical explanation for frequency dispersion in the form a Constant Phase Element(CPE). CPEs are used in equivalent circuits to fit impedance data exhibiting frequency dispersion; however, the physical significance of the CPE is not always understood. Finite element models were used to simulate the impedance response of electrodes with surface roughness, a distribution of capacitance, and a distribution of reactivity to determine if surface heterogeneity. A characteristic length for surface roughness was established that is based on the period and the roughness factor of the roughness. The characteristic length for a distribution of capacitance was the period of the distribution. A distribution of reactivity associated with a single step reaction did not cause frequency dispersion; however, when a two step reaction coupled by an adsorbed intermediate was considered, frequency dispersion occurred at low and high frequencies. ( en )
General Note:
In the series University of Florida Digital Collections.
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Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2017.
Local:
Adviser: ORAZEM,MARK E.
Local:
Co-adviser: JIANG,PENG.
Statement of Responsibility:
by Christopher L Alexander.

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Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
LD1780 2017 ( lcc )

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IMPEDANCESPECTROSCOPY:THEINFLUENCEOFSURFACEHETEROGENEITYANDAPPLICATIONTOCORROSIONMONITORINGOFBRIDGETENDONSByCHRISTOPHERL.ALEXANDERADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2017

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c2017ChristopherL.Alexander

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ACKNOWLEDGMENTSIgratefullyacknowledgenancialsupportfromtheFloridaDepartmentofTransportation(ContractBDV31-977-35,RonaldSimmons,technicalmonitor).Theopinionsandndingsinthisdissertationarethoseoftheauthorandnotnecessarilythoseofthefundingagency.Iwouldliketothankmyadviser,MarkOrazemforhiscontinuedmentorshipandsupport.IwouldalsoliketothankBernardTribolletandVincentVivierfortheirassistanceaswellasmyresearchgroupfortheirsupportandguidance. 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 3 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 LISTOFSYMBOLS .................................... 16 ABSTRACT ........................................ 18 CHAPTER 1INTRODUCTION .................................. 20 2ELECTROCHEMICALIMPEDANCESPECTROSCOPY ............ 23 2.1GraphicalRepresentation ............................ 23 2.2Analysis&Interpretation ........................... 25 2.3FrequencyDispersion .............................. 26 2.3.1GeometryEect ............................. 26 2.3.2NormalDistribution ........................... 28 2.3.3SurfaceHeterogeneity .......................... 28 3CORROSIONPROBLEMSINEXTERNALBRIDGETENDONS ....... 31 3.1PrecastSegmentallyConstructedBridges ................... 31 3.2CorrosionIssuesinPost-tensioningSystems ................. 32 3.3MethodsofCorrosionDetectioninBridgeTendons ............. 33 3.4IndirectImpedanceSpectroscopy ....................... 35 4PROOFOFCONCEPTFORCORROSIONDETECTIONUSINGINDIRECTIMPEDANCE ..................................... 38 4.1Conventional3-electrodeConguration .................... 38 4.2RegressionFit .................................. 41 4.3CorrosionDetectionUsingIndirectImpedance ................ 44 5FINITEELEMENTSIMULATIONSANDINTERPRETATION ........ 52 5.1MathematicalDevelopment .......................... 52 5.2JusticationofBoundaryConditions ..................... 54 5.3Results&Analysis ............................... 56 5.3.1ExperimentalDataFitting ....................... 57 5.3.2DeterminationofSteelSensingArea .................. 57 5.3.3EquivalentCircuit ............................ 60 5.3.4AnalogueCircuit ............................ 66 4

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5.3.5InuenceofElectrodeConguration .................. 69 5.3.6SensitivitytoSteelPolarizationResistance .............. 73 6FEASIBILITYOFINDIRECTIMPEDANCEFORPOST-TENSIONEDTENDONS 77 6.1Methods ..................................... 77 6.2ExperimentalResults&Analysis ....................... 80 6.2.1TexasA&MMockBridge ........................ 80 6.2.2RinglingCausewayBridge ....................... 81 6.3SimulationResults ............................... 83 6.4Discussion .................................... 87 7INFLUENCEOFROUGHNESSONIMPEDANCE ................ 89 7.1MathematicalDevelopment .......................... 90 7.2ImpedanceCalculations ............................ 93 7.3ResultsandDiscussion ............................. 94 7.3.1InuenceofRoughnessonaDiskElectrodeWithinanInsulatedPlane ................................... 95 7.3.2InuenceofSurfaceRoughnessonaRecessedElectrode ....... 104 7.3.2.1Eectofporegeometry ................... 111 7.3.2.2Transitionfromaroughelectrodetoaporouselectrode .. 112 8INFLUENCEOFCAPACITANCEDISTRIBUTIONONIMPEDANCE .... 116 8.1ImpedanceCalculations ............................ 117 8.2ResultsandDiscussion ............................. 119 8.2.1CapacitanceDistributiononRecessedElectrodes ........... 120 8.2.2CapacitanceDistributiononDiskElectrodes ............. 127 9INFLUENCEOFREACTIVITYDISTRIBUTIONONIMPEDANCE ..... 135 9.1MathematicalDevelopment&Impedancecalculations ............ 137 9.2Results ...................................... 139 9.3Discussion .................................... 141 10INFLUENCEOFNONUNIFORMREACTIONRATESASSOCIATEDWITHREACTIONSCOUPLEDBYANADSORBEDINTERMEDIATEONIMPEDANCE 144 10.1MathematicalDevelopment .......................... 144 10.2Finite-ElementModel .............................. 149 10.3Results ...................................... 151 10.3.1DiskElectrode .............................. 151 10.3.2RecessedElectrode ........................... 156 10.4Discussion .................................... 160 5

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11CONCLUSIONS ................................... 163 11.1IndirectImpedanceAppliedtoExternalPost-TensionedTendons ...... 163 11.2InuenceofSurfaceHeterogeneityinImpedanceMeasurements ...... 164 12SUGGESTIONSFORFUTUREWORK ...................... 166 REFERENCES ....................................... 171 BIOGRAPHICALSKETCH ................................ 177 6

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LISTOFTABLES Table page 4-1Regressionparametersandstandarderrorforequivalentcircuitttoconventionalthreeelectrodeimpedancefortheactiveandpassivecases. ............ 44 10-1SimulationParameters. ................................ 151 7

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LISTOFFIGURES Figure page 2-1Schematicrepresentationofthecalculationofthetransferfunctionforasinusoidalinputatfrequency!.ThetimelagbetweenthetwosignalsistandtheperiodofthesignalsisT.[57] ................................ 24 2-2ImpedanceofaresistorandcapacitorinparallelinNyquistformat.[57] ..... 24 3-1Schematicrepresentationshowingthecurrentpathsforaconductivestrandaresistivematerial ................................... 35 4-1Schematicshowingtheconventional3{electrodeimpedancemeasurementonacylindricalelectrochemicalcellinwhichtheelectrolyteisgroutandtheworkingelectrodeisacouponofthesteelstrand. ...................... 39 4-2Schematicshowingtheimpressedcurrenttechniqueforacylindricalelectrochemicalcellinwhichtheelectrolyteisgroutandtheworkingelectrodeisacouponofthesteelstrand. .................................... 39 4-3Conventional3{electrodeimpedanceofasteeldiskelectrodeingroutbeforeoneofthespecimens(corroded)wasforcedtocorrode. .............. 40 4-4Conventional3{electrodeimpedanceofasteeldiskelectrodeingroutafteroneofthespecimens(corroded)wasforcedtocorrode. ................. 41 4-5Imagesofthesteeldiskelectrodeextractedfromthegrout ............ 42 4-6Circuitdiagramusedtotthepassivecaseimpedanceofthesteelandgroutinterface. ........................................ 42 4-7ImpedanceofthepassivesteeldiskelectrodeingroutttedwiththecircuitinFigure4-6. ....................................... 43 4-82ft.fabricatedtendonwith10electrodelocationsformeasuringtheindirectimpedanceandschematicofthefourelectrodemeasurement. ........... 44 4-9Indirectimpedanceoffabricatedtendonwithandwithoutsteel ......... 46 4-10Schematicrepresentationoftheimpressedcurrenttechniqueusedtoforcecorrosion. 46 4-11IndirectimpedanceinNyquistformatofafabricatedtendonwithpassivesteelandelectrodelocationasaparameter. ....................... 47 4-12IndirectimpedanceinNyquistformatofafabricatedtendonwithcorrodedsteelandelectrodelocationasaparameter. ....................... 48 4-13Imagesofthesteelsurfacedirectlybeneatheachelectrodeforthecorrodedcase. 49 4-14Imagesofthesteelsurfacedirectlybeneatheachelectrodeforthecorrodedcase. 50 8

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5-1Currentandpotentialdistributionofa1cmsquare10Ohm-mresistivitygroutmodelwithcurrentinjectingelectrodesplacedontheverticalsides ....... 54 5-2Simulatedrealimpedanceofasafunctionoffrequencyofa1cmsquaregroutmodelwithcurrentinjectingelectrodesplacedontheverticalsides. ....... 55 5-3Currentandpotentialdistributionatthelowfrequencylimitofa1cmsquaregroutmodelwitha0.25cmradiussteelplacedinthecenterandcurrentinjectingelectrodesplacedontheverticalsides. ....................... 55 5-4Currentandpotentialdistributionatthehighfrequencylimitofa1cmsquaregroutmodelwitha0.25cmradiussteelcircleplacedinthecenterandcurrentinjectingelectrodesplacedontheverticalsides. .................. 56 5-5Simulatedimpedanceofa1cmsquaregroutmodelwitha0.25cmradiussteelcircleplacedinthecenterandcurrentinjectingelectrodesplacedontheverticalsides .......................................... 56 5-6Meshofthe3Dtendonmodel. ............................ 57 5-7Simulatedimpedanceresultscomparedtotheexperimentalresultswithanelectrodecongurationof1357. ................................. 58 5-8Simulatedimpedanceresultscomparedtotheexperimentalresultswithanelectrodecongurationof2356. ................................. 58 5-9Tendonmodelwithlocallycorrodingsectioninthecenterofthesteel. ...... 58 5-10Schematicrepresentationofthesystemgeometryforareferenceelectrodespacingof4cm. ........................................ 59 5-11Simulatedindirectimpedanceofa2ftmodeltendoncontaining1steelstrandforapassivecase,alocallycorrodingcasesof4cmatthemidpointofthesteelstrand,andauniformlycorrodingsteelforareferenceelectrodespacingof4cm 59 5-12Schematicrepresentationofthesystemgeometryforareferenceelectrodespacingof4cm. ........................................ 60 5-13Simulatedindirectimpedanceofa2ftmodeltendoncontaining1steelstrandforapassivecase,andalocallycorrodingcasesof4cmatthemidpointofthesteelstrand ...................................... 60 5-14Equivalentcircuitdiagramusedtorepresenttheindirectimpedance. ...... 61 5-15Cutplaneusedtodeterminetheoscillatingcurrentthroughthegrout. ..... 61 5-16Magnitudeoftheserieslocalimpedance ...................... 63 5-17Theohmicimpedanceofasegment ......................... 64 9

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5-18SimulatedindirectimpedanceandequivalentcircuitimpedancecalculatedusingEquation5{17inNyquistformat. .......................... 65 5-19Reducedanaloguecircuitusedtorepresenttheindirectimpedance. ....... 66 5-20Simulatedparallelohmicimpedance. ........................ 66 5-21Schematicshowingtheeectiveareaofpolarizedsteel. .............. 67 5-22Seriespathsimulatedimpedanceandseriessimulatedohmicimpedance. .... 68 5-23Calculatedequivalentcircuitimpedance. ...................... 68 5-24TheseriesohmicimpedanceinNyquistformatwiththespacingbetweenreferenceelectrodesasaparameter. .............................. 69 5-25TheparallelohmicimpedancescaledbythehighfrequencylimitoftherealpartoftheparallelohmicimpedanceinNyquistformatwiththedistancebetweenreferenceelectrodeasaparameter. ......................... 70 5-26TheseriesohmicimpedanceinNyquistformatwiththedistancebetweentheworkingandcounterelectrodeasaparameter. ................... 71 5-27TheparallelohmicimpedanceinNyquistformatwiththedistancebetweentheworkingandcounterelectrodeasaparameter. ................... 71 5-28Thesimulatedindirectimpedancescaledbytheohmicresistancewithelectrodespacingasaparameter.Threesimulationswereperformedforchangesinreferenceelectrodespacingandtheotherthreewereforchangingthespacingbetweentheworkingandcounterelectrode. ......................... 72 5-29CalculatedimpedanceofanR-Ccircuit. ...................... 74 5-30ThesimulatedindirectimpedanceinNyquistformatwithRpasaparameter. .. 75 5-31TheseriesohmicimpedanceinNyquistformatwithRpasaparameter. ..... 75 5-32TheparallelohmicimpedanceinNyquistformatwithRpasaparameter. .... 76 6-1Thecross-sectionoftheRinglingBridgetendon.Thenumbersindicatethelocationsoftheelectrodes. ................................... 78 6-2AnimageoftheinsideofthemockbridgebuiltatTexasA&M. ......... 79 6-3Experimentalsetupoftheindirectimpedancemeasurement.PhotographtakenattheTexasA&Mmockbridge.TheGamryReference600potentiostatisthewhite/blueboxinthecenterofthephotograph. .................. 79 6-4ExperimentalimpedanceinNyquistformatmeasuredatdierentsectionsoftheTexasA&Mbridgetendons. ........................... 80 10

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6-5ExperimentalimpedanceinNyquistformatmeasuredatlocation16VWoftheTexasA&Mmockbridgetendons. .......................... 82 6-6ExperimentalimpedanceinNyquistformatmeasuredatlocation16SToftheTexasA&Mmockbridgetendons. .......................... 83 6-7ExperimentalimpedanceinNyquistformatmeasuredonanextractedtendonfromtheRinglingCausewayBridge ......................... 84 6-8ExperimentalimpedanceinNyquistformatmeasuredonanextractedtendonfromtheRinglingCausewayBridgeatlocation4 ................. 84 6-9ExperimentalimpedanceinNyquistformatmeasuredonanextractedtendonfromtheRinglingCausewayBridge ......................... 85 6-10FiniteelementrepresentationoftheRinglingBridgetendonshowninFigure6-1. 86 6-11Simulatedindirectimpedancefora2ft.cylindricaltendonwith6steelstrandswithcorrosionstateasaparameter. ......................... 86 6-12Simulatedindirectimpedancefora2ft.cylindricaltendonwith6steelstrandswithcorrosionstateasaparameter. ......................... 87 7-1Schematicrepresentationshowingthenite-elementmeshusedforthediskelectrodesimulations ...................................... 93 7-2Schematicrepresentationshowingthenite-elementmeshusedfortherecesseddiskelectrodesimulations .............................. 94 7-3Imaginarypartofthecalculatedglobalimpedanceforaroughdiskelectrodewithroughnessfactorasaparameter ........................ 95 7-4Calculatedglobalimpedanceasafunctionoffrequencyforaroughdiskelectrodewithroughnessfactorasaparameter ........................ 97 7-5Calculatedglobalimpedanceasafunctionoffrequencyforaroughdiskelectrodewithroughnessfactorasaparameter ........................ 98 7-6Phaseanglesobtainedfromequations7{23,9{15,and8{10fortheimpedancepresentedinFigure7-3foraroughdiskelectrodeasafunctionfrequency.Theroughnessfactorwasfr=2andtheroughnessperiodwasP=40m. ..... 99 7-7Imaginary{impedance{derived{phaseanglecalculatedfromtheimpedancepresentedinFigure7-3foraroughdiskelectrode ....................... 100 7-8Schematicrepresentationshowingthemannerinwhichtheroughnessperiodwasvariedforaxedroughnessfactorequalto2.Thethreecongurationshavethesamesurfacearea. ................................ 102 11

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7-9Theimaginary{impedance{derived{phaseanglecalculatedfromimpedancesimulationsofaroughdiskelectrode ............................... 102 7-10Imaginary{impedance{derived{phaseanglecalculatedfromthesimulatedimpedanceofaroughrecessedelectrodewiththeroughnessperiodasaparameter ..... 105 7-11ThecurrentpathsobtainedasRef~iexp(jwt)gataxedtime ........... 106 7-12Calculatedcurrentdensityatapeakandtroughforarecessedelectrodeasafunctionoffrequency. ................................. 107 7-13TheratioofthecalculatedeectivecapacitanceandtheinputcapacitanceasafunctionofdimensionlessfrequencyKfrwiththeroughnessfactorasaparameterforrecesseddiskelectrodes. ............................. 108 7-14ThecharacteristicfrequencyassociatedwithdimensionlessfrequencyK(f2rP=r0)=1 ............................................ 109 7-15Theimaginary{impedancephaseanglecalculatedfromimpedancesimulationsofaroughrecessedelectrode ............................. 110 7-16Theimaginary{impedance{derivedphaseangleforaroughelectrodewithrectangulargrooveswiththewidthofthegroovesasaparameters. .............. 112 7-17PhaseangleforaroughelectrodewithV-shapedgrooveswithaspacebetweenthem .......................................... 113 7-18Theimaginaryimpedancederivedphaseangleofroughelectrodesasafunctionoffrequencywiththedepthofgroovesandroughnessfactorasparameters. ... 114 7-19Theimaginaryimpedancederivedphaseangleofroughelectrodesasafunctionofdimensionlessfrequencywiththedepthofgroovesandroughnessfactorasparameters. ...................................... 115 8-1Finite-elementmeshforthediskelectrodesimulations. .............. 117 8-2CapacitancedistributionasafunctionofradialpositionbasedonasquarewaverepresentedbyaFourierserieswithaperiodof60m. .............. 118 8-3TheimpedanceinNyquistformatofarecesseddiskelectrodewiththesquarewavecapacitancedistributionshowninFigure8-2andtheperiodofdistributionasaparameter. .................................... 120 8-4Thesimulatedimpedanceasafunctionoffrequencyofarecesseddiskelectrodewithasquarewavecapacitancedistributionandtheperiodofdistributionasaparameter. ....................................... 121 8-5Thecurrentpathsnearthesurfaceofarecessedelectrodeexhibitingasquare{wavedistributionofcapacitanceobtainedasq ~i2r+~i2j. .................. 123 12

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8-6Normalcurrentdistributionattheelectrodesurfaceduetoanonuniformcapacitancedistributionwithaperiodof60mofarecessedelectrodeasafunctionofradialposition. ........................................ 124 8-7Imaginary{impedance{derivedphaseanglecalculatedfromtheimpedancedatainFigure8-4. ..................................... 125 8-8Theratioofthecalculatedeectivecapacitanceandthesurface{averagedinputcapacitanceasafunctionofdimensionlessfrequencyKP=r0withtheperiodofthedistributionasaparameterforrecesseddiskelectrodes. ............ 127 8-9Normalcurrentdistributionatadiskelectrodesurfaceasafunctionofradialposition:a)currentdistributionat10mHz. .................... 128 8-10Imaginary{impedance{derivedphaseangleforadiskelectrodewithinaninsulatingplane:a)phaseangleasafunctionoffrequency. .................. 129 8-11Imaginary{impedance{derivedphaseanglesvaluescalculatedfromtheimpedancedatawiththediskradiusasaparameter. ...................... 131 8-12Imaginary{impedance{derivedphaseanglecalculatedfromimpedancedataondiskelectrodeswitharadialdistributionofcapacitance .............. 132 8-13Theratioofthecalculatedeectivecapacitanceandthesurface{averagedinputcapacitanceasafunctionofdimensionlessfrequencyKP=r0withtheperiodofthedistributionasaparameterfordiskelectrodeswithinaninsulatingplane. 133 8-14ThefrequencyKP=r0=1atwhichthesurfaceheterogeneityinuencestheimpedanceasafunctionofdistributionperiodanddiskradiuswith=hC0iasaparameter. 134 9-1ReactivitydistributionasafunctionofradialpositionbasedonasquarewaverepresentedbyaFourierserieswithaperiodof60m. .............. 137 9-2Globalimpedancescaledbytheohmicresistanceofarecessedelectrodewithasquare{wavedistributionofcapacitanceasafunctionoffrequencywiththeperiodandaveragedcharge{transferresistanceasparameters. .......... 140 9-3Imaginary{impedance{derivedphaseangleforarecessedelectrodewiththeperiodandaveragedcharge{transferresistanceasparameters. .............. 142 10-1Equivalentcircuitdiagramforarecessedelectrodewithreactionscoupledbyanadsorbedintermediate. ................................ 148 10-2CalculatedimpedancebasedonanequivalentcircuitwiththeohmicresistanceinserieswithadoublelayercapacitorinparallelwiththefaradaicimpedancecalculatedfromEquation(10{21). .......................... 148 10-3Thedomainfortheniteelementsimulations. ................... 149 10-4Steadystatecurrentdensityasafunctionofsurfaceoverpotential. ........ 152 13

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10-5Surface-averagedAasafunctionofpotentialwithdiskradiusasaparameter. 152 10-6Surfacepotentialasafunctionofradialposition. ................. 153 10-7Steady-stateinterfacialparametersasafunctionofradialposition:A)A;B)B;C)Rt .......................................... 153 10-8SimulatedimpedanceresponseinNyquistformatofadiskelectrode.Thesolidlinerepresentstheglobalimpedanceresponse.TheDashedlinerepresentsthesurface-averagedinterfacialimpedance.Thedottedlinerepresentstheinterfacialimpedance. ...................................... 154 10-9Theimaginarypartoftheohmicimpedancescaledbytheohmicresistanceasafunctionoffrequencywithradiusasaparameter. ................. 154 10-10TheadsorptionimpedancescaledbytheB=Awithdiskradiusasaparameter.Thedashedlinerepresentstheresultswithouttheeectofdiskgeometry. ... 155 10-11Thederivativeofthelogarithmoftheimaginarypartoftheadsorptionimpedancewithrespecttothelogarithmoffrequencyasafunctionoffrequencywiththeradiusasaparameter. ................................ 156 10-12EectiverateconstantforReaction10{1asafunctionofradialposition. .... 157 10-13EectiverateconstantforReaction10{2asafunctionofradialposition. .... 157 10-14Imaginaryohmicimpedancescaledbytheohmicresistanceasafunctionoffrequencywiththeperiodofthedistributionasaparameter. ........... 158 10-15Thederivativeofthelogarithmoftheimaginarypartoftheadsorptionimpedancewithrespecttothelogarithmoffrequencyasafunctionoffrequencywiththeperiodasaparameter. ................................ 158 10-16ThesimulatedimpedanceinNyquistformatscaledbythechargetransferresistanceRtwithsteadystatepotentialasaparameter. ................... 159 10-17Thederivativeofthelogarithmoftheimaginarypartoftheadsorptionimpedancewithrespecttothelogarithmoffrequencyasafunctionoffrequencywithsteady-statepotentialasaparameter. ............................... 160 10-18Thederivativeofthelogarithmoftheimaginarypartoftheadsorptionimpedancewithrespecttothelogarithmoffrequencyasafunctionofdimensionlessfrequencywithsteady-statepotentialasaparameter. ..................... 160 10-19Parametersofthefaradaicimpedancedeterminedfromtheimpedancesimulationsscaledbythesurfaceaveragedparametersasafunctionofsteadystatepotential. 161 12-1Thesimulatedglobalimpedanceofarotatingdiskelectrodewithasilverredoxcouplewithandwithoutcoupledchargingandfaradaiccurrents. ......... 167 14

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12-2Thesimulatedglobalimpedanceofarotatingdiskelectrodewithasilverredoxcouplewithandwithoutcoupledchargingandfaradaiccurrents. ......... 167 12-3Thesimulatedglobalimpedanceofarotatingdiskelectrodewithasilverredoxcouplewithandwithoutcoupledchargingandfaradaiccurrents. ......... 168 12-4Thesimulatedglobalimpedanceofarotatingdiskelectrodewithasilverredoxcouplewithandwithoutcoupledchargingandfaradaiccurrents. ......... 168 12-5Themagnitudeoftheoscillatingcurrentdensityobtainedfromthesilverredoxsimulationsasafunctionofdimensionlessfrequencyforcoupledanduncoupledcases. ......................................... 169 12-6Themagnitudeoftheoscillatingcurrentdensityobtainedfromtheferro/ferricyanideredoxsimulationsasafunctionofdimensionlessfrequencyforcoupledanduncoupledcases. ......................................... 170 15

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LISTOFSYMBOLS Roman A adsorptionimpedanceparameter,)]TJ /F5 7.97 Tf 6.58 0 Td[(1s)]TJ /F5 7.97 Tf 6.59 0 Td[(1cm)]TJ /F5 7.97 Tf 6.58 0 Td[(2 B adsorptionimpedanceparameter,s)]TJ /F5 7.97 Tf 6.58 0 Td[(1 C capacitance,F/cm2orF(1F=1C=V) Cdl double-layercapacitance,F/cm2orF(1F=1C=V) ci volumetricconcentrationofspeciesi,mol/cm3 f frequency,f=!=2,Hz F Faraday'sconstant,96,487C/equiv fr roughnessfactor,truesurfacearea/geometricarea 0 exchangecurrentdensity,mA/cm2 iF faradaiccurrentdensity,mA/cm2 K dimensionlessfrequencyassociatedwithadiskelectrode L inductance,H(1H=1Vs2=C) lc characteristiclength,cm P periodofasquare-waveradialdistribution electrolytepotential,V Q CPEcoecient,s=cm2 R resistance,cm2or(1=1Vs=C) r radialcoordinate,cm r0 radiusofadiskelectrode,s=cm2 R universalgasconstant,8.3143J/molK Re ohmicresistance,cm2or(1=1Vs=C) Rp polarizationresistance,cm2or(1=1Vs=C) Rt charge-transferresistance,cm2or(1=1Vs=C) T temperature,K V perturbationpotential,V 16

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Z globalimpedance,cm2 z localimpedance,cm2 Z0 globalinterfacialimpedance,cm2 Zads adsorptionimpedance,cm2 Ze globalohmicimpedance,cm2 ZF faradaicimpedance,cm2 Greek exponentialparameterforaCPE )]TJ ET 0 G 0 g BT /F1 11.955 Tf 41.85 -203.22 Td[(maximumsurfacecoverage,mol/cm2 surfacecoverage,mol/cm2 conductivity,S/cm phaseangle,degree resistivity,cm angularfrequency,rad/s GeneralNotation ImfXg imaginarypartofX RefXg realpartofX X steady-stateortime-averagedpartofX(t) z complexconjugateofacomplexnumberz,z=zr)]TJ /F3 11.955 Tf 11.95 0 Td[(zjofX(t) hXi surface-averagedvalueofX ~X oscillatingpartofX(t) Subscripts a pertainingtoanodicreactions c pertainingtocathodicreactions j imaginary r real 17

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyIMPEDANCESPECTROSCOPY:THEINFLUENCEOFSURFACEHETEROGENEITYANDAPPLICATIONTOCORROSIONMONITORINGOFBRIDGETENDONSByChristopherL.AlexanderMay2017Chair:MarkE.OrazemMajor:ChemicalEngineeringAnindirectimpedancetechniqueisproposedasawaytodetectcorrosionwithinexternalpost-tensionedbridgetendons.Non-destructivetechniquesareneededtodeterminetheintegrityofthesteelwithouthavingtobreakopenthetendon.Theindirectimpedanceaimstoextracttheimpedanceofthesteel-groutinterfacefromtheimpedancemeasuredatthesurfaceofthegrout.However,theelectrodecongurationrequiredfortheindirectimpedancemeasurementyieldsgeometry-inducedfrequencydispersion.Therefore,thebiggestobstacleintheapplicationofthistechniqueisthelackofareliablewayofinterpretingthedata.Bench-topexperimentswereperformedonfabricatedtendonswithandwithoutinducedcorrosiontoshowthattheindirectimpedancewassensitivetothepropertiesofthesteelandgroutinterface.Finite-elementmodelswereusedtosimulatetheindirectimpedanceanddeterminehowthegeometryoftheunusualelectrodecongurationinuencestheimpedance.Frequencydispersionmayalsobearesultofsurfaceheterogeneityoftheelectrode.TheinuenceofsurfaceheterogeneityonimpedancemeasurementswasexploredtodetermineifsurfacedistributionscouldprovideaphysicalexplanationforfrequencydispersionobservedexperimentallyasConstantPhaseElements(CPE).Finite-elementmodelswereusedtosimulatetheimpedanceresponseofelectrodeswithsurfaceroughness,adistributionofcapacitance,andadistributionofreactivity.Similartothegeometry 18

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inducedfrequencydispersionobservedforadiskelectrode,surfaceroughnessandadistributionofcapacitancecausedfrequencydispersionathighfrequenciesandcouldbedenedintermsofacharacteristiclength.Asthecharacteristiclengthdecreases,thefrequencydispersionshiftstohigherfrequencies.Thecharacteristiclengthforsurfaceroughnesswasbasedontheperiodandtheroughnessfactor.Thecharacteristiclengthforadistributionofcapacitancewastheperiodofthedistribution.Adistributionofreactivityassociatedwithasinglestepreactiondidnotcausefrequencydispersion;however,whenatwostepreactioncoupledbyanadsorbedintermediatewasconsidered,frequencydispersionoccurredatbothlowandhighfrequencies. 19

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CHAPTER1INTRODUCTIONElectrochemicalimpedancespectroscopy(EIS)isanexperimentalcharacterizationmethodthatmaybeusedtoassessthephysicalprocessesthatoccurattheinterfaceofanelectrodeandanelectrolyte.Whilemeasuringtheimpedanceofanelectrochemicalsystemisrelativelytrivial,interpretingtheresultscanbecomplicatedbythepresenceoffrequencydispersion.Frequencydispersionissometimesanunavoidableaspectofimpedancemeasurementsandhasmultiplecontributingfactorssuchasgeometry,surfaceheterogeneity,homogeneousreactions,aswellasanomalousdiusionprocesses.Workpresentedinthisdissertationisfocusedon2projects.Intherstpart,EISwasusedtoestablishawayofdetectingcorrosioninexternalpost-tensionedbridgetendonswhichareusedinsegmentallyconstructedbridges.OneofthemajorapplicationsofEISistodeterminethepolarizationresistanceofametalliquidinterfacewhichisinverselyrelatedtothecorrosionrate.EIShasbeenappliedtoassessthecorrosivestateofsteelwithinconcrete.DeteriorationofAmericasinfrastructurehascostbillionsofdollarsayeartomitigate.11percentofthenationsbridgeshavebeendeemedstructurallydecientwhileanother13percentarefunctionallyobsolete.Corrosionofsteelwithinbridgeshasbeenoneofthemajorcontributingfactors.Posttensionedtendonsconsistofhelicallywound7-wirepre-stressingsteelstrandsthataresurroundedbyacementitiousgroutcontainedwithinaHighDensityPolyethylene(HDPE)duct.Thegroutprovidesanalkalineenvironmenttopreventthesteelfromcorroding,andtheductpreventstheintrusionofaggressivecontaminants.Despitethesepreventativemeasures,therehavebeenmanycasesinwhichcorrosionhasstilloccurred.Thecurrentmethodofensuringthatthesteelcontainedwithintendonsisnotcorrodingistocutthroughthetendonandvisuallyinspectthesteel.Anondestructivealternativeistouseanindirectimpedancemeasurementtoassessthepropertiesofsteelcontainedwithinthegroutwithoutadirectconnectiontothesteel.Bymeasuringtheimpedanceusinganarrayof4electrodeson 20

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thesurfaceofthegrout,thepropertiesofthesteelmaybesampled.However,extractingthetrueimpedanceofthesteelfromtheimpedanceofthewholesystemisdicult.Themajorobstacletocommercializationofthistechniqueisdeterminingawaytoaccountforthefrequencydispersioncausedbythegeometryofthemeasurement.Inthesecondpart,workwasdonetoestablishanunderstandingofwhatfactorscontributetofrequencydispersioninEISmeasurementsandtouncoverthephysicaloriginoftheconstant-phaseelement(CPE).Thisdissertationaddressesfrequencydispersionduetogeometryaswellassurfaceheterogeneity.Jorcinetal.usedlocalimpedancespectroscopytoshowthatfrequencydispersioncanarisefromsurfaceornormaldistributionsoftimeconstants.Thefrequencydispersionassociatedwithadistributionoftimeconstantsnormaltotheelectrodesurfaceiswellestablished;whereas,thefrequencydispersionassociatedwithasurfacedistributionisnotwellunderstood.Hirschornetal.[ 24 25 ]showedthatapower{lawdistributionofresistivitythroughalmyieldsCPEbehavior.Thepower{law{modelapproachhasbeenusedsuccessfullytoextractalmcapacitanceandassociatedparametersforavarietyofsystems,includingoxidesonsteel,[ 58 ]humanskin,[ 58 74 ]andpolymercoatings.[ 48 52 ]Intermsofsurfacedistributions,Brugetal.[ 12 ]developedanexpressionforthecapacitanceextractedfromaCPEcausedbyasurfacedistributionofcapacitance.Cordoba-Torresetal.[ 15 ]showedthattheBrugmodelexplainedthecorrelationobservedbetweenCPEparametersandQfortwoexperimentalconditions:thecorrosionofpolycrystallineironandthedepositionofCaCO3scaleongoldelectrodes.Theresultswereattributedtoadistributionoftimeconstantsassociatedwithsurfaceheterogeneity.Theexactnatureofthesurfaceheterogeneitywasnotidentied.Insubsequentwork,Cordoba-Torresetal.[ 16 ]suggestedthattheCPEbehaviorresultsfromenergeticdistributionsratherthangeometricheterogeneityorroughness.Inthisdissertation,abriefhistoryandreviewofelectrochemicalimpedancespectroscopywillbepresentedinChapter 2 alongwithadescriptionofhowfrequency 21

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dispersioncancomplicatedatainterpretation.Backgroundknowledgeonexternalpost-tensionedtendonsandcorrosionmechanismsofsteelinconcretewillbepresentedinChapter 3 .Aproofofconceptforanindirectimpedancespectroscopymethodtodetectcorrosioninpost-tensionedbridgetendonswillbepresentedinChapter 4 .AnexperimentalproofofconceptforindirectimpedancemeasurementswillbepresentedinChapter 5 .ExperimentalresultsofeldapplicationsoftheindirectimpedancetechniquewillbepresentedinChapter 6 .Theinuenceofsurfaceroughness,adistributionofcapacitance,andadistributionofcharge-transferresistancewillbepresentedinChapters 7 8 ,and 9 respectively.Wuetal.showedthatthegeometryofadiskelectrodeembeddedwithinaninsulatingplanewithreactionsinvolvingadsorbedintermediatescausesfrequencydispersionatlowfrequencies.Chapter 10 willpresenttheinuenceofheterogeneousreactionsratesassociatedwithreactionscoupledbyanadsorbedintermediate.ConclusionsofthisworkareprovidedinChapter 11 .SuggestionsforfutureworkwillbepresentedinChapter 12 22

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CHAPTER2ELECTROCHEMICALIMPEDANCESPECTROSCOPYElectrochemicalimpedancespectroscopyisapowerfulexperimentaltechniquethatmaybeusedtostudyinterfacialphenomenasuchascharging,electrochemicalreactions,anddiusionprocesses.TheinterfaceisinterrogatedwithanapplicationofasinusoidalperturbationofeitherpotentialorcurrentasisshowninFigure 2-1 Theinputsignalshownisapotentialperturbationasafunctionoftime.Theperturbationmaybeimposedinadditiontoasteadystatevalueofpotential Vwhichallowsthemodulationofpotentialatdierentpointsalongthepolarizationcurve.Theresponseismonitoredusuallywiththeuseofareferenceelectrodetohelpreducetheohmiceectandisolatetheimpedanceoftheworkingelectrodeelectrolyteinterface.Iftheinputsignalispotentialthentheresponsewillbecurrent.Themagnitudeofthetime-domainresponsesignalwillalwaysbelessthanthemagnitudeoftheinputsignalindicatingtheresistivepropertiesoftheinterfaceandifthereareanycapacitivefeaturestheresponsesignalwilllagtheinputsignalbyatimeoft.Themagnitudeoftheimpedanceiscalculatedastheratioofthemagnitudeoftheinputandtheoutputsignalsandthephaseiscalculatedas2t=T,whereTrepresentstheperiodoftheinputsignal.Therealandimaginarypartsoftheimpedancearefunctionsofthephaseandmagnitudeexpressedas Zr=jZ(!)jcos(')(2{1)and Zj=jZ(!)jsin(')(2{2)respectivelyandthetotalcompleximpedanceisthesumofthetwoparts. 2.1GraphicalRepresentationTheimpedanceismeasuredoverarangeoffrequenciesandistraditionallyplottedinNyquistorBodeformat.AnexampleoftheimpedanceofaresistorandcapacitorinparallelisshowninFigure 2-2 inNyquistformat.TheNyquistformatdisplaysthe 23

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Figure2-1. Schematicrepresentationofthecalculationofthetransferfunctionforasinusoidalinputatfrequency!.ThetimelagbetweenthetwosignalsistandtheperiodofthesignalsisT.[ 57 ] Figure2-2. ImpedanceofaresistorandcapacitorinparallelinNyquistformat.[ 57 ] 24

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negativeoftheimaginarypartoftheimpedanceasafunctionoftherealpart.Therealpartoftheimpedanceisnormallyassociatedwithresistivepropertiesofthesystemandtheimaginarypartisrepresentativeofinductiveorcapacitivefeatures.Forthiscircuit,thelowfrequencylimitoftheimpedanceisequaltotheresistanceRt.Thecapacitancemaybedeterminedfromthecharacteristicfrequencywhichislocatedatthepeakofthesemi-circle.OneofthedisadvantagesoftheNyquistrepresentationisthatsomefeaturesmaybelostathighfrequencies.TheBodeformatdisplaysthephaseandthemagnitudeoftheimpedanceeachasafunctionofthelogarithmofthefrequencywhichhastheadvantageofshowingthefrequencydependenceofthesystem.Thephasemaybecalculatedfromtherealandimaginarypartsoftheimpedanceastan)]TJ /F5 7.97 Tf 6.59 -0.01 Td[(1(Zj=Zr).However,incaseswherethereisalargeohmicresistance,theBoderepresentationismisconceivingathigherfrequencies.Anohmicresistancecorrectedphaseanglemaybecalculatedas'adj=tan)]TJ /F5 7.97 Tf 6.59 0 Td[(1(Zj=(Zr)]TJ /F3 11.955 Tf 12.29 0 Td[(Re))whichprovidesamoreaccuraterepresentationofthesystem. 2.2Analysis&InterpretationInformationobtainedfromimpedancemeasurementsmayinclude,but,isnotlimitedtotheinterfacialcapacitance,kineticparametersassociatedwithelectrochemicalreactionsacrosstheinterface,aswellasdiusionparameters.TheKramers-Kronigrelationsareoftenusedtotestforconsistencybetweentherealandimaginarypartstoensuretheconditionsoflinearity,stability,andcausalitywereadheredto.Equivalentcircuitsincorporatingelementssuchasresistors,capacitors,anddiusionelementsareoftenusedtottheresultsandextractphysicallymeaningfulparameters.Thereareoftenmultipleequivalentcircuitsthatmaytthedatawhichcomplicatestheinterpretation.Analternativeistottheimpedancewithamathematicalmodelwhichaccountsfortheexpectedphysicsofthesystemunderstudy. 25

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2.3FrequencyDispersionInmostcases,theimpedancedatacannotbetusingacircuitcomposedoffrequency-independentparameterssuchasresistorsandcapacitorsduetothepresenceoffrequencydispersion.Constant-phaseelements(CPE)areoftenusedinplaceofcapacitorstoaccountforthefrequencydispersion.Theearliestaccountoftheconstant-phaseelementwaspresentedbyHugoFrickein1932.[ 22 ]Undertheassumptionthatthecapacitancewasfrequencydependent,herealizedthatthecapacitancecouldbeexpressedasafunctionofthefrequencyraisedtoapowersuchthatC=f(!)]TJ /F4 7.97 Tf 6.59 0 Td[().TheimpedanceofaCPEmaybeexpressedas ZCPE=1 (j!)Q(2{3)wherewhen=1,Qrepresentstheinterfacialcapacitance,andwhen<1thephysicalmeaningofQisnotclear.However,manyphysicalsourcesoffrequencydispersionhavebeenidentiedinelectrochemicalsystemsincludingcellgeometry,normaldistributionsofresistivity,diusionofspecies,andporouselectrodes.Otherproposedsourcesoffrequencydispersionincludesurfaceheterogeneityandthecouplingofchargingandfaradaiccurrents.Thebackgroundfortheinuenceofgeometry,anormaldistributionoftimeconstants,andsurfaceheterogeneityonimpedancemeasurementsispresentedbelow. 2.3.1GeometryEectTheelectrochemicalimpedancemaybeconsideredtobecomposedoftwopartsincludinganinterfacialcomponentandanohmiccomponent.Theinterfacialimpedanceisassociatedwiththekineticandchargingpropertiesoftheinterfaceandtheohmiccomponentisrepresentativeoftheelectrolyteconductivity.Inanelectrochemicalcellinwhichthecurrentdistributionisuniformacrosstheelectrodesurface,thesolutionresistanceisinseriestotheinterfacialimpedancesuchthattheinterfacialimpedancemaybedeterminedsimplybysubtractingouttheohmicresistance.However,incaseswherethecurrentdistributionisnonuniformacrosstheelectrodesurface,frequencydispersion 26

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occursmakingtheseparationoftheohmiceectandtheinterfacialimpedancemuchmorecomplicated.Foradiskelectrodeembeddedwithinaninsulatingplanewiththecounterelectrodelocatedinnitelyfarawayfromtheworkingelectrode,theohmicresistancemaybeexpressedasR=1=4r0.[ 50 ]Adimensionlessfrequencybasedontheradiusofthedisk,r0,aswellastheratiooftheelectrolyteconductivityandthedoublelayercapacitance,C0=,maybeexpressedas K=!C0r0 (2{4)suchthatfrequencydispersionisinducedatK1.Thecharacteristicfrequencyassociatedwiththefrequencydispersionisthen fc= 2C0r0(2{5)whereastheradiusofthediskincreasesthefrequencydispersionoccursinalowerfrequencyrange.NewmanaccountedforthefrequencydispersioncausedbythegeometryofthediskelectrodebyexpressingtheimpedanceasRe;e+1 j!C0;einwhichRe;eandC0;ewerefrequencydependentterms.Huangetal.usedniteelementmodelstosimulatetheimpedanceofablockingdiskelectrodeandshowedthattheohmiccontributionisactuallyacomplextermwithrealandimaginarypartsinwhichtheoverallimpedancemaybeexpressedasZ=Ze+Z0.AtfrequenciesbelowK=1,Zebehavesasapureresistor.Asthegeometry-inducedfrequencydispersionappearsonlyathighfrequencies,[ 28 ]theinuenceofgeometrycannotprovideanexplanationforCPEbehaviorthatmayextendovermanydecadesoffrequencyforblockingelectrodes.Onephysicalexplanationforacomplexohmicimpedanceistheradialcomponentofthelocalcurrentdensitybecomingafunctionoftheradiallocation.Forinstance,thelocalcurrentdensityonadiskelectrodehasaradialcomponentwhichisdependentontheradialpositionandalsodecreaseswithaxialposition.[ 10 ]Arecessedelectrode,whichhasauniformlocalcurrentdensitydoesnot 27

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havearadialcomponentandthereforetheohmiccontributiontotheimpedanceisapureresistance. 2.3.2NormalDistributionSomemetals,whenplacedinalkalinemedia,formaprotectiveoxidelayerseveralnanometersinthicknesswhichpreventscorrosionoftheunderlyingmetal.Thepassivelayervariesinresistivityinthedirectionnormaltotheelectrodesurfacewiththemostresistiveregionbeingclosesttotheelectrode.Hirschornetal.[ 24 25 ]showedthatapowerlawdistributionofresistivitynormaltothesurfaceofanelectrodecausesconstant-phaseelementbehavioroverabroadrangeoffrequencies.AmathematicalexpressionwasformulatedwhichallowstheestimationofthelmthicknessformtheparametersoftheCPEandalsorelatestheparametersoftheCPEtoaneectivecapacitance. 2.3.3SurfaceHeterogeneityInearlyexperimentsonsolidelectrodes,micro{scalesurfaceroughnesswasbelievedtocontributetofrequencydispersioninelectrochemicalmeasurements.[ 19 ]BorisovaandErshlerwerethersttoobservethatroughnessinuencedelectrochemicalmeasurements.[ 11 ]Theyfoundthattheextentoffrequencydispersionwasreducedbymeltingametalelectrodeandlettingitcooltoformintoadroplet,suggestingthatthesmoothersurfaceledtoamoreidealresponse.Followingthedevelopmentoffractaltheory,[ 43 ]therewasanattempttocorrelatethefractaldimensionsofthesurfacetotheCPEexponent.[ 45 55 ]Fractalgeometrywasshowntocausefrequencydispersion,howeveracorrelationbetweenthefractaldimensionandvariancefromidealitycouldnotbefound.Kantetal.[ 31 36 37 38 70 ]usedaformalismdescribedinreference[ 32 ]todescribetheinuenceoffractalroughnessonimpedanceresponse,buttheirmathematicalapproachdidnotprovideinformationonthefrequencydispersionassociatedwithCPEbehavior.Similarly,theDebye-FalkenhagentreatmentoftheelectricaldoublelayerbySinghandKantrevealedonlylow-frequencyinuenceofroughnessonimpedanceresponse.[ 67 ]Pajkossyshowedexperimentallythat 28

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annealingcanreducethedegreeoffrequencydispersion,eventhoughtheroughnessofthesurfaceremainedthesame,andtherebyconcludedthatthefrequencydispersioncannotbeduetothegeometriceectsolelybutmayalsohaveacontributionofatomic{scaleheterogeneities.[ 60 ]Janschetal.[ 29 ]usedexperimentsonthreegoldelectrodeswithdierentsurfacestructureandroughnesstoshowthatCPEbehaviorcouldnotbeattributedtoelectroderoughness.Emmanuelusedananalytic{continuationmethodtocalculatetheimpedanceofa2DHullcellandsimulatedtheeectofalineardistributionofsolutionresistanceassumingauniformcapacitance.[ 20 ]Theresultsofhisworkshowedthatfrequencydispersionoccurredathighfrequencieswhilethecalculationsyieldedidealbehavioratlowfrequencies.Anotherproposedcauseoffrequencydispersionisthesurfacedistributionofcapacitance[ 41 ].Brugetal.developedaformulawhichrelatestheparametersoftheCPEtoaneectivedoublelayercapacitance[ 12 ].Theypostulatedthatthedistributionwasduetoavariationincapacitanceacrosstheelectrodesurface.In1978,LeekandHampson[ 41 ]calculatedthefrequencydispersionusingacircuitelementladderwithdierentcapacitancevaluespersurfaceareaofelectrode.Theyconcludedthatsurfaceheterogeneitywasanimportantfactorcontributingtofrequencydispersion.In1992,PajkossyandNyikos[ 61 ]usedaself-similarcapacitancedistributiontoshowthatfrequencydispersionintheformofconstant{phase{elementbehaviorcannotbeexplainedbyacapacitancedistributionsincetheCPEbehaviorisonlyobservedforphysicallyimpossiblevariationsincapacitance.Alsoin1992,KurtkyaanddeLevie[ 39 ]conductednumericalsimulationstoexplainthatthefrequencydispersionduetononuniformcapacitanceiscausedbytheshiftofthecurrentlineswithfrequencyfromlocationsofhighimpedancetolocationsoflowimpedance.Whilemuchhasbeendonetodeterminethefactorscontributingtofrequencydispersioninimpedancemeasurements,thereisstillalackofunderstandingofthephysicalsignicanceofconstantphaseelementbehavioroveralargerangeoffrequencies 29

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whenasurfacelmisnotpresent.Themostcurrentproposedexplanationisasurfacedistributionoftimeconstants.Toeitherconrmorrejectthishypothesis,niteelementsimulationswereperformedondiskelectrodesincorporatingvariousformsofsurfaceheterogeneity.Theresultsarepresentedinlaterchaptersfortheinuenceofroughness,adistributionofcapacitance,aswellasadistributionofreactivity.Theinuenceofcoupledchargingandfaradaiccurrentswillalsobeexplored. 30

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CHAPTER3CORROSIONPROBLEMSINEXTERNALBRIDGETENDONSBridgetendonsincorporatinghighstrengthsteelstrandsareusedinsegmentalbridgeconstructiontolinkconcretesegmentstogetherwhichformthebridgespan.Corrosionofthesteelwithinthetendonsposesamajorthreattothestructuralintegrityofabridge.Abriefdescriptionofsegmentalbridgeconstructionispresentedaswellasbackgroundonpost-tensioning. 3.1PrecastSegmentallyConstructedBridgesPrecastsegmentalbridgesareconstructedbyconnectingprecastconcretememberstoformthespanofthebridgebetweenpiersorcolumnswhichholdthespanup.Thetermprecastmeansthattheconcretesegmentsarebuiltpriortobeingsetinplace.Thesegmentsactaspuzzlepieceswhichgreatlyexpeditestheconstructiontime.Thesegmentsarefastenedtogetherwiththeuseoflongitudinalpost-tensionedtendons.Twoclassicationsoftendonsmaybeused,includingexternalandinternaltendons.Internaltendonsareplacedwithintheconcretesegmentsthroughholeswhicharepre-formedintotheconcretesegments.Externaltendonsareusuallyplacedwithintheinneropeningsofthesegmentsbutareexternaltotheconcrete.Thetendonsconsistofmultiple7-wirepre-stressingstrandscontainedwithinaHigh-densityPolyethylene(HDPE)duct.Theyruncontinuouslythroughdeviatorblockswhichhelpformtheproleofthetendon.Thetendonsmayeitherbebondedorunbondedmeaninggroutisusedtollthespacebetweentheductandthesteel(bonded)orisleftempty(unbonded).Thetendonsarecalledpost-tensionedbecauseaftertheyareinplacethesteelwithinthetendonispulledintotensionusingahighstrengthjackshown.Stretchingthesteelstrandsforcestheconcretesegmentsintocompression.Concreteisabrittlematerialthatcracksundertensionbutcanwithstandlargecompressiveforceswithoutanystructuraldamage.Therefore,steelreinforcementisusedtowithstandanytensileforcesthestructuremayexperience.However,forthetensileloadtobetransferredtothesteelundernormalor 31

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unstressedreinforcment,theconcretehastocrackrst.Bystressingthesteelwithinthetendonsandforcingtheconcreteintocompression,thetensileforcesaretransferredtothesteelpriortotheconcretecrackingwhichgreatlyimprovesthedurabilityofthebridgeandallowsforlongerbridgespans.Theendsofthetendonsareanchoreddownatbulkheadsandstressedafterwhichtheductislledwithcementitiousgrout.[ 75 ]Thealkalinegroutisintendedtoprovideprotectionagainstcorrosionbut,dueto,possiblevoidsinthegroutandareasofimpropermixing,casesofseverecorrosionhaveoccurred. 3.2CorrosionIssuesinPost-tensioningSystemsDespitetheuseofgrouttopreventcorrosion,therehavebeenmanycasesinwhichcorrosionhasstilloccurred.Someofthecausesofcorrosionwithinpost-tensionedtendonsincludevoidsinthegrout,groutbleedwater,cracksintheduct,andgroutsegregation.Voidsinthegroutcanbecausedbytheadsorptionofbleedwaterandgroutsegregationisusuallycausedbyimpropergroutmixingprocedures(Rafols2013).Sincepost-tensioningtechnologyisstillrelativelynew,theseproblemswerenotevidentuntilthe1980s.Therstinstanceoccurredin1980whenthesouthernouterroofoftheBerlinCongressHallcollapsed23yearsafteritwasconstructed.[ 1 ]Soonthereafter,twobridgeswerefoundwithsimilarseriouscorrosionissues{theTafFawrBridgeonA470inWales,EnglandandtheAngelRoadBridgeontheA406NorthCircularinLondon,England.[ 64 ]Ultimately,theTafFawrBridgewasdemolishedin1986whiletheAngelRoadBridgewassignicantlyretrottedin1982.In1985,thesingle-spansegmentalpost-tensionedYnys-y-GwasBridgeinWalescollapsedasaresultofcorrosionoflongitudinaltendonsatitssegmentedjoints.Thisstructurewasonly32yearsold,andtherehadbeennopreviousindicationofdistresspriortocollapse.[ 76 ]In1992,theBritishDepartmentofTransportationconductedastudyonthesecorrosionissuesandconcludedthattherewasnomethodthatcouldguaranteecompletecorrosionprevention.Laterthatyear,post-tensionedbridgeswereeectivelybannedintheUnitedKingdom.[ 42 ] 32

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TheUnitedKingdomwasnottheonlycountrywithpost-tensionedbridgeissues.Thepost-tensionedMelleBridge,whichwasbuiltinBelgiumin1956,collapsedin1992.Inthisinstance,thebridgehadbeeninspected,loadtested,re-waterproofed,declaredadequate,andjustrestoredtoservicetwoyearspriortoitscollapse.[ 1 ]Morerecently,theSaintStefanoBridgeinItaly[ 65 ]andtheLowesMotorSpeedwayfootbridgeinNorthCarolina[ 13 ]collapsedduetosimilarcorrosion-relatedfailures.Corrosioninpost-tensionedbridgesisamajorconcerninFloridaaswell.Therstreportedpost-tensionedcorrosionissuewasatthe18-yearoldNilesChannelBridgeintheKeys.[ 66 ],[ 2 ]Similarissueswerereportedatthe7-yearoldMidBayBridgeintheWesternPanhandle[ 17 ],[ 75 ]andthe15-yearoldSunshineSkywayBridgeinTampa.[ 2 ]AnumberofstudieswerecommissionedbytheFloridaDepartmentofTransportation(FDOT)toaddresstheseissues.AnimportantconclusionfromthestudyrelatedtotheMidBayBridgewasthatanon-destructivetechniquefortestingcorrosionandcorrosion-riskinthesepost-tensionedmemberswasrequired. 3.3MethodsofCorrosionDetectioninBridgeTendonsIn2006,FDOTandresearchersattheUniversityofFloridatriedtodevelopanon{destructivetechniqueforcorrosiondetectioninpost-tensionedmembers.[ 46 ]Thestudyhingedonndingair-voidsand/orentrainedwaterinthegroutmatrixbecausethesevariableshavebeenshowntoleadtocorrosion.Anumberofmethodswereusedinthisstudyincludinggroundpenetratingradar,impactecho,ultrasonicsoundwaves,andgamma-rayspectroscopy.Resultsfromthisstudyindicatethat,ofthesemethods,onlygamma-rayspectroscopyshowedanyrealpromiseasapossiblesolution.However,spectroscopyresultswerepreliminary(atbest),andwerebasedonlyonalimitednumberoflaboratory-preparedsamples.Furthermore,eld-implementationofsuchasystemappearedtobeunlikelysinceithingeduponusinganHPGedetectorwhichrequiredliquidnitrogen(at77K;-196oC; 33

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-321oF).Whileinvestigatorsfromthepreviousstudyrecommendeddesigningabettergamma-raydetector,specicsaboutexactlyhowthiswastobedonewereneveraddressed.TheFederalHighwayAdministrationidentiedmainmagneticuxasapossiblenondestructivemethodforexternaltendonswhichisstillinneedofdevelopment[ 23 ].Highpoweredmagnetsareusedtoinduceastaticmagneticeldinthetendonandthemagneticux,whichisafunctionofsteelcross-sectionalareaismonitoredtodetectfractures[ 72 ].Ultrasonictomographyhasbeenusedtodetectvoidswithintheinternaltendonsbysendingultrasoundwavesandmeasuringthetimeforthemtobetransmitted.Dierencesinthedensityofmedialeadtolongertransmissiontimes.[ 44 ]Whilethesemethodsareusefulinidentifyingproblemareas,onlyelectrochemicaltechniquessuchasimpedancespectroscopycanyieldactualcorrosionrates.Inapplicationtoreinforcedconcrete,manymethodshavebeendevelopedtoestimatethecorrosionrateoftheembeddedsteel.ThemostnotableofthesetechniquesistheLinearPolarizationResistance(LPR)methodinwhichasmallover-potentialisappliedtothereinforcingsteelandthecurrentresponseismonitored.Thepolarizationresistanceofthesteelisestimatedbydividingthepotentialbythecurrentresponse.Withtherelationship, icorr=B=ARp(3{1)developedbySternandGearythesteelpolarizationresistance,Rp,isusedtoestimatethecorrosionratebasedontheTafelslope,B.However,theinherentassumptionisthatthecorrosionreactionfollowsTafelkinetics.Also,theLPRmethodrequiresaconnectiontothesteeltopolarizeitbutinreinforcedstructuresaccesstothesteelcanonlybeprovidedbycuttingthroughtheconcrete.Toavoidthis,researchhasbeendonetodevelopawaytoindirectlypolarizethesteelwithoutanelectricalconnection.Anindirectmethodhasbeenexploredinwhichanelectriceldisappliedtothesurfaceoftheconcreteandtheinducedcurrentpulseindirectlypolarizesthesteel.[ 8 ]Analternativetothepulsemethodiselectrochemical 34

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Figure3-1. Schematicrepresentationshowingthecurrentpathsforahighlyconductivestrandembeddedinamoreresistivematerial.A)interfacialimpedanceatthepositiveelectrode;B)ohmicresistanceanddielectricresponseoftheresistivematerialbetweenthepositiveelectrodeandthestrand;C)interfacialimpedanceatthesurfaceofthestrand;D)ohmicresistanceofthestrand;E)interfacialimpedanceatthesurfaceofthestrand;F)ohmicresistanceanddielectricresponseoftheresistivematerialbetweenthenegativeelectrodeandthestrand;andG)interfacialimpedanceatthenegativeelectrode.TheletterHsigniestheohmicresistanceordielectricresponseoftheresistivematerialbetweenthepositiveandnegativeelectrodes. impedancespectroscopy,whichusesasinusoidalcurrentorpotentialperturbationappliedtotheconcretesurfaceatarangeoffrequenciestoindirectlypolarizethesteel.[ 33 47 ]Monteiroetal.[ 47 ]reportedusingindirectimpedancespectroscopytodeterminethelocationandtheconditionofsteelrebarwithinconcreteslabs.Theywereabletoqualitativelydeterminethatthemeasuredsurfaceimpedancewasafunctionofthecorrosionstateofthesteelaswellastheresistanceoftheconcrete.[ 47 ] 3.4IndirectImpedanceSpectroscopyTheindirectimpedancetechniquerequiresa4electrodearrayconsistingofaworkingandcounterelectrode(outertwoelectrodesofthearray)andtworeferenceelectrodes(innertwoelectrodesofthearray).AschematicrepresentationoftheindirectimpedancemeasurementisshowninFigure 3-1 Anacperturbationisappliedbetweentheworkingandcounterelectrodesandthereferenceelectrodesmeasurethepotentialresponse.Thecurrentfollowsthepathofleastresistancebetweentheworkingandcounterelectrode_Dependingonthegeometryofthespecimenandthefrequencyofthe 35

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perturbation,acertainfractionofthecurrentwillgosolelythroughthegroutwhiletherestwillenterthesteel.Thereforethesteelisindirectlypolarizedduetotheacperturbationatthesurface.Theelectricalimpedancemeasuredbetweenthepositiveandnegativecontactsisinuencedbytheinterfacialimpedanceatthepositiveelectrode(A);theresistivityofthematerialbetweenthepositiveelectrodeandthestrand(B);theinterfacialimpedanceatthesurfaceofthestrand(CandE);theohmicresistanceofthestrand(D);theresistivityofthematerialbetweenthepositiveelectrodeandthestrand(F);andtheinterfacialimpedanceatthenegativeelectrode(G).Thecurrentthatowsonlythroughtheresistivematerialwillbeinuencedbytheresistivityofthematerialbetweenthepositiveandnegativeelectrodes(H).Useofafour-electrodecongurationforpotentialmeasurement(indicatedinFigure 3-1 byarrowsconnectedtoavoltmeter)eliminatestheinuenceoftheinterfacialimpedancesatthepositiveandnegativeelectrodes.Thefour-pointprobecongurationcausestheimpedanceresponsetoreecttheconditionofthegroutandthestrandwithouttheconfoundinginuenceofthecurrentinjectionpoints.Oneofthecomplicationsintheapplicationoftheindirectimpedancetechniquetopost-tensionedtendonsisthedicultyinextractingtheimpedanceofthesteelandgroutinterface.Anequationtoestimatethepolarizationresistanceofthesteelfromtheresistanceoftheelectrolyteandtheresistanceofthecellmeasuredwiththesteelpresentwasproposed.Theequationwasderivedbasedontheassumptionthatthecurrentranparalleltothesteel.Andradeetal.[ 7 9 ]usedresultsofniteelementmodelstoproposeananaloguecircuitthataccountsforthepolarizationbehaviorofthesteelandthepropertiesofthemortarinwhichthesteelisembedded.Theydeterminedthatsinceaportionofthecurrentgoesthroughtheconcreteinparalleltothesteelandanotherportiontravelsthroughthegroutandthenentersthesteel,theconcreteresistancehasbothaparallelandseriescomponent.However,theywerenotabletodeterminetheexactcircuittofullydescribetheindirectmethod.Keddametal.[ 33 ] 36

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showedthatatlowfrequencies(i.e.,thed.c.limit)thecurrentowsparalleltothesteel;whereas,athighfrequencies,thecurrententersthesteelperpendicularly.Theyfoundthatthezero-frequencylimitoftheimpedancescaledbytheresistivityofthegroutisafunctionofthepolarizationresistance,similarlyscaledbytheresistivityofthegrout.Thisrelationshipisindependentofthegroutresistivityforadenedgeometryandelectrodeconguration.Thedisadvantageofusingequivalentcircuitsistheinabilitytoaccountfortheeectsofnonuniformcurrentdistributionandill-denedgeometries.Huangetal.[ 28 ]showedthattheimpedanceofadiskelectrodeisinuencedbythegeometryofthediskathighfrequencies.Theohmiccontributiontotheimpedancebecomescomplexathighfrequenciesduetothenonuniformcurrentdistribution.Thereforetheuseofequivalentcircuitstottheimpedanceofdiskelectrodesisonlyvalidbelowacharacteristicfrequencydependentontheradiusofthedisk.Thenonuniformcurrentdistributionofanindirectimpedancemeasurementmaycausesimilarissues.Theobjectiveofourresearchistouseindirectimpedancetomonitortheintegrityofthesteelaswellasthegrout.Todothis,theappropriateelectrodecongurationsandspacingpertendongeometrywillneedtobedetermined.Also,theappropriateelectrodematerialsandcontactschemestothegroutaswellastheappropriatefrequencyrangeandperturbationmodetousewillneedtobedetermined.Theothermajorobjectiveistodevelopameansofinterpretingtheindirectimpedanceresponseanddeterminefromthatmeasurementifthesteeliscorroding. 37

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CHAPTER4PROOFOFCONCEPTFORCORROSIONDETECTIONUSINGINDIRECTIMPEDANCEAproofofconceptwasestablishedforusingindirectimpedancespectroscopytodetecttheexistenceofcorrosioninpost-tensionedtendons.Thisdevelopmentwassupportedbyacombinationofbench-topexperimentsperformedondiskelectrodesingroutandsynthetictendons.Thisworkshowedthatindirectimpedancecanbeusedtoobservedierencesintheimpedanceanddierentiatebetweenactiveandpassivesteel.Sensitivitytolocalizedcorrosionwasfoundtobegreatestwhenacurrentinjectionelectrodeispositioneddirectlyabovethecorrodingmetal. 4.1Conventional3-electrodeCongurationConventionalimpedanceexperimentswereconductedwithadiskelectrodeembeddedingrouttodeterminetheimpedancebehaviorofthesteelwithintheenvironmentofthegrout.Theimpedancewasanalyzedwithmathematicalmodelswhichincludephysicalparameterstodescribethebehaviorofthesteelandgroutinterface.Oneofthenecessaryparametersneededtodevelopareliableinterpretationtechniqueoftheindirectimpedanceistheimpedanceofthesteelandgroutinterfaceinlocationsofpassiveandactivelycorrodingsteel.Cellsconguredwith3electrodesweremadeofsmallplasticcylinderscontaininggroutastheelectrolyteandtheimpedancewasmeasuredacrossthesteelandgroutinterface.A3-inrodofsteelwascutfromthekingwireofthesteelstrandandwasinsertedintothegroutasshowninFigure 4-1 .Heatshrinktubingandduct-tapewasusedtoinsulatethesidesofthesteelrodsuchthatonlythecrosssectionoftherodwasexposedtothegroutintheformofadiskelectrodewithinaninsulatingplane.Astainless{steelwiremeshwasusedasthecounterelectrodeandasolidAg{AgClelectrodewasusedasthereference.Fourcellsweremadeandtwoofthemwereforcedtocorrode.AschematicoftheimpressedcurrenttechniqueispresentedinFigure 4-2 .Aconstant20-Vpotentialwasappliedbetweenthesteelrodandthestainlesssteelmeshfor1week.Theimpedancewas 38

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Figure4-1. Schematicshowingtheconventional3{electrodeimpedancemeasurementonacylindricalelectrochemicalcellinwhichtheelectrolyteisgroutandtheworkingelectrodeisacouponofthesteelstrand. Figure4-2. Schematicshowingtheimpressedcurrenttechniqueforacylindricalelectrochemicalcellinwhichtheelectrolyteisgroutandtheworkingelectrodeisacouponofthesteelstrand. 39

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Figure4-3. Conventional3{electrodeimpedanceofasteeldiskelectrodeingroutbeforeoneofthespecimens(corroded)wasforcedtocorrode. measuredbeforeandaftertheapplicationoftheimpressedcurrent.TheimpedanceresultsoftwoofthespecimensarepresentedinFigure 4-3 inwhichoneislabeledasthecontrolandtheotheristheto-be-corrodedspecimen.Theresultsshouldberepresentativeoftheimpedanceofthesteelandgroutinterface.Theimpedancewasmeasuredatfrequenciesbetween500Hzand10mHz.Athighfrequenciesthereisadepressedcapacitivearcfollowedbyastraightlineatanangleatlowerfrequencies.The-highfrequencybehaviorinbothcasesisalmostidenticalwhiletheslopeoftheimpedanceisslightlylargerforthecontrolspecimen.TheimpedanceofacorrodingspecimeniscomparedtothecontrolcaseinFigure 4-4 .Afterforcingoneofthespecimenstocorrode,theimpedancedecreaseddrastically.Theohmicresistanceofthecorrodedspecimenincreasedwhichcanbeexplainedbythereductionofwaterthatoccursduetothecathodicreactionwhichincreasestheresistivity.Theimpedanceofthecontrolspecimendidnotchangesignicantly.Thegureinset 40

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Figure4-4. Conventional3{electrodeimpedanceofasteeldiskelectrodeingroutafteroneofthespecimens(corroded)wasforcedtocorrode. showsamagniedviewofthecorrodedspecimenimpedancewhichcontainsahighfrequencytailandasmallcapacitivearcatlowerfrequencies.Themagnitudeoftheimpedancedecreasedbyafactorof50afterforcingcorrosion.Thesteelrodsfromthecontrolandcorrodedspecimenswereremovedtoviewthesteelsurface.ImagesofthesteelsurfacearepresentedinFigure 4-5 .Thecontrolcase,Figure 4-5A ,hasashinysurfaceandareasofresidualgroutwhichhaveadheredtothesurface.ThecorrodedcaseshowninFigure 4-5B hasreddish-browncorrosionproductsimilartowhatwasobservedonthesteelremovedfromthetendonsaftertheinducedcorrosion.Inthiscasethecorrosionismuchmoreuniformandadvanced. 4.2RegressionFitAnequivalentcircuitmodelwasdevelopedtottheimpedanceresultsofthe3-electrodemeasurementstobuildanunderstandingoftheimpedanceofthesteelandgroutinterface.TheequivalentcircuitisbaseduponaphysicalmodeloftheinterfacewhichispresentedinFigure 4-6 .Wehaveassumedtheinterfacecomprisesaporouslm 41

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A BFigure4-5. Imagesofthesteeldiskelectrodeextractedfromthegrout:a)Passivecaseb)Corrodedcase. Figure4-6. Circuitdiagramusedtotthepassivecaseimpedanceofthesteelandgroutinterface. 42

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Figure4-7. ImpedanceofthepassivesteeldiskelectrodeingroutttedwiththecircuitinFigure 4-6 duetoanunevenoxidelayer.TheresistancetocurrentowthroughtheporeisdescribedbyRspandtheimpedanceoftheinterfacebetweenthesolutionandthebaremetalisdescribedasaresistorandconstant{phaseelementinparallelwithparametersRt,Q2and2.Theimpedanceofthelmwasmodeledasaconstant{phaseelementwithparametersQ1and1assumingthatthelmhasadistributionofresistivitynormaltotheelectrodesurface.ThettingresultsarepresentedinFigure 4-7 inNyquistformatforthecontrolcase.OriginLabnon{linearcomplexregressionanalysiswithmodulusweightingwasusedtottheparameterstothedata.ThettingresultsarepresentedinTable 4-1 alongwiththestandarderrorforeachparameter.Theoretically,Rtshouldbeverylargeforthepassivecaseandshoulddecreaseasthecorrosionrateincreases.Thesamecircuitmodelwasappliedtothecorrodedcasebuttheresultswerenotrealistic.Therefore,a 43

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Table4-1. Regressionparametersandstandarderrorforequivalentcircuitttoconventionalthreeelectrodeimpedancefortheactiveandpassivecases. Passive Re2069.0cm2Rsp31.570.86kcm2Rt532k26cm2Q13.34e-40.04e-4Sscm210.860.01Q22.13e-40.01Sscm220.790.01 Figure4-8. 2ft.fabricatedtendonwith10electrodelocationsformeasuringtheindirectimpedanceandschematicofthefourelectrodemeasurement. measurementmodel[ 3 ]wasusedtodeterminetheextrapolatedvalueofthezerofrequencylimitoftherealimpedancewhichsigniesthelowestpossiblevalueofthepolarizationresistance.AseriesofVoightelementsarettotheimpedancesequentiallyuntilthetcannotbeimprovedwiththeadditionofanotherelement.Theextrapolatedvalueofthelowfrequencylimitoftheimpedanceisthenbasedontheelementwhichhasthelargesttimeconstant.Thevalueobtainedwasapproximately15kwhichisroughly30timeslessthanthecharge-transferresistanceobtainedforthepassivecase. 4.3CorrosionDetectionUsingIndirectImpedanceExperimentaltendonswereconstructedtoresemblea2-ft.sectionofa3-in.diametertendon.AtendonisshowninFigure 4-8 inwhichaschematicdiagramofapotentiostatshowstheindirectimpedanceset-up.A4{electrodearrayisusedinwhichanaccurrentis 44

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passedbetweentheoutertwoelectrodes(workingandcounter)andthepotentialresponseismeasuredbetweenthetwoinnerelectrodes(reference1andreference2).Fortheinitialeaseofinterpretation,tendonsectionsweremadeeitherwithonesteelstrandlocatedattheaxisofthecylindricalductorwithjustgrout.Aproprietarygrout(Sika)wasusedasthelleranditwasmixedaccordingtomanufacturer'sspecicationsandallowedtocureforatleast30days.Ten1-cmdiameterholesweredrilledandtappedintotheductsothattheelectrodescouldbescrewedintoplacetherebypreventingthegroutfromdryingoutatthesepoints.Anin-houseelectrolytegelwasusedastheelectricalcontactbetweenthetitaniumelectrodesandthegrout.Thegelwasmadebymixingcarboxyl{methylcellulosepolymerintoanelectrolytesolutionof1MKSO4untilconsistentlyviscous.Dimensionallystable0.5cmdiametertitaniumrodscoatediniridium-oxidewereusedastheelectrodes.Impedancemeasurementsonthetendonswithoutthesteelwerecomparedtothemeasurementsperformedonthetendonswithonesteelstrand.TheresultsareshowninFigure 4-9 .TheimpedanceofeachcaseispresentedinNyquistformatinFigure 4-9A .Therealpartoftheimpedanceofthegroutonlycaseismuchlargerthanthetendonwiththesteelbuthaslittlecapacitiveresponse.Theimpedancemeasuredonthetendonwiththesteelhasamuchsmallerohmicimpedancebuthasalargercapacitiveloopindicatingthattheindirectimpedancemustsamplethepropertiesofthesteel.TherealpartoftheimpedanceisshowninFigure 4-9B asafunctionoffrequency.Therealimpedanceofthetendonwithoutsteelisconstantwithfrequencywhichmeanstheimpedanceofthebehavesasaresistor,andshouldreecttheresistivityofthegrout.Corrosionwasforcedononeofthetendonscontainingsteelusinganimpressedcurrentmethod.AschematicoftheimpressedcurrenttechniqueisshowninFigure 4-10 .Apowersourcewasusedtoapplyaxedpotentialdierencebetweenthesteelandtwoofthetitaniumelectrodessuchthatcorrosionwouldbeinducedonthesteelsurface 45

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A BFigure4-9. Indirectimpedanceoffabricatedtendonwithandwithoutsteel:a)inNyquistformat,b)therealimpedanceasafunctionoffrequency. Figure4-10. Schematicrepresentationoftheimpressedcurrenttechniqueusedtoforcecorrosion. 46

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Figure4-11. IndirectimpedanceinNyquistformatofafabricatedtendonwithpassivesteelandelectrodelocationasaparameter. preferentiallybeneaththetwoelectrodes.Aconstantpotentialof20Vwasappliedfor1week.Theaveragecurrentthatowedduringthattimewasapproximately150A/s.TheindirectimpedanceresultsarepresentedinFigure 4-11 withthelocationoftheelectrodesasaparameter.Thenumbersdistinguishingeachmeasurementcorrespondtothelocationoftheelectrodesinorderofworking,reference1,reference2,andcounterelectrodes.Theimpedanceateachlocationalongthetendoncontainsacapacitivearcandanohmicresistancebetween125and155.Thereislittlevariationintheimpedancebetweeneachlocationindicatingthattheimpedanceofthesteelinthecontrolcaseisforthemostpartuniform.TheimpedanceresultsofthecorrodedcasearepresentedinFigure 4-12 withthelocationoftheelectrodesasaparameter.Thesamenumberingschemeoftheelectrodesasinthecorrodedcasewasused.Inthecorrodedcase,theimpedancevariedsignicantlywithlocation,indicatingthattheimpedanceofthesteelwasnotuniform.Theimpedanceatlocationsofcorrosionshouldbesmallerthantheimpedanceofpassivelocations.ThegureinsetinFigure 4-12 showsamagniedviewofthe3smallestimpedances.Thelocationofthesmallestimpedanceoccurredatlocations1{2{3{4,5{6{7{8,and7{8{9{10. 47

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Figure4-12. IndirectimpedanceinNyquistformatofafabricatedtendonwithcorrodedsteelandelectrodelocationasaparameter. Accordingtothelocationofthecathodesintheimpressedcurrenttechnique,thecorrosionshouldbelocatedbeneathelectrodes5and6.Toverifythis,thesteelwasextractedfromthetendonbybreakingapartthegrout.Magniedimagesweretakenalongthesurfaceofthesteeltodeterminewherethemostseverecorrosionwaspresent.ImagesarepresentedinFigures 4-13 and 4-14 ofthesteelsurfacelocatedbeneatheachelectrode.Atlocations2,6,and10thesurfaceofthesteelissmoothanddoesnotshowanysignsofcorrosion.Thelocationswiththemostcorrosionwerebeneathelectrodes1,3,5,and7.Therefore,theindirectimpedancemeasurementsshouldsignifycorrosionatlocations1{2{3{4,3{4{5{6,5{6{7{8,7{8{9{10.Threeoutoffouroftheofthelocationsdidhavesmallimpedancesbuttheimpedanceatlocation3{4{5{6,forsomereason,hadanimpedancemoresimilartoapassivestate.Atsteellocation5,Figure 4-13E ,therewas 48

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A B C D E FFigure4-13. Imagesofthesteelsurfacedirectlybeneatheachelectrodeforthecorrodedcase. 49

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A B C DFigure4-14. Imagesofthesteelsurfacedirectlybeneatheachelectrodeforthecorrodedcase. acorrosionpitthatformed.Thecorrosionproductwasintheformofaredish{brownrust.Someofthecorrosionproductwasincidentallyscratchedobytheremovalofthegrout.Theindirectimpedancewassmallbutalsoincludedaninductiveloopatlowfrequenciesusuallyassociatedwithamodulatinglm.Atsteellocation4therewasasmalllongitudinalcrackinthesteelwhichispartiallylledwithgroutindicatingthatthecrackwastherepriortoconstructingthetendon. 50

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Theindirectimpedancemeasurementsonacorrodedtendonandacontroltendonshowedtheabilitytodetectcorrosion.Theresultspresentedhereindicatethatonlythelocationofthesteeldirectlybeneaththeworkingelectrodeissensed.Finiteelementsimulationstodeterminethelocationofthesteelthatissensedandtodevelopawayofinterpretingthemeasurementsaredescribedinthenextchapter. 51

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CHAPTER5FINITEELEMENTSIMULATIONSANDINTERPRETATIONFiniteelementsimulationsoftheindirectimpedancemeasurementwereperformedtouncoverthecontributionofthegroutresistivitytoimpedanceaswellasdeterminethelocationofthesteelthataparticularcongurationsensed.Theindirectimpedancewasfoundtoincludetwoseparatecontributionsofthegroutresistivity.Thereisanohmicimpedanceassociatedwiththegroutthatisparalleltothesteelandanotherohmicimpedanceassociatedwiththegroutthatisinseriestothesteel.Theparallelcomponentwasmuchlargerthantheseriescomponent,andtheimpedancedecreasedwithdecreasedfrequency,whereas,theseriescomponentincreasedwithdecreasedfrequency.Byunderstandingtheexactinuencethegrouthasontheindirectimpedance,amethodmaybedevisedtoextractthepropertiesofthesteelanddeterminethecorrosionrate. 5.1MathematicalDevelopmentToaidintheinterpretationoftheexperimentalresults,aniteelementmodelwasdevelopedtosimulatetheindirectimpedance.Huangetal.[ 28 ]explainedtheuseoflinearkineticsastheboundaryconditiononadiskelectrodebasedonthederivationsofNewman[ 50 ]andNisancioglu.[ 53 54 ]Thenormalcurrentdensityatthesurfaceoftheelectrodecanbeexpressedintermsofafaradaicreactionandachargingcurrentas i=C@(V)]TJ /F1 11.955 Tf 11.95 0 Td[() @t+(a+c)i0F RT(V)]TJ /F1 11.955 Tf 11.96 0 Td[()=)]TJ /F3 11.955 Tf 9.3 0 Td[(@ @y(5{1)Theoscillatingcurrentdensitymaybeexpressedinthefrequencydomainas ~i=j!C(~V)]TJ /F1 11.955 Tf 13.26 3.02 Td[(~)+(a+c)i0F RT(~V)]TJ /F1 11.955 Tf 13.26 3.02 Td[(~)(5{2)withtheuseoftherelationship i= i+Re~iej!t(5{3)wherethecurrentisexpressedastheadditionofasteady-stateandanoscillatingterm.InEquation 5{2 ,~Visthepotentialperturbation,and~isthecomplexoscillatingpotential 52

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withintheelectrolyte.Fortheindirectimpedancesimulationboththeworkingandcounterelectrodeboundaryconditionsweresetasoscillatingcurrentswithapositivepotentialperturbationappliedtotheworkingelectrodeandanegativeoneappliedtothecounterelectrode.Asimilarboundaryconditionwassetatthesteelbutwiththepotentialsettozerosuchthatallotherpotentialswouldbeinreferencedtothesteel.Thesteelwasmodeledforanactivecorrosioncaseandapassiveblockingelectrodecase.Theactivecase,Equation 5{4 ,isexpressedastheadditionofthechargingandfaradaiccurrent,i.e., ~i=j!C()]TJ /F1 11.955 Tf 10.6 3.03 Td[(~)+(a+c)i0F RT()]TJ /F1 11.955 Tf 10.6 3.03 Td[(~)(5{4)TheoscillatingpotentialsolutionisfoundbysolvingLaplace'sequationwiththegivenfrequency-dependentboundaryconditions.Withtheuseofpotentialprobestheimpedanceissimulatedasthequotientofthepotentialdierencebetweentworeferenceprobesandthecurrentperturbationappliedbetweenthecurrent-injectingelectrodesexpressedas Z=~Vref1)]TJ /F1 11.955 Tf 13.74 3.02 Td[(~Vref2 ~I(5{5)Thecharge-transferresistanceforlinearkineticscanbeexpressedintermsoftheexchangecurrentdensityas Rt=RT i0(a+c)(5{6)whichisthesameexpressionusedtoestimatethepolarizationresistanceofthesteelinthecorrodingcase.The3Dpotentialdistributionwasdeterminedundertheassumptionofuniformelectrolyteconductivity,andtheindirectimpedancewassimulated.Theactivecase,Equation 5{7 ,isexpressedastheadditionofthechargingandfaradaiccurrent. ~i=j!C()]TJ /F1 11.955 Tf 10.6 3.02 Td[(~)+(a+c)i0F RT()]TJ /F1 11.955 Tf 10.6 3.02 Td[(~)(5{7)ThepassivecaseismodeledusingaConstant-Phase-Element(CPE)withanimpedanceof ZCPE=1 (j!)Q(5{8) 53

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Figure5-1. Currentandpotentialdistributionofa1cmsquare10Ohm-mresistivitygroutmodelwithcurrentinjectingelectrodesplacedontheverticalsides wherethephaseangleisindependentoffrequency.When=1,Qhasunitsofcapacitance.Whendoesnotequalunitythesystemhasadistributionoftimeconstantsorsurfaceheterogeneityeithernormalorparalleltothesurface.[ 57 ]TheexpressionusedtorepresentblockingbehavioratthesteelforthenormalcurrentdensityofaCPEis ~i=)]TJ /F1 11.955 Tf 10.81 3.15 Td[(~!Qhcos( 2)+jsin( 2)i(5{9)TheoscillatingpotentialsolutionisfoundbysolvingLaplace'sequationwiththegivenfrequency-dependentboundaryconditions. 5.2JusticationofBoundaryConditionsA2Dsquareofuniformconductivity,Figure 5-1 ,wasmodeledtoconrmtheoscillatingcurrentboundaryconditions.Theverticalsidesofthesquareactedasthecurrent-injectingelectrodes.ThepotentialdistributionisshownbythecolorgradientinFigure 5-1 andthecurrentpathisshownbythehorizontalredlines.The2-ptimpedancewassimulatedbydividingthepotentialdierencebetweentheelectrodesbythetotalcurrentcrossingoneoftheelectrodeboundaries.Atallfrequenciestherealimpedance,Figure 5-2 ,istheresistivityoftheelectrolytemultipliedbythedistancebetweentheelectrodesanddividedbythecrosssectionalarea.Theimaginaryimpedanceiszerosincethegroutismodeledasahomogenousmaterialwithaconstantresistancewithoutanydielectricproperties. 54

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Figure5-2. Simulatedrealimpedanceofasafunctionoffrequencyofa1cmsquaregroutmodelwithcurrentinjectingelectrodesplacedontheverticalsides. Figure5-3. Currentandpotentialdistributionatthelowfrequencylimitofa1cmsquaregroutmodelwitha0.25cmradiussteelplacedinthecenterandcurrentinjectingelectrodesplacedontheverticalsides. AsteelcircularelementwasinsertedintothegroutmodelwiththeboundaryconditiondescribedbyEquation 5{7 .Thecharge-transferresistancewassettoandthedoublelayercapacitancewas.Atlowfrequencies,Figure 5-3 ,thesteelbehavesasanopencircuitduetothedominanceofthecharge-transferresistanceandrepelsthecurrent.Athighfrequencies,Figure 5-4 ,itbehavesasaclosed-circuitandthecurrententersitnormaltothesurface.TheseresultsareconsistentwiththoseofKeddametal.[ 33 ]TheNyquistplotofthesimulatedimpedance,Figure 5-5 ,isacapacitivelooprepresentativeofaresistorandacapacitorinparallel.Thisisasimplemodelthatshowstheconceptofindirectimpedance,andconrmstheuseoftheoscillatingboundaryconditions. 55

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Figure5-4. Currentandpotentialdistributionatthehighfrequencylimitofa1cmsquaregroutmodelwitha0.25cmradiussteelcircleplacedinthecenterandcurrentinjectingelectrodesplacedontheverticalsides. Figure5-5. Simulatedimpedanceofa1cmsquaregroutmodelwitha0.25cmradiussteelcircleplacedinthecenterandcurrentinjectingelectrodesplacedontheverticalsides Athree-dimensional,60cmlongcylindricalsectionofatendonwasmodeled,withandwithoutsteeltosimulatetheimpedanceofapost-tensionedtendon.Thesteelstrandis0.625cminradiusandislocatedalongthelongitudinalaxisofthecylinder.Alldimensionsofthemodelweremadetomatchthefabricatedtendons.Themeshofthemodel,Figure 5-6 ,iscomposedoffreetetrahedralelementswhichdecreaseinsizeattheelectrodeboundaries.Boundarylayerelementswereaddedtothesteelandelectrodeboundaries.Referenceelectrodeswereplacedalongthesurfacetoanalyzethepotentialdistributionalongthesurface. 5.3Results&AnalysisThesimulatedindirectimpedanceispresentedforaniteelementmodeltendonwithonesteelstrand.TheparametersusedwereRt=11:8kcm2,C=20F=cm2, 56

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Figure5-6. Meshofthe3Dtendonmodel. and=125m.Anequivalentcircuitispresentedtottheimpedancebasedontheohmicimpedanceofthegroutandtheinterfacialimpedanceofthesteel.Asimpliedanaloguecircuitispresentedwhichreducestheequivalentcircuittothreecomponents.Asensitivityanalysisoftheparameterstochangesinsteelpolarizationresistanceaswellasthedistancebetweenthemeasurementelectrodesisalsopresented. 5.3.1ExperimentalDataFittingThepassivecasesimulationwasusedtoiterativelyttheexperimentalimpedancebyrstestimatingtheresistivityofthegroutas125-manditeratingtheCPEparametersatthesteel.Qwasfoundtobe0:9Ssandwas0.9.Thesimulatedimpedanceisshowntotexperimentaldataof2electrodecongurationsatlowfrequencies,providedinFigures 5-7 and 5-8 .Thettingoftheindirectimpedancefortwodierentelectrodecongurationsisavalidationofourniteelementmodelinitsabilitytoreplicateexperimentalmeasurementsandthereforemaybeusedtoestablishaninterpretationprocedure. 5.3.2DeterminationofSteelSensingAreaTheareaofsteelthatissensedforeachelectrodearrangementisrequiredfortheinterpretation.Theindirectimpedancewassimulatedfor3dierentcases:afullypassivesteelstrand,auniformlycorrodingsteelstrand,andapassivestrandwithalocalizedareaofcorrosionatthecenterindicatedbytheredsectioninFigure 5-9 .Twenty-ve 57

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Figure5-7. Simulatedimpedanceresultscomparedtotheexperimentalresultswithanelectrodecongurationof1357. Figure5-8. Simulatedimpedanceresultscomparedtotheexperimentalresultswithanelectrodecongurationof2356. Figure5-9. Tendonmodelwithlocallycorrodingsectioninthecenterofthesteel. 58

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Figure5-10. Schematicrepresentationofthesystemgeometryforareferenceelectrodespacingof4cm. Figure5-11. Simulatedindirectimpedanceofa2ftmodeltendoncontaining1steelstrandforapassivecase,alocallycorrodingcasesof4cmatthemidpointofthesteelstrand,andauniformlycorrodingsteelforareferenceelectrodespacingof4cm electrodepointswereplacedalongthesurfacesuchthatmultipleelectrodecongurationscouldbesimulatedatonce.InonesetofsimulationstheelectrodeswereplacedwiththecenterlineofthearraydirectlyovertheactivesiteasisshowninFigure 5-10 .Thedistancebetweenthetworeferenceelectrodeswasvariedfrom32cmto4cmtodetermineifthereisamaximumdistanceinwhichthecorrosioncouldnotbedetected.Whentheelectrodeswerespacedat32cm,thedierencebetweenthepassivecaseandthelocallycorrodingcasewasextremelysmall.Astheelectrodesweremovedclosertogether,thedierenceincreased,but,evenwhenthereferenceelectrodeswereplacedjustabovetheactivelocation,Figure 5-11 ,theimpedanceofthelocallycorrodingcasealonedidnotindicatethepresenceofcorrosion.However,thepresenceofasmalldierenceevenwhenthereferenceelectrodedistanceis32cmindicatesthatthepolarizedsteelareaextendsfaroutfromtheelectrodepoints,butthelocationofsteelmostsensedisnotatthecenterlineoftheelectrodearray.Intheanothersetofsimulations,theelectrodeswereequallyspacedat4cmandweremovedalongthetendontomimicalikelyprocedureforaeldapplication.Inthiscase,whenthemidpointoftheelectrodearraywas18cmleftoftheactivelocation,the 59

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Figure5-12. Schematicrepresentationofthesystemgeometryforareferenceelectrodespacingof4cm. Figure5-13. Simulatedindirectimpedanceofa2ftmodeltendoncontaining1steelstrandforapassivecase,andalocallycorrodingcasesof4cmatthemidpointofthesteelstrandwiththeelectrodesequallyspacedat4cm.Thecenterlineoftheelectrodearrayis6cmrightofthecenterlineofthe2fttendonandtheworkingelectrodeisdirectlyoverthecorrodingarea. impedanceofthepassiveandtheactivelycorrodingcaseshowedonlysmalldierencesatlowerfrequencies.Whentheelectrodeswerelocated10cmfromthecorrodingsection,thedierencebecamemoreapparent.However,whenoneofthereferenceelectrodeswasdirectlyoverthesiteofcorrosion,thedierencediminished.Themostprominentdierenceoccurredwhenthecurrentinjectionelectrodewasdirectlyoverthesiteofcorrosion,asisshowninFigure 5-12 .ThesimulatedimpedanceresultsforthiscongurationareshowninFigure 5-13 inNyquistformat.Thepassivecaseshowsonedepressedcapacitiveloopwhilethecorrodingcaseshowstwooverlappingtimeconstants. 5.3.3EquivalentCircuitDuringtheindirectimpedancemeasurement,currentowsthroughthegroutaswellasthroughthesteel.Ifthecontributionofthegroutresistivitytotheindirectimpedancecanbedetermined,thenthetotalsteelimpedancemaybeextracted.Allpreviousresearchershaveattemptedtoaddressthecontributionofthegroutresistivitytotheindirectimpedancewiththeuseofresistorsinseriesorparalleltotheimpedanceof 60

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Figure5-14. Equivalentcircuitdiagramusedtorepresenttheindirectimpedance. Figure5-15. Cutplaneusedtodeterminetheoscillatingcurrentthroughthegrout. thesteel.However,duetothenonuniformpotentialdistributionalongthesurfaceofthesteelandthroughouttheresistivematerial,thecontributionofthegroutmustbeintheformofanohmicimpedancewithrealandimaginaryparts.TheappropriateequivalentcircuitmodelforanindirectimpedancemeasurementisshowninFigure 5-14 .Therearetwoprimarycurrentpathsfromtheworkingelectrodetothecounterelectrode.Thecurrentmayeitherrunparalleltothesteelorinseriestothesteel.Sincesomeofthecurrentcantakeonepathwhiletheresttakestheother,thesetwopathsmustbeinparallel.Theimpedanceoftheparallelpathmaybeexpressedas Zparallel=~Vref1)]TJ /F1 11.955 Tf 13.74 3.02 Td[(~Vref2 ~icutplane(5{10)inwhichthepotentialdierencebetweenthetworeferenceelectrodesisdividedbythetotalcurrentthroughaplanelocatedatthemidpointoftheelectrodearrayasshowninFigure 5-15 .Theplaneonlyincludesthecross-sectionoftheresistivematerialandnot 61

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thesteel.Theseriespathimpedancemustcontaintheimpedancethroughthegroutaswellastheinterfacialimpedanceofthesteel.Duetothenonuniformpotentialdistributionthroughoutthegroutandalongthesteelsurfacetheseriespathimpedancemustbeexpressedas Zs=Z600Z36001 ze(x;)+z0(x;)ddx)]TJ /F5 7.97 Tf 6.59 0 Td[(1(5{11)inwhichthesumofthelocalohmicandinterfacialimpedancesasafunctionofpositiononthesurfaceofthesteelareintegratedinaparallelfashion.AdiagramshowingthelocalohmicandinterfacialimpedancescongurationsisprovidedinFigure 5-14 .Thelocalohmicimpedanceiscalculatedas ze(x;)=~Vref)]TJ /F1 11.955 Tf 13.25 3.03 Td[(~0(x;) ~i0(x;)(5{12)whichisbasedonthepotentialdierenceofthereferenceelectrodeandthepotentialonapointofthesteel.ThemodulusoftheohmicandinterfacialimpedancesispresentedinFigure 5-16 asafunctionofaxiallocationfor=0.Thesolidlinesrepresentthelocalohmicimpedanceandthedashedlinesrepresentthelocalinterfacialimpedance.Theinterfacialimpedanceisuniformalongthesteelsurfaceandincreaseswithfrequency.Theohmicimpedanceisnonuniformalongthelengthofthesteelandreachesaminimumattheworkingandcounterelectrodelocations.Theohmicimpedanceoutsidetheelectrodearrayincreaseswithincreasesinfrequency.Athighfrequencies,theseriespathimpedanceismostlycomprisedoftheohmicimpedancecontribution.Atlowfrequenciestheinterfacialimpedanceisonthesameorderastheohmicimpedanceatlocationsneartheworkingandcounterelectrodes.Forsimplicity,theseriespathimpedancewascalculatedbysegmentingthesteelsurfaceinto1cmby90sectionsandcalculatingtheglobalohmicandinterfacialimpedancesofeachsection.Theseriesimpedancewasthenevaluatedastheparallelcombinationoftheglobalimpedancesbetweenthereferenceelectrodeandasegmentof 62

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Figure5-16. Magnitudeoftheserieslocalohmicimpedance(solidlines)andthelocalinterfacialimpedance(dashedlines)asafunctionofsteelpositionwithfrequencyasaparameter. thesteelsurface.Theohmicimpedancebetweenthereferenceelectrodeandasegmentofsteelmaybeexpressedas Ze;segment=~Vref1)]TJ /F1 11.955 Tf 13.75 3.03 Td[(~Vavg;segment ~isegment(5{13)Theohmicimpedanceofthersthalfofthesteelsegmentsiscalculatedbasedon~Vref1andthesecondhalfofthesegmentsiscalculatedusing~Vref2.Itisnecessarytosegmentthesteeltoaccountforthelargevariationinpotentialalongthesteelsurfacewhichresultsinadistributionoflocalohmicimpedance.TheohmicimpedanceinNyquistformatatthreesegmentslocatedoutsidetheelectrodearrayandonelocatedinsidethearrayisshowninFigure 5-17 .Theohmicimpedanceat1cm,showninFigure 5-17A ,isinductiveandatlocations16cm(Figure 5-17B )and17cm(Figure 5-17C ),whichareclosertotheelectrodearray,theimpedanceis 63

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A B C DFigure5-17. Theohmicimpedanceofasegmentlocatedat:a)1,b)16,c)17,d)26. capacitiveathighfrequenciesandinductiveatlowfrequencies.Theohmicimpedanceatlocation26cm(Figure 5-17D ),whichislocatedwithintheelectrodearray,iscapacitive.Thetotalseriesimpedanceforeachsectionofsteelmaybeexpressedas Zsegment=Ze;segment+Z0;segment(5{14)whereZ0;segmentrepresentstheinterfacialimpedanceandmaybeexpressedas Z0;segment=~Vavg;segment ~isegment(5{15) 64

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Figure5-18. SimulatedindirectimpedanceandequivalentcircuitimpedancecalculatedusingEquation 5{17 inNyquistformat. whichisthequotientoftheaverageoscillatingpotentialofasegmentandtheoscillatingcurrentthroughsegment.Thecompleteseriesimpedancemaybeexpressedas Zseries=1 Psegments11 Zsegment(5{16)whichisaparallelcombinationoftheseriesimpedancesforeachsegment.Theindirectimpedancemaybeexpressedas Zindirect=ZseriesZparallel Zseries+Zparallel(5{17)representingaparallelcombinationoftheseriespathimpedanceandtheparallelpathimpedance.AcomparisonofthesimulatedindirectimpedanceandtheimpedancecalculatedusingEquation 5{17 isshowninFigure 5-18 .Thetwoimpedancesarenearlyidentical.Anyerrorbetweenthesimulatedindirectimpedanceandthecircuitimpedanceisduetotheaveragingofthepotentialalongthesurfaceofeachsteelsegment.Theerrordecreaseswithdecreasesinthesizeofthesegments.Thebreakdownoftheindirectimpedancepresentedheremayalsobeextendedtoaccountformultiplesteelstrands.Theseries 65

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Figure5-19. Reducedanaloguecircuitusedtorepresenttheindirectimpedance. Figure5-20. Simulatedparallelohmicimpedance. ohmicimpedancetoeachstrandmaybecalculatedandeachofthemwouldbeaddedinparallel. 5.3.4AnalogueCircuitDuetothenonuniformcurrentdistributionimposedbytheunusualgeometryoftheelectrodecongurationinanindirectimpedancemeasurement,thecontributionofthegroutresistivitytotheoverallindirectimpedanceisintheformofanohmicimpedanceandhasrealandimaginaryparts.Therefore,anequivalentcircuitcontaininglinearelementsthatproperlydescribesthesystemisnotfeasible.AmoreappropriatecircuitexpressesthecontributionofthegroutresistivityintermsofanohmicimpedancesuchasthecircuitshowninFigure 5-19 .TheparallelpathimpedanceisthesameasexpressedinEquation 5{10 fortheequivalentcircuit.TheparallelohmicimpedanceispresentedinFigure 5-20 .Theparallelimpedanceisinductive 66

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Figure5-21. Schematicshowingtheeectiveareaofpolarizedsteel. andlargeincomparisontotheoverallimpedance.AsdescribedinEquation 5{14 theseriespathimpedancecontainstheseriesohmiccontributionofthegroutaswellasthesteelinterfacialimpedance.Tosubtractthetotalsteelinterfacialimpedancefromtheseriespathimpedance,theeectivepolarizedareaofsteelneedstobeestimated.Technically,theentiresteelsurfaceispolarizedsincethepotentialeverywherealongthesteelisnon-zero.However,thefurtherlocationofthesteelsegmentisfromthecurrent-injectingelectrodes,thelargertheohmiccontributionforthatsegmentisandthelessinuencethesteelinterfacialimpedanceofthatsegmenthas.Theeectiveareaofpolarizedsteelwasdeterminedbydecreasingthepolarizationresistanceofasegmentbyafactorof10andassessinghowtheoverallimpedancechanged.Ifthelowfrequencylimitoftheindirectimpedancechangedbymorethanonepercent,thesegmentofsteelwascategorizedaspolarized.TheeectiveareaofpolarizedsteelisshowninFigure 5-21 .Theeectiveareaofsteelthatispolarizedduringtheindirectimpedancedidnotincludetheouter10cmofthesteelstrand.Theeectiveseriesohmicimpedance,showninFigure 5-22 ,wasfoundbysubtractingthetotalinterfacialimpedancebasedontheeectivepolarizedareaofsteelfromtheseriespathimpedance,alsopresentedinFigure 5-22 .Theseriesandparallelohmicimpedancesmakeitdiculttointerprettheindirectimpedance.However,withouttakingintoaccount 67

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Figure5-22. Seriespathsimulatedimpedanceandseriessimulatedohmicimpedance. Figure5-23. Simulatedindirectimpedanceandtheimpedancecalculatedfromacircuitcontainingresistorsinsteadoftheseriesandparallelohmicimpedances. thevariationinohmicimpedanceasafunctionoffrequency,thepolarizationresistanceofthesteelwillbeoverestimatedindicatingthatthecorrosionrateofthesteelisactuallyslowerthanisthecase.Asanexample,acircuitinwhichtheseriesandparallelohmicimpedanceswereexpressedasresistorswasttotheindirectimpedance,showninFigure 5-23 .Theregressionyieldedanestimatedtotalpolarizationresistanceof313.1,whereas,theactualtotalpolarizationresistancebasedontheeectiveareaofsteelwas71.2.Without 68

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Figure5-24. TheseriesohmicimpedanceinNyquistformatwiththespacingbetweenreferenceelectrodesasaparameter. accountingforthecomplexnatureoftheohmiccontributionstotheindirectimpedancemeasurement,thepolarizationresistancewasoverestimatedbyafactorofmorethanfour. 5.3.5InuenceofElectrodeCongurationTheohmicimpedanceparametersvarywithchangesintheplacementofthemeasurementelectrodes.Simulationswereperformedinwhichthedistancebetweenthereferenceelectrodeswasincreasedwhilethedistancebetweentheworkingelectrodeandcounterelectrodewereheldxed.Simulationswerealsoperformedinwhichthedistancebetweentheworkingelectrodeswasincreasedasthedistancebetweenthereferenceelectrodeswereheldxed.Theeectiveseriesandparallelohmicimpedanceswerecalculatedforeachcase.TheseriesohmicimpedanceisshowninFigure 5-24 inNyquistformatwiththedistancebetweenthetworeferenceelectrodeasaparameter.Thedistancebetweentheworkingelectrodeandcounterelectrodewasxedat14-in.whilethedistancebetweenthereferenceselectrodeswasincreasedfrom2-in.to10-in.byincrementsof4-in.Theresultsindicatethattheseriesohmicimpedanceisastrongfunctionofthedistancebetweentheworkingelectrodesandthereferenceelectrodes.Whenthisdistanceislarge,theseriesohmicimpedanceisinductiveasisshowninthecaseswherethereferenceelectrode 69

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Figure5-25. TheparallelohmicimpedancescaledbythehighfrequencylimitoftherealpartoftheparallelohmicimpedanceinNyquistformatwiththedistancebetweenreferenceelectrodeasaparameter. spacingis2-in.and6-in.Whenthedistancebetweenthereferenceelectrodesis10-in.andthedistancebetweentheworkingelectrodeandthereferenceelectrodeis2-in.,theseriesohmicimpedanceiscapacitive.Anotherinterestingfeatureisatlowfrequenciestheseriesohmicimpedanceisnegativewhichwouldleadtolargedistortionsintheoverallindirectimpedance.TheparallelohmicimpedanceisshowninFigure 5-25 forthesamesetofsimulationspresentedinFigure 5-24 .TheresultsareshowninNyquistformatbutscaledbythehighfrequencylimitoftheimpedancesincetheparallelimpedancevariedsignicantlywithchangesinthereferenceelectrodespacing.Theparallelohmicimpedancewasinductiveforeachcaseandthesizeoftheinductiveloopincreasedwithincreasesinthereferenceelectrodespacing.Also,theshapeoftheinductiveloopbecamemoredeformedwithlargerdistancebetweenreferenceelectrodes.TheseriesohmicimpedanceisshowninFigure 5-26 inNyquistformatwiththedistancebetweentheworkingandcounterelectrodesasaparameterwhilethedistancebetweenthereferenceelectrodeswasxedat2in:Theresultsaresimilartothecasewhen 70

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Figure5-26. TheseriesohmicimpedanceinNyquistformatwiththedistancebetweentheworkingandcounterelectrodeasaparameter. Figure5-27. TheparallelohmicimpedanceinNyquistformatwiththedistancebetweentheworkingandcounterelectrodeasaparameter. thedistancebetweenreferenceelectrodeswasincreased,whichindicatesthattheseriesohmicimpedanceismostsensitivetothedistancebetweentheworkingelectrodeandthereferenceelectrode.TheparallelohmicimpedanceisshowninFigure 5-27 inNyquistformatwiththedistanceforthesameelectrodespacingsaspresentedinFigure 5-26 .Themagnitudeoftheparallelohmicimpedancedoesnotchangemuchwithchangesinthedistancebetween 71

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Figure5-28. Thesimulatedindirectimpedancescaledbytheohmicresistancewithelectrodespacingasaparameter.Threesimulationswereperformedforchangesinreferenceelectrodespacingandtheotherthreewereforchangingthespacingbetweentheworkingandcounterelectrode. theworkingandcounterelectrodesindicatingthattheparallelcomponentoftheohmicimpedanceismostsensitivetothedistancebetweenthereferenceelectrodes.TheoverallindirectimpedanceisshowninFigure 5-28 scaledbytheohmicresistanceforvariouselectrodecongurationstodeterminewhichelectrodecongurationyieldstheleastamountoffrequencydispersion.Themostfrequencydispersionisobservedwhenthedistancebetweentheworkingandcounterelectrodeismuchlargerthanthedistancebetweenthetworeferenceelectrodes.Theleastamountoffrequencydispersionisobservedwhentheelectrodeswereequallyspacedoutatadistanceof2in:Theindirectimpedancewouldalsochangeasthedepthofthesteelfromtheelectrodeschanges.However,iftheelectrodeareplacedsuchthatthedistancebetweeneachelectrodesis 72

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uniformandthatdistanceisclosetothedepthofthesteelthefrequencydispersionwillbeminimalforsystemswithonlyonesteelstrand. 5.3.6SensitivitytoSteelPolarizationResistanceSimulationswereperformedinwhichthesteelpolarizationresistancewasincreasedtodeterminethesensitivityoftheindirectimpedancetothesteelcorrosionstate.ThevariationinthesteelandgroutinterfacialimpedanceisshowninFigure 5-29 withthepolarizationresistanceasaparameter.Thecapacitancewasheldconstant.Asthepolarizationresistanceisincreased,theinterfacialimpedancebecomesmorecapacitive.ThecorrespondingindirectimpedanceisshowninFigure 5-30 inNyquistformat.Theindirectimpedanceincreasesasthepolarizationresistanceofthesteelincreasesshowingthattheindirectimpedanceissensitivetothesteelcondition.Theoverallgoalistobeabletomeasuretheindirectimpedanceandsomehowdeterminetheinterfacialimpedancesuchthatthecorrosionratemaybeestimated.Theseriesandparallelohmicimpedancecontributionstotheindirectimpedancemakestheextractionoftheinterfacialimpedancedicult.However,iftheseparameterscanbeestimatedfromthegeometryofthesystem,estimatingthesteelpolarizationresistancemaybefeasible.Theseriesandparallelohmicimpedanceswerecalculatedasafunctionofsteelpolarizationresistancetodeterminehowdependenttheohmiccontributionoftheindirectimpedanceisonthesteelimpedance.TheseriesohmicimpedanceisshowninFigure 5-31 withthepolarizationresistanceasaparameter.Asthepolarizationresistanceincreases,theseriesohmicimpedancealsoincreasesandbecomesmoreinductive.However,thehigh-frequencylimitdoesnotchange.TheparallelohmicimpedanceisshowninFigure 5-32 withthepolarizationresistanceasaparameter.Aswiththeseriesohmicimpedance,theparallelcomponentalsoincreasesasthepolarizationincreases.However,thelow-frequencylimitoftherealpartoftheparallelohmicimpedanceismorewelldened.Thehigh-frequencylimitoftheparallelohmicimpedancealsodoesnotchange.Sincetheohmiccomponentsoftheindirectimpedancearefunctionsof 73

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Figure5-29. TheinterfacialimpedanceforacircuitwithRpinparallelwithC0withRpasaparameter. 74

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Figure5-30. ThesimulatedindirectimpedanceinNyquistformatwithRpasaparameter. Figure5-31. TheseriesohmicimpedanceinNyquistformatwithRpasaparameter. 75

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Figure5-32. TheparallelohmicimpedanceinNyquistformatwithRpasaparameter. thesteelinterfacialimpedance,itwouldbediculttopredicttheseparameterssolelyfromthegeometry.Nevertheless,sincethehigh-frequencylimitsarenotdependentontheinterfacialimpedance,theohmicresistanceoftheindirectimpedancecoupledwithknowledgeofthesystemgeometrymaybeusedtoestimatethehigh-frequencylimitsoftheseriesandparallelohmicimpedances. 76

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CHAPTER6FEASIBILITYOFINDIRECTIMPEDANCEFORPOST-TENSIONEDTENDONSThischapterpresentsexperimentalresultsfromindirectimpedancemeasurementsperformedonanextractedtendonsectionfromtheRinglingCausewayBridgeaswellmeasurementsperformedontheTexasA&Mmockbridgesection.Apreliminaryanalysiswasdonetoestimatethepolarizationresistanceofthesteelbyttingthedatawithasimpleequivalentcircuit.However,aswasshowninthepreviouschapter,thepolarizationresistanceofthesteelissignicantlyoverestimatedusingthismethod.SimulationsofatendonwiththesamegeometryandsteelplacementasoneofthetendonsfromtheRinglingCausewayBridgewereperformedtocomparewithexperimentalresults. 6.1MethodsTheexternalpost-tensionedtendonsontheRinglingCausewayBridgeshowedsignsofcorrosionfewerthan7yearsafterconstruction.Thetendonsthatwerereplacedwerecutinto3or4ft.sectionsandstoredattheFDOTStateMaterialsOce.Indirectimpedancemeasurementswereperformedononeofthesectionsthatshowedcorrosion.Animageofthecross-sectionisshowninFigure 6-1 .Thetendoncontained22strandsofsteelwhicharerandomlydispersed.Therewerevisiblesignsofgroutsegregationandcorrosionofoneofthesteelstrands.Measurementswereperformedat6dierentlocationsaroundthetendontodetermineifcorrosioncouldbedetected.TheexternaltendonsontheTexasA&Mmockbridge,showninFigure 6-2 ,wereconstructedwithvariouslevelsofcorrosion.CorrosionwasinducedonthesteelbysubmergingpartsofthesteelstrandsinanHClbathsolutionpriortoconstructingthetendons.Someofthestrandswereseveredcompletelyandtheendsweresubmergedintotheacidicsolution.Afterconstruction,thetendonsandthebridgeitselfwerediscretizedinto1ft.sectionsandlabeledalphabetically.Eachtendonlocationspecicationincludedthetendonnumberaswellasthesectiondesignation(i.e.,13-JK).Indirectimpedancemeasurementswereperformedatlocationsinwhichallthesteelstrandsweresevered 77

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Figure6-1. Thecross-sectionoftheRinglingBridgetendon.Thenumbersindicatethelocationsoftheelectrodes. withcorrodedends,thesteelstrandswerepartiallycorroded,orallthestrandswerenotcorroded.TheexperimentalsetupisshowninFigure 6-3 withtheleadsofthepotentiostatconnectedtotheelectrodes.FoursmallholesweredrilledintotheHDPEducttoprovideaccesstothegrout.Dimensionallystable0.5cmdiametertitaniumrodscoatediniridium-oxidewereinsertedintotheholesandusedastheelectrodes.Anin-houseconductivegelconsistingofa1Msodiumsulphatesolutionandcarboxyl-methylcelluloseasapolymergellingagentwasusedtomakeanelectricalconnectionbetweentheelectrodeandthegrout.Impedancemeasurementscanbetakenbyeithermodulatingpotential,referredtoaspotentiostaticmodulation,orcurrent,referredtoasgalvanostaticmodulation.Potentiostaticmodulationwasfoundtobemorereproducibleandlessnoisythangalvanostaticmodulationmeasurements.TheimpedancewasmeasuredwithaGamryreference600potentiostatoverarangeoffrequencies.Real-timeLissajous 78

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Figure6-2. AnimageoftheinsideofthemockbridgebuiltatTexasA&M. Figure6-3. Experimentalsetupoftheindirectimpedancemeasurement.PhotographtakenattheTexasA&Mmockbridge.TheGamryReference600potentiostatisthewhite/blueboxinthecenterofthephotograph. 79

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Figure6-4. ExperimentalimpedanceinNyquistformatmeasuredatdierentsectionsoftheTexasA&Mbridgetendons. plotsaredisplayedonthecomputerscreenwhilemeasuringeachfrequency.Afterthemeasurementswereperformed,theholesweresealedbyusingHDPEweldingtechnique. 6.2ExperimentalResults&AnalysisResultsarepresentedforimpedancemeasurementsperformedontendonsconstructedforthemockbridgesectionatTexasA&Maswellastendonsextractedfromtheringlingcausewaybridge.FiniteelementsimulationswereperformedforamodelwhichincludedthegeometryofoneofthetendonsextractedfromtheRinglingbridgetoassesstheinuenceoftheplacement. 6.2.1TexasA&MMockBridgeResultsfromtheTexasA&MmockbridgearepresentedinFigure 6-4 fordierentlocationsasmarkedonthebridge.Theindirectimpedanceisafunctionofthegroutresistivity,thesteel-groutinterfacialimpedance,andthelocationofthesteelstrands.Measurementsweretakenatthetopofthetendon,unlessthedesignationendsin2in 80

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whichtheyweretakenalongtheside.Corrosionassociatedwithsegregationofgroutismostlikelytooccurnearthetopofthetendonsincethegroutnearthetopwillhaveahigherconductivity.Ifthereisasteelstrandlocatedneartheelectrodes,therealimpedancewillbenegativeathighfrequenciesduetothepotentialdistribution,asisthecaseforlocation16CD.Thehigh-frequencylimitoftherealimpedancerepresentstheohmicresistancewhichcanbeassociatedwiththeresistanceofcurrentowbetweenthegroutandthesteel.Theohmicresistanceisalsoagoodmeasureofthedepthofthesteel.Thesignicantlylargerohmicresistancemeasuredatsection16VWmaybeattributedtopropertiesofthegrout,suchasvoids.Sections16VWand16SThadthesmallestimpedancevalues.Thesmallerimpedancevaluessuggestthatcorrosionwaspresent.Theindirectimpedanceforsection16VWisshownseparatelyinFigure 6-5 .ThemeasurementwastwithasimplecircuitwithaseriesandaparallelresistoraccountingfortheohmiccontributionandthesteelimpedancewasexpressedasaRpC0circuitwhereRpisthepolarizationresistanceofthesteelwhichisinverselyrelatedtothecorrosionrate.BasedonthecircuitttingRp=292:9.Thegureinsetshowsthehighfrequencybehavior.Theresultsshowaverysmallhighfrequencycapacitiveloopwhichoverlapsthelowerfrequencydataandmaybeasignofcorrosion.Theindirectimpedanceforsection16STisshownseparatelyinFigure 6-6 .Thehigh{frequencybehaviorforsection16STshowsaninductivefeatureandsignicanthigh{frequencynoisewhichhasnotbeenobservedinourbench-topworkornumericalsimulations.ThecircuitdidnotttheimpedancewellbutRpwasestimatedtobe316:5. 6.2.2RinglingCausewayBridgeTheindirectimpedanceresultsfromtheRinglingBridgetendonareshowninFigure 6-7 .ThenumberscorrespondtothelocationofthemeasurementshowninFigure 6-1 .Measurementscouldnotbemadeatlocation2becausethesteelstrandwasexposedfromthegroutwhentheholesweredrilled.Thesmallestimpedancewasfoundforthelocation 81

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Figure6-5. ExperimentalimpedanceinNyquistformatmeasuredatlocation16VWoftheTexasA&Mmockbridgetendons. ofthecorrodedstrand,shownasnumber4inFigure2.Asmallerimpedanceisexpectedforcorrodingsteelastheimpedanceassociatedwithcorrosionofsteelissmallerthantheimpedanceassociatedwithpassivatedsteel.Theimpedanceatlocation4isshownseparatelyinFigure 6-8 .Therearetwocapacitiveloopswhichoverlap,indicatingthepresenceoftwotimeconstants.Thehigher-frequencytimeconstantisassociatedwiththecorrosionreaction.Theresultsindicatethatcorrosioncanbedetectedifitislocateddirectlybeneaththearrayofelectrodes.However,ifthecorrosionislocatedonasteelstrandnotnearthemeasurementelectrodes,itmaygoundetected.Anexperimentwassetuptotrytoovercomethisobstacle.Insteadoftheelectrodesbeingplacedalongtheaxisofthetendon,theelectrodeswereplacedcircumferentiallyinhopesthatthiselectrodecongurationwasmoresensitivetocorrosion.Theimpedance 82

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Figure6-6. ExperimentalimpedanceinNyquistformatmeasuredatlocation16SToftheTexasA&Mmockbridgetendons. resultsareshowninFigure 6-9 withthenumberdesignationindicatingthelocationofthe4electrodescorrespondingtothediagraminFigure 6-1 .TheresultsaresimilartotheresultsinFigure 6-7 inthatthereisonlyoneparticularmeasurement,whentheelectrodeswereatlocations5-4-6-1,thatwasabletodetectthecorrosionshownbythesignicantlysmallerimpedance.Therefore,thecorrosiondetectioncapabilitiesoftheindirectimpedancemeasurementareconnedtocaseswherethecorrosionispresentrightbeneaththemeasurementelectrodes. 6.3SimulationResultsThegeometryofthetendonextractedfromtheRinglingBridgewasmodeledintoaniteelementstimulation.ThemodelgeometryisshowninFigure 6-10 includingthesamelocationsofelectrodeplacementastheexperimentalcase.Impedancesimulationswereperformedwiththeelectrodearrayplacedalongtheaxisatoneofthedesignatedlocations.Allofthesteelstrandsweremodeledtobepassive.Thesimulatedimpedance 83

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Figure6-7. ExperimentalimpedanceinNyquistformatmeasuredonanextractedtendonfromtheRinglingCausewayBridgewiththelocationoftheelectrodesasaparameter.ThenumbersinparenthesescorrespondtothelocationofthetendonshowninFigure 6-1 Figure6-8. ExperimentalimpedanceinNyquistformatmeasuredonanextractedtendonfromtheRinglingCausewayBridgeatlocation4 84

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Figure6-9. ExperimentalimpedanceinNyquistformatmeasuredonanextractedtendonfromtheRinglingCausewayBridgewiththelocationoftheelectrodesasaparameter.ThenumbersindesignationcorrespondstotheelectrodelocationsdescribedinFigure 6-1 ispresentedinFigure 6-11 inNyquistformatwiththelocationoftheelectrodesasaparameter.Duetothelocationofthesteelstrandsinrelationtothelocationoftheelectrodeprobes,thesimulatedimpedancehadinductivefeaturesathighfrequencies.Thesesimulationsshowthattheindirectimpedanceissensitivetosteelpositionanditmaybeimportanttohaveageneralsenseofthelocationofthesteelstrandstoformulateareasonableestimateofthecorrosionrate.Simulationswerealsoperformedinwhichtheelectrodeswereplacedcircumferentiallyaroundthetendonaswasdoneexperimentally.Thesimulationsweredonewithallthesteelstrandsettoapassiveboundaryconditionsaswellasascenarioinwhichoneofthesteelstrandswascorroding,shownbythelocationoftheredcircleinFigure 6-10 .ThesimulationresultsarepresentedinFigure 6-12 fortheallpassivecase,representedbythesolidlines,andwhenoneofthestrandsiscorroding,shownbythedottedlines. 85

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Figure6-10. FiniteelementrepresentationoftheRinglingBridgetendonshowninFigure 6-1 Figure6-11. Simulatedindirectimpedancefora2ft.cylindricaltendonwith6steelstrandswithcorrosionstateasaparameter. 86

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Figure6-12. Simulatedindirectimpedancefora2ft.cylindricaltendonwith6steelstrandswithcorrosionstateasaparameter. Whentheelectrodeprobeswereplacedatlocations1-6-4-5thedierencebetweenthepassivecaseimpedanceandthecorrodingcasewasnotextreme.Thedierencewasmorepronouncedwhentheelectrodeprobeswereplacedatlocations2-1-6-4.Whentheelectrodesareplacedlongitudinallyalongthetendontheindirectimpedancemeasurementismostsensitivewhenthecorrosionisdirectlybeneaththeworkingorcounterelectrode.Thesameisstilltruewhentheelectrodesareplacedcircumferentiallyaroundthetendon. 6.4DiscussionThroughtheexperienceofperformingindirectimpedancemeasurementsontheTexasA&MmockbridgeandthetendonsextractedfromtheRinglingbridge,somedicultieswererealized.Iftheelectrodesareplacedatalocationwherethesteelisexposedfromthegrout,itisnotpossibletoobtainanimpedancemeasurement.Ifthisoccursintheeld,theelectrodearrayshouldbeshiftedtoadierentlocationaroundthetendon.TheresultsfromtheTexasA&MbridgedidnotshowasmuchofthecapacitiveloopasthemeasurementstakenfromtheRinglingtendon.Infact,themagnitudeoftheimpedancealsovariedsignicantlybetweenthetwocases.Thismaybeduetotheuseofdierentgroutswhichmayhavedierentmaterialpropertieswhichposesamajorchallengeindevisingageneralinterpretationprocedure.Also,sincetheindirectimpedanceisa 87

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functionofthesteellocation,whichwouldbediculttodetermineapriori,anyanalysisaimedatestimatingacorrosionrateofthesteelwouldhavetoincludeawayofaccountingforthis.Theexperimentsandsimulationsbothshowedthattheindirectimpedanceisahighlylocalizedtechniqueasitisonlycapableofdetectingcorrosionoratleastshowingsignicantsignsofcorrosionifitislocateddirectlybeneaththeworkingorcounterelectrodes.Otherwise,themeasurementdoesnotshowqualitativesignsofcorrosion.Therewasnoadvantagetoplacingtheelectrodescircumferentiallyaroundthetendoninsteadoflongitudinally.Therefore,ifameasurementistakenatthetopofthetendonandthecorrosionoccursinthecenter,itwilllikelygoundetected.However,corrosionwasshowntooccurinthelocationsofdecientgroutformedatelevatedlocationsofthetendonincaseswhereexcesswaterwasusedtomixthegrout.[ 63 ]Whiletherearemanydicultiestotheapplicationofindirectimpedancetocorrosiondetectionintendons,therearestillclearadvantagesoverexistingtechnologies.Theprocedurefortheindirectimpedancemeasurementisrelativelysimpleanddoesnotrequireheavyequipment.Measurementsareperformedinapproximately20minutes,andtheholesthataredrilledcanbesealedifdesired.Theinterpretationoftheindirectimpedanceresultsisstillaworkinprogress.Finiteelementsimulationswereusedtodeterminethesensitivityoftheindirectimpedancetocorrosion.Wefullyunderstandhowtheimpedanceofthegroutcontributestooverallimpedanceandwithmoreresearch,areliablemethodtoestimatingthecorrosionratemaybeachieved. 88

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CHAPTER7INFLUENCEOFROUGHNESSONIMPEDANCEElectricalcircuitsinvokingconstant{phaseelements(CPE)areoftenusedtotimpedancedataarisingfromabroadrangeofexperimentalsystems.TheimpedanceforablockingelectrodeshowingCPEbehaviormaybeexpressedintermsofohmicresistanceReandCPEparametersandQas Z=Re+1 (j!)Q(7{1)where!istheangularfrequencyinunitsofs)]TJ /F5 7.97 Tf 6.59 0 Td[(1.When=1,thesystemisdescribedbyasingletime{constant,andtheparameterQhasunitsofcapacitance;otherwise,Qhasunitsofs/cm2orF/s(1)]TJ /F4 7.97 Tf 6.59 0 Td[()cm2[ 57 ].WhileCPEsarecommonlyusedtoimprovethettoexperimentalimpedancespectra,theexplanationfortheiroriginhasbeenlargelyconjectural.Recently,Jorcinetal.[ 30 ]haveproposedthattheCPEcouldhave,asanorigin,adistributionoftimeconstantsthroughalmoralonganelectrodesurface.Theysupportedtheirhypothesiswithlocalimpedancemeasurements.AninterpretationforCPEbehaviorcausedbyanormaldistributionwasdevelopedintheformofapower{lawdistributionofresistivity.[ 24 25 ]Thepower{lawmodelapproachhasbeenusedsuccessfullytoextractalmcapacitanceandassociatedparametersforavarietyofsystems,includingoxidesonsteel,[ 58 ]humanskin,[ 58 74 ]andpolymercoatings.[ 48 ][ 51 ]Brugetal.[ 12 ]developedanexpressionforthecapacitanceextractedfromaCPEcausedbyasurfacedistributionofcapacitanceorchargetransferresistance.TheBrugmodelwasshowntoapplyfortheapparentCPEbehaviorassociatedwithgeometry-inducednonuniformcurrentandpotentialdistributionsonadiskelectrode.[ 26 27 ]Asthegeometry-inducedCPEbehaviorappearsonlyathighfrequency,[ 28 ]thediskgeometrycannotprovideanexplanationforCPEbehaviorcausedbysurfacedistributionsthatmayextendovermanydecadesoffrequency. 89

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SeveralotherpotentialcausesforsurfaceCPEbehaviorhavebeenproposed,includingporouselectrodes,[ 19 40 69 ]specicadsorption,[ 62 ]surfaceroughness,[ 35 59 ]andevenheterogeneityonanatomicscale.[ 68 ]Since1950,electroderoughnessemergedasaleadingexplanationforCPEbehavior.WhileroughnesswasinitiallyconsideredtocauseCPEbehavior,recentworkhasquestionedthispremise.Theobjectiveofthisworkistoexplore,byuseofnite{elementmodels,whethersurfaceroughnesscouldprovideavalidphysicalexplanationforCPEbehaviorassociatedwithasurfacedistributionoftimeconstants.Anessentialquestioninthisanalysisiswhetherelectrochemicalimpedancespectroscopymeasurementsencompassthefrequencyatwhichroughnessisexpectedtoinuencetheimpedanceresponse. 7.1MathematicalDevelopmentThepotentialdistributionwithinanelectrolytewithuniformcompositionisgovernedbyLaplace'sequation, r2=0:(7{2)Forthepresentwork,axisymmetriccylindricalcoordinateswereusedinwhichthepotentialwasassumedtobeindependentoftheangularcoordinate.Thepotentialwasseparatedintosteady-stateandoscillatingcomponentsas =+Ren~exp(j!t)o(7{3)whererepresentsthesteady-stateportionand~representsthecomplexoscillatingportionthatisafunctionoffrequencyandpositionbutindependentoftime.Thesamerelationshipforthepotentialappliedtotheelectrodecanbeexpressedas V=V+Ren~Vexp(j!t)o(7{4)Foracapacitivesystem,thesteadystatepotentialisequaltozero;thus,onlytheoscillatingcomponentwascalculated.Atr=0,forally,thesymmetrycondition 90

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@~=@y=0applied.Farfromtheelectrode,asp r2+y2!1,~!0.Thisconditionplacesthecounterelectrodeinnitelyfarfromtheworkingelectrode.FollowingNewman[ 50 ]andHuangetal.[ 28 ],theelectrodewasassumedtobeideallypolarized.Thenormaluxatthesurfaceoftheelectrodewasexpressedas i=C0@(V)]TJ /F1 11.955 Tf 11.96 0 Td[(0) @t=)]TJ /F3 11.955 Tf 9.29 0 Td[(@ @yy=0(7{5)whereC0isthecapacitanceontheelectrodesurface,istheconductivityoftheelectrolyte,and0isthepotentialoutsidethediusepartoftheelectrodedoublelayer.Theoscillatingcurrentdensitywasconvertedtothefrequencydomainsuchthat ~i=j!C0(~V)]TJ /F1 11.955 Tf 13.25 3.02 Td[(~)(7{6)where~Visthepotentialperturbationattheelectrodeand~isthecomplexoscillatingpotentialwithintheelectrolyte.Frequencyortime-constantdispersionmayoccurforanyelectrochemicalcelldesignthatdoesnotprovideauniformcurrentdistributionacrosstheworkingelectrodesurface.Thisformoffrequencydispersionmaybereferredtoasgeometry-inducedfrequencydispersion,inwhichthecharacteristicfrequencyatwhichtime-constantdispersionisevidentisinverselyrelatedtoacharacteristiclength.Adimensionlessfrequency,K,maybedenedas K=!C0lc (7{7)suchthatfrequencydispersionisinducedatK1.Thecorrespondingcharacteristicfrequencymaybeexpressedas fc= 2C0lc(7{8)Asthecharacteristiclengthincreases,thefrequencydispersionshiftstolowerfrequencies.Therefore,thecharacteristiclengthshouldbeminimizedtoavoidfrequencydispersioninexperiments.Inthecaseofablockingdiskelectrodewithinaninsulatingplane,thecharacteristiclengthistheradiusofthedisk.Sincetheohmicresistanceofadisk 91

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electrodeinunitsofcm2isgivenby[ 49 ] Re=r0 4(7{9)itmaybeconvenienttoexpressthecharacteristicfrequencyastheratiooftheinterfacialimpedanceandtheohmicresistanceas K=Re Z0(7{10)Theinterfacialimpedanceforablockingelectrodecanbeexpressedas Z0=1 !C0(7{11)suchthatthedimensionlessfrequencyis K= 4!C0r0 (7{12)Thisformulationisconvenientforablockingdiskelectrode,however,itdoesnotapplytoothergeometriessuchasaringorarectanglewheretheohmicresistanceisnotproportionaltothecharacteristiclength.Therefore,forthesakeofconsistency,theappropriateformulationistheonepresentedbyHuangetal.[ 28 ]inwhichthedimensionlessfrequencywasdenedtobe K=!C0r0 (7{13)Theinclusionofthe=4factordoesnotaectsignicantlythepredictionofthecharacteristicfrequencybutitisnotapplicabletoothergeometriesotherthanthediskelectrode.Theroughnessoftheelectrodewasquantiedintermsoftheroughnessfactororrugosity,fr,obtainedbydividingthetruepolarizableareabythegeometricarea.[ 71 ]Asmoothelectrodehasaroughnessfactorofunity;whereas,roughnessfactorsmuchgreaterthanunitymaybeattributedtoporouselectrodes.Forauniformlyroughsurfacewith 92

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Figure7-1. Schematicrepresentationshowingthenite-elementmeshusedforthediskelectrodesimulations:a)entiredomain;b)groovedsurfaceoftheelectrode;c)detailofthegroovedelectrode. concentricV{shapedgrooves,theroughnessfactormaybeexpressedasfr=1=cos,whererepresentsthecontactanglebetweentheroughsurfaceandasmoothplane. 7.2ImpedanceCalculationsThreediskelectrodecongurationsweresimulated,includingacompletelysmoothelectrodewithinaninsulatingplane,aroughelectrodewithinaninsulatingplane,andarecessedroughelectrode.Thecalculationswereperformedinaxisymmetriccylindricalcoordinatesforaquarterofacircledomain,showninFigure 10-3 (a),whichrepresentstheelectrolyte.Thecounterelectrodewaslocatedatp r2+y2=500cm,makingthedomainsizeatleast1000timeslargerthantheradiusofthedisk.Afreetriangularmeshwasusedwithagreaterdensityofelementsneartheworkingelectrode.ThegroovedsurfaceofthediskisshowninFigure 10-3 (b),andanexpandedviewofthegroovesis 93

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Figure7-2. Schematicrepresentationshowingthenite-elementmeshusedfortherecesseddiskelectrodesimulations:a)portionofthedomainemphasizingtherecessedelectrode;b)therecessedelectrode;andc)detailofthegroovedelectrode. showninFigure 10-3 (c).TheangleusedforthecalculationofroughnessfactorisalsoshowninFigure 10-3 (c).Theelectrodewasplacedinthey=0planecenteredatr=0suchthattheanglebetweentheelectrodesurfaceandtheinsulatingplanewas180degrees.Thecorrespondingprimarycurrentdistributionapproachesinnityattheperipheryofthedisk.ArecessedelectrodewasconguredsuchthattheinsulatingplanewasperpendiculartotheelectrodesurfaceasshowninFigure 7-2 .Thedepthoftherecesswasthreetimesthediskdiameter,ensuringthattheprimarycurrentdistributionwasuniformacrosstheelectrodesurface.AroughsurfacewassimulatedinbothcasesbyaddingconcentricV{shapedgroovestotheelectrodesurface,asisillustratedinmagniedviewinFigure 7-2 (c).Aroughrecessedelectrodewasusedtoisolatetheeectoftheroughsurfaceontheimpedance;whereas,theroughelectrodeembeddedwithinaninsulatingplaneshowedthecoupledeectoftheroughnessandthenonuniformcurrentdistributionassociatedwiththediskgeometry. 7.3ResultsandDiscussionTheresultsofimpedancesimulationsarepresentedtoshowtheinuenceofsurfaceroughnessonelectrodesembeddedwithinaninsulatingplane.Resultsfromarecessed 94

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A BFigure7-3. Imaginarypartofthecalculatedglobalimpedanceforaroughdiskelectrodewithroughnessfactorasaparameter:a)dimensionalimpedanceasafunctionoffrequency;b)dimensionlessimpedanceasafunctionofdimensionlessfrequencyKfr. electrodearethenusedtodistinguishbetweentheeectoftheroughnessandtheinuenceofdiskgeometry. 7.3.1InuenceofRoughnessonaDiskElectrodeWithinanInsulatedPlaneTheimpedanceofsmoothandroughdiskelectrodeswithinaninsulatingplanewassimulatedtoshowthecoupledeectofsurfaceroughnessandnonuniformcurrentdistributionscausedbydiskgeometry.ThecalculatedimaginarypartoftheimpedanceisprovidedinFigure 7-3A asafunctionoffrequencywiththeroughnessfactorasaparameter.Theradiuswas0.24cm.Thesimulatedimaginaryimpedancefortheroughelectrodesatlowfrequencieswassmallerthantheimpedanceofthesmoothelectrodeanddecreasedastheroughnessfactorwasincreased.Thereducedimpedanceoftheelectrodecouldbeattributedtothelargerpolarizablesurfacearea.Conversely,theimpedanceoftheroughelectrodewaslargerthantheimpedanceofthesmoothelectrodeathighfrequencies.Theimaginarypartofthesimulatedglobalimpedance,adjustedfortheroughnessoftheelectrode,isprovidedinFigure 7-3B asafunctionofdimensionlessfrequency.The 95

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dimensionlessfrequencyforarough{diskelectrodecanbeexpressedas Kfr=!C0frr0 (7{14)TheimpedanceisexpressedasZj=frr0wheretheimpedanceZjisscaledbyunitareawithunitsof-cm2.Thetotalsurfaceareaofaroughelectrodeisfrr20.Theuseoftheroughnessfactorinthescalingisconrmedbythesuperpositionoftheimpedanceatlowfrequencieswherethecurrentreachesallpartsoftheroughsurface.Bodephaseplotsareoftenusedtorepresentimpedancedata.Thephasewascalculatedfromtherealandimaginarypartsoftheimpedanceas '=arctanZj Zr(7{15)ThemagnitudeandphaseanglecorrespondingtotheimpedancepresentedinFigure 7-3 areexpressedinFigure 7-4 asfunctionsoffrequencywiththeroughnessfactorasaparameter.Thephaseanglewaszeroatthehigh-frequencylimitand-90atthelow{frequencylimit.Theinectionpointcorrespondedtothecharacteristicfrequency.OneofthedisadvantagesoftheBodephaseplotisthatthecontributionoftheohmicresistancemayconcealimportantfeatures.Theohmic-resistance-correctedmodulusandphaseangle,[ 56 ] jZadjj=q (Zr)]TJ /F3 11.955 Tf 11.95 0 Td[(Re)2+Z2j(7{16)and 'adj=arctanZj Zr)]TJ /F3 11.955 Tf 11.96 0 Td[(Re(7{17)respectively,areshowninFigure 7-5 .Theohmic{resistance{correctedphaseangle,inparticular,emphasizesthephaseresponseoftheelectrode.Theohmicresistanceusedinequations 7{16 and 9{15 wasobtainedfromthehigh-frequencylimitofthecalculatedimpedance.Theadjusted{phaseangleshowedtwodeviationsfromanidealcapacitiveresponse.However,thedistinctionbetweenthetwotimeconstantswasnotclear. 96

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A BFigure7-4. Calculatedglobalimpedanceasafunctionoffrequencyforaroughdiskelectrodewithroughnessfactorasaparameter:a)magnitude;b)phase. Aclearerseparationoffeatureswasobtainedbyuseofanimaginary{impedance{derived{phaseangle,whichisbasedonlyontheimaginarypartoftheimpedance,i.e., 'dZj=dlog(Zj) dlog(f)90(7{18)Thederivativeofthelogarithmoftheimaginaryimpedancewithrespecttothelogarithmofthefrequency,usedpreviouslytoestimatetheexponentforaconstant-phaseelement,[ 56 ] 97

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A BFigure7-5. Calculatedglobalimpedanceasafunctionoffrequencyforaroughdiskelectrodewithroughnessfactorasaparameter:a)ohmic{resistance{correctedmagnitude;b)ohmic{resistance{correctedphaseangle. 98

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isexpressedhereasaphaseangle.Acomparisonbetweentheimaginary{impedance{derivedphaseangleandtheadjustedphaseangleispresentedinFigure 7-6 foraroughness Figure7-6. Phaseanglesobtainedfromequations 7{23 9{15 ,and 8{10 fortheimpedancepresentedinFigure 7-3 foraroughdiskelectrodeasafunctionfrequency.Theroughnessfactorwasfr=2andtheroughnessperiodwasP=40m. factorof2andaroughnessperiodof40m.Theimaginary{impedance{derivedphaseangleshowsclearerdelineationbetweenthetwodeviationsascomparedtotheohmic{resistance{correctedphaseangle.Theimaginary{impedance{derived{phaseangleispresentedinFigure 7-7A asafunctionoffrequencywiththeroughnessfactorasaparameter.Theresultsforthesmoothelectrodeshowednon{idealbehavioratfrequenciesgreaterthan800Hzduetothenonuniformcurrentdistributioncausedbytheelectrodeconguration[ 27 ].Fortheroughelectrodes,twodistinctdeviationsfromidealbehaviorwereobserved.Thelower-frequencydeviationfora0.24cmradiuselectrodewitharoughnessfactorof2occurredatfrequenciesequaltoandgreaterthan300Hz,whichis500Hzlessthanthatobservedforthesmoothelectrode.Themagnitudeofthedierenceinfrequencybetweenthelower-frequencydeviationoftheroughelectrodecomparedtothesmoothelectrodeisproportionaltotheroughnessfactor.Thehigher-frequencydeviationoccurredatroughly20kHz. 99

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A BFigure7-7. Imaginary{impedance{derived{phaseanglecalculatedfromtheimpedancepresentedinFigure 7-3 foraroughdiskelectrode:a)Imaginary{impedance{derived{phaseangleasafunctionoffrequency;b)Imaginary{impedance{derived{phaseangleasafunctionofdimensionlessfrequencyKfr. 100

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Thecharacteristicfrequencyatwhichthecurrentandpotentialdistributionsonasmoothdiskelectrodewithinaninsulatingplanebegintoinuencetheimpedanceresponsemaybeexpressedas[ 28 ] fc= 2C0r0(7{19)Foraroughelectrode,theradiusmustbemodiedtoincludetheeectofsurfaceroughness.Therefore,thecharacteristicfrequencyatwhicharoughdiskelectrodeinuencestheimpedanceresponseisexpressedas fc= 2C0frr0(7{20)whichisslightlysmallerthanthecharacteristicfrequencyassociatedwiththeinuenceofthenonuniformcurrentdistributions.Equation( 7{20 )accountsforthecoupledeectofsurfaceroughnessanddiskelectrodegeometry.Thedimensionlessresults,presentedinFigure 7-7B inwhichthecharacteristiclengthincludedtheroughnessfactor,showedthattheinitialdeviationcouldbeattributedtothecoupledeectofdiskgeometryandsurfaceroughnessasitoccurredatKfr=1.Themagnitudeofthehigher{frequencydeviationincreasedastheroughnessfactorincreased,butthecharacteristicfrequencywasnotaltered.Thisprovidesindicationthatthehigher{frequencydeviationmayalsobedependentontheroughnessfactor.Aseriesofsimulationswereperformedinwhichthedepthandperiodoftheroughnesswasincreasedsuchthattheroughnessfactorwasxedat2,asillustratedinFigure 7-8 .Theimaginary{impedance{derived{phaseanglecorrespondingtotheimpedanceresponseonelectrodeswithvariousroughnessperiodsispresentedinFigure 7-9A inwhichthedimensionlessfrequencywasmodiedbytheroughnessfactor.Theresultsshowthatastheroughnessdepthincreased,thecharacteristicfrequencyoftheroughnessdecreased,butthemagnitudeofthedeviationfromidealityremainedconsistent.Also,theportionoftheplotthatissuperposedisthelow{frequencydeviation 101

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Figure7-8. Schematicrepresentationshowingthemannerinwhichtheroughnessperiodwasvariedforaxedroughnessfactorequalto2.Thethreecongurationshavethesamesurfacearea. A B CFigure7-9. Theimaginary{impedance{derived{phaseanglecalculatedfromimpedancesimulationsofaroughdiskelectrodewithinaninsulatingplanewiththeroughnessperiodasaparameterandtheroughnessfactorheldconstant:a)Phaseangleasafunctionoffrequencymodiedbytheroughnessfactor;b)Phaseangleasafunctionof!ReC0inwhichtheohmicresistancewascalculatedfromrecessedelectrodesimulations(seeequation 7{21 );c)Phaseangleasafunctionofdimensionlessfrequencyderivedfromtheapproximatedohmicresistance. 102

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representativeofthecoupledeectofthenonuniformcurrentdistributionduetothediskgeometryandthesurfaceroughness.TheresultsofFigure 7-9A areprovidedinFigure 7-9B asafunctionofthedimensionlessfrequency,Equation 7{10 .Theohmicresistanceassociatedwiththeroughgrooveswasdeterminedfromthefrequencyatwhichdispersionoccurswhichcanbeobtainedfromrecessedelectrodesimulationsas Re=1 2fcC0(7{21)Inthiscase,thesuperpositionoccurredatthehigherfrequencydeviation,indicatingthattheohmicresistanceobtainedfromrecessedelectrodesimulationsisalsoapplicabletodiskelectrodeswithinaninsulatingplane.Theohmicresistanceofthegroovesisdependentontheshapeofthegroovesandthereforecanonlyapplytoaparticulargeometry.Anapproximationoftheohmicresistanceoftheroughgroovesthatisgeneralenoughtobeindependentofgeometrywasobservedtobeafunctionoftheroughnessfactorandthewidthorperiodofthegroovesexpressedas Ref2rP (7{22)Thecorrespondingdimensionlessfrequencymaybeexpressedas Kf2rP=r0=!C0f2rP (7{23)inwhichthecharacteristiclengthofsurfaceroughnessisexpressedas lc;rough=f2rP(7{24)accordingtoequation 7{7 .Theimaginary|impedance|derivedphaseangleisshowninFigure 7-9C asafunctionofthedimensionlessfrequencycorrespondingtotheapproximatedohmicresistance.TheapproximationoftheohmicresistanceshowedthesamesuperpositionofthehigherfrequencydeviationasdidtheresultsinFigure 7-9B inwhichthetrueohmicresistancewasused.Similarsuperpositionwasobserved 103

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forsimulationsofgroovesseparatedbysmoothsurfacesandforrectangularindentations.Whileitisinterestingthattheohmicresistanceassociatedwiththeroughgroovessomehowcorrespondstothedimensionlessfrequency,themostsignicantndingisthecharacteristiclengthofthesurfaceroughnesswhichdirectlyexplainsthefrequencyatwhichdispersionisobserved.Moreworkwillneedtobedonetofullyunderstandtherelationshipbetweenohmicresistanceandthecharacteristiclengthinregardstofrequencydispersion. 7.3.2InuenceofSurfaceRoughnessonaRecessedElectrodeSimulationswereperformedonaroughrecessedelectrodetoisolatetheeectofthesurfaceroughnessfromthatofthediskgeometry.Theimaginary{impedance{derivedphaseangle,presentedinFigure 7-10A asafunctionofdimensionlessfrequencywiththeroughnessperiodasaparameter,showsthatthefrequenciesatwhichdeviationfromanidealresponseoccurreddecreasedasthedepthoftheroughnessincreased.Thelengthscaleusedtomakethefrequencydimensionlesswastheproductoftheroughnessfactorandtheradiusofthedisk.Theimaginary{impedance{derived{phaseangleispresentedinFigure 7-10B asafunctionofthedimensionlessfrequencythatisbasedontheacharacteristicdimensionf2rP.Theresultsforthreedierentroughnessperiods,seeninFigure 7-10A ,superposewhenplottedasafunctionofKf2rP=r0.Theproductoftheroughnessfactorandthegeometricdiskradiusisthecharacteristicdimensionforaroughdisk;whereastheradiusofthediskisthecharacteristiclengthforasmoothelectrode.Theperiodoftheroughnessmultipliedbythesquareoftheroughnessfactoristheappropriatecharacteristiclengthassociatedwiththeroughnessitself.ThedeviationfromanidealresponseofaroughrecessedelectrodeathighfrequencymaybeexplainedbythenonuniformcurrentandpotentialdistributionalongtheelectrodesurfacepresentedinFigure 7-11 .TheresultsforK=10)]TJ /F5 7.97 Tf 6.59 0 Td[(5arepresentedinFigure 7-11A andtheresultsatK=103arepresentedinFigure 7-11B .Thestreamlinesrepresentthe 104

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A BFigure7-10. Imaginary{impedance{derived{phaseanglecalculatedfromthesimulatedimpedanceofaroughrecessedelectrodewiththeroughnessperiodasaparameter:a)asafunctionofdimensionlessfrequencyKfr;andb)asafunctionofdimensionlessfrequencyKf2rP=r0. 105

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A BFigure7-11. ThecurrentpathsobtainedasRef~iexp(jwt)gataxedtime,t:a)atK=10)]TJ /F5 7.97 Tf 6.59 0 Td[(5;b)atK=103.Thepotentialdistributionwithintheelectrolyteadjacenttotheroughsurfaceisshownbythefalse{colorrepresentation. currentdistributionandthepotentialdistributionisshownasafalsecolorgradient.Atlowfrequencies,thepotentialvarieduniformlywiththedepthoftheroughnesssuchthatthecurrentlineswereparalleltothey{axis.Thepotentialdistributionathighfrequenciesfollowedthesurfaceproleandthecurrentdidnotreachthedeepestpartsoftheroughsurface.Thechangeincurrentdistributionwithfrequencycausedtheobservedfrequencydispersion.ThelocalcurrentdensityatapeakandatroughofoneofthegroovesoftheelectrodesurfaceispresentedinFigure 7-12 asafunctionoffrequencyforarecessedelectrode.Thecurrentwasverysmallatlowfrequencies,asexpectedforablockingelectrode,andincreasedwithfrequencyinthesamemannerforboththepeakandtrough.Athighfrequencies,thecurrentatthepeakincreasedsharply;whereas,thecurrentatthetrough 106

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Figure7-12. Calculatedcurrentdensityatapeakandtroughforarecessedelectrodeasafunctionoffrequency. droppedtozero.Thefrequencyatwhichthecurrentatthepeakandthetroughdivergedcorrespondedtothefrequencyatwhichthesurfaceroughnessinuencedtheimpedance.Theeectivecapacitanceoftheelectrode/electrolyteinterfacemaybedeterminedfromtheimaginarypartoftheimpedanceas Ce=)]TJ /F1 11.955 Tf 9.29 0 Td[(1 !Zj(7{25)TheratiooftheeectivecapacitanceandtheinputcapacitanceisprovidedinFigure 7-13 fortherecessedelectrodeasafunctionofdimensionlessfrequency,Kfr,withtheroughnessfactorasaparameter.Thecurrentreachedallpartsoftheroughelectrodeatlowfrequencies,makingthecapacitanceratioequaltotheroughnessfactor.Athighfrequencies,thepenetrationdepthofthecurrentwassmallerthanthedepthoftheroughness,andthecapacitanceratiodecreasedduetothenonuniformcurrentdistributionassociatedwiththecombinedeectsofthediskgeometryandthesurfaceroughness.Theeectivecapacitanceofthesmoothdisk,fr=1,wasequaltotheinputcapacitance,C0,forallofthefrequenciessimulated. 107

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Figure7-13. TheratioofthecalculatedeectivecapacitanceandtheinputcapacitanceasafunctionofdimensionlessfrequencyKfrwiththeroughnessfactorasaparameterforrecesseddiskelectrodes. Aninherentdicultyindeterminingthecharacteristicfrequencyassociatedwithroughnessisthattheshapeoftheroughnessgreatlyinuencestheohmicresistance[ 59 ].Nevertheless,accordingtothegeometry-independentohmicresistanceexpressedinEquation 7{22 ,thecharacteristicfrequencyatwhichroughnessbeginstoinuencetheimpedancemayberoughlyestimatedas fc= 2C0f2rP(7{26)whichmaybecomparedtothecharacteristicfrequencydevelopedforaroughdiskgeometryinequation 7{20 .Thefrequencyatwhichtime{constantdispersionwasobservedispresentedinFigure 7-14 asafunctionoff2rPforthreedierentratiosofelectrolyteconductivityandinterfacialcapacitance.Thecharacteristiclengthfrr0associatedwithadiskwithinaninsulatingplaneisalsoprovided.The104cm/sconductivitytocapacitanceratiomaybeobtainedforacapacitanceC0=1F/cm2,representativeofanoxidelayer,andconductivity=0.01S/cm,representinga0.1MNaClsolution.Thevalue=C0=103 108

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Figure7-14. Thecharacteristicfrequencies,fc,associatedwithdimensionlessfrequenciesK(f2rP=r0)=1orK(fr)=1atwhichthesurfaceroughnessinuencestheimpedanceasafunctionofeithertheroughnessdepthortheproductofroughnessfactorandelectroderadiuswith=C0asaparameter. cm/smaycorrespondtoacapacitanceC0=10F/cm2representativeofadoublelayeronaninertmetal,andaconductivity,=0.01S/cm,againindicativeofa0.1MNaClsolution.Foraninertroughdiskelectrodewitharoughnessfactorof5andanaveragewidthorperiodofroughnessequalto4m(approximatelyhalfoftheaverageparticlesizeofNo.1000gritpaper),frequencydispersionduetoroughnesswillinitiateatafrequencyof16kHzfor=C0=103.Iftheelectroderadiusis0.5cm,frequencydispersionduetothediskgeometrywithinaninsulatingplanewilloccurat65Hz,whichismuchlowerthanthefrequencyatwhichtheeectofsurfaceroughnessisobserved.Theresultspresentedheredemonstratethatsurfaceroughnessonsolidelectrodesinuencestheimpedanceresponsewhencoupledwiththeeectofnonuniformcurrentdistributionsduetodiskelectrodegeometry.Forsmallroughnessfactors,roughnessbyitselfcausesfrequencydispersiononlyatfrequenciesgreaterthanthatduetothegeometryofdiskelectrodes. 109

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Figure7-15. Theimaginary{impedance{derived{phaseanglescalculatedfromimpedancesimulationsofaroughrecessedelectrode(fr=2andP=40m),aroughdiskelectrodewithinaninsulatingplane(fr=2andP=40m),andasmoothdiskelectrodewithinaninsulatingplane. However,sincetheroughnessfactorisaparametercontainedwithinthecharacteristicdimensionofthediskgeometryandthesurfaceroughness,therewillexistcasesinwhichtheroughnessofthediskwillcausefrequencydispersionatfrequencieslowerthantheeectofthediskgeometry.Porouselectrodesprovideanexampleinwhichtheroughnessfactorcanbeontheorderof1000andtheeectofdiskgeometrywouldnotbeseen.Specically,whenfrPisgreaterthanr0,frequencydispersionduetothesurfaceroughnesswillcontrol.ComparisonsareprovidedinFigure 7-15 fortheimaginary{impedance{derived{phaseangleassociatedwithasmoothdiskelectrode,aroughdiskelectrodewithinaninsulatingplanewithfr=2andP=40m,andaroughrecesseddiskelectrodewithfr=2andP=40m.ThegeometryofthesmoothdiskelectrodewithinaninsulatedplanecauseddeviationfromidealbehavioratKfr=1duetothenonuniformcurrentdistribution.Thelinerepresentingtherough,recessedelectrodeshowedfrequencydispersionatKfr=20,whichisnumericallyequaltoKf2rP=r0=1.Theroughelectrodewithintheinsulating 110

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planeshowedtheeectofboththediskgeometryandtheroughness.TheroughnesseectoccurredatKfr=20andthediskgeometryeectwasobservedatKfr=K=1. 7.3.2.1EectofporegeometrySincetheshapeoftheroughnessinuencestheroughnessfactor,othergeometriesweremodeledtoensuretheuseofthecharacteristicdimensionforroughness.RectangulargroovesaswellasseparatedV-shapedgroovesweresimulated.Thespacingbetweenthegroovescanbeaccommodatedbyreplacingf2rwithfrfpwherefpisthesurfaceareaofthegroovedividedbytheareaofthegroovemouth.Therefore,thecharacteristicfrequencyatwhichtheroughnesswillbegintoinuencetheimpedanceis fc==2C0frfpP(7{27)Theimaginary{impedance{derivedphaseangleforaroughelectrodewithrectangulargroovesisshowninFigure 7-16 withthewidthofthegroovesasaparameter.Figure 7-16A showstheresultsasafunctionoffrequencyandFigure 7-16B showstheresultsasafunctionofdimensionlessfrequencyinwhichthecharacteristicdimensionwasexpressedasfrfpP.Whentheresultsareshownasafunctionoffrequency,thefrequencyatwhichdisherisonbeginstooccurincreasesasthewidthofthegroovesincrease.Whentheresultsareshownasafunctionofdimensionlessfrequencytheplotssuperposeandfrequencydispersionbeginsatadimensionlessfrequencyof1.SimilarsimulationswereperformedforaroughelectrodewithV{shapedgrooveswithaspacebetweenthem.Theimaginary{impedance{derivedphaseangleisshowninFigure 7-17 withthewidthofthegroovesasaparameter.Figure 7-17A showstheresultsasafunctionoffrequencyandFigure 7-17B showstheresultsasafunctionofdimensionlessfrequencyinwhichthecharacteristicdimensionwasexpressedasfrfpP.Onceagain,whentheresultsareshownasafunctionofdimensionlessfrequency,thefrequencydispersionforeachwidthoccursatdimensionlessfrequencyequalto1.Thus,thecharacteristic 111

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A BFigure7-16. Theimaginary{impedance{derivedphaseangleforaroughelectrodewithrectangulargrooveswiththewidthofthegroovesasaparameters. dimensionforroughnesspresentedhereisgeneralandcanbeappliedtodierentshapesofroughness. 7.3.2.2TransitionfromaroughelectrodetoaporouselectrodePorouselectrodesareextremelyroughelectrodesusuallycontainingacomplicatednetworkofelectro{activepores.Roughnessfactorsforporouselectrodescanbe1000orgreater.DeLeviedevelopedthetheoryofporouselectrodesbyderivinganexpression 112

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A BFigure7-17. Theimaginary{impedance{derivedphaseangleforaroughelectrodewithV-shapedgrooveswithaspacebetweenthemandthewidthofthegroovesasaparameter. 113

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Figure7-18. Theimaginaryimpedancederivedphaseangleofroughelectrodesasafunctionoffrequencywiththedepthofgroovesandroughnessfactorasparameters. fortheimpedanceofasinglecylindricalpore.[ 18 ]Ablockingporouselectrodewillhaveaphaseangleof45athighfrequencies.Thetransitionfromaroughelectrodetoaporouselectrodemaybeshownbyincreasingthedepthoftherectangulargroovesuntilthesimulatedresponsereectsaporousbehavior.ShowninFigure 7-18 istheimaginary{impedance{derivedphaseangleasafunctionoffrequencyforincreasinglyroughelectrodes,characterizedbythedepthoftherectangulargroovesaswellastheroughnessfactor.Asthedepthofthegroovesisincreased,thefrequencydispersionoccursatlowerfrequencies.Themoreinterestingfeature,however,istheincreaseinthemagnitudeofthefrequencydispersionatfrequenciesbeyondtheinitialoccurrenceoffrequencydispersion.Asthedepthoftherectangulargroovesandthustheroughnessfactorincreases,thephaseangleapproachesalimitof45thereforeexhibitingaporouselectroderesponse.Ifthesameresultsarepresentedasafunctionofthedimensionless 114

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Figure7-19. Theimaginaryimpedancederivedphaseangleofroughelectrodesasafunctionofdimensionlessfrequencywiththedepthofgroovesandroughnessfactorasparameters. frequency,showninFigure 7-19 ,theplotssuperposeatdimensionlessfrequencyequalto1andclearlyshowtheincreaseinfrequencydispersionasitapproachesaphaseangleof45athigherfrequencies.Therefore,thecharacteristicdimensionprovidedforroughelectrodesmayalsobeusedtodescribethefrequencyatwhichaporouselectroderesponseshouldoccur. 115

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CHAPTER8INFLUENCEOFCAPACITANCEDISTRIBUTIONONIMPEDANCEImpedancemeasurementsonsolidelectrodesoftenyieldconstant{phase{element(CPE)behavioroveralargerangeoffrequencies.ACPEdescribesadistributionoftimeconstantswhichcanbephysicallycategorizedasoccurringeithernormaltooralongtheelectrodesurface[ 30 ].Hirschornetal.[ 24 25 ]usedameasurementmodelincorporatingaseriesofelementstoshowthatapower-lawdistributionofresistivitythroughanoxidelmgivesrisetoCPEbehavior.Thismodelhasbeenusedtodeterminethelmthicknessofoxidesonstainlesssteelandaluminumelectrodes[ 58 ].Ithasalsobeenusedtodeterminephysicalpropertiesofhumanskin[ 58 74 ]andpolymercoatings[ 48 ].Distributionsofsolutionresistanceareobservedinplanardiskelectrodeexperiments.Huangetal.showedthatthegeometryofadiskelectrodewithinaninsulatingplaneleadstofrequencydispersiononblockingelectrodesatfrequenciesgreaterthandimensionlessfrequency!C0r0==1[ 28 ].Jorcinetal.[ 30 ]conrmedtheseresultsexperimentallywiththeuseoflocalelectrochemicalimpedancespectroscopy.Surfaceroughnessoftheelectrode,oncebelievedtocontributetothecauseforfrequencydispersion[ 11 ],willalsoyieldadistributionofohmicresistance.Alexanderetal.[ 4 ]showedthatsurfaceroughnessatthemicronscaleleadstofrequencydispersionatfrequencieslargerthanthoseduetothediskgeometryiftheradiusofthediskisgreaterthanthecharacteristiclength,frP,wherefristheroughnessfactorandPistheperiodoftheroughness.Theperiodoftheroughnessmayrepresenttheaveragewidthoftheroughgrooves.Thediskradiuswillinmostcasesbegreaterthanthischaracteristicdimensionexceptincasesofverylargeroughnessfactorssuchasthosethatyieldporouselectrodebehavior.ThisresultcontradictsthetheorythatsurfaceroughnessmaybeasourceofCPEbehavioroveralargerangeoffrequencies. 116

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A BFigure8-1. Finite-elementmeshforthediskelectrodesimulations.a)Axi-symmetricdomainusedtorepresentdiskelectrodeexperiments;b)meshshowingadetailedviewofarecessedelectrode.TheouterdomainisshowninFigure 10-3 Theobjectiveofthisworkistoexplore,byuseofnite-elementmodels,whetherasurfacedistributionofcapacitancecouldprovideavalidphysicalexplanationforCPEbehavioroverabroadrangeoffrequenciesandtodeterminethecharacteristicfrequencyanddimensionassociatedwiththisformofsurfaceheterogeneity. 8.1ImpedanceCalculationsTwodiskelectrodecongurationsweresimulated,includingadiskelectrodeembeddedwithinaninsulatingplane,andarecesseddiskelectrode.ThecalculationswereperformedinCOMSOLMultiphysics4.3usingcylindricalcoordinatesforaquarterofacircledomain,providedinFigure 10-3 ,whichrepresentedtheelectrolyte.Calculationswereperformedforelectroderadiiof0.12cm,0.24cm,and0.48cm.Thecounterelectrodewaslocatedatp r2+y2=500cm,makingthedomainsizeatleast1000timeslargerthantheradiusofthedisk(seeAlexanderetal.[ 4 ]).Thisgeometrywaschosentoensurethatthecounterelectrodecouldbeconsideredtobeinnitelyfarfromtheworkingelectrode. 117

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Figure8-2. CapacitancedistributionasafunctionofradialpositionbasedonasquarewaverepresentedbyaFourierserieswithaperiodof60m. Afreetriangularmeshwasusedwithagreaterdensityofelementsneartheworkingelectrode.AcapacitancedistributionwassimulatedbyaFourierseriesthatrepresentedasquarewavedistribution,asshowninFigure 8-2 asafunctionofradialposition.TheFourierseries[ 21 ]wasexpressedas C0(r)=hCi+1Xn=1(Cmax)]TJ /F3 11.955 Tf 11.95 0 Td[(Cmin)cos(2nPr)(8{1)wheretheconstantsCmaxandCminrepresentedthemaximumandminimumcapacitance.TheaveragecapacitanceoftheelectrodesurfacehCiwascalculatedas hC0i=2 r20Zr00C0(r)rdr(8{2)wherer0istheradiusofthedisk.Thecapacitanceasafunctionoftheradialpositionwasintegratedoverthesurfaceoftheelectrodeandthendividedbytheareaoftheelectrode.TheperiodPofthesquarewavemayberepresentativeofanelementalgrainsizeinwhichthecapacitanceisassumedtoberelativelyuniformacrossagrainandthenjumptoanothervalueatanadjacentgrain.Simulationswereperformedwith5,10,20,and40termsoftheFourierseries.Numericalartifactswereobservedathighfrequencieswhentheserieswastruncatedat5and10terms.Asmoretermswereaddedtotheseriestheartifactsshiftedtohigherfrequenciesandwith20termsintheseriesthisbehaviorwas 118

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eliminatedfromthesimulatedfrequencyrange.TheFourierserieswasthereforetruncatedafter20terms.Theelectrodewasassumedtobepurelycapacitive,i.e.,thecontributionsofslowreactionkineticsandmass-transferwereneglected.ThepotentialdistributionwithintheelectrolytedomainwassolvedusingLaplace'sequation r2=0:(8{3)Thepotentialcomprisessteady-stateandoscillatingcomponentsandmaybeexpressedas =+Ren~exp(j!t)o(8{4)Thepotentialoftheelectrodesurfacemayalsobeexpressedinthesamemanneras V=V+Ren~Vexp(j!t)o(8{5)Thenormaloscillatingcurrentdensityatthesurfaceoftheelectrodewasexpressedas ~i=j!C0(r)(~V)]TJ /F1 11.955 Tf 13.26 3.02 Td[(~)(8{6)where~Visthepotentialperturbationattheelectrodeand~isthecomplexoscillatingpotentialwithintheelectrolyte.Thevalueofthepotentialperturbationwillnotaecttheimpedanceresponse,but,forallofthestimulationspresentedinthiswork,avalueof10mVwasused.Theoscillatingpotentialwasconstrainedtobeequaltozeroatthecounterelectrode.Theimpedancewascalculatedas Z(!)=~V ~i(8{7)foraspeciedrangeoffrequencies. 8.2ResultsandDiscussionImpedancesimulationsarepresentedtoshowtheinuenceofaheterogeneoussurfacecapacitanceondiskelectrodes.Resultsareshownrstforarecessedelectrode 119

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Figure8-3. TheimpedanceinNyquistformatofarecesseddiskelectrodewiththesquarewavecapacitancedistributionshowninFigure 8-2 andtheperiodofdistributionasaparameter. congurationandthenadiskelectrodewithinaninsulatingplane.Dimensionlessresultsareusedtodeterminethecharacteristiclengthsandfrequenciesassociatedwitharadially-periodicdistributionofcapacitance. 8.2.1CapacitanceDistributiononRecessedElectrodesArecessedelectrodemodelwasusedtoisolatetheeectofsurfaceheterogeneityontheimpedanceresponseofblockingelectrodes.ThesimulatedimpedanceresponseofarecesseddiskelectrodewiththesquarewavecapacitancedistributionshowninFigure 8-2 ispresentedinFigure 8-3 asafunctionoffrequencywiththeperiodofdistributionasaparameter.TheminimumandmaximumcapacitancevaluesexpressedinEquation 9{5 weresetto1F/cm2and10F/cm2respectivelytorepresentasurfacewithanoxidelmandabaremetalsurface.Theconductivityofthesolutionwas10)]TJ /F5 7.97 Tf 6.58 0 Td[(5S/cm.Theimpedancewasrepresentativeofanidealcapacitoratfrequenciesbelow100kHz,shownbytheverticallinesthatareperpendiculartotherealaxis.Thegureinsetshowsamagniedviewoftheimpedanceathighfrequencies.Astheperiodofthecapacitancedistributionincreases,thefrequencyatwhichdispersionbeginsdecreases. 120

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A BFigure8-4. Thesimulatedimpedanceasafunctionoffrequencyofarecesseddiskelectrodewithasquarewavecapacitancedistributionandtheperiodofdistributionasaparameter:a)therealpartoftheimpedance;b)theimaginarypartoftheimpedance. ThesimulatedrealimpedanceasafunctionoffrequencyispresentedinFigure 8-4A withtheperiodofthecapacitancedistributionasaparameter.Thelow{frequencylimitoftherealimpedanceincreasedastheperiodofthedistributionincreased.The 121

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high{frequencylimit,whichrepresentstheohmicresistance,wasunaectedbythedistributionofcapacitance.Theimaginaryimpedancemaybeexpressedasafunctionofthesurface{averagedcapacitanceas hZi=)]TJ /F1 11.955 Tf 9.3 0 Td[(1 !hC0i(8{8)Similarexpressionsmaybeformedforthemaximumandminimumvaluesofthecapacitance.Theimpedancecalculatedfromsurface{averagedaswellasthemaximumandminimumcapacitancevaluesarecomparedtothesimulatedimaginaryimpedanceinFigure 8-4B asafunctionoffrequencywiththeperiodofthedistributionasaparameter.Theimaginaryimpedanceatlowfrequenciescoincidedwiththeimpedancebasedonthesurface{averagedcapacitance.Deviationfromthesurface{averagedimpedanceoccurredathighfrequencieswheretheimpedanceasymptoticallyapproachedtheimpedanceassociatedwiththeminimumcapacitancevalue.Thedeviationfromtheexpectedimpedanceresponsemaybeexplainedbythecurrentandpotentialdistributionalongtheelectrodesurface,whichispresentedinFigure 8-5 .Inthecaseofblockingelectrodes,thesolutionresistancecontrolsthecurrentdistributionathighfrequencieswhiletheinterfacialimpedancecontrolsatlowfrequencies.ThepotentialdistributionatlowfrequenciesisrepresentedbythecolorgradientinFigure 8-5A .Thestreamlinescorrespondedtothepathofthemodulusoftheoscillatingcurrentexpressedas j~i(!)j=q ~ir2+~ij2(8{9)inwhich~irand~ijrepresenttherealandimaginarypartsoftheoscillatingcurrent.Thefalsecolorrepresentationofpotentialdistributionwasadjustedtoemphasizethevariationsneartheelectrodesurface.Theregionsoftheelectrodesurfacewithhighercapacitancehavealowerimpedanceandthecurrentowsmoreeasilythroughthesepoints.Despitethenonuniformcurrentdistribution,theimpedanceresponseatlowfrequencieswasindicativeofapurecapacitorwithavalueoftheaveragedsurface 122

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A BFigure8-5. Thecurrentpathsnearthesurfaceofarecessedelectrodeexhibitingasquare{wavedistributionofcapacitanceobtainedasq ~i2r+~i2j:a)at10mHz;b)at100kHz.Thepotentialdistributionwithintheelectrolyteadjacenttotheroughsurfaceisshownbythefalse{colorrepresentation. capacitance.Thepotentialdistributionalongtheelectrodesurfaceat100kHz,presentedinFigure 8-5B ,wasmoreuniform.ThemodulusoftheoscillatingcurrentdensityalongtheelectrodesurfaceispresentedinFigure 8-6 asafunctionofradialposition.Thelow{frequencycurrentresponse,presentedinFigure 8-6A ,showedavariationofcurrentproportionaltothevariationinsurfacecapacitanceandsmallvaluesofcurrent.Thehighfrequencyresponse,presentedinFigure 8-6B ,showedmuchhighervaluesofcurrenthoweverthedistributionwasmuchmoreuniformwithonlyvariationsatlocationswherethecapacitancechangesfromone 123

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A BFigure8-6. Normalcurrentdistributionattheelectrodesurfaceduetoanonuniformcapacitancedistributionwithaperiodof60mofarecessedelectrodeasafunctionofradialposition:a)currentdistributionat10mHz;b)currentdistributionatthehighfrequencylimitofthesimulationsf=100kHz. valuetotheotherwhichcanbeattributedtothenitenumberoftermsintheFourierseries.AphaseangledependentonlyontheimaginarypartoftheimpedancewasdenedbyAlexanderetal.[ 4 ]as 'dZj=dlog(Zj) dlog(f)90:(8{10)Ascomparedtootherdenitionsofphaseangle,theimaginary{impedance{derivedphaseangleismoresensitivetotheonsetoffrequencydispersion.Theimaginary{impedance{derivedphaseangleispresentedinFigure 8-7A asafunctionoffrequencywiththeperiodofdistributionasaparameter.Theeectofavaryingcapacitancealongthesurfaceof 124

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A BFigure8-7. Imaginary{impedance{derivedphaseanglecalculatedfromtheimpedancedatainFigure 8-4 :a)phaseangleasafunctionoffrequency;b)phaseangleasafunctionofdimensionlessfrequencybasedontheaveragedcapacitanceandtheperiodofthedistribution. 125

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theelectrodedidnotinuencetheimaginaryimpedanceatlowfrequenciesasindicatedbyaphaseangle,dZj=-90.However,frequencydispersiondidoccurinallcasesathighfrequencieswithaminimumphaseangleofapproximately)]TJ /F1 11.955 Tf 9.3 0 Td[(50.Thedeviationfromtheexpectedcapacitivebehavioroccurredatlowerfrequenciesastheperiodofthedistributionincreased.Iftheperiodofthedistributionistakenasthegrainsizewithinapolycrystallinesurface,grainsizesontheorderof1mandlessshouldnotleadtofrequencydispersionatfrequencieslessthan1kHz.AdimensionlessfrequencycanbeexpressedforthedistributionofcapacitancebyamendingEquation 7{7 toincludethesurface{averagedcapacitanceexpressedas K=!hC0ilc;Cap(8{11)wherelc;Caprepresentsthecharacteristiclengthforthecapacitancedistribution.Sincethefrequencydispersionshiftedtolowerfrequencieswithincreasesintheperiodofthedistribution,oneobviouschoiceforthecharacteristiclengthassociatedwithadistributionofcapacitanceistheperiodofthedistribution.Theimaginary{impedance{derivedphaseangleispresentedinFigure 8-7B asafunctionofdimensionlessfrequency.Theresultsweresuperposed,conrmingthattheperiodistheappropriatecharacteristiclengthtousetodescribesurfaceheterogeneityofcapacitance.Thecharacteristicfrequencyatwhichanon-uniformcapacitancebegantoinuencetheimpedancewasdeterminedtobeKP=r0=1.Sinceimpedanceisoftenusedtomeasureinterfacialcapacitance,itisimportanttoensurethatadistributionofcapacitancealongthesurfacedoesnotcomplicatetheuseofthistechnique.Theeectivecapacitanceoftheelectrode/electrolyteinterfacemaybedeterminedfromtheimaginarypartoftheimpedanceas Ce=)]TJ /F1 11.955 Tf 9.29 0 Td[(1 !Zj(8{12) 126

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Figure8-8. Theratioofthecalculatedeectivecapacitanceandthesurface{averagedinputcapacitanceasafunctionofdimensionlessfrequencyKP=r0withtheperiodofthedistributionasaparameterforrecesseddiskelectrodes. Theratioofthecalculatedcapacitanceandthesurface{averagedcapacitanceisshowninFigure 8-8 asafunctionofdimensionlessfrequencywiththeperiodofthedistributionasaparameter.Inallcases,thesimulatedcapacitancecloselymatchedthesurface{averagedcapacitanceoftheelectrodeatlowfrequencies. 8.2.2CapacitanceDistributiononDiskElectrodesTheimpedanceofadiskelectrodewithinaninsulatingplanecontainingaheterogenoussurfacecapacitancewassimulatedtoshowthecoupledeectofcapacitancedistributionandnon-uniformcurrentdistributionsduetodiskgeometry.Thegoalofthesesimulationswastodetermineiftheeectofcapacitancedistributioninuencestheimpedanceatfrequencieslowerthantheeectofdiskgeometry.Thegeometryofadiskelectrodeembeddedwithinaninsulatingplanecausesfrequencydispersionathighfrequenciesduetothenon-uniformcurrentdistributionathighfrequency.Asthefrequencyincreasestoinnity,themagnitudeoftheoscillatingcomponentofthemodulatedcurrentattheperipheryofthediskapproachesinnitywhilethecurrentatthecenterofthediskremainsnite.Thesimulatedcurrentdistribution 127

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A BFigure8-9. Normalcurrentdistributionatadiskelectrodesurfaceasafunctionofradialposition:a)currentdistributionat10mHz;b)currentdistributionat100kHz. atanelectrodesurfacecontainingthecapacitancedistributioninFigure 8-2 isprovidedasafunctionofradialpositionfor10mHzinFigure 8-9A andfor100kHzinFigure 8-9B .Thecurrentdistributionatlowfrequenciesresembledthesquare{wavedistributionfeatures.Athighfrequencies,theeectofthediskgeometryovershadowedtheeectofthecapacitancedistributionsincethecurrentattheperipheryapproachedinnity.Theimaginary{impedance{derivedphaseangleisprovidedinFigure 8-10A asafunctionoffrequency.Atlowfrequenciesthephaseanglewasconstantwithavalueequalto)]TJ /F1 11.955 Tf 9.3 0 Td[(90.Frequencydispersionbecameapparentatapproximately1Hzforallperiodsofcapacitancedistributionduetothegeometryofthedisk.Athigherfrequencies,dispersionduetothenonuniformcapacitancedistributionoccurredsuchthatincreasesintheperiod 128

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A BFigure8-10. Imaginary{impedance{derivedphaseangleforadiskelectrodewithinaninsulatingplane:a)phaseangleasafunctionoffrequency;b)phaseangleasafunctionofdimensionlessfrequencybasedontheaveragedcapacitance. 129

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ofthedistributioncausedthedeviationtoshifttolowerfrequencies.ThesameresultsarepresentedinFigure 8-10B asafunctionofdimensionlessfrequencyinwhichtheperiodofthedistributionwasusedasthecharacteristiclength.Thefrequencydispersionduetothesurfaceheterogeneitywassuperposed,indicatingthattheperiodofthecapacitancedistributionistheappropriatecharacteristiclength.TheresultsobtainedforarecessedelectrodeprovidedinFigure 8-7 showonlythefrequencydispersionduetothesurfacedistributionofcapacitance.Theimaginary{impedance{derivedphaseanglesarepresentedinFigure 8-11A asafunctionoffrequencywiththeradiusofthediskasaparameter.Theperiodofthedistributionwasxedat60m.Changesintheradiusoftheelectrodeinuencedonlythefrequencydispersionassociatedwiththegeometryofthedisk.TheimaginaryimpedancederivedphaseangleisshowninFigure 8-11B asafunctionofdimensionlessfrequency,Equation 7{8 .Thefrequencydispersionassociatedwithdiskgeometrywassuperposedandthefrequencydispersionassociatedwiththecapacitancedistributionwasnot.Simulationswereperformedtoexplorewhethertheamplitudeofthecapacitancedistributionmayinuencethefrequencyatwhichtheimpedanceisinuenced.Theimaginary{impedance{derived{phaseanglewascalculatedfromsimulationsonplanardiskelectrodeswithdierentamplitudesofcapacitancedistributionandarepresentedinFigure 8-12 asafunctionoffrequency.Astheamplitudeofthedistributionincreased,thedeviationfrom)]TJ /F1 11.955 Tf 9.3 0 Td[(90shiftedtolowerfrequencies.However,whenthefrequencyismadedimensionlesswiththeuseoftheperiodasthecharacteristiclengthandtheaveragevalueofthecapacitance,showninFigure 8-12B ,thecharacteristicfrequencyatwhichdispersionoccurreddidnotchangewithamplitudeandonlythemagnitudeofthedispersionincreased.TheratioofthecalculatedcapacitanceandtheaveragecapacitanceispresentedasafunctionofdimensionlessfrequencyinFigure 8-13 withtheperiodofthedistributionasaparameter.Inallcases,thesimulatedcapacitancecloselymatchedtheaveraged{surface 130

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A BFigure8-11. Imaginary{impedance{derivedphaseanglesvaluescalculatedfromtheimpedancedatawiththediskradiusasaparameter:a)imaginary{impedance{derivedphaseangleasafunctionoffrequency;b)imaginary{impedance{derivedphaseangleasafunctionofdimensionlessfrequencyK=!hC0ir0=. 131

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A BFigure8-12. Imaginary{impedance{derivedphaseanglecalculatedfromimpedancedataondiskelectrodeswitharadialdistributionofcapacitancewiththeamplitudeofthesquarewaveasaparameter:a)phaseangleasafunctionoffrequency;b)phaseangleasafunctionofdimensionlessfrequencybasedontheaveragedcapacitanceandtheperiodofthesquarewaveasthecharacteristiclength. 132

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Figure8-13. Theratioofthecalculatedeectivecapacitanceandthesurface{averagedinputcapacitanceasafunctionofdimensionlessfrequencyKP=r0withtheperiodofthedistributionasaparameterfordiskelectrodeswithinaninsulatingplane. capacitanceoftheelectrodeatlowfrequencies,indicatedbyacapacitanceratiovalueequalto1.Athighfrequencies,aninitialdecreaseofthecapacitanceratiowascausedbythenonuniformcurrentdistributiononthedisksurface.Atadimensionlessfrequencyequaltoone,afurtherdecreasewasobservedduetothecapacitanceheterogeneity.Thecharacteristicfrequencyatwhichfrequencydispersionoccursforaplanardiskelectrodeexhibitingadistributionofcapacitancemaybeexpressedas fc;r0=2 2hC0ir0(8{13)wherehC0irepresentsthesurface-averagedcapacitance.Thecharacteristicfrequencyatwhichfrequencydispersionbeginsduetothecapacitancedistributionmaybeexpressedas fc;P= 2hC0iP(8{14)inwhichtheperiodofthedistributionisthecharacteristiclengthassociatedwiththedistribution.ThefrequencyatwhichdispersionbeginsispresentedinFigure 8-14 asafunctionofboththediskradiusaswellastheperiodofthecapacitancedistribution 133

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Figure8-14. ThefrequencyKP=r0=1atwhichthesurfaceheterogeneityinuencestheimpedanceasafunctionofdistributionperiodanddiskradiuswith=hC0iasaparameter. withtheratio,=hC0i,asaparameter.Theperiodofthedistributionmaybeassociatedwiththeaveragewidthofgrainsizeswithinanoncrystallinesurface.Fora0.5cmradiuselectrodeinasystemwith=hC0i=1cm/s(correspondingforexample,tohC0i=20F/cm2anda0.16mMsodiumchlorideconcentration)andanaveragegrainsizeof1m,thefrequencyatwhichdispersionoccursduetothediskgeometryis400mHzwhilethefrequencyatwhichdispersionoccursduetothenonuniformcapacitanceis1.6kHz.Thefrequencydispersionduetothecapacitancedistributionwillalwaysoccuratgreaterfrequenciesthanthefrequencydispersionduetothediskgeometry. 134

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CHAPTER9INFLUENCEOFREACTIVITYDISTRIBUTIONONIMPEDANCEElectricalcircuitsinvokingconstant-phaseelements(CPE),i.e., Z=Re+Rt 1+(j!)QRt(9{1)areoftenusedtotimpedancedataarisingfromabroadrangeofexperimentalsystems.Equation( 9{1 )isexpressedintermsofanohmicresistanceRe,acharge{transferresistanceRtandCPEparametersandQ.When=1,theparameterQhasunitsofcapacitance;otherwise,Qhasunitsofs/cm2orF/s(1)]TJ /F4 7.97 Tf 6.58 0 Td[()cm2[ 57 ].ThephysicaloriginoftheCPEhasbeendescribedasbeingassociatedwithadistributionoftimeconstantsthroughalmoralonganelectrodesurface.Thefrequencydispersionassociatedwithadistributionoftimeconstantsnormaltotheelectrodesurfaceiswellestablished;whereas,thefrequencydispersionassociatedwithasurfacedistributionislesswellunderstood.Hirschornetal.[ 24 25 ]showedthatapower{lawdistributionofresistivitythroughalmyieldsCPEbehavior.Thepower{law{modelapproachhasbeenusedsuccessfullytoextractalmcapacitanceandassociatedparametersforavarietyofsystems,includingoxidesonsteel,[ 58 ]humanskin,[ 58 74 ]andpolymercoatings.[ 48 52 ]Brugetal.[ 12 ]developedanexpressionforthecapacitanceextractedfromaCPEcausedbyasurfacedistributionofcapacitance.Cordoba-Torresetal.[ 15 ]showedthattheBrugmodelexplainedthecorrelationobservedbetweenCPEparametersandQfortwoexperimentalconditions:thecorrosionofpolycrystallineironandthedepositionofCaCO3scaleongoldelectrodes.Theresultswereattributedtoadistributionoftimeconstantsassociatedwithsurfaceheterogeneity.Theexactnatureofthesurfaceheterogeneitywasnotidentied.Insubsequentwork,Cordoba-Torresetal.[ 16 ]suggestedthattheCPEbehaviorresultsfromenergeticdistributionsratherthangeometricheterogeneityorroughness. 135

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RingelectrodeswerestudiedbyChenetal.[ 14 ]todeterminethecharacteristicfrequencyatwhichfrequencydispersionisinducedduetothegeometryofthering.Thecurrentneartheinnerandouteredgesofaringelectrodeapproachesinnity.Chenetal.[ 14 ]determinedempiricallyanapproximationofthecharacteristiclengthtobe `c;ring=r2)]TJ /F3 11.955 Tf 11.96 0 Td[(r1 1+(r1=r2)2(9{2)wherethewidthoftheringwasscaledbyatermwhichapproached0.5asr1approachedr2andunityasr1tendsto0.Thisexpressionworksbestforintermediatevaluesofr1=r2.Alexanderetal.[ 4 ]identiedthecharacteristicdimensionassociatedwithroughnessexpressedas `c;rough=f2rP(9{3)inwhichPwasdenedastheperiodoftheroughnessortheaveragewidthofaroughgroove.Theyshowedthatroughnessconsistentwithevenpoorlypolishedelectrodesdoesnotleadtotime{constantdispersioninthefrequencyrangeusedforelectrochemicalmeasurements.Alexanderetal.[ 5 ]showedthatacapacitancedistributiongaverisetofrequencydispersion,butthattheeectwasseenatfrequencieshigherthanthatassociatedwiththediskgeometry.Thecharacteristiclengthforaperiodicdistributionasafunctionoftheradialcoordinatewastheperiodofthedistributionsuchthat `c;cap=P(9{4)and,astheperioddecreased,thefrequencydispersionoccurredathigherfrequencies.TheobjectofthepresentworkistoexplorethetypeofsurfaceheterogeneitythatcanleadtoaBrug{styleCPE.Aswaspreviouslydescribed,surfaceroughnessandgeometricconsiderationsdidnotleadtoCPEbehavior.Nordidadistributionofcapacitance.ThispaperaddressesthequestionofhowadistributionofreactivityaectstheimpedanceresponseandwhetherthisformofheterogeneitycanleadtoaCPEresponse. 136

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Figure9-1. ReactivitydistributionasafunctionofradialpositionbasedonasquarewaverepresentedbyaFourierserieswithaperiodof60m. 9.1MathematicalDevelopment&ImpedancecalculationsAreactivitydistributionwassimulatedbyacharge-transferresistancedistribution.AFourierserieswasusedtorepresentasquarewave,showninFigure 9-1 asafunctionofradialposition.TheFourierserieswasexpressedas Rt(r)=hRti+1Xn=1(Rt;max)]TJ /F3 11.955 Tf 11.96 0 Td[(Rt;min)cos(2nP)(9{5)wheretheconstantsRt;maxandRt;minrepresentedthemaximumandminimumscaledvaluesofreactivity.TheaveragereactivityoftheelectrodesurfacehRtiwascalculatedas hRti=2 r20Zr00Rt(r)rdr(9{6)inwhichthereactivityasafunctionoftheradialpositionwasintegratedoverthesurfaceoftheelectrodeanddividedbytheareaoftheelectrode.Theperiodofthesquarewavemayrepresentanelementalgrainsizeinwhichthereactivityisassumedtoberelativelyuniformacrossagrainandtothenjumptoanothervalueatanadjacentgrain.TheFourierserieswastruncatedafter20terms.ThepotentialdistributionwithinanelectrolytedomainwithuniformcompositionisgovernedbyLaplace'sequation r2=0(9{7) 137

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Atwo-dimensionalaxialsymmetriccoordinatesystemwasusedinwhichthepotentialwasassumedtobeindependentoftheangularcoordinate.Thepotentialwasseparatedintosteady-stateandoscillatingcomponentsas =+Ren~exp(j!t)o(9{8)whererepresentsthesteady-stateportionand~representsthecomplexoscillatingportionthatisafunctionoffrequencyandpositionbutindependentoftime.Thesamerelationshipforthepotentialappliedtotheelectrodecanbeexpressedas V=V+Ren~Vexp(j!t)o(9{9)Atr=0,forally,thesymmetrycondition@~=@r=0wasassumedtoapply.Farfromtheelectrode,asp r2+y2!1,~!0.Thisconditionplacesthecounterelectrodeinnitelyfarfromtheworkingelectrode.Thenormaluxatthesurfaceoftheelectrodeexhibitingfaradaicreactionswasexpressedas i=C0@(V)]TJ /F1 11.955 Tf 11.96 0 Td[(0) @t+(a+c)i0F RT(V)]TJ /F1 11.955 Tf 11.96 0 Td[(0)=)]TJ /F3 11.955 Tf 9.3 0 Td[(@ @yy=0(9{10)whereC0isthecapacitanceontheelectrodesurface,istheconductivityoftheelectrolyte,and0isthepotentialoutsidethediusepartoftheelectrodedoublelayer.[ 27 ]Theoscillatingcurrentdensitywasconvertedtothefrequencydomainsuchthat ~i=j!C0(~V)]TJ /F1 11.955 Tf 13.26 3.02 Td[(~)+(a+c)i0F RT(~V)]TJ /F1 11.955 Tf 13.26 3.02 Td[(~)(9{11)where~Visthepotentialperturbationattheelectrodeand~isthecomplexoscillatingpotentialwithintheelectrolyte.Thecharge-transferresistancemaybeexpressedas Rt=RT (a+c)i0F(9{12) 138

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Equation 9{11 wasthenexpressedas ~i=j!C0(~V)]TJ /F1 11.955 Tf 13.26 3.02 Td[(~)+1 Rt(~V)]TJ /F1 11.955 Tf 13.26 3.02 Td[(~)(9{13)Inordertoobservedeviationsfromanexpectedresponseitisusefultoexpressdataindimensionlessform.Throughoutthispaper,theimpedanceisscaledbytheohmicresistance.Simulationswereperformedforbothdiskandrecesseddiskelectrodes.In1963,Kelman[ 34 ]solvedasteady-statediusionproblemforowthroughacylindricalporeintoaninnitedomain.WestandNewman[ 73 ]appliedhissolutiontoformulateamathematicalequivalenttotheohmicresistanceofarecessedelectrodeas Re;rec=r0 4+l +r0ha(l=r0) (9{14)wherelisthedepthoftherecess.Thersttermrepresentstheohmicresistancetoadiskelectrodeandthesecondtermistheresistancethroughthecylindricalrecessedregion.ThethirdtermrepresentsacorrectionfactorwhichKelmansolvedusingBesselfunctionsandasymptoticmatching.WestandNewmandeterminedfromKelman'ssolutionsthatha(l=r0)/(l=r0)ln(l=r0)asl=rr0approaches0.Theupperlimitha(l=r0)is0.011andappliesfor(l=r0)greaterthan0.5.Forthepresentsimulations,(l=r0)=3,suchthatha(l=r0)=0:011 9.2ResultsTheeectofanonuniformdistributionofreactivitymaybeobservedinFigure 9-2A forarecessedelectrode.Thesimulatedrealimpedancewasscaledbytheohmicresistanceoftherecessedelectrode.Theaveragecharge{transferresistanceandtheperiodofthedistributionweretreatedtobeparameters.Thelargestperiodsimulated,P=60m,showedthelargestvariationinthelow-frequencyasymptoteoftherealimpedance,usedtodeterminethecharge-transferresistance.However,thevariationwasminusculeandwouldnotgreatlyinuencetheinterpretationoftheimpedance.TheimaginaryimpedancescaledbytheohmicresistanceoftherecessedelectrodeispresentedinFigure 9-2B asafunction 139

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A BFigure9-2. Globalimpedancescaledbytheohmicresistanceofarecessedelectrodewithasquare{wavedistributionofcapacitanceasafunctionoffrequencywiththeperiodandaveragedcharge{transferresistanceasparameters:a)realimpedance;b)imaginaryimpedance. 140

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offrequencyinlog-logscale.ThevariationsintherealimpedanceshowninFigure 9-2A arenotvisibleintheplotformatofFigure 9-2B .Anadjustedphaseanglemaybeexpressedas 'adj=arctanZj Zr)]TJ /F3 11.955 Tf 11.96 0 Td[(Re(9{15)wheretheohmicresistanceissubtractedfromtherealpartoftheimpedance.TheadjustedphaseangleispresentedinFigure 9-3A asafunctionoffrequencyfortheresultspresentedinFigure 9-2 .Theresultsdonotshowsignsoffrequencydispersion.Thelowandhigh{frequencylimitsofthephaseanglewere0and-90respectively,indicatingapurelyRCresponse.Forafaradaicreaction,thedimensionlessfrequencydependsonthecharge{transferresistanceofthereactiontakingplaceanddoesnotinvolvetheohmicresistance.Therefore,thedimensionlessfrequencymaybeexpressedas KRt=Re=!C0hRti(9{16)inwhichthesurface{averagedcharge{transferresistanceisscaledbytheimpedanceassociatedwiththeinterfacialcapacitance.Theadjusted{phaseangleispresentedinFigure 9-3B asafunctionofdimensionlessfrequency.Allofthesimulateddataassociatedwithdierentcombinationsofperiodsandaveragedcharge{transferresistancesaresuperposedandtheinectionpoint,orthepointatwhichthephaseanglewasequalto)]TJ /F1 11.955 Tf 9.3 0 Td[(45occurredatdimensionlessfrequencyKRt=Re=1. 9.3DiscussionThenonuniformreactivityloweredtheglobaleectivecharge-transferresistancewithincreasesintheperiodofthedistribution.Thisbehaviorisnotassociatedwithfrequencydispersionbecause,asshowninFigure 9-3B ,thephaseanglecalculationssuperposedwiththosedeterminedfromanelectrodewithauniformcharge-transferresistance.FrequencydispersionisdependentontheReCtimeconstant.Allgeometriceectsincludingthediskandringgeometryaswellasroughnessinuencetheohmicresistance,Re.The 141

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A BFigure9-3. Imaginary{impedance{derivedphaseangleforarecessedelectrodewiththeperiodandaveragedcharge{transferresistanceasparameters:a)phaseangleasafunctionoffrequency;b)phaseangleasafunctionofdimensionlessfrequencybasedontheaveragedcharge{transferresistance. 142

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capacitancedistributiondirectlyeectsthecapacitance,C.ThereactivitydistributiondoesnotinuencetheimpedancesignicantlybecauseneitherRenorCareaected.Foreachformofsurfaceheterogeneitythereexistsacharacteristicfrequencyatwhichdispersionisinduced.Thecharacteristicfrequencyisinverselyproportionaltothecharacteristiclengthassociatedwiththetypeofsurfaceheterogeneity.Asthecharacteristiclengthdecreases,thefrequencydispersionshiftstohigherfrequencies.Thereactivitydistributiondidnotproduceafrequencydispersionandthereforeacharacteristiclengthisnotapplicable.However,astheperiodofthedistributionincreased,thedierenceintheasymptoticvalueoftherealimpedanceatlowfrequencyincreased.Foradiskelectrodewithinaninsulatingplane,thecharacteristiclengthistheradiusofthedisk.Anapproximatecharacteristiclengthofaringelectrodecanbeexpressedasafunctionofthewidthofthering.Forsurfaceroughness,thecharacteristiclengthistheproductofthesquareoftheroughnessfactorandtheperiodofthesurfaceirregularities.Thecharacteristiclengthwhichdescribesadistributionofcapacitanceistheperiodofthedistribution.Inallofthesecases,thecharacteristiclengthissmall,leadingtofrequencydispersionsthatoccurattheupperlimitoroutsidethemeasurablefrequencyrange.Brugderivedaformulawhichrelatestheparametersofaconstant{phaseelementtoaneectivecapacitanceunderthepremisetheconstant{phaseelementwasduetoasurfacedistributionoftimeconstants.WhiletheBrugformulahasbeenusedtoexplainthecorrelationbetweenCPEparametersandQ,theresultspresentedheresuggestasurfacedistributionofohmicresistance,capacitanceorreactivitywillnotcauseconstant{phaseelementbehavioroverawiderangeoffrequencies. 143

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CHAPTER10INFLUENCEOFNONUNIFORMREACTIONRATESASSOCIATEDWITHREACTIONSCOUPLEDBYANADSORBEDINTERMEDIATEONIMPEDANCETheinuenceofreactionscoupledbyanadsorbedintermediateonimpedancemeasurementsofdiskelectrodeswasexploredtodeterminehowasurfacedistributionofrateconstantsforreactionscoupledbyanadsorbedintermediateinuencetheimpedance.Finiteelementmodelswereusedtosimulatetheimpedanceofdiskandrecesseddiskelectrodeswithandwithoutsurfaceheterogeneityofreactionrates.Resultsfromnite{elementsimulationsondiskandrecesseddiskelectrodesshowthattherearetwocomponentsthatgiverisetofrequencydispersion.Frequencydispersionoccursduetononuniformcurrentdistributionswhichleadstoacomplexohmicimpedance.Frequencydispersionalsooccursduetothepotentialdependenceofthefaradaicimpedance.Simulationresultsshowthatfordiskelectrodesembeddedwithinaninsulatedplane,theinterfacialfrequencydispersionmaybereducedwiththeuseofsmallelectrodes. 10.1MathematicalDevelopmentThesystemunderconsiderationinvolvestworeactionscoupledbyanadsorbedion.Intherststep, M!Xads++e)]TJ /F1 11.955 Tf 166.12 -4.94 Td[((10{1)wherethemetalreactstoformaadsorbediononthesurface,anelectronisreleased.Inthesecondstep, Xads+!P2++e)]TJ /F1 11.955 Tf 161.84 -4.94 Td[((10{2)wheretheadsorbediondesorbs,anotherelectronisreleased.Thetotalfaradaiccurrentmaybeexpressedasthesumofthecurrentsassociatedwithreactions( 10{1 )and( 10{2 ) iF=iM+iX:(10{3) 144

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UndertheassumptionofTafelkinetics,thecurrentassociatedwiththerststepmaybeexpressedas iM=KM(1)]TJ /F3 11.955 Tf 11.95 0 Td[()exp[bM(V)]TJ /F1 11.955 Tf 11.96 0 Td[(0)](10{4)andisdependentontheavailablesurfacethatisnotcoveredbytheadsorbedspecies.Asimilarexpressionmaybeexpressedforthesecondstepas iX=KXexp[bX(V)]TJ /F1 11.955 Tf 11.95 0 Td[(0)](10{5)whereKMandKXaretheeectiverateconstantswithunitsofcurrentdensity,isthefractionalsurfacecoverageoftheadsorbedintermediate,bMandbXareconstantswhichcanbeexpressedintermsoddtransferfunctions,MandX,as b=F RT:(10{6)TheelectrodepotentialisrepresentedbyVand0isthepotentialinsolutionadjacenttotheelectrodesurface.Alloscillatingvariablesmaybeexpressedasasumofasteady-stateandtime-dependentcomponentsas X= X+RefeXexp(j!t)g(10{7)inwhicheXisacomplexvariabledependentonfrequency.Therefore,thefaradaiccurrentmaybeexpressedas iF= iF+RefeiFexp(j!t)g(10{8)inwhicheiFistheoscillatingcomponentofthefaradaiccurrent.Thetotalsteady-statefaradaiccurrentmaybeexpressedas iF=KM(1)]TJ ET q 0.478 w 158.66 -562.49 m 165.38 -562.49 l S Q BT /F3 11.955 Tf 158.66 -569.31 Td[()exp[bM( V)]TJ ET q 0.478 w 234.88 -559.47 m 243.33 -559.47 l S Q BT /F1 11.955 Tf 234.88 -569.31 Td[(0)]+KX exp[bX( V)]TJ ET q 0.478 w 357.34 -559.47 m 365.8 -559.47 l S Q BT /F1 11.955 Tf 357.34 -569.31 Td[(0)](10{9) 145

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andisdependentonthesteady-statesurfacecoverage .Thesteady-statesurfacecoveragemaybederivedbytakingtherateofchangeinsurfacecoverageovertimeexpressedas @ @t= iM)]TJ ET q 0.478 w 247.06 -54.13 m 251.05 -54.13 l S Q BT /F3 11.955 Tf 247.06 -63.64 Td[(iX )]TJ /F3 11.955 Tf 7.31 0 Td[(F=0(10{10)yielding =KMexp[bM( V)]TJ ET q 0.478 w 276.32 -113.57 m 284.77 -113.57 l S Q BT /F1 11.955 Tf 276.32 -123.41 Td[(0)] KMexp[bM( V)]TJ ET q 0.478 w 218.45 -130.9 m 226.9 -130.9 l S Q BT /F1 11.955 Tf 218.45 -140.74 Td[(0)]+KXexp[bX( V)]TJ ET q 0.478 w 334.19 -130.9 m 342.65 -130.9 l S Q BT /F1 11.955 Tf 334.19 -140.74 Td[(0)](10{11)Thecharge-transferresistanceassociatedwitheachreactionstepisinverselyrelatedtothesteadystatecurrentas Rt;M=1 bM iM(10{12)and Rt;X=1 bX iX(10{13)respectively.Thetotalcharge-transferresistanceistheparallelcombinationofthecharge-transferresistancesofeachreactionexpressedas 1 Rt=1 Rt;M+1 Rt;X(10{14)Theoscillatingfaradaiccurrentdensityisderivedbydierentiatingthesteadycurrentwithrespecttopotentialandsurfaceconcentrationas eiF=@ iF @V eV+@ iF @ Ve(10{15)or eiF=1 Rt(eV)]TJ /F6 11.955 Tf 12.57 3.03 Td[(f0)+KXexp[bX( V)]TJ ET q 0.478 w 241.35 -499.7 m 249.8 -499.7 l S Q BT /F1 11.955 Tf 241.35 -509.55 Td[(0)])]TJ /F3 11.955 Tf 11.96 0 Td[(KMexp[bM( V)]TJ ET q 0.478 w 360.11 -499.7 m 368.56 -499.7 l S Q BT /F1 11.955 Tf 360.11 -509.55 Td[(0)]e(10{16)Following )]TJ 8.51 8.09 Td[(d dt=iM F)]TJ /F3 11.955 Tf 13.16 8.09 Td[(iX F(10{17)theoscillatingcomponentofthesurfacecoveragecanbeexpressedas )]TJ /F3 11.955 Tf 7.31 0 Td[(Fj!e=1 Rt;M)]TJ /F1 11.955 Tf 20.92 8.08 Td[(1 Rt;XeV)]TJ /F6 11.955 Tf 11.95 9.68 Td[()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(KXexp)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(bX V+KMexp)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(bM Ve(10{18) 146

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yielding e=0BB@1 Rt;M)]TJ /F1 11.955 Tf 20.92 8.09 Td[(1 Rt;X )]TJ /F3 11.955 Tf 7.31 0 Td[(Fj!+)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(KXexp)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(bX V+KMexp)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(bM V1CCAeV(10{19)Thefaradaicadmittancemaybederivedbytakingtheratiooftheoscillatingfaradaiccurrentandtheelectrodepotential,i.e.,, YF=eiF eV1 Rt+)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(R)]TJ /F5 7.97 Tf 6.59 0 Td[(1t;M)]TJ /F3 11.955 Tf 11.95 0 Td[(R)]TJ /F5 7.97 Tf 6.59 0 Td[(1t;X(KXexp[bX( V)]TJ ET q 0.478 w 256.66 -129.23 m 265.11 -129.23 l S Q BT /F1 11.955 Tf 256.66 -139.07 Td[(0)])]TJ /F3 11.955 Tf 11.95 0 Td[(KMexp[bM( V)]TJ ET q 0.478 w 375.42 -129.23 m 383.88 -129.23 l S Q BT /F1 11.955 Tf 375.42 -139.07 Td[(0)]) )]TJ /F3 11.955 Tf 7.31 0 Td[(Fj!+KXexp[bX( V)]TJ ET q 0.478 w 243.61 -147.93 m 252.07 -147.93 l S Q BT /F1 11.955 Tf 243.61 -157.77 Td[(0)]+KMexp[bM( V)]TJ ET q 0.478 w 362.18 -147.93 m 370.64 -147.93 l S Q BT /F1 11.955 Tf 362.18 -157.77 Td[(0)(10{20)Equation( 10{20 )maybeexpressedastoafaradaicimpedanceexpressedas ZF=1 Rt+A j!+B)]TJ /F5 7.97 Tf 6.58 0 Td[(1(10{21)inwhichvariablesAandBareexpressedas A=)]TJ /F3 11.955 Tf 5.47 -9.68 Td[(R)]TJ /F5 7.97 Tf 6.59 0 Td[(1t;M)]TJ /F3 11.955 Tf 11.95 0 Td[(R)]TJ /F5 7.97 Tf 6.59 0 Td[(1t;X(KXexp[bX( V)]TJ ET q 0.478 w 233.25 -272.68 m 241.71 -272.68 l S Q BT /F1 11.955 Tf 233.25 -282.52 Td[(0)])]TJ /F3 11.955 Tf 11.95 0 Td[(KMexp[bM( V)]TJ ET q 0.478 w 352.01 -272.68 m 360.47 -272.68 l S Q BT /F1 11.955 Tf 352.01 -282.52 Td[(0)]) )]TJ /F3 11.955 Tf 7.31 0 Td[(F(10{22)and B=KXexp[bX( V)]TJ ET q 0.478 w 218.64 -333.82 m 227.09 -333.82 l S Q BT /F1 11.955 Tf 218.64 -343.66 Td[(0)]+KMexp[bM( V)]TJ ET q 0.478 w 337.2 -333.82 m 345.66 -333.82 l S Q BT /F1 11.955 Tf 337.2 -343.66 Td[(0) )]TJ /F3 11.955 Tf 7.32 0 Td[(F(10{23)respectively.Theimpedanceofarecesseddiskelectrodeforthissystemwouldincludeanohmicresistanceinserieswithadoublelayercapacitanceandthefaradaicimpedanceinparallel,asshowninFigure 10-1 .Equation( 10{21 )appearsastheparallelcombinationofRtandanadsorptionimpedanceexpressedas Zads=B+j! A(10{24)Equation( 10{24 )isamathematicalexpressionthatcannotbeexpressedasanequivalentcircuitwithdenedpassiveelementswhenA<0.ThecalculatedimpedanceforarecessedelectrodeisshowninFigure 10-2 forcaseswhereA=0,A<0,andwhenA>0.WhenA=0,thefaradaicimpedancereducestoZF=Rtandtheinterfacialimpedanceonlyexhibitsonetimeconstant.WhenA<0,theinterfacialimpedanceshowstwocapacitivesemicirclesandwhen 147

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Figure10-1. Equivalentcircuitdiagramforarecessedelectrodewithreactionscoupledbyanadsorbedintermediate. Figure10-2. CalculatedimpedancebasedonanequivalentcircuitwiththeohmicresistanceinserieswithadoublelayercapacitorinparallelwiththefaradaicimpedancecalculatedfromEquation( 10{21 ). A>0,theinterfacialimpedancehasahigh-frequencycapacitiveloopandlow-frequencyinductiveloopsuchthatthelow-frequencylimitforthecapacitiveloopisRe+Rt.Thecharacteristicfrequencyofthehigh-frequencyloopisbasedontheRtC0timeconstant.Thecharacteristicfrequencyofthelow-frequencyloopisbasedonthetimeconstant=1=(B+ARt).ForallvaluesofA,thesameexpressionprovidesthelow-frequencylimitfortheimpedance. 148

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Figure10-3. Thedomainfortheniteelementsimulations. 10.2Finite-ElementModelTheimpedancewassimulatedbysolvingLaplace'sequationforthepotentialdistributionintheelectrolytedomain,expressedas r2=0:(10{25)Recallingequation( 10{7 ),thepotentialmaybeexpressedas = +Refeexp(j!t)g:(10{26)Theelectrolytedomaincompriseda2-Daxisymmetricquarterofacircledomain,showninFigure 10-3 .Thecounterelectrodewassetasthecurvedboundarywiththeconditionthat =0forthesteady-statesolutionande=0fortheoscillatingcondition.Theworkingelectrodewasplacedonther-axiscenteredatr=0.Theelectrolytedomainwas2000timeslargerthantheradiusofthedisk.Theboundaryconditionattheworkingelectrodeforthesteady-statecasewasexpressedasequation( 10{9 ).Theboundary 149

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conditionattheworkingelectrodefortheoscillatingcasewasexpressedas ei=(j!C0+R)]TJ /F5 7.97 Tf 6.59 0 Td[(1t;M+R)]TJ /F5 7.97 Tf 6.59 0 Td[(1t;X)(eV)]TJ /F6 11.955 Tf 12.57 3.02 Td[(f0)+KXexp(bX V)e)]TJ /F3 11.955 Tf 11.95 0 Td[(KMexp(bM V)e(10{27)whereemaybeexpressedas e=R)]TJ /F5 7.97 Tf 6.59 0 Td[(1t;M+R)]TJ /F5 7.97 Tf 6.59 0 Td[(1t;X )]TJ /F3 11.955 Tf 7.31 0 Td[(Fj!+(KMexp(bM V)+KXexp(bX V))(eV)]TJ /F6 11.955 Tf 12.57 3.02 Td[(f0)(10{28)Equations 10{25 10{27 ,andtheboundaryconditionatthecounterelectrodeweresolvedforarangeoffrequencies,andtheglobalimpedancewascalculatedastheratiooftheappliedperturbationeVandthetotaloscillatingcurrentatthediskelectrode,ei.Thelocalinterfacialimpedancemaybecalculatedbyprobingtheoscillatingpotentialintheelectrolyteatthedisksurfacee0andmaybeexpressedas z0(r)=eV)]TJ /F6 11.955 Tf 12.57 3.02 Td[(f0(r) ei(r)(10{29)Theglobalinterfacialimpedancemaybefoundbyintegratingthelocalinterfacialimpedanceacrossthesurfaceofthewholedisk.TheglobalohmicimpedancemaybecalculatedasthedierencebetweentheglobalimpedanceandtheglobalinterfacialimpedanceasZe=Z)]TJ /F3 11.955 Tf 11.95 0 Td[(Z0.Thepotentialdistributionforadiskisnonuniformsuchthattheinterfacialimpedance,whichincludespotential-dependentparameters,maynotrepresentthesurface-averagedproperties.Therefore,theglobalinterfacialimpedanceiscomparedinthepresentworktotheimpedancecalculatedwithsurfaceaveragedparameterswherethesurface{averagedinterfacialimpedancemaybeexpressedas hZ0i=j!C0+1 hRti+hAi j!+hBi)]TJ /F5 7.97 Tf 6.59 0 Td[(1(10{30)withtheinterfacialparametersA,B,andRtcalculatedbasedontheirlocalvaluesandaveragedoverthesurfaceofthedisk. 150

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Table10-1. SimulationParameters.SymbolMeaningValueunits bMMF/RT40V)]TJ /F5 7.97 Tf 6.59 0 Td[(1bXXF/RT10V)]TJ /F5 7.97 Tf 6.59 0 Td[(1KMEectivereactionconstantforReaction277.2Acm)]TJ /F5 7.97 Tf 6.59 0 Td[(2KXEectivereactionconstantforReaction30.19Acm)]TJ /F5 7.97 Tf 6.59 0 Td[(2 VSteady-stateelectrodepotential-0.038VeVPerturbationofelectrodepotential10mV)]TJ 59.7 0 Td[(Maximumsurfacecoverageofintermediate210)]TJ /F5 7.97 Tf 6.59 0 Td[(6mol/cm2Electrolyteconductivity0.0485S/cmC0Double-layercapacitance20F/cm2 10.3ResultsFinite-elementsimulationswereusedtosimulatetheimpedanceofdiskandrecesseddiskelectrodeswithreactionscoupledbyanadsorbedintermediate.Thediskgeometryproducesaradialpotentialdependencewhichinuencestheimpedanceresponse.Therecessedelectrodewasusedtoanalyzetheeectofheterogenousreactionratesontheimpedanceresponsewithouttheeectofthediskgeometry.Steady-stateresultsarepresentedtoshowthevariationofsurfacepropertiesontheelectrodes,andtheimpedanceresultsarepresentedtoshowtheeectofthesurfacedistribution.Forthesakeofbrevity,resultsareonlyshownfornegativevaluesofA. 10.3.1DiskElectrodeTheanglebetweenthediskelectrodesurfaceandtheinsulatingplaneis180whichleadstoagreatercurrentdensityandalowerpotentialattheedgesofthediskthanatthecenterofthedisk.Finite-elementsimulationswereperformedtosolveforthesteady-statepotentialdistributionthroughouttheelectrolyteadjacenttothediskelectrode.TheparametersusedinthesimulationsarepresentedinTable 10-1 andwerechosentomaximizefrequencydispersionduetothediskgeometry.Thesurface-averagedcurrentdensityisshowninFigure 10-4 asafunctionofthesurfaceoverpotential.Thecurrentapproachedzeroatmorenegativepotentials.Thesurface-averagedvalueofhAiispresentedinFigure 10-5 asafunctionof 151

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Figure10-4. Steadystatecurrentdensityasafunctionofsurfaceoverpotential. Figure10-5. Surface-averagedAasafunctionofpotentialwithdiskradiusasaparameter. potentialwithelectroderadiusasaparameter.IncreasesinthediskradiuscausedthepointatwhichhAi=0toshifttomoreanodicpotentials.Thesteady-statepotentialattheelectrodesurfaceforadiskelectrodeisshowninFigure 10-6 .Thepotentialattheedgeofthediskissignicantlylowerthanthecenter.Theparametersassociatedwiththeinterfacialimpedancearepotentialdependentexceptforthedoublelayercapacitancewhichwasassumedtobeindependentofpotential.Therefore,theinterfacialparameters,showninFigure 10-7 alsovaryalongthedisksurface.Adecreasesandbecomesmorenegativeattheedgesofthedisk,whileBincreases.Thecharge-transferresistancealsodecreasesattheedgesofthedisk. 152

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Figure10-6. Surfacepotentialasafunctionofradialposition. A B CFigure10-7. Steady-stateinterfacialparametersasafunctionofradialposition:A)A;B)B;C)Rt 153

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Figure10-8. SimulatedimpedanceresponseinNyquistformatofadiskelectrode.Thesolidlinerepresentstheglobalimpedanceresponse.TheDashedlinerepresentsthesurface-averagedinterfacialimpedance.Thedottedlinerepresentstheinterfacialimpedance. Figure10-9. Theimaginarypartoftheohmicimpedancescaledbytheohmicresistanceasafunctionoffrequencywithradiusasaparameter. ThesimulatedglobalimpedanceZscaledbytheohmicresistanceRe=r0 4isshowninFigure 10-8 forhAi=)]TJ /F1 11.955 Tf 9.3 0 Td[(0:288.TheglobalimpedanceiscomparedtothesummationoftheinterfacialimpedanceandtheohmicresistanceZ0+Retoshowthefrequencydispersioncausedbytheohmicimpedance.Thistermiscomparedtothesurface-averagedinterfacialimpedance,equation( 10{30 ),toshowthattheinterfacialimpedancealsoincludesfrequencydispersionduetothepotentialdependenceoftheinterfacialparameters.TheimaginarypartoftheohmicimpedancescaledbytheohmicresistanceisshowninFigure 10-9 asafunctionoffrequencywiththeradiusofthediskasaparameter.The 154

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Figure10-10. TheadsorptionimpedancescaledbytheB=Awithdiskradiusasaparameter.Thedashedlinerepresentstheresultswithouttheeectofdiskgeometry. globalimpedancethatismeasuredonadiskelectrodeincludestheohmicimpedance,whichdoesnothaverelevantphysicalmeaningtotheinterfacialimpedance.Therefore,presenceofanohmicimpedancedistortsthemeasuredimpedanceandcomplicatesinterpretation.Whiletheohmicimpedanceincreaseswithdecreasesinthesizeofthedisk,thehigh-andlow-frequencytimeconstantsarenotsignicantlychanged.Thefrequencydispersionassociatedwiththeinterfacialimpedanceisconnedtothefaradaicpart.Theamountoffrequencydispersionmayberevealedbyanalysisofthecapacitivepartofthefaradaicimpedancereferredtointhispaperastheadsorptionimpedance,equation( 10{24 ).TheabsolutevalueoftheimaginarypartoftheadsorptionimpedancescaledbytheratioB=AispresentedinFigure 10-10 asafunctionoftherealpartoftheadsorptionimpedance,alsoscaledbytheB=A,withtheradiusofthediskasaparameter.Theimaginarypartoftheimpedanceisshowninlog-scaletorevealthevariationinimpedanceatlowfrequencies,showninthelower-left-handcornerofthegure.Thefrequencydispersionoftheadsorptionimpedanceincreasesasthediskradiusincreases. 155

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Figure10-11. Thederivativeofthelogarithmoftheimaginarypartoftheadsorptionimpedancewithrespecttothelogarithmoffrequencyasafunctionoffrequencywiththeradiusasaparameter. ThederivativeofthelogarithmoftheimaginarypartoftheadsorptionimpedancewithrespecttothelogarithmoffrequencyisshownasafunctionoffrequencyinFigure 10-11 withtheradiusofthediskasaparameter.Frequencydispersionisshownforvalueswhichdeviatefromunity.Therefore,astheradiusofthediskelectrodeincreases,theamountoffrequencydispersionassociatedwiththeinterfacialimpedanceincreases.Thetotalamountoffrequencydispersioncontainedwithintheglobalimpedanceisacombinationoftheinterfacialdispersionandthedispersioncausedbytheohmicimpedance.Theohmicimpedanceincreasesastheradiusofthediskdecreasesandtheinterfacialdispersiondecreasesastheradiusofthediskdecreases.Theoverallfrequencydispersionisminimizedbyusingsmalldiskelectrodesbutunlikethecaseforablockingdiskelectrode,thecharacteristicfrequencyassociatedwiththefrequencydispersionisnotdependentonthesizeofthedisk. 10.3.2RecessedElectrodeTherateconstantsassociatedwiththereactionmechanismmayvaryalongthesurfaceofapolycrystallineelectrode.Todeterminetheeectofthisformofsurfaceheterogeneity,arecessedelectrodewasusedtoanalyzetheeectontheimpedanceresponsewithouttheeectofthediskgeometry.Therateconstantsassociatedwiththe 156

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Figure10-12. EectiverateconstantforReaction 10{1 asafunctionofradialposition. Figure10-13. EectiverateconstantforReaction 10{2 asafunctionofradialposition. rstandsecondelementarystepsareshowninFigures 10-12 and 10-13 asafunctionofradialposition.Therateconstantswerevariedaroundameanvaluefollowingasquare-wavedistribution.ThesquarewaveformwascalculatedusingaFourierserieswithaspeciedperiodandamplitude.TheimaginarypartoftheohmicimpedancescaledbytheohmicresistanceisshowninFigure 10-14 withtheperiodoftheheterogeneityasaparameter.Theohmicimpedanceincreasedastheperiodofthedistributionincreased,butthecharacteristicfrequencywasnotchanged.Therefore,thefrequencyatwhichthedispersionoccursmustbedependentonsomeotherparameter. 157

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Figure10-14. Imaginaryohmicimpedancescaledbytheohmicresistanceasafunctionoffrequencywiththeperiodofthedistributionasaparameter. Figure10-15. Thederivativeofthelogarithmoftheimaginarypartoftheadsorptionimpedancewithrespecttothelogarithmoffrequencyasafunctionoffrequencywiththeperiodasaparameter. ThederivativeofthelogarithmoftheimaginarypartoftheadsorptionimpedancewithrespecttothederivativeofthelogarithmoffrequencyisshownasafunctionoffrequencyinFigure 10-15 withtheperiodofthedistributionasaparameter.Thecharacteristicfrequencyassociatedwiththeinterfacialfrequencydispersiondidnotchangewithchangesintheperiodofthedistribution,andtheamountoffrequencydispersionincreasedslightlywithdecreasesintheperiodofthedistribution.Thecharacteristicfrequencyassociatedwiththesurfaceheterogeneitymustbebasedontheinterfacialparameters. 158

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Figure10-16. ThesimulatedimpedanceinNyquistformatscaledbythechargetransferresistanceRtwithsteadystatepotentialasaparameter. Todeterminethevalidityofthisconcept,simulationswereperformedatdierentpotentials,therefore,changingthevaluesoftheinterfacialparameters.ThesimulatedimpedanceisshowninFigure 10-16 ,scaledbythesurface-averagedcharge-transferresistance.Mostofthefrequencydispersionisconnedtothefaradaicimpedancewhichisrepresentedbythelowfrequencycapacitiveloop.At V=0:1V,aninductivefeatureisobservedthatisassociatedwiththevariationofAalongthesurfacebetweenpositiveandnegativevalues.Asthepotentialincreased,thefrequencydispersionbecamemoresignicantandthelow-frequencyloopbroadened,reectingthepresenceoftwodistincttimeconstants.Thereisminimalfrequencydispersionassociatedwiththehighfrequencytimeconstant.ThederivativeofthelogarithmoftheimaginarypartoftheadsorptionimpedancewithrespecttothederivativeofthelogarithmoffrequencyisshownasafunctionoffrequencyinFigure 10-17 with Vasaparameter.Asthepotentialisshiftedintheanodicdirection,thefrequencydispersionoccursathigherfrequenciesandthemagnitudeofthedispersionincreases.ThederivativeofthelogarithmoftheimaginarypartoftheadsorptionimpedancewithrespecttothelogarithmoffrequencyisshowninFigure 10-18 asafunctionofadimensionlessfrequencybasedontheinterfacialparametersA;B;andRtwith Vasaparameter.Thedimensionlessfrequencywasderivedbyscalingtheimpedancebythe 159

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Figure10-17. Thederivativeofthelogarithmoftheimaginarypartoftheadsorptionimpedancewithrespecttothelogarithmoffrequencyasafunctionoffrequencywithsteady-statepotentialasaparameter. Figure10-18. Thederivativeofthelogarithmoftheimaginarypartoftheadsorptionimpedancewithrespecttothelogarithmoffrequencyasafunctionofdimensionlessfrequencywithsteady-statepotentialasaparameter. characteristicfrequencyassociatedwiththefaradaicimpedance.Thisshowsthatthecharacteristicfrequencyassociatedwiththefrequencydispersionisstronglydependentonthecharacteristicfrequencyofthefaradaicimpedance. 10.4DiscussionForsystemswithasingle-stepreaction,surfaceheterogeneityoftherateconstantdidnotinducefrequencydispersion.[ 6 ]Interestingly,thefrequencydispersionassociatedwiththeinterfacialimpedanceforsystemswithreactionscoupledbyanadsorbedintermediate 160

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Figure10-19. Parametersofthefaradaicimpedancedeterminedfromtheimpedancesimulationsscaledbythesurfaceaveragedparametersasafunctionofsteadystatepotential. onlyoccurredatlowfrequencieswherethefaradaicimpedanceisseen.Inordertoreliablyinterprettheimpedanceresultsforsystemswithadsorbedintermediates,thecharacteristicfrequenciesandlimitsoftheimpedanceshouldreectthesurface-averagedinterfacialparameters.Thecharacteristicfrequenciesandthehigh-andlow-frequencylimitsoftheimpedanceweredeterminedusingameasurementmodelinwhichaseriesofVoigtelementswasusedtottheimpedance.Thecharacteristicfrequencyofthehigh-frequencytimeconstantwasusedtocalculateRt.TheadsorptionparametersAandBwerecalculatedfromthelow-frequencylimitoftheimpedanceaswellasthecharacteristicfrequencyofthelow-frequencyloop.EachparameterisshowninFigure 10-19 scaledbythesurfaceaveragevalueasafunctionofsteadystatepotential.Thecharge-transferresistancescaledbythesurfaceaveragevaluedidnotvarymuchfromunityforallofthepotentialssimulated.Atlowpotentials,thevalueofAextractedfromthesimulationswasinagreementwiththesurfaceaveragedvalue.However,atlargerpotentialsthevaluedeterminedfromthesimulateddatawasalmost20percentlessthanthesurfaceaveragedvalue.ThevalueofBfromthesimulatedimpedancecannotbereliablydeterminedfromtheimpedanceastherewasarangeinerrorof20{80percentfromthesurfaceaveragedvalue.Therefore,forthe 161

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mostreliableinterpretation,thecharge-transferresistancemaybedeterminedfromthehigh-frequencytimeconstant.However,thefaradaicimpedanceparametersAandBmustbeestimatedwithcautionfromthelow-frequencytimeconstant. 162

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CHAPTER11CONCLUSIONS 11.1IndirectImpedanceAppliedtoExternalPost-TensionedTendonsIndirectimpedancewasshowntobecapableofmonitoringthecorrosionactivityinpost-tensionedtendons.Throughproofofconceptexperimentsinwhichtendonsectionswerefabricatedwithonesteelstrandandforcedtocorrode,itwasdeterminedthattheindirectimpedanceisqualitativelysensitivetothecorrosionrateofthesteel.Finiteelementmodelswereusedtosimulatetheindirectimpedanceresponseanddeterminethattheresistivityofthegroutcontributestotheoverallimpedanceintwoways.Thereisaseriesohmiccomponentassociatedwiththecurrentthatentersthesteelandthereisalsoaparallelohmiccomponentrepresentativeofthecurrentthatowsparalleltothesteel.Sincethecurrentdistributionchangeswithfrequencyandisnonuniformthroughoutthegrouttheohmiccomponentsmustbeexpressedascomplexvariables.Aniteelementmodelofatendoncontainingonesteelstrandwasusedtogainanunderstandingofthecomponentsthatcontributetotheindirectimpedance.Itwasshownthatthereisaparallelohmicimpedancecomponent,aswellas,aseriesohmicimpedancewhichfullyaccountsforthecontributionofthegroutresistivitytotheimpedance.Theparallelohmicimpedanceismuchlargerthantheseriesohmicimpedanceandisinductivewhiletheseriescomponentiscapacitive.Iftheohmicimpedancesareexpressedasresistors,thepolarizationresistanceofthesteelwillbegreatlyoverestimatedandcorrosionmaygoundetected.Theseriesandparallelohmicimpedanceschangewithsteellocation,resistivityofthegrout,andthepropertiesofthesteelwhichmakesitdiculttoextracttheinterfacialimpedancefromtheindirectimpedancedata.Finiteelementsimulationswerealsoperformedtodeterminehowtheohmicimpedancecomponentschangewithvariationsinelectrodespacingandsteelpolarizationresistance.Theresultsshowedthatwhentheelectrodesareequallyspacedapartatadistancerelativelysimilartothedepthofthesteel,frequencydispersionwas 163

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minimized.Changesinthesteelpolarizationresistancealsoinuencetheohmicimpedancecomponents.Asthepolarizationresistanceincreasesandtheinterfacialimpedanceismorecapacitive,theseriesohmicimpedancebecomesmoreinductive.Theparallelohmicimpedanceisinductiveforallpolarizationresistancesbutincreasesinmagnitudeasthepolarizationresistanceincreases.ExperimentsperformedontendonswithmultiplesteelstrandsincludingatendonextractedfromtheRinglingbridgeaswellastendonsonthemockbridgesectionatTexasA&Mbroughtupsomechallenges.Specically,itwasshownthatcorrosioncouldgoundetectedifitisnotpresentdirectlyundertheworkingorcounterelectrodes.Forexample,ifthereare10steelstrandsandoneofthestrandsinthecenterofthetendoniscorroding,itwouldbeextremelydiculttodetectthisfromtheindirectimpedancemeasurement.Therefore,anyadoptionofindirectimpedanceasasuitablecorrosiondetectiontechniquewouldrequiremultiplemeasurementsatmultiplelocations. 11.2InuenceofSurfaceHeterogeneityinImpedanceMeasurementsThepresenceoffrequencydispersionintheformofaConstantPhaseElementinimpedancemeasurementshasforalongtimebeenattributedtovariationoftheimpedancealongthesurfaceoftheelectrode.Finiteelementsimulationsonadiskelectrodeshowthatfrequencydispersionoccursatfrequenciesgreaterthanacharacteristicfrequencywhichisinverselyrelatedtothecharacteristiclengthofthedisk,r0.Ifthedisksurfaceisroughthecharacteristiclengthbecomesfrr0andthereisalsoacharacteristiclengthassociatedwiththeroughnessitself,whichisf2rP.Thefrequencydispersionassociatedwiththeroughnesswillonlyoccuratlowerfrequenciesthantheinuenceofthediskgeometryif`c;rough>`c;roughdisk.Thiswillonlyoccurforlargeroughnessfactorssuchasinthecaseofporouselectrodes.Therefore,roughnesscannotbeusedasaphysicalexplanationofCPEbehavior.Thetransitionfromaroughelectrodetoaporouselectrodewasshownthroughimpedancesimulationsandthecharacteristiclengthusedtodescriberoughnesswasalsoapplicabletoporouselectrodes. 164

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Brug[ 12 ]attributedtheCPEresponsetoasurfacedistributionofcapacitanceandusedtheparametersoftheCPEtoderiveanexpressionforaneectivecapacitance.Finiteelementsimulationsforadiskelectrodeincorporatingaperiodicvariationofcapacitancealongtheradialaxisshowedthatthereisalsoacharacteristicdimensionassociatedwiththisformofheterogeneitywhichistheperiod,P,ofthedistribution.Therefore,adistributionofcapacitancecannotbeaphysicalexplanationofCPEbehavior.Similarsimulationsperformedwithadistributionofcharge-transferresistanceforasinglestepreactiondidnotyieldfrequencydispersion.However,whenthereactionmechanismwassettoatwo-stepreactioncoupledbyanadsorbedintermediate,frequencydispersionoccurredathighfrequenciesassociatedwiththeRtC0timeconstant,aswellas,atlowfrequenciesassociatedwiththefaradaicimpedance.Thisformofsurfacedistributionwasnotdependentonacharacteristiclengthbutinsteadontheparametersofthefaradaicimpedance.Thefrequencydispersionatlowfrequencyhinderstheabilitytoextractphysicallysignicantparametersfromtheimpedance.However,theuseofsmallelectrodeswillminimizetheinuenceoftheheterogeneity. 165

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CHAPTER12SUGGESTIONSFORFUTUREWORKWhileafullunderstandingofthecomponentswhichcontributetotheindirectimpedancemeasurementhasbeenachieved,ithasnotbeenclearhowthecorrosionrateofthesteelmaybeestimated.Futureworkshouldbeperformedtodevelopwaysofestimating,atleast,thelimitsoftheseriesandparallelohmicimpedancesbasedongrouttosteelvolumeratio,relativepositionofsteelstrands,andthegroutresistivity.Ifthiscanbeachieved,thepolarizationresistanceofthesteelcanbeestimated.ThephysicalmeaningofaCPEthatisnotassociatedwithanormaldistributionoftimeconstantsstilleludesus.Futureworkshouldfocusonunderstandingtheinuenceofcoupledfaradaicandchargingcurrents.AniteelementmodelanalysisoftheinuenceofcoupledchargingandfaradaiccurrentsispresentedbyWuetal.[ 77 ]ontheimpedanceresponseofrotatingdiskelectrodes.TheanalysisrequiredtheuseofasimpliedmodelofthedoublelayercapacitancewhichwasbasedontheGouy-Chapman-Sternmodel.Ion-specicadsorptionwasneglected.Thesimulationsperformedshowedthatthecouplingofchargingandfaradaiccurrentsforarotatingdiskelectrodeinducefrequencydispersioninthehigh-frequencyloopassociatedwiththefaradaicandchargingprocesses.However,therateconstantusedinherworkweretoolarge.Preliminaryniteelementsimulationsforarotatingdiskelectrodewereperformedfortwosystemsincludingtheredoxcoupleferro/ferricyanide Fe(CN)4)]TJ /F5 7.97 Tf -4.23 -9.3 Td[(6Fe(CN)3)]TJ /F5 7.97 Tf -4.24 -9.3 Td[(6+(e))]TJ /F1 11.955 Tf 131.73 -4.93 Td[((12{1)andthesilverredoxcouple, AgAg++(e))]TJ /F1 11.955 Tf 162.92 -4.94 Td[((12{2)respectively.Thesolutionsusedinthesimulationswere0.1MK3Fe(CN)6and0.1MK4Fe(CN)6withasupportingelectrolyteof1MKCl.Forthesilvercase,a0.1MAgNO3withasupportingelectrolyteofKNO3wasconsidered. 166

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Figure12-1. Thesimulatedglobalimpedanceofarotatingdiskelectrodewithasilverredoxcouplewithandwithoutcoupledchargingandfaradaiccurrents.Therateconstantsofka=1:7510)]TJ /F5 7.97 Tf 6.59 0 Td[(9A/cm2andkc=5:71109Acm/molwereused. Figure12-2. Thesimulatedglobalimpedanceofarotatingdiskelectrodewithasilverredoxcouplewithandwithoutcoupledchargingandfaradaiccurrents.Therateconstantsofka=1:7510)]TJ /F5 7.97 Tf 6.59 0 Td[(7A/cm2andkc=5:71107Acm/molwereused. ThesimulatedglobalimpedanceforthesilvercaseisshowninFigure 12-1 whichcomparesthecoupledanduncoupledcasesoffaradaicandchargingcurrentsforka=1:7510)]TJ /F5 7.97 Tf 6.58 0 Td[(9A/cm2andkc=5:71109Acm/mol.Thecouplingofchargingandfaradaiccurrentscauseddepressionofthehighfrequencysemi-circle.Changingtherateconstantstoincreasethesizeofthehigh-frequencyloop,makesthefrequencydispersionmorevisible.Thesimulatedglobalimpedanceforrateconstantska=1:7510)]TJ /F5 7.97 Tf 6.59 0 Td[(7A/cm2andkc=5:71107Acm/molisshowninFigure 12-2 .Thefrequencydispersionwasstillconnedtothehighfrequencyloop.Similarsimulationswereperformedfortheferro/ferricyanidecase.ThesimulatedglobalimpedanceisshowninFigure 12-3 .Inthiscase,thedierencebetweenthe 167

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Figure12-3. Thesimulatedglobalimpedanceofarotatingdiskelectrodewithasilverredoxcouplewithandwithoutcoupledchargingandfaradaiccurrents.Therateconstantsofka=3:4Acm/molandkc=2:64104Acm/molwereused. Figure12-4. Thesimulatedglobalimpedanceofarotatingdiskelectrodewithasilverredoxcouplewithandwithoutcoupledchargingandfaradaiccurrents.Therateconstantsofka=310)]TJ /F5 7.97 Tf 6.58 0 Td[(5Acm/molandkc=0:01Acm/molwereused. coupledandtheuncoupledcasesisnegligible.Withlargerrateconstants,theeectisstillverysmallasshowninFigure 12-4 .Futureworkshouldbedonetoexplorewhycoupledcurrentsinthesilverredoxcouplecaseinducedfrequencydispersionwhileitdidnotfortheferro/ferricyanidecase.Itmaybeusefultocomparethecomponentsofoscillatingcurrentdensityforboththecoupledanduncoupledcases.Forexample,themagnitudeoftheoscillatingcurrentdensityisshowninFigure 12-5 asafunctionofdimensionlessfrequencyforthesilverredoxsystem.Thetotaloscillatingcurrentdensityisshownaswellasthefaradaicandchargingcomponentsforboththecoupledanduncoupledcases.Thefaradaiccurrentforthecoupledcaseismuchlargerthantheuncoupledcase 168

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Figure12-5. Themagnitudeoftheoscillatingcurrentdensityobtainedfromthesilverredoxsimulationsasafunctionofdimensionlessfrequencyforcoupledanduncoupledcases.Alsoshownarethefaradaicandchargingcomponentsofthecurrentdensity. athighfrequencies,whilethechargingcurrentisonlyslightlylargerinthecoupledcasethantheuncoupledcaseatlowfrequencies.Thegureinsetshowsamagniedviewofthetransitionzonewherethetotalcurrentswitchesfromfaradaiccontrolledtochargingcontrolled.Thereisacleardierencewithinthetransitionzonebetweenthecoupledanduncoupledcases.Themagnitudeoftheoscillatingcurrentdensityfortheferro/ferricyanidesimulationisshowninFigure 12-6 asafunctionofdimensionlessfrequency.Inthiscase,despitethedierencesinthecoupledanduncoupledchargingandfaradiccurrentsathighandlowfrequencies,thetotalcurrentofthecoupledcasematchesthetotalcurrentoftheuncoupledcaseatallfrequencies.Amorein-depthanalysisshouldbedonetouncoverthefactorsthatcontributetobothofthesescenarios. 169

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Figure12-6. Themagnitudeoftheoscillatingcurrentdensityobtainedfromthesilverredoxsimulationsasafunctionofdimensionlessfrequencyforcoupledanduncoupledcases.Alsoshownarethefaradaicandchargingcomponentsofthecurrentdensity. 170

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REFERENCES [1] \DurablePost-tensionedConcreteStructures."Tech.Rep.47,TheConcreteSociety,1996. [2] A.A.Sags,R.H.Hoehne,S.C.Kranc.\InitialDevelopmentofMethodsforAssessingConditionofPost-tensionedTendonsofSegmentalBridges."Tech.rep.,UniversityofSouthFlorida,2000. [3] Agarwal,Pankaj,Orazem,M.E.,andGarca-Rubio,L.H.\MeasurementModelsforElectrochemicalImpedanceSpectroscopy:I.DemonstrationofApplicability."JournaloftheElectrochemicalSociety139(1992).7:1917{1927. [4] Alexander,ChristopherL.,Tribollet,Bernard,andOrazem,MarkE.\ContributionofSurfaceDistributionstoConstant-Phase-Element(CPE)Behavior:1.InuenceofRoughness."ElectrochimicaActa173(2015):416{424. [5] |||.\ContributionofSurfaceDistributionstoConstant-Phase-Element(CPE)Behavior:2.Capacitance."ElectrochimicaActa(2015):submitted. [6] |||.\ContributionofSurfaceDistributionstoConstant-Phase-Element(CPE)Behavior:2.Reactivity."ElectrochimicaActa(2015):inpreparation. [7] Andrade,C.andMartinez,I.\MetalCorrosionRateDeterminationofDierentSolutionsandReinforcedConcreteSpecimensbyMeansofaNoncontactingCorrosionMethod."NanostructuredMaterialsandNanotechnologyIi66(2010).5:056001{1{056001{10. [8] Andrade,C.,Martnez,I.,andCastellote,M.\Feasibilityofdeterminingcorrosionratesbymeansofstraycurrent-inducedpolarisation."JournalofAppliedElectro-chemistry38(2008):1467{1476. [9] Andrade,C,Sanchez,J,Martinez,I,andRebolledo,Nuria.\AnalogueCircuitoftheInductivePolarizationResistance."ElectrochimicaActa56(2011).4:1874{1880. [10] Blanc,C.,Orazem,M.E.,Pbre,N.,Tribollet,B.,Vivier,V.,andWu,S.\TheOriginoftheComplexCharacteroftheOhmicImpedance."ElectrochimicaActa55(2010):6313{6321. [11] Borisova,T.andErshler,B.\DeterminationoftheZeroVoltagePointsofSolidMetalsfromMeasurementsoftheCapacityoftheDoubleLayer."ZhurnalFizicheskoiKhimii24(1950):337{344. [12] Brug,G.J.,vandenEeden,A.L.G.,Sluyters-Rehbach,M.,andSluyters,J.H.\TheAnalysisofElectrodeImpedancesComplicatedbythePresenceofaConstantPhaseElement."JournalofElectroanalyticalChemistry176(1984):275{295. [13] Cederquist,SallyCole.\MotorSpeedwayBridgeCollapseCausedbyCorrosion."MaterialsPerformance39(2000).7:18{19. 171

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[14] Chen,Yu-Min,Alexander,ChristopherL,Cleveland,Christopher,andOrazem,MarkE.\InuenceofGeometry-InducedFrequencyDispersionontheImpedanceofRingElectrodes."ElectrochimicaActa(2017). [15] Cordoba-Torres,P.,Mesquita,T.J.,Devos,O.,Tribollet,B.,Roche,V.,andNogueira,R.P.\OntheIntrinsicCouplingbetweenConstant-PhaseElementParametersandQinElectrochemicalImpedanceSpectroscopy."ElectrochimicaActa72(2012):172{178. [16] Cordoba-Torres,Pedro,Mesquita,ThiagoJ.,andNogueira,RicardoP.\RelationshipbetweentheOriginofConstant-PhaseElementBehaviorinElectrochemicalImpedanceSpectroscopyandElectrodeSurfaceStructure."TheJournalofPhys-icalChemistryC119(2015):4136{4147. [17] Corven,J.\Mid-Baybridgepost-tensioningevaluation."Tech.rep.,FloridaDepartmentofTransportation,2001. [18] deLevie,R.\OnPorousElectrodesinElectrolyteSolutions.IV."ElectrochimicaActa9(1964).9:1231{1245. [19] deLevie,Robert.\ElectrochemicalResponsesofPorousandRoughElectrodes."AdvancesinElectrochemistryandElectrochemicalEngineering.ed.PaulDelahay,vol.6.NewYork:Interscience,1967.329{397. [20] Emmanuel,Bosco.\ComputationofACresponsesofArbitraryElectrodeGeometriesfromtheCorrespondingSecondaryCurrentDistributions:AMethodbasedonAnalyticContinuation."JournalofElectroanalyticalChemistry605(2007):89{97. [21] Fong,C.F.ChanMan,Kee,D.De,andKalomi,P.N.AdvancedMathematicsforAppliedandPureSciences.Amsterdam,TheNetherlands:GordonandBreachSciencePublishers,1997. [22] Fricke,H.\TheTheoryofElectrolyticPolarization."PhilosophicalMagazine14(1932):310{318. [23] Ghorbanpoor,A.\Magnetic-BasedNDEofPrestressedandPost-TensionedConcreteMembersTheMFLSystem."Tech.rep.,UniveristyofWisconsin-Milwaukee,2000. [24] Hirschorn,Bryan,Orazem,MarkE.,Tribollet,Bernard,Vivier,Vincent,Frateur,Isabelle,andMusiani,Marco.\Constant-Phase-ElementBehaviorCausedbyResistivityDistributionsinFilms:1.Theory."JournaloftheElectrochemicalSociety157(2010):C452{C457. [25] |||.\Constant-Phase-ElementBehaviorCausedbyResistivityDistributionsinFilms:2.Applications."JournaloftheElectrochemicalSociety157(2010):C458{C463. [26] |||.\DeterminationofEectiveCapacitanceandFilmThicknessfromCPEParameters."ElectrochimicaActa55(2010):6218{6227. 172

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[27] Huang,VickyMei-Wen,Vivier,Vincent,Orazem,MarkE.,Pebere,Nadine,andTribollet,Bernard.\TheApparentCPEBehaviorofaDiskElectrodewithFaradaicReactions."JournaloftheElectrochemicalSociety154(2007):C99{C107. [28] |||.\TheApparentCPEBehaviorofanIdeallyPolarizedDiskElectrode:AGlobalandLocalImpedanceAnalysis."JournaloftheElectrochemicalSociety154(2007):C81{C88. [29] Jansch,Thomas,Wallauer,Jens,andRoling,Bernhard.\InuenceofElectrodeRoughnessonDoubleLayerFormationinIonicLiquids."TheJournalofPhysicalChemistryC119(2015):4620{4626. [30] Jorcin,Jean-Baptiste,Orazem,MarkE.,Pebere,Nadine,andTribollet,Bernard.\CPEAnalysisbyLocalElectrochemicalImpedanceSpectroscopy."ElectrochimicaActa51(2006):1473{1479. [31] Kant,Rama,Kumar,Rajesh,andYadav,VivekK.\TheoryofAnomalousDiusionImpedanceofRealisticFractalElectrode."JournalofPhysicalChemistryC112(2008):4019{4023. [32] Kant,RamaandRangarajan,S.K.\DiusiontoRoughInterfaces:FiniteChargeTransferRates."JournalofElectroanalyticalChemistry396(1995):285{301. [33] Keddam,M.,Novoa,X.R.,andVivier,V.\TheConceptofFloatingElectrodeforContact-LessElectrochemicalMeasurements:ApplicationtoReinforcingSteel-BarCorrosioninConcrete."CorrosionScience51(2009):17951801. [34] Kelman,R.B.\Steady-StateDiusionThroughaFinitePoreintoanInniteReservoir:AnExactSolution."TheBulletinofMathematicalBiophysics27(1965).1:57{65. [35] Kerner,ZsoltandPajkossy,Tamas.\ImpedanceofRoughCapacitiveElectrodes:TheRoleofSurfaceDisorder."JournalofElectroanalyticalChemistry448(1998):139{142. [36] Kumar,RajeshandKant,Rama.\GeneralizedWarburgImpedanceonRealisticSelf-AneFractals:ComparativeStudyofStatisticallyCorrugatedandIsotropicRoughness."JournalofChemicalSciences121(2009):579{588. [37] |||.\TheoryofGeneralizedGerischerAdmittanceofRealisticFractalElectrode."JournalofPhysicalChemistryC113(2009):19558{19567. [38] |||.\AdmittanceofDiusionLimitedAdsorptionCoupledtoReversibleChargeTransferonRoughandFiniteFractalElectrodes."ElectrochimicaActa95(2013):275{287. [39] Kurtyka,BogdananddeLevie,Robert.\Frequencydispersionassociatedwithanon-homogeneousinterfacialcapacitance."JournalofElectroanalyticalChemistry322(1992).1:63{77. 173

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[40] Lasia,Andrzej.\ImpedanceofPorousElectrodes."JournalofElectroanalyticalChemistry397(1995):27{33. [41] Leek,RalphandHampson,NoelA.\Thedispersionofdouble-layercapacitancewithfrequencyI.Smoothsolidelectrodes."SurfaceTechnology7(1978).2:151{155. [42] Lewis,A.\AMoratoriumLifted."Concrete30(1996).6:25{27. [43] Mandelbrot,BenoitB.TheFractalGeometryofNature.Freeman,SanFranscisco,1982. [44] Martin,J.,Broughton,K.J.,Giannopolous,A.,Hardy,M.S.A.,andForde,M.C.\Ultrasonictomographyofgroutedductpost-tensionedreinforcedconcretebridgebeams."NDT&EInternational34(2001):107{103. [45] Mehaute,A.LeandCrepy,G.\IntroductiontoTransferandMotioninFractalMedia:TheGeometryofKinetics."SolidStateIonics9(1983):17{30. [46] MinchinJr,REdward.\IdenticationandDemonstrationofaTechnologyAdaptabletoLocatingWaterinPost-tensionedBridgeTendons."Tech.rep.,2006. [47] Monteiro,PauloJMandMorrison,HF.\Non-destructiveMethodofDeterminingthePositionandConditionofReinforcingSteelinConcrete."1999.USPatent5,855,721. [48] Musiani,M.,Orazem,M.E.,Pbre,N.,Tribollet,B.,andVivier,V.\DeterminationofResistivityProlesinAnti-corrosionCoatingsfromConstant-Phase-ElementParameters."ProgressinOrganicCoatings77(2014):2076{2083. [49] Newman,JohnS.\ResistanceforFlowofCurrenttoaDisk."JournaloftheElectrochemicalSociety113(1966).5:501{502. [50] |||.\FrequencyDispersioninCapacityMeasurementsataDiskElectrode."JournaloftheElectrochemicalSociety117(1970):198{203. [51] Nguyen,AnhSon,Musiani,Marco,Orazem,MarkE.,Pbre,Nadine,Tribollet,Bernard,andVivier,Vincent.\ImpedanceAnalysisoftheDistributedResistivityofCoatingsinDryandWetConditions."ElectrochimicaActa(2015).0:InPress. [52] Nguyen,AnhSon,Musiani,Marco,Orazem,MarkE.,Pebere,Nadine,Tribollet,Bernard,andVivier,Vincent.\ImpedanceAnalysisoftheDistributedResistivityofCoatingsinDryandWetConditions."ElectrochimicaActa(2015):inpress. [53] Nisancioglu,K.\TheoreticalProblemsRelatedtoOhmicResistanceCompensation."TheMeasurementandCorrectionofElectrolyteResistanceinElectrochemicalTests.eds.L.L.ScribnerandS.R.Taylor,STP1056.Philadelphia,PA:AmericanSocietyforTestingandMaterials,1990,61{77. 174

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[54] Nisancioglu,K.,Gartland,P.O.,Dahl,T.,andSander,E.\RoleofSurfaceStructureandFlowRateonthePolarizationofCathodicallyProtectedSteelinSeawater."NanostructuredMaterialsandNanotechnologyIi43(1987):710{718. [55] Nyikos,L.andPajkossy,T.\fractalDimensionandFractionalPowerFrequency-DependentImpedanceOfBlockingElectrodes."ElectrochimicaActa30(1985).11:1533{1540. [56] Orazem,MarkE.,Pebere,Nadine,andTribollet,Bernard.\EnhancedGraphicalRepresentationofElectrochemicalImpedanceData."JournaloftheElectrochemicalSociety153(2006):B129{B136. [57] Orazem,MarkE.andTribollet,Bernard.ElectrochemicalImpedanceSpectroscopy.Hoboken,NJ:JohnWiley&Sons,2008. [58] Orazem,MarkE.,Tribollet,Bernard,Vivier,Vincent,Marcelin,Sabrina,Pebere,Nadine,Bunge,AnnetteL.,White,ErickA.,Riemer,DouglasP.,Frateur,Isabelle,andMusiani,Marco.\InterpretationofDielectricPropertiesforMaterialsshowingConstant-Phase-Element(CPE)ImpedanceResponse."JournaloftheElectrochemicalSociety160(2013):C215{C225. [59] Pajkossy,T.\ImpedanceofRoughCapacitiveElectrodes."JournalofElectroanalyticChemistry364(1994):111{125. [60] Pajkossy,Tamas.\ImpedanceSpectroscopyatInterfacesofMetalsandAqueousSolutions-SurfaceRoughness,CPEandRelatedIssues."SolidStateIonics176(2005):1997{2003. [61] Pajkossy,TamasandNyikos,Lajos.\Impedanceofplanarelectrodeswithscale-invariantcapacitancedistribution."JournalofElectroanalyticalChemistry332(1992).1:55{61. [62] Pajkossy,Tamas,Wandlowski,Thomas,andKolb,DieterM.\ImpedanceAspectsofAnionAdsorptiononGoldSingleCrystalElectrodes."JournalofElectroanalyticalChemistry414(1996):209{220. [63] Permeh,Samanbar,Vigneshwaran,KK,andLau,Kingsley.\CorrosionofPost-TensionedTendonswithDecientGrout."(2016). [64] Proverbio,EandBonaccorsi,LM.\FailureofPrestressingSteelInducedbyCreviceCorrosioninPrestressedConcreteStructures."Proeedingsof9thInternationalConferenceonDurabilityofMaterialsandComponents.2002. [65] Proverbio,G.,E./Ricciardi.\Failureofa40YearsOldPostTensionedBridgenearSeaside."Eurocorr:PastSuccess-FutureChallenges.2000. [66] RodneyG.Powers,YashPaulVirman,AlbertoA.Sags.\CorrosionofPost-tensionedTendonsinFloridaBridges."Tech.rep.,FloridaDepartmentofTransportation,1999. 175

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[67] Singh,MaibamBirlaandKant,Rama.\Debye-FalkenhagenDynamicsofElectricDoubleLayerinPresenceofElectrodeHeterogeneities."JournalofElectroanalyticalChemistry704(2013):197{207. [68] |||.\TheoryofAnomalousDynamicsofElectricDoubleLayeratHeterogeneousandRoughElectrodes."J.Phys.Chem.C.118(2014):5122{5133. [69] Song,H.K.,Hwang,H.Y.,Lee,K.H.,andDao,L.H.\TheEectofPoreSizeDistributionontheFrequencyDispersionofPorousElectrodes."ElectrochimicaActa45(2000):2241{2257. [70] Srivastav,ShrutiandKant,Rama.\Anomalouswarburgimpedance:Inuenceofuncompensatedsolutionresistance."JournalofPhysicalChemistryC115(2011):12232{12242. [71] Trassati,S.andParsons,R.\InterphasesinSystemsofConductingPhases."PureandAppliedChemistry58(1986):437{454. [72] Virmani,PaulY.\LiteratureReviewofChlorideThresholdValuesforGroutedPost-TensionedTendons."Tech.rep.,FederalHighwayAdministration,2012. [73] West,AlanC.andNewman,JohnS.\CurrentDistributionsonRecessedElectrodes."JournaloftheElectrochemicalSociety138(1991).6:1620{1625. [74] White,ErickA.,Orazem,MarkE.,andBunge,AnnetteL.\CharacterizationofDamagedSkinbyImpedanceSpectroscopy:ChemicalDamagebyDimethylSulfoxide."PharmaceuticalResearch30(2013):2607{2624. [75] WiliamH.Hartt,SivaVenugopalan.\CorrosionEvaluationofPost-TensionedTendosnontheMidBayBridgeinDestin,Florida."FDOT1(2002):995{1006. [76] Woodward,RJandWilliams,FW.\CollapseofYns-Y-gwasBridge,."ProceedingsoftheInstitutionofCivilEngineers84(1988).4:635{669. [77] Wu,Shao-Ling,Orazem,MarkE.,Tribollet,Bernard,andVivier,Vincent.\TheInuenceofCoupledFaradaicandChargingCurrentsonImpedanceSpectroscopy."ElectrochimicaActa131(2014):3{12. 176

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BIOGRAPHICALSKETCHChristopherAlexandergraduatedfromtheUniversityofSouthFloridawithaBSinCivilEngineeringin2010.Duringhistimethereheledagroupofundergraduateresearchers,workeddirectlywiththepresidentasanocialstudentrepresentative,andtutoredstudentsinmath.Heobtainedamaster'sdegreeinCivilEngineeringfromtheUniversityofFloridain2013andaPhDinChemicalEngineeringunderMarkOrazem,aDistinguishedProfessorandexpertonelectrochemicalengineeringandimpedancespectroscopy.Hisresearchwasprimarilyfocusedondevelopingadevicethatcannon-destructivelydetectcorrosioninbridgetendonsusingElectrochemicalImpedanceSpectroscopy.Heworkedonseveralsideprojectsincludingassessingtheroleofsurfaceheterogeneityonimpedancespectroscopy,simulatingtheimpedanceresponseofacontinuousglucosemonitorfordiabetespatients,anddeterminingtheinuenceofcoupledchargingandfaradiccurrentonimpedancespectroscopy.HisworkontheinuenceofsurfaceheterogeneitywasrecognizedasanovelcontributiontotheeldandwaspublishedinaspecialissueofElectrochmicaActa. 177