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APPLICATIONOFMEASUREMENTMODELS TO IMPEDANCESPECTROSCOPY By PANKAJAGARWAL ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 1994 Copyright1994 by PankajAgarwal ACKNOWLEDGMENTS TheauthorthanksProfessorMarkE. Orazem,ProfessorLuisH. Garcia-Rubio andProfessorOscarD.Crisallefortheirguidanceandsupport;Dr.ClaudeDeslouisand Dr. Bernard Tribollet ofCNRS, Paris, France fortheirguidance and support inthe electro-hydrodynamicalimpedancework;ProfessorJ.RossMacdonaldoftheUniversity ofNorthCarolina,ChapelHillforhisguidanceintheinitialregressionscarriedoutinthis work;researchgroupmembersAndrewN.Jansen,OliverC.MoghissiandPaulT.Wojcik for providing some ofthe experimental dataused in this work; and research group membersStevenL.CarsonandJ.MatthewEstebanfortheircontributions. ThisworkwassupportedinpartbyGatesEnergyProductsInc.,OfficeofNaval ResearchandtheNationalScienceFoundationandtheElectrochemicalSociety. iii TABLEOFCONTENTS ACKNOWLEDGMENTS iii LISTOFTABLES vii LISTOFFIGURES viii ABSTRACT xviii CHAPTERS INTRODUCTION 1 1 1.1AFrameworkforModelIdentification 2 1.1.1OptimalDesignofExperiments 3 1.1.2CharacterizationofMeasurements 4 1.1.3ModelIdentification 6 1.2NewDirectionsforStatisticalAnalysis 6 2IMPEDANCESPECTROSCOPY:EXPERIMENTALDESIGN 9 2.1TheoryofImpedanceMeasurement 9 2.1.1Single-SineTechnique 10 2.2Instrumentation 13 2.2.1TheElectrochemicalCell 14 2.2.2Potentiostat 15 2.2.3FrequencyResponseAnalyzer 16 2.2.4ComputerInterface 16 2.3 Software 17 2.3.1MenuOptions 18 2.3.2TypicalExperimentalResults 26 2.4Conclusions 27 3MEASUREMENTMODELS:DEMONSTRATIONOFAPPLICABILITY 41 3.1Introduction 41 3.2MeasurementModels 42 3.3ApproachforDemonstrationofApplicability 44 IV 3.4ResultsandDiscussion 46 3.4.1SequentialConstructionofaModel:Circuit1 46 3.4.2ApplicationtoDiffusionSystems:Circuit2 49 3.4.3ApplicationtoInductiveSystems:Circuit3 50 3.4.4ApplicationtoConstantPhaseElements:Circuit4 51 3.4.5ApplicationtoCoatedMetals:Circuit5 51 3.4.6InterpretationofParameterValues 52 3.5Conclusions 52 4MEASUREMENTMODELS:DETERMINATIONOFERRORSTRUCTURE 71 4.1ClassificationofMeasurementErrors 71 4.2DeterminationofStochasticErrors 73 4.2.1StationarySystems 73 4.2.2Non-StationarySystems 74 4.2.3FilteringTechnique 76 4.3ModelfortheErrorStructure 77 4.4ApplicationsoftheErrorStructure 81 4.5Conclusions 83 5MEASUREMENTMODELS:CONSISTENCYWITHKRAMERS-KRONIG RELATIONS 100 5.1TheKramers-KronigRelations 101 5.2ReviewofMethodsforDeterminingConsistency 102 5.2.1DirectIntegrationofKramers-KronigRelation 103 5.2.2ExperimentalChecksforConsistency 107 5.2.3RegressionofCircuitAnalogues 108 5.3MethodBasedonMeasurementModel 108 5.3.1Algorithm 109 5.3.2Monte-CarloSimulation 114 5.3.3ApplicationtoExperimentalData 116 5.4Conclusion 119 6METALHYDRIDES:ELECTROCHEMICALIMPEDANCEEXPERIMENTS ANDANALYSIS 130 6.1Background 130 6.2ElectrochemicalImpedanceExperiments 134 6.2.1LaNis 136 6.2.2MmNij 137 6.3Analysis 137 6.3.1MeasurementModel 138 6.3.2ProcessModel 141 6.4Conclusions 148 v 1 7MEASUREMENTMODELS:APPLICATIONTOELECTRO- HYDRODYNAMICALIMPEDANCESPECTROSCOPY 193 7.1PrinciplesofEHD 193 7.2EHDExperiments 196 7.3MeasurementModelAnalysis 197 7.4ProcessModelforEHD 198 7.5RegressionofProcessModeltoData 199 7.6Conclusions 201 8CONCLUSIONS 216 9SUGGESTIONSFORFUTUREWORK 218 APPENDIXA 220 FORTRANCODEFORMEASUREMENTMODEL 220 APPENDIXB 244 FORTRANCODEFORMONTE-CARLOSIMULATION 244 APPENDIXC 251 FORTRANCODEFORPROCESSMODEL 25 REFERENCES 265 BIOGRAPHICALSKETCH 271 vi LISTOFTABLES Tables 2.1:ThemeasuringresistorsavailableonSI1286[18] 28 2.2:ThemeasuringresistorsavailableonPAR273A[19] 28 3.1:Valuesforcircuitparameters 54 3.2:TheresultsoftheRegressionoftheVoigtMeasurementmodel(Fig.3.1)tothe circuitsshowninFig3.2 55 4.1:ParameterEstimatesfortheErrorStructureModel(Equation4.14)Obtainedby RegressiontoreplicateDataforn-GaAs 84 7.1:SummaryoftheexperimentalconditionsusedtocollectEHDdata 202 7.2:ExperimentalconditionsusedtocollectdatashowninFigure7.2 203 Vll 1 LISTOFFIGURES Figures 1.1:Flowdiagramforthemeasuementandcharacterizationofamaterial-electrodesystem [7](reproducedwithpermissionfromtheauthor) 7 1.2:Flowdiagramforimpedancespectroscopyproposedinthiswork 8 2.1:Small-signalanalysisofanelectrochemicalnon-linearsystem[4](reproducedwith permissionfromtheauthor) 29 2.2:Principleofcorrelationanalysis[14](reproducedwithpermissionfromthe author) 30 2.3:a)2-terminalelectrochemicalcell,b)3-terminalelectrochemicalcell,c)4-terminal electrochemicalcell[11](reproducedwithpermissionfromtheauthor) 3 2.4:Schematicoftheexperimentalsetup 32 2.5:Controlfrontpaneloftheimpedancesoftware(PotentiostaticMode) 33 2.6:Controlfrontpaneloftheimpedancesoftware(GalvanostaticMode) 34 2.7:DatacollectionfrontpanelforFigure2.5 35 2.8:DatacollectionfrontpanelforFigure2.6 36 2.9:TypicaldatafileforanexperimentwithsettingsfromFigure2.5 37 2.10:TypicaldatafileforanexperimentwithsettingsfromFigure2.6 38 2.11:ThesolidlinerepresentstheabsolutevaluefortheexperimentshowninFigure2.5. Thedashedlinerepresentsthemeasuringresistor 39 2.12:a)ThesolidlinerepresentstheabsolutevaluefortheexperimentshowninFigure 2.6.Thedashedlinerepresentsthemeasuringresistor,b)Thecirclesrepresentthe perturbationcurrentamplitudeasafunctionoffrequency 40 3.1:TheVoigtelectricalcircuitanaloguecorrespondingtothemeasurementmodelused inthiswork(equation(3.1)) 56 vm 3.2:Electricalcircuitanaloguesfor:1)hydrogenevolutiononLaNi5[24];2)corrosionof icraornboinn0st.e5elMiHn23SwOt4%atNathCelcsoorlruotsiioonnnpeoatrentthiealco[r3r0]o;si4o)ncpoortreonstiiaoln[o2f9]a;m3o)dceolrrpiotsionof electrodein0.5MNaCl[31];and5)corrosionofapaintedmetal[32] 57 3.3:RegressionofameasurementmodelwithoneVoigtelementtosyntheticdata(o) obtainedfromCircuit1(hydrogenevolutiononLaNis[24]).Solidlinesinallfigures aretheresultsofthemodelfit. Theresidualsumofsquaresforthefitwas37.15...58 3.4:RegressionofameasurementmodelwithtwoVoigtelementstosyntheticdata(o) obtainedfromCircuit1(hydrogenevolutiononLaNi5[24]).Theresidualsumof squaresforthefitwas0.5588.Solidlinesinallfiguresaretheresultsofthemodelfit. Thelinespresentedinthelowertwofiguresarethemodelfitand itsdeconvolution intothecontributionofeachVoigtelement 59 3.5:RegressionofameasurementmodelwiththreeVoigtelementstosyntheticdata(o) obtainedfromCircuit1(hydrogenevolutiononLaNi5[24]).Theresidualsumof squaresforthefitwas2.112x10'4.Solidlinesinallfiguresaretheresultsofthe modelfit.Thelinespresentedinthelowertwofiguresarethemodelfitandits deconvolutionintothecontributionofeachVoigtelement 60 3.6:RegressionofameasurementmodelwithfourVoigtelementstosyntheticdata(o) obtainedfromCircuit1(hydrogenevolutiononLaNis[24]).Theresidualsumof squaresforthefitwas1.772xlO'7.Solidlinesinallfiguresaretheresultsofthe modelfit.Thelinespresentedinthelowertwofiguresarethemodelfitandits deconvolutionintothecontributionofeachVoigtelement 61 3.7:RegressionofameasurementmodelwithfiveVoigtelementstosyntheticdata(o) obtainedfromCircuit1(hydrogenevolutiononLaNi5[24]).Theresidualsumof squaresforthefitwas4.915x10'10.Solidlinesinallfiguresaretheresultsofthe model 62 3.8:Normalizedsumofsquaresfortheregressionofameasurementmodeltosynthetic dataobtainedfromCircuit1(hydrogenevolutiononLaNi5[24])asafunctionofthe numberofVoigttimeconstantsemployedinthemodel:(o)regressiontosynthetic data;(A)regressiontosyntheticdatawithrandomnoiseadded;and(dashedline) normalizednoiselevel 63 3.9:Residualerrors(o)asafunctionoffrequencyfortheregressionofameasurement modelwithfiveVoigtelementstosyntheticdataobtainedfromCircuit1(hydrogen evolutiononLaNis[24]) 64 3.10:RegressionofameasurementmodelwiththreeVoigtelementstosyntheticdata(o) obtainedfromCircuit2withaddedrandomnoise(5%amplitude). Theresidualsum IX ofsquaresforthefitwas2.424xl0'2.Solidlinesinallfiguresaretheresultsofthe modelfit.Thelinespresentedinthelowertwofiguresarethemodelfitandits deconvolutionintothecontributionofeachVoigtelement 65 3.11:Residualerrorsasafunctionoffrequencyfortheregressionofameasurement modelwiththreeVoigtelementstosyntheticdata(o)obtainedfromCircuit2with addedrandomnoise(5%maximumamplitude) 66 3.12:RegressionofameasurementmodelwithfourVoigtelementstosyntheticdata(o) obtainedfromCircuit2(corrosionofcarbonsteelin3wt%NaClsolutionnearthe corrosionpotential[29]).Theresidualsumofsquaresforthefitwas3.763xlO'6 Solidlinesinallfiguresaretheresultsofthemodelfit.Thelinespresentedinthe lowertwofiguresarethemodelfitanditsdeconvolutionintothecontributionof eachVoigtelement 67 3.13:RegressionofameasurementmodelwiththreeVoigtelementstosyntheticdata(o) obtainedfromCircuit3(corrosionofironin0.5MH2SO4atthecorrosionpotential [30]).Theresidualsumofsquaresforthefitwas3.221xl0'14.Solidlinesinall figuresaretheresultsofthemodelfit.Thelinespresentedinthelowertwofigures arethemodelfitanditsdeconvolutionintothecontributionofeach Voigtelement 68 3.14:RegressionofameasurementmodelwitheightVoigtelementstosyntheticdata(o) obtainedfromCircuit4(corrosionofamodelpitelectrodein0.5MNaCl[30]).The residualsumofsquaresforthefitwas1.491x10'2.Solidlinesinallfiguresarethe resultsofthemodelfit.Thelinespresentedinthelowertwofiguresarethemodelfit anditsdeconvolutionintothecontributionofeachVoigtelement 69 3.15:RegressionofameasurementmodelwithtwoVoigtelementstosyntheticdata(o) obtainedfromCircuit5(corrosionofapaintedmetal[32]).Theresidualsumof squaresforthefitwas2.055x1O'14.Solidlinesinallfiguresaretheresultsofthe modelfit.Thelinespresentedinthelowertwofiguresarethemodelfitandits deconvolution 70 4.1:Fivereplicatemeasurementsforann-GaAssinglecrystalheldat320K 85 4.2:Unfilteredstandarddeviation forGaAs singlecrystalat320K. Circlesrepresentthe realandthetrianglesrepresenttheimaginarypartofthestandarddeviation 86 4.3:SixreplicateImpedancemeasurementsforCopperinalkalinesolution 87 4.4:Realresidualerrorsfortheregressionofasinglemeasurementmodeltothedata showninFigure4.3 88 x