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Mathematical Models for Impedance Spectroscopy

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Title:
Mathematical Models for Impedance Spectroscopy
Creator:
Harding, Morgan S
Publisher:
University of Florida
Publication Date:
Language:
English

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Chemical Engineering
Committee Chair:
ORAZEM,MARK E
Committee Co-Chair:
HAGELIN WEAVER,HELENA ELISABETH
Committee Members:
ZIEGLER,KIRK JEREMY
PATRICK,ERIN E
VIVIER,VINCENT

Subjects

Subjects / Keywords:
eis
mathematical-models

Notes

General Note:
Electrochemical impedance spectroscopy is a popular transient technique used to characterize electrochemical systems. Mathematical models for impedance spectroscopy help gain insight into impedance results for a variety of electrochemical systems. Convection and diffusion play a role in the impedance results of a reacting electrochemical system. A finite Schmidt number analysis for convective-diffusion impedance on both a rotating disk and a submerged impinging jet electrode is presented. A mathematical model is developed for the impedance response associated with the coupled homogeneous chemical and heterogeneous electrochemical reactions. The model includes a homogeneous reaction in the electrolyte where species AB reacts reversibly to form AB and B+ and B+ reacts electrochemically on a rotating disk electrode to produce B. An analytic expression for velocity was employed that combined a three-term velocity expansion near the electrode surface to a three-term expansion that applied far from the electrode. The resulting convective diffusion impedance has two asymmetric capacitive loops, one associated with convective diffusion impedance the other with the homogeneous reaction. For an infinitely fast homogeneous reaction, the system is shown to behave as though AB is the electroactive species. A modified Gerischer impedance was found to provide a good fit to the simulated data. A mathematical model was also developed for the impedance response of a glucose oxidase enzyme-based electrochemical biosensor. The model accounts for a glucose limiting membrane GLM, which controls the amount of glucose participating in the enzymatic reaction. The glucose oxidase was assumed to be immobilized within a thin film adjacent to the electrode. In the glucose oxidase layer, a process of enzymatic catalysis transforms the glucose into peroxide, which can be detected electrochemically. This system may be considered to be a special case of the coupled homogeneous and heterogeneous reactions addressed by Levich. The influence of coupled faradaic and charging currents on impedance spectroscopy was simulated and analyzed by Wu et al. Preliminary experimental results do not perfectly agree with simulations. Further analysis of the experimental results will be conducted along with more finite-element three-dimensional simulations and a finite-difference one-dimensional simulation to further characterize and understand this phenomenon.

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UFRGP
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All applicable rights reserved by the source institution and holding location.
Embargo Date:
5/31/2018

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MATHEMATICALMODELSFORIMPEDANCESPECTROSCOPYByMORGANS.HARDINGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2017

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c2017MorganS.Harding

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Tomyparents,KC,IonandEric

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ACKNOWLEDGMENTSIthankmyadvisor,ProfessorMarkOrazem,forallhissupportandguidancethroughoutmyPh.D.Iwouldnothavemadeitthroughsuchamentallydemandingandintellectuallychallengingprogramwithouthisconstantmentorship.Iwouldalsoliketothankmyfamily.TheyencouragedmetotakeSTEMcoursesthroughoutmyeducation,whichisthereasonIamwhereIamtoday.Iwouldliketothankmysister,KC,forhersupportandlightcompetitionthroughoutourentirelives.Iwouldliketothankmyfriendsfromhighschool,undergrad,andgradschool.IwouldliketothankEricforbeingmypartner,mybestfriend,myrock,andmybiggestsupporter.IwouldliketothankIon,mydog,formakingmylifebetter.IwouldliketothankDr.VincentVivierforguidingmeinmyworkinhislabinthesummerof2016inParis,France.Heisanextraordinaryexperimentalistandhasagreatsenseofhumor.IwouldliketothankDr.BernardTribolletforhismentorshipandhelp.Iwouldliketothankmylabmatesforalltheircompanionshipandhelp. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 LISTOFCODES ...................................... 14 LISTOFSYMBOLS .................................... 14 ABSTRACT ........................................ 18 CHAPTER 1INTRODUCTION .................................. 20 2BACKGROUND ................................... 24 2.1ElectrochemicalImpedanceSpectroscopy ................... 24 2.1.1ModelingElectrochemicalImpedanceSpectroscopy ......... 27 2.1.2RepresentationofElectrochemicalImpedanceSpectroscopy ..... 28 2.1.3CharacteristicFrequencyinElectrochemicalImpedanceSpectroscopy 30 2.2NumericalSimulationsforElectrochemicalImpedanceSpectroscopy .... 31 2.2.1Finite-DierenceMethods ....................... 31 2.2.2ConvergenceMethods .......................... 31 2.3ImpedanceModelsforRotatingDiskElectrode ............... 32 2.3.1FluidowforaRotatingDisk ..................... 32 2.3.2ConvectiveDiusionImpedanceModels ................ 37 2.3.3ImpedanceModelswithHomogeneousReactions ........... 37 2.4ImpedanceModelsforContinuousGlucoseMonitors ............. 40 3INFLUENCEOFFINITESCHMIDTNUMBERONTHEIMPEDANCEOFDISKELECTRODES ................................ 43 3.1ElectrochemicalMathematicalEquations ................... 43 3.2RotatingDiskElectrode ............................ 44 3.2.1FluidVelocityProleforRotatingDiskElectrode .......... 45 3.2.2MathematicalDevelopmentforRotatingDiskElectrode ....... 45 3.2.3NumericalMethodsforRotatingDiskElectrode ........... 49 3.2.4Convective-DiusionImpedanceforaRotatingDiskElectrode ... 49 3.3SubmergedImpingingJetElectrode ...................... 54 3.3.1VelocityExpansionforaSubmergedImpingingJet ......... 54 3.3.2MathematicalDevelopmentforImpingingJetElectrode ....... 55 3.3.3NumericalMethodsforImpingingJetElectrode ........... 56 5

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3.3.4Convective-DiusionImpedanceforaSubmergedImpingingJetElectrode ................................. 56 4CONVECTIVE-DIFFUSIONIMPEDANCEWITHHOMOGENEOUSCHEMICALREACTIONS ..................................... 62 4.1MathematicalDevelopmentforConvectiveDiusionandHomogeneousReaction ..................................... 62 4.1.1GoverningEquations .......................... 62 4.1.2HomogeneousReaction ......................... 63 4.1.3VelocityExpression ........................... 64 4.1.4ImpedancewithHomogeneousChemicalReactions .......... 65 4.1.4.1Diusionimpedance ..................... 67 4.1.4.2Overallimpedance ...................... 68 4.1.5NumericalMethods ........................... 69 4.1.5.1Accuracyofnumericalmethods ............... 70 4.1.5.2Couplingdomainswithdierentmeshsize ......... 71 4.2ImpedanceforConvective-DiusionandHomogeneousReaction ...... 72 4.3DiscussionforConvectiveDiusionandHomogeneousReaction ...... 80 4.3.1FastHomogeneousReaction ...................... 80 4.3.2ModiedGerischerImpedance ..................... 81 4.3.2.1Mathematicaldescription .................. 81 4.3.2.2ModiedGerischerimpedance ................ 83 5IMPEDANCERESPONSEFORCONTINUOUSGLUCOSEMONITORS ... 87 5.1MathematicalDevelopmentfortheContinuousGlucoseMonitor ...... 87 5.1.1GoverningEquationsforCGMSystem ................ 89 5.1.2HomogeneousEnzymaticReactionsforCGMSystem ........ 90 5.1.2.1DiusionimpedanceforCGMsystem ............ 97 5.1.2.2OverallimpedanceforCGMsystem ............. 98 5.1.3NumericalMethodsforCGM ...................... 99 5.2CGMResultsandDiscussion ......................... 101 5.2.1HomogeneousReactionRateInuenceontheCGM ......... 102 5.2.2InuenceofOxygenConcentrationontheCGM ........... 108 6CONCLUSIONS ................................... 116 7FUTUREWORK ................................... 119 7.1FutureInvestigationofContinuousGlucoseMonitorCode ......... 119 7.1.1CGMParameterStudy ......................... 119 7.1.2OverallImpedanceAnalysis ...................... 121 7.2InuenceofCoupledFaradaicandChargingCurrentsonEIS ........ 122 7.2.1HistoryofCoupledChargingandFaradaicCurrents ......... 122 7.2.2Constant-PhaseElements ........................ 124 7.2.3ElectrochemicalInstrumentation .................... 125 6

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7.2.4CVCurvesandEISExperimentalResults .............. 125 APPENDIX ABAND ......................................... 131 BCODESFORROTATINGDISKELECTRODE .................. 141 CCODESFORIMPINGINGJETELECTRODE .................. 151 DCODESFORCONVECTIVEDIFFUSIONIMPEDANCEWITHHOMOGENEOUSREACTION ...................................... 161 D.1InputforConvectiveDiusionImpedancewithHomogeneousReactionsCode ....................................... 161 D.2Steady-StateConvectiveDiusionImpedancewithHomogeneousReactionsCode ....................................... 163 D.3OscillatingConvectiveDiusionImpedancewithHomogeneousReactionsCode ....................................... 185 ECODESFORCONTINUOUSGLUCOSEMONITOR .............. 203 E.1InputlesfortheContinuousGlucoseMonitor ................ 203 E.2Steady-StateContinuousGlucoseMonitorCode ............... 205 E.3OscillatingContinuousGlucoseMonitorCode ................ 236 REFERENCES ....................................... 273 BIOGRAPHICALSKETCH ................................ 281 7

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LISTOFTABLES Table page 3-1Errorinconvectivediusionimpedanceforarotatingdiskelectrode ....... 53 3-2Errorinconvectivediusionimpedanceforaimpingingjetelectrode ...... 58 4-1Speciesandassociatedparametervaluesforthesystem .............. 73 4-2Systemandkineticparametervaluesforthesystem ................ 73 4-3Fittingparametersfoundfromregression ...................... 85 5-1Speciesandassociatedparametervaluesforthesystem .............. 91 5-2Systemparametervaluesforthecontinuousglucosemonitorsimulations .... 101 5-3Kineticparametervaluesforsystem1 ........................ 102 5-4Kineticparametervaluesforsystem2 ........................ 102 5-5Kineticparametervaluesforsystem3 ........................ 102 7-1Valuesforkineticparameterstudy ......................... 121 8

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LISTOFFIGURES Figure page 1-1NumberofarticlesmentioningElectrochemicalImpedanceSpectroscopy .... 21 2-1Sinusoidalperturbationofanelectrochemicalsystem ............... 25 2-2Schematicrepresentationofthecalculationofatransferfunction ........ 25 2-3Reactingcircuitusedtomodelimpedance ..................... 27 2-4Boderepresentationofimpedancedata ....................... 28 2-5Representationofimpedancedataasafunctionoffrequency ........... 29 2-6Nyquistplotofimpedancedata ........................... 29 2-7Velocityexpansionforuidowofarotatingdiskelectrode ........... 35 2-8Velocityexpansionsandinterpolationvelocityforarotatingdiskelectrode ... 36 3-1Rotatingdiskelectrodeow ............................. 44 3-2Oscillatingdimensionlessconcentrationsforarotatingdiskelectrode ...... 50 3-3ContributionstotheimpedanceforniteSchmidtnumber ............ 52 3-4Dimensionlessdiusionimpedanceforarotatingdiskelectrode .......... 53 3-5Submergedimpingingjetow ............................ 54 3-6Oscillatingdimensionlessconcentrationsforasubmergedimpingingjetelectrode 57 3-7ContributionstotheimpedanceforniteSchmidtnumber ............ 59 3-8Dimensionlessdiusionimpedanceforasubmergedimpingingjetelectrode ... 61 4-1Adiagramofanelectrochemicalreactioncoupledbytheinuenceofachemicalreaction ........................................ 63 4-2Electricalcircuitrepresentationoftheoverallelectrodeimpedance ........ 69 4-3One-dimensionalschematicrepresentationshowingtwodissimilarmeshsizes .. 70 4-4GraphicalevidenceofH2accuracy ......................... 71 4-5Polarizationcurvecalculatedforsystemparameters ................ 72 4-6Calculatedsteady-stateconcentrationdistributionsandhomogeneousreactioncorrespondingtodierentfractionsofthelimitingcurrent ............ 74 9

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4-7Polarizationcurvesandsteady-stateconcentrationofSpeciesB+withhomogeneousreactionrateasaparamater ............................. 75 4-8Dimensionlessconvective-diusionimpedance ................... 75 4-9Thedimensionlessconvective-diusionimpedancefordierentfractionsofthelimitingcurrent .................................... 76 4-10Theabsolutevalueoftheimaginarypartofthedimensionlessconvective-diusionimpedance ....................................... 77 4-11Theoverallimpedancecorrespondingtofractionsofthelimitingcurrent .... 78 4-12Theoverallimpedanceformass-transfer-limitedcurrentandzerocurrentwithanexpandedfrequencyrange ............................ 79 4-13Dimensionlessconvective-diusionimpedancewheretheoscillatingconcentrationofABweretakenintoaccount ............................ 81 4-14Simulateddimensionlessconvective-diusionimpedanceandregressiontting 84 4-15Reaction-layerthicknessobtainedfromregressionparameters ........... 85 5-1Glucosesensorinsertedundertheskin. ....................... 88 5-2LayersandoverallreactioninGlucoseSensor. ................... 88 5-3Representationoftheglucoseoxidasereaction. ................... 89 5-4CircuitdiagramofhowtheoverallimpedancewillbemodeledfortheCGM .. 98 5-5OnedimensionalschematicshowingthreedissimilarmeshsizesfortheCGM .. 100 5-6Polarizationcurvecalculatedfromdierenthomogeneousreactionrates ..... 103 5-7Calculatedsteady-stateconcentrationdistributionsfordierenthomogeneousreactionratesforthecontinuousglucosemonitor ................. 104 5-8Reactionprolescalculatedfordierenthomogeneousreactionratesforthecontinuousglucosemonitor .................................... 105 5-9CalculateddimensionlessdiusionimpedancesforthecontinuousglucosemonitorwithhomogeneousreactionratecorrespondingtoSystem1 ............ 106 5-10CalculateddimensionlessdiusionimpedancesforthecontinuousglucosemonitorwithhomogeneousreactionratecorrespondingtoSystem2 ............ 107 5-11CalculateddimensionlessdiusionimpedancesforthecontinuousglucosemonitorwithhomogeneousreactionratecorrespondingtoSystem3 ............ 108 5-12Polarizationcurvecalculatedfromdierentbulkoxygenconcentrations ..... 109 10

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5-13Calculatedsteady-stateconcentrationsfordierentbulkoxygenconcentrations 110 5-14Reactionprolecalculatedfordierentbulkoxygenconcentrations ....... 111 5-15Calculateddimensionlessdiusionimpedanceforthecontinuousglucosemonitorwithoxygenconcentrationof510)]TJ /F3 7.9701 Tf 6.587 0 Td[(8mol=cm3 ................... 113 5-16Calculateddimensionlessdiusionimpedanceforthecontinuousglucosemonitorwithoxygenconcentrationof510)]TJ /F3 7.9701 Tf 6.587 0 Td[(9mol=cm3 ................... 114 5-17Calculateddimensionlessdiusionimpedanceforthecontinuousglucosemonitorwithoxygenconcentrationof510)]TJ /F3 7.9701 Tf 6.587 0 Td[(10mol=cm3 .................. 114 5-18CalculateddimensionlessdiusionimpedancefortheCGMwithdierentoxygenconcentrations ..................................... 115 7-1Dimensionlessdiusionimpedancesfordierentkineticparameters ....... 120 7-2Schematicrepresentationillustratingthecontributionofthereactingspeciestothechargingoftheelectrode-electrolyteinterface ................. 123 7-3Schematicrepresentationoftime-constantdistribution .............. 124 7-4EISexperimentsetup ................................. 126 7-5CVcurveswithdierentsweeprates ........................ 127 7-6PolarizationCurveswithdierentrotationrates .................. 128 7-7Impedancespectrumwitharotationspeedof500RPM .............. 129 7-8Adjustedphaseanglewitharotationspeedof500RPM ............. 130 A-1MatrixdeningBAND ................................ 132 11

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LISTOFCODES page A.1MATLABBANDandMATINVcode ...................... 134 A.2FORTRANBANDCode ............................. 137 A.3FORTRANMATINVCode ........................... 139 B.1FiniteSchmidtConvectionDiusionTerm0forrotatingdiskelectrode ... 142 B.2FiniteSchmidtConvectionDiusionTerm1forrotatingdiskelectrode ... 145 B.3FiniteSchmidtConvectionDiusionTerm2forrotatingdiskelectrode ... 148 C.1FiniteSchmidtConvectionDiusionTerm0foranimpingingjetelectrode 152 C.2FiniteSchmidtConvectionDiusionTerm1foranimpingingjetelectrode 155 C.3FiniteSchmidtConvectionDiusionTerm2foranimpingingjetelectrode 158 D.1InputlefortheConvectiveDiusionwithHomogeneousReaction ..... 162 D.2PotentialinputlefortheConvectiveDiusionwithHomogeneousReaction 162 D.3SteadyStateConvectiveDiusionwithHomogeneousReactionMainProgram 164 D.4Steady-StateConvectiveDiusionwithHomogeneousReactionSubroutinetoCreatetheVelocityProle .......................... 167 D.5Steady-StateConvectiveDiusionwithHomogeneousReactionSubroutinefortheElectrodeBoundaryCondition ...................... 169 D.6SteadyStateConvectiveDiusionwithHomogeneousReactionSubroutinefortheReactionRegion ............................. 172 D.7SteadyStateConvectiveDiusionwithHomogeneousReactionSubroutinefortheCoupler .................................. 174 D.8SteadyStateConvectiveDiusionwithHomogeneousReactionSubroutinefortheInnerRegion ............................... 178 D.9SteadyStateConvectiveDiusionwithHomogeneousReactionSubroutinefortheBulkBoundaryCondition ........................ 181 D.10Matlabcodetoplotresultsfromsteady-statesolutions ............ 182 D.11Matlabcodetocreateandplotpolarizationcurve ............... 184 D.12OscillatingConvectiveDiusionwithHomogeneousReactionMainProgram 186 D.13OscillatingConvectiveDiusionwithHomogeneousReactionSubroutinefortheElectrodeBoundaryCondition ....................... 190 D.14OscillatingConvectiveDiusionwithHomogeneousReactionSubroutinefortheElectrodeBoundaryCondition ....................... 192 D.15OscillatingConvectiveDiusionwithHomogeneousReactionSubroutinefortheElectrodeBoundaryCondition ....................... 194 D.16OscillatingConvectiveDiusionwithHomogeneousReactionSubroutinefortheElectrodeBoundaryCondition ....................... 197 D.17OscillatingConvectiveDiusionwithHomogeneousReactionSubroutinefortheElectrodeBoundaryCondition ....................... 199 D.18Matlabcodetocreateanddimensionlessdiusionimpedanceandoverallimpedance ..................................... 200 E.1InputlefortheContinuousGlucoseMonitorCode .............. 204 E.2InputleforthepotentialoftheContinuousGlucoseMonitorCode ..... 204 E.3Steady-StateContinuousGlucoseMonitorMainProgram ........... 206 12

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E.4Steady-StateContinuousGlucoseMonitorfortheElectrodeBoundaryCondition 210 E.5Steady-StateContinuousGlucoseMonitorSubroutinefortheReactionRegion 214 E.6Steady-StateContinuousGlucoseMonitorSubroutinefortheFirstCoupler 218 E.7Steady-StateContinuousGlucoseMonitorSubroutinefortheInnerRegion 223 E.8Steady-StateContinuousGlucoseMonitorSubroutinefortheSecondCoupler 227 E.9Steady-StateContinuousGlucoseMonitorSubroutineforGLMRegion ... 231 E.10Steady-StateContinuousGlucoseMonitorSubroutinefortheBulkBoundaryCondition ..................................... 234 E.11OscillatingContinuousGlucoseMonitorMainProgram ............ 237 E.12OscillatingContinuousGlucoseMonitorfortheElectrodeBoundaryCondition 242 E.13OscillatingContinuousGlucoseMonitorSubroutinefortheReactionRegion 247 E.14OscillatingContinuousGlucoseMonitorSubroutinefortheFirstCoupler .. 251 E.15OscillatingContinuousGlucoseMonitorSubroutinefortheInnerRegion .. 256 E.16OscillatingContinuousGlucoseMonitorSubroutinefortheSecondCoupler 260 E.17OscillatingContinuousGlucoseMonitorSubroutinefortheGLMRegion .. 265 E.18OscillatingContinuousGlucoseMonitorSubroutinefortheBulkBoundaryCondition ..................................... 268 E.19Matlabcodetocreateanddimensionlessdiusionimpedanceandoverallimpedance ..................................... 270 13

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LISTOFSYMBOLS Roman A constantusedforthevelocityexpansionforfarfromtheelectrodeofarotatingdisk,A=0:92486353 a constantusedforthevelocityexpansionforclosetotheelectrodeofarotatingdisk,a=0:510232618867 B constantusedforthevelocityexpansionforfarfromtheelectrodeofarotatingdisk,B=1:20221175 b constantusedforthevelocityexpansionforclosetotheelectrodeofarotatingdisk,b=)]TJ /F1 11.9552 Tf 9.298 0 Td[(0:615922014399 C capacitance,F/cm2orF(1F=1C=V) Cdl double-layercapacitance,F/cm2orF(1F=1C=V) ci volumetricconcentrationofspeciesi,mol/cm3 Di diusioncoecientforspeciesi,cm2=s F dimensionlessradialcomponentofvelocityforlaminarowtoadiskelectrode F Faradaysconstant,96,487C/equiv f frequency,f=!=2,Hz f parameterinvelocityinterpolation,equation( 2{41 ) G dimensionlessangularcomponentofvelocityforlaminarowtoadiskelectrode GA Gluconicacidspeciesinacontinuousglucosemonitor G Glucosespeciesincontinuousglucosemonitor GOx Glucoseoxidaseenzyme,oxidizedversion,inacontinuousglucosemonitor GOx)]TJ /F1 11.9552 Tf 11.956 0 Td[(H2O2 Glucoseoxidasecomplex,participatinginthesecondenzymaticregenerationstep,inacontinuousglucosemonitor GOx2 Glucoseoxidaseenzyme,reducedversion,inacontinuousglucosemonitor GOx2)]TJ /F1 11.9552 Tf 11.956 0 Td[(GA Glucoseoxidasecomplex,participatingintherstenzymaticreaction,inacontinuousglucosemonitor 14

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H dimensionlessaxialcomponentofvelocityforlaminarowtoadiskelectrode H meshsizeofinnerregionofhomogeneousreactioncodeandofouter(GLM)regionofCGMcode h meshsizeinageneralnitedierenceexpression HH meshsizeofreactionregionofhomogeneousreactioncodeandofinnerregionofCGMcode HHH meshsizeofreactionregionofCGMcode H2O2 Hydrogenperoxidemolecule,H2O2,inacontinuousglucosemonitor I Current,I I amplitudeofsinusoidalcurrentsignal j complexnumber,p )]TJ /F1 11.9552 Tf 9.298 0 Td[(1 K dimensionlessfrequencyassociatedwiththegeometryofadiskelectrode K rateconstantforelectrochemicalreactionthatexcludestheexponentialdependenceonpotential kb backwardrateconstantforachemicalreaction Keq equilibriumrateconstantforachemicalreaction,Keq=kf=kb kf forwardrateconstantforachemicalreaction L inductance,H(1H=1Vs2=C) Ni uxofspeciesi,mol=cm2s O2 Oxygenmolecule,O2,inacontinuousglucosemonitor Q CPEcoecient,s=cm2 r radialdirectionincylindricalcoordinatesforrotatingdisk R universalgasconstant,8.3143J=molK Re electrolyteorohmicresistance,orcm2 Ri homogeneousreactionofspeciesi,mol=cm2=s Rt charge-transferresistance,cm2 R resistance,cm2or(1=1Vs=C) 15

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Sc Schmidtnumber,Sc==Di,dimensionless T temperature,K t time,s T Period V Potential,V v velocity,cm=s V amplitudeofsinusoidalpotentialsignal y axialcoordinateincylindricalcoordinatesforrotatingdisk Z impedance,orcm2,orifnoted,dimensionless zi chargeassociatedwithspeciesi ZD diusionimpedance,orcm2 ZF faradaicimpedance,orcm2 Greek constantusedforthevelocityexpansionforfarfromtheelectrodeofarotatingdisk,=0:88447411 CPEcomponent,dimensionless parameterinvelocityinterpolation,equation( 2{42 ) functionoftheforwardandbackwardrateofachemicalreaction,equation( 2{43 ) N Nernstdiusionlayerthickness r reactionlayerthickness dimensionlessposition,=yp = 0 parameterinvelocityinterpolation,equation( 2{42 ) uidviscosity,g=cms i mobilityofspeciesi,mol=cm2=s=J kinematicviscosity,==;cm2=s dimensionlessposition uiddensity,g=cm3 16

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angulardirectionincylindricalcoordinatesforrotatingdisk phasedierencebetweenthepotentialandcurrent electrostaticpotential,V r gradientofelectrostaticpotential,negativeelectriceld,V=cm rotationspeed,s)]TJ /F3 7.9701 Tf 6.587 0 Td[(1 angularfrequency,!=2f,s)]TJ /F1 11.9552 Tf 7.085 -4.338 Td[(1 GeneralNotation ImfXg imaginarypartofX RefXg realpartofX X steady-stateortime-averagedpartofX(t) Subscripts IJ impingingjet I pertainingtocurrent i pertainingtochemicalspeciesi j imaginary r real r pertainingtoradialcomponent pertainingtotheangularcomponent i pertainingtopotential y pertainingtoaxialcomponent 17

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyMATHEMATICALMODELSFORIMPEDANCESPECTROSCOPYByMorganS.HardingMay2017Chair:MarkE.OrazemMajor:ChemicalEngineeringDiusionorconvectivediusioncaninuencetheimpedanceresponseofasystemassociatedwithelectrochemicalreactions.Analysisforconvective-diusionimpedanceonbotharotatingdiskandasubmergedimpingingjetelectrodearepresentedforniteSchmidtnumbers.Theconvective-diusionimpedancesimulationswereperformedinMATLAB,andtheimpedanceiscalculatedusingtheoscillatingconcentrationofthereactingspecies.Thedevelopmentofmodelsforconvective-diusionimpedanceservedasafoundationforthestudyofsystemsinwhichhomogeneousreactionsinuencetheimpedanceofelectrochemicalsystems.Amathematicalmodelwasdevelopedfortheimpedanceresponseassociatedwithcoupledhomogeneouschemicalandheterogeneouselectrochemicalreactions.ThemodelincludedahomogeneousreactionintheelectrolytewherespeciesABreactsreversiblytoformA-andB+,andB+reactselectrochemicallyonarotatingdiskelectrodetoproduceB.Thismodelprovidesanextensiontotheliteraturebyusinganonlinearexpressionforthehomogenousreactionanduniquediusioncoecientsforeachspecies.Theresultingconvective-diusionimpedancehadtwoasymmetriccapacitiveloops,oneassociatedwithconvective-diusionimpedance,theotherwiththehomogeneousreaction.Eventhoughtheassumptionofalinearexpressionforthehomogeneousreactionwasrelaxed,amodiedGerischerimpedancewasfoundtoprovideagoodttothesimulateddata.ThemodelwasdevelopedinFORTRAN,andasteady-statesolutioncontainingfourvariables 18

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wassolvedfollowedbyasolutioninthefrequency-domaininvolvingeightvariables.TheoscillatingconcentrationofB+wasusedtoobtaintheimpedancespectrum.Thedevelopmentofamathematicalmodelfortheimpedanceresponseofglucoseoxidaseelectrochemicalbiosensorsrepresentsanextensionofthemodeldevelopedforasinglehomogenousreaction.Inthebiosensor,aprocessofenzymaticcatalysistransformsglucoseintohydrogenperoxide,whichcanbedetectedelectrochemically.Thismodelprovidesanextensiontotherelevantliteraturebyconsideringfourenzymaticreactions,twoofwhicharenonlinearexpressions,concentrationsoftheenzymeinanoxidizedandreducedform,andtheconcentrationofenzymecomplexesformedasintermediatesintheenzymaticreaction.AFORTRANcodewasusedtosolvethesteady-stateequationsfor12variableswhichwereusedsubsequentlytosolvethe24frequency-domainequations.Asbefore,theoscillatingconcentrationoftheelectroactivespecies,hydrogenperoxideinthiscase,wasusedtoobtaintheimpedanceresults. 19

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CHAPTER1INTRODUCTIONElectrochemicalimpedancespectroscopy(EIS)playsanimportantroleinelectrochemistryandelectrochemicalengineering.Interpretationofimpedancespectroscopymeasurementsrequiresaknowledgeofreactionsandtransportpropertiesandhowtheseeectanelectrochemicalinterface.EISisnondestructiveandnoninvasivetothesamplesbeingtested.SinceEISisatransienttechnique,moredetailsofanelectrochemicalsystemcanbeextractedthenusingsteady-statemethodsalone.Thepopularityofimpedancespectroscopyhasgrowninrecentyears.ThenumberofjournalarticlesreferencingEISisincreasingdramaticallyandisapproximatelyanexponentialgrowth,aspresentedinFigure 1-1 .Impedancemeasurementsareperformedbyinputtingasmallsinusoidaloscillatingperturbationofpotential(orcurrent)toanelectrochemicalsystemandcapturingthecurrent(orpotential)response.Toobtainanimpedancespectrum,theinputoscillationsaredoneoverarangeoffrequencies.Forasimulation,atypicalfrequencyrangegoesfromadimensionlessfrequencyof10)]TJ /F3 7.9701 Tf 6.587 0 Td[(5to105.Foranexperiment,atypicalfrequencyrangecouldbefrom1mHzto1GHz.Arotatingdiskelectrode(RDE)embeddedinaninsulatorisacommonelectrodetousewhenconductingEISexperiments.Thedevelopeduidowpatternsinarotatingdiskelectrodesystemarewellknown.EISsimulationsallowinterpretationofsteady-stateandimpedancedataonalevelgreaterthancanbeachievedwithexperimentsalone.ItisnotalwayspracticaltoconductEISexperiments.ElectrochemicalsystemscanexhibitmanybehaviorsthatoverlapinEISexperimentalresults.SimulatedEISexperimentsallowtheseconictingbehaviorstobesplitupanddiscovertheirindividualinuenceontheoverallimpedanceresults.Inarotatingdiskelectrode,forexample,theuidowisunderstood,seesection 2.3.1 ,andthekineticsofanelectrochemicalsystemcanbestudiedsucientlywithoutcomplicatingthesimulationwithmoredimensions.Aone-dimensionalmodelusingNewman'sBAND 20

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Figure1-1. NumberofjournalarticlesthatreferenceelectrochemicalimpedancespectroscopyusingEngineeringVillagersearchengine.Thekeywordsusedwere(impedanceoradmittance)and(electrochemical),journalarticlesonly.ReprintedwithpermissionfromOrazemandTribollet[ 3 ]. algorithm,discussedinlengthinAppendix A ,isanidealplatformtostudymanyelectrochemicalsystems.Inthisdissertation,fundamentalconceptsandnumericalmethodsforEISaredescribedinChapter 2 .Backgroundinformationoftheconvective-diusionequationforEISalongwithinformationforsystemswithconvection,diusion,andahomogeneousreactioneectingthesystemarealsodiscussed.Avelocityanalysisforrotatingdiskelectrodes,includinganinterpolationformulatodescribetheuidowoveralargerangeisdiscussed.Thischapteralsocontainshistoryoftheenzymeglucoseoxidaseanditsuseinanelectrochemicalsensortomeasurebloodglucoselevelsfortype1diabetics.Thecurrentunderstandingofthekineticsinaglucosesensorarealsodiscussed.Continuousglucosemonitors(CGMs)areembeddedbio-sensors,withalayerofglucoseoxidasesurroundingtheelectrode,thatmonitorbloodglucoselevelsconstantly.TheuseofEIStomonitorthestateofhealthofaCGMisdiscussedinChapter 2 21

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Theresultsfortheconvective-diusionequationforanelectrochemicalsystemwithareactingspeciesonarotatingdiskelectrode[ 4 ]andonasubmergedimpingingjetelectrode[ 5 ]havebeenpublished.Havingatabulatedsetofdimensionlessimpedancedataforthethree-expansiontermsdependentontheSchmidtnumbercanbeincrediblyuseful.Forexample,thediusioncoecientcanbeobtainedafterconductinganexperimentandttingtoamodelfortheconvective-diusionequation.ThemathematicaldevelopmentandsimulationresultsforaniteSchmidtnumbersforboththerotatingdiskelectrodeandsubmergedimpingingjetelectrodearepresentedinChapter 3 .TheexpressionforaninniteSchmidtnumber,asimplifyingassumption,andtheerrorforassuminganinniteSchmidtnumberforcaseswhereaniteSchmidtnumberisnecessaryisalsodiscussed.ThecodesthatwereusedtoobtaintheresultsinthischapterarelistedinAppendices B and C .Theconvective-diusionequationisfurthercomplicatedbyconsideringahomogeneous,orchemical,reaction.Unlikewhennohomogeneousreactionispresent,thesteady-statesimulationisrequiredtoobtaintheimpedance,sothesolutionbecomespurelynumerical.Consideringtheinuenceofahomogenousreactionontheimpedanceresultsalsocreatesthenecessitytoaddavelocityexpansiontothesystem.Thisexpansionandtheinterpolationbetweentheexpansionforclosetotheelectrodearepresentedinmoredetailinsection 2.3.1 .ThepresentationofthemathematicalmodelsandresultsforasystemeectedbyafaradaicandhomogeneousreactionsonarotatingdiskelectrodearepresentedinChapter 4 .TheFORTRANcodesusedtoobtainthesteady-stateandoscillatingsolutionstotheconvective-diusionequationwithhomogenousreactionsaswellastheMATLABcodetoanalyzetheresultsarepresentedinAppendix D .Amathematicalmodelwasdevelopedfortheimpedanceresponseofaglucoseoxidaseenzyme-basedelectrochemicalbiosensor.Themodelaccountsforaglucoselimitingmembrane(GLM),whichcontrolstheamountofglucoseparticipatingintheenzymaticreaction.Theglucoseoxidasewasassumedtobeimmobilizedwithinathinlmadjacent 22

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totheelectrode.Intheglucoseoxidaselayer,aprocessofenzymaticcatalysistransformstheglucoseintoperoxide,whichcanbedetectedelectrochemically.Thebackgroundmaterialandliteraturereviewrelatedtotheglucosebiosensorispresentedinsection 2.3.3 .ThemathematicalworkupandresultsfortheimpedanceresponseofaCGMarepresentedinChapter 5 .FORTRANcodesusedtosolvethesteadyandoscillatingequationsforthebiosensoralongwiththeMATLABcodeusedtoplottheresultsarelocatedin E .AfurtherparameteranalysisfortheimpedanceresponseofaCGMisneededtofullyunderstandtheinuenceofparametersontheCGM.AdetailedworkupofthissuggestedfutureworkispresentedinChapter 7 .ExperimentswereconductedtotrytovalidatethecouplingoffaradaicandchargingcurrentsonEIS.ThebackgroundtounderstandthisphenomenonaswellassimulationsconductedbyWuetal.[ 1 ]andthepreliminaryresultsarepresentedinChapter 7 23

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CHAPTER2BACKGROUNDElectrochemicalImpedanceSpectroscopy(EIS)isapopulartechniquetocharacterizeelectrodeprocesses.ApredominatefeatureofEISisthatitisnon-invasiveandtherefordoesnotdestroythesystem.Electrochemicalimpedancespectroscopysimulationsandmodelshelpdevelopagreaterunderstandingofelectrochemicalsystems. 2.1ElectrochemicalImpedanceSpectroscopyImpedancemeasurementsareconductedinthetimedomainandareperformedbyapplyingasmallamplitudesinusoidalperturbationofpotentialtoanelectrochemicalsystemandmeasuringtheoutputresponseofcurrent.Apolarizationcurveisthesteady-statecurrentresponseasafunctionofpotentialforanelectrochemicalsystem.ApointonapolarizationcurveischosentoexaminemoreandobtainEISresults.Asinusoidalperturbationaboutthatpointisimputed,forarangeoffrequencies,andtheoutputismeasuredinordertoobtaintheEISresults,showninFigure 2-1 .Theinputsignalisrepresentedby V(t)= V+j4Vjcos(!t)(2{1)andtheoutputas I(t)= I+j4Ijcos(!t+')(2{2)Thesteady-statetermsare Vand Iandthemagnitudeoftheoscillatingpartofthesignalisrepressedbyj4Vjandj4Ij.Theindependentvariable!istheangularfrequency,!=2f,andiscommonlychosenaspointsperdecadeinimpedancemeasurements.Thephaselagbetweeninputandoutputisrepresentedby'andtistime.Thecalculationofthetransferfunctionatagivenfrequency!ispresentedschematicallyinFigure 2-2 .Analternativewaytowriteequations( 2{1 )and( 2{2 )is V(t)= V+RefeVexp(j!t)g(2{3) 24

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Figure2-1. Sinusoidalperturbationofanelectrochemicalsystematsteadystate,whereV(!)andI(!)representthepotentialandcurrent,madeupofasteady-stateandoscillatingpart,atthefrequency!withaphasedierenceof'[ 2 ]. Figure2-2. Schematicrepresentationofthecalculationofthetransferfunctionforasinusoidalinputatfrequency!.Thetimelagbetweenthetwosignalsis4tandtheperiodofthesignalsisT.[ 3 ] 25

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and I(t)= I+RefeIexp(j!t)g(2{4)whereeVandeIarecomplexquantitiesthatarefunctionsoffrequencybutindependentoftime.Thecomplexvalueisrepresentedasj,whichequalsp )]TJ /F1 11.9552 Tf 9.298 0 Td[(1;jisusedinsteadofitoavoidconfusionwithcurrentdensity.InEIStheoscillatingmagnitudeissucientlysmallsotheresponseislinearandwillhavethesameformoftheinputandoccuratthesamefrequency.Thereisaphaselagbetweentheinputandoutput,whichisrepresentedby'.Thephaselaginunitsofradianscanbeobtainedas '(!)=24t T(2{5)If4t=0orif4t=Tthephaseangleisequaltozero.AsshowninFigure 2-2 ,theoutputlagstheinput,andthephaseanglehasapositivevalue.Theoutputresponseisacomplexquantity,withrealandimaginarycomponents,thatisafunctionoffrequency.Impedanceisthereforeacomplexvaluedenedastheratioofpotentialandcurrent,seeequation( 2{6 ). Z(!)=eV(!) eI(!)=Zr+jZj(2{6)whereeV(!)andeI(!)comefromequations( 2{3 )and( 2{4 )andarerepresentedas eV(!)=j4Vjexp(j'V)(2{7)and eI(!)=j4Ijexp(j'I)(2{8)wherethephasedierencebetweenthepotentialandcurrentis'='V)]TJ /F5 11.9552 Tf 12.151 0 Td[('I.Inequation( 2{6 ),ZrandZjaretherealandimaginarypartsoftheimpedance,respectively. 26

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Figure2-3. SchematicrepresentationofcircuitelementsusedtomodelEISresults. 2.1.1ModelingElectrochemicalImpedanceSpectroscopyIfthereisnophasedierencebetweenpotentialandcurrent,thentheimpedanceisarealnumberasaresistance. Zres(!)=R(2{9)Ifthephasedierencelagsby90thentheimpedanceiscompletelyimaginary,andacapacitance. Zcap(!)=1 j!C(2{10)A90lagcanalsorepresentaninductor. Zind(!)=j!L(2{11)Fromthis,impedancecanbemodeledusingcircuitelements.AcommoncircuitforimpedanceisshowninFigure 2-3 .Therstresistor,Re,istheohmicresistance,andrepresentstheresistanceintheelectrolyte.TheparallelcombinationofaresistorandacapacitoriscalledaVoigtelement,orRCelement.TheresistorintheRCelementrepresentsthecharge-transfer-resistance,Rtandthecapacitorrepresentsasurfacecapacitance,commonlyadouble-layer-capacitance,C.Followingtheapproachofusingcircuitelements,equations( 2{9 )-( 2{11 )canbeusedtorepresentormodelimpedance. 27

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A BFigure2-4. BoderepresentationofimpedancedatashowninFigure 2-6 :a)magnitudeandb)phaseangle. TheimpedanceforFigure 2-3 canberepresentedinequationformby Z(!)=Re+Rt 1+j!CRt(2{12)TherealpartoftheimpedanceforFigure 2-3 canberepresentedby Zr(!)=Re+Rt 1+!2C2R2t(2{13)andtheimaginaryimpedanceby Zj(!)=)]TJ /F5 11.9552 Tf 9.299 0 Td[(j!CR2t 1+!2C2R2t(2{14) 2.1.2RepresentationofElectrochemicalImpedanceSpectroscopyImpedanceisacomplexvalueandfrequentlyshowninacomplexplane,calledaNyquistplot.Thedataispresentedasalocusofpoints,whereeachpointrepresentsadierentfrequencymeasurement.TheNyquistplothidesthefrequencydependence,henceitisadvantageoustolabelsomepointswiththeircorrespondingfrequency.Figure 2-6 showsatypicalNyquistplot,withtheimaginarypartoftheimpedanceontheyaxisandtherealpartoftheimpedanceonthexaxis.Inordertoshowimpedanceisafunction 28

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A BFigure2-5. Representationofimpedancedataasafunctionoffrequencyfor:a)realpartofimpedanceb)imaginarypartofimpedance. Figure2-6. NyquistplotofimpedancedatacorrespondingtoaRCcircuitandCPEcircuitofRt=100cm2,Re=10cm2andCo=20F=cm2 29

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withrespecttofrequencyaBodeplotisused.ABodeplotshowsmagnitude,equation( 2{15 ),andphaseangle,equation( 2{16 ),asafunctionoffrequency,showninFigure 2-4 jZj=q Z2r+Z2j(2{15) '=tanh)]TJ /F3 7.9701 Tf 6.586 0 Td[(1Zj Zr(2{16)Anothercommonwaytodisplayimpedancedataistherealimpedanceandimaginaryimpedanceasafunctionoffrequency,showninFigure 2-5 .Therealimpedanceandimaginaryimpedance,calculatedfromequations( 2{13 )and( 2{14 ),arepresentedinalogscaleasafunctionoffrequency,showninFigures 2-5A and 2-5B ,respectively. 2.1.3CharacteristicFrequencyinElectrochemicalImpedanceSpectroscopyThecharacteristicfrequencyofanimpedancespectrumisthemaximumoftheabsolutevalueoftheimaginarypartoftheimpedance.Muchcanbelearnedaboutasystemfromitscharacteristicfrequency.Forexample,forthedatapresentedinFigures 2-4 2-6 hasacharacteristicfrequencyequalto fRC=1=(2RtC)(2{17)ThischaracteristicfrequencycanbefoundforasystemwithaRCcircuit,asshowninFigure 2-3 .Forarotatingdiskelectrode,areactingelectrochemicalsysteminuencedbyconvection,thedimensionlesscharacteristicfrequencyisequalto2.5.Thedimensionlessfrequencyforthissystem,aectedbyconvectionanddiusion,is K=!2i Di(2{18)whereiisthecharacteristiclengthformasstransferandDiisthediusioncoecientofspeciesi.Thecharacteristicfrequencycanhelpanexperimentalistestimatethecapacitanceofasystemormeasurethediusioncoecientofaspecies.Thisisderivedlateristhedissertation,whenequation( 3{23 )isusedtonon-dimensionalizethe 30

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convective-diusionequation.Bycombiningequations( 3{23 )and( 3{22 ),equation( 2{18 )isobtainedforthelargestabsolutevalueoftheimaginaryimpedance.AcircuitdiagramforthiselementisshowninFigure 4-2 ,wherethefaradaicimpedanceelementismadeupofthesumofthechargetransferresistance,Rtandthediusionimpedance. 2.2NumericalSimulationsforElectrochemicalImpedanceSpectroscopyItisnotalwayspracticaltoconductEISexperiments.ElectrochemicalsystemscanexhibitmanybehaviorsthatoverlapinEISexperimentalresults.SimulatedEISexperimentsallowtheseconictingbehaviorstobesplitupanddiscovertheirindividualinuenceontheoverallimpedanceresults.One-dimensionalmodelsarepracticalforelectrochemicalsystems.Inarotatingdiskelectrode,forexample,theuidowisunderstood,seesection 2.3.1 ,andthekineticsofanelectrochemicalsystemcanbestudiedsucientlywithoutcomplicatingthesimulationwithmoredimensions.Aone-dimensionalmodelusingNewman'sBANDalgorithm,discussedinlengthinAppendix A ,isanidealplatformtostudyelectrochemicalsystems. 2.2.1Finite-DierenceMethodsDierentialequationsdescribingthebehaviorinanelectrochemicalsystemcanbesolvedusingnite-dierencemethods.Adierentialequationcanbediscretizedusinganite-dierencemethod.Forexamplearstderivativecanbewrittenas dc dy=c(x+h))]TJ /F5 11.9552 Tf 11.955 0 Td[(c(x)]TJ /F5 11.9552 Tf 11.955 0 Td[(h) 2h+O(h2)(2{19)andasecondderivativecanbewrittenas d2c dy2=c(x+h))]TJ /F1 11.9552 Tf 11.955 0 Td[(2c(x)+c(x)]TJ /F5 11.9552 Tf 11.955 0 Td[(h) h2+O(h2)(2{20)wheretheaccuracyofthediscretizationistheorderofthemeshsquared. 2.2.2ConvergenceMethodsFinite-dierencemethodsinconjunctionwithNewman'sBANDmethodareusedinChapters 4 and 5 .Atechniquetohelptheconvergenceofthecodewasrstdeveloped 31

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byOrazem[ 6 ].Thetechniqueisreferredtoas"BIGs".Afterthedierentialequationhasbeendiscretized,thebiggesttermintheequationismultipliedbyEBIG=110)]TJ /F3 7.9701 Tf 6.587 0 Td[(10andiscomparedtoG(I).G(I)representsthediscritizedequationsetequaltozero.IftheabsolutevalueofG(I)islessthantheabsolutevalueofthebiggesttermmultipliedbyEBIG,thenG(I)issettozero.Thenon-linearcodeconvergeseasierwiththeimplementationofBIGs. 2.3ImpedanceModelsforRotatingDiskElectrodeDuetothepopularityoftherotatingdiskelectrode,theimpedanceresponseforelectrochemicalsystemswitharotatingdiskelectrodehasbeenexaminedbymanyintheelectrochemicalcommunity[ 7 8 9 10 11 12 13 14 15 16 ].Thepopularityofarotatingdiskelectrodeisattributedtowellknownhydrodynamicconditionsandsmallandsimpleexperimentalsetup. 2.3.1FluidowforaRotatingDiskThesteadyowcreatedbyaninnitediskrotatingataconstantangularvelocityinauidwithconstantphysicalpropertieswasrststudiedbyvonKarman.[ 17 ]Theuidsareconsideredtobeincompressible.Thevelocityonthesurfaceofthediskwasconsideredtobe vr=0(2{21) v=r(2{22)and vy=0(2{23)Theboundaryconditionfortheangularvelocityexpressesthattheuidvelocityonthediskisbeingdraggedbythediskwiththespeedofthevelocityintheangulardirection.Becauseoftherotatingdiskthereisacentrifugaleectthatmovestheuidtowardsthediskintheaxialdirectionandoutfromthediskintheradialdirection.Incylindrical 32

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coordinates,theequationofcontinuityis 1 r@ @r(rvr)+1 r@v @+@vy @y=0(2{24)TheNavier-Stokesequationrepresentstheuidowofconstantdensityandviscosityandisshownbelow @v @t+vrv=)]TJ /F1 11.9552 Tf 10.586 8.088 Td[(1 rp+r2v+g(2{25)whereisthekinematicviscosityincm2=s,gisgravity,visthemassaveragedvelocity,tistime,isthedensity,andpisthethermodynamicpressure.Thepressureandgravitycomponentcanbewrittenasadynamicpressure rP=rp)]TJ /F5 11.9552 Tf 11.955 0 Td[(g(2{26)whichcanbeusedtore-writetheNavier-Stokesequation @v @t+vrv=)]TJ /F1 11.9552 Tf 10.586 8.088 Td[(1 rP+r2v(2{27)Thecomponentsoftheequationofmotionare@vr @t+vr@vr @r+v r@vr @)]TJ /F5 11.9552 Tf 13.15 8.088 Td[(v2 r+vy@vr @y=)]TJ /F1 11.9552 Tf 10.586 8.088 Td[(1 @P @r+@ @r1 r@ @r(rvr)+1 r2@2vr @2)]TJ /F1 11.9552 Tf 15.39 8.088 Td[(2 r2@v @+@2vr @y2 (2{28)fortheradialcomponent@v @t+vr@v @r+v r@v @+vvr r+vy@v @y=)]TJ /F1 11.9552 Tf 13.386 8.088 Td[(1 r@P @+@ @r1 r@ @r(rv)+1 r2@2v @2)]TJ /F1 11.9552 Tf 15.391 8.088 Td[(2 r2@vr @+@2v @y2 (2{29)theangularcomponent@vy @t+vr@vy @r+v r@vy @+vy@vy @y=)]TJ /F1 11.9552 Tf 10.587 8.087 Td[(1 @P @y+1 r@ @rr@vy @r+1 r2@2vy @2+@2vy @y2 (2{30) 33

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andtheaxialcomponent.Thesolutionwassoughtbyusingaseparationofvariablesusingadimensionlessdistance =yp =(2{31)anddimensionlessradialvelocity vr=rF()(2{32)dimensionlessangularvelocity v=rG()(2{33)anddimensionlessaxialvelocity vy=p H()(2{34)whereistherotationspeedinrad=s,yrepresentstheaxialdistance,rtheradialdistanceandtheangulardistancefromtheelectrodeincm.Forthesystemillustratedhereavalueof=0.01cm2=sand=209.4rad=swhichcorrespondsto2000RPMwereused.TheNavier-Stokesequationscanbesolvednumericallywhenequations( 2{32 ),( 2{33 ),and( 2{34 )areinserted.AsshownbyCochran[ 18 ],thevariablesF,GandHcanbewrittenastwosetsofseriesexpansions,oneclosetotheelectrode( 0),andonewhere!1.Forsmallvaluesof,orclosetotheelectrode,theexpansionsare F=a)]TJ /F1 11.9552 Tf 13.151 8.087 Td[(1 22)]TJ /F1 11.9552 Tf 13.151 8.087 Td[(1 3b3+:::(2{35) G=1+b+1 33+:::(2{36)and H=)]TJ /F5 11.9552 Tf 9.299 0 Td[(a2+1 33+b 64+:::(2{37)wherea=0.5102326189andb=-0.6159220144.AgraphofF,GandHclosetotheelectrodeversusdimensionlessdistanceisshowninFigure 2-7A 34

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A BFigure2-7. Velocityexpansionforuidowofarotatingdiskelectrodefora)smallvaluesofandb)largevaluesof Farfromtheelectrode,whenislarge,theexpansionequationsbecome F=Aexp()]TJ /F5 11.9552 Tf 9.299 0 Td[())]TJ /F5 11.9552 Tf 13.151 8.087 Td[(A2+B2 2exp()]TJ /F1 11.9552 Tf 9.298 0 Td[(2)+A(A2+B2) 44exp()]TJ /F1 11.9552 Tf 9.298 0 Td[(3)+:::(2{38) G=Bexp()]TJ /F5 11.9552 Tf 9.299 0 Td[())]TJ /F5 11.9552 Tf 13.151 8.088 Td[(A(A2+B2) 124exp()]TJ /F1 11.9552 Tf 9.299 0 Td[(3)+:::(2{39) H=)]TJ /F5 11.9552 Tf 9.299 0 Td[(+2A exp()]TJ /F5 11.9552 Tf 9.299 0 Td[())]TJ /F5 11.9552 Tf 13.15 8.088 Td[(A2+B2 23exp()]TJ /F1 11.9552 Tf 9.298 0 Td[(2))]TJ /F5 11.9552 Tf 13.15 8.088 Td[(A(A2+B2) 65exp()]TJ /F1 11.9552 Tf 9.299 0 Td[(3)+:::(2{40)where=0:88447441,A=0:93486353,andB=1:2021175.Theexpansionforwhenislargeforeachspeciesareplottedversusdimensionlesspositionin 2-7B .Tosolvethemasschargeconservationequations,avelocityproleisrequiredtodescribetheuidowinthewholedomain.Aweightingfunction,showninequation 2{42 ,andaninterpolationfunction,equation 2{41 ,areusedtoaccomplishthis,rstdevelopedbyWu[ 2 ].ThisinterpolationfunctionissimilartotheFermi-Diracfunctionappliedinquantummechanicsfordescribingthedistributionoffermions. vi=(1)]TJ /F5 11.9552 Tf 11.956 0 Td[(f)vi;!0+fvi;!1(2{41) 35

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A B CFigure2-8. Velocityexpansionsandinterpolationvelocityforarotatingdiskelectrodeforsmallvaluesofandlargevaluesoffora)ther{direction,b)the{direction,andc)they{direction wherei,canber,y,ortorepresentthedirectionofthevelocityowand f=1 1+e)]TJ /F7 7.9701 Tf 6.586 0 Td[(()]TJ /F7 7.9701 Tf 6.587 0 Td[(0)(2{42)Inthepresentcalculation,weused=20and0=1:25astheinterpolationconstants.Theaxialcomponentsofthedimensionlessvelocityasafunctionofdimensionlessdistancefromtheelectrodesurfaceareshowningure 2-8C .Theradialandangulardimensionlessvelocitiesareshowningures 2-8B and 2-8A .Thevelocityexpression 36

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applyingtheinterpolationfunctionsatisesthevelocitiesforsmallandlargevaluesofandshowsatransitioninthemediumdistancefromthedisksurface.Theinterpolationfunctionneedstobeasaccurateaspossibleneartheelectrodeandthiscreatesanon{smoothtransitionfromthenearexpansiontothefarexpansion.Thisnon{smoothtransitionisnotsignicantbecausetheimpedancedependsonlyontheresultsveryclosetotheelectrode.Therefore,theuseofthevelocityproleinequation 2{42 onaRDEisjustied. 2.3.2ConvectiveDiusionImpedanceModelsAone-dimensionalnumericalmodelfortheimpedanceofarotatingdiskwithaniteSchmidtnumberwaspresentedbyTribolletandNewman[ 4 ].Theyusedatwotermexpansionintheaxialdirectiontodescribethemotioninthesystem. 2.3.3ImpedanceModelswithHomogeneousReactionsThecouplingofhomogeneouschemicalreactionsandheterogeneouselectrochemicalreactionshasdrawnsubstantialinterestoverthepast80years.KouteckyandLevich[ 19 20 21 ]developedasteady-statemodelforahomogeneousreactioncoupledwithanelectrochemicalreactiononarotatingdiskelectrode.Thehomogenousreactionwasassumedtobelinear,alldiusioncoecientswereassumedtobeequal,andtheSchmidtnumberwasassumedinnite.KouteckyandLevichdenedacharacteristicdimensionforthehomogeneousreaction,termedthethicknessofthekineticlayerandrepresentedby r=r D (2{43)where=kf+kb,kfandkbarerespectivelytheforwardandbackwardrateconstantsofthelinearhomogeneousreaction,andDisthediusioncoecient.Bosscheetal.[ 22 ]describenite-dierencecalculationsunderassumptionofasteadystateforanelectrochemicalsystemcontrolledbydiusion,migration,convection,andnonlinearhomogeneousreactionkinetics.Theirconvectiontermusedathree-termexpansionappropriateforpositionsclosetotheelectrodesurface[ 17 ].Deslouisetal.[ 23 ] 37

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usedasubmergedimpingingjetcelltomeasureinterfacialpHduringthereductionofdissolvedoxygeninthepresenceofcarbonate.Theiranalysisconsideredthehomogeneousreactioninvolvingwaterandhydroxide,bicarbonate,andcarbonateions.Remitaetal.[ 24 ]haveshownthat,foradeaeratedaqueouselectrolytecontainingdissolvedcarbondioxide,hydrogenevolutionisenhancedbythehomogeneousdissociationofCO2.Tranetal.[ 25 ]demonstratedthathomogeneousdissociationofaceticacidenhancescathodicreductionofhydroniumions.Smith[ 26 27 ]usedACPolarographytostudydierentlinearrst-orderhomogeneousreactionmechanisms,includingpreceding,following,andcatalyticchemicalreactionscoupledwithelectrochemicalreactions.JurczakowskiandPolczynski[ 28 ]developedanACmodelwithcoupledhomogenousandheterogeneousreactionsaccountingforcaseswherediusioncoecientsarenotconsideredtobeequal.Theabovemechanismsassumedsimpliedhomogeneousreactions,withamaximumoftwospeciesconsidered.Usingchronopotentiometry,DelahayandBerzins[ 29 ]showedthatcadmiumcyanidecomplexesundergoadissociationbeforeelectroreductiononamercuryelectrode.SaveantandVianello[ 30 ]devisedthetheoreticalapproachoftheECmechanisminthecaseofcyclicvoltametryandproposedakineticzonediagramrepresentingthevariousregimesofcompetition.Fromthetheoreticaldevelopmentofpolarographyatastationaryelectrode,NicholsonandShain[ 31 ]developeddiagnosticcriteriaforfollowing,preceding,andcatalyticchemicalreactionscoupledwithchargetransfer.In1951,Gerischer[ 32 ]publishedtherstformaltreatmentofcoupledchemicalandelectrochemicalreactionsundersteady-stateandoscillatingsteady-state(impedance)conditions.Heconsideredalinearhomogeneousreaction,equaldiusioncoecients,andaNernststagnantdiusionlayer.AsummaryofGerischer'sderivationispresentedbyLasia[ 33 ].Recently,PototskayaaandGichan[ 34 ]extendedtheGerischerimpedancetoaccountforaroughenedelectrodeandnon-identicaldiusioncoecients.FollowingGerischer,PototskayaaandGichaninvokedlinearhomogeneouskineticsandaNernststagnant 38

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diusionlayer.ChapmanandAntaeno[ 35 ]discusstheuseoforthogonalcollocationasameanstoexploretheinuenceofnonlinearhomogeneousreactionsontheimpedanceresponsewithinaNernststagnantdiusionlayer.LevartandSchuhmann[ 11 ]developedamodelfortheconvectivediusionimpedanceofadiskelectrodeunderassumptionofaniteSchmidtnumberinthepresenceofahomogeneouschemicalreaction.Thediusioncoecientsofthesubstancesinvolvedinthereactionwereassumedequal,andalinearexpressionforthehomogeneousreactionwasassumed.TribolletandNewman[ 36 ]describeamodelforconcentratedsolutions,basedontheStefan-Maxwellequations,withprovisionforanarbitrarynumberofsimultaneoushomogeneousandheterogeneousreactions.TheirmodelwasemployedbyHauserandNewman[ 37 38 ]todescribetheinuenceofhomogenousconsumptionofcuprousionontheimpedanceresponseassociatedwithdissolutionofarotatingcopperdiskelectrodeunderassumptionoflinearhomogeneouskineticsandaninniteSchmidtnumber.Vazquez-ArenasandPritzker[ 39 40 ]developedamodelforthedepositionofcobaltionsonarotatingcobaltdiskundertheassumptionthathomogeneousreactionswereequilibrated.AGerischer-typeimpedancehasbeenusedtotmanyelectrodeprocesses,includingsolidoxidefuelcellsystems[ 41 42 43 ],oxideelectrodesystems[ 44 ],mixedconductingsolidelectrolytesystems[ 45 ],systemswithboundaryconditionsonadisorderedboundary[ 46 ]andelectrocatalyticsystemsinuencedbythehydrogenevolutionreaction[ 47 ].Coupledelectrochemicalandenzymatichomogeneousreactionsarealsoinvolvedinsensorsusedtomonitorglucoseconcentrationsformanagementofdiabetes[ 48 49 50 ].TheobjectofthepresentworkwastorelaxtheassumptionsimplicitinpreviouslypublishedpapersbydevelopingamodelfortheconvectivediusionimpedanceresponseofarotatingdiskelectrodeforaniteSchmidtnumber,un-equaldiusioncoecients,andusingnonlinearexpressionsforthekineticsofhomogeneousreactions.TheresultingimpedanceresponseswerecomparedtoahybridGerischerimpedancethataccountsfor 39

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linearhomogeneousreactionkinetics,equaldiusioncoecients,andaniteSchmidtnumberfortheconvectivediusionpartoftheexpression. 2.4ImpedanceModelsforContinuousGlucoseMonitorsContinuousglucosemonitors(CGMs)areusedtoconstantlymeasurethebloodglucoselevelinatype1diabeticperson.AnarticialpancreasiscomposedofaclosedloopsystemwhereaCGMislinkedtoaninsulinpump.CGMsarenotreliableafter7daysandelectrochemicalimpedancespectroscopyisproposedtomonitorthestateofhealthofaCGM.TheenzymeusedinCGMsisglucoseoxidase.Muller[ 51 ]discoveredtheenzymenotatinfromAspergillusnigerandPenicilliumglaucumin1928andnameditglucoseoxidase.Thenamenotatinstuckandcanstillbereferencedinmedicaljournals[ 52 53 ]butrecentpublicationsallrefertotheenzymeasglucoseoxidase.Heshowedthatglucosewasoxidizedbyglucoseoxidaseandcreatedaproductofgluconicacid.FrankeandLorenz[ 54 ]foundthathydrogenperoxidewasalsoaproductoftheoxidationofglucosebyglucoseoxidaseaswellassuggestedthattheenzymeglucoseoxidasecontainedaavoprotein.TheglucoseoxidaseenzymecatalyzestheoxidationofD-glucosetoD-glucono--lactonewhichchangesbyanon-enzymaticreactiontogluconicacidwiththepresenceofwater[ 55 ].Inthe1960'sabigscienticpushwasmadetounderstandthekineticsofglucoseoxidase.NakamuraandOgura[ 56 ]lookedattheactionofglucoseoxidase,namelythattheproductD-glucono--lactonecanactasaninhibitor.Glucosewasfoundtoreactmuchfasterthenothersugarsinthepresenceofglucoseoxidase[ 57 ].NakamuraandOguraandBrightandGibson[ 58 59 ]believedpartoftheoxidationofglucosebygluconicacidwaspartiallyreversible.AnoverviewofGlucoseOxidaseandtheuseofglucoseoxidaseinfoodsciencerelatedareaswaspublishedbyBanker[ 60 ]. 40

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Themechanismfortheenzymaticreactionconsumingglucoseusingtheglucoseoxidaseenzymetoproducehydrogenperoxideusedinthisdissertationwas G+GOxkf1)439()222()439(! )351()]TJ /F5 11.9552 Tf 3.321 -8.302 Td[(kb1GOx2)]TJ /F1 11.9552 Tf 11.955 0 Td[(GAkf2)439()222()439(!GA+GOx2(2{44) GOx2+O2kf3)439()222()439(! )351()]TJ /F5 11.9552 Tf 3.321 -8.303 Td[(kb3GOx)]TJ /F1 11.9552 Tf 11.955 0 Td[(H2O2kf4)439()222()439(!GOx+H2O2(2{45)whereGrepresentsglucose,GOxistheoxidizedformofglucoseoxidase,GOx2-GAistherstenzymecomplexthatiscreatedbyglucoseandGOx,GOx2isthereducedformofglucoseoxidase,GAisgluconicacid,O2isoxygen,GOx)]TJ /F1 11.9552 Tf 11.955 0 Td[(H2O2isthesecondenzymecomplexformed,andH2O2ishydrogenperoxide.Enzymaticreactions,fromreactanttoproduct,iscommonlythoughtofasanirreversibleprocess.Byexplicitlyconsideringtheenzymecomplexwecansaythatthereactionfromreactanttoproductisirreversiblebutwithareversiblerststep.Withtherevolutionofcoatingelectrodeswithenzymes[ 49 61 62 ]itbecamepossibletocoatanelectrodewithglucoseoxidaseandthenmeasuretheproductofH2O2electrochemicallyandthusmeasurebloodglucoselevels.Tofurthertheunderstandingofanenzymecoatedelectrodethekineticsofimmobilizedglucoseoxidase[ 63 ]anddiusioncoecientsinhydrogelsforoxygen,hydrogenperoxideandglucose[ 64 ]werestudiedbyvanStroe-Biezenetal.TemperaturedependenceofdiusioncoecientsofoxygeninwaterwerestudiedbyHanandBartels[ 65 ].Somestudieslookatadirectelectricalconnectiontotheglucoseoxidaseenzyme[ 50 66 ].Since1974,thearticialpancreashasbeenacknowledgedbythegreaterscienticcommunitytobeapossiblewaytomanagetype1diabetes[ 67 ].In2013and2008,casestudiesshowedthatasubcutaneouscontinuousglucosemonitorsensorscouldbeimplantedfor14days[ 68 ]and28days[ 69 ]inpatientswithtype1diabetes.However,commerciallyavailablesubcutaneouscontinuousglucosemonitorsaregenerallychanged 41

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everysevendaystoavoidunreliableresults.Somereasonsforfailurearesensoranalysisfailure[ 70 ],cellbasedmetabolicbarrierslikemacrophages[ 71 ],andredbloodcellclotsormetabolicsinks[ 72 73 ].ThepotentialofimpedancespectroscopytomonitorthestateofhealthofsubcutaneouscontinuousglucosemonitorsensorshasnotbeenfullyexploredandnowitissuggestedthatthisisaplausiblewaytoseeiftheCGMisstillworking.ThestatusofimplantedbiologicalsensorshasbeenstudiedusingEIS[ 74 75 76 ]. 42

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CHAPTER3INFLUENCEOFFINITESCHMIDTNUMBERONTHEIMPEDANCEOFDISKELECTRODESTherotatingdiskandsubmergedimpingingjetelectrodesarecommonlyusedinelectrochemicalexperiments.Inelectrochemicalsystemswherethereisnoconvectiveow,temperatureandconcentrationgradientscancreateanaturalconvection,whichishardtocharacterize.Characterizingtheuidowinanelectrochemicalsystemhasadvantages;theoryandmathematicalmodelsareeasiertoworkwithandsimulationsofthesystemaremucheasierthanwithnaturalconvection.Simulationsofarotatingdiskandasubmergedimpingingjetuseaconvective-diusionimpedancedependentontheSchmidtnumber,Sc==Di.WhentheSchmidtnumberisconsideredinnitetheproblemcanbesolvedanalytically.ThischapterexplorescaseswhentheSchmidtnumberisequaltoinnityaswellasanitevalueforbothsystems. 3.1ElectrochemicalMathematicalEquationsTheuxdensityofeachspeciesinaninnitelydiluteelectrochemicalsystemisgivenby Ni=)]TJ /F5 11.9552 Tf 9.299 0 Td[(ziuiFcir)]TJ /F5 11.9552 Tf 11.955 0 Td[(Dirci+civ(3{1)andhasunitsofmol=(cm2s)[ 77 ].Therstterminequation( 3{1 )representsmigrationofthespeciesi,whichcanoccurifanelectriceldispresentandifthespeciesischarged.Thesecondtermrepresentsthediusionofthespecies.Itaccountsfordeviationsfromtheaveragevelocityifdiusionisduetoaconcentrationgradient,rciandDiisthediusioncoecientofspeciesi.Thelastterm,convection,isduetothebulkvelocity,v.Inequation( 3{1 ),FisFaraday'sconstantwithavalueof96,487C/equiv,istheelectrostaticpotentialwherethegradientisthenegativeelectriceld,andzirepresentsthenumberofprotonchargescarriedbyanion.Themobilityofthespecies,ui,isrelatedtothediusioncoecientusingtheNernst-Einsteinequation Di=RTui(3{2) 43

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Figure3-1. Rotatingdiskelectrodeowpatternshowingathree-dimensionalowtrajectoryleadingtoanetowtowardsthediskandintheradialdirection[ 3 ]. whereRistheuniversalgasconstantequalto8.3143J/molKandTistemperatureinKelvin.Substitutionofthemobilityintoequation( 3{1 )gives Ni=Di)]TJ /F5 11.9552 Tf 9.298 0 Td[(ziF RTcir)-222(rci+civ(3{3)Amaterialbalanceisneededtoobtainthemassuxattheinterface @ci @t=rNi+Ri(3{4)whereRiisthehomogeneousreaction.Eachspeciesinthesystemwillhavetheirownformofequation( 3{4 ). 3.2RotatingDiskElectrodeAschematicoftheowinarotatingdiskelectrodesystemisshowninFigure 3-1 .Theuidisspiraledtowardtheelectrodeandanetvelocityoccursperpendiculartotheelectrodeandintheradialdirection. 44

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3.2.1FluidVelocityProleforRotatingDiskElectrodeFollowingsection( 2.3.1 )theuidowforarotatingdiskelectrodecanbedescribedbyavelocityexpansionintheaxialdirection,seeequation( 2{37 ).WithoutahomogeneousreactionoccurringinanRDEsystemthereisnoneedforavelocityexpansionthatgoesfurtherthanthatdescribedbyequation( 2{37 ).Thismeansthevelocityproleforthesystemisdescribedbythecombinationofequation( 2{34 ),theaxialvelocityandequation( 2{37 ),theseriesexpansionfortheaxialdirectionforsmallvaluesof,toobtain vy=p ()]TJ /F5 11.9552 Tf 9.299 0 Td[(a2+1 33+b 64+:::)(3{5) 3.2.2MathematicalDevelopmentforRotatingDiskElectrodeElectrochemicalimpedancespectroscopyinvolvestheperturbationofanelectrochemicalsystemwithasmallsinusoidalsignalandthenrecordingtheoutput.Alloscillatingquantities,suchasconcentrationandpotential,canbewrittenintheform X=X+Ren~Xexp(j!t)o(3{6)wheretheover-barrepresentsthesteady-statevalue,jistheimaginarynumber,!istheangularfrequencyandthetilderepresentsacomplexoscillatingcomponent.Thegoverningequationformasstransferofarotatingdiskintheabsenceofhomogeneousreactionbecomes @ci @t=rNi(3{7)ThedivergenceoperatorofNiincylindricalformis rNi=1 r@(rNr;i) @r+1 r@(N;i) @+@(Ny;i) @y(3{8)whereristheradialcomponent,istheangularcomponent,andyistheaxialcomponent. 45

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Whenapplyingequation( 3{8 )to( 3{7 )andassumingantisymmetricowthegoverningequationbecomes @ci @t=1 r@(rNr;i) @r+@(Ny;i) @y(3{9)Fluxintheydirection,excludingmigration,is Ny;i=)]TJ /F5 11.9552 Tf 9.298 0 Td[(Di@ci @y+civy(3{10)anduxintherdirection,excludingmigration,is Nr;i=civr(3{11)Whensubstitutingtheuxexpressions,equations( 3{11 )and( 3{10 ),fortherandydirectionsrespectively,into( 3{9 )thegoverningequationisnow @ci @t=)]TJ /F1 11.9552 Tf 10.494 8.088 Td[(1 r@ @r(rcivr))]TJ /F5 11.9552 Tf 16.219 8.088 Td[(@ @y()]TJ /F5 11.9552 Tf 9.298 0 Td[(Di@ci @y+civy)(3{12)Aftercalculatingthederivativesin( 3{12 )andinsertingthersttermofthevelocityexpansionforvy vy=)]TJ /F5 11.9552 Tf 9.299 0 Td[(ay3=2 1=2(3{13)andvr vr=ray3=2 1=2(3{14)wherea=0:5102326189,istherotationspeedandisthekinematicviscosity,thegoverningequationbecomes @ci @t=)]TJ /F5 11.9552 Tf 9.298 0 Td[(ci r2ra3=2 1=2y+ci2a3=2 1=2y)]TJ /F10 11.9552 Tf 11.955 0 Td[(vy@ci @y+Di@2ci @y2(3{15)Thersttwotermsin( 3{15 )cancelandthenalgoverningequationforconvectivediusionbecomes @ci @t=)]TJ /F5 11.9552 Tf 9.299 0 Td[(vy@ci @y+Di@2ci @y2(3{16) 46

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Iftheexpansionsusedmorethanthersttermtheywouldalsocausecancelationandthenalgoverningequationforconvectivediusionremainsthesame.Inordertoobtainimpedancetheoscillatingconcentrationsofthereactingspeciesinanelectrochemicalsystemisneeded.Whenapplyingequation( 3{6 )to( 3{16 )wegetanequationwithseparablesteady-stateandoscillatingvariables j!~ciej!t+vydci dy+vyd~ci dyej!t)]TJ /F5 11.9552 Tf 11.955 0 Td[(Did2ci dt)]TJ /F5 11.9552 Tf 11.955 0 Td[(Did2~ci dtej!t=0(3{17)Theequationforthesteady-statevariablesis vydci dy)]TJ /F5 11.9552 Tf 11.955 0 Td[(Did2ci dt=0(3{18)andtheequationfortheoscillatingvariablesis j!~ciej!t+vyd~ci dyej!t)]TJ /F5 11.9552 Tf 11.955 0 Td[(Did2~ci dtej!t=0(3{19)Theexponentialtermej!tin( 3{19 )canbecanceled,thuseliminatingtheexplicitdependenceontime. j!~ci+vyd~ci dy)]TJ /F5 11.9552 Tf 11.955 0 Td[(Did2~ci dt=0(3{20)Theconvective-diusionequationismadedimensionlessbyadimensionlessposition =y=i(3{21)whereiisdenedas i=3Di 1=3=3 a1=31 Sc1=3ir (3{22)adimensionlessconcentrationi=~ci=~ci(0),andadimensionlessfrequencygivenby Ki=! 9 a2Di1=3=! 9 a21=3Sc1=3(3{23)Aftersubstitution,thedimensionlessconvective-diusionequationis d2i d2+32di d)]TJ /F1 11.9552 Tf 11.955 0 Td[(jKii=0(3{24) 47

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Undertheassumptionthattheaxialvelocityisgivenbythethree-termexpressionvalidclosetotheelectrodesurface,equation( 3{5 ),andismadedimensionlessbyequation( 3{21 ),equation( 3{24 )maybeexpressedas d2i d2+ 32)]TJ /F8 11.9552 Tf 11.955 16.857 Td[(3 a41=33 Sc1=3i)]TJ /F5 11.9552 Tf 13.588 8.088 Td[(b 63 a5=34 Sc2=3i!di d)]TJ /F1 11.9552 Tf 11.955 0 Td[(jKii=0(3{25)Fromequation( 3{25 ),threecoupledequationsforaniteSchmidtnumbercanbeobtainedusing i(;Sci;K)=i;0(;K)+i;1(;K) Sc1=3i+i;2(;K) Sc2=3i+:::(3{26)Therstisequation,notdependentoftheSchmidtnumber,is d2i;0 d2+32di;0 d)]TJ /F1 11.9552 Tf 11.956 0 Td[(jKii;0=0(3{27)Theothertwoequations,dependentonSc)]TJ /F3 7.9701 Tf 6.586 0 Td[(1=3andSc)]TJ /F3 7.9701 Tf 6.586 0 Td[(2=3are d2i;1 d2+32di;1 d)]TJ /F1 11.9552 Tf 11.955 0 Td[(jKii;1=3 a41=33di;0 d(3{28)and d2i;2 d2+32di;2 d)]TJ /F1 11.9552 Tf 11.955 0 Td[(jKii;2=b 63 a5=34di;0 d+3 a41=33di;1 d(3{29)wheretermsorderSc)]TJ /F3 7.9701 Tf 6.586 0 Td[(1andgreaterareneglected.Theboundaryconditionsare i;0!0;i;1!0;i;2!0as!1(3{30)and i;0=1;i;1=0;i;2=0as=0(3{31)Thevalueof1;0(0)waschosenarbitrarilybecausethegoverningequationsfortheimpedanceresponsearelinear,evenwhenthesteady-stateproblemisnon-linear.AstheSchmidtnumberapproachesinnite,equation( 3{27 )describesthesystem.ForniteSchmidtnumbers,solutionsofequations( 3{27 )-( 3{29 )areneededtoobtainaccuratevaluesfordiusionimpedance. 48

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Thediusionimpedanceforarotatingdiskelectrode,foraniteSchmidtnumber,isgivenby )]TJ /F1 11.9552 Tf 21.991 8.088 Td[(1 0i(0)=1 0i;0(0)+0i;1(0)Sci)]TJ /F3 7.9701 Tf 6.586 0 Td[(1=3+0i;2(0)Sci)]TJ /F3 7.9701 Tf 6.587 0 Td[(2=3(3{32)whichcanbeexpressedas )]TJ /F1 11.9552 Tf 21.991 8.087 Td[(1 0i(0)=)]TJ /F1 11.9552 Tf 22.493 8.087 Td[(1 0i;0(0)+0i;1(0) (0i;0(0))21 Sc1=3i)]TJ /F1 11.9552 Tf 25.149 8.087 Td[(1 0i;0(0)0i;1(0) 0i;0(0)2)]TJ /F5 11.9552 Tf 13.151 9.167 Td[(0i;2(0) 0i;0(0)1 Sc2=3i(3{33)suchthat Z(0)=)]TJ /F1 11.9552 Tf 22.492 8.088 Td[(1 0i;0(0)(3{34) Z(1)=0i;1(0) (0i;0(0))2(3{35)and Z(2)=)]TJ /F1 11.9552 Tf 22.493 8.088 Td[(1 0i;0(0)0i;1(0) 0i;0(0)2)]TJ /F5 11.9552 Tf 13.15 9.168 Td[(0i;2(0) 0i;0(0)(3{36)Theconvective-diusionimpedanceisobtaineddirectlyasafunctionoftheSchmidtnumberfrom )]TJ /F1 11.9552 Tf 21.991 8.088 Td[(1 0i(0)=Z(0)+Z(1) Sc1=3i+Z(2) Sc2=3i(3{37) 3.2.3NumericalMethodsforRotatingDiskElectrodeMATLABisapowerfulsimulationenvironmentchosentosolvethecomplex,non-linearcoupleddierentialequationsofanelectrochemicalsystem.ThealgorithmBANDdevelopedbyNewman[ 77 ]in1968inFortran,wasconvertedintoaMATLABcodeandiscalledintheprogramsdevelopedtosolveelectrochemicaldierentialequations.BANDisdescribedindetailinAppendix A .TheMATLABcodeswrittentosolveequation( 3{25 )areshowninAppendix B 3.2.4Convective-DiusionImpedanceforaRotatingDiskElectrodeThedimensionlessoscillatingconcentration,i=~ci=~ci(0),isacomplexquantityandresultsfori;0,i;1andi;2arepresentedinFigure 3-2 .TheseresultspresentedinFigure 3-2A wereobtainedfromsolvingforequation( 3{27 ).TheresultspresentedinFigure 49

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A B CFigure3-2. Oscillatingdimensionlessconcentrationsforarotatingdiskelectrodeoverarangeoffrequenciesforboththerealandimaginarycomponentsfora)i;0,b)i;1,andc)i;2.ThevaluesofK,arrowindicatingincreasingvaluesofK,rangedfrom10)]TJ /F3 7.9701 Tf 6.587 0 Td[(2to102with10pointsperdecade. 3-2B camefromsolvingequation( 3{28 ),however,theseresultsdependonthesolutionofequation( 3{27 ).TheresultspresentedinFigure 3-2C camefromsolvingequation( 3{29 )andisdependentonthesolutionsforbothequation( 3{27 )and( 3{28 ).ItisclearfromdecreasingmagnitudeoftheoscillatingconcentrationvaluesinFigure 3-2A to 3-2C thatthersttermisthemostimportantfordeterminingtheimpedanceandimportancedecreaseswithdependenceontheSchmidtnumber. 50

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ThecontributiontotheimpedanceforeachtermispresentedinFigure 3-3 .Theseplotswerecreatedfromequations( 3{34 )to( 3{36 ).TheNyquistplotofallthreeZivaluesshowsclearlythatZ0resemblesthehyperbolictangentmodelusedtodescribediusionthroughalm(noconvectiveterm).ThecharacteristicfrequencyforZ1isabouthalfthatofZ0andthecharacteristicfrequencyforZ2isevensmaller.Thiscanbeseenbetterwhenlookingattheimaginaryimpedanceasafunctionoffrequency,presentedinFigure 3-3C .TherealimpedanceasafunctionoffrequencyaswellastheNyquistplotbothshowthemagnitudeofthedimensionlessdiusionimpedanceismuchlargerforZ0thenforZ1orZ2.AcomparisonoftheimpedanceresponseobtainedundertheassumptionofaninniteSchmidtnumberandaniteSchmidtnumber(Sc=1000andSc=100)ispresentedinFigure 3-4 .ThesolutionsgeneratedwereinagreementwithOrazemandTribollet[ 5 ].Thelowfrequencyvaluesshowthegreatestdierence,seeFigure 3-4A ,betweentheassumptionofusinganinniteSchmidtnumber.ThedierencesinlowfrequencyvaluesarealsoshowninFigrues 3-4B and 3-4C .TheerroratlowfrequencyofthedimensionlessdiusionimpedanceforarotatingdiskelectrodewithdierentSchmidtnumbersisshowninTable 3-1 .Thepercenterrorwascalculatedusing %error=)]TJ /F1 11.9552 Tf 24.682 8.088 Td[(1 0i;r(0)!!0)]TJ /F5 11.9552 Tf 11.955 0 Td[(Z(0);r(!!0))]TJ /F1 11.9552 Tf 24.682 8.088 Td[(1 0i;r(0)!!0(3{38)whereZ(0);ristherealcomponentofequation( 3{34 ).IftheinniteSchmidtnumberanalysisisusedtocalculatetheconvective-diusionimpedanceandtheSchmidtnumberinthesystemisontheorderof100,thelowfrequencypartoftheimpedancewillbeinaccurateupto6.62%.WithaSchmidtnumberof1000theerrorisstillasgreatas3.03%.Thedimensionlessconvective-diusionimpedanceanalyticalexpression(foraninniteSchmidtnumber)shouldnotbeusedforsmallvaluesoftheSchmidtnumberespeciallywhenmasstransferplaysalargeroleintheelectrochemicalsystem. 51

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A B CFigure3-3. ContributionstotheimpedancefromseriesexpansionforniteSchmidtnumber,Z(0),Z(1),andZ(2)shownina)Nyquistform,b)realpartofthedimensionlessdiusionimpedanceasafunctionoffrequency,andc)imaginarypartofthedimensionlessdiusionimpedanceasafunctionoffrequency.calculatedfromequations( 3{34 )-( 3{36 ). 52

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A B CFigure3-4. DimensionlessdiusionimpedanceobtainedforarotatingdiskunderassumptionofinniteSchmidtnumberandforaniteSchmidtnumberequalto100and1000showninA)Nyquistform,B)realpartofthedimensionlessdiusionimpedanceasafunctionoffrequency,andC)imaginarypartofthedimensionlessdiusionimpedanceasafunctionoffrequency. Table3-1. ErroratlowfrequencyinconvectivediusionimpedanceforarotatingdiskelectrodewhencomparedtousinginniteSchmidtnumberanalysis Sc%Error 100001.4010003.031006.62508.39 53

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Figure3-5. Submergedimpingingjetowdiagrama)schematicillustrationandb)identicationofowregimes[ 3 ]. 3.3SubmergedImpingingJetElectrodeThesubmergedimpingingjetelectrode,Figure 3-5 ,thoughlesspopularthentherotatingdiskelectrode,isalsoveryattractivetocharacterizeelectrochemicalsystems.Withastagnationregion,showninFigure 3-5 B,theconvectivediusionofuidtowardstheelectrodeisuniform,similartotherotatingdiskelectrode.Foranelectrodethatisentirelyinthestagnationregion,themass-transferrateisuniform.Incontrasttotherotatingdisk,theelectrodeisstationaryandisthereforesuitableforinsituobservation[ 3 ]. 3.3.1VelocityExpansionforaSubmergedImpingingJetTheuidowwithinthestagnationregionoftheelectrodeinanimpingingjetcelliswell-dened[ 78 79 80 81 82 83 ].Thestagnationregionisdenedtobetheregionsurroundingthestagnationpointinwhichtheaxialvelocity,givenby vy=)]TJ 9.299 8.298 Td[(p aIJ()(3{39)isindependentoftheradialvelocity vr=aIJ 2r() d(3{40) 54

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whereaIJisahydrodynamicconstantthatisdependentongeometryanduidvelocity,yandraretheaxialandradialdirections,isthekinematicviscosityandisastreamfunctiongivenintermsofadimensionlessaxialposition =yr aIJ (3{41)as[ 78 ] ()=1:3522)]TJ /F1 11.9552 Tf 13.151 8.088 Td[(1 33+7:288810)]TJ /F3 7.9701 Tf 6.587 0 Td[(36+:::(3{42)Thestagnationregionextendstoaradialdistancethatisapproximatelythesizeoftheimpingingjetnozzle[ 79 ]. 3.3.2MathematicalDevelopmentforImpingingJetElectrodeUndertheassumptionthattheelectrodeisuniformlyaccessible,theequationgoverningmasstransferinthefrequencydomainisgivenby d2i d2+ 32)]TJ /F8 11.9552 Tf 11.955 16.857 Td[(3 1:35241=33 Sc1=3i!di d)]TJ /F1 11.9552 Tf 11.955 0 Td[(jKii=0(3{43)where Ki=! aIJ9 1:35221=3Sc1=3i(3{44)Theaxialdirectionismadedimensionlessusingequation( 3{21 ).Thedimensionlessdistancefortheimpingingjetissetby i=3 1:3521=3r aIJSc1=3i(3{45)andtheoscillatingconcentrationismadedimensionlessinthesamewayasintherotatingdiskelectrodecasepreviouslydescribed,i=eci=eci(0).Asdonefortherotatingdiskelectrode,theimpingingjetconvective-diusionequationcanbeexpressedasaseriesexpansion,seeequation( 3{26 ),whereasolutiontoeachoscillatingconcentrationcanbefoundfrom d2i;0 d2+32di;0 d)]TJ /F1 11.9552 Tf 11.956 0 Td[(jKii;0=0(3{46) 55

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d2i;1 d2+32di;1 d)]TJ /F1 11.9552 Tf 11.955 0 Td[(jKii;1=3 1:35241=33di;0 d(3{47)and d2i;2 d2+32di;2 d)]TJ /F1 11.9552 Tf 11.955 0 Td[(jKii;2=3 1:35241=33di;1 d(3{48)TheindividualcontributionstoimpedanceseparatedbyinuenceofSchmidtnumberare Z(0)=)]TJ /F1 11.9552 Tf 22.492 8.088 Td[(1 0i;0(0)(3{49) Z(1)=0i;1(0) (0i;0(0))2(3{50)and Z(2)=)]TJ /F1 11.9552 Tf 22.493 8.088 Td[(1 0i;0(0)0i;1(0) 0i;0(0)2)]TJ /F5 11.9552 Tf 13.15 9.168 Td[(0i;2(0) 0i;0(0)(3{51)Theconvective-diusionimpedanceisobtaineddirectlyasafunctionoftheSchmidtnumberfrom )]TJ /F1 11.9552 Tf 21.991 8.088 Td[(1 0i(0)=Z(0)+Z(1) Sc1=3i+Z(2) Sc2=3i(3{52) 3.3.3NumericalMethodsforImpingingJetElectrodeSimilaritytotherotatingdiskelectrode,MATLABwasthechosensimulationenvironmenttosolvethecomplex,non-linearcoupleddierentialequationsforanimpingingjetelectrode.TheMATLAB-converted-BANDalgorithmiscalledintheprogramsdevelopedtosolveelectrochemicaldierentialequations.BANDisdescribedindetailinAppendix A .TheMATLABcodeswrittentosolveequation( 3{43 )areshowninAppendix C 3.3.4Convective-DiusionImpedanceforaSubmergedImpingingJetElec-trodeAsfortherotatingdiskelectrode,thedimensionlessoscillatingconcentrationfortheimpingingjetelectrode,i=~ci=~ci(0),isacomplexquantityandresultsfori;0,i;1andi;2(fromequations( 3{46 )-( 3{48 ))arepresentedinFigure 3-6 .TheresultspresentedinFigure 3-6A wereobtainedfromsolvingforequation( 3{46 ).Theresultspresented 56

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A B CFigure3-6. Oscillatingdimensionlessconcentrationsforasubmergedimpingingjetelectrodeoverarangeoffrequenciesforboththerealandimaginarycomponentsfora)i;0,b)i;1,andc)i;2.ThevaluesofK,arrowindicatingincreasingvaluesofK,rangedfrom10)]TJ /F3 7.9701 Tf 6.587 0 Td[(2to102with10pointsperdecade. inFigure 3-6B camefromsolvingequation( 3{47 ),however,theseresultsdependonthesolutionofequation( 3{46 ).TheresultspresentedinFigure 3-6C camefromsolvingequation( 3{48 )andaredependentonthesolutionsforbothequation( 3{46 )and( 3{47 ).TheoscillatingconcentrationpresentedinFigure 3-6A isthesameasfortherotatingdiskelectrode,Figure 3-2A .Thesecondandthirdoscillatingconcentrationplots,Figures 3-6B and 3-6C ,however,showanevensmallermagnitudethanthesecondandthirdoscillating 57

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Table3-2. ErroratlowfrequencyinconvectivediusionimpedanceforasubmergedimpingingjetelectrodewhencomparedtousinginniteSchmidtnumberanalysis Sc%Error 100000.3810000.831001.82502.31 concentrationsfortherotatingdiskelectrode.Thisimpliesthatthesecondandthirdtermsinthevelocityexpansionwillhavelessimpactontheimpedanceresults.ThecontributiontotheimpedanceforeachtermispresentedinFigure 3-7 .Theseplotswerecreatedfromequations( 3{49 )to( 3{51 ).TheNyquistplotofallthreeZivaluesshowsclearlythatZ0resembles,butisnotidenticalto,thehyperbolictangentmodelusedtodescribediusionthroughalm(noconvectiveterm).ThecharacteristicfrequencyforZ1isabouthalfthatofZ0andthecharacteristicfrequencyforZ2isevensmaller.Thiscanbeseenbetterwhenlookingattheimaginaryimpedanceasafunctionoffrequency,presentedinFigure 3-7C .TherealimpedanceasafunctionoffrequencyaswellastheNyquistplotbothshowthemagnitudeofthedimensionlessdiusionimpedanceismuchlargerforZ0thenforZ1orZ2.TheimpedancesthatdependontheSchmidtnumber,Z1orZ2,areevensmallerthenfortherotatingdiskelectrodesystem.AcomparisonoftheimpedanceresponseobtainedundertheassumptionofaninniteSchmidtnumberandaniteSchmidtnumber(Sc=1000andSc=100)ispresentedinFigure 3-8 .Thelowfrequencyvaluesshowthegreatestdierence,seeFigure 3-8A ,betweentheassumptionofusinganinniteSchmidtnumber.ThedierencesinlowfrequencyvaluesarealsoshowninFigrues 3-8B and 3-8C .Thedierencesinlowfrequencyarelessthenwhenusingarotatingdiskelectrode.TheerroratlowfrequencyofthedimensionlessdiusionimpedanceforarotatingdiskelectrodewithdierentSchmidtnumbersisshowninTable 3-2 .Thepercenterrorwascalculatedusingequation( 3{38 ). 58

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A B CFigure3-7. ContributionstotheimpedancefromseriesexpansionforniteSchmidtnumber,Z(0),Z(1),andZ(2)shownina)Nyquistform,b)realpartofthedimensionlessdiusionimpedanceasafunctionoffrequency,andc)imaginarypartofthedimensionlessdiusionimpedanceasafunctionoffrequency.calculatedfromequations( 3{49 )-( 3{51 ). 59

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IftheinniteSchmidtnumberanalysisisusedtocalculatetheconvective-diusionimpedanceandtheSchmidtnumberinthesystemisontheorderof100,thelowfrequencypartoftheimpedancewillbeinaccurateupto1.82%.WithaSchmidtnumberof1000theerroris0.83%.Thedimensionlessconvective-diusionimpedanceanalyticalexpression(foraninniteSchmidtnumber)iswilllikelybewithin2%oftheexactanswer. 60

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A B CFigure3-8. DimensionlessdiusionimpedanceobtainedforanimpingingjetelectrodeundertheassumptionofinniteSchmidtnumberandforaniteSchmidtnumberequalto100and1000showninA)Nyquistform,B)realpartofthedimensionlessdiusionimpedanceasafunctionoffrequency,andC)imaginarypartofthedimensionlessdiusionimpedanceasafunctionoffrequency. 61

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CHAPTER4CONVECTIVE-DIFFUSIONIMPEDANCEWITHHOMOGENEOUSCHEMICALREACTIONSAmathematicalmodelwasdevelopedfortheimpedanceresponseassociatedwiththecoupledhomogeneouschemicalandheterogeneouselectrochemicalreactions.ThemodelincludesahomogeneousreactionintheelectrolytewherespeciesABreactsreversiblytoformA)]TJ /F1 11.9552 Tf 10.987 -4.338 Td[(andB+andB+reactselectrochemicallyonarotatingdiskelectrodetoproduceB.Theresultingconvectivediusionimpedancehastwoasymmetriccapacitiveloops,oneassociatedwithconvectivediusionimpedancetheotherwiththehomogeneousreaction.Foraninnitelyfasthomogeneousreaction,thesystemisshowntobehaveasthoughABistheelectroactivespecies.AmodiedGerischerimpedancewasfoundtoprovideagoodttothesimulateddata. 4.1MathematicalDevelopmentforConvectiveDiusionandHomogeneousReactionAmathematicalmodelispresentedbelowfortheimpedanceresponseassociatedwiththecouplingofhomogeneousandheterogeneouselectrochemicalreactions. 4.1.1GoverningEquationsTheuxdensityofspeciesiinadiluteelectrolyteandinabsenceofmigrationwasexpressedas Ni=)]TJ /F5 11.9552 Tf 9.299 0 Td[(Dirci+civ(4{1)whereDiisthediusioncoecientincm2=s,ciistheconcentrationofspeciesiinmol=cm3,andvisthemassaveragedvelocityincm=s.[ 77 ]Foranaxisymmetricrotating-diskelectrode,theconvective-diusionequationwithhomogeneousreactionwasexpressedincylindricalcoordinatesas @ci @t+vy@ci @y=Di@2ci @y2+Ri(4{2)whereRiistherateofproductionofspeciesibyhomogeneousreactions,yistheaxialdirectionincmmeasuredawayfromtheelectrodesurface,andvyisthevelocityintheaxialdirection. 62

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Figure4-1. Adiagramofanelectrochemicalreactioncoupledbytheinuenceofahomogeneouschemicalreaction. 4.1.2HomogeneousReactionAdiagramofanelectrochemicalreactioncoupledwithachemicalreactionisshowninFigure 4-1 .wherethereactionsmaybeexpressedas ABkf)312()312(! )]TJ /F5 11.9552 Tf 3.321 -8.302 Td[(kbA)]TJ /F1 11.9552 Tf 9.742 -4.936 Td[(+B+(4{3)wherekfandkbarerateconstantsforthehomogeneousreaction.ThespeciesB+wasassumedtobeelectroactiveandconsumedattheelectrodefollowing B++e)]TJ /F2 11.9552 Tf 10.405 -4.937 Td[(!B(4{4)Thecorrespondingcurrentdensityfromtheelectrochemicalreaction,whichisdependentonconcentrationandpotential,wasexpressedas iB+=)]TJ /F5 11.9552 Tf 9.299 0 Td[(KB+cB+(0)exp()]TJ /F5 11.9552 Tf 9.299 0 Td[(bB+V)(4{5)whereKB+istherateconstant,bB+isthetransfercoecient,cB+(0)istheconcentrationofB+attheelectrodesurfaceandVistheappliedpotentialreferencedtoanelectrodeinnitelyfarfromtheelectrode.Astheelectroactivespeciesisconsumed,aconcentrationgradientofB+mustexistneartheelectrodesurface. 63

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Farfromtheelectrodesurface,thespeciesAB,A)]TJ /F1 11.9552 Tf 7.085 -4.338 Td[(,andB+areassumedtobeequilibrated;thus, Keq=kf kb=cA)]TJ /F1 11.9552 Tf 6.752 -.299 Td[((1)cB+(1) cAB(1)(4{6)Thereactiontermwasexpressedas RA)]TJ /F1 11.9552 Tf 10.073 -.299 Td[(=RB+=)]TJ /F5 11.9552 Tf 9.298 0 Td[(RAB=kfcAB(y))]TJ /F5 11.9552 Tf 11.955 0 Td[(kbcA)]TJ /F1 11.9552 Tf 6.752 -.299 Td[((y)cB+(y)(4{7)wherekfhasunitsofinversetimeandkbhasunitsofinverseconcentrationpertime,i.e.,cm3=molsCombinationofequations( 4{6 )and( 4{7 )yields RA)]TJ /F1 11.9552 Tf 10.073 -.299 Td[(=RB+=)]TJ /F5 11.9552 Tf 9.298 0 Td[(RAB=kb(KeqcAB(y))]TJ /F5 11.9552 Tf 11.955 0 Td[(cA)]TJ /F1 11.9552 Tf 6.753 -.299 Td[((y)cB+(y))(4{8)Inequation( 4{8 ),kbwasindependentlyexploredandKeqwasassumedtobeconstantandcalculatedfrombulkconcentrationsofspeciesinvolvedinthehomogeneousreaction.Theformofequation( 4{8 )makesthesteady-stateproblemnonlinear. 4.1.3VelocityExpressionThevelocityproleusedinequation( 4{2 )iscreatedfromequation( 2{34 ).Itincludestheinterpolationfunction,equation( 2{41 ),withweightingparameterf,equation( 2{42 ),tocreateafunctionthatusesthevelocityexpansionforclosetotheelectrode,equation( 2{37 ),andfarfromtheelectrode,equation( 2{40 ).Thenalvelocityexpressionincludingallthedetailsaboveis vy=1)]TJ /F1 11.9552 Tf 54.48 8.088 Td[(1 1+e)]TJ /F7 7.9701 Tf 6.586 0 Td[((yp =)]TJ /F7 7.9701 Tf 6.587 0 Td[(0)p )]TJ /F5 11.9552 Tf 11.955 0 Td[(ay2 +1 3y3 3+b 6y4 (4{9) +1 1+e)]TJ /F7 7.9701 Tf 6.587 0 Td[((yp =)]TJ /F7 7.9701 Tf 6.587 0 Td[(0)p ()]TJ /F5 11.9552 Tf 11.955 0 Td[(+2A exp)]TJ /F5 11.9552 Tf 11.955 0 Td[(yr )]TJ /F5 11.9552 Tf 20.457 8.087 Td[(A2+B2 23exp)]TJ /F1 11.9552 Tf 11.955 0 Td[(2yr )]TJ /F5 11.9552 Tf 13.151 8.087 Td[(A(A2+B2) 65exp)]TJ /F1 11.9552 Tf 11.955 0 Td[(3yr )Theaxialvelocityisshowningure 2-8C ,wherethesolidlinerepresentsthevelocityexpressionabove. 64

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4.1.4ImpedancewithHomogeneousChemicalReactionsTheconservationequationforeachspeciesmaybewrittenas @cAB @t+vy@cAB @y=DAB@2cAB @y2)]TJ /F5 11.9552 Tf 11.956 0 Td[(kb(KeqcAB(y))]TJ /F5 11.9552 Tf 11.955 0 Td[(cA)]TJ /F1 11.9552 Tf 6.752 -.299 Td[((y)cB+(y))(4{10)forAB, @cA)]TJ ET q .4782 w 76.204 -129.023 m 101.167 -129.023 l S Q BT /F5 11.9552 Tf 83.157 -140.212 Td[(@t+vy@cA)]TJ ET q .4782 w 128.72 -129.023 m 153.684 -129.023 l S Q BT /F5 11.9552 Tf 134.719 -140.212 Td[(@y=DA)]TJ /F5 11.9552 Tf 7.948 7.789 Td[(@2cA)]TJ ET q .4782 w 194.636 -129.023 m 224.331 -129.023 l S Q BT /F5 11.9552 Tf 200.532 -140.212 Td[(@y2+kb(KeqcAB(y))]TJ /F5 11.9552 Tf 11.955 0 Td[(cA)]TJ /F1 11.9552 Tf 6.752 -.299 Td[((y)cB+(y))(4{11)forA)]TJ /F1 11.9552 Tf 7.085 -4.338 Td[(,and @cB+ @t+vy@cB+ @y=DB+@2cB+ @y2+kb(KeqcAB(y))]TJ /F5 11.9552 Tf 11.955 0 Td[(cA)]TJ /F1 11.9552 Tf 6.753 -.299 Td[((y)cB+(y))(4{12)forB+.Theboundaryconditionsfarfromtheelectrodewere ci!ci(1)fory!1(4{13)andtheboundaryconditionsattheelectrodesurfacewere @ci @yy=0=0fory=0(4{14)forthenon{reactingspecies,and FDB+@cB+ @yy=0=iB+fory=0(4{15)forthereactingspeciesB+.Theconcentrationsofeachspecieswererepresentedintermsofsteady-stateandoscillatingtermsas[ 3 5 ] ci= ci+Refeciexp(j!t)g(4{16)Theresultingequationsgoverningthesteady-stateare vy@ cAB @y=DAB@2 cAB @y2)]TJ ET q .4782 w 284.367 -570.953 m 293.376 -570.953 l S Q BT /F5 11.9552 Tf 284.367 -580.796 Td[(RAB(4{17) vy@ cA)]TJ ET q .4782 w 173.306 -618.846 m 198.269 -618.846 l S Q BT /F5 11.9552 Tf 179.304 -630.036 Td[(@y=DA)]TJ /F5 11.9552 Tf 7.948 7.789 Td[(@2 cA)]TJ ET q .4782 w 239.221 -618.846 m 268.917 -618.846 l S Q BT /F5 11.9552 Tf 245.117 -630.036 Td[(@y2+ RA)]TJ /F1 11.9552 Tf 135.601 -.299 Td[((4{18) 65

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and vy@ cB+ @y=DB+@2 cB+ @y2+ RB+(4{19)where RA)]TJ /F1 11.9552 Tf 10.073 -.299 Td[(= RB+=)]TJ ET q .4782 w 180.923 -80.311 m 189.932 -80.311 l S Q BT /F5 11.9552 Tf 180.923 -90.154 Td[(RAB=kb(Keq cAB(y))]TJ ET q .4782 w 300.774 -83.333 m 305.812 -83.333 l S Q BT /F5 11.9552 Tf 300.774 -90.154 Td[(cA)]TJ /F1 11.9552 Tf 6.752 -.299 Td[((y) cB+(y))(4{20)Theequationsgoverningthefrequencydomainwere j!ecAB+vy@ecAB @y=DAB@2ecAB @y2)]TJ /F8 11.9552 Tf 14.114 3.022 Td[(eRAB(4{21) j!ecA)]TJ /F1 11.9552 Tf 9.409 -.299 Td[(+vy@ecA)]TJ ET q .4782 w 195.278 -194.451 m 220.242 -194.451 l S Q BT /F5 11.9552 Tf 201.277 -205.641 Td[(@y=DA)]TJ /F5 11.9552 Tf 7.948 7.789 Td[(@2ecA)]TJ ET q .4782 w 261.194 -194.451 m 290.889 -194.451 l S Q BT /F5 11.9552 Tf 267.09 -205.641 Td[(@y2+eRA)]TJ /F1 11.9552 Tf 113.727 -.299 Td[((4{22)and j!ecB++vy@ecB+ @y=DB+@2ecB+ @y2+eRB+(4{23)where eRA)]TJ /F1 11.9552 Tf 16.715 -.299 Td[(=eRB+=)]TJ /F8 11.9552 Tf 11.458 3.022 Td[(eRAB (4{24) =kfecAB(y))]TJ /F5 11.9552 Tf 11.955 0 Td[(kb cA)]TJ /F1 11.9552 Tf 6.753 -.299 Td[((y)ecB+(y))]TJ /F5 11.9552 Tf 11.956 0 Td[(kbecA)]TJ /F1 11.9552 Tf 6.752 -.299 Td[((y) cB+(y)andtermsO(~c2)andgreaterhavebeenneglected.Equations( 4{21 )-( 4{24 )arecomplex,coupled,andlinearandaredependentonthesteady-statesolution.Theboundaryconditionsfortheoscillatingconcentrationswere eci=0fory!1(4{25)foreachspecies @eci @yy=0=0fory=0(4{26)fortheABandA)]TJ /F1 11.9552 Tf 7.085 -4.339 Td[(,and ecB+(0)=1fory=0(4{27) 66

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forthereactingspeciesB+.ThevalueofecB+(0)waschosenarbitrarilybecausethegoverningequationsfortheimpedanceresponsearelinear,evenwhenthesteady-stateproblemisnon-linear. 4.1.4.1DiusionimpedanceTheoscillatingcurrentdensityassociatedwithB+wasexpressedasaTaylorseriesexpansionaboutthesteady-state eiB+=@iB+ @VcB+(0)eV+@iB+ @cB+(0)VecB+(0)(4{28)whereeVandecB+(0)areassumedtohaveasmallmagnitudesuchthatthehigherordertermscanbeneglected.TheuxexpressionforB+intheabsenceofmigrationyieldsasecondequationfortheoscillatingcurrentdensityas eiB+=FDB+decB+ dyy=0(4{29)Equation( 4{28 )wasdividedbyequation( 4{29 ),yielding 1=@iB+ @VcB+(0)eV eiB++@iB+ @cB+(0)VecB+(0) FDB+decB+ dyy=0(4{30)Thefaradaiccontributiontotheimpedanceisdenedby ZF;B+(!)=eV eiB+(4{31)where!istheangularfrequency.Thus,byrearrangingequation( 4{30 ),thefaradaiccontributiontotheimpedancewasexpressedas ZF;B+(!)=Rt;B++ZD;B+(!)(4{32)where,fromtherespectivederivativesofequation( 4{5 ),thechargetransferresistancewasgivenas Rt;B+=1 KB+bB+ cB+(0)exp)]TJ /F2 11.9552 Tf 5.48 -9.683 Td[()]TJ /F5 11.9552 Tf 9.298 0 Td[(bB+ V(4{33) 67

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andthediusionimpedancewasgivenas ZD;B+(!)=Rt;B+KB+exp)]TJ /F2 11.9552 Tf 5.479 -9.683 Td[()]TJ /F5 11.9552 Tf 9.299 0 Td[(bB+ V FDB+0BBB@)]TJ /F8 11.9552 Tf 17.796 8.088 Td[(ecB+(0) decB+ dyy=01CCCA(4{34)Theconcentrationdistributionsrequiredtoassessthediusionimpedance,equation( 4{34 ),wereobtainedforeachfrequencyfromthenumericalsolutionofequations( 4{21 )-( 4{24 ).Thedimensionlessdiusionimpedanceisgivenby )]TJ /F1 11.9552 Tf 19.242 8.088 Td[(1 0B+(K)=1 N;B+0BBB@)]TJ /F8 11.9552 Tf 17.796 8.088 Td[(ecB+(0) decB+ dyy=01CCCA(4{35)whereN;B+isthediusion-layerthicknessusingtheNernsthypothesis,shownas N;B+=\(4=3)3 a1=31 Sc1=3B+r (4{36)andtheSchmidtnumberisdenedas,ScB+==DB+.Equation( 4{35 )isafunctionofadimensionlessfrequency K=!B+ DB+(4{37)where B+=3 a1=31 Sc1=3B+r (4{38)ThevariablesKandB+areusedtonon-dimenionalizetheconvective-diusionequationforaninniteSchmidtnumber.[ 3 5 ] 4.1.4.2OverallimpedanceTheoverallimpedancecanberepresentedbythecircuitshowninFigure 4-2 .Anohmicresistanceisinserieswiththeparallelcontributionofthefaradaicimpedance,equation( 4{32 ),anddouble-layercapacitance,yieldingamathematicalexpressionfortheoverallimpedanceas 68

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Figure4-2. Electricalcircuitrepresentationoftheelectrodeimpedanceassociatedwiththefaradaicimpedancegivenasequation( 4{32 ). Z(!)=Re+ZF;B+ 1+j!ZF;B+Cdl(4{39)Thecontributiontotheoverallimpedancefromthefaradaicimpedancecalculatedfromequation( 4{32 )andthedouble-layercapacitanceexhibitedthreeloopsintheimpedanceresponse. 4.1.5NumericalMethodsThecouplednonlineardierentialequationsweresolvedrstunderthesteady-statecondition.Thentheoscillatingconditionwassolvedwiththesteady-stateresultsasaninput.Alltheequationswerelinearized,formulatedinnitedierenceformandsolvednumericallyusingNewman'sBANDmethodcoupledwithNewton{Raphsoniteration[ 77 ].ForinputvaluesshowninTables 4-1 and 4-2 ,thesteady-stateconcentrationsofAB,A)]TJ /F1 11.9552 Tf 7.085 -4.339 Td[(,andB+aswellasthevalueofthehomogeneousreactionratewereobtained.Thesesteady-statevaluesalongwiththeoriginalinputvalueswereusedtoobtaintheoscillatingvaluesonconcentrationofthereactingspeciesB+.Foraspectrumofdimensionlessfrequencies,fromK=110)]TJ /F3 7.9701 Tf 6.586 0 Td[(5to1105with20pointsperdecade,theoscillatingconcentrationsofthereactingspeciesobtainedneartheelectrodewereimputedintoequation( 4{35 )toobtainadimensionlessdiusionimpedanceandequation( 4{34 )toobtainadiusionimpedance. 69

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Figure4-3. One-dimensionalschematicrepresentationshowingtwodissimilarmeshsizes.HHisthesmallermeshsizeandHisthelargermeshsize.J=KJistheinterfaceregion. 4.1.5.1AccuracyofnumericalmethodsToobtainasolutionthatisthemostaccurate,thenitedierenceerrorsandround-oerrorsneedtobeminimized.Neartheelectrodesurfacetheuxofthereactingspeciesischangingsodramaticallyitisnecessarytohaveasmallermeshtoobtainanaccurateanswer.Farfromtheelectrodetheconcentrationsarenotchangingandthereforethesamesmallmeshwouldresultinunnecessaryroundoerrors.Tomitigateerrorswehavesolvedthesystemusingtwomeshsizes.Asmallmeshsize,HH,isusedneartheelectrodeandalargemeshsize,H,isusedfarawayfromtheelectrode.Withtheuseoftwodissimilarmeshsizes,itisnecessarythatamethodofcouplingbedevelopedtoallowthecomputationtransitionfromoneregiontoanotherwhileretainingtheaccuracyofthenitedierencecalculation.Suchtransitionisperformedbyacouplingsubroutineataninterfacedesignatedatj=KJ.AvisualrepresentationoftwomeshsizeregionsisshowninFigure 4-3 .Thesmallnumberofnodes,withNJ=12,isexaggeratedlysmalltoshowthedierencesofmeshregions,theactualcodeusesNJ=12;000nodes.Agraphshowingtheaccuracyofthesimulations,whichisontheorderofthemeshsizesquared,isshowninFigure 4-4 .TheconcentrationofB+isshownwithrespecttodierentsquaredmeshvalues.Thesimulationsfordierentmeshsizes,rangingfrom3000pointsto12000points,areaccuratetosevenordersofmagnitude.TheR2valueofthelineartis1.0andtheinterceptisvalueiscalculatedas2:13466784110)]TJ /F3 7.9701 Tf 6.587 0 Td[(31:010)]TJ /F3 7.9701 Tf 6.587 0 Td[(14. 70

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Figure4-4. GraphicalevidenceofH2accuracyinnitedierencesimulation.TheconcentrationofB+isshownvsdierentsquaredmeshvalues. 4.1.5.2CouplingdomainswithdierentmeshsizeThedierentmeshsizespresentedinFigure 4-3 needtobeaddresseddierentlythenthebulkequations( 4{17 )to( 4{19 ).Theuxatthecouplerpoint,j=KJ,shouldbeequalwhenapproachfromthesmallermeshorthelargermesh NijKJ)]TJ /F3 7.9701 Tf 6.586 0 Td[(1!KJ=NijKJ+1!KJ(4{40)Toaccomplishthis,thematerialbalanceequationwaswrittenforKJ)]TJ /F1 11.9552 Tf 11.813 0 Td[(1=4andKJ+1=4 dci dtKJ)]TJ /F3 7.9701 Tf 6.586 0 Td[(1=4=rNijKJ)]TJ /F3 7.9701 Tf 6.586 0 Td[(1=4+RijKJ)]TJ /F3 7.9701 Tf 6.587 0 Td[(1=4(4{41)and dci dtKJ+1=4=rNijKJ+1=4+RijKJ+1=4(4{42)Thederivativeofuxwrittenincylindricalcoordinates,assumingaxisymmetricow,is rNi=1 r@(rNr;i) @r+@(Ny;i) @y(4{43) 71

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Figure4-5. PolarizationcurvecalculatedforsystemparameterspresentedinTables 4-1 and 4-2 .Labeledpotentialvaluesatfractionsofthelimitingcurrentcorrespondtosteady-stateconcentrationprolespresentedinFigure( 4-6 ). re-writingthematerialbalanceequationinadierenceofuxesgives dci dtKJ)]TJ /F3 7.9701 Tf 6.587 0 Td[(1=4=)]TJ /F8 11.9552 Tf 9.298 16.857 Td[(1 r@(rNr;i) @rKJ)]TJ /F3 7.9701 Tf 6.587 0 Td[(1=4)]TJ /F8 11.9552 Tf 11.955 16.857 Td[(NijKJ)]TJ /F5 11.9552 Tf 11.955 0 Td[(NijKJ)]TJ /F3 7.9701 Tf 6.587 0 Td[(1=2 HH=2+RijKJ)]TJ /F3 7.9701 Tf 6.587 0 Td[(1=4(4{44)and dci dtKJ+1=4=)]TJ /F8 11.9552 Tf 9.298 16.857 Td[(1 r@(rNr;i) @rKJ+1=4)]TJ /F8 11.9552 Tf 11.955 16.857 Td[(NijKJ+1=2)]TJ /F5 11.9552 Tf 11.955 0 Td[(NijKJ H=2+RijKJ+1=4(4{45)Rearrangingequations( 4{44 )and( 4{45 )forNijKJandthensettingthemequaltoeachotherprovidesthenecessaryrelationshipforthecoupler,whenj=KJ. 4.2ImpedanceforConvective-DiusionandHomogeneousReactionThepolarizationcurvecorrespondingtotheparametersgiveninTables 4-1 and 4-2 ispresentedinFigure 4-5 .Theheterogeneousreactionrateincreasesasthepotentialbecomesmorenegative,reachingamass-transfer-limitedplateauforpotentialssmallerthan)]TJ /F1 11.9552 Tf 9.298 0 Td[(2:2V.Thelargemagnitudeofthemass-transfer-limitedcurrentdensitymaybeattributedtotheproductionofB+bythehomogeneousreaction.Labeledfractionsofthemass-transfer-limitedcurrentdensityvaluesinFigure 4-5 correspondtosteady-stateconcentrationprolesandthereactionprolepresentedin 72

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Table4-1. Speciesandassociatedparametervaluesforthesystem Speciesci(1);mol=cm3ziDi;cm2=s AB0.0101:68410)]TJ /F3 7.9701 Tf 6.587 0 Td[(5A)]TJ /F1 11.9552 Tf 52.553 -4.338 Td[(0.0001-11:95710)]TJ /F3 7.9701 Tf 6.587 0 Td[(5B+0.000111:90210)]TJ /F3 7.9701 Tf 6.587 0 Td[(5 Table4-2. Systemandkineticparametervaluesforthesystem ParameterValueUnits Diskrotationrate,2;000rpmKinematicviscosity,0.01cm2=sHomogeneousequilibriumconstant,Keq10)]TJ /F3 7.9701 Tf 6.586 0 Td[(6mol=cm3Homogeneousrateconstant,kb,107cm3=molsHeterogeneousrateconstant,KB+210)]TJ /F3 7.9701 Tf 6.586 0 Td[(12A=cm2Heterogeneousconstant,bB+19.9V)]TJ /F3 7.9701 Tf 6.586 0 Td[(1 Figure 4-6 .Concentrationswerescaledbythesumofallthespeciesinvolvedinthehomogeneousreaction,co=cA)]TJ /F1 11.9552 Tf 9.706 -.299 Td[(+cB++cAB,toemphasizetherelativechangesinvaluesaswellastheoverallconcentrationintheelectrolyte.TheconcentrationofAB,showninFigure 4-6A ,decreasestoavaluethatis94percentofcoatapotentialcorrespondingtoavalueinthemass-transfer-limitedcurrentrange.Incontrast,theconcentrationofA)]TJ /F1 11.9552 Tf 10.987 -4.339 Td[(showninFigure 4-6B reachesavaluethatisalmost5timesitsbulkvalueatthemass-transfer-limitedcurrentdensity.ThenormalizedconcentrationdistributionofB+ispresentedinFigure 4-6C .TheconcentrationofB+attheelectrodesurfaceapproachesavalueofzeroasthemass-transfer-limitedcurrentdensityisapproached.ThesharpprolethatappearsclosetotheelectrodesurfaceinFigure 4-6C isconsistentwithalargecurrentdensity.Toemphasizetheproleneartheelectrode,therateofthehomogeneousreactionispresentedinFigure 4-6D asafunctionofpositiononalogarithmicscale.TheinuenceofthehomogeneousreactionrateconstantoncurrentdensitycanbeseeninFigure 4-7A ,whereahomogeneousrateconstantofkb=109cm3=molsyieldsamass-transfer-limitedcurrentdensitythatisalmost20timeslargerthanthatintheabsenceofhomogeneousreactions.TheconcentrationsofB+correspondingtohalfthemass-transfer-limitedcurrentfordierenthomogeneousreactionrateconstantsandarepresentedinFigure 4-7B asfunctionsofposition.Intheabsenceofhomogeneousreaction, 73

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A B C DFigure4-6. Calculatedsteady-stateconcentrationdistributionsandhomogeneousreactioncorrespondingtosystemparameterspresentedinTables 4-1 and 4-2 atdierentfractionsofthelimitingcurrent;a)B+b)A)]TJ /F1 11.9552 Tf 10.987 -4.338 Td[(c)ABc)RB+ theconcentrationproleisthatexpectedforarotatingdiskelectrode[ 77 ].Theslopeattheelectrode{electrolyteinterfacebecomeslargerasthehomogeneousrateconstantincreases.Theconvective-diusionimpedancecorrespondingtothesteady-stateresultspresentedinFigure 4-6 arepresentedinFigure 4-8 .Forallpotentials,twoasymmetriccapacitiveloopsareseeninFigure 4-8 ascomparedtoasingleloopintraditionalconvective-diusionimpedance.Thelow-frequencyloop,forallcases,hasacharacteristicfrequency,K=2:5,whichisinagreementwiththecharacteristicfrequencyassociatedwithdiusionintheabsenceofhomogeneousreactions.Thelow-frequencyloopdecreased 74

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A BFigure4-7. CalculationscorrespondingtosystemparameterspresentedinTables 4-1 and 4-2 withhomogeneousrateconstantkbasaparameter:a)polarizationcurveb)concentrationdistributionforB+athalfthemass-transfer-limitingcurrent. Figure4-8. Dimensionlessconvective-diusionimpedanceforthesystempresentedinFigure 4-6 takenatfractionsofthelimitingcurrentdensity. insizeasthepotentialcorrespondingtothemass-transfer-limitedcurrentdensitywasapproached.Thehigh-frequencyloop,presentforallfractionsofthemass-transfer-limitedcurrentdensity,correspondstothehomogeneousreactionanddecreasedslightlyandbecamebetterdenedatmorecathodicpotentials.TheindividualimpedancediagramsfromFigure 4-8 arepresentedseparatelyinFigure 4-9 75

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Figure4-9. Thedimensionlessconvective-diusionimpedancecorrespondingtoFigures 4-5 and 4-6 fordierentvaluesofi=ilim:a)0;b)1/4;c)1/2;e)3/4;andf)1.Thedimensionless-diusionimpedancesarepresentedinaclockwisefashion,goingfromthemostanodicpotentialtothemostcathodicpotential.Alineindicatesthepointonthepolarizationcurve(d)correspondingtoeachNyquistdiagram.Dimensionlessfrequencies,K=!2i=Di,arelabeledatcharacteristicfrequenciesaswellasatrelevantdecadepoints. 76

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Figure4-10. Theabsolutevalueoftheimaginarypartofthedimensionlessconvective-diusionimpedanceforthesystempresentedinFigure 4-6 asafunctionofpotentialstakenatfractionsofthelimitingcurrentasaparameter.Thecharacteristicfrequencyforconvective-diusion,atK=2:5,andtheapproximatecharacteristicfrequencyassociatedwiththehomogeneousreaction,atK=1000,areillustratedwithdashedverticallines. Eachdimensionlessdiusionimpedancecorrespondstoapointonthepolarizationcurve,whichisillustratedinFigure 4-9 byaline.Thedimensionless-diusionimpedancesarepresentedinaclockwisefashion,goingfromthemostanodicpotentialtothemostcathodicpotential.Thehigh-frequencyloopislessdenedformoreanodicpotentials,asshowninFigures 4-9 (a)and 4-9 (b).Atmorenegativepotentials,thehigh-frequencyloopresemblesaGerischerimpedanceelement,asshowninFigures 4-9 (c), 4-9 (e)and 4-9 (f).Thecharacteristicfrequencyforthehigh-frequencyloopisontheorderofK=1000,suggestingthatthecharacteristicdimensionforthereactionismuchsmallerthantheNernstdiusion-layerthickness.ThetwocharacteristicfrequenciesarebetterillustratedinFigure 4-10 ,wheretheabsolutevalueofthedimensionlessimaginaryimpedanceisplottedasafunctionofdimensionlessfrequency.Thecharacteristicfrequencyforconvective-diusion,atK=2:5,andtheapproximatecharacteristicfrequencyassociatedwiththehomogeneousreaction,atK=1000,areillustratedwithdashedverticallinesinFigure 4-10 77

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Figure4-11. TheoverallimpedancecorrespondingtoFigure 4-9 withanohmicresistanceof10cm2andadouble-layercapacitanceof20F=cm2fordierentvaluesofi=ilim:a)0;b)1/4;d)1/2;e)3/4;andf)1.Alineindicatesthepointonthepolarizationcurve(c)correspondingtoeachNyquistdiagram.Characteristicfrequenciesarelabeled. 78

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Figure4-12. TheoverallimpedancepresentedcorrespondingtoFigures 4-11 (a)and 4-11 (f)withanexpandedfrequencyrange. Thecellimpedancewascalculatedfollowingequation( 4-2 )andFigure 4-2 ,undertheassumptionsthatRe=10cm2,Cdl=20F=cm2,andthefaradaicimpedancewasgivenbyequation( 4{39 ).TheresultsarepresentedinFigure 4-11 .Eachimpedancediagramisconnectedtothecorrespondingcurrentdensityonthepolarizationcurvebyaline.Theimpedancediagramsarealsopresentedinaclockwisefashion,fromthemostanodicpotentialtothemostcathodicpotential.Thehigh-frequencyloopspresentedinFigures 4-11 (b), 4-11 (d),and 4-11 (e)decreasedinsizewithincreasingfractionofthelimitingcurrentdensity,indicatingthatthehighfrequencyloopcanbeattributedtothecharge-transferresistance.Thelow-frequencyloopshaveacharacteristicfrequency,K=2:5,thatcorrespondstoconvective-diusionimpedance.Theloopatintermediatefrequenciescanbeattributedtothehomogenousreaction.Atpotentialsof)]TJ /F1 11.9552 Tf 9.298 0 Td[(0:5Vand)]TJ /F1 11.9552 Tf 9.299 0 Td[(3:0V,theimpedanceresponseappearsalmostcapacitiveasmightbeexpectedwhenthecurrentisnotsensitivetoappliedpotential.TheexpandedfrequencyrangeshowninFigure 4-12 indicatedthattheresponsehassomecurvature,butthiscurvatureisevidentonlyatverylowfrequencies,generallyconsideredtobeunaccessibleexperimentally. 79

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4.3DiscussionforConvectiveDiusionandHomogeneousReactionSimulationsofthesteady-stateconcentrationsandreactiondistributionandconvective-diusionimpedanceforarotatingdiskelectrodeinuencedbyhomogenousreactiongiveinsighttotheimpedanceresponseofanelectrochemicalsysteminuencedbyhomogenousreaction.Thediusionimpedanceshavetwocapacitiveloops.Thepresentsectionprovidesadevelopmentofkineticandmass-transferexpressionsforfasthomogeneousreactionsandtheapplicationofGerischerimpedanceasameanstointerpretexperimentaldataconformingtotheassumptionsassociatedwiththepresentmathematicalmodel. 4.3.1FastHomogeneousReactionInthelimitofaninnitelyfasthomogeneousreaction,reactions( 4{3 )and( 4{4 )canbeexpressedas AB+e)]TJ /F2 11.9552 Tf 10.405 -4.937 Td[(!A)]TJ /F1 11.9552 Tf 9.741 -4.937 Td[(+B+(4{46)whereABisconsideredtobetheelectroactivespecies.ThedimensionlessdiusionimpedancebasedontheoscillatingconcentrationecABisgivenby )]TJ /F1 11.9552 Tf 19.37 8.088 Td[(1 0AB=1 N;AB0BBB@)]TJ /F8 11.9552 Tf 17.796 8.088 Td[(ecAB(0) decAB dyy=01CCCA(4{47)Theconvective-diusionimpedancegivenasequation( 4{47 )ispresentedinFigure 4-13 withhomogeneousrateconstantasaparameter.Forkb=108cm3/mols,thedimensionlessdiusionimpedanceisverylargeandhastheappearanceofadistortedsemi-circle.Forkb=1040cm3/mols,thedimensionlessdiusionimpedancetakestheappearanceofausualdimensionlessconvective-diusionimpedance,withacharacteristicfrequencyofK=2:5,alow-frequencylimitof1:0392,andananglewithrespecttotherealaxisof45degreesatthehigh-frequencylimit.Theseresultsconrmthat,forkb=1040cm3/mols,reactions( 4{3 )and( 4{4 )canbeexpressedasreaction( 4{46 ). 80

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Figure4-13. Dimensionlessconvective-diusionimpedancewheretheoscillatingconcentrationofABweretakenintoaccountinequation( 4{47 )andlargevaluesofthehomogeneousreactionrateconstantwereused. 4.3.2ModiedGerischerImpedanceTheconvective-diusionimpedancespresentedinFigure 4-9 resemblesaGerischerimpedance.ThemathematicaldevelopmentfortheGerischerimpedanceissummarizedinthefollowingsection.AmodiedGerischerImpedancewasregressedtosimulationresultstoexplorethemannerinwhichsimulatiuonsmaybeemployedtoextractmeaningfulparameters,eveniftheassumptionsimplicitintheGerischermodelarenotsatised. 4.3.2.1MathematicaldescriptionUndertheassumptionsthatthediusioncoecientsforABandB+areequal,thatconvectionmaybeignored,andthattheconcentrationofA)]TJ /F1 11.9552 Tf 10.986 -4.338 Td[(issucientlylargetobeconsideredconstant,GerischerdevelopedananalyticexpressionforthediusionimpedanceassociatedwithaheterogeneousreactioninuencedbyahomogeneousreactioninaNernststagnantdiusionlayer.TherateofproductionofspeciesABandB+by 81

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reaction( 4{3 )wasexpressedas RB+=)]TJ /F5 11.9552 Tf 9.298 0 Td[(RAB=kfcAB(y))]TJ /F5 11.9552 Tf 11.955 0 Td[(kbcB+(y)(4{48)TheconservationequationsforspeciesABandB+maybeexpressedas @cAB @t=D@2cAB @y2)]TJ /F5 11.9552 Tf 11.956 0 Td[(kfcAB+kbcB+(4{49)and @cB+ @t=D@2cB+ @y2+kfcAB)]TJ /F5 11.9552 Tf 11.956 0 Td[(kbcB+(4{50)respectively,where D=DAB=DB+(4{51)Thesumofequations( 4{49 )and( 4{50 )yields @ @t(cAB+cB+)=D@2 @y2(cAB+cB+)(4{52)Afteralgebraicmanipulation,thedierencebetweenequations( 4{49 )and( 4{50 )maybeexpressedas @ @tcAB)]TJ /F5 11.9552 Tf 13.687 8.088 Td[(cB+ Keq=D@2 @y2cAB)]TJ /F5 11.9552 Tf 13.686 8.088 Td[(cB+ Keq)]TJ /F5 11.9552 Tf 11.955 0 Td[(kcAB)]TJ /F5 11.9552 Tf 13.686 8.088 Td[(cB+ Keq(4{53)wherek=kf+kbandKeq=kf=kb.Asequations( 4{52 )and( 4{53 )arelinear,thesolutionfortheconvective-diusionimpedancedoesnotrequireasolutionforthesteady-state.Equations( 4{52 )and( 4{53 )inthefrequencydomainweresolvedfortheboundaryconditions ecAB()=ecB+()=0(4{54) ecB+(0)=1(4{55)and decAB dyy=0=0(4{56) 82

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Thedimensionlessdiusionimpedancemaybeexpressedas )]TJ /F1 11.9552 Tf 16.585 8.088 Td[(1 0B+=)]TJ /F1 11.9552 Tf 10.494 8.088 Td[(1 ecB+(0) decB+ dyy=0 (4{57) =1 Keq+1tanhr (j!+k)2 D r (j!+k)2 D+Keq Keq+1tanhr j!2 D r j!2 D (4{58) or )]TJ /F1 11.9552 Tf 19.242 8.087 Td[(1 0B+=1 Keq+1tanhp jK+kdim p jK+kdim+Keq Keq+1tanhp jK p jK(4{59)whereK=!2N=Dandkdim=k2N=D.Regressionofequation( 4{59 )tosimulateddiusionimpedancesyieldedunsatisfactoryresults.Thus,amodiedGerischerimpedancewasintroducedinwhichthesecondtermwasreplacedbyanexpressionfortheconvectivediusionimpedanceofadiskelectrodeunderassumptionofaniteSchmidtnumber,i.e., )]TJ /F1 11.9552 Tf 19.242 8.088 Td[(1 0B+=1 Keq+1tanhp jK+kdim p jK+kdim+Keq Keq+1)]TJ /F1 11.9552 Tf 9.299 0 Td[(1 0CD(4{60)where)]TJ /F1 11.9552 Tf 9.299 0 Td[(1=0CDcanbeobtainedusingK=!2B+=DB+andalookuptablegeneratedfromthesolutionoftheconvective-diusionimpedancewithouthomogeneousreaction[ 3 4 5 ]. 4.3.2.2ModiedGerischerimpedanceALevenberg{MarquardtregressionwasusedinOrigin2017rtoregressequation( 4{60 )tothedatashowninFigure 4-8 .Theextractedparameterswerekdim,KEq,andtheSchmidtnumber.AcomparisonbetweenthetandthesimulationresultsareshowninFigure 4-14 asNyquistandBodeplots.FittingparametersarelistedinTable 4-3 .TheparameterKeqwasfoundtovarywithpotentialandrateconstantkb,eventhoughtheequilibriumconstantusedforthesimulationsandreportedinTable 4-2 wasunchanged.ThisresultisconsistentwiththeobservationthattheassumptionsemployedfortheGerischerimpedancearemuchmorerestrictivethanthoseusedforthenumerical 83

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A B CFigure4-14. Simulateddimensionlessconvective-diusionimpedancefromFigure 4-8 andregressionttingusingequation( 4{60 )forkb=107atdierentfractionsofthelimitingcurrentshownina)Nyquistform,andBodeplotwithb)Magnitudeandc)Phase. simulations.TheregressedvalueofKeqservestoweightthehomogeneousreactionandconvectivediusionloopsasshowninequation( 4{60 ).Followingthedenitionofkdim=k2=Dandequation( 2{43 ),arelationshipbetweenthereactionthicknessrandtheNernstdiusionlayerthicknesswasfoundtobe r=N;B+r 1 kdim(4{61) 84

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Table4-3. Fittingparametersfoundfromregressionusingequation( 4{60 )andFigure 4-14 Potential,Vi=ilimkbkdimKeqScr,m )]TJ /F1 11.9552 Tf 9.299 0 Td[(0:501071975:50:990:0057521850:9830:014)]TJ /F1 11.9552 Tf 9.298 0 Td[(1:650.051072004:60:790:0042528870:9760:011)]TJ /F1 11.9552 Tf 9.298 0 Td[(1:75490.251072303:10:340:00165281010:9100:006)]TJ /F1 11.9552 Tf 9.298 0 Td[(1:82620.501073013:00:130:00085021340:7950:004)]TJ /F1 11.9552 Tf 9.298 0 Td[(1:850.591073333:20:100:00064941510:7560:003)]TJ /F1 11.9552 Tf 9.298 0 Td[(1:89380.751073923:20:060:00054471680:6970:003)]TJ /F1 11.9552 Tf 9.299 0 Td[(2:00.951074743:50:030:0003386780:6340:002)]TJ /F1 11.9552 Tf 9.299 0 Td[(3:01.01074933:60:030:00033681890:6210:002)]TJ /F1 11.9552 Tf 9.299 0 Td[(1:76740.50106210:40:350:0058249782:9830:027)]TJ /F1 11.9552 Tf 9.299 0 Td[(1:89510.501085240390:040:00026521780:1970:001)]TJ /F1 11.9552 Tf 9.299 0 Td[(1:96460.50109988886660:010:00035541430:0440:0001 A BFigure4-15. Reaction-layerthicknessobtainedfromequation( 4{61 )withregressionparametersobtainedbyregressingequation( 4{60 )tothesimulatedconvective-diusionimpedance:a)reaction-layerthicknessandcurrentdensityasafunctionofpotentialandb)reaction-layerthicknessasafunctionofhomogeneousreactionrate. Thereactionthicknessesobtainedfromequation( 4{61 )areshowninFigure 4-15 .Thereactiondistributionwasscaledbythecorrespondingvaluesaty!0toemphasizetherelativechangesinvalues.Astheheterogeneousreactionbecomesbiggerthereactionthicknessdecreases.Errorbarsareshownandwerecalculatedusingalinearpropagationoferrorsfromregressedvaluesofkdim.ThisworkdemonstratesthatamodiedGerischer 85

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impedancemaybeusedtoprovidemeaningfulparametersforthegeneralcaseforarotatingdiskinwhichdiusioncoecientsarenotequalandthehomogeneousreactionisnotlinear. 86

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CHAPTER5IMPEDANCERESPONSEFORCONTINUOUSGLUCOSEMONITORSAmathematicalmodelwasdevelopedfortheimpedanceresponseofaglucoseoxidaseenzyme-basedelectrochemicalbiosensor.AschematicoftheglucosesensorinsertedintotheinterstitialuidispresentedinFigure 5-1 .ThemodelaccountsforaglucoselimitingmembraneGLM,whichcontrolstheamountofglucoseparticipatingintheenzymaticreaction.BetweentheGLMandtheelectrodeistheenzymeglucoseoxidaselayer(GOX).Theglucoseoxidasewasassumedtobeimmobilizedwithinathinlmadjacenttotheelectrode.Intheglucoseoxidaselayer,aprocessofenzymaticcatalysistransformstheglucoseintohydrogenperoxide,whichcanbedetectedelectrochemically,presentedinFigure 5-2 .ThissystemwasconsideredtobeaspecialcaseofthecoupledhomogeneousandheterogeneousreactionsaddressedbyLevich[ 21 ].Themodeldevelopmentrequiredtwosteps.Thenonlinearcoupleddierentialequationsgoverningthissystemweresolvedundertheassumptionofasteadystate.Thesteady-stateconcentrationsresultingfromthesteady-statesimulationwereusedinthesolutionofthelinearizedsetofdierentialequationsdescribingthesinusoidalsteadystate.Theenzymaticcatalysiswastreatedintermsoftwohomogeneousreactions,oneconsumingtheglucoseoxidaseandforminggluconicacid,andtheotherregeneratingtheglucoseoxidaseandformingtheperoxide.Wittetal.[ 84 ]cameupwitharepresentationoftheGOXreaction.ThereactionofD-glucono--lactonetogluconicacidisconsideredtobeextremelyfastandthereforeisnotrepresentedinthismodel. 5.1MathematicalDevelopmentfortheContinuousGlucoseMonitorAmathematicalmodelispresentedbelowfortheimpedanceresponseassociatedwithaCGMincludingthecouplingoftheenzymaticandelectrochemicalreactions.Theglucoseoxidaselayer(GOX)overtheelectrodewasconsideredtobe7mthickandthe 87

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Figure5-1. Schematicshowingglucosesensorinsertedundertheskinintotheinterstitialuid.Glucosediusesthroughthebloodvesselsintotheinterstitialuidwhereitcanbemeasuredwiththeglucosesensor. Figure5-2. LayersintheGlucoseSensor,theglucoselimitingmembrane(GLM)limitstheamountofglucosefromtheinterstitialuidtodiusetotheglucoseoxidaselayer(GOX).TheglucosereactswiththeGOXlayerwithoxygentoformgluconicacidandhydrogenperoxide.Thehydrogenperoxidereactselectrochemicallyandthecurrentoftheelectrochemicalreactionisproportionaltotheconcentrationofglucose. 88

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Figure5-3. Representationoftheglucoseoxidasereaction.TheproductoftherstenzymaticreactionisD-glucono--lactonewhichcombineschemicallywithwatertoformgluconicacid.Theproductofthesecondenzymaticreactionishydrogenperoxide.RecreatedfromWittetal.[ 84 ].Themodelusedinthisdissertationassumesanenzymecomplexisformedandtheformationstepisreversible,andthereactionoftheenzymecomplextotheproductsisirreversible. glucoselimitingmembranelayer(GLM)ontopoftheenzymelayerwasconsideredtobe15mthick. 5.1.1GoverningEquationsforCGMSystemTheuxdensityofspeciesiinadiluteelectrolyteandinabsenceofmigrationandconvectionwasexpressedas Ni=)]TJ /F5 11.9552 Tf 9.298 0 Td[(Dirci(5{1)Forasystemwherethehomogeneousreactionoccursinahydrogeltheconvectivediusionequationwithenzymaticreactionwasexpressedas @ci @t=Di@2ci @y2+Ri(5{2) 89

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whereRiistherateofproductionofspeciesibyhomogeneousreactions.Theoverallenzymatic(homogeneous)reactionwas G+GOxkf1)439()222()439(! )351()]TJ /F5 11.9552 Tf 3.321 -8.302 Td[(kb1GOx2)]TJ /F1 11.9552 Tf 11.955 0 Td[(GAkf2)439()222()439(!GA+GOx2(5{3)whichrepresentstheenzymaticcatalysisreactionand GOx2+O2kf3)439()222()439(! )351()]TJ /F5 11.9552 Tf 3.321 -8.302 Td[(kb3GOx)]TJ /F1 11.9552 Tf 11.955 0 Td[(H2O2kf4)439()222()439(!GOx+H2O2(5{4)whichreporesentsthemediation/regenerationreaction.ThespeciesaredenedinTable 5-1 5.1.2HomogeneousEnzymaticReactionsforCGMSystemTheenzymatichomogeneousreactions,fromequations( 5{3 )and( 5{4 ),wereexpressedasindividualreactionsas G+GOxkf1)439()222()439(! )351()]TJ /F5 11.9552 Tf 3.32 -8.302 Td[(kb1GOx2)]TJ /F1 11.9552 Tf 11.956 0 Td[(GA(5{5) GOx2)]TJ /F1 11.9552 Tf 11.955 0 Td[(GAkf2)439()222()439(!GA+GOx2(5{6) GOx2+O2kf3)439()222()439(! )351()]TJ /F5 11.9552 Tf 3.321 -8.302 Td[(kb3GOx)]TJ /F1 11.9552 Tf 11.955 0 Td[(H2O2(5{7)and GOx)]TJ /F1 11.9552 Tf 11.955 0 Td[(H2O2kf4)439()222()439(!GOx+H2O2(5{8)Theseequationsrepresenttheenzymaticreactionoftheglucoseoxidaseenzymeturningglucoseintohydrogenperoxide.Onemoleofglucoseandonemoleculeofoxygencreateonemoleofgluconicacidasabyproductandonemoleofhydrogenperoxideasthedesired 90

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Table5-1. Speciesandassociatedparametervaluesforthesystem Speciesci(1);mol=cm3Di;cm2=sAbbreviation Glucose(C6H12O6)5:5507510)]TJ /F3 7.9701 Tf 6.587 0 Td[(67:210)]TJ /F3 7.9701 Tf 6.587 0 Td[(6GGlucoseOxidase(GOx-FAD)0:510)]TJ /F3 7.9701 Tf 6.587 0 Td[(30GOxGluconicAcid(C6H12O7)1:010)]TJ /F3 7.9701 Tf 6.587 0 Td[(207:210)]TJ /F3 7.9701 Tf 6.587 0 Td[(6GAEnzymeComplex(GOx)]TJ /F1 11.9552 Tf 11.955 0 Td[(FADH2)]TJ /F1 11.9552 Tf 11.955 0 Td[(GA)0:510)]TJ /F3 7.9701 Tf 6.587 0 Td[(30GOx2-GAGlucoseOxidase(GOx)]TJ /F1 11.9552 Tf 11.955 0 Td[(FADH2)0:510)]TJ /F3 7.9701 Tf 6.587 0 Td[(31:010)]TJ /F3 7.9701 Tf 6.587 0 Td[(20GOx2Oxygen(O2)3.12510)]TJ /F3 7.9701 Tf 6.587 0 Td[(92:4610)]TJ /F3 7.9701 Tf 6.586 0 Td[(5O2EnzymeComplex2(GOx)]TJ /F1 11.9552 Tf 11.955 0 Td[(FAD)]TJ /F1 11.9552 Tf 11.955 0 Td[(H2O2)1:9020:5)]TJ /F3 7.9701 Tf 6.586 0 Td[(30GOx-H2O2HydrogenPeroxide(H2O2)0:510)]TJ /F3 7.9701 Tf 6.587 0 Td[(30H2O2 product.Eachspeciesandtheirbulkvalues,diusioncoecients,andanabbreviationusedinthemathematicalworkupareshowninTable 5-1 .Hydrogenperoxideiselectroactiveandwasconsumedattheelectrode.Theelectrochemicalreactionwasexpressedas H2O2!2H++O2+2e)]TJ /F1 11.9552 Tf 150.187 -4.937 Td[((5{9)Oxygenwasaproductoftheelectrochemicalreaction,whichhelpsfueltheenzymaticreaction.Thecorrespondingcurrent,whichisdependentonconcentrationandpotential,wasexpressedas iH2O2=KH2O2cH2O2(0)exp(bH2O2V)(5{10)Astheelectroactivespeciesisconsumed,aconcentrationgradientofH2O2mustexistneartheelectrodesurface.Thefourreactiontermscorrespondingtoreactions( 5{5 )-( 5{8 )wereexpressedas R1=kf1cG(y)cGOx(y))]TJ /F5 11.9552 Tf 11.956 0 Td[(kb1cGOx2)]TJ /F3 7.9701 Tf 6.587 0 Td[(GA(y)(5{11) R2=kf2cGOx2)]TJ /F3 7.9701 Tf 6.587 0 Td[(GA(y)(5{12) R3=kf3cO2(y)cGOx2(y))]TJ /F5 11.9552 Tf 11.955 0 Td[(kb3cGOx)]TJ /F3 7.9701 Tf 6.587 0 Td[(H2O2(y)(5{13) R4=kf4cGOx)]TJ /F3 7.9701 Tf 6.587 0 Td[(H2O2(y)(5{14) 91

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Theconservationequationforeachspeciesmaybewrittenas @cG @t=DG@2cG @y2)]TJ /F5 11.9552 Tf 11.956 0 Td[(R1(5{15) @cGOx @t=)]TJ /F5 11.9552 Tf 9.298 0 Td[(R1+R4(5{16) @cGOx2)]TJ /F3 7.9701 Tf 6.586 0 Td[(GA @t=R1)]TJ /F5 11.9552 Tf 11.956 0 Td[(R2(5{17) @cGA @t=DGA@2cGA @y2+R2(5{18) @cO2 @t=DO2@2cO2 @y2)]TJ /F5 11.9552 Tf 11.955 0 Td[(R3(5{19) @cGOx2 @t=)]TJ /F5 11.9552 Tf 9.298 0 Td[(R3+R2(5{20) @cGOx)]TJ /F3 7.9701 Tf 6.586 0 Td[(H2O2 @t=R3)]TJ /F5 11.9552 Tf 11.955 0 Td[(R4(5{21) @cH2O2 @t=DH2O2@2cH2O2 @y2+R4(5{22)Equations( 5{11 )-( 5{14 ),representingthereactionsoccurringinthesystem,alongwithequations( 5{15 )-( 5{22 ),representingtheconcentrationsofthevariousspeciesaresolvedforthesteady-stateandfrequency-domainvariables.Thissolutionfollowsthatofsection 4.1.2 .Theboundaryconditionsfarfromtheelectrodewere ci!ci(1)fory!1(5{23) 92

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andtheboundaryconditionsattheelectrodesurfacewere @ci @yy=0=0fory=0(5{24)forthenon-reactingspecies,and FDB+@cH2O2 @yy=0=iH2O2fory=0(5{25)forthereactingspecies,hydrogenperoxide.Splittingtheconcentrationvariablesusingequation( 4{16 )wecanseparateoutthesteady-state.Thesteady-stateequationsrepresentingthenon-enzymaticspecieswere @ cG @t=DG@2 cG @y2)]TJ ET q .4782 w 274.505 -229.243 m 283.514 -229.243 l S Q BT /F5 11.9552 Tf 274.505 -239.086 Td[(R1(5{26) @ cGA @t=DGA@2 cGA @y2+ R2(5{27) @ cO2 @t=DO2@2 cO2 @y2)]TJ ET q .4782 w 280.772 -305.842 m 289.78 -305.842 l S Q BT /F5 11.9552 Tf 280.772 -315.685 Td[(R3(5{28)and @ cH2O2 @t=DH2O2@2 cH2O2 @y2+ R4(5{29)andthesteady-stateequationsrepresentingtheenzymaticspeciesandenzymaticcomplexeswere @ cGOx @t=)]TJ ET q .4782 w 241.801 -438.332 m 250.809 -438.332 l S Q BT /F5 11.9552 Tf 241.801 -448.175 Td[(R1+ R4(5{30) @ cGOx2)]TJ /F3 7.9701 Tf 6.587 0 Td[(GA @t= R1)]TJ ET q .4782 w 277.311 -470.395 m 286.319 -470.395 l S Q BT /F5 11.9552 Tf 277.311 -480.238 Td[(R2(5{31) @ cGOx2 @t=)]TJ ET q .4782 w 243.918 -502.457 m 252.926 -502.457 l S Q BT /F5 11.9552 Tf 243.918 -512.3 Td[(R3+ R2(5{32)and @ cGOx)]TJ /F3 7.9701 Tf 6.587 0 Td[(H2O2 @t= R3)]TJ ET q .4782 w 279.399 -556.749 m 288.408 -556.749 l S Q BT /F5 11.9552 Tf 279.399 -566.592 Td[(R4(5{33)andthesteady-statereactionequationswere R1=kf1 cG(y) cGOx(y))]TJ /F5 11.9552 Tf 11.955 0 Td[(kb1 cGOx2)]TJ /F3 7.9701 Tf 6.587 0 Td[(GA(y)(5{34) 93

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R2=kf2 cGOx2)]TJ /F3 7.9701 Tf 6.587 0 Td[(GA(y)(5{35) R3=kf3 cO2(y) cGOx2(y))]TJ /F5 11.9552 Tf 11.955 0 Td[(kb3 cGOx)]TJ /F3 7.9701 Tf 6.587 0 Td[(H2O2(y)(5{36) R4=kf4 cGOx)]TJ /F3 7.9701 Tf 6.587 0 Td[(H2O2(y)(5{37)Allthederivativeswithrespecttotimeareequaltozero.AndthereactiontermsareequaltozerointheGLMlayer.TosatisfyBANDandthenumberofequationsandvariables,equations( 5{30 )-( 5{33 )arerewritten,andonebecameamassbalanceequation,withtheassumptionthatthetotalenzymespeciesconcentrationisconstant. 0=)]TJ ET q .4782 w 228.499 -200.849 m 237.508 -200.849 l S Q BT /F5 11.9552 Tf 228.499 -210.692 Td[(R1+ R4(5{38) 0= cGOx(1)+ cGOx2)]TJ /F3 7.9701 Tf 6.586 0 Td[(GA(1)+ cGOx2(1)+ cGOx)]TJ /F3 7.9701 Tf 6.586 0 Td[(H2O2(1) (5{39) )]TJ ET q .4782 w 134.541 -278.586 m 139.579 -278.586 l S Q BT /F5 11.9552 Tf 134.541 -285.407 Td[(cGOx(y))]TJ ET q .4782 w 187.63 -278.586 m 192.668 -278.586 l S Q BT /F5 11.9552 Tf 187.63 -285.407 Td[(cGOx2)]TJ /F3 7.9701 Tf 6.586 0 Td[(GA(y))]TJ ET q .4782 w 264.525 -278.586 m 269.563 -278.586 l S Q BT /F5 11.9552 Tf 264.525 -285.407 Td[(cGOx2(y))]TJ ET q .4782 w 321.847 -278.586 m 326.885 -278.586 l S Q BT /F5 11.9552 Tf 321.847 -285.407 Td[(cGOx)]TJ /F3 7.9701 Tf 6.586 0 Td[(H2O2 0= R1)]TJ ET q .4782 w 252.106 -311.427 m 261.114 -311.427 l S Q BT /F5 11.9552 Tf 252.106 -321.27 Td[(R2(5{40)and 0= R3)]TJ ET q .4782 w 252.106 -365.718 m 261.114 -365.718 l S Q BT /F5 11.9552 Tf 252.106 -375.561 Td[(R4(5{41)Theformofequations( 5{34 )and( 5{36 )madethecoupledsteady-statedierentialequationsnon-linear.Thesteady-stateequationsweresolvedinBANDinasimilarfashionasinChapter 4 .Theoscillatingequationswere @ecG @t=DG@2ecG @y2)]TJ /F8 11.9552 Tf 14.114 3.022 Td[(eR1(5{42) @ecGA @t=DGA@2ecGA @y2+eR2(5{43) @ecO2 @t=DO2@2ecO2 @y2)]TJ /F8 11.9552 Tf 14.114 3.022 Td[(eR3(5{44)and @ecH2O2 @t=DH2O2@2ecH2O2 @y2)]TJ /F8 11.9552 Tf 14.114 3.022 Td[(eR4(5{45) 94

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forthenon-enzymaticconcentrationsand @ecGOx @t=)]TJ /F8 11.9552 Tf 11.457 3.022 Td[(eR1+eR4(5{46) @ecGOx2)]TJ /F3 7.9701 Tf 6.586 0 Td[(GA @t=eR1)]TJ /F8 11.9552 Tf 14.115 3.022 Td[(eR2(5{47) @ecGOx2 @t=)]TJ /F8 11.9552 Tf 11.457 3.022 Td[(eR3+eR2(5{48)and @ecGOx)]TJ /F3 7.9701 Tf 6.586 0 Td[(H2O2 @t=eR3)]TJ /F8 11.9552 Tf 14.114 3.022 Td[(eR4(5{49)fortheenzymaticspeciesandenzymecomplexes.Andtheequationsfortheoscillatingreactiontermswere eR1=kf1ecG(y) cGOx(y)+kf1 cG(y)ecGOx(y))]TJ /F5 11.9552 Tf 11.955 0 Td[(kb1ecGOx2)]TJ /F3 7.9701 Tf 6.586 0 Td[(GA(y)(5{50) eR2=kf2ecGOx2)]TJ /F3 7.9701 Tf 6.587 0 Td[(GA(y)(5{51) eR3=kf3ecO2(y) cGOx2(y)+kf3 cO2(y)ecGOx2(y)kb3ecGOx)]TJ /F3 7.9701 Tf 6.586 0 Td[(H2O2(y)(5{52) eR4=kf4ecGOx)]TJ /F3 7.9701 Tf 6.587 0 Td[(H2O2(y)(5{53)andtermswithec2wereneglected.Equations( 5{42 )-( 5{53 )arecoupledandlinearbutaredependentonthesteady-statesolutionthatshowsupinequations( 5{50 )and( 5{52 ).Theboundaryconditionsfortheoscillatingconcentrationswere eci=0fory!1(5{54)foreachspecies @eci @yy=0=0fory=0(5{55)fortheallthespeciesexceptforhydrogenperoxideandoxygen.Theboundaryconditionforthereactingspecies(hydrogenperoxide)was ecH2O2(0)=1fory=0(5{56) 95

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ThevalueofecH2O2(0)waschosenarbitrarilybecausethegoverningequationsfortheimpedanceresponsearelinear,evenwhenthesteady-stateproblemisnon-linear.TheboundaryconditionforecO2,alsotechnicallyareactingspecies,cannotbethesameasforhydrogenperoxidebecausetheproblemwouldbeoverspecied.Followingreactionstoichiometry, )-222(rNH2O2jJ=1=rNO2jJ=1(5{57)asshowninequation( 5{9 ).Inamannersimilartothatdevelopedinsection 4.1.5.2 ,amaterialbalanceatthequartermodeforeachspecieswastakenas dcH2O2 dtJ+1=4=rNH2O2jJ+1=4+R4jJ+1=4(5{58)and dcO2 dtJ+1=4=rNO2jJ+1=4)]TJ /F5 11.9552 Tf 11.955 0 Td[(R3jJ+1=4(5{59)re-writingthematerialbalanceequationinadierenceofuxesgave dcH2O2 dtJ+1=4=)]TJ /F8 11.9552 Tf 9.298 16.857 Td[(NH2O2jJ+1=2)]TJ /F5 11.9552 Tf 11.956 0 Td[(NH2O2jJ HHH=2+R4jJ+1=4(5{60)and dcO2 dtJ+1=4=)]TJ /F8 11.9552 Tf 9.299 16.857 Td[(NO2jJ+1=2)]TJ /F5 11.9552 Tf 11.955 0 Td[(NO2jJ HHH=2)]TJ /F5 11.9552 Tf 11.955 0 Td[(R3jJ+1=4(5{61)Seesection 5.1.3 forclaricationonthemeshsize,HHH.Withtherelationshipbetweentheuxofhydrogenperoxideandoxygenattheelectrodesurface,equation( 5{57 ),andrearrangementofequations( 5{60 )and( 5{60 ),analexpressionfortheoscillatingboundaryconditionofO2is 0=)]TJ /F1 11.9552 Tf 10.494 8.088 Td[(HHH 2dcH2O2 dtJ+1=4)]TJ /F1 11.9552 Tf 13.15 8.088 Td[(HHH 2dcO2 dtJ+1=4+NH2O2jJ+1=2+NO2jJ+1=2 (5{62) +HHH 2R4jJ+1=4)]TJ /F1 11.9552 Tf 13.15 8.087 Td[(HHH 2R3jJ+1=4 96

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5.1.2.1DiusionimpedanceforCGMsystemTheoscillatingcurrentassociatedwithB+,wasexpressedas eiH2O2=@iH2O2 @VcH2O2(0)eV+@iH2O2 @cH2O2(0)VecH2O2(0)(5{63)TheuxexpressionforH2O2yieldsasecondequationfortheoscillatingcurrentdensityas eiH2O2=FDH2O2decH2O2 dyy=0(5{64)Equation( 5{63 )wasdividedbyequation( 5{64 ),yielding 1=@iH2O2 @VcH2O2(0)eV eiH2O2+@iH2O2 @cH2O2(0)VecH2O2(0) FDH2O2decH2O2 dyy=0(5{65)Thus,theimpedancewasexpressedas ZF;H2O2=Rt;H2O2+ZD;H2O2(5{66)where,fromtherespectivederivativesofequation( 5{10 ), Rt;H2O2=1 KH2O2bH2O2 cH2O2(0)exp)]TJ /F2 11.9552 Tf 5.48 -9.684 Td[()]TJ /F5 11.9552 Tf 9.298 0 Td[(bH2O2 V(5{67)and ZD;H2O2=Rt;H2O2KH2O2exp)]TJ /F2 11.9552 Tf 5.48 -9.683 Td[()]TJ /F5 11.9552 Tf 9.298 0 Td[(bH2O2 V FDH2O20BBB@)]TJ /F8 11.9552 Tf 17.797 8.087 Td[(ecH2O2(0) decH2O2 dyy=01CCCA(5{68)Theconcentrationdistributionsrequiredtoassessthediusionimpedance,equation( 5{68 ),wereobtainedforeachfrequencyfromthenumericalsolutionofequations( 5{42 )-( 5{53 ).Thedimensionlessdiusionimpedanceis 1 0H2O2=1 GOx0BBB@)]TJ /F8 11.9552 Tf 17.796 8.088 Td[(ecH2O2(0) decH2O2 dyy=01CCCA(5{69)whereGOxisthethicknessoftheglucoseoxidaselayer. 97

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Figure5-4. Circuitdiagramofhowtheoverallimpedancewillbemodeledwiththesimulatedfaradaicimpedance,equation( 4{32 ). 5.1.2.2OverallimpedanceforCGMsystemTheoverallimpedancewasexpressedbyacircuitmodelshowninFigure 5-4 .AnohmicresistanceisinserieswithRCcircuitsfortheGOXandGLMlayersandtheparallelcontributionofthefaradaicimpedance,equation( 5{68 ),anddouble-layercapacitance.ToapproximatetheinuenceoftheGOXandGLMlayersthecapacitanceforeachlayerisexpressedas CGOX="GOX"o GOX(5{70)and CGLM="GLM"o GLM(5{71)where"isthedielectricconstantofthemediumandisapproximatedas20fortheGOXandGLM.ThedistancefortheGOXlayerandtheGLMlayer,GOXandGLM,are7mand15mrespectively.Thepermittivityofavacuum,"o,is8:854210)]TJ /F3 7.9701 Tf 6.586 0 Td[(14F/cm.ThecapacitancefortheGOXandGLMare0.002and0.001F=cm2,respectively.Theresistanceforeachlayerisrepresentedby RGOX=GOXGOX(5{72)and RGLM=GLMGLM(5{73)whereistheresistivitywhichistheinverseofconductivity.Conductivitywasapproximatedasbeingthesameasaphosphatebueredsaline(PBS)solution,0.01)]TJ /F3 7.9701 Tf 6.586 0 Td[(1cm)]TJ /F3 7.9701 Tf 6.587 0 Td[(1. 98

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Theresistanceforeachlayeris0.058cm2forGOXand0.125cm2forGLM.Thecharacteristicfrequencyisapproximatedby fc=1 2RlayerClayer(5{74)Thecharacteristicfrequencyforbothsystems,afterinputtingthevaluesofcapacitanceandresistancecalculatedabove,isestimatedasgreaterthen1GHz,whichisoutsidethetypicalfrequencyrangeforimpedancespectroscopy.ThismeansFigure 5-4 canberepresentedmathematicallybyequation( 5{75 ).Thefaradaicimpedance,ZF;H2O2comesfromequation( 5{66 ). Z=Re+RGOx+RGLM+ZF;H2O2 1+j!ZF;B+Cdl(5{75) 5.1.3NumericalMethodsforCGMThecouplednon-lineardierentialequationsweresolvedfollowingthenumericalapproachdescribedinSection 4.1.5 .BANDisdescribedinmoredetailinAppendix A .Thesecondderivativeswerediscretizedby( A{2 )andrstderivativesby( A{3 ).ForinputvaluesshowninTables 5-1 and 5-2 thesteady-stateconcentrationsofthenonenzymaticandenzymaticconcentrationsaswellasthevalueofthehomogeneousreactionswereobtained.Thesesteady-statevaluesalongwiththeoriginalinputvalueswereusedtoobtaintheoscillatingvaluesonconcentrationofthereactingspeciesH2O2.Foraspectrumofdimensionlessfrequencies,fromK=1E)]TJ /F1 11.9552 Tf 12.181 0 Td[(5to1E5with20pointsperdecade,theoscillatingconcentrationsofthereactingspeciesobtainedneartheelectrodewereimputedintoequation( 5{69 )toobtainadimensionlessdiusionimpedanceandequation( 5{68 )toobtainadiusionimpedance.Toobtainasolutionthatisthemostaccurate,thenitedierenceerrorsandround-oerrorsneedtobeminimized.Neartheelectrodesurfacetheuxofthereactingspeciesischangingsodramaticallyitisnecessarytohaveasmallermeshtoobtainanaccurateanswer.Farfromtheelectrodetheconcentrationsarenotchangingasmuchand 99

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Figure5-5. Onedimensionalschematicshowingthreedissimilarmeshsizes.HHHisthesmallestmeshsizeandHHisthemediummeshsize,andbothareintheGOXlayer.J=KJistheinterfaceintheGOXlayer.HisthelargestmeshsizeandisusedthroughouttheGLMlayer.J=IJistheinterfacebetweentheGOXandGLMlayers. thereforethesamesmallmeshwouldresultinunnecessaryroundoerrors.Evenfurtherfromtheelectrode,intheGLMregion,theonlyinuenceonconcentrationisdiusion,soanevenlargermeshsizeisnecessaryinthisregion.Tomitigateerrorswehavesolvedthesystemusingthreemeshsizes.Asmallmeshsize,HHH,isusedneartheelectrodeandalargemeshsize,HH,isusedfarawayfromtheelectrodebutstillintheGOXregion.Thelargestmeshsize,H,isusedforintheGLMregionwherenoreactionsareoccurring.Withtheuseofmultipledissimilarmeshsizes,itisnecessarythatamethodofcouplingbedevelopedtoallowthecomputationtransitionfromoneregiontoanotherwhileretainingtheaccuracyofthenitedierencecalculation.Suchtransitionisperformedbyacouplingsubroutineataninterfacedesignatedatj=KJintheGOXregionandj=IJattheGOX-GLMinterface.AvisualrepresentationofthethreemeshsizeregionsisshowninFigure 5-5 .ThecouplerintheGOXregionofthecodeiscouplingtwodierentmeshsizeswiththesamegoverningequations.Theuxononesideofthecoupleratj=KJ,isequaltotheuxontheothersideofthecoupler.Thedivergenceofuxofeachsideisconsidered. rNijKJ)]TJ /F3 7.9701 Tf 6.587 0 Td[(1=4=NijKJ)]TJ /F3 7.9701 Tf 6.587 0 Td[(1=2)]TJ /F5 11.9552 Tf 11.955 0 Td[(NijKJ 2HHH+RijKJ)]TJ /F3 7.9701 Tf 6.587 0 Td[(1=4=0(5{76) 100

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Table5-2. Systemparametervaluesforthecontinuousglucosemonitorsimulations ParameterValueUnits DistanceoftheReactionLayer,Y10.0004cmDistanceoftheInnerLayer,Y20.0003cmDistanceoftheGLMLayer,Y30.0015cmPorosityfactorGOX,GOX0.8dimensionlessPorosityfactorGLMsmallmolecules,GLM)]TJ /F1 11.9552 Tf 11.955 0 Td[(small0.42dimensionlessPorosityfactorGLMlargemolecules,GLM)]TJ /F1 11.9552 Tf 11.955 0 Td[(large0.169dimensionlessSolubilitycoecientH2O2,H2O20.32dimensionlessSolubilitycoecientO2,O20.11dimensionlessSolubilitycoecientGlucose,G0.0176dimensionlessHeterogeneousRateConstant,KH2O21A=cm2HeterogeneousConstant,bH2O237.42V)]TJ /F3 7.9701 Tf 6.587 0 Td[(1 and rNijKJ+1=4=NijKJ)]TJ /F5 11.9552 Tf 11.955 0 Td[(NijKJ+1=2 2HH+RijKJ+1=4=0(5{77)Re-writingequations( 5{76 )and( 5{77 )forNijKJandsettingequalweobtain Dici(KJ+1))]TJ /F5 11.9552 Tf 11.955 0 Td[(ci(KJ) HH)]TJ /F5 11.9552 Tf 11.955 0 Td[(Dici(KJ))]TJ /F5 11.9552 Tf 11.955 0 Td[(ci(KJ)]TJ /F1 11.9552 Tf 11.955 0 Td[(1) HHH+RijKJ)]TJ /F3 7.9701 Tf 6.586 0 Td[(1=4)]TJ /F5 11.9552 Tf 11.956 0 Td[(RijKJ+1=4=0(5{78)torepresentthej=KJcouplernode.Thesecondcoupler,representingj=IJ,isverysimilartotherstone.BecausetherearenoenzymesintheGLM,thereisnoreactiontermintheGLM.TheequationrepresentingthecouplerbetweentheGOXregionandGLMregionwas Dici(IJ+1))]TJ /F5 11.9552 Tf 11.955 0 Td[(ci(IJ) H)]TJ /F5 11.9552 Tf 11.955 0 Td[(Dici(IJ))]TJ /F5 11.9552 Tf 11.955 0 Td[(ci(IJ)]TJ /F1 11.9552 Tf 11.955 0 Td[(1) HH+RijIJ)]TJ /F3 7.9701 Tf 6.586 0 Td[(1=4=0(5{79) 5.2CGMResultsandDiscussionThesteady-stateandimpedanceresultsfromthemathematicalmodelforacontinuousglucosemonitorarepresented.SomeparameterswerekeptconstantforallsimulationsandtheseparameterarepresentedinTable 5-2 .Thevariablenames,bulkvalueconcentrationsanddiusioncoecientswerepresentedearlierinthechapterinTable 5-1 .Therstsetofresultsareforsystemswherethechangingparametersarethehomogeneous(orenzymatic)reactionrates.Thedierentsystemslookedat,titledSystem 101

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Table5-3. Kineticparametervaluesforsystem1 ParameterValueUnits HomogeneousRateConstant1,kf1,106cm3=mol=sHomogeneousEquilibriumConstant1,Keq110)]TJ /F3 7.9701 Tf 6.586 0 Td[(2mol=cm3HomogeneousRateConstant2,kf2,102cm3=mol=sHomogeneousRateConstant3,kf3,106cm3=mol=sHomogeneousEquilibriumConstant3,Keq310)]TJ /F3 7.9701 Tf 6.586 0 Td[(2mol=cm3HomogeneousRateConstant3,kf4,102cm3=mol=s Table5-4. Kineticparametervaluesforsystem2 ParameterValueUnits HomogeneousRateConstant1,kf1,105cm3=mol=sHomogeneousEquilibriumConstant1,Keq1102mol=cm3HomogeneousRateConstant2,kf2,103cm3=mol=sHomogeneousRateConstant3,kf3,105cm3=mol=sHomogeneousEquilibriumConstant3,Keq3102mol=cm3HomogeneousRateConstant3,kf4,103cm3=mol=s 1,System2andSystem3allhavevaryingkineticparameters,presentedinTables 5-3 5-5 .Thesecondsetofresultsshowtheaectsofbulkoxygenconcentrationonthesteady-stateandimpedanceresults.Theoxygenconcentrationvariedfrom510)]TJ /F3 7.9701 Tf 6.587 0 Td[(9mol=cm3,thevaluegiveninTable 5-1 ,toanorderofmagnitudelarger,510)]TJ /F3 7.9701 Tf 6.586 0 Td[(8mol=cm3,andanorderofmagnitudesmaller,510)]TJ /F3 7.9701 Tf 6.586 0 Td[(10mol=cm3. 5.2.1HomogeneousReactionRateInuenceontheCGMThepolarizationcurveforasystemdescribedinTable 5-2 usingdierenthomogeneousreactionratesisshowninFigure 5-6 .ThethreecurvescorrespondtodierenthomogeneousreactionratesandareinTables 5-3 5-4 ,and 5-5 .TheinsetgraphisazoomedinviewtoshowthebehaviorofSystem1.Astheoverallhomogeneousreactionincreases,from Table5-5. Kineticparametervaluesforsystem3 ParameterValueUnits HomogeneousRateConstant1,kf1,108cm3=mol=sHomogeneousEquilibriumConstant1,Keq11mol=cm3HomogeneousRateConstant2,kf2,107cm3=mol=sHomogeneousRateConstant3,kf3,108cm3=mol=sHomogeneousEquilibriumConstant3,Keq31mol=cm3HomogeneousRateConstant3,kf4,107cm3=mol=s 102

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Figure5-6. PolarizationcurvecalculatedfromdatainTable 5-2 .ThedierenthomogenousreactionratesareinTables 5-3 5-4 ,and 5-5 System1toSystem3,thecurrentresponseisalsogreater.Thedierenthydrogellayersontopoftheelectrodecomplicatethesystem.Speciesdonothavethesamemobilityinahydrogelastheywouldinamoretraditionalelectrolyte.TheporosityfactorsinTable 5-2 accountforageneralporosityfactorintheGOXlayerandtwodierentporosityfactorsintheGLMlayer,oneforthesmallmolecules,oxygenandhydrogenperoxideandoneforlargemolecules,glucoseandgluconicacid.ThelargemoleculeshaveamuchmorehinderedmobilitythroughtheGLMthemsmallmoleculesandthatisreectedinthemuchsmallerporosityfactor.Theporosityeectsthesystemanywhereadiusioncoecientiscalled.Thediusioncoecientismultipliedbytheappropriateporosityfactortothe1.5power.Becausethemoleculeswillbeowinginthebulkinanaqueouselectrolyteandthenintoahydrogel,apartitioncoecient(or 103

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A B C D E FFigure5-7. Calculatedsteady-stateconcentrationdistributionscorrespondingtosystemparameterspresentedinTables 5-1 and 5-2 ;a)Glucoseb)Oxygenc)HydrogenPeroxided)GluconicAcide)GlucoseOxidaseEnzymeinreducedandoxidizedformsf)GlucoseOxidaseEnzymecomplexformedinbothenzymaticreactionswiththedierentsystemsbeingdescribedbydierenthomogeneousreactionspresentedinTables 5-3 5-4 ,and 5-5 104

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Figure5-8. ReactionprolecalculatedfordierenthomogeneousreactionratesfromdatainTable 5-2 andcorrespondingwiththesteady-stateconcentrationsinFigure 5-7 .ThedierenthomogenousreactionratesareinTables 5-3 5-4 ,and 5-5 solubilityfactor)ismultipliedbythebulkvaluetoshowthechangeofconcentration.Thesteady-stateresultscorrespondingtothepolarizationcurve,Figure 5-6 aredisplayedinFigure 5-7 .Theglucoseconcentration,Figure 5-7A ,decreasesdramaticallyassoonasitdiusesintotheglucoselimitingmembraneduetothesolubilitycoecientandreactsintheGOXlayer.Withalargeenoughhomogeneousreactionrate,System3,theglucoseconcentrationiscompletelyconsumedasitgetsclosertotheelectrodesurface.Theoxygenconcentration,Figure 5-7B ,isalsoeectedbyasolubilitycoecientsotheconcentrationinthesubcutaneousuidissignicantlymorethenisrightinsidetheGLM.Similartotheglucoseconcentrationforthelargerhomogeneousreactionrate,System3,theoxygenconcentrationiscomplectlyconsumedforthislargerhomogeneousreaction 105

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Figure5-9. CalculateddimensionlessdiusionimpedancesfortheCGMwithdierenthomogeneousreactionratescorrespondingtosystemparameterspresentedinTables 5-1 and 5-2 forSystem1describedinTable 5-3 rate.TheoxygenandhydrogenperoxideareallowedtoowthroughtheGLMeasierthanthelargermolecules.Oxygenisalsocreatedintheelectrochemicalreactionattheelectrodesurfacewhichisshownforallthreesystems.Thehydrogenperoxide,Figure 5-7C ,isproducedintheGOXlayerandreactsontheelectrodesurface,however,someofthehydrogenperoxidediusesintotheGLM.Gluconicacid,Figure 5-7D ,isalsoformedintheenzymaticreactionbutdoesnotreactintheelectrochemicalreactionandthereforehasamuchlargerconcentration,anddiusesintothebodythroughtheGLM.Theconcentrationproleoftheglucoseoxidaseenzymeintheoxidizedform(solidlines)andthereducedform(dashedlines)arepresentedinFigure 5-7E .TheenzymecomplexconcentrationprolesarepresentedinFigure 5-7F .Theoverallconcentrationoftheenzymeinthereducedandoxidizedformaswellastheconcentrationoftheenzymecomplexesatanydistancefromtheelectrodeisaconstant.Theenzyme 106

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Figure5-10. CalculateddimensionlessdiusionimpedancesfortheCGMwithdierenthomogeneousreactionratescorrespondingtosystemparameterspresentedinTables 5-1 and 5-2 forSystem2describedinTable 5-4 complexconcentrationprolesfollowthesameshapeasthereactionprole,Figure 5-8 .Thereactionprolesshowdierentbehaviors.ForSystem3,thereactionspikesinthecenteroftheGOXlayer.ThereactionproleforSystem2islargestneartheelectrodesurface.System1showsasmallerreactionvaluethentheothertwosystems,andtheproleisatthroughouttheGOXregion.AllthesystemsshownoreactionoccurringintheGLM.ThereisalsonoconcentrationoftheenzymeineitherofitsformsoreitherenzymecomplexintheGLM.ThediusionimpedanceforSystems1-3arepresentedinFigures 5-9 5-11 .Thediusionimpedanceisnotdramaticallydierentforanyofthesystems.System1,presentedinFigure 5-9 ,istheonlydiusionimpedancewithaninductivefeature.TherstloopresemblesaGerischerimpedanceandhasacharacteristicfrequencyof10.Theinductiveloophasacharacteristicfrequencyof0.1.Forsystem2,showninFigure 5-10 107

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Figure5-11. CalculateddimensionlessdiusionimpedancesfortheCGMwithdierenthomogeneousreactionratescorrespondingtosystemparameterspresentedinTables 5-1 and 5-2 forSystem3describedinTable 5-5 theimpedancelookslikeaGerischerimpedanceathighfrequencyandatlowfrequencyhasatailwithtwomoretimeconstants.Thersttimeconstantisatadimensionlessfrequencyof12.6.Thesecondtimeconstantcouldbeattributedtoconvective-diusionimpedance.Thethirdtimeconstantoccursatverylowfrequency,adimensionlessfrequencyof2:210)]TJ /F3 7.9701 Tf 6.587 0 Td[(5.ThediusionimpedanceforSystem3ispresentedinFigure 5-11 whichstronglyresemblestheshapeofSystem2.Therstlooplookslikeatraditionalconvective-diusionimpedanceandcharacteristicfrequencyis20.Thelowfrequencyloopshaveacharacteristicfrequencyof0.1and3:110)]TJ /F3 7.9701 Tf 6.587 0 Td[(4. 5.2.2InuenceofOxygenConcentrationontheCGMOtherpropertiesbesidesthehomogeneousreactionratecanbeexplored.Theeectofoxygenconcentrationinthebulk(subcutaneousuid)wasstudiedforthehomogeneouskineticpropertiesofSystem2,Table 5-4 .Theoxygenconcentrationinthebulkvariedfromastandardvalueof510)]TJ /F3 7.9701 Tf 6.587 0 Td[(9mol=cm3,usedforthehomogeneousreaction 108

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Figure5-12. PolarizationcurvecalculatedfromdatainTable 5-2 andcorrespondingwiththesteady-stateconcentrationsinFigure 5-13 .ThehomogenousreactionratesareinTable 5-4 .Theoxygenconcentrationvariesfrom5E-8,5E-9and5E-10mol=cm3. ratestudy,toavalueanorderofmagnitudelargerandsmaller,510)]TJ /F3 7.9701 Tf 6.587 0 Td[(8mol=cm3and510)]TJ /F3 7.9701 Tf 6.587 0 Td[(10mol=cm3,respectively.Thepolarizationcurve,seeFigure 5-12 ,showsthatmoreoxygeninthebulkcorrespondstoahighercurrentandlesstoasmallercurrent.Themiddlecurve,correspondingtothestandardvalueof510)]TJ /F3 7.9701 Tf 6.586 0 Td[(9mol=cm3andSystem2kineticparameters,Table 5-4 ,isthesamedataplottedintheabovesection,section 5.2.1 .Thesteady-stateresultscorrespondingtothepolarizationcurve,Figure 5-12 aredisplayedinFigure 5-13 .Theglucoseconcentrationproles,Figure 5-13A showthathavingahigherconcentrationofoxygenallowsmoreglucosetobeconsumedintheglucoseoxidaselayer.Theoxygenconcentrationproles,Figure 5-13B ,havethesameshaperegardless 109

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A B C D E FFigure5-13. Calculatedsteady-stateconcentrationdistributionscorrespondingtosystemparameterspresentedinTables 5-1 5-2 and;a)Glucoseb)Oxygenc)HydrogenPeroxided)GluconicAcide)GlucoseOxidaseEnzymeinreducedandoxidizedformsf)GlucoseOxidaseEnzymecomplexformedinbothenzymaticreactionswiththedierentconcentrationsofoxygen 110

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Figure5-14. ReactionprolecalculatedfromdatainTable 5-2 andcorrespondingwiththesteady-stateconcentrationsinFigure 5-13 .ThehomogenousreactionratesareinTables 5-4 ofthebulkoxygenconcentration.Theglucoseandoxygen,aspreviouslyshowninFigure 5-7 ,haveadramaticconcentrationdierenceinsidetheCGMandoutsidetheCGMintheinterstitialuidduetothepartitioncoecients.Thehydrogenperoxideprolesshowthatthehydrogenperoxideisconsumedattheelectrodeandarealsothesameshaperegardlessoftheoxygenconcentration,seeFigure 5-13C .Gluconicacid,Figure 5-13D ,abyproductintheenzymaticreaction,isproducedmorewithalargerbulkoxygenconcentration.Theenzymeconcentrationprole,boththereducedandoxidizedforms,arepresentedinFigure 5-13E andtheenzymecomplexconcentrationprolesarepresentedinFigure 5-13F .Theenzymecomplexconcentrationprolehasthesameshapeasthereactionprole,seeFigure 5-14 .Whentheoxygenconcentrationissmallerthereactionproleshowsa 111

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spikeattheelectrodesurface.Astheoxygenconcentrationinthebulkisincreasedto510)]TJ /F3 7.9701 Tf 6.586 0 Td[(8mol=cm3,thespikeinthereactionproleisattheGOX{GLMinterface.ThedimensionlessdiusionimpedancesforthedierentoxygenconcentrationsareshowninFigures 5-15 5-17 .Thediusionimpedanceforthelargestconcentrationofoxygen,Figure 5-15 ,showsonetimeconstantat0.56andisamuchlargerimpedancethentheotherbulkoxygenconcentrations.Fortheimpedancesforboththeoxygenconcentrationof510)]TJ /F3 7.9701 Tf 6.587 0 Td[(9mol=cm3and510)]TJ /F3 7.9701 Tf 6.587 0 Td[(10mol=cm3,inFigures 5-16 and 5-17 ,thecharacteristicfrequencyisat12.5.ThedataforalltheimpedanceswithdierentbulkoxygenconcentrationsarepresentedinFigure 5-18 ,anditisclearthattheloweroxygenconcentrationsoverlapintheirdiusionimpedanceathighfrequenciesandareslightlydierent,butvisuallyverysimilar,atlowfrequencies.Thiscomparisonalsomakesitmoreclearthattheimpedanceforthehighbulkconcentrationofoxygenismuchbiggerthentheothertwoimpedances. 112

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Figure5-15. CalculateddimensionlessdiusionimpedancefortheCGMwithoxygenconcentrationof510)]TJ /F3 7.9701 Tf 6.586 0 Td[(8mol=cm3correspondingtosystemparameterspresentedinTables 5-1 and 5-2 andkineticparametersshowninTable 5-4 113

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Figure5-16. CalculateddimensionlessdiusionimpedancefortheCGMwithoxygenconcentrationof510)]TJ /F3 7.9701 Tf 6.586 0 Td[(9mol=cm3correspondingtosystemparameterspresentedinTables 5-1 and 5-2 andkineticparametersshowninTable 5-4 Figure5-17. CalculateddimensionlessdiusionimpedancefortheCGMwithoxygenconcentrationof510)]TJ /F3 7.9701 Tf 6.586 0 Td[(10mol=cm3correspondingtosystemparameterspresentedinTables 5-1 and 5-2 andkineticparametersshowninTable 5-4 114

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Figure5-18. CalculateddimensionlessdiusionimpedancefortheCGMwithdierentoxygenconcentrationsandcorrespondingtosystemparameterspresentedinTables 5-1 and 5-2 andkineticparametersshowninTable 5-4 115

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CHAPTER6CONCLUSIONSMathematicalmodelsweredevelopedfortheimpedanceresponseforelectrochemicalsystems.Diusionorconvectivediusioncaninuencetheimpedanceresponseofasystemassociatedwithelectrochemicalreactions.Therstmodelsdevelopedwereforconvective-diusionforarotatingdiskelectrodeandasubmergedimpingingjetelectrode,inChapter 3 .TheinuenceofaniteSchmidtnumberanalysisimpactedthedimensionlessdiusionimpedanceatlowfrequency.TheerroroftheimpedanceresponseusinganiteSchmidtnumberanalysiswhencomparedtoainniteSchmidtnumberanalysiswasgreaterforarotatingdiskelectrodethenforasubmergedimpingingjetelectrode.TheerrorforasystemwithaSchmidtnumberof100wasupto6.62%forarotatingdiskelectrodeandupto1.82%forasubmergedimpingingjetelectrode.Theconvective-diusionimpedancesimulationswereperformedinMATLAB,andtheimpedanceiscalculatedusingtheoscillatingconcentrationofthereactingspecies.Thedevelopmentofmodelsforconvective-diusionimpedanceservedasafoundationforthestudyofsystemsinwhichhomogeneousreactionsinuencetheimpedanceofelectrochemicalsystems.Thesecondmodelexploredaconvective-diusionrotatingdiskelectrodewhereahomogenousreactioninuencesthesystem.ThemodelincludedahomogeneousreactionintheelectrolytewherespeciesABreactsreversiblytoformA-andB+,andB+reactselectrochemicallyonarotatingdiskelectrodetoproduceB.Ananalyticexpressionforvelocitywasemployedthatcombinedathree-termvelocityexpansionneartheelectrodesurfacetoathree-termexpansionthatappliedfarfromtheelectrode.ThenonlinearexpressionforthehomogeneousreactionwasemployedinwhichtheconcentrationsofbothA-andB+wereassumedtobedependentonposition.Thismodelprovidesanextensiontotheliteraturebyusinganonlinearexpressionforthehomogenousreactionanduniquediusioncoecientsforeachspecies.Theresulting 116

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convective-diusionimpedancehadtwoasymmetriccapacitiveloop.Thelowfrequencyloopisassociatedwithconvective-diusionimpedancewithacharacteristicfrequencyof2.5,inagreementwiththeimpedanceforaconvective-diusionsystemintheabsenceofahomogeneousreaction.Theotherloop,atahigherfrequency,isassociatedwiththehomogeneousreaction.Foraninnitelyfasthomogeneousreaction,thesystemisshowntobehaveasthoughABistheelectroactivespecies.Eventhoughtheassumptionofalinearexpressionforthehomogeneousreactionwasrelaxed,amodiedGerischerimpedancewasfoundtoprovideagoodttothesimulateddata.ThemodelwasdevelopedinFORTRAN,andasteady-statesolutioncontainingfourvariableswassolvedfollowedbyasolutioninthefrequency-domaininvolvingeightvariables.TheoscillatingconcentrationofB+wasusedtoobtaintheimpedancespectrum.Thedevelopmentofthismodelchemical/electrochemicalsystemguideddevelopmentofmodelsfortheenzyme-basedbiosensors.Acontinuousglucosemonitorisarealworldapplicationofahomogeneousreactioninuencingtheelectrochemicalreaction.Thissystemincludedadiusionthroughahydrogellikemediumwhereglucosewouldreactinanenzymaticreactiontoformhydrogenperoxide,whichcanbedetectedelectrochemically.ThemodelaccountsforaglucoselimitingmembraneGLM,whichcontrolstheamountofglucoseparticipatingintheenzymaticreaction,andaglucoseoxidaseenzymelayer.Theglucoseoxidasewasassumedtobeimmobilizedwithinathinlmadjacenttotheelectrode.Intheglucoseoxidaselayer,aprocessofenzymaticcatalysistransformstheglucoseintohydrogenperoxide.Theelectrochemicalreactionproducedacurrentresponsethatcorrespondstotheoverallconcentrationofglucoseinthesubcutaneousuid.Themodeldevelopmentrequiredtwosteps.Thenonlinearcoupleddierentialequationsgoverningthissystemweresolvedundertheassumptionofasteadystate.Thesteady-stateconcentrationsresultingfromthesteady-statesimulationwereusedinthesolutionofthelinearizedsetofdierentialequationsdescribingthesinusoidalsteadystate.Theenzymaticcatalysis 117

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wastreatedintermsoffourhomogeneousreactions.AFORTRANcodewasusedtosolvethesteady-stateequationsfor12variableswhichwereusedsubsequentlytosolvethe24frequency-domainequations.Asbefore,theoscillatingconcentrationoftheelectroactivespecies,hydrogenperoxideinthiscase,wasusedtoobtaintheimpedanceresults. 118

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CHAPTER7FUTUREWORK 7.1FutureInvestigationofContinuousGlucoseMonitorCodeAFORTRANcodewascreatedtoinvestigatetheimpedanceresponseofacontinuousglucosemonitor.InitialsimulationsandresultsarediscussedinChapter 5 .Amoreextensiveparameterstudyofthekineticparametersisstillnecessarytounderstandthesystem.Otherparametersbesideshomogeneouskineticswillalsoneedtobeexplored.Theoverallimpedanceneedstobeexaminedaswell. 7.1.1CGMParameterStudyAnextensiveparameterstudyneedstooccurtofullyunderstandthecapabilitiesofthecodeaswellastheinuenceofparametersonthesystem.Aninitialaggressivestudyofthekineticparametersisinprogress.TheparametersbeinginvestigatedareshowninTable 7-1 ,whereSystems1-3arethesystemsdescribedinsection 5.2.1 .Theparameterswerechosenarbitrarilyduetothemillionsofcombinationsofkineticparametersthatcouldbetested.Afteranalysisoftheparameterslistedmoreparameterscanbechosenbasedontheknowledgegainedfromthisinitialstudy.Thedimensionlessdiusionimpedanceforall10systemsnotedinTable 7-1 areshowninFigure 7-1 .Fromthesediusionimpedancestherearethreedistinctshapes.Systems1and6showaninductiveloopatlowfrequency,presentedinFigures 7-1A and 7-1F .TwocapacitiveloopsareobservedforSystems2,3and10,showninFigures 7-1B 7-1C ,and 7-1J .Thelastdistinctshapeisasinglecapacitiveloop,whichwasobservedforSystems4,5,7,8,and9,presentedinFigures 7-1D 7-1E 7-1G 7-1H ,and 7-1I .Thesedierentshapesofdiusionimpedanceneedtobeexaminedfurthertounderstandtheirinuenceoftheoverallimpedance.Otherkineticparametersmayleadtomoreshapesofdiusionimpedanceandwillbeanalyzedinthefuture.FurtheranalysisofotherparametersintheCGMthatstillneedtobeexploredinclude 119

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A B C D E F G H I JFigure7-1. Dimensionlessdiusionimpedancesfordierentkineticparameters;A)System1B)System2C)System3D)System4E)System5F)System6G)System7H)System8I)System9J)System10 120

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Table7-1. Valuesforkineticparameterstudywheretheforwardratesallhaveunitsofcm3=mol=sandequilibriumrateshaveunitsofmol=cm3. Systemkf1Keq1kf2kf3Keq3kf4 110610)]TJ /F3 7.9701 Tf 6.587 0 Td[(210210610)]TJ /F3 7.9701 Tf 6.587 0 Td[(21022105102103105102103310810110710810110741051021031031021015106101103103101101610610110310610110371061011031021011018103101103102101101910310210510310210510108101107105101104 1. Heterogeneousreactionrates 2. Bulkconcentrationsofglucose 3. Bulkconcentrationsofoxygen 4. Activityofglucoseoxidase 5. LayerthicknessofGOX 6. LayerthicknessofGLM 7. Porosityfactorsforalldiusingspecies 8. Diusioncoecientsofallspecies 9. PartitioncoecientsforallspeciesExploringtheaectsoftheparameterswillgiveadeeperunderstandingofhowtheCGMworksaswellasunderstandingtheinuenceofparametersontheimpedance. 7.1.2OverallImpedanceAnalysisTheoverallimpedanceanalysisstillneedstobedone.Therearepossibleproblemswithconductingtheoverallimpedanceanalysisatthepotentialusedinthisdissertationbecausewhenoperatingatthemass-transfer-limitingcurrentthecurrentwillnothavealargemagnitude,(4I),sotheimpedanceisextremelylargeandhardtocharacterize. 121

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Conductinganimpedanceanalysisathalfthemass-transfer-limitingcurrentmayshowmoreinterestingandcharacterizablefeatures. 7.2InuenceofCoupledFaradaicandChargingCurrentsonEISTheinuenceofcoupledfaradaicandchargingcurrentsonimpedancespectroscopywassimulatedandanalyzedbyWuetal.[ 1 ].Preliminaryexperimentalresultsdonotperfectlyagreewithsimulations.Furtheranalysisoftheexperimentalresultswillbeconductedalongwithmorenite-elementthree-dimensionalsimulationsandanite-dierenceone-dimensionalsimulationtofurthercharacterizeandunderstandthisphenomenon. 7.2.1HistoryofCoupledChargingandFaradaicCurrentsInthe1960s,acontroversyemergedoverthecorrectmethodfordeterminingmodelsforimpedanceresponse.Thecontroversycenteredonwhetherfaradaicandchargingcurrentsinanelectrochemicalsystemshouldbeconsideredtobecoupledorseparate.Modelsforimpedancetypicallyassumethatfaradaicandchargingcurrentsarenotcoupled.ApreviousmemberofDr.Orazem'sgroup,Shao-LingWu,showedforarotatingdiskelectrode,thatcouplingchargingandfaradaiccurrentsresultsinfrequencydispersion.Thiseectcouldbedistinguishedfromthefrequencydispersionknowntobecausedbythediskgeometry.Thegoalofmyworkistodevelopaone-dimensionalmodelwhichwillhavelowercomputationalcomplexitythanthemodelsforadiskelectrodeandwillisolatethefrequencydispersioncausedbycoupledchargingandfaradaiccurrents.Thisworkwillfacilitateexplorationoftheinuenceofdierentmodelsforthedoublelayer.Duringthe1960'sacontroversyappearedintheelectrochemicalliteratureoverthecorrectmethodforderivingmodelsforimpedance.Sluyterstreatedthetotalpassageofcurrentthroughanelectrodeassimplytheadditionoffaradaicandchargingcurrent[ 85 ].ThisapproachwascriticizedbyDelahay,whosaidthatthefaradaicandchargingcurrentscouldnotbeconsideredseparately[ 86 87 88 ].Hebelievedthattheuxofthereactingspeciesshouldcontributetothefaradaiccurrentandalsotochargingthedoublelayer.Inspite 122

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Figure7-2. Schematicrepresentationillustratingthecontributionofthereactingspeciestothechargingoftheelectrode-electrolyteinterfacecorrespondingto:a)thecasewitha-prioriseparation(APS);andb)thecasewithnoa-prioriseparation(NAPS).TakenfromWuetal.[ 1 ] ofDelahay'sobjections,impedancemodelstodayrelyontheassumptionthatfaradaicandchargingcurrentareseparable[ 5 89 90 ].TheSluytersapproachprevailedinlargepartbecauseanalyticsolutionsarepossibleformodelsthatallowdecouplingoffaradaicandchargingcurrents.Totreatthereactingspeciesascontributingtoboththefaradaicreactionandchargingthesurfacerequirescouplinganexplicitmodelofthedoublelayertotheconvectivediusionequationsforeachionicspecies.Suchmodelsrequirenumericalsimulations.NisanciogluandNewmandevelopedaframeworkforcouplingthechargingandfaradaiccurrents[ 91 ].Theysuggestedthatthecouplingonlyhasasignicanteectforwellsupportedelectrolytes.Wuetal.followedNissancioglu'sandNewman'sapproachintheirmodelforarotatingdiskelectrode[ 1 ].Theyfoundthatcouplingcausesafrequencydispersion,eveninthepresenceofawellsupportedelectrolyte.Thedierencebetweena-prioriseparation(APS)andnoa-prioriseparation(NAPS)ofthefaradaicandchargingcurrentinanelectrochemicalsystemisshownschematicallyinFigure 7-2 [ 1 ].WithAPS,theuxofthereactingspeciescontributesonlytothefaradaicreaction,Figure 7-2 (a).IntheNAPScase,thereactingspeciescontributetoboththefaradaicreactionand,alongwiththeinertspecies,tothechargingcurrent,Figure 7-2 (b). 123

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Figure7-3. Schematicrepresentationoftime-constantdistribution.WhereA)isalongtheareaofelectrode-electrolyteinterface;andB)isalongthedirectionnormaltotheelectrodesurface[ 2 ]. 7.2.2Constant-PhaseElementsAconstant-phaseelement(CPE)isacircuitelementthatdisplaysaconstantphaseangle.Thiscouldbearesistor,capacitororinductor.However,theelectricaldouble-layeratanelectrodedoesnotgenerallybehaveasapurecapacitance,butratheranimpedancedisplayingafrequencyphaseangledierentfrom90degrees[ 92 ].TheimpedanceforaCPEis ZCPE=1 (j!)Q(7{1)whereandQareconstants.When=1,Qhasunitsofcapacitance.When6=1,Qhasunitsofs=cm2.Frequently,0:5<<1foranelectrochemicalinterfaceofarealcell.Non-idealbehaviorleadingtoaCPEcanbeattributedtothefrequencyortime-constantdistribution.Thefrequencydispersioncanoccuralongtheareaoftheelectrodeoralongthedirectionnormaltotheelectrodesurface.Thesurfacedistributioncanarisefrom 124

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surfacedierences[ 93 ].Anormaldistributioncanbeattributedtocompositionofoxidelayers[ 94 ]ortoporosity[ 95 ].Figure 7-3 showsthesurfaceandnormaldistributionsusingequivalentcircuits,whereRtisthecharge-transferresistance,C0isthedouble-layercapacitance,andRfandCfaretheresistanceandcapacitanceofanoxidelm.Thevariationsofreactionreactivityanddouble-layercapacitanceattheelectrode{electrolyteinterfacecauseafrequencyortime-constantdistributionattheelectrodesurface.Variationsoflmpropertiesinanoxidelayercancauseafrequencydistributiontoo.ThesedistributionsareobservedduringtheimpedancemeasurementsintheformofaCPE.ThepresenceofaCPEbehavior,isverycommonevenforahomogenousandsmoothsurface. 7.2.3ElectrochemicalInstrumentationApictureoftheelectrochemicalinstrumentationusedtoconducttheEISexperimentsispresentedinFigure 7-4 .AAg/AgClelectrodewasusedasthereferenceelectrode.Theworkingelectrodewaseitherplatinumorglassycarbonandthecounterelectrodewasaplatinummesh. 7.2.4CVCurvesandEISExperimentalResultsTheCVcurvesinFigure 7-5 showsresultsforanon-polished(Figure 7-5A )andpolished(Figure 7-5B )glassycarbonelectrode,respectively,fora10mMFe(CN)6(II)=(III)and0.5MKClelectrolyte.Thesweepratevariedfrom10mV/sto10V/s.Thelargestsweeprateof10V/sgavethebiggestcurrentresponse.ThepolarizationcurvesfordierentrotationratesareshowninFigure 7-6 .Thepolarizationcurvesweretakenatasweeprateof50mV/sforrotationratesfrom300RPMto2400RPM.Thenon-polishedelectrode,Figure 7-6A ,approachedthemass-transfer-limitedcurrentslowerthenforthepolishedelectrode,Figure 7-6B .Forbothelectrodes,thehighertherotationspeedsledtohighercurrentresponse,butthenon-polishedelectrodehadhighercurrentsthenthepolishedelectrode.Theimpedancefor500RPMforfractionsofthelimitingcurrentforboththenon-polishedelectrodeandpolishedelectrodearepresentedinFigure 7-7 .The 125

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Figure7-4. ElectrochemicalinstrumentationusedtoconducttheEISexperiments non-polishedelectrode,Figure 7-7A ,hadimpedancesthatweredoubletheimpedancesforthepolishedelectrode,Figure 7-7B .Thehigh-frequencycapacitiveloopwaslargerforthenon-polishedelectrode.Anotherwaytolookattheimpedancedataisintheformofanadjustedphaseangleplot,presentedinFigure 7-8 126

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A BFigure7-5. CVcurveswithdierentsweepratesfrom10mV/sto10V/sfortheA)non-polishedandB)polishedglassycarbonelectrodewitha10mMFe(CN)6(II)=(III)and0.5MKClelectrolyte. 127

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A BFigure7-6. Polarizationcurveswithdierentrotationratesfrom300RPMto2400RPMforthe;A)non-polishedandB)polishedglassycarbonelectrodewitha10mMFe(CN)6(II)=(III)and0.5MKClelectrolyte. 128

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A BFigure7-7. Impedancespectrumwitharotationspeedof500RPMforA)non-polishedandB)polishedglassycarbonelectrodewitha10mMFe(CN)6(II)=(III)and0.5MKClelectrolyte. 129

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A BFigure7-8. Adjustedphaseanglewitharotationspeedof500RPMforA)non-polishedandB)polishedglassycarbonelectrodewitha10mMFe(CN)6(II)=(III)and0.5MKClelectrolyte. 130

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APPENDIXABANDElectrochemicalsystemscanbemodeledmathematicallyasboundaryvalueproblemswhichconsistofsetsofnon-linear,coupled,second-orderdierentialequations[ 96 ].ThealgorithmBANDwasdevelopedinFortranbyNewman.[ 77 ]Alinearsetofcoupled,second-orderdierentialequationscanbewrittenas nXk=1ai;k(x)d2ck(x) dx2+bi;k(x)dck(x) dx+di;k(x)ck(x)=gi(x)(A{1)Therearenequationsoftheformofequation( A{1 ),wherekistheindexrepresentingthedependentvariableandiistheequationnumber.Acentralnitedierenceapproximationtothederivativesappearinginequation( A{1 ),accuratetotheorderh2are d2ck dx2=ck(xj+h)+ck(xj)]TJ /F5 11.9552 Tf 11.955 0 Td[(h))]TJ /F1 11.9552 Tf 11.956 0 Td[(2ck(xj) h2+O(h2)(A{2)and dck dx=ck(xj+h))]TJ /F5 11.9552 Tf 11.955 0 Td[(ck(xj)]TJ /F5 11.9552 Tf 11.955 0 Td[(h) 2h+O(h2)(A{3)forasecondderivativeandrstderivative,respectively.Equations( A{2 )and( A{3 )aretheapproximationstothederivativesatthepointxjinthemeshusedfornumericalsolution.Thesizebetweenmeshpointsish.Equations( A{2 )and( A{3 )gointoequation( A{1 )toform nXk=1Ai;k(J)Ck(J)]TJ /F1 11.9552 Tf 11.956 0 Td[(1)+Bi;k(J)Ck(J)+Di;k(J)Ck(J+1)=Gi(j)(A{4)whichisevaluatedatpositionxj)]TJ /F3 7.9701 Tf 6.586 0 Td[(1=xj)]TJ /F5 11.9552 Tf 12.087 0 Td[(h.Bi;k(j)isthecoecient,inequationi,atthepositionxjofthedependentvariableck.Thecoecientsinequation(( A{4 ))are Ai;k(j)=ai;k(xj))]TJ /F5 11.9552 Tf 13.15 8.087 Td[(h 2bi;k(xj)(A{5) Bi;k(j)=)]TJ /F1 11.9552 Tf 9.298 0 Td[(2ai;k(xj))]TJ /F5 11.9552 Tf 11.955 0 Td[(h2di;k(xj)(A{6) 131

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FigureA-1. MatrixdeningBAND[ 96 ]. Di;k(j)=ai;k(xj)+h 2bi;k(xj)(A{7)and Gi(j)=h2gi(xj)(A{8)Asimpleboundarycondition,forthesolutionofasystemwithNJnodesis nXk=1ei;k(x)Ck=fi(x)(A{9)fortherstmeshpoint,atJ=1,canbewritten nXk=1Bi;k(1)Ck(1)=Gi(1)(A{10)andthelastmeshpoint,atJ=NJ,canbewritten nXk=1Bi;k(NJ)Ck(NJ)=Gi(NJ)(A{11)Thesegoverningdierenceequations,equations( A{4 ),( A{10 ),and( A{11 ),canbewrittenconvenientlyinmatrixform(seeFigure A-1 ).Oras MT=Z(A{12)wheretheMisannbynmatrixandTandZaren-dimensionalvectors.Equation( A{12 )canbesolvedbydecomposingMintotriangularmatrices.Theupperandlower 132

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trianglematriceswillbedenotedUandLso MT=LUT=Z(A{13)Letting UT=(A{14)equation( A{13 )becomes L=Z(A{15)TheunknownTcanbesolvedbyequation( A{14 )afterL,Uandareknown.Forwardandbackwardsubstitutionscanbemadetosolveequations( A{15 )and( A{14 ).ThesegoverningmatrixequationscanbesolvedwithNewman'sBANDalgorithm.ThebehaviorofBANDwasstudiedbyCurtisetal.[ 97 ]andWhite[ 96 ].Curtisetal.comparedtheBANDmethodtothedeBoormethod.SometimestheBANDalgorithmsignalsthematrixasfalselysingularandtheerror\DETERMINANT=0ATJ=2"iscalledandinthiscaseitisbettertorewritethenitedierenceequationsoruseadierentmethodforsolvingtheproblem.ThefollowingcodesinthisAppendixareNewman'sBANDalgorithmandmatrixinversioncodewritteninMATLABandthenFORTRAN.TheMATLABcodewasusedformodelsofconvective-diusionimpedance,Chapter 3 ,andtheFORTRANcodewasusedforalltheothermodelsinthisdissertation,explainedinlengthinChapters 4 and 5 133

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CodeA.1.MATLABBANDandMATINVcode 1 % 2 % MATLAB version of BAND 3 % The following is a version of BAND ( and MATINV ) obtained by 4 % translating the FORTRAN version in Newman 5 % 6 7 function [y]=band(J) % i e J = I from test band 8 global ABCDEGXYNPOINTSNNJNP1DETERMINANT 9 % 10 % Case where J negative 11 % 12 if (J)]TJ /F11 9.9626 Tf 8.081 0 Td[(2)<0 % i e image point ( I =1) outside boundary ( left hand side ) 13 14 NP1=N+1; % I runs from 1: NPOINTS and NPOINTS = NJ while N = # eqn 15 for I=1:N 16 D(I,2N+1)=G(I); 17 for L=1:N 18 LPN=L+N; 19 D(I,LPN)=X(I,L); 20 end 21 end 22 % 23 % Calling Matinv 24 % 25 matinv(N,2N+1); 26 if (DETERMINANT==0) 27 sprintf ('DETERMINANT=0atJ=%6.3f',J) 28 return % stod tidligere break 29 else 30 for K=1:N 31 E(K,NP1,1)=D(K,2N+1); 32 for L=1:N 33 E(K,L,1)=)]TJ /F11 9.9626 Tf 6.342 0 Td[(D(K,L); 34 LPN=L+N; 35 X(K,L)=)]TJ /F11 9.9626 Tf 6.342 0 Td[(D(K,LPN); 36 end 37 end 38 return 39 end 40 end 41 % 42 % Case where J )]TJ /F11 9.9626 Tf 7.49 0 Td[(2 equal to zero ( I =2, i e first real point ) 43 % 44 if ((J)]TJ /F11 9.9626 Tf 8.081 0 Td[(2)==0) 45 46 for I=1:N 47 for K=1:N 48 for L=1:N 49 D(I,K)=D(I,K)+A(I,L)X(L,K); 50 end 51 end 52 end 53 end 54 % 55 % Case where J )]TJ /F11 9.9626 Tf 7.49 0 Td[(2 is greater than zero 56 % 134

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57 if (J)]TJ /F11 9.9626 Tf 6.651 0 Td[(NJ)<0 58 59 for I=1:N 60 D(I,NP1)=)]TJ /F11 9.9626 Tf 6.238 0 Td[(G(I); 61 for L=1:N 62 D(I,NP1)=D(I,NP1)+A(I,L)E(L,NP1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1); 63 for K=1:N 64 B(I,K)=B(I,K)+A(I,L)E(L,K,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1); 65 end 66 end 67 end 68 else 69 70 for I=1:N 71 for L=1:N 72 G(I)=G(I))]TJ /F11 9.9626 Tf 6.411 0 Td[(Y(I,L)E(L,NP1,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(2); 73 for M=1:N 74 A(I,L)=A(I,L)+Y(I,M)E(M,L,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(2); 75 end 76 end 77 end 78 for I=1:N 79 D(I,NP1)=)]TJ /F11 9.9626 Tf 6.238 0 Td[(G(I); 80 for L=1:N 81 D(I,NP1)=D(I,NP1)+A(I,L)E(L,NP1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1); 82 for K=1:N 83 B(I,K)=B(I,K)+A(I,L)E(L,K,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1); 84 end 85 end 86 end 87 end 88 % 89 % Calling Matinv 90 % 91 92 matinv(N,NP1); 93 if DETERMINANT~=0 94 95 for K=1:N 96 for M=1:NP1 97 E(K,M,J)=)]TJ /F11 9.9626 Tf 6.342 0 Td[(D(K,M); 98 end 99 end 100 else 101 sprintf ('DETERMINANT=0atJ=%6.3f',J) 102 return % stod tidligere break 103 end 104 105 if (J)]TJ /F11 9.9626 Tf 6.651 0 Td[(NJ)<0 106 107 return 108 else 109 110 for K=1:N 111 112 C(K,J)=E(K,NP1,J); 113 end 114 for JJ=2:NJ 135

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115 116 M=NJ)]TJ /F11 9.9626 Tf 13.708 0 Td[(JJ+1; 117 for K=1:N 118 C(K,M)=E(K,NP1,M); 119 for L=1:N 120 C(K,M)=C(K,M)+E(K,L,M)C(L,M+1); 121 end 122 end 123 end 124 for L=1:N 125 126 for K=1:N 127 C(K,1)=C(K,1)+X(K,L)C(L,3); 128 end 129 end 130 end 131 132 return 133 134 % 136

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CodeA.2.FORTRANBANDCode 1 C SUBROUTINE BAND ( J ) 2 3 SUBROUTINE BAND(J) 4 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.808 0 Td[(Z) 5 6 DIMENSION E(4,5,100001) 7 COMMON /BAB/A(4,4),B(4,4),C(4,100001),D(4,9),G(4),X(4,4),Y(4,4) 8 COMMON /NSN/N,NJ 9 k 10 SAVE E,NP1 11 101 FORMAT (15HDETERM=0ATJ=,I4) 12 IF (J)]TJ /F11 9.9626 Tf 8.081 0 Td[(2)1,6,8 13 1NP1=N+1 14 DO 2I=1,N 15 D(I,2N+1)=G(I) 16 DO 2L=1,N 17 LPN=L+N 18 2D(I,LPN)=X(I,L) 19 CALL MATINV(N,2N+1,DETERM) 20 21 IF (DETERM)4,3,4 22 3 PRINT 101,J 23 4 DO 5K=1,N 24 E(K,NP1,1)=D(K,2N+1) 25 DO 5L=1,N 26 E(K,L,1)=)]TJ /F11 9.9626 Tf 6.047 0 Td[(D(K,L) 27 LPN=L+N 28 5X(K,L)=)]TJ /F11 9.9626 Tf 6.047 0 Td[(D(K,LPN) 29 RETURN 30 31 6 DO 7I=1,N 32 DO 7K=1,N 33 DO 7L=1,N 34 7D(I,K)=D(I,K)+A(I,L)X(L,K) 35 8 IF (J)]TJ /F11 9.9626 Tf 6.651 0 Td[(NJ)11,9,9 36 9 DO 10I=1,N 37 DO 10L=1,N 38 G(I)=G(I))]TJ /F11 9.9626 Tf 6.411 0 Td[(Y(I,L)E(L,NP1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(2) 39 DO 10M=1,N 40 10A(I,L)=A(I,L)+Y(I,M)E(M,L,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(2) 41 11 DO 12I=1,N 42 D(I,NP1)=)]TJ /F11 9.9626 Tf 5.943 0 Td[(G(I) 43 DO 12L=1,N 44 D(I,NP1)=D(I,NP1)+A(I,L)E(L,NP1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1) 45 DO 12K=1,N 46 12B(I,K)=B(I,K)+A(I,L)E(L,K,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1) 47 48 CALL MATINV(N,NP1,DETERM) 49 IF (DETERM)14,13,14 50 13 PRINT 101,J 51 14 DO 15K=1,N 52 DO 15M=1,NP1 53 15E(K,M,J)=)]TJ /F11 9.9626 Tf 6.047 0 Td[(D(K,M) 54 IF (J)]TJ /F11 9.9626 Tf 6.651 0 Td[(NJ)20,16,16 55 16 DO 17K=1,N 56 17C(K,J)=E(K,NP1,J) 137

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57 DO 18JJ=2,NJ 58 M=NJ)]TJ /F11 9.9626 Tf 7.435 0 Td[(JJ+1 59 DO 18K=1,N 60 C(K,M)=E(K,NP1,M) 61 DO 18L=1,N 62 18C(K,M)=C(K,M)+E(K,L,M)C(L,M+1) 63 DO 19L=1,N 64 DO 19K=1,N 65 19C(K,1)=C(K,1)+X(K,L)C(L,3) 66 67 20 RETURN 68 69 END 138

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CodeA.3.FORTRANMATINVCode 1 C SUBROUTINE MATINV 2 3 SUBROUTINE MATINV(N,M,DETERM) 4 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.808 0 Td[(Z) 5 6 COMMON /BAB/A(4,4),B(4,4),C(4,100001),D(4,9),G(4),X(4,4),Y(4,4) 7 COMMON /NSN/NTEMP,NJ 8 9 DIMENSION ID(4) 10 11 DETERM=1.01 12 DO 1I=1,N 13 1ID(I)=0 14 DO 18NN=1,N 15 BMAX=1.1 16 DO 6I=1,N 17 IF (ID(I). NE .0) GO TO 6 18 BNEXT=0.0 19 BTRY=0.0 20 DO 5J=1,N 21 IF (ID(J). NE .0) GO TO 5 22 IF (DABS(B(I,J)). LE .BNEXT) GO TO 5 23 BNEXT=DABS(B(I,J)) 24 IF (BNEXT. LE .BTRY) GO TO 5 25 BNEXT=BTRY 26 BTRY=DABS(B(I,J)) 27 JC=J 28 5 CONTINUE 29 30 IF (BNEXT. GE .BMAXBTRY) GO TO 6 31 BMAX=BNEXT/BTRY 32 IROW=I 33 JCOL=JC 34 6 CONTINUE 35 36 IF (ID(JC). EQ .0) GO TO 8 37 DETERM=0.0 38 RETURN 39 40 8ID(JCOL)=1 41 IF (JCOL. EQ .IROW) GO TO 12 42 DO 10J=1,N 43 SAVE =B(IROW,J) 44 B(IROW,J)=B(JCOL,J) 45 10B(JCOL,J)= SAVE 46 47 DO 11K=1,M 48 SAVE =D(IROW,K) 49 D(IROW,K)=D(JCOL,K) 50 11D(JCOL,K)= SAVE 51 52 12F=1.0/B(JCOL,JCOL) 53 DO 13J=1,N 54 13B(JCOL,J)=B(JCOL,J)F 55 DO 14K=1,M 56 14D(JCOL,K)=D(JCOL,K)F 139

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57 DO 18I=1,N 58 IF (I. EQ .JCOL) GO TO 18 59 F=B(I,JCOL) 60 DO 16J=1,N 61 16B(I,J)=B(I,J))]TJ /F11 9.9626 Tf 6.895 0 Td[(FB(JCOL,J) 62 DO 17K=1,M 63 17D(I,K)=D(I,K))]TJ /F11 9.9626 Tf 6.895 0 Td[(FD(JCOL,K) 64 18 CONTINUE 65 66 RETURN 67 END 140

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APPENDIXBCODESFORROTATINGDISKELECTRODEThisappendixcontainsthedierentMATLABcodesusedtosolvetheconvective-diusionequationforarotatingdiskelectrodeinChapter 3 .TherstMATLABcodeistosolvetheinniteSchmidtconvective-diusionequationortheterm0niteSchmidtconvective-diusionequation.ThesecondMATLABcodesolvesforterm1oftheniteSchmidtconvective-diusionequation.ThelastMATLABcodesolvesforterm2oftheniteSchmidtconvective-diusionequation.Thesolutionstoeachcodeareimputedintoequation( 3{32 )toobtainasolutionfortheniteSchmidtconvective-diusionequation.Themathematicaldevelopmentoftheequationsinsertedintothefollowingcodesisdiscussedinsection 3.2 141

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CodeB.1.FiniteSchmidtConvectionDiusionTerm0forrotatingdiskelectrode 1 % This problem solves for the first term of the convective )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 348.369 -35.608 Td[(diffusion equation with a finite schmidt number 2 % d ^2( theta0 ) / d ( xi ) ^2+3 xi ^2 d ( theta0 ) / d ( xi ) )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 264.483 -57.65 Td[(j K theta0 =0 3 4 clc ; close all ; clear all ; 5 format longE; 6 global ABCDGXYNNJ 7 8 h=0.01; % Step )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 198.934 -123.777 Td[(size 9 logK=)]TJ /F11 9.9626 Tf 8.856 0 Td[(2:.05:4; % Frequency range 0.01 to 10000 10 K=10.^logK; 11 K 12 x=0:h:10; % Range of x 13 NJ= length (x); 14 NJ 15 ZZd= zeros (1, length (K)); % Place holder for values of diffusion impedance 16 17 c 0=1; % Boundary condition for x =0 18 c e=0; % Boundary condition for x = end ( NJ ) 19 tol=0; % Absolute tolerance 20 21 % Define initial guess 22 conc(1,1:NJ)=0.5; % Conc (1,)]TJ /F11 9.9626 Tf 8.579 0 Td[() is the real component of theta0 23 conc(2,1:NJ)=0.5; % Conc (2,)]TJ /F11 9.9626 Tf 8.579 0 Td[() is the imaginary component of theta0 24 N=2; 25 26 for kk=1: length (K) 27 28 error =1; 29 jcount=0; 30 jcountmax=4; % Number of iterations the program will allow to converge 31 while error >=tol; 32 X= zeros (N,N); 33 Y=X; 34 35 jcount=jcount+1; 36 for J=1:NJ; 37 A= zeros (N,N); 38 B=A; 39 D=A; 40 % Boundary condition at J =1 41 if J==1; 42 G(1)=conc(1,J))]TJ /F11 9.9626 Tf 8.393 0 Td[(c 0; % Real equation so BC at electrode surface is 1 43 B(1,1)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1; 44 45 G(2)=conc(2,J))]TJ /F11 9.9626 Tf 8.236 0 Td[(0; % Imaginary equation so BC at electrode surface is 0 46 B(2,2)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1; 47 48 error = abs (G(1))+ abs (G(2)); 49 50 % Region between boundaries 51 elseif (J<=(NJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)); 142

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52 G(1)=conc(1,J+1))]TJ /F11 9.9626 Tf 7.793 0 Td[(2conc(1,J)+conc(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+h^2K(kk)conc(2,J)+1.5h^3( J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2(conc(1,J+1))]TJ /F11 9.9626 Tf 8.066 0 Td[(conc(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)); 53 A(1,1)=)]TJ /F11 9.9626 Tf 7.937 0 Td[(1+1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 54 B(1,1)=2; 55 D(1,1)=)]TJ /F11 9.9626 Tf 7.92 0 Td[(1)]TJ /F11 9.9626 Tf 7.92 0 Td[(1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 56 B(1,2)=)]TJ /F11 9.9626 Tf 7.085 0 Td[(h^2K(kk); 57 58 G(2)=conc(2,J+1))]TJ /F11 9.9626 Tf 7.793 0 Td[(2conc(2,J)+conc(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))]TJ /F11 9.9626 Tf 7.38 0 Td[(h^2K(kk)conc(1,J)+1.5h^3( J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2(conc(2,J+1))]TJ /F11 9.9626 Tf 8.066 0 Td[(conc(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)); 59 A(2,2)=)]TJ /F11 9.9626 Tf 7.937 0 Td[(1+1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 60 B(2,2)=2; 61 D(2,2)=)]TJ /F11 9.9626 Tf 7.92 0 Td[(1)]TJ /F11 9.9626 Tf 7.92 0 Td[(1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 62 B(2,1)=h^2K(kk); 63 64 error = error + abs (G(1))+ abs (G(2)); 65 66 % Boundary condition at J = NJ 67 else 68 G(1)=conc(1,J))]TJ /F11 9.9626 Tf 8.236 0 Td[(0; % BC at far away point is always 0 69 B(1,1)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1; 70 71 G(2)=conc(2,J))]TJ /F11 9.9626 Tf 8.236 0 Td[(0; % BC at far away point is always 0 72 B(2,2)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1; 73 74 error = error + abs (G(1))+ abs (G(2)); 75 end ; 76 77 band(J) 78 end ; 79 conc=conc+C; % C comes from band ( J ) program 80 81 if jcount>=jcountmax, break end ; 82 end ; 83 84 theta0=complex(conc(1,:),conc(2,:)); % Creating theta from conc (1,)]TJ /F11 9.9626 Tf 8.579 0 Td[() and conc (2,)]TJ /F11 9.9626 Tf 8.579 0 Td[() values 85 dtheta0= gamma (4/3)()]TJ /F11 9.9626 Tf 9.185 0 Td[(theta0(3)+4theta0(2))]TJ /F11 9.9626 Tf 7.793 0 Td[(3theta0(1))/(2h) % dtheta accurate to h ^2 86 87 88 Zd=)]TJ /F11 9.9626 Tf 7.491 0 Td[(1/dtheta0; % turning theta into Impedance 89 90 % str = sprintf (' K = % g : Zd = %.10 e + j %.10 e ', K ( kk ) real ( Zd ) imag ( Zd ) ) ; disp ( str ) ; 91 % str = sprintf (' dtheta0 = % g ', dtheta0 ) ; disp ( str ) ; 92 % str = sprintf ('%.10 e ', K ( kk ) ) ; disp ( str ) ; 93 % str = sprintf ('%.10 e ', real ( Zd ) ) ; disp ( str ) ; 94 % str = sprintf ('%.10 e ', imag ( Zd ) ) ; disp ( str ) ; 95 96 figure (1) 97 plot (x,conc(1,:),')]TJ /F11 9.9626 Tf 8.196 0 Td[(r') 98 hold on; 99 plot (x,conc(2,:),')]TJ /F11 9.9626 Tf 7.656 0 Td[(g') 100 axis ([02)]TJ /F11 9.9626 Tf 8.235 0 Td[(.41]); 101 legend ('theta r','theta i'); 102 title ('DimensionlessConcentrationawayfromElectrodeSurface'); 103 xlabel ('length'); 104 ylabel ('theta'); 143

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105 106 ZZd(kk)=Zd; 107 Ktemp=K(kk); 108 fname= sprintf ('theta0 %i',kk); 109 save (fname,'theta0','Ktemp','h'); 110 end 111 112 fname= sprintf ('zero order %i.txt',NJ); 113 fid= fopen (fname,'wt'); 114 fprintf (fid,'%14.6e;%14.6enn',h,h^2); 115 for kk=1: length (K); 116 fprintf (fid,'%12.6e;%20.12e;%20.12e;nn',K(kk), real (ZZd(kk)), imag (ZZd(kk)) ); 117 end ; 118 fclose (fid); 119 120 figure (2) 121 plot ( real (ZZd),)]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 96.218 -199.315 Td[(imag (ZZd),')]TJ /F11 9.9626 Tf 8.081 0 Td[(ks'); hold on; axis equal; 122 Zfilm= tanh ( sqrt (jK))./ sqrt (jK); 123 plot ( real (Zfilm),)]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 108.173 -221.357 Td[(imag (Zfilm),')142()]TJ /F11 9.9626 Tf 13.594 0 Td[(b'); 124 legend ('MatLabData','FiniteFilmThicknesstanh(sqrt(jK))/sqrt(jK)'); 125 title ('Nyquistplot'); 126 xlabel ('RealpartofImpedance'); 127 ylabel ('ImaginarypartofImpedance'); 128 129 figure (3) 130 loglog (K,)]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 66.33 -298.506 Td[(imag (ZZd),')142()]TJ /F11 9.9626 Tf 13.732 0 Td[(k'); hold on; 131 loglog (K, real (ZZd),')142()]TJ /F11 9.9626 Tf 13.593 0 Td[(b'); 132 legend (')]TJ /F11 9.9626 Tf 8.108 0 Td[(Imaginary','Real'); 133 title ('ImpedancevsFrequency'); 134 xlabel ('DimensionlessFrequency'); 135 ylabel ('Impedance'); 144

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CodeB.2.FiniteSchmidtConvectionDiusionTerm1forrotatingdiskelectrode 1 % This problem solves for the first term of the convective )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 348.369 -35.608 Td[(diffusion equation with a finite schmidt number 2 3 clc ; close all ; clear all ; 4 format longE; 5 global ABCDGXYNNJ 6 7 h=0.01; % Step )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 198.934 -112.756 Td[(size 8 logK=)]TJ /F11 9.9626 Tf 8.856 0 Td[(2:.05:4; % Frequency range 0.01 to 10000 9 K=10.^logK; 10 x=0:h:10; % Range of x 11 NJ= length (x); 12 ZZd= zeros (1, length (K)); % Place holder for values of diffusion impedance 13 aa=0.51023; 14 bb=)]TJ /F11 9.9626 Tf 8.487 0 Td[(0.61592; 15 cc=(3/aa^4)^(1/3); 16 17 c 0=1; % Boundary condition for x =0 18 c e=0; % Boundary condition for x = end ( NJ ) 19 tol=0; % Absolute tolerance 20 21 % Define initial guess 22 conc(1,1:NJ)=0.5; % Conc (1,)]TJ /F11 9.9626 Tf 8.579 0 Td[() is the real component of theta0 23 conc(2,1:NJ)=0.5; % Conc (2,)]TJ /F11 9.9626 Tf 8.579 0 Td[() is the imaginary component of theta0 24 N=2; 25 26 for kk=1: length (K) 27 fname= sprintf ('theta0 %i',kk); 28 load (fname); 29 flag =0; if ( abs (Ktemp)]TJ /F11 9.9626 Tf 5.977 0 Td[(K(kk))~=0); flag =3; quit ; end ; 30 31 error =1; 32 jcount=0; 33 jcountmax=4; % Number of iterations the program will allow to converge 34 while error >=tol; 35 X= zeros (N,N); 36 Y=X; 37 38 jcount=jcount+1; 39 for J=1:NJ; 40 A= zeros (N,N); 41 B=A; 42 D=A; 43 % Boundary condition at J =1 44 if J==1; 45 G(1)=conc(1,J); % Real equation so BC at electrode surface is 0 46 B(1,1)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1; 47 48 G(2)=conc(2,J); % Imaginary equation so BC at electrode surface is 0 49 B(2,2)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1; 50 51 error = abs (G(1))+ abs (G(2)); 52 53 % Region between boundaries 145

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54 elseif (J<=(NJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)); 55 G(1)=conc(1,J+1))]TJ /F11 9.9626 Tf 7.793 0 Td[(2conc(1,J)+conc(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+h^2K(kk)conc(2,J)+1.5h^3( J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2(conc(1,J+1))]TJ /F11 9.9626 Tf 8.066 0 Td[(conc(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))... 56 )]TJ /F11 9.9626 Tf 8.192 0 Td[(cc(h^4)((J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^3)( real (theta0(J+1)))]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 306.426 -45.019 Td[(real (theta0(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)))/2; 57 A(1,1)=)]TJ /F11 9.9626 Tf 7.937 0 Td[(1+1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 58 B(1,1)=2; 59 D(1,1)=)]TJ /F11 9.9626 Tf 7.92 0 Td[(1)]TJ /F11 9.9626 Tf 7.92 0 Td[(1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 60 B(1,2)=)]TJ /F11 9.9626 Tf 7.085 0 Td[(h^2K(kk); 61 62 G(2)=conc(2,J+1))]TJ /F11 9.9626 Tf 7.793 0 Td[(2conc(2,J)+conc(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))]TJ /F11 9.9626 Tf 7.38 0 Td[(h^2K(kk)conc(1,J)+1.5h^3( J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2(conc(2,J+1))]TJ /F11 9.9626 Tf 8.066 0 Td[(conc(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))... 63 )]TJ /F11 9.9626 Tf 8.192 0 Td[(cc(h^4)((J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^3)( imag (theta0(J+1)))]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 305.436 -133.188 Td[(imag (theta0(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)))/2; 64 A(2,2)=)]TJ /F11 9.9626 Tf 7.937 0 Td[(1+1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 65 B(2,2)=2; 66 D(2,2)=)]TJ /F11 9.9626 Tf 7.92 0 Td[(1)]TJ /F11 9.9626 Tf 7.92 0 Td[(1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 67 B(2,1)=h^2K(kk); 68 69 error = error + abs (G(1))+ abs (G(2)); 70 71 % Boundary condition at J = NJ 72 else 73 G(1)=conc(1,J))]TJ /F11 9.9626 Tf 8.236 0 Td[(0; % BC at far away point is always 0 74 B(1,1)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1; 75 76 G(2)=conc(2,J))]TJ /F11 9.9626 Tf 8.236 0 Td[(0; % BC at far away point is always 0 77 B(2,2)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1; 78 79 error = error + abs (G(1))+ abs (G(2)); 80 end ; 81 82 band(J) 83 end ; 84 conc=conc+C; % C comes from band ( J ) program 85 86 if jcount>=jcountmax, break end ; 87 end ; 88 89 theta1=complex(conc(1,:),conc(2,:)); % Creating theta from conc (1,)]TJ /F11 9.9626 Tf 8.579 0 Td[() and conc (2,)]TJ /F11 9.9626 Tf 8.579 0 Td[() values 90 dtheta1= gamma (4/3)()]TJ /F11 9.9626 Tf 9.185 0 Td[(theta1(3)+4theta1(2))]TJ /F11 9.9626 Tf 7.793 0 Td[(3theta1(1))/(2h); % dtheta accurate to h ^2 91 dtheta0= gamma (4/3)()]TJ /F11 9.9626 Tf 9.185 0 Td[(theta0(3)+4theta0(2))]TJ /F11 9.9626 Tf 7.793 0 Td[(3theta0(1))/(2h); 92 93 Zd=dtheta1/(dtheta0^2); % turning theta into Impedance 94 95 % str = sprintf (' K = %.10 e : Zd = %.10 e + j %.10 e ', K ( kk ) real ( Zd ) imag ( Zd ) ) ; disp ( str ) ; 96 % str = sprintf (' dtheta1 = % g ', dtheta1 ) ; disp ( str ) ; 97 98 str= sprintf ('%.10e',K(kk)); disp (str); 99 str= sprintf ('%.10e', real (Zd)); disp (str); 100 str= sprintf ('%.10e', imag (Zd)); disp (str); 101 102 figure (1) 103 plot (x,conc(1,:),')]TJ /F11 9.9626 Tf 8.196 0 Td[(r') 104 hold on; 105 plot (x,conc(2,:),')]TJ /F11 9.9626 Tf 7.656 0 Td[(g') 106 axis ([02)]TJ /F11 9.9626 Tf 8.235 0 Td[(.41]); 146

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107 legend ('theta r','theta i'); 108 title ('DimensionlessConcentrationawayfromElectrodeSurface'); 109 xlabel ('length'); 110 ylabel ('theta'); 111 112 ZZd(kk)=Zd; 113 Ktemp=K(kk); 114 fname= sprintf ('theta1 %i',kk); 115 save (fname,'theta1','Ktemp','h'); 116 end 117 118 fname= sprintf ('first order %i.txt',NJ); 119 fid= fopen (fname,'wt'); 120 fprintf (fid,'%14.6e;%14.6enn',h,h^2); 121 for kk=1: length (K); 122 fprintf (fid,'%12.6e;%20.12e;%20.12e;nn',K(kk), real (ZZd(kk)), imag (ZZd(kk)) ); 123 end ; 124 fclose (fid); 125 126 figure (2) 127 plot ( real (ZZd),)]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 96.218 -243.4 Td[(imag (ZZd),')]TJ /F11 9.9626 Tf 8.081 0 Td[(ks'); hold on; axis equal; 128 title ('Nyquistplot'); 129 xlabel ('RealpartofImpedance'); 130 ylabel ('ImaginarypartofImpedance'); 131 132 figure (3) 133 semilogx (K,)]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 78.285 -309.527 Td[(imag (ZZd),')142()]TJ /F11 9.9626 Tf 13.732 0 Td[(k'); hold on; 134 semilogx (K, real (ZZd),')142()]TJ /F11 9.9626 Tf 13.594 0 Td[(b'); 135 legend (')]TJ /F11 9.9626 Tf 8.108 0 Td[(Imaginary','Real'); 136 title ('ImpedancevsFrequency'); 137 xlabel ('DimensionlessFrequency'); 138 ylabel ('Impedance'); 147

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CodeB.3.FiniteSchmidtConvectionDiusionTerm2forrotatingdiskelectrode 1 % This problem solves for the first term of the convective )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 348.369 -35.608 Td[(diffusion equation with a finite schmidt number 2 3 clc ; close all ; clear all ; 4 format longE; 5 global ABCDGXYNNJ 6 7 h=0.01; % Step )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 210.889 -112.756 Td[(size 8 logK=)]TJ /F11 9.9626 Tf 8.856 0 Td[(2:.05:4; % Frequency range 0.01 to 10000 9 K=10.^logK; 10 x=0:h:10; % Range of x 11 NJ= length (x); 12 ZZd= zeros (1, length (K)); % Place holder for values of diffusion impedance 13 aa=0.51023; 14 bb=)]TJ /F11 9.9626 Tf 8.487 0 Td[(0.61592; 15 cc=(3/aa^4)^(1/3); 16 17 c 0=1; % Boundary condition for x =0 18 c e=0; % Boundary condition for x = end ( NJ ) 19 tol=0; % Absolute tolerance 20 21 % Define initial guess 22 conc(1,1:NJ)=0.5; % Conc (1,)]TJ /F11 9.9626 Tf 8.579 0 Td[() is the real component of theta0 23 conc(2,1:NJ)=0.5; % Conc (2,)]TJ /F11 9.9626 Tf 8.579 0 Td[() is the imaginary component of theta0 24 N=2; 25 26 for kk=1: length (K) 27 fname= sprintf ('theta0 %i',kk); % load solution for Theta )]TJ /F11 9.9626 Tf 7.49 0 Td[(0 28 load (fname); 29 flag =0; if ( abs (Ktemp)]TJ /F11 9.9626 Tf 5.977 0 Td[(K(kk))~=0); flag =3; quit ; end ; 30 fname= sprintf ('theta1 %i',kk); % load solution for Theta )]TJ /F11 9.9626 Tf 7.49 0 Td[(1 31 load (fname); 32 flag =0; if ( abs (Ktemp)]TJ /F11 9.9626 Tf 5.977 0 Td[(K(kk))~=0); flag =3; quit ; end ; 33 34 error =1; 35 jcount=0; 36 jcountmax=4; % Number of iterations the program will allow to converge 37 while error >=tol; 38 X= zeros (N,N); 39 Y=X; 40 41 jcount=jcount+1; 42 for J=1:NJ; 43 A= zeros (N,N); 44 B=A; 45 D=A; 46 % Boundary condition at J =1 47 if J==1; 48 G(1)=conc(1,J); % Real equation so BC at electrode surface is 0 49 B(1,1)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1; 50 51 G(2)=conc(2,J); % Imaginary equation so BC at electrode surface is 0 52 B(2,2)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1; 53 148

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54 error = abs (G(1))+ abs (G(2)); 55 56 % Region between boundaries 57 elseif (J<=(NJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)); 58 G(1)=conc(1,J+1))]TJ /F11 9.9626 Tf 7.793 0 Td[(2conc(1,J)+conc(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+h^2K(kk)conc(2,J)+1.5h^3( J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2(conc(1,J+1))]TJ /F11 9.9626 Tf 8.066 0 Td[(conc(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))... 59 )]TJ /F11 9.9626 Tf 7.454 0 Td[(bb(3/aa)^(5/3)(h^5)((J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1)^4)( real (theta0(J+1)))]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 384.136 -78.082 Td[(real (theta0(J )]TJ /F11 9.9626 Tf 8.081 0 Td[(1)))/12... 60 )]TJ /F11 9.9626 Tf 8.192 0 Td[(cc(h^4)((J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^3)( real (theta1(J+1)))]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 306.426 -100.125 Td[(real (theta1(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)))/2; 61 A(1,1)=)]TJ /F11 9.9626 Tf 7.937 0 Td[(1+1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 62 B(1,1)=2; 63 D(1,1)=)]TJ /F11 9.9626 Tf 7.92 0 Td[(1)]TJ /F11 9.9626 Tf 7.92 0 Td[(1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 64 B(1,2)=)]TJ /F11 9.9626 Tf 7.085 0 Td[(h^2K(kk); 65 66 G(2)=conc(2,J+1))]TJ /F11 9.9626 Tf 7.793 0 Td[(2conc(2,J)+conc(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))]TJ /F11 9.9626 Tf 7.38 0 Td[(h^2K(kk)conc(1,J)+1.5h^3( J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2(conc(2,J+1))]TJ /F11 9.9626 Tf 8.066 0 Td[(conc(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))... 67 )]TJ /F11 9.9626 Tf 7.454 0 Td[(bb(3/aa)^(5/3)(h^5)((J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1)^4)( imag (theta0(J+1)))]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 383.145 -188.294 Td[(imag (theta0(J )]TJ /F11 9.9626 Tf 8.081 0 Td[(1)))/12... 68 )]TJ /F11 9.9626 Tf 8.192 0 Td[(cc(h^4)((J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^3)( imag (theta1(J+1)))]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 305.436 -210.336 Td[(imag (theta1(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)))/2; 69 A(2,2)=)]TJ /F11 9.9626 Tf 7.937 0 Td[(1+1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 70 B(2,2)=2; 71 D(2,2)=)]TJ /F11 9.9626 Tf 7.92 0 Td[(1)]TJ /F11 9.9626 Tf 7.92 0 Td[(1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 72 B(2,1)=h^2K(kk); 73 74 error = error + abs (G(1))+ abs (G(2)); 75 76 % Boundary condition at J = NJ 77 else 78 G(1)=conc(1,J))]TJ /F11 9.9626 Tf 8.236 0 Td[(0; % BC at far away point is always 0 79 B(1,1)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1; 80 81 G(2)=conc(2,J))]TJ /F11 9.9626 Tf 8.236 0 Td[(0; % BC at far away point is always 0 82 B(2,2)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1; 83 84 error = error + abs (G(1))+ abs (G(2)); 85 end ; 86 87 band(J) 88 end ; 89 conc=conc+C; % C comes from band ( J ) program 90 91 if jcount>=jcountmax, break end ; 92 end ; 93 94 theta2=complex(conc(1,:),conc(2,:)); % Creating theta from conc (1,)]TJ /F11 9.9626 Tf 8.579 0 Td[() and conc (2,)]TJ /F11 9.9626 Tf 8.579 0 Td[() values 95 dtheta0= gamma (4/3)()]TJ /F11 9.9626 Tf 9.185 0 Td[(theta0(3)+4theta0(2))]TJ /F11 9.9626 Tf 7.793 0 Td[(3theta0(1))/(2h); 96 dtheta1= gamma (4/3)()]TJ /F11 9.9626 Tf 9.185 0 Td[(theta1(3)+4theta1(2))]TJ /F11 9.9626 Tf 7.793 0 Td[(3theta1(1))/(2h); % dtheta accurate to h ^2 97 dtheta2= gamma (4/3)()]TJ /F11 9.9626 Tf 9.185 0 Td[(theta2(3)+4theta2(2))]TJ /F11 9.9626 Tf 7.793 0 Td[(3theta2(1))/(2h); 98 99 Zd=)]TJ /F11 9.9626 Tf 7.828 0 Td[((1/dtheta0)((dtheta1/dtheta0)^2)]TJ /F11 9.9626 Tf 8.872 0 Td[(dtheta2/dtheta0); % turning theta into Impedance 100 ZZd(kk)=Zd; 101 102 % str = sprintf (' K = %.10 e : Zd = %.10 e + j %.10 e ', K ( kk ) real ( Zd ) imag ( Zd ) ) ; disp ( str ) ; 103 % str = sprintf (' dtheta2 = % g ', dtheta2 ) ; disp ( str ) ; 149

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104 105 str= sprintf ('%.10e',K(kk)); disp (str); 106 str= sprintf ('%.10e', real (Zd)); disp (str); 107 str= sprintf ('%.10e', imag (Zd)); disp (str); 108 109 figure (1) 110 plot (x,conc(1,:),')]TJ /F11 9.9626 Tf 8.196 0 Td[(r') 111 hold on; 112 plot (x,conc(2,:),')]TJ /F11 9.9626 Tf 7.656 0 Td[(g') 113 axis ([02)]TJ /F11 9.9626 Tf 8.235 0 Td[(.41]); 114 legend ('theta r','theta i'); 115 title ('DimensionlessConcentrationawayfromElectrodeSurface'); 116 xlabel ('length'); 117 ylabel ('theta'); 118 119 120 end 121 122 fname= sprintf ('first order %i.txt',NJ); 123 fid= fopen (fname,'wt'); 124 fprintf (fid,'%14.6e;%14.6enn',h,h^2); 125 for kk=1: length (K); 126 fprintf (fid,'%12.6e;%20.12e;%20.12e;nn',K(kk), real (ZZd(kk)), imag (ZZd(kk)) ); 127 end ; 128 fclose (fid); 129 130 figure (2) 131 plot ( real (ZZd),)]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 96.218 -320.548 Td[(imag (ZZd),')]TJ /F11 9.9626 Tf 8.081 0 Td[(ks'); hold on; axis equal; 132 title ('Nyquistplot'); 133 xlabel ('RealpartofImpedance'); 134 ylabel ('ImaginarypartofImpedance'); 135 136 figure (3) 137 semilogx (K,)]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 78.285 -386.675 Td[(imag (ZZd),')142()]TJ /F11 9.9626 Tf 13.732 0 Td[(k'); hold on; 138 semilogx (K, real (ZZd),')142()]TJ /F11 9.9626 Tf 13.594 0 Td[(b'); 139 legend (')]TJ /F11 9.9626 Tf 8.108 0 Td[(Imaginary','Real'); 140 title ('ImpedancevsFrequency'); 141 xlabel ('DimensionlessFrequency'); 142 ylabel ('Impedance'); 150

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APPENDIXCCODESFORIMPINGINGJETELECTRODEThisappendixcontainsthedierentMATLABcodesusedtosolvetheconvective-diusionequationforasubmergedimpingingjetelectrodeinChapter 3 .TherstMATLABcodeistosolvetheinniteSchmidtconvective-diusionequationortheterm0niteSchmidtconvective-diusionequation.ThesecondMATLABcodesolveforterm1oftheniteSchmidtconvective-diusionequation.ThelastMATLABcodesolvesforterm2oftheniteSchmidtconvective-diusionequation.Thesolutionstoeachcodeareimputedintoequation( 3{32 )toobtainasolutionfortheniteSchmidtconvective-diusionequation.Themathematicaldevelopmentoftheequationsinsertedintothefollowingcodesisdiscussedinsection 3.3 151

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CodeC.1.FiniteSchmidtConvectionDiusionTerm0foranimpingingjetelectrode 1 % This problem solves for the first term of the convective )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 348.369 -35.608 Td[(diffusion equation with a finite schmidt number 2 % d ^2( theta0 ) / d ( xi ) ^2+3 xi ^2 d ( theta0 ) / d ( xi ) )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 264.483 -57.65 Td[(j K theta0 =0 3 4 clc ; close all ; clear all ; 5 format longE; 6 global ABCDGXYNNJ 7 8 h=0.01; % Step )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 210.889 -123.777 Td[(size 9 logK=)]TJ /F11 9.9626 Tf 8.856 0 Td[(2:.05:4; % Frequency range 0.01 to 10000 10 K=10.^logK; 11 x=0:h:10; % Range of x 12 NJ= length (x); 13 ZZd= zeros (1, length (K)); % Place holder for values of diffusion impedance 14 15 c 0=1; % Boundary condition for x =0 16 c e=0; % Boundary condition for x = end ( NJ ) 17 tol=0; % Absolute tolerance 18 19 conc real= zeros ( length (x), length (K)); 20 conc imag= zeros ( length (x), length (K)); 21 22 % Define initial guess 23 conc(1,1:NJ)=0.5; % Conc (1,)]TJ /F11 9.9626 Tf 8.579 0 Td[() is the real component of theta0 24 conc(2,1:NJ)=0.5; % Conc (2,)]TJ /F11 9.9626 Tf 8.579 0 Td[() is the imaginary component of theta0 25 N=2; 26 27 for kk=1: length (K) 28 29 error =1; 30 jcount=0; 31 jcountmax=4; % Number of iterations the program will allow to converge 32 while error >=tol; 33 X= zeros (N,N); 34 Y=X; 35 36 jcount=jcount+1; 37 for J=1:NJ; 38 A= zeros (N,N); 39 B=A; 40 D=A; 41 % Boundary condition at J =1 42 if J==1; 43 G(1)=conc(1,J))]TJ /F11 9.9626 Tf 8.393 0 Td[(c 0; % Real equation so BC at electrode surface is 1 44 B(1,1)=)]TJ /F11 9.9626 Tf 7.859 .001 Td[(1; 45 46 G(2)=conc(2,J))]TJ /F11 9.9626 Tf 8.236 0 Td[(0; % Imaginary equation so BC at electrode surface is 0 47 B(2,2)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1; 48 49 error = abs (G(1))+ abs (G(2)); 50 51 % Region between boundaries 52 elseif (J<=(NJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)); 152

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53 G(1)=conc(1,J+1))]TJ /F11 9.9626 Tf 7.793 0 Td[(2conc(1,J)+conc(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+h^2K(kk)conc(2,J)+1.5h^3( J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2(conc(1,J+1))]TJ /F11 9.9626 Tf 8.066 0 Td[(conc(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)); 54 A(1,1)=)]TJ /F11 9.9626 Tf 7.937 0 Td[(1+1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 55 B(1,1)=2; 56 D(1,1)=)]TJ /F11 9.9626 Tf 7.92 0 Td[(1)]TJ /F11 9.9626 Tf 7.92 0 Td[(1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 57 B(1,2)=)]TJ /F11 9.9626 Tf 7.085 0 Td[(h^2K(kk); 58 59 G(2)=conc(2,J+1))]TJ /F11 9.9626 Tf 7.793 0 Td[(2conc(2,J)+conc(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))]TJ /F11 9.9626 Tf 7.38 0 Td[(h^2K(kk)conc(1,J)+1.5h^3( J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2(conc(2,J+1))]TJ /F11 9.9626 Tf 8.066 0 Td[(conc(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)); 60 A(2,2)=)]TJ /F11 9.9626 Tf 7.937 0 Td[(1+1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 61 B(2,2)=2; 62 D(2,2)=)]TJ /F11 9.9626 Tf 7.92 0 Td[(1)]TJ /F11 9.9626 Tf 7.92 0 Td[(1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 63 B(2,1)=h^2K(kk); 64 65 error = error + abs (G(1))+ abs (G(2)); 66 67 % Boundary condition at J = NJ 68 else 69 G(1)=conc(1,J))]TJ /F11 9.9626 Tf 8.236 0 Td[(0; % BC at far away point is always 0 70 B(1,1)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1; 71 72 G(2)=conc(2,J))]TJ /F11 9.9626 Tf 8.236 0 Td[(0; % BC at far away point is always 0 73 B(2,2)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1; 74 75 error = error + abs (G(1))+ abs (G(2)); 76 end ; 77 78 band(J) 79 end ; 80 conc=conc+C; % C comes from band ( J ) program 81 82 if jcount>=jcountmax, break end ; 83 end ; 84 85 theta0=complex(conc(1,:),conc(2,:)); % Creating theta from conc (1,)]TJ /F11 9.9626 Tf 8.579 0 Td[() and conc (2,)]TJ /F11 9.9626 Tf 8.579 0 Td[() values 86 dtheta0= gamma (4/3)()]TJ /F11 9.9626 Tf 9.185 0 Td[(theta0(3)+4theta0(2))]TJ /F11 9.9626 Tf 7.793 0 Td[(3theta0(1))/(2h); % dtheta accurate to h ^2 87 88 Zd=)]TJ /F11 9.9626 Tf 7.491 0 Td[(1/dtheta0; % turning theta into Impedance 89 90 % str = sprintf (' K = % g : Zd = %.10 e + j %.10 e ', K ( kk ) real ( Zd ) imag ( Zd ) ) ; disp ( str ) ; 91 % str = sprintf (' dtheta0 = % g ', dtheta0 ) ; disp ( str ) ; 92 93 str= sprintf ('%.10e',K(kk)); disp (str); 94 str= sprintf ('%.10e', real (Zd)); disp (str); 95 str= sprintf ('%.10e', imag (Zd)); disp (str); 96 97 figure (1) 98 plot (x,conc(1,:),')]TJ /F11 9.9626 Tf 8.196 0 Td[(r') 99 hold on; 100 plot (x,conc(2,:),')]TJ /F11 9.9626 Tf 7.656 0 Td[(g') 101 axis ([02)]TJ /F11 9.9626 Tf 8.235 0 Td[(.41]); 102 legend ('theta r','theta i'); 103 title ('DimensionlessConcentrationawayfromElectrodeSurface'); 104 xlabel ('length'); 105 ylabel ('theta'); 153

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106 107 for kkk=1: length (x) 108 conc real(kkk,kk)=(conc(1,kkk)); 109 conc imag(kkk,kk)=(conc(2,kkk)); 110 end 111 112 ZZd(kk)=Zd; 113 114 Ktemp=K(kk); 115 fname= sprintf ('theta0 %i',kk); 116 save (fname,'theta0','Ktemp','h'); 117 end 118 119 fname= sprintf ('zero order %i.txt',NJ); 120 fid= fopen (fname,'wt'); 121 fprintf (fid,'%14.6e;%14.6enn',h,h^2); 122 for kk=1: length (K); 123 fprintf (fid,'%12.6e;%20.12e;%20.12e;nn',K(kk), real (ZZd(kk)), imag (ZZd(kk)) ); 124 end ; 125 fclose (fid); 126 127 figure (2) 128 plot ( real (ZZd),)]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 96.218 -265.442 Td[(imag (ZZd),')]TJ /F11 9.9626 Tf 8.081 0 Td[(ks'); hold on; axis equal; 129 Zfilm= tanh ( sqrt (jK))./ sqrt (jK); 130 plot ( real (Zfilm),)]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 108.173 -287.484 Td[(imag (Zfilm),')142()]TJ /F11 9.9626 Tf 13.594 0 Td[(b'); 131 legend ('MatLabData','FiniteFilmThicknesstanh(sqrt(jK))/sqrt(jK)'); 132 title ('Nyquistplot'); 133 xlabel ('RealpartofImpedance'); 134 ylabel ('ImaginarypartofImpedance'); 135 136 str= sprintf ('%.10e', real (Zfilm)); disp (str); 137 138 figure (3) 139 loglog (K,)]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 66.33 -386.675 Td[(imag (ZZd),')142()]TJ /F11 9.9626 Tf 13.732 0 Td[(k'); hold on; 140 loglog (K, real (ZZd),')142()]TJ /F11 9.9626 Tf 13.593 0 Td[(b'); 141 legend (')]TJ /F11 9.9626 Tf 8.108 0 Td[(Imaginary','Real'); 142 title ('ImpedancevsFrequency'); 143 xlabel ('DimensionlessFrequency'); 144 ylabel ('Impedance'); 145 146 realZfilm= real (Zfilm); 147 imagZfilm= imag (Zfilm); 148 149 realZZd= real (ZZd)'; 150 imagZZd= imag (ZZd)'; 154

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CodeC.2.FiniteSchmidtConvectionDiusionTerm1foranimpingingjetelectrode 1 % This problem solves for the second term of the convective )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 354.347 -35.608 Td[(diffusion equation with a finite schmidt number 2 3 clc ; close all ; clear all ; 4 format longE; 5 global ABCDGXYNNJ 6 7 h=0.01; % Step )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 210.889 -112.756 Td[(size 8 logK=)]TJ /F11 9.9626 Tf 8.856 0 Td[(2:.05:4; % Frequency range 0.01 to 10000 9 K=10.^logK; 10 x=0:h:10; % Range of x 11 NJ= length (x); 12 ZZd= zeros (1, length (K)); % Place holder for values of diffusion impedance 13 aa=0.51023; 14 bb=)]TJ /F11 9.9626 Tf 8.487 0 Td[(0.61592; 15 cc=(3/(1.352)^4)^(1/3); 16 17 c 0=1; % Boundary condition for x =0 18 c e=0; % Boundary condition for x = end ( NJ ) 19 tol=0; % Absolute tolerance 20 21 conc real= zeros ( length (x), length (K)); 22 conc imag= zeros ( length (x), length (K)); 23 24 % Define initial guess 25 conc(1,1:NJ)=0.5; % Conc (1,)]TJ /F11 9.9626 Tf 8.579 0 Td[() is the real component of theta0 26 conc(2,1:NJ)=0.5; % Conc (2,)]TJ /F11 9.9626 Tf 8.579 0 Td[() is the imaginary component of theta0 27 N=2; 28 29 for kk=1: length (K) 30 fname= sprintf ('theta0 %i',kk); 31 load (fname); 32 flag =0; if ( abs (Ktemp)]TJ /F11 9.9626 Tf 5.977 0 Td[(K(kk))~=0); flag =3; quit ; end ; 33 34 error =1; 35 jcount=0; 36 jcountmax=4; % Number of iterations the program will allow to converge 37 while error >=tol; 38 X= zeros (N,N); 39 Y=X; 40 41 jcount=jcount+1; 42 for J=1:NJ; 43 A= zeros (N,N); 44 B=A; 45 D=A; 46 % Boundary condition at J =1 47 if J==1; 48 G(1)=conc(1,J); % Real equation so BC at electrode surface is 0 49 B(1,1)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1; 50 51 G(2)=conc(2,J); % Imaginary equation so BC at electrode surface is 0 52 B(2,2)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1; 53 155

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54 error = abs (G(1))+ abs (G(2)); 55 56 % Region between boundaries 57 elseif (J<=(NJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)); 58 G(1)=conc(1,J+1))]TJ /F11 9.9626 Tf 7.793 0 Td[(2conc(1,J)+conc(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+h^2K(kk)conc(2,J)+1.5h^3( J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2(conc(1,J+1))]TJ /F11 9.9626 Tf 8.066 0 Td[(conc(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))... 59 )]TJ /F11 9.9626 Tf 8.192 0 Td[(cc(h^4)((J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^3)( real (theta0(J+1)))]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 306.426 -78.082 Td[(real (theta0(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)))/2; 60 A(1,1)=)]TJ /F11 9.9626 Tf 7.937 0 Td[(1+1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 61 B(1,1)=2; 62 D(1,1)=)]TJ /F11 9.9626 Tf 7.92 0 Td[(1)]TJ /F11 9.9626 Tf 7.92 0 Td[(1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 63 B(1,2)=)]TJ /F11 9.9626 Tf 7.085 0 Td[(h^2K(kk); 64 65 G(2)=conc(2,J+1))]TJ /F11 9.9626 Tf 7.793 0 Td[(2conc(2,J)+conc(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))]TJ /F11 9.9626 Tf 7.38 0 Td[(h^2K(kk)conc(1,J)+1.5h^3( J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2(conc(2,J+1))]TJ /F11 9.9626 Tf 8.066 0 Td[(conc(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))... 66 )]TJ /F11 9.9626 Tf 8.192 0 Td[(cc(h^4)((J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^3)( imag (theta0(J+1)))]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 305.436 -166.252 Td[(imag (theta0(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)))/2; 67 A(2,2)=)]TJ /F11 9.9626 Tf 7.937 0 Td[(1+1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 68 B(2,2)=2; 69 D(2,2)=)]TJ /F11 9.9626 Tf 7.92 0 Td[(1)]TJ /F11 9.9626 Tf 7.92 0 Td[(1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 70 B(2,1)=h^2K(kk); 71 72 error = error + abs (G(1))+ abs (G(2)); 73 74 % Boundary condition at J = NJ 75 else 76 G(1)=conc(1,J))]TJ /F11 9.9626 Tf 8.236 0 Td[(0; % BC at far away point is always 0 77 B(1,1)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1; 78 79 G(2)=conc(2,J))]TJ /F11 9.9626 Tf 8.236 0 Td[(0; % BC at far away point is always 0 80 B(2,2)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1; 81 82 error = error + abs (G(1))+ abs (G(2)); 83 end ; 84 85 band(J) 86 end ; 87 conc=conc+C; % C comes from band ( J ) program 88 89 if jcount>=jcountmax, break end ; 90 end ; 91 92 theta1=complex(conc(1,:),conc(2,:)); % Creating theta from conc (1,)]TJ /F11 9.9626 Tf 8.579 0 Td[() and conc (2,)]TJ /F11 9.9626 Tf 8.579 0 Td[() values 93 dtheta1= gamma (4/3)()]TJ /F11 9.9626 Tf 9.185 0 Td[(theta1(3)+4theta1(2))]TJ /F11 9.9626 Tf 7.793 0 Td[(3theta1(1))/(2h); % dtheta accurate to h ^2 94 dtheta0= gamma (4/3)()]TJ /F11 9.9626 Tf 9.185 0 Td[(theta0(3)+4theta0(2))]TJ /F11 9.9626 Tf 7.793 0 Td[(3theta0(1))/(2h); 95 96 Zd=dtheta1/(dtheta0^2); % turning theta into Impedance 97 98 % str = sprintf (' K = %.10 e : Zd = %.10 e + j %.10 e ', K ( kk ) real ( Zd ) imag ( Zd ) ) ; disp ( str ) ; 99 % str = sprintf (' dtheta1 = % g ', dtheta1 ) ; disp ( str ) ; 100 str= sprintf ('%.10e',K(kk)); disp (str); 101 str= sprintf ('%.10e', real (Zd)); disp (str); 102 str= sprintf ('%.10e', imag (Zd)); disp (str); 103 104 figure (1) 105 plot (x,conc(1,:),')]TJ /F11 9.9626 Tf 8.196 0 Td[(r') 106 hold on; 156

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107 plot (x,conc(2,:),')]TJ /F11 9.9626 Tf 7.656 0 Td[(g') 108 axis ([02)]TJ /F11 9.9626 Tf 8.235 0 Td[(.41]); 109 legend ('theta r','theta i'); 110 title ('DimensionlessConcentrationawayfromElectrodeSurface'); 111 xlabel ('length'); 112 ylabel ('theta'); 113 114 for kkk=1: length (x) 115 conc real(kkk,kk)=(conc(1,kkk)); 116 conc imag(kkk,kk)=(conc(2,kkk)); 117 end 118 119 ZZd(kk)=Zd; 120 Ktemp=K(kk); 121 fname= sprintf ('theta1 %i',kk); 122 save (fname,'theta1','Ktemp','h'); 123 end 124 125 fname= sprintf ('first order %i.txt',NJ); 126 fid= fopen (fname,'wt'); 127 fprintf (fid,'%14.6e;%14.6enn',h,h^2); 128 for kk=1: length (K); 129 fprintf (fid,'%12.6e;%20.12e;%20.12e;nn',K(kk), real (ZZd(kk)), imag (ZZd(kk)) ); 130 end ; 131 fclose (fid); 132 133 figure (2) 134 plot ( real (ZZd),)]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 96.218 -320.548 Td[(imag (ZZd),')]TJ /F11 9.9626 Tf 8.081 0 Td[(ks'); hold on; axis equal; 135 title ('Nyquistplot'); 136 xlabel ('RealpartofImpedance'); 137 ylabel ('ImaginarypartofImpedance'); 138 139 figure (3) 140 semilogx (K,)]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 78.285 -386.675 Td[(imag (ZZd),')142()]TJ /F11 9.9626 Tf 13.732 0 Td[(k'); hold on; 141 semilogx (K, real (ZZd),')142()]TJ /F11 9.9626 Tf 13.594 0 Td[(b'); 142 legend (')]TJ /F11 9.9626 Tf 8.108 0 Td[(Imaginary','Real'); 143 title ('ImpedancevsFrequency'); 144 xlabel ('DimensionlessFrequency'); 145 ylabel ('Impedance'); 146 147 realZZd= real (ZZd)'; 148 imagZZd= imag (ZZd)'; 157

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CodeC.3.FiniteSchmidtConvectionDiusionTerm2foranimpingingjetelectrode 1 % This problem solves for the third term of the convective )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 348.369 -35.608 Td[(diffusion equation with a finite schmidt number 2 3 clc ; close all ; clear all ; 4 format longE; 5 global ABCDGXYNNJ 6 7 h=0.01; % Step )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 210.889 -112.756 Td[(size 8 logK=)]TJ /F11 9.9626 Tf 8.856 0 Td[(2:.05:4; % Frequency range 0.01 to 10000 9 K=10.^logK; 10 x=0:h:10; % Range of x 11 NJ= length (x); 12 ZZd= zeros (1, length (K)); % Place holder for values of diffusion impedance 13 aa=0.51023; 14 bb=)]TJ /F11 9.9626 Tf 8.487 0 Td[(0.61592; 15 cc=(3/(1.352)^4)^(1/3); 16 17 c 0=1; % Boundary condition for x =0 18 c e=0; % Boundary condition for x = end ( NJ ) 19 tol=0; % Absolute tolerance 20 21 conc real= zeros ( length (x), length (K)); 22 conc imag= zeros ( length (x), length (K)); 23 24 25 % Define initial guess 26 conc(1,1:NJ)=0.5; % Conc (1,)]TJ /F11 9.9626 Tf 8.579 0 Td[() is the real component of theta0 27 conc(2,1:NJ)=0.5; % Conc (2,)]TJ /F11 9.9626 Tf 8.579 0 Td[() is the imaginary component of theta0 28 N=2; 29 30 for kk=1: length (K) 31 fname= sprintf ('theta0 %i',kk); % load solution for Theta )]TJ /F11 9.9626 Tf 7.49 0 Td[(0 32 load (fname); 33 flag =0; if ( abs (Ktemp)]TJ /F11 9.9626 Tf 5.977 0 Td[(K(kk))~=0); flag =3; quit ; end ; 34 fname= sprintf ('theta1 %i',kk); % load solution for Theta )]TJ /F11 9.9626 Tf 7.49 0 Td[(1 35 load (fname); 36 flag =0; if ( abs (Ktemp)]TJ /F11 9.9626 Tf 5.977 0 Td[(K(kk))~=0); flag =3; quit ; end ; 37 38 error =1; 39 jcount=0; 40 jcountmax=4; % Number of iterations the program will allow to converge 41 while error >=tol; 42 X= zeros (N,N); 43 Y=X; 44 45 jcount=jcount+1; 46 for J=1:NJ; 47 A= zeros (N,N); 48 B=A; 49 D=A; 50 % Boundary condition at J =1 51 if J==1; 52 G(1)=conc(1,J); % Real equation so BC at electrode surface is 0 53 B(1,1)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1; 54 158

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55 G(2)=conc(2,J); % Imaginary equation so BC at electrode surface is 0 56 B(2,2)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1; 57 58 error = abs (G(1))+ abs (G(2)); 59 60 % Region between boundaries 61 elseif (J<=(NJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)); 62 G(1)=conc(1,J+1))]TJ /F11 9.9626 Tf 7.793 0 Td[(2conc(1,J)+conc(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+h^2K(kk)conc(2,J)+1.5h^3( J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2(conc(1,J+1))]TJ /F11 9.9626 Tf 8.066 0 Td[(conc(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))... 63 )]TJ /F11 9.9626 Tf 8.192 0 Td[(cc(h^4)((J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^3)( real (theta1(J+1)))]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 306.426 -122.167 Td[(real (theta1(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)))/2; 64 A(1,1)=)]TJ /F11 9.9626 Tf 7.937 0 Td[(1+1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 65 B(1,1)=2; 66 D(1,1)=)]TJ /F11 9.9626 Tf 7.92 0 Td[(1)]TJ /F11 9.9626 Tf 7.92 0 Td[(1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 67 B(1,2)=)]TJ /F11 9.9626 Tf 7.085 0 Td[(h^2K(kk); 68 69 G(2)=conc(2,J+1))]TJ /F11 9.9626 Tf 7.793 0 Td[(2conc(2,J)+conc(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))]TJ /F11 9.9626 Tf 7.38 0 Td[(h^2K(kk)conc(1,J)+1.5h^3( J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2(conc(2,J+1))]TJ /F11 9.9626 Tf 8.066 0 Td[(conc(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))... 70 )]TJ /F11 9.9626 Tf 8.192 0 Td[(cc(h^4)((J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^3)( imag (theta1(J+1)))]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 305.436 -210.336 Td[(imag (theta1(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)))/2; 71 A(2,2)=)]TJ /F11 9.9626 Tf 7.937 0 Td[(1+1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 72 B(2,2)=2; 73 D(2,2)=)]TJ /F11 9.9626 Tf 7.92 0 Td[(1)]TJ /F11 9.9626 Tf 7.92 0 Td[(1.5h^3(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)^2; 74 B(2,1)=h^2K(kk); 75 76 error = error + abs (G(1))+ abs (G(2)); 77 78 % Boundary condition at J = NJ 79 else 80 G(1)=conc(1,J))]TJ /F11 9.9626 Tf 8.236 0 Td[(0; % BC at far away point is always 0 81 B(1,1)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1; 82 83 G(2)=conc(2,J))]TJ /F11 9.9626 Tf 8.236 0 Td[(0; % BC at far away point is always 0 84 B(2,2)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1; 85 86 error = error + abs (G(1))+ abs (G(2)); 87 end ; 88 89 band(J) 90 end ; 91 conc=conc+C; % C comes from band ( J ) program 92 93 if jcount>=jcountmax, break end ; 94 end ; 95 96 theta2=complex(conc(1,:),conc(2,:)); % Creating theta from conc (1,)]TJ /F11 9.9626 Tf 8.579 0 Td[() and conc (2,)]TJ /F11 9.9626 Tf 8.579 0 Td[() values 97 dtheta0= gamma (4/3)()]TJ /F11 9.9626 Tf 9.185 0 Td[(theta0(3)+4theta0(2))]TJ /F11 9.9626 Tf 7.793 0 Td[(3theta0(1))/(2h); 98 dtheta1= gamma (4/3)()]TJ /F11 9.9626 Tf 9.185 0 Td[(theta1(3)+4theta1(2))]TJ /F11 9.9626 Tf 7.793 0 Td[(3theta1(1))/(2h); % dtheta accurate to h ^2 99 dtheta2= gamma (4/3)()]TJ /F11 9.9626 Tf 9.185 0 Td[(theta2(3)+4theta2(2))]TJ /F11 9.9626 Tf 7.793 0 Td[(3theta2(1))/(2h); 100 101 Zd=)]TJ /F11 9.9626 Tf 7.828 0 Td[((1/dtheta0)((dtheta1/dtheta0)^2)]TJ /F11 9.9626 Tf 8.872 0 Td[(dtheta2/dtheta0); % turning theta into Impedance 102 ZZd(kk)=Zd; 103 104 str= sprintf ('%.10e',K(kk)); disp (str); 105 str= sprintf ('%.10e', real (Zd)); disp (str); 106 str= sprintf ('%.10e', imag (Zd)); disp (str); 159

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107 108 figure (1) 109 plot (x,conc(1,:),')]TJ /F11 9.9626 Tf 8.196 0 Td[(r') 110 hold on; 111 plot (x,conc(2,:),')]TJ /F11 9.9626 Tf 7.656 0 Td[(g') 112 axis ([02)]TJ /F11 9.9626 Tf 8.235 0 Td[(.41]); 113 legend ('theta r','theta i'); 114 title ('DimensionlessConcentrationawayfromElectrodeSurface'); 115 xlabel ('length'); 116 ylabel ('theta'); 117 118 for kkk=1: length (x) 119 conc real(kkk,kk)=(conc(1,kkk)); 120 conc imag(kkk,kk)=(conc(2,kkk)); 121 end 122 123 end 124 125 fname= sprintf ('first order %i.txt',NJ); 126 fid= fopen (fname,'wt'); 127 fprintf (fid,'%14.6e;%14.6enn',h,h^2); 128 for kk=1: length (K); 129 fprintf (fid,'%12.6e;%20.12e;%20.12e;nn',K(kk), real (ZZd(kk)), imag (ZZd(kk)) ); 130 end ; 131 fclose (fid); 132 133 figure (2) 134 plot ( real (ZZd),)]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 96.218 -320.548 Td[(imag (ZZd),')]TJ /F11 9.9626 Tf 8.081 0 Td[(ks'); hold on; axis equal; 135 title ('Nyquistplot'); 136 xlabel ('RealpartofImpedance'); 137 ylabel ('ImaginarypartofImpedance'); 138 139 figure (3) 140 semilogx (K,)]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 78.285 -386.675 Td[(imag (ZZd),')142()]TJ /F11 9.9626 Tf 13.732 0 Td[(k'); hold on; 141 semilogx (K, real (ZZd),')142()]TJ /F11 9.9626 Tf 13.594 0 Td[(b'); 142 legend (')]TJ /F11 9.9626 Tf 8.108 0 Td[(Imaginary','Real'); 143 title ('ImpedancevsFrequency'); 144 xlabel ('DimensionlessFrequency'); 145 ylabel ('Impedance'); 146 147 realZZd= real (ZZd)'; 148 imagZZd= imag (ZZd)'; 160

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APPENDIXDCODESFORCONVECTIVEDIFFUSIONIMPEDANCEWITHHOMOGENEOUSREACTIONThisappendixcontainsthedierentFORTRANcodesthatproducedtheresultsforthesolutionoftheconvective-diusionequationwithhomogeneousreactioninChapter 4 .ThisappendixalsohastheMATLABcodestoplotthesteady-stateresults,toplotapolarizationcurvefromthesteady-stateresultsatdierentpotentialsandacodetocreatetheimpedancefromtheresultsoftheoscillatingFORTRANcode. D.1InputforConvectiveDiusionImpedancewithHomogeneousReactionsCodeThefollowingcodesaretheinputlesfortheconvectivediusionequationwithahomogeneousreaction.Theinputlesarebrokenintotwoles.Therstinputlehasalltheinputparametersexcepttheinputpotential.Theinputcodehasthenumberofspeciesbeingsolved,thetotalnumberofpoints,thenumberofpointsuntilthecoupler,thedistanceofthereactionregionincm,thedistanceoftheinnerlayerincm.Theinputlealsoincludestherotationspeedoftherotatingdisk,inrpm,andthekinematicviscosityofthesolution,incm2=s.Therateconstantsintheinputlearetheequilibriumrateofthehomogeneousreaction,inmol=cm3,thebackwardrateofthehomogeneousreaction,incm3=mols,therateconstantfortheheterogeneousreactionofthereactingspecies,inAcm=mol,andthetafelkineticsvaluefortheheterogeneousreaction,in1=V.TheinputleincludestheerrorallowedfortheBIGvalues,whichisdiscussedinsection 2.2.2 .Andtheendoftheinputlehasthespecicvaluestodescribeeachspeciesinthesystem,includingdiusioncoecientsincm2=s,thechargeofthespecies,acharacternametoidentifythespecies,andtheconcentrationvalueinthebulk,inmol=cm3.Thesecondinputlecontainthepotential,involts,usedintheheterogeneousreaction.Thepotentialisinitsowninputletoallowapolarizationcurvetobecalculatedeasier. 161

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1 4 2 12001 3 8001 4 0.004 5 0.024 6 2000. 7 0.01 8 1.E)]TJ /F11 9.9626 Tf 7.49 0 Td[(6 9 1.E7 10 2.E)]TJ /F11 9.9626 Tf 7.804 0 Td[(12 11 19.9 12 1.E)]TJ /F11 9.9626 Tf 7.804 0 Td[(12 13 1.684E)]TJ /F11 9.9626 Tf 7.49 0 Td[(50.AB0.01 14 1.957E)]TJ /F11 9.9626 Tf 7.49 0 Td[(5)]TJ /F11 9.9626 Tf 8.236 0 Td[(1.A)]TJ /F11 9.9626 Tf 14.012 0 Td[(0.0001 15 1.902E)]TJ /F11 9.9626 Tf 7.49 0 Td[(5+1.B+0.0001 16 17 18 C line 1 is the number of species 19 C line 2 is NJ 20 C line 3 is KJ number of points in the reaction 21 C line 4 is the distance of the inner reaction layer in cm (5 um ) 22 C line 5 is the distance of the rest of the inner layer in cm (10 um ) 23 C line 6 is the rotation speed in rpm 24 C line 7 is the kinematic viscosity cm ^2/ s 25 C line 8 is the equilibrium rate of rxn mol / cm ^3 26 C line 9 is the backward rate of reaction cm ^3/( mol s ) 27 C line 10 is the rate constant for the flux of the reacting species 28 C line 11 is the tafel b value for the flux of the reacting species 29 C line 12 is the error allowed for the BIGs 30 C lines 13)]TJ /F11 9.9626 Tf 8.065 0 Td[(15 specify values used to describe each species in the system CodeD.1.InputlefortheConvectiveDiusionwithHomogeneousReaction 1 )]TJ /F11 9.9626 Tf 8.321 0 Td[(0.5 CodeD.2.PotentialinputlefortheConvectiveDiusionwithHomogeneousReaction 162

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D.2Steady-StateConvectiveDiusionImpedancewithHomogeneousReactionsCodeThissectioncontainsthesteady-stateFORTRANcodesusedtosolvetheconvectivediusionequationwithahomogeneousreaction.ThemathematicalworkupforequationsusedinthesecodesareinChapter 4 .TheFORTRANcodesarefollowedbytwoMATLABcodes.TherstMATLABcodetakestheoutputfromthesteady-stateFORTRANcodeandplotsthedatatomoreeasilyvisualizeresults.ThesecondMATLABcodecreatesapolarizationcurvebyrunningtheexecutablecreatedfromtheFORTRANcodefordierentinputpotentialsandplottingthecurrentasafunctionofpotential.Therstsectioninthecode,calledCONVDIFF,isthemainprogram,whichoutlinestheglobalvariablesandsetsupcallinglestosaveoverasoutputlesaswellascallingtheinputles.Thenthesubroutinesthatarecalledinthemainprogramaresubsequentlyshown.ThesubroutineVELOCITYcreatesthevelocityproleforthecode.Itwasnecessarytocreatethevelocityproleusingthetwodierentmeshsizes,schematicallyshownin 4-3 .TheVELOCITYsubroutinealsondstheexactvalueofthevelocityatthehalfmeshpointsoneithersideofthecoupler,atJ=KJ.ThesubroutineBC1solvestheboundaryconditionattheelectrodesurface.ThesubroutinesREACTIONandINNERsolvethenonlinearcoupleddierentialequationsusingmeshsizesHHandH,respectively.ThesubroutineCOUPLER,setstheuxatKJequalusingtwouxexpressionequations.Themathematicsbehindthecouplerarediscussedinsection 4.1.5.2 .FinallythesubroutineBCNJsolvestheboundaryconditioninthebulk.BANDandMATINV,in A ,arecalledinordertosolvethesteady-statesolution. 163

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CodeD.3.SteadyStateConvectiveDiusionwithHomogeneousReactionMainProgram 1 C Convective Diffusion Equation with Homogeneous Reaction 2 C 3 species system 3 C SPECIES 1 = AB SPECIES 2 = A )]TJ /F11 9.9626 Tf 8.469 0 Td[(, SPECIES 3 = B + 4 C Species 3 is the reacting species 5 C This is the steady state solution only 6 C It should be ran prior to cdh os for 7 C The input file is the same for both 8 9 PROGRAM CONVDIFF 10 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 11 COMMON /BAB/A(4,4),B(4,4),C(4,100001),D(4,9),G(4),X(4,4),Y(4,4) 12 COMMON /NSN/N,NJ 13 COMMON /VAR/CONC(3,100001),RXN(100001),DIFF(3),H,EBIG,IJ,KJ,HH 14 COMMON /BCI/VEL1,FLUX 15 COMMON /VET/VEL(100001),Y1,VELH12,VELHH12 16 COMMON /VETT/VELNEARH(100001),VELFARH(100001),FVELH(100001) 17 COMMON /VEKK/VELNEARHH(100001),VELFARHH(100001),FVELHH(100001) 18 COMMON /EXTRA/Z(6),REF(6) 19 COMMON /BUL/CBULK(6),JCOUNT 20 COMMON /RTE/rateb,equilib 21 22 CHARACTER REF6 23 24 102 FORMAT (/30HTHENEXTRUNDID NOT CONVERGE) 25 103 FORMAT ('Error=',E16.6/(1X,'Species=',A6,2X,'CatElectrode=', 26 1E12.5,2X,'CatBulk=',E12.5)) 27 104 FORMAT (A6) 28 105 FORMAT (E12.7) 29 333 FORMAT (4x,'AB'12x,'A)]TJ /F11 9.9626 Tf 9.003 0 Td[(',12x,'B+',12x,'RXN') 30 300 FORMAT (20x,'G(1)'14x,'G(2)',14x,'G(3)',14x,'RXN') 31 301 FORMAT (5x,'J='I5,8E18.9) 32 334 FORMAT (4(E25.15,5X)) 33 302 FORMAT ('Iteration='I4) 34 305 FORMAT (E20.12,3X,E20.12,3X,E20.12,3X,E20.12) 35 36 OPEN ( UNIT =13, FILE ='cdh out.txt') 37 CLOSE ( UNIT =13, STATUS ='DELETE') 38 OPEN ( UNIT =13, FILE ='cdh out.txt') 39 40 OPEN ( UNIT =16, FILE ='VEL.txt') 41 CLOSE ( UNIT =16, STATUS ='DELETE') 42 OPEN ( UNIT =16, FILE ='VEL.txt') 43 44 OPEN (12, FILE ='cdh G out.txt') 45 CLOSE (12, STATUS ='DELETE') 46 OPEN (12, FILE ='cdh G out.txt') 47 WRITE (12,300) 48 49 OPEN ( UNIT =17, FILE ='VEL12.txt') 50 CLOSE ( UNIT =17, STATUS ='DELETE') 51 OPEN ( UNIT =17, FILE ='VEL12.txt') 52 53 OPEN (14, file ='cdh in.txt', status ='old') 54 READ (14,)N,NJ,KJ,Y1,Y2,RPM,ANU,equilib,rateb,AKB,BB,EBIG 55 READ (14,)(DIFF(I),Z(I),REF(I),CBULK(I),I=1,(N)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 56 164

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57 OPEN (11, file ='potential in.txt', status ='old') 58 READ (11,)V 59 60 C Constants 61 F=96487. 62 ROT=RPM23.141592653589793/60 63 PRINT ,'ROT=',ROT 64 C H =( ANU / ROT ) (1./2.) (3. DIFF (3) /(0.51023 ANU ) ) (1./3.) 65 C 1 1.00 E )]TJ /F11 9.9626 Tf 7.49 0 Td[(3 66 67 PRINT ,'Y2=',Y2 68 H=Y2/(NJ)]TJ /F11 9.9626 Tf 6.559 0 Td[(KJ) 69 PRINT ,'H=',H 70 71 PRINT ,'Y1=',Y1 72 HH=Y1/(KJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1) 73 PRINT ,'HH=',HH 74 75 DELTA=(ANU/ROT)(1./2.)(3.DIFF(3)/(0.51023ANU))(1./3.)5 76 PRINT ,'DELTA=',DELTA 77 78 OPEN (15, FILE ='cdh ssvalues out.txt') 79 CLOSE (15, STATUS ='DELETE') 80 OPEN (15, FILE ='cdh ssvalues out.txt') 81 337 FORMAT (I2/I7/I7/E15.8/E15.8/E15.8/E15.8/E15.8/E15.4/E15.4/E15.4) 82 write (15,337)N,NJ,KJ,H,HH,RPM,ANU,DIFF(3),AKB,BB,V 83 84 c CALL THE SUBROUTINE TO CALCULATE THE VELOCITY 85 CALL VELOCITY(ROT,ANU) 86 OPEN (17, FILE ='VEL12.txt') 87 CLOSE (17, STATUS ='DELETE') 88 OPEN (17, FILE ='VEL12.txt') 89 338 FORMAT (E20.13/E20.13) 90 write (17,338)VELH12,VELHH12 91 92 C Create flux of the reacting species constants 93 FLUX=)]TJ /F11 9.9626 Tf 5.777 0 Td[(AKB exp ()]TJ /F11 9.9626 Tf 7.112 0 Td[(BBV)/F/Z(3) 94 PRINT ,'FLUX=',FLUX 95 96 DO 21J=1,NJ 97 DO 21I=1,N)]TJ /F11 9.9626 Tf 7.49 0 Td[(1 98 C(I,J)=0.0 99 21CONC(I,J)=CBULK(I) 100 JCOUNT=0 101 TOL=1.E)]TJ /F11 9.9626 Tf 7.992 0 Td[(15NNJ/1000 102 PRINT ,'TOL=',TOL 103 22JCOUNT=JCOUNT+1 104 AMP=0.0 105 J=0 106 DO 23I=1,N 107 DO 23K=1,N 108 Y(I,K)=0.0 109 23X(I,K)=0.0 110 24J=J+1 111 DO 25I=1,N 112 G(I)=0.0 113 DO 25K=1,N 114 A(I,K)=0.0 165

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115 B(I,K)=0.0 116 25D(I,K)=0.0 117 118 IF (J. EQ .1) CALL BC1(J) 119 IF (J. GT .1. AND .J. LT .KJ) CALL REACTION(J) 120 IF (J. EQ .KJ) CALL COUPLER(J) 121 IF (J. GT .KJ. AND .J. LT .NJ) CALL INNER(J) 122 IF (J. EQ .NJ) CALL BCNJ(J) 123 CALL BAND(J) 124 125 AMP=AMP+DABS(G(1))+DABS(G(2))+DABS(G(3))+DABS(G(4)) 126 127 IF (J. LT .NJ) GO TO 24 128 129 PRINT ,'ERROR=',AMP 130 131 DO 16K=1,NJ 132 RXN(K)=RXN(K)+C(4,K) 133 DO 16I=1,N)]TJ /F11 9.9626 Tf 7.49 0 Td[(1 134 IF (C(I,K). LT .)]TJ /F11 9.9626 Tf 8.712 0 Td[(0.999CONC(I,K))C(I,K)=)]TJ /F11 9.9626 Tf 8.214 0 Td[(0.999CONC(I,K) 135 IF (C(I,K). GT .999.CONC(I,K))C(I,K)=999.CONC(I,K) 136 CONC(I,K)=CONC(I,K)+C(I,K) 137 16 CONTINUE 138 139 WRITE (12,302)(JCOUNT) 140 141 c If the error is less then the tolerance finish program 142 IF (DABS(AMP). LT .DABS(TOL)) GO TO 15 143 144 c If the error is greater then tolerance do another iteration 145 33 IF (JCOUNT. LE .19) GO TO 22 146 print 102 147 148 15 PRINT 103,AMP,(REF(I),CONC(I,1),CONC(I,NJ),I=1,N)]TJ /F11 9.9626 Tf 8.08 0 Td[(1) 149 150 PRINT ,'JCOUNT=',JCOUNT 151 152 DO 19J=1,NJ 153 BIG=RXN(J) 154 BIG2=1.0E)]TJ /F11 9.9626 Tf 7.804 0 Td[(40 155 19 IF ( ABS (BIG). LE .BIG2)RXN(J)=0.0 156 157 DO 18J=1,NJ 158 DO 18I=1,N)]TJ /F11 9.9626 Tf 7.49 -.001 Td[(1 159 BIG=CONC(I,J) 160 BIG2=1.0E)]TJ /F11 9.9626 Tf 7.804 0 Td[(40 161 18 IF ( ABS (BIG). LE .BIG2)CONC(I,J)=0.0 162 163 WRITE (13,334)(CONC(1,J),CONC(2,J),CONC(3,J),RXN(J),J=1,NJ) 164 165 END PROGRAM CONVDIFF 166

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CodeD.4.Steady-StateConvectiveDiusionwithHomogeneousReactionSubroutinetoCreatetheVelocityProle 1 SUBROUTINE VELOCITY(ROT,ANU) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 COMMON /NSN/N,NJ 4 COMMON /VAR/CONC(3,100001),RXN(100001),DIFF(3),H,EBIG,IJ,KJ,HH 5 COMMON /VET/VEL(100001),Y1,VELH12,VELHH12 6 COMMON /VETT/VELNEARH(100001),VELFARH(100001),FVELH(100001) 7 COMMON /VEKK/VELNEARHH(100001),VELFARHH(100001),FVELHH(100001) 8 COMMON /BCI/VEL1,FLUX 9 10 305 FORMAT (E20.12,3X,E20.12,3X,E20.12,3X,E20.12) 11 12 C Create a term for ( rot / nu ) ^(1/2) 13 ROTNU=(ROT/ANU)(1./2.) 14 PRINT ,'ROTNU=',ROTNU 15 c Create term for 2 A / c in term 2 of expansion 16 FAR2=20.92486353/0.88447411 17 c Create term for ( A ^2+ B ^2) /2 c ^2, in term 3 of expansion 18 FAR3=(0.924863532.+1.202211752.)/(2.0.884474113.) 19 c Create term for A ( A ^2+ B ^2) /6 c ^5, in term 4 of expansion 20 FAR4=0.92486353(0.924863532.+1.202211752.)/ 21 1(6.0.884474115.) 22 C Create a term for ( rot nu ) ^(1/2) 23 ROTNU2=(ROTANU)(1./2.) 24 25 26 C CREATE VELOCITY PROFILE 27 DO 221I=1,KJ 28 VELNEARHH(I)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(ROT(3./2.)0.510232618867(HH(I)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))2/ANU(.5) 29 1+((1./3.)(HH(I)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))3.ROT2/ANU) 30 2+(()]TJ /F11 9.9626 Tf 8.754 0 Td[(0.615922014399/6.)(HH(I)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))4.ROT(2.5)/(ANU(1.5))) 31 VELFARHH(I)=)]TJ /F11 9.9626 Tf 5.958 0 Td[(ROTNU20.88447411 32 1+ROTNU2FAR2 EXP ()]TJ /F11 9.9626 Tf 8.724 0 Td[(0.88447411(HH(I)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))ROTNU) 33 2)]TJ /F11 9.9626 Tf 6.253 0 Td[(ROTNU2FAR3 EXP ()]TJ /F11 9.9626 Tf 8.727 0 Td[(20.88447411(HH(I)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))ROTNU) 34 3+ROTNU2FAR4 EXP ()]TJ /F11 9.9626 Tf 8.727 0 Td[(30.88447411(HH(I)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))ROTNU) 35 FVELHH(I)=1/(1+( EXP ()]TJ /F11 9.9626 Tf 8.223 0 Td[(20ROTNU(((HH(I)]TJ /F11 9.9626 Tf 8.08 0 Td[(1)))]TJ /F11 9.9626 Tf 8.614 0 Td[(1.25(1/ROTNU)))))) 36 VEL(I)=((1)]TJ /F11 9.9626 Tf 7.12 0 Td[(FVELHH(I))VELNEARHH(I))+(FVELHH(I)VELFARHH(I)) 37 221 CONTINUE 38 39 VEL(1)=0. 40 VEL1=(0.5)VEL(2) 41 C PRINT VEL1 =', VEL1 42 WRITE (16,305)(VELNEARHH(I),VELFARHH(I),FVELHH(I),VEL(I),I=1,KJ) 43 44 C CREATE VELOCITY PROFILE 45 DO 222I=1,(NJ)]TJ /F11 9.9626 Tf 6.559 0 Td[(KJ)+1 46 VELNEARH(I)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(ROT(3./2.)0.510232618867(Y1+(H(I)))2/ANU(.5) 47 1+((1./3.)(Y1+(H(I)))3.ROT2/ANU) 48 2+(()]TJ /F11 9.9626 Tf 8.754 .001 Td[(0.615922014399/6.)(Y1+(H(I)))4.ROT(2.5)/(ANU(1.5))) 49 VELFARH(I)=)]TJ /F11 9.9626 Tf 5.958 0 Td[(ROTNU20.88447411 50 1+ROTNU2FAR2 EXP ()]TJ /F11 9.9626 Tf 8.724 0 Td[(0.88447411(Y1+(H(I)))ROTNU) 51 2)]TJ /F11 9.9626 Tf 6.253 0 Td[(ROTNU2FAR3 EXP ()]TJ /F11 9.9626 Tf 8.727 0 Td[(20.88447411(Y1+(H(I)))ROTNU) 52 3+ROTNU2FAR4 EXP ()]TJ /F11 9.9626 Tf 8.727 0 Td[(30.88447411(Y1+(H(I)))ROTNU) 53 FVELH(I)=1/(1+( EXP ()]TJ /F11 9.9626 Tf 8.223 0 Td[(20ROTNU(((Y1+(H(I))))]TJ /F11 9.9626 Tf 8.614 0 Td[(1.25(1/ROTNU)))))) 54 VEL(KJ+I)=((1)]TJ /F11 9.9626 Tf 7.225 0 Td[(FVELH(I))VELNEARH(I))+(FVELH(I)VELFARH(I)) 167

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55 222 CONTINUE 56 57 WRITE (16,305)(VELNEARH(I)]TJ /F11 9.9626 Tf 6.559 0 Td[(KJ),VELFARH(I)]TJ /F11 9.9626 Tf 6.559 0 Td[(KJ),FVELH(I)]TJ /F11 9.9626 Tf 6.559 0 Td[(KJ), 58 1VEL(I),I=KJ+1,NJ) 59 60 VELNEARHH12=)]TJ /F11 9.9626 Tf 5.777 0 Td[(ROT(3./2.)0.510232618867(HH(FLOAT(KJ))]TJ /F11 9.9626 Tf 8.539 0 Td[(1.5))2 61 1/ANU(.5) 62 1+((1./3.)(HH(FLOAT(KJ))]TJ /F11 9.9626 Tf 8.54 0 Td[(1.5))3.ROT2/ANU) 63 2+(()]TJ /F11 9.9626 Tf 8.754 0 Td[(0.615922014399/6.)(HH(FLOAT(KJ))]TJ /F11 9.9626 Tf 8.539 0 Td[(1.5))4.ROT(2.5)/ 64 3(ANU(1.5))) 65 VELFARHH12=)]TJ /F11 9.9626 Tf 5.958 0 Td[(ROTNU20.88447411 66 1+ROTNU2FAR2 EXP ()]TJ /F11 9.9626 Tf 8.724 0 Td[(0.88447411(HH(FLOAT(KJ))]TJ /F11 9.9626 Tf 8.539 0 Td[(1.5))ROTNU) 67 2)]TJ /F11 9.9626 Tf 6.253 0 Td[(ROTNU2FAR3 EXP ()]TJ /F11 9.9626 Tf 8.727 0 Td[(20.88447411(HH(FLOAT(KJ))]TJ /F11 9.9626 Tf 8.54 0 Td[(1.5))ROTNU) 68 3+ROTNU2FAR4 EXP ()]TJ /F11 9.9626 Tf 8.727 0 Td[(30.88447411(HH(FLOAT(KJ))]TJ /F11 9.9626 Tf 8.54 0 Td[(1.5))ROTNU) 69 FVELHH12=1./(1.+( EXP ()]TJ /F11 9.9626 Tf 8.566 0 Td[(20.ROTNU(((HH(FLOAT(KJ))]TJ /F11 9.9626 Tf 8.54 0 Td[(1.5)))]TJ ET 0.95 0.95 0.92 RG 0.95 0.95 0.92 rg 0 G 0 g 0.95 0.95 0.92 RG 0.95 0.95 0.92 rg q 0 -180.579 468 11.021 re f Q 0 G 0 g 0.95 0.95 0.92 RG 0.95 0.95 0.92 rg 0 G 0 g 0.5 0.5 0.5 RG 0.5 0.5 0.5 rg BT /F11 9.9626 Tf -14.944 -177.273 Td[(70 11.25(1/ROTNU)))))) 71 VELHH12=((1.)]TJ /F11 9.9626 Tf 7.905 0 Td[(FVELHH12)VELNEARHH12)+(FVELHH12VELFARHH12) 72 VELHH122=(VEL(KJ)+VEL(KJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/2. 73 PRINT ,'VELHH122=',VELHH122 74 PRINT ,'HH(FLOAT(KJ))]TJ /F11 9.9626 Tf 8.32 0 Td[(1.5',HH(FLOAT(KJ))]TJ /F11 9.9626 Tf 8.539 0 Td[(1.5) 75 76 VELNEARH12=)]TJ /F11 9.9626 Tf 5.777 0 Td[(ROT(3./2.).510232618867(Y1+(H(0.5)))2/ANU(.5) 77 1+((1./3.)(Y1+(H(0.5)))3.ROT2/ANU) 78 2+(()]TJ /F11 9.9626 Tf 8.755 0 Td[(0.615922014399/6.)(Y1+(H(0.5)))4.ROT(2.5)/ 79 3(ANU(1.5))) 80 VELFARH12=)]TJ /F11 9.9626 Tf 5.958 0 Td[(ROTNU20.88447411 81 1+ROTNU2FAR2 EXP ()]TJ /F11 9.9626 Tf 8.724 0 Td[(0.88447411(Y1+(H(0.5)))ROTNU) 82 2)]TJ /F11 9.9626 Tf 6.253 0 Td[(ROTNU2FAR3 EXP ()]TJ /F11 9.9626 Tf 8.727 0 Td[(20.88447411(Y1+(H(0.5)))ROTNU) 83 3+ROTNU2FAR4 EXP ()]TJ /F11 9.9626 Tf 8.727 0 Td[(30.88447411(Y1+(H(0.5)))ROTNU) 84 FVELH12=1/(1+( EXP ()]TJ /F11 9.9626 Tf 8.223 0 Td[(20ROTNU(((Y1+(H(0.5))))]TJ /F11 9.9626 Tf 8.614 0 Td[(1.25(1/ROTNU)))))) 85 VELH12=((1)]TJ /F11 9.9626 Tf 7.664 0 Td[(FVELH12)VELNEARH12)+(FVELH12VELFARH12) 86 VELH122=(VEL(KJ)+VEL(KJ+1))/2. 87 PRINT ,'VELH122=',VELH122 88 PRINT ,'Y1+(H(0.5)',Y1+(H(0.5)) 89 90 PRINT ,'VELKJ=',VEL(KJ) 91 92 RETURN 93 END 168

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CodeD.5.Steady-StateConvectiveDiusionwithHomogeneousReactionSubroutinefortheElectrodeBoundaryCondition 1 SUBROUTINE BC1(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 4 COMMON /BAB/A(4,4),B(4,4),C(4,100001),D(4,9),G(4),X(4,4),Y(4,4) 5 COMMON /NSN/N,NJ 6 COMMON /VAR/CONC(3,100001),RXN(100001),DIFF(3),H,EBIG,IJ,KJ,HH 7 COMMON /BCI/VEL1,FLUX 8 COMMON /RTE/rateb,equilib 9 10 301 FORMAT (5x,'J='I5,8E18.9) 11 12 C For AB non reacting species 13 G(1)=2.DIFF(1)(CONC(1,J+1))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(1,J))/HH2. 14 1)]TJ /F11 9.9626 Tf 6.849 0 Td[(VEL1/HH(CONC(1,J+1))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(1,J)) 15 2)]TJ /F11 9.9626 Tf 8.54 0 Td[((3.RXN(J)+RXN(J+1))/4. 16 B(1,1)=2.DIFF(1)/HH2.)]TJ /F11 9.9626 Tf 8.071 0 Td[(VEL1/HH 17 D(1,1)=)]TJ /F11 9.9626 Tf 7.986 0 Td[(2.DIFF(1)/HH2.+VEL1/HH 18 B(1,4)=+0.75 19 D(1,4)=+0.25 20 21 BIG= ABS (2.DIFF(1)(CONC(1,J+1))/HH2.) 22 BIG2= ABS (2.DIFF(1)CONC(1,J)/HH2.) 23 IF (BIG2. GT .BIG)BIG=BIG2 24 BIG3= ABS (VEL1/HHCONC(1,J+1)) 25 IF (BIG3. GT .BIG)BIG=BIG3 26 BIG4= ABS (VEL1/HHCONC(1,J)) 27 IF (BIG4. GT .BIG)BIG=BIG4 28 IF ( ABS ()]TJ /F11 9.9626 Tf 8.539 0 Td[(3.RXN(J)/4.). GT .BIG)BIG= ABS ()]TJ /F11 9.9626 Tf 8.539 0 Td[(3.RXN(J)/4.) 29 IF ( ABS ()]TJ /F11 9.9626 Tf 6.746 0 Td[(RXN(J+1)/4.). GT .BIG)BIG= ABS ()]TJ /F11 9.9626 Tf 6.746 0 Td[(RXN(J+1)/4.) 30 IF ( ABS (G(1)). LT .BIGEBIG)G(1)=0 31 32 C For A + non reacting species 33 G(2)=2.DIFF(2)(CONC(2,J+1))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(2,J))/HH2. 34 1)]TJ /F11 9.9626 Tf 6.849 0 Td[(VEL1/HH(CONC(2,J+1))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(2,J)) 35 2+(3.RXN(J)+RXN(J+1))/4. 36 B(2,2)=2.DIFF(2)/HH2.)]TJ /F11 9.9626 Tf 8.071 0 Td[(VEL1/HH 37 D(2,2)=)]TJ /F11 9.9626 Tf 7.986 0 Td[(2.DIFF(2)/HH2.+VEL1/HH 38 B(2,4)=)]TJ /F11 9.9626 Tf 8.081 0 Td[(0.75 39 D(2,4)=)]TJ /F11 9.9626 Tf 8.081 0 Td[(0.25 40 41 BIG= ABS (2.DIFF(2)(CONC(2,J+1))/HH2.) 42 BIG2= ABS (2.DIFF(2)CONC(2,J)/HH2.) 43 IF (BIG2. GT .BIG)BIG=BIG2 44 BIG3= ABS (VEL1/HHCONC(2,J+1)) 45 IF (BIG3. GT .BIG)BIG=BIG3 46 BIG4= ABS (VEL1/HHCONC(2,J)) 47 IF (BIG4. GT .BIG)BIG=BIG4 48 IF ( ABS ()]TJ /F11 9.9626 Tf 8.539 .001 Td[(3.RXN(J)/4.). GT .BIG)BIG= ABS ()]TJ /F11 9.9626 Tf 8.539 .001 Td[(3.RXN(J)/4.) 49 IF ( ABS ()]TJ /F11 9.9626 Tf 6.746 0 Td[(RXN(J+1)/4.). GT .BIG)BIG= ABS ()]TJ /F11 9.9626 Tf 6.746 0 Td[(RXN(J+1)/4.) 50 IF ( ABS (G(2)). LT .BIGEBIG)G(2)=0 51 52 C For B )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 79.145 -621.595 Td[(reacting species 53 G(3)=2.DIFF(3)(CONC(3,J+1))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(3,J))/HH2. 54 1)]TJ /F11 9.9626 Tf 6.848 0 Td[(VEL1/HH(CONC(3,J+1))]TJ /F11 9.9626 Tf 6.019 0 Td[(CONC(3,J)) 169

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55 2+FLUXCONC(3,J)/(HH/2.) 56 3+(3.RXN(J)+RXN(J+1))/4. 57 B(3,3)=2.DIFF(3)/HH2.)]TJ /F11 9.9626 Tf 8.071 0 Td[(VEL1/HH)]TJ /F11 9.9626 Tf 6.11 0 Td[(FLUX/(HH/2.) 58 D(3,3)=)]TJ /F11 9.9626 Tf 7.986 0 Td[(2.DIFF(3)/HH2.+VEL1/HH 59 B(3,4)=)]TJ /F11 9.9626 Tf 8.081 0 Td[(0.75 60 D(3,4)=)]TJ /F11 9.9626 Tf 8.081 0 Td[(0.25 61 62 BIG= ABS (2.DIFF(3)(CONC(3,J+1))/HH2.) 63 BIG2= ABS (2.DIFF(3)CONC(3,J)/HH2.) 64 IF (BIG2. GT .BIG)BIG=BIG2 65 BIG3= ABS (VEL1/HHCONC(3,J+1)) 66 IF (BIG3. GT .BIG)BIG=BIG3 67 BIG4= ABS (VEL1/HHCONC(3,J)) 68 IF (BIG4. GT .BIG)BIG=BIG4 69 BIG5= ABS ()]TJ /F11 9.9626 Tf 7.079 0 Td[(FLUXCONC(3,J)/(HH/2.)) 70 IF (BIG5. GT .BIG)BIG=BIG5 71 IF ( ABS ()]TJ /F11 9.9626 Tf 8.539 0 Td[(3.RXN(J)/4.). GT .BIG)BIG= ABS ()]TJ /F11 9.9626 Tf 8.539 0 Td[(3.RXN(J)/4.) 72 IF ( ABS ()]TJ /F11 9.9626 Tf 6.746 0 Td[(RXN(J+1)/4.). GT .BIG)BIG= ABS ()]TJ /F11 9.9626 Tf 6.746 0 Td[(RXN(J+1)/4.) 73 IF ( ABS (G(3)). LT .BIGEBIG)G(3)=0 74 75 C For Reaction term 76 IF (rateb. EQ .0) GO TO 214 77 EPS=RXN(J)/(ratebCONC(2,J)CONC(3,J)) 78 IF ( ABS (EPS). GT .0.2) GO TO 214 79 80 REM= LOG (1.+EPS) 81 IF ( ABS (REM). LT .1.0E)]TJ /F11 9.9626 Tf 8.213 0 Td[(09)REM=EPS(1.)]TJ /F11 9.9626 Tf 8.249 0 Td[(EPS(.5)]TJ /F11 9.9626 Tf 8.249 0 Td[(EPS(1./3.)]TJ /F11 9.9626 Tf 8.532 0 Td[(EPS/4.))) 82 83 G(4)= LOG (equilibCONC(1,J)/CONC(2,J)/CONC(3,J)))]TJ /F11 9.9626 Tf 5.831 0 Td[(REM 84 B(4,1)=)]TJ /F11 9.9626 Tf 7.986 0 Td[(1./CONC(1,J) 85 B(4,2)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(((RXN(J)/(ratebCONC(3,J)CONC(2,J)2.)) 86 1(1./(1.+RXN(J)/(ratebCONC(2,J)CONC(3,J)))))+1./CONC(2,J) 87 B(4,3)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(((RXN(J)/(ratebCONC(2,J)CONC(3,J)2.)) 88 1(1./(1.+RXN(J)/(ratebCONC(2,J)CONC(3,J)))))+1./CONC(3,J) 89 B(4,4)=1./(ratebCONC(3,J)CONC(2,J) 90 1(1.+RXN(J)/(ratebCONC(2,J)CONC(3,J)))) 91 92 BIG= ABS (REM) 93 BIG2= ABS ( LOG (equilibCONC(1,J)/CONC(2,J)/CONC(3,J))) 94 IF (BIG2. GT .BIG)BIG=BIG2 95 IF ( ABS (G(4)). LT .BIGEBIG)G(4)=0 96 GO TO 212 97 98 214G(4)=)]TJ /F11 9.9626 Tf 5.777 -.001 Td[(RXN(J)+rateb(equilibCONC(1,J))]TJ /F11 9.9626 Tf 6.018 -.001 Td[(CONC(2,J)CONC(3,J)) 99 B(4,1)=)]TJ /F11 9.9626 Tf 8.058 0 Td[(ratebequilib 100 B(4,2)=ratebCONC(3,J) 101 B(4,3)=ratebCONC(2,J) 102 B(4,4)=1. 103 104 BIG= ABS (ratebequilibCONC(1,J)) 105 BIG2= ABS (ratebCONC(2,J)CONC(3,J)) 106 IF (BIG2. GT .BIG)BIG=BIG2 107 BIG3= ABS (ratebequilibCONC(1,J)) 108 IF (BIG3. GT .BIG)BIG=BIG3 109 IF ( ABS (RXN(J)). GT .BIG)BIG= ABS (RXN(J)) 110 IF ( ABS (G(4)). LT .BIGEBIG)G(4)=0 111 CONTINUE 112 170

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113 212 WRITE (12,301)J,(G(K),K=1,N) 114 115 RETURN 116 END 171

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CodeD.6.SteadyStateConvectiveDiusionwithHomogeneousReactionSubroutinefortheReactionRegion 1 SUBROUTINE REACTION(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 COMMON /BAB/A(4,4),B(4,4),C(4,100001),D(4,9),G(4),X(4,4),Y(4,4) 4 COMMON /NSN/N,NJ 5 COMMON /VAR/CONC(3,100001),RXN(100001),DIFF(3),H,EBIG,IJ,KJ,HH 6 COMMON /RTE/rateb,equilib 7 COMMON /VET/VEL(100001),Y1,VELH12,VELHH12 8 301 FORMAT (5x,'J='I5,8E18.9) 9 10 C For AB 11 G(1)=DIFF(1)(CONC(1,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.CONC(1,J)+CONC(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH2. 12 1)]TJ /F11 9.9626 Tf 6.522 0 Td[(VEL(J)(CONC(1,J+1))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/(2.HH))]TJ /F11 9.9626 Tf 6.072 0 Td[(RXN(J) 13 B(1,1)=2.DIFF(1)/HH2. 14 D(1,1)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(1)/HH2.+VEL(J)/(2.HH) 15 A(1,1)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(1)/HH2.)]TJ /F11 9.9626 Tf 7.745 0 Td[(VEL(J)/(2.HH) 16 B(1,4)=+1. 17 18 BIG= ABS (DIFF(1)CONC(1,J+1)/HH2.) 19 BIG2= ABS ()]TJ /F11 9.9626 Tf 8.539 0 Td[(1.DIFF(1)2.CONC(1,J)/HH2.) 20 IF (BIG2. GT .BIG)BIG=BIG2 21 BIG3= ABS (DIFF(1)CONC(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)/HH2.) 22 IF (BIG3. GT .BIG)BIG=BIG3 23 BIG4= ABS ()]TJ /F11 9.9626 Tf 8.539 0 Td[(1.VEL(J)CONC(1,J+1)/(2.HH)) 24 IF (BIG4. GT .BIG)BIG=BIG4 25 BIG5= ABS (VEL(J)CONC(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)/(2.HH)) 26 IF (BIG5. GT .BIG)BIG=BIG5 27 IF ( ABS (RXN(J)). GT .BIG)BIG= ABS (RXN(J)) 28 IF ( ABS (G(1)). LT .BIGEBIG)G(1)=0 29 30 C For A )]TJ ET 0 G 0 g 0.95 0.95 0.92 RG 0.95 0.95 0.92 rg 0 G 0 g 0.95 0.95 0.92 RG 0.95 0.95 0.92 rg q 0 -393.457 468 11.021 re f Q 0 G 0 g 0.95 0.95 0.92 RG 0.95 0.95 0.92 rg 0 G 0 g 0.5 0.5 0.5 RG 0.5 0.5 0.5 rg BT /F11 9.9626 Tf -14.944 -390.151 Td[(31 G(2)=DIFF(2)(CONC(2,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.CONC(2,J)+CONC(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH2. 32 1)]TJ /F11 9.9626 Tf 6.522 0 Td[(VEL(J)(CONC(2,J+1))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/(2.HH)+RXN(J) 33 B(2,2)=2.DIFF(2)/HH2. 34 D(2,2)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(2)/HH2.+VEL(J)/(2.HH) 35 A(2,2)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(2)/HH2.)]TJ /F11 9.9626 Tf 7.745 0 Td[(VEL(J)/(2.HH) 36 B(2,4)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 37 38 BIG= ABS (DIFF(2)CONC(2,J+1)/HH2.) 39 BIG2= ABS ()]TJ /F11 9.9626 Tf 8.539 0 Td[(1.DIFF(2)2.CONC(2,J)/HH2.) 40 IF (BIG2. GT .BIG)BIG=BIG2 41 BIG3= ABS (DIFF(2)CONC(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)/HH2.) 42 IF (BIG3. GT .BIG)BIG=BIG3 43 BIG4= ABS ()]TJ /F11 9.9626 Tf 8.539 0 Td[(1.VEL(J)CONC(2,J+1)/(2.HH)) 44 IF (BIG4. GT .BIG)BIG=BIG4 45 BIG5= ABS (VEL(J)CONC(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)/(2.HH)) 46 IF (BIG5. GT .BIG)BIG=BIG5 47 IF ( ABS (RXN(J)). GT .BIG)BIG= ABS (RXN(J)) 48 IF ( ABS (G(2)). LT .BIGEBIG)G(2)=0 49 50 C For B )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 71.926 -599.553 Td[(The reacting species 51 G(3)=DIFF(3)(CONC(3,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.CONC(3,J)+CONC(3,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH2. 52 1)]TJ /F11 9.9626 Tf 6.522 0 Td[(VEL(J)(CONC(3,J+1))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(3,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/(2.HH)+RXN(J) 53 B(3,3)=2DIFF(3)/HH2. 54 D(3,3)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(3)/HH2.+VEL(J)/(2.HH) 172

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55 A(3,3)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(3)/HH2.)]TJ /F11 9.9626 Tf 7.745 0 Td[(VEL(J)/(2.HH) 56 B(3,4)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 57 BIG= ABS (DIFF(3)CONC(3,J+1)/HH2.) 58 BIG2= ABS ()]TJ /F11 9.9626 Tf 8.539 0 Td[(1.DIFF(3)2.CONC(3,J)/HH2.) 59 IF (BIG2. GT .BIG)BIG=BIG2 60 BIG3= ABS (DIFF(3)CONC(3,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)/HH2.) 61 IF (BIG3. GT .BIG)BIG=BIG3 62 BIG4= ABS ()]TJ /F11 9.9626 Tf 8.539 0 Td[(1.VEL(J)CONC(3,J+1)/(2.HH)) 63 IF (BIG4. GT .BIG)BIG=BIG4 64 BIG5= ABS (VEL(J)CONC(3,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)/(2.HH)) 65 IF (BIG5. GT .BIG)BIG=BIG5 66 IF ( ABS (RXN(J)). GT .BIG)BIG= ABS (RXN(J)) 67 IF ( ABS (G(3)). LT .BIGEBIG)G(3)=0 68 69 C For Reaction term 70 IF (rateb. EQ .0) GO TO 210 71 EPS=RXN(J)/(ratebCONC(2,J)CONC(3,J)) 72 IF ( ABS (EPS). GT .0.2) GO TO 210 73 REM= LOG (1.+EPS) 74 IF ( ABS (REM). LT .1.0E)]TJ /F11 9.9626 Tf 8.214 0 Td[(09)REM=EPS(1.)]TJ /F11 9.9626 Tf 8.249 0 Td[(EPS(.5)]TJ /F11 9.9626 Tf 8.248 0 Td[(EPS(1./3.)]TJ /F11 9.9626 Tf 8.532 0 Td[(EPS/4.))) 75 G(4)= LOG (equilibCONC(1,J)/CONC(2,J)/CONC(3,J)))]TJ /F11 9.9626 Tf 5.831 0 Td[(REM 76 B(4,1)=)]TJ /F11 9.9626 Tf 7.986 0 Td[(1./CONC(1,J) 77 B(4,2)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(((RXN(J)/(ratebCONC(3,J)CONC(2,J)2.)) 78 1(1./(1.+RXN(J)/(ratebCONC(2,J)CONC(3,J)))))+1./CONC(2,J) 79 B(4,3)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(((RXN(J)/(ratebCONC(2,J)CONC(3,J)2.)) 80 1(1./(1.+RXN(J)/(ratebCONC(2,J)CONC(3,J)))))+1./CONC(3,J) 81 B(4,4)=1./(ratebCONC(3,J)CONC(2,J) 82 1(1.+RXN(J)/(ratebCONC(2,J)CONC(3,J)))) 83 BIG= ABS (REM) 84 BIG2= ABS ( LOG (equilibCONC(1,J)/CONC(2,J)/CONC(3,J))) 85 IF (BIG2. GT .BIG)BIG=BIG2 86 IF ( ABS (G(4)). LT .BIGEBIG)G(4)=0 87 GO TO 212 88 89 210G(4)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(J)+rateb(equilibCONC(1,J))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(2,J)CONC(3,J)) 90 B(4,1)=)]TJ /F11 9.9626 Tf 8.058 0 Td[(ratebequilib 91 B(4,2)=ratebCONC(3,J) 92 B(4,3)=ratebCONC(2,J) 93 B(4,4)=1. 94 BIG= ABS (ratebequilibCONC(1,J)) 95 BIG2= ABS (ratebCONC(2,J)CONC(3,J)) 96 IF (BIG2. GT .BIG)BIG=BIG2 97 BIG3= ABS (ratebequilibCONC(1,J)) 98 IF (BIG3. GT .BIG)BIG=BIG3 99 IF ( ABS (RXN(J)). GT .BIG)BIG= ABS (RXN(J)) 100 IF ( ABS (G(4)). LT .BIGEBIG)G(4)=0 101 CONTINUE 102 103 c SAVE G OUT DATA 104 212 DO 11I=2,20 105 11 If (I. EQ .J) WRITE (12,301)J,(G(K),K=1,N) 106 DO 12I=100,120 107 12 If (I. EQ .J) WRITE (12,301)J,(G(K),K=1,N) 108 IF (J. EQ .KJ/2) THEN 109 WRITE (12,301)J,(G(K),K=1,N) 110 END IF 111 RETURN 112 END 173

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CodeD.7.SteadyStateConvectiveDiusionwithHomogeneousReactionSubroutinefortheCoupler 1 SUBROUTINE COUPLER(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 COMMON /BAB/A(4,4),B(4,4),C(4,100001),D(4,9),G(4),X(4,4),Y(4,4) 4 COMMON /NSN/N,NJ 5 COMMON /VAR/CONC(3,100001),RXN(100001),DIFF(3),H,EBIG,IJ,KJ,HH 6 COMMON /RTE/rateb,equilib 7 COMMON /VET/VEL(100001),Y1,VELH12,VELHH12 8 9 301 FORMAT (5x,'J='I5,8E18.9) 10 11 PRINT ,'VELH12=',VELH12 12 PRINT ,'VELHH12=',VELHH12 13 14 COEFF1H=DIFF(1)/(H) 15 COEFF1HH=DIFF(1)/(HH) 16 COEFF2H=DIFF(2)/(H) 17 COEFF2HH=DIFF(2)/(HH) 18 COEFF3H=DIFF(3)/(H) 19 COEFF3HH=DIFF(3)/(HH) 20 21 PRINT ,'AB H=',H 22 PRINT ,'AB HH=',HH 23 PRINT ,'kj=',KJ 24 25 C For AB 26 G(1)=COEFF1H(CONC(1,J+1))]TJ /F11 9.9626 Tf 6.019 0 Td[(CONC(1,J)) 27 1)]TJ /F11 9.9626 Tf 6.866 0 Td[(VELH12(CONC(1,J+1)+CONC(1,J))/2. 28 2+(H/2.)(CONC(1,J+1)+3.CONC(1,J))/4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 29 3)]TJ /F11 9.9626 Tf 7.831 0 Td[((H/2.)(RXN(J+1)+3.RXN(J))/4. 30 4)]TJ /F11 9.9626 Tf 6.399 0 Td[(COEFF1HH(CONC(1,J))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 31 5+VELHH12(CONC(1,J)+CONC(1,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/2. 32 6+(HH/2.)(CONC(1,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1)+3.CONC(1,J))/4.(VEL(J))]TJ /F11 9.9626 Tf 6.715 0 Td[(VELHH12)/(HH/2.) 33 7)]TJ /F11 9.9626 Tf 7.831 0 Td[((HH/2.)(RXN(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.RXN(J))/4. 34 B(1,1)=COEFF1H+VELH12/2.)]TJ /F11 9.9626 Tf 8.54 0 Td[((H/2.)3./4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 35 1+COEFF1HH)]TJ /F11 9.9626 Tf 6.42 0 Td[(VELHH12/2.)]TJ /F11 9.9626 Tf 8.54 0 Td[((HH/2.)3./4.(VEL(J))]TJ /F11 9.9626 Tf 6.716 0 Td[(VELHH12)/(HH/2.) 36 D(1,1)=)]TJ /F11 9.9626 Tf 6.195 0 Td[(COEFF1H+VELH12/2.)]TJ /F11 9.9626 Tf 8.539 0 Td[((H/2.)/4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 37 A(1,1)=)]TJ /F11 9.9626 Tf 6.18 0 Td[(COEFF1HH)]TJ /F11 9.9626 Tf 6.42 0 Td[(VELHH12/2.)]TJ /F11 9.9626 Tf 8.539 0 Td[((HH/2.)/4.(VEL(J))]TJ /F11 9.9626 Tf 6.715 0 Td[(VELHH12)/(HH/2.) 38 B(1,4)=(H/2.)(3./4.)+(HH/2.)(3./4.) 39 D(1,4)=(H/2.)(1./4.) 40 A(1,4)=(HH/2.)(1./4.) 41 42 BIG= ABS (COEFF1HCONC(1,J+1)) 43 BIG2= ABS (COEFF1HCONC(1,J)) 44 IF (BIG2. GT .BIG)BIG=BIG2 45 BIG3= ABS ()]TJ /F11 9.9626 Tf 7.072 0 Td[(COEFF1HHCONC(1,J)) 46 IF (BIG3. GT .BIG)BIG=BIG3 47 BIG4= ABS ()]TJ /F11 9.9626 Tf 7.072 0 Td[(COEFF1HHCONC(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 48 IF (BIG4. GT .BIG)BIG=BIG4 49 BIG5= ABS ((H/2.)(RXN(J+1)/4.)) 50 IF (BIG5. GT .BIG)BIG=BIG5 51 BIG6= ABS ((H/2.)(3.RXN(J))/4.) 52 IF (BIG6. GT .BIG)BIG=BIG6 53 BIG7= ABS ((HH/2.)(RXN(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)/4.)) 54 IF (BIG7. GT .BIG)BIG=BIG7 174

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55 BIG8= ABS ((HH/2.)(3.RXN(J))/4.) 56 IF (BIG8. GT .BIG)BIG=BIG8 57 BIG9= ABS (VELH12(CONC(1,J))/2.) 58 IF (BIG9. GT .BIG)BIG=BIG9 59 BIG10= ABS (VELH12(CONC(1,J+1))/2.) 60 IF (BIG10. GT .BIG)BIG=BIG10 61 BIG11= ABS (VELHH12(CONC(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/2.) 62 IF (BIG11. GT .BIG)BIG=BIG11 63 BIG12= ABS (VELHH12(CONC(1,J))/2.) 64 IF (BIG12. GT .BIG)BIG=BIG12 65 BIG13= ABS ((H/2.)CONC(1,J+1)/4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.)) 66 IF (BIG12. GT .BIG)BIG=BIG13 67 BIG14= ABS ((H/2.)3./4.CONC(1,J)(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.)) 68 IF (BIG12. GT .BIG)BIG=BIG14 69 BIG15= ABS ((HH/2.)CONC(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)/4.(VEL(J))]TJ /F11 9.9626 Tf 6.716 0 Td[(VELHH12)/(HH/2.)) 70 IF (BIG12. GT .BIG)BIG=BIG15 71 BIG16= ABS ((HH/2.)3./4.CONC(1,J)(VEL(J))]TJ /F11 9.9626 Tf 6.715 0 Td[(VELHH12)/(HH/2.)) 72 IF (BIG12. GT .BIG)BIG=BIG16 73 IF ( ABS (G(1)). LT .BIGEBIG)G(1)=0 74 75 PRINT ,'A H=',H 76 PRINT ,'A HH=',HH 77 78 C For A )]TJ ET 0 G 0 g 0.95 0.95 0.92 RG 0.95 0.95 0.92 rg 0 G 0 g 0.95 0.95 0.92 RG 0.95 0.95 0.92 rg q 0 -279.77 468 11.021 re f Q 0 G 0 g 0.95 0.95 0.92 RG 0.95 0.95 0.92 rg 0 G 0 g 0.5 0.5 0.5 RG 0.5 0.5 0.5 rg BT /F11 9.9626 Tf -14.944 -276.463 Td[(79 G(2)=COEFF2H(CONC(2,J+1))]TJ /F11 9.9626 Tf 6.019 0 Td[(CONC(2,J)) 80 1)]TJ /F11 9.9626 Tf 6.866 0 Td[(VELH12(CONC(2,J+1)+CONC(2,J))/2. 81 2+(H/2.)(CONC(2,J+1)+3.CONC(2,J))/4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 82 3+(H/2.)(RXN(J+1)+3.RXN(J))/4. 83 4)]TJ /F11 9.9626 Tf 6.399 0 Td[(COEFF2HH(CONC(2,J))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 84 5+VELHH12(CONC(2,J)+CONC(2,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/2. 85 6+(HH/2.)(CONC(2,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1)+3.CONC(2,J))/4.(VEL(J))]TJ /F11 9.9626 Tf 6.715 0 Td[(VELHH12)/(HH/2.) 86 7+(HH/2.)(RXN(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.RXN(J))/4. 87 B(2,2)=COEFF2H+VELH12/2.)]TJ /F11 9.9626 Tf 8.54 0 Td[((H/2.)3./4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 88 1+COEFF2HH)]TJ /F11 9.9626 Tf 6.42 0 Td[(VELHH12/2.)]TJ /F11 9.9626 Tf 8.54 0 Td[((HH/2.)3./4.(VEL(J))]TJ /F11 9.9626 Tf 6.716 0 Td[(VELHH12)/(HH/2.) 89 D(2,2)=)]TJ /F11 9.9626 Tf 6.195 0 Td[(COEFF2H+VELH12/2.)]TJ /F11 9.9626 Tf 8.539 0 Td[((H/2.)/4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 90 A(2,2)=)]TJ /F11 9.9626 Tf 6.18 0 Td[(COEFF2HH)]TJ /F11 9.9626 Tf 6.42 0 Td[(VELHH12/2.)]TJ /F11 9.9626 Tf 8.539 0 Td[((HH/2.)/4.(VEL(J))]TJ /F11 9.9626 Tf 6.715 0 Td[(VELHH12)/(HH/2.) 91 B(2,4)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((H/2.)(3./4.))]TJ /F11 9.9626 Tf 7.832 0 Td[((HH/2.)(3./4.) 92 D(2,4)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((H/2.)(1./4.) 93 A(2,4)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(1./4.) 94 95 BIG= ABS (COEFF2HCONC(2,J+1)) 96 BIG2= ABS (COEFF2HCONC(2,J)) 97 IF (BIG2. GT .BIG)BIG=BIG2 98 BIG3= ABS ()]TJ /F11 9.9626 Tf 7.072 -.001 Td[(COEFF2HHCONC(2,J)) 99 IF (BIG3. GT .BIG)BIG=BIG3 100 BIG4= ABS ()]TJ /F11 9.9626 Tf 7.072 0 Td[(COEFF2HHCONC(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 101 IF (BIG4. GT .BIG)BIG=BIG4 102 BIG5= ABS ((H/2.)(RXN(J+1)/4.)) 103 IF (BIG5. GT .BIG)BIG=BIG5 104 BIG6= ABS ((H/2.)(3.RXN(J))/4.) 105 IF (BIG6. GT .BIG)BIG=BIG6 106 BIG7= ABS ((HH/2.)(RXN(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)/4.)) 107 IF (BIG7. GT .BIG)BIG=BIG7 108 BIG8= ABS ((HH/2.)(3.RXN(J))/4.) 109 IF (BIG8. GT .BIG)BIG=BIG8 110 BIG9= ABS (VELH12(CONC(2,J))/2.) 111 IF (BIG9. GT .BIG)BIG=BIG9 112 BIG10= ABS (VELH12(CONC(2,J+1))/2.) 175

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113 IF (BIG10. GT .BIG)BIG=BIG10 114 BIG11= ABS (VELHH12(CONC(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/2.) 115 IF (BIG11. GT .BIG)BIG=BIG11 116 BIG12= ABS (VELHH12(CONC(2,J))/2.) 117 IF (BIG12. GT .BIG)BIG=BIG12 118 BIG13= ABS ((H/2.)/4.CONC(2,J+1)(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.)) 119 IF (BIG12. GT .BIG)BIG=BIG13 120 BIG14= ABS ((H/2.)3./4.CONC(2,J)(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.)) 121 IF (BIG12. GT .BIG)BIG=BIG14 122 BIG15= ABS ((HH/2.)/4.CONC(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)(VEL(J))]TJ /F11 9.9626 Tf 6.716 0 Td[(VELHH12)/(HH/2.)) 123 IF (BIG12. GT .BIG)BIG=BIG15 124 BIG16= ABS ((HH/2.)3./4.CONC(2,J)(VEL(J))]TJ /F11 9.9626 Tf 6.715 0 Td[(VELHH12)/(HH/2.)) 125 IF (BIG12. GT .BIG)BIG=BIG16 126 IF ( ABS (G(2)). LT .BIGEBIG)G(2)=0 127 128 C For B )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 71.926 -177.273 Td[(The reacting species 129 G(3)=COEFF3H(CONC(3,J+1))]TJ /F11 9.9626 Tf 6.019 0 Td[(CONC(3,J)) 130 1)]TJ /F11 9.9626 Tf 6.866 0 Td[(VELH12(CONC(3,J+1)+CONC(3,J))/2. 131 2+(H/2.)(CONC(3,J+1)+3.CONC(3,J))/4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 132 3+(H/2.)(RXN(J+1)+3.RXN(J))/4. 133 4)]TJ /F11 9.9626 Tf 6.399 0 Td[(COEFF3HH(CONC(3,J))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(3,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 134 5+VELHH12(CONC(3,J)+CONC(3,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/2. 135 6+(HH/2.)(CONC(3,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1)+3.CONC(3,J))/4.(VEL(J))]TJ /F11 9.9626 Tf 6.715 0 Td[(VELHH12)/(HH/2.) 136 7+(HH/2.)(RXN(J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.RXN(J))/4. 137 B(3,3)=COEFF3H+VELH12/2.)]TJ /F11 9.9626 Tf 8.54 0 Td[((H/2.)3./4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 138 1+COEFF3HH)]TJ /F11 9.9626 Tf 6.42 0 Td[(VELHH12/2.)]TJ /F11 9.9626 Tf 8.54 0 Td[((HH/2.)3./4.(VEL(J))]TJ /F11 9.9626 Tf 6.716 0 Td[(VELHH12)/(HH/2.) 139 D(3,3)=)]TJ /F11 9.9626 Tf 6.195 0 Td[(COEFF3H+VELH12/2.)]TJ /F11 9.9626 Tf 8.539 0 Td[((H/2.)/4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 140 A(3,3)=)]TJ /F11 9.9626 Tf 6.18 0 Td[(COEFF3HH)]TJ /F11 9.9626 Tf 6.42 0 Td[(VELHH12/2.)]TJ /F11 9.9626 Tf 8.539 0 Td[((HH/2.)/4.(VEL(J))]TJ /F11 9.9626 Tf 6.715 0 Td[(VELHH12)/(HH/2.) 141 B(3,4)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((H/2.)(3./4.))]TJ /F11 9.9626 Tf 7.832 0 Td[((HH/2.)(3./4.) 142 D(3,4)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((H/2.)(1./4.) 143 A(3,4)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(1./4.) 144 145 BIG= ABS (COEFF3HCONC(3,J+1)) 146 BIG2= ABS (COEFF3HCONC(3,J)) 147 IF (BIG2. GT .BIG)BIG=BIG2 148 BIG3= ABS ()]TJ /F11 9.9626 Tf 7.072 0 Td[(COEFF3HHCONC(3,J)) 149 IF (BIG3. GT .BIG)BIG=BIG3 150 BIG4= ABS ()]TJ /F11 9.9626 Tf 7.072 0 Td[(COEFF3HHCONC(3,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 151 IF (BIG4. GT .BIG)BIG=BIG4 152 BIG5= ABS ((H/2.)(RXN(J+1)/4.)) 153 IF (BIG5. GT .BIG)BIG=BIG5 154 BIG6= ABS ((H/2.)(3.RXN(J))/4.) 155 IF (BIG6. GT .BIG)BIG=BIG6 156 BIG7= ABS ((HH/2.)(RXN(J)]TJ /F11 9.9626 Tf 8.081 -.001 Td[(1)/4.)) 157 IF (BIG7. GT .BIG)BIG=BIG7 158 BIG8= ABS ((HH/2.)(3.RXN(J))/4.) 159 IF (BIG8. GT .BIG)BIG=BIG8 160 BIG9= ABS (VELH12(CONC(3,J))/2.) 161 IF (BIG9. GT .BIG)BIG=BIG9 162 BIG10= ABS (VELH12(CONC(3,J+1))/2.) 163 IF (BIG10. GT .BIG)BIG=BIG10 164 BIG11= ABS (VELHH12(CONC(3,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/2.) 165 IF (BIG11. GT .BIG)BIG=BIG11 166 BIG12= ABS (VELHH12(CONC(3,J))/2.) 167 IF (BIG12. GT .BIG)BIG=BIG12 168 BIG13= ABS ((H/2.)/4.CONC(3,J+1)(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.)) 169 IF (BIG12. GT .BIG)BIG=BIG13 170 BIG14= ABS ((H/2.)3./4.CONC(3,J)(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.)) 176

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171 IF (BIG12. GT .BIG)BIG=BIG14 172 BIG15= ABS ((HH/2.)/4.CONC(3,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)(VEL(J))]TJ /F11 9.9626 Tf 6.716 0 Td[(VELHH12)/(HH/2.)) 173 IF (BIG12. GT .BIG)BIG=BIG15 174 BIG16= ABS ((HH/2.)3./4.CONC(3,J)(VEL(J))]TJ /F11 9.9626 Tf 6.715 0 Td[(VELHH12)/(HH/2.)) 175 IF (BIG12. GT .BIG)BIG=BIG16 176 IF ( ABS (G(3)). LT .BIGEBIG)G(3)=0 177 178 C For Reaction term 179 IF (rateb. EQ .0) GO TO 210 180 EPS=RXN(J)/(ratebCONC(2,J)CONC(3,J)) 181 IF ( ABS (EPS). GT .0.2) GO TO 210 182 183 REM= LOG (1.+EPS) 184 IF ( ABS (REM). LT .1.0E)]TJ /F11 9.9626 Tf 8.213 0 Td[(09)REM=EPS(1.)]TJ /F11 9.9626 Tf 8.249 0 Td[(EPS(.5)]TJ /F11 9.9626 Tf 8.249 0 Td[(EPS(1./3.)]TJ /F11 9.9626 Tf 8.532 0 Td[(EPS/4.))) 185 186 G(4)= LOG (equilibCONC(1,J)/CONC(2,J)/CONC(3,J)))]TJ /F11 9.9626 Tf 5.831 0 Td[(REM 187 B(4,1)=)]TJ /F11 9.9626 Tf 7.986 0 Td[(1./CONC(1,J) 188 B(4,2)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(((RXN(J)/(ratebCONC(3,J)CONC(2,J)2.)) 189 1(1./(1.+RXN(J)/(ratebCONC(2,J)CONC(3,J)))))+1./CONC(2,J) 190 B(4,3)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(((RXN(J)/(ratebCONC(2,J)CONC(3,J)2.)) 191 1(1./(1.+RXN(J)/(ratebCONC(2,J)CONC(3,J)))))+1./CONC(3,J) 192 B(4,4)=1./(ratebCONC(3,J)CONC(2,J) 193 1(1.+RXN(J)/(ratebCONC(2,J)CONC(3,J)))) 194 195 BIG= ABS (REM) 196 BIG2= ABS ( LOG (equilibCONC(1,J)/CONC(2,J)/CONC(3,J))) 197 IF (BIG2. GT .BIG)BIG=BIG2 198 IF ( ABS (G(4)). LT .BIGEBIG)G(4)=0 199 GO TO 212 200 201 210G(4)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(J)+rateb(equilibCONC(1,J))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(2,J)CONC(3,J)) 202 B(4,1)=)]TJ /F11 9.9626 Tf 8.058 0 Td[(ratebequilib 203 B(4,2)=ratebCONC(3,J) 204 B(4,3)=ratebCONC(2,J) 205 B(4,4)=1. 206 207 BIG= ABS (ratebequilibCONC(1,J)) 208 BIG2= ABS (ratebCONC(2,J)CONC(3,J)) 209 IF (BIG2. GT .BIG)BIG=BIG2 210 BIG3= ABS (ratebequilibCONC(1,J)) 211 IF (BIG3. GT .BIG)BIG=BIG3 212 IF ( ABS (RXN(J)). GT .BIG)BIG= ABS (RXN(J)) 213 IF ( ABS (G(4)). LT .BIGEBIG)G(4)=0 214 CONTINUE 215 216 212 WRITE (12,301)J,(G(K),K=1,N) 217 RETURN 218 END 177

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CodeD.8.SteadyStateConvectiveDiusionwithHomogeneousReactionSubroutinefortheInnerRegion 1 SUBROUTINE INNER(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 4 COMMON /BAB/A(4,4),B(4,4),C(4,100001),D(4,9),G(4),X(4,4),Y(4,4) 5 COMMON /NSN/N,NJ 6 COMMON /VAR/CONC(3,100001),RXN(100001),DIFF(3),H,EBIG,IJ,KJ,HH 7 COMMON /VET/VEL(100001),Y1,VELH12,VELHH12 8 COMMON /RTE/rateb,equilib 9 10 301 FORMAT (5x,'J='I5,8E18.9) 11 C For AB 12 G(1)=DIFF(1)(CONC(1,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.CONC(1,J)+CONC(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/H2. 13 1)]TJ /F11 9.9626 Tf 6.522 0 Td[(VEL(J)(CONC(1,J+1))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/(2.H))]TJ /F11 9.9626 Tf 6.072 0 Td[(RXN(J) 14 B(1,1)=2DIFF(1)/H2 15 D(1,1)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(1)/H2+VEL(J)/(2.H) 16 A(1,1)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(1)/H2)]TJ /F11 9.9626 Tf 7.315 0 Td[(VEL(J)/(2.H) 17 B(1,4)=+1. 18 19 BIG= ABS (DIFF(1)CONC(1,J+1)/H2.) 20 BIG2= ABS ()]TJ /F11 9.9626 Tf 8.539 0 Td[(1.DIFF(1)2.CONC(1,J)/H2.) 21 IF (BIG2. GT .BIG)BIG=BIG2 22 BIG3= ABS (DIFF(1)CONC(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)/H2.) 23 IF (BIG3. GT .BIG)BIG=BIG3 24 BIG4= ABS ()]TJ /F11 9.9626 Tf 8.539 0 Td[(1.VEL(J)CONC(1,J+1)/(2.H)) 25 IF (BIG4. GT .BIG)BIG=BIG4 26 BIG5= ABS (VEL(J)CONC(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)/(2.H)) 27 IF (BIG5. GT .BIG)BIG=BIG5 28 IF ( ABS (G(1)). LT .BIGEBIG)G(1)=0 29 30 C For A )]TJ ET 0 G 0 g 0.95 0.95 0.92 RG 0.95 0.95 0.92 rg 0 G 0 g 0.95 0.95 0.92 RG 0.95 0.95 0.92 rg q 0 -393.457 468 11.021 re f Q 0 G 0 g 0.95 0.95 0.92 RG 0.95 0.95 0.92 rg 0 G 0 g 0.5 0.5 0.5 RG 0.5 0.5 0.5 rg BT /F11 9.9626 Tf -14.944 -390.151 Td[(31 G(2)=DIFF(2)(CONC(2,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.CONC(2,J)+CONC(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/H2. 32 1)]TJ /F11 9.9626 Tf 6.522 0 Td[(VEL(J)(CONC(2,J+1))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/(2.H)+RXN(J) 33 B(2,2)=2DIFF(2)/H2 34 D(2,2)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(2)/H2+VEL(J)/(2.H) 35 A(2,2)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(2)/H2)]TJ /F11 9.9626 Tf 7.315 0 Td[(VEL(J)/(2.H) 36 B(2,4)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 37 38 BIG= ABS (DIFF(2)CONC(2,J+1)/H2.) 39 BIG2= ABS ()]TJ /F11 9.9626 Tf 8.539 0 Td[(1.DIFF(2)2.CONC(2,J)/H2.) 40 IF (BIG2. GT .BIG)BIG=BIG2 41 BIG3= ABS (DIFF(2)CONC(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)/H2.) 42 IF (BIG3. GT .BIG)BIG=BIG3 43 BIG4= ABS ()]TJ /F11 9.9626 Tf 8.539 0 Td[(1.VEL(J)CONC(2,J+1)/(2.H)) 44 IF (BIG4. GT .BIG)BIG=BIG4 45 BIG5= ABS (VEL(J)CONC(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)/(2.H)) 46 IF (BIG5. GT .BIG)BIG=BIG5 47 IF ( ABS (G(2)). LT .BIGEBIG)G(2)=0 48 49 C For B )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 71.926 -588.532 Td[(The reacting species 50 G(3)=DIFF(3)(CONC(3,J+1))]TJ /F11 9.9626 Tf 7.793 0 Td[(2CONC(3,J)+CONC(3,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/H2 51 1)]TJ /F11 9.9626 Tf 6.522 0 Td[(VEL(J)(CONC(3,J+1))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(3,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/(2.H)+RXN(J) 52 B(3,3)=2DIFF(3)/H2 53 D(3,3)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(3)/H2+VEL(J)/(2.H) 54 A(3,3)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(3)/H2)]TJ /F11 9.9626 Tf 7.315 0 Td[(VEL(J)/(2.H) 178

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55 B(3,4)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 56 57 BIG= ABS (DIFF(3)CONC(3,J+1)/H2.) 58 BIG2= ABS ()]TJ /F11 9.9626 Tf 8.539 0 Td[(1.DIFF(3)2.CONC(3,J)/H2.) 59 IF (BIG2. GT .BIG)BIG=BIG2 60 BIG3= ABS (DIFF(3)CONC(3,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)/H2.) 61 IF (BIG3. GT .BIG)BIG=BIG3 62 BIG4= ABS ()]TJ /F11 9.9626 Tf 8.539 0 Td[(1.VEL(J)CONC(3,J+1)/(2.H)) 63 IF (BIG4. GT .BIG)BIG=BIG4 64 BIG5= ABS (VEL(J)CONC(3,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)/(2.H)) 65 IF (BIG5. GT .BIG)BIG=BIG5 66 IF ( ABS (G(3)). LT .BIGEBIG)G(3)=0 67 68 C For Reaction term 69 IF (rateb. EQ .0) GO TO 210 70 EPS=RXN(J)/(ratebCONC(2,J)CONC(3,J)) 71 C PRINT EPS2 =', EPS 72 IF ( ABS (EPS). GT .0.2) GO TO 210 73 REM= LOG (1.+EPS) 74 C PRINT REM2 =', REM 75 IF ( ABS (REM). LT .1.0E)]TJ /F11 9.9626 Tf 8.214 0 Td[(09)REM=EPS(1.)]TJ /F11 9.9626 Tf 8.249 0 Td[(EPS(.5)]TJ /F11 9.9626 Tf 8.248 0 Td[(EPS(1./3.)]TJ /F11 9.9626 Tf 8.532 0 Td[(EPS/4.))) 76 G(4)= LOG (equilibCONC(1,J)/CONC(2,J)/CONC(3,J)))]TJ /F11 9.9626 Tf 5.831 0 Td[(REM 77 B(4,1)=)]TJ /F11 9.9626 Tf 7.986 0 Td[(1./CONC(1,J) 78 B(4,2)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(((RXN(J)/(ratebCONC(3,J)CONC(2,J)2.)) 79 1(1./(1.+RXN(J)/(ratebCONC(2,J)CONC(3,J)))))+1./CONC(2,J) 80 B(4,3)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(((RXN(J)/(ratebCONC(2,J)CONC(3,J)2.)) 81 1(1./(1.+RXN(J)/(ratebCONC(2,J)CONC(3,J)))))+1./CONC(3,J) 82 B(4,4)=1./(ratebCONC(3,J)CONC(2,J) 83 1(1.+RXN(J)/(ratebCONC(2,J)CONC(3,J)))) 84 85 BIG= ABS (REM) 86 BIG2= ABS ( LOG (equilibCONC(1,J)/CONC(2,J)/CONC(3,J))) 87 IF (BIG2. GT .BIG)BIG=BIG2 88 IF ( ABS (G(4)). LT .BIGEBIG)G(4)=0 89 GO TO 209 90 91 210G(4)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(J)+rateb(equilibCONC(1,J))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(2,J)CONC(3,J)) 92 B(4,1)=)]TJ /F11 9.9626 Tf 8.058 0 Td[(ratebequilib 93 B(4,2)=ratebCONC(3,J) 94 B(4,3)=ratebCONC(2,J) 95 B(4,4)=1. 96 97 BIG= ABS (ratebequilibCONC(1,J)) 98 BIG2= ABS (ratebCONC(2,J)CONC(3,J)) 99 IF (BIG2. GT .BIG)BIG=BIG2 100 BIG3= ABS (ratebequilibCONC(1,J)) 101 IF (BIG3. GT .BIG)BIG=BIG3 102 IF ( ABS (RXN(J)). GT .BIG)BIG= ABS (RXN(J)) 103 IF ( ABS (G(4)). LT .BIGEBIG)G(4)=0 104 CONTINUE 105 106 209 IF (J. EQ .2) THEN 107 WRITE (12,301)J,(G(K),K=1,N) 108 ELSE IF (J. EQ .3) THEN 109 WRITE (12,301)J,(G(K),K=1,N) 110 ELSE IF (J. EQ .(NJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) THEN 111 WRITE (12,301)J,(G(K),K=1,N) 112 END IF 179

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113 RETURN 114 END 180

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CodeD.9.SteadyStateConvectiveDiusionwithHomogeneousReactionSubroutinefortheBulkBoundaryCondition 1 SUBROUTINE BCNJ(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 COMMON /BAB/A(4,4),B(4,4),C(4,100001),D(4,9),G(4),X(4,4),Y(4,4) 4 COMMON /NSN/N,NJ 5 COMMON /VAR/CONC(3,100001),RXN(100001),DIFF(3),H,EBIG,IJ,KJ,HH 6 COMMON /BUL/CBULK(6),JCOUNT 7 COMMON /RTE/rateb,equilib 8 301 FORMAT (5x,'J='I5,8E18.9) 9 10 29 DO 14I=1,N)]TJ /F11 9.9626 Tf 7.49 0 Td[(1 11 G(I)=CBULK(I))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(I,J) 12 14B(I,I)=1.0 13 DO 121I=1,N)]TJ /F11 9.9626 Tf 7.491 0 Td[(1 14 IF ( ABS (CONC(I,J)). GT .BIG)BIG= ABS (CONC(I,J)) 15 121 IF ( ABS (G(I)). LT .BIGEBIG)G(I)=0 16 17 C For Reaction term 18 IF (rateb. EQ .0) GO TO 207 19 EPS=RXN(J)/(ratebCONC(2,J)CONC(3,J)) 20 IF ( ABS (EPS). GT .0.2) GO TO 207 21 REM= LOG (1.+EPS) 22 IF ( ABS (REM). LT .1.0E)]TJ /F11 9.9626 Tf 8.214 0 Td[(09)REM=EPS(1.)]TJ /F11 9.9626 Tf 8.249 0 Td[(EPS(.5)]TJ /F11 9.9626 Tf 8.248 0 Td[(EPS(1./3.)]TJ /F11 9.9626 Tf 8.532 0 Td[(EPS/4.))) 23 G(4)= LOG (equilibCONC(1,J)/CONC(2,J)/CONC(3,J)))]TJ /F11 9.9626 Tf 5.831 0 Td[(REM 24 B(4,1)=)]TJ /F11 9.9626 Tf 7.986 0 Td[(1./CONC(1,J) 25 B(4,2)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(((RXN(J)/(ratebCONC(3,J)CONC(2,J)2)) 26 1(1/(1+RXN(J)/(ratebCONC(2,J)CONC(3,J)))))+1/CONC(2,J) 27 B(4,3)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(((RXN(J)/(ratebCONC(2,J)CONC(3,J)2)) 28 1(1/(1+RXN(J)/(ratebCONC(2,J)CONC(3,J)))))+1/CONC(3,J) 29 B(4,4)=1./(ratebCONC(3,J)CONC(2,J) 30 1(1.+RXN(J)/(ratebCONC(2,J)CONC(3,J)))) 31 BIG= ABS (REM) 32 BIG2= ABS ( LOG (equilibCONC(1,J)/CONC(2,J)/CONC(3,J))) 33 IF (BIG2. GT .BIG)BIG=BIG2 34 IF ( ABS (G(4)). LT .BIGEBIG)G(4)=0 35 GO TO 206 36 37 207G(4)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(J)+rateb(equilibCONC(1,J))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(2,J)CONC(3,J)) 38 B(4,1)=)]TJ /F11 9.9626 Tf 8.058 0 Td[(ratebequilib 39 B(4,2)=ratebCONC(3,J) 40 B(4,3)=ratebCONC(2,J) 41 B(4,4)=1. 42 BIG= ABS (ratebequilibCONC(1,J)) 43 BIG2= ABS (ratebCONC(2,J)CONC(3,J)) 44 IF (BIG2. GT .BIG)BIG=BIG2 45 BIG3= ABS (ratebequilibCONC(1,J)) 46 IF (BIG3. GT .BIG)BIG=BIG3 47 IF ( ABS (RXN(J)). GT .BIG)BIG= ABS (RXN(J)) 48 IF ( ABS (G(4)). LT .BIGEBIG)G(4)=0 49 50 CONTINUE 51 206 WRITE (12,301)J,(G(K),K=1,N) 52 PRINT ,'ITERATION=',JCOUNT 53 RETURN 54 END 181

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CodeD.10.Matlabcodetoplotresultsfromsteady-statesolutions 1 % Steady State 2 clc ; close all ; clear all ; 3 format longE; 4 5 % Read constant values used in the Fortran code 6 M= dlmread ('cdh ssvalues out.txt'); 7 8 N=M(1); 9 NJ=M(2); 10 KJ=M(3); 11 H=M(4); 12 HH=M(5); 13 RPM=M(6); 14 ANU=M(7); 15 DiffB=M(8); 16 AKB=M(9); 17 BB=M(10); 18 POT=M(11); 19 20 ROT=RPM23.141592653589793/60; 21 22 % Read the velocity profile stuff 23 V= dlmread ('VEL.txt'); 24 25 % Read the steady state values for CB 26 Bss1= dlmread ('cdh out.txt'); 27 Bss=Bss1(:,3); 28 29 % Other constants 30 F=96487; 31 alpha=0.51023ROT^(3./2.)/ANU^(0.5); 32 33 % Create rates 34 ee=)]TJ /F11 9.9626 Tf 6.144 0 Td[(BBPOT; 35 i=)]TJ /F11 9.9626 Tf 7.617 0 Td[(Bss(1)AKB exp (ee); 36 37 % Create y values for plotting 38 y= zeros (NJ,1); 39 40 far=HH(KJ)]TJ /F11 9.9626 Tf 8.08 0 Td[(1); 41 y1=0:HH:far; 42 43 far1=H(NJ)]TJ /F11 9.9626 Tf 6.559 0 Td[(KJ); 44 y2=y1(KJ):H:y1(KJ)+far1; 45 46 for i=1:KJ)]TJ /F11 9.9626 Tf 7.491 0 Td[(1 47 y(i)=y1(i); 48 end 49 for i=KJ:NJ 50 y(i)=y2(i)]TJ /F11 9.9626 Tf 6.559 0 Td[(KJ+1); 51 end 52 53 figure (1) % Plot velocity 54 plot (y,V(:,1),')]TJ /F11 9.9626 Tf 7.38 0 Td[(b'); hold on; 55 plot (y,V(:,2),')]TJ /F11 9.9626 Tf 7.518 0 Td[(k'); 56 plot (y,V(:,4),')]TJ /F11 9.9626 Tf 8.196 0 Td[(r'); 182

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57 title ('VelocityProfileforH'); 58 xlabel ('Length,cm'); 59 ylabel ('Velocity,cm/s'); 60 61 figure (2) 62 plot (y,Bss1(:,3),')]TJ /F11 9.9626 Tf 8.196 0 Td[(r'); hold on; 63 title ('SteadyStateConcentrationofB+awayfromElectrodeSurface'); 64 xlabel ('Length,cm'); 65 ylabel ('Concentration,moles/cm3'); 66 67 figure (3) 68 plot (y,Bss1(:,1),')]TJ /F11 9.9626 Tf 7.38 0 Td[(b'); hold on; 69 title ('ABSteadyStateConcentrationawayfromElectrodeSurface'); 70 xlabel ('Length,cm'); 71 ylabel ('Concentration,moles/cm3'); 72 73 figure (4) 74 plot (y,Bss1(:,2),')]TJ /F11 9.9626 Tf 7.656 0 Td[(g'); hold on; 75 title ('A)]TJ /F11 9.9626 Tf 13.734 0 Td[(SteadyStateConcentrationawayfromElectrodeSurface'); 76 xlabel ('Length,cm'); 77 ylabel ('Concentration,moles/cm3'); 78 79 figure (5) 80 plot (y,Bss1(:,4),')]TJ /F11 9.9626 Tf 7.518 0 Td[(k'); hold on; 81 title ('ReactiontermawayfromElectrodeSurface'); 82 xlabel ('Length,cm'); 83 ylabel ('RateofReaction'); 183

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CodeD.11.Matlabcodetocreateandplotpolarizationcurve 1 clc ; close all ; clear all ; 2 format longE; 3 4 h=0.05; % Step )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 192.956 -68.671 Td[(size 5 V=)]TJ /F11 9.9626 Tf 8.358 0 Td[(2.5:h:)]TJ /F11 9.9626 Tf 9.014 0 Td[(1.; % Potential range 6 Current= length (V); % Current to be saved 7 8 for k=1: length (V); 9 fileID= fopen ('potential in.txt','w'); 10 fprintf (fileID,'%8.3f',V(k)); 11 12 % Run the executable 13 system('cdh ss.exe') 14 pause (0.01) % in seconds 15 16 % Read constant values used in the Fortran code 17 M= dlmread ('cdh ssvalues out.txt'); 18 19 N=M(1); 20 NJ=M(2); 21 KJ=M(3); 22 H=M(4); 23 HH=M(5); 24 RPM=M(6); 25 ANU=M(7); 26 DiffB=M(8); 27 AKB=M(9); 28 BB=M(10); 29 POT=M(11); 30 31 ROT=RPM23.141592653589793/60; 32 33 % Read the steady state values for CB 34 Bss1= dlmread ('cdh out.txt'); 35 Bss=Bss1(:,3); 36 37 % Other constants 38 F=96487; 39 alpha=0.51023ROT^(3./2.)/ANU^(0.5); 40 41 % Create rates 42 ee=)]TJ /F11 9.9626 Tf 6.144 0 Td[(BBPOT; 43 i=)]TJ /F11 9.9626 Tf 7.617 0 Td[(Bss(1)AKB exp (ee); 44 45 % Check to make sure current is correct 46 i2=)]TJ /F11 9.9626 Tf 6.601 0 Td[(FDiffB(()]TJ /F11 9.9626 Tf 9.075 0 Td[(Bss(3)+4Bss(2))]TJ /F11 9.9626 Tf 7.793 0 Td[(3Bss(1))/(2H)); 47 48 % Save current name 49 Current(k)=i; 50 end 51 52 figure (1) 53 plot (V,Current,')]TJ /F11 9.9626 Tf 8.108 0 Td[(.b'); hold on; 54 title ('PolarizationCurve'); 184

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D.3OscillatingConvectiveDiusionImpedancewithHomogeneousReactionsCodeThissectioncontainstheoscillatingFORTRANcodesusedtosolvetheconvectivediusionequationwithahomogeneousreaction.Itreadsthesteady-stateinputandoutputlesinordertosolvefortheoscillatingconcentrations.ThemathematicalworkupforthesecodesareinChapter 4 .TheFORTRANcodesarefollowedbyaMATLABcodethatreadtheoscillatingconcentrationofthereactingspeciesandcreatesthedimensionlessdiusion-impedanceandanoverallimpedance.Therstsectioninthecode,calledCONVDIFFOSCILLATING,isthemainprogram,whichoutlinestheglobalvariablesandsetsupcallinglestosaveoverasoutputlesaswellascallingtheinputles.Thenthesubroutines,allsubroutinescorrespondtothesubroutinesinthesteady-statecode,thatarecalledinthemainprogramareallshown.Theyarethesametitledsubroutinesasthesteadystate. 185

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CodeD.12.OscillatingConvectiveDiusionwithHomogeneousReactionMainProgram 1 C Convective Diffusion Equation with Homogeneous Reaction 2 C 3 species system 3 C SPECIES 1 = AB SPECIES 2 = A )]TJ /F11 9.9626 Tf 8.469 0 Td[(, SPECIES 3 = B + 4 C Species 3 is the reacting species 5 C This is the unsteady state solution that will eventually lead to 6 c the impedance 7 8 C This should be ran after cdh ss for 9 C The input file is the same for both of these 10 11 PROGRAM CONVDIFFOSCILLATING 12 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 13 COMMON /BAT/A(8,8),B(8,8),C(8,100001),D(8,17),G(8),X(8,8),Y(8,8) 14 COMMON /NST/N,NJ 15 COMMON /CON/C1(2,100001),C2(2,100001),C3(2,100001),RXN(2,100001) 16 COMMON /RTE/rateb,equilib,H,EBIG,HH,KJ 17 COMMON /BCI/VEL1,FLUX,omega 18 COMMON /CAR/CONCSS(3,100001),CBULK(3),DIFF(3),Z(3),REF(3) 19 COMMON /VAR/RXNSS(100001),VELNEAR(100001),VELFAR(100001) 20 COMMON /FRE/CB(2010,100001),FREQ(100001),VEL(100001),FVEL(100001) 21 COMMON /VEL12/VELH12,VELHH12 22 CHARACTER REF6 23 24 102 FORMAT (/30HTHENEXTRUNDID NOT CONVERGE) 25 103 FORMAT ('Error=',E16.6/(1X,'Species=',A6,2X,'ConcatElectrode=', 26 1E12.5,2X,'ConcatBulk=',E12.5)) 27 334 FORMAT (4(E25.15,5X)) 28 305 FORMAT (E20.12,3X,E20.12,3X,E20.12,3X,E20.12) 29 335 FORMAT (8(E25.15,5X)) 30 336 FORMAT (1000(E25.15,1X)) 31 339 FORMAT (1000(E25.15,1X)) 32 301 FORMAT (5x,'J='I5,8E18.9) 33 302 FORMAT ('Iteration='I4) 34 35 C Read input values used in steady state 36 open (10, file ='cdh in.txt', status ='old') 37 read (10,)N,NJ,KJ,Y1,Y2,RPM,ANU,equilib,rateb,AKB,BB,EBIG 38 read (10,)(DIFF(I),Z(I),REF(I),CBULK(I),I=1,(N)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 39 40 open (18, file ='potential in.txt', status ='old') 41 read (18,)V 42 43 C Read steady state values from previous file 44 OPEN ( UNIT =11, FILE ='cdh out.txt') 45 READ (11,334)(CONCSS(1,I),CONCSS(2,I),CONCSS(3,I),RXNSS(I),I=1,NJ) 46 47 C Read velocity values from previous file 48 OPEN ( UNIT =12, FILE ='VEL.txt') 49 READ (12,305)(VELNEAR(I),VELFAR(I),FVEL(I),VEL(I),I=1,NJ) 50 51 OPEN ( UNIT =13, FILE ='cdh os out.txt') 52 CLOSE ( UNIT =13, STATUS ='DELETE') 53 OPEN ( UNIT =13, FILE ='cdh os out.txt') 54 55 OPEN (14, FILE ='cdh G out.txt') 56 CLOSE (14, STATUS ='DELETE') 186

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57 OPEN (14, FILE ='cdh G out.txt') 58 59 OPEN (15, FILE ='cdh B out.txt') 60 CLOSE (15, STATUS ='DELETE') 61 OPEN (15, FILE ='cdh B out.txt') 62 63 OPEN (16, FILE ='cdh values out.txt') 64 CLOSE (16, STATUS ='DELETE') 65 OPEN (16, FILE ='cdh values out.txt') 66 67 OPEN (17, FILE ='k values out.txt') 68 CLOSE (17, STATUS ='DELETE') 69 OPEN (17, FILE ='k values out.txt') 70 71 C Constants 72 F=96487. 73 ROT=RPM23.141592653589793/60 74 75 PRINT ,'Y2=',Y2 76 H=Y2/(NJ)]TJ /F11 9.9626 Tf 6.559 0 Td[(KJ) 77 PRINT ,'H=',H 78 79 PRINT ,'Y1=',Y1 80 HH=Y1/(KJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1) 81 PRINT ,'HH=',HH 82 83 C Create flux of the reacting species constants 84 FLUX=)]TJ /F11 9.9626 Tf 5.777 0 Td[(AKB exp ()]TJ /F11 9.9626 Tf 7.112 0 Td[(BBV)/F/Z(3) 85 PRINT ,'FLUX=',FLUX 86 87 C Create charge transfer resistance 88 RTB=1/(AKBBBCONCSS(3,1) EXP ()]TJ /F11 9.9626 Tf 7.112 0 Td[(BBV)) 89 PRINT ,'ChargeTransferResistance=',RTB 90 91 C Create delta 92 delta=(3.DIFF(3)/.51023/ANU)(1./3.)(ANU/ROT)(1./2.) 93 94 N=2N 95 PRINT ,'N=',N 96 337 FORMAT (I2/I7/I7/E15.8/E15.8/E15.8/E15.8/E15.8/E15.8/E15.8/ 97 1E15.8/E15.8/E15.8) 98 write (16,337)N,NJ,KJ,H,HH,V,AKB,BB,RTB,DIFF(3),delta,ROT,ANU 99 100 VEL1=0.25VEL(2) 101 102 C The number of points for frequency 103 NPTS=241 104 PRINT ,'NPTS=',NPTS 105 106 c Create range for the dimensionless frequency 107 DO 261I=1,NPTS 108 FREQ(I)=10.()]TJ /F11 9.9626 Tf 8.67 0 Td[(5.+0.05(I)]TJ /F11 9.9626 Tf 8.656 0 Td[(1.)) 109 261 WRITE (17,339),FREQ(I) 110 111 112 C c The number of points for frequency 113 C NPTS =13 114 C PRINT NPTS =', NPTS 187

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115 C 116 C c Create range for the dimensionless frequency 117 C DO 261 I =1, NPTS 118 C FREQ ( I ) =10.()]TJ /F11 9.9626 Tf 8.665 0 Td[(3.+0.5( I )]TJ /F11 9.9626 Tf 8.657 0 Td[(1.) ) 119 C 261 WRITE (17,339) FREQ ( I ) 120 121 DO 19nf=1,NPTS 122 c DO 19 nf =1,3 123 124 PRINT ,'FREQ(NF)=',FREQ(NF) 125 omega=FREQ(NF)DIFF(3)/(delta)2 126 127 PRINT ,'omega=',omega 128 340 FORMAT (E12.6) 129 write (17,340),omega 130 131 C Start actual code 132 DO 20J=1,NJ 133 DO 20I=1,N 134 20C(I,J)=0.0 135 DO 21J=1,NJ 136 DO 21K=1,2 137 C1(K,J)=0.0 138 C2(K,J)=0.0 139 C3(K,J)=0.0 140 21RXN(K,J)=0.0 141 JCOUNT=0 142 TOL=1.E)]TJ /F11 9.9626 Tf 7.992 0 Td[(10NNJ 143 22JCOUNT=JCOUNT+1 144 AMP=0.0 145 J=0 146 DO 23I=1,N 147 DO 23K=1,N 148 Y(I,K)=0.0 149 23X(I,K)=0.0 150 24J=J+1 151 DO 25I=1,N 152 G(I)=0.0 153 DO 25K=1,N 154 A(I,K)=0.0 155 B(I,K)=0.0 156 25D(I,K)=0.0 157 158 IF (J. EQ .1) CALL BC1(J) 159 IF (J. GT .1. AND .J. LT .KJ) CALL REACTION(J) 160 IF (J. EQ .KJ) CALL COUPLER(J) 161 IF (J. GT .KJ. AND .J. LT .NJ) CALL INNER(J) 162 IF (J. EQ .NJ) CALL BCNJ(J) 163 CALL BAND(J) 164 165 AMP=DABS(G(1))+DABS(G(2))+DABS(G(3))+DABS(G(4))+DABS(G(5)) 166 1+DABS(G(6))+DABS(G(7))+DABS(G(8)) 167 168 IF (J. LT .NJ) GO TO 24 169 170 PRINT ,'ERROR=',AMP 171 172 DO 16K=1,NJ 188

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173 DO 16I=1,2 174 C1(I,K)=C1(I,K)+C(I,K) 175 C2(I,K)=C2(I,K)+C(I+2,K) 176 C3(I,K)=C3(I,K)+C(I+4,K) 177 RXN(I,K)=RXN(I,K)+C(I+6,K) 178 16 CONTINUE 179 180 WRITE (14,302)(JCOUNT) 181 182 IF (DABS(AMP). LT .DABS(TOL)) GO TO 15 183 184 IF (JCOUNT. LE .4) GO TO 22 185 print 102 186 187 15 CONTINUE 188 PRINT ,'JCOUNT=',JCOUNT 189 190 PRINT ,'nf1=',nf 191 192 DO 18I=1,2 193 DO 18J=1,NJ 194 BIG=C3(I,J) 195 BIG2=1.0E)]TJ /F11 9.9626 Tf 7.804 0 Td[(40 196 18 IF ( ABS (BIG). LE .BIG2)C3(I,J)=0.0 197 198 WRITE (13,335)(C1(1,J),C1(2,J),C2(1,J),C2(2,J),C3(1,J),C3(2,J), 199 1RXN(1,J),RXN(2,J),J=1,NJ) 200 201 DO 19J=1,NJ 202 CB(2nf)]TJ /F11 9.9626 Tf 8.236 0 Td[(1,J)=C3(1,J) 203 19CB(2nf,J)=C3(2,J) 204 205 c for some reason nf is one greater then necessary 206 PRINT ,'nf2=',nf 207 208 C DO 17 I =1,2 nf )]TJ /F11 9.9626 Tf 7.491 0 Td[(2 209 nf=nf)]TJ /F11 9.9626 Tf 7.49 0 Td[(1 210 DO 17J=1,NJ 211 17 WRITE (15,336)(CB(I,J),I=1,2nf) 212 213 338 FORMAT (I5) 214 write (16,338)nf 215 216 217 XI=(ROT/ANU)(0.5)(3DIFF(3)/.51023/ANU)()]TJ /F11 9.9626 Tf 8.901 0 Td[(1./3.) 218 PRINT ,'XI/Y=',XI 219 PRINT ,'delta=',delta 220 PRINT ,'ROT=',ROT 221 PRINT ,'ANU=',ANU 222 PRINT ,'DIFF(3)=',DIFF(3) 223 224 END PROGRAM CONVDIFFOSCILLATING 189

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CodeD.13.OscillatingConvectiveDiusionwithHomogeneousReactionSubroutinefortheElectrodeBoundaryCondition 1 SUBROUTINE BC1(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 COMMON /BAT/A(8,8),B(8,8),C(8,100001),D(8,17),G(8),X(8,8),Y(8,8) 4 COMMON /NST/N,NJ 5 COMMON /CON/C1(2,100001),C2(2,100001),C3(2,100001),RXN(2,100001) 6 COMMON /RTE/rateb,equilib,H,EBIG,HH,KJ 7 COMMON /BCI/VEL1,FLUX,omega 8 COMMON /CAR/CONCSS(3,100001),CBULK(3),DIFF(3),Z(3),REF(3) 9 COMMON /VAR/RXNSS(100001),VELNEAR(100001),VELFAR(100001) 10 COMMON /FRE/CB(2010,100001),FREQ(100001),VEL(100001),FVEL(100001) 11 12 301 FORMAT (5x,'J='I5,16E15.6) 13 14 C For AB non reacting species 15 G(1)=omega(3.C1(1,J)+C1(1,J+1))/4. 16 1+2.DIFF(1)(C1(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C1(1,J))/HH2. 17 1)]TJ /F11 9.9626 Tf 6.849 0 Td[(VEL1(C1(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C1(1,J))/HH 18 2)]TJ /F11 9.9626 Tf 8.54 0 Td[((3.RXN(1,J)+RXN(1,J+1))/4. 19 B(1,1)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega3./4.+2.DIFF(1)/HH2.)]TJ /F11 9.9626 Tf 8.071 0 Td[(VEL1HH 20 D(1,1)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega1./4.)]TJ /F11 9.9626 Tf 8.902 0 Td[(2.DIFF(1)/HH2.+VEL1HH 21 B(1,7)=+3./4. 22 D(1,7)=+1./4. 23 24 G(2)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega(3.C1(2,J)+C1(2,J+1))/4. 25 1+2.DIFF(1)(C1(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C1(2,J))/HH2. 26 2)]TJ /F11 9.9626 Tf 6.849 0 Td[(VEL1(C1(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C1(2,J))/HH 27 3)]TJ /F11 9.9626 Tf 8.54 0 Td[((3.RXN(2,J)+RXN(2,J+1))/4. 28 B(2,2)=omega3./4.+2.DIFF(1)/HH2.)]TJ /F11 9.9626 Tf 8.072 0 Td[(VEL1HH 29 D(2,2)=omega1./4.)]TJ /F11 9.9626 Tf 8.902 0 Td[(2.DIFF(1)/HH2.+VEL1HH 30 B(2,8)=+3./4. 31 D(2,8)=+1./4. 32 33 C For A + non reacting species 34 G(3)=omega(3.C2(2,J)+C2(2,J+1))/4. 35 1+2.DIFF(2)(C2(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C2(1,J))/HH2. 36 1)]TJ /F11 9.9626 Tf 6.849 0 Td[(VEL1(C2(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C2(1,J))/HH 37 2+(3.RXN(2,J)+RXN(2,J+1))/4. 38 B(3,3)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega3./4.+2.DIFF(2)/HH2.)]TJ /F11 9.9626 Tf 8.071 0 Td[(VEL1HH 39 D(3,3)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega1./4.)]TJ /F11 9.9626 Tf 8.902 0 Td[(2.DIFF(2)/HH2.+VEL1HH 40 B(3,7)=)]TJ /F11 9.9626 Tf 8.401 0 Td[(3./4. 41 D(3,7)=)]TJ /F11 9.9626 Tf 8.401 0 Td[(1./4. 42 43 G(4)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega(3.C2(2,J)+C2(2,J+1))/4. 44 1+2.DIFF(2)(C2(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C2(2,J))/HH2. 45 2)]TJ /F11 9.9626 Tf 6.849 0 Td[(VEL1(C2(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C2(2,J))/HH 46 3+(3.RXN(2,J)+RXN(2,J+1))/4. 47 B(4,4)=omega3./4.+2.DIFF(2)/HH2.)]TJ /F11 9.9626 Tf 8.072 0 Td[(VEL1HH 48 D(4,4)=omega1./4.)]TJ /F11 9.9626 Tf 8.902 .001 Td[(2.DIFF(2)/HH2.+VEL1HH 49 B(4,8)=)]TJ /F11 9.9626 Tf 8.401 0 Td[(3./4. 50 D(4,8)=)]TJ /F11 9.9626 Tf 8.401 0 Td[(1./4. 51 52 C For B )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 79.145 -621.595 Td[(reacting species 53 G(5)=1)]TJ /F11 9.9626 Tf 7.166 0 Td[(C3(1,J) 54 B(5,5)=1. 190

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55 56 G(6)=C3(2,J) 57 B(6,6)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 58 59 C For Reaction term 60 G(7)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(1,J)+ratebequilibC1(1,J))]TJ /F11 9.9626 Tf 8.353 0 Td[(ratebCONCSS(2,J)C3(1,J) 61 1)]TJ /F11 9.9626 Tf 8.353 0 Td[(ratebCONCSS(3,J)C2(1,J) 62 B(7,1)=)]TJ /F11 9.9626 Tf 8.058 0 Td[(ratebequilib 63 B(7,3)=ratebCONCSS(3,J) 64 B(7,5)=ratebCONCSS(2,J) 65 B(7,7)=+1. 66 67 G(8)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(2,J)+ratebequilibC1(2,J))]TJ /F11 9.9626 Tf 8.353 0 Td[(ratebCONCSS(2,J)C3(2,J) 68 1)]TJ /F11 9.9626 Tf 8.353 0 Td[(ratebCONCSS(3,J)C2(2,J) 69 B(8,2)=)]TJ /F11 9.9626 Tf 8.058 0 Td[(ratebequilib 70 B(8,4)=ratebCONCSS(3,J) 71 B(8,6)=ratebCONCSS(2,J) 72 B(8,8)=+1. 73 74 WRITE (14,301)J,(G(K),K=1,N) 75 76 RETURN 77 END 191

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CodeD.14.OscillatingConvectiveDiusionwithHomogeneousReactionSubroutinefortheElectrodeBoundaryCondition 1 SUBROUTINE REACTION(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 COMMON /BAT/A(8,8),B(8,8),C(8,100001),D(8,17),G(8),X(8,8),Y(8,8) 4 COMMON /NST/N,NJ 5 COMMON /CON/C1(2,100001),C2(2,100001),C3(2,100001),RXN(2,100001) 6 COMMON /RTE/rateb,equilib,H,EBIG,HH,KJ 7 COMMON /BCI/VEL1,FLUX,omega 8 COMMON /CAR/CONCSS(3,100001),CBULK(3),DIFF(3),Z(3),REF(3) 9 COMMON /VAR/RXNSS(100001),VELNEAR(100001),VELFAR(100001) 10 COMMON /FRE/CB(2010,100001),FREQ(100001),VEL(100001),FVEL(100001) 11 12 301 FORMAT (5x,'J='I5,16E15.6) 13 14 C For AB 15 G(1)=omegaC1(2,J) 16 1+DIFF(1)(C1(1,J+1))]TJ /F11 9.9626 Tf 8.32 0 Td[(2.C1(1,J)+C1(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH2. 17 2)]TJ /F11 9.9626 Tf 6.522 0 Td[(VEL(J)(C1(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C1(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/(2.HH) 18 3)]TJ /F11 9.9626 Tf 6.072 0 Td[(RXN(1,J) 19 B(1,1)=2.DIFF(1)/HH2. 20 A(1,1)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(1)/HH2.)]TJ /F11 9.9626 Tf 7.745 0 Td[(VEL(J)/(2.HH) 21 D(1,1)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(1)/HH2.+VEL(J)/(2.HH) 22 B(1,2)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 23 B(1,7)=+1. 24 25 G(2)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC1(1,J) 26 1+DIFF(1)(C1(2,J+1))]TJ /F11 9.9626 Tf 8.32 0 Td[(2.C1(2,J)+C1(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH2. 27 2)]TJ /F11 9.9626 Tf 6.522 0 Td[(VEL(J)(C1(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C1(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/(2.HH) 28 3)]TJ /F11 9.9626 Tf 6.072 0 Td[(RXN(2,J) 29 B(2,2)=2.DIFF(1)/HH2. 30 A(2,2)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(1)/HH2.)]TJ /F11 9.9626 Tf 7.745 0 Td[(VEL(J)/(2.HH) 31 D(2,2)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(1)/HH2.+VEL(J)/(2.HH) 32 B(2,1)=omega 33 B(2,8)=+1. 34 35 C For A )]TJ ET 0 G 0 g 0.95 0.95 0.92 RG 0.95 0.95 0.92 rg 0 G 0 g 0.95 0.95 0.92 RG 0.95 0.95 0.92 rg q 0 -448.563 468 11.021 re f Q 0 G 0 g 0.95 0.95 0.92 RG 0.95 0.95 0.92 rg 0 G 0 g 0.5 0.5 0.5 RG 0.5 0.5 0.5 rg BT /F11 9.9626 Tf -14.944 -445.256 Td[(36 G(3)=omegaC2(2,J)+DIFF(2)(C2(1,J+1))]TJ /F11 9.9626 Tf 8.32 0 Td[(2.C2(1,J)+C2(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH2. 37 1)]TJ /F11 9.9626 Tf 6.522 0 Td[(VEL(J)(C2(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C2(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/(2.HH)+RXN(1,J) 38 B(3,3)=2.DIFF(2)/HH2. 39 A(3,3)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(2)/HH2.)]TJ /F11 9.9626 Tf 7.745 0 Td[(VEL(J)/(2.HH) 40 D(3,3)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(2)/HH2.+VEL(J)/(2.HH) 41 B(3,4)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 42 B(3,7)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 43 44 G(4)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC2(1,J)+DIFF(2)(C2(2,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C2(2,J)+C2(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH2. 45 1)]TJ /F11 9.9626 Tf 6.522 0 Td[(VEL(J)(C2(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C2(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/(2.HH)+RXN(2,J) 46 B(4,4)=2.DIFF(2)/HH2. 47 A(4,4)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(2)/HH2.)]TJ /F11 9.9626 Tf 7.745 0 Td[(VEL(J)/(2.HH) 48 D(4,4)=)]TJ /F11 9.9626 Tf 6.802 .001 Td[(DIFF(2)/HH2.+VEL(J)/(2.HH) 49 B(4,3)=omega 50 B(4,8)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 51 52 C For B + 53 G(5)=omegaC3(2,J))]TJ /F11 9.9626 Tf 7.097 0 Td[(DIFF(3)(C3(1,J+1))]TJ /F11 9.9626 Tf 8.32 0 Td[(2.C3(1,J)+C3(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH2. 54 1)]TJ /F11 9.9626 Tf 6.522 0 Td[(VEL(J)(C3(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C3(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/(2.HH)+RXN(1,J) 192

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55 B(5,5)=2.DIFF(3)/HH2. 56 A(5,5)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(3)/HH2.)]TJ /F11 9.9626 Tf 7.745 0 Td[(VEL(J)/(2.HH) 57 D(5,5)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(3)/HH2.+VEL(J)/(2.HH) 58 B(5,6)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 59 B(5,7)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 60 61 G(6)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC3(1,J))]TJ /F11 9.9626 Tf 7.098 0 Td[(DIFF(3)(C3(2,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C3(2,J)+C3(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH2. 62 1)]TJ /F11 9.9626 Tf 6.522 0 Td[(VEL(J)(C3(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C3(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/(2.HH)+RXN(2,J) 63 B(6,6)=2.DIFF(3)/HH2. 64 A(6,6)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(3)/HH2.)]TJ /F11 9.9626 Tf 7.745 0 Td[(VEL(J)/(2.HH) 65 D(6,6)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(3)/HH2.+VEL(J)/(2.HH) 66 B(6,5)=omega 67 B(6,8)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 68 69 C For Reaction term 70 G(7)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(1,J)+ratebequilibC1(1,J))]TJ /F11 9.9626 Tf 8.353 0 Td[(ratebCONCSS(2,J)C3(1,J) 71 1)]TJ /F11 9.9626 Tf 8.353 0 Td[(ratebCONCSS(3,J)C2(1,J) 72 B(7,1)=)]TJ /F11 9.9626 Tf 8.058 0 Td[(ratebequilib 73 B(7,3)=ratebCONCSS(3,J) 74 B(7,5)=ratebCONCSS(2,J) 75 B(7,7)=+1. 76 77 G(8)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(2,J)+ratebequilibC1(2,J))]TJ /F11 9.9626 Tf 8.353 0 Td[(ratebCONCSS(2,J)C3(2,J) 78 1)]TJ /F11 9.9626 Tf 8.353 0 Td[(ratebCONCSS(3,J)C2(2,J) 79 B(8,2)=)]TJ /F11 9.9626 Tf 8.058 0 Td[(ratebequilib 80 B(8,4)=ratebCONCSS(3,J) 81 B(8,6)=ratebCONCSS(2,J) 82 B(8,8)=+1. 83 84 212 WRITE (12,301)J,(G(K),K=1,N) 85 86 RETURN 87 END 193

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CodeD.15.OscillatingConvectiveDiusionwithHomogeneousReactionSubroutinefortheElectrodeBoundaryCondition 1 SUBROUTINE COUPLER(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 COMMON /BAT/A(8,8),B(8,8),C(8,100001),D(8,17),G(8),X(8,8),Y(8,8) 4 COMMON /NST/N,NJ 5 COMMON /CON/C1(2,100001),C2(2,100001),C3(2,100001),RXN(2,100001) 6 COMMON /RTE/rateb,equilib,H,EBIG,HH,KJ 7 COMMON /BCI/VEL1,FLUX,omega 8 COMMON /CAR/CONCSS(3,100001),CBULK(3),DIFF(3),Z(3),REF(3) 9 COMMON /VAR/RXNSS(100001),VELNEAR(100001),VELFAR(100001) 10 COMMON /VEL12/VELH12,VELHH12 11 COMMON /FRE/CB(2010,100001),FREQ(100001),VEL(100001),FVEL(100001) 12 301 FORMAT (5x,'J='I5,8E15.6) 13 14 C For AB real 15 G(1)=HH/2omega(C1(2,J+1)+3.C1(2,J))/4. 16 1+H/2omega(C1(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C1(2,J))/4. 17 2+DIFF(1)/H(C1(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C1(1,J)) 18 3)]TJ /F11 9.9626 Tf 7.097 0 Td[(DIFF(1)/HH(C1(1,J))]TJ /F11 9.9626 Tf 7.084 0 Td[(C1(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 19 4)]TJ /F11 9.9626 Tf 6.866 0 Td[(VELH12(C1(1,J+1)+C1(1,J))/2. 20 5+VELHH12(C1(1,J)+C1(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/2. 21 6+(H/2.)(C1(1,J+1)+3.C1(1,J))/4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 22 7+(HH/2.)(C1(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C1(1,J))/4.(VEL(J))]TJ /F11 9.9626 Tf 6.715 0 Td[(VELHH12)/(HH/2.) 23 8)]TJ /F11 9.9626 Tf 7.831 0 Td[((H/2.)(RXN(1,J+1)+3.RXN(1,J))/4. 24 9)]TJ /F11 9.9626 Tf 7.831 0 Td[((HH/2.)(RXN(1,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1)+3.RXN(1,J))/4. 25 B(1,1)=DIFF(1)/H+DIFF(1)/HH)]TJ /F11 9.9626 Tf 7.859 0 Td[((H/2.)3./4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 26 1)]TJ /F11 9.9626 Tf 7.832 0 Td[((HH/2.)3./4.(VEL(J))]TJ /F11 9.9626 Tf 6.715 0 Td[(VELHH12)/(HH/2.) 27 D(1,1)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(1)/H+(VEL(J)+VEL(J+1))/4. 28 1)]TJ /F11 9.9626 Tf 7.832 0 Td[((H/2.)/4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 29 A(1,1)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(1)/HH)]TJ /F11 9.9626 Tf 7.86 0 Td[((VEL(J)+VEL(J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/4. 30 1)]TJ /F11 9.9626 Tf 7.832 0 Td[((HH/2.)/4.(VEL(J))]TJ /F11 9.9626 Tf 6.716 0 Td[(VELHH12)/(HH/2.) 31 B(1,2)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2omega(3./4.))]TJ /F11 9.9626 Tf 6.411 0 Td[(H/2omega(3./4.) 32 D(1,2)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2omega(1./4.) 33 A(1,2)=)]TJ /F11 9.9626 Tf 6.116 0 Td[(H/2omega(1./4.) 34 B(1,7)=(H/2.)3./4.+(HH/2.)(1./4.) 35 D(1,7)=(H/2.)3./4. 36 A(1,7)=(HH/2.)(1./4.) 37 C For AB imaginary 38 G(2)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2omega(C1(1,J+1)+3.C1(1,J))/4. 39 1)]TJ /F11 9.9626 Tf 6.411 0 Td[(H/2omega(C1(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C1(1,J))/4. 40 2+DIFF(1)/H(C1(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C1(2,J)) 41 3)]TJ /F11 9.9626 Tf 7.097 0 Td[(DIFF(1)/HH(C1(2,J))]TJ /F11 9.9626 Tf 7.084 0 Td[(C1(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 42 4)]TJ /F11 9.9626 Tf 7.831 0 Td[((VEL(J)+VEL(J+1))/2.(C1(2,J+1)+C1(2,J))/2. 43 5+(VEL(J)+VEL(J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/2.(C1(2,J)+C1(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/2. 44 6+(H/2.)(C1(2,J+1)+3.C1(2,J))/4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 45 7+(HH/2.)(C1(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C1(2,J))/4.(VEL(J))]TJ /F11 9.9626 Tf 6.715 0 Td[(VELHH12)/(HH/2.) 46 8)]TJ /F11 9.9626 Tf 7.831 0 Td[((H/2.)(RXN(2,J+1)+3.RXN(2,J))/4. 47 9)]TJ /F11 9.9626 Tf 7.831 0 Td[((HH/2.)(RXN(2,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1)+3.RXN(2,J))/4. 48 B(2,2)=DIFF(1)/H+DIFF(1)/HH)]TJ /F11 9.9626 Tf 7.859 .001 Td[((H/2.)3./4.(VELH12)]TJ /F11 9.9626 Tf 6.227 .001 Td[(VEL(J))/(H/2.) 49 1)]TJ /F11 9.9626 Tf 7.832 0 Td[((HH/2.)3./4.(VEL(J))]TJ /F11 9.9626 Tf 6.715 0 Td[(VELHH12)/(HH/2.) 50 D(2,2)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(1)/H+(VEL(J)+VEL(J+1))/4. 51 1)]TJ /F11 9.9626 Tf 7.832 0 Td[((H/2.)/4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 52 A(2,2)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(1)/HH)]TJ /F11 9.9626 Tf 7.86 0 Td[((VEL(J)+VEL(J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/4. 53 1)]TJ /F11 9.9626 Tf 7.832 0 Td[((HH/2.)/4.(VEL(J))]TJ /F11 9.9626 Tf 6.716 0 Td[(VELHH12)/(HH/2.) 54 B(2,1)=+HH/2omega(3./4.)+H/2omega(3./4.) 194

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55 D(2,1)=+HH/2omega(1./4.) 56 A(2,1)=+H/2omega(1./4.) 57 B(2,8)=(H/2.)3./4.+(HH/2.)(1./4.) 58 D(2,8)=(H/2.)3./4. 59 A(2,8)=(HH/2.)(1./4.) 60 61 C For A )]TJ /F11 9.9626 Tf 8.468 0 Td[(, real 62 G(3)=HH/2omega(C2(2,J+1)+3.C2(2,J))/4. 63 1+H/2omega(C2(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C2(2,J))/4. 64 2+DIFF(2)/H(C2(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C2(1,J)) 65 3)]TJ /F11 9.9626 Tf 7.097 0 Td[(DIFF(2)/HH(C2(1,J))]TJ /F11 9.9626 Tf 7.084 0 Td[(C2(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 66 4)]TJ /F11 9.9626 Tf 7.831 0 Td[((VEL(J)+VEL(J+1))/2.(C2(1,J+1)+C2(1,J))/2. 67 5+(VEL(J)+VEL(J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/2.(C2(1,J)+C2(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/2. 68 6+(H/2.)(C2(1,J+1)+3.C2(1,J))/4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 69 7+(HH/2.)(C2(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C2(1,J))/4.(VEL(J))]TJ /F11 9.9626 Tf 6.715 0 Td[(VELHH12)/(HH/2.) 70 8+(H/2.)(RXN(1,J+1)+3.RXN(1,J))/4. 71 9+(HH/2.)(RXN(1,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1)+3.RXN(1,J))/4. 72 B(3,3)=DIFF(2)/H+DIFF(2)/HH)]TJ /F11 9.9626 Tf 7.859 0 Td[((H/2.)3./4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 73 1)]TJ /F11 9.9626 Tf 7.832 0 Td[((HH/2.)3./4.(VEL(J))]TJ /F11 9.9626 Tf 6.715 0 Td[(VELHH12)/(HH/2.) 74 D(3,3)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(2)/H+(VEL(J)+VEL(J+1))/4. 75 1)]TJ /F11 9.9626 Tf 7.832 0 Td[((H/2.)/4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 76 A(3,3)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(2)/HH)]TJ /F11 9.9626 Tf 7.86 0 Td[((VEL(J)+VEL(J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/4. 77 1)]TJ /F11 9.9626 Tf 7.832 0 Td[((HH/2.)/4.(VEL(J))]TJ /F11 9.9626 Tf 6.716 0 Td[(VELHH12)/(HH/2.) 78 B(3,4)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2omega(3./4.))]TJ /F11 9.9626 Tf 6.411 0 Td[(H/2omega(3./4.) 79 D(3,4)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2omega(1./4.) 80 A(3,4)=)]TJ /F11 9.9626 Tf 6.116 0 Td[(H/2omega(1./4.) 81 B(3,7)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((H/2.)3./4.)]TJ /F11 9.9626 Tf 8.823 0 Td[((HH/2.)(1./4.) 82 D(3,7)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((H/2.)3./4. 83 A(3,7)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(1./4.) 84 C For A )]TJ /F11 9.9626 Tf 8.468 0 Td[(, imaginary 85 G(4)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2omega(C2(1,J+1)+3.C2(1,J))/4. 86 1)]TJ /F11 9.9626 Tf 6.411 0 Td[(H/2omega(C2(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C2(1,J))/4. 87 2+DIFF(2)/H(C2(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C2(2,J)) 88 3)]TJ /F11 9.9626 Tf 7.097 0 Td[(DIFF(2)/HH(C2(2,J))]TJ /F11 9.9626 Tf 7.084 0 Td[(C2(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 89 4)]TJ /F11 9.9626 Tf 7.831 0 Td[((VEL(J)+VEL(J+1))/2.(C2(2,J+1)+C2(2,J))/2. 90 5+(VEL(J)+VEL(J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/2.(C2(2,J)+C2(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/2. 91 6+(H/2.)(C2(2,J+1)+3.C2(2,J))/4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 92 7+(HH/2.)(C2(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C2(2,J))/4.(VEL(J))]TJ /F11 9.9626 Tf 6.715 0 Td[(VELHH12)/(HH/2.) 93 8+(H/2.)(RXN(2,J+1)+3.RXN(2,J))/4. 94 9+(HH/2.)(RXN(2,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1)+3.RXN(2,J))/4. 95 B(4,4)=DIFF(2)/H+DIFF(2)/HH)]TJ /F11 9.9626 Tf 7.859 0 Td[((H/2.)3./4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 96 1)]TJ /F11 9.9626 Tf 7.832 0 Td[((HH/2.)3./4.(VEL(J))]TJ /F11 9.9626 Tf 6.715 0 Td[(VELHH12)/(HH/2.) 97 D(4,4)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(2)/H+(VEL(J)+VEL(J+1))/4. 98 1)]TJ /F11 9.9626 Tf 7.832 -.001 Td[((H/2.)/4.(VELH12)]TJ /F11 9.9626 Tf 6.227 -.001 Td[(VEL(J))/(H/2.) 99 A(4,4)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(2)/HH)]TJ /F11 9.9626 Tf 7.86 0 Td[((VEL(J)+VEL(J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/4. 100 1)]TJ /F11 9.9626 Tf 7.832 0 Td[((HH/2.)/4.(VEL(J))]TJ /F11 9.9626 Tf 6.716 0 Td[(VELHH12)/(HH/2.) 101 B(4,3)=+HH/2omega(3./4.)+H/2omega(3./4.) 102 D(4,3)=+HH/2omega(1./4.) 103 A(4,3)=+H/2omega(1./4.) 104 B(4,8)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((H/2.)3./4.)]TJ /F11 9.9626 Tf 8.823 0 Td[((HH/2.)(1./4.) 105 D(4,8)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((H/2.)3./4. 106 A(4,8)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(1./4.) 107 108 C For B +, real 109 G(5)=HH/2omega(C3(2,J+1)+3.C3(2,J))/4. 110 1+H/2omega(C3(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C3(2,J))/4. 111 2+DIFF(3)/H(C3(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C3(1,J)) 112 3)]TJ /F11 9.9626 Tf 7.097 0 Td[(DIFF(3)/HH(C3(1,J))]TJ /F11 9.9626 Tf 7.084 0 Td[(C3(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 195

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113 4)]TJ /F11 9.9626 Tf 7.831 0 Td[((VEL(J)+VEL(J+1))/2.(C3(1,J+1)+C3(1,J))/2. 114 5+(VEL(J)+VEL(J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/2.(C3(1,J)+C3(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/2. 115 6+(H/2.)(C3(1,J+1)+3.C3(1,J))/4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 116 7+(HH/2.)(C3(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C3(1,J))/4.(VEL(J))]TJ /F11 9.9626 Tf 6.715 0 Td[(VELHH12)/(HH/2.) 117 8+(H/2.)(RXN(1,J+1)+3.RXN(1,J))/4. 118 9+(HH/2.)(RXN(1,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1)+3.RXN(1,J))/4. 119 B(5,5)=DIFF(3)/H+DIFF(3)/HH)]TJ /F11 9.9626 Tf 7.859 0 Td[((H/2.)3./4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 120 1)]TJ /F11 9.9626 Tf 7.832 0 Td[((HH/2.)3./4.(VEL(J))]TJ /F11 9.9626 Tf 6.715 0 Td[(VELHH12)/(HH/2.) 121 D(5,5)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(3)/H+(VEL(J)+VEL(J+1))/4. 122 1)]TJ /F11 9.9626 Tf 7.832 0 Td[((H/2.)/4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 123 A(5,5)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(3)/HH)]TJ /F11 9.9626 Tf 7.86 0 Td[((VEL(J)+VEL(J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/4. 124 1)]TJ /F11 9.9626 Tf 7.832 0 Td[((HH/2.)/4.(VEL(J))]TJ /F11 9.9626 Tf 6.716 0 Td[(VELHH12)/(HH/2.) 125 B(5,6)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2omega(3./4.))]TJ /F11 9.9626 Tf 6.411 0 Td[(H/2omega(3./4.) 126 D(5,6)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2omega(1./4.) 127 A(5,6)=)]TJ /F11 9.9626 Tf 6.116 0 Td[(H/2omega(1./4.) 128 B(5,7)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((H/2.)3./4.)]TJ /F11 9.9626 Tf 8.823 0 Td[((HH/2.)(1./4.) 129 D(5,7)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((H/2.)3./4. 130 A(5,7)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(1./4.) 131 C For B +, imaginary 132 G(6)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2omega(C3(1,J+1)+3.C3(1,J))/4. 133 1)]TJ /F11 9.9626 Tf 6.411 0 Td[(H/2omega(C3(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C3(1,J))/4. 134 2+DIFF(3)/H(C3(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C3(2,J)) 135 3)]TJ /F11 9.9626 Tf 7.097 0 Td[(DIFF(3)/HH(C3(2,J))]TJ /F11 9.9626 Tf 6.549 0 Td[(C(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 136 4)]TJ /F11 9.9626 Tf 7.831 0 Td[((VEL(J)+VEL(J+1))/2.(C3(2,J+1)+C3(2,J))/2. 137 5+(VEL(J)+VEL(J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/2.(C3(2,J)+C3(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/2. 138 6+(H/2.)(C3(2,J+1)+3.C3(2,J))/4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 139 7+(HH/2.)(C3(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C3(2,J))/4.(VEL(J))]TJ /F11 9.9626 Tf 6.715 0 Td[(VELHH12)/(HH/2.) 140 8+(H/2.)(RXN(2,J+1)+3.RXN(2,J))/4. 141 9+(HH/2.)(RXN(2,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1)+3.RXN(2,J))/4. 142 B(6,6)=DIFF(3)/H+DIFF(3)/HH)]TJ /F11 9.9626 Tf 7.859 0 Td[((H/2.)3./4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 143 1)]TJ /F11 9.9626 Tf 7.832 0 Td[((HH/2.)3./4.(VEL(J))]TJ /F11 9.9626 Tf 6.715 0 Td[(VELHH12)/(HH/2.) 144 D(6,6)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(3)/H+(VEL(J)+VEL(J+1))/4. 145 1)]TJ /F11 9.9626 Tf 7.832 0 Td[((H/2.)/4.(VELH12)]TJ /F11 9.9626 Tf 6.227 0 Td[(VEL(J))/(H/2.) 146 A(6,6)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(3)/HH)]TJ /F11 9.9626 Tf 7.86 0 Td[((VEL(J)+VEL(J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/4. 147 1)]TJ /F11 9.9626 Tf 7.832 0 Td[((HH/2.)/4.(VEL(J))]TJ /F11 9.9626 Tf 6.716 0 Td[(VELHH12)/(HH/2.) 148 B(6,5)=+HH/2omega(3./4.)+H/2omega(3./4.) 149 D(6,5)=+HH/2omega(1./4.) 150 A(6,5)=+H/2omega(1./4.) 151 B(6,8)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((H/2.)3./4.)]TJ /F11 9.9626 Tf 8.823 0 Td[((HH/2.)(1./4.) 152 D(6,8)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((H/2.)3./4. 153 A(6,8)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(1./4.) 154 155 C For Reaction term 156 G(7)=)]TJ /F11 9.9626 Tf 5.777 -.001 Td[(RXN(1,J)+ratebequilibC1(1,J))]TJ /F11 9.9626 Tf 8.353 -.001 Td[(ratebCONCSS(2,J)C3(1,J) 157 1)]TJ /F11 9.9626 Tf 8.353 0 Td[(ratebCONCSS(3,J)C2(1,J) 158 B(7,1)=)]TJ /F11 9.9626 Tf 8.058 0 Td[(ratebequilib 159 B(7,3)=ratebCONCSS(3,J) 160 B(7,5)=ratebCONCSS(2,J) 161 B(7,7)=+1. 162 G(8)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(2,J)+ratebequilibC1(2,J))]TJ /F11 9.9626 Tf 8.353 0 Td[(ratebCONCSS(2,J)C3(2,J) 163 1)]TJ /F11 9.9626 Tf 8.353 0 Td[(ratebCONCSS(3,J)C2(2,J) 164 B(8,2)=)]TJ /F11 9.9626 Tf 8.058 0 Td[(ratebequilib 165 B(8,4)=ratebCONCSS(3,J) 166 B(8,6)=ratebCONCSS(2,J) 167 B(8,8)=+1. 168 212 WRITE (12,301)J,(G(K),K=1,N) 169 RETURN 170 END 196

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CodeD.16.OscillatingConvectiveDiusionwithHomogeneousReactionSubroutinefortheElectrodeBoundaryCondition 1 SUBROUTINE INNER(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 COMMON /BAT/A(8,8),B(8,8),C(8,100001),D(8,17),G(8),X(8,8),Y(8,8) 4 COMMON /NST/N,NJ 5 COMMON /CON/C1(2,100001),C2(2,100001),C3(2,100001),RXN(2,100001) 6 COMMON /RTE/rateb,equilib,H,EBIG,HH,KJ 7 COMMON /BCI/VEL1,FLUX,omega 8 COMMON /CAR/CONCSS(3,100001),CBULK(3),DIFF(3),Z(3),REF(3) 9 COMMON /VAR/RXNSS(100001),VELNEAR(100001),VELFAR(100001) 10 COMMON /FRE/CB(2010,100001),FREQ(100001),VEL(100001),FVEL(100001) 11 12 301 FORMAT (5x,'J='I5,16E15.6) 13 14 C For AB 15 G(1)=omegaC1(2,J) 16 1+DIFF(1)(C1(1,J+1))]TJ /F11 9.9626 Tf 8.32 0 Td[(2.C1(1,J)+C1(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/H2. 17 2)]TJ /F11 9.9626 Tf 6.522 0 Td[(VEL(J)(C1(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C1(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/(2.H) 18 3)]TJ /F11 9.9626 Tf 6.072 0 Td[(RXN(1,J) 19 B(1,1)=2.DIFF(1)/H2. 20 A(1,1)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(1)/H2.)]TJ /F11 9.9626 Tf 7.745 0 Td[(VEL(J)/(2.H) 21 D(1,1)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(1)/H2.+VEL(J)/(2.H) 22 B(1,2)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 23 B(1,7)=+1. 24 25 G(2)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC1(1,J) 26 1+DIFF(1)(C1(2,J+1))]TJ /F11 9.9626 Tf 8.32 0 Td[(2.C1(2,J)+C1(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/H2. 27 2)]TJ /F11 9.9626 Tf 6.522 0 Td[(VEL(J)(C1(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C1(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/(2.H) 28 3)]TJ /F11 9.9626 Tf 6.072 0 Td[(RXN(2,J) 29 B(2,2)=2.DIFF(1)/H2. 30 A(2,2)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(1)/H2.)]TJ /F11 9.9626 Tf 7.745 0 Td[(VEL(J)/(2.H) 31 D(2,2)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(1)/H2.+VEL(J)/(2.H) 32 B(2,1)=omega 33 B(2,8)=+1. 34 35 C For A )]TJ ET 0 G 0 g 0.95 0.95 0.92 RG 0.95 0.95 0.92 rg 0 G 0 g 0.95 0.95 0.92 RG 0.95 0.95 0.92 rg q 0 -448.563 468 11.021 re f Q 0 G 0 g 0.95 0.95 0.92 RG 0.95 0.95 0.92 rg 0 G 0 g 0.5 0.5 0.5 RG 0.5 0.5 0.5 rg BT /F11 9.9626 Tf -14.944 -445.256 Td[(36 G(3)=omegaC2(2,J)+DIFF(2)(C2(1,J+1))]TJ /F11 9.9626 Tf 8.32 0 Td[(2.C2(1,J)+C2(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/H2. 37 1)]TJ /F11 9.9626 Tf 6.522 0 Td[(VEL(J)(C2(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C2(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/(2.H)+RXN(1,J) 38 B(3,3)=2.DIFF(2)/H2. 39 A(3,3)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(2)/H2.)]TJ /F11 9.9626 Tf 7.745 0 Td[(VEL(J)/(2.H) 40 D(3,3)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(2)/H2.+VEL(J)/(2.H) 41 B(3,4)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 42 B(3,7)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 43 44 G(4)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC2(1,J)+DIFF(2)(C2(2,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C2(2,J)+C2(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/H2. 45 1)]TJ /F11 9.9626 Tf 6.522 0 Td[(VEL(J)(C2(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C2(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/(2.H)+RXN(2,J) 46 B(4,4)=2.DIFF(2)/H2. 47 A(4,4)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(2)/H2.)]TJ /F11 9.9626 Tf 7.745 0 Td[(VEL(J)/(2.H) 48 D(4,4)=)]TJ /F11 9.9626 Tf 6.802 .001 Td[(DIFF(2)/H2.+VEL(J)/(2.H) 49 B(4,3)=omega 50 B(4,8)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 51 52 C For B + 53 G(5)=omegaC3(2,J))]TJ /F11 9.9626 Tf 7.097 0 Td[(DIFF(3)(C3(1,J+1))]TJ /F11 9.9626 Tf 8.32 0 Td[(2.C3(1,J)+C3(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/H2. 54 1)]TJ /F11 9.9626 Tf 6.522 0 Td[(VEL(J)(C3(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C3(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/(2.H)+RXN(1,J) 197

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55 B(5,5)=2.DIFF(3)/H2. 56 A(5,5)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(3)/H2.)]TJ /F11 9.9626 Tf 7.745 0 Td[(VEL(J)/(2.H) 57 D(5,5)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(3)/H2.+VEL(J)/(2.H) 58 B(5,6)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 59 B(5,7)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 60 61 G(6)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC3(1,J))]TJ /F11 9.9626 Tf 7.098 0 Td[(DIFF(3)(C3(2,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C3(2,J)+C3(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/H2. 62 1)]TJ /F11 9.9626 Tf 6.522 0 Td[(VEL(J)(C3(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C3(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/(2.H)+RXN(2,J) 63 B(6,6)=2.DIFF(3)/H2. 64 A(6,6)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(3)/H2.)]TJ /F11 9.9626 Tf 7.745 0 Td[(VEL(J)/(2.H) 65 D(6,6)=)]TJ /F11 9.9626 Tf 6.802 0 Td[(DIFF(3)/H2.+VEL(J)/(2.H) 66 B(6,5)=omega 67 B(6,8)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 68 69 C For Reaction term 70 G(7)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(1,J)+ratebequilibC1(1,J))]TJ /F11 9.9626 Tf 8.353 0 Td[(ratebCONCSS(2,J)C3(1,J) 71 1)]TJ /F11 9.9626 Tf 8.353 0 Td[(ratebCONCSS(3,J)C2(1,J) 72 B(7,1)=)]TJ /F11 9.9626 Tf 8.058 0 Td[(ratebequilib 73 B(7,3)=ratebCONCSS(3,J) 74 B(7,5)=ratebCONCSS(2,J) 75 B(7,7)=+1. 76 77 G(8)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(2,J)+ratebequilibC1(2,J))]TJ /F11 9.9626 Tf 8.353 0 Td[(ratebCONCSS(2,J)C3(2,J) 78 1)]TJ /F11 9.9626 Tf 8.353 0 Td[(ratebCONCSS(3,J)C2(2,J) 79 B(8,2)=)]TJ /F11 9.9626 Tf 8.058 0 Td[(ratebequilib 80 B(8,4)=ratebCONCSS(3,J) 81 B(8,6)=ratebCONCSS(2,J) 82 B(8,8)=+1. 83 84 IF (J. EQ .KJ/2) THEN 85 WRITE (14,301)J,(G(K),K=1,N) 86 ELSE IF (J. EQ .(NJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(2)) THEN 87 WRITE (14,301)J,(G(K),K=1,N) 88 ELSE IF (J. EQ .(NJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) THEN 89 WRITE (14,301)J,(G(K),K=1,N) 90 END IF 91 92 RETURN 93 END 198

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CodeD.17.OscillatingConvectiveDiusionwithHomogeneousReactionSubroutinefortheElectrodeBoundaryCondition 1 SUBROUTINE BCNJ(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 COMMON /BAT/A(8,8),B(8,8),C(8,100001),D(8,17),G(8),X(8,8),Y(8,8) 4 COMMON /NST/N,NJ 5 COMMON /CON/C1(2,100001),C2(2,100001),C3(2,100001),RXN(2,100001) 6 COMMON /RTE/rateb,equilib,H,EBIG,HH,KJ 7 COMMON /BCI/VEL1,FLUX,omega 8 COMMON /CAR/CONCSS(3,100001),CBULK(3),DIFF(3),Z(3),REF(3) 9 COMMON /VAR/RXNSS(100001),VELNEAR(100001),VELFAR(100001) 10 COMMON /FRE/CB(2010,100001),FREQ(100001),VEL(100001),FVEL(100001) 11 12 301 FORMAT (5x,'J='I5,16E15.6) 13 14 C For AB non reacting species 15 29G(1)=C1(1,J) 16 B(1,1)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 17 18 G(2)=C1(2,J) 19 B(2,2)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 20 21 C For A + non reacting species 22 G(3)=C2(1,J) 23 B(3,3)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 24 25 G(4)=C2(2,J) 26 B(4,4)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 27 28 C For B )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 79.145 -357.087 Td[(reacting species 29 G(5)=C3(1,J) 30 B(5,5)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 31 32 G(6)=C3(2,J) 33 B(6,6)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 34 35 C For Reaction term 36 G(7)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(1,J)+ratebequilibC1(1,J))]TJ /F11 9.9626 Tf 8.353 0 Td[(ratebCONCSS(2,J)C3(1,J) 37 1)]TJ /F11 9.9626 Tf 8.353 0 Td[(ratebCONCSS(3,J)C2(1,J) 38 B(7,1)=)]TJ /F11 9.9626 Tf 8.058 0 Td[(ratebequilib 39 B(7,3)=ratebCONCSS(3,J) 40 B(7,5)=ratebCONCSS(2,J) 41 B(7,7)=+1. 42 43 G(8)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(2,J)+ratebequilibC1(2,J))]TJ /F11 9.9626 Tf 8.353 0 Td[(ratebCONCSS(2,J)C3(2,J) 44 1)]TJ /F11 9.9626 Tf 8.353 0 Td[(ratebCONCSS(3,J)C2(2,J) 45 B(8,2)=)]TJ /F11 9.9626 Tf 8.058 0 Td[(ratebequilib 46 B(8,4)=ratebCONCSS(3,J) 47 B(8,6)=ratebCONCSS(2,J) 48 B(8,8)=+1. 49 50 WRITE (12,301)J,(G(K),K=1,N) 51 52 RETURN 53 END 199

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CodeD.18.Matlabcodetocreateanddimensionlessdiusionimpedanceandoverallimpedance 1 % Inserting concentration data from Fortran 2 clc ; close all ; clear all ; 3 format longE; 4 5 % Read the unsteady state data at each frequency 6 B= dlmread ('cdh B out.txt'); 7 8 % Read constant values used in the Fortran code 9 M= dlmread ('cdh values out.txt'); 10 11 N=M(1); 12 NJ=M(2); 13 KJ=M(3); 14 H=M(4); 15 HH=M(5); 16 V=M(6); 17 AKB=M(7); 18 BB=M(8); 19 RTB=M(9); 20 DiffB=M(10); 21 delta=M(11); 22 rot=M(12); 23 anu=M(13); 24 nf=M(14); 25 26 % Read frequency points Kw = omega KK = K 27 K= dlmread ('k values out.txt'); 28 K=K'; 29 for n=1:nf 30 Kw(n)=K(n+nf); 31 KK(n)=K(n); 32 end 33 34 deltan= gamma (4/3)delta; 35 36 % Read the steady state values for CB 37 Css= dlmread ('cdh out.txt'); 38 39 % Other constants 40 F=96487; 41 42 % Create y values for plotting 43 y= zeros (NJ,1); 44 45 far=HH(KJ)]TJ /F11 9.9626 Tf 8.08 0 Td[(1); 46 y1=0:HH:far; 47 48 far1=H(NJ)]TJ /F11 9.9626 Tf 6.559 .001 Td[(KJ); 49 y2=y1(KJ):H:y1(KJ)+far1; 50 51 for i=1:KJ)]TJ /F11 9.9626 Tf 7.491 0 Td[(1 52 y(i)=y1(i); 53 end 54 for i=KJ:NJ 200

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55 y(i)=y2(i)]TJ /F11 9.9626 Tf 6.559 0 Td[(KJ+1); 56 end 57 58 % Create complex numbers from unsteady state data 59 CB= zeros (NJ,nf); 60 for n=1:nf 61 for i=1:NJ 62 CB(i,n)=complex(B(i,2n)]TJ /F11 9.9626 Tf 8.08 0 Td[(1),B(i,2n)); 63 end 64 end 65 66 % Calculate the impedance 67 Zdfront=(RTBAKB exp ()]TJ /F11 9.9626 Tf 7.113 0 Td[(BBV))/(FDiffB); 68 Zd= zeros (1,nf); 69 for i=1:nf 70 Zd(i)=Zdfront()]TJ /F11 9.9626 Tf 7.249 0 Td[(CB(1,i)/(()]TJ /F11 9.9626 Tf 7.555 0 Td[(CB(3,i)+4CB(2,i))]TJ /F11 9.9626 Tf 7.793 0 Td[(3CB(1,i))/(2HH))); 71 end 72 73 Zdd= zeros (1,nf); 74 for i=1:nf 75 Zdd(i)=()]TJ /F11 9.9626 Tf 6.695 0 Td[(CB(1,i)/(()]TJ /F11 9.9626 Tf 7.555 0 Td[(CB(3,i)+4CB(2,i))]TJ /F11 9.9626 Tf 7.793 0 Td[(3CB(1,i))/(2HH)))/deltan; 76 end 77 78 Zf= zeros (1,nf); 79 for i=1:nf 80 Zf(i)=RTB+Zd(i); 81 end 82 83 Zi= zeros (1,nf); 84 for n=1:nf 85 Zi(n)=10+Zf(n)/(1+1iKw(n)20E)]TJ /F11 9.9626 Tf 7.804 0 Td[(6Zf(n)); 86 end 87 88 figure (1) 89 plot (y,Css(:,1),')]TJ /F11 9.9626 Tf 7.38 0 Td[(b'); hold on; 90 plot (y,Css(:,3),')]TJ /F11 9.9626 Tf 8.196 0 Td[(r'); 91 plot (y,Css(:,2),')]TJ /F11 9.9626 Tf 7.518 0 Td[(k'); 92 % axis ([0 0.1 5 e )]TJ /F11 9.9626 Tf 7.491 0 Td[(5 10.1 e )]TJ /F11 9.9626 Tf 8.656 0 Td[(5]) ; 93 % legend (' SS C Glucose ',' SS C H2O2 ',' SS C GOX ') ; 94 title ('SteadyStateConcentrationawayfromElectrodeSurface'); 95 xlabel ('Length,cm'); 96 ylabel ('Concentration,moles/cm3'); 97 98 figure (2) 99 plot ( real (Zdd),)]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 96.218 -496.887 Td[(imag (Zdd),')]TJ /F11 9.9626 Tf 8.081 0 Td[(ks'); hold on; axis equal; 100 % legend (' MatLab Data ',' Finite Film Thickness tanh ( sqrt ( j K ) ) / sqrt ( j K ) ') ; 101 title ('DiffusionImpedanceNyquistplot'); 102 xlabel ('RealpartofImpedance'); 103 ylabel ('ImaginarypartofImpedance'); 104 realdd= real (Zdd); 105 imagdd= imag (Zdd); 106 107 realf= real (Zf); 108 imagf= imag (Zf); 109 110 figure (4) 111 plot ( real (Zi),)]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 90.24 -629.141 Td[(imag (Zi),')]TJ /F11 9.9626 Tf 8.081 0 Td[(ks'); hold on; axis equal; 112 % legend (' MatLab Data ',' Finite Film Thickness tanh ( sqrt ( j K ) ) / sqrt ( j K ) ') ; 201

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113 title ('OverallImpedanceNyquistplot'); 114 xlabel ('RealpartofImpedance'); 115 ylabel ('ImaginarypartofImpedance'); 116 real = real (Zi); 117 imag = imag (Zi); 118 119 impedance= zeros (nf,2); 120 impedance(:,1)= real '; 121 impedance(:,2)= imag '; 122 123 impedancef= zeros (nf,2); 124 impedancef(:,1)=realf'; 125 impedancef(:,2)=imagf'; 126 127 impedancedd= zeros (nf,2); 128 impedancedd(:,1)=realdd'; 129 impedancedd(:,2)=imagdd'; 202

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APPENDIXECODESFORCONTINUOUSGLUCOSEMONITORThisappendixcontainsthedierentFORTRANcodesthatproducedtheresultsforthesolutionoftheconvective-diusionequationwithhomogeneousreactionforthecontinuousglucosemonitor.ThisappendixalsohastheMATLABcodestoplotthesteady-stateresults,toplotapolarizationcurvefromthesteadystateresultsandacodetocreatetheimpedancefromtheresultsoftheoscillatingFORTRANcode. E.1InputlesfortheContinuousGlucoseMonitorThefollowingcodesaretheinputlesforthecontinuousglucosemonitor.Theinputcodehasthenumberofspeciesbeingsolved,thetotalnumberofpoints,thenumberofpointsuntiltherstcoupler,thenumberofpointsuntilthesecondcoupler,thedistanceofthereactionregionincm,thedistanceoftheinnerlayerincm,andthedistanceintheGLMlayerincm.Therateconstantsintheinputlearetheequilibriumrates,incm3=mol,oftworeactionsandtheforwardrateofreactionfortworeversiblereactions,incm3=mols,andtwoirreversiblereactions,in1=s,andtherateconstantfortheheterogeneousreactionofthereactingspecies,inAcm=mol,andthetafelkineticsvaluefortheheterogeneousreaction,in1=V.TheinputleincludestheerrorallowedfortheBIGvalues,whichisdiscussedinsection 2.2.2 .Andtheendoftheinputlehasthespecicvaluestodescribeeachspeciesinthesystem,includingdiusioncoecientsincm2=s,thecharge,acharactername,andtheconcentrationvalueinthebulkinmol=cm3.Thesecondinputlecontainsthepotentialusedtocontroltherateoftheheterogeneousreaction.Thepotentialisinitsowninputletoallowapolarizationcurvetobecalculatedeasier. 203

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CodeE.1.InputlefortheContinuousGlucoseMonitorCode 1 12 2 40001 3 39001 4 29001 5 0.0004 6 0.0003 7 0.0015 8 0.8 9 0.42 10 0.169 11 0.32 12 0.11 13 0.0176 14 1.E8 15 10. 16 1.E9 17 1.E8 18 10. 19 1.E9 20 1. 21 37.42 22 1.E)]TJ /F11 9.9626 Tf 7.804 0 Td[(14 23 7.2E)]TJ /F11 9.9626 Tf 7.491 0 Td[(6GL5.55075E)]TJ /F11 9.9626 Tf 7.491 0 Td[(6 24 0.GOx.5E)]TJ /F11 9.9626 Tf 7.491 0 Td[(3 25 7.2E)]TJ /F11 9.9626 Tf 7.491 0 Td[(6GA1.E)]TJ /F11 9.9626 Tf 7.804 0 Td[(20 26 0.GOx2.5E)]TJ /F11 9.9626 Tf 7.49 0 Td[(3 27 2.46E)]TJ /F11 9.9626 Tf 7.49 0 Td[(5O23.125E)]TJ /F11 9.9626 Tf 7.49 0 Td[(9 28 1.83E)]TJ /F11 9.9626 Tf 7.49 0 Td[(5H2O21.E)]TJ /F11 9.9626 Tf 7.804 0 Td[(20 29 0.CX)]TJ /F11 9.9626 Tf 6.041 0 Td[(GOX2.5E)]TJ /F11 9.9626 Tf 7.49 0 Td[(3 30 0.CX)]TJ /F11 9.9626 Tf 5.586 0 Td[(GOX.5E)]TJ /F11 9.9626 Tf 7.491 0 Td[(3 31 32 C line 1 is the number of species 33 C line 2 is the number of points NJ 34 C line 3 is the point where the domains split value of IJ 35 C line 4 is the point where the reaction layer is value of KJ 36 C line 5 is the distance of the inner reaction later in cm (1 um ) 37 C line 6 is the distance of the inner GOx layer in cm (6 um ) 38 C line 7 is the distance of the outer GLM layer in cm (15 um ) 39 C line 8 is the porosity factor of the inner layer and reaction layer 40 C line 9 is the porosity factor of the outer layer for small species 41 C line 10 is the porosity factor of the outer layer for large species 42 C line 11 is the solubility coefficient of H2O2 43 C line 12 is the solubility coefficient of O2 44 c line 13 is the solubility coefficient of Glucose 45 C line 14 is the ratef1 of rxn1 mol / cm ^3 46 C line 15 is the equilib1 of rxn1 cm ^3/( mol s ) 47 C line 16 is the ratef2 of rxn2 mol / cm ^3 48 C line 17 is the ratef3 of rxn3 mol / cm ^3 49 C line 18 is the equilib3 of rxn3 cm ^3/( mol s ) 50 C line 19 is the ratef3 of rxn4 mol / cm ^3 51 C line 20 is the rate constant ( K ) for the flux of the reacting species A / cm2 cm3 / mol 52 C line 20 is the tafel b value for the flux of the reacting species 53 C line 21 is the error allowed for the BIGs 54 C lines 22)]TJ /F11 9.9626 Tf 8.065 0 Td[(39 specify values used to describe each species in the system 204

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CodeE.2.InputleforthepotentialoftheContinuousGlucoseMonitorCode 1 0.2 E.2Steady-StateContinuousGlucoseMonitorCodeThissectioncontainsthesteady-stateFORTRANcodesusedtosolvethecontinuousglucosemonitorequations.ThemathematicalworkupforthesecodesareinChapter 5 .TheFORTRANcodesarefollowedbyaMATLABcode.TheMATLABcodetakestheoutputfromthesteady-stateFORTRANcodeandplotsthedata.Therstsectioninthecode,calledCONVDIFF,isthemainprogram,whichoutlinestheglobalvariablesandsetsupcallinglestosaveoverasoutputlesaswellascallingtheinputles.Thenthesubroutinesthatarecalledinthemainprogramareallshown.ThesubroutineBC1solvestheboundaryconditionattheelectrodesurface.ThesubroutinesREACTIONandINNERsolvethenonlinearcoupleddierentialequationsintheGOxlayerusingmeshsizesHHHandHH,respectively.ThesubroutineCOUPLER1setstheuxatKJequalusingtwouxexpressionequations.ThesubroutineOUTERsolvesthedierentialequationsintheGLMlayerusingmeshsizeHwherethehomogeneousreactionisnotconsidered.ThesubroutineCOUPLER2setstheuxatIJequalusingtwouxexpressionequations.Themathematicsbehindthecouplersarediscussedinchapter 5 section 5.1.3 .FinallythesubroutineBCNJsolvestheboundaryconditioninthebulk.BANDandMATINV,in A ,arecalledinordertosolvethesteady-statesolution. 205

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CodeE.3.Steady-StateContinuousGlucoseMonitorMainProgram 1 C Convective Diffusion Equation with Homogeneous Reaction 2 C Enzyme kinetics added 3 C 8 species system 4 C SPECIES 1 = glucose SPECIES 2 = GOx )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 256.308 -68.671 Td[(FAD SPECIES 3 = Gluconic acid 5 C SPECIES 4 = GOx )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 130.915 -79.692 Td[(FADH2 SPECIES 5 = O2 SPECIES 6 = H2O2 6 C SPECIES 7 = GOx )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 130.915 -90.714 Td[(FADH2 )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 166.54 -90.714 Td[(GA SPECIES 8 = GOx )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 286.196 -90.714 Td[(FAD )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 310.583 -90.714 Td[(H2O2 7 C Species 6 is the reacting species 8 C This is the steady state solution only 9 C It should be ran prior to cdhgox os for 10 C The input file is the same for both 11 C This version of the code is reversible normal kinetics for reactions 1 and 3 12 C Reactions 2 and 4 are irreversible 13 14 PROGRAM CONVDIFF 15 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 16 COMMON /BAB/A(12,12),B(12,12),C(12,80001),D(12,25),G(12),X(12,12) 17 1,Y(12,12) 18 COMMON /NSN/N,NJ 19 COMMON /VAR/CONC(8,80001),RXN(4,80001),DIFF(8),H,EBIG,HH,IJ 20 COMMON /VARR/HHH,KJ 21 COMMON /POR/POR1,POR2,PORGLU 22 COMMON /BCI/FLUX 23 COMMON /RTE/ratef1,equilib1,ratef2,ratef3,equilib3,ratef4 24 COMMON /BUL/CBULK(8),PARH2O2,PAR02,PARGLUCOSE,JCOUNT 25 COMMON /EXTRA/REF(8) 26 CHARACTER REF8 27 28 29 102 FORMAT (/30HTHENEXTRUNDID NOT CONVERGE) 30 103 FORMAT ('Error=',E16.6/(1X,'Species=',A6,2X,'CatElectrode=', 31 1E12.5,2X,'CatBulk=',E12.5)) 32 300 FORMAT (18x,'Glucose'14x,'GOx',14x,'GA',14x,'GOx2',14x,'O2',14x, 33 1'H2O2',14x,'CX)]TJ /F11 9.9626 Tf 6.484 0 Td[(GOx2',14x,'CX)]TJ /F11 9.9626 Tf 6.14 0 Td[(GOx',14x,'RXN1',14x,'RXN2',14x, 34 1'RXN3',14x,'RXN4') 35 301 FORMAT (5x,'J='I5,12E18.9) 36 334 FORMAT (12(E25.15,5X)) 37 302 FORMAT ('Iteration='I4) 38 39 OPEN ( UNIT =13, FILE ='cdhgox out.txt') 40 CLOSE ( UNIT =13, STATUS ='DELETE') 41 OPEN ( UNIT =13, FILE ='cdhgox out.txt') 42 43 OPEN (12, FILE ='cdhgox G out.txt') 44 CLOSE (12, STATUS ='DELETE') 45 OPEN (12, FILE ='cdhgox G out.txt') 46 WRITE (12,300) 47 48 open (14, file ='cdhgox in.txt', status ='old') 49 106 FORMAT (I2/I7/I7/E15.4/E15.4/E15.4/E15.4/E15.4/E15.4/E15.4/E15.4 50 1/E15.4/E15.4/E15.4/E15.4/E15.4/E15.4) 51 print ,'doesthiswork' 52 read (14,)N,NJ,IJ,KJ,Y1,Y2,Y3,POR1,POR2,POR3,PARH2O2,PAR02, 53 1PARGLUCOSE,ratef1,equilib1,ratef2,ratef3,equilib3,ratef4, 54 2AKB,BB,EBIG 55 read (14,)(DIFF(I),REF(I),CBULK(I),I=1,(N)]TJ /F11 9.9626 Tf 8.081 0 Td[(4)) 206

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56 POR1=POR1(1.5) 57 POR2=POR2(1.5) 58 PORGLU=POR3(1.5) 59 60 open (16, file ='pot in.txt', status ='old') 61 read (16,)V 62 63 c open (16, file =' O2 in txt ', status =' old ') 64 c 305 FORMAT ( E25 .5) 65 c read (16,305) CBULK (5) 66 67 C Constants 68 F=96487. 69 70 c THIS IS SPACING FOR OUTER LAYER BCNJ 71 H=Y3/(NJ)]TJ /F11 9.9626 Tf 7.943 0 Td[(IJ) 72 PRINT ,'H=',H 73 PRINT ,'Y3=',Y3 74 PRINT ,'NJ)]TJ /F11 9.9626 Tf 7.943 0 Td[(IJ=',NJ)]TJ /F11 9.9626 Tf 7.942 0 Td[(IJ 75 76 c THIS IS SPACING FOR INNER LAYER 77 HH=(Y2)/(IJ)]TJ /F11 9.9626 Tf 6.559 0 Td[(KJ) 78 PRINT ,'HH=',HH 79 PRINT ,'Y2=',Y2 80 PRINT ,'IJ)]TJ /F11 9.9626 Tf 6.559 0 Td[(KJ=',IJ)]TJ /F11 9.9626 Tf 6.559 0 Td[(KJ 81 82 c THIS IS SPACING FOR REACTION LAYER 83 HHH=(Y1)/(KJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1) 84 PRINT ,'Y1=',Y1 85 PRINT ,'KJ)]TJ /F11 9.9626 Tf 7.112 0 Td[(1=',KJ)]TJ /F11 9.9626 Tf 7.491 0 Td[(1 86 PRINT ,'HHH=',HHH 87 88 89 OPEN (15, FILE ='cdhgox ssvalues out.txt') 90 CLOSE (15, STATUS ='DELETE') 91 OPEN (15, FILE ='cdhgox ssvalues out.txt') 92 337 FORMAT (I2/I7/I7/I7/E25.15/E25.15/E25.15/E15.8/E15.8/E15.4/E15.4 93 1/E15.4/E15.4/E25.15) 94 WRITE (15,337)N,NJ,IJ,KJ,H,HH,HHH,DIFF(6),AKB,BB,V,POR1 95 96 C Create flux of the reacting species constants 97 FLUX=)]TJ /F11 9.9626 Tf 5.777 0 Td[(AKB exp (BBV)/F/2. 98 PRINT ,'FLUX=',FLUX 99 100 c THIS IS THE MAIN PART OF THE PROGRAM 101 DO 21J=1,NJ 102 RXN(1,J)=0.00001 103 RXN(2,J)=0.00001 104 RXN(3,J)=0.00001 105 RXN(4,J)=0.00001 106 DO 21I=1,N)]TJ /F11 9.9626 Tf 7.49 0 Td[(4 107 C(I,J)=0.0 108 21CONC(I,J)=CBULK(I) 109 JCOUNT=0 110 TOL=1.E)]TJ /F11 9.9626 Tf 7.992 0 Td[(10NNJ/1.E10 111 PRINT ,'TOL=',TOL 112 22JCOUNT=JCOUNT+1 113 AMP=0.0 207

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114 J=0 115 DO 23I=1,N 116 DO 23K=1,N 117 Y(I,K)=0.0 118 23X(I,K)=0.0 119 24J=J+1 120 DO 25I=1,N 121 G(I)=0.0 122 DO 25K=1,N 123 A(I,K)=0.0 124 B(I,K)=0.0 125 25D(I,K)=0.0 126 127 IF (J. EQ .1) CALL BC1(J) 128 IF (J. GT .1. AND .J. LT .KJ) CALL REACTION(J) 129 IF (J. EQ .KJ) CALL COUPLER1(J) 130 IF (J. GT .KJ. AND .J. LT .IJ) CALL INNER(J) 131 IF (J. EQ .IJ) CALL COUPLER2(J) 132 IF (J. GT .IJ. AND .J. LT .NJ) CALL OUTER(J) 133 IF (J. EQ .NJ) CALL BCNJ(J) 134 CALL BAND(J) 135 136 AMP=AMP+DABS(G(1))+DABS(G(2))+DABS(G(3))+DABS(G(4))+DABS(G(5)) 137 1+DABS(G(6))+DABS(G(7))+DABS(G(8))+DABS(G(9))+DABS(G(10)) 138 2+DABS(G(11))+DABS(G(12)) 139 140 IF (J. LT .NJ) GO TO 24 141 142 PRINT ,'ERROR=',AMP 143 144 DO 16K=1,NJ 145 RXN(1,K)=RXN(1,K)+C(9,K) 146 RXN(2,K)=RXN(2,K)+C(10,K) 147 RXN(3,K)=RXN(3,K)+C(11,K) 148 RXN(4,K)=RXN(4,K)+C(12,K) 149 DO 16I=1,N)]TJ /F11 9.9626 Tf 7.49 0 Td[(4 150 IF (C(I,K). LT .)]TJ /F11 9.9626 Tf 8.712 0 Td[(0.999CONC(I,K))C(I,K)=)]TJ /F11 9.9626 Tf 8.214 0 Td[(0.999CONC(I,K) 151 IF (C(I,K). GT .999.CONC(I,K))C(I,K)=999.CONC(I,K) 152 CONC(I,K)=CONC(I,K)+C(I,K) 153 16 CONTINUE 154 155 156 WRITE (12,302)(JCOUNT) 157 158 c If the error is less then the tolerance finish program 159 IF (DABS(AMP). LT .DABS(TOL)) GO TO 15 160 161 c If the error is greater then tolerance do another iteration 162 33 IF (JCOUNT. LE .19) GO TO 22 163 print 102 164 165 15 PRINT 103,AMP,(REF(I),CONC(I,1),CONC(I,NJ),I=1,N)]TJ /F11 9.9626 Tf 8.08 0 Td[(4) 166 167 PRINT ,'JCOUNT=',JCOUNT 168 169 WRITE (13,334)(CONC(1,J),CONC(2,J),CONC(3,J),CONC(4,J),CONC(5,J), 170 1CONC(6,J),CONC(7,J),CONC(8,J),RXN(1,J),RXN(2,J),RXN(3,J), 171 2RXN(4,J),J=1,NJ) 208

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172 173 C WRITE (13,334) ( CONC (1, J ) CONC (2, J ) CONC (3, J ) CONC (4, J ) CONC (5, J ) 174 C 1 CONC (6, J ) CONC (7, J ) CONC (8, J ) J =1, NJ ) 175 176 END PROGRAM CONVDIFF 209

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CodeE.4.Steady-StateContinuousGlucoseMonitorfortheElectrodeBoundaryCondition 1 SUBROUTINE BC1(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 COMMON /BAB/A(12,12),B(12,12),C(12,80001),D(12,25),G(12),X(12,12) 4 1,Y(12,12) 5 COMMON /NSN/N,NJ 6 COMMON /VAR/CONC(8,80001),RXN(4,80001),DIFF(8),H,EBIG,HH,IJ 7 COMMON /VARR/HHH,KJ 8 COMMON /POR/POR1,POR2,PORGLU 9 COMMON /BCI/FLUX 10 COMMON /RTE/ratef1,equilib1,ratef2,ratef3,equilib3,ratef4 11 COMMON /BUL/CBULK(8),PARH2O2,PAR02,PARGLUCOSE,JCOUNT 12 13 301 FORMAT (5x,'J='I5,12E18.9) 14 15 C For Glucose being consumed only 16 G(1)=2.POR1DIFF(1)(CONC(1,J+1))]TJ /F11 9.9626 Tf 6.019 0 Td[(CONC(1,J))/HHH2. 17 2)]TJ /F11 9.9626 Tf 8.54 0 Td[((3.RXN(1,J)+RXN(1,J+1))/4. 18 B(1,1)=2.POR1DIFF(1)/HHH2. 19 D(1,1)=)]TJ /F11 9.9626 Tf 7.986 0 Td[(2.POR1DIFF(1)/HHH2. 20 B(1,9)=+0.75 21 D(1,9)=+0.25 22 23 BIG= ABS (2.POR1DIFF(1)(CONC(1,J+1))/HHH2.) 24 PRINT ,"BIG=",BIG 25 BIG2= ABS (2.POR1DIFF(1)()]TJ /F11 9.9626 Tf 6.874 0 Td[(CONC(1,J))/HHH2.) 26 PRINT ,"BIG2=",BIG2 27 IF (BIG2. GT .BIG)BIG=BIG2 28 PRINT ,"BIG new=",BIG 29 IF ( ABS ()]TJ /F11 9.9626 Tf 8.539 0 Td[(3.RXN(1,J)/4.). GT .BIG)BIG= ABS ()]TJ /F11 9.9626 Tf 8.54 0 Td[(3.RXN(1,J)/4.) 30 IF ( ABS ()]TJ /F11 9.9626 Tf 6.746 0 Td[(RXN(1,J+1)/4.). GT .BIG)BIG= ABS ()]TJ /F11 9.9626 Tf 6.745 0 Td[(RXN(1,J+1)/4.) 31 PRINT ,"G(1)=", ABS (G(1)) 32 PRINT ,"BIGEBIG=",BIGEBIG 33 IF ( ABS (G(1)). LT .BIGEBIG)G(1)=0 34 35 C For GOx enzyme 36 G(2)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(1,J)+RXN(4,J) 37 B(2,9)=+1. 38 B(2,12)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 39 40 IF ( ABS (RXN(1,J)). GT .BIG)BIG= ABS (RXN(1,J)) 41 IF ( ABS (RXN(4,J)). GT .BIG)BIG= ABS (RXN(4,J)) 42 IF ( ABS (G(2)). LT .BIGEBIG)G(2)=0 43 44 C For Gluconic Acid being produced only 45 G(3)=2.POR1DIFF(3)(CONC(3,J+1))]TJ /F11 9.9626 Tf 6.019 0 Td[(CONC(3,J))/HHH2. 46 2+(3.RXN(2,J)+RXN(2,J+1))/4. 47 B(3,3)=2.POR1DIFF(3)/HHH2. 48 D(3,3)=)]TJ /F11 9.9626 Tf 7.986 .001 Td[(2.POR1DIFF(3)/HHH2. 49 B(3,10)=)]TJ /F11 9.9626 Tf 8.08 0 Td[(0.75 50 D(3,10)=)]TJ /F11 9.9626 Tf 8.081 0 Td[(0.25 51 52 BIG= ABS (2.POR1DIFF(3)(CONC(3,J+1))/HHH2.) 53 BIG2= ABS (2.POR1DIFF(3)()]TJ /F11 9.9626 Tf 6.874 0 Td[(CONC(3,J))/HHH2.) 54 IF (BIG2. GT .BIG)BIG=BIG2 210

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55 IF ( ABS (3.RXN(2,J)/4.). GT .BIG)BIG= ABS (3.RXN(2,J)/4.) 56 IF ( ABS (RXN(2,J+1)/4.). GT .BIG)BIG= ABS (RXN(2,J+1)/4.) 57 IF ( ABS (G(3)). LT .BIGEBIG)G(3)=0 58 59 C For GOx2 enzyme 60 G(4)=CBULK(2)+CBULK(4)+CBULK(7)+CBULK(8))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(2,J))]TJ /F11 9.9626 Tf 6.019 0 Td[(CONC(4,J) 61 1)]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(7,J))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(8,J) 62 B(4,2)=+1. 63 B(4,4)=+1. 64 B(4,7)=+1. 65 B(4,8)=+1. 66 67 BIG= ABS (CBULK(2)) 68 IF ( ABS (CBULK(4)). GT .BIG)BIG= ABS (CBULK(4)) 69 IF ( ABS (CBULK(7)). GT .BIG)BIG= ABS (CBULK(7)) 70 IF ( ABS (CBULK(8)). GT .BIG)BIG= ABS (CBULK(8)) 71 IF ( ABS (CONC(2,J)). GT .BIG)BIG= ABS (CONC(2,J)) 72 IF ( ABS (CONC(4,J)). GT .BIG)BIG= ABS (CONC(4,J)) 73 IF ( ABS (CONC(7,J)). GT .BIG)BIG= ABS (CONC(7,J)) 74 IF ( ABS (CONC(8,J)). GT .BIG)BIG= ABS (CONC(8,J)) 75 IF ( ABS (G(4)). LT .BIGEBIG)G(4)=0 76 77 C For O2 being consumed only 78 G(5)=2.POR1DIFF(5)(CONC(5,J+1))]TJ /F11 9.9626 Tf 6.019 0 Td[(CONC(5,J))/HHH2. 79 1)]TJ /F11 9.9626 Tf 6.406 0 Td[(FLUXCONC(6,J)/(HHH/2.) 80 2)]TJ /F11 9.9626 Tf 8.539 0 Td[((3.RXN(3,J)+RXN(3,J+1))/4. 81 B(5,5)=2.POR1DIFF(5)/HHH2. 82 D(5,5)=)]TJ /F11 9.9626 Tf 7.986 0 Td[(2.POR1DIFF(5)/HHH2. 83 B(5,11)=+0.75 84 D(5,11)=+0.25 85 B(5,6)=+FLUX/(HHH/2.) 86 87 BIG= ABS (2.POR1DIFF(5)(CONC(5,J+1))/HHH2.) 88 BIG2= ABS (2.POR1DIFF(5)()]TJ /F11 9.9626 Tf 6.874 0 Td[(CONC(5,J))/HHH2.) 89 IF (BIG2. GT .BIG)BIG=BIG2 90 IF ( ABS ()]TJ /F11 9.9626 Tf 8.539 0 Td[(3.RXN(3,J)/4.). GT .BIG)BIG= ABS ()]TJ /F11 9.9626 Tf 8.54 0 Td[(3.RXN(3,J)/4.) 91 IF ( ABS ()]TJ /F11 9.9626 Tf 6.746 0 Td[(RXN(3,J+1)/4.). GT .BIG)BIG= ABS ()]TJ /F11 9.9626 Tf 6.745 0 Td[(RXN(3,J+1)/4.) 92 IF ( ABS (G(5)). LT .BIGEBIG)G(5)=0 93 94 C For H2O2 reacting species 95 G(6)=2.POR1DIFF(6)(CONC(6,J+1))]TJ /F11 9.9626 Tf 6.019 0 Td[(CONC(6,J))/HHH2. 96 2+FLUXCONC(6,J)/(HHH/2.) 97 3+(3.RXN(4,J)+RXN(4,J+1))/4. 98 B(6,6)=2.POR1DIFF(6)/HHH2.)]TJ /F11 9.9626 Tf 7.628 -.001 Td[(FLUX/(HHH/2.) 99 D(6,6)=)]TJ /F11 9.9626 Tf 7.986 0 Td[(2.POR1DIFF(6)/HHH2. 100 B(6,12)=)]TJ /F11 9.9626 Tf 8.08 0 Td[(0.75 101 D(6,12)=)]TJ /F11 9.9626 Tf 8.081 0 Td[(0.25 102 103 BIG= ABS (2.POR1DIFF(6)(CONC(6,J+1))/HHH2.) 104 BIG2= ABS (2.POR1DIFF(6)()]TJ /F11 9.9626 Tf 6.874 0 Td[(CONC(6,J))/HHH2.) 105 IF (BIG2. GT .BIG)BIG=BIG2 106 IF ( ABS (3.RXN(4,J)/4.). GT .BIG)BIG= ABS (3.RXN(4,J)/4.) 107 IF ( ABS (RXN(4,J+1)/4.). GT .BIG)BIG= ABS (RXN(4,J+1)/4.) 108 IF ( ABS (G(6)). LT .BIGEBIG)G(6)=0 109 110 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.393 -618.12 Td[(GOx2 enzyme 111 G(7)=RXN(1,J))]TJ /F11 9.9626 Tf 6.072 0 Td[(RXN(2,J) 112 B(7,9)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 211

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113 B(7,10)=1. 114 115 IF ( ABS (RXN(1,J)). GT .BIG)BIG= ABS (RXN(1,J)) 116 IF ( ABS (RXN(2,J)). GT .BIG)BIG= ABS (RXN(2,J)) 117 IF ( ABS (G(7)). LT .BIGEBIG)G(7)=0 118 119 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.131 -78.082 Td[(GOx enzyme 120 G(8)=RXN(3,J))]TJ /F11 9.9626 Tf 6.072 0 Td[(RXN(4,J) 121 B(8,11)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 122 B(8,12)=1. 123 124 IF ( ABS (RXN(3,J)). GT .BIG)BIG= ABS (RXN(3,J)) 125 IF ( ABS (RXN(4,J)). GT .BIG)BIG= ABS (RXN(4,J)) 126 IF ( ABS (G(8)). LT .BIGEBIG)G(8)=0 127 128 C REACTION1 129 214G(9)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(1,J)+ratef1(CONC(1,J)CONC(2,J))]TJ /F11 9.9626 Tf 7.832 0 Td[((CONC(7,J)/equilib1)) 130 B(9,1)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef1CONC(2,J) 131 B(9,2)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef1CONC(1,J) 132 B(9,7)=ratef1/equilib1 133 B(9,9)=+1. 134 135 BIG= ABS (RXN(1,J)) 136 BIG2= ABS (ratef1CONC(1,J)CONC(2,J)) 137 IF (BIG2. GT .BIG)BIG=BIG2 138 BIG3= ABS (ratef1(CONC(7,J)/equilib1)) 139 IF (BIG3. GT .BIG)BIG=BIG3 140 IF ( ABS (G(9)). LT .BIGEBIG)G(9)=0 141 142 C REACTION2 143 215G(10)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(2,J)+ratef2CONC(7,J) 144 B(10,7)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef2 145 B(10,10)=+1. 146 147 BIG= ABS (RXN(2,J)) 148 BIG2= ABS (ratef2CONC(7,J)) 149 IF (BIG2. GT .BIG)BIG=BIG2 150 IF ( ABS (G(10)). LT .BIGEBIG)G(10)=0 151 152 C REACTION3 153 216G(11)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(3,J)+ratef3(CONC(4,J)CONC(5,J))]TJ /F11 9.9626 Tf 7.831 0 Td[((CONC(8,J)/equilib3)) 154 B(11,4)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONC(5,J) 155 B(11,5)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONC(4,J) 156 B(11,8)=ratef3/equilib3 157 B(11,11)=+1. 158 159 BIG= ABS (RXN(3,J)) 160 BIG2= ABS (ratef3CONC(4,J)CONC(5,J)) 161 IF (BIG2. GT .BIG)BIG=BIG2 162 BIG3= ABS (ratef3(CONC(8,J)/equilib3)) 163 IF (BIG3. GT .BIG)BIG=BIG3 164 IF ( ABS (G(11)). LT .BIGEBIG)G(11)=0 165 166 C REACTION4 167 217G(12)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(4,J)+ratef4CONC(8,J) 168 B(12,8)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef4 169 B(12,12)=+1. 170 212

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171 BIG= ABS (RXN(4,J)) 172 BIG2= ABS (ratef4CONC(8,J)) 173 IF (BIG2. GT .BIG)BIG=BIG2 174 IF ( ABS (G(12)). LT .BIGEBIG)G(12)=0 175 176 212 WRITE (12,301)J,(G(K),K=1,N) 177 178 RETURN 179 END 213

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CodeE.5.Steady-StateContinuousGlucoseMonitorSubroutinefortheReactionRegion 1 SUBROUTINE REACTION(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 COMMON /BAB/A(12,12),B(12,12),C(12,80001),D(12,25),G(12),X(12,12) 4 1,Y(12,12) 5 COMMON /NSN/N,NJ 6 COMMON /VAR/CONC(8,80001),RXN(4,80001),DIFF(8),H,EBIG,HH,IJ 7 COMMON /VARR/HHH,KJ 8 COMMON /POR/POR1,POR2,PORGLU 9 COMMON /RTE/ratef1,equilib1,ratef2,ratef3,equilib3,ratef4 10 COMMON /BUL/CBULK(8),PARH2O2,PAR02,PARGLUCOSE,JCOUNT 11 12 301 FORMAT (5x,'J='I5,12E18.9) 13 14 C For Glucose being consumed only 15 G(1)=POR1DIFF(1)(CONC(1,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.CONC(1,J)+CONC(1,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/HHH2. 16 2)]TJ /F11 9.9626 Tf 6.072 0 Td[(RXN(1,J) 17 B(1,1)=2.POR1DIFF(1)/HHH2. 18 D(1,1)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(1)/HHH2. 19 A(1,1)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(1)/HHH2. 20 B(1,9)=+1. 21 22 BIG= ABS (POR1DIFF(1)(CONC(1,J+1))/HHH2.) 23 BIG2= ABS (POR1DIFF(1)()]TJ /F11 9.9626 Tf 8.565 0 Td[(2.CONC(1,J))/HHH2.) 24 IF (BIG2. GT .BIG)BIG=BIG2 25 BIG3= ABS (POR1DIFF(1)(CONC(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HHH2.) 26 IF (BIG3. GT .BIG)BIG=BIG3 27 IF ( ABS ()]TJ /F11 9.9626 Tf 6.746 0 Td[(RXN(1,J)). GT .BIG)BIG= ABS ()]TJ /F11 9.9626 Tf 6.746 0 Td[(RXN(1,J)) 28 IF ( ABS (G(1)). LT .BIGEBIG)G(1)=0 29 30 C For GOx enzyme 31 G(2)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(1,J)+RXN(4,J) 32 B(2,9)=+1. 33 B(2,12)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 34 35 IF ( ABS (RXN(1,J)). GT .BIG)BIG= ABS (RXN(1,J)) 36 IF ( ABS (RXN(4,J)). GT .BIG)BIG= ABS (RXN(4,J)) 37 IF ( ABS (G(2)). LT .BIGEBIG)G(2)=0 38 39 C For Gluconic Acid being produced only 40 G(3)=POR1DIFF(3)(CONC(3,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.CONC(3,J)+CONC(3,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/HHH2. 41 2+RXN(2,J) 42 B(3,3)=2.POR1DIFF(3)/HHH2. 43 D(3,3)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(3)/HHH2. 44 A(3,3)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(3)/HHH2. 45 B(3,10)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 46 47 BIG= ABS (POR1DIFF(3)(CONC(3,J+1))/HHH2.) 48 BIG2= ABS (POR1DIFF(3)()]TJ /F11 9.9626 Tf 8.565 0 Td[(2.CONC(3,J))/HHH2.) 49 IF (BIG2. GT .BIG)BIG=BIG2 50 BIG3= ABS (POR1DIFF(3)(CONC(3,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HHH2.) 51 IF (BIG3. GT .BIG)BIG=BIG3 52 IF ( ABS (RXN(2,J)). GT .BIG)BIG= ABS (RXN(2,J)) 53 IF ( ABS (G(3)). LT .BIGEBIG)G(3)=0 54 55 C For GOx2 enzyme 56 G(4)=CBULK(2)+CBULK(4)+CBULK(7)+CBULK(8))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(2,J))]TJ /F11 9.9626 Tf 6.019 0 Td[(CONC(4,J) 214

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57 1)]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(7,J))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(8,J) 58 B(4,2)=+1. 59 B(4,4)=+1. 60 B(4,7)=+1. 61 B(4,8)=+1. 62 63 BIG= ABS (CBULK(2)) 64 IF ( ABS (CBULK(4)). GT .BIG)BIG= ABS (CBULK(4)) 65 IF ( ABS (CBULK(7)). GT .BIG)BIG= ABS (CBULK(7)) 66 IF ( ABS (CBULK(8)). GT .BIG)BIG= ABS (CBULK(8)) 67 IF ( ABS (CONC(2,J)). GT .BIG)BIG= ABS (CONC(2,J)) 68 IF ( ABS (CONC(4,J)). GT .BIG)BIG= ABS (CONC(4,J)) 69 IF ( ABS (CONC(7,J)). GT .BIG)BIG= ABS (CONC(7,J)) 70 IF ( ABS (CONC(8,J)). GT .BIG)BIG= ABS (CONC(8,J)) 71 IF ( ABS (G(4)). LT .BIGEBIG)G(4)=0 72 73 C For O2 being consumed only 74 G(5)=POR1DIFF(5)(CONC(5,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.CONC(5,J)+CONC(5,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/HHH2. 75 2)]TJ /F11 9.9626 Tf 6.072 0 Td[(RXN(3,J) 76 B(5,5)=2.POR1DIFF(5)/HHH2. 77 D(5,5)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(5)/HHH2. 78 A(5,5)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(5)/HHH2. 79 B(5,11)=+1. 80 81 BIG= ABS (POR1DIFF(5)(CONC(5,J+1))/HHH2.) 82 BIG2= ABS (POR1DIFF(5)()]TJ /F11 9.9626 Tf 8.565 0 Td[(2.CONC(5,J))/HHH2.) 83 IF (BIG2. GT .BIG)BIG=BIG2 84 BIG3= ABS (POR1DIFF(5)(CONC(5,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HHH2.) 85 IF (BIG3. GT .BIG)BIG=BIG3 86 IF ( ABS ()]TJ /F11 9.9626 Tf 6.746 0 Td[(RXN(3,J)). GT .BIG)BIG= ABS ()]TJ /F11 9.9626 Tf 6.746 0 Td[(RXN(3,J)) 87 IF ( ABS (G(5)). LT .BIGEBIG)G(5)=0 88 89 C For H2O2 reacting species 90 G(6)=POR1DIFF(6)(CONC(6,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.CONC(6,J)+CONC(6,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/HHH2. 91 2+RXN(4,J) 92 B(6,6)=2.POR1DIFF(6)/HHH2. 93 D(6,6)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(6)/HHH2. 94 A(6,6)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(6)/HHH2. 95 B(6,12)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 96 97 BIG= ABS (POR1DIFF(6)(CONC(6,J+1))/HHH2.) 98 BIG2= ABS (POR1DIFF(6)()]TJ /F11 9.9626 Tf 8.565 0 Td[(2.CONC(6,J))/HHH2.) 99 IF (BIG2. GT .BIG)BIG=BIG2 100 BIG3= ABS (POR1DIFF(6)(CONC(6,J)]TJ /F11 9.9626 Tf 8.081 -.001 Td[(1))/HHH2.) 101 IF (BIG3. GT .BIG)BIG=BIG3 102 IF ( ABS (RXN(4,J)). GT .BIG)BIG= ABS (RXN(4,J)) 103 IF ( ABS (G(6)). LT .BIGEBIG)G(6)=0 104 105 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.393 -540.971 Td[(GOx2 enzyme 106 G(7)=RXN(1,J))]TJ /F11 9.9626 Tf 6.072 0 Td[(RXN(2,J) 107 B(7,9)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 108 B(7,10)=1. 109 110 IF ( ABS (RXN(1,J)). GT .BIG)BIG= ABS (RXN(1,J)) 111 IF ( ABS (RXN(2,J)). GT .BIG)BIG= ABS (RXN(2,J)) 112 IF ( ABS (G(7)). LT .BIGEBIG)G(7)=0 113 114 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.131 -640.162 Td[(GOx enzyme 215

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115 G(8)=RXN(3,J))]TJ /F11 9.9626 Tf 6.072 0 Td[(RXN(4,J) 116 B(8,11)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 117 B(8,12)=1. 118 119 IF ( ABS (RXN(3,J)). GT .BIG)BIG= ABS (RXN(3,J)) 120 IF ( ABS (RXN(4,J)). GT .BIG)BIG= ABS (RXN(4,J)) 121 IF ( ABS (G(8)). LT .BIGEBIG)G(8)=0 122 123 124 C REACTION1 125 214G(9)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(1,J)+ratef1(CONC(1,J)CONC(2,J))]TJ /F11 9.9626 Tf 7.832 0 Td[((CONC(7,J)/equilib1)) 126 B(9,1)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef1CONC(2,J) 127 B(9,2)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef1CONC(1,J) 128 B(9,7)=ratef1/equilib1 129 B(9,9)=+1. 130 131 BIG= ABS (RXN(1,J)) 132 BIG2= ABS (ratef1CONC(1,J)CONC(2,J)) 133 IF (BIG2. GT .BIG)BIG=BIG2 134 BIG3= ABS (ratef1(CONC(7,J)/equilib1)) 135 IF (BIG3. GT .BIG)BIG=BIG3 136 IF ( ABS (G(9)). LT .BIGEBIG)G(9)=0 137 138 C REACTION2 139 215G(10)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(2,J)+ratef2CONC(7,J) 140 B(10,7)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef2 141 B(10,10)=+1. 142 143 BIG= ABS (RXN(2,J)) 144 BIG2= ABS (ratef2CONC(7,J)) 145 IF (BIG2. GT .BIG)BIG=BIG2 146 IF ( ABS (G(10)). LT .BIGEBIG)G(10)=0 147 148 C REACTION3 149 216G(11)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(3,J)+ratef3(CONC(4,J)CONC(5,J))]TJ /F11 9.9626 Tf 7.831 0 Td[((CONC(8,J)/equilib3)) 150 B(11,4)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONC(5,J) 151 B(11,5)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONC(4,J) 152 B(11,8)=ratef3/equilib3 153 B(11,11)=+1. 154 155 BIG= ABS (RXN(3,J)) 156 BIG2= ABS (ratef3CONC(4,J)CONC(5,J)) 157 IF (BIG2. GT .BIG)BIG=BIG2 158 BIG3= ABS (ratef3(CONC(8,J)/equilib3)) 159 IF (BIG3. GT .BIG)BIG=BIG3 160 IF ( ABS (G(11)). LT .BIGEBIG)G(11)=0 161 162 C REACTION4 163 217G(12)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(4,J)+ratef4CONC(8,J) 164 B(12,8)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef4 165 B(12,12)=+1. 166 167 BIG= ABS (RXN(4,J)) 168 BIG2= ABS (ratef4CONC(8,J)) 169 IF (BIG2. GT .BIG)BIG=BIG2 170 IF ( ABS (G(12)). LT .BIGEBIG)G(12)=0 171 172 c SAVE G OUT DATA 216

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173 212 DO 11I=2,13 174 11 If (I. EQ .J) WRITE (12,301)J,(G(K),K=1,N) 175 IF (J. EQ .KJ/2) THEN 176 WRITE (12,301)J,(G(K),K=1,N) 177 ELSE IF (J. EQ .(KJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) THEN 178 WRITE (12,301)J,(G(K),K=1,N) 179 ELSE IF (J. EQ .(KJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(2)) THEN 180 WRITE (12,301)J,(G(K),K=1,N) 181 ELSE IF (J. EQ .(KJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(3)) THEN 182 WRITE (12,301)J,(G(K),K=1,N) 183 END IF 184 185 RETURN 186 END 217

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CodeE.6.Steady-StateContinuousGlucoseMonitorSubroutinefortheFirstCoupler 1 SUBROUTINE COUPLER1(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 COMMON /BAB/A(12,12),B(12,12),C(12,80001),D(12,25),G(12),X(12,12) 4 1,Y(12,12) 5 COMMON /NSN/N,NJ 6 COMMON /VAR/CONC(8,80001),RXN(4,80001),DIFF(8),H,EBIG,HH,IJ 7 COMMON /VARR/HHH,KJ 8 COMMON /POR/POR1,POR2,PORGLU 9 COMMON /RTE/ratef1,equilib1,ratef2,ratef3,equilib3,ratef4 10 COMMON /BUL/CBULK(8),PARH2O2,PAR02,PARGLUCOSE,JCOUNT 11 12 C DIMENSION COEFF1 COEFF3 COEFF5 COEFF6 13 14 301 FORMAT (5x,'J='I5,12E18.9) 15 16 COEFF1HH=POR1DIFF(1)/(HH) 17 COEFF1HHH=POR1DIFF(1)/(HHH) 18 COEFF3HH=POR1DIFF(3)/(HH) 19 COEFF3HHH=POR1DIFF(3)/(HHH) 20 COEFF5HH=POR1DIFF(5)/(HH) 21 COEFF5HHH=POR1DIFF(5)/(HHH) 22 COEFF6HH=POR1DIFF(6)/(HH) 23 COEFF6HHH=POR1DIFF(6)/(HHH) 24 25 C For Glucose being consumed only 26 G(1)=COEFF1HH(CONC(1,J+1))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(1,J)) 27 1)]TJ /F11 9.9626 Tf 6.325 0 Td[(COEFF1HHH(CONC(1,J))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 28 2)]TJ /F11 9.9626 Tf 7.831 0 Td[((HH/2.)(RXN(1,J+1)+3.RXN(1,J))/4. 29 3)]TJ /F11 9.9626 Tf 7.831 0 Td[((HHH/2.)(RXN(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.RXN(1,J))/4. 30 B(1,1)=COEFF1HH+COEFF1HHH 31 D(1,1)=)]TJ /F11 9.9626 Tf 6.104 0 Td[(COEFF1HH 32 A(1,1)=)]TJ /F11 9.9626 Tf 6.03 0 Td[(COEFF1HHH 33 B(1,9)=+(HH/2.)(3./4.)+(HHH/2.)(3./4.) 34 D(1,9)=+(HH/2.)(1./4.) 35 A(1,9)=+(HHH/2.)(1./4.) 36 37 BIG= ABS (COEFF1HHCONC(1,J+1)) 38 BIG2= ABS (COEFF1HHCONC(1,J)) 39 IF (BIG2. GT .BIG)BIG=BIG2 40 BIG3= ABS ()]TJ /F11 9.9626 Tf 6.998 0 Td[(COEFF1HHHCONC(1,J)) 41 IF (BIG3. GT .BIG)BIG=BIG3 42 BIG4= ABS ()]TJ /F11 9.9626 Tf 6.998 0 Td[(COEFF1HHHCONC(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 43 IF (BIG4. GT .BIG)BIG=BIG4 44 BIG5= ABS ((HH/2.)(RXN(1,J+1)/4.)) 45 IF (BIG5. GT .BIG)BIG=BIG5 46 BIG6= ABS ((HH/2.)(3.RXN(1,J))/4.) 47 IF (BIG6. GT .BIG)BIG=BIG6 48 BIG7= ABS ((HHH/2.)(RXN(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)/4.)) 49 IF (BIG7. GT .BIG)BIG=BIG7 50 BIG8= ABS ((HHH/2.)(3.RXN(1,J))/4.) 51 IF (BIG8. GT .BIG)BIG=BIG8 52 IF ( ABS (G(1)). LT .BIGEBIG)G(1)=0 53 54 C For GOx enzyme 55 G(2)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(1,J)+RXN(4,J) 56 B(2,9)=+1. 218

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57 B(2,12)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 58 59 BIG= ABS (RXN(1,J)) 60 BIG2= ABS (RXN(4,J)) 61 IF (BIG2. GT .BIG)BIG=BIG2 62 IF ( ABS (G(2)). LT .BIGEBIG)G(2)=0 63 64 C For Gluconic Acid being produced only 65 G(3)=COEFF3HH(CONC(3,J+1))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(3,J)) 66 1)]TJ /F11 9.9626 Tf 6.325 0 Td[(COEFF3HHH(CONC(3,J))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(3,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 67 2+(HH/2.)(RXN(2,J+1)+3.RXN(2,J))/4. 68 3+(HHH/2.)(RXN(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.RXN(2,J))/4. 69 B(3,3)=COEFF3HH+COEFF3HHH 70 D(3,3)=)]TJ /F11 9.9626 Tf 6.104 0 Td[(COEFF3HH 71 A(3,3)=)]TJ /F11 9.9626 Tf 6.03 0 Td[(COEFF3HHH 72 B(3,10)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(3./4.))]TJ /F11 9.9626 Tf 7.832 0 Td[((HHH/2.)(3./4.) 73 D(3,10)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(1./4.) 74 A(3,10)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HHH/2.)(1./4.) 75 76 BIG= ABS (COEFF3HHCONC(3,J+1)) 77 BIG2= ABS (COEFF3HHCONC(3,J)) 78 IF (BIG2. GT .BIG)BIG=BIG2 79 BIG3= ABS ()]TJ /F11 9.9626 Tf 6.998 0 Td[(COEFF3HHHCONC(3,J)) 80 IF (BIG3. GT .BIG)BIG=BIG3 81 BIG4= ABS ()]TJ /F11 9.9626 Tf 6.998 0 Td[(COEFF3HHHCONC(3,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 82 IF (BIG4. GT .BIG)BIG=BIG4 83 BIG5= ABS ((HH/2.)(RXN(2,J+1)/4.)) 84 IF (BIG5. GT .BIG)BIG=BIG5 85 BIG6= ABS ((HH/2.)(3.RXN(2,J))/4.) 86 IF (BIG6. GT .BIG)BIG=BIG6 87 BIG7= ABS ((HHH/2.)(RXN(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)/4.)) 88 IF (BIG7. GT .BIG)BIG=BIG7 89 BIG8= ABS ((HHH/2.)(3.RXN(2,J))/4.) 90 IF (BIG8. GT .BIG)BIG=BIG8 91 IF ( ABS (G(3)). LT .BIGEBIG)G(3)=0 92 93 C For GOx2 enzyme 94 G(4)=CBULK(2)+CBULK(4)+CBULK(7)+CBULK(8))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(2,J))]TJ /F11 9.9626 Tf 6.019 0 Td[(CONC(4,J) 95 1)]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(7,J))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(8,J) 96 B(4,2)=+1. 97 B(4,4)=+1. 98 B(4,7)=+1. 99 B(4,8)=+1. 100 101 BIG= ABS (CBULK(2)) 102 IF ( ABS (CBULK(4)). GT .BIG)BIG= ABS (CBULK(4)) 103 IF ( ABS (CBULK(7)). GT .BIG)BIG= ABS (CBULK(7)) 104 IF ( ABS (CBULK(8)). GT .BIG)BIG= ABS (CBULK(8)) 105 IF ( ABS (CONC(2,J)). GT .BIG)BIG= ABS (CONC(2,J)) 106 IF ( ABS (CONC(4,J)). GT .BIG)BIG= ABS (CONC(4,J)) 107 IF ( ABS (CONC(7,J)). GT .BIG)BIG= ABS (CONC(7,J)) 108 IF ( ABS (CONC(8,J)). GT .BIG)BIG= ABS (CONC(8,J)) 109 IF ( ABS (G(4)). LT .BIGEBIG)G(4)=0 110 111 C For O2 being consumed only 112 G(5)=COEFF5HH(CONC(5,J+1))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(5,J)) 113 1)]TJ /F11 9.9626 Tf 6.325 0 Td[(COEFF5HHH(CONC(5,J))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(5,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 114 2)]TJ /F11 9.9626 Tf 7.831 0 Td[((HH/2.)(RXN(3,J+1)+3.RXN(3,J))/4. 219

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115 3)]TJ /F11 9.9626 Tf 7.831 0 Td[((HHH/2.)(RXN(3,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.RXN(3,J))/4. 116 B(5,5)=COEFF5HH+COEFF5HHH 117 D(5,5)=)]TJ /F11 9.9626 Tf 6.104 0 Td[(COEFF5HH 118 A(5,5)=)]TJ /F11 9.9626 Tf 6.03 0 Td[(COEFF5HHH 119 B(5,11)=+(HH/2.)(3./4.)+(HHH/2.)(3./4.) 120 D(5,11)=+(HH/2.)(1./4.) 121 A(5,11)=+(HHH/2.)(1./4.) 122 123 BIG= ABS (COEFF5HHCONC(5,J+1)) 124 BIG2= ABS (COEFF5HHCONC(5,J)) 125 IF (BIG2. GT .BIG)BIG=BIG2 126 BIG3= ABS ()]TJ /F11 9.9626 Tf 6.998 0 Td[(COEFF5HHHCONC(5,J)) 127 IF (BIG3. GT .BIG)BIG=BIG3 128 BIG4= ABS ()]TJ /F11 9.9626 Tf 6.998 0 Td[(COEFF5HHHCONC(5,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 129 IF (BIG4. GT .BIG)BIG=BIG4 130 BIG5= ABS ((HH/2.)(RXN(3,J+1)/4.)) 131 IF (BIG5. GT .BIG)BIG=BIG5 132 BIG6= ABS ((HH/2.)(3.RXN(3,J))/4.) 133 IF (BIG6. GT .BIG)BIG=BIG6 134 BIG7= ABS ((HHH/2.)(RXN(3,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)/4.)) 135 IF (BIG7. GT .BIG)BIG=BIG7 136 BIG8= ABS ((HHH/2.)(3.RXN(3,J))/4.) 137 IF (BIG8. GT .BIG)BIG=BIG8 138 IF ( ABS (G(5)). LT .BIGEBIG)G(5)=0 139 140 C For H2O2 reacting species 141 G(6)=COEFF6HH(CONC(6,J+1))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(6,J)) 142 1)]TJ /F11 9.9626 Tf 6.325 0 Td[(COEFF6HHH(CONC(6,J))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(6,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 143 2+(HH/2.)(RXN(4,J+1)+3.RXN(4,J))/4. 144 3+(HHH/2.)(RXN(4,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.RXN(4,J))/4. 145 B(6,6)=COEFF6HH+COEFF6HHH 146 D(6,6)=)]TJ /F11 9.9626 Tf 6.104 0 Td[(COEFF6HH 147 A(6,6)=)]TJ /F11 9.9626 Tf 6.03 0 Td[(COEFF6HHH 148 B(6,12)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(3./4.))]TJ /F11 9.9626 Tf 7.832 0 Td[((HHH/2.)(3./4.) 149 D(6,12)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(1./4.) 150 A(6,12)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HHH/2.)(1./4.) 151 152 BIG= ABS (COEFF6HHCONC(6,J+1)) 153 BIG2= ABS (COEFF16HHCONC(6,J)) 154 IF (BIG2. GT .BIG)BIG=BIG2 155 BIG3= ABS ()]TJ /F11 9.9626 Tf 6.998 0 Td[(COEFF6HHHCONC(6,J)) 156 IF (BIG3. GT .BIG)BIG=BIG3 157 BIG4= ABS ()]TJ /F11 9.9626 Tf 6.998 0 Td[(COEFF6HHHCONC(6,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 158 IF (BIG4. GT .BIG)BIG=BIG4 159 BIG5= ABS ((HH/2.)(RXN(4,J+1)/4.)) 160 IF (BIG5. GT .BIG)BIG=BIG5 161 BIG6= ABS ((HH/2.)(3.RXN(4,J))/4.) 162 IF (BIG6. GT .BIG)BIG=BIG6 163 BIG7= ABS ((HHH/2.)(RXN(4,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)/4.)) 164 IF (BIG7. GT .BIG)BIG=BIG7 165 BIG8= ABS ((HHH/2.)(3.RXN(4,J))/4.) 166 IF (BIG8. GT .BIG)BIG=BIG8 167 IF ( ABS (G(6)). LT .BIGEBIG)G(6)=0 168 169 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.393 -607.098 Td[(GOx2 enzyme 170 G(7)=RXN(1,J))]TJ /F11 9.9626 Tf 6.072 0 Td[(RXN(2,J) 171 B(7,9)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 172 B(7,10)=1. 220

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173 174 IF ( ABS (RXN(1,J)). GT .BIG)BIG= ABS (RXN(1,J)) 175 IF ( ABS (RXN(2,J)). GT .BIG)BIG= ABS (RXN(2,J)) 176 IF ( ABS (G(7)). LT .BIGEBIG)G(7)=0 177 178 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.131 -67.061 Td[(GOx enzyme 179 G(8)=RXN(3,J))]TJ /F11 9.9626 Tf 6.072 0 Td[(RXN(4,J) 180 B(8,11)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 181 B(8,12)=1. 182 183 IF ( ABS (RXN(3,J)). GT .BIG)BIG= ABS (RXN(3,J)) 184 IF ( ABS (RXN(4,J)). GT .BIG)BIG= ABS (RXN(4,J)) 185 IF ( ABS (G(8)). LT .BIGEBIG)G(8)=0 186 187 188 C REACTION1 189 214G(9)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(1,J)+ratef1(CONC(1,J)CONC(2,J))]TJ /F11 9.9626 Tf 7.832 0 Td[((CONC(7,J)/equilib1)) 190 B(9,1)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef1CONC(2,J) 191 B(9,2)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef1CONC(1,J) 192 B(9,7)=ratef1/equilib1 193 B(9,9)=+1. 194 195 BIG= ABS (RXN(1,J)) 196 BIG2= ABS (ratef1CONC(1,J)CONC(2,J)) 197 IF (BIG2. GT .BIG)BIG=BIG2 198 BIG3= ABS (ratef1(CONC(7,J)/equilib1)) 199 IF (BIG3. GT .BIG)BIG=BIG3 200 IF ( ABS (G(9)). LT .BIGEBIG)G(9)=0 201 202 C REACTION2 203 215G(10)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(2,J)+ratef2CONC(7,J) 204 B(10,7)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef2 205 B(10,10)=+1. 206 207 BIG= ABS (RXN(2,J)) 208 BIG2= ABS (ratef2CONC(7,J)) 209 IF (BIG2. GT .BIG)BIG=BIG2 210 IF ( ABS (G(10)). LT .BIGEBIG)G(10)=0 211 212 C REACTION3 213 216G(11)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(3,J)+ratef3(CONC(4,J)CONC(5,J))]TJ /F11 9.9626 Tf 7.831 0 Td[((CONC(8,J)/equilib3)) 214 B(11,4)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONC(5,J) 215 B(11,5)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONC(4,J) 216 B(11,8)=ratef3/equilib3 217 B(11,11)=+1. 218 219 BIG= ABS (RXN(3,J)) 220 BIG2= ABS (ratef3CONC(4,J)CONC(5,J)) 221 IF (BIG2. GT .BIG)BIG=BIG2 222 BIG3= ABS (ratef3(CONC(8,J)/equilib3)) 223 IF (BIG3. GT .BIG)BIG=BIG3 224 IF ( ABS (G(11)). LT .BIGEBIG)G(11)=0 225 226 C REACTION4 227 217G(12)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(4,J)+ratef4CONC(8,J) 228 B(12,8)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef4 229 B(12,12)=+1. 230 221

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231 BIG= ABS (RXN(4,J)) 232 BIG2= ABS (ratef4CONC(8,J)) 233 IF (BIG2. GT .BIG)BIG=BIG2 234 IF ( ABS (G(12)). LT .BIGEBIG)G(12)=0 235 236 212 WRITE (12,301)J,(G(K),K=1,N) 237 RETURN 238 END 222

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CodeE.7.Steady-StateContinuousGlucoseMonitorSubroutinefortheInnerRegion 1 SUBROUTINE INNER(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 COMMON /BAB/A(12,12),B(12,12),C(12,80001),D(12,25),G(12),X(12,12) 4 1,Y(12,12) 5 COMMON /NSN/N,NJ 6 COMMON /VAR/CONC(8,80001),RXN(4,80001),DIFF(8),H,EBIG,HH,IJ 7 COMMON /POR/POR1,POR2,PORGLU 8 COMMON /RTE/ratef1,equilib1,ratef2,ratef3,equilib3,ratef4 9 COMMON /BUL/CBULK(8),PARH2O2,PAR02,PARGLUCOSE,JCOUNT 10 COMMON /VARR/HHH,KJ 11 12 301 FORMAT (5x,'J='I5,12E18.9) 13 14 C For Glucose being consumed only 15 G(1)=POR1DIFF(1)(CONC(1,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.CONC(1,J)+CONC(1,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/HH2. 16 2)]TJ /F11 9.9626 Tf 6.072 0 Td[(RXN(1,J) 17 B(1,1)=2.POR1DIFF(1)/HH2. 18 D(1,1)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(1)/HH2. 19 A(1,1)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(1)/HH2. 20 B(1,9)=+1. 21 22 BIG= ABS (POR1DIFF(1)(CONC(1,J+1))/HH2.) 23 BIG2= ABS (POR1DIFF(1)()]TJ /F11 9.9626 Tf 8.565 0 Td[(2.CONC(1,J))/HH2.) 24 IF (BIG2. GT .BIG)BIG=BIG2 25 BIG3= ABS (POR1DIFF(1)(CONC(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH2.) 26 IF (BIG3. GT .BIG)BIG=BIG3 27 IF ( ABS ()]TJ /F11 9.9626 Tf 6.746 0 Td[(RXN(1,J)). GT .BIG)BIG= ABS ()]TJ /F11 9.9626 Tf 6.746 0 Td[(RXN(1,J)) 28 IF ( ABS (G(1)). LT .BIGEBIG)G(1)=0 29 30 C For GOx enzyme 31 G(2)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(1,J)+RXN(4,J) 32 B(2,9)=+1. 33 B(2,12)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 34 35 BIG= ABS (RXN(1,J)) 36 BIG2= ABS (RXN(4,J)) 37 IF (BIG2. GT .BIG)BIG=BIG2 38 IF ( ABS (G(2)). LT .BIGEBIG)G(2)=0 39 40 C For Gluconic Acid being produced only 41 G(3)=POR1DIFF(3)(CONC(3,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.CONC(3,J)+CONC(3,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/HH2. 42 2+RXN(2,J) 43 B(3,3)=2.POR1DIFF(3)/HH2. 44 D(3,3)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(3)/HH2. 45 A(3,3)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(3)/HH2. 46 B(3,10)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 47 48 BIG= ABS (POR1DIFF(3)(CONC(3,J+1))/HH2.) 49 BIG2= ABS (POR1DIFF(3)()]TJ /F11 9.9626 Tf 8.565 0 Td[(2.CONC(3,J))/HH2.) 50 IF (BIG2. GT .BIG)BIG=BIG2 51 BIG3= ABS (POR1DIFF(3)(CONC(3,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH2.) 52 IF (BIG3. GT .BIG)BIG=BIG3 53 IF ( ABS (RXN(2,J)). GT .BIG)BIG= ABS (RXN(2,J)) 54 IF ( ABS (G(3)). LT .BIGEBIG)G(3)=0 55 56 C For GOx2 enzyme 223

PAGE 224

57 G(4)=CBULK(2)+CBULK(4)+CBULK(7)+CBULK(8))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(2,J))]TJ /F11 9.9626 Tf 6.019 0 Td[(CONC(4,J) 58 1)]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(7,J))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(8,J) 59 B(4,2)=+1. 60 B(4,4)=+1. 61 B(4,7)=+1. 62 B(4,8)=+1. 63 64 BIG= ABS (CBULK(2)) 65 IF ( ABS (CBULK(4)). GT .BIG)BIG= ABS (CBULK(4)) 66 IF ( ABS (CBULK(7)). GT .BIG)BIG= ABS (CBULK(7)) 67 IF ( ABS (CBULK(8)). GT .BIG)BIG= ABS (CBULK(8)) 68 IF ( ABS (CONC(2,J)). GT .BIG)BIG= ABS (CONC(2,J)) 69 IF ( ABS (CONC(4,J)). GT .BIG)BIG= ABS (CONC(4,J)) 70 IF ( ABS (CONC(7,J)). GT .BIG)BIG= ABS (CONC(7,J)) 71 IF ( ABS (CONC(8,J)). GT .BIG)BIG= ABS (CONC(8,J)) 72 IF ( ABS (G(4)). LT .BIGEBIG)G(4)=0 73 74 C For O2 being consumed only 75 G(5)=POR1DIFF(5)(CONC(5,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.CONC(5,J)+CONC(5,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/HH2. 76 2)]TJ /F11 9.9626 Tf 6.072 0 Td[(RXN(3,J) 77 B(5,5)=2.POR1DIFF(5)/HH2. 78 D(5,5)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(5)/HH2. 79 A(5,5)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(5)/HH2. 80 B(5,11)=+1. 81 82 BIG= ABS (POR1DIFF(5)(CONC(5,J+1))/HH2.) 83 BIG2= ABS (POR1DIFF(5)()]TJ /F11 9.9626 Tf 8.565 0 Td[(2.CONC(5,J))/HH2.) 84 IF (BIG2. GT .BIG)BIG=BIG2 85 BIG3= ABS (POR1DIFF(5)(CONC(5,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH2.) 86 IF (BIG3. GT .BIG)BIG=BIG3 87 IF ( ABS ()]TJ /F11 9.9626 Tf 6.746 0 Td[(RXN(3,J)). GT .BIG)BIG= ABS ()]TJ /F11 9.9626 Tf 6.746 0 Td[(RXN(3,J)) 88 IF ( ABS (G(5)). LT .BIGEBIG)G(5)=0 89 90 C For H2O2 reacting species 91 G(6)=POR1DIFF(6)(CONC(6,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.CONC(6,J)+CONC(6,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/HH2. 92 2+RXN(4,J) 93 B(6,6)=2.POR1DIFF(6)/HH2. 94 D(6,6)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(6)/HH2. 95 A(6,6)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(6)/HH2. 96 B(6,12)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 97 98 BIG= ABS (POR1DIFF(6)(CONC(6,J+1))/HH2.) 99 BIG2= ABS (POR1DIFF(6)()]TJ /F11 9.9626 Tf 8.565 0 Td[(2.CONC(6,J))/HH2.) 100 IF (BIG2. GT .BIG)BIG=BIG2 101 BIG3= ABS (POR1DIFF(6)(CONC(6,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH2.) 102 IF (BIG3. GT .BIG)BIG=BIG3 103 IF ( ABS (RXN(4,J)). GT .BIG)BIG= ABS (RXN(4,J)) 104 IF ( ABS (G(6)). LT .BIGEBIG)G(6)=0 105 106 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.393 -551.993 Td[(GOx2 enzyme 107 G(7)=RXN(1,J))]TJ /F11 9.9626 Tf 6.072 0 Td[(RXN(2,J) 108 B(7,9)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 109 B(7,10)=1. 110 111 IF ( ABS (RXN(1,J)). GT .BIG)BIG= ABS (RXN(1,J)) 112 IF ( ABS (RXN(2,J)). GT .BIG)BIG= ABS (RXN(2,J)) 113 IF ( ABS (G(7)). LT .BIGEBIG)G(7)=0 114 224

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115 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.131 -11.955 Td[(GOx enzyme 116 G(8)=RXN(3,J))]TJ /F11 9.9626 Tf 6.072 0 Td[(RXN(4,J) 117 B(8,11)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 118 B(8,12)=1. 119 120 IF ( ABS (RXN(3,J)). GT .BIG)BIG= ABS (RXN(3,J)) 121 IF ( ABS (RXN(4,J)). GT .BIG)BIG= ABS (RXN(4,J)) 122 IF ( ABS (G(8)). LT .BIGEBIG)G(8)=0 123 124 125 C REACTION1 126 214G(9)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(1,J)+ratef1(CONC(1,J)CONC(2,J))]TJ /F11 9.9626 Tf 7.832 0 Td[((CONC(7,J)/equilib1)) 127 B(9,1)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef1CONC(2,J) 128 B(9,2)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef1CONC(1,J) 129 B(9,7)=ratef1/equilib1 130 B(9,9)=+1. 131 132 BIG= ABS (RXN(1,J)) 133 BIG2= ABS (ratef1CONC(1,J)CONC(2,J)) 134 IF (BIG2. GT .BIG)BIG=BIG2 135 BIG3= ABS (ratef1(CONC(7,J)/equilib1)) 136 IF (BIG3. GT .BIG)BIG=BIG3 137 IF ( ABS (G(9)). LT .BIGEBIG)G(9)=0 138 139 C REACTION2 140 215G(10)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(2,J)+ratef2CONC(7,J) 141 B(10,7)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef2 142 B(10,10)=+1. 143 144 BIG= ABS (RXN(2,J)) 145 BIG2= ABS (ratef2CONC(7,J)) 146 IF (BIG2. GT .BIG)BIG=BIG2 147 IF ( ABS (G(10)). LT .BIGEBIG)G(10)=0 148 149 C REACTION3 150 216G(11)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(3,J)+ratef3(CONC(4,J)CONC(5,J))]TJ /F11 9.9626 Tf 7.831 0 Td[((CONC(8,J)/equilib3)) 151 B(11,4)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONC(5,J) 152 B(11,5)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONC(4,J) 153 B(11,8)=ratef3/equilib3 154 B(11,11)=+1. 155 156 BIG= ABS (RXN(3,J)) 157 BIG2= ABS (ratef3CONC(4,J)CONC(5,J)) 158 IF (BIG2. GT .BIG)BIG=BIG2 159 BIG3= ABS (ratef3(CONC(8,J)/equilib3)) 160 IF (BIG3. GT .BIG)BIG=BIG3 161 IF ( ABS (G(11)). LT .BIGEBIG)G(11)=0 162 163 C REACTION4 164 217G(12)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(4,J)+ratef4CONC(8,J) 165 B(12,8)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef4 166 B(12,12)=+1. 167 168 BIG= ABS (RXN(4,J)) 169 BIG2= ABS (ratef4CONC(8,J)) 170 IF (BIG2. GT .BIG)BIG=BIG2 171 IF ( ABS (G(12)). LT .BIGEBIG)G(12)=0 172 225

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173 174 c SAVE G OUT DATA 175 212 DO 11I=2,13 176 11 If (I. EQ .J) WRITE (12,301)J,(G(K),K=1,N) 177 IF (J. EQ .IJ/2) THEN 178 WRITE (12,301)J,(G(K),K=1,N) 179 ELSE IF (J. EQ .(KJ+1)) THEN 180 WRITE (12,301)J,(G(K),K=1,N) 181 ELSE IF (J. EQ .(KJ+2)) THEN 182 WRITE (12,301)J,(G(K),K=1,N) 183 ELSE IF (J. EQ .(KJ+3)) THEN 184 WRITE (12,301)J,(G(K),K=1,N) 185 ELSE IF (J. EQ .(KJ+4)) THEN 186 WRITE (12,301)J,(G(K),K=1,N) 187 ELSE IF (J. EQ .(IJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) THEN 188 WRITE (12,301)J,(G(K),K=1,N) 189 ELSE IF (J. EQ .(IJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(2)) THEN 190 WRITE (12,301)J,(G(K),K=1,N) 191 ELSE IF (J. EQ .(IJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(3)) THEN 192 WRITE (12,301)J,(G(K),K=1,N) 193 END IF 194 195 RETURN 196 END 226

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CodeE.8.Steady-StateContinuousGlucoseMonitorSubroutinefortheSecondCoupler 1 SUBROUTINE COUPLER2(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 COMMON /BAB/A(12,12),B(12,12),C(12,80001),D(12,25),G(12),X(12,12) 4 1,Y(12,12) 5 COMMON /NSN/N,NJ 6 COMMON /VAR/CONC(8,80001),RXN(4,80001),DIFF(8),H,EBIG,HH,IJ 7 COMMON /POR/POR1,POR2,PORGLU 8 COMMON /RTE/ratef1,equilib1,ratef2,ratef3,equilib3,ratef4 9 COMMON /BUL/CBULK(8),PARH2O2,PAR02,PARGLUCOSE,JCOUNT 10 11 301 FORMAT (5x,'J='I5,12E18.9) 12 13 COEFF1H=PORGLUDIFF(1)/H 14 COEFF1HH=POR1DIFF(1)/HH 15 COEFF3H=PORGLUDIFF(3)/H 16 COEFF3HH=POR1DIFF(3)/HH 17 COEFF5H=POR2DIFF(5)/H 18 COEFF5HH=POR1DIFF(5)/HH 19 COEFF6H=POR2DIFF(6)/H 20 COEFF6HH=POR1DIFF(6)/HH 21 22 C For Glucose being consumed only 23 G(1)=COEFF1H(CONC(1,J+1))]TJ /F11 9.9626 Tf 6.019 0 Td[(CONC(1,J)) 24 1)]TJ /F11 9.9626 Tf 6.399 0 Td[(COEFF1HH(CONC(1,J))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 25 3)]TJ /F11 9.9626 Tf 7.831 0 Td[((HH/2.)(RXN(1,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1)+3.RXN(1,J))/4. 26 B(1,1)=COEFF1H+COEFF1HH 27 D(1,1)=)]TJ /F11 9.9626 Tf 6.195 0 Td[(COEFF1H 28 A(1,1)=)]TJ /F11 9.9626 Tf 6.104 0 Td[(COEFF1HH 29 B(1,9)=+(HH/2.)(3./4.) 30 A(1,9)=+(HH/2.)(1./4.) 31 32 BIG= ABS (COEFF1HCONC(1,IJ+1)) 33 BIG2= ABS (COEFF1HCONC(1,IJ)) 34 IF (BIG2. GT .BIG)BIG=BIG2 35 BIG5= ABS (COEFF1HHCONC(1,IJ)) 36 IF (BIG5. GT .BIG)BIG=BIG5 37 BIG6= ABS (COEFF1HHCONC(1,IJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 38 IF (BIG6. GT .BIG)BIG=BIG6 39 BIG7= ABS (3(HH/2.)RXN(1,J)/4) 40 IF (BIG7. GT .BIG)BIG=BIG7 41 BIG8= ABS ((HH/2.)RXN(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)/4) 42 IF (BIG8. GT .BIG)BIG=BIG8 43 IF ( ABS (G(1)). LT .BIGEBIG)G(1)=0 44 45 C For GOx enzyme 46 G(2)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(1,J)+RXN(4,J) 47 B(2,9)=+1. 48 B(2,12)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 49 50 BIG= ABS (RXN(1,J)) 51 BIG2= ABS (RXN(4,J)) 52 IF (BIG2. GT .BIG)BIG=BIG2 53 IF ( ABS (G(2)). LT .BIGEBIG)G(2)=0 54 55 C For Gluconic Acid being produced only 56 G(3)=COEFF3H(CONC(3,J+1))]TJ /F11 9.9626 Tf 6.019 0 Td[(CONC(3,J)) 227

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57 1)]TJ /F11 9.9626 Tf 6.399 0 Td[(COEFF3HH(CONC(3,J))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(3,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 58 2+(HH/2.)(RXN(2,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1)+3.RXN(2,J))/4. 59 B(3,3)=COEFF3H+COEFF3HH 60 D(3,3)=)]TJ /F11 9.9626 Tf 6.195 0 Td[(COEFF3H 61 A(3,3)=)]TJ /F11 9.9626 Tf 6.104 0 Td[(COEFF3HH 62 B(3,10)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(3./4.) 63 A(3,10)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(1./4.) 64 65 BIG= ABS (COEFF3HCONC(3,IJ+1)) 66 BIG2= ABS (COEFF3HCONC(3,IJ)) 67 IF (BIG2. GT .BIG)BIG=BIG2 68 BIG5= ABS (COEFF3HHCONC(3,IJ)) 69 IF (BIG5. GT .BIG)BIG=BIG5 70 BIG6= ABS (COEFF3HHCONC(3,IJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 71 IF (BIG6. GT .BIG)BIG=BIG6 72 BIG7= ABS (3.(HH/2.)RXN(2,J)/4.) 73 IF (BIG7. GT .BIG)BIG=BIG7 74 BIG8= ABS ((HH/2.)RXN(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)/4.) 75 IF (BIG8. GT .BIG)BIG=BIG8 76 IF ( ABS (G(3)). LT .BIGEBIG)G(3)=0 77 78 C For GOx2 enzyme 79 G(4)=CBULK(2)+CBULK(4)+CBULK(7)+CBULK(8))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(2,J))]TJ /F11 9.9626 Tf 6.019 0 Td[(CONC(4,J) 80 1)]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(7,J))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(8,J) 81 B(4,2)=+1. 82 B(4,4)=+1. 83 B(4,7)=+1. 84 B(4,8)=+1. 85 86 BIG= ABS (CBULK(2)) 87 IF ( ABS (CBULK(4)). GT .BIG)BIG= ABS (CBULK(4)) 88 IF ( ABS (CBULK(7)). GT .BIG)BIG= ABS (CBULK(7)) 89 IF ( ABS (CBULK(8)). GT .BIG)BIG= ABS (CBULK(8)) 90 IF ( ABS (CONC(2,J)). GT .BIG)BIG= ABS (CONC(2,J)) 91 IF ( ABS (CONC(4,J)). GT .BIG)BIG= ABS (CONC(4,J)) 92 IF ( ABS (CONC(7,J)). GT .BIG)BIG= ABS (CONC(7,J)) 93 IF ( ABS (CONC(8,J)). GT .BIG)BIG= ABS (CONC(8,J)) 94 IF ( ABS (G(4)). LT .BIGEBIG)G(4)=0 95 96 C For O2 being consumed only 97 G(5)=COEFF5H(CONC(5,J+1))]TJ /F11 9.9626 Tf 6.019 0 Td[(CONC(5,J)) 98 1)]TJ /F11 9.9626 Tf 6.399 0 Td[(COEFF5HH(CONC(5,J))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(5,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 99 2)]TJ /F11 9.9626 Tf 7.831 0 Td[((HH/2.)(RXN(3,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1)+3.RXN(3,J))/4. 100 B(5,5)=COEFF5H+COEFF5HH 101 D(5,5)=)]TJ /F11 9.9626 Tf 6.195 0 Td[(COEFF5H 102 A(5,5)=)]TJ /F11 9.9626 Tf 6.104 0 Td[(COEFF5HH 103 B(5,11)=(HH/2.)(3./4.) 104 A(5,11)=(HH/2.)(1./4.) 105 106 BIG= ABS (COEFF5HCONC(5,IJ+1)) 107 BIG2= ABS (COEFF5HCONC(5,IJ)) 108 IF (BIG2. GT .BIG)BIG=BIG2 109 BIG5= ABS (COEFF5HHCONC(5,IJ)) 110 IF (BIG5. GT .BIG)BIG=BIG5 111 BIG6= ABS (COEFF5HHCONC(5,IJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 112 IF (BIG6. GT .BIG)BIG=BIG6 113 BIG7= ABS (3.(HH/2.)RXN(3,J)/4.) 114 IF (BIG7. GT .BIG)BIG=BIG7 228

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115 BIG8= ABS ((HH/2.)RXN(3,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)/4.) 116 IF (BIG8. GT .BIG)BIG=BIG8 117 IF ( ABS (G(5)). LT .BIGEBIG)G(5)=0 118 119 C For H2O2 reacting species 120 G(6)=COEFF6H(CONC(6,J+1))]TJ /F11 9.9626 Tf 6.019 0 Td[(CONC(6,J)) 121 1)]TJ /F11 9.9626 Tf 6.399 0 Td[(COEFF6HH(CONC(6,J))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(6,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 122 2+(HH/2.)(RXN(4,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1)+3.RXN(4,J))/4. 123 B(6,6)=COEFF6H+COEFF6HH 124 D(6,6)=)]TJ /F11 9.9626 Tf 6.195 0 Td[(COEFF6H 125 A(6,6)=)]TJ /F11 9.9626 Tf 6.104 0 Td[(COEFF6HH 126 B(6,12)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(3./4.) 127 A(6,12)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(1./4.) 128 129 BIG= ABS (COEFF6HCONC(6,IJ+1)) 130 BIG2= ABS (COEFF6HCONC(6,IJ)) 131 IF (BIG2. GT .BIG)BIG=BIG2 132 BIG5= ABS (COEFF6HHCONC(6,IJ)) 133 IF (BIG5. GT .BIG)BIG=BIG5 134 BIG6= ABS (COEFF6HHCONC(6,IJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) 135 IF (BIG6. GT .BIG)BIG=BIG6 136 BIG7= ABS (3.(HH/2.)RXN(4,J)/4.) 137 IF (BIG7. GT .BIG)BIG=BIG7 138 BIG8= ABS ((HH/2.)RXN(4,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)/4.) 139 IF (BIG8. GT .BIG)BIG=BIG8 140 IF ( ABS (G(6)). LT .BIGEBIG)G(6)=0 141 142 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.393 -309.527 Td[(GOx2 enzyme 143 G(7)=RXN(1,J))]TJ /F11 9.9626 Tf 6.072 0 Td[(RXN(2,J) 144 B(7,9)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 145 B(7,10)=1. 146 147 IF ( ABS (RXN(1,J)). GT .BIG)BIG= ABS (RXN(1,J)) 148 IF ( ABS (RXN(2,J)). GT .BIG)BIG= ABS (RXN(2,J)) 149 IF ( ABS (G(7)). LT .BIGEBIG)G(7)=0 150 151 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.131 -408.717 Td[(GOx enzyme 152 G(8)=RXN(3,J))]TJ /F11 9.9626 Tf 6.072 0 Td[(RXN(4,J) 153 B(8,11)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 154 B(8,12)=1. 155 156 IF ( ABS (RXN(3,J)). GT .BIG)BIG= ABS (RXN(3,J)) 157 IF ( ABS (RXN(4,J)). GT .BIG)BIG= ABS (RXN(4,J)) 158 IF ( ABS (G(8)). LT .BIGEBIG)G(8)=0 159 160 161 C REACTION1 162 214G(9)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(1,J)+ratef1(CONC(1,J)CONC(2,J))]TJ /F11 9.9626 Tf 7.832 0 Td[((CONC(7,J)/equilib1)) 163 B(9,1)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef1CONC(2,J) 164 B(9,2)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef1CONC(1,J) 165 B(9,7)=ratef1/equilib1 166 B(9,9)=+1. 167 168 BIG= ABS (RXN(1,J)) 169 BIG2= ABS (ratef1CONC(1,J)CONC(2,J)) 170 IF (BIG2. GT .BIG)BIG=BIG2 171 BIG3= ABS (ratef1(CONC(7,J)/equilib1)) 172 IF (BIG3. GT .BIG)BIG=BIG3 229

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173 IF ( ABS (G(9)). LT .BIGEBIG)G(9)=0 174 175 C REACTION2 176 215G(10)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(2,J)+ratef2CONC(7,J) 177 B(10,7)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef2 178 B(10,10)=+1. 179 180 BIG= ABS (RXN(2,J)) 181 BIG2= ABS (ratef2CONC(7,J)) 182 IF (BIG2. GT .BIG)BIG=BIG2 183 IF ( ABS (G(10)). LT .BIGEBIG)G(10)=0 184 185 C REACTION3 186 216G(11)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(3,J)+ratef3(CONC(4,J)CONC(5,J))]TJ /F11 9.9626 Tf 7.831 0 Td[((CONC(8,J)/equilib3)) 187 B(11,4)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONC(5,J) 188 B(11,5)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONC(4,J) 189 B(11,8)=ratef3/equilib3 190 B(11,11)=+1. 191 192 BIG= ABS (RXN(3,J)) 193 BIG2= ABS (ratef3CONC(4,J)CONC(5,J)) 194 IF (BIG2. GT .BIG)BIG=BIG2 195 BIG3= ABS (ratef3(CONC(8,J)/equilib3)) 196 IF (BIG3. GT .BIG)BIG=BIG3 197 IF ( ABS (G(11)). LT .BIGEBIG)G(11)=0 198 199 C REACTION4 200 217G(12)=)]TJ /F11 9.9626 Tf 5.777 0 Td[(RXN(4,J)+ratef4CONC(8,J) 201 B(12,8)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef4 202 B(12,12)=+1. 203 204 BIG= ABS (RXN(4,J)) 205 BIG2= ABS (ratef4CONC(8,J)) 206 IF (BIG2. GT .BIG)BIG=BIG2 207 IF ( ABS (G(12)). LT .BIGEBIG)G(12)=0 208 209 212 WRITE (12,301)J,(G(K),K=1,N) 210 RETURN 211 END 230

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CodeE.9.Steady-StateContinuousGlucoseMonitorSubroutineforGLMRegion 1 SUBROUTINE OUTER(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 COMMON /BAB/A(12,12),B(12,12),C(12,80001),D(12,25),G(12),X(12,12) 4 1,Y(12,12) 5 COMMON /NSN/N,NJ 6 COMMON /VAR/CONC(8,80001),RXN(4,80001),DIFF(8),H,EBIG,HH,IJ 7 COMMON /POR/POR1,POR2,PORGLU 8 COMMON /RTE/ratef1,equilib1,ratef2,ratef3,equilib3,ratef4 9 10 301 FORMAT (5x,'J='I5,12E18.9) 11 12 C For Glucose being consumed only 13 G(1)=PORGLUDIFF(1)(CONC(1,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.CONC(1,J)+CONC(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/H2. 14 B(1,1)=2.PORGLUDIFF(1)/H2. 15 D(1,1)=)]TJ /F11 9.9626 Tf 5.79 0 Td[(PORGLUDIFF(1)/H2. 16 A(1,1)=)]TJ /F11 9.9626 Tf 5.79 0 Td[(PORGLUDIFF(1)/H2. 17 18 BIG= ABS (PORGLUDIFF(1)(CONC(1,J+1))/H2.) 19 BIG2= ABS (PORGLUDIFF(1)()]TJ /F11 9.9626 Tf 8.565 0 Td[(2.CONC(1,J))/H2.) 20 IF (BIG2. GT .BIG)BIG=BIG2 21 BIG3= ABS (PORGLUDIFF(1)(CONC(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/H2.) 22 IF (BIG3. GT .BIG)BIG=BIG3 23 IF ( ABS (G(1)). LT .BIGEBIG)G(1)=0 24 25 C For GOx enzyme 26 G(2)=CONC(2,J) 27 B(2,2)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 28 29 c BIG = ABS ( CONC (2, J ) ) 30 c IF ( ABS ( G (2) ) LT BIG EBIG ) G (2) =0 31 32 C For Gluconic Acid being produced only 33 G(3)=PORGLUDIFF(3)(CONC(3,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.CONC(3,J)+CONC(3,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/H2. 34 B(3,3)=2.PORGLUDIFF(3)/H2. 35 D(3,3)=)]TJ /F11 9.9626 Tf 5.79 0 Td[(PORGLUDIFF(3)/H2. 36 A(3,3)=)]TJ /F11 9.9626 Tf 5.79 0 Td[(PORGLUDIFF(3)/H2. 37 38 BIG= ABS (PORGLUDIFF(3)(CONC(3,J+1))/H2.) 39 BIG2= ABS (PORGLUDIFF(3)()]TJ /F11 9.9626 Tf 8.565 0 Td[(2.CONC(3,J))/H2.) 40 IF (BIG2. GT .BIG)BIG=BIG2 41 BIG3= ABS (PORGLUDIFF(3)(CONC(3,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/H2.) 42 IF (BIG3. GT .BIG)BIG=BIG3 43 IF ( ABS (G(3)). LT .BIGEBIG)G(3)=0 44 45 C For GOx2 enzyme 46 G(4)=CONC(4,J) 47 B(4,4)=)]TJ /F11 9.9626 Tf 7.859 .001 Td[(1. 48 49 c BIG = ABS ( CONC (4, J ) ) 50 c IF ( ABS ( G (4) ) LT BIG EBIG ) G (4) =0 51 52 C For O2 being consumed only 53 G(5)=POR2DIFF(5)(CONC(5,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.CONC(5,J)+CONC(5,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/H2. 54 B(5,5)=2.POR2DIFF(5)/H2. 55 D(5,5)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR2DIFF(5)/H2. 56 A(5,5)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR2DIFF(5)/H2. 231

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57 58 BIG= ABS (POR2DIFF(5)(CONC(5,J+1))/H2.) 59 BIG2= ABS (POR2DIFF(5)()]TJ /F11 9.9626 Tf 8.565 0 Td[(2.CONC(5,J))/H2.) 60 IF (BIG2. GT .BIG)BIG=BIG2 61 BIG3= ABS (POR2DIFF(5)(CONC(5,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/H2.) 62 IF (BIG3. GT .BIG)BIG=BIG3 63 IF ( ABS (G(5)). LT .BIGEBIG)G(5)=0 64 65 C For H2O2 reacting species 66 G(6)=POR2DIFF(6)(CONC(6,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.CONC(6,J)+CONC(6,J)]TJ /F11 9.9626 Tf 8.08 0 Td[(1))/H2. 67 B(6,6)=2.POR2DIFF(6)/H2. 68 D(6,6)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR2DIFF(6)/H2. 69 A(6,6)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR2DIFF(6)/H2. 70 71 BIG= ABS (POR2DIFF(6)(CONC(6,J+1))/H2.) 72 BIG2= ABS (POR2DIFF(6)()]TJ /F11 9.9626 Tf 8.565 0 Td[(2.CONC(6,J))/H2.) 73 IF (BIG2. GT .BIG)BIG=BIG2 74 BIG3= ABS (POR2DIFF(6)(CONC(6,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/H2.) 75 IF (BIG3. GT .BIG)BIG=BIG3 76 IF ( ABS (G(6)). LT .BIGEBIG)G(6)=0 77 78 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.393 -243.4 Td[(GOx2 enzyme complex 79 G(7)=CONC(7,J) 80 B(7,7)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 81 82 c BIG = ABS ( CONC (7, J ) ) 83 c IF ( ABS ( G (7) ) LT BIG EBIG ) G (7) =0 84 85 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.131 -320.548 Td[(GOx enzyme complex 86 G(8)=CONC(8,J) 87 B(8,8)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 88 89 c BIG = ABS ( CONC (8, J ) ) 90 c IF ( ABS ( G (8) ) LT BIG EBIG ) G (8) =0 91 92 C For Reaction 1 Enzymatic Catalysis 93 G(9)=RXN(1,J) 94 B(9,9)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 95 96 c BIG = ABS ( RXN (1, J ) ) 97 c IF ( ABS ( G (9) ) LT BIG EBIG ) G (9) =0 98 99 C For Reaction 2 100 G(10)=RXN(2,J) 101 B(10,10)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 102 103 c BIG = ABS ( RXN (2, J ) ) 104 c IF ( ABS ( G (10) ) LT BIG EBIG ) G (10) =0 105 106 C For Reaction 3 Meditation / regeneration 107 G(11)=RXN(3,J) 108 B(11,11)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 109 110 c BIG = ABS ( RXN (3, J ) ) 111 c IF ( ABS ( G (11) ) LT BIG EBIG ) G (11) =0 112 113 C For Reaction 4 114 G(12)=RXN(4,J) 232

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115 B(12,12)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 116 117 c BIG = ABS ( RXN (4, J ) ) 118 c IF ( ABS ( G (12) ) LT BIG EBIG ) G (12) =0 119 120 c SAVE G OUT DATA 121 IF (J. EQ .(IJ+(NJ)]TJ /F11 9.9626 Tf 7.942 0 Td[(IJ)/2)) THEN 122 WRITE (12,301)J,(G(K),K=1,N) 123 ELSE IF (J. EQ .(NJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) THEN 124 WRITE (12,301)J,(G(K),K=1,N) 125 ELSE IF (J. EQ .(IJ+1)) THEN 126 WRITE (12,301)J,(G(K),K=1,N) 127 END IF 128 129 RETURN 130 END 233

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CodeE.10.Steady-StateContinuousGlucoseMonitorSubroutinefortheBulkBoundaryCondition 1 SUBROUTINE BCNJ(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 COMMON /BAB/A(12,12),B(12,12),C(12,80001),D(12,25),G(12),X(12,12) 4 1,Y(12,12) 5 COMMON /NSN/N,NJ 6 COMMON /VAR/CONC(8,80001),RXN(4,80001),DIFF(8),H,EBIG,HH,IJ 7 COMMON /POR/POR1,POR2,PORGLU 8 COMMON /BUL/CBULK(8),PARH2O2,PAR02,PARGLUCOSE,JCOUNT 9 10 301 FORMAT (5x,'J='I5,12E18.9) 11 12 C For Glucose being consumed only 13 G(1)=PARGLUCOSECBULK(1))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(1,J) 14 B(1,1)=1.0 15 16 BIG= ABS (PARGLUCOSECBULK(1)) 17 IF ( ABS (CONC(1,J)). GT .BIG)BIG= ABS (CONC(1,J)) 18 IF ( ABS (G(1)). LT .BIGEBIG)G(1)=0 19 20 C For GOx enzyme 21 G(2)=CONC(2,J) 22 B(2,2)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 23 24 BIG= ABS (CONC(2,J)) 25 IF ( ABS (G(2)). LT .BIGEBIG)G(2)=0 26 27 C For Gluconic Acid being produced only 28 G(3)=PARGLUCOSECBULK(3))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(3,J) 29 B(3,3)=1.0 30 31 BIG= ABS (PARGLUCOSECBULK(3)) 32 IF ( ABS (CONC(3,J)). GT .BIG)BIG= ABS (CONC(3,J)) 33 IF ( ABS (G(3)). LT .BIGEBIG)G(3)=0 34 35 C For GOx2 enzyme 36 G(4)=CONC(4,J) 37 B(4,4)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 38 39 BIG= ABS (CONC(4,J)) 40 IF ( ABS (G(4)). LT .BIGEBIG)G(4)=0 41 42 C For O2 being consumed only 43 G(5)=PAR02CBULK(5))]TJ /F11 9.9626 Tf 6.019 0 Td[(CONC(5,J) 44 B(5,5)=1.0 45 46 BIG= ABS (PARO2CBULK(5)) 47 IF ( ABS (CONC(5,J)). GT .BIG)BIG= ABS (CONC(5,J)) 48 IF ( ABS (G(5)). LT .BIGEBIG)G(5)=0 49 50 C For H2O2 reacting species 51 G(6)=PARH2O2CBULK(6))]TJ /F11 9.9626 Tf 6.018 0 Td[(CONC(6,J) 52 B(6,6)=1.0 53 54 BIG= ABS (PARH2O2CBULK(6)) 234

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55 IF ( ABS (CONC(6,J)). GT .BIG)BIG= ABS (CONC(6,J)) 56 IF ( ABS (G(6)). LT .BIGEBIG)G(6)=0 57 58 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.393 -45.019 Td[(GOx2 enzyme complex 59 G(7)=CONC(7,J) 60 B(7,7)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 61 62 BIG= ABS (CONC(7,J)) 63 IF ( ABS (G(7)). LT .BIGEBIG)G(7)=0 64 65 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.131 -122.167 Td[(GOx enzyme complex 66 G(8)=CONC(8,J) 67 B(8,8)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 68 69 BIG= ABS (CONC(8,J)) 70 IF ( ABS (G(8)). LT .BIGEBIG)G(8)=0 71 72 C For Reaction 1 Enzymatic Catalysis 73 G(9)=RXN(1,J) 74 B(9,9)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 75 76 BIG= ABS (RXN(1,J)) 77 IF ( ABS (G(9)). LT .BIGEBIG)G(9)=0 78 79 C For Reaction 2 80 G(10)=RXN(2,J) 81 B(10,10)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 82 83 BIG= ABS (RXN(2,J)) 84 IF ( ABS (G(10)). LT .BIGEBIG)G(10)=0 85 86 C For Reaction 3 Meditation / regeneration 87 G(11)=RXN(3,J) 88 B(11,11)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 89 90 BIG= ABS (RXN(3,J)) 91 IF ( ABS (G(11)). LT .BIGEBIG)G(11)=0 92 93 C For Reaction 4 94 G(12)=RXN(4,J) 95 B(12,12)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 96 97 BIG= ABS (RXN(4,J)) 98 IF ( ABS (G(12)). LT .BIGEBIG)G(12)=0 99 C SAVE G OUT DATA 100 206 WRITE (12,301)J,(G(K),K=1,N) 101 PRINT ,'ITERATION=',JCOUNT 102 RETURN 103 END 235

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E.3OscillatingContinuousGlucoseMonitorCodeThissectioncontainstheoscillatingFORTRANcodesusedtosolvetheconvectivediusionequationwithahomogeneousreactionforacontinuousglucosemonitor.Itreadsthesteady-stateinputleinordertosolvefortheoscillatingconcentrations.ThemathematicalworkupforthesecodesareinChapter 5 .TheFORTRANcodesarefollowedbyaMATLABcodethatreadstheoscillatingconcentrationofthereactingspeciesandcreatesthedimensionlessdiusion-impedance.Therstsectioninthecode,calledCONVDIFFOSCILLATING,isthemainprogram,whichoutlinestheglobalvariablesandsetsupcallinglestosaveoverasoutputlesaswellascallingtheinputles.Thenthesubroutinesthatarecalledinthemainprogramareallshown.Theyarethesametitledsubroutinesasthesteadystate. 236

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CodeE.11.OscillatingContinuousGlucoseMonitorMainProgram 1 C Convective Diffusion Equation with Homogeneous Reaction 2 C Enzyme kinetics added 3 C 6 species system 4 C SPECIES 1 = glucose SPECIES 2 = GOx )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 256.308 -68.671 Td[(FAD SPECIES 3 = Gluconic acid 5 C SPECIES 4 = GOx )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 130.915 -79.692 Td[(FADH2 SPECIES 5 = O2 SPECIES 6 = H2O2 6 C SPECIES 7 = GOx )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 130.915 -90.714 Td[(FADH2 )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 166.54 -90.714 Td[(GA SPECIES 8 = GOx )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 286.196 -90.714 Td[(FAD )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 310.583 -90.714 Td[(H2O2 7 C Species 6 is the reacting species 8 C This is the unsteady state solution that will eventually lead to 9 c the impedance 10 11 C This should be ran after cdhgox ss for 12 C The input file is the same for both of these 13 14 C cd C :n Morgan n FORTRAN2016 n3 DomainsComplex 15 C gfortran )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 97.358 -189.904 Td[(static cdhgox os for )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 227.649 -189.904 Td[(o cdhgox os exe 16 17 PROGRAM CONVDIFFOSCILLATING 18 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 19 COMMON /BAT/A(24,24),B(24,24),C(24,80001),D(24,49),G(24), 20 1X(24,24),Y(24,24) 21 COMMON /NST/N,NJ 22 23 COMMON /VAR/CONCSS(8,80001),RXNSS(4,80001),DIFF(8) 24 COMMON /VARR/PORGLU,HHH,KJ 25 COMMON /CON/C1(2,80001),C2(2,80001),C3(2,80001),C4(2,80001), 26 1C5(2,80001),C6(2,80001),C7(2,80001),C8(2,80001), 27 2RXN1(2,80001),RXN2(2,80001),RXN3(2,80001),RXN4(2,80001) 28 COMMON /RTE/ratef1,equilib1,ratef2,ratef3,equilib3,ratef4 29 COMMON /OTH/POR1,POR2,H,EBIG,HH,IJ 30 COMMON /BCI/FLUX,omega 31 COMMON /BUL/CBULK(8),PARH2O2,PAR02,PARGLUCOSE,JCOUNT 32 COMMON /DELT/DELTA1,DELTA2,FREQ(400),CB(2010,80001) 33 COMMON /EXTRA/Z(8),REF(8) 34 CHARACTER REF6 35 36 102 FORMAT (/30HTHENEXTRUNDID NOT CONVERGE) 37 103 FORMAT ('Error=',E16.6/(1X,'Species=',A6,2X,'ConcatElectrode=', 38 1E12.5,2X,'ConcatBulk=',E12.5)) 39 334 FORMAT (12(E25.15,5X)) 40 305 FORMAT (E20.12,3X,E20.12,3X,E20.12,3X,E20.12) 41 335 FORMAT (16(E25.15,5X)) 42 336 FORMAT (1000(E25.15,1X)) 43 339 FORMAT (1000(E16.9,1X)) 44 301 FORMAT (5x,'J='I5,16E15.6) 45 302 FORMAT ('Iteration='I4) 46 47 C Read input values used in steady state 48 open (10, file ='cdhgox in.txt', status ='old') 49 read (10,)N,NJ,IJ,KJ,Y1,Y2,Y3,POR1,POR2,POR3,PARH2O2,PAR02, 50 1PARGLUCOSE,ratef1,equilib1,ratef2,ratef3,equilib3,ratef4, 51 2AKB,BB,EBIG 52 read (10,)(DIFF(I),REF(I),CBULK(I),I=1,(N)]TJ /F11 9.9626 Tf 8.081 0 Td[(4)) 53 POR1=POR1(1.5) 54 POR2=POR2(1.5) 55 PORGLU=POR3(1.5) 56 237

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57 open (16, file ='pot in.txt', status ='old') 58 read (16,)V 59 60 C Read steady state values from previous file 61 OPEN ( UNIT =11, FILE ='cdhgox out.txt') 62 READ (11,334)(CONCSS(1,I),CONCSS(2,I),CONCSS(3,I),CONCSS(4,I), 63 1CONCSS(5,I),CONCSS(6,I),CONCSS(7,I),CONCSS(8,I),RXNSS(1,I), 64 2RXNSS(2,I),RXNSS(3,I),RXNSS(4,I),I=1,NJ) 65 66 OPEN ( UNIT =13, FILE ='cdhgox os out.txt') 67 CLOSE ( UNIT =13, STATUS ='DELETE') 68 OPEN ( UNIT =13, FILE ='cdhgox os out.txt') 69 70 OPEN (14, FILE ='cdhgox G out.txt') 71 CLOSE (14, STATUS ='DELETE') 72 OPEN (14, FILE ='cdhgox G out.txt') 73 74 OPEN (15, FILE ='cdhgox H2O2 out.txt') 75 CLOSE (15, STATUS ='DELETE') 76 OPEN (15, FILE ='cdhgox H2O2 out.txt') 77 78 OPEN (16, FILE ='cdhgox values out.txt') 79 CLOSE (16, STATUS ='DELETE') 80 OPEN (16, FILE ='cdhgox values out.txt') 81 82 OPEN (17, FILE ='kgox values out.txt') 83 CLOSE (17, STATUS ='DELETE') 84 OPEN (17, FILE ='kgox values out.txt') 85 86 C Constants 87 F=96487. 88 89 c THIS IS SPACING FOR OUTER LAYER BCNJ 90 H=Y3/(NJ)]TJ /F11 9.9626 Tf 7.943 0 Td[(IJ) 91 92 c THIS IS SPACING FOR INNER LAYER BC1 93 HH=Y2/(IJ)]TJ /F11 9.9626 Tf 6.559 0 Td[(KJ) 94 95 c THIS IS SPACING FOR REACTION LAYER 96 HHH=Y1/(KJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1) 97 98 C Create flux of the reacting species constants 99 FLUX=)]TJ /F11 9.9626 Tf 5.777 0 Td[(AKB exp (BBV)/F/2. 100 PRINT ,'FLUX=',FLUX 101 102 C Create charge transfer resistance 103 RTB=1./(AKBBBCONCSS(6,1) EXP (BBV)) 104 PRINT ,'ChargeTransferResistance=',RTB 105 106 N=2N 107 PRINT ,'N=',N 108 109 337 FORMAT (I2/I7/I7/I7/10(E15.8/)E15.8) 110 write (16,337)N,NJ,IJ,KJ,H,HH,HHH,V,AKB,BB,DIFF(6),RTB,POR1 111 112 C The number of points for frequency 113 NPTS=241 114 PRINT ,'NPTS=',NPTS 238

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115 c Create range for the dimensionless frequency 116 DO 261I=1,NPTS 117 FREQ(I)=10.()]TJ /F11 9.9626 Tf 8.67 0 Td[(5.+0.05(I)]TJ /F11 9.9626 Tf 8.656 0 Td[(1.)) 118 261 WRITE (17,339)FREQ(I) 119 120 121 C The number of points for frequency 122 C NPTS =13 123 C PRINT NPTS =', NPTS 124 C Create range for the dimensionless frequency 125 C DO 261 I =1, NPTS 126 C FREQ ( I ) =10.()]TJ /F11 9.9626 Tf 8.665 0 Td[(3.+0.5( I )]TJ /F11 9.9626 Tf 8.656 0 Td[(1.) ) 127 C 261 WRITE (17,339) FREQ ( I ) 128 129 DO 19nf=1,NPTS 130 C DO 19 nf =1,3 131 132 133 PRINT ,'FREQ(NF)=',FREQ(NF) 134 omega=FREQ(NF)DIFF(6)POR1/(Y1+Y2)2 135 136 IF (ratef1. LT .1E)]TJ /F11 9.9626 Tf 8.214 0 Td[(10)omega=FREQ(NF)DIFF(6)POR1/(Y1+Y2+Y3)2 137 138 PRINT ,'omega=',omega 139 340 FORMAT (E12.6) 140 write (17,340)omega 141 142 C Start actual code 143 DO 20J=1,NJ 144 DO 20I=1,N 145 20C(I,J)=0.0 146 DO 21J=1,NJ 147 DO 21K=1,2 148 C1(K,J)=0.0 149 C2(K,J)=0.0 150 C3(K,J)=0.0 151 C4(K,J)=0.0 152 C5(K,J)=0.0 153 C6(K,J)=0.0 154 C7(K,J)=0.0 155 C8(K,J)=0.0 156 RXN1(K,J)=0.0 157 RXN2(K,J)=0.0 158 RXN3(K,J)=0.0 159 21RXN4(K,J)=0.0 160 JCOUNT=0 161 TOL=1.E)]TJ /F11 9.9626 Tf 7.992 0 Td[(10NNJ/1000000 162 22JCOUNT=JCOUNT+1 163 AMP=0.0 164 J=0 165 DO 23I=1,N 166 DO 23K=1,N 167 Y(I,K)=0.0 168 23X(I,K)=0.0 169 24J=J+1 170 DO 25I=1,N 171 G(I)=0.0 172 DO 25K=1,N 239

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173 A(I,K)=0.0 174 B(I,K)=0.0 175 25D(I,K)=0.0 176 177 IF (J. EQ .1) CALL BC1(J) 178 IF (J. GT .1. AND .J. LT .KJ) CALL REACTION(J) 179 IF (J. EQ .KJ) CALL COUPLER1(J) 180 IF (J. GT .KJ. AND .J. LT .IJ) CALL INNER(J) 181 IF (J. EQ .IJ) CALL COUPLER2(J) 182 IF (J. GT .IJ. AND .J. LT .NJ) CALL OUTER(J) 183 IF (J. EQ .NJ) CALL BCNJ(J) 184 CALL BAND(J) 185 186 AMP=DABS(G(1))+DABS(G(2))+DABS(G(3))+DABS(G(4))+DABS(G(5)) 187 1+DABS(G(6))+DABS(G(7))+DABS(G(8))+DABS(G(9))+DABS(G(10)) 188 2+DABS(G(11))+DABS(G(12))+DABS(G(13))+DABS(G(14)) 189 3+DABS(G(15))+DABS(G(16)) 190 191 IF (J. LT .NJ) GO TO 24 192 PRINT ,'ERROR=',AMP 193 194 DO 16K=1,NJ 195 DO 16I=1,2 196 C1(I,K)=C1(I,K)+C(I,K) 197 C2(I,K)=C2(I,K)+C(I+2,K) 198 C3(I,K)=C3(I,K)+C(I+4,K) 199 C4(I,K)=C4(I,K)+C(I+6,K) 200 C5(I,K)=C5(I,K)+C(I+8,K) 201 C6(I,K)=C6(I,K)+C(I+10,K) 202 C7(I,K)=C7(I,K)+C(I+12,K) 203 C8(I,K)=C8(I,K)+C(I+14,K) 204 RXN1(I,K)=RXN1(I,K)+C(I+16,K) 205 RXN2(I,K)=RXN2(I,K)+C(I+18,K) 206 RXN3(I,K)=RXN1(I,K)+C(I+20,K) 207 RXN4(I,K)=RXN2(I,K)+C(I+22,K) 208 16 CONTINUE 209 210 WRITE (14,302)(JCOUNT) 211 212 IF (DABS(AMP). LT .DABS(TOL)) GO TO 15 213 214 IF (JCOUNT. LE .4) GO TO 22 215 print 102 216 217 15 CONTINUE 218 PRINT ,'JCOUNT=',JCOUNT 219 220 PRINT ,'nf1=',nf 221 222 DO 18I=1,2 223 DO 18J=1,NJ 224 BIG=C6(I,J) 225 BIG2=1.0E)]TJ /F11 9.9626 Tf 7.804 0 Td[(40 226 18 IF ( ABS (BIG). LE .BIG2)C6(I,J)=0.0 227 228 c WRITE (13,335) ( C1 (1, J ) C1 (2, J ) C2 (1, J ) C2 (2, J ) C3 (1, J ) C3 (2, J ) 229 c 1 C4 (1, J ) C4 (2, J ) C5 (1, J ) C5 (2, J ) C6 (1, J ) C6 (2, J ) C6 (1, J ) 230 c 2 C6 (2, J ) C6 (1, J ) C6 (2, J ) RXN1 (1, J ) RXN1 (2, J ) RXN2 (1, J ) 240

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231 c 3 RXN2 (2, J ) RXN3 (1, J ) RXN3 (2, J ) RXN4 (1, J ) RXN4 (2, J ) J =1, NJ ) 232 233 DO 19J=1,NJ 234 CB(2nf)]TJ /F11 9.9626 Tf 8.236 0 Td[(1,J)=C6(1,J) 235 19CB(2nf,J)=C6(2,J) 236 237 c for some reason nf is one greater then necessary 238 PRINT ,'nf2=',nf 239 240 C DO 17 I =1,2 nf )]TJ /F11 9.9626 Tf 7.491 0 Td[(2 241 nf=nf)]TJ /F11 9.9626 Tf 7.49 0 Td[(1 242 DO 17J=1,NJ 243 17 WRITE (15,336)(CB(I,J),I=1,2nf) 244 245 338 FORMAT (I5) 246 write (16,338)nf 247 248 PRINT ,'DIFF(6)=',DIFF(6) 249 250 END PROGRAM CONVDIFFOSCILLATING 241

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CodeE.12.OscillatingContinuousGlucoseMonitorfortheElectrodeBoundaryCondition 1 SUBROUTINE BC1(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 COMMON /BAT/A(24,24),B(24,24),C(24,80001),D(24,49),G(24), 4 1X(24,24),Y(24,24) 5 COMMON /NST/N,NJ 6 7 COMMON /VAR/CONCSS(8,80001),RXNSS(4,80001),DIFF(8) 8 COMMON /VARR/PORGLU,HHH,KJ 9 COMMON /CON/C1(2,80001),C2(2,80001),C3(2,80001),C4(2,80001), 10 1C5(2,80001),C6(2,80001),C7(2,80001),C8(2,80001), 11 2RXN1(2,80001),RXN2(2,80001),RXN3(2,80001),RXN4(2,80001) 12 COMMON /RTE/ratef1,equilib1,ratef2,ratef3,equilib3,ratef4 13 COMMON /OTH/POR1,POR2,H,EBIG,HH,IJ 14 COMMON /BCI/FLUX,omega 15 COMMON /BUL/CBULK(8),PARH2O2,PAR02,PARGLUCOSE,JCOUNT 16 COMMON /DELT/DELTA1,DELTA2,FREQ(400),CB(2010,80001) 17 18 19 301 FORMAT (5x,'J='I5,24E15.6) 20 21 C BOUNDARY CONDITION AT THE ELECTRODE J =1 22 C For Glucose being consumed only 23 G(1)=omega(3.C1(2,J)+C1(2,J+1))/4. 24 1+2.POR1DIFF(1)(C1(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C1(1,J))/HHH2. 25 2)]TJ /F11 9.9626 Tf 8.54 0 Td[((3.RXN1(1,J)+RXN1(1,J+1))/4. 26 B(1,1)=+2.POR1DIFF(1)/HHH2. 27 D(1,1)=)]TJ /F11 9.9626 Tf 7.986 0 Td[(2.POR1DIFF(1)/HHH2. 28 B(1,2)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega(3./4.) 29 D(1,2)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega(1./4.) 30 B(1,17)=+3./4. 31 D(1,17)=+1./4. 32 33 G(2)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega(3.C1(1,J)+C1(1,J+1))/4. 34 1+2.POR1DIFF(1)(C1(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C1(2,J))/HHH2. 35 2)]TJ /F11 9.9626 Tf 8.54 0 Td[((3.RXN1(2,J)+RXN1(2,J+1))/4. 36 B(2,2)=+2.POR1DIFF(1)/HHH2. 37 D(2,2)=)]TJ /F11 9.9626 Tf 7.986 0 Td[(2.POR1DIFF(1)/HHH2. 38 B(2,1)=omega(3./4.) 39 D(2,1)=omega(1./4.) 40 B(2,18)=+3./4. 41 D(2,18)=+1./4. 42 43 C For GOx enzyme 44 G(3)=omega(3.C2(2,J)+C2(2,J+1))/4. 45 1)]TJ /F11 9.9626 Tf 8.54 0 Td[((3.RXN1(1,J)+RXN1(1,J+1))/4. 46 2+(3.RXN4(1,J)+RXN4(1,J+1))/4. 47 B(3,4)=)]TJ /F11 9.9626 Tf 7.232 .001 Td[(omega(3./4.) 48 D(3,4)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega(1./4.) 49 B(3,17)=+3./4. 50 D(3,17)=+1./4. 51 B(3,23)=)]TJ /F11 9.9626 Tf 8.401 0 Td[(3./4. 52 D(3,23)=)]TJ /F11 9.9626 Tf 8.401 0 Td[(1./4. 53 54 G(4)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega(3.C2(1,J)+C2(1,J+1))/4. 55 1)]TJ /F11 9.9626 Tf 8.54 0 Td[((3.RXN1(2,J)+RXN1(2,J+1))/4. 56 2+(3.RXN4(2,J)+RXN4(2,J+1))/4. 242

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57 B(4,3)=omega(3./4.) 58 D(4,3)=omega(1./4.) 59 B(4,18)=+3./4. 60 D(4,18)=+1./4. 61 B(4,24)=)]TJ /F11 9.9626 Tf 8.401 0 Td[(3./4. 62 D(4,24)=)]TJ /F11 9.9626 Tf 8.401 0 Td[(1./4. 63 64 C For Gluconic Acid being produced only 65 G(5)=omega(3.C3(2,J)+C3(2,J+1))/4. 66 1+2.POR1DIFF(3)(C3(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C3(1,J))/HHH2. 67 2+(3.RXN2(1,J)+RXN2(1,J+1))/4. 68 B(5,5)=+2.POR1DIFF(3)/HHH2. 69 D(5,5)=)]TJ /F11 9.9626 Tf 7.986 0 Td[(2.POR1DIFF(3)/HHH2. 70 B(5,6)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega(3./4.) 71 D(5,6)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega(1./4.) 72 B(5,19)=)]TJ /F11 9.9626 Tf 8.401 0 Td[(3./4. 73 D(5,19)=)]TJ /F11 9.9626 Tf 8.401 0 Td[(1./4. 74 75 G(6)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega(3.C3(1,J)+C3(1,J+1))/4. 76 1+2.POR1DIFF(3)(C3(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C3(2,J))/HHH2. 77 2+(3.RXN2(2,J)+RXN2(2,J+1))/4. 78 B(6,6)=+2.POR1DIFF(3)/HHH2. 79 D(6,6)=)]TJ /F11 9.9626 Tf 7.986 0 Td[(2.POR1DIFF(3)/HHH2. 80 B(6,5)=omega(3./4.) 81 D(6,5)=omega(1./4.) 82 B(6,20)=)]TJ /F11 9.9626 Tf 8.401 0 Td[(3./4. 83 D(6,20)=)]TJ /F11 9.9626 Tf 8.401 0 Td[(1./4. 84 85 C For GOx2 enzyme 86 G(7)=omega(3.C4(2,J)+C4(2,J+1))/4. 87 1+(3.RXN2(1,J)+RXN2(1,J+1))/4. 88 2)]TJ /F11 9.9626 Tf 8.54 0 Td[((3.RXN3(1,J)+RXN3(1,J+1))/4. 89 B(7,8)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega(3./4.) 90 D(7,8)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega(1./4.) 91 B(7,19)=)]TJ /F11 9.9626 Tf 8.401 0 Td[(3./4. 92 D(7,19)=)]TJ /F11 9.9626 Tf 8.401 0 Td[(1./4. 93 B(7,21)=+3./4. 94 D(7,21)=+1./4. 95 96 G(8)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega(3.C4(1,J)+C4(1,J+1))/4. 97 1+(3.RXN2(2,J)+RXN2(2,J+1))/4. 98 2)]TJ /F11 9.9626 Tf 8.54 0 Td[((3.RXN3(2,J)+RXN3(2,J+1))/4. 99 B(8,7)=omega(3./4.) 100 D(8,7)=omega(1./4.) 101 B(8,20)=)]TJ /F11 9.9626 Tf 8.401 0 Td[(3./4. 102 D(8,20)=)]TJ /F11 9.9626 Tf 8.401 0 Td[(1./4. 103 B(8,22)=+3./4. 104 D(8,22)=+1./4. 105 106 C For O2 being consumed only 107 G(9)=(HHH/2.)omega(3.C6(2,J)+C6(2,J+1))(1./4.) 108 1+(HHH/2.)omega(3.C5(2,J)+C5(2,J+1))(1./4.) 109 2+(POR1DIFF(6)/HHH)(C6(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C6(1,J)) 110 3+(POR1DIFF(5)/HHH)(C5(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C5(1,J)) 111 4+(HHH/2.)(3.RXN4(1,J)+RXN4(1,J+1))(1./4.) 112 5)]TJ /F11 9.9626 Tf 7.831 0 Td[((HHH/2.)(3.RXN3(1,J)+RXN3(1,J+1))(1./4.) 113 B(9,9)=POR1DIFF(5)/HHH 114 D(9,9)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(5)/HHH 243

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115 B(9,10)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HHH/2.)omega(3./4.) 116 D(9,10)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HHH/2.)omega(1./4.) 117 B(9,11)=POR1DIFF(6)/HHH 118 D(9,11)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR1DIFF(6)/HHH 119 B(9,12)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HHH/2.)omega(3./4.) 120 D(9,12)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HHH/2.)omega(1./4.) 121 B(9,23)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HHH/2.)(3./4.) 122 D(9,23)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HHH/2.)(1./4.) 123 B(9,21)=(HHH/2.)(3./4.) 124 D(9,21)=(HHH/2.)(1./4.) 125 126 G(10)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HHH/2.)omega(3.C6(1,J)+C6(1,J+1))(1./4.) 127 1)]TJ /F11 9.9626 Tf 7.831 0 Td[((HHH/2.)omega(3.C5(1,J)+C5(1,J+1))(1./4.) 128 2+(POR1DIFF(6)/HHH)(C6(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C6(2,J)) 129 3+(POR1DIFF(5)/HHH)(C5(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C5(2,J)) 130 4+(HHH/2.)(3.RXN4(2,J)+RXN4(2,J+1))(1./4.) 131 5)]TJ /F11 9.9626 Tf 7.831 0 Td[((HHH/2.)(3.RXN3(2,J)+RXN3(2,J+1))(1./4.) 132 B(10,10)=POR1DIFF(5)/HHH 133 D(10,10)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR1DIFF(5)/HHH 134 B(10,9)=(HHH/2.)omega(3./4.) 135 D(10,9)=(HHH/2.)omega(1./4.) 136 B(10,12)=POR1DIFF(6)/HHH 137 D(10,12)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR1DIFF(6)/HHH 138 B(10,11)=(HHH/2.)omega(3./4.) 139 D(10,11)=(HHH/2.)omega(1./4.) 140 B(10,24)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HHH/2.)(3./4.) 141 D(10,24)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HHH/2.)(1./4.) 142 B(10,22)=(HHH/2.)(3./4.) 143 D(10,22)=(HHH/2.)(1./4.) 144 145 C For H2O2 reacting species 146 G(11)=1.)]TJ /F11 9.9626 Tf 7.786 0 Td[(C6(1,J) 147 B(11,11)=1. 148 149 G(12)=C6(2,J) 150 B(12,12)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 151 152 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.393 -419.738 Td[(GOx2 enzyme 153 G(13)=omega(3.C7(2,J)+C7(2,J+1))/4. 154 1+(3.RXN1(1,J)+RXN1(1,J+1))/4. 155 2)]TJ /F11 9.9626 Tf 8.54 0 Td[((3.RXN2(1,J)+RXN2(1,J+1))/4. 156 B(13,14)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega(3./4.) 157 D(13,14)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega(1./4.) 158 B(13,17)=)]TJ /F11 9.9626 Tf 8.4 -.001 Td[(3./4. 159 D(13,17)=)]TJ /F11 9.9626 Tf 8.4 0 Td[(1./4. 160 B(13,19)=+3./4. 161 D(13,19)=+1./4. 162 163 G(14)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega(3.C7(1,J)+C7(1,J+1))/4. 164 1+(3.RXN1(2,J)+RXN1(2,J+1))/4. 165 2)]TJ /F11 9.9626 Tf 8.54 0 Td[((3.RXN2(2,J)+RXN2(2,J+1))/4. 166 B(14,13)=omega(3./4.) 167 D(14,13)=omega(1./4.) 168 B(14,18)=)]TJ /F11 9.9626 Tf 8.4 0 Td[(3./4. 169 D(14,18)=)]TJ /F11 9.9626 Tf 8.4 0 Td[(1./4. 170 B(14,20)=+3./4. 171 D(14,20)=+1./4. 172 244

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173 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.131 -11.955 Td[(GOx enzyme 174 G(15)=omega(3.C8(2,J)+C8(2,J+1))/4. 175 1+(3.RXN3(1,J)+RXN3(1,J+1))/4. 176 2)]TJ /F11 9.9626 Tf 8.54 0 Td[((3.RXN4(1,J)+RXN4(1,J+1))/4. 177 B(15,16)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega(3./4.) 178 D(15,16)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega(1./4.) 179 B(15,21)=)]TJ /F11 9.9626 Tf 8.4 0 Td[(3./4. 180 D(15,21)=)]TJ /F11 9.9626 Tf 8.4 0 Td[(1./4. 181 B(15,23)=+3./4. 182 D(15,23)=+1./4. 183 184 G(16)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega(3.C8(1,J)+C8(1,J+1))/4. 185 1+(3.RXN3(2,J)+RXN3(2,J+1))/4. 186 2)]TJ /F11 9.9626 Tf 8.54 0 Td[((3.RXN4(2,J)+RXN4(2,J+1))/4. 187 B(16,15)=omega(3./4.) 188 D(16,15)=omega(1./4.) 189 B(16,22)=)]TJ /F11 9.9626 Tf 8.4 0 Td[(3./4. 190 D(16,22)=)]TJ /F11 9.9626 Tf 8.4 0 Td[(1./4. 191 B(16,24)=+3./4. 192 D(16,24)=+1./4. 193 194 C REACTION1 195 G(17)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN1(1,J)+ratef1CONCSS(2,J)C1(1,J) 196 1+ratef1CONCSS(1,J)C2(1,J) 197 2)]TJ /F11 9.9626 Tf 7.084 0 Td[(C7(1,J)/equilib1 198 B(17,1)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef1CONCSS(2,J) 199 B(17,3)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef1CONCSS(1,J) 200 B(17,13)=+1./equilib1 201 B(17,17)=+1. 202 203 G(18)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN1(2,J)+ratef1CONCSS(2,J)C1(2,J) 204 1+ratef1CONCSS(1,J)C2(2,J) 205 2)]TJ /F11 9.9626 Tf 7.084 0 Td[(C7(2,J)/equilib1 206 B(18,2)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef1CONCSS(2,J) 207 B(18,4)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef1CONCSS(1,J) 208 B(18,14)=+1./equilib1 209 B(18,18)=+1. 210 211 C REACTION2 212 G(19)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN2(1,J)+ratef2C7(1,J) 213 B(19,13)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef2 214 B(19,19)=+1. 215 216 G(20)=)]TJ /F11 9.9626 Tf 6.194 -.001 Td[(RXN2(2,J)+ratef2C7(2,J) 217 B(20,14)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef2 218 B(20,20)=+1. 219 220 C REACTION3 221 G(21)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN3(1,J)+ratef3CONCSS(4,J)C5(1,J) 222 1+ratef3CONCSS(5,J)C4(1,J) 223 2)]TJ /F11 9.9626 Tf 7.084 0 Td[(C8(1,J)/equilib3 224 B(21,7)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONCSS(5,J) 225 B(21,9)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONCSS(4,J) 226 B(21,15)=+1./equilib3 227 B(21,21)=+1. 228 229 G(22)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN3(2,J)+ratef3CONCSS(4,J)C5(2,J) 230 1+ratef3CONCSS(5,J)C4(2,J) 245

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231 2)]TJ /F11 9.9626 Tf 7.084 0 Td[(C8(2,J)/equilib3 232 B(22,8)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONCSS(5,J) 233 B(22,10)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef3CONCSS(4,J) 234 B(22,16)=+1./equilib3 235 B(22,22)=+1. 236 237 C REACTION4 238 G(23)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN4(1,J)+ratef4C8(1,J) 239 B(23,15)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef4 240 B(23,23)=+1. 241 242 G(24)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN4(2,J)+ratef4C8(2,J) 243 B(24,16)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef4 244 B(24,24)=+1. 245 246 WRITE (14,301)J,(G(K),K=1,N) 247 248 RETURN 249 END 246

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CodeE.13.OscillatingContinuousGlucoseMonitorSubroutinefortheReactionRegion 1 SUBROUTINE REACTION(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 COMMON /BAT/A(24,24),B(24,24),C(24,80001),D(24,49),G(24), 4 1X(24,24),Y(24,24) 5 COMMON /NST/N,NJ 6 7 COMMON /VAR/CONCSS(8,80001),RXNSS(4,80001),DIFF(8) 8 COMMON /VARR/PORGLU,HHH,KJ 9 COMMON /CON/C1(2,80001),C2(2,80001),C3(2,80001),C4(2,80001), 10 1C5(2,80001),C6(2,80001),C7(2,80001),C8(2,80001), 11 2RXN1(2,80001),RXN2(2,80001),RXN3(2,80001),RXN4(2,80001) 12 COMMON /RTE/ratef1,equilib1,ratef2,ratef3,equilib3,ratef4 13 COMMON /OTH/POR1,POR2,H,EBIG,HH,IJ 14 COMMON /BCI/FLUX,omega 15 COMMON /BUL/CBULK(8),PARH2O2,PAR02,PARGLUCOSE,JCOUNT 16 COMMON /DELT/DELTA1,DELTA2,FREQ(400),CB(2010,80001) 17 18 301 FORMAT (5x,'J='I5,24E15.6) 19 20 C For Glucose being consumed only 21 G(1)=omegaC1(2,J) 22 1+POR1DIFF(1)(C1(1,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C1(1,J)+C1(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HHH2. 23 3)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN1(1,J) 24 B(1,1)=2.POR1DIFF(1)/HHH2. 25 A(1,1)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(1)/HHH2. 26 D(1,1)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(1)/HHH2. 27 B(1,2)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 28 B(1,17)=+1. 29 30 G(2)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC1(1,J) 31 1+POR1DIFF(1)(C1(2,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C1(2,J)+C1(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HHH2. 32 3)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN1(2,J) 33 B(2,2)=2.POR1DIFF(1)/HHH2. 34 A(2,2)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(1)/HHH2. 35 D(2,2)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(1)/HHH2. 36 B(2,1)=omega 37 B(2,18)=+1. 38 39 C For GOx enzyme 40 G(3)=omegaC2(2,J) 41 1)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN1(1,J) 42 2+RXN4(1,J) 43 B(3,4)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 44 B(3,17)=+1. 45 B(3,23)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 46 47 G(4)=)]TJ /F11 9.9626 Tf 7.232 .001 Td[(omegaC2(1,J) 48 1)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN1(2,J) 49 2+RXN4(2,J) 50 B(4,3)=omega 51 B(4,18)=+1. 52 B(4,24)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 53 54 C For Gluconic Acid being produced only 55 G(5)=omegaC3(2,J) 56 1+POR1DIFF(3)(C3(1,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C3(1,J)+C3(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HHH2. 247

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57 3+RXN2(1,J) 58 B(5,5)=2.POR1DIFF(3)/HHH2. 59 A(5,5)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(3)/HHH2. 60 D(5,5)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(3)/HHH2. 61 B(5,6)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 62 B(5,19)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 63 64 G(6)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC3(1,J) 65 1+POR1DIFF(3)(C3(2,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C3(2,J)+C3(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HHH2. 66 3+RXN2(2,J) 67 B(6,6)=2.POR1DIFF(3)/HHH2. 68 A(6,6)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(3)/HHH2. 69 D(6,6)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(3)/HHH2. 70 B(6,5)=omega 71 B(6,20)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 72 73 C For GOx2 enzyme 74 G(7)=omegaC4(2,J) 75 1+RXN2(1,J) 76 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN3(1,J) 77 B(7,8)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 78 B(7,19)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 79 B(7,21)=+1. 80 81 G(8)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC4(1,J) 82 1+RXN2(2,J) 83 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN3(2,J) 84 B(8,7)=omega 85 B(8,20)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 86 B(8,22)=+1. 87 88 C For O2 being consumed only 89 G(9)=omegaC5(2,J) 90 1+POR1DIFF(5)(C5(1,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C5(1,J)+C5(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HHH2. 91 3)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN3(1,J) 92 B(9,9)=2.POR1DIFF(5)/HHH2. 93 A(9,9)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(5)/HHH2. 94 D(9,9)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(5)/HHH2. 95 B(9,10)=)]TJ /F11 9.9626 Tf 7.233 0 Td[(omega 96 B(9,21)=+1. 97 98 G(10)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC5(1,J) 99 1+POR1DIFF(5)(C5(2,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C5(2,J)+C5(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HHH2. 100 3)]TJ /F11 9.9626 Tf 6.488 -.001 Td[(RXN3(2,J) 101 B(10,10)=2.POR1DIFF(5)/HHH2. 102 A(10,10)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR1DIFF(5)/HHH2. 103 D(10,10)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR1DIFF(5)/HHH2. 104 B(10,9)=omega 105 B(10,22)=+1. 106 107 C For H2O2 reacting species 108 G(11)=omegaC6(2,J) 109 1+POR1DIFF(6)(C6(1,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C6(1,J)+C6(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HHH2. 110 3+RXN4(1,J) 111 B(11,11)=2.POR1DIFF(6)/HHH2. 112 A(11,11)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR1DIFF(6)/HHH2. 113 D(11,11)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR1DIFF(6)/HHH2. 114 B(11,12)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 248

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115 B(11,23)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 116 117 G(12)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC6(1,J) 118 1+POR1DIFF(6)(C6(2,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C6(2,J)+C6(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HHH2. 119 3+RXN4(2,J) 120 B(12,12)=2.POR1DIFF(6)/HHH2. 121 A(12,12)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR1DIFF(6)/HHH2. 122 D(12,12)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR1DIFF(6)/HHH2. 123 B(12,11)=omega 124 B(12,24)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 125 126 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.393 -133.188 Td[(GOx2 enzyme 127 G(13)=omegaC7(2,J) 128 1+RXN1(1,J) 129 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN2(1,J) 130 B(13,14)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 131 B(13,17)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 132 B(13,19)=+1. 133 134 G(14)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC7(1,J) 135 1+RXN1(2,J) 136 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN2(2,J) 137 B(14,13)=omega 138 B(14,18)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 139 B(14,20)=+1. 140 141 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.393 -298.506 Td[(GOx2 enzyme 142 G(15)=omegaC8(2,J) 143 1+RXN3(1,J) 144 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN4(1,J) 145 B(15,16)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 146 B(15,21)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 147 B(15,23)=+1. 148 149 G(16)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC8(1,J) 150 1+RXN3(2,J) 151 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN4(2,J) 152 B(16,15)=omega 153 B(16,22)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 154 B(16,24)=+1. 155 156 C REACTION1 157 G(17)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN1(1,J)+ratef1CONCSS(2,J)C1(1,J) 158 1+ratef1CONCSS(1,J)C2(1,J) 159 2)]TJ /F11 9.9626 Tf 7.084 0 Td[(C7(1,J)/equilib1 160 B(17,1)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef1CONCSS(2,J) 161 B(17,3)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef1CONCSS(1,J) 162 B(17,13)=+1./equilib1 163 B(17,17)=+1. 164 165 G(18)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN1(2,J)+ratef1CONCSS(2,J)C1(2,J) 166 1+ratef1CONCSS(1,J)C2(2,J) 167 2)]TJ /F11 9.9626 Tf 7.084 0 Td[(C7(2,J)/equilib1 168 B(18,2)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef1CONCSS(2,J) 169 B(18,4)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef1CONCSS(1,J) 170 B(18,14)=+1./equilib1 171 B(18,18)=+1. 172 249

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173 C REACTION2 174 G(19)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN2(1,J)+ratef2C7(1,J) 175 B(19,13)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef2 176 B(19,19)=+1. 177 178 G(20)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN2(2,J)+ratef2C7(2,J) 179 B(20,14)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef2 180 B(20,20)=+1. 181 182 C REACTION3 183 G(21)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN3(1,J)+ratef3CONCSS(4,J)C5(1,J) 184 1+ratef3CONCSS(5,J)C4(1,J) 185 2)]TJ /F11 9.9626 Tf 7.084 0 Td[(C8(1,J)/equilib3 186 B(21,7)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONCSS(5,J) 187 B(21,9)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONCSS(4,J) 188 B(21,15)=+1./equilib3 189 B(21,21)=+1. 190 191 G(22)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN3(2,J)+ratef3CONCSS(4,J)C5(2,J) 192 1+ratef3CONCSS(5,J)C4(2,J) 193 2)]TJ /F11 9.9626 Tf 7.084 0 Td[(C8(2,J)/equilib3 194 B(22,8)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONCSS(5,J) 195 B(22,10)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef3CONCSS(4,J) 196 B(22,16)=+1./equilib3 197 B(22,22)=+1. 198 199 C REACTION4 200 G(23)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN4(1,J)+ratef4C8(1,J) 201 B(23,15)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef4 202 B(23,23)=+1. 203 204 G(24)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN4(2,J)+ratef4C8(2,J) 205 B(24,16)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef4 206 B(24,24)=+1. 207 208 c SAVE G OUT DATA 209 DO 11I=2,13 210 11 If (I. EQ .J) WRITE (14,301)J,(G(K),K=1,N) 211 IF (J. EQ .IJ/2) THEN 212 WRITE (14,301)J,(G(K),K=1,N) 213 ELSE IF (J. EQ .(IJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) THEN 214 WRITE (14,301)J,(G(K),K=1,N) 215 ELSE IF (J. EQ .(IJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(2)) THEN 216 WRITE (14,301)J,(G(K),K=1,N) 217 ELSE IF (J. EQ .(IJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(3)) THEN 218 WRITE (14,301)J,(G(K),K=1,N) 219 END IF 220 221 RETURN 222 END 250

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CodeE.14.OscillatingContinuousGlucoseMonitorSubroutinefortheFirstCoupler 1 SUBROUTINE COUPLER1(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 COMMON /BAT/A(24,24),B(24,24),C(24,80001),D(24,49),G(24), 4 1X(24,24),Y(24,24) 5 COMMON /NST/N,NJ 6 7 COMMON /VAR/CONCSS(8,80001),RXNSS(4,80001),DIFF(8) 8 COMMON /VARR/PORGLU,HHH,KJ 9 COMMON /CON/C1(2,80001),C2(2,80001),C3(2,80001),C4(2,80001), 10 1C5(2,80001),C6(2,80001),C7(2,80001),C8(2,80001), 11 2RXN1(2,80001),RXN2(2,80001),RXN3(2,80001),RXN4(2,80001) 12 COMMON /RTE/ratef1,equilib1,ratef2,ratef3,equilib3,ratef4 13 COMMON /OTH/POR1,POR2,H,EBIG,HH,IJ 14 COMMON /BCI/FLUX,omega 15 COMMON /BUL/CBULK(8),PARH2O2,PAR02,PARGLUCOSE,JCOUNT 16 COMMON /DELT/DELTA1,DELTA2,FREQ(400),CB(2010,80001) 17 18 301 FORMAT (5x,'J='I5,24E15.6) 19 20 C For Glucose being consumed only 21 G(1)=HH/2.omega(C1(2,J+1)+3.C1(2,J))/4. 22 1+HHH/2.omega(C1(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C1(2,J))/4. 23 2+POR1DIFF(1)(C1(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C1(1,J))/HH 24 3)]TJ /F11 9.9626 Tf 6.571 0 Td[(POR1DIFF(1)(C1(1,J))]TJ /F11 9.9626 Tf 7.085 0 Td[(C1(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HHH 25 8)]TJ /F11 9.9626 Tf 7.831 0 Td[((HH/2.)(RXN1(1,J+1)+3.RXN1(1,J))/4. 26 9)]TJ /F11 9.9626 Tf 7.831 0 Td[((HHH/2.)(RXN1(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.RXN1(1,J))/4. 27 B(1,1)=POR1DIFF(1)/HH+POR1DIFF(1)/HHH 28 D(1,1)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(1)/HH 29 A(1,1)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(1)/HHH 30 B(1,2)=)]TJ /F11 9.9626 Tf 5.742 0 Td[(HHH/2.omega(3./4.))]TJ /F11 9.9626 Tf 6.162 0 Td[(HH/2.omega(3./4.) 31 D(1,2)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2.omega(1./4.) 32 A(1,2)=)]TJ /F11 9.9626 Tf 5.742 0 Td[(HHH/2.omega(1./4.) 33 B(1,17)=(HH/2.)(3./4.)+(HHH/2.)(3./4.) 34 D(1,17)=(HH/2.)(1./4.) 35 A(1,17)=(HHH/2.)(1./4.) 36 37 G(2)=)]TJ /F11 9.9626 Tf 5.742 0 Td[(HHH/2.omega(C1(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C1(1,J))/4. 38 1)]TJ /F11 9.9626 Tf 6.162 0 Td[(HH/2.omega(C1(1,J+1)+3.C1(1,J))/4. 39 2+POR1DIFF(1)(C1(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C1(2,J))/HH 40 3)]TJ /F11 9.9626 Tf 6.571 0 Td[(POR1DIFF(1)(C1(2,J))]TJ /F11 9.9626 Tf 7.085 0 Td[(C1(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HHH 41 4)]TJ /F11 9.9626 Tf 7.831 0 Td[((HH/2.)(RXN1(2,J+1)+3.RXN1(2,J))/4. 42 5)]TJ /F11 9.9626 Tf 7.831 0 Td[((HHH/2.)(RXN1(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.RXN1(2,J))/4. 43 B(2,2)=POR1DIFF(1)/HH+POR1DIFF(1)/HHH 44 D(2,2)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(1)/HH 45 A(2,2)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(1)/HHH 46 B(2,1)=+HHH/2.omega(3./4.)+HH/2.omega(3./4.) 47 D(2,1)=+HH/2.omega(1./4.) 48 A(2,1)=+HHH/2.omega(1./4.) 49 B(2,18)=(HH/2.)(3./4.)+(HHH/2.)(3./4.) 50 D(2,18)=(HH/2.)(1./4.) 51 A(2,18)=(HHH/2.)(1./4.) 52 53 C For GOx enzyme 54 G(3)=omegaC2(2,J) 55 1)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN1(1,J) 56 2+RXN4(1,J) 251

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57 B(3,4)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 58 B(3,17)=+1. 59 B(3,23)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 60 61 G(4)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC2(1,J) 62 1)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN1(2,J) 63 2+RXN4(2,J) 64 B(4,3)=omega 65 B(4,18)=+1. 66 B(4,24)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 67 68 C For Gluconic Acid being produced only 69 G(5)=HHH/2.omega(C3(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C3(2,J))/4. 70 1+HH/2.omega(C3(2,J+1)+3.C3(2,J))/4. 71 2+POR1DIFF(3)(C3(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C3(1,J))/HH 72 3)]TJ /F11 9.9626 Tf 6.571 0 Td[(POR1DIFF(3)(C3(1,J))]TJ /F11 9.9626 Tf 7.085 0 Td[(C3(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HHH 73 4+(HH/2.)(RXN2(1,J+1)+3.RXN2(1,J))/4. 74 5+(HHH/2.)(RXN2(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.RXN2(1,J))/4. 75 B(5,5)=POR1DIFF(3)/HH+POR1DIFF(3)/HHH 76 D(5,5)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(3)/HH 77 A(5,5)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(3)/HHH 78 B(5,6)=)]TJ /F11 9.9626 Tf 5.742 0 Td[(HHH/2.omega(3./4.))]TJ /F11 9.9626 Tf 6.162 0 Td[(HH/2.omega(3./4.) 79 D(5,6)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2.omega(1./4.) 80 A(5,6)=)]TJ /F11 9.9626 Tf 5.742 0 Td[(HHH/2.omega(1./4.) 81 B(5,19)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(3./4.))]TJ /F11 9.9626 Tf 7.832 0 Td[((HHH/2.)(3./4.) 82 D(5,19)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(1./4.) 83 A(5,19)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HHH/2.)(1./4.) 84 85 G(6)=)]TJ /F11 9.9626 Tf 5.742 0 Td[(HHH/2.omega(C3(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C3(1,J))/4. 86 1)]TJ /F11 9.9626 Tf 6.162 0 Td[(HH/2.omega(C3(1,J+1)+3.C3(1,J))/4. 87 2+POR1DIFF(3)(C3(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C3(2,J))/HH 88 3)]TJ /F11 9.9626 Tf 6.571 0 Td[(POR1DIFF(3)(C3(2,J))]TJ /F11 9.9626 Tf 7.085 0 Td[(C3(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HHH 89 4+(HH/2.)(RXN2(2,J+1)+3.RXN2(2,J))/4. 90 5+(HHH/2.)(RXN2(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.RXN2(2,J))/4. 91 B(6,6)=POR1DIFF(3)/HH+POR1DIFF(3)/HHH 92 D(6,6)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(3)/HH 93 A(6,6)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(3)/HHH 94 B(6,5)=+HHH/2.omega(3./4.)+HH/2.omega(3./4.) 95 D(6,5)=+HH/2.omega(1./4.) 96 A(6,5)=+HHH/2.omega(1./4.) 97 B(6,20)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(3./4.))]TJ /F11 9.9626 Tf 7.832 0 Td[((HHH/2.)(3./4.) 98 D(6,20)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(1./4.) 99 A(6,20)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HHH/2.)(1./4.) 100 101 C For GOx2 enzyme 102 G(7)=omegaC4(2,J) 103 1+RXN2(1,J) 104 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN3(1,J) 105 B(7,8)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 106 B(7,19)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 107 B(7,21)=+1. 108 109 G(8)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC4(1,J) 110 1+RXN2(2,J) 111 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN3(2,J) 112 B(8,7)=omega 113 B(8,20)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 114 B(8,22)=+1. 252

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115 116 C For O2 being consumed only 117 G(9)=HHH/2.omega(C5(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C5(2,J))/4. 118 1+HH/2.omega(C5(2,J+1)+3.C5(2,J))/4. 119 2+POR1DIFF(5)(C5(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C5(1,J))/HH 120 3)]TJ /F11 9.9626 Tf 6.571 0 Td[(POR1DIFF(5)(C5(1,J))]TJ /F11 9.9626 Tf 7.085 0 Td[(C5(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HHH 121 4)]TJ /F11 9.9626 Tf 7.831 0 Td[((HH/2.)(RXN3(1,J+1)+3.RXN3(1,J))/4. 122 5)]TJ /F11 9.9626 Tf 7.831 0 Td[((HHH/2.)(RXN3(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.RXN3(1,J))/4. 123 B(9,9)=POR1DIFF(5)/HH+POR1DIFF(5)/HHH 124 D(9,9)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(5)/HH 125 A(9,9)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(5)/HHH 126 B(9,10)=)]TJ /F11 9.9626 Tf 5.743 0 Td[(HHH/2.omega(3./4.))]TJ /F11 9.9626 Tf 6.162 0 Td[(HH/2.omega(3./4.) 127 D(9,10)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2.omega(1./4.) 128 A(9,10)=)]TJ /F11 9.9626 Tf 5.743 0 Td[(HHH/2.omega(1./4.) 129 B(9,21)=(HH/2.)(3./4.)+(HHH/2.)(3./4.) 130 D(9,21)=(HH/2.)(1./4.) 131 A(9,21)=(HHH/2.)(1./4.) 132 133 G(10)=)]TJ /F11 9.9626 Tf 5.743 0 Td[(HHH/2.omega(C5(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C5(1,J))/4. 134 1)]TJ /F11 9.9626 Tf 6.162 0 Td[(HH/2.omega(C5(1,J+1)+3.C5(1,J))/4. 135 2+POR1DIFF(5)(C5(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C5(2,J))/HH 136 3)]TJ /F11 9.9626 Tf 6.571 0 Td[(POR1DIFF(5)(C5(2,J))]TJ /F11 9.9626 Tf 7.085 0 Td[(C5(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HHH 137 4)]TJ /F11 9.9626 Tf 7.831 0 Td[((HH/2.)(RXN3(2,J+1)+3.RXN3(2,J))/4. 138 5)]TJ /F11 9.9626 Tf 7.831 0 Td[((HHH/2.)(RXN3(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.RXN3(2,J))/4. 139 B(10,10)=POR1DIFF(5)/HH+POR1DIFF(5)/HHH 140 D(10,10)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR1DIFF(5)/HH 141 A(10,10)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR1DIFF(5)/HHH 142 B(10,9)=+HHH/2.omega(3./4.)+HH/2.omega(3./4.) 143 D(10,9)=+HH/2.omega(1./4.) 144 A(10,9)=+HHH/2.omega(1./4.) 145 B(10,22)=(HH/2.)(3./4.)+(HHH/2.)(3./4.) 146 D(10,22)=(HH/2.)(1./4.) 147 A(10,22)=(HHH/2.)(1./4.) 148 149 C For H2O2 reacting species 150 G(11)=HHH/2.omega(C6(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C6(2,J))/4. 151 1+HH/2.omega(C6(2,J+1)+3.C6(2,J))/4. 152 2+POR1DIFF(6)(C6(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C6(1,J))/HH 153 3)]TJ /F11 9.9626 Tf 6.571 0 Td[(POR1DIFF(6)(C6(1,J))]TJ /F11 9.9626 Tf 7.085 0 Td[(C6(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HHH 154 4+(HH/2.)(RXN4(1,J+1)+3.RXN4(1,J))/4. 155 5+(HHH/2.)(RXN4(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.RXN4(1,J))/4. 156 B(11,11)=POR1DIFF(6)/HH+POR1DIFF(6)/HHH 157 D(11,11)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR1DIFF(6)/HH 158 A(11,11)=)]TJ /F11 9.9626 Tf 6.277 -.001 Td[(POR1DIFF(6)/HHH 159 B(11,12)=)]TJ /F11 9.9626 Tf 5.743 0 Td[(HHH/2.omega(3./4.))]TJ /F11 9.9626 Tf 6.163 0 Td[(HH/2.omega(3./4.) 160 D(11,12)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2.omega(1./4.) 161 A(11,12)=)]TJ /F11 9.9626 Tf 5.743 0 Td[(HHH/2.omega(1./4.) 162 B(11,23)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(3./4.))]TJ /F11 9.9626 Tf 7.832 0 Td[((HHH/2.)(3./4.) 163 D(11,23)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(1./4.) 164 A(11,23)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HHH/2.)(1./4.) 165 166 G(12)=)]TJ /F11 9.9626 Tf 5.743 0 Td[(HHH/2.omega(C6(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C6(1,J))/4. 167 1)]TJ /F11 9.9626 Tf 6.162 0 Td[(HH/2.omega(C6(1,J+1)+3.C6(1,J))/4. 168 2+POR1DIFF(6)(C6(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C6(2,J))/HH 169 3)]TJ /F11 9.9626 Tf 6.571 0 Td[(POR1DIFF(6)(C6(2,J))]TJ /F11 9.9626 Tf 7.085 0 Td[(C6(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HHH 170 4+(HH/2.)(RXN4(2,J+1)+3.RXN4(2,J))/4. 171 5+(HHH/2.)(RXN4(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.RXN4(2,J))/4. 172 B(12,12)=POR1DIFF(6)/HH+POR1DIFF(6)/HHH 253

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173 D(12,12)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR1DIFF(6)/HH 174 A(12,12)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR1DIFF(6)/HHH 175 B(12,11)=+HHH/2.omega(3./4.)+HH/2.omega(3./4.) 176 D(12,11)=+HH/2.omega(1./4.) 177 A(12,11)=+HHH/2.omega(1./4.) 178 B(12,24)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(3./4.))]TJ /F11 9.9626 Tf 7.832 0 Td[((HHH/2.)(3./4.) 179 D(12,24)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(1./4.) 180 A(12,24)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HHH/2.)(1./4.) 181 182 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.393 -111.146 Td[(GOx2 enzyme 183 G(13)=omegaC7(2,J) 184 1+RXN1(1,J) 185 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN2(1,J) 186 B(13,14)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 187 B(13,17)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 188 B(13,19)=+1. 189 190 G(14)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC7(1,J) 191 1+RXN1(2,J) 192 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN2(2,J) 193 B(14,13)=omega 194 B(14,18)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 195 B(14,20)=+1. 196 197 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.393 -276.463 Td[(GOx2 enzyme 198 G(15)=omegaC8(2,J) 199 1+RXN3(1,J) 200 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN4(1,J) 201 B(15,16)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 202 B(15,21)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 203 B(15,23)=+1. 204 205 G(16)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC8(1,J) 206 1+RXN3(2,J) 207 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN4(2,J) 208 B(16,15)=omega 209 B(16,22)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 210 B(16,24)=+1. 211 212 C REACTION1 213 G(17)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN1(1,J)+ratef1CONCSS(2,J)C1(1,J) 214 1+ratef1CONCSS(1,J)C2(1,J) 215 2)]TJ /F11 9.9626 Tf 7.084 0 Td[(C7(1,J)/equilib1 216 B(17,1)=)]TJ /F11 9.9626 Tf 8.386 -.001 Td[(ratef1CONCSS(2,J) 217 B(17,3)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef1CONCSS(1,J) 218 B(17,13)=+1./equilib1 219 B(17,17)=+1. 220 221 G(18)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN1(2,J)+ratef1CONCSS(2,J)C1(2,J) 222 1+ratef1CONCSS(1,J)C2(2,J) 223 2)]TJ /F11 9.9626 Tf 7.084 0 Td[(C7(2,J)/equilib1 224 B(18,2)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef1CONCSS(2,J) 225 B(18,4)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef1CONCSS(1,J) 226 B(18,14)=+1./equilib1 227 B(18,18)=+1. 228 229 C REACTION2 230 G(19)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN2(1,J)+ratef2C7(1,J) 254

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231 B(19,13)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef2 232 B(19,19)=+1. 233 234 G(20)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN2(2,J)+ratef2C7(2,J) 235 B(20,14)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef2 236 B(20,20)=+1. 237 238 C REACTION3 239 G(21)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN3(1,J)+ratef3CONCSS(4,J)C5(1,J) 240 1+ratef3CONCSS(5,J)C4(1,J) 241 2)]TJ /F11 9.9626 Tf 7.084 0 Td[(C8(1,J)/equilib3 242 B(21,7)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONCSS(5,J) 243 B(21,9)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONCSS(4,J) 244 B(21,15)=+1./equilib3 245 B(21,21)=+1. 246 247 G(22)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN3(2,J)+ratef3CONCSS(4,J)C5(2,J) 248 1+ratef3CONCSS(5,J)C4(2,J) 249 2)]TJ /F11 9.9626 Tf 7.084 0 Td[(C8(2,J)/equilib3 250 B(22,8)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONCSS(5,J) 251 B(22,10)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef3CONCSS(4,J) 252 B(22,16)=+1./equilib3 253 B(22,22)=+1. 254 255 C REACTION4 256 G(23)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN4(1,J)+ratef4C8(1,J) 257 B(23,15)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef4 258 B(23,23)=+1. 259 260 G(24)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN4(2,J)+ratef4C8(2,J) 261 B(24,16)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef4 262 B(24,24)=+1. 263 264 WRITE (14,301)J,(G(K),K=1,N) 265 266 RETURN 267 END 255

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CodeE.15.OscillatingContinuousGlucoseMonitorSubroutinefortheInnerRegion 1 SUBROUTINE INNER(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 COMMON /BAT/A(24,24),B(24,24),C(24,80001),D(24,49),G(24), 4 1X(24,24),Y(24,24) 5 COMMON /NST/N,NJ 6 7 COMMON /VAR/CONCSS(8,80001),RXNSS(4,80001),DIFF(8) 8 COMMON /VARR/PORGLU,HHH,KJ 9 COMMON /CON/C1(2,80001),C2(2,80001),C3(2,80001),C4(2,80001), 10 1C5(2,80001),C6(2,80001),C7(2,80001),C8(2,80001), 11 2RXN1(2,80001),RXN2(2,80001),RXN3(2,80001),RXN4(2,80001) 12 COMMON /RTE/ratef1,equilib1,ratef2,ratef3,equilib3,ratef4 13 COMMON /OTH/POR1,POR2,H,EBIG,HH,IJ 14 COMMON /BCI/FLUX,omega 15 COMMON /BUL/CBULK(8),PARH2O2,PAR02,PARGLUCOSE,JCOUNT 16 COMMON /DELT/DELTA1,DELTA2,FREQ(400),CB(2010,80001) 17 18 301 FORMAT (5x,'J='I5,24E15.6) 19 20 C For Glucose being consumed only 21 G(1)=omegaC1(2,J) 22 1+POR1DIFF(1)(C1(1,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C1(1,J)+C1(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH2. 23 3)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN1(1,J) 24 B(1,1)=2.POR1DIFF(1)/HH2. 25 A(1,1)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(1)/HH2. 26 D(1,1)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(1)/HH2. 27 B(1,2)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 28 B(1,17)=+1. 29 30 G(2)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC1(1,J) 31 1+POR1DIFF(1)(C1(2,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C1(2,J)+C1(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH2. 32 3)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN1(2,J) 33 B(2,2)=2.POR1DIFF(1)/HH2. 34 A(2,2)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(1)/HH2. 35 D(2,2)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(1)/HH2. 36 B(2,1)=omega 37 B(2,18)=+1. 38 39 C For GOx enzyme 40 G(3)=omegaC2(2,J) 41 1)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN1(1,J) 42 2+RXN4(1,J) 43 B(3,4)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 44 B(3,17)=+1. 45 B(3,23)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 46 47 G(4)=)]TJ /F11 9.9626 Tf 7.232 .001 Td[(omegaC2(1,J) 48 1)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN1(2,J) 49 2+RXN4(2,J) 50 B(4,3)=omega 51 B(4,18)=+1. 52 B(4,24)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 53 54 C For Gluconic Acid being produced only 55 G(5)=omegaC3(2,J) 56 1+POR1DIFF(3)(C3(1,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C3(1,J)+C3(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH2. 256

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57 3+RXN2(1,J) 58 B(5,5)=2.POR1DIFF(3)/HH2. 59 A(5,5)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(3)/HH2. 60 D(5,5)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(3)/HH2. 61 B(5,6)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 62 B(5,19)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 63 64 G(6)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC3(1,J) 65 1+POR1DIFF(3)(C3(2,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C3(2,J)+C3(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH2. 66 3+RXN2(2,J) 67 B(6,6)=2.POR1DIFF(3)/HH2. 68 A(6,6)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(3)/HH2. 69 D(6,6)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(3)/HH2. 70 B(6,5)=omega 71 B(6,20)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 72 73 C For GOx2 enzyme 74 G(7)=omegaC4(2,J) 75 1+RXN2(1,J) 76 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN3(1,J) 77 B(7,8)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 78 B(7,19)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 79 B(7,21)=+1. 80 81 G(8)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC4(1,J) 82 1+RXN2(2,J) 83 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN3(2,J) 84 B(8,7)=omega 85 B(8,20)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 86 B(8,22)=+1. 87 88 C For O2 being consumed only 89 G(9)=omegaC5(2,J) 90 1+POR1DIFF(5)(C5(1,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C5(1,J)+C5(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH2. 91 3)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN3(1,J) 92 B(9,9)=2.POR1DIFF(5)/HH2. 93 A(9,9)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(5)/HH2. 94 D(9,9)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(5)/HH2. 95 B(9,10)=)]TJ /F11 9.9626 Tf 7.233 0 Td[(omega 96 B(9,21)=+1. 97 98 G(10)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC5(1,J) 99 1+POR1DIFF(5)(C5(2,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C5(2,J)+C5(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH2. 100 3)]TJ /F11 9.9626 Tf 6.488 -.001 Td[(RXN3(2,J) 101 B(10,10)=2.POR1DIFF(5)/HH2. 102 A(10,10)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR1DIFF(5)/HH2. 103 D(10,10)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR1DIFF(5)/HH2. 104 B(10,9)=omega 105 B(10,22)=+1. 106 107 C For H2O2 reacting species 108 G(11)=omegaC6(2,J) 109 1+POR1DIFF(6)(C6(1,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C6(1,J)+C6(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH2. 110 3+RXN4(1,J) 111 B(11,11)=2.POR1DIFF(6)/HH2. 112 A(11,11)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR1DIFF(6)/HH2. 113 D(11,11)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR1DIFF(6)/HH2. 114 B(11,12)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 257

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115 B(11,23)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 116 117 G(12)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC6(1,J) 118 1+POR1DIFF(6)(C6(2,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C6(2,J)+C6(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH2. 119 3+RXN4(2,J) 120 B(12,12)=2.POR1DIFF(6)/HH2. 121 A(12,12)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR1DIFF(6)/HH2. 122 D(12,12)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR1DIFF(6)/HH2. 123 B(12,11)=omega 124 B(12,24)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 125 126 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.393 -133.188 Td[(GOx2 enzyme 127 G(13)=omegaC7(2,J) 128 1+RXN1(1,J) 129 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN2(1,J) 130 B(13,14)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 131 B(13,17)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 132 B(13,19)=+1. 133 134 G(14)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC7(1,J) 135 1+RXN1(2,J) 136 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN2(2,J) 137 B(14,13)=omega 138 B(14,18)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 139 B(14,20)=+1. 140 141 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.393 -298.506 Td[(GOx2 enzyme 142 G(15)=omegaC8(2,J) 143 1+RXN3(1,J) 144 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN4(1,J) 145 B(15,16)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 146 B(15,21)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 147 B(15,23)=+1. 148 149 G(16)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC8(1,J) 150 1+RXN3(2,J) 151 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN4(2,J) 152 B(16,15)=omega 153 B(16,22)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 154 B(16,24)=+1. 155 156 C REACTION1 157 G(17)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN1(1,J)+ratef1CONCSS(2,J)C1(1,J) 158 1+ratef1CONCSS(1,J)C2(1,J) 159 2)]TJ /F11 9.9626 Tf 7.084 0 Td[(C7(1,J)/equilib1 160 B(17,1)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef1CONCSS(2,J) 161 B(17,3)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef1CONCSS(1,J) 162 B(17,13)=+1./equilib1 163 B(17,17)=+1. 164 165 G(18)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN1(2,J)+ratef1CONCSS(2,J)C1(2,J) 166 1+ratef1CONCSS(1,J)C2(2,J) 167 2)]TJ /F11 9.9626 Tf 7.084 0 Td[(C7(2,J)/equilib1 168 B(18,2)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef1CONCSS(2,J) 169 B(18,4)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef1CONCSS(1,J) 170 B(18,14)=+1./equilib1 171 B(18,18)=+1. 172 258

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173 C REACTION2 174 G(19)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN2(1,J)+ratef2C7(1,J) 175 B(19,13)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef2 176 B(19,19)=+1. 177 178 G(20)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN2(2,J)+ratef2C7(2,J) 179 B(20,14)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef2 180 B(20,20)=+1. 181 182 C REACTION3 183 G(21)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN3(1,J)+ratef3CONCSS(4,J)C5(1,J) 184 1+ratef3CONCSS(5,J)C4(1,J) 185 2)]TJ /F11 9.9626 Tf 7.084 0 Td[(C8(1,J)/equilib3 186 B(21,7)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONCSS(5,J) 187 B(21,9)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONCSS(4,J) 188 B(21,15)=+1./equilib3 189 B(21,21)=+1. 190 191 G(22)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN3(2,J)+ratef3CONCSS(4,J)C5(2,J) 192 1+ratef3CONCSS(5,J)C4(2,J) 193 2)]TJ /F11 9.9626 Tf 7.084 0 Td[(C8(2,J)/equilib3 194 B(22,8)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONCSS(5,J) 195 B(22,10)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef3CONCSS(4,J) 196 B(22,16)=+1./equilib3 197 B(22,22)=+1. 198 199 C REACTION4 200 G(23)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN4(1,J)+ratef4C8(1,J) 201 B(23,15)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef4 202 B(23,23)=+1. 203 204 G(24)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN4(2,J)+ratef4C8(2,J) 205 B(24,16)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef4 206 B(24,24)=+1. 207 208 c SAVE G OUT DATA 209 DO 11I=2,13 210 11 If (I. EQ .J) WRITE (14,301)J,(G(K),K=1,N) 211 IF (J. EQ .IJ/2) THEN 212 WRITE (14,301)J,(G(K),K=1,N) 213 ELSE IF (J. EQ .(IJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) THEN 214 WRITE (14,301)J,(G(K),K=1,N) 215 ELSE IF (J. EQ .(IJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(2)) THEN 216 WRITE (14,301)J,(G(K),K=1,N) 217 ELSE IF (J. EQ .(IJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(3)) THEN 218 WRITE (14,301)J,(G(K),K=1,N) 219 END IF 220 221 RETURN 222 END 259

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CodeE.16.OscillatingContinuousGlucoseMonitorSubroutinefortheSecondCoupler 1 SUBROUTINE COUPLER2(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 COMMON /BAT/A(24,24),B(24,24),C(24,80001),D(24,49),G(24), 4 1X(24,24),Y(24,24) 5 COMMON /NST/N,NJ 6 7 COMMON /VAR/CONCSS(8,80001),RXNSS(4,80001),DIFF(8) 8 COMMON /VARR/PORGLU,HHH,KJ 9 COMMON /CON/C1(2,80001),C2(2,80001),C3(2,80001),C4(2,80001), 10 1C5(2,80001),C6(2,80001),C7(2,80001),C8(2,80001), 11 2RXN1(2,80001),RXN2(2,80001),RXN3(2,80001),RXN4(2,80001) 12 COMMON /RTE/ratef1,equilib1,ratef2,ratef3,equilib3,ratef4 13 COMMON /OTH/POR1,POR2,H,EBIG,HH,IJ 14 COMMON /BCI/FLUX,omega 15 COMMON /BUL/CBULK(8),PARH2O2,PAR02,PARGLUCOSE,JCOUNT 16 COMMON /DELT/DELTA1,DELTA2,FREQ(400),CB(2010,80001) 17 18 301 FORMAT (5x,'J='I5,24E15.6) 19 20 C For Glucose being consumed only 21 G(1)=H/2.omega(C1(2,J+1)+3.C1(2,J))/4. 22 1+HH/2.omega(C1(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C1(2,J))/4. 23 2+PORGLUDIFF(1)(C1(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C1(1,J))/H 24 3)]TJ /F11 9.9626 Tf 6.571 0 Td[(POR1DIFF(1)(C1(1,J))]TJ /F11 9.9626 Tf 7.085 0 Td[(C1(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH 25 9)]TJ /F11 9.9626 Tf 7.831 0 Td[((HH/2.)(RXN1(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.RXN1(1,J))/4. 26 B(1,1)=PORGLUDIFF(1)/H+POR1DIFF(1)/HH 27 D(1,1)=)]TJ /F11 9.9626 Tf 5.79 0 Td[(PORGLUDIFF(1)/H 28 A(1,1)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(1)/HH 29 B(1,2)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2.omega(3./4.))]TJ /F11 9.9626 Tf 6.411 0 Td[(H/2.omega(3./4.) 30 D(1,2)=)]TJ /F11 9.9626 Tf 6.116 0 Td[(H/2.omega(1./4.) 31 A(1,2)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2.omega(1./4.) 32 B(1,17)=(HH/2.)(1./4.) 33 A(1,17)=(HH/2.)(1./4.) 34 35 G(2)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2.omega(C1(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C1(1,J))/4. 36 1)]TJ /F11 9.9626 Tf 6.411 0 Td[(H/2.omega(C1(1,J+1)+3.C1(1,J))/4. 37 2+PORGLUDIFF(1)(C1(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C1(2,J))/H 38 3)]TJ /F11 9.9626 Tf 6.571 0 Td[(POR1DIFF(1)(C1(2,J))]TJ /F11 9.9626 Tf 7.085 0 Td[(C1(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH 39 5)]TJ /F11 9.9626 Tf 7.831 0 Td[((HH/2.)(RXN1(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.RXN1(2,J))/4. 40 B(2,2)=PORGLUDIFF(1)/H+POR1DIFF(1)/HH 41 D(2,2)=)]TJ /F11 9.9626 Tf 5.79 0 Td[(PORGLUDIFF(1)/H 42 A(2,2)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(1)/HH 43 B(2,1)=+HH/2.omega(3./4.)+H/2.omega(3./4.) 44 D(2,1)=+H/2.omega(1./4.) 45 A(2,1)=+HH/2.omega(1./4.) 46 B(2,18)=(HH/2.)(3./4.) 47 A(2,18)=(HH/2.)(1./4.) 48 49 C For GOx enzyme 50 G(3)=omegaC2(2,J) 51 1)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN1(1,J) 52 2+RXN4(1,J) 53 B(3,4)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 54 B(3,17)=+1. 55 B(3,23)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 56 260

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57 G(4)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC2(1,J) 58 1)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN1(2,J) 59 2+RXN4(2,J) 60 B(4,3)=omega 61 B(4,18)=+1. 62 B(4,24)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 63 64 C For Gluconic Acid being produced only 65 G(5)=HH/2.omega(C3(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C3(2,J))/4. 66 1+H/2.omega(C3(2,J+1)+3.C3(2,J))/4. 67 2+PORGLUDIFF(3)(C3(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C3(1,J))/H 68 3)]TJ /F11 9.9626 Tf 6.571 0 Td[(POR1DIFF(3)(C3(1,J))]TJ /F11 9.9626 Tf 7.085 0 Td[(C3(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH 69 5+(HH/2.)(RXN2(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.RXN2(1,J))/4. 70 B(5,5)=PORGLUDIFF(3)/H+POR1DIFF(3)/HH 71 D(5,5)=)]TJ /F11 9.9626 Tf 5.79 0 Td[(PORGLUDIFF(3)/H 72 A(5,5)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(3)/HH 73 B(5,6)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2.omega(3./4.))]TJ /F11 9.9626 Tf 6.411 0 Td[(H/2.omega(3./4.) 74 D(5,6)=)]TJ /F11 9.9626 Tf 6.116 0 Td[(H/2.omega(1./4.) 75 A(5,6)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2.omega(1./4.) 76 B(5,19)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(3./4.) 77 A(5,19)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(1./4.) 78 79 G(6)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2.omega(C3(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C3(1,J))/4. 80 1)]TJ /F11 9.9626 Tf 6.411 0 Td[(H/2.omega(C3(1,J+1)+3.C3(1,J))/4. 81 2+PORGLUDIFF(3)(C3(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C3(2,J))/H 82 3)]TJ /F11 9.9626 Tf 6.571 0 Td[(POR1DIFF(3)(C3(2,J))]TJ /F11 9.9626 Tf 7.085 0 Td[(C3(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH 83 5+(HH/2.)(RXN2(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.RXN2(2,J))/4. 84 B(6,6)=PORGLUDIFF(3)/H+POR1DIFF(3)/HH 85 D(6,6)=)]TJ /F11 9.9626 Tf 5.79 0 Td[(PORGLUDIFF(3)/H 86 A(6,6)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(3)/HH 87 B(6,5)=+HH/2.omega(3./4.)+H/2.omega(3./4.) 88 D(6,5)=+H/2.omega(1./4.) 89 A(6,5)=+HH/2.omega(1./4.) 90 B(6,20)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(3./4.) 91 A(6,20)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(1./4.) 92 93 C For GOx2 enzyme 94 G(7)=omegaC4(2,J) 95 1+RXN2(1,J) 96 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN3(1,J) 97 B(7,8)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 98 B(7,19)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 99 B(7,21)=+1. 100 101 G(8)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC4(1,J) 102 1+RXN2(2,J) 103 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN3(2,J) 104 B(8,7)=omega 105 B(8,20)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 106 B(8,22)=+1. 107 108 C For O2 being consumed only 109 G(9)=HH/2.omega(C5(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C5(2,J))/4. 110 1+H/2.omega(C5(2,J+1)+3.C5(2,J))/4. 111 2+POR2DIFF(5)(C5(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C5(1,J))/H 112 3)]TJ /F11 9.9626 Tf 6.571 0 Td[(POR1DIFF(5)(C5(1,J))]TJ /F11 9.9626 Tf 7.085 0 Td[(C5(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH 113 5)]TJ /F11 9.9626 Tf 7.831 0 Td[((HH/2.)(RXN3(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.RXN3(1,J))/4. 114 B(9,9)=POR2DIFF(5)/H+POR1DIFF(5)/HH 261

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115 D(9,9)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR2DIFF(5)/H 116 A(9,9)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR1DIFF(5)/HH 117 B(9,10)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2.omega(3./4.))]TJ /F11 9.9626 Tf 6.411 0 Td[(H/2.omega(3./4.) 118 D(9,10)=)]TJ /F11 9.9626 Tf 6.116 0 Td[(H/2.omega(1./4.) 119 A(9,10)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2.omega(1./4.) 120 B(9,21)=(HH/2.)(3./4.) 121 A(9,21)=(HH/2.)(1./4.) 122 123 G(10)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2.omega(C5(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C5(1,J))/4. 124 1)]TJ /F11 9.9626 Tf 6.411 0 Td[(H/2.omega(C5(1,J+1)+3.C5(1,J))/4. 125 2+POR2DIFF(5)(C5(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C5(2,J))/H 126 3)]TJ /F11 9.9626 Tf 6.571 0 Td[(POR1DIFF(5)(C5(2,J))]TJ /F11 9.9626 Tf 7.085 0 Td[(C5(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH 127 5)]TJ /F11 9.9626 Tf 7.831 0 Td[((HH/2.)(RXN3(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.RXN3(2,J))/4. 128 B(10,10)=POR2DIFF(5)/H+POR1DIFF(5)/HH 129 D(10,10)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR2DIFF(5)/H 130 A(10,10)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR1DIFF(5)/HH 131 B(10,9)=+HH/2.omega(3./4.)+H/2.omega(3./4.) 132 D(10,9)=+H/2.omega(1./4.) 133 A(10,9)=+HH/2.omega(1./4.) 134 B(10,22)=(HH/2.)(3./4.) 135 A(10,22)=(HH/2.)(1./4.) 136 137 C For H2O2 reacting species 138 G(11)=HH/2.omega(C6(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C6(2,J))/4. 139 1+H/2.omega(C6(2,J+1)+3.C6(2,J))/4. 140 2+POR2DIFF(6)(C6(1,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C6(1,J))/H 141 3)]TJ /F11 9.9626 Tf 6.571 0 Td[(POR1DIFF(6)(C6(1,J))]TJ /F11 9.9626 Tf 7.085 0 Td[(C6(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH 142 5+(HH/2.)(RXN4(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.RXN4(1,J))/4. 143 B(11,11)=POR2DIFF(6)/H+POR1DIFF(6)/HH 144 D(11,11)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR2DIFF(6)/H 145 A(11,11)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR1DIFF(6)/HH 146 B(11,12)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2.omega(3./4.))]TJ /F11 9.9626 Tf 6.411 0 Td[(H/2.omega(3./4.) 147 D(11,12)=)]TJ /F11 9.9626 Tf 6.116 0 Td[(H/2.omega(1./4.) 148 A(11,12)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2.omega(1./4.) 149 B(11,23)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(3./4.) 150 A(11,23)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(1./4.) 151 152 G(12)=)]TJ /F11 9.9626 Tf 5.867 0 Td[(HH/2.omega(C6(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.C6(1,J))/4. 153 1)]TJ /F11 9.9626 Tf 6.411 0 Td[(H/2.omega(C6(1,J+1)+3.C6(1,J))/4. 154 2+POR2DIFF(6)(C6(2,J+1))]TJ /F11 9.9626 Tf 7.085 0 Td[(C6(2,J))/H 155 3)]TJ /F11 9.9626 Tf 6.571 0 Td[(POR1DIFF(6)(C6(2,J))]TJ /F11 9.9626 Tf 7.085 0 Td[(C6(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/HH 156 5+(HH/2.)(RXN4(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)+3.RXN4(2,J))/4. 157 B(12,12)=POR2DIFF(6)/H+POR1DIFF(6)/HH 158 D(12,12)=)]TJ /F11 9.9626 Tf 6.277 -.001 Td[(POR2DIFF(6)/H 159 A(12,12)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR1DIFF(6)/HH 160 B(12,11)=+HH/2.omega(3./4.)+H/2.omega(3./4.) 161 D(12,11)=+H/2.omega(1./4.) 162 A(12,11)=+HH/2.omega(1./4.) 163 B(12,24)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(3./4.) 164 A(12,24)=)]TJ /F11 9.9626 Tf 7.461 0 Td[((HH/2.)(1./4.) 165 166 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.393 -574.035 Td[(GOx2 enzyme 167 G(13)=omegaC7(2,J) 168 1+RXN1(1,J) 169 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN2(1,J) 170 B(13,14)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 171 B(13,17)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 172 B(13,19)=+1. 262

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173 174 G(14)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC7(1,J) 175 1+RXN1(2,J) 176 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN2(2,J) 177 B(14,13)=omega 178 B(14,18)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 179 B(14,20)=+1. 180 181 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.393 -100.125 Td[(GOx2 enzyme 182 G(15)=omegaC8(2,J) 183 1+RXN3(1,J) 184 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN4(1,J) 185 B(15,16)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 186 B(15,21)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 187 B(15,23)=+1. 188 189 G(16)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC8(1,J) 190 1+RXN3(2,J) 191 2)]TJ /F11 9.9626 Tf 6.488 0 Td[(RXN4(2,J) 192 B(16,15)=omega 193 B(16,22)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 194 B(16,24)=+1. 195 196 C REACTION1 197 G(17)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN1(1,J)+ratef1CONCSS(2,J)C1(1,J) 198 1+ratef1CONCSS(1,J)C2(1,J) 199 2)]TJ /F11 9.9626 Tf 7.084 0 Td[(C7(1,J)/equilib1 200 B(17,1)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef1CONCSS(2,J) 201 B(17,3)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef1CONCSS(1,J) 202 B(17,13)=+1./equilib1 203 B(17,17)=+1. 204 205 G(18)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN1(2,J)+ratef1CONCSS(2,J)C1(2,J) 206 1+ratef1CONCSS(1,J)C2(2,J) 207 2)]TJ /F11 9.9626 Tf 7.084 0 Td[(C7(2,J)/equilib1 208 B(18,2)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef1CONCSS(2,J) 209 B(18,4)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef1CONCSS(1,J) 210 B(18,14)=+1./equilib1 211 B(18,18)=+1. 212 213 C REACTION2 214 G(19)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN2(1,J)+ratef2C7(1,J) 215 B(19,13)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef2 216 B(19,19)=+1. 217 218 G(20)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN2(2,J)+ratef2C7(2,J) 219 B(20,14)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef2 220 B(20,20)=+1. 221 222 C REACTION3 223 G(21)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN3(1,J)+ratef3CONCSS(4,J)C5(1,J) 224 1+ratef3CONCSS(5,J)C4(1,J) 225 2)]TJ /F11 9.9626 Tf 7.084 0 Td[(C8(1,J)/equilib3 226 B(21,7)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONCSS(5,J) 227 B(21,9)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONCSS(4,J) 228 B(21,15)=+1./equilib3 229 B(21,21)=+1. 230 263

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231 G(22)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN3(2,J)+ratef3CONCSS(4,J)C5(2,J) 232 1+ratef3CONCSS(5,J)C4(2,J) 233 2)]TJ /F11 9.9626 Tf 7.084 0 Td[(C8(2,J)/equilib3 234 B(22,8)=)]TJ /F11 9.9626 Tf 8.386 0 Td[(ratef3CONCSS(5,J) 235 B(22,10)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef3CONCSS(4,J) 236 B(22,16)=+1./equilib3 237 B(22,22)=+1. 238 239 C REACTION4 240 G(23)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN4(1,J)+ratef4C8(1,J) 241 B(23,15)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef4 242 B(23,23)=+1. 243 244 G(24)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN4(2,J)+ratef4C8(2,J) 245 B(24,16)=)]TJ /F11 9.9626 Tf 8.385 0 Td[(ratef4 246 B(24,24)=+1. 247 248 WRITE (14,301)J,(G(K),K=1,N) 249 250 RETURN 251 END 264

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CodeE.17.OscillatingContinuousGlucoseMonitorSubroutinefortheGLMRegion 1 SUBROUTINE OUTER(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 COMMON /BAT/A(24,24),B(24,24),C(24,80001),D(24,49),G(24), 4 1X(24,24),Y(24,24) 5 COMMON /NST/N,NJ 6 7 COMMON /VAR/CONCSS(8,80001),RXNSS(4,80001),DIFF(8) 8 COMMON /VARR/PORGLU,HHH,KJ 9 COMMON /CON/C1(2,80001),C2(2,80001),C3(2,80001),C4(2,80001), 10 1C5(2,80001),C6(2,80001),C7(2,80001),C8(2,80001), 11 2RXN1(2,80001),RXN2(2,80001),RXN3(2,80001),RXN4(2,80001) 12 COMMON /RTE/ratef1,equilib1,ratef2,ratef3,equilib3,ratef4 13 COMMON /OTH/POR1,POR2,H,EBIG,HH,IJ 14 COMMON /BCI/FLUX,omega 15 COMMON /BUL/CBULK(8),PARH2O2,PAR02,PARGLUCOSE,JCOUNT 16 COMMON /DELT/DELTA1,DELTA2,FREQ(400),CB(2010,80001) 17 18 301 FORMAT (5x,'J='I5,24E15.6) 19 20 C For Glucose being consumed only 21 G(1)=omegaC1(2,J) 22 1+PORGLUDIFF(1)(C1(1,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C1(1,J)+C1(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/H2. 23 B(1,1)=2.PORGLUDIFF(1)/H2. 24 A(1,1)=)]TJ /F11 9.9626 Tf 5.79 0 Td[(PORGLUDIFF(1)/H2. 25 D(1,1)=)]TJ /F11 9.9626 Tf 5.79 0 Td[(PORGLUDIFF(1)/H2. 26 B(1,2)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 27 28 G(2)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC1(1,J) 29 1+PORGLUDIFF(1)(C1(2,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C1(2,J)+C1(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/H2. 30 B(2,2)=2.PORGLUDIFF(1)/H2. 31 A(2,2)=)]TJ /F11 9.9626 Tf 5.79 0 Td[(PORGLUDIFF(1)/H2. 32 D(2,2)=)]TJ /F11 9.9626 Tf 5.79 0 Td[(PORGLUDIFF(1)/H2. 33 B(2,1)=omega 34 35 C For GOx enzyme 36 G(3)=C2(1,J) 37 B(3,3)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 38 39 G(4)=C2(2,J) 40 B(4,4)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 41 42 C For Gluconic Acid being produced only 43 G(5)=omegaC3(2,J) 44 1+PORGLUDIFF(3)(C3(1,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C3(1,J)+C3(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/H2. 45 B(5,5)=2.PORGLUDIFF(3)/H2. 46 A(5,5)=)]TJ /F11 9.9626 Tf 5.79 0 Td[(PORGLUDIFF(3)/H2. 47 D(5,5)=)]TJ /F11 9.9626 Tf 5.79 .001 Td[(PORGLUDIFF(3)/H2. 48 B(5,6)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 49 50 G(6)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC3(1,J) 51 1+PORGLUDIFF(3)(C3(2,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C3(2,J)+C3(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/H2. 52 B(6,6)=2.PORGLUDIFF(3)/H2. 53 A(6,6)=)]TJ /F11 9.9626 Tf 5.79 0 Td[(PORGLUDIFF(3)/H2. 54 D(6,6)=)]TJ /F11 9.9626 Tf 5.79 0 Td[(PORGLUDIFF(3)/H2. 55 B(6,5)=omega 56 265

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57 C For GOx2 enzyme 58 G(7)=C4(1,J) 59 B(7,7)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 60 61 G(8)=C4(2,J) 62 B(8,8)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 63 64 C For O2 being consumed only 65 G(9)=omegaC5(2,J) 66 1+POR2DIFF(5)(C5(1,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C5(1,J)+C5(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/H2. 67 B(9,9)=2.POR2DIFF(5)/H2. 68 A(9,9)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR2DIFF(5)/H2. 69 D(9,9)=)]TJ /F11 9.9626 Tf 6.276 0 Td[(POR2DIFF(5)/H2. 70 B(9,10)=)]TJ /F11 9.9626 Tf 7.233 0 Td[(omega 71 72 G(10)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC5(1,J) 73 1+POR2DIFF(5)(C5(2,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C5(2,J)+C5(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/H2. 74 B(10,10)=2.POR2DIFF(5)/H2. 75 A(10,10)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR2DIFF(5)/H2. 76 D(10,10)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR2DIFF(5)/H2. 77 B(10,9)=omega 78 79 C For H2O2 reacting species 80 G(11)=omegaC6(2,J) 81 1+POR2DIFF(6)(C6(1,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C6(1,J)+C6(1,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/H2. 82 B(11,11)=2.POR2DIFF(6)/H2. 83 A(11,11)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR2DIFF(6)/H2. 84 D(11,11)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR2DIFF(6)/H2. 85 B(11,12)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omega 86 87 G(12)=)]TJ /F11 9.9626 Tf 7.232 0 Td[(omegaC6(1,J) 88 1+POR2DIFF(6)(C6(2,J+1))]TJ /F11 9.9626 Tf 8.321 0 Td[(2.C6(2,J)+C6(2,J)]TJ /F11 9.9626 Tf 8.081 0 Td[(1))/H2. 89 B(12,12)=2.POR2DIFF(6)/H2. 90 A(12,12)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR2DIFF(6)/H2. 91 D(12,12)=)]TJ /F11 9.9626 Tf 6.277 0 Td[(POR2DIFF(6)/H2. 92 B(12,11)=omega 93 94 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.393 -419.738 Td[(GOx2 enzyme 95 G(13)=C7(1,J) 96 B(13,13)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 97 98 G(14)=C7(2,J) 99 B(14,14)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 100 101 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.131 -496.887 Td[(GOx enzyme 102 G(15)=C8(1,J) 103 B(15,15)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 104 105 G(16)=C8(2,J) 106 B(16,16)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 107 108 C REACTION1 109 G(17)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN1(1,J) 110 B(17,17)=+1. 111 112 G(18)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN1(2,J) 113 B(18,18)=+1. 114 266

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115 C REACTION2 116 G(19)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN2(1,J) 117 B(19,19)=+1. 118 119 G(20)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN2(2,J) 120 B(20,20)=+1. 121 122 C REACTION3 123 G(21)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN3(1,J) 124 B(21,21)=+1. 125 126 G(22)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN3(2,J) 127 B(22,22)=+1. 128 129 C REACTION4 130 G(23)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN4(1,J) 131 B(23,23)=+1. 132 133 G(24)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN4(2,J) 134 B(24,24)=+1. 135 136 c SAVE G OUT DATA 137 IF (J. EQ .(IJ+(NJ)]TJ /F11 9.9626 Tf 7.942 0 Td[(IJ)/2)) THEN 138 WRITE (14,301)J,(G(K),K=1,N) 139 ELSE IF (J. EQ .(NJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1)) THEN 140 WRITE (14,301)J,(G(K),K=1,N) 141 END IF 142 143 RETURN 144 END 267

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CodeE.18.OscillatingContinuousGlucoseMonitorSubroutinefortheBulkBoundaryCondition 1 SUBROUTINE BCNJ(J) 2 IMPLICIT DOUBLE PRECISION (A)]TJ /F11 9.9626 Tf 6.365 0 Td[(H,O)]TJ /F11 9.9626 Tf 6.807 0 Td[(Z) 3 COMMON /BAT/A(24,24),B(24,24),C(24,80001),D(24,49),G(24), 4 1X(24,24),Y(24,24) 5 COMMON /NST/N,NJ 6 7 COMMON /VAR/CONCSS(8,80001),RXNSS(4,80001),DIFF(8) 8 COMMON /VARR/PORGLU,HHH,KJ 9 COMMON /CON/C1(2,80001),C2(2,80001),C3(2,80001),C4(2,80001), 10 1C5(2,80001),C6(2,80001),C7(2,80001),C8(2,80001), 11 2RXN1(2,80001),RXN2(2,80001),RXN3(2,80001),RXN4(2,80001) 12 COMMON /RTE/ratef1,equilib1,ratef2,ratef3,equilib3,ratef4 13 COMMON /OTH/POR1,POR2,H,EBIG,HH,IJ 14 COMMON /BCI/FLUX,omega 15 COMMON /BUL/CBULK(8),PARH2O2,PAR02,PARGLUCOSE,JCOUNT 16 COMMON /DELT/DELTA1,DELTA2,FREQ(400),CB(2010,80001) 17 18 301 FORMAT (5x,'J='I5,24E15.6) 19 20 DO 42I=1,2 21 C For Glucose being consumed only 22 G(I)=C1(I,J) 23 B(I,I)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 24 C For GOx enzyme 25 G(2+I)=C2(I,J) 26 B(2+I,2+I)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 27 C For Gluconic Acid being produced only 28 G(4+I)=C3(I,J) 29 B(4+I,4+I)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 30 C For GOx2 enzyme 31 G(6+I)=C4(I,J) 32 B(6+I,6+I)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 33 C For O2 being consumed only 34 G(8+I)=C5(I,J) 35 B(8+I,8+I)=)]TJ /F11 9.9626 Tf 7.86 0 Td[(1. 36 C For H2O2 reacting species 37 G(10+I)=C6(I,J) 38 B(10+I,10+I)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 39 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.393 -478.32 Td[(GOx2 enzyme 40 G(12+I)=C7(I,J) 41 B(12+I,12+I)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 42 C For CX )]TJ ET 0 G 0 g 0 0.5 0.2 RG 0 0.5 0.2 rg BT /F11 9.9626 Tf 77.131 -511.383 Td[(GOx enzyme 43 G(14+I)=C8(I,J) 44 42B(14+I,14+I)=)]TJ /F11 9.9626 Tf 7.859 0 Td[(1. 45 46 C REACTION1 47 G(17)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN1(1,J) 48 B(17,17)=+1. 49 50 G(18)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN1(2,J) 51 B(18,18)=+1. 52 53 C REACTION2 54 G(19)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN2(1,J) 268

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55 B(19,19)=+1. 56 57 G(20)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN2(2,J) 58 B(20,20)=+1. 59 60 C REACTION3 61 G(21)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN3(1,J) 62 B(21,21)=+1. 63 64 G(22)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN3(2,J) 65 B(22,22)=+1. 66 67 C REACTION4 68 G(23)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN4(1,J) 69 B(23,23)=+1. 70 71 G(24)=)]TJ /F11 9.9626 Tf 6.194 0 Td[(RXN4(2,J) 72 B(24,24)=+1. 73 74 WRITE (14,301)J,(G(K),K=1,N) 75 76 RETURN 77 END 269

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CodeE.19.Matlabcodetocreateanddimensionlessdiusionimpedanceandoverallimpedance 1 % Inserting concentration data from Fortran 2 3 clc ; close all ; clear all ; 4 format longE; 5 6 % Read the unsteady state data at each frequency 7 B= dlmread ('cdhgox H2O2 out.txt'); 8 9 Bss1= dlmread ('cdhgox out.txt'); 10 Bss=Bss1(:,6); 11 12 % Read constant values used in the Fortran code 13 M= dlmread ('cdhgox values out.txt'); 14 15 N=M(1); 16 NJ=M(2); 17 IJ=M(3); 18 KJ=M(4); 19 H=M(5); 20 HH=M(6); 21 HHH=M(7); 22 V=M(8); 23 AKB=M(9); 24 BB=M(10); 25 DiffH2O2=M(11); 26 RTB=M(12); 27 POR1=M(13); 28 nf=M(14); 29 30 % Read frequency points Kw = omega KK = K 31 K= dlmread ('kgox values out.txt'); 32 K=K'; 33 Kw= zeros (1,nf); 34 KK= zeros (1,nf); 35 for n=1:nf 36 Kw(n)=K(n+nf); 37 KK(n)=K(n); 38 end 39 40 % Read the steady state values for CB 41 Css= dlmread ('cdhgox out.txt'); 42 43 % Other constants 44 F=96487; 45 46 % Create y values for plotting 47 y= zeros (NJ,1); 48 49 far=HHH(KJ)]TJ /F11 9.9626 Tf 8.081 0 Td[(1); 50 y1=0:HHH:far; 51 52 far1=HH(IJ)]TJ /F11 9.9626 Tf 6.558 0 Td[(KJ); 53 y2=y1(KJ):HH:y1(KJ)+far1; 54 270

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55 far2=H(NJ)]TJ /F11 9.9626 Tf 7.942 0 Td[(IJ); 56 y3=y2(IJ)]TJ /F11 9.9626 Tf 6.558 0 Td[(KJ+1):H:y2(IJ)]TJ /F11 9.9626 Tf 6.558 0 Td[(KJ+1)+far2; 57 58 for k=1:KJ)]TJ /F11 9.9626 Tf 7.491 0 Td[(1 59 y(k)=y1(k); 60 end 61 for k=KJ:IJ)]TJ /F11 9.9626 Tf 7.49 0 Td[(1 62 y(k)=y2(k)]TJ /F11 9.9626 Tf 6.559 0 Td[(KJ+1); 63 end 64 for k=IJ:NJ 65 y(k)=y3(k)]TJ /F11 9.9626 Tf 7.942 0 Td[(IJ+1); 66 end 67 68 % Create K2 from K1 69 KK2= zeros (1,nf); 70 for n=1:nf 71 KK2(n)=KK(n)(far2/far1)^2; 72 end 73 74 % Create complex numbers from unsteady state data 75 CB= zeros (NJ,nf); 76 for n=1:nf 77 for k=1:NJ 78 CB(k,n)=complex(B(k,2n)]TJ /F11 9.9626 Tf 8.08 0 Td[(1),B(k,2n)); 79 end 80 end 81 82 figure (1) 83 plot (CB,Css(:,1),')]TJ /F11 9.9626 Tf 7.38 0 Td[(b'); hold on; 84 85 % Calculate the impedance 86 Zdd= zeros (1,nf); 87 for k=1:nf 88 Zdd(k)=()]TJ /F11 9.9626 Tf 6.695 0 Td[(CB(1,k)/(()]TJ /F11 9.9626 Tf 7.555 0 Td[(CB(3,k)+4CB(2,k))]TJ /F11 9.9626 Tf 7.793 0 Td[(3CB(1,k))/(2HHH)))/((far+far1)); 89 end 90 91 Zdfront=(RTBAKB exp (BBV))/(2FDiffH2O2POR1); 92 Zd= zeros (1,nf); 93 for k=1:nf 94 Zd(k)=Zdfront()]TJ /F11 9.9626 Tf 7.249 0 Td[(CB(1,k)/(()]TJ /F11 9.9626 Tf 7.555 0 Td[(CB(3,k)+4CB(2,k))]TJ /F11 9.9626 Tf 7.793 0 Td[(3CB(1,k))/(2HHH))); 95 end 96 97 Zf= zeros (1,nf); 98 for k=1:nf 99 Zf(k)=RTB+Zd(k); 100 end 101 102 Zo= zeros (1,nf); 103 for n=1:nf 104 Zo(n)=10+Zf(n)/(1+(1iKw(n)20e)]TJ /F11 9.9626 Tf 7.804 0 Td[(6Zf(n))); 105 end 106 107 % Oscillating concentration 108 o=[1234]; 109 ci1= zeros (NJ, length (o)); 110 ci2= zeros (NJ, length (o)); 111 dimfreq=[61101144181]; 112 for k=1:NJ 271

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113 for l=1: length (o) 114 t(l)=o(l) pi /2; 115 ci1(k,l)= real (CB(k,141) exp (1it(l))); 116 ci2(k,l)= real (CB(k,101) exp (1it(l))); 117 ci3(k,l)= real (CB(k,61) exp (1it(l))); 118 end 119 end 120 121 figure (1) 122 plot (y,Css(:,1),')]TJ /F11 9.9626 Tf 7.38 0 Td[(b'); hold on; 123 plot (y,Css(:,6),')]TJ /F11 9.9626 Tf 8.196 0 Td[(r'); 124 plot (y,Css(:,2),')]TJ /F11 9.9626 Tf 7.518 0 Td[(k'); 125 legend ('SSCGlucose','SSCH2O2','SSCGOX'); 126 title ('SteadyStateConcentrationawayfromElectrodeSurface'); 127 xlabel ('Length,cm'); 128 ylabel ('Concentration,moles/cm3'); 129 130 figure (2) 131 plot ( real (Zdd),)]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 96.218 -210.336 Td[(imag (Zdd),')]TJ /F11 9.9626 Tf 8.081 0 Td[(ks'); hold on; axis equal; 132 title ('Nyquistplot'); 133 xlabel ('RealpartofImpedance'); 134 ylabel ('ImaginarypartofImpedance'); 135 reald= real (Zdd); 136 imagd= imag (Zdd); 137 138 real = real (Zo); 139 imag = imag (Zo); 140 141 figure (3) 142 plot ( real ,)]TJ ET 0 0.1 0.5 RG 0 0.1 0.5 rg BT /F11 9.9626 Tf 66.234 -331.569 Td[(imag ,')]TJ /F11 9.9626 Tf 8.081 0 Td[(ks'); hold on; axis equal; 143 title ('OverallImpedanceNyquistplot'); 144 145 impedanceo= zeros (nf,2); 146 impedanceo(:,1)= real '; 147 impedanceo(:,2)= imag '; 148 149 impedancedd= zeros (nf,2); 150 impedancedd(:,1)=reald'; 151 impedancedd(:,2)=imagd'; 272

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BIOGRAPHICALSKETCHMorganHardingreceivedherbachelorsofChemicalEngineeringattheUniversityofIdahoinMay2012.ShethenbecameaPh.D.studentattheUniversityofFloridastudyingElectrochemicalImpedanceSpectroscopyinProfessorMarkOrazemsgroup.ShewasaveryactivestudentwhileattheUniversityofFloridaasasocialchairandpresidentoftheGraduateAssociationofChemicalEngineers,Vice-ChairandChairoftheEngineeringGraduateStudentCouncilandChairoftheUniversityofFlorida'sStudentChapteroftheElectrochemicalSociety.ShewasalsoamemberoftheChemicalEngineeringStudentSafetyCouncil,UniversityofFlorida'sStudentChapteroftheAmericanSocietyforEngineeringEducation,andtheOrganizationforGraduateStudentAdvancementandProfessionalDevelopment.MorganwonaC200scholarshipinFebruary2016andwasabletousethewinningstofundasummerabroadexperiencein2016conductingresearchwithDr.VincentViverinParis,FranceatthePierreandMarieCurieUniversity.DuringhertimeattheUniversityofFlorida,MorganwasmarriedtoDr.EricHazelbaker,afellowUniversityofFloridaChemicalEngineeringPh.D.graduate,inDecember2016.ShereceivedherPh.D.fromtheUniversityofFloridainthespringof2017. 281