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

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Distributed Time-Constant Impedance Responses Interpreted in Terms of Physically Meaningful Properties DISTRIBUTED TIME-CONSTANT IMPEDANCE RESPONSES INTERPRETED IN TERMS OF PHYSICALLY MEANINGFUL PROPERTIES By BRYAN D. HIRSCHORN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010 c 2010 Bryan D. Hirschorn 2 To my wife, Erin Hirschorn, and my daughter, Sienna Hirschorn 3 ACKNOWLEDGMENTS I thank my advisor, Professor Mark E. Orazem, for his expertise, insight, and support in this work. His patience with students is unmatched and he always takes the time to listen and offer his assistance. His interest, thoroughness, and attention to detail has significantly improved my study and has helped push me to realize my capabilities and to improve my weaknesses. I thank Dr. Bernard Tribollet, Dr. Isabelle Frateur, Dr. Marco Musiani, and Dr. Vincent Vivier for their help developing and improving this study. The extension of their areas of research broadened the scope of this work. I also thank my committee members Professor Jason Weaver, Professor Anuj Chauhan, and Professor Juan Nino. I thank my wife, my mother, and my father for their love and encouragement during my studies. 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS..................................4 LIST OF TABLES......................................8 LIST OF FIGURES.....................................9 LIST OF SYMBOLS.................................... 17 ABSTRACT......................................... 20 CHAPTER 1 INTRODUCTION................................... 23 2 LITERATURE REVIEW............................... 28 2.1 The Constant-Phase Element......................... 28 2.1.1 Surface Distributions.......................... 28 2.1.2 Normal Distributions.......................... 30 2.2 Determination of Capacitance from the CPE................. 31 2.3 Errors Associated with Nonlinearity...................... 32 2.4 Linearity and the Kramers-Kronig Relations................. 35 3 OPTIMIZATION OF SIGNAL-TO-NOISE RATIO UNDER A LINEAR RESPONSE 37 3.1 Circuit Models Incorporating Faradaic Reactions.............. 37 3.2 Numerical Solution of Nonlinear Circuit Models............... 39 3.3 Simulation Results............................... 41 3.3.1 Errors in Assessment of Charge-Transfer Resistance........ 41 3.3.2 Optimal Perturbation Amplitude.................... 42 3.3.3 Experimental Assessment of Linearity................ 46 3.3.4 Frequency Dependence of the Interfacial Potential......... 47 3.3.5 Optimization of the Input Signal.................... 52 3.3.6 Potential-Dependent Capacitance................... 54 3.4 Conclusions................................... 55 4 THE SENSITIVITY OF THE KRAMERS-KRONIG RELATIONS TO NONLIN- EAR RESPONSES.................................. 56 4.1 Application of the Kramers-Kronig Relations................. 57 4.2 Simulation Results............................... 58 4.3 The Applicability of the Kramers-Kronig Relations to Detecting Nonlin- earity...................................... 64 4.3.1 Influence of Transition Frequency................... 64 4.3.2 Application to Experimental Systems................. 69 4.4 Conclusions................................... 72 5 5 CHARACTERISTICS OF THE CONSTANT-PHASE ELEMENT......... 74 6 THE CAPACITIVE RESPONSE OF ELECTROCHEMICAL SYSTEMS..... 79 6.1 Capacitance of the Diffuse Layer....................... 79 6.2 Capacitance of a Dielectric Layer....................... 81 6.3 Calculation of Capacitance from Impedance Spectra............ 83 6.3.1 Single Time-Constant Responses................... 83 6.3.2 Surface Distributions.......................... 84 6.3.3 Normal Distributions.......................... 87 6.4 Conclusions................................... 89 7 ASSESSMENT OF CAPACITANCE-CPE RELATIONS TAKEN FROM THE LITERATURE..................................... 91 7.1 Surface Distributions.............................. 91 7.2 Normal Distributions.............................. 95 7.2.1 Niobium................................. 95 7.2.2 Human Skin............................... 99 7.2.3 Films with an Exponential Decay of Resistivity............ 102 7.3 Application of the Young Model to Niobium and Skin............ 106 7.4 Conclusions................................... 110 8 CPE BEHAVIOR CAUSED BY RESISTIVITY DISTRIBUTIONS IN FILMS... 112 8.1 Resistivity Distribution............................. 112 8.2 Impedance Expression............................. 118 8.3 Discussion................................... 122 8.3.1 Extraction of Physical Parameters................... 122 8.3.2 Comparison to Young Model...................... 123 8.3.3 Variable Dielectric Constant...................... 125 8.4 Conclusions................................... 126 9 APPLICATION OF THE POWER-LAW MODEL TO EXPERIMENTAL SYS- TEMS......................................... 127 9.1 Method..................................... 127 9.2 Results and Discussion............................ 128 9.2.1 Aluminum Oxide (Large ρ0 and Small ρδ)............... 130 9.2.2 Stainless Steel (Finite ρ0 and Small ρδ)................ 133 9.2.3 Human Skin (Power Law with Parallel Path)............. 136 9.3 Discussion................................... 139 9.4 Conclusions................................... 141 10 CPE BEHAVIOR CAUSED BY SURFACE DISTRIBUTIONS OF OHMIC RE- SISTANCE...................................... 142 10.1 Mathematical Development.......................... 142 6 10.2 Disk Electrodes................................. 148 10.2.1 Increase of resistance with increasing radius............ 148 10.2.2 Decrease of resistance with increasing radius............ 148 10.3 Conclusions................................... 152 11 OVERVIEW OF CAPACITANCE-CPE RELATIONS................ 153 12 CONCLUSIONS................................... 156 13 SUGGESTIONS FOR FUTURE WORK...................... 158 13.1 CPE Behavior Caused by Surface Distributions of Reactivity........ 158 13.1.1 Mathematical Development...................... 159 13.1.2 Interpretation.............................. 161 13.2 CPE Behavior Caused by Normal Distributions of Properties....... 164 APPENDIX A PROGRAM CODE FOR LARGE AMPLITUDE PERTURBATIONS....... 165 B PROGRAM CODE FOR NORMAL DISTRIBUTIONS.............. 170 C PROGRAM CODE FOR SURFACE DISTRIBUTIONS.............. 173 D PROGRAM CODE FOR DISK ELECTRODE DISTRIBUTION.......... 175 REFERENCES....................................... 179 BIOGRAPHICAL SKETCH................................ 188 7 LIST OF TABLES Table page 4-1 Simulation results used to explore the role of the Kramers-Kronig relations for 2 nonlinear systems with parameters: ∆U = 100 mV, Cdl = 20 µF/cm , Ka = 2 −1 Kc = 1 mA/cm , ba = bc = 19 V , and V = 0 V................... 58 7-1 CPE parameters, resistance, effective capacitance, and thickness of oxide films formed on a Nb disk electrode in 0.1 M NH4F solution (pH 2) as a func- tion of the anodization potential........................... 97 7-2 Thickness of oxide films developed on a Nb electrode, as a function of the anodization potential. Comparison of values deduced from impedance data with those from the literature............................. 98 7-3 CPE parameters, resistance, effective capacitance, and thickness for heat- stripped human stratum corneum in 50 mM buffered CaCl2 electrolyte as a function of immersion time. Data taken from Membrino.1 ............ 100 7-4 Physical properties obtained by matching the high-frequency portion of the impedance response given in Figure 7-5 for heat-stripped stratum corneum in 50 mM buffered CaCl2 electrolyte as a function of immersion time........ 109 8 LIST OF FIGURES Figure page 3-1 Circuit models with Faradaic reaction: a) non-Ohmic Faradaic system; b) Ohmic Faradaic system; and c) Ohmic constant charge-transfer resistance system... 38 3-2 Calculated impedance response with applied perturbation amplitude as a pa- 2 2 rameter. The system parameters were Re = 1 Ωcm , Ka = Kc = 1 mA/cm , −1 2 ba = bc = 19 V , Cdl = 20 µF/cm , and V = 0 V, giving rise to a linear charge- 2 transfer resistance Rt;0 = 26 Ωcm .......................... 42 3-3 The error in the low-frequency impedance asymptote associated with use of a large amplitude potential perturbation....................... 46 3-4 Lissajous plots with perturbation amplitude and frequency as parameters. 2 2 The system parameters were Re = 0 Ωcm , Ka = Kc = 1 mA/cm , ba = bc = −1 2 19 V , Cdl = 20 µF/cm , and V = 0 V, giving rise to a linear charge-transfer 2 resistance Rt;0 = 26 Ωcm .............................. 47 3-5 Maximum variation of the interfacial potential signal as a function of frequency 2 2 for parameters ∆U = 100 mV, Re = 1 Ωcm , Cdl = 20 µF/cm , and Rt;0 = 26 Ωcm2........................................ 48 3-6 Maximum variation of the interfacial potential signal as a function of frequency 2 2 2 for parameters ∆U = 100 mV, Re = 1 Ωcm , Cdl = 20 µF/cm , Rt;0 = 26 Ωcm . The solid curve is ∆Vmax resulting from the numeric simulation. The dashed 2 curve is ∆V predicted from equation (3–9) using Rt;obs = 19 Ωcm , which de- 2 creases from the linear value, Rt;0 = 26 Ωcm , due to the large input pertur- bation.......................................... 49 2 3-7 Calculated results for parameters ∆U = 100 mV, Cdl = 20 µF/cm , and Rt;0 = 26 Ωcm2 with Ohmic resistance as a parameter. a) Maximum variation of the interfacial potential signal as a function of frequency; and b) the correspond- ing Lissajous plots at a frequency of 0.016 Hz................... 50 3-8 Inflection point of ∆Vmax is located at the transitional
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