Contents. I Background 1. Contents... Preface... Acknowledgments... The Blind Men and the Elephant... xxi. History of Impedance Spectroscopy...

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3 Contents Contents Preface Acknowledgments v xv xix The Blind Men and the Elephant xxi History of Impedance Spectroscopy xxv I Background 1 1 Complex Variables Why Imaginary Numbers? Terminology The Imaginary Number Complex Variables Conventions for Notation in Impedance Spectroscopy Operations Involving Complex Variables Multiplication and Division of Complex Numbers Complex Variables in Polar Coordinates Properties of Complex Variables Elementary Functions of Complex Variables Exponential Logarithmic Polynomial Problems

4 vi CONTENTS 2 Differential Equations Linear First-Order Differential Equations Homogeneous Linear Second-Order Differential Equations Nonhomogeneous Linear Second-Order Differential Equations Partial Differential Equations by Similarity Transformations Differential Equations with Complex Variables Problems Statistics Definitions Expectation and Mean Variance, Standard Deviation, and Covariance Normal Distribution Probability Central-Limit Theorem Error Propagation Linear Systems Nonlinear Systems Hypothesis Tests Terminology Student s t-test for Equality of Mean F-test for Equality of Variance Chi-Squared Test for Equality of Variance Problems Electrical Circuits Passive Electrical Circuits Circuit Elements Parallel and Series Combinations Fundamental Relationships Nested Circuits Mathematical Equivalence of Circuits Graphical Representation of Circuit Response Problems Electrochemistry Resistors and Electrochemical Cells Equilibrium in Electrochemical Systems Polarization Behavior for Electrochemical Systems Zero Current Kinetic Control Mass-Transfer Control

5 CONTENTS vii 5.4 Definitions of Potential Rate Expressions Law of Mass Action Generalized Electrode Kinetics Transport Processes Primary Current and Potential Distributions Application to Blocking Electrodes Secondary Current and Potential Distributions Tertiary Current and Potential Distributions Mass-Transfer-Controlled Current Distributions Potential Contributions Ohmic Potential Drop Surface Overpotential Concentration Overpotential Capacitance Contributions Double-Layer Capacitance Dielectric Capacitance Problems Electrochemical Instrumentation The Ideal Operational Amplifier Elements of Electrochemical Instrumentation Electrochemical Interface Potentiostat Galvanostat Potentiostat for EIS Measurement Problems II Experimental Considerations Experimental Methods Steady-State Polarization Curves Transient Response to a Potential Step Analysis in Frequency Domain Lissajous Analysis Phase-Sensitive Detection (Lock-in Amplifier) Single-Frequency Fourier Analysis Multiple-Frequency Fourier Analysis Comparison of Measurement Techniques Lissajous Analysis Phase-Sensitive Detection (Lock-in Amplifier) Single-Frequency Fourier Analysis Multiple-Frequency Fourier Analysis

6 viii CONTENTS 7.5 Specialized Techniques Transfer Function Analysis Local Electrochemical Impedance Spectroscopy Problems Experimental Design Cell Design Reference Electrodes Flow Configurations Current Distribution Experimental Considerations Frequency Range Linearity Modulation Technique Oscilloscope Instrumentation Parameters Improve Signal-to-Noise Ratio Reduce Bias Errors Improve Information Content Problems III Process Models Equivalent Circuit Analogs General Approach Current Addition Impedance at the Corrosion Potential Partially Blocked Electrode Potential Addition Electrode Coated with an Inert Porous Layer Electrode Coated with Two Inert Porous Layers Problems Kinetic Models Electrochemical Reactions Reaction Dependent on Potential Only Reaction Dependent on Potential and Mass Transfer Coupled Reactions Dependent on Potential and Surface Coverage Reactions Dependent on Potential, Surface Coverage, and Transport 176 Problems

7 CONTENTS ix 11 Diffusion Impedance Uniformly Accessible Electrode General Mathematical Framework Stagnant Diffusion Layer Diffusion through a Solid Film Region of Film Diffusion Control Film Impedance Response Coupled Diffusion Impedance Rotating Disk Fluid Flow Mass Transfer Classification of Models for Convective Diffusion Submerged Impinging Jet Fluid Flow Mass Transfer Rotating Cylinders Problems Semiconducting Systems Semiconductor Physics Electrons and Holes as Species Doping Deep-Level States Shockley-Read-Hall Processes Interfaces Steady-State Models Mass Transfer Space-Charge Region Application to Electrolyte Semiconductor Junctions Impedance Models Equivalent Electrical Circuits Mott-Schottky Analysis Problems Time-Constant Dispersion Constant-Phase Elements D and 3-D Distributions Determination of Capacitance Limitations to the Use of the CPE Convective Diffusion Impedance at Small Electrodes Analysis Local Diffusion Convective Impedance Global Convective Diffusion Impedance

8 x CONTENTS 13.3 Geometry-Induced Current and Potential Distributions Mathematical Development Global and Local Impedances Porous Electrodes Oxide Layers Problems Generalized Transfer Functions Multi-Input/Multi-Output Systems Current or Potential Are the Output Quantity Current or Potential Are the Input Quantity Experimental Quantities Transfer Functions Involving Exclusively Electrical Quantities Ring-Disk Impedance Measurements Multifrequency Measurements for Double-Layer Studies Transfer Functions Involving Nonelectrical Quantities Thermoelectrochemical (TEC) Transfer Function Photoelectrochemical Impedance Measurements Electrogravimetry Impedance Measurements Problems Electrohydrodynamic Impedance Hydrodynamic Transfer Function Mass-Transport Transfer Function Asymptotic Solution for Large Schmidt Numbers Asymptotic Solution for High Frequencies Kinetic Transfer Function for Simple Electrochemical Reactions Interface with a 2-D or 3-D Insulating Phase Partially Blocked Electrode Rotating Disk Electrode Coated by a Porous Film Problems IV Interpretation Strategies Methods for Representing Impedance Impedance Format Complex-Impedance-Plane Representation Bode Representation Electrolyte-Resistance-Corrected Bode Representation Impedance Representation Admittance Format Admittance-Plane Representation Admittance Representation

9 CONTENTS xi Electrolyte-Resistance-Corrected Representation Complex-Capacitance Format Complex-Capacitance-Plane Representation Complex-Capacitance Representation Effective Capacitance Problems Preliminary Graphical Methods Application to a Randles Circuit Traditional Representation of Data Phase Angle and Modulus Corrected for Ohmic Resistance Real and Imaginary Components Effective High-Frequency Capacity or CPE Coefficient Application to Blocking Electrodes Nyquist and Bode Representations Imaginary Component Effective CPE Coefficient Overview Problems Model-Based Graphical Methods Mass Transfer Scaled Plots of Impedance Asymptotic Behavior at Low-Frequency Reaction Kinetics: Arrhenius Relations Mott-Schottky Plots Problems Complex Nonlinear Regression Concept Objective Functions Formalism of Regression Strategies Linear Regression Nonlinear Regression Regression Strategies for Nonlinear Problems Gauss-Newton Method Method of Steepest Descent Levenberg-Marquardt Method Downhill Simplex Strategies Influence of Data Quality on Regression Presence of Stochastic Errors in Data Ill-Conditioned Regression Caused by Stochastic Noise Ill-Conditioned Regression Caused by Insufficient Range. 375

10 xii CONTENTS 19.6 Initial Estimates for Regression Regression Statistics Confidence Intervals for Parameter Estimates Statistical Measure of the Regression Quality Problems Assessing Regression Quality Methods to Assess Regression Quality Quantitative Methods Qualitative Methods Application of Regression Concepts Finite-Diffusion-Length Model Measurement Model Convective-Diffusion-Length Model Problems V Statistical Analysis Error Structure of Impedance Measurements Error Contributions Stochastic Errors in Impedance Measurements Stochastic Errors in Time-Domain Signals Transformation from Time Domain to Frequency Domain Stochastic Errors in Frequency Domain Bias Errors Instrument Artifacts Ancillary Parts of the System under Study Nonstationary Behavior Time Scales in Impedance Spectroscopy Measurements Incorporation of Error Structure Measurement Models for Error Identification Stochastic Errors Bias Errors Problems The Kramers-Kronig Relations Mathematical Origin Background Application of Cauchy s Theorem Transformation from Real to Imaginary Transformation from Imaginary to Real Application of the Kramers-Kronig Relations The Kramers-Kronig in an Expectation Sense

11 CONTENTS xiii Transformation from Real to Imaginary Transformation from Imaginary to Real Methods for Application Direct Integration of the Kramers-Kronig Relations Experimental Assessment of Consistency Regression of Process Models Regression of Measurement Models Problems VI Overview An Integrated Approach to Impedance Spectroscopy Flowcharts for Regression Analysis Integration of Measurements, Error Analysis, and Model Impedance Measurements Integrated with Error Analysis Process Models Developed Using Other Observations Regression Analysis in Context of Error Structure Application Problems VII Reference Material 461 A Complex Integrals A.1 Definition of Terms A.2 Cauchy-Riemann Conditions A.3 Complex Integration A.3.1 Cauchy s Theorem A.3.2 Improper Integrals of Rational Functions Problems B Tables of Reference Material C List of Examples List of Symbols References Index