Electrochemical Impedance Spectroscopy

Mark E. Orazem is a Distinguished Professor of Chemical Engineering at the University of Florida, adjunct professor at the Beijing University of Chemical Technology, a Fellow of the Electrochemical Society, past President of the International Society of Electrochemistry, and recipient of the 2012 ECS Linford Award for Outstanding Teaching. He organized the 6th International Symposium on Electrochemical Impedance Spectroscopy and teaches short courses on impedance spectroscopy for industry and for The Electrochemical Society. Bernard Tribollet is Director of Research Emeritus at the Laboratory for Interfaces and Electrochemical Systems (LISE) at the University of Pierre and Marie Curie and adjunct professor at the Beijing University of Chemical Technology. He instructs an annual short course at his university on impedance spectroscopy. He is a Fellow of The Electrochemical Society, Treasurer of the International Society of Electrochemistry, and organized the 2010 Annual Meeting of the International Society of Electrochemistry held in Nice, France.

• Preface to the Second Edition xviiPreface to the First Edition xixAcknowledgments xxiiiThe Blind Men and the Elephant xxvA Brief Introduction to Impedance Spectroscopy xxixHistory of Impedance Spectroscopy xxxviiI Background 11 Complex Variables 31.1 Why Imaginary Numbers? 31.2 Terminology 41.3 Operations Involving Complex Variables 51.4 Elementary Functions of Complex Variables 16Problems 222 Differential Equations 252.1 Linear First-Order Differential Equations 252.2 Homogeneous Linear Second-Order Differential Equations 292.3 Nonhomogeneous Linear Second-Order Differential Equations 322.4 Chain Rule for Coordinate Transformations 362.5 Partial Differential Equations by Similarity Transformations 382.6 Differential Equations with Complex Variables 42Problems 433 Statistics 453.1 Definitions 453.2 Error Propagation 533.3 Hypothesis Tests 59Problems 704 Electrical Circuits 734.1 Passive Electrical Circuits 734.2 Fundamental Relationships 794.3 Nested Circuits 804.4 Mathematical Equivalence of Circuits 824.5 Graphical Representation of Circuit Response 82Problems 855 Electrochemistry 875.1 Resistors and Electrochemical Cells 875.2 Polarization Behavior for Electrochemical Systems 905.3 Definitions of Potential 1065.4 Rate Expressions 1075.5 Transport Processes 1115.6 Potential Contributions 1175.7 Capacitance Contributions 1205.8 Further Reading 124Problems 1256 Electrochemical Instrumentation 1276.1 The Ideal Operational Amplifier 1276.2 Elements of Electrochemical Instrumentation 1296.3 Electrochemical Interface 131Problems 135II Experimental Considerations 1377 Experimental Methods 1397.1 Steady-State Polarization Curves 1397.2 Transient Response to a Potential Step 1407.3 Analysis in Frequency Domain 1417.4 Comparison of Measurement Techniques 1547.5 Specialized Techniques 155Problems 1608 Experimental Design 1638.1 Cell Design 1638.2 Experimental Considerations 1688.3 Instrumentation Parameters 181Problems 186III Process Models 1879 Equivalent Circuit Analogs 1899.1 General Approach 1899.2 Current Addition 1909.3 Potential Addition 196Problems 20110 Kinetic Models 20310.1 General Mathematical Framework 20310.2 Electrochemical Reactions 20510.3 Multiple Independent Electrochemical Reactions 21810.4 Coupled Electrochemical Reactions 22110.5 Electrochemical and Heterogeneous Chemical Reactions 229Problems 23511 Diffusion Impedance 23711.1 Uniformly Accessible Electrode 23811.2 Porous Film 23911.3 Rotating Disk 24911.4 Submerged Impinging Jet 25911.5 Rotating Cylinders 26211.6 Electrode Coated by a Porous Film 26411.7 Impedance with Homogeneous Chemical Reactions 27111.8 Dynamic Surface Films 280Problems 29012 Impedance of Materials 29112.1 Electrical Properties of Materials 29112.2 Dielectric Response in Homogeneous Media 29212.3 Cole-Cole Relaxation 29512.4 Geometric Capacitance 29512.5 Dielectric Response of Insulating Non-Homogenous Media 29712.6 Mott-Schottky Analysis 298Problems 30513 Time-Constant Dispersion 30713.1 Transmission Line Models 30713.2 Geometry–Induced Current and Potential Distributions 32513.3 Electrode Surface Property Distributions 33713.4 Characteristic Dimension for Frequency Dispersion 35813.5 Convective Diffusion Impedance at Small Electrodes 35913.6 Coupled Charging and Faradaic Currents 36513.7 Exponential Resistivity Distributions 378Problems 38114 Constant–Phase Elements 38314.1 Mathematical Formulation for a CPE 38314.2 When is a Time–Constant Distribution a CPE? 38414.3 Origin of Distributions Resulting in a CPE 38814.4 Approaches for Extracting Physical Properties 38914.5 Limitations to the Use of the CPE 404Problems 40615 Generalized Transfer Functions 40915.1 Multi-Input/Multi-Output Systems 40915.2 Transfer Functions Involving Exclusively Electrical Quantities 41715.3 Transfer Functions Involving Nonelectrical Quantities 422Problems 42916 Electrohydrodynamic Impedance 43116.1 Hydrodynamic Transfer Function 43316.2 Mass-Transport Transfer Function 43616.3 Kinetic Transfer Function for Simple Electrochemical Reactions 44116.4 Interface with a 2-D or 3-D Insulating Phase 442Problems 454IV Interpretation Strategies 45517 Methods for Representing Impedance 45717.1 Impedance Format 45917.2 Admittance Format 46817.3 Complex-Capacitance Format 47417.4 Effective Capacitance 478Problems 48218 Graphical Methods 48318.1 Based on Nyquist Plots 48418.2 Based on Bode Plots 49118.3 Based on Imaginary Part of the Impedance 49518.4 Based on Dimensionless Frequency 49618.5 System–Specific Applications 50218.6 Overview 512Problems 51519 Complex Nonlinear Regression 51719.1 Concept 51719.2 Objective Functions 51919.3 Formalism of Regression Strategies 52119.4 Regression Strategies for Nonlinear Problems 52419.5 Influence of Data Quality on Regression 52719.6 Initial Estimates for Regression 53319.7 Regression Statistics 533Problems 53620 Assessing Regression Quality 53920.1 Methods to Assess Regression Quality 53920.2 Application of Regression Concepts 540Problems 555V Statistical Analysis 55721 Error Structure of Impedance Measurements 55921.1 Error Contributions 55921.2 Stochastic Errors in Impedance Measurements 56021.3 Bias Errors 56621.4 Incorporation of Error Structure 57021.5 Measurement Models for Error Identification 572Problems 58322 The Kramers-Kronig Relations 58522.1 Methods for Application 58522.2 Mathematical Origin 59022.3 The Kramers-Kronig in an Expectation Sense 601Problems 605VI Overview 60723 An Integrated Approach to Impedance Spectroscopy 60923.1 Flowcharts for Regression Analysis 60923.2 Integration of Measurements, Error Analysis, and Model 61023.3 Application 613Problems 619VII Reference Material 621A Complex Integrals 623A.1 Definition of Terms 623A.2 Cauchy-Riemann Conditions 625A.3 Complex Integration 627Problems 633B Tables of Reference Material 635C List of Examples 637List of Symbols 643References 655Index 684