Intro -- Electrochemical Impedance Spectroscopy -- Contents -- Preface to the Second Edition -- Preface to the First Edition -- Acknowledgments -- The Blind Men and the Elephant -- A Brief Introduction to Impedance Spectroscopy -- History of Impedance Spectroscopy -- I Background -- 1 Complex Variables -- 1.1 Why Imaginary Numbers? -- 1.2 Terminology -- 1.2.1 The Imaginary Number -- 1.2.2 Complex Variables -- 1.2.3 Conventions for Notation in Impedance Spectroscopy -- 1.3 Operations Involving Complex Variables -- 1.3.1 Multiplication and Division of Complex Numbers -- 1.3.2 Complex Variables in Polar Coordinates -- 1.3.3 Properties of Complex Variables -- 1.4 Elementary Functions of Complex Variables -- 1.4.1 Exponential -- 1.4.2 Logarithmic -- 1.4.3 Polynomial -- Problems -- 2 Differential Equations -- 2.1 Linear First-Order Differential Equations -- 2.2 Homogeneous Linear Second-Order Differential Equations -- 2.3 Nonhomogeneous Linear Second-Order Differential Equations -- 2.4 Chain Rule for Coordinate Transformations -- 2.5 Partial Differential Equations by Similarity Transformations -- 2.6 Differential Equations with Complex Variables -- Problems -- 3 Statistics -- 3.1 Definitions -- 3.1.1 Expectation and Mean -- 3.1.2 Variance, Standard Deviation, and Covariance -- 3.1.3 Normal Distribution -- 3.1.4 Probability -- 3.1.5 Central Limit Theorem -- 3.2 Error Propagation -- 3.2.1 Linear Systems -- 3.2.2 Nonlinear Systems -- 3.3 Hypothesis Tests -- 3.3.1 Terminology -- 3.3.2 Student's t-Test for Equality of Mean -- 3.3.3 F-Test for Equality of Variance -- 3.3.4 Chi-Squared Test for Goodness of Fit -- Problems -- 4 Electrical Circuits -- 4.1 Passive Electrical Circuits -- 4.1.1 Circuit Elements -- Response to a Sinusoidal Signal -- Impedance Response of Passive Circuit Elements -- 4.1.2 Parallel and Series Combinations.

4.2 Fundamental Relationships -- 4.3 Nested Circuits -- 4.4 Mathematical Equivalence of Circuits -- 4.5 Graphical Representation of Circuit Response -- Problems -- 5 Electrochemistry -- 5.1 Resistors and Electrochemical Cells -- 5.2 Polarization Behavior for Electrochemical Systems -- 5.2.1 Zero Current -- Equilibrium -- Nonequilibrium -- 5.2.2 Kinetic Control -- 5.2.3 Mixed-Potential Theory -- 5.2.4 Mass-Transfer Control -- 5.3 Definitions of Potential -- 5.4 Rate Expressions -- 5.5 Transport Processes -- 5.5.1 Primary Current and Potential Distributions -- 5.5.2 Secondary Current and Potential Distributions -- 5.5.3 Tertiary Current and Potential Distributions -- 5.5.4 Mass-Transfer-Controlled Current Distributions -- 5.6 Potential Contributions -- 5.6.1 Ohmic Potential Drop -- 5.6.2 Surface Overpotential -- 5.6.3 Concentration Overpotential -- 5.7 Capacitance Contributions -- 5.7.1 Double-Layer Capacitance -- 5.7.2 Dielectric Capacitance -- 5.8 Further Reading -- Problems -- 6 Electrochemical Instrumentation -- 6.1 The Ideal Operational Amplifier -- 6.2 Elements of Electrochemical Instrumentation -- 6.3 Electrochemical Interface -- 6.3.1 Potentiostat -- 6.3.2 Galvanostat -- 6.3.3 Potentiostat for EIS Measurement -- Problems -- II Experimental Considerations -- 7 Experimental Methods -- 7.1 Steady-State Polarization Curves -- 7.2 Transient Response to a Potential Step -- 7.3 Analysis in Frequency Domain -- 7.3.1 Lissajous Analysis -- 7.3.2 Phase-Sensitive Detection (Lock-in Amplifier) -- 7.3.3 Single-Frequency Fourier Analysis -- 7.3.4 Multiple-Frequency Fourier Analysis -- 7.4 Comparison of Measurement Techniques -- 7.4.1 Lissajous Analysis -- 7.4.2 Phase-Sensitive Detection (Lock-in Amplifier) -- 7.4.3 Single-Frequency Fourier Analysis -- 7.4.4 Multiple-Frequency Fourier Analysis -- 7.5 Specialized Techniques -- 7.5.1 Transfer-Function Analysis.

7.5.2 Local Electrochemical Impedance Spectroscopy -- Global Impedance -- Local Impedance -- Local Interfacial Impedance -- Local Ohmic Impedance -- Global Interfacial Impedance -- Global Ohmic Impedance -- Problems -- 8 Experimental Design -- 8.1 Cell Design -- 8.1.1 Reference Electrodes -- 8.1.2 Flow Configurations -- Rotating Disk -- Disk under Submerged Impinging Jet -- Rotating Cylinders -- Rotating Hemispherical Electrode -- 8.1.3 Current Distribution -- 8.2 Experimental Considerations -- 8.2.1 Frequency Range -- 8.2.2 Linearity -- 8.2.3 Modulation Technique -- 8.2.4 Oscilloscope -- 8.3 Instrumentation Parameters -- 8.3.1 Improve Signal-to-Noise Ratio -- 8.3.2 Reduce Bias Errors -- Nonstationary Effects -- Instrument Bias -- 8.3.3 Improve Information Content -- Problems -- III Process Models -- 9 Equivalent Circuit Analogs -- 9.1 General Approach -- 9.2 Current Addition -- 9.2.1 Impedance at the Corrosion Potential -- 9.2.2 Partially Blocked Electrode -- 9.3 Potential Addition -- 9.3.1 Electrode Coated with an Inert Porous Layer -- 9.3.2 Electrode Coated with Two Inert Porous Layers -- Problems -- 10 Kinetic Models -- 10.1 General Mathematical Framework -- 10.2 Electrochemical Reactions -- 10.2.1 Potential Dependent -- 10.2.2 Potential and Concentration Dependent -- Charge-Transfer Resistance -- Diffusion Impedance -- Cell Impedance -- 10.3 Multiple Independent Electrochemical Reactions -- 10.4 Coupled Electrochemical Reactions -- 10.4.1 Potential and Surface Coverage Dependent -- 10.4.2 Potential, Surface Coverage, and Concentration Dependent -- 10.5 Electrochemical and Heterogeneous Chemical Reactions -- Problems -- 11 Diffusion Impedance -- 11.1 Uniformly Accessible Electrode -- 11.2 Porous Film -- 11.2.1 Diffusion with Exchange of Electroactive Species -- 11.2.2 Diffusion without Exchange of Electroactive Species -- 11.3 Rotating Disk.

11.3.1 Fluid Flow -- 11.3.2 Steady-State Mass Transfer -- 11.3.3 Convective Diffusion Impedance -- 11.3.4 Analytic and Numerical Solutions -- Nernst Hypothesis -- Assumption of an Infinite Schmidt Number -- Treatment of a Finite Schmidt Number -- 11.4 Submerged Impinging Jet -- 11.4.1 Fluid Flow -- 11.4.2 Steady-State Mass Transfer -- 11.4.3 Convective Diffusion Impedance -- 11.5 Rotating Cylinders -- 11.6 Electrode Coated by a Porous Film -- 11.6.1 Steady-State Solutions -- 11.6.2 Coupled Diffusion Impedance -- 11.7 Impedance with Homogeneous Chemical Reactions -- 11.8 Dynamic Surface Films -- 11.8.1 Mass Transfer in the Salt Layer -- 11.8.2 Mass Transfer in the Electrolyte -- 11.8.3 Oscillating Film Thickness -- 11.8.4 Faradaic Impedance -- Problems -- 12 Impedance of Materials -- 12.1 Electrical Properties of Materials -- 12.2 Dielectric Response in Homogeneous Media -- 12.3 Cole-Cole Relaxation -- 12.4 Geometric Capacitance -- 12.5 Dielectric Response of Insulating Nonhomogeneous Media -- 12.6 Mott-Schottky Analysis -- Problems -- 13 Time-Constant Dispersion -- 13.1 Transmission Line Models -- 13.1.1 Telegrapher's Equations -- 13.1.2 Porous Electrodes -- 13.1.3 Pore-in-Pore Model -- 13.1.4 Thin-Layer Cell -- 13.2 Geometry-Induced Current and Potential Distributions -- 13.2.1 Mathematical Development -- Blocking Electrode -- Blocking Electrode with CPE Behavior -- Electrode with Faradaic Reactions -- Electrode with Faradaic Reactions Coupled by Adsorbed Intermediates -- 13.2.2 Numerical Method -- 13.2.3 Complex Ohmic Impedance at High Frequencies -- 13.2.4 Complex Ohmic Impedance at High and Low Frequencies -- 13.3 Electrode Surface Property Distributions -- 13.3.1 Electrode Roughness -- Influence of Roughness on a Disk Electrode -- Influence of Surface Roughness on a Recessed Electrode -- 13.3.2 Capacitance.

Capacitance Distribution on Recessed Electrodes -- Capacitance Distribution on Disk Electrodes -- 13.3.3 Reactivity -- 13.4 Characteristic Dimension for Frequency Dispersion -- 13.5 Convective Diffusion Impedance at Small Electrodes -- 13.5.1 Analysis -- 13.5.2 Local Convective Diffusion Impedance -- Low-Frequency Solution -- High-Frequency Solution -- 13.5.3 Global Convective Diffusion Impedance -- 13.6 Coupled Charging and Faradaic Currents -- 13.6.1 Theoretical Development -- Mass Transport in Dilute Solutions -- Coupled Faradaic and Charging Currents -- Double-Layer Model -- Decoupled Faradaic and Charging Currents -- 13.6.2 Numerical Method -- Steady-State Calculations -- Double-Layer Properties -- Impedance Calculations -- 13.6.3 Consequence of Coupled Charging and Faradaic Currents -- 13.7 Exponential Resistivity Distributions -- Problems -- 14 Constant-Phase Elements -- 14.1 Mathematical Formulation for a CPE -- 14.2 When Is a Time-Constant Distribution a CPE? -- 14.3 Origin of Distributions Resulting in a CPE -- 14.4 Approaches for Extracting Physical Properties -- 14.4.1 Simple Substitution -- 14.4.2 Characteristic Frequency: Normal Distribution -- 14.4.3 Characteristic Frequency: Surface Distribution -- 14.4.4 Power-Law Distribution -- Bounds for Resistivity -- Comparative Analysis -- 14.5 Limitations to the Use of the CPE -- Problems -- 15 Generalized Transfer Functions -- 15.1 Multi-input/Multi-output Systems -- 15.1.1 Current or Potential Are the Output Quantity -- 15.1.2 Current or Potential Are the Input Quantity -- 15.1.3 Experimental Quantities -- 15.2 Transfer Functions Involving Exclusively Electrical Quantities -- 15.2.1 Ring-Disk Impedance Measurements -- 15.2.2 Multifrequency Measurements for Double-Layer Studies -- 15.3 Transfer Functions Involving Nonelectrical Quantities.