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 xvii

Preface to the First Edition xix

Acknowledgments xxiii

The Blind Men and the Elephant xxv

A Brief Introduction to Impedance Spectroscopy xxix

History of Impedance Spectroscopy xxxvii

I Background 1

1 Complex Variables 3

1.1 Why Imaginary Numbers? 3

1.2 Terminology 4

1.3 Operations Involving Complex Variables 5

1.4 Elementary Functions of Complex Variables 16

Problems 22

2 Differential Equations 25

2.1 Linear First-Order Differential Equations 25

2.2 Homogeneous Linear Second-Order Differential Equations 29

2.3 Nonhomogeneous Linear Second-Order Differential Equations 32

2.4 Chain Rule for Coordinate Transformations 36

2.5 Partial Differential Equations by Similarity Transformations 38

2.6 Differential Equations with Complex Variables 42

Problems 43

3 Statistics 45

3.1 Definitions 45

3.2 Error Propagation 53

3.3 Hypothesis Tests 59

Problems 70

4 Electrical Circuits 73

4.1 Passive Electrical Circuits 73

4.2 Fundamental Relationships 79

4.3 Nested Circuits 80

4.4 Mathematical Equivalence of Circuits 82

4.5 Graphical Representation of Circuit Response 82

Problems 85

5 Electrochemistry 87

5.1 Resistors and Electrochemical Cells 87

5.2 Polarization Behavior for Electrochemical Systems 90

5.3 Definitions of Potential 106

5.4 Rate Expressions 107

5.5 Transport Processes 111

5.6 Potential Contributions 117

5.7 Capacitance Contributions 120

5.8 Further Reading 124

Problems 125

6 Electrochemical Instrumentation 127

6.1 The Ideal Operational Amplifier 127

6.2 Elements of Electrochemical Instrumentation 129

6.3 Electrochemical Interface 131

Problems 135

II Experimental Considerations 137

7 Experimental Methods 139

7.1 Steady-State Polarization Curves 139

7.2 Transient Response to a Potential Step 140

7.3 Analysis in Frequency Domain 141

7.4 Comparison of Measurement Techniques 154

7.5 Specialized Techniques 155

Problems 160

8 Experimental Design 163

8.1 Cell Design 163

8.2 Experimental Considerations 168

8.3 Instrumentation Parameters 181

Problems 186

III Process Models 187

9 Equivalent Circuit Analogs 189

9.1 General Approach 189

9.2 Current Addition 190

9.3 Potential Addition 196

Problems 201

10 Kinetic Models 203

10.1 General Mathematical Framework 203

10.2 Electrochemical Reactions 205

10.3 Multiple Independent Electrochemical Reactions 218

10.4 Coupled Electrochemical Reactions 221

10.5 Electrochemical and Heterogeneous Chemical Reactions 229

Problems 235

11 Diffusion Impedance 237

11.1 Uniformly Accessible Electrode 238

11.2 Porous Film 239

11.3 Rotating Disk 249

11.4 Submerged Impinging Jet 259

11.5 Rotating Cylinders 262

11.6 Electrode Coated by a Porous Film 264

11.7 Impedance with Homogeneous Chemical Reactions 271

11.8 Dynamic Surface Films 280

Problems 290

12 Impedance of Materials 291

12.1 Electrical Properties of Materials 291

12.2 Dielectric Response in Homogeneous Media 292

12.3 Cole-Cole Relaxation 295

12.4 Geometric Capacitance 295

12.5 Dielectric Response of Insulating Non-Homogenous Media 297

12.6 Mott-Schottky Analysis 298

Problems 305

13 Time-Constant Dispersion 307

13.1 Transmission Line Models 307

13.2 Geometry–Induced Current and Potential Distributions 325

13.3 Electrode Surface Property Distributions 337

13.4 Characteristic Dimension for Frequency Dispersion 358

13.5 Convective Diffusion Impedance at Small Electrodes 359

13.6 Coupled Charging and Faradaic Currents 365

13.7 Exponential Resistivity Distributions 378

Problems 381

14 Constant–Phase Elements 383

14.1 Mathematical Formulation for a CPE 383

14.2 When is a Time–Constant Distribution a CPE? 384

14.3 Origin of Distributions Resulting in a CPE 388

14.4 Approaches for Extracting Physical Properties 389

14.5 Limitations to the Use of the CPE 404

Problems 406

15 Generalized Transfer Functions 409

15.1 Multi-Input/Multi-Output Systems 409

15.2 Transfer Functions Involving Exclusively Electrical Quantities 417

15.3 Transfer Functions Involving Nonelectrical Quantities 422

Problems 429

16 Electrohydrodynamic Impedance 431

16.1 Hydrodynamic Transfer Function 433

16.2 Mass-Transport Transfer Function 436

16.3 Kinetic Transfer Function for Simple Electrochemical Reactions 441

16.4 Interface with a 2-D or 3-D Insulating Phase 442

Problems 454

IV Interpretation Strategies 455

17 Methods for Representing Impedance 457

17.1 Impedance Format 459

17.2 Admittance Format 468

17.3 Complex-Capacitance Format 474

17.4 Effective Capacitance 478

Problems 482

18 Graphical Methods 483

18.1 Based on Nyquist Plots 484

18.2 Based on Bode Plots 491

18.3 Based on Imaginary Part of the Impedance 495

18.4 Based on Dimensionless Frequency 496

18.5 System–Specific Applications 502

18.6 Overview 512

Problems 515

19 Complex Nonlinear Regression 517

19.1 Concept 517

19.2 Objective Functions 519

19.3 Formalism of Regression Strategies 521

19.4 Regression Strategies for Nonlinear Problems 524

19.5 Influence of Data Quality on Regression 527

19.6 Initial Estimates for Regression 533

19.7 Regression Statistics 533

Problems 536

20 Assessing Regression Quality 539

20.1 Methods to Assess Regression Quality 539

20.2 Application of Regression Concepts 540

Problems 555

V Statistical Analysis 557

21 Error Structure of Impedance Measurements 559

21.1 Error Contributions 559

21.2 Stochastic Errors in Impedance Measurements 560

21.3 Bias Errors 566

21.4 Incorporation of Error Structure 570

21.5 Measurement Models for Error Identification 572

Problems 583

22 The Kramers-Kronig Relations 585

22.1 Methods for Application 585

22.2 Mathematical Origin 590

22.3 The Kramers-Kronig in an Expectation Sense 601

Problems 605

VI Overview 607

23 An Integrated Approach to Impedance Spectroscopy 609

23.1 Flowcharts for Regression Analysis 609

23.2 Integration of Measurements, Error Analysis, and Model 610

23.3 Application 613

Problems 619

VII Reference Material 621

A Complex Integrals 623

A.1 Definition of Terms 623

A.2 Cauchy-Riemann Conditions 625

A.3 Complex Integration 627

Problems 633

B Tables of Reference Material 635

C List of Examples 637

List of Symbols 643

References 655

Index 684