Electrochemical Impedance Spectroscopy

Electrochemical Impedance Spectroscopy

THE ELECTROCHEMICAL SOCIETY SERIES ECS-The Electrochemical Society 65 South Main Street Pennington, NJ 08534-2839 http://www.electrochem.org A complete list of the titles in this series appears at the end of this volume.

Electrochemical Impedance Spectroscopy Second Edition Mark E. Orazem University of Florida Bernard Tribollet Université Pierre et Marie Curie

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Dedicated to our families: Jennifer and Athéna, Françoise, Julie, and Benjamin

Contents Preface to the Second Edition............................ xxi Preface to the First Edition.............................. xxiii Acknowledgments.................................... xxvii The Blind Men and the Elephant.......................... xxix A Brief Introduction to Impedance Spectroscopy................. xxxiii History of Impedance Spectroscopy......................... xli I Background 1 1 Complex Variables................................. 3 1.1 Why Imaginary Numbers?.......................... 3 1.2 Terminology.................................. 4 1.2.1 The Imaginary Number....................... 4 1.2.2 Complex Variables.......................... 4 1.2.3 Conventions for Notation in Impedance Spectroscopy..... 5 1.3 Operations Involving Complex Variables................. 5 1.3.1 Multiplication and Division of Complex Numbers........ 6 1.3.2 Complex Variables in Polar Coordinates.............. 9 1.3.3 Properties of Complex Variables.................. 15 1.4 Elementary Functions of Complex Variables................ 15

viii CONTENTS 1.4.1 Exponential.............................. 15 1.4.2 Logarithmic.............................. 17 1.4.3 Polynomial.............................. 21 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 3.1.1 Expectation and Mean........................ 45 3.1.2 Variance, Standard Deviation, and Covariance.......... 45 3.1.3 Normal Distribution......................... 49 3.1.4 Probability............................... 50 3.1.5 Central Limit Theorem........................ 52 3.2 Error Propagation............................... 55 3.2.1 Linear Systems............................ 55 3.2.2 Nonlinear Systems.......................... 56 3.3 Hypothesis Tests............................... 59 3.3.1 Terminology.............................. 60 3.3.2 Student s t-test for Equality of Mean................ 61 3.3.3 F-Test for Equality of Variance................... 64 3.3.4 Chi-Squared Test for Goodness of Fit............... 70 Problems.................................... 71 4 Electrical Circuits.................................. 75 4.1 Passive Electrical Circuits.......................... 75 4.1.1 Circuit Elements........................... 75 Response to a Sinusoidal Signal................... 77 Impedance Response of Passive Circuit Elements........ 79 4.1.2 Parallel and Series Combinations.................. 79

CONTENTS ix 4.2 Fundamental Relationships......................... 81 4.3 Nested Circuits................................ 83 4.4 Mathematical Equivalence of Circuits................... 84 4.5 Graphical Representation of Circuit Response............... 85 Problems.................................... 87 5 Electrochemistry.................................. 89 5.1 Resistors and Electrochemical Cells..................... 89 5.2 Polarization Behavior for Electrochemical Systems............ 92 5.2.1 Zero Current............................. 93 Equilibrium.............................. 94 Nonequilibrium............................ 94 5.2.2 Kinetic Control............................ 95 5.2.3 Mixed-Potential Theory....................... 97 5.2.4 Mass-Transfer Control........................ 104 5.3 Definitions of Potential............................ 107 5.4 Rate Expressions............................... 109 5.5 Transport Processes.............................. 113 5.5.1 Primary Current and Potential Distributions........... 114 5.5.2 Secondary Current and Potential Distributions.......... 116 5.5.3 Tertiary Current and Potential Distributions........... 116 5.5.4 Mass-Transfer-Controlled Current Distributions......... 117 5.6 Potential Contributions............................ 119 5.6.1 Ohmic Potential Drop........................ 119 5.6.2 Surface Overpotential........................ 119 5.6.3 Concentration Overpotential.................... 120 5.7 Capacitance Contributions.......................... 122 5.7.1 Double-Layer Capacitance...................... 122 5.7.2 Dielectric Capacitance........................ 126 5.8 Further Reading................................ 126 Problems.................................... 127 6 Electrochemical Instrumentation......................... 129 6.1 The Ideal Operational Amplifier...................... 129 6.2 Elements of Electrochemical Instrumentation............... 131 6.3 Electrochemical Interface........................... 133 6.3.1 Potentiostat.............................. 134 6.3.2 Galvanostat.............................. 135

x CONTENTS 6.3.3 Potentiostat for EIS Measurement................. 135 Problems.................................... 137 II Experimental Considerations 139 7 Experimental Methods............................... 141 7.1 Steady-State Polarization Curves...................... 141 7.2 Transient Response to a Potential Step................... 142 7.3 Analysis in Frequency Domain....................... 143 7.3.1 Lissajous Analysis.......................... 144 7.3.2 Phase-Sensitive Detection (Lock-in Amplifier).......... 151 7.3.3 Single-Frequency Fourier Analysis................. 152 7.3.4 Multiple-Frequency Fourier Analysis............... 155 7.4 Comparison of Measurement Techniques................. 156 7.4.1 Lissajous Analysis.......................... 156 7.4.2 Phase-Sensitive Detection (Lock-in Amplifier).......... 156 7.4.3 Single-Frequency Fourier Analysis................. 156 7.4.4 Multiple-Frequency Fourier Analysis............... 156 7.5 Specialized Techniques............................ 157 7.5.1 Transfer-Function Analysis..................... 157 7.5.2 Local Electrochemical Impedance Spectroscopy......... 158 Global Impedance.......................... 160 Local Impedance........................... 160 Local Interfacial Impedance..................... 160 Local Ohmic Impedance....................... 161 Global Interfacial Impedance.................... 161 Global Ohmic Impedance...................... 161 Problems.................................... 162 8 Experimental Design................................ 165 8.1 Cell Design................................... 165 8.1.1 Reference Electrodes......................... 165 8.1.2 Flow Configurations......................... 167 Rotating Disk............................. 167 Disk under Submerged Impinging Jet............... 168 Rotating Cylinders.......................... 168 Rotating Hemispherical Electrode................. 168 8.1.3 Current Distribution......................... 169

CONTENTS xi 8.2 Experimental Considerations........................ 170 8.2.1 Frequency Range........................... 170 8.2.2 Linearity................................ 170 8.2.3 Modulation Technique........................ 181 8.2.4 Oscilloscope.............................. 184 8.3 Instrumentation Parameters......................... 184 8.3.1 Improve Signal-to-Noise Ratio................... 184 8.3.2 Reduce Bias Errors.......................... 186 Nonstationary Effects........................ 186 Instrument Bias............................ 186 8.3.3 Improve Information Content.................... 187 Problems.................................... 188 III Process Models 191 9 Equivalent Circuit Analogs............................ 193 9.1 General Approach............................... 193 9.2 Current Addition............................... 195 9.2.1 Impedance at the Corrosion Potential............... 195 9.2.2 Partially Blocked Electrode..................... 196 9.3 Potential Addition.............................. 201 9.3.1 Electrode Coated with an Inert Porous Layer........... 201 9.3.2 Electrode Coated with Two Inert Porous Layers......... 202 Problems.................................... 205 10 Kinetic Models................................... 207 10.1 General Mathematical Framework..................... 207 10.2 Electrochemical Reactions.......................... 209 10.2.1 Potential Dependent......................... 209 10.2.2 Potential and Concentration Dependent.............. 213 Charge-Transfer Resistance..................... 216 Diffusion Impedance......................... 217 Cell Impedance............................ 220 10.3 Multiple Independent Electrochemical Reactions............. 222 10.4 Coupled Electrochemical Reactions..................... 226 10.4.1 Potential and Surface Coverage Dependent............ 226 10.4.2 Potential, Surface Coverage, and Concentration Dependent............................... 231

xii CONTENTS 10.5 Electrochemical and Heterogeneous Chemical Reactions........ 234 Problems.................................... 240 11 Diffusion Impedance................................ 243 11.1 Uniformly Accessible Electrode....................... 244 11.2 Porous Film.................................. 245 11.2.1 Diffusion with Exchange of Electroactive Species........ 245 11.2.2 Diffusion without Exchange of Electroactive Species....... 251 11.3 Rotating Disk................................. 256 11.3.1 Fluid Flow............................... 256 11.3.2 Steady-State Mass Transfer..................... 259 11.3.3 Convective Diffusion Impedance.................. 261 11.3.4 Analytic and Numerical Solutions................. 262 Nernst Hypothesis.......................... 262 Assumption of an Infinite Schmidt Number........... 263 Treatment of a Finite Schmidt Number.............. 264 11.4 Submerged Impinging Jet.......................... 266 11.4.1 Fluid Flow............................... 266 11.4.2 Steady-State Mass Transfer..................... 268 11.4.3 Convective Diffusion Impedance.................. 268 11.5 Rotating Cylinders.............................. 269 11.6 Electrode Coated by a Porous Film..................... 271 11.6.1 Steady-State Solutions........................ 272 11.6.2 Coupled Diffusion Impedance................... 277 11.7 Impedance with Homogeneous Chemical Reactions........... 278 11.8 Dynamic Surface Films............................ 291 11.8.1 Mass Transfer in the Salt Layer................... 292 11.8.2 Mass Transfer in the Electrolyte................... 294 11.8.3 Oscillating Film Thickness...................... 295 11.8.4 Faradaic Impedance......................... 297 Problems.................................... 300 12 Impedance of Materials.............................. 303 12.1 Electrical Properties of Materials...................... 303 12.2 Dielectric Response in Homogeneous Media............... 304 12.3 Cole Cole Relaxation............................. 307 12.4 Geometric Capacitance............................ 307 12.5 Dielectric Response of Insulating Nonhomogeneous Media....... 309

CONTENTS xiii 12.6 Mott Schottky Analysis........................... 311 Problems.................................... 317 13 Time-Constant Dispersion............................. 319 13.1 Transmission Line Models.......................... 319 13.1.1 Telegrapher s Equations....................... 321 13.1.2 Porous Electrodes........................... 322 13.1.3 Pore-in-Pore Model.......................... 330 13.1.4 Thin-Layer Cell............................ 333 13.2 Geometry-Induced Current and Potential Distributions......... 337 13.2.1 Mathematical Development..................... 338 Blocking Electrode.......................... 338 Blocking Electrode with CPE Behavior............... 339 Electrode with Faradaic Reactions................. 339 Electrode with Faradaic Reactions Coupled by Adsorbed Intermediates................. 341 13.2.2 Numerical Method.......................... 342 13.2.3 Complex Ohmic Impedance at High Frequencies........ 342 13.2.4 Complex Ohmic Impedance at High and Low Frequencies.............................. 346 13.3 Electrode Surface Property Distributions.................. 349 13.3.1 Electrode Roughness......................... 349 Influence of Roughness on a Disk Electrode............ 351 Influence of Surface Roughness on a Recessed Electrode.......................... 355 13.3.2 Capacitance.............................. 360 Capacitance Distribution on Recessed Electrodes........ 361 Capacitance Distribution on Disk Electrodes........... 365 13.3.3 Reactivity............................... 369 13.4 Characteristic Dimension for Frequency Dispersion........... 369 13.5 Convective Diffusion Impedance at Small Electrodes.......... 370 13.5.1 Analysis................................ 371 13.5.2 Local Convective Diffusion Impedance.............. 373 Low-Frequency Solution....................... 374 High-Frequency Solution...................... 374 13.5.3 Global Convective Diffusion Impedance............. 374 13.6 Coupled Charging and Faradaic Currents................. 377 13.6.1 Theoretical Development...................... 378 Mass Transport in Dilute Solutions................. 378

xiv CONTENTS Coupled Faradaic and Charging Currents............. 379 Double-Layer Model......................... 380 Decoupled Faradaic and Charging Currents........... 382 13.6.2 Numerical Method.......................... 383 Steady-State Calculations...................... 383 Double-Layer Properties....................... 383 Impedance Calculations....................... 384 13.6.3 Consequence of Coupled Charging and Faradaic Currents................................ 385 13.7 Exponential Resistivity Distributions.................... 389 Problems.................................... 392 14 Constant-Phase Elements............................. 395 14.1 Mathematical Formulation for a CPE.................... 395 14.2 When Is a Time-Constant Distribution a CPE?.............. 396 14.3 Origin of Distributions Resulting in a CPE................. 399 14.4 Approaches for Extracting Physical Properties.............. 401 14.4.1 Simple Substitution.......................... 402 14.4.2 Characteristic Frequency: Normal Distribution.......... 402 14.4.3 Characteristic Frequency: Surface Distribution.......... 404 14.4.4 Power-Law Distribution....................... 406 Bounds for Resistivity........................ 413 Comparative Analysis........................ 413 14.5 Limitations to the Use of the CPE...................... 415 Problems.................................... 418 15 Generalized Transfer Functions.......................... 421 15.1 Multi-input/Multi-output Systems..................... 421 15.1.1 Current or Potential Are the Output Quantity.......... 425 15.1.2 Current or Potential Are the Input Quantity........... 427 15.1.3 Experimental Quantities....................... 429 15.2 Transfer Functions Involving Exclusively Electrical Quantities..... 429 15.2.1 Ring Disk Impedance Measurements............... 429 15.2.2 Multifrequency Measurements for Double-Layer Studies................................. 431 15.3 Transfer Functions Involving Nonelectrical Quantities.......... 434 15.3.1 Thermoelectrochemical (TEC) Transfer Function......... 434 15.3.2 Photoelectrochemical Impedance Measurements......... 438

CONTENTS xv 15.3.3 Electrogravimetry Impedance Measurements........... 439 Problems.................................... 441 16 Electrohydrodynamic Impedance......................... 443 16.1 Hydrodynamic Transfer Function...................... 445 16.2 Mass-Transport Transfer Function..................... 448 16.2.1 Asymptotic Solution for Large Schmidt Numbers........ 451 16.2.2 Asymptotic Solution for High Frequencies............ 451 16.3 Kinetic Transfer Function for Simple Electrochemical Reactions.................................... 453 16.4 Interface with a 2-D or 3-D Insulating Phase................ 454 16.4.1 Partially Blocked Electrode..................... 455 16.4.2 Rotating Disk Electrode Coated by a Porous Film........ 457 Steady-State Solutions........................ 458 AC and EHD Impedances...................... 459 Problems.................................... 464 IV Interpretation Strategies 465 17 Methods for Representing Impedance...................... 467 17.1 Impedance Format.............................. 467 17.1.1 Complex-Impedance-Plane Representation............ 470 17.1.2 Bode Representation......................... 473 17.1.3 Ohmic-Resistance-Corrected Bode Representation........ 475 17.1.4 Impedance Representation..................... 476 17.2 Admittance Format.............................. 478 17.2.1 Admittance-Plane Representation................. 480 17.2.2 Admittance Representation..................... 481 17.2.3 Ohmic-Resistance-Corrected Representation........... 483 17.3 Complex-Capacitance Format........................ 483 17.4 Effective Capacitance............................. 488 Problems.................................... 491 18 Graphical Methods................................. 493 18.1 Based on Nyquist Plots............................ 494 18.1.1 Characteristic Frequency....................... 494 18.1.2 Superposition............................. 499

xvi CONTENTS Mass Transfer............................. 499 Evolution of Active Area...................... 501 18.2 Based on Bode Plots.............................. 501 18.2.1 Ohmic-Resistance-Corrected Phase................. 504 18.2.2 Ohmic-Resistance-Corrected Magnitude............. 505 18.3 Based on Imaginary Part of the Impedance................ 505 18.3.1 Evaluation of Slopes......................... 505 18.3.2 Calculation of Derivatives...................... 506 18.4 Based on Dimensionless Frequency..................... 506 18.4.1 Mass Transport............................ 507 18.4.2 Geometric Contribution....................... 507 18.5 System-Specific Applications........................ 512 18.5.1 Effective CPE Coefficient...................... 512 18.5.2 Asymptotic Behavior for Low-Frequency Mass Transport............................... 516 18.5.3 Arrhenius Superposition....................... 518 18.5.4 Mott Schottky Plots......................... 521 18.5.5 High-Frequency Cole Cole Plots.................. 522 18.6 Overview.................................... 523 Problems.................................... 525 19 Complex Nonlinear Regression.......................... 527 19.1 Concept..................................... 527 19.2 Objective Functions.............................. 529 19.3 Formalism of Regression Strategies..................... 530 19.3.1 Linear Regression........................... 530 19.3.2 Nonlinear Regression........................ 532 19.4 Regression Strategies for Nonlinear Problems............... 534 19.4.1 Gauss Newton Method....................... 534 19.4.2 Method of Steepest Descent..................... 534 19.4.3 Levenberg Marquardt Method................... 534 19.4.4 Downhill Simplex Strategies.................... 535 19.5 Influence of Data Quality on Regression.................. 536 19.5.1 Presence of Stochastic Errors in Data................ 537 19.5.2 Ill-Conditioned Regression Caused by Stochastic Noise.................................. 537 19.5.3 Ill-Conditioned Regression Caused by Insufficient Range................................. 538 19.6 Initial Estimates for Regression....................... 541

CONTENTS xvii 19.7 Regression Statistics............................. 542 19.7.1 Confidence Intervals for Parameter Estimates.......... 542 19.7.2 Statistical Measure of the Regression Quality........... 543 Problems.................................... 544 20 Assessing Regression Quality........................... 545 20.1 Methods to Assess Regression Quality................... 545 20.1.1 Quantitative Methods........................ 545 20.1.2 Qualitative Methods......................... 546 20.2 Application of Regression Concepts.................... 546 20.2.1 Finite-Diffusion-Length Model................... 548 Quantitative Assessment...................... 549 Visual Inspection........................... 549 20.2.2 Measurement Model......................... 552 Quantitative Assessment...................... 553 Visual Inspection........................... 553 20.2.3 Convective-Diffusion-Length Model................ 555 Quantitative Assessment...................... 557 Visual Inspection........................... 558 Problems.................................... 563 V Statistical Analysis 565 21 Error Structure of Impedance Measurements................. 567 21.1 Error Contributions.............................. 567 21.2 Stochastic Errors in Impedance Measurements.............. 568 21.2.1 Stochastic Errors in Time-Domain Signals............. 569 21.2.2 Transformation from Time Domain to Frequency Domain................................ 571 21.2.3 Stochastic Errors in Frequency Domain.............. 571 21.3 Bias Errors................................... 575 21.3.1 Instrument Artifacts......................... 575 21.3.2 Ancillary Parts of the System under Study............ 576 21.3.3 Nonstationary Behavior....................... 576 21.3.4 Time Scales in Impedance Spectroscopy Measurements..... 576 21.4 Incorporation of Error Structure....................... 579 21.5 Measurement Models for Error Identification............... 581 21.5.1 Stochastic Errors........................... 583

xviii CONTENTS 21.5.2 Bias Errors............................... 586 Problems.................................... 593 22 The Kramers Kronig Relations.......................... 595 22.1 Methods for Application........................... 595 22.1.1 Direct Integration of the Kramers Kronig Relations....... 597 22.1.2 Experimental Assessment of Consistency............. 598 22.1.3 Regression of Process Models.................... 598 22.1.4 Regression of Measurement Models................ 599 22.2 Mathematical Origin............................. 600 22.2.1 Background.............................. 600 22.2.2 Application of Cauchy s Theorem................. 604 22.2.3 Transformation from Real to Imaginary.............. 605 22.2.4 Transformation from Imaginary to Real.............. 607 22.2.5 Application of the Kramers Kronig Relations.......... 609 22.3 The Kramers Kronig Relations in an Expectation Sense......... 610 22.3.1 Transformation from Real to Imaginary.............. 611 22.3.2 Transformation from Imaginary to Real.............. 612 Problems.................................... 614 VI Overview 615 23 An Integrated Approach to Impedance Spectroscopy............ 617 23.1 Flowcharts for Regression Analysis..................... 617 23.2 Integration of Measurements, Error Analysis, and Model........ 618 23.2.1 Impedance Measurements Integrated with Error Analysis................................ 619 23.2.2 Process Models Developed Using Other Observations...... 620 23.2.3 Regression Analysis in Context of Error Structure........ 621 23.3 Application.................................. 621 Problems.................................... 627 VII Reference Material 629 A Complex Integrals.................................. 631 A.1 Definition of Terms.............................. 631 A.2 Cauchy Riemann Conditions........................ 634

CONTENTS xix A.3 Complex Integration............................. 635 A.3.1 Cauchy s Theorem.......................... 635 A.3.2 Improper Integrals of Rational Functions............. 639 Problems.................................... 641 B Tables of Reference Material........................... 643 C List of Examples.................................. 645 List of Symbols..................................... 651 References........................................ 663 Author Index....................................... 691 Subject Index...................................... 697

Preface to the Second Edition We are gratified by the response to the first edition of our book. We thank all those who wrote us to alert us to errors and have sought to make appropriate corrections. There are other major changes in the second edition: We increased significantly the number of examples and homework problems. We adopted a coherent system of notation in which the math italic font is reserved for variables. We added a brief introduction in the front matter that provides a somewhat qualitative introduction to electrochemical impedance spectroscopy. Within the Background part of the book, Chapters 1, 2, 3, and 5 have been greatly expanded. Mathematical development in the rest of the book is now referred to examples in Chapters 1 and 2, and the reader is referred as needed to electrochemical principles in Chapter 5. The Process Models part of the book has been greatly expanded. The consequences of the circuit models presented in Chapter 9 are described in detail. An example is introduced to show how impedance measurements at potentials slightly above and below the open-circuit potential may be used to determine the role of anodic and cathodic impedances in model development. The chapter demonstrates the use of superposition of impedance data to determine the influence of a partial coverage of the electrode surface by a thin dielectric. The presentation of models based on kinetics has been streamlined in Chapter 10, and the number of cases treated has been increased. This chapter now includes discussion of coupled electrochemical and heterogeneous chemical reactions. The presentation of diffusion impedance in Chapter 11 has been improved. The impedance associated with diffusion through a film now includes diffusion with and without exchange of electroactive species. Other new topics

xxii PREFACE TO THE SECOND EDITION include convective diffusion to an electrode coated by a porous film, impedance with homogeneous chemical reactions, and dynamic surface films in which the film thickness is modulated due to competing dissolution and growth mechanisms. The chapter on semiconducting systems has been replaced by a more general Chapter 12 on the impedance of materials. The new topics include electrical properties of materials, dielectric response in homogeneous media, Cole Cole relaxation, and geometric capacitance. Chapter 13 on time-constant dispersion has been expanded. More examples are provided for the discussion of transmission line models, and a more thorough discussion is provided for the effects of geometry-induced current and potential distributions. Treatments of electrode surface property distributions and coupled charging and faradaic currents have been introduced. In response to recent developments in the understanding of constant-phase elements, a new Chapter 14 has been introduced. This chapter addresses important topics, such as methods to determine whether a given time-constant distribution is a CPE, the origin of distributions resulting in a CPE, and approaches for extracting physical properties. The discussion of graphical methods in the Interpretation Strategies part of the book has been reorganized into two chapters: Chapter 17 provides methods for representing impedance data, and Chapter 18 describes graphical methods. The role of complex capacitance representation is expanded in both chapters. In the Statistical Analysis part of the book, Chapter 22 on the Kramers Kronig relations has been reorganized. Mark E. Orazem Gainesville, Florida Bernard Tribollet Paris, France February 2017

Preface to the First Edition This book is intended for use both as a professional reference and as a textbook suitable for training new scientists and engineers. As a textbook, this work is suitable for graduate students in a variety of disciplines, including electrochemistry, materials science, physics, electrical engineering, and chemical engineering. As these audiences have very different backgrounds, a portion of the book reviews material that may be known to some students but not to others. There are many short courses offered on impedance spectroscopy, but formal courses on the topic are rarely offered in university settings. Accordingly, this textbook is designed to accommodate both directed and independent learning. Organization The textbook has been prepared in seven parts: Part I Background This part provides material that may be covered selectively, depending on the background of the students. The subjects covered include complex variables, differential equations, statistics, electrical circuits, electrochemistry, and instrumentation. The coverage of these topics is limited to what is needed to understand the core of the textbook, which is covered in the subsequent parts. Part II Experimental Considerations This part introduces methods used to measure impedance and other transfer functions. The chapters in this section are intended to provide an understanding of frequencydomain techniques and the approaches used by impedance instrumentation. This understanding provides a basis for evaluating and improving experimental design. The material covered in this section is integrated with the discussion of experimental errors and noise. The extension of impedance spectroscopy to other transfer-function techniques is developed in Part III.

xxiv PREFACE TO THE FIRST EDITION Part III Process Models This part demonstrates how deterministic models of impedance response can be developed from physical and kinetic descriptions. When possible, correspondence is drawn between hypothesized models and electrical circuit analogues. The treatment includes electrode kinetics, mass transfer, solid-state systems, time-constant dispersion, models accounting for two- and three-dimensional interfaces, generalized transfer functions, and a more specific example of a transfer-function technique in which the rotation speed of a disk electrode is modulated. Part IV Interpretation Strategies This part describes methods for interpretation of impedance data, ranging from graphical methods to complex nonlinear regression. The material covered in this section is integrated with the discussion of experimental errors and noise. Bias errors are shown to limit the frequency range useful for regression analysis, and the variance of stochastic errors is used to guide the weighting strategy used for regression. Part V Statistical Analysis This part provides a conceptual understanding of stochastic, bias, and fitting errors in frequency-domain measurements. A major advantage of measurements in the frequency domain is that real and imaginary parts of the response must be internally consistent. The expression of this consistency takes different forms that are known collectively as the Kramers Kronig relations. The Kramers Kronig relations and their application to spectroscopy measurements are described. Measurement models, used to assess the error structure, are described and compared with process models used to extract physical properties. Part VI Overview The final chapter in this book provides a philosophy for electrochemical impedance spectroscopy that integrates experimental observation, model development, and error analysis. This approach is differentiated from the usual sequential model development for given impedance spectra by its emphasis on obtaining supporting observations to guide model selection, use of error analysis to guide regression strategies and experimental design, and use of models to guide selection of new experiments. These concepts are illustrated with examples taken from the literature. This chapter is intended to illustrate that selection of models, even those based on physical principles, requires both error analysis and additional experimental verification.

PEDAGOGICAL APPROACH xxv Part VII Reference Material The reference material includes an appendix on complex integration needed to follow the derivation of the Kramers Kronig relations, a list of tables, a list of examples, a list of symbols, and a list of references. Pedagogical Approach The material is presented in a manner that facilitates sequential development of understanding and expertise either in a course or in self-study. Illustrative examples are interspersed throughout the text to show how the principles described are applied to common impedance problems. These examples are in the form of questions, followed by the solution to the question posed. The student can attempt to solve the problem before reading how the problem is solved. Homework problems, suitable either for self-study or for study under direction of an instructor, are developed for each chapter. Important equations and relations are collected in tables, which can be easily accessed. Important concepts are identified and set aside at the bottom of pages as they appear in the text. Readily identifiable icons are used to distinguish examples and important concepts. As can be found in any field, the notation used in the impedance spectroscopy literature is inconsistent. In treatments of diffusion impedance, for example, the symbol θ is used to denote the dimensionless oscillating concentration variable; whereas, the symbol θ used in kinetic studies denotes the fractional surface coverage by a reaction intermediate. Compromises were necessary to create a consistent notation for this book. For example, the dimensionless oscillating concentration variable was given the symbol θ, and γ was used to denote the fractional surface coverage by a reaction intermediate. As discussed in Section 1.2.3, the book deviates from the IUPAC convention for the notation used to denote the imaginary number and the real and imaginary parts of impedance. This book is intended to provide a background and training suitable for application of impedance spectroscopy to a broad range of applications, such as corrosion, biomedical devices, semiconductors and solid-state devices, sensors, batteries, fuel cells, electrochemical capacitors, dielectric measurements, coatings, electrochromic materials, analytical chemistry, and imaging. The emphasis is on generally applicable fundamentals rather than on detailed treatment of applications. The reader is referred to other sources for discussion of specific applications of impedance. 1 4 Remember! 0.1 The elephant at the left is used to identify important concepts for each chapter. It is intended to remind the student of the parable of the blind men and the elephant.

xxvi PREFACE TO THE FIRST EDITION The active participation in related short courses demonstrates a rising interest in impedance spectroscopy. As discussed in the preliminary section on the history of the technique, the number of papers published that mention use of electrochemical impedance spectroscopy has increased dramatically over the past 10 years. Nevertheless, the question may be raised: Why teach a full semester-long course on impedance spectroscopy? It is, after all, just an experimental technique. In our view, impedance spectroscopy represents the confluence of a significant number of disciplines, and successful training in the use and interpretation of impedance requires a coherent education in the application of each of these disciplines to the subject. In addition to learning about impedance spectroscopy, the student will gain a better understanding of a general philosophy of scientific inquiry. Mark E. Orazem Gainesville, Florida Bernard Tribollet Paris, France July 2008

Acknowledgments The authors met for the first time in 1981 in the research group of John Newman at the University of California, Berkeley. Mark Orazem was a graduate student, and Bernard Tribollet was a visiting scientist on sabbatical leave from the Centre National de la Recherche Scientifique (CNRS) in Paris. We have maintained a fruitful collaboration ever since, and our careers, as well as the content of this book, build on the foundation we received from John. We owe an additional debt of gratitude to many people, including: Our families, to whom this book is dedicated and who have embraced our collaboration over the years. We appreciate their immense patience and support. The Electrochemical Society (ECS), which encouraged Mark Orazem to teach the ECS short course on impedance spectroscopy on an biannual basis, thus allowing us to test the pedagogical approach developed in this book. The CNRS, which provided financial support for a sabbatical year spent by Mark Orazem in Paris in 2001/2002. The University of Florida, which provided funds to Mark Orazem for a second sabbatical in 2012. Graduate students, who provided suggestions and content. Bryan Hirschorn, Vicky Huang, J. Patrick McKinney, Sunil Roy, and Shao-Ling Wu contributed to the first edition. Christopher Alexander, Ya-Chiao Chang, Yu-Min Chen, Christopher Cleveland, Arthur Dizon, Salim Erol, Ming Gao, Morgan Harding, Yuelong Huang, Rui Kong, and Chen You read various iterations of the second edition, helping with examples, solving homework problems, and identifying errors in the text. Research conducted by an undergraduate student, Katherine Davis, contributed to the text. Hubert Cachet, Sandro Cattarin, Isabelle Frateur, and Nadine Pébère, who read different chapters of the first edition and provided comments, corrections, and suggestions.

xxviii ACKNOWLEDGMENTS Michel Keddam, who suggested the presentation of the historical perspective in terms of the categories presented in Table 1. Max Yaffe of Gamry Instruments, who provided assistance with the chapter on the Kramers Kronig relations. Christopher Brett, who graciously provided a last technical review before publication of the first edition. Mary Yess of the ECS, who has given us unyielding support for this project. Dinia Agarwala of the ECS, who designed the cover of both editions. Amy Hendrickson, TeXnology Inc., who provided the LaTeX expertise needed to fine tune the visual appearance of the text. Laura Carlson, our copy editor, who taught us the proper usage of hyphens, endashes, and em-dashes. Bob Esposito, our editor from John Wiley & Sons, who helped us through the many steps associated with publishing a book and reassured us that it was good to be considered difficult authors. Melissa Yanuzzi, Senior Production Editor, who managed the final stages of publication. Our many colleagues and friends who assured us that there would be a demand for this book. While we have received help and support from many people, the remaining errors and omissions in this text are ours. We will gratefully receive corrections and suggestions from our readers to be implemented in possible future editions of this book.