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EB

EB

ONLINE

2nd edition

Hoboken, New Jersey : John Wiley & Sons, Inc., [2017]

1 online zdroj

ISBN 9781119363682 (e-kniha)

ISBN 9781118527399 (vázáno)

The Electrochemical Society series

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001479510

Preface to the Second Edition XX1 // 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 // 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 r-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 // 4 2 Fundamental Relationships 81 // 4.3 Nested Circuits 83 // 4.4 Mathematical Equivalence of Circuits 84 // 4 5 Graphical Representation of Circuit Response 84 // 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 // 6.3.3 Potentiostat for EIS Measurement I35 // 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 // 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 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 // 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 // Nemst 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 // 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 328 // 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 // 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 Lise 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 // 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 484 // 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 // 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 539 // 19.6 Initial Estimates for Regression 541 // 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 543 // 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 // 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 // Reference Material 629 // A Complex Integrals 631 // A.l Definition of Terms 631 // A.2 Cauchy-Riemann Conditions 634 // A.3 Complex Integration 635 // A.3.1 Cauchy’s Theorem 635 // A.3.2 Improper Integrals of Rational Functions 639 // Problems 641 // ? Tables of Reference Material 643 // C List of Examples 645 // List of Symbols 651 // References 663 // Author Index 691 // Subject Index 697

(OCoLC)957656496