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Evgenij Barsoukov, PhD, is a TI Fellow and the Head of Algorithm Development at the Battery Management unit of Texas Instruments. His research focuses on impedance spectroscopy-based modelling to improve battery monitoring and charging technology.
J. Ross Macdonald, DSc, is the William Rand Kenan, Jr. Professor Emeritus of Physics at the University of North Carolina. His research uses impedance spectroscopy to help analyze the electrical response of high-resistivity ionically conducting solid materials.
Preface to the Third Edition xi
Preface to the Second Edition xiii
Preface to the First Edition xv
Contributors to the Third Edition xvii
Chapter 1 Fundamentals of Impedance Spectroscopy 1 J. Ross Macdonald and William B. Johnson 1
1.1 Background, Basic Definitions, and History 1
1.1.1 The Importance of Interfaces 1
1.1.2 The Basic Impedance Spectroscopy Experiment 2
1.1.3 Response to a Small-Signal Stimulus in the Frequency Domain 3
1.1.4 Impedance-Related Functions 5
1.1.5 Early History 6
1.2 Advantages and Limitations 7
1.2.1 Differences between Solid-State and Aqueous Electrochemistry 9
1.3 Elementary Analysis of Impedance Spectra 10
1.3.1 Physical Models for Equivalent Circuit Elements 10
1.3.2 Simple RC Circuits 11
1.3.3 Analysis of Single Impedance Arcs 12
1.4 Selected Applications of IS 16
Chapter 2 Theory 21 Ian D. Raistrick, J. Ross Macdonald, and Donald R. Franceschetti 21
2.1 The Electrical Analogs of Physical and Chemical Processes 21
2.1.1 Introduction 21
2.1.2 The Electrical Properties of Bulk Homogeneous Phases 23
2.1.2.1 Introduction 23
2.1.2.2 Dielectric Relaxation in Materials with a Single Time Constant 23
2.1.2.3 Distributions of Relaxation Times 27
2.1.2.4 Conductivity and Diffusion in Electrolytes 34
2.1.2.5 Conductivity and Diffusion: A Statistical Description 36
2.1.2.6 Migration in the Absence of Concentration Gradients 38
2.1.2.7 Transport in Disordered Media 40
2.1.3 Mass and Charge Transport in the Presence of Concentration Gradients 45
2.1.3.1 Diffusion 45
2.1.3.2 Mixed Electronic-Ionic Conductors 49
2.1.3.3 Concentration Polarization 50
2.1.4 Interfaces and Boundary Conditions 51
2.1.4.1 Reversible and Irreversible Interfaces 51
2.1.4.2 Polarizable Electrodes 52
2.1.4.3 Adsorption at the Electrode-Electrolyte Interface 54
2.1.4.4 Charge Transfer at the Electrode-Electrolyte Interface 56
2.1.5 Grain Boundary Effects 60
2.1.6 Current Distribution: Porous and Rough Electrodes-The Effect of Geometry 62
2.1.6.1 Current Distribution Problems 62
2.1.6.2 Rough and Porous Electrodes 63
2.2 Physical and Electrochemical Models 67
2.2.1 The Modeling of Electrochemical Systems 67
2.2.2 Equivalent Circuits 67
2.2.2.1 Unification of Immittance Responses 67
2.2.2.2 Distributed Circuit Elements 69
2.2.2.3 Ambiguous Circuits 75
2.2.3 Modeling Results 79
2.2.3.1 Introduction 79
2.2.3.2 Supported Situations 80
2.2.3.3 Unsupported Situations: Theoretical Models 84
2.2.3.4 Equivalent Network Models 96
2.2.3.5 Unsupported Situations: Empirical and Semiempirical Models 97
Chapter 3 Measuring Techniques and Data Analysis 107 Michael C. H. McKubre, Digby D. Macdonald, Brian Sayers, and J. Ross Macdonald 107
3.1 Impedance Measurement Techniques 107
3.1.1 Introduction 107
3.1.2 Frequency Domain Methods 108
3.1.2.1 Audio Frequency Bridges 108
3.1.2.2 Transformer Ratio Arm Bridges 110
3.1.2.3 Berberian-Cole Bridge 112
3.1.2.4 Considerations of Potentiostatic Control 115
3.1.2.5 Oscilloscopic Methods for Direct Measurement 116
3.1.2.6 Phase-Sensitive Detection for Direct Measurement 118
3.1.2.7 Automated Frequency Response Analysis 119
3.1.2.8 Automated Impedance Analyzers 122
3.1.2.9 The Use of Kramers-Kronig Transforms 124
3.1.2.10 Spectrum Analyzers 126
3.1.3 Time-Domain Methods 128
3.1.3.1 Introduction 128
3.1.3.2 Analog-to-Digital Conversion 129
3.1.3.3 Computer Interfacing 133
3.1.3.4 Digital Signal Processing 135
3.1.4 Conclusions 138
3.2 Commercially Available Impedance Measurement Systems 139
3.2.1 General Measurement Techniques 139
3.2.1.1 Current-to-Voltage (I-E) Conversion Techniques 139
3.2.1.2 Measurements Using 2-, 3-, or 4-Terminal Techniques 144
3.2.1.3 Measurement Resolution and Accuracy 146
3.2.1.4 Single Sine and FFT Measurement Techniques 148
3.2.2 Electrochemical Impedance Measurement Systems 152
3.2.2.1 System Configuration 152
3.2.2.2 Why Use a Potentiostat? 152
3.2.2.3 Multi-electrode Techniques 153
3.2.2.4 Effects of Connections and Input Impedance 154
3.2.2.5 Verification of Measurement Performance 155
3.2.2.6 Floating Measurement Techniques 156
3.2.2.7 Multichannel Techniques 157
3.2.3 Materials Impedance Measurement Systems 157
3.2.3.1 System Configuration 157
3.2.3.2 Measurement of Low Impedance Materials 158
3.2.3.3 Measurement of High Impedance Materials 158
3.2.3.4 Reference Techniques 159
3.2.3.5 Normalization Techniques 159
3.2.3.6 High Voltage Measurement Techniques 160
3.2.3.7 Temperature Control 160
3.2.3.8 Sample Holder Considerations 161
3.3 Data Analysis 161
3.3.1 Data Presentation and Adjustment 161
3.3.1.1 Previous Approaches 161
3.3.1.2 Three-Dimensional Perspective Plotting 162
3.3.1.3 Treatment of Anomalies 164
3.3.2 Data Analysis Methods 166
3.3.2.1 Simple Methods 166
3.3.2.2 Complex Nonlinear Least Squares 167
3.3.2.3 Weighting 168
3.3.2.4 Which Impedance-Related Function to Fit? 169
3.3.2.5 The Question of "What to Fit" Revisited 169
3.3.2.6 Deconvolution Approaches 169
3.3.2.7 Examples of CNLS Fitting 170
3.3.2.8 Summary and Simple Characterization Example 172
Chapter 4 Applications of Impedance Spectroscopy 175
4.1 Characterization of Materials 175 N. Bonanos, B. C. H. Steele, and E. P. Butler 175
4.1.1 Microstructural Models for Impedance Spectra of Materials 175
4.1.1.1 Introduction 175
4.1.1.2 Layer Models 176
4.1.1.3 Effective Medium Models 183
4.1.1.4 Modeling of Composite Electrodes 191
4.1.2 Experimental Techniques 194
4.1.2.1 Introduction 194
4.1.2.2 Measurement Systems 195
4.1.2.3 Sample Preparation: Electrodes 199
4.1.2.4 Problems Associated with the Measurement of Electrode Properties 201
4.1.3 Interpretation of the Impedance Spectra of Ionic Conductors and Interfaces 203
4.1.3.1 Introduction 203
4.1.3.2 Characterization of Grain Boundaries by IS 204
4.1.3.3 Characterization of Two-Phase Dispersions by IS 215
4.1.3.4 Impedance Spectra of Unusual Two-Phase Systems 218
4.1.3.5 Impedance Spectra of Composite Electrodes 219
4.1.3.6 Closing Remarks 224
4.2 Characterization of the Electrical Response of Wide-Range-Resistivity Ionic and Dielectric Solid Materials by Immittance Spectroscopy 224 J. Ross Macdonald 224
4.2.1 Introduction 224
4.2.2 Types of Dispersive Response Models: Strengths and Weaknesses 225
4.2.2.1 Overview 225
4.2.2.2 Variable-Slope Models 226
4.2.2.3 Composite Models 227
4.2.3 Illustration of Typical Data Fitting Results for an Ionic Conductor 233
4.2.4 Utility and Importance of Poisson-Nernst-Planck (PNP) Fitting Models 239
4.2.4.1 Introduction 239
4.2.4.2 Selective History of PNP Work 240
4.2.4.3 Exact PNP Responses at All Four Immittance Levels 243
4.3 Solid-State Devices 247 William B. Johnson, Wayne L. Worrell, Gunnar A. Niklasson, Sara Malmgren, Maria Strømme, and S. K. Sundaram 247
4.3.1 Electrolyte-Insulator-Semiconductor (EIS) Sensors 248
4.3.2 Solid Electrolyte Chemical Sensors 254
4.3.3 Photoelectrochemical Solar Cells 258
4.3.4 Impedance Response of Electrochromic Materials and Devices 263
4.3.4.1 Introduction 263
4.3.4.2 Materials 265
4.3.4.3 Theoretical Background 266
4.3.4.4 Experimental Results on Single Materials 270
4.3.4.5 Experimental Results on Electrochromic Devices 280
4.3.4.6 Conclusions and Outlook 280
4.3.5 Fast Processes in Gigahertz-Terahertz Region in Disordered Materials 281
4.3.5.1 Introduction 281
4.3.5.2 Lunkenheimer-Loidl Plot and Scaling of the Processes 282
4.3.5.3 Dynamic Processes 285
4.3.5.4 Final Remarks 292
4.4 Corrosion of Materials 292 Michael C. H. McKubre, Digby D. Macdonald, and George R. Engelhardt 292
4.4.1 Introduction 292
4.4.2 Fundamentals 293
4.4.3 Measurement of Corrosion Rate 293
4.4.4 Harmonic Analysis 297
4.4.5 Kramers-Kronig Transforms 303
4.4.6 Corrosion Mechanisms 306
4.4.6.1 Active Dissolution 306
4.4.6.2 Active-Passive Transition 308
4.4.6.3 The Passive State 312
4.4.7 Reaction Mechanism Analysis of Passive Metals 324
4.4.7.1 The Point Defect Model 324
4.4.7.2 Prediction of Defect Distributions 334
4.4.7.3 Optimization of the PDM on the Impedance Data 335
4.4.7.4 Sensitivity Analysis 339
4.4.7.5 Extraction of PDM Parameters from EIS Data 343
4.4.7.6 Simplified Method for Expressing the Impedance of a Stationary Barrier Layer 349
4.4.7.7 Comparison of Simplified Model with Experiment 355
4.4.7.8 Summary and Conclusions 359
4.4.8 Equivalent Circuit Analysis 360
4.4.8.1 Coatings 365
4.4.9 Other Impedance Techniques 366
4.4.9.1 Electrochemical Hydrodynamic Impedance (EHI) 366
4.4.9.2 Fracture Transfer Function (FTF) 368
4.4.9.3 Electrochemical Mechanical Impedance 370
4.5 Electrochemical Power Sources 373 Evgenij Barsoukov, Brian E. Conway, Wendy G. Pell, and Norbert Wagner 373
4.5.1 Special Aspects of Impedance Modeling of Power Sources 373
4.5.1.1 Intrinsic Relation between Impedance Properties and Power Source Performance 373
4.5.1.2 Linear Time-Domain Modeling Based on Impedance Models: Laplace Transform 374
4.5.1.3 Expressing Electrochemical Model Parameters in Electrical Terms, Limiting Resistances, and Capacitances of Distributed Elements 376
4.5.1.4 Discretization of Distributed Elements, Augmenting Equivalent Circuits 379
4.5.1.5 Nonlinear Time-Domain Modeling of Power Sources Based on Impedance Models 381
4.5.1.6 Special Kinds of Impedance Measurement Possible with Power Sources: Passive Load Excitation and Load Interrupt 384
4.5.2 Batteries 386
4.5.2.1 Generic Approach to Battery Impedance Modeling 386
4.5.2.2 Lead-Acid Batteries 396
4.5.2.3 Nickel-Cadmium Batteries 398
4.5.2.4 Nickel-Metal Hydride Batteries 399
4.5.2.5 Li-ion Batteries 400
4.5.3 Nonideal Behavior Developed in Porous Electrode Supercapacitors 406
4.5.3.1 Introduction 406
4.5.3.2 Equivalent Circuits and Representation of Electrochemical Capacitor Behavior 409
4.5.3.3 Impedance and Voltammetry Behavior of Brush Electrode Models of Porous Electrodes 417
4.5.3.4 Deviations from Ideality 421
4.5.4 Fuel Cells 424
4.5.4.1 Introduction 424
4.5.4.2 Alkaline Fuel Cells (AFCs) 437
4.5.4.3 Polymer Electrolyte Fuel Cells (PEFCs) 443
4.5.4.4 The Solid Oxide Fuel Cells (SOFCs) 454
4.6 Dielectric Relaxation Spectroscopy 459 C. M. Roland 459
4.6.1 Introduction 459
4.6.2 Dielectric Relaxation 460
4.6.2.1 Ion Conductivity 462
4.6.2.2 Dielectric Modulus 467
4.6.2.3 Use of Impedance Function in Dielectric Relaxation Experiments 467
4.6.2.4 Summary 472
4.7 Electrical Structure of Biological Cells and Tissues: Impedance Spectroscopy, Stereology, and Singular Perturbation Theory 472 Robert S. Eisenberg 472
4.7.1 Impedance Spectroscopy of Biological Structures Is a Platform Resting on Four Pillars 474
4.7.1.1 Anatomical Measurements 474
4.7.1.2 Impedance Measurements 475
4.7.1.3 Measurement Difficulties 476
4.7.1.4 Future Measurements 476
4.7.1.5 Interpreting Impedance Spectroscopy 477
4.7.1.6 Fitting Data 477
4.7.1.7 Results 477
4.7.1.8 Future Perspectives 478 Acronym and Model Definitions 479 References 481
Index 517
J. Ross Macdonald1 and William B. Johnson2
1(William R. Kenan, Jr., Professor of Physics, Emeritus), Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC, USA
2W. L. Gore & Associates, Elkton, MD, USA
Since the end of World War II, we have witnessed the development of solid-state batteries as rechargeable energy storage devices with high power density; a revolution in high-temperature electrochemical sensors in environmental, industrial, and energy efficiency control; and the introduction of fuel cells to avoid the Carnot inefficiency inherent in noncatalytic energy conversion. The trend away from corrosive aqueous solutions and toward solid-state technology was inevitable in electrochemical energy engineering, if only for convenience and safety in bulk handling. As a consequence, the characterization of systems with solid-solid or solid-liquid interfaces, often involving solid ionic conductors and frequently operating well above room temperature, has become a major concern of electrochemists and materials scientists.
At an interface, physical properties-crystallographic, mechanical, compositional, and, particularly, electrical-change precipitously, and heterogeneous charge distributions (polarizations) reduce the overall electrical conductivity of a system. Proliferation of interfaces is a distinguishing feature of solid-state electrolytic cells, where not only is the junction between electrode and electrolyte considerably more complex than in aqueous cells but also the solid electrolyte is commonly polycrystalline. Each interface will polarize in its unique way when the system is subjected to an applied potential difference. The rate at which a polarized region will change when the applied voltage is reversed is characteristic of the type of interface: slow for chemical reactions at the triple-phase contacts between atmosphere, electrode, and electrolyte, appreciably faster across grain boundaries in the polycrystalline electrolyte. The emphasis in electrochemistry has consequently shifted from a time/concentration dependency to frequency-related phenomena, a trend toward small-signal alternating current (ac) studies. Electrical double layers and their inherent capacitive reactances are characterized by their relaxation times or more realistically by the distribution of their relaxation times. The electrical response of a heterogeneous cell can vary substantially depending on the species of charge present, the microstructure of the electrolyte, and the texture and nature of the electrodes.
Impedance spectroscopy (IS) is a relatively new and powerful method of characterizing many of the electrical properties of materials and their interfaces with electronically conducting electrodes. It may be used to investigate the dynamics of bound or mobile charge in the bulk or interfacial regions of any kind of solid or liquid material: ionic, semiconducting, mixed electronic-ionic, and even insulators (dielectrics). Although we shall primarily concentrate in this monograph on solid electrolyte materials-amorphous, polycrystalline, and single crystal in form-and on solid metallic electrodes, reference will be made, where appropriate, to fused salts and aqueous electrolytes and to liquid metal and high-molarity aqueous electrodes as well. We shall refer to the experimental cell as an electrode-material system. Similarly, although much of the present work will deal with measurements at room temperature and above, a few references to the use of IS well below room temperature will also be included. A list of acronym and model definitions appears at the end of this work.
In this chapter we aim to provide a working background for the practical materials scientist or engineer who wishes to apply IS as a method of analysis without needing to become a knowledgeable electrochemist. In contrast to the subsequent chapters, the emphasis here will be on practical, empirical interpretations of materials problems, based on somewhat oversimplified electrochemical models. We shall thus describe approximate methods of data analysis of IS results for simple solid-state electrolyte situations in this chapter and discuss more detailed methods and analyses later. Although we shall concentrate on intrinsically conductive systems, most of the IS measurement techniques, data presentation methods, and analysis functions and methods discussed herein apply directly to lossy dielectric materials as well.
Electrical measurements to evaluate the electrochemical behavior of electrode and/or electrolyte materials are usually made with cells having two identical electrodes applied to the faces of a sample in the form of a cylinder or parallelepiped. However, if devices such as chemical sensors or living cells are investigated, this simple symmetrical geometry is often not feasible. Vacuum, a neutral atmosphere such as argon, or an oxidizing atmosphere is variously used. The general approach is to apply an electrical stimulus (a known voltage or current) to the electrodes and observe the response (the resulting current or voltage). It is virtually always assumed that the properties of the electrode-material system are time invariant and it is one of the basic purposes of IS to determine these properties, their interrelations, and their dependences on such controllable variables as temperature, oxygen partial pressure, applied hydrostatic pressure, and applied static voltage or current bias.
A multitude of fundamental microscopic processes take place throughout the cell when it is electrically stimulated and, in concert, lead to the overall electrical response. These include the transport of electrons through the electronic conductors, the transfer of electrons at the electrode-electrolyte interfaces to or from charged or uncharged atomic species that originate from the cell materials and its atmospheric environment (oxidation or reduction reactions), and the flow of charged atoms or atom agglomerates via defects in the electrolyte. The flow rate of charged particles (current) depends on the ohmic resistance of the electrodes and the electrolyte and on the reaction rates at the electrode-electrolyte interfaces. The flow may be further impeded by band structure anomalies at any grain boundaries present (particularly if second phases are present in these regions) and by point defects in the bulk of all materials. We shall usually assume that the electrode-electrolyte interfaces are perfectly smooth, with a simple crystallographic orientation. In reality of course, they are jagged, full of structural defects and electrical short and open circuits, and they often contain a host of adsorbed and included foreign chemical species that influence the local electric field.
There are three different types of electrical stimuli that are used in IS. First, in transient measurements a step function of voltage [V(t)?=?V0 for t?>?0, V(t)?=?0 for t?<?0] may be applied at t?=?0 to the system, and the resulting time-varying current i(t) measured. The ratio V0/i(t), often called the indicial impedance or the time-varying resistance, measures the impedance resulting from the step function voltage perturbation at the electrochemical interface. This quantity, although easily defined, is not the usual impedance referred to in IS. Rather, such time-varying results are generally Fourier or Laplace transformed into the frequency domain, yielding a frequency-dependent impedance. If a Fourier transform is used, a distortion arising because of the non-periodicity of excitation should be corrected by using windowing. Such transformation is only valid when |V0| is sufficiently small that system response is linear. The advantages of this approach are that it is experimentally easily accomplished and that the independent variable, voltage, controls the rate of the electrochemical reaction at the interface. Disadvantages include the need to perform integral transformation of the results and the fact that the signal-to-noise ratio differs between different frequencies, so the impedance may not be well determined over the desired frequency range.
Second, a signal ?(t) composed of random (white) noise may be applied to the interface and measure the resulting current. Again, one generally Fourier transforms the results to pass into the frequency domain and obtain an impedance. This approach offers the advantage of fast data collection because only one signal is applied to the interface for a short time. The technique has the disadvantages of requiring true white noise and then the need to carry out a Fourier analysis. Often a microcomputer is used for both the generation of white noise and the subsequent analysis. Using a sum of well-defined sine waves as excitation instead of white noise offers the advantage of a better signal-to-noise ratio for each desired frequency and the ability to analyze the linearity of system response.
Third, the most common and standard one is to measure impedance by applying a single-frequency voltage or current to the interface and measuring the phase shift and amplitude, or real and imaginary parts, of the...
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