IMPEDANCE SPECTROSCOPY

IMPEDANCE SPECTROSCOPY Applications to Electrochemical and Dielectric Phenomena

Vadim F. Lvovich

) WILEY A JOHN WILEY & SONS, INC., PUBLICATION

Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada

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Library of Congress Cataloging-in-Publication Data:

Lvovich, Vadim F., 1967-Impedance spectroscopy with application to electrochemical and dielectric phenomena / Vadim F.

Lvovich. p. cm.

Includes index. ISBN 978-0-470-62778-5 (hardback)

1. Impedance spectroscopy. 2. Electrochemistry. I. Title. QD116.I57L86 2012 543'.4—dc23 2011028940

Printed in the United States of America.

10 9 8 7 6 5 4 3 2 1

Contents

Preface ix

1. Fundamentals of electrochemical impedance spectroscopy 1

1.1. Concept of complex impedance 1

1.2. Complex dielectric, modulus, and impedance

data representations 6

1.3. Electrochemical experiment: charge and material transport 14

1.4. Fundamental ambiguity of impedancespectroscopy analysis . . . 19

2. Graphical representation of impedance spectroscopy data 23

2.1. Nyquist and Bode representation of complex impedance data for ideal electrical circuits 23

2.2. Dielectric data representation 32

3. Equivalent-circuit elements and modeling of the impedance phenomenon 37

3.1. Ideal circuit elements 37

3.2. Nonideal circuit elements 39

3.3. Circuit models for systems with two and more time constants .. 44

4. Examples of ideal equivalent circuit models 49

4.1. Basic R-C circuit 49

4.2. Basic R|C circuit 51

4.3 Randies RS0L- RCT\CDL circuit 53

4.4. Debye dielectric relaxation (R1-C1)\C circuit 54

5. Impedance representation of bulk-material and electrode processes 59

5.1. Uncompensated impedance Z0HM 60

5.2. Bulk-media impedance—RS0L, RBULK and CBUUC 61

5.3. Electrochemical double-layer capacitance CD1 69

V

VI Contents

5.4. Electrochemical charge-transfer resistance RCT 73

5.5. Electrochemical sorption impedance ZS0RP 77

5.6. Mass-transport impedance 78

5.7. Mixed charge-transfer, homogeneous, and diffusion-controlled kinetics 87

6. Distributed impedance models

6.1. Distributed RBULK |CBULK - R|NT|CPEDL circuit model 97

6.2. General impedance models for distributed electrode processes 102

6.3. Identification of frequency ranges for conductivity and permittivity measurements 108

7. Impedance analysis of complex systems 113

7.1. Dielectric analysis of highly resistive composite materials with particle conduction 113

7.2. Dielectric analysis of ionic colloidal suspensions 123

7.3. AC electrokinetics and dielectrophoretic spectroscopy of colloidal suspensions 130

7.4. Specific adsorption and multistep heterogeneous kinetics 145

7.5. Impedance kinetics studies on porous electrodes 152

8. Impedance Instrumentation, testing, and data validation 163

8.1. Impedance test equipment 163

8.2. Single-sine impedance equipment—lock-in amplifier

and frequency-response analyzer 165

8.3. Multiple-sine impedance equipment 169

8.4. Electrochemical cells 172

8.5. Linearity, causality, stability, consistency, and error

analysis of impedance measurements 187

8.6. Complex nonlinear least-squares regression fitt ing 194

8.7. Practical approach to experimental impedance data collection and analysis 196

9. Selected examples of impedance-analysis applications: electroactive polymer films 205

9.1. The field of electroactive polymers 205

9.2. Impedance analysis of electrochemically active polymerfilms . 207

9.3. EIS models of conducting polymerfilms 208

9.4. The future of electroactive polymers 214

Contents VÜ

10. Selected examples of EIS analysis applications:

industrial colloids and lubricants 219

10.1. The field of industrial colloids and lubricants 219

10.2. Physical and chemical properties of lubricants 222

10.3. Degradation modes of lubricants 224

10.4. Impedance analysis of lubricants 227

10.5. Equivalent-circuit model of lubricants 242

11. Selected examples of EIS analysis applications: cell suspensions, protein adsorption, and implantable biomedical devices 247

11.1. The field of biomedical impedance applications 247

11.2. Analysis of biological suspensions by dielectric, impedance,

and AC electrokinetic methods 251

11.3. Impedance analysis of protein-adsorption kinetics 266

11.4. Impedance monitoring of implanted devices 271

12. Selected examples of impedance-analysis applications 281

12.1. Impedance analysis of insulating films and coatings 281

12.2. Impedance analysis of metallic paints 285

12.3. Electrorheological fluids and charged suspensions 289

12.4. Impedance of metal-oxide films and alloys 296

12.5. Li-ion kinetics in alkaline batteries 298

12.6. Impedance analysis of polymer electrolyte fuel cells 302

12.7. Impedance corrosion monitoring 309

13. Impedance-spectroscopy modifications 319

13.1. AC voltammetry 319

13.2. Potentiodynamicand Fourier-transform

impedance spectroscopy 321

13.3. Nonlinear higher-harmonic impedance analysis 323

13.4. Local EIS 327

13.5. Scanning photo-induced impedance microscopy 328

14. Conclusions and perspectives of EIS 333

Abbreviations and Symbols 335

Index 345

Preface

Since its conceptual introduction in the late 19th century, the impedance spec-troscopy has undergone a tremendous evolution into a rich and vibrant mul-tidisciplinary science. Over the last decade Electrochemical Impedance Spec-troscopy (EIS) has become established as one of the most popular analytical tools in materials research. The technique is being widely and effectively ap-plied to a large number of important areas of materials research and analysis, such as corrosion studies and corrosion control; monitoring of properties of electronic and ionic conducting polymers, colloids and coatings; measure-ments in energy storage, batteries, and fuel cells-related systems; biological analysis and biomedical sensors; measurements in semiconductors and solid electrolytes; studies of electrochemical kinetics, reactions and processes. Im-pedance spectroscopy is a powerful technique for investigating electrochemi-cal systems and processes. EIS allows to study, among others, such processes as adsorption, charge- and mass-transport, and kinetics of coupled sequential and parallel reactions.

In a broader sense, EIS is an extraordinarily versatile, sensitive, and infor-mative technique broadly applicable to studies of electrochemical kinetics at electrode-media interfaces and determination of conduction mechanisms in various materials through bound or mobile electronic, ionic, semiconductor, and mixed charges. Impedance analysis is fundamentally based on a relative-ly simple electrical measurement that can be automated and remotely con-trolled. Its main strength lies in its ability to interrogate relaxation phenomena whose time constants ranging over several orders of magnitude from minutes down to microseconds. In contrast to other analytical techniques, EIS is non-invasive technique that can be used for on-line analysis and diagnostics. The method offers the most powerful on-line and off-line analysis of the status of electrodes, monitors and probes in many different complex time- and space-resolved processes that occur during electrochemical experiments. For in-stance, the EIS technique has been broadly practiced in the development of sensors for monitoring rates of materials' degradation, such as metal corro-sion and biofouling of implantable medical devices.

EIS is useful as an empirical quality-control procedure that can also be employed to interpret fundamental electrochemical and electronic processes. Experimental impedance results can be correlated with many practically use-ful chemical, physical, mechanical, and electrical variables. With the current availability of ever evolving automated impedance equipment covering broad frequency and potential ranges, the EIS studies have become increasingly popular as more and more electrochemists, material scientists, and engineers

IX

X Preface

understand the theoretical basis for impedance spectroscopy and gain skill in the impedance data interpretation.

The impedance technique appears destined to play an increasingly impor-tant role in fundamental and applied electrochemistry and material science in the coming years. However, broader practical utilization of EIS has been hin-dered by the lack of comprehensive and cohesive explanation of the theory, measurements, analysis techniques, and types of acquired data for different investigated systems. These factors may be connected with the fact that exist-ing literature reviews of EIS are very often difficult to understand by non-spe-cialists. As will be shown later, the ambiguity of impedance data interpreta-tion and the establishment of direct relationships with practical physical, chemical, electrical, and mechanical parameters constitute the main disadvan-tages of the technique. These general weaknesses are amplified especially when considering a great variety of practical impedance applications, where a practical investigator or researcher often has to decide if any of the previously known impedance response models and their interpretations are even remote-ly applicable to the problem in hand. EIS data demonstrate the investigated system's response to applied alternating or direct electrical fields. It becomes the investigators' responsibility to convert the electrical data into parameters of interest, whether it is a concentration of bioanalyte, corrosion rate of metal surfaces, performance characteristics of various components of a fuel cell, or rate of oxidative decomposition of polymer films.

As industrial scientist and engineer with career encompassing multiple senior technology and product development positions in leading R&D divi-sions in Specialty Chemicals, Electronics, BioMedical, and Aerospace indus-trial corporations, the author has learned over the years to greatly appreciate the investigative power and flexibility of EIS and impedance-based devices in commercial product development. This book was born out of acute need to catalog and explain multiple variations of the EIS data characteristics encoun-tered in many different practical applications. Although the principle behind the method remains the same, the impedance phenomena investigation in dif-ferent systems presents a widely different data pattern and requires signifi-cant variability in the experimental methodology and interpretation strategy to make sense of the results. The EIS experimental data interpretation for both unknown experimental systems, and well-known systems investigated by other (non-electrochemical) means is widely acknowledged to be the main source of the method's application challenge, often listed at the main impedi-ment to the method's broader penetration into scientific and technological markets. This book attempts to at least partially standardize the catalog of EIS responses across many practically encountered fields of use and to present a coherent approach to the analysis of experimental results.

This book is intended to serve as a reference on the topic of practical ap-plications of impedance spectroscopy, while also addressing some of the most basic aspects of EIS theory. The theory of the impedance spectroscopy has been presented in great details and with remarkable skill in well-received monographs by J. R. MacDonald, and recently by M. Orazem and B. Tribollet, as well as in many excellent review chapters referenced in this book. There

Preface XI

are a number of short courses, several monographs and many independent publications on the impedance spectroscopy. However, the formal courses on the topic are rarely offered in the university settings. At the same time, there is a significant worldwide need to offer independent, direct and comprehen-sive training on practical applications of the impedance analysis to many in-dustrial scientists and engineers relatively unfamiliar with the EIS theory but eager to apply impedance analysis to address their everyday product devel-opment technological challenges. This manuscript emphasizes practical ap-plications of the impedance spectroscopy. This book in based around a cata-logue representing a typical impedance data for large variety of established, emerging, and non-conventional experimental systems; relevant mathemati-cal expressions; and physical and chemical interpretation of the experimental results. Many of these events are encountered in the field by industrial scien-tists and engineers in electrochemistry, physical and analytical chemistry, and chemical engineering.

This book attempts to present a balance of theoretical considerations and practical applications for problem solving in several of the most widely used fields where electrochemical impedance spectroscopy analysis is being em-ployed. The goal was to produce a text that would be useful to both the novice and the expert in EIS. It is primarily intended for industrial researchers (mate-rial scientists, analytical and physical chemists, chemical engineers, material researchers), and applications scientists, wishing to understand how to cor-rectly make impedance measurements, interpret the results, compare results with expected previously published results form similar chemical systems, and use correct mathematical formulas to verify the accuracy of the data. A majority of these individuals reside in the specialty chemicals, polymers, col-loids, electrochemical renewable energy and power sources, material science, electronics, biomedical, pharmaceutical, personal care and other smaller in-dustries. The book intends to provide a working background for the practical scientist or engineer who wishes to apply EIS as a method of analysis without needing to become an expert electrochemist. With that in mind, both some-what oversimplified electrochemical models and in-depth analysis of specific topics of common interest are presented. The manuscript covers many of the topics needed to help readers identify whether EIS may be an appropriate method for their particular practical application or research problem. A num-ber of practical examples and graphical representations of the typical data in the most common practical experimental systems are presented. In that re-spect the book may also be addressed to students and researchers who may found the presented catalog of impedance phenomenological data and the relevant discussions to be of assistance in their introduction to theoretical and practical aspects of electrochemical research.

Starting with general principles, the book emphasizes practical applica-tions of the electrochemical impedance spectroscopy to separate studies of bulk solution and interfacial processes, using of different electrochemical cells and equipment for experimental characterization of different systems. The monograph provides relevant examples of characterization of large variety of materials in electrochemistry, such as polymers, colloids, coatings, biomedical

XÜ Preface

species, metal oxides, corroded metals, solid-state devices, and electrochemi-cal power sources. The book covers many of the topics needed to help readers identify whether impedance spectroscopy may be an appropriate method for their particular research problem.

This book incorporates the results of the last two decades of research on the theories and applications of impedance spectroscopy, including more de-tailed reviews of the impedance methods applications in industrial colloids, biomedical sensors and devices, and supercapacitive polymeric films. The book is organized so each chapter stands on its own. The book should assist readers to quickly grasp how to apply their new knowledge of impedance spectroscopy methods to their own research problems through the use of fea-tures such as:

• Equations and circuit diagrams for the most widely used impedance models and applications

• Figures depicting impedance spectra of typical materials and devices • Theoretical considerations for dealing with modeling, equivalent cir-

cuits, and equations in the complex domain • Best measurement methods for particular systems and alerts to poten-

tial sources of errors • Review of impedance instrumentation • Review of related techniques and impedance spectroscopy modifica-

tions • Extensive references to the scientific literature for more information on

particular topics and current research

It is hoped that the more advanced reader will also find this book valuable as a review and summary of the literature on the subject. Of necessity, com-promises have been made between depth, breadth of coverage, and reason-able size. Many of the subjects such as mathematical fundamentals, statistical and error analysis, and a number of topics on electrochemical kinetics and the method theory have been exceptionally well covered in the previous manu-scripts dedicated to the impedance spectroscopy. Similarly the book has not been able to accommodate discussions on many techniques that are useful but not widely practiced. While certainly not nearly covering the whole breadth of the impedance analysis universe, the manuscript attempts to provide both a convenient source of EIS theory and applications, as well as illustrations of applications in areas possibly unfamiliar to the reader. The approach is first to review the fundamentals of electrochemical and material transport processes as they are related to the material properties analysis by impedance / modu-lus / dielectric spectroscopy (Chapter 1), discuss the data representation (Chapter 2) and modeling (Chapter 3) with relevant examples (Chapter 4). Chapter 5 discusses separate components of the impedance circuit, and Chap-ters 6 and 7 present several typical examples of combining these components into practically encountered complex distributed systems. Chapter 8 is dedi-cated to the EIS equipment and experimental design. Chapters 9 through 12

Preface XÜi

are dedicated to detailed discussions of impedance analysis applications to specific experimental systems, representing both well-studied and emerging fields. Chapter 13 offers a brief review of EIS modifications and closely related analytical methods.

I owe thanks to many others who have helped with this project. I am espe-cially grateful to John Wiley & Sons, Inc. and Lone Wolf Enterprises, Ltd. for their conscientious assistance with many details of preparation and produc-tion. Over the years many valuable comments and encouragement have been provided by colleagues through the electrochemical community who assured that there would be a demand for this book. I also would like to thank my wife Laura and my son William for affording me the time and freedom required to undertake such a project.

CHAPTER

Fundamentals of Electrochemical

Impedance Spectroscopy

1.1. Concept of complex impedance

The concept of electrical impedance was first introduced by Oliver Heaviside in the 1880s and was soon afterward developed in terms of vector diagrams and complex numbers representation by A. E. Kennelly and C. P. Steinmetz [1, p. 5]. Since then the technique has gained in exposure and popularity, pro-pelled by a series of scientific advancements in the field of electrochemistry, improvements in instrumentation performance and availability, and increased exposure to an ever-widening range of practical applications.

For example, the development of the double-layer theory by Frumkin and Grahame led to the development of the equivalent circuit (EC) modeling ap-proach to the representation of impedance data by Randies and Warburg. Ex-tended studies of electrochemical reactions coupled with diffusion (Gerisher) and adsorption (Eppelboin) phenomena, effects of porous surfaces on electro-chemical kinetics (de Levie), and nonuniform current and potential distribu-tion dispersions (Newman) all resulted in a tremendous expansion of imped-ance-based investigations addressing these and other similar problems [1]. Along with the development of electrochemical impedance theory, more elab-orate mathematical methods for data analysis came into existence, such as Kramers-Kronig relationships and nonlinear complex regression [1, 2]. Trans-formational advancements in electrochemical equipment and computer tech-nology that have occurred over the last 30 years allowed for digital automated

1

1

2 Chapter One

impedance measurements to be performed with significantly higher quality, better control, and more versatility than what was available during the early years of EIS. One can argue that these advancements completely revolution-ized the field of impedance spectroscopy (and in a broader sense the field of electrochemistry), allowing the technique to be applicable to an exploding universe of practical applications. Some of these applications, such as dielec-tric spectroscopy analysis of electrical conduction mechanisms in bulk poly-mers and biological cell suspensions, have been actively practiced since the 1950s [3,4]. Others, such as localized studies of surface corrosion kinetics and analysis of the state of biomedical implants, have come into prominence only relatively recently [5, 6, 7, 8].

In spite of the ever-expanding use of EIS in the analysis of practical and experimental systems, impedance (or complex electrical resistance, for a lack of a better term) fundamentally remains a simple concept. Electrical resistance R is related to the ability of a circuit element to resist the flow of electrical cur-rent. Ohm's Law (Eq. 1-1) defines resistance in terms of the ratio between in-put voltage V and output current I:

V R=- (1-1)

I

While this is a well-known relationship, its use is limited to only one cir-cuit element—the ideal resistor. An ideal resistor follows Ohm's Law at all current, voltage, and AC frequency levels. The resistor's characteristic resis-tance value R [ohm] is independent of AC frequency, and AC current and voltage signals though the ideal resistor are "in phase" with each other. Let us assume that the analyzed sample material is ideally homogeneous and com-pletely fills the volume bounded by two external current conductors ("elec-trodes") with a visible area A that are placed apart at uniform distance d, as shown in Figure 1-1. When external voltage V is applied, a uniform current I passes through the sample, and the resistance is defined as:

R=P~ (1-2)

where p [ohm cm] is the characteristic electrical resistivity of a material, representing its ability to resist the passage of the current. The inverse of resis-tivity is conductivity σ [1 / (ohm cm)] or [Sm/cm], reflecting the material's ability to conduct electrical current between two bounding electrodes.

An ideal resistor can be replaced in the circuit by another ideal element that completely rejects any flow of current. This element is referred as an "ide-al" capacitor (or "inductor"), which stores magnetic energy created by an ap-plied electric field, formed when two bounding electrodes are separated by a non-conducting (or "dielectric") medium. The AC current and voltage signals though the ideal capacitor are completely "out of phase" with each other, with current following voltage. The value of the capacitance presented in Far-ads [F] depends on the area of the electrodes A, the distance between the

Fundamentals of Electrochemical Impedance Spectroscopy 3

electrodes d, and the properties of the dielectric reflected in a "relative permit-tivity" parameter ε as:

C = ^ (1-3) a

where ε0 = constant electrical permittivity of a vacuum (8.85 10~14F/cm). The relative permittivity value represents a characteristic ability of the ana-lyzed material to store electrical energy. This parameter (often referred to as simply "permittivity" or "dielectric") is essentially a convenient multiplier of the vacuum permittivity constant ε0 that is equal to a ratio of the material's permittivity to that of the vacuum. The permittivity values are different for various media: 80.1 (at 20°C) for water, between 2 through 8 for many poly-mers, and 1 for an ideal vacuum. A typical EIS experiment, where analyzed material characteristics such as conductivity, resistivity, and permittivity are determined, is presented in Figure 1-1.

Impedance is a more general concept than either pure resistance or ca-pacitance, as it takes the phase differences between the input voltage and out-put current into account. Like resistance, impedance is the ratio between volt-age and current, demonstrating the ability of a circuit to resist the flow of electrical current, represented by the "real impedance" term, but it also re-flects the ability of a circuit to store electrical energy, reflected in the "imagi-nary impedance" term. Impedance can be defined as a complex resistance encountered when current flows through a circuit composed of various resis-tors, capacitors, and inductors. This definition is applied to both direct current (DC) and alternating current (AC).

FIGURE 1-1 Fundamental impedance experiment

4 Chapter One

phase-shift

FIGURE 1-2 Impedance experiment: sinusoidal voltage input V at a single frequency/and current response /

In experimental situations the electrochemical impedance is normally measured using excitation AC voltage signal V with small amplitude VA (ex-pressed in volts) applied at frequency/(expressed in Hz or 1/sec). The voltage signal V (t), expressed as a function of time f, has the form:

V(f) = VA sin(2;#i) = VA sin(cof) (1-4)

In this notation a "radial frequency" co of the applied voltage signal (ex-pressed in radians/second) parameter is introduced, which is related to the applied AC frequency/as ω = 2 π / .

In a linear or pseudolinear system, the current response to a sinusoidal voltage input will be a sinusoid at the same frequency but "shifted in phase" (either forward or backward depending on the system's characteristics)—that is, determined by the ratio of capacitive and resistive components of the out-put current (Figure 1-2). In a linear system, the response current signal I(t) is shifted in phase (φ) and has a different amplitude, IA:

I(t) = 1 sin(cof + (|>) (1-5)

An expression analogous to Ohm's Law allows us to calculate the complex impedance of the system as the ratio of input voltage V(t) and output mea-sured current I(t):

Z* V(t) _ VAsin(p)t) _ sin((or) 1(f) ~ ί sin(oot + <|>) ~ Λ3ίη(ωί + φ)

(1-6)

The impedance is therefore expressed in terms of a magnitude (abso-lute value), ZA= \Z\, and a phase shift, φ. If we plot the applied sinusoidal voltage signal on the x-axis of a graph and the sinusoidal response signal I(t) on the y-axis, an oval known as a "Lissajous figure" will appear (Figure 1-3A).

Fundamentals of Electrochemical Impedance Spectroscopy 5

I *

Ι+ΔΙ m

-*■ co=°°

V+AV A V+AV B " ZREAL ZREAL

FIGURE 1-3 Impedance data representations: A. Lissajous figure; B. Com-plex impedance plot

Analysis of Lissajous figures on oscilloscope screens was the accepted method of impedance measurement prior to the availability of lock-in amplifiers and frequency response analyzers. Modern equipment allows automation in ap-plying the voltage input with variable frequencies and collecting the output impedance (and current) responses as the frequency is scanned from very high (MHz-GHz) values where timescale of the signal is in micro- and nano-seconds to very low frequencies (μΗζ) with timescales of the order of hours.

Using Euler's relationship: exp(j0) = cos<j> + jsin((> (1-7)

it is possible to express the impedance as a complex function. The poten-tial V(t) is described as:

V(t) = VA e'°* (1-8)

and the current response as:

/(f) =/ (,/"*-* (1-9)

The impedance is then represented as a complex number that can also be expressed in complex mathematics as a combination of "real," or in-phase (ZREAI), and "imaginary," or out-of-phase (Z/M), parts (Figure 1-3B):

V z * = y = Z A ^ =ZA (cos0 + jsin<|)) = ZREAL + jZu (1-10)

and the phase angle <f> at a chosen radial frequency ω is a ratio of the imag-inary and real impedance components:

tan0 or φ =arctan( ——) (i-ii)

6 Chapter One

1.2. Complex dielectric, modulus, and impedance data representations

In addition to the AC inputs such as voltage amplitude VA and radial frequen-cy ω, impedance spectroscopy also actively employs DC voltage modulation (which is sometimes referred to as "offset voltage" or "offset electrochemical potential") as an important tool to study electrochemical processes. Alterna-tive terms, such as "dielectric spectroscopy" or "modulus spectroscopy," are often used to describe impedance analysis that is effectively conducted only with AC modulation in the absence of a DC offset voltage (Figure 1-4).

Dielectric analysis measures two fundamental characteristics of a materi-al—permittivity ε and conductivity σ (or resistivity p)—as functions of time, temperature, and AC radial frequency ω. As was discussed above, permittiv-ity and conductivity are two parameters characteristic of respective abilities of analyzed material to store electrical energy and transfer electric charge. Both of these parameters are related to molecular activity. For example, a "dielec-tric" is a material whose capacitive current (out of phase) exceeds its resistive (in phase) current. An "ideal dielectric" is an insulator with no free charges that is capable of storing electrical energy. The Debye Equation (Eq. 1-12) re-lates the relative permittivity ε to a concept of material polarization density P [C/m2], or electrical dipole moment [C/m] per unit volume [m3], and the applied electric field V:

P=(e - l ) e 0 V (1-12)

Depending on the investigated material and the frequency of the applied electric field, determined polarization can be electronic and atomic (very small translational displacement of the electronic cloud in THz frequency range), orientational or dipolar (rotational moment experienced by permanently po-lar molecules in kHz-MHz frequency range), and ionic (displacement of ions with respect to each other in Hz-kHz frequency region).

The dielectric analysis typically presents the permittivity and conductivity material properties as a combined "complex permittivity" ε* parameter, which is analogous to the concept of complex impedance Z* (Figure 1-4A). Just as complex impedance can be represented by its real and imaginary components, complex permittivity is a function of two parameters—"real" permittivity (of-ten referred to as "permittivity" or "dielectric constant") ε' and imaginary per-mittivity (or "loss factor") ε" as:

ε* = ε ' - /ε" (1-13)

In dielectric material ε' represents the alignment of dipoles, which is the energy storage component that is an inverse equivalent of ZM. ε" represents the ionic conduction component that is an inverse equivalent of ZREAL. Both real permittivity and loss factor can be calculated from sample resistance R, conductivity σ, resistivity p, and capacitance C measured in a fundamental experimental setup (Figure 1-1) as:

, Cd „ d σ 1 /·, i„\ ε = and ε = = = (1-14)

ε0Λ RA(OE0 ωε0 ρωε0

Fundamentals of Electrochemical Impedance Spectroscopy 7

Permittivity and conductivity values and their relative contributions to the measured voltage to current ratio (impedance) are often dependent on the material's temperature, external AC frequency, and magnitude of the applied voltage. In fact, real permittivity is often not quite appropriately referred to as the "dielectric constant," the parameter that should always be specified at a standard AC frequency (usually about 100 kHz) and temperature conditions (typically 25CC) and therefore is not exactly "constant." The concepts of "con-ductivity" and "resistivity" for a chosen material are also vague. These pa-rameters have to be specified at standard temperature conditions and be care-fully measured with full consideration of the impedance dependence on the applied electric field. In practice these rules are often not followed. For in-stance, conductivity of many solutions is often measured by hand-held meters operating at an arbitrary frequency around ~ 1kHz, which, as will be shown in Section 6-3, may or may not be appropriate conditions for many materials even when they belong to the same family (such as aqueous solutions). Alter-natively, operating at much higher frequencies may result in the measurement being dominated by the out-of-phase capacitive impedance, which is a func-tion of the sample's dielectric constant and not of its conductivity. For instance, saline (p = 100 ohm cm, ε = 80) is a conductor below 250MHz and a capacitor above 250MHz.

Dielectric spectroscopy, although using the same type of electrical infor-mation as impedance spectroscopy, is logically different in its analysis and approach to data representation. Dielectric response is based on a concept of "energy storage" and resulting "relaxation" per release of this energy by the system's individual components. Initially the concept of dielectric relaxation was introduced by Maxwell and expanded by Debye, who used it to describe the time required for dipolar molecules to reversibly orient themselves in the external AC electric field. In the experiments of Debye a step function excita-tion was applied to the system, and the system was allowed to "relax" to equi-librium after the excitation was removed. The time required for that process to take place was called "relaxation time" τ = 1/2π/ that is inversely related to "critical relaxation frequency"/. Dielectric spectroscopy measures relaxation times by detecting frequency dependence of complex permittivity ε* and de-termining/ values from positions of the peaks in the ε* =/(/) plot as the input voltage signal is scanned over the experimental AC frequency range.

Dispersion, or frequency dependence, according to the laws of relaxation, is the corresponding frequency domain expression of complex permittivity ε* as a function of radial (or cycling) frequency ω. For example, as the applied frequency ω is increased, a steplike decrease in complex permittivity is ob-served due to the fact that polarized molecules that are fully aligned with each change in direction of the AC field at lower frequencies cannot follow the high-er frequency field at each direction reversal (Figure 1-4B). As the high-frequen-cy AC field changes direction faster, these molecules "relax" to nonaligned positions where they cannot store energy. Large nonpolar molecules typically lose their orientation with the field at low Hz frequencies and have relaxation times on the order of seconds. Smaller and more polar ionic species "relax" at kHz-MHz frequencies and show millisecond to microsecond relaxation times. The components of a sample typically have high permittivity (capacitance)

8 Chapter One

values at low frequencies where more different types of molecules can com-pletely align with the field and store the maximum possible energy and are being effectively charged as dielectric dipoles. At high frequencies fewer di-poles store energy, and the total measured capacitance and permittivity of the system are low. Comparative analysis of dielectric dependencies of pure com-ponents presents an opportunity to identify and separate these species in com-plex mixed systems based on the AC frequency dependence of their dielectric response ε* = f(co).

The above frequency dependence of dielectric material properties, such as capacitance C(w), permittivity ε'(ω), and conductivity σ(ω), can be expressed by Debye's "single relaxation" model [3, p. 65]. The Debye model is a popular representation used to illustrate bulk relaxation processes in ideal dielectrics, such as highly resistive polymers, where it is assumed that there is no conduc-tion (or "loss") through the bulk material as the sample resistance R is infi-nitely high and conductivity σ —> 0 [3, 4]. This model is a classical representa-tion of a simple dielectric or fully capacitive experimental system, where transition occurs from high-frequency permittivity ε^ (or capacitance C2) to low-frequency permittivity eLf (or capacitance Cj) where Cj > C2 (Section 4-4). In the Debye model the response is ideally capacitive at both high and low frequency extremes, with the transition between the two regimes character-ized by permittivity increment Δε = eLf -Em. It is usually expressed for a me-dium permittivity ε as a function of the AC field's radial frequency ω, where τ is the characteristic "relaxation time" of the system. Expansion of Equation 1-13 leads to:

ε * (ω) =ε - /ε = ε + where ε (ω) = ε + — and ε (ω) = — (1-15) " 1 + /ωτ ~ 1 + ω τ 1 + ω τ

Complex permittivity of a more realistic "lossy" dielectric where non-zero parallel DC conductivity σ(ω) exists can be represented on the basis of a more complex Havriliak-Negami model. This model also accounts for non-ideali-ties of both capacitive and resistive components accounting for the asymme-try and broadness of the dielectric dispersion curve and resulting frequency-dependent conductivity σ(ω) and permittivity ε(ω) contributions:

ε*(ω) = Δε ε +-(1 + 0'ω*)Ύ

■ ; ^σ(ω) ν ε ο ω

(1-16)

Where: τ = 1/2π£. = characteristic relaxation time, ω = radial frequency, N = parameter that defines the frequency dependence of the conductivity term (typically N —> 1 and equals the slope of the low-frequency increase in £*=f((ü) or ε" =/(ω) plot due to the low-frequency conduction through the system, as shown in Figure 1-4B), a and β = shape parameters accounting for symmetric and asymmetric broadening of the relaxation peak.

In addition to the Debye model for dielectric bulk materials, other dielec-tric relaxations expressed according to Maxwell-Wagner or Schwartz "interra-cial" mechanisms exist. For example, the Maxwell-Wagner "interfacial" polar-ization concept deals with processes at the interfaces between different components of an experimental system. Maxwell-Wagner polarization occurs

Fundamentals of Electrochemical Impedance Spectroscopy 9

FIGURE 1-4 Representations of complex impedance data as function of AC frequency: A. impedance and phase angle; B. permittivity and conductivity; C. modulus

10 Chapter One

either at inner boundary layers separating two dielectric components of a sample or more often at an interface between the sample and an external ideal electrical conductor (electrode). In both cases this leads to a significant separa-tion of charges over a considerable distance. This contribution of Maxwell-Wagner polarization to dielectric loss can be orders of magnitude larger than the molecular fluctuation's dielectric response described by the Debye mecha-nism. Maxwell-Wagner interfacial effects are prominent in electrochemical studies dealing with heterogeneous interfacial kinetics.

The Debye model is primarily describing a bulk material "dielectric" re-sponse. Traditional dielectric spectroscopy has found a significant use in the characterization of multicomponent resistive materials with mixed or particle-based conduction mechanisms, such as polymers, nonpolar organics (lubri-cants), and moderately resistive aqueous (cellular) colloids. However, there are relatively few practical cases of nearly ideal dielectric materials where more than one well-resolved dielectric relaxations can be identified at a con-stant temperature. The Maxwell-Wagner electrode-sample interface phenom-enon and other interfacial effects (such as double layer charging and Faradaic kinetics) that result in apparent high interfacial capacitance masking settled capacitance changes in bulk material at frequencies below -10 kHz are viewed as severe restricting factors in studies of dielectric materials. With the excep-tion of extreme cases, such as ion-free insulating media with polarized particle conduction (such as electrorheological fluids) or subfreezing sample tempera-tures, the frequency range relatively free of the effects of interfacial polariza-tion is often limited to high kHz-GHz. In such a limited frequency range at a constant temperature the appearance of several types of conducting species showing significant frequency dependence, not interfering with one another and present in a significant and balanced range of concentrations, is a rare oc-currence. Hence, dielectric analysis often relies not on the AC frequency but on wide temperature modulation at a few selected AC frequencies as the pri-mary interrogation mode to extract details of sample analytical information.

Another representation of the dielectric properties of analyzed media is complex modulus M* (Figure 1-4C). The modulus is the inverse of complex permittivity ε* and can also be expressed as a derivative of complex imped-ance Z*:

M * =^7 = M ' - JM" = A06»2* = -ωε»2™ + ;«KOZRML (1-17)

Fundamentally, complex electrochemical impedance (Z*), modulus (M*), and permittivity (ε*) parameters are all determined by applying an AC poten-tial at a variable frequency and measuring output current through the sample (Figure 1-1). In a broader sense dielectric, modulus, and impedance analysis represent the same operational principles and can be referred to as subsets of a universal broadband electrochemical impedance spectroscopy (UBEIS). This technique analyzes both resistive and capacitive components of the AC cur-rent signal response, containing the excitation frequency and its harmonics. This current signal output can be analyzed as a sum of sinusoidal functions (a Fourier series). Depending on applied AC frequency and voltage, the output

Fundamentals of Electrochemical Impedance Spectroscopy 11

current can be supported by various conductive mechanisms through the analyzed system. These conductive mechanisms can be related to a single pro-cess, or be supported by a combination of various ionic, electronic, and parti-cle conductors and their relative concentrations. The conduction process oc-curs both in bulk sample and at the interface between the sample and the electrodes, where a series of electron-exchanging reactions may take place. Critical relaxation frequencies can be determined from peak positions in com-plex impedance, modulus and permittivity plots.

The same data can be presented in modulus, dielectric, and impedance domains (Figure 1-4) and can be converted directly between the domains us-ing expressions based on Maxwell equations. Although the data representing the electrochemical relaxation phenomenon fundamentally contain the same information and are independent of the chosen representation method, pre-senting the data sepa-rately as dielectric, impedance, and modulus plots often allows extracting additional useful information about the analyzed system. For instance, localized relaxations result in the peaks appearing at different frequencies in complex impedance or complex modulus vs. frequency plots, whereas long-range fundamental conductivity results in exact overlapping of the modulus and impedance peaks. Also, as will be shown later, these data representations have different resolving capabilities to present the results as a function of the applied experimental conditions.

Nevertheless, historically a differentiation exists between "dielectric" and "impedance" spectroscopies. Traditional dielectric analysis has been applied primarily to the analysis of bulk "dielectric" properties of polymers, plastics, composites, and nonaqueous fluids with very high bulk material resistance. The dielectric method is characterized by using higher AC voltage amplitudes, temperature modulation as an independent variable, lack of DC voltage per-turbation, and often operating frequencies above 1 kHz or measurements at several selected discrete frequencies [2, p. 33].

Traditional impedance spectroscopy is preoccupied with investigating charge and material-exchange ("electrochemical") kinetic processes that occur at electrode-sample interfaces. This technique actively employs DC modula-tion just as most electrochemical techniques do, typically uses low AC voltage amplitudes to maintain linearity of signal response, and employs wide fre-quency ranges from MHz to μΗζ. Unlike dielectric spectroscopy, the analyti-cal aspect of traditional impedance analysis is based not on bulk material in-vestigations but rather on quantification of sample species through relevant interfacial impedance parameters. The majority of traditional EIS studies em-phasize involved analysis of Faradaic and double-layer interfacial kinetics and the effects of DC potential modulation. As opposed to the dielectric spec-troscopy experimental approach, traditional impedance analysis often at-tempts to minimize or keep constant the effects of bulk material impedance and related dielectric relaxations due to capacitance effects of energy-storage components in the system. That is typically achieved by operating in highly conductive samples (such as aqueous solutions with supporting electrolytes) with a total bulk solution resistance of just several ohms and negligible ca-pacitive effects. The impedance response of the electrode-solution interface,

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which is located in series with the bulk-solution impedance, can be easily de-termined at relatively low, typically Hertzian, AC frequencies. As demonstrat-ed in Figure 1-4A, the impedance response of the conducting species and me-dium properties are qualitatively reflected in an integrative manner. Therefore the appearance of additional interfacial impedance at low frequencies results in an increase in the measured impedance to reflect a combined effect of both bulk and interfacial impedances. However, if the bulk impedance is kept very low by operating in highly conductive media, the total measured low-fre-quency impedance effectively becomes equal to the interfacial impedance.

The impedance output is fundamentally determined by the characteristics of an electrical current conducted through the system and is based on the con-cept that the obtained data represents the sample's "least impedance to the current." A significant portion of this book is dedicated to analysis of various possible parallel and sequential conducting mechanisms through the ana-lyzed systems of interest. In any given experimental system there are always several competing paths for the current to travel through a sample. The cur-rent, however, chooses one or several closely matched predominant "paths of least resistance" between two electrical conductors (electrodes) under applied conditions such as AC frequency and voltage amplitude, DC voltage, elec-trode geometry and configuration, sample composition and concentration of main conducting species, temperature, pressure, convection, and external magnetic fields. Only the conduction through this predominant path of least resistance, or, to be more exact, the path of the least impedance, is measured by the EIS.

For example, for a sample represented by a parallel combination of a ca-pacitor C and a resistor R (defined there as R/C), at high frequencies ω the impedance to current is the lowest through the capacitive component where impedance is inversely proportional to the frequency, as Z * ~ (coC)~, and therefore is smaller than the impedance of the finite resistor R. At lower fre-quencies the opposite becomes true—the capacitive impedance component becomes large, and the current predominantly flows through the resistor; the total measured impedance reflects the resistance value as Z * ~ R. The detected impedance output is determined by measuring the current passing through the least impeding segment of the circuit. The characteristic parameters of this ideal system can therefore be determined from the total impedance response Z* = f(co) as pure capacitance C at higher frequencies and pure resistance R at lower frequencies. Characteristic relaxation frequency fc corresponds to a val-ue where switching between the two conduction mechanisms occurs. The in-verse of this frequency is characteristic "relaxation time" T=l/ooc for this circuit. For example, in aqueous conductive solution with permittivity of -80 and bulk resistance ~ 1 ohm, it is easy to determine that at all frequencies un-der fc~ 1 GHz the current is conducted through a very small bulk solution resistor, and the capacitive characteristics are not contributing to the measured impedance signal. Similar results were shown for the above example illustrat-ing current conduction through a saline solution.

These examples represent a circuit composed of ideal capacitive and resis-tive electrical components. The path of least impedance through a real-life sample placed between two conducting metal electrodes can be represented

Fundamentals of Electrochemical Impedance Spectroscopy 13

by a combination of chemical and mechanical elements that can only to some degree be approximated by these ideal electrical elements. These conducting venues through the sample may be the most plentiful and mobile electrons, ions, and particles; double-layer capacitive charging effects, specific adsorp-tion, charge transfer "resistance" to tunneling electrons crossing the interface between the sample and the electrodes, and transport of discharging species to the surface of the electrode through diffusion layer concentration gradients at the electrode/sample interfaces. To interpret the EIS results, scientists and engineers have to intelligently devise the experiment to extract the needed analytical information from the experimentally obtained impedance. This in-terpretation includes initial development of the relationships between the electrical parameters representing the path of least impedance to the current and the "real life" investigated chemical and mechanical components that re-sult in the measured impedance response. Through these relationships it may be possible to analyze the actual chemical, physical, and mechanical processes inside the analyzed system.

To the great benefit of analytical chemists concerned with the accurate analysis of concentrations of analytes and material scientists preoccupied with sample properties, the mechanisms of transporting current are largely deter-mined by limited types of species that are present in the analyzed sample at significant levels. In that respect UBEIS is indeed a "spectroscopy," where a combination of carefully conducted experiments, some degree of initial pre-conceived knowledge of the analyzed sample, selection of experimental con-ditions, and careful interpretation of results are required to determine the sample composition and the mechanisms responsible for the impedance re-sponse, quantify them, and develop a mechanical model of the entire system's physical and chemical response to applied external electric field.

With that in mind, one cannot always determine "all" species present in the experimental system, as not "all" species may participate in the conduc-tion through the path of least impedance, even if their response is dependent on the controlled experimental parameters, such as AC frequency. As with many other spectroscopy methods, UBEIS has the advantage of often being able to separate and identify several types of species inside an experimental sample, as various species may create a path of least impedance through the sample at different (and variable) experimental conditions. However, many potential analytes may still not be detectable, as they do not conduct current or are present in such low concentrations that their contribution to the overall conduction process may be negligible. However, they may have a dispropor-tionably strong activity at the electrode interface. The same general limitations are present in optical and other "spectroscopies" based on "detection of the most prominent contributors" principle. It is the task of good experimental scientists to develop experimental designs that would provide the necessary information about the task at hand.

As will be further discussed, current traveling through a sample has to overcome successive impedances of two fundamentally different regions. One of them is the "bulk" of the sample, and the other is the "interface" where the sample meets the current conducting electrodes. Electrochemistry in gen-eral and the UBEIS in particular are often used to analyze both bulk sample

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conduction mechanisms and interfacial processes, where electron transfer, mass transport, and adsorption are often present. EIS analysis has often treat-ed the bulk and interfacial processes separately [2, p. 28]. The analysis is achieved on the basis of selective responses of bulk and interfacial processes to sampling AC frequencies. The peaks appearing in the impedance spectrum can be described according to the theory of dielectric/impedance relaxations. Again, as in the case of any other spectroscopy method, the subject of the UBEIS analysis is the detection and interpretation of these peaks.

1.3. Electrochemical experiment: charge and material transport

In the basic electrochemical experiment there are at least two electrodes bounding a sample (such as electrolyte solution) with external potential (volt-age) V difference applied between the two (Figure 1-5). Under the influence of the electric field the current / passes through a complete circuit, transported by electrons in the metal conductors, the electrodes, and ionic migration in the electrolyte. The electrons are free to move in the conduction bands inside the metal. A current that seems to flow quickly is not due to a "very fast" speed of the electrons in metal conductors (which is, in fact, rather slow at ~ 0.3 mm/ sec), but because there are different electrons entering and leaving a conduc-tor. In principle, when an electron is supplied to a wire end, "another" electron is coming out of the other end.

Ionic liquid (electrolyte), such as water with added ionic salts, is one of the common samples investigated by EIS. An average ion electrophoretic migra-tion velocity in aqueous solution is ~ 10 mm/sec. As was shown above, the resistance to ionic migration current in the aqueous bulk solution within the frequency range of a typical impedance measurement can be simplified by a

FIGURE 1-5 The basic electrochemical experiment wi th material transport directions