Electrochemical Impedance Spectroscopy

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3 Electrochemical Impedance Spectroscopy

4 THE ELECTROCHEMICAL SOCIETY SERIES ECS-The Electrochemical Society 65 South Main Street Pennington, NJ A complete list of the titles in this series appears at the end of this volume.

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

6 Copyright 2017 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) , fax (978) , or on the web at Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) , fax (201) , or online at Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) , outside the United States at (317) or fax (317) Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at Library of Congress Cataloging-in-Publication Data: Names: Orazem, Mark E. Tribollet, Bernard. Title: Electrochemical impedance spectroscopy / Mark E. Orazem, University of Florida, Bernard Tribollet, Université Pierre et Marie Curie. Description: 2nd edition. Hoboken, New Jersey : John Wiley & Sons, Inc., [2017] Includes bibliographical references and index. Identifiers: LCCN (print) LCCN (ebook) ISBN (cloth) ISBN (pdf) ISBN (epub) Subjects: LCSH: Impedance spectroscopy. Classification: LCC QD116.I57 O (print) LCC QD116.I57 (ebook) DDC 543/.6--dc23 LC record available at Printed in the United States of America

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

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9 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 Why Imaginary Numbers? Terminology The Imaginary Number Complex Variables Conventions for Notation in Impedance Spectroscopy Operations Involving Complex Variables Multiplication and Division of Complex Numbers Complex Variables in Polar Coordinates Properties of Complex Variables Elementary Functions of Complex Variables

10 viii CONTENTS Exponential Logarithmic Polynomial Problems Differential Equations Linear First-Order Differential Equations Homogeneous Linear Second-Order Differential Equations Nonhomogeneous Linear Second-Order Differential Equations Chain Rule for Coordinate Transformations Partial Differential Equations by Similarity Transformations Differential Equations with Complex Variables Problems Statistics Definitions Expectation and Mean Variance, Standard Deviation, and Covariance Normal Distribution Probability Central Limit Theorem Error Propagation Linear Systems Nonlinear Systems Hypothesis Tests Terminology Student s t-test for Equality of Mean F-Test for Equality of Variance Chi-Squared Test for Goodness of Fit Problems Electrical Circuits Passive Electrical Circuits Circuit Elements Response to a Sinusoidal Signal Impedance Response of Passive Circuit Elements Parallel and Series Combinations

11 CONTENTS ix 4.2 Fundamental Relationships Nested Circuits Mathematical Equivalence of Circuits Graphical Representation of Circuit Response Problems Electrochemistry Resistors and Electrochemical Cells Polarization Behavior for Electrochemical Systems Zero Current Equilibrium Nonequilibrium Kinetic Control Mixed-Potential Theory Mass-Transfer Control Definitions of Potential Rate Expressions Transport Processes Primary Current and Potential Distributions Secondary Current and Potential Distributions Tertiary Current and Potential Distributions Mass-Transfer-Controlled Current Distributions Potential Contributions Ohmic Potential Drop Surface Overpotential Concentration Overpotential Capacitance Contributions Double-Layer Capacitance Dielectric Capacitance Further Reading Problems Electrochemical Instrumentation The Ideal Operational Amplifier Elements of Electrochemical Instrumentation Electrochemical Interface Potentiostat Galvanostat

12 x CONTENTS Potentiostat for EIS Measurement Problems II Experimental Considerations Experimental Methods Steady-State Polarization Curves Transient Response to a Potential Step Analysis in Frequency Domain Lissajous Analysis Phase-Sensitive Detection (Lock-in Amplifier) Single-Frequency Fourier Analysis Multiple-Frequency Fourier Analysis Comparison of Measurement Techniques Lissajous Analysis Phase-Sensitive Detection (Lock-in Amplifier) Single-Frequency Fourier Analysis Multiple-Frequency Fourier Analysis Specialized Techniques Transfer-Function Analysis Local Electrochemical Impedance Spectroscopy Global Impedance Local Impedance Local Interfacial Impedance Local Ohmic Impedance Global Interfacial Impedance Global Ohmic Impedance Problems Experimental Design Cell Design Reference Electrodes Flow Configurations Rotating Disk Disk under Submerged Impinging Jet Rotating Cylinders Rotating Hemispherical Electrode Current Distribution

13 CONTENTS xi 8.2 Experimental Considerations Frequency Range Linearity Modulation Technique Oscilloscope Instrumentation Parameters Improve Signal-to-Noise Ratio Reduce Bias Errors Nonstationary Effects Instrument Bias Improve Information Content Problems III Process Models Equivalent Circuit Analogs General Approach Current Addition Impedance at the Corrosion Potential Partially Blocked Electrode Potential Addition Electrode Coated with an Inert Porous Layer Electrode Coated with Two Inert Porous Layers Problems Kinetic Models General Mathematical Framework Electrochemical Reactions Potential Dependent Potential and Concentration Dependent Charge-Transfer Resistance Diffusion Impedance Cell Impedance Multiple Independent Electrochemical Reactions Coupled Electrochemical Reactions Potential and Surface Coverage Dependent Potential, Surface Coverage, and Concentration Dependent

14 xii CONTENTS 10.5 Electrochemical and Heterogeneous Chemical Reactions Problems Diffusion Impedance Uniformly Accessible Electrode Porous Film Diffusion with Exchange of Electroactive Species Diffusion without Exchange of Electroactive Species Rotating Disk Fluid Flow Steady-State Mass Transfer Convective Diffusion Impedance Analytic and Numerical Solutions Nernst Hypothesis Assumption of an Infinite Schmidt Number Treatment of a Finite Schmidt Number Submerged Impinging Jet Fluid Flow Steady-State Mass Transfer Convective Diffusion Impedance Rotating Cylinders Electrode Coated by a Porous Film Steady-State Solutions Coupled Diffusion Impedance Impedance with Homogeneous Chemical Reactions Dynamic Surface Films Mass Transfer in the Salt Layer Mass Transfer in the Electrolyte Oscillating Film Thickness Faradaic Impedance Problems Impedance of Materials Electrical Properties of Materials Dielectric Response in Homogeneous Media Cole Cole Relaxation Geometric Capacitance Dielectric Response of Insulating Nonhomogeneous Media

15 CONTENTS xiii 12.6 Mott Schottky Analysis Problems Time-Constant Dispersion Transmission Line Models Telegrapher s Equations Porous Electrodes Pore-in-Pore Model Thin-Layer Cell Geometry-Induced Current and Potential Distributions Mathematical Development Blocking Electrode Blocking Electrode with CPE Behavior Electrode with Faradaic Reactions Electrode with Faradaic Reactions Coupled by Adsorbed Intermediates Numerical Method Complex Ohmic Impedance at High Frequencies Complex Ohmic Impedance at High and Low Frequencies Electrode Surface Property Distributions Electrode Roughness Influence of Roughness on a Disk Electrode Influence of Surface Roughness on a Recessed Electrode Capacitance Capacitance Distribution on Recessed Electrodes Capacitance Distribution on Disk Electrodes Reactivity Characteristic Dimension for Frequency Dispersion Convective Diffusion Impedance at Small Electrodes Analysis Local Convective Diffusion Impedance Low-Frequency Solution High-Frequency Solution Global Convective Diffusion Impedance Coupled Charging and Faradaic Currents Theoretical Development Mass Transport in Dilute Solutions

16 xiv CONTENTS Coupled Faradaic and Charging Currents Double-Layer Model Decoupled Faradaic and Charging Currents Numerical Method Steady-State Calculations Double-Layer Properties Impedance Calculations Consequence of Coupled Charging and Faradaic Currents Exponential Resistivity Distributions Problems Constant-Phase Elements Mathematical Formulation for a CPE When Is a Time-Constant Distribution a CPE? Origin of Distributions Resulting in a CPE Approaches for Extracting Physical Properties Simple Substitution Characteristic Frequency: Normal Distribution Characteristic Frequency: Surface Distribution Power-Law Distribution Bounds for Resistivity Comparative Analysis Limitations to the Use of the CPE Problems Generalized Transfer Functions Multi-input/Multi-output Systems Current or Potential Are the Output Quantity Current or Potential Are the Input Quantity Experimental Quantities Transfer Functions Involving Exclusively Electrical Quantities Ring Disk Impedance Measurements Multifrequency Measurements for Double-Layer Studies Transfer Functions Involving Nonelectrical Quantities Thermoelectrochemical (TEC) Transfer Function Photoelectrochemical Impedance Measurements

17 CONTENTS xv Electrogravimetry Impedance Measurements Problems Electrohydrodynamic Impedance Hydrodynamic Transfer Function Mass-Transport Transfer Function Asymptotic Solution for Large Schmidt Numbers Asymptotic Solution for High Frequencies Kinetic Transfer Function for Simple Electrochemical Reactions Interface with a 2-D or 3-D Insulating Phase Partially Blocked Electrode Rotating Disk Electrode Coated by a Porous Film Steady-State Solutions AC and EHD Impedances Problems IV Interpretation Strategies Methods for Representing Impedance Impedance Format Complex-Impedance-Plane Representation Bode Representation Ohmic-Resistance-Corrected Bode Representation Impedance Representation Admittance Format Admittance-Plane Representation Admittance Representation Ohmic-Resistance-Corrected Representation Complex-Capacitance Format Effective Capacitance Problems Graphical Methods Based on Nyquist Plots Characteristic Frequency Superposition

18 xvi CONTENTS Mass Transfer Evolution of Active Area Based on Bode Plots Ohmic-Resistance-Corrected Phase Ohmic-Resistance-Corrected Magnitude Based on Imaginary Part of the Impedance Evaluation of Slopes Calculation of Derivatives Based on Dimensionless Frequency Mass Transport Geometric Contribution System-Specific Applications Effective CPE Coefficient Asymptotic Behavior for Low-Frequency Mass Transport Arrhenius Superposition Mott Schottky Plots High-Frequency Cole Cole Plots Overview Problems Complex Nonlinear Regression Concept Objective Functions Formalism of Regression Strategies Linear Regression Nonlinear Regression Regression Strategies for Nonlinear Problems Gauss Newton Method Method of Steepest Descent Levenberg Marquardt Method Downhill Simplex Strategies Influence of Data Quality on Regression Presence of Stochastic Errors in Data Ill-Conditioned Regression Caused by Stochastic Noise Ill-Conditioned Regression Caused by Insufficient Range Initial Estimates for Regression

19 CONTENTS xvii 19.7 Regression Statistics Confidence Intervals for Parameter Estimates Statistical Measure of the Regression Quality Problems Assessing Regression Quality Methods to Assess Regression Quality Quantitative Methods Qualitative Methods Application of Regression Concepts Finite-Diffusion-Length Model Quantitative Assessment Visual Inspection Measurement Model Quantitative Assessment Visual Inspection Convective-Diffusion-Length Model Quantitative Assessment Visual Inspection Problems V Statistical Analysis Error Structure of Impedance Measurements Error Contributions Stochastic Errors in Impedance Measurements Stochastic Errors in Time-Domain Signals Transformation from Time Domain to Frequency Domain Stochastic Errors in Frequency Domain Bias Errors Instrument Artifacts Ancillary Parts of the System under Study Nonstationary Behavior Time Scales in Impedance Spectroscopy Measurements Incorporation of Error Structure Measurement Models for Error Identification Stochastic Errors

20 xviii CONTENTS Bias Errors Problems The Kramers Kronig Relations Methods for Application Direct Integration of the Kramers Kronig Relations Experimental Assessment of Consistency Regression of Process Models Regression of Measurement Models Mathematical Origin Background Application of Cauchy s Theorem Transformation from Real to Imaginary Transformation from Imaginary to Real Application of the Kramers Kronig Relations The Kramers Kronig Relations in an Expectation Sense Transformation from Real to Imaginary Transformation from Imaginary to Real Problems VI Overview An Integrated Approach to Impedance Spectroscopy Flowcharts for Regression Analysis Integration of Measurements, Error Analysis, and Model Impedance Measurements Integrated with Error Analysis Process Models Developed Using Other Observations Regression Analysis in Context of Error Structure Application Problems VII Reference Material 629 A Complex Integrals A.1 Definition of Terms A.2 Cauchy Riemann Conditions

21 CONTENTS xix A.3 Complex Integration A.3.1 Cauchy s Theorem A.3.2 Improper Integrals of Rational Functions Problems B Tables of Reference Material C List of Examples List of Symbols References Author Index Subject Index

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23 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

24 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

25 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.

26 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.

27 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.

28 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

29 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 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.

30 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.