• Author / Uploaded
• Yu Shu Hearn

• Categories
• Dieléctrico
• Condensador
• Electroquímica
• Electricidad
• Ciencias fisicas

Citation preview

Electrochemical Supercapacitors Scientific Fundamentals and Technological Applications

Electrochemical Supercapacitors Scientific Fundamentals and Technological Applications 8. E. Conway Fellow of the Royal Society of Canada University of Ottawa Ottawa, Ontario, Canada

Springer Science+Business Media, LLC

LIbrary of Congress CatalogIng-In-PublIcatIon Data

Conway, B. E. ElectrochemIcal supercapacitors , scientific fundamentals and technological applications I B.E. Conway. cm. p. Includes bibliographical references and index. 1. Storage batteries. 2. Electrolytic capacitors. double layer. 1. Title. TK2941.C66 1999 621.31·2424--dc21

3. Electric 98-48209 CIP

ISBN 978-1-4757-3060-9 ISBN 978-1-4757-3058-6 (eBook) DOI 10.1007/978-1-4757-3058-6

© 1999 Springer Science+Business Media New York Originally published by Kluwer Academic / Plenum Publishers, New York in 1999. Softcover reprint ofthe hardcover 1st edition 1999

10987654321 A C.I.P. record for this book is available from the Library of Congress. All rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher

To my son, Dr. Adrian and his sons, Alexander and the "Little B"

Foreword

The first model for the distribution of ions near the surface of a metal electrode was devised by Helmholtz in 1874. He envisaged two parallel sheets of charges of opposite sign located one on the metal surface and the other on the solution side, a few nanometers away, exactly as in the case of a parallel plate capacitor. The rigidity of such a model was allowed for by Gouy and Chapman independently, by considering that ions in solution are subject to thermal motion so that their distribution from the metal surface turns out diffuse. Stern recognized that ions in solution do not behave as point charges as in the Gouy-Chapman treatment, and let the center of the ion charges reside at some distance from the metal surface while the distribution was still governed by the Gouy-Chapman view. Finally, in 1947, D. C. Grahame transferred the knowledge of the structure of electrolyte solutions into the model of a metal/solution interface, by envisaging different planes of closest approach to the electrode surface depending on whether an ion is solvated or interacts directly with the solid wall. Thus, the Gouy-Chapman-Stern-Grahame model of the so-called electrical double layer was born, a model that is still qualitatively accepted, although theoreticians have introduced a number of new parameters of which people were not aware 50 years ago. Irrespective of the structural details, it has long been accepted that a double layer exists at the electrode/electrolyte solution boundary, which governs adsorption phenomena and influences charge transfer reaction rates, and where electrostatic energy is stored as in a capacitor a few molecular diameters thick. Nevertheless, the existence of a double layer has always been inferred from indirect observations of related properties and quantities, but never directly vii

viii

Foreword

probed, so much that it was compared to the Arab Phoenix: "Everybody says it exists, nobody knows where it is." This until recently, when it was realized that the energy stored per unit surface area of an electrode is noticeable per se and becomes technologically very interesting with the introduction of new materials with an exceptionally extended active surface: especially treated carbons, some transition metal oxides, electrosynthesized conducting polymers. The interfacial capacity is further increased if the purely capacitive charge is supplemented by a Faradaic charge related to bidimensional redox reactions or tridimensional intercalation processes. "Supercapacitors" are devices that store electrical energy on the basis of the above phenomena and that can be discharged at a much higher rate than conventional batteries. They have aroused interest for various applications, including electric vehicles, in particular cars as well as trains. I should say that in spite of our awareness of the principles, supercapacitors have appeared on the scientific scene rather suddenly, or at least this has been the impression ofthose who have realized that something was happening at the technological level. Of the many examples we can produce of innovations developed in technology first and then "discovered" from a fundamental point of view, supercapacitors furnish an authoritative example ofthe reverse: a technological innovation pushed by fundamental knOWledge. The situation is now that fundamental researchers know everything of the electrical double layer but ignore its application to supercapacitors, while engineers know of supercapacitors but may ignore the fundamentals of their operation. This monograph comes at an opportune time to fill this gap, with a balanced presentation of fundamentals aimed at applications, and applications related to fundamental principles. B. E. Conway has worked for more than 50 years in almost all areas of electrochemistry, particularly interfacial electrochemistry. He is therefore a "veteran" in the field, being the first to realize the potentialities of some materials for their double layer energy. This volume offers what cannot be found in any other work for its comprehensiveness, exhaustiveness, and focus. For the first time a highly theoretical topic, the electrical double layer at electrodes, is shown to manifest itself in highly technological applications. It is with a real sense of pride that electrochemists in the near future will press the accelerator in their electric car knowing that certain performances are possible only thanks to the discharge of the "socalled" (but is it indeed there?) electrical double layer of which technologists have long maintained "electrochemistry can do without it." The content of this book is useful both for scientists working in fundamental research and technologists, in particular those interested in electrochemical energy conversion and chemistry and physics of electrified interfaces, as well as for engineers working in the field of electrochemical power sources and electrical energy storing devices. They will find the book an invaluable source of in-

Foreword

ix

formation and inspiration. For the way the topics are presented, people working in the area of materials chemistry and physics will find this book of great general interest in view of the typical dependence of the performance of supercapacitors on the structure of materials.

Milan, Italy

Sergio Trasatti

Preface

Systems for electrochemical energy production originated with Volta's discovery in 1800 of "voltaic electricity" and were developed in various forms during the nineteenth century. Toward the end of that period, reversibly chargeable batteries for electrical energy storage and utilization became a major development in applied electrochemistry and during the present century have been improved to a high state of the art. They also represent a large fraction of the economic activity in industrial electrochemistry. In relatively recent years, but originating with Becker's patent in 1957, a new type of electrochemically reversible energy storage system has been developed that uses the capacitance associated with charging and discharging of the double layer at electrode interfaces or, complementarily, the pseudocapacitance associated with electrosorption processes or surface redox reactions. In the first case, large interfacial capacities of many tens of farads per gram of active electrode material can be achieved at high-area carbon powders, fibers, or felts, while, in the second case, large pseudocapacitances can be developed at certain high-area oxides or conducting polymers where extents of Faradaic charge (Q) transfer are functionally related to the potential of the electrode (V), giving rise to a derivative corresponding to a capacitance dQ/dV. These large specific-value capacitors, especially of the double-layer type, are perceived as electrical energy storage systems that can offer high power-density in discharge and recharge, and cycle lives on the order of 1 to 106, many times those of conventional batteries. A variety of uses of such electrochemical or socalled "supercapacitors" are now recognized and a new direction of power-source development, complementary to that of batteries, is well established.

as

xi

xii

Preface

An important aspect of this monograph is that it gives a comprehensi ve account of the electrochemical science and technology of these capacitor systems. An attempt is made to present a self-contained and unified treatment of the field, including essential details of the background science (e.g. of double-layer capacitance and the origins of pseudocapacitance, the electrolyte solutions used in electrochemical capacitors) as well as basic concepts of electrode kinetics and interfacial electrochemistry, dielectric polarization theory, porous electrodes, and conducting polymer materials that give rise to large specific capacitances. In this way, understanding and study of the material presented in this volume will not require frequent reference to other textbooks of physical. chemistry or electrochemistry . The text contains many illustrative diagrams and cross-references between chapters, and includes many literature references. For the convenience of the reader, three or four diagrams have been duplicated from one chapter or another to avoid the necessity of seeking earlier or later pages in the volume where cross-referenced material is cited. The author's work in this field originated with a research contract between Continental Group Inc. and the University of Ottawa's Electrochemistry Group. We would like to acknowledge here the work carried out by Drs. H. AngersteinKozlowska, V. Birss, J. Wojtowicz, and Visiting Professor S. Hadzi-lordanov (University of Skopje) with Mr. Dwight Craig (electrical engineer) of Continental Group in the period 1975 to 1981. More recently, new work in this field is being carried out at the University of Ottawa and is supported by the Natural Sciences and Engineering Research Council of Canada. For this work, acknowledgment is made to Dr. W. 1. Pell and Mr. T. C. Liu. Special thanks are due to Dr. B. V. Tilak of Occidental Chemical Corp., N.Y., for his critical reading of the manuscript before its submission for publication, and for his suggestions for additions and revisions. Appreciation is expressed to Dr. Tilak and Dr. S. Sarangapani (ICET Inc., Norwood, Mass.) for their detailed examination of Chapter 20 on technology development, and in particular for their suggestions for the best systematic organization of the manifold aspects of the subject treated in that chapter. Thanks are also due to Drs. S. GoUesfeld (Los Alamos National Laboratory) and J. Miller (1. M. Inc., Shaker Heights, Ohio) for reading the chapters on conducting-polymer capacitors and ac impedance, respectively. We are grateful to Dr. Miller for permission to reproduce some of his computer-generated graphs and data on ac impedance evaluation of capacitors. The author is also most grateful to Drs. S. P. Wolsky and N. Marincic for their permission to draw on various diagrams and tables from the proceedings of papers presented at the seminars on electrochemical capacitors held at Deerfield Beach and Boca Raton, Fla, over the period 1991 to 1997, under the auspices of Florida Educational Seminars Inc. (abbreviated as FES in the text).

Preface

xiii

Finally, special thanks are due to Denise Angel, who typed, with great efficiency and accuracy, all the chapters of this volume in several drafts, exercising literacy and care that would be difficult to match. Grateful thanks are also due to Eva Szabo for drafting most of the diagrams. Ottawa, Canada

B. E. Conway

Contents

Chapter 1

Chapter 2

Introduction and Historical Perspective 1.1. Historical Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Scope of the Monograph. . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Similarities and Differences between Supercapacitors and Batteries for Storing Electrical Energy 2.1. Introduction................................. 2.1.1. Energy Storage Systems. . . . . . . . . . . . . . . .. 2.1.2. Modes of Electrical Energy Storage by Capacitors and Batteries. . . . . . . . . . . . . . . .. 2.2. Faradaic and Non-Faradaic Processes. . . . . . . . . . . . . 2.2.1. Non-Faradaic......................... 2.2.2. Faradaic............................. 2.3. Types of Capacitors and Types of Batteries. . . . . . .. 2.3.1. Distinguishable Systems. . . . . . . . . . . . . . . .. 2.3.2. Cell Design and Equivalent Circuits. . . . . .. 2.4. Differences of Densities of Charge Storage in Capacitors and Batteries. . . . . . . . . . . . . . . . . . . . . . . 2.4.1. Electron Densities per Atom or Molecule . .. 2.4.2. Comparison of Energy Densities Attainable in Electrochemical Capacitors and Batteries. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.5. Comparison of Capacitor and Battery Charging Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xv

1 8 9

11 11 12 13 14 14 15 15 17 18 18

19 20

xvi

Contents

2.6. Comparison of Charge and Discharge Behavior of Electrochemical Capacitors and Battery Cells Evaluated by Cyclic Voltammetry . . . . . . . . . . . . . .. 2.7. Li Intercalation Electrodes-A Transition Behavior. .. 2.8. Charging of a Nonideally Polarizable Capacitor Electrode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.9. Comparative Summary of Properties of Electrochemical Capacitors and Batteries. . . . . . . . .. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. General Reading References. . . . . . . . . . . . . . . . . . .. Chapter 3

Chapter 4

Energetics and Elements of the Kinetics of Electrode Processes 3.1. Introduction................................ 3.2. Energetics of Electrode Processes. . . . . . . . . . . . . .. 3.3. Energy Factors in Relation to Electrode Potential.. 3.4. Kinetics of Electrode Reactions at Metals. . . . . . .. 3.4.1. Currents and Rate Equations. . . . . . . . . . .. 3.4.2. Linearization of the Butler-Volmer Equation for Near-Equilibrium Conditions (low Yf) . . . . . . . . . . . . . • . . . . . . . . . . . . . .. 3.5. Graphical Representation of the Exchange Current Density, im and Behavior Near Equilibrium. . . . . . .. 3.6. Onset of Diffusion Control in the Kinetics of Electrode Processes. . . . . . . . . . . . . . . . . . . . . . . . . .. 3.7. Kinetics when Steps Following an Initial Electron Transfer Are Rate Controlling. . . . . . . . . . . . . . . . . .. 3.8. Double-Layer Effects in Electrode Kinetics. . . . . . .. 3.9. Electrical Response Functions Characterizing Capacitative Behavior of Electrodes . . . . . . . . . . . . .. 3.10. Instruments and Cells for Electrochemical Characterization of Capacitor Behavior . . . . . . . . . .. 3.10.1. Cells and Reference Electrodes. . . . . . . . . .. 3.10.2. Instruments......... . . . . . . . . . . . . . . . . .. 3.10.3. Two-Electrode Device Measurements. . . . .. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. General Reading References. . . . . . . . . . . . . . . . . . .. Elements of Electrostatics Involved in Treatment of Double Layers and Ions at Capacitor Electrode Interphases 4.1. Introduction........... . . . . . . . . . . . . . . . . . . . . .. 4.2. Electrostatic Principles . . . . . . . . . . . . . . . . . . . . . . ..

22 25 28 29 31 31

33 34 37 41 41

45 46 48 50 51 53 59 59 61 63 64 64

67 68

Contents

xvii

4.2.1.

4.3. 4.4. 4.5. 4.6. 4.7. 4.8. 4.9.

4.10.

4.11. 4.12.

Chapter 5

Coulomb's Law: Electric Potential and Field, and the Significance of the Dielectric Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.2.1.1. Units........................ 4.2.1.2. Dielectric Constant. . . . . . . . . . . .. 4.2.1.3. Electrostatic Potential, Field, and Force. . . . . . . . . . . . . . . . . . . . . . .. 4.2.1.4. Potential ifJ and Field E at an Ion .. Lines of Force and Field Intensity-A Theorem. . .. Capacity of a Condenser or Capacitor. . . . . . . . . . . .. Field Due to a Surface of Charges: Gauss's Relation Poisson's Equation: Charges in a 3-Dimensional Medium.................................... The Energy of a Charge. . . . . . . . . . . . . . . . . . . . . . .. Electric Tension in a Dielectric in a Field ......... , Electric Polarization Responses at the Molecular Level. . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . .. 4.9.1. Atoms and Molecules in Fields: Electronic Polarization. . . . . . . . . . . . . . . . . . . . . . . . . .. 4.9.2. Interaction of a Permanent Dipole with a Field 4.9.2.1. Uniform Field. . . . . . . . . . . . . . . . .. 4.9.2.2. Nonuniform Field. . . . . . . . . . . . . .. 4.9.2.3. Forces on a Quadrupole in a Field.. Atoms and Molecules in Fields: Dielectric Properties and Dielectric Polarization . . . . . . . . . . . . . . . . . . . .. 4.10.1. Dielectrics...................... . . . . .. 4.10.2. Polarization of Solvent Molecules in Double-Layer and Ion Fields. . . . . . . . . . . .. 4.10.3. Dipole Moments of Complex Molecules. . .. Electric Polarization in Dielectrics . . . . . . . . . . . . . .. Energy and Entropy Stored by a Capacitor. . . . . . . .. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. General Reading References. . . . . . . . . . . . . . . . . . ..

Behavior of Dielectrics in Capacitors and Theories of Dielectric Polarization 5.1. Introduction................................. 5.2. Definitions and Relation of Capacitance to Dielectric Constant of the Dielectric Medium . . . . .. 5.3. Electric Polarization of Dielectrics in a Field. . . . . .. 5.4. Formal Electrostatic Theory of Dielectrics. . . . . . . ..

68 68 70 71 72 73 74 74 75 76 77 78 78 79 79 79 80 81 81 81 82 83 83 86 86

87 88 91 92

Contents

xviii

5.5. 5.6. 5.7. 5.8. 5.9.

Chapter 6

Chapter 7

Dielectric Behavior Due to Induced, Distortional Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 98 Dielectric Polarization in a Simple Condensed Phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 98 Dielectric Polarization in a System of Noninteracting but Orientable Dipoles. . . . . . . . . . . . . . . . . . . . . . . .. 99 Dielectric Polarization of Strongly Interacting Dipoles (High Dielectric Constant Solvents) . . . . . .. 100 Dielectric Behavior of the Sol vent in the Double Layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 102 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 104

The Double Layer at Capacitor Electrode Interfaces: Its Structure and Capacitance 6.1. Introduction................................. 6.2. Models and Structures of the Double Layer. . . . . . .. 6.3. Two-Dimensional Density of Charges in the Double Layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.4. Ionic Charge Density and Interionic Distances on the Solution Side of the Double Layer. . . . . . . . . . . . . .. 6.5. Electron-Density Variation: "Jellium" Model. . . . . .. 6.6. Electric Field across the Double Layer. . . . . . . . . . .. 6.7. Double-Layer Capacitance and the Ideally Polarizable Electrode. . . . . . . . . . . . . . . . . . . . . . . . .. 6.8. Equivalent Circuit Representation of Double-Layer Electrical Behavior. . . . . . . . . . . . . . . . . . . . . . . . . .. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Theoretical Treatment and Modeling of the Double Layer at Electrode Interfaces 7.1. Early Models ................................ 7.2. Treatment of the Diffuse Layer. . . . . . . . . . . . . . . . .. 7.3. Capacitance of the Diffuse Part of the Double Layer ...................................... 7.4. Ion Adsorption and the Treatment of the Compact or Helmholtz Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7.4.1. Stern's Treatment ...................... 7.4.2. Quasi-Chemical Aspect of Anion Adsorption 7.5. The Solvent as Dielectric of the Double-Layer Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7.5.1. General.............................. 7.5.2. Types of Solvents that Constitute the Double-Layer Interphase. . . . . . . . . . . . . . ..

105 108 114 116 117 119 121 123 124

125 127 129 133 133 135 136 136 137

Contents

xix

7.5.3.

Dielectric Constant in the Double-Layer Interphase. . . . . . . . . . . . . . . . . . . . . . . . . . .. 7.5.4. Electrostatic Polarization of Water as Solvent in the Double Layer. . . . . . . . . . . . .. 7.5.5. Molecular-Level Treatments of Solvent Dipole Orientation at Charged Interfaces. . .. 7.5.5.1. Two-State Dipole Orientation Treatments ................. " 7.5.5 .2. Cluster Models for Water Adsorption and Orientation . . . . .. 7.5.6. H-Bonded Lattice Models ............... 7.5.7. Spontaneous Orientation of Water at Electrode Surfaces Due to Chemisorption. .. 7.5.8. Solvent Adsorption Capacitance at Solid Metals ............................... 7.5.9. Recent Modeling Calculations. . . . . . . . . . .. 7.6. The Metal Electron Contribution to Double-Layer Capacitance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7.6.1. Origin of the Metal Contribution. . . . . . . . .. 7.6.2. Profile of Electron Density at Electrode Surfaces ............................ " 7.7. The Potential Profile across the Diffuse Layer. . . . .. 7.8. The Double Layer in Pores of a Porous Capacitor Electrode ................................. " References ................................ " General Reading References. . . . . . . . . . . . . . . . . . .. Chapter 8

Chapter 9

Behavior of the Double Layer in Nonaqueous Electrolytes and Nonaqueous Electrolyte Capacitors 8.1. Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8.2. Fundamental Aspects of Double-Layer Capacitance Behavior in Nonaqueous Solvent Media. . . . . . . . . .. 8.3. Comparative Double-Layer Capacitance Behavior in Several Nonaqueous Solutions. . . . . . . . . . . . . . . . .. 8.4. General Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . .. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

138 139 141 141 143 148 149 151 152 156 156 157 160 161 165 168

169 170 176 180 180

The Double Layer and Surface Functionalities at Carbon 9.1. Introduction ................................. 183 9.1.1. Historical............................ 183 9.1.2. Carbon Materials for Electrochemical Capacitors. . . . . . . . . . . . . . . . . . . . . . . . . . .. 185

xx

Contents

9.2. Surface Properties and Functionalities of Carbon Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9.3. Double-Layer Capacitance of Carbon Materials. . . .. 9.4. Oxidation of Carbon. . . . . . . . . . . . . . . . . . . . . . . . .. 9.5. Surface Specificity of Double-Layer Capacitance Behavior at Carbon and Metals. . . . . . . . . . . . . . . . .. 9.6. Double-Layer Capacitance at Edge and Basal Planes of Graphite .................................. 9.7. Materials Science Aspects of Carbon Materials for Conditioned Double-Layer Capacitors ............ 9.7.1. Heat and Chemical Treatments of Carbon Materials for Capacitors. . . . . . . . . . . . . . . .. 9.7.2. Research Requirements for Carbon Materials in Electrochemical Capacitors . . . . . . . . . . .. 9.7.3. Electron Spin Resonance Characterization of Free Radicals at Carbon Surfaces. . . . . . . . .. 9.8. Interaction of Oxygen with Carbon Surfaces. . . . . .. 9.9. Electronic Work Function and Surface Potentials of Carbon Surfaces .............................. 9.10. Intercalation Effects. . . . . . . . . . . . . . . . . . . . . . . . . .. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. General Reading References. . . . . . . . . . . . . . . . . . .. Chapter 10

Electrochemical Capacitors Based on Pseudocapacitance 10.1. Origins of Pseudocapacitance . . . . . . . . . . . . . . . . . .. 10.2. Theoretical Treatments of Pseudocapacitance (C¢) .. 10.2.1. Types of Treatment. .................... 10.2.2. Electrosorption Isotherm Treatment of Pseudocapacitance: A Thermodynamic Approach ............................ 10.3. Kinetic Theory of Pseudocapacitance. . . . . . . . . . . .. 10.3.1. Electrode Kinetics under Linearly Time-Variant Potential. . . . . . . . . . . . . . . . .. 10.3.2. Evaluation of Characteristic Peak Current and Peak Potential Quantities. . . . . . . . . . . .. 10.3.3. Transition between Reversibility and Irreversibility ... . . . . . . . . . . . . . . . . . . . . .. 10.3.4. Relation to Behavior under dc Charge and Discharge Conditions. . . . . . . . . . . . . . . . . .. 10.4. Potential Ranges of Significant Pseudocapacitances . . . . . . . . . . . . . . . . . . . . . . . . . ..

186 193 196 198 199 203 203 208 209 212 213 217 219 220

221 224 224

224 236 236 239 241 243 246

Contents

xxi

10.5. Origin of Redox and Intercalation Pseudocapacitances . . . . . . . . . . . . . . . . . . . . . . . . . .. 10.6. Pseudocapacitance Associated with Specific Adsorption of Anions and the Phenomenon of Partial Charge Transfer. . . . . . . . . . . . . . . . . . . . . . .. 10.7. Pseudocapacitance Behavior at High-Area Carbon Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10.8. Procedures for Distinguishing Pseudocapacitance (CI/J) from Double-Layer Capacitance (Cdl ) . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. General Reading References. . . . . . . . . . . . . . . . . . .. Chapter 11

The Electrochemical Behavior of Ruthenium Oxide (RU02) as a Material for Electrochemical Capacitors 11.1. Historical Aspects . . . . . . . . . . . . . . . . . . . . . . . . . .. 11.2. Introduction...... . . . . . . . . . . . . . . . . . . . . . . . . .. 11.3. Formation of Ru02 Films that Have Capacitative Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11.4. The Transition from Monolayer to Multilayer Electrochemical Formation of Ru02. . . . . . . . . . . .. 11.5. States and Chemical Constitution of Electrochemically and Thermochemically Formed RU02 for Capacitors ................... . . . . .. 11.6. Mechanism of Charging and Discharging RU02. . .. 11.7. Oxidation States Involved in Voltammetry of RU02 and Ir0 2 Electrodes. . . . . . . . . . . . . . . . . . . .. 11.7.1. Oxidation States and Redox Mechanisms .. 11.7.2. Charging in Inner and Outer Surface Regions of RU02 Films. . . . . . . . . . . . . . . .. 11.8. Conclusions on Mechanisms of Charging RU02 Capacitor Materials. . . . . . . . . . . . . . . . . . . . . . . . .. 11.9 . Weight Changes on Charge and Discharge. . . . . . .. 11.10. dc and ac Response Behavior of Ru02 Electrochemical Capacitor Electrodes ............ 11.11. Other Oxide Films Exhibiting Redox Pseudocapacitance Behavior. . . . . . . . . . . . . . . . . .. 11.12. Surface Analysis and Structure of RU02-Ti0 2 Films ..................................... 11.13. Impedance Behavior of Ru02-Ti0 2 Composite Electrodes ................... . . . . . . . . . . . . .. 11.14. Use and Behavior ofIr02 . . . . . . . . . . . . . . . . . . . ..

248

253 255 255 256 257

259 264 265 267

270 276 277 277 279 282 284 285 286 290 292 293

xxii

Contents

11.15. Comparative Oxide Film Behavior at Transition Metal Electrodes. . . . . . . . . . . . . . . . . . . . . . . . . . .. 293 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 295 General Reading References. . . . . . . . . . . . . . . . . .. 297 Chapter 12 Capacitance Behavior of Films of Conducting, Electrochemically Reactive Polymers 12.1. Introduction and General Electrochemical Behavior 12.2. Chemistry of the Polymerization Processes ....... 12.3. General Behavior in Relation to Pseudocapacitance ........................... 12.4. Forms of Cyclic Voltammograms for Conducting Polymers .................................. 12.5. Classification of Capacitor Systems Based on Conducting Polymer Active Materials ........... 12.6. Complementary Studies Using Other Procedures ... 12.7. Ellipsometric Studies of Conducting Polymer Film Growth and Redox Pseudocapacitative Behavior ... 12.8. Other Developments on Conducting Polymer Capacitors ................................. References ................................. General Reading References and Tabulations Chapter 13 The Electrolyte Factor in Supercapacitor Design and Performance: Conductivity, Ion Pairing and Solvation 13.1. Introduction ................................ 13.2. Factors Determining the Conductance of Electrolyte Solutions. . . . . . . . . . . . . . . . . . . . . . . .. 13.3. Electrolyte Conductance and Dissociation ........ 13.4. Mobility of the Free (Dissociated) Ions . . . . . . . . .. 13.5. Role of the Dielectric Constant and Donicity of the Solvent in Dissociation and Ion Pairing . . . . . . . . .. 13.6. Favored Electrolyte-Solvent Systems ......... .. 13.6.1. Aqueous Media. . . . . . . . . . . . . . . . . . . . . .. 13.6.2. Nonaqueous Media. . . . . . . . . . . . . . . . . . .. 13.6.3. Molten Electrolytes ............. . . . . .. 13.7. Properties of Solvents and Solutions for Nonaqueous Electrochemical Capacitor Electrolytes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13.8. Relation of Electrolyte Conductivity to Electrochemically Available Surface Area and Power Performance of Porous Electrode Supercapacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

299 304 312 314 320 322 327 331 332 334

335 337 338 343 344 345 345 347 350

351

360

Contents

xxiii

13.9. Separation of Cations and Anions on Charge and Its Effect on the Electrolyte's Local Conductivity ..... 13.10. The Ion Solvation Factor ...................... 13.11. Compilations of Solution Properties. . . . . . . . . . . .. 13.12. Appendix: Selection of Experimental Data on Properties of Electrolyte Solutions in Nonaqueous Solvents and Their Mixtures. . . . . . . . . . . . . . . . . .. 13.12.1. Summary Tables ..................... 13.12.2. Some Graphically Represented Data from the Literature ...................... " 13.12.3. Selected Tabulations. . . . . . . . . . . . . . . . .. 13.12.4. Conductivities......... . . . . . . . . . . . . .. References ....................... _. . . . . . . .. General Reading References ................. " Chapter 14 Electrochemical Behavior at Porous Electrodes; Applications to Capacitors 14.1. Introduction... . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14.2. Charging and Frequency Response of RC Networks 14.3. General Theory of Electrochemical Behavior of Porous Electrodes .......................... " 14.3.1. System Requirements ................. " 14.3.2. The de Levie Model and its Treatment. . . .. 14.3.3. Configuration of Double Layers in Porous Electrodes. . . . . . . . . . . . . . . . . . . . . . . . . . .. 14.4. Porous Electrode Interfaces as Fractal Surfaces ..... 14.5. Atom Densities in Surfaces and Bulk of Fine Particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14.6. Pore Size and Pore-Size Distribution ............. 14.7. Real Area and Double-Layer Capacitance ......... 14.8. Electro-osmotic Effects in Porous Electrodes. . . . . .. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Chapter 15 Energy Density and Power Density of Electrical Energy Storage Devices 15.1. Ragone Plots of Power Density vs. Energy Density.. 15.2. Energy Density and Power Density, and Their Relationship. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15.2.1. General Considerations ......... . . . . . . .. 15.2.2. Power Density ..... . . . . . . . . . . . . . . . . . .. 15.2.3. Relation to Energy Density .............. 15.2.4. Power and Energy Density Relationships for Capacitors. . . . . . . . . . . . . . . . . . . . . . . . . . ..

361 362 365

366 366 366 366 373 374 375

377 380 383 383 383 403 405 406 408 411 415 416

417 421 421 425 427 433

xxiv

Contents

15.2.5. Power Density Rating of a Capacitor. ...... 15.3. Power Limitation Due to Concentration Polarization 15.4. Relation between C-Rate Specification and Power Density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15.4.1. Formal Definition ...................... 15.4.2. Significance of C-Rate in Battery and Capacitor Discharge. . . . . . . . . . . . . . . . . . .. 15.5. Optimization of Energy Density and Power Density 15.5.1. Capacitor-Battery Hybrid Systems ....... 15.5.2. Condition for Maximum Power Delivery .. 15.5.3. Test Modes. . . . . . . . . . . . . . . . . . . . . . . . .. 15.5.4. Constant Power Discharge Regime for a Capacitor ........................... , 15.5.5. Effects of Temperature ................. 15.6. The Entropy Component of the Energy Held by a Charged Capacitor. . . . . . . . . . . . . . . . . . . . . . . . . .. 15.7. Energy Density of Electrolytic Capacitors ....... , 15.8. Some Application Aspects of Power-Density Factors .................................... 15.9. Energy Storage by Flywheel Systems. . . . . . . . . . .. References ................................. Chapter 16 AC Impedance Behavior of Electrochemical Capacitors and Other Electrochemical Systems 16.1. Introduction ................................. 16.2. Elementary Introductory Principles Concerning Impedance Behavior ......................... , 16.2.1. Alternating Current and Voltage Relationships .... . . . . . . . . . . . . . . . . . . . .. 16.2.2. Root-mean-square and Average Currents in ac Studies . . . . . . . . . . . . . . . . . . . . . . . . . . .. 16.3. Origin of the Semicircular Form of Complex-Plane Plots for Z" vs. Z' over a Range of Frequencies ... , 16.3.1. Impedance Relationships as a Function of Frequency ........................... 16.3.2. Time Constant and Characteristic Frequency Wr . . . . . . . . . . . . . . . . . . . . . . .. 16.4. Significance of RC Time Constants . . . . . . . . . . . .. 16.4.1. Transient Currents and Voltages ........ , 16.4.2. Formal Significance of the RC Product as a Time Constant. ...................... , 16.5. Measurement Techniques .....................

436 440 443 443 444 448 448 452 456 459 462 463 464 468 474 475

479 486 486 489 491 491 496 497 497 501 502

Contents

xxv

16.5.1. ACBridges ........................... 16.5.2. Lissajous Figures ...................... 16.5.3. Phase-Sensitive Detection Using Lock-in Amplifiers. . . . . . . . . . . . . . . . . . . . . . . . . . .. 16.5.4. Digital Frequency-Response Analyzers (Solartron and Other Instruments) ......... 16.6. Kinetic and Mechanistic Approach to Interpretation of Impedance Behavior of Electrochemical Systems 16.6.1. Procedures and Role of Diffusion Control. ., 16.6.2. Principles of the Kinetic Analysis Method .. 16.6.3. Example of the Kinetic Analysis of ac Behavior of the Cathodic H2 Evolution Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 16.6.4. Relation to Linear-Sweep Modulation and Cyclic Voltammetry .................... 16.6.4.1. Methodology ................. 16.6.4.2. Response-Current Behavior .... " 16.6.4.3. Relation between Response Currents in Cyclic Voltammetry and Alternating Voltage Modulation ................. " 16.6.5. Impedance of a Pseudocapacitance ........ References .................................. Chapter 17 Treatments ofImpedance Behavior of Various Circuits and Modeling of Double-Layer Capacitor Frequency Response 17.1. Introduction and Types of Equivalent Circuits ...... 17 .2. Equivalent Series Resistance. . . . . . . . . . . . . . . . . . .. 17.2.1. Significance of esr . . . . . . . . . . . . . . . . . . . .. 17.2.2. Impedance Limits for Some Commercial Capacitors Due to esr . . . . . . . . . . . . . . . . . .. 17.3. Impedance Behavior of Selected Equivalent Circuit Models ..................................... 17.4. Discharge of a Capacitor with esr into a Load Resistance, RL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17.5. Simulation of Porous Electrode Frequency Response by Multielement RC Equivalent Circuits . . . . . . . . .. 17.6. Impedance Behavior of a Redox Pseudocapacitance .. 17.7. Electrochemistry at Porous Electrodes ............ References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

502 503 504 505 506 506 509

510 513 513 513

515 518 524

525 528 528 530 532 538 547 549 555 556

xxvi

Chapter 18

Contents

Self-Discharge of Electrochemical Capacitors in Relation to that at Batteries 18.1. Introduction.......... . . . . . . . . . . . . . . . . . . . . .. 18.2. Practical Phenomenology of Self-Discharge ....... 18.3. Self-Discharge Mechanisms ................... 18.4. Methodologies for Self-Discharge Measurements .. 18.5. Self-Discharge by Activation-Controlled Faradaic Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 18.6. Slope Parameters for Decline of Potential on Self-Discharge .............................. 18.7. Comparison with a Regular Capacitor Discharging through an Ohmic Leakage Resistance. . . . . . . . . .. 18.8. Self-Discharge under Diffusion Control .......... 18.9. Charging of a Nonideally Polarizable Electrode .... 18.10. Self-Discharge of Double-Layer-Type Supercapacitor Devices. . . . . . . . . . . . . . . . . . . . . .. 18.11. Time-Dependent Redistribution of Charge in Nonuniformly Charged Porous Electrodes ........ 18.12. Temperature Effects on Self-Discharge . . . . . . . . .. 18.13. Self-Discharge of a Pseudocapacitance ........... 18.14. Examples of Experimental Measurements on Self-Discharge of Carbon Capacitors and Carbon Fiber Electrodes. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 18.14.1. Introduction ........................ 18.14.2. Potential Decay (Self-Discharge) and Recovery in Terms of a Faradaic Process ............................ 18.14.3. Self-Discharge Behavior of a Commercial Capacitor. . . . . . . . . . . . . . . . . . . . . . . . . .. 18.15. Self-Discharge and Potential Recovery Behavior at an Ru02 Electrode. . . . . . . . . . . . . . . . . . . . . . . . . .. 18.15.1. Background ........................ 18.15.2. Potential Decay (Self-Discharge) and Recovery in Relation to Charge and Discharge Curves. . . . . . . . . . . . . . . . . . .. 18.15.3. Model for Potential Recovery ........ " 18.15.4. Quasi-Reversible Potentials of Ru02 after Self-Discharge ...................... 18.16. Self-Discharge in a Stack ..................... References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

557 557 559 561 562 567 568 569 573 574 575 578 579

582 582

583 584 586 586

587 591 592 595 595

Contents

Chapter 19 Practical Aspects of Preparation and Evaluation of Electrochemical Capacitors 19.1. Introduction....... . . . . . . . . . . . . . . . . . . . . . . . . .. 19.2. Preparation of Electrodes for Small Aqueous Carbon-Based Capacitors for Testing Materials ... " 19.3. Preparation of RuOx Capacitor Electrodes . . . . . . . .. 19.4. Preparation of RuOx Capacitors with a Polymer Electrolyte Membrane (U.S. Patent 5,136,477) ..... 19.5. Assembly of Capacitors. . . . . . . . . . . . . . . . . . . . . . .. 19.6. Experimental Evaluation of Electrochemical Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19.6.1. Cyclic Voltammetry .................... 19.6.2. Impedance Measurements ............... 19.6.3. Constant Current Charge or Discharge. . . .. 19.6.4. Constant Potential Charge or Discharge .... 19.6.5. Constant Power Charge or Discharge ...... 19.6.6. Leakage Current and Self-Discharge Behavior ................ 19.7. Other Test Procedures ......................... References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Chapter 20 Technology Development 20.1. Introduction................................ 20.2. Development of the Technology of Electrochemical Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20.2.1. Classes of Capacitors. . . . . . . . . . . . . . . . .. 20.3. Summaries of Device Developments and Technology Advances ........................ 20.4. Materials Requirements ....................... 20.4.1. Electrodes.......................... 20.4.2. Carbon Electrode Materials ............ 20.4.3. Activation Procedures for Carbon Particles and Fibers ................... 20.4.4. Oxide, Redox-Pseudocapacitance Systems 20.4.5. Conducting-Polymer Electrodes ........ 20.4.6. Electrolyte Systems ................. , 20.4.7. Practical Design Aspects ............. , 20.4.8. Capacitor Stacking. . . . . . . . . . . . . . . . . .. 20.4.9. Bipolar Electrode Arrangements. . . . . . .. 20.4.10. Current Distribution in Capacitor Devices 20.4.11. Scale-up Factors. . . . . . . . . . . . . . . . . . . .. 20.5. State of the Art. . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

xxvii

597 598 599 600 600 602 602 602 603 605 605 605 606 606 609 610 610 612 613 613 615 615 618 618 618 620 620 622 623 625 627

Contents

xxviii

20.6. 20.7. 20.8.

20.9. 20.10. 20.11. 20.12. 20.13. 20.14.

20.15. 20.16. 20.17. 20.18.

Chapter 21

20.5.1. Electrode Development. . . . . . . . . . . . . . .. 20.5.2. Ruthenium Oxide Materials. . . . . . . . . . . .. 20.5.3. Other Embodiments. . . . . . . . . . . . . . . . . .. Self-Discharge: Phenomenological Aspects. . . . . .. Thermal Management ........................ Other Variables that Affect Capacitor Performance 20.8.1. Temperature Dependence of Capacitance and Capacitor Performance . . . . . . . . . . . .. 20.8.2. Constant Current versus Constant Potential Charging Modes. . . . . . . . . . . . . . . . . . . . .. 20.8.3. Rate Effects on Charge or Discharge. . . . .. Safety and Health Hazards in the Use of Electrochemical Capacitors. . . . . . . . . . . . . . . . . . .. Recent Advances in the Use of Materials ......... Usage Basis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Commercial Development and Testing ........... Capacitor-Battery Hybrid Application for Electric Vehicle Drive Systems . . . . . . . . . . . . . . . . . . . . . .. Market Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20.14.1. Electrochemical Capacitors in the Capacitor Market . . . . . . . . . . . . . . . . . . .. 20.14.2. Market Status and Future Opportunities .. Technology Summary Based on Patent Literature.. Energy Storage by High-Voltage Electrostatic Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Concluding Summary ........................ Appendix on Information Sources .............. References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. General Reading References ...................

627 634 635 641 643 644 644 648 649 649 651 655 658 663 666 666 667 667 668 670 671 673 674

Patent Survey .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 675

Index ...................................................... 685

Electrochemical Supercapacitors Scientific Fundamentals and Technological Applications

Chapter 1

Introduction and Historical Perspective

1.1. HISTORICAL OVERVIEW

The discovery of the possibility of storing an electrical charge on surfaces arose from phenomena associated with the rubbing of amber in ancient times. Of course, the origin of such effects was not understood until the mideighteenth century in the period when the physics of so-called "static electricity" was being investigated and various "electrical machines" were being developed, such as the Electrophorus and the Wimshurst machine. (Excellent examples of these, as well as Leyden jars, can be seen in the Museum of Science in Florence.) Understanding of electricity at the molecular electronic level did not begin until 140 years later, starting indirectly with the work of Michael Faraday and later with that of J. J. Thomson and of Millikan on the electron. In relation to such historic investigations, the development of the Leyden jar, and the discovery of the principle of charge separation and charge storage on the two surfaces of the Leyden jar, separated by a layer of glass, was of major significance for the physics of electricity and later for electrical technology, electronics, and electrochemical engineering. The discovery of the Leyden jar, referred to in early works and in technological applications up to the middle of this century as the "condenser," is variously attributed either to Dean Kleist at Leyden or almost simultaneously to Musschenbroek at Kamin, Pomerania. In later terminology, the device in various embodiments is referred to as a "capacitor" and its capability (in the extensive sense) for charge storage per volt, is referred to as its "capacitance," C. The word "capacity," used in battery terminology to indicate the extent of Faradaic charge storage (in units of coulombs or watt-hours) should not be confused with "capacitance" (given in units of farads), which applies to capacitors. 1

2

Chapter 1

The original Leyden jar consisted of a glass phial containing an aqueous acidic electrolyte as a conductor which was contacted by an immersed electrode, with a metal foil coating on the outside surface of the glass phial, thus providing a dielectric material in between-the glass. Later improvements, leading to modern systems, used glass plates as dielectric materials with metal foil on each surface, or rigid metal plates separated by vacuum or air (air condensers), or by mica or polystyrene insulating films. Various forms of condensers, as illustrated in the Encyclopaedia Britannica, are shown in Fig. 1.1 and include an example of a Leyden jar. For many years there was much uncertainty about the nature of electricity after the separate discoveries of "animal electricity" by Galvani in 1776 and "voltaic electricity" by Volta in 1800, and it remained poorly understood for a long time, notwithstanding the work by Faraday (including his discovery of the chemical equivalence of electrical charge). It was not until the discoveries by J. J. Thomson' on the charge-to-mass ratio of the ubiquitously produced negative charge carriers arising in the ionization of low-pressure gases (first investigated by Crookes 2 ), and the work by Millikan 3 and by Townsend4 on the absolute value of the charges borne by such particles that a modern view of the nature of electricity could be proposed in terms of accumulation or deficiency of such charges, and the dynamics of their motion. In 1881 Johnstone Stoney5 coined the name "electron" (from the Greek eAliKrOV for amber) for such negatively charged particles, which are the natural units of electricity, the electron charge being 4.80 x 10- 10 electrostatic units (esu) of charge or 1.60 X 10-20 absolute electromagnetic units (emu) of charge. It is of historical interest that Faraday himself failed to reach the conclusion that his laws relating the extent of a charge's passage (current x time == coulombs) to quantitatively determinable extents of chemical change associated with the electrolysis of aqueous acid and metal-salt solutions, implied an atomic unit of electricity. However, the importance of his laws is in no way diminished by that difficulty because the laws clearly demonstrated in a quantitative way the equivalence of electrical charge to the extent of a generated chemical change that was dependent on the chemical identity of the element concerned and its equivalent weight or oxidation state in solution. It was only much later, in his Faraday Lecture of 1881, that von Helmholti reached the key conclusion that Faraday's laws implied that a fundamental unit of electrical charge was universally involved. This paved the way for the development of the quantitative and more fundamental science of electrochemistry and for a quantitative science of the electrical nature of matter. This work was further elucidated through spectroscopy and the theoretical advances of Bohr, Sommerfeld, Schroedinger, and Heisenberg on electron energy states in atoms and molecules, and the significance of ionization and its relation to solvolysis in solution and the state of solid salts 7 .

Introduction and Historical Perspective

3

Although the experimental phenomenology of electrical charging of surfaces, including those of the Leyden jar condenser, was well understood in the mid- and later part of the eighteenth century, the full physical significance of charging or discharging processes at the plates of capacitors could not have been at all fully understood until the atomic nature of electricity, the electron, was characterized and its properties directly determined. 1,3,4 Similarly, the charging or discharging of capacitors by a flow of electrical charges in wires could not have been understood until the electron theory of metals had been developed and the mechanism of current flow in metal conductors was understood. Nor for that matter could the physical and chemical significance of the charging of amber be understood until satisfactory ideas about the ionization of molecules and macromolecules (in that case, through frictional or triboelectrical effects) had been formulated, partly through the results on spectroscopic ionization limits of electron energy states in molecules or atoms. Thus, the mechanisms of electrical charge storage in capacitors remained poorly understood at the atomic physical level until some 140 years after the development of the Leyden jar capacitor and related contemporary electrical machines. Nevertheless, it was Faraday who had some of the first (in principle) correct ideas about polarization in dielectrics and the significance of dielectric strain and lines of (electrical) force in the dielectric materials of charged condensers. At this point it should be stated that the charging of metallic plates of a capacitor involves production of an excess (negative plate) or deficiency (positive plate) of the density of the delocalized electron plasma of the metal over a short distance (ca. 0.1-0.2 nm, the Thomas-Fermi screening distance) from the formal outer surface of the metal plate. However, each plate has its own electric potential (an equipotential) throughout its material, except very near its surface. Hence, local charge density variation within the so-called "Thomas-Fermi screening distance" has to arise according to the Poisson relation that expresses the field gradient in terms of the local space charge density and the Gauss relation that expresses the field as a function of surface charge density (see Chapter 4). At an insulator such as amber, the excess charge density that arises upon charging has a different origin that is associated with localized molecular ionization (localized oxidation) of the insulator material at its surface, or in some cases is due to negative ionization by localized uptake of electrons at electron acceptor sites on the surface (localized reduction). These latter phenomena are the subject of "triboelectricity." The principle that electrical energy can be stored in a charged capacitor was known since 1745; at a voltage difference, V, established between the plates accommodating charges +q and -q, the stored energy, G, is 112 CV 2 or 112 qV, G being a Gibbs (free) energy which increases as the square ofV.

4

Chapter 1

~I

FIGURE 1.1. Photographs of various designs and arrangements of condensers (capacitor devices), including first the Leyden jar. (Reproduced from Encyclopaedia Britannica, 1929 edition, with permission). (1) A Leyden jar, the simplest type of condenser. (2) A battery of Moscicki glass condenser tubes-an improvement on the Leyden jar, introduced about 1904 by Moscicki. (3) The transmitter at one of the stations of the British Broadcasting Corp., showing two air dielectric condensers in the lower compartment (center and right-hand side). (4) Ship's radio transmitter, showing mica dielectric condenser on right side on shelf over generator. (5) Mica dielectric condenser of special construction for shortwave radio stations. (6) Condenser gallery of the Rugby radio station of the British post office. (7) Interior of the bell box used with British office telephone instrument, showing paper dielectric condenser (bottom of box). (8) Top row: several patterns of fixed condensers. Bottom row: several types of variable condensers commonly used as tuning capacitors in old radios.

•

"i.

•

·\r ~.!~ · ,~~

-

~

en

CD

Ic'

3

'tI

e!.

C'j'

~...

:I:

a.

III :J

o·:J

c:

!l

i

5"

6

Chapter 1

The utilization of this principle to store electrical energy for practical purposes, as in a cell or battery of cells seems to have been first proposed and claimed as an original development in the patent granted to Becker in 1957.8 The patent described electrical energy storage by means of the charge held in the interfacial double layer at a porous carbon material perfused with an aqueous electrolyte. The principle involved was charging of the capacitance, Cd!> of the double layer, which arises at all solid/electrolyte interfaces, such as metal, semiconductor, and colloid surfaces (and also at the phase boundary between two immiscible electrolyte solutions9 ). Carbon is an element almost uniquely suited for fabrication of electrochemical capacitors of the double-layer type. It exists in several, well-known allotropic forms-diamond, the fullerenes, and graphite; the latter and glassy carbon can be generated in the form of high-area fibers or felts. Amorphous carbons and carbon black are available as high specific-area powders. The fiber or felt materials are particularly convenient for formation of electrode structures having good mechanical integrity, while the high-area powders are more difficult to handle. However, glassy carbon, graphite, and carbon black materials are convenient for forming high-area electrode structures, often on a support matrix. From an electrochemical point of view, carbon is relatively, though not entirely, unreactive and thus has a potential voltage range of almost ideal polarizability (Chapters 6 and 7), approaching 1.0 V in aqueous solution and possibly up to 3.5 V in nonaqueous media. After Becker, the Sohio Corporation in Cleveland, Ohio, also utilized the double-layer capacitance of high-area carbon materials, but in a nonaqueous solvent containing a dissolved tetraalkylarnmonium salt electrolyte. Such systems provide higher operating voltages (3.4-4.0 V) owing to the greater decomposition voltage of nonaqueous electrolytes than those for aqueous ones. Thus they can accommodate higher charge densities and provide larger specific energy storage since the storable energy increases with the square of the voltage attainable on charge. A different principle, originating from ref. 14, was utilized and developed in 1975 onward to 1981 by ConwaylO in Ottawa, under contract with Continental Group. This was based on the concept of D. Craig that was developed at Hooker Corp. Here, in one type of system, the large, so-called "pseudocapacitance," CIP' associated with the potential dependence of extents of electrochemical adsorption of H or monolayer levels of electrodeposition of some base metals (Pb, Bi, Cu) at Pt or Au was used ll as a basis for an energy-storing capacitor. In another type of system, the pseudocapacitance associated with solid oxide redox systems was used, especially that developed over some 1.4 V (practical range 1.2 V) in aqueous H 2S04 at RU02 filmsy-15 This system approaches almost ideal capacitative behavior, with a large degree of reversibility between charge and discharge, and multiple cyclability over some lOS cycles (see Chapter 11).

Introduction and Historical Perspective

7

Work on the latter type of system has been continued by Pinnacle Research Corp., Cupertino (now at Los Gatos, California), in the former laboratories of Continental Group. Useful military applications have been developed, but the Ru materials required are too expensive for the development of a large-scale capacitor, e.g., for use in hybrid systems with batteries for electric-vehicle motive power. Pseudocapacitance arises whenever, for thermodynamic reasons, there is some continuous dependence of a charge, q, passed Faradaically in oxidation or reduction, upon the electrode potential, V. Thus, a derivative dq/dV can arise that corresponds to a pseudocapacitance that is directly measurable, or utilizable, as a capacitance (Chapter 10). The large capacitances (on the order of several or more farads per gram) that can be developed with the RuOz film system and also with the carbon double-layer-type capacitors led to the terms "supercapacitor" or "ultracapacitor" being coined, respectively, for these two types of high specific capacitance devices. Recently it has been suggested that the more general term "electrochemical capacitors" be used to refer to these systems. However, this name should not be confused with "electrolytic capacitor," the latter term referring to the well-known, moderately high-capacitance device (on the order of tens of millifarads) that is based on a thin-film oxide dielectric formed electrolytically with a gel electrolyte on such metals as Ta, Zr, Ti, or AI. The key practical factor that allows very high capacitances, on the order of farads or tens of farads rather than millifarads, to be achieved in a small volume, say 1 cm3, is the utilization of high-area materials such as activated porous carbons for which real areas are up to 1000 to 2000 m 2 g-l. Similarly, with the Ru02 pseudocapacitance system, the material, which is a hydrous oxide, has a quasi-3-dimensional, electronically conducting structure, giving accessibility to protons and electrons 15 that are involved in two or three (Chapter 11) successive, reversible oxidation or reduction steps in charge or discharge, respectiVely. The continuous dependence of the extents of oxidation or reduction on electrode potential (over 1.4 V), with corresponding passage of charge, leads to the high specific redox pseudocapacitance of this material,16 which is usually coupled with an appreciable double-layer capacitance component (see Chapter 11). The use of high-area carbon or oxide redox systems has led to the commercial production of practical high-capacitance electrochemical capacitor devices such as that developed (the Gold Capacitor) by Matsushita Electric Industrial Co. (Osaka, Japan) and by Pinnacle Research; the latter has been developed mainly for military applications. The commercial products are designed to provide standby power for random access memory (RAM) devices or telephone equipment, as power sources for operating activators, and as elements for long time-constant circuits, etc. An attractive technology employing Ru02 in a thin film applied to a Nafion membrane, or a powder treated with Nafion, has been developed by Giner, Inc. (Waltham, Massachusetts) and gives high specific ca-

8

Chapter 1

pacitance. The design avoids a liquid electrolyte and is analogous to membrane electrolyte fuel cell electrodes. Further details are described in Chapter 11. Recent opportunities for the use and development of larger scale capacitors arise from the possibility of using them in hybrid configurations with secondary batteries in electric vehicle power systems.

1~.

SCOPEOFTHEMONOGRAPH

It is the aim of this volume to give a comprehensive and to a large extent self-contained account of the development of electrochemical capacitors, covering both the fundamental science (physics, chemistry and electrical engineering principles involved) and, at the end of the book, to provide an overview of recent technology development. Some "tutorial" aspects are also included (Chapters 3 and 5) to provide necessary background information and principles of electrochemistry, e.g., on topics such as the double layer, electrostatics, and electrode kinetics. Chapter 2 gives an overview of similarities and differences between electrochemical capacitor and battery systems for electrical energy storage, with stress on the differences between non-Faradaic and Faradaic electrochemical processes that are involved in these two types of devices. Considerable detailed attention is given to the essential topics of doublelayer capacitance and Faradaic pseudocapacitance. Chapters 6, 7, and 8 give the basic conceptual and theoretical background concerning the phenomenon of double-layer capacitance and its modeling. Chapter 10 gives an account of the complementary phenomenon of pseudocapacitance associated with electrochemical adsorption and redox processes involving charge transfer. Chapter 12 discusses the electrical behavior of conducting polymers as capacitor materials. The electrical response behavior of capacitors to ac and dc, and pulse potential modulation signals is treated (Chapters 4, 16, and 17) since the results of such procedures provide the principal means of characterizing capacitor behavior, both fundamentally and in technological evaluation. Basic information on the principles of electrostatics (Chapter 4) and of ac impedance spectroscopy is covered. Material is also included on the characterization of various high-area carbon preparations (Chapters 9 and 14), and of Ru02 (Chapter 11) used for capacitor fabrication. General aspects of energy-density vs. power-density relations (Ragone plots) are treated (Chapter 15), including the results of computer-simulation evaluations of the effects of ohmic and Tafel (kinetic) polarization effects. Since the former effects are closely related to properties of the electrolyte solutions used in electrochemical capacitors of both the aqueous and

Introduction and Historical Perspective

9

nonaqueous, aprotic solution type, Chapter 13 discusses electrolyte solutions and their properties. Chapter 18 covers the important practical problem of self-discharge on open-circuit and is followed by a summary of procedures for preparation and evaluation of electrochemical capacitors (Chapter 19). The volume concludes with an extensive survey of recent technology developments (Chapter 20) in the field. It is taken mainly from presentations made at seminars and conferences on electrochemical capacitors and batteries over the past 7 years. Finally, the principal patent literature is surveyed.

REFERENCES 1. 1. J. Thomson, Phil. Mag., 5,346 (1903); see also J. J. Thomson, The Electron in Chemistry, Franklin Institute Lectures, Chapman and Hall, London (1923). 2. W. Crookes (1879), quoted by S. Glasstone in Textbook of Physical Chemistry, Van Nostrand, New York (1940). 3. R. A. Millikan, Phys. Rev., 2, 143 (1913). 4. 1. S. Townsend, Electricity in Gases (1879), quoted by Glasstone as in Ref. 2. 5. G. Johnstone Stoney, Phil. Mag.,ll, 381 (1881); Sci. Trans. Roy. Soc. Dublin, 4,583 (1891). 6. H. von Helmholtz, 1. Chern. Soc., Lond., 39, 277 (1881). 7. B. E. Conway, Ionic Hydration in Chemistry and Biophysics, Elsevier, Amsterdam (1981). 8. H. E. Becker, U.S. patent 2,800,616 (to General Electric Co.) (1957). 9. Z. Samec, 1. Electroanal. Chern., 103, I (1979). 10. B. E. Conway, 1. Electrochem. Soc., 138, 1539 (1991). 11. B. E. Conway and H. A. Kozlowska, Acct. Chern. Res., 14,49 (1981). 12. S. Hadzi-Jordanov, H. A. Kozlowska, M. Vukovic, and B. E. Conway, 1. Electrochem. Soc., 125, 1471 (1978). 13. S. Hadzi-lordanov, B. E. Conway, and H. A. Kozlowska, 1. Electroanal. Chern., 60, 359 (1975). 14. S. Trasatti and G. Buzzanca, 1. Electroanal. Chern., 29, App. 1 (1971). 15. R. Galizzioli, F. Tantardini, and S. Trasatti, 1. Appl. Electrochem., 4, 57 (1974). 16. B. E. Conway, in Proc. Symp. on Electrochemical Capacitors, F. M. Delnick and M. Tomkiewicz, eds., vol. 95-29, 15, Electrochem. Society, Pennington, N.J. (1996).

Chapter 2

Similarities and Differences between Supercapacitors and Batteries for Storing Electrical Energy

2.1. INTRODUCTION

2.1.1. Energy Storage Systems

A modern technological society demands the use and storage of energy on a major scale, employing large and small systems for that purpose. Energy stored as potential energy is involved in hydroelectric systems through the hydrostatic "head" of water behind dams; it is also stored in a potential sense in fuels (e.g., coal, oil, and cryogenic hydrogen) and becomes available, albeit with rather poor efficiency, through combustion utilizing steam-piston, steam-turbine, and internal combustion engines of various kinds as energy transduction devices. Energy may also be stored as rotational kinetic energy in flywheels. Electrical energy can be stored in two fundamentally different ways: (1) indirectly in batteries as potentially available chemical energy requiring Faradaic oxidation and reduction (see Sections 2.4.2 and 2.5) ofthe electrochemically active reagents to release charges that can perform electrical work when they flow between two electrodes having different electrode potentials (i.e., across the voltage difference between the poles of battery cells); and (2) directly, in an electrostatic way, as negative and positive electric charges on the plates of a capacitor, a process known as non-Faradaic electrical energy storage. The efficiency of these two modes of storing electrical energy is usually substantially 11

12

Chapter 2

larger than that of fuel combustion systems, which are limited by thermodynamic Carnot cycle considerations while electrochemical systems usually involve more reversible processes, with direct conversion of potentially available chemical energy to free (or Gibbs) energy, tJ.G. 2.1.2. Modes of Electrical Energy Storage by Capacitors and Batteries

An important difference arises between the reversibility of Faradaic and non-Faradaic systems [(1) and (2)]. In energy storage by capacitors, only an excess and a deficiency of electron charges on the capacitor plates have to be established on charge and the reverse on discharge; no chemical changes are involved. However, with storage of electrochemical energy in battery cells through Faradaic reactions, chemical interconversions of the anode and cathode materials must take place, usually with phase changes. Although the overall energy change can be conducted in a relatively reversible thermodynamic way, the charge and discharge processes in a battery often involve irreversibility in interconversion of the chemical electrode reagents; thus the cycle life of battery cells is usually restricted to one thousand to several thousand discharge and recharge cycles, depending on the type of battery. By contrast, a hardware capacitor has an almost unlimited cyclability since no chemical and phase changes are involved in its charging and discharging. Ordinary capacitors have, however, a very small amount of charge storage unless they are large, i.e., they have a low energy density for electrical energy storage. However, charged electrode/solution interfaces contain double layers (Chapters 6 and 7) that have capacitances of about 16 to 50 flF cm- 2 ; hence, with the sufficiently large accessible electrode areas that are realizable with high-area carbon powders, felts, and aerogels, very large double-layer capacitances on the order of 10 to 100 F per gram can be achieved. It is the practical realization of this possibility in recent years (but originating in early technological development some 35 years ago at Sohio) that has led to the relatively new field of electrochemical capacitors. These are now actively progressing as energy storage devices to complement batteries. Because the charging and discharging of such double-layer capacitors involves no chemical phase and composition changes which, in batteries, lead to materials irreversibility, such capacitors have a high degree ofrecyclability, on the order of 105_106 times. Only electrons need to be moved to and from the electrode surfaces through the external circuit, and cations and anions of the electrolyte transported within the solution to the charged interfaces. It is for these reasons that capacitor charging and discharging processes are highly reversible. In the cyclic voltammetry of such systems (see Chapter 10), the charging and discharging voltammograms are almost mirror images of one another, while for battery processes they are rarely of this kind (see Fig. 2.3 later in this chap-

Supercapacitors and Batteries

13

ter). This is a major and characteristic difference between battery and capacitor electrical energy storage systems. It must be emphasized at the outset that there has never been an aim or projection of a possible substitution of batteries by supercapacitors; rather, opportunities arise for complementary operation of electrochemical capacitors that are electrically coupled in discharge and recharge with batteries, while other kinds of applications especially favor capacitor-type behavior, e.g., for backup power systems. Also, there are stand-alone opportunities for using multiply rechargeable electrochemical capacitors in a variety of independent functions, as briefly indicated in Chapter 1. General aspects of the electrochemistry and technology of batteries are to be found, for example, in Refs. 1 and 2. In the early stages of the development of electrochemical capacitor technology and related fundamental work, there was some confusion in the electrochemical engineering field and at symposia about the differences between the properties and operation of a battery and a supercapacitor, and what advantages one might have over the other. This confusion may have been assisted by some groups calling the capacitor devices "ultracapacitors" and others, "supercapacitors," the latter as originated by the Ottawa group in 1975. The present preferred name, proposed by Burke, as referred to in Chapter 1, is now more scientific and generic, namely "electrochemical capacitors." This chapter identifies and explains some of the similarities and differences between electrochemical capacitors and batteries in relation to the electrochemical processes that are involved in their discharge and recharge cycling, and in their potential uses as electrical energy storage devices. In particular, the fundamentally different mechanisms of charge storage that are normally involved will be emphasized, along with the consequent, usually different, relations between the extents of charge accommodated at the electrodes and the electric potential differences (cell voltage) between pairs of electrodes having conjugate, ±, polarities. One of the main and kinetically significant differences between capacitors and batteries is that the electrodes of the latter usually undergo substantial phase changes during discharge and recharge (minimally though for the intercalation systems), which lead to kinetic and thermodynamic irreversibility. On the other hand, capacitors of the double-layer type require only electrostatic charge accommodation with virtually no phase change, though some small but significant reversible electrostriction of the electrolyte can arise upon charging.

2.2. FARADAIC AND NON-FARADAIC PROCESSES There is a general and fundamental difference between the mechanisms of operation of electrochemical capacitors and battery cells: for the double-layer

14

Chapter 2

type of capacitor, the charge storage process is non-Faradaic, i.e., ideally no electron transfer takes place across the electrode interface and the storage of electric charge and energy is electrostatic. In battery-type processes, the essential process is Faradaic, i.e., electron transfer does take place across the double layers, with a consequent change of oxidation state, and hence the chemistry of the electro active materials. Intermediate situations arise where Faradaic charge transfer occurs, but owing to special thermodynamic conditions that apply, the potential, V, of the electrode is some continuous function of the quantity of charge, q, passed so that a derivative, dq/dV, arises. This is equivalent to and measurable as a capacitance and is designated as a pseudocapacitance, as explained in Chapter 10. A somewhat different situation arises when chemisorption of ions or molecules takes place with partial charge transfer,3-5 e.g., in a process such as M + A- ~ MlA(1-J)- + oe(in M)

(2.1)

Such a reaction at the surface of an electrode M usually gives rise to a potentialdependent pseudocapacitance (see again Chapter 10) and the quantity Oe is related to the so-called "electrosorption valence" treated in Refs. 3, 4, and 5. To summarize, the important differences in the charge storage processes are as follows: 2.2.1. Non-Faradaic

The charge accumulation is achieved electrostatically by positive and negative charges residing on two interfaces separated by a vacuum or a molecular dielectric (the double layer or, e.g., a film of mica, a space of air or an oxide film, as in electrolytic capacitors). 2.2.2. Faradaic

The charge storage is achieved by an electron transfer that produces chemicalor oxidation state changes in the electroactive materials according to Faraday's laws (hence the term) related to electrode potential. Pseudocapacitance can arise in some cases. The energy storage is indirect and is analogous to that in a battery. In a battery cell, every electron charge is Faradaically neutralized by charge transfer, resulting in a change of oxidation stage of some redoxelectroactive reagent, e.g., NP+·02-·0H- + e + H+ ~ NF+·20H-

in the cathode of an Ni-Cd battery. 1

(2.2)

Supercapacitors and Batteries

15

In a capacitor, actual electron charges, either in excess or deficiency, are accumulated on the electrode plates with lateral repulsion and no involvement of redox chemical changes. (However, in some cases of double-layer charging, some partial electron transfer does occur, giving rise to pseudocapacitance, e.g., when chemisorption of electron-donative anions such as cr, Be 1-, or CNS-, takes place as illustrated in Eq. (2.1). The electrons involved in double-layer charging are the delocalized conduction-band electrons of the metal or carbon electrode, while the electrons involved in Faradaic battery-type processes are transferred to or from valence-electron states (orbitals) of the redox cathode or anode reagent, although they may arrive in or depart from the conduction-band states of the electronically conducting support material. In certain cases, the Faradaically reactive battery material itself is metallically conducting (e.g., PbOb some sulfides, RU02), or else is a well-conducting semiconductor and a proton conductor, e.g., Ni·O·OH.

2.3. TYPES OF CAPACITORS AND TYPES OF BATIERIES 2.3.1. Distinguishable Systems

Table 2.1 contains a summary of the types of capacitors and their mode of energy storage: electrostatic or Faradaic, the latter in the case when pseudocapacitance arises (Chapter 10). Types d and f, which are printed in bold, are the principal kinds treated in this volume. Normally, capacitors function as elements of electronic circuits or communications equipment, or as ballast for starting electric motors or electric discharge TABLE 2.1. Types of Capacitors and Mode of Energy Storage Type (a) Vacuum (b) Dielectric (c) Oxide electrolytic (thin film) (d) Double-layer

(e) Colloidal electrolyte (f) Redox oxide film

(g) Redox polymer film (h) Soluble redox system

Basis of charge or energy storage Electrostatic Electrostatic Electrostatic Electrostatic (charge separation at double-layer at electrode interface) Electrostatic (special double-layer system) Faradaic charge transfer (pseudocapacitance) Faradaic charge transfer (pscudocapacitance) Faradaic charge transfer (pseudocapacitance)

Examples Mica, Mylar, paper Ta205, Al 20 3 C preparations, powders, fibers Undeveloped

polyaniline, polythiophenes

Fe(CN)~--Fe(CN)~-, V 2+N 3+N0 2+

16

Chapter 2

tubes (fluorescent lights). As has been explained earlier, devices of very large capacitance are now available for storing electric energy in various applications. Table 2.2 summarizes the types of batteries currently extant. 1,2 These are generally classified as primary (nonrechargeable) or secondary (multiply rechargeable). The discharge or recharge mechanism is mainly Faradaic, although all electrode interfaces exhibit a double-layer capacitance that is reversibly chargeable. For batteries the latter mechanism accommodates about 2-5% of the total charge accepted. In a different class from the battery systems listed in Table 2.2 are fuel cells in which the anode and cathode (02 or air) reactants are supplied on a continuous basis from external reservoirs, and the electrode surfaces provide an interface for either electrocatalytic oxidation or reduction of the reagents supplied. The primary metal-air cells are operated as semi-fuel cells, but the "fuel" is an easily oxidizable base metal and a gas-diffusion catalyzed air or O 2 cathode is employed. Such cells using Al are not rechargeable except by mechanical replacement of the metal anodes. However, if Zn is used, electrochemical recharging is possible, but requires a bifunctional catalyzed counter electrode capable of evolving H2 on recharge or reducing O2 (air) on discharge.

TABLE 2.2. Types of Battery and Mode of Energy Storage Type Primary Leclanche, zinc-Mn02 Alkaline, zinc-Mn02 Mg-AgCI Mg-PbCI 2 Li-SOCI2 and other cathodes Li-CFx AI-air (catalyzed) Zn-air (catalyzed) Secondary Lead acid, Pb-Pb0 2 Nickel-cadmium, Ni·O·OH-Cd Nickel-hydrogen, Ni·O·OH-metal hydride Nickel-zinc, Ni·O·OH-Zn Mercuric oxide-zinc, HgO-Zn Silver oxide(s)-zinc, AgO-Zn Zinc-air (catalyzed) Li-TiS2 Li-MoS2 Li-Mn02 Li-Co0 2 Li-C-Co0 2 and other cathodes Li-iron sulfides Na-S

Basis of charge or energy storage

Faradaic

Faradaic

Faradaic (exhibiting intercalative pseudocapacitance)

Supercapacitors and Batteries

17

TABLE 2.3. Materials for Constructing Electrochemical Capacitors and Batteries Batteries Pb-PbCI 2 Pb-Pb0 2 Pb-PbS0 4 Cd-Cd(OHh Ni(OHh-Ni·O·OH-··· MnOrMn(OHh Zn-ZnO-ZnCYzAg-Ag20 or Ag-AgO

Electrochemical capacitors Carbon (double-layer capacitors), H on Pt (UPD) Underpotential-deposited Pb on Au Ru02

lr0 2 Modified cheaper transition metal oxides, metal nitrides H in Pd, LaNi s(?) Conducting polymer electrodes, e.g., polyanilines, polythiophenes

Hg-HgO Li-SOCI 2 Li-S02 Li-CFx Intermediate transitional systems: Li-TiS2; Li-MoS2; Li-Mo02; Li-Mn02; Li-C002; Li-C

The general question of experimental and theoretical constraints in the choice of materials for electrochemical capacitors was examined in an article by Sarangapani, Tilak and Chen. 6 Such factors as accessibility of the active surface, the shapes and reversibility of volt ammo grams, and the components of the redox capacitance at Ru02 (Chapter 11) needed to account for its almost constant value over a 1.4-V range are discussed, together with the question of the stability of materials in overcharging and extended cycling. Table 2.3 lists the principal materials used to construct batteries and electrochemical capacitors. 2.3.2. Cell Design and Equivalent Circuits

As discussed and illustrated in Chapters 16 and 17, an electrochemical capacitor, like a battery cell, requires two electrodes, one of which is charged negatively with respect to the other, the charge storage and separation being electrostatic. At each ofthe two electrodes, double-layer electron and ion charge separations are established across the electrode interfaces. The macroequivalent circuit (in the absence of self-discharge processes) is represented by two capacitances linked in series with ohmic resistances representing the resistivities of the solution and the separator, as in a battery. In the usual case where the electrodes are high-area, porous matrices, a further microequivalent circuit is needed (see

Chapter 2

18

Chapters 14 and 16) to represent the electrical behavior of the internal surfaces and the electrolyte-containing interstices. As with batteries, bipolar electrode configurations can be fabricated for higher voltage series combinations; these diminish the internal resistance of the device, but require that the edges of the electrodes be carefully sealed to the case in order to avoid shunt, i.e., leakage currents. Such systems optimize power density. The equivalent circuit for most battery-type charge storage systems involves simply a Faradaic resistance (R F ) that represents the potential dependence of the reciprocal of the rate of the oxidation and reduction charge transfer process; it is in parallel with (an always significant) double-layer capacitance, Cdl • Under some high-rate discharge or recharge conditions, some diffusion control may arise, in which case RF is in series with a so-called Warburg C-R impedance element written as W. In addition, with some rechargeable battery systems (e.g., of the intercalation type), a pseudocapacitance element may also be required to represent the impedance behavior of the Faradaic process (Chapters 10 and 17). For both electrochemical capacitors and battery cells, a solution resistance element, Rs' in series with the Faradaic impedance, ZF, is usually necessary in order to fully represent the charging or discharging behavior. In fact, Rs is usually very important in the evaluation and performance of capacitors and batteries for high-rate discharge applications, and is an important influence on the ac impedance spectrum (see Chapters 16 and 17) of the device. In the absence of self-discharge processes, or any parallel pseudocapacitance, the macroequivalent circuit of an electrochemical capacitor involves only a solution resistance (Rs) and a double-layer capacitance Cd1 • However, Rs and Cd1 have distributed components in the microequivalent circuit in the case of high-area, porous electrode materials, as treated in Chapter 14. The same applies to battery electrodes fabricated in porous material configurations, a procedure and design that are commonly adopted in battery technology; then complex microequivalent circuits also apply, but with the inclusion of the essential RF components.

2.4. DIFFERENCES OF DENSITIES OF CHARGE STORAGE IN CAPACITORS AND BATTERIES

2.4.1. Electron Densities per Atom or Molecule In the double layer at plane electrodes, charge densities of about 16 to 50 f.1,C cm- 2 are commonly realizable. Taking an average value of 30 f.1,F cm- 2 or 30 f.1,C V-I, and an atom density of ca. 10 15 cm- 2 at a smooth electrode sur-

Supercapacitors and Batteries

19

face, it is easily seen that the charge accommodated per atom will be 3011 0 15 f..lc. This is equivalent to 30 x (10- 61105) x 6 x 1023110 15 electrons per atom where 105 and 6 x 1023 are taken approximately as the Faraday constant and Avogadro's number, respectively. The above quotient works out to be 0.18 electrons per atom stored as a double-layer charge at 1 V for the example chosen, but this is the average delocalized charge distribution associated with conduction-band electrons. By contrast, in most battery processes, redox reactions involving usually one or two valence electron charges per atom (sometimes three for Al or Bi) or molecule of electroactive reactant are involved. Thus, electrochemical capacitor charge storage involves, per atom of active available surface area, only about 20 or 10% (respectively) of that involved (indirectly) or available with battery redox materials. Hence the energy densities available with capacitors are usually substantially smaller than those with batteries. This is a well-recognized limitation, but it is usually compensated by much larger cyclability (cycle life) and often better power density attainable from electrochemical capacitors. Of course, because the energy density of a capacitor increases with the square of the voltage on charge, a substantial improvement in energy density is attained by using nonaqueous electrolytes instead of aqueous ones, in which the decomposition voltage can be increased to 3 or 3.5 V or more per cell unit. Elsewhere in this volume (Chapter 14) it will be shown that with large-area carbon materials (e.g., 1000 m2 g-I), very large specific capacitances of 300 F g-I can (theoretically) arise. It is then of interest to compare energy densities attainable at double-layer capacitors with those realized in an Ni-Cd battery.

2.4.2. Comparison of Energy Densities Attainable in Electrochemical Capacitors and Batteries

For a capacitor electrode of 1000 m2 g-1 operating at 1 V and having a specific double-layer capacitance of 30 f..lF cm- 2 (e.g., for carbon blacks), the total capacitance is 300 F g-l, as mentioned earlier. At 1 V, the energy, G, stored is G = (112) CV2 = 112 x 300 x 12 = 150 Wsec g-1 or J g-1 or 150 kJ kg-I, theoretically. Equivalently, in watt-hours (1 Wh =3600 J), the above energy density is about 42 Wh kg-I. (In practice, it will usually be substantially less than this figure owing to the inaccessibility of the electrolyte solution to the fine pores of the porous electrode structure in the case of carbon powders or felts, and the weight of the packaging structure and electrolyte.) A comparison with an Ni-Cd battery can be made as follows: The molecular weight of Ni·O·OH is 92 and the equivalent weight of Cd is 112/2, i.e., 56. The electron number is 2 for Cd (1 for the equivalent weight of 56) and approximately 1 for Ni ·O·OH being reduced to Ni(OH)2. (The oxidation state of Ni in

20

Chapter 2

charged nickel oxide electrodes can approach higher values at low temperatures and in strong aqueous KOH.) For a working voltage of 1.2, the ideal energy density (~G = -zFE) will be 1.2 x 1 x 105/148 J g-I == 8.1 X 105 J kg-I = 810 kJ kg-I where 105 is approximately F, the Faraday constant in coulombs mole-I. With I Wh = 3600 J, the (theoretical) energy density for Ni-Cd (electroactive materials alone) is then 225 Wh kg-I. Thus an electrochemical double-layer capacitor electrode (charged to I V working potential) would have about 20% of the energy density of an Ni-Cd battery electrode, both figures being based on ideal, theoretical performance. A double-layer capacitor based on nonaqueous electrolyte technology would have a substantially larger energy density than that for I-V aqueous solution charging. In an actual two-electrode capacitor device, with one electrode worked against the other, the energy density will be about (112)2 smaller, namely, about 10 Wh kg-I in aqueous electrolyte, since each electrode can be discharged down to only about half its initial voltage following ± charging.

2.5. COMPARISON OF CAPACITOR AND BATTERY CHARGING CURVES

As shown later in this chapter, the energy of the charging of a capacitor to a plate voltage difference of V is (1/2) CV 2 • It is an electrostatic free energy (Gibbs energy), G. Its entropy component S (= [H - G]IT, where H is the enthalpy of charging) arises from the temperature dependence of the permittivity of the dielectric, here the dielectric constant of the double layer. For a battery process, the maximum Gibbs energy is the product of charge Q and the difference of potential, M, between the Nernstian reversible potentials of the two electrodes, i.e., G = Q·M. For the capacitor case, for a charge Q accommodated, G is (1/2) QV. For a given electrode potential difference, M = V, in the two cases it is then evident that the energy stored by a two-electrode cell accommodating a given Faradaic charge Q at voltage M, = V, is twice that stored in a capacitor charged with the same Q to the same voltage. This difference can be understood in the following way: In the process of charging a pure double-layer capacitor, as explained in Chapter 4, every additional element of charge that is added has to do electrical work (Gibbs energy) against the charge density already accumulated on the plates, progressively increasing the interelectrode potential difference. In a battery cell being charged, a thermodynamic potential (ideally) exists independent of the extent of charge Q added, so long as the two components (reduced and oxidized forms) of the electroactive material remain coexisting. Thus the potential difference (electromotive force, emf) of the battery cell is ideally constant throughout the discharge or recharge half-cycles, so that G is Q·M

Supercapacitors and Batteries \

"

21

---_ .. -.. -_ .. --- _.. - _.... -_ .... --.- .. -...

-----~-------------------.-----,

....... - .. __ ..... --

Recharge

Discharge

'.

~

'\: IDEAL \1

~

BATIERY

..J

S

IZ

W

__ IDEAL CAPACITOR

I-

oa.

,. I I

: State of : charge indication I I

-41---

CHARGE / DISCHARGE -----l~_

FIGURE 2.1. Difference of discharge and recharge relationships for a capacitor and a battery: potential as a function of state of charge, Q.

rather than Q·ll2 M (or 1I2Y). This difference can be illustrated by the discharge curves shown schematically in Fig. 2.1 where the voltage on the capacitor declines linearly (for potential-independent Cdl ) with the extent of charge, while that for an ideal battery remains constant* as long as two phases remain and are in equilibrium (upper curves in Fig. 2.1). The decline of the capacitor voltage arises formally and phenomenologically since C = QIVor V = QIC; therefore dVldQ = lie. The ideal battery cell voltages on discharge and recharge, as a function of state of charge, are shown as parallel lines of zero slope4 in the upper parts of the diagram. These two lines differ as a result of any cathodic and anodic polarization (including so-called ohmic IR potential drop due to internal or solution resistance) arising in the discharge or recharge half-cycles. In the sloping discharge and recharge line for the capacitor (Fig. 2.1), there will also be significant IR drop, depending on the discharge or recharge rates, that is, the discharge line will actually be somewhat separated from the recharge line by a voltage difference equal to 2IR. The significance of the difference between the energy density for the capacitor and a battery cell charged to the same potential can be illustrated by ref-

*In practice, there is usually some decline ofvoJtage in a battery with increasing extent of discharge. In certain battery systems e.g., LirriSz , Li/CoOz, and metal hydride anodes, this decline is larger and occurs for fundamental thermodynamic reasons connected with the form of the sorption isotherms.

Chapter 2

22

erence to the working diagram shown in Fig. 2.2. Here the voltage vs. charge relations for a capacitor and an ideal (constant voltage) battery are compared. In actuality, most batteries, with the exception of Li-SOCI2, do exhibit some decline of voltage with decreasing state of charge, but this is for reasons quite different from that in the case of capacitor discharge. The line VB is the voltage across the battery cell as a function of charge Q held during a constant rate charge, Vc is the plot of voltage across a capacitor being charged up to the same final voltage, VB' with accommodation of the same accumulated charge, Q. However, since the line Vc is for capacitor charging, it has a slope of lie (see Fig. 2.1). The integrals VQ under the two lines of the working diagram correspond to the energy of charging. Thus it is seen that VC'dQ is half VB·dQ, i.e., the energy of charging the capacitor to a terminal voltage VB with charge Q is half that for charging the battery with the same charge also to VB' The superimposed hatched and slashed areas in Fig. 2.2 correspond to this difference of energy stored. As shown mathematically in Chapter 4, the factor of half in the charging energy of a capacitor arises for the same reason that it enters into the Born equation for the self-energy of charging of spherical ions in vacuo or in a dielectric (see Chapter 4).

J

J

2.6. COMPARISON OF CHARGE AND DISCHARGE BEHAVIOR OF ELECTROCHEMICAL CAPACITORS AND BATIERY CELLS EVALUATED BY CYCLIC VOLTAMMETRY

As shown in Chapter 3, a capacitor subjected to a cyclic regime of linear potential change with time at a rate dVldt = s generates a response current ± I = C(±S). This procedure provides a convenient and sensitive method for characterization of both double-layer and pseudocapacitance behavior of electrochemical capacitors. Under conditions where the response is reversible to positive and negative-moving sweeps (s having + and - values), the resulting voltammogram for one direction of potential sweep is the mirror image of that generated for the opposite direction of sweep, provided there is no involvement of diffusion control. This is a useful criterion for reversibility in capacitative or pseudocapacitative charge and discharge processes and is a fundamental characteristic of pure capacitative behavior. It corresponds, at all states of charge across the cyclic voltammogram or along the charging curve, to an equilibrium situation being maintained provided that (see Chapter 10) the voltammogram is not scanned at too high a rate or the discharge curve at too high a current density. This is a very important and basic matter to understand; it distinguishes capacitor and battery behavior in a fundamental way.

Supercapacitors and Batteries

23

v

o

..... dQ~

Full charge

Q

Es =!VsdQ; Ec =/VCdQ ; Ec=t Es VC=q/C ; Ec=tQV= ~ cf

~Battery

~ Capacitor f

FIGURE 2.2. Working diagram illustrating storage of energy E = V·dQ for a battery at ideally constant charging voltage V = VB, and for a capacitor with progressively changing voltage, V = Vc, leading to EB =QVB or Ec =(t)QVc.

In contrast, battery-type processes are rarely reversible in the above sense and a substantially different range of potentials is required for oxidation of the active material compared with that for its reduction. The cyclic voltammogram is then asymmetric and no mirror-image appearance is manifested. An example of the contrast between a reversible, capacitative cyclic voltammogram (for the pseudocapacitance at Ru02) and an irreversible one (for the battery cell system Pb-PbCI 2) is shown in Figs. 2.3(a) and 2.3(b). For RU02 [Fig. 2.3(a)], the profile of positive charging currents over the potential range 0.05 to 1.40 V vs. the reversible H2 electrode potential (RHE) is almost the mirror im-

Chapter 2

24

a) RU02 film u;

..0 :::J

6.25

VI

N

IE u

O~-+--+--r~--~-+--+--+--~-r~--;--+--r----

.....

xl>

to some value tJ> ± eV to achieve the condition of balance [Figs. 3.1(a) and 3.1(b)] required for facile electron transfer to take place at the potential V. If I is the ionization potential of the reactant in the reduced form and !J..Gs is the change of its solvation energy upon electron transfer, then the energetic condition for the process to occur in the direction of donation of charge is 7- 9 : (3.2)

Chapter 3

38 Vac--r---.

Redox reoction 0+ e(M)+ R (non - matching energy levels)

o

R

(a) Vac --.--,,-

o

R

0+ e(M) ..... R at electrode potential V (matching energy levels)

CONDITION FOR e-TRANSFER CP±eV-l+S~O

FIGURE 3.1. Conditions for energy mismatching (a) and energy matching (b) at the electrode Fermi level for facile electron transfer to occur in a radiationless process (based on Ref. 3 and later representations by Gerischer).

for the redox process Ox + e ~ Red; flG s in Eq. (3.2) is positive for a decrease of net charge. When the result of electron transfer is the production of an intermediate chemisorbed with energy A (A being negative), condition (3.1) becomes9

l/> - eV - 1- flG s + A == 0

(3.3)

39

Energetics and Elements of the Kinetics of Electrode Processes

if the charge transfer is to a cation [e.g., H30+ in the H2 evolution reaction or for underpotential deposition of H from H30+ or H 20] where A is the chemisorption energy of H at the electrode metal in the H2 evolution reaction or in UPD below the H2 reversible potential. The latter case corresponds to the development of a potential-dependent coverage by adsorbed H, leading to an adsorption pseudocapacitance (see Chapter 10). For cathodic H2 evolution, the electrodeposited H is an overpotentially deposited (OPD) adsorbed intermediate in the H2 evolution reaction. Changes of A from one metal to another for a given process (e.g., the HER) provide the principal basis for dependence of the kinetics of the electrode process on the metal. They are recognized as the origin of the electrocatalysis associated with a reaction in which the first step is electron transfer with the formation of an adsorbed intermediate. 9 In the case of the HER, this effect is manifested in the dependence of the log of the exchange current density, ia> on metal properties 12,13 such as (/J (Fig. 3.2). However, for reasons peculiar to electrochemistry, reaction rate constants cannot depend on (/J under conditions of currents experimentally measured at controlled potentials (and referred to the potential of some reference electrode) since (/J quantities cancel out around the interfaces of the measuring circuit, as explained earlier. Hence relations such as those in Fig. 3.2 must arise from some other factor, as discussed in Refs. 13 and

8 N

------=-

•••

Ti

.Nb

IE

u

Cu __;'w

.Mo

,TO

.Au

AQ,W-eF AQ e"~ Ni

0

CI'

Bi

Mo

LOG [CURRENT DENSITY] FIGURE 3.8. Schematic Tafel relations for alternative rate-controlling stages and chemisorption conditions (OH« I or OH ~ I) in the HER: (a) rate-controlling H+ discharge; (b) rate-controlling H desorption by a second H+ discharge; (c) rate-controlling H desorption by catalytic H + H recombination (Tafel's mechanism).

where the first term on the rhs ofEq. (3.24) is the logarithmic derivative ofthe adsorption isotherm, O(V) (e.g., for H in the H2 evolution reaction), and the second term arises from the potential dependence of the electron charge transfer rate through the barrier symmetry factor (fJ =:: 0.5) or electrochemical Brjijnsted factor. 16 Special cases arise when d In OldV =FIRT or d In OldV =0 (0 ~ 1). In the case of underpotential deposition (e.g. of H), no continuous Faradaic currents pass (i.e., fJFIRT = 0 in Eq. 3.24), and adsorption pseudocapacitance arises because 0 (e.g., 0H) is a thermodynamic function of V over some defined potential range, about 0.05 to 0.35 V, depending on the substrate metal, which is usually a noble one.

3.S. DOUBLE-LAYER EFFECTS IN ELECTRODE KINETICS In the kinetic equation (Eq. 3.9) for i, the term for concentration ofthe reacting species (for example, H+ discharge) was written as CH,+s' the concentration of the reacting proton (or some other ionic reactive species). In terms of the structure of the double layer at the electrode interface (Chapter 6), the local concentration of reacting cation (here aqueous H+) has to be expressed in terms of the bulk concentration of that species CH,+b' From the theory of ion distribution in the double layer (Chapter 6), the local concentration at the outer Helmholtz plane is given by

52

Chapter 3

(3.25) where IfIl is the potential at the inner limit of the diffuse double layer (i.e., at the Helmholtz compact layer) relative to the potential in the bulk of the solution in the absence of net ionic currents that could lead to an ohmic iR drop. This is the potential that is derived from the theory of the diffuse double layer according to Gouy and to Chapman (Chapter 6, references 7 and 9, respectively). (In the case where the ions are significantly chemisorbed [usually anions at positively charged electrode surfaces or near the potential of zero charge, pzc, at negatively charged surfaces as well], the potentiallfll is appreciably modified.) In the kinetic equation (Eq. 3.9), CH,+s must be substituted according to Eq. (3.25). In addition, in Eq. 3.7, when double-layer effects are to be properly included in the rate equation for i, account must be taken of the fact that it is the local interfacial potential difference V - 1fIl> rather than V (or 11), that has to be written in the rate equation. When these two effects are included (in the absence of specific chemisorption of the reacting ions), the full electrode kinetic rate equation for the example of proton discharge becomes from Eq. (3.7):

. exp -[P(V - IfIl)FIRT]

(3.26)

In dilute solutions, the theory for ion distribution in the region near and up to the compact Helmholtz layer of the double layer formally gives an approximate solution for the ionic concentration dependence of IfIl for dilute solutions (ionic strength < ca. 0.1 M) as: (3.27) Note that for a cathodic process (i.e., one involving discharge of cations), IfIl is a negative potential (relative to the bulk solution potential) that becomes numerically smaller with increasing ionic strength. The effective thickness of the diffuse double layer (the Debye length) becomes smaller in proportion, approximately, to the log of the ionic strength at relatively low ionic concentrations. The basis of double-layer effects in electrode kinetics was thoroughly worked out by Frumkin and his school in the 1930s and by Gierst in more specialized ways in the 1950s and 1960s (see Refs. 11 and 20). For the case of specific (chemisorbed) ions in the double layer, usually polarizable and electron-donative anions, the double-layer effects cannot be treated in a generalized way, i.e., without individual evaluations of their adsorption behavior and their effects on the local potential difference across the compact Helmholtz

Energetics and Elements of the Kinetics of Electrode Processes

53

layer region of the electrode/solution interphase. * A first attempt to include such specific adsorption effects in the theory of the double layer was, however, made in 1924 by Stern and later by Grahame, as described in Chapter 6. The original theories of Gouy and of Chapman were nonspecific in this regard since they treated only the diffuse part of the double layer. The specificity of ion adsorption effects, especially those involving anions, is especially manifested in the double-layer capacitance behavior at electrode interfaces as a function of electrode potential. Examples based on the results of Grahame 21 will be shown in Chapter 6 and are important in determining electrochemical capacitor behavior in the case of the doublelayer types of capacitor device in aqueous or nonaqueous media, using higharea powdered or filamentous (fibrous) carbon preparations. It is to be understood that this is an elementary and brief account of the energetics and kinetics of electrode processes, such as may be useful as a basis for understanding the principles that may govern the involvement of Faradaic electrode processes in electrochemical capacitor behavior. This applies particularly for cases where overcharge, overdischarge, self-discharge, and Faradaic pseudocapacitance charging processes arise in this field. General source references on electrode kinetics are listed separately later.

3.9. ELECTRICAL RESPONSE FUNCTIONS CHARACTERIZING CAPACITATIVE BEHAVIOR OF ELECTRODES

In the field of electrochemical capacitor development and testing, and on the fundamental side of the subject, it is usually necessary to be able to quantitatively evaluate capacitance and its dependence on various experimental variables by direct instrumental measurements. All electrode interfaces with solutions or solid electrolytes exhibit a double-layer capacitance, as treated in Chapters 6 and 7, and in addition some exhibit a pseudocapacitance as treated in Chapters 10 and 11. Several simple and elementary criteria for characterizing the electrical response of an electroactive material as behaving capacitatively are as follows: 1. In an interfacial charging process at constant current density, i, the potential difference, ~ V, developed across the capacitor plates changes linearly with time as the charge supplied by i builds up across the interface, i.e. C=~q/~V

(3.28)

*The term interphase is distinguished from interface in order to recognize the 3-dimensional nature of the former as a region of finite thickness residing on the 2-dimensional interface defined by the boundary of the electrode surface with the solution.

Chapter 3

54

and Aq

c=

=f i·dt

f i· dtlAV= i· AtlAV

(3.29)

(3.30)

over some time interval At. Equation (3.30), as written, applies if the capacitance, C, is constant with potential. Often experimentally it is not, so AV deviates from a linear dependence on time at constant current. C is thus obtained as the reciprocal of the slope of the relation between AV and At (Fig. 3.9), the time elapsed to some point on the AV vs. At curve or differentially, at some point on it, as (dAV I dtr l when C is not constant with AV. The measurable dependence of AVon time at a constant current (i.e., on charge passed) is commonly referred to as the "charging curve." This applies to a so-called "ideally polarizable" electrode (see Chapter 6) where the current i simply passes charge into the interface without any Faradaic processes of transfer of charge across the double layer taking place, leading to chemical change at the electrode surface. As long as that condition obtains, the electrode interface remains ideally polarizable and i is purely a double-layer charging current, i dl . However, in practice, the charging current may be maintained in a voltage range across the interface where Faradaic decomposition of the solution begins to take place at that electrode (and then also at the counter electrode). The dou-

, ••- - - Onset of non-idealpolarizability. iF>Q

lIe

a TIME (AT CONSTANT i)

FIGURE 3.9. Constant current charging curve for an electrode interface having a double-layer capacitance C.

Energetics and Elements of the Kinetics of Electrode Processes

55

ble layer continues to be increasingly charged with the rise of Ll V, but the current i then becomes divided into two components, i d1 and iF, where iF is the Faradaic current, which increases exponentially with Ll V when Ll V exceeds a value corresponding to a thermodynamic reversible potential for solution decomposition (e.g., H2 evolution) at that electrode. This increase of iF follows a Tafel relation in the overpotential, Yf, involved, as discussed earlier and represented by Eq. (3.13). Then the charging situation can be written as (3.31)

i = C(dVldt) + iF

where C(dVldt) = i d1 and iF has the form i F = io exp [aYfFIR1J, which is the Tafel equation in exponential form (Eq. 3.7), with a the transfer coefficient. The charging curve then has the form shown in Fig. 3.9 (for constant C) and as Ll V increases beyond its value corresponding to solution decomposition, the Ll V vs. Lll relation increasingly deviates from linearity as the iF component of i becomes progressively larger, i.e., a greater leakage current across the double layer passes until the Faradaic process is the main charge-carrying component, iF» i d1 • Then the electrode is behaving in a nonideally polarizable way. The situation of transition from double-layer charging to mainly Faradaic decomposition of the electrolyte as the potential is raised is illustrated in Fig. 3.9. Region ab is for pure double-layer charging; region bc includes an increasing component of the passage of charge in a parallel Faradaic reaction. ab is the region of ideal polarizability of the interface, prior to decomposition of the solution over region bc. Figure 3.10 shows schematically the relative values of the

LL

o

~

UJ Z

oc... o== o

l

onset of non-ideal polarizability, iF 0

---:....-

T

!Z UJ

Constant i = idl+ iF

a:: a::

;:)

o

POTENTIAL FIGURE 3.10. Components of the constant charging current of Fig. 3.9 at an electrode interface when a Faradaic partial current, iF, passes in parallel with the double-layer charging current, idl.

56

Chapter 3

components i d1 and iF of the total (constant) i as a function of increasing potential as the interphase becomes increasingly nonideally polarizable, corresponding to decomposition of the solution. Quite generally, a charging current is C dVldt (since C dV == dq) where dV/dt is the rate of change in potential with time in the absence of any time effects (e.g., deactivation) in the kinetics of a Faradaic reaction when iF becomes > 0. In the first ascending region of Fig. 3.9, where iF = 0, dVldt at a constant current gives directly the reciprocal of C. In the region where iF becomes> 0, C dV/dt still gives the non-Faradaic component of i, which can be evaluated if C has remained constant during the charging process with the increase of potential beyond the solution decomposition limit. Eventually, a steady state is reached where iF ---t i as dVldt ---t and then i d1 ---t 0, i.e., the charge transfer is entirely Faradaic at a sufficiently high steady potential or overvoltage for the Faradaic reaction. The double layer can also be charged potentiostatically in a potential step or potentiodynamically in a linear voltage sweep as described later. 2. In a sequence of potential steps, JV, the charge, Jq, flows into the interface to an extent determined by CJV where C is the capacitance over the potential range JV. A current transient it is generated (Fig. 3.11) by the pulse JV over a small but finite response time and Jq is the integral of it over that time interval. The capacitance can be calculated from the recorded Jq response for the pulse JV, or the sum of Jqs over a sum of sequential JVs. In electroanalysis, this procedure is known as chronoamperometry. As in the first case of constant-current charging (chronopotentiometry), as potential is increased toward and through the solution decomposition limit or if a large-amplitude potential step, AV, is applied, the transient current response it contains transient components of doublelayer charging current and of Faradaic current. The response time, Jr, in which the transient current it passes, is ideally very short but in practice it can be tens of microseconds or some milliseconds, depending on the impedance characteristics of the cell and the measurement system. The time integral of the current-response transient, itr, in Fig. 3.11 gives the charge that has flowed into the electrode interface as a result of the application of the potential step, JV or AV. If the potential step covers some range of potential where a Faradaic current, iF, also flows (upper curve in Fig. 3.11), then the charge passed by this parallel current component must be allowed for in calculating the charge entering the double-layer capacitance. In some types of electrode-process measurements, a double-pulse procedure developed by Gerischer is employed, first to charge up the double layer quickly and then to observe the kinetic response of an electrode reaction to a second, subsequent pulse. 3. In a so-called linear potential-sweep (potentiodynamic) experiment, the applied potential (measured with respect to a reference electrode) is varied line-

°

Energetics and Elements of the Kinetics of Electrode Processes

57

Potential- step

I •

w (J) z oD..

V2 ..J

W

§

tzw

oD..

(J)

a:: a:: a::

;:)

o

I:!!

v,

"" , ,,

iF>O "-

',iF"O ......

o

TIME (order of !IS or ms)

FIGURE 3.11. Current-response transient for charging an electrode interface under the influence of a step of potential from VI to V2.

arly with time in a three-electrode cell and the resulting time-dependent response current (Fig. 3.12) is registered by an analog (X-Y recorder or oscilloscope) or digital (transient recorder, computer, or digital oscilloscope) recording instrument. In such a voltage sweep experiment at a sweep rate s = dVldt (= constant), the capacitative charging current, i, is i

= C (dVldt) = Cs

(3.32)

or (3.33) C=ils In a sweep-reversal experiment (cyclic voltammetry) at constant is, the current response profile is ideally a rectangle along the time-potential axis when C is constant (Fig. 3.12, line a) or if it is not, then the differential profile of C is generated, often with a peaked structure (line b in Fig. 3.12). 4. In a self-discharge experiment through a load resistance, R, the time dependence of potential is (see Chapter 18) Vet) = Va exp [-tIRC]

(3.34)

from which C can be evaluated; and 5. In an ac impedance experiment (Chapters 16 and 17), the (imaginary) component of impedance is

Z'= lIjwC

(3.35)

58

Chapter 3

a

.

VI

u

.!!..

H

u

:.0 0

t-" Z

c ct

w

II·exs) ~

i

I i

I I

i

t

+ i

a:: a::

::J U

w

I I

0

0

w

a::

__

I I

Z

a..

VI

I

CJ)

CJ)

i

I

u

:c0

•

.&;.

"0

--

IV2

-......

--

.-_

....

Sweep - signal

vO) _ -

... ..... -"

U

-!

oil(

< .................

..

.~

....

ct

VI

U II

'C

I

POTENTIAL,V I time) I

t

>1

I I

I

I

1.::EILl

l~ '0

1% ,0

t~ ,ILl jl3 ,0

tg:

b

0

C

~

w

a:: a:: => U

0

w

V>

POTENTIAL, V ( time)

Z

0

a..

V>

w

a::

u

:c0 .&;.

0

u

FIGURE 3.12. Cyclic voltammogram for an interface having a potential-independent capacitance (C = ils) (curve a) and potential-dependent capacitance (curve b),

and ideally the real component Z' is infinity for an ideally polarizable electrode, Most capacitor systems, especially those of the electrochemical type, have a complex R-C equivalent circuit of distributed Rand C due to porosity (see Chapter 14) and/or complex electrochemical reactions, and hence do not have a simple impedance behavior as a function of frequency, In some cases, the im-

Energetics and Elements of the Kinetics of Electrode Processes

59

pedance behaves like that of a transmission line having a 45° phase angle over a wide frequency range instead of 90° for ideal capacitance. Finite or significant R limits power output performance as discussed in Chapter 15.

3.10. INSTRUMENTS AND CELLS FOR ELECTROCHEMICAL CHARACTERIZATION OF CAPACITOR BEHAVIOR

3.10.1. Cells and Reference Electrodes

Angerstein-Kozlowska22 has given a detailed account of various experimental procedures for electrochemical experimentation, including the design of cells and preparation of solutions, especially techniques for measurements in very pure solutions and on clean electrode surfaces. Almost all packaged electrochemical capacitors or batteries are sealed two-electrode (or a stacked series of two-electrode-) systems. However, electrochemical test and evaluation information is often required on individual electrodes (the positive or the negative one) of the device. In this case, a third, unpolarized, electrode is required as a reference probe. Provision for such measurements must then be made by using a three-electrode or three-compartment cell of the type shown in Fig. 3.13. Alternatively, a third (reference) electrode probe can be inserted between two working plate-type electrodes in a cell of the type shown in Fig. 3.14. Then, during tests, the potentials of each working electrode on charge, discharge, or open circuit (for self-discharge measurements) can be separately recorded against the potential of an appropriately compatible reference electrode. Some commonly available reference electrodes are as follows: The Hz-H+-H20 or Hz-OH--H20 electrode at platinized Pt. The PdH electrode at Pd containing sorbed H. The dynamic hydrogen electrode where H2 is generated in situ at low current density at a small platinized Pt electrode. Hg-Hg2Clz-CI-, reversible to Cl- ion. Hg-Hg2S04-S0~-, reversible to SO~- ion. Hg-HgO-OH-, reversible to OW ion at pH > 9. Ag-AgX-X-, reversible to halide, X-, ions. Ag-AgOH film-OH-, reversible to OW ion. Fe-FeF2 film or Cu-CuF2 film for P- ion solution. Ag-Ag+-AgCI0 4 in nonaqueous solvents. Glass electrode, sensitive to pH in aq. solutions. Ion-selective electrode (Orion Co.) sensitive to various specific ions in solution.

60

Chapter 3

FIGURE 3.13. Three-compartment electrochemical cell for evaluation of electrode behavior against a reference electrode.

Energetics and Elements of the Kinetics of Electrode Processes

61

Ref.

±

-

iii

I.

1:1

I-

~

Plate-type cell in plastic frame (Schematic)

I

I I I

J)

I I

Ii

"I I

~ ~

~

I

1111

~separator

FIGURE 3.14. Plate-type test cell with arrangement allowing insertion of a reference electrode probe.

When a reference electrode is used, potential measurements or other connections must be made into or through a high input impedance (ca. 106 ohms) recording device in order to minimize current drain from the reference electrode to below several microamperes. This ensures that the reference electrode maintains its proper reference potential to within a millivolt, i.e., it does not itself become polarized on account of its inclusion in the measurement circuit. A typical three-electrode measurement circuit is shown in Fig. 3.15.

3.10.2. Instruments

The various electrochemical instrumental procedures available for studying the behavior of capacitor devices and batteries, and electrode processes in general, are well treated in the monographs by Delahay (General Reading Ref. 1) and by Bard and Faulkner (General Reading Ref. 6). Basically, for the evaluation of electrical response functions of electrochemical capacitors, the following equipment is required: 1. A potentiostat [e.g., EG and G (PAR), Hokuto or Wenking] that is capable of addressing and controlling cells up to 5 V. Some higher voltage options

62

Chapter 3

FUNCTION - GENERATOR (5)

POTENTIOSTAT R

C~

W

~

-

;;;;,CELl I

-'V"",.N'y

(

,

\ C I ',/ '\ \ I

)

x- y

1-0 RECORDER '-9

~

\., //--'" / "

I

~

{, W ,

,,- ..... 'R'

,

f

.I

-

-

POTENTIOMETER r--

FIGURE 3.1S. Operating electric circuit for potentiostatic or potentiodynarnic measurements with a function generator using a three-electrode system.

are available. Driving output voltages up to SO V are sometimes required. For more ambitious testing of larger devices, higher power potentiostats are necessary. 2. A function generator capable of generating single-step, single-ramp, and repeated step and ramp potential forms is needed to drive the potentiostat for transient measurements and cyclic voltammetry. Most function generators also provide sinusoidal output signals at various frequencies and amplitudes. 3. A controlled current generator for dc charging and discharging experiments. Computer-controlled, multichannel systems are now available (e.g., from Mackor or Arbin) with the capability of repetitively recording the charge and discharge curves of potential with state of charge. 4. A digital oscilloscope with diskette recording, e.g., from Nicolet Corp. S. Computer interfacing and appropriate software for data acquisition and processing. Electrochemical experiments provide ideal opportunities for computer control of the instruments themselves, and for recording and processing the resulting data. Potential and current are the conjugate variables of electrode processes, coupled with charge, the integral of the current passed during a controlled time interval (also digitally controllable down to the microsecond level or less). These variables can be precisely controlled and recorded in a digital

Energetics and Elements of the Kinetics of Electrode Processes

63

manner. A variety of software is commercially available and "in lab" personal software is not difficult to design and generate. 6. A system for determining the frequency response of the capacitor device over a wide range of frequencies from 0.001 or 0.01 Hz up to 100 kHz, or sometimes more. This enables the real (ohmic) and imaginary (capacitative) components of the impedance of the system to be determined over the frequency range scanned (see Chapters 16 and 17). This is a widely used procedure and is essential for full evaluation of capacitor performance and the behavior of capacitor devices in the various circuits in which they are utilized. Excellent instruments are now available (e.g., from Schulumberger Solartron or EG and G), together with the required software for complex-plane impedance and phase-angle plots. 7. A coulometer; this is useful for automatically recording the charge passed into or out from an electrochemical energy storage system over a given recorded time interval. Instruments giving digital or analog outputs are available. Coulometric measurements are specially relevant to evaluating the performance of electrochemical capacitors, coupled with recordings of cell voltages (see item 3). 8. A calorimeter or thermocouple recording system may also be a useful item of instrumentation. Especially for large capacitor devices, the problem of heat management becomes a significant factor in practical operation of capacitors at large current densities. This becomes of greater significance when scaleup of small devices to larger ones is under consideration. 9. Other equipment: various more specialized equipment is required for some of the more fundamental research operations in the field. For example, Gottesfeld (see Chapter 12) has profitably used the techniques of ellipsometry (see Chapter 12) and quartz microbalance gravimetry (to nanogram sensitivities) to examine the development and growth of conducting polymer films on substrates for electrochemical capacitor fabrication. Also, surface spectroscopy (e.g., at C) has been useful. 10. Of course, other regular minor equipment such as portable multirange digital ammeters and voltmeters is a daily requirement in an electrochemical testing laboratory. In the case of fabrication of nonaqueous double-layer capacitors, a good dry box (Vacuum Atmospheres Corp.) is a necessity, or better, on a larger scale, a dry room.

3.10.3. Two-Electrode Device Measurements

Most packaged electrochemical capacitor devices are two-terminal systems, so no reference electrode can be used as a third potential probe without making an opening in the device's case. Testing is therefore conducted mainly by recording charge and discharge curves at controlled currents at various tem-

64

Chapter 3

peratures and by potentiometric recording of open-circuit potentials or float currents (Chapter 18) when self-discharge rates are being evaluated. Under these conditions, of course, an overall evaluation of the two-electrode system is obtained so that information on the behavior of each electrode interface is not available from such measurements. However, in practice, useful performance information on the charge and discharge capacity, energy density, and power density of capacitor devices is easily obtainable by means of two-terminal measurements, but is less informative in a fundamental direction than data obtained at individual electrodes in three-electrode cells.

REFERENCES 1. L. Pearce Williams, Michnel Faraday: A Biography, Chapman and Hall, London (1965). 2. B. E. Conway, in Electrochemistry, Past and Present, Chapter II, ACS Symposium Series 390, American Chemical Society, Washington, D.C. (1989). 3. 1. Weiss, Proc. Roy. Soc., London A222, 128 (1954). 4. R. A. Marcus, 1. Chem. Phys., 24, 966 (1956). 5. R. A. Marcus, Ann. Rev. Phys. Chem., 15, 15 (1964). 6. R. Parsons, and 1. O'M. Bockris, Trans. Faraday Soc., 47, 914 (1951). 7. R. W. Gurney, Proc. Roy. Soc., London, AI34, 137 (1931); A138, 378 (1932). 8. F. P. Bowden, Trans. Faraday Soc., 28, 368 (1932). 9. 1. A. V. Butler, Proc. Roy. Soc., London, A157, 423 (1936). 10. B. E. Conway, Ionic Hydration in Chemistry and Biophysics, Chapter 13, p. 260, Elsevier, Amsterdam (1981). 11. B. E. Conway, Theory and Principles ofElectrode Processes, Ronald Press, New York (1964). 12. R. Parsons, Trans. Faraday Soc., 341053 (1958). 13. B. E. Conway and 1. O'M. Bockris, 1. Chem. Phys., 26, 532 (1957). 14. A. A. Balandin, Z. Phys. Chem., B2, 289 (1929). 15. S. Trasatti, 1. Electroanal. Chem., 39, 183 (1977). 16. 1. W. Schultze and F. D. Koppitz, Electrochim. Acta, 21, 327, 337 (1976). 17. B. E. Conway, in Progress in Reaction Kinetics, G. Porter, ed., vol. 4, Chapter 10, Pergamon, Oxford (1967). 18. B. Losev, quoted by L. Antropov, Theoretical Electrochemistry, p. 395, Mir Pub!., Moscow, 1972. 19. K. 1. Vetter, Electrochemical Kinetics, Springer-Verlag, Berlin (1965). 20. L. Gierst, in Electrochemical Society Symposium on Electrode Processes, E. Yeager and P. Delahay, eds., Electrochemical Society, Pennington, N.J. (1961). 21. D. C. Grahame, Chem. Rev., 41, 441 (1947). 22. H. Angerstein-Kozlowska, in Comprehensive Treatise of Electrochemistry, E. Yeager, J. O'M. Bockris, B. E. Conway, and S. Sarangapani, eds., vol. 9, Chapter 2, Plenum, New York (1984).

GENERAL READING REFERENCES 1. P. Delahay, New Instrumental Methods in Electrochemistry, Wiley-Interscience, New York (1954).

Energetics and Elements of the Kinetics of Electrode Processes 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

65

P. Delahay, Double-Layer and Electrode Kinetics, Interscience, New York (1965). K. 1. Vetter, Electrochemical Kinetics, Springer-Verlag, Berlin (1965). B. E. Conway, Theory and Principles of Electrode Kinetics, Ronald Press, New York (1964). 1. O'M. Bockris and A. K. N. Reddy, Modem Electrochemistry, vols. 1 and 2, Plenum, New York (1970); Second edition, Plenum, New York (1998). A. J. Bard and L. R. Faulkner, Electrochemical Methods, Wiley, New York (1980). B. E. Conway, Ionic Hydration in Chemistry and Biophysics, Elsevier, Amsterdam (1981). K. B. Oldham and 1. E. Myland, Fundamentals of Electrochemical Science, Academic Press, New York (1994). J. O'M. Bockris, in Modem Aspects of Electrochemistry, 1. O'M. Bockris, ed., vol. 1, Chapter 4, Butterworths, London (1954). W. Schmickler, Interfacial Electrochemistry, Springer-Verlag, Berlin (1993). J. Koryta, J. Dvorak, and J. Bohackova, Electrochemistry, Methuen, London (1972). L. Antropov, Theoretical Electrochemistry, Mir Publ., Moscow (1975). E. Gileadi, Electrode Kinetics for Chemists, Chemical Engineers and Materials Scientists, VCH Publishers, New York (1993). E. Gileadi, E. Kirowa-Eisner, and J. Penciner, Interfacial Electrochemistry, an Experimental Approach, Addison-Wesley, Reading, Mass. (1975). Southampton Electrochemistry Group, Instrumental Methods in Electrochemistry, Ellis Horwood, Chichester, UK (1985).

Chapter 4

Elements of Electrostatics Involved in Treatment of Double Layers and Ions at Capacitor Electrode Interphases

4.1. INTRODUCTION

The electrochemistry of double layers, and the ions and solvent molecules constituting them, involves the electrostatic energies and molecular or ionic distributions of these species in high interphasial fields. At charged electrode interphases in double layers, the electric fields can become as high as 107 V cm- 1, and similarly in the solvation shells of ions. The electrostatics of such interphases are concerned with: 1. the energies of individual charges and molecular electric dipoles, and of their interactions; 2. the motions of individual charges (ions) in fields; 3. the configuration of groups of charges on ionized complex molecules (e.g., ionic centers arising at conducting polymers) currently being developed as pseudocapacitors; 4. the orientational movement of electric dipoles in homogeneous fields; 5. the translational motion of dipoles in inhomogeneous fields; 6. the interaction of solvent dipoles with electric fields, and quadrupoles with field gradients; and 7. the effective local dielectric coefficient in the double-layer interphase. 67

68

Chapter 4

Magnetic interactions are much weaker and are only important in spectroscopy, in some solids, and in transition metal ion properties. When a charged electrode interface is in contact with an ionic solution, there is an accumulation of ions of one sign or the other, forming a double layer (Chapter 6) and causing an orientation of dipoles of the solvent. Since ions in the double layer are normally situated in a liquid solvent in contact with the electrodes, the electric polarization of the medium will also be involved in determining the properties of the charged particles at or near an electrode interface. In order to provide a basis for discussing the details of such questions in the theory of the double layer and electrochemical capacitors, we present in this chapter some elementary principles of electrostatics.

4.2. ELECTROSTATIC PRINCIPLES 4.2.1. Coulomb's Law: Electric Potential and Field, and the Significance of the Dielectric Constant

Perhaps the two most basic laws in physics and chemistry that determine the short-range properties of substances are Coulomb's law of electrostatic force, F, between two charges q1 and qz, distant r from one another, and the corresponding law of interaction that determines the force between two magnetic dipoles. A third basic relation is the Pauli exclusion principle, which places restrictions on quantized energy states in assemblies of electrons in atoms or molecules and is hence fundamental in defining interatomic forces arising from electronic interactions in molecules. Analogous effects occur in metals, giving rise to their band structure. Coulomb's law has been experimentally verified as an inverse square law like that for gravitation. The general relation for the force F between two charges in a vacuum, is given by (4.1)

where F is an attractive force when q1 and qz are of opposite sign and a repulsive one when they are of like sign. The constant k depends on the units in which the charges and distance are expressed and the force F is also dependent on the electrical properties of the medium in which the charges q1 and qz are situated when it is not a vacuum. 4.2.1.1. Units

In the electrostatic system of units, the unit charge is defined by Coulomb's law as that charge which repels a like charge I cm distant from it in vacuo with

Treatment of Double Layers and Ions

69

a force of 1 dyne or equivalently in newtons. The value of the proportionality constant k in Eq. (4.1) in this system of units is simply 1. The electronic charge that arises in many calculations in the electrochemistry of double layers and ionic solutions has a value of 4.80 x 10- 10 esu in this system of units, or the Faraday constant in coulombs per gram-equivalent divided by Avogadro's number in the electromagnetic units system, giving the results in coulombs as 1.6021 X 10- 19 • In the so-called "rationalized" rnksa (meter, kilogram, second, ampere) system of units, k has the value 8.9876 X 109 N mZC-z, i.e., in newtons meterZ per coulombz where F, the force, is measured in newtons, r is in meters, and ql and q2 are in coulombs. In this system of units, the constant k is written

k = 1I4nKo

(4.2)

where KO is the permittivity of the vacuum. The rationalized system is introduced so that certain fundamental relations in electrostatics, such as Gauss's and Poisson's equations, can be written without the 4n factor which otherwise enters into them. The value of KO is

KO = 8.85435

X

10-12 C2N-I m-Z or farads m- I

(4.3)

In some texts the symbol eo is used for the permittivity of free space, but the symbol Ko is preferred here in order to avoid confusion with the static (zero frequency) dielectric constant of a medium, also usually denoted by eo or eO. KO is also sometimes referred to as the diabattivity of free space. Electrostatic calculations in the rationalized units system thus depend on forces calculated from Fr=QIQzI4nKoe,z or Ur=QIQ2/4nKoBr from corresponding energies. The charges are written QI, Qz here in coulombs to distinguish them from q], q2 in Eq. (4.1), where the units are to be taken in esu.1t is often, however, simpler to perform such calculations in the esu system with e then taken simply as the dielectric constant (see later discussion) of the medium; then forces appear as dynes and energies as ergs per particle. Conversion to joules mol- I is easily achieved by multiplying by NAil 07 , where NA is Avogadro's number, noting that 107 ergs = 1 joule. Itis obvious that the two systems must give the same results with the appropriate interconversion of units. Thus a unit charge in coulombs is 3 x 109 times that in esu, or 1 C = 3 X 109 esu. The conversion factor 3 x 109 (actually more exactly 2.998 x J09) originates from the speed oflight in a vacuum, 2.998 x 1010 cm S-I, and is 1110 of this figure, corresponding to the coulomb being a practical unit that is 1110 of the value of the absolute electromagnetic unit of charge, which is itself defined in terms of the magnetic field produced in a circular coil of one turn having a radius of 1 cm, when a unit charge per second (current) is flowing.

70

Chapter 4

A comparison of calculations in the two systems of units for the energy of two like unit charges 1 cm apart enables KO to be identified. In the esu system, for example, for ql = q2 = 1 esu in vacuo (s = 1): (4.4)

In the emu system, using SI units, ql = q2 = (1(esu»/(10c(ms- 1» = (1 erg)/(3 x 109) coulomb, where c is the velocity oflight (2.998 x 108 m s-I). The energy Ur=lem must be the same in either system of units, but with r taken in meters and ql or q2 in coulombs in the SI system. Hence

= (113 x 1092)2 J (r = 10-2 m 4nKo10-

q

'1

= q = 113 X 2

109 coulombs) (4.5)

Therefore, equating the result in joules from the two systems, it is seen that Ur=1 em

= 10-7 J =10-16 /9 X 4nKo J

(4.6)

or KO

10-16

=-10- 7 /9 x 4n = 8.85435 x 10- 12 farad m- 1

(4.7)

as mentioned earlier in this chapter (Eq. 4.3). The significance of Ko also follows from the important definition of capacitance (see Section 4.4) since C =(As)/(4nd) where d is the thickness of the dielectric medium (dielectric constant s) between two parallel plates of area A. In order for C to be correctly evaluated in terms of A and d in the emu system (C in farads or coulombs per volt), the relation must be written C = ASKOld with KO = 8.85435 X 10- 12 farads m- 1. 4.2.1.2. Dielectric Constant

The dielectric constant, s, of a medium is measured by the capacitance C of a condenser containing the medium, relative to that of the same condenser in vacuo. As defined earlier, the capacity of the condenser in vacuo is sof4nd per unit area of the plates separated by a distance d. In a material medium, the capacitance is ASKOld per unit area where KO is the permittivity of the medium. The dielectric constant S is defined as the relative permittivity, KIKo, and hence has the value 1 for free space and is a pure number for any material medium. The physical significance of S in terms of the

71

Treatment of Double Layers and Ions

electrical properties of molecules of the medium is treated later. In terms of Eq. (4.1), it is seen that the quantity e characterizes the attenuation of the coulomb force of interaction between two (or more) charges by the material medium in which they are separated.

4.2.1.3. Electrostatic Potential, Field, and Force

For a charge Q, the electric potential ¢J at a distance d from the charge is defined as the work done to bring a unit charge from (charge-free) infinity to the point at distance d. At any distance x from the charge Q, the unit charge experiences an electrostatic force given by

qx1

(4.8)

-F(x)=-eX2

If the unit charge is progressively moved from against this force so that the work or energy

00

V(X)

to d, increasing work is done expended is given by

(4.9)

This is defined as the potential ¢J due to q at distance d; thus ¢J(d)

=q / ed

or, in general,

¢J(x)

=q / eX

(4.10)

The corresponding electric field E is defined as the gradient of the potential. For the above case,

E = d¢J/dx = -q/ex2

(4.11)

Hence field varies as the inverse square of a distance x from a charge while potential varies as the inverse first power of this distance (Fig. 4.1). From the form of Coulomb's law, it is evident that the force on a charge q' in the field of q is given by

F=_lxq'

(4.12)

eX2

i.e., force is the product of the charge multiplied by the field due to another (or other) charge(s). This formula provides the basis for the definition of the unit charge given earlier.

72

Chapter 4

Potential

cp

Distance x

FIGURE 4.1. Relation between electric potential and electric field as functions of distance, x, from an ionic charge (schematic).

4.2.1.4. Potential ¢J and Field E at an Ion The field is a vector quantity since it depends on the direction of the gradient of the potential ifJ. The potential, on the other hand, is simply a scalar quantity having no direction. It can, however, have positive or negative magnitude, depending on the sign of q generating the potential ifJ. Potentials generated at a given point from a distribution of charges can be algebraically added [Fig. 4.2(a)], but fields at a given point originating from various charges must be vectorically summed, i.e., with due consideration of their directions or their components in given directions [Fig. 4.2(b)]. Strictly speaking, the dielectric constant e may also depend on the direction of the field if the medium concerned is electrically anisotropic, Le., in the case of certain crystals or liquid-crystals, or organic molecules that are birefringent. The resultant field vector can usually be easily constructed for a 2-dimensional array of charges [Fig. 4.2(b)]. For the general case, 2 or 3-dimensional trigonometry is required. For orthogonal vectors, the results are easily calculated by means of Pythagoras's theorem. More generally, the field may be expressed in terms of the unit vector r where E = qrlu?, and the result in terms of the vector sum or, within limits, the vector integral. The field calculation discussed here enters into the calculation of forces and fields caused by assemblies of oriented dipoles in liquids around ions or charge arrays on polyions or surfaces.

Treatment of Double Layers and Ions

73

CPA =CPI + 4>2 T 4>3

4>A

_...s..+ _q_ - Ed, Ed 2

(0 ) -qEd 3

(b)

q2

Field Resultant

E,.z.~ =E1 + EZ+IE3

FIGURE 4.2. (a) Scalar summation of potentials due to a distribution of charges at various distances dl. d2. d3 from point A. (b) Vectorial summation of fields to derive resultant field at a given point due to the distribution of charges shown in (a).

4.3. LINES OF FORCE AND FIELD INTENSITY-A THEOREM

Field intensity gives a measure of the so-called "force" associated with the action of the field on a unit charge. One "line" per square centimeter is defined as existing in classical electrostatics for each dyne per esu of field strength. Lines of forces correspond to the tangential force experienced by a charge in the field at the particular point concerned. Density of lines of force corresponds to the flux and electrification. The concept of lines of force originated with Faraday. The force at ron a unit charge due to q is q/r2 (8 = 1), i.e., the field intensity. This field can be represented by drawing q/r2lines of force per square centime-

74

Chapter 4

ter at r. The total of such lines at a distance is the line density multiplied by the total cross-section at r, i.e.,

(;zz )x (4nr2) =4nq

(4.13)

i.e., a unit charge has 4 n lines offorce associated with it.

4.4. CAPACITY OF A CONDENSER OR CAPACITOR

The capacity C is defined as C =q/!l¢ or, differentially, C(d) =dq/d¢. The surface charge density, (J, on the plates of the capacitor of area A can be defined as ±(J = ±q/A and the field E is E = !l¢/d where d is the distance between the plates, i.e., C =A(JlEd. Now, from the above theorem, the charge (J is associated with 4n lines of force and this is the field E (= 4n(J) since (J refers to 1 cm- 2. Then A(J

A

Ae

C=--=-or-4n(Jd 4nd 4nd

(4.14)

with a dielectric present. In the rationalized system, C = AeKr/d, as earlier. Equation (4.14) forms the basis for treating other problems (described later) and the experimental determination of dielectric constant. It is also used in determining the electric capacitance of a system of separated plus and minus charges in double layers at interfaces of colloids and electrodes in electrochemical capacitors.

4.5. FIELD DUE TO A SURFACE OF CHARGES: GAUSS'S RELATION

When the charge distribution is a uniform one on a plane (charged plate or electrode), the resultant field takes a simple form. This is important in electrochemistry, for the field due to an electrode corresponds precisely to this case if the fluid adjacent to the plate is assumed to be a uniform dielectric. Similarly, the field due to a colloid interface may be treated in terms of this relation in limiting cases (neglecting discreteness of charge). This theorem relates the field due to charge density on a surface (e.g., a plane metal electrode) to the electric flux through the surface. Consider the normal induction In across an element of surface JS at a distance r from a charge q (Fig. 4.3). It was shown earlier that the total flux of electric induction across a closed surface, s, is 4n times the sum of the charges enclosed by that surface, i.e.

75

Treatment of Double Layers and Ions

FIGURE 4.3. Model for discussion of the basis of Gauss's theorem in terms of normal induction across an element of sectional surface "S through a surface Be at distance r from a charge q.

fIn' ds =4nq

(4.15)

Let q be the enclosed charge and the flux of induction will be In . ds, and this will be I cos e·ds and be equal to the field (q/?-) X (the shaded circle area, BM). However ,the area BM/?- measures the solid angle Je. Then the flux of induction across the area Be is qde. Hence the total flux is

f qde =q f de =-q . 4n

(4.16)

However, this total induction is eE. Hence E= -4nq/e

(4.17)

which is a form of Gauss's theorem. If the plate has two surfaces (i.e., it is not a bulk conductor on one side), the result is E = -2nq/e.

4.6. POISSON'S EQUATION: CHARGES IN A 3-DlMENSIONAL MEDIUM

Gauss's equation is useful for relating thefield caused by a charged surface to the charge density on that surface, as at an electrode. A related expression, Poisson's equation, expresses field gradients in terms of the space charge density, p, existing in a medium. Poisson's equation is of value in dealing with problems in ionic solutions or ion distributions near charged interfaces, e.g., at double layers at electrodes. Normally a net space charge cannot exist without a corresponding field directed to where the space charge is smaller or zero. In the language of vectors in electrostatics, Poisson's equation is expressed as the divergence (div) of the gradient (grad) of ljJ, i.e., how the gradient of ljJ, namely,

76

Chapter 4

the field components d¢Jldx, d¢Jldy, or d¢Jldz, increase or decrease in the various orthogonal directions of the Cartesian coordinate system. In terms of p and the dielectric constant I'. of the medium in which the space charge resides, Poisson's equation is written div

(I'.

grad ¢J) = -4np

(4.18)

In terms of the x, y, Z coordinate system, and for the case where I'. is not dependent on the field components d¢Jldx, d¢Jldy or d¢Jldz, the above equation is written in terms of the partial field derivatives (4.19)

Usually the series of partial second derivatives is represented by the symbol V2, "del squared" (the Laplace operator), so that (4.20) The mathematical operations required to obtain the Poisson equation can be complex, depending on the degree of rigor involved in its derivation, but a simple treatment gives the required result by application of Gauss's theorem (Eq. 4.17) (General Ref. 1).

4.7. THE ENERGY OF A CHARGE

The idea that a charge is associated with an energy is less familiar than the concept of field or potential due to a charge. As with all energy quantities, it is necessary to specify with respect to what reference state the energy is measured. In the case of a charge, the energy originates from the work associated with building up that charge (e.g., q) by adding elements of charge Jq brought from infinity. Hence the "energy of the charge" is the work done in the building up of that charge from an initial value of zero to its final value q, with increments of charge, Jq, transported from reference distance 00 where the potential due to the charge built up at any stage in the charging process is zero. Let Aq be the charge established at a given stage of the charging of the initially uncharged particle, where A is a fraction 0 < A < 1. The electrical work done, oWe, in bringing up the next element of charge oq is (Aq·Jq)la if a is the radius of the particle being charged. The total work done, We' for transfer of all the charges Jq (i.e., from A =0 to A = 1) is given by

Treatment of Double Layers and Ions

77 q

W =f Aq·dq e a o

(4.21)

The integral is evaluated by regarding the fraction A. as being progressively increased from 0 to 1, so that the integration is made with respect to A. rather than q, i.e., because,jq can be written as q.,j A.. Then 1

1

Aq·qdA. n 2 f We= f -a-=~ A·dA. =q2/2a o 0

(4.22)

The same result may also be obtained by considering the charging energy (1I2)C(11¢J)2 of a spherical particle treated as a conductor having a capacitance C equal to its radius, a. If the charging is carried out in a medium of dielectric constant, e. (4.23) A similar concept of "self-energy" applies to a capacitor which has an energy of 1I2CV2 or 1I2C(I1¢Jf

4.8. ELECTRIC TENSION IN A DIELECTRIC IN A FIELD

In terms of the electrical work of charging a capacitor [We = 1I2C(11¢J)2], it follows from the derivation ofEq. (4.14) that

W'=('8~)Ad

(4.24)

where E is the field 11¢J/d. Ad is the volume of dielectric in the capacitor so that

eE2/Sn: must be the electric pressure or really the internal tension associated with the field E, which follows from considering the units of the left- and righthand sides of Eq. (4.24). Equation (4.24) also provides a way of evaluating the self-energy of an ion by integrating the P·dV work experienced by the dielectric from the ion's periphery at a to 00. Thus

eW =f. 4n:r2 . dr =q2/2ea Sn: ~

W

e

a

noting that E at r for the charge q is qle?

(4.25)

78

Chapter 4

4.9. ELECTRIC POLARIZATION RESPONSES AT THE MOLECULAR LEVEL

The electric polarization in a dielectric (i.e., the solvent in the interphasial double layer), originates at the molecular level in the dielectric material. We consider here the electric polarization in individual atoms and molecules and then the interaction of a permanent dipole in a field such as exists in a charged double layer.

4.9.1. Atoms and Molecules in Fields: Electronic Polarization

From a sufficient distance, an atom presents an electrically homogeneous aspect to an applied electric field. However, the field will produce an attraction or repulsion of the electrons of the atom with respect to its positive nucleus and modify the initial, normal charge distribution. The resulting distribution corresponds to an induced electric dipole (Fig. 4.4) of moment J1i' The dipole moment of such an induced charge distribution is defined as e x t, i.e., the length t corresponding to the hypothetical displacement of a unit of electronic charge. The induced electric moment, J1i' is proportional to the field E: fli

= a.,E

(4.26)

where u e is called the "electronic polarizability." It depends on the number and type (a or n) of electrons in the molecule and the presence or otherwise of conjugation. In fact ue has the units of volume, of magnitude 10-24 cm3 . In addition to displacing the electron charge, an electric field can modify the degeneracy of electronic orbitals and lead to spectroscopic line splitting called the Stark effect (linear and quadratic). Also, a vibrational Stark effect in molecules (e.g., CO) is observable in double layers at electrodes where CO is adsorbed.

o

Atom in zero field

Induced dipole moment fL i-et

Atom in field E

FIGURE 4.4. Electric polarization of an atom or molecular particle in a field leading to an induced dipole moment, j1.i.

79

Treatment of Double Layers and Ions

As in the case of charging an ion, work is required to produce the induced polarization, given by: (4.27)

This work, Wi' is done on the atom or molecule; hence the positive sign for Wi' The induced dipole, fli, however, interacts with the field E that produced it with an energy -fl iE or -aeE2. Hence the total energy of induced polarization is

-aeE212.

4.9.2. Interaction of a Permanent Dipole with a Field

4.9.2.1. Uniform Field

A permanent dipole interacts with a field E experiencing (1) an induction of the moment fli determined by its a e ; (2) an interaction with the field, flE cos where is the angle of orientation of the dipole to the field [Fig. 4.5(a)]; and (3) a torque if -:f. 0 [Fig. 4.5(b)]. If -:f. 0, fli and fl must be added vectorially (Fig. 4.6) to give a resultant /lR along the field direction (/lR =/li + /l); also the torque tends to align the dipole eventually in the direction of E. The torque arises because the interaction of the charges ±e on the ends of the dipole with E produces a couple. Alternatively, the energy of a dipole, -flE cos is minimized in a field as cos ~ 1 (e ~ 0). The torque is responsible for the libratory motion of dipoles in a field, e.g., at an ion in its solvation shell. Alignment of a dipole in a field is normally incomplete at finite temperatures owing to thermal fluctuations in a gas or liquid. The average extent of alignment depends on the ratio of flE cos e to kT. Only at high E or low kT is the alignment almost complete. At 298 K, E must be > 2 x 104 esu or 6 x 106 V cm- 1 if flE for H 20 dipoles is to be > kT.

e,

e

e

e

e,

e

4.9.2.2. Nonuniform Field

In a nonuniform field, a dipole not only tends to become aligned with the field but to be translated down the field gradient to minimize its energy (i.e., to maximize the numerical value of flE). The movement of the dipole continues as E increases (e.g., near an ion where the field varies as ±ze/B?, or near an electrode surface in its double layer) until the dipole experiences some sufficient contact repulsive force. This is the reason for the internal compressional force near ions in a polar medium or in a double layer. This leads to "electrostriction" or a decrease in volume of the dielectric medium near ions.

Chapter 4

80

8

Field direction

------------------------.~

(t)

ILl 1-'=

cose a

Force -eE

Field E

Force ~eE

--b

FIGURE 4.5. Interaction of a permanent dipole of moment, fl., with an electric field E: (a) Interaction energy with the field, fl.E cos O. (b) Torque resulting in orientation.

4.9.2.3. Forces on a Quadrupole in a Field

Many polyatomic molecules (e.g., H20, CO2 ) have charge distributions that cannot be represented by only a single dipole. In these cases, more complex interactions with a field arise from the quadrupolar as well as the dipolar nature of such molecules. Quadrupole interactions are important in solvation and in inhomogeneous fields in the double layers at electrochemical capacitors. Note that a quadrupole experiences an energy only with an inhomogeneous field, i.e., with afield gradient.

Treatment of Double Layers and Ions 1. Similarly, capacitance of a given geometrical configuration of plate electrodes is enhanced by the factor t: (> 1) over that for a vacuum (t: = 1) and correspondingly more (free) energy can be stored for a given applied voltage, Ll V, when t: > 1, the energy being 112 CCLl V)2 (Chapter 2). For two capacitors having identical geometries, the difference of stored free energy for t: > 1 (real dielectric) and t: = 1 (vacuum) is (5.12)

The fundamental question regarding dielectrics is then how and in what state is this extra energy stored in a capacitor having a dielectric of constant t: > 1 in relation to that for a vacuum (t: = 1), for a given voltage difference between the plates? This question is fundamental to the materials dependence of the capacitance of capacitors utilizing various dielectric media, including electro-

Behavior of Dielectrics in Capacitors and Theories of Dielectric Polarization

91

chemical double-layer capacitors utilizing various solvents. The answer is the electric polarization behavior of the dielectric material in the field E corresponding to the voltage difference between the plates, Ll V, divided by the separation distance d: E = Ll V/d.

5.3. ELECTRIC POLARIZATION OF DIELECTRICS IN A FIELD

The term "polarization" can be understood as describing the extent to which polarity is induced in a given unit volume of dielectric medium by the field E. Microscopically, in a related way, polarity or extra polarity is introduced by E in individual molecules or in assemblies of them. In advanced theories of dielectric polarization and the properties of dielectrics, the essential requirement has been the development of various models at different levels of approximation and detail for relating molecular electrical properties to bulk electric polarization of liquid (or gaseous) media, i.e., for assemblies of interacting polar molecules. This becomes a statistical-mechanical problem. The quantitative relations between this polarization and the field E (and thence K or e) for various types of dielectric media are the essence of the theory of dielectrics and the behavior of solvent molecules in the double layer at capacitor electrodes. Two kinds of approach have been made: 1. The first involves treatments in which the dielectric and its polarization are examined in terms of bulk properties. 2. The second involves treatments in which such bulk properties are evaluated by modeling at the molecular or molecular-assembly level. This latter treatment involves principles of statistical mechanics in order to relate averaged molecular properties in assemblies to bulk-phase behavior at finite temperatures. Since the capacitance of a capacitor of given geometrical proportions depends directly on the K or e of its dielectric medium, the interpretation of the dielectric constant is of main interest in this field of capacitor behavior, especially the dielectric behavior of the double-layer interphase at electrodes in solution. In the case of ordinary hardware capacitors, their C depends only on the geometrical parameters and K or e of the medium separating the plates. With electrolytic capacitors, the situation is essentially the same except that the dielectric is an anodically formed oxide film of an appropriate metal (e.g., Ta, Zr, Ti, AI) that can be formed under controlled electrolytic conditions (anodization). Its thickness (to which C is inversely related) is determined by the magnitude ofthe anodic-forming voltage and the time, t, for which it is applied. (An

92

Chapter 5

inverse "log" law in time usually applies or a "parabolic" law for some conditions; see Ref. 3.) The situation with double-layer capacitors is different since at each of the capacitor ± electrodes a double layer exists across which there is a drop in potential (see Fig. 6.3, Chapter 6) between the electrode plate and the electrolyte solution; the dielectric of this double molecular capacitor is composed of only one or two molecular layers of solvent, including solvated anions or cations in that solvent layer, and it is polarized by the field associated with the differences in potential between the metal and solution, especially that part which falls across the first 0.3 to 0.6 nm (Helmholtz layer) of the solution near the electrode surface. Interfacial dielectric properties of this very thin film have to be formulated (see Chapter 7) to account for the magnitude of the double-layer capacitance (16 to 50 JiF cm- 2) and its dependence on electrode potential, the sign of the charge of the electrode plate, and the nature of the solvent and the electrolyte solute. Some of the principal aspects of the molecular and ionic modeling of the double layer and its effective average dielectric properties are given in Chapters 6 and 7. In order to provide the background for the discussion in those Chapters, we return now to the more general theory of bulk dielectrics in order to explain the concepts of polarization and polarizability more completely, and their relation to experimentally measurable dielectric constants and molar polarizabilities. These are fundamental matters that lie at the bases of capacitance behavior in electrostatics, and in the phenomenology of double-layer capacitance and electrical behavior of supercapacitors.

5.4. FORMAL ELECTROSTATIC THEORY OF DIELECTRICS

Apart from the invention and use of the Leyden jar (Chapter 1), Faraday was the first to consider the state of dielectrics in charged capacitors. In fact it was in correspondence with the polymath, Whewell of Cambridge, that the term "dielectric" was suggested (December 1836) and it has remained in use ever since. Faraday! initiated ideas of "polarization" due to electric fields and recognized the induction of electric charges,with polarization being induced by one polarized or polar particle into or onto another. He also recognized concepts such as specific inductive capacity (8), lines of inductive force, and the strain introduced in an electrified dielectric medium by an imposed field. Thus Faraday recognized some of the properties of dielectrics and the nature of electric inductive polarization before the middle of the nineteenth century. Of course, it was many years later before these ideas became more exactly and mathematically formulated by workers such as Mosotti, Clausius, Coulomb, Frohlich, Debye, and Kirkwood in terms of molecular polar models. Unfortunately, as is well

Behavior of Dielectrics in Capacitors and Theories of Dielectric Polarization

93

known, 1 despite his depth of scientific perception and his inventiveness, Faraday never developed his concepts and interpretations of physical phenomena in mathematical terms. The electrostatic behavior of two identical configurations of capacitor plates is considered here. One capacitor [Fig. 5.1(a)] has a vacuum dielectric, the other [Fig. 5.1 (b)] a real dielectric medium having 8 > 1. An external field Eo is applied and is operative between the plates generating or arising from plus and minus charge densities on the plates, equally and oppositely. For a given voltage difference, ~ V, between the plates or a corresponding field ~ Vld, charge densities 0'1 ±qtfA or 0'2 ±q21A appear, respectively, on the plates of the two capacitors. 0'1 and 0'2 differ on account of the difference in 8 for the two cases; 81 = 1; 82 > 1. This difference corresponds to an induced charge density ±p, which appears at the plates, changing the original a value [Fig. 5.1(b)]. When the dielectric medium is present [Fig. 5.1(b)], the field between the plates is reduced from the vacuum value, Eo, to a diminished value E.

=

=

(0)

+

Field Eo +

.-

Field Eo

-

~

Field E~

+

+ +

Vacuum

+

...

+CT

-CT

Dielectric

(b)

/ ..........

....---

/--

External field

'......--"......,, Field E«E o) ,,..---"',/" ""

,

-"""-, --,- -,

Eo

1/--

. . . . . /,/"

+p

,

, I

Eo

t

-p

FIGURE 5.1. Fields E between charged capacitor plates: (a) in vacuum; (b) in the presence of a dielectrically polarizable medium. Original charge density ±a; induced charge density ±p.

ChapterS

94

In tenns of the dielectric polarization induced by the external applied field Ll V/d, = E, the dielectric medium in case (b) is regarded as composed of a density of induced dipole moments oriented in the direction of E. This results [see Fig. 5.2(a)] in opposing the respective charge densities on the two plates of the capacitor by the presence of the plus and minus ends of the bulk polarization set up by the field in the dielectric between the plates. This polarization can be expressed fonnally as an induced dipole moment per unit volume of dielectric, designated P [Fig. 5 .2(b)]. Between the plates, the plus and minus ends of the induced dipole elements cancel out; only at the interfaces of the dielectric at the plates do the plus and minus ends of the induced polarity lead to surface charge densities induced at the boundaries of the dielectric. These determine the capacitance of the capacitor, increasing it over the induction of charge density by the field Ll Vld in a vacuum (I: = 1).

a

+ + + + + +

+ b

+

p

+ + + 1 cm2

+ - -.....~E---...•• FIGURE 5.2. Schematic diagram of induction of polarity in a dielectric medium by an externally applied field. (a) Induced charges or orientation of polar molecules by an external field Eo in capacitor. (b) Resulting polarization vector, P, corresponding to induction of polarity, -, +, on the faces of a centimeter cube.

Behavior of Dielectrics in Capacitors and Theories of Dielectric Polarization

95

The origins of the polarization thus induced in the dielectric are fourfold: 1. The first is a redistribution of electronic charge density (Fig. 5.3), mainly of valence electrons, within the individual atoms or molecules of the dielectric material; it is characterized by the so-called "electronic polarizability," denoted by a, of the molecules. This a depends on the number of polarizable electrons and thus on the volume of the molecule (a has the units of cm3 mol-I, on the order of 10-24 ) and on the type(s) of orbitals involved in the molecules, e.g., (J or 7r. a for 7r orbitals is greater than a for (J orbitals; for example, with benzonitrile C6H5 . C=N, the polarizability is anisotropic, along rather than across the molecule. 2. There is a slight change in the atom positions or bond lengths of the molecule, and hence its polarity. This is called "atom polarization." 3. Third, for intrinsically polar molecules such as H 20, HCI, HF, and CH3CN, a larger degree of polarization per unit volume is set up within the dielectric due to an orientational response (Fig. 5.4) of the already existing dipoles of the molecules of the dielectric medium. This is a statistical effect, evaluated in important work by Debye and by Langevin,4 and is the main source of dielectric polarization in fluids of intrinsically polar molecules. Since for ordinary fields of e.g., 10-100 V cm- 1 there is a small but significant net average orientation of dipoles, this has an influence on the induced electronic polarization in the molecules of the dielectric, depending in an average way on their momentary orientations in the field disturbed by thermal fluctuations in the bulk fluid. This effect does not of course arise with atomic fluids or 3-dimensionally symmetric molecules (e.g., CH4 , CCI 4 ) except to a small extent through dipole induction by the experimental field. For intrinsically polar molecules, polarization of the dipole orientation is normally substantially larger than the polarizations of types (1) and (2).

Field E

fL~

8+-8-

.....

. .. ..

fJ-i = a E FIGURE 5.3. Redistribution of electronic charge density in a polarizable molecule in an electric field (induced polarity).

96

ChapterS Average moment.. ~

,,/ \

-- -

-----

, . Field

• -,/.-- • -IfI'

-..

(0) ~E dq/dE; such types of quantities always give more resolved information than corresponding integral relations, e.g., total charge q plotted vs. the electrode potential E at which it has been accumulated, or q divided by E at a given electrode potential. That is why cyclic voltammetry and ac impedance measurements are especially valuable and preferred in studies of the double layer and the resulting electrochemical capacitor devices; they both give differential information. A typical relation between

The Double Layer at Capacitor Electrode Interfaces a

/

107

Electrolyte

~

- ;,-+ +-+- ~+ - -+- ~ +- +-+

-~ ++ +-+ -~ +- +-+ -~ ++ +-+ + -+ - +-~ + -+ - +-~ + -+ - +-

-:/

+ - + -- - +- - + + - + -- -+-+- +-+-+-+- - - -+-+-+-+- +- - .;

Porous electrode

Separator

Porous electrode

Configuration of an electrochemical capacitor requiring two electrodes and thus two double-layers b

Electrolyte

,/

~ + + +

+ + + + + + Separator

Potential profile across an electrochemical capacitor on discharge FIGURE 6.1. Diagrams of electric potential profiles in an electrochemical capacitor comprising a double layer at each of two electrodes: (a) charged capacitor at open circuit, (b) capacitor passing current on discharge with IR drops.

108

Chapter 6

f( y)

y

\

J,/f(dY/dX) \ \ \

\

x

\

"....

FIGURE 6.2. The relation between arbitrary integral signal curve x,y, (e.g., for a double-layer charge as a function of potential) and its differential coefficient, dyldx (e.g., for differential capacitance).

an arbitrary integral curve and its differential, illustrating this point, is shown schematically in Fig. 6.2. It is appropriate first to describe the structure of the double layer, i.e., of the interphasial region of the metal/solution boundary. The term interphasial region! is deliberately chosen over the word "interfacial" (see Chapter 3, section 3.8) since the boundary region is really a 3-dimensional one, albeit very thin, some 0.3 to 0.5 nm in thickness, rather than the 2-dimensional one implied if the term "interfacial" were used. This means that the concepts of 3-dimensional phases can be used to describe the double layer, such as surface concentration or surface excesses, e.g., of ions of the electrolyte and solvent molecules.!,2

6.2. MODELS AND STRUCTURES OF THE DOUBLE LAYER

It is logical to describe these models in the order in which they were historically proposed in the literature, which is also the order of their progressive approach toward a correct description of the structure of the interphase at electrode surfaces. The concept of a double layer corresponds to a model consisting of two array layers of opposite charges, separated by a small distance having atomic dimensions, and facing each other, as on the plates of a two-plate hardware capacitor. This model was adopted by von Helmholtz3 to describe his perception of the distribution of opposite charges, quasi-2-dimensionally, first at the interface of colloidal particles. It is illustrated in Fig. 6.3(a), which shows its compact structure, and is referred to as the Helmholtz double-layer mode1. 3,4

The Double Layer at Capacitor Electrode Interfaces

¢..

"-S· (19+) + (11-)

¢..

¢..

+

109

+

+ _.+ -+ + + "'S IT..

(0)

-

+

(b)

+ -

+

+

+

+

-

fs

01 FFUSE LAYER

0--

=8-

+

+

+

HELMHOLTZ LAYER

(c)

FIGURE 6.3. Models of the double layer: (a) Helmholtz model, (b) Gouy point-charge model (specific charges a per unit area as indicated for anions and cations as an example), (c) Stem model for finite ion size with thermal distribution, combining Helmholtz and Gouy models.

In the original model for colloid interfaces, the charges on the surface side of the double layer arise either from acid-base ionization, as with proteins or polyelectrolytes, or on account of the adsorption of ions, as at lyophobic colloids. On the solution side of the double layer, counterions of opposite sign of charge accumulate to balance the charge on the colloid, forming a double-layer array of positive and negative charges. The Helmholtz model was later adapted to the case of electrode interfaces where, on the metal side, a controllable surface density of excess negative or positive charge can arise that corresponds to an excess or deficiency of electron charges of the delocalized electron plasma of the metal. Owing to the high free electron (e) density in the metal (approximately 1 e per atom), any net charge density of electrons at the surface is strongly screened, so the gradient of electron density at a charged metal interface is highly localized over a distance of only 0.05 to 0.2 nm, the so-called Thomas-Fermi screening distance (Fig. 6.4). Because the wave function amplitudes of the conduction-band electrons retain significant but diminishing magnitudes outside the formal electrode surface plane, there is significant spillover of electron density into the double layer on the solution side of the interface5,6 and the effect is potential dependent. In the case of p- or n-semiconductors, the charge-carrier (hole or electron) densities are, however, very much smaller than in metals by a factor varying from about 10-4 to 10-15. As a consequence, there is a distribution of the charge carriers away from the interface (Fig. 6.5) but extending into the bulk of the semiconductor over a relatively large distance that is inversely related to the charge-carrier density (cf. the situation for ion distribution in an electrolyte in the solution side of the interface, Section 6.3). This distribution of charge carriers within semiconductors, near their interfaces, follows mathematically exactly

Chapter 6

110

Thomas - Fermi screening distance

..

Appro)!.. 1e per atom in bulk

~

---

Free electron density distribution

SOLUTION

METAL

I

\,/Electran overspill .... - - - - to zero

I

~Nominol surface plane t Locus of atomic nuclei

FIGURE 6.4. Electron overspill profiles at an electrode surface illustrating Fermi-Thomas screening length in electron space charge over narrow region at a metal surface (schematic).

0 ~

1 cr

9

-----

~9>

n space - charge / region

/

/

SEMICONDUCTOR

e IONIC SOLUTION

FIGURE 6.5. A space charge distribution of charge carriers within an n-type semiconductor near its surface that is like ion distribution in the diffuse layer in solution.

The Double Layer at Capacitor Electrode Interfaces

111

the same form as for ion distribution in the diffuse layer on the solution side (Gouy-Chapman theory for dilute ionic solutions). Some time after von Helmholtz's model was proposed, it became realized that ions on the solution side of the double layer would not remain static in a compact array as in Fig. 6.3(a) but would be subject to the effects of thermal fluctuation 7 according to the Boltzmann principle. 8 This latter effect would depend on the extent to which the electrostatic energy Ue (together with any chemisorption energy Uc ) of the ions' interactions with the charged metal surface exceeded, or were exceeded by, the average thermal energy, kT, at temperature, T, K, i.e., the ratio (Ue + UJlkT. Gouy7 introduced this thermal fluctuation factor into a modified representation of the double layer in which the counterions conjugate to the metal surface's electron charge were envisaged as a 3-dimensional diffusely distributed population of cations and anions [Fig. 6.3(b)] of the electrolyte having a net charge density equal and opposite to the virtually 2-dimensional electron excess or deficit charge on the metal surface. In this model, the ions were assumed to be point charges. Historically, this was an important restriction since it led to a failure of Gouy' s model on account of (1) an incorrect potential profile and local field near the electrode surface and (2) consequently a too-large capacitance being predicted, that quantity being defined as the rate of change of net ionic charge on the solution side with the change of metal-solution potential difference across the interphase. The interphasial capacitance associated with this model is commonly referred to as the "diffuse" double-layer capacitance. A full mathematical treatment of the Gouy diffuse-layer model was given in some detail by Chapman in 1913 9 (see Section 6.3), based on the combined application of Boltzmann's energy distribution equationS and Poisson's equation lO for the relation between ionic space charge density in the interphasial region to the second derivative of electric potential, IJI, with respect to distance from the electrode surface. It is interesting to note that the mathematics and principles used by Chapman anticipated the approach taken by Debye and Huckel in 1923 11 in determining ion distribution in three dimensions around a given ion in their treatment of activity coefficients and conductance of electrolytes. Later it was used by Onsager,12 in an improved treatment of the conductivity of electrolytes. In both Chapman's and Debye and Huckel' s treatments of ionic charge distribution, the key equation resulting from the combination of Boltzmann's energy distribution function and Poisson's electrostatic equation lO has been referred to as the "Poisson-Boltzmann" equation. It is also utilized in the treatment of band profiles and space charge effects in semiconductors. 13 The serious problem with the Gouy-Chapman treatment, overestimation of the double-layer capacitance, was overcome by Stern in 1924 14 in the next

112

Chapter 6

stage of development of the theory of double layers. In his model and calculations it was recognized that the inner region of the ion distribution could be treated in terms of an adsorption process according to Langmuir's adsorption isotherm, and the region beyond this inner layer, into the solution, could be validly treated in terms of a diffuse region of distributed ionic charge [Fig. 6.3(b)] as treated by Gouy 7 and by Chapman. 9 In addition, if the ions were recognized as having finite size, including the annular thickness of their hydration shells (their so-called Gurney cosphere radii 15 ), it was easy to define a geometrical limit to the compact region of adsorption of ions at the electrode surface [Fig. 6.3(c)]. This is taken to correspond to a Helmholtz type of compact double layer having a capacitance CH , while the remaining ionic charge density beyond this compact ion array is referred to as the "diffuse" region of the double layer, having a capacitance Cdiff . Cdiff and CH are conjugate components of the overall double-layer capacitance, Cd], related by the equation 1 Cd!

1 CH

1 Cdiff

-=-+--

(6.1)

corresponding to a series relation between CH and Cdiff according to an equivalent circuit:

-----II I

I 1-1- -

On account of the reciprocal form of the terms of Eq. (6.1), it will be seen that the Cd! will be determined by the smaller of the two components, CH and Cdiff . This is of considerable importance in determining the properties of the double layer and its capacitance as a function of electrode potential and ionic concentration of the solution. The original paper by Stern!4 in Zeitschrift for Elektrochemie (1924) is somewhat obscurely written, but Parsons, in an important article in 1954,4 presented a much clearer version of this treatment in which the limit for a distinction between the Helmholtz compact layer and the diffuse layer beyond it [Fig. 6.3(c)] can be understood in terms of the distance of closest approach of counter anions or counter cations to the metal electrode surface. By introducing a distance of closest approach of finite-sized ions and thus geometrically defining a compact Helmholtz inner region of the double layer, the problem of a far too high capacitance that arises in the Gouy-Chapman treatment is automatically avoided. This difficulty arises since the capacitance of two separated arrays of charges increases inversely as their separation distance, so very large capacitance values would arise in the limit of infinitesimally small (point charge) ions very closely approaching the electrode surface.

113

The Double Layer at Capacitor Electrode Interfaces

The Stern theory of the double layer remained a good basis for general interpretations of electrode interface phenomena, including double-layer effects in electrode kinetics 16 until the detailed work of Grahame in the 1940s on the double layer capacitance at the mercury electrode in aqueous electrolyte solutions reported in various papers, particularly in the seminal source review paper in Chemical Reviews in 1947.17 Its semicentenary was celebrated at a special anniversary Electrochemical Society symposium in May 1997. Grahame's work emphasized the great significance of the specificity of double-layer capacitance behavior at Hg to the nature of the cations and anions of the electrolyte, particularly the size, polarizability, and electron-pair donor properties of the anions of the electrolyte. This led Grahame to make an important distinction between an inner and an outer Helmholtz layer in the interphase which correspond to the different distances of closest approach that can arise for anions vis a vis cations at the electrode surface. This difference of distance of closest approach is mainly caused by the fact that most common cations are smaller than common anions (Table 6.1) and retain solvation shells due to strong ion-solvent dipole interaction.1O·18.23 Thus, the Gfahame model (Fig. 6.6) consists of three distinguishable regions: the inner Helmholtz layer. the outer Helmholtz layer, and always a diffuse ion distribution region. At extremes of polarization (i.e., for high positive or high negative charge densities), one or the other of the Helmholtz layer regions dominates, with a population of anions or cations, corresponding to such polarizations. Because anion distances of closest approach are usually smaller than hydrated cation distances of closest approach, the inner layer capacitance at positively charged electrode surfaces is usually about twice that at a corresponding negatively charged surface, (16-25 f.1,F cm- 2), though this depends on the metal and the ions of the electrolyte, and the solvent. These aspects of double-layer capacitance behavior are of great significance for understanding the properties of double-layer supercapacitors and the magnitude of capacitance that can be achieved per square centimeter over various ranges of potential and at various electrode materials. TABLE 6.1.

Li+ Na+ K+ Rb+ Cs+

NH! H30+

Pauling Crystal Ionic Radii of Alkali and Halide Ions (nm) 0.060 0.095 0.133 0.148 0.169 0.139 0.138

F ClBr[-

0.136 0.181 0.195 0.216

Notes: See Ref. 10 for other scales of ionic radii. Source: Reprinted with permission from f. Am. Chern. Soc., 49, 771 (1972). Copyright American Chemical Society.

114

Chapter 6

GOUY-CHAPMAN

I

I

I I

1.

/

NEUTRAL MOLECULE

I1//~

1 c..OUTER'HELMHOLTZ LAYER

t"-----lNNER ....

:2.

FIGURE 6.6. General representation of the structure of the double layer showing different regions for adsorption of hydrated cations and less hydrated anions (Grahame model!?), together with solvent molecules and an adsorbed neutral molecule.

6.3. TWO-DIMENSIONAL DENSITY OF CHARGES IN THE DOUBLE LAYER

The range of values of accumulated electronic charges on the metal side of interfaces of double layers at electrodes in aqueous solutions extends from about + 30 flC cm-2 at potentials positive to the potential-of-zero charge (pzc), to about -20 flC cm- 2 at corresponding negative potentials. This range depends on the electrode material and the electrolyte ions in the solution and the solvent, as well as the temperature and pressure (in the case of high applied pressures). For the above figures, it is of interest to calculate the range of excess charge per atom of an assumed planar interface. We can deduce that the number n of atoms having a diameter of 3 A (0.3 nm) in, say, a square array [i.e., in a (100) lattice configuration of particles in the surface of the electrode], is n = 1/(3 x 10-8)2 = 1.1 X 10 15 . Per atom, the excess charge densities in C cm- 2 correspond then to

The Double Layer at Capacitor Electrode Interfaces

30 X 10-6

20 X 10-6

115

.

+ - - - - or - 1.11 x 10 15 ' respectIvely, 1.11 x 1015 and thus to +

30 X 10-6 X 6.04 X 1023 15 11.11 x 10 x 96,500

=+0.17 e atom-lor -0.11 e atom-I

These figures are shown just to give an idea of the extent to which electron deficiency or electron excess can arise on the electrode surface of the double layer. These charge densities depend on the electrode potential, E, and hence can be controlled or modulated by direct, alternating, or pulsed voltage. At capacitor plates, as also at the interfaces of two-electrode double-layer electrochemical capacitors, the surface charge densities (plus and minus charges) arise on account of the application of a potential difference, till, between the two electrodes. In response to the applied till, electrons from one surface are driven through the external circuit containing the polarizing device (a power supply, a battery, or a regenerative braking dynamo in an electric vehicle hybrid system) to the other surface, establishing a difference of sign of charge density between the two plates. In an electrochemical capacitor, the respective plus and minus charge densities on the two plates are matched by net equal and opposite accumulations of respective negative (anion) and positive (cation) charge densities in the interphasial regions of the solution [Fig. 6.3(c)] over distances from ca. 1 to 100 nm into the bulk solutior., depending on total ionic concentration. Note also that when an interphasial potential difference is generated through a Nernstian thermodynamic equilibrium (as for the electrode potentials of a pair of battery cell electrodes on open circuit), a double layer is also spontaneously set up at each electrode interface but is not generated by charge flow from or to an external source of electric charge; the electrode equilibration processes generate their own double layers. Upon charge or discharge of the battery, the two double-layers will become more, or less, charged, depending on the direction of current flow and the changes (if any) of electrode potentials resulting from charging or discharging. Their dependence on the electrode potential E corresponds to the development and manifestation of the double-layer capacitance, Cdl = dqldE, or integrally Cdl = q/E or !l.qltill. It is of interest chemically that the above extents of controllable variation of surface electron excess or deficiency are comparable with the local changes of electronic charge density that can arise chemically on various conjugated aromatic ring structures such as benzene or naphthalene, owing to the presence of substituents such as -OH, -S03H-, -CH3, -NHz, and -COOH, where charge density changes arise on account of electronic inductive and resonance effects.

116

Chapter 6

6.4. IONIC CHARGE DENSITY AND INTERIONIC DISTANCES ON THE SOLUTION SIDE OF THE DOUBLE LAYER

A complementary variation of charge density on the solution side of the electrode interface arises through adjustment of the distribution of cations and anions ofthe electrolyte in response to changes of electron density, qM' on the metal or carbon surface, as was illustrated in Figs. 6.3(a), 6.3(b), 6.3(c) and 6.6. This adjustment arises in the inner or outer Helmholtz layer regions of the structure of the electrode/solution interphase [Figs. 6.3(a) and 6.6] and in a coupled way in the diffuse-layer region [Figs. 6.3(b) and 6.6]. The average interionic distances in the Helmholtz region of the double layer can be easily evaluated and correspond to the numbers of charges per square centimeter or the e/atom numbers. For the two data derived above, the time-average, square-lattice, interionic spacings would be 0.73 and 0.89 nm, respectively. Of course the Helmholtz layer is continually in a state of thermal fluctuation, as noted by Gouy7 and by Chapman,9 and some of the ionic charge (depending on ionic strength) will be distributed in the diffuse part of the double layer. At a hexagonally close-packed atomic surface [i.e., in a (111) lattice configuration], the number of atoms cm- 2 is 21-Y3 times larger than on a (100) surface of atoms having the same diameter. On the basal plane of graphite crystals, the carbon atoms are arranged in such a hexagonal lattice, with interatomic distances of about 0.11 nm. The interatomic distance normal to the hexagonallattice (i.e., between the lattice planes) is substantially larger. Similarly, the electronic properties (e.g., work function) of the basal plane are very different from those of edge sections in which intercalation processes can also occur, e.g., accommodation of Li, K, and F2 • The average 2-dimensional interionic distance in the Helmholtz layer (a square-lattice approximation, which is sufficient for the purposes of this discussion) will be inversely proportional to the square root of the total cation + anion surface excess of ions, r= r_,i + r+,i' However, according to Grahame,17 the location planes (distances of closest approach) of adsorbed anions and cations in the Helmholtz compact region of the double layer will be different on account of the differences of closest approach of anions and cations to the electrode surface owing to specific electronic effects in the chemisorption of anions and to the usually more strongly developed coordination structure of hydrated cations than of comparable anions. lO,l7 Taking into account the hydration of adsorbed ions in the double layer [e.g., the hydrated ion radii ofK+ and F are about 0.133 (reF or re,d nm] plus the 0.276-nm diameter of hydrating water molecules, it is seen that there is rather little free space (or free interionic distance) between the ions in the Helmholtz layer when their hydration radii are taken into account. In the case of the

The Double Layer at Capacitor Electrode Interfaces

117

relatively concentrated aqueous acid (H 2S04) or base (KOH) solutions that are used as electrolytes for a number of electrochemical capacitor embodiments, it is clear that there must be very little free water in the interphase that is not influenced by the ions either in the bulk electrolyte or in the Helmholtz layer. 19 This is an important but little realized aspect of double~layer modeling and has significant practical consequences, in that there may be local changes of solvent activity and balance during charging or discharging processes.

6.5. ELECTRON-DENSITY VARIATION: "JELLlUM" MODEL

It is to be emphasized that charging the double layer involves, on the metal (or carbon) side of the electrode/solution interface, only changes in the density and distribution of the de localized electrons of the metal-electron plasma (approximately one delocalized electron per atom). This is treated in recent works according to the so-called "jellium" model in which the lattice of "ions" of the metal containing the free conduction electrons is regarded as a structureless jellylike distribution. (In practice, at some carbon electrode materials, a change in potential can also cause modification of surface functionalities owing to a Faradaic partial current that corresponds to some pseudocapacitance charging.) The surface of a conductor can be regarded in two ways: (1) as the relatively fixed (apart from intralattice vibrations) atoms of the material or (2) electrically, as the moveable plane of emergent electron densitlO at the boundary of the atomic lattice (Fig. 6.4), depending on electrode potential (Fig. 6.7). The wave theory of electrons leads to a concept and model of electron distribution at metal or carbon surfaces. The wave functions of the delocalized conduction-band electrons spill over at the discontinuity of the lattice at its nominal surface (see, e.g., Fig. 2 in Chapter 2 of Ref. 20), leading to a significant but exponentially decreasing probability of finding electron density beyond the nominal atomic surface of the metal. This spillover effect is enhanced by negative polarization of the electrode, corresponding to the situation in Fig. 6.7. In a vacuum, under the influence of an electric field of an appropriate polarity, cold (field) emission 2o of electrons into the vacuum can in fact be promoted. This well-known phenomenon is a good example of the spillover or spill out effect. The redistribution of this electron density relative to the "jellium edge" at the surface of the metal with changes of potential was illustrated in Fig. 6.7 (cf. Fig. 6.4). At relatively positive potentials to the potential-of-zero charge, the electron density boundary retreats inward at the electrode surface relative to the plane of the centers of surface metal atoms. At potentials relatively negative to the pzc, the electron density becomes pushed outward toward the inner region of the solution boundary, where it will interact more easily with cations and sol-

Chapter 6

118

Doublelayer

I"

.\

Bulk electron density

~tozero

METAL

tt

SOLUTION

Nom;nol metal surface plane LOCus of centers of metal surface atoms

FIGURE 6.7. Schematic profiles of emergent electron density at a metal at three potentials: one positive (1), one negative (3) to the potential-of-zero charge, and one (2) at the potential-of-zero charge. (After Lang and Kohn ref. 5).

vent molecules in the compact region of the double layer. This model has been envisaged and treated by Lang and Kohn,5 and more recently by Amokrane and Badiali. 6 More details are discussed in Chapter 7. The variation of the locus of the electron density overspill with changing electrode potential is important for modeling such processes as solvent-dipole adsorption and orientation, and the chemisorption of ions, especially anions, each of which factors determines, among others, the capacitance of conductor/solution interfaces as in a double-layer supercapacitor. In energy terms, this modification of surface-region electron density by a changing potential corresponds to effective changes of the electronic work function, cP, of the metal (when it is an electrode) or its electron affinity (-cP), which is one of the bases of the effects of changing potential on the kinetics of electrode processes l8 (Chapter 3) and on the chemisorption of ions. The overall structure of the electrode/solution interphase is seen to be quite complex. Its construction in terms of surface-chemical, ion hydration, and double-layer aspects is illustrated schematically in Fig. 6.8.

The Double Layer at Capacitor Electrode Interfaces

119

METAL/ELECTROLYTE INTERFACE _ _ _ _ .--------OUTER HELMHOLTZ LAYE.R

DIFFUSE LAV'ER

•

FIGURE 6.8. Overall construction of the electrode/solution interphase illustrating its surfacechemical and double-layer aspects. XM and Xd are the intrinsic surface potentials of the metal and adsorbed dipoles, respectively.

6.6. ELECTRIC FIELD ACROSS THE DOUBLE LAYER

Electric fields arise whenever there is a separation of electric charges. This is a matter related to Coulomb's law and Poisson's equation,1O and is a fundamental property of electrical behavior of the universe or physical systems. At a plane metal electrode interface with a potential difference of, say, 1 V across an ideally polarizable electrode double layer, the field E will be approximately E = 1.0/3.8

X

10-8 V cm- 1

(6.2)

for a double-layer thickness of 3.8 A (0.38 nm). The latter figure corresponds to the distance of closest approach of simple hydrated cations, e.g., Na+. The actual value of the thickness of the double layer will depend on the crystal ionic radius of the ion and the thickness of its time-average hydration or solvation shell.

120

Chapter 6

From this it is seen that E has a very high value-some 2.9 x 107 V cm- I . Sometimes nonelectrochemists are puzzled as to how such enormous fields can exist without electrical breakdown. Thus, in an ordinary capacitor, e.g., with a polystyrene dielectric, fields cannot be sustained beyond ca. 5000 Vern-I. However, in the double layer, the behavior is quite different since there is no bulk dielectric in the normal sense associated with dielectrics in regular capacitors. Only the water of hydration of the ions and the monolayer film of adsorbed solvent water at the electrode interface (Fig. 6.6) constitute the dielectric of the double-layer capacitance. The situation is analogous to that on the atomic scale in, e.g., a molecular dipole such as HCl or H 20 where the internal, interatomic field is also on the order of 107 Vern-I. Similarly, the interion (±) local fields in an ionic crystal are of about the same magnitude. No charge transfer breakdown by passage of charges between the ions can occur since they are very stable in their regular ionic states, Na+ and Cl-, e.g., in a rock-salt crystal. In the double layer, leakage currents across it can arise only when thermodynamically and kinetically, electron charge transfer processes are allowed beyond certain critical potentials and corresponding interphasial fields, as outlined earlier. The interphasial field, E, across the double layer can also be calculated in a different way by employing the electrostatic equation of Gauss for the field generated normal to a metal plate charged to a density of q charges per square centimeter.

E=-4nq/e

(6.3)

where e is the dielectric constant (coefficient) ofthe medium in which the field is established by the charge density q. In the double layer, e is probably on the order 6 but varies with the field. Earlier we showed that the maximum charge density sustainable at an Hg electrode in aqueous electrolyte is about 0.17 e/atom of the surface. The electron charge is 4.8 x 10- 10 electrostatic units (esu, see Chapter 4). Then we find E=

4n x 4.8 x 10-10 6

x 0.17 x 3 x 10 15 esu cm- I

(6.4)

where the factor 3 x 1015 is approximately the number of metal atoms per square centimeter in a close-packed Hg surface. Thus E =5 x 105 esu cm-I. In practical units of V cm- 1, E is then found to be 5 x 105 X 300 V cm- 1, the factor 300 being from the definition 1 V = 1 esu/300. The resulting figure for E, depending on the value to be assigned for e, is seen to be of the same magnitude as that derived above by dividing a typical potential difference value across the double layer by an appropriate thickness value d. A very high value for E again results and of

The Double Layer at Capacitor Electrode Interfaces

121

course has to be consistent with the first approximate way of estimating the field. The agreement or disagreement obviously depends on the value chosen for e in Eq. (6.4). More sophisticated calculations ofE would take into account the microscopic distribution of charges of ions and associated solvent dipoles, and the electron overspill from the metal toward the solution side of the interface in the whole interphasial region (Fig. 6.8) constituting the double layer.

6.7. DOUBLE-LAYER CAPACITANCE AND THE IDEALLY POLARIZABLE ELECTRODE

The significance of double-layer capacitance at electrode interfaces has to be understood in terms of the conditions under which charge separation takes place between the electrode metal surface and the solvated ions in the solution, near the metal interface; in particular in relation to Grahame's concept of an "ideally polarizable" metal interface. l7 This concept of the ideally polarizable electrode was implicit in the early work of Bowden and Rideal,2l based on the observation of the change in potential of a mercury electrode with time in response to a constant charging current (galvanostatic charging). It was more clearly and definitively described in the work by Grahame 17 : an ideally polarizable electrode is one where changes of potential due to flow of charge to or from the electrode cause only changes of charge density on the metal and conjugately of ion density on the solution side of the electrode interface, leading to charging of the resulting double layer. The essential aspect of the ideally polarizable electrode is that with changes of potential, charges flow from the external circuit and within the solution only to charge the double layer (-+), with no charges passing across the double-layer interphase, i.e., through some Faradaic reaction. Such a charged interface is at electrostatic equilibrium at a given potential rather than Faradaically at a Nernstian thermodynamic equilibrium, though one of the important contributions of Grahame's paper17 was to show how double-layer properties could be treated thermodynamically, for example, how surface charge densities in relation to surface excesses of cations or anions could be derived as a function of electrode potential from differential capacitance measurements. The Hg electrode in aqueous electrolyte solutions, ideally NaF, NaOH, or Na2S04, comes close to the above requirements for an ideally polarizable interphase, i.e., one with zero Faradaic current passing across the double layer. Hg is almost ideally polarizable over the potential range +0.23 V to -0.9 V (RHE). At potentials more negative than -0.9 V, significant Faradaic charge transfer currents pass owing to the reaction of solvent water decomposition, giving rise toH 2:

122

Chapter 6

(6.5) At potentials approaching 0.23 V or higher Hg becomes electrochemically oxidized according to the reaction (e.g., in KCI solution) 2Hg + 2CI- ~ Hg 2Cl 2 + 2e

(6.6)

or in alkaline solution at a different potential Hg + 20H- ~ HgO + H20 + 2e

(6.7)

Thermodynamically, cathodic decomposition of H 20 in the standard-state H+ ion concentration (activity) can commence at, or negative to, 0.0 V standard hydrogen electrode (SHE), but in practice, electrocatalysis for H2 evolution at Hg via the consecutive steps: H 20 + Hg + e

~

HgH + OH-

(6.8)

followed by HgH + H 20 + e

~

Hg + H2 + OH-

(6.9)

is very poor (exchange current density, io = 10- 13 A cm-2) so that significant currents for Faradaic charge transfer across the double layer (corresponding to a leakage current across the interphase) do not begin to pass until potentials more negative (vs. RHE or SHE) than ca. -0.9 to -1.0 V are attained. Then between +0.23 V and ca. -0.9 V, only double-layer charging currents effectively pass so that within that range the interphase approaches ideal polarizability with an almost ideal nonleaky double-layer capacitance being then manifested which can be experimentally characterized by means of transient charging curves or ac impedance. The electrode interphase is then capacitorlike in its electrical behavior, so that charging energy can be stored. Gold is another metal that exhibits almost ideal polarizability over a certain range of potentials: in this case between -0.2 V and + 1.30 V in aqueous H 2S04 solution. Beyond + 1.30 V, surface oxidation begins, eventually leading to O 2 evolution on a gold-oxide film while below ca. -0.2 V, H2 evolution leakage currents become significant (> 10-5 A cm- 2). Within the above limits, the electrode interphase behaves like an almost ideal capacitor but with significantly potential-dependent capacitance, as is also the case at Hg. 4 ,17 It should be noted that with excursions of potential, positively or negatively, beyond the potential limits for solvent-electrolyte decomposition, further double-layer charging (or discharging) still occurs so that the total current density passing is then the sum of a double-layer charging component, idb and

123

The Double Layer at Capacitor Electrode Interfaces

a Faradaic solution decomposition component, iF, i.e., i = i dl + iF, as discussed in Chapter 3, Section 3.8. In the presence of other solutes or impurities that have thermodynamic oxidation and reduction potentials that lie between the potential limits for water (or other solvent) decomposition, Faradaic currents can also pass in parallel with the double-layer charging current. Such Faradaic currents can obey the Tafel equation with respect to variation of the electrode potential, E (or overpotential, 1/, i.e., log iF is proportional to 1/ or E); or they may be diffusion controlled, depending on how their io values compare with their diffusion-limited maximum current densities, e.g., when some impurities are present at low concentrations. With modern impedance spectroscopy equipment, now there is usually no problem in distinguishing double-layer charging processes from Faradaic ones, including the onset of diffusion control in the latter. 6.B. EQUIVALENT CIRCUIT REPRESENTATION OF DOUBLE-LAYER ELECTRICAL BEHAVIOR

The electrode interphases referred to here can usefully be represented by equivalent circuits (Chapter 17); that for an ideally polarized electrode is simply a capacitance (a in the diagram) which may, however, have a potential-dependent value. In the case where a Faradaic process may also pass a current that is parallel with the double-layer charging current, the equivalent circuit b in the diagram applies, with an equivalent Faradaic leakage resistance RF (Eq. 23 in Chapter 3). RF is usually exponentially dependent on electrode potential, E, but, for small excursions, Ll V, of potential (see Chapter 16), it is approximately linear in Ll V or overpotential, 1/. Its variation with electrode potential can be indicated by so-called "micropolarization" experiments or by observing changes in the diameters of Z" vs. Z' plots in the complex-plane representation of impedance measurements at various constant electrode potentials (Chapters 16 and 17).

Cdl

b

The behaviors of equivalent circuits (a) and (b) are easily distinguished by the difference between their impedance spectra with variation of frequency; thus (a) is purely capacitative while (b) has a maximum capacitative impedance for a given value of Cdl and RF (for a particular potential) at a certain frequency.

124

ChapterS

The Faradaic leakage resistance, RFo in circuit b is very important as the basis of self-discharge (see Chapter 18) in electrochemical capacitors and in battery cells. Its role will be analyzed in Chapter 18. In the case of electrodes that are base metals, nonideal polarizability usually occurs because of the possibility of anodic corrosion or oxide film formation at potentials already near the H2 reversible potential; this leads to Faradaic leakage currents in parallel with double-layer charging. In addition, such metals usually have larger exchange current density for H2 evolution from water, so they cannot be polarized very far cathodically to the RHE or SHE potentials without appreciable H2 evolution currents arising.

REFERENCES 1. E. A. Guggenheim, J. Chern. Phys., 4, 689 (1936). 2. N. K. Adam, The Physics and Chemistry of Surfaces, 3rd ed., Chapter 3, p. 107, Oxford University Press, Oxford (1941). 3. H. von Helmholtz, Ann. Phys. (Leipzig), 89, 211 (1853). 4. R. Parsons, Chapter 4 in Modem Aspects of Electrochemistry, J. O'M. Bockris and B. E. Conway, eds., vol. 1, Chapter 4, Butterworths, London (1954). 5. N. D. Lang and W. Kohn, Phys. Rev., Bl, 4555 (1970); B3, 1215 (1971). 6. S. Amokrane and J. P. Badiali, in Modem Aspects of Electrochemistry, J. O'M. Bockris, B. E. Conway, and R. White, eds., vol. 22, Chapter 1, Plenum, New York (1992). 7. G. Gouy, Ann. Phys., Paris, 7, 129 (1917); J. de Phys., 9, 457 (1910). 8. R. H. Fowler and E. A. Guggenheim, Statistical Thermodynamics, p. 77, Cambridge University Press (1939). 9. D. L. Chapman, Phil. Mag., 25, 475 (1913). 10. B. E. Conway, Ionic Hydration in Chemistry and Biophysics, Chapter 10, Elsevier, Amsterdam (1981). 11. P. Debye and E. Hiickel, Phys. Zeit., 24, 185 (1923). 12. L. Onsager, Phys. Zeit., 27, 388 (1926); 28, 277 (1928). 13. M. Green, in Modem Aspects of Electrochemistry, J. O'M. Bockris, ed., vol. 2, Chapter 5, p. 343, Butterworths, London (1961). 14. O. Stem, Zeit. Elektrochem., 30, 508 (1924). 15. R. W. Gurney, Ionic Processes in Solution, Dover, New York (1940). 16. A. N. Frumkin, Zeit. Phys. Chem., AI64, 121 (1933); Acta Physicochim., USSR, 6, 502 (1937). 17. D. C. Grahame, Chem. Rev., 41, 441 (1947). 18. J. A. V. Butler, Proc. Roy. Soc. Lond.,A 157, 423 (1936). 19. B. E. Conway, J. Electroanal. Chem., 123, 81 (1981) 20. B. E. Conway, Theory and Principles of Electrode Processes, Ronald Press, New York (1964). 21. F. P. Bowden and E. K. Ridea1, Proc. Roy. Soc., Lond., AI07, 486 (1925); A114, 103 (1927); A119, 680; 686 (1928). 22. J. A. V. Butler, Electrocapillarity, Methuen, London (1940). 23. J. D. Bernal and R. H. Fowler, J. Chem. Phys., 1, 515 (1933).

Chapter 7

Theoretical Treatment and Modeling of the Double Layer at Electrode Interfaces

7.1. EARLY MODELS

In order to provide a fundamental basis for understanding the properties and behavior of double-layer types of capacitor devices, this chapter gives a broad account of the theoretical treatments of the structure and capacitance of the double layer at electrode interfaces. This topic has been one of major activity and interest in electrochemistry for about a hundred years, and has now found substantial technological applications. In 1997, the Electrochemical Society Sponsored a major symposium on the double-layer to recognize the 50th anniversary of Grahame's seminal paper! in Chemical Reviews (1947). Electrostatic and thermodynamic treatments of the double layer are based on a model in which the interphasial region between an electrode and an ionic solution is ideally polarizable (Chapter 6), i.e., a potential difference can be established between a metal electrode surface and the inner boundary of an electrolyte solution without Faradaic charge transfer processes taking place. The interphase is then ideally capacitative and the electrode is referred to! as an ideally polarizable one. The interphase then has a pure double-layer capacitance that ideally is frequency independent in an ac evaluation of that capacitance. This is the ideal requirement for an electrochemical capacitor device. In practice, some frequency dependence is commonly observed, i.e., the phase angle for the double-layer capacitance may not have the ideal value of 90° at all frequencies, and potential-dependent dc leakage can also occur (Chapter 18). Deviations from ideal capacitative behavior can arise when there is some 125

126

Chapter 7

dielectric loss associated with the solvent orientation polarization in the doublelayer dielectric (at very high frequencies) and/or when there are some slow anionic chemisorption processes that lead to lower frequency losses. In either case, there is energy dissipation in the charging and discharging cycles at an incompletely polarizable electrode interface. Another source of nonideal behavior is distribution of the double-layer capacitance over a porous electrode surface (Chapter 14). In a practical electrochemical capacitor device, frequency dependence of the overall capacitance is generally observed and is due, in addition to the "porous electrode" effect, to coupling with other equivalent series resistance (esr) components. As emphasized by Grahame, l the double layer at an ideally polarizable electrode is in a state of electrostatic equilibrium, in contrast to other types of electrode interphases across which free charge transfer takes place (nonpolarizable interphases). This corresponds to an electrochemical thermodynamic equilibrium, ideally of a Nernstian kind. Theoretical treatments of the structure (Chapter 6) and properties of the double layer can be conveniently considered in a hierarchy of four sections: 1. The first is the diffuse part of the double layer [Fig. 6.3(b), Chapter 6], beyond (toward the solution) the inner contact layer of ions (solvated or otherwise) that defines the so-called "Helmholtz layer" [Fig. 6.3(a), Chapter 6]. Historically, this was the first theoretical examination of double-layer properties in work by Gouy and by Chapman. 2,3 2. The second is the compact or Helmholtz layer,4 which became treated in terms of adsorption of ions as part of Stem's combination5 of the Helmholtz and Gouy models, taking into account the finite sizes of the ions of the electrolyte, which determine their distances of closest approach to the electrode surface [Fig. 6.3(c), Chapter 6]. Later, Grahame 1 treated the compact region in terms of two layers, one for anions located at their distances of closest approach to the electrode metal, and the other for hydrated cations located at their somewhat longer distances of closest approach, which are determined by their hydration (or solvation) radii, as shown in Chapter 6. The two regions distinguished by Grahame are referred to as the "inner Helmholtz layer" (or plane) and the "outer Helmholtz layer," (or plane), respectively. 3. Work on the two classically distinguished regions of the double layer in (1) and (2), was followed by important treatments of the role of adsorbed solvent dipoles and their potential (field)-dependent orientation in the compact regions of the double layer6- 9 in determining the capacitance. 4. Finally, in relatively recent years, attention has been paid to the potential-dependent spillover of electron density from the formal surface of

The Double Layer at Electrode Interfaces

127

the metal toward (or away from) the solution and compact layers, depending on the sign of charge of the electrode metal, and its influence on the interphasial capacitance of the electrode. 10-12 The above factors in double-layer behavior are interactive and determine the overall capacitance of the electrode interphase and its dependence on electrode potential, the types of ions in the electrolyte solute, and the solvent in which they are dissolved. Such an interphasial system is obviously complex and has been modeled according to various levels of recognition of this complexity. These are treated in the following sections.

7.2. TREATMENT OF THE DIFFUSE LAYER

For completeness and for historical reasons, in discussing the models of the double layer, we first outline the electrostatic statistical treatment of the diffuse part of the double layer. In practice, however, the behavior of the diffuse-layer capacitance is of less significance for the properties of electrochemical capacitors than is that of the inner layer since the latter dominates the overall interfacial properties at the high concentrations of electrolyte commonly employed in capacitors to minimize equivalent series resistance or internal pore resistance. However, treatments of the diffuse part of the double layer have formed a cornerstone in our understanding of double-layer properties, and an accurate theory of diffuse-layer capacitance is an essential requirement for evaluating compact layer capacitance. This latter mainly determines the capacitance of doublelayer-type devices, through application of the reciprocal sum relationship (Eqs. 7.11 and 7.12) to experimental data on overall capacity. Direct theoretical treatment of the inner layer itself, although included in Stern's paper,5 has been slow to develop in reliable form, owing to the complexity of the modeling involved (see later discussion). The treatment of the diffuse-layer ion distribution [Fig. 6.3(b), Chapter 6] and its dependence on the surface charge density, qM, on the metal, which is related to the latter's electrode potential relative to the potential at which the electrode bears a zero charge (the potential-of-zero charge), proceeds in the following way: Poisson's electrostatic equation is used in conjunction with the Boltzmann distribution law4 and applied to the electrostatic energy experienced by both cations and anions owing to the interaction of their charges with the potential that results from the electrode surface charge or, more particularly, to the gradient of the electric field near the electrode. If IfIr is the potential at a distance r from the electrode surface, the local concentration of cations and anions, c+ and c, at r will be4

128

Chapter 7

(7.1) where cO terms refer to the bulk average stoichiometric concentrations of cations and anions related by c+z+ = IczJ, and z+ and z_ are the charge numbers of the cations and anions, respectively. The space charge density pat r is then

_

Pr - 4 ed?-

J 4 kTe/flr] + z_ec_ eX1J z_e/flr] eX1kT 0

(7.2)

if only electrostatic interactions determine the ion distribution. As in the Debye-Hiickel theory, a further relation is available between /fir and Pr by use of Poisson's equation relating field gradient to space charge density, P, but expressed for a one-dimensional electrostatic distribution of ions in the form (7.3) since Br may be a function of r in the double layer as first treated by Conway, Bockris, and Ammar. 13 ,14 Substitution of Pr from Eq. (7.2) into Eq. (7.3) gives (7.4) Using the identity (7.5)

in Eq. (7.4), and integrating with Br ='ii, a mean constant value of the dielectric constant B in the double layer (since B is not very sensitive to variation of r except very close to the electrode surface), gives

(7.6)

for the boundary conditions d/flr1dr -7 0, r -7 00; and /fir -7 /fIs, the potential in the bulk of the solution where Pr =O. The term -1 in the braces in Eq. (7.6) arises from the integration constant, i.e., when r -7 00 , exp[(/fIr - /fIs)lkT] = 1 for d/flr1dr=0.

The Double Layer at Electrode Interfaces

129

At this point in the treatment of the diffuse double layer it is assumed that Gauss's relation (7.7) can be applied near the electrode surface bearing the charge density, qM, at the distance of closest approach, a, of the ions to the electrode surface. Sa is the local value of the dielectric constant, s, in that region. Sa is taken as a mean value though Sa may depend on potential (field 13 ) and will probably have a different mean value on each side of the pzc. Also, solvent dipole orientation (cf. dielectric saturation effects) at the electrode surface6- 9 leads to a lower value of S in the region r::e a than further away from the electrode. From Egs. (7.6) and (7.7), qM is obtained with the above assumptions as

e,

(7.8)

C (=

For a symmetrical electrolyte, when Iz±1 = z+ = IzJ (= z, say) and c± = c+ = c, say),

_ (2kTCi]112 . h [Ze(lf/a - If/S)] n sm 2kT

qM-

(7.9)

noting that when qM is positive If/a - If/s will also be positive. It is seen that Eg. (7.9) is a relation between the qM and the potential difference between the compact layer plane (If/a) and the solution (If/s). Therefore differentiation of qM with respect to that difference will give a capacitance quantity, the capacity of the diffuse part of the double layer.

7.3. CAPACITANCE OF THE DIFFUSE PART OF THE DOUBLE LAYER

If the ions are point charges as considered by Gouy, If/a becomes identical with C= C2 (fromEq. 7.10 with 'l/a= '1/1), or when C2 » Cj, C = C 1; the first inequality is valid when '1/1 - 'l/s is quite small, and the second when it is large. It is important to note that it is the value of the smaller capacity contribution that mainly determines the overall value of C since the two contributions are in a series relationship (Eq. 7.12), as discussed in Chapter 6 (Eq. 6.1). Apart from giving a quite accurate representation of the diffuse-layer capacitance, the Gouy-Chapman theory is important because it provides the means of evaluating the inner-layer capacitance, C 1, which is of greater interest, from the overall C measured, through Eq. (7.12). In fact, at the rather high concentrations of electrolytes employed in aqueous double-layer capacitors, it is the compact or Helmholtz layer capacitance that mainly determines the capacitance of a double-layer electrochemical capacitor device. Equation (7.13) represents the data for capacitance at the mercury/solution interface (e.g., for a KCI solution!), if C 1 = 18 f1F cm-2 is assumed when cations preferentially popUlate the interface region (l/JM - 'l/s negative) and C, = 38 f1F cm- 2 when the anions dominate in the double layer (l/JM - 'l/s positive). Typical capacity-potential profiles for a mercury electrode are shown in Fig. 7.1 (from Ref. 1). The capacity contribution C 1 is referred to as the capacity of the Helmholtz layer and C2 is that of the diffuse layer or ionic atmosphere region. Further divisions of C! into contributions from the inner or the outer Helmholtz layers,' referred to earlier, will be discussed later. In Eq. (7.13) it will not generally be validl,15 to assume that the same potential '1/1 applies to the position of closest approach for anions and for cations. Owing to the more polarizable nature of anions and their specific affinity as adsorbed surface ligands for some metal surfaces, anions may approach more closely than hydrated cations at a given potential, so that '1/1 will not have quite the same significance for anions as for cations. It is convenient to call this potential'l/2 for the region of closer approach and specific adsorption, i.e., chemisorption associated with anions. This matter is now examined in more detail according to the theory developed by Stern5 and reviewed by Parsons. 15

132

Chapter 7

NaGI

NoF 46

0.1 M

Hump

28

20

0.001 M

12

N

~Influence of

Ie

u

Cditt.

LL.

::i... .......

4

W U

Z

0.8

1 M), a rather different model consisting of a quasi-lattice of anions, cations, and solvent molecules near the electrode interface over distances ~l nm has to be adopted 15 and a diffuse layer, in the original sense, ceases to exist. At an electrode interface distributed within a microporous structure, the configuration of the distribution of electrode potential into the solution phase is much more complex than at an infinite planar electrode. 72 The problem arises on account ofthe thickness,76 i.e., the Debye length, (K- i ) of the diffuse part of the double layer in relation to the width or space available for the ion distribution to be set up within the pores of the electrode. In the original Debye-Hiickel theory,74,75 the Poisson-Boltzmann equation relating the second-derivative operator, V21f!, to the ionic space charge density became transformed to a second-order differential equation, V21f! = K21f!, where K2 involves the ionic strength I of the solution,16 i.e., K is square-root in I. It was also shown that K- i had the dimensions of distance, namely, the effective radius of the ionic atmosphere distribution around any given ion. A similar significance applies to the effective thickness of the diffuse-layer ionic atmosphere in double-layer theory, which is inversely related to a function of the ionic strength of the electrolyte. From this it can be appreciated that in very fine pores of a porous electrode matrix, say 2-10 nm in extent (diameter), the diffuse region of the double layer in dilute electrolyte solutions could extend all the way across the pore diameter and overlap72 with the diffuse layer ion distribution at the other side of the pore (Fig. 7.14). This situation then becomes similar to that envisaged and mathematically treated by Verwey and Overbeeck in their monograph (see general reading reference 1) on the theory of the stability oflyophobic colloids, and the fall of potential with distance in the diffuse layer73 between the inner boundaries of the double layer is then less than for free, noninterfering interfaces. The ion distributions then overlap, so that the charge density and potential equations become much more complex 72 than those given in the Gouy-Chapman theory.

The Double Layer at Electrode Interfaces

163

..J

is developed. Its behavior is exactly the same as that of the ionic diffuse-layer capacitance outside the carbon/solution interface, except for the dielectric constant of the medium in which the charges reside. The same Gouy-Chapman mathematics (Chapter 7) apply to the treatment of this space charge region, employing the Poisson-Boltzmann equation. Further details are described in Section 9.6. Since Csc is in a series relation with the other components of capacitance, it adds to them in a reciprocal manner and is hence only significant in the overall C when it (Csc ) is small; thus: 1

1 CH

1

1

Cdiff

Csc

-=-+--+C

(9.1)

Normally, graphitic carbon materials would not be expected to exhibit semiconducting behavior since the free electron density is sufficiently large for

196

Chapter 9

such materials to behave in an almost metallic way (metalloid behavior). However, there are some experimental indications 1,2 for the basal-plane from double-layer capacity measurements that semiconductor properties are in fact exhibited. Because of the asymmetric structure of crystalline graphite in three dimensions, there will be corresponding differences in the electronic work function of the basal plane and edge sections. According to general principles of interfacial electrochemical science,33 it follows that there must be related differences of potential-of-zero charge. This then leads to specific differences in anion adsorption as a function of graphite-crystal electrode potential, on the basal plane relative to edge sections. Of course, in powdered graphite preparations, these differences in properties will be unevenly distributed, causing heterogeneity in surface properties and interfacial electrochemistry.

9.4. OXIDATION OF CARBON

A very large amount of work has been done on oxidation of carbon including of course the ultimate oxidation processes, combustion to CO and CO 2, Electrochemically, oxidation is much more restricted, especially at ordinary temperatures, but fuel cells employing a coal slurry have been investigated in a number of works and applications. For carbon-based double-layer capacitors, the state of superficial oxidation especially at edge planes, is of most significance because this can determine the charge accommodation per gram and the corresponding capacitance. It can also influence the self-discharge characteristics (Chapter 18) of carbon capacitors and how carbon materials can be conditioned by heating them to elevated temperatures in a vacuum, nitrogen, or hydrogen, and sometimes in water vapor. One of the earliest conferences devoted to the state of carbon was the Faraday Discussion in 1937 on chemical reactions involving solids, published in the Transactions of the Faraday Society. This volume included a classic paper by Meyer34 on surface reactions of graphite with oxygen, carbon dioxide, and water vapor; and other papers on the characterization of surface groups,? (e.g., ketonic, ketenic, and quinonoid) on carbon, including the possibility of formation of surface peroxide groups due to direct combination with dioxygen molecules, either bridged or end-on.? At elevated temperatures (1450 DC), some reaction to form surface groups [e.g., a-diketo functions (ortho-quinone)] occurs with release of H 2, though some rupture points at edges lead to decomposition of such groups. Under conditions of anodic electrolysis, the a-diketo grouping on graphite, which corresponds to an ortho-quinone structure, is capable of being oxidized. 32 At low current density in H 2S04, the maximum volume of CO generated is

197

The Double Layer and Surface Functionalities at Carbon

equal to that of CO 2, This process occurs at rupture points on the edges of the graphite planes, with breaking of the outer C-C bonds; a new keto group is formed in the graphite structure while the outer keto group is oxidized to a carboxyl group. Sihvonen proposed further complex surface-edge chemical reactions?2 Figure 9.5 is a schematic diagram of various reaction product groupings at the surface of an element of the graphite structure. As indicated earlier, these structures also determine the electrochemical interfacial state of graphitic carbon materials and their double-layer properties, including pzc values and specific adsorption of ions. It must be emphasized, however, that a variety of high specific-area carbon preparations used in fabricating electrochemical double-layer capacitors are not graphitic in structure, though some of the surface groupings referred to here may be present. In other work, the role played by surface oxides in the oxidation of carbon was treated by Strickland-Constable,? who emphasized the significance of :::;C=O groups on the surface, like those in the structure of acetone. He concluded that the oxidized state of carbon surfaces consists of covalently bound surface compound species. This conclusion was in agreement with Schilow,35 who showed that when outgassed charcoal was treated with molecular 2, surface compound species developed that had both acidic and basic properties, but such species corresponded to only a small fraction of the bound oxygen. In this work it was indicated that amorphous carbon was still mainly graphitic, but con-

°

FIGURE 9.5. Surface-edge chemical-oxidation structures and related reaction products developed at graphite. After Sihvonen;32 in the original paper, Roman numerals identified various reactions and functionalities at the surface that are referred to in that paper.

198

Chapter 9

sisted of microcrystallites of that structure. These papers provided some of the earliest ideas on surface or edge functionalization of carbon materials. Somewhat surprisingly, Lambert36 and others found that the oxidation of carbon treated with iron or manganese compounds is partly inhibited by catalysis of the decomposition of surface oxide function ali ties that are otherwise stable. This early work on the oxidation of carbon provided an important basis for understanding the state of carbon (graphite) surfaces that had been exposed to oxygen or electrolytically oxidized. Later work has been thoroughly reviewed in Kinoshita's monograph,3 Carbon: Electrochemical and Physicochemical Properties, referred to earlier. This volume is a major source-book on the properties of various carbon materials, their electrochemical, chemical and surface reactivity, their characterization and the commercial sources of various modifications of carbon, some of which are important for electrochemical capacitor fabrication or primary battery production.

9.5. SURFACE SPECIFICITY OF DOUBLE·LAYER CAPACITANCE BEHAVIOR AT CARBON AND METALS

The behavior of double layers at electrode interfaces is known to be very surface specific (Chapters 6 and 7). This situation applies to the compact layer, which can involve short-range adsorptive interactions between ions ofthe electrolyte (especially anions) and surface atoms or, in the case of carbon materials, surface functionalities usually involving the oxygen species referred to earlier. In addition, specific interactions among these surface species and/or the atoms of the underlying electrode material surface, and with the solvent molecules in the double layer, are involved. Both the intrinsic surface properties of given crystal faces of the electrode material and any surface functionalities present determine33 the electronic work function,

..........

o

0.4

+-

c

Q)

+-

o

Langmuir

case I t

0... 0.2

.....

-4

,.

/.---

.

~.

~

I

'

o

4

8

12

16

20

9 FIGURE 10.2. Widths (AV1I2) at half-height of C,p vs. V profiles as a function of the lateral interaction parameter, g. (From Conway and Gileadi 1 and Kozlowska et al? Copyright 1962 and 1977 Royal Society of Chemistry.)

therm equation will contain two exponential terms, exp -[fOIRT] and exp[VFIRT] (i.e., in combination, exp[(VF - jfJ)IRTJ. This illustrates how VF must be increased as/is increased positively in order to maintain a given value of (VF - /O)IRT, which then determines the value of the configurational term9 01(1 - 0) of the isotherm function (Eq. 10.9). If the g value is negative, up to g = -4, the opposite conclusion applies; the electrosorption is easier with increasing coverage due to attractive interactions in the adlayer, so that a relatively lower potential V is required to attain a given coverage, and the effective potential range required to reach almost full coverage (0 --7 1) is then diminished. Because the required charge (ql) is still the same for monolayer formation, the resulting linear-sweep voltammogram must be sharper than for the Langmuir case, g or/ =0 (see Fig. 10.3). Note that the integral under the voltammogram at a sweep rate s = dVldt gives the charge passed: V2

V2

f i(V)· dt= f i(V)· dV/s

VI

VI

(10.11)

230

Chapter 10

0 .0 1.0

0.8

e

b

9 -8

0.6

0.4

0 .2

0.0

-0.2

0.0

0.2

OA

0..6

Potential/Volts FIGURE 10.3. (a) Ctp vs. electrode potential profiles for an electrosorption process involving negative g values, i.e., lateral attractions. (b) Related isotherms for coverage, B, as a function of potential for positive and negative g values.

Electrochemical Capacitors Based on Pseudocapacitance

231

for a sweep between potential limits V2 and VI. Then since i(V)/s is a pseudocapacitance C¢J' the integral is equivalent to charging the C¢J from potential VI to V2. The charge passed is then C¢J (V2 - VI) or f~2 C¢J . dV since C¢J is rarely constant with changing potential, especially for elec'trosorption processes. I - 7 Experimentally, the pseudocapacitance behavior represented by the above equations for C¢J can be directly recorded by means of cyclic voltammetry since for a sweep rate, dV/dt,

C¢J· dVldt= i

(10.12)

or C¢J =iI(dVldt)

(10.13)

where i is the recorded response current density for a system exhibiting pseudocapacitance, addressed by a potential that varies linearly in time t. Thus linearsweep or cyclic voltammetry is the most convenient technique for accurately recording C¢J behavior, including situations commonly encountered where C¢J is far from constant and the CV may exhibit more than one current maximum in the swept potential range. This situation is common on polycrystalline surfaces where several maxima in C¢J vs. V profiles are observable in UPD studies (Fig. lOA); it is interesting that even for well-prepared single-crystal surfaces [e.g., ofPt(lOO)], more than one peak in the C¢J profile can arise as for H [Fig. 10.4(a)]. Similarly, multiple peaks arise in the UPD ofPb adatoms on single-crystal surfaces of Au [Fig. lO.4(b)]. In evaluation of pseudocapacitance by means of cyclic voltammetry, it must be remembered that there will always arise components of response currents owing to the ubiquitous presence of the double-layer capacitance. This will lead to a double-layer charging current, Cdl (dVldt), that is proportional to the sweep rate; this current is normally 5 to 10% of the (maximum) pseudocapacitance charging currents when C¢J is appreciable. The double-layer charging component can usually be distinguished from the pseudocapacitance component by its different frequency dependence; thus, the double-layer capacitance response is usually maintained at up to 105_106 Hz in a well-designed measurement system while pseudocapacitances are often dispersed in the range 103-104 Hz due to electrode kinetic rate limitations. Writing the isotherm function, Eq. (10.9), logarithmically, i.e. In 0/(1- 0) + gO =In KCH+ + VFIRT

(10.14)

232

Chapter 10

.... I

E u

ci

:1.

::::::

J

2

100

a FIGURE 10.4. (a) Cyclic voltammetry profiles for UPO of H adatoms on clean Pt single-crystal surfaces: I, (111); 2, (tOO); and 3, (110) surface. (b) Cyclic voltammetry profiles for UPO of Pb adatorns at indicated single-crystal surfaces of Au. [(a) From Clavilier et aI., 1. Electroanal. Chern., 295, 333 (1990); (b) from Engelsman et al. 18]

it is seen that differentiation W.r.t. V will give rise l to two reciprocal capacitance components 2 : RT - . d 1n[8/(1- 8)] qlF

(lO.15a)

and RT qlF' g

(1O.15b)

The first corresponds to the reciprocal of a Langmuir-type capacitance and the second to the reciprocal of a Temkin-type capacitance but originating for physi-

cal reasons lO different from those envisaged by Temkin,14 namely, surface site heterogeneity. Since the terms ofEqs. (1O.15a) and (1 0.15b) arise from addition in Eq. (10.14), they can be regarded as corresponding to two capacitance contributions in series; that is, a configurational9 capacitance contribution from the Oil - 0 term ofthe isotherm and an interaction contribution from the exp(-gO) term ofEq. (10.9). Obviously, as g ~ 0, the adsorption behavior reverts to that represented by a Langmuir isotherm. For intermediate values of 0 (e.g., 0.3 < 0 < 0.7) and appreciable values of g (e.g. ~ 10), 0 becomes approximately linear in V, a result also corresponding formally to Temkin's isotherm. This corresponds to a potential range over which C¢ is approximately constant with changing potential, a condition that is desirable if pseudocapacitative electrochemical capacitor devices are to be practically useful over a reasonable charging voltage range. Later in this chapter it is shown that a Nernst-type redox equation has the same form as the electrochemical Langmuir isotherm equation. 2 Hence, upon

234

Chapter 10

differentiation it also leads to a pseudocapacitance having the same properties as those represented by Eq. (10.6); such pseudocapacitance is measurable and can be large. However, like that arising for chemisorption processes, it can be strongly potential dependent, with a large maximum when concentrations (or activities) of "Red" and "Ox" species are equal. Conway and KU 15 discovered another interesting system that exhibits redox pseudocapacitance over a potential range of ca. 0.8 V-FeS2 (the mineral pyrite, "fool's gold"). This is a ferrous salt of the disulfide (persulfide) ion, -S-S-. It can be reversibly reduced in a 2e reaction S~-

+ 2e ~ 2S 2-

Cathod ic

1.389

..

(10.16)

10

(bl

9

.....

............. ,

'~ - 200

r U

c

0.2

:.=

c

Q)

0

0.1

a..

0.0

electrochem'lcall electrochemical reaction reaction reversible I irreve sible 0.05L---------~--------~0~--------2~------~4 -4 -2

log ( :1) FIGURE 10.10. Peak potential values for linear-sweep voltammograms for various slkl values (based on Gileadi and Srinivasan6 and Kozlowska, Klinger, and Conway\

sweep rate up to which complete reversibility is maintained) to irreversible behavior when S > So or s » So. The latter authors also considered the effect of the introduction of interaction effects, g > 0 or g < 0, on the shapes of the voltammetry (C¢) curves (Fig. 10.11). They showed the behavior of cyclic voltammograms when sweep reversals were conducted from various potentials in the sweeps corresponding to incomplete coverages, e < 1, and the resulting shapes of the current response curves for the opposite directions of sweep after successive sweep reversals from those potentials. Examples are shown in Figs. 10.12 and 10.13.

10.3.4. Relation to Behavior under dc Charge and Discharge Conditions

The results of these various calculations by Gileadi et a1. 5•6 and by Conway with Kozlowska and Klinger7 are relevant to the performance of electrochemical capacitors that utilize pseudocapacitance rather than double-layer capacitance for charge (energy) storage. Although the calculations have been performed for cases relevant to the characterization of the pseudocapacitance by means of linear-sweep voltammetry, the conclusions apply equivalently to dc charging or discharging at various rates where Tafel-type polarization can set in at elevated charge or discharge rates when the net currents for such processes

244

Chapter 10

0.6

....

N

Ie ()

C/I

ec u.. ..,

g=10 1- =10

0.4

0.2

""C

0.0

0.4

g M

U

Potential (Volts)

0.2 0.4 0.6 0.8

1.0

FIGURE 10.11. Effect of interaction effects on cyclic yoltammograms for a process displaced from equilibrium. s > so. (From Conway and Klinger, unpublished.)

exceed the exchange current density for the reversible process, i.e., for conditions where either the forward or the reverse terms of Eq. (10.17) become dominant on charge or discharge. Such changes of conditions are exactly the same, in principle, as those that arise in the polarization of regular, continuous Faradaic reactions as the overvoltage is progressively increased from zero to appreciable values. Thus, the behavior of surface processes involving underpotential deposition is closely analogous to that of regular Faradaic reactions except that the finite availability of surface sites (1 - ~ 0 as ~ 1) and a corresponding finite value of ql restricts the possibility of continuously (and exponentially) increasing currents as the potential is raised since the processes take place 2-dimensionally on a finitely limited density of surface sites, about 1015 cm- 2• Note also that a surface (UPD) electrochemical reaction that is proceeding under equilibrium or quasi-equilibrium conditions has a current that is determined at all potentials in the sweep, not by an overvoltage, but by a potential defined by the equilibrium adsorption isotherm, e.g., Eq. (10.7). Thus the latter is a form of the Nernst equation for relative extents of vacancy (1 - e) and oc-

e

e

Electrochemical Capacitors Based on Pseudocapacitance

t=)O-3

245

: =10- 2

1.6

0.8

0

Ie

N

-0.'

0.2

0.4

-0.1

./.

\

f\' /0.2

0.4

0.8

(,) f/)

1.6

~ ........

2.4

e

"0

-

'0.

(b)

(0)

.1..-10 k -

t=10-1

1.6

U

0.8

0

L.z

-0.'

'Y

0.5

,0/I

0.8

1.6

l

jJ) :r,(00.1 0.'

Vp.v'o

(c)

W I I

(d)

POTENTIALS/V

e

FIGURE 10.12. Anodic and cathodic CV profiles for various initial cathodic values and for various degrees of reversibility of reaction (I). The values of slk are (a) 10-3, (b) 10-2, (c) 10- 1, and (d) 10; g = 0 for all cases. Reprinted from H. A. Kozlowska, 1. Klinger, and B. E. Conway, 1. Electroanal. Chern., 75, 45 (1977), with permission from Elsevier Science.

cupancy (B) of surface sites; each state of relative occupancy, characterized by the ratio, B/(l - B) is associated with its own characteristic equilibrium potential and as the potential applied experimentally is varied, the charge passes to provide the necessary occupancy (or vacancy) of sites, like charging a capacitance. Only if the rate of potential change, s, exceeds the rates at which occupancy or vacancy of the adsorption sites (or states of oxidation and reduction in a redox

246

Chapter 10

1.0

~ =10 g=2

0.5

N

IE

6

III

/

u

"C

...

0

0

I.L. ..........

r or » C dl . If the Faradaic resistance, R F , which decouples C¢ from C dl is small or becomes small at high potentials, then usually when C¢ is significant, it cannot be distinguished; C¢ and Cd! then effectively add in parallel.

256

Chapter 10

In the case of a porous electrode (Chapter 14), however, the distinction between Cdl and CtP , when the latter is significant, is much more difficult owing to the distributed nature of both CtP and Cdl throughout the porous matrix, with a progressively increasing electrolyte resistance down pores (Chapter 17) in a series-parallel arrangement with the C components. For such reasons, it is usually difficult to distinguish a CtP component in carbon double-layer capacitors, although it is probably significant owing to reactive surface functional groups (Chapter 9). The discussion given here concerning the distinction between double-layer and pseudocapacitance in experimental measurements has implicitly assumed that the processes involved in double-layer charging and Faradaic passage of a charge in charging the pseudocapacitance at the same electrode are independent. This, however, is not necessarily true and there can in fact be some coupling between double-layer charging and simultaneous Faradaic processes. Although it is a somewhat arcane topic, this matter has been thoroughly discussed and treated in a review by Parsons.z5 It can be a significant factor when quasiFaradaic processes such as anion-specific adsorption with partial charge transfer12,13 are involved. However, more detailed examination of this problem is beyond the scope of this chapter, so readers are referred to Ref. 25 and the literature cited there.

REFERENCES 1. B. E. Conway and E. Gileadi, Trans. Faraday Soc., 58, 2493 (1962). 2. E. Gileadi and B. E. Conway, in Modem Aspects of Electrochemistry, J. O'M. Bockris and B. E. Conway, eds., vol. 3, Chapter 2, Butterworths, London (1965). 3. E. Gileadi, Electrosorption, Plenum, New York (1967). 4. A. Eucken and B. Weblus, Zeit. Elektrochem., 55, 114 (1951). 5. E. Gileadi and B. E. Conway, 1. Chem. Phys., 31, 716 (1964). 6. E. Gileadi and S. Srinivasan, Electrochim. Acta, 11, 321 (1966). 7. H. A. Kozlowska, 1. Klinger, and B. E. Conway, 1. Electroanal. Chem., 75, 45 (1977). 8. M. Boudart, 1. Amer. Chem. Soc., 72, 3566 (1952). 9. T. L. Hill, Introduction to Statistical Thermodynamics, Addison-Wesley, Reading, Mass. (1960). 10. 1. G. N. Thomas, Trans. Faraday Soc., 57, 1603 (1961). 11. S. K. Rangarajan, Specialist Periodical Reports (Chemical Society, London), vol. 7, p. 203 (1980). 12. W. Lorenz and G. Salie, Zeit. Phys. Chem., N.F., 29, 390,408 (1961). 13. 1. W. Schultze and F. D. Koppitz, Electrochim. Acta, 21, 327, 337 (1976). 14. M. I. Temkin, Zhur. Fiz. Khim., 15, 296 (1941). 15. B. E. Conway and J. Ku, 1. Coli. Interfacial Sci., 52, 542 (1973). 16. H. A. Kozlowska and B. E. Conway, 1. Electroanal. Chem., 95,1 (1979). 17. 1. O'M. Bockris and H. Kita, 1. Electrochem. Soc., 108, 676 (1961). 18. K. Engelsman, W. J. Lorenz, and E. Schmidt, 1. Electroanal. Chem., 114, 1 (1980). 19. A. Rudge, 1. Davey, I. Raistrick, and S. Gottesfeld, 1. Power Sources, 47, 89 (1994).

Electrochemical Capacitors Based on Pseudocapacitance

257

20. B. E. Conway, Electrochim. Acta, 38, 1249 (1993). 21. U. Sacken, M. Sc. thesis, physics, Univ. British Columbia, Vancouver, Canada (1980). 22. Y. Chabre, Extended Abstracts, Sixth International Meeting on Li Batteries, Wilhelm University, Munster, Germany, p. 459, section IV-C-02 (1992). 23. 1. Thompson, 1. Electrochem. Soc., 126,608 (1979). 24. K. Kinoshita, Carbon, Wiley, New York (1988). 25. R. Parsons, in Advances in Electrochemistry and Electrochemical Engineering, P. Delahay and C. W. Tobias, eds., vol. 7, p. 177, Interscience, New York (1970).

GENERAL READING REFERENCES 1. E. Gileadi, Chapter J in Electrode Kinetics for Chemists, Chemical Engineers and Materials Scientists, VCH Publishers, New York (1993). 2. D. G. Grahame, Chem Rev., 41, 441 (1947).

Chapter 11

The Electrochemical Behavior of Ruthenium Oxide (Ru02) as a Material for Electrochemical Capacitors

11.1 HISTORICAL ASPECTS

Ruthenium is an element giving rise to many compounds having interesting and often unique properties (e.g., its complexes with N-containing ligands and its oxide, Ru02) which are usually nonstoichiometric. The element was discovered by K. K. Klaus of the University of Kazan in 1844 and named ruthenium after Ruthenia, the place where it was found. The concept and use of RU021-7 as an electrochemical supercapacitor material can be traced to the paper by Trasatti and Buzzanca6 on ruthenium dioxide as a new interesting electrode material. Ru02 had been developed earlier as an excellent electrocatalytic surface, in combination with Ti0 2 on Ti substrates, through the Beer patents on electrodes for anodic Cl 2 evolution in the chlor-alkali process, which operates at about 95°C in brine. These so-called "dimensionally stable anodes" (DSA®) now provide the ideal, extensively used anode materials for electrolytic production of C1 2. High specific-area RU02 films can be prepared thermochemically from RuCl 3 or (NH4h RuCl 6 painted onto Ti substrates, sometimes with the addition of titanium isopropoxide or TiCI 3. Thermochemical decomposition, leading to RU02 or to some RU02 + Ti0 2 composite, is carried out between 350 and 550°C (see Ref. 4). In Chapter 10 on the topic of pseudocapacitance in general, it was shown how a reversible redox process, represented thermodynamically by the Nernst 259

260

Chapter 11

equation with regard to its electrode potential, can give rise to a pseudocapacitance over a range of potentials. This is the main basis of the pseudocapacitance developed at RU02 in anodic and cathodic cycling; it also originates with a simple redox couple (e.g., Fe(CN)t-Fe(CN)~-) in a thin-layer cell or anchored on a polymer as shown in Chapter 10. However, the latter type of system gives appreciable pseudocapacitance only over about a 120-mV potential range, whereas RU02 gives relatively constant and appreciable capacitance over a 1.4V range. 5,6 Later in this chapter we suggest how this important difference may originate in terms of several overlapping redox processes within the l.4-V potential range. Ruthenium oxide films can also be formed at Ru metal electrodes by potential cycling over about a 1.4-V range above the hydrogen reversible potential, e.g., in aqueous sulfuric acid. Initially monolayer oxide formation and reduction take place (Fig. 11.1), like that at Pt. However, upon continuous cycling, the oxide film progressively grows (Fig. 11.2) since it is not reduced to Ru metal at the least positive potential of the sweeps (+0.05 V, RHE), as is the initial monolayer at Ru or Pt. Ruthenium dioxide films thus formed give a cyclic voltammogram having the shape shown in Fig. 11.2 and a corresponding charge and discharge diagram closely similar to that for a capacitor. 5- 7 It was on this basis that Trasatti and Buzzanca6 recognized the interesting pseudocapacitative behavior of this Ru02 film material, since an almost rectangular, mirror-image cyclic voltammogram (Fig. 11.2) is characteristic of a capacitance. A useful but brief review on RU02 as a electrochemical capacitor material for electrical energy storage has recently been published by Trasatti (see general reading ref. 1). The Ru02 material formed thermochemically on Ti also gives a voltammogram similar to that in Fig. 11.2. Especially with the RU02 films thermally formed on Ti with Ti0 2, or with Ta205, the cycle life in capacitor charge and discharge between 0.05 and 1.2 V, or even to 1.40 V (RHE) under some conditions is remarkable, allowing cycling over 105 times with little degradation. What small degradation may take place arises due to RuO~- formation at the high potential ends of the anodic voltage excurSIOns. By integration ofthe I vs. E voltammogram curves, the charges, q, for supposed oxidation and reduction of Ru atoms in the thermally prepared Ru02 films were related4 to the fraction (%) of Ru atoms involved in solid-phase redox reactions (Table 11.1). This fraction is actually quite small and tended to decrease with increasing temperature of calcination between 350 and 450°C. The q values in Table 11.1 cannot be related only to surface redox processes; some bulk-phase reactions must also take place and some double-layer charging must of course also be appreciable. This conclusion has been confirmed by

The Electrochemical Behavior of Ruthenium Oxide

___- r

---

./"""

261

---

/

t

"

0 cu

0 cu

~ u ;;; .5 £J j

'" ";

..z

0

Irreversible region

E u

Revers ible req ion

I

as will be seen later (see Refs. 16 and 17). Returning to quasi-thermodynamic potentials, Pell, Liu, and Conway26 recently found that either a thermochemically or electrochemically formed (hence a hydrous film) film of RU02 decayed in potential (on open circuit; see Chapter 18 and Refs. 29 and 30) after anodic polarization to 1.4 V, down to 0.75 V, while, after discharge to 0.1 V, it recovered in potential to almost that value, which remained stable for many hours. Thus 0.75 V seems to correspond to a well-defined redox potential; it happens to correspond (at pH -0) to the line in the Pourbaix pH potential diagram for Ru for equilibrium between Ru and RU203, which is an interesting result but one inconsistent with the difficulty of reducing Ru oxides back to the metallic Ru state, the problem referred to in Eq. (11.3). Although superficially (e.g., in the cyclic voltammograms) the thermally and electrochemically grown Ru02 films behave similarly, Pell, Liu, and Conway26 found that the pH dependence of their stationary potentials was reproducibly quite different, being -30 mV for the former film and -60 m V for the latter. By contrast, Daghetti, Lodi, and Trasatti31 found also a value of -60 mV for the thermally formed oxide film. The two values found in the work of Ref. 26 indicate a qualitative difference between the thermodynamically potential-determining processes that are involved at the initially anhydrous and the electrochemically formed hydrous oxide films. Thus the -60-mV slope (corrected for the pH dependence of the RHE) requires a process involving an equal number of electrons and protons, while the -30-mV slope requires a ratio of elH+ of 2. A similar recovery behavior (see Ref. 26) at thermally formed RU02 had been observed earlier by Arikado et aI.,27 but the potential approached was ca. 0.5 V rather than 0.75. In this work, a role for proton diffusion in the recovery process was proposed. In the voltammogram for RU02 in 1 M aqueous HCI04 at 60 m V s-l, two conjugate (anodic and cathodic) pairs of peaks are observed (but not well resolved) at ca. 1.14 and 0.6 V (RHE) which were attributed5 to the RU20rRu02

272

Chapter 11

and RU20rRu couples. However, as remarked earlier, Ru02 or other lower oxides are not returned to the Ruo state in the cathodic sweep except when only a monolayer of Ru surface oxide has been formed 7 as at Pt. 12 In fact, it is precisely the incomplete reduction of RU02 back to Ru on each sweep that is believed to be the origin of the phenomenon of electrochemical growth of RU02 films on potential cycling from 0.05 to 1.4 V. The increase in thickness of the RU02 film is up to several microns, as is visible under the electron microscope or even under the optical interference microscope. 13 Under potentiodynamic conditions, the initial stages of oxide film formation are typical of an irreversible, slow anodic process, as also found at pt. 12 Some coverage by surface oxide can be established in an anodic sweep up to a certain potential, say, 1.0 V. Upon arrest of the sweep, growth continues at constant potential; then when the anodic sweep is restarted, the original i-V voltammogram is only picked up at some 50 to 100 mV higher potentials. This means that the oxide that is formed at one (higher) potential (V2) in a sweep can also be formed upon holding in time at a lower potential VI' < V2 . Galizzioli, Tantardini, and Trasatti5 concluded that not all the Ru atoms in the Ru02lattice were redox active, giving rise to pseudocapacitance; that is, in a potential range of 1.35 to 0.3 V, the formal process Ru4+ + e -4 Ru 3+ (in the oxide) was considered to be restricted to a number of atoms corresponding to the degree of nonstoichiometry. Coupled with this change of oxidation state (transfer of electrons) is a corresponding injection or withdrawal of protons to maintain a charge balance through conversion of 0 2- ions to OH-. Possibly these coupled redox and proton-exchange processes take place only in the nearsuiface region 26 of RU02, especially when it has been thermochemically prepared. Electrochemically prepared RU02 films are hydrous in nature, so that a more facile proton penetration, coupled with redox changes of oxidation state, can take place, as suggested earlier. The three-dimensional and near-surface (see Refs. 32 and 33) chemical constitution of thermally formed RU02 was investigated34 using depth profiling by means of sputtering. Oxygen species are rich in the surface region but decrease in concentration toward the bulk, where Ru also decreases (Fig. 11.5). Cl- ion is adventitiously present and increases somewhat toward the bulk; it arises from the RuC1 3 used to make the thermochemically generated RU02 film. A new charge storage mechanism involving protonation ofRu02 (presumably coupled with reduction) was recently claimed by Zheng and Jow. 16 However, it does not seem any different from the basis of operation of the redox pseudocapacitance of RU02 recognized in various, much earlier papers 4- 7,27 and more recently by Sarangapani et al.,9-21 as well as with oxides of Ni. 35 Good specific capacitance was obtained with an RU02 . xH 20 material made by a solgel precipitation process, followed by heating between 25 and 400°C. Values of capacitance were up to 350 F g-I while, with the hydrous form, up to 720 F g-I

273

The Electrochemical Behavior of Ruthenium Oxide 1.6

0 ...J W

>=

Z

1.2

0

>-

0:

u

W .0

(/)

•

\ I/)

so; see Chapter 10) or at overcharge potentials. As emphasized earlier, mirror-image cyclic voltammetry behavior is essential for a pseudocapacitor system to behave electrically like a regular capacitor or, more particularly, like a double-layer capacitance. This kind of requirement is well demonstrated by the behavior of a variety of electroactive conductive polymers, hence their potential value as cheap electrochemical capacitor materials, as recognized in the recent literature.l0.ll.13.14.36.40

12.4. FORMS OF CYCLIC VOLTAMMOGRAMS FOR CONDUCTING POLYMERS

Figure 12.3 showed the cyclic voltammogram for polyaniline in dilute aqueous H 2S04, The voltammogram, as reported in work from various laboratories, including our own in which we obtained the curves of Fig. 12.3, typically exhibits three or four peaks over the 0.8-V range of electrochemical activity. This behavior is reminiscent of the voltammetry behavior of underpotential deposition processes at noble metals, e.g., H on Pt or Pb on Au, even on singlecrystal surfaces of those metals. The explanation, in those cases, excluding polycrystallinity, is that successive 2-dimensional array structures of differing geometries arise to minimize free energy (Gibbs energy) as the lattice of vacant sites on the metal surface is progressively filled up. Scanning tunneling microscopy allows direct imaging at atomic resolution of such metal adatom 2-dimensional array structures. This type of process is obviously inapplicable at the electroactive polymer films. Possibly the progression of the development of positive charges (on anodic polarization) as the potential is changed is not random nor is the associated counter anion binding (ion association). However, it is interesting that films formed from o-anisidine (Fig. 12.4) show only one somewhat asymmetric peak, as was found in Conway's work with Gu_Ping. 41 In the case of p-toluidine (pmethylaniline), almost no current response arises in cyclic voltammetry except that expected residually from double-layer charging. This behavior is easier to explain since blocking of the para position on the aromatic ring structure (preventing head-to-tail coupling) sterically inhibits polymerization.

Capacitance Behavior of Polymer Films

315

ANILINE

IZ

w

a::

0:

POTENTIAL

:::>

u

FIGURE 12.3. Sequence of voltammograms for polyaniline oxidation and re-reduction in successive cycles as the film builds up on a gold electrode in aqueous H2S04. (From the author's laboratory with Gu Ping; cf. Gholamian. 6)

o-ANISIDINE

IZ

w

0: ~~----------------~~---------/~------0: POTENTIAL

:::>

u

FIGURE 12.4. Voltammogram for a poly-o-anisidine film grown on a gold electrode in aqueous H2S04. (From the author's laboratory, with Gu Ping.)

316

Chapter 12

Interesting results related to this question were found by D' Aprano and Leclerc,39 who showed voltammograms for poly(2-methylaniline) (PMA), poly(2methoxyaniline) (PMOA), poly(2-methoxy-5-methylaniline) (PMOMA), and poly(2,5-dimethoxyaniline) (PDMOA) in 1 M aqueous ReI. These authors also recorded the conductivity of films ofthese materials (Fig. 12.5) and found maxima at 0.42 V (vs. SeE) for PANI, PMA, and PABA; 0.35 V for PMOA and PMOMA; and 0.28 V for PDMOA, the films being virtually nonconductive in the fully reduced (leucoemeraldine) or fully oxidized (pernigraniline) states in the case of PANI. They also examined the optical (electronic) spectra of these PMOMA with the change in the oxidation state (Fig. 12.6). In this work some attempt was made to explain the multiplicity of peaks observed in the voltammetry of PANI; it was suggested that they may arise as a result of some degradation products formed during the electrochemical polymerization and/or side-couplings (branching defects in the linear polymerization). The selectivity of head-to-tail (isosteric) couplings was diminished by methoxy substitution on the ring. The range of appreciable anodic and cathodic electroactivity ofthese polymers was about 0.5 to 0.7 V, which is similar to that for PANI itself. A different basis for explanation of the multiple current peaks observed with PANI films was offered by Jiang, Zhang, and Xiang. 16 Peaks these authors designated a and d at 0.13 V and -0.04 V (vs. SeE) may relate to protonation and deprotonation processes (I, II) while they thought that peaks designated as band cat 0.42 and 0.35 V may be associated with electron transfer coupled with anion ion pairing with the charged chain (II, III). Some of the anodic and cathodic separation between the respective peaks may be due to an iR drop in the film and/or to some kinetic irreversibility. Another and possibly more correct explanation is that the multiplicity of the peaks for PANI arises simply from a stepwise oxidation to the first radical cation and thence to the dication (bipolaron), based on electron-proton or on electron-counterion coupling, depending on the acidity or the counterion' s mobility, i.e., its Stokes's law diameter. Alternatively, another probable reason is that some component of artha coupling could lead to the minor peak(s) observed between the two main ones, as suggested by Genies et aI.9 An interesting and original variant was prepared and investigated by Gottesfeld et al. 14,42: poly-3-(4-fluorophenyl)-thiophene (PFPT) in an electrolyte of tetramethylammonium trifluoromethanesulfonate (TMATFMS) in acetonitrile. Structures of the PFPT (formula 3) and related nonfluorinated repeating (y) groups (formulas 1 and 2) are shown below. The cyclic voltammogram of PFPT (Fig. 12.7) is strikingly different from that for PANI in that it exhibits two widely separated regions of almost reversible electroactivity that correspond to both positive and negative charge injection (coupled with ion association). The voltage range within each

Capacitance Behavior of Polymer Films

317

0.006r-----------------~~----------_,

PMOMA

'E 0.004 u

If)

"~

0.002

PDMOA

0.3

IE

u 0.2

If)

"~

0.1

0.0 -0.2

0.6

FIGURE 12.5. Conductivities of PMOMA and PDOMA conducting polymer films as a function of electrode potential. (From d' Aprano and Leclerc. 39 Reproduced by permission of The Electrochemical Society, Inc.) 1.0.-------...,-----.-----,------,------,

PMOMA ::J

d

......

w u

.....

................

Z Ac) is the equivalent conductivity at infinite dilution, lim(Klzc) as c ~ 0 (see Table 13.1 for aqueous Hel and acetic acids as examples). Substitution of a in Eq. (13.2) by A/Ao leads to Ostwald's dilution law by which the applicability of the Arrhenius relation for a in Eq. (13.2) can be tested in various ways for weak electrolytes. Such plots give distinguishable relations for weak and strong electrolytes (Fig. 13.2) where, for the latter, the Arrhenius equation, a = A/Ao, does not apply. For the solutions employed as electrolytes for nonaqueous solvent doublelayer capacitors, there is significant weak electrolyte behavior so that a is appreciably less than its value, which is near 1 in aqueous solutions. This usually leads to larger esr values for nonaqueous solution devices than for aqueous ones using the same electrode materials and cell geometries. Also the Ao values are usually different. In strongly dissociating solvents such as water, Kc is relatively large and a ~ 1, so that the concentration of species (13.3) is small. However, Ac is still found to progressively decrease with c in relation to Ao for infinite dilution. This is due to the long-range electrostatic attractions between the

340

Chapter 13

TABLE 13.1. Test of Ostwald's Dilution Law (based on Eq. 13.2) HCI in water. t = 25°C, cI03 (mol/liter]

Ao = 426.16

Ac

0.028408 0.081181 0.17743 0.31836 0.59146 0.75404 1.5768 1.8766

Acetic acid in water, t = 25°C C.Ao = 390.71 K(c)

425.13 424.87 423.94 432.55 422.54 421.78 420.00 419.76

0.0116 0.02666 0.03355 0.05139 0.05995 0.07169 0.1059 0.1212

cI03 (mollliter) 0.028014 0.15321 1.02831 2.41400 5.91153 12.829 50.000 52.303

Ac 210.38 112.05 48.146 32.217 20.962 14.375 7.358 7.202

K(c) X

5

10

1.760 1.767 1.781 1.789 1.789 1.803 1.808 1.811

free,dissociated ions (it also occurs with the ions arising from weakly dissociated salts, but the effect on A is much less than that due to small K or Iowa), which also depends on the dielectric constant, e, of the solvent. Up to moderate concentrations, Ac decreases with the square root of c, an effect treated in the well-known theory of Debye and Huckel (1923), and later by Onsager (1926). The different effects of an increasing concentration of a dissolved salt or acid on equivalent conductivity in the case of weakly dissociated (or ion paired) electrolytes compared with strongly dissociated salts for which long-range interactions are more important are illustrated in Fig. 13.2 and are usually easily distinguishable experimentally. Intermediate cases, of course, arise.

___ A ~"" ,

, \, ,, ' ,, ,

CD

FIGURE 13.2. Comparative conductivity behavior of a strongly and a weakly dissociated (or ionpaired) electrolyte as a function of concentration (schematic).

The Electrolyte Factor in Supercapacitor Design

341

Electrolytes of a given type (e.g., alkali halides) exhibit a wide range of conductivities, depending on the nature of the solvent and also on the radii and charges of the ions. However, it has been recognized that this variation with solvent can be described in terms oftwo general types of solvents designated "leveling" and "non leveling" or "discriminating." A leveling solvent is one (e.g., especially water) that provides strong solvation and a tendency for complete dissociation or minimum ion pairing. Such solvents are usually those that have high dielectric constants, often with hydrogen-bonded structures with large dipole moments, and lead to less differentiation (hence leveling) of conductivities due to ion pairing. The nonleveling solvents lead to a wider differentiation of conductivities, mainly on account of solvent-specific ion pairing that is related to ion size and solvation, and to the dielectric constant and donicity of the solvent. In the case of tetraalkylammonium salts, which are commonly used as electrolytes for nonaqueous solvent capacitors, different principles apply: First, their extents of ion pairing are usually less at appreciable concentrations than those for inorganic salts owing to their relatively large radii. Second, their alkyl groups tend to interact well with organic solvents whereas they behave as hydrophobic ions in water, although they are often quite soluble. A number of new electrolytes with good conductivities and degrees of dissociation have been developed and used in recent years, partly in the fuel-cell field. They are of interest also for Li battery systems and electrochemical capacitors. 5 They can be looked upon as trifluoromethylsulfonic derivatives of the isoelectronic molecules H2 0, NH 3 , and CH4 . The structures of the anions of the acids are, respectively:

The electron-withdrawing CF3·S02 functions bonded to OH, NH, or CH promote strong acidity of the Hs on those 0, N, or C groups. The latter two are referred to as imide- and methide-type ions. They have good conductivities as salts in both aqueous and nonaqueous solvent media, and have also been employed as acid fuel cell electrolytes. In addition, low melting point alkyl pyridinium and alkyl imidazolium salts offer interesting possibilities as nonaqueous electrolytes having good conductivities. Some specific conductances at 22°C of these and other salts of the imidazolium cations are listed in Table 13.2 together with their van der Waals volumes, Vc and Va' for the cations and anions. Data for aqueous 3.9 M H 2 S04 and 0.65 M tetraethylammonium-Bf4 in propylene carbonate are shown for comparison; they are mostly lower than the values for the latter two electrolytes. In the organic molten salt series, 1m and Me represent the imide and methide coanions.

342

TABLE 13.2.

Chapter 13

Specific Conductivities of Various Solvent-Free Ionic Liquids at 22°C

Melt EMI AICI4 EMIIm EMIMe DMPIAICI 4 DMPIIm DMPIMe 3.9MH2S04 0.65 M TEA BF4fPC

15.0 7.0 1.3 7.1 2.5 0.5 575 10.6

118 118 118 152 152 152

113 144 206 113 114 206

Source: From Koch et al. 5 aVe = van der Waals volume of the cation. b Ve = van der Waals volume of the anion. Key to abbreviations: EM! = ethyl, methyl imidazolium; DMP! - dimethyl, n-propyl imidazolium cation. TEA = tetraethyl ammonium cation.

Double-layer capacitance values for some of these salts were determined at the dropping mercury electrode5 ; values between 5 and 151lF cm-2 were recorded (Table 13.3) but are appreciably potential dependent, like data for regular aqueous electrolytes at mercury. Comparative data for the capacitance at activated Spectracarb carbon having a BET surface area of 2000 m2 g-l are listed in Table 13.4. Practical cell capacitances with Spectracarb 2220 electrodes gave values on the order of 23 to 29 F g-l in the molten imidazolium imide electrolyte or in that electrolyte in 20 to 40% benzene [specific conductances from 7.0 (molten electrolyte) to 11.9 mS cm- 1 in benzene solutions]. Some declines of capacitance by about 20-30% over 5000 cycles were observed. At activated carbon electrodes, the capacitance per gram realized was much less than expected theoretically on the basis of the intrinsic double-layer capacitance. This discrepancy was attributed to the presence of micropores

TABLE 13.3.

Double-Layer Capacitance at the Dropping Hg Electrode for Various Electrolytes

Electrol yte 0.1 M KCI 3.0MH2S04 1.0 M TEA BFiPC DMPI AICI4 EMllm EMI 1m: C6H6 a Source: From Koch et al. 5 a60:40 vol. %; 1m = (CF3 . S02):z~r.

2.65 2.50 2.57 2.35 3.43 3.43

15 10

7 5 12 12

24 21 15 3.4 26 22

343

The Electrolyte Factor in Supercapacitor Design

TABLE 13.4. Double-Layer Capacitance at Hg and Activated Carbon for Various Electrolytes Electrolyte 0.1 M KCI 3.0 M H 2S0 4

1 M TEA BF4IPC

DMPI AIC14 EMllm EMI 1m + C 6H 6 c

C dl a (int)C,uF cm-2)

Cdl bTheory 480 420 300

24 21 15 3.4 26 22

240 120

68 520 438

100

Source: From Koch et al. 5 aAtHg. bAt Spectracarb 2220 with BET surface area of 2000 m2 g-l. '60:40 vol. %. aVe = van der Waals volume of the cation. b Va = van der Waals volume of the anion.

and/or associated poor wettability, i.e., unsuitable contact angles between the molten electrolytes and elements of the carbon surface.

13.4. MOBILITY OF THE FREE (DISSOCIATED) IONS

Movement of free ions in a solvent under an electric field E is treated in terms of a classical hydrodynamic relation, Stokes's law, involving the viscosity of the solvent, Y/: the force on ion =zeE =hydrodynamic resistance (6ny/ ri(s»v where ri(s) is the radius of the solvated ion and V is the velocity of the ion under field E. vIE is the mobility of the ion calculated per unit field, usually V cm- I . Hence

vIE = zeI6ny/ri(s)

(13.4 )

for an ion of charge ± ze. The role of solvation in determining the ion's effective size in solution, and hence its mobility, is illustrated by the fact that the order of increasing mobility in the alkali-metal ion series is Li+ < Na+ < K+ < Rb+ < Cs+, yet the crystal ionic radii vary in the opposite order, Cs+> Lt, etc. Equation (13.4) predicts that (viE) x Y/ should be a constant for the mobility of an ion in various solvents having different viscosities. This relation is referred to as "Walden's rule." However, it is not followed very precisely (see the later examples) owing to (1) variation of the ion's solvation radius ri(s) in various solvents; and because (2) the relevant Y/ is an effective viscosity applicable locally near the ion as it moves, and orientationally polarizes and relaxes solvent molecules (dipoles) in its path (Fuoss-Zwanzig effect) owing to its local field. This

344

Chapter 13

causes dissipation of extra energy corresponding to a frictional effect in the fluid-a component of the viscous force encountered by the moving ion. However, Walden's rule applies approximately provided that in a series of solvents having different viscosities, the dielectric constants, e, are not so low that ion pairing begins to dominate the behavior. When that occurs, serious deviations from Walden's rule arise, as they also do in mixed solvents because of selective solvation. In practice, Eq. (13.4) indicates that conductance of an electrolyte will be better in a low-viscosity solvent than in a higher one, other things being equal. However, this is not always the case since viscosity is related to the molecular interaction and dipole moment (electric polarity) of the solvent. The latter two factors also determine the dielectric constant, s, of the solvent; a higher s diminishes ion pairing (see later discussion) and improves Ac for a given salt concentration. Hence, although lower '1 improves the actual mobility of the free ions, the lower s that is usually associated with 10w-11 solvents (because intermolecular interactions are weak) tends to diminish Ac for the above reason concerning ion pairing. The use of mixed solvents (i.e., the addition of a low '1 solvent to a high '1, high s solvent) often achieves optimization of conductance, an effect that has been utilized in optimizing the specifications of nonaqueous electrochemical capacitors. In liquid mixtures, viscosities are rarely linear in the mole fraction of the components 6- 9 owing to specificities of intermolecular interaction, except for a pair of solvents of closely related structure, e.g., benzene and toluene. Examples are shown in the figures later in this chapter.

13.5. ROLE OF THE DIELECTRIC CONSTANT AND DONICITY OF THE SOLVENT IN DISSOCIATION AND ION PAIRING

The dielectric constant of a solvent determines the interaction energy, U, between ions at some distance x between each other (Chapter 4): (13.5) where Zl and Z2 are the charge numbers for the ions. U is an attractive energy, tending to diminish interionic free motion when Zl and Z2 are opposite in sign. Equation (13.5) applies especially to long-range interactions as treated in the theory of Debye and Huckel (1923). When x becomes comparable or equal to the sum of the ionic radii or solvated-ion radii ri,s (Eq. 13.4), an ion-paired situation, (Eq. 13.3), arises and the energy U is then

The Electrolyte Factor in Supercapacitor Design

345

(13.6) When the (negative) energy U in the above equation is > or > > kT, the extent of ion pairing is substantial (a« 1). Similarly, a usually increases with a rise in temperature, leading to improved conductance. This of course is a common problem with supercapacitors and batteries, i.e., their internal resistance increases at lower temperatures and their power delivery capability (Chapter 15) becomes substantially diminished. In some cases (Fig. 13.1), when solvated ions are paired their solvation shells become shared (a case of solvent-shared ion pairs), so that x < r;,s,+ + r;,s,_. In poorly solvating solvents, the plus and minus ions come into a contact situation [Fig. 13.1(c)] (contact ion pairs, Fuoss). Then, the maximum plus/minus interaction arises electrostatically but is counterbalanced by a substantialloss of solvation energy of the two ions. In these close-encounter situations, which lead to diminution of the conductance of an electrolyte, the normal value of B for the solvent becomes inapplicable and a lower local value due to dielectric saturation should be used for the region between and near the ions. Thus ion-pairing equilibrium constants for an electrolyte in various solvents do not follow simply a liB relation although there is often a general qualitative trend. For example, three solvents having the same nominal B (=20) (e.g., acetic anhydride, n-butyronitrile, and trimethylphosphate) can exhibit quite different ion-pairing effects for a given electrolyte solute, with consequent substantial differences between the Ac vs. cor c I12 relations, the latter corresponding to the behavior of fully dissociated electrolytes (Debye-Hiickel effect). However, for the short-range interaction effects involved in ion pairing, it is often the lone-pair electron donicity that is more significant than the B (Table 13.5). In the cases of the three solvents above, the respective donicity numbers are quite different, namely, 10.5, 16.6, and 23 in a range of 0 (reference) to 38.8 for a variety of solvents (Table 13.5). The ion pairing in various solvents can be quantitatively characterized by infrared, Raman, and NMR spectroscopies, often in a specific way.4,II,I2 A summary of factors determining electrolyte conductance is given in Table 13.6.

13.6. FAVORED ELECTROLYTE-SOLVENT SYSTEMS

13.6.1. Aqueous Media

Here the choice for supercapacitor electrolytes, for obvious reasons, is sulfuric acid or KOH for carbon-type double-layer capacitors using an aqueous medium. However, the decomposition voltage limit is then theoretically 1.23 V

346

Chapter 13 TABLE 13.5.

Solvent Donicity Numbers and Dielectric Constants (s)

Solvent 1,2-Dichloroethane (ref. solvent) Sulfuryl chloride Thionyl chloride Acetyl chloride Tetrachloroethylene carbonate Benzyl chloride Nitromethane (NM) Dichloroethylene carbonate (DEC) Nitrobenzene (NB) Aceticanhydride (AA) Phosphorus oxychloride Benzonitrile (BN) Selenium oxychloride Acetonitrile (AN) Sulfolane (TMS) Propanediol-l,2-carbonate (PDe or PC) Benzylcyanide (nitrile) Ethylene sulfite (ES) iso-Butyronitrile Propionitrile Ethylene carbonate (EC) Phenylphosphonic difluoride Methylacetate y-Butryronitrile Acetone Ethyl acetate (ETAC) Water Phenylphosphonic dichloride Diethyl ether Tetrahydrofuran (THF) Diphenylphosphonic chloride Trimethylphosphate (TMP) Tributylphosphate (TMP) Dimethylformamide (DMF) N,N-Dimethylacetamide (DMA) Dimethylsulfoxide (DMSO) N,N-Diethylformamide N,N-Diethylacetamide Pyridine (PY) Hexamethylphosphoramide (HMPA) Hydrazine Ethylenediamine Ethylamine t-Butylamine Ammonia Trithylamine Source: From Gutmann. lO Most e data are for 298 K.

DN 0.1 0.4 0.7 0.8 2.3 2.7 3.2 4.4 10.5 11.7 11.9 12.2 14.1 14.8 15.1 15.1 15.3 15.4 16.1 16.4 16.4 16.5 16.6 17.0 17.1 18.0 18.5 19.2 20.0 22.4 23.0 23.7 26.6 27.8 29.8 30.9 32.2 33.1 38.8 44 55 55.5 57.0 59.0 61.0

e 10.1 10.0 9.2 15.8 9.2 23.0 35.9 31.6 34.8 20.7 14.0 25.2 46.0 38.0 42.0 69.0 18.4 41.0 20.4 27.7 89.1 27.9 6.7 20.3 20.7 6.0 81.0 26.0 4.3 7.6 20.6 6.8 36.1 38.9 45.0

12.3 30.0

The Electrolyte Factor in Supercapacitor Design

347

TABLE 13.6. Summary of Factors Determining Conductance of Electrolyte Solutions 1. 2. 3. 4. 5. 6. 7. 8. 9.

Solubility of the salt or acid Degree of dissociation or extent of cation and anion pairing in solution Dielectric constant of the bulk solvent Electron pair donicity of the solvent molecules Mobility of the free, dissociated ions Viscosity of the solvent Solvation of the free ions and the radii of the solvated ions Temperature coefficient of viscosity and of ion-pairing equilibria Dielectric relaxation time of the solvent

or practically, in kinetic terms, 1.3-1.4 V. As for battery systems (lead-acid, NiCd), relatively concentrated electrolytes are required to minimize the esr and maximize power capability. However, strong solutions of acid are much more corrosive than strong solutions ofKOH or NaOH, so the latter may be preferred for some embodiments. These hydroxide electrolytes are very soluble in water and, because of the OH- anion, have very good conductivities. Corrosion of the hardware components is an important factor to consider in design and reliability, and with regard to the self-discharge characteristics of the capacitor device (Chapter 18). Both the acid and alkaline electrolytes have advantageously high equivalent conductivities in aqueous medium owing to the special mechanism of proton transport that determines their conductance (proton hopping). 13.6.2. Nonaqueous Media

The use of nonaqueous electrolytes for electrochemical capacitors is, in principle, preferred since higher operating voltages, V, can be utilized due to the larger decomposition limits (Fig. 13.3) of such electrolyte solutions. Since the stored energy increases as V2 (Chapter 2), this is an obvious advantage over aqueous systems. With nonaqueous media, a much wider variety of electrolyte-solvent systems is available and has been extensively investigated. Tetraalkylammonium salts have been preferred electrolytes owing to their usually good solubility in nonaqueous solvents and moderately good conductivity. Use of such salts avoids the possibility of alkali-metal deposition on the cathode electrode of a capacitor upon adventitious overcharge. However, such electrolytes are quite expensive and must (like the solvent) be very pure and dry to avoid 2H2 + O 2 formation on charge and subsequent recombination shuttle reactions leading to self-discharge. Also, they can decompose on strong overcharge, usually at the negative electrode.

348

Chapter 13

(a)

pore

Cdl

liquid electrolyte

(b)

FIGURE 13.3. (a) Transmission-line model (Chapter 14) for a single pore. Re and Rs are the ohmic resistance of the electrolyte and the solid electrode material, respectively, is the electrode Faradaic impedance and Cdl is the double-layer capacitance. (b) When RI = Rs = 0, the equivalent circuit in (a) becomes simplified to that shown as (b). (See Chapters 14 and 16).

Z,

The choice of solvents usually follows the same principles as for nonaqueous Li battery systems, although stability to reaction with Li is no longer a critical factor. However, aprotic structures are still needed to avoid H discharge at the cathodes. With suitable solvents and R4N+ electrolytes, operating potential differences of double-layer-type supercapacitors can reach ca. 4.0 V. Since, as was mentioned, capacitative energy storage increases as V2, this is one of the principal advantages of a nonaqueous double-layer supercapacitor. However, this has to be balanced against higher cost, lower conductivity (greater esr), smaller power capability, more complex production conditions to maintain dryness, and greater potential for degradation than for the aqueous system types. Some comparative information on the conductivities of various ion-conducting media is provided in Table 13.7. The resistivity (esr) factor becomes amplified in importance when the conductivity in the pores of a porous electrode is involved, as in the transmission-line model (Fig. 13.3) of such a matrix (see Section 13.8).

The Electrolyte Factor in Supercapacitor Design

349

TABLE 13.7. Conductivities of Various Materials that Conduct by an lon-Transport Mechanism Type of electrolyte Aqueous Aqueous Mixed-aqueous Mixed-aqueous Nonaqueous Nonaqueous Nonaqueous High-temperature molten salt Ambient temperature molten salt Polycrystalline Crystalline materials Crystalline materials Dispersed phase Inorganic glasses Solid polymer electrolyte Solid polymer electrolyte Solid polymer electrolyte Immobilized electrolyte solution

Materials 5.68 M HC1IH 2O 2.04 m MgCI21H2O 1.806 m MgC1 21H2O/ EtOH(20.3 wt.%) 1.547 m MgC1 21H2O/ EtOHl(39.8 wt.%) 1.362 m MgCl 2IEtOH 0.662 m LiCIOiPC 1.652 m LiCIOiPCI DME(28.2 wt.%PC) LiCVKCI eutectic

K

(mS/cm)

Remarks

849 160.3 69.3

Maximum Maximum Maximum

37.9

Maximum

23.0 5.420 14.38

Maximum Maximum Maximum

1615.0

(45rC)

EMIClAICl 3 X(A103) = .5

22.6

RbCu16I7CI13 RbAg4I5-single crystal LiI LiI(AI 20 3) LiULi 2SIP2S5 (45/371l8)(wt.%) LiCF3S03IPE0 6 LiCl4lPClPoly(vinylidene fluoride) ECIPClLiCI041PAN (38/33/8/21 )(mol%) I M LiCI0 4IPC Aerosil 200

340 260,270 0.01,0.042 10 to 0.1 I to 2 0.01 2

Cu+ conductor Ag+ conductor Li+ conductor

1.7

20°C

5.8

22°C

Li + conductor 60°C 100°C

Source: From Barthel and Gores. 13 Notes: Maximum =maximum of the K(m) function, see Eq. (13.8); EtOH =ethanol; PC =propylene carbonate; DME =dimethoxy ethane; EMIC = l-methyl-3-ethyl immidazolium cholride, EC =ethylene carbonate; PAN = poly(acrylonitrile). Original references given in Ref. 13.

Another factor, also important in Li battery technology, that is involved in the choice of electrolyte systems for capacitors is the decomposition limit. Some examples are given in Table 13.8. A diagrammatic representation is provided in Fig. 13.4. Section 13.7 gives more details. Suitable solvents are mainly from the following three classes13: 1. High dielectric constant dipolar aprotic solvents such as organic carbonates (Ee,pC); 2. Lower dielectric constant but high donor-number solvents such as ethers (DME, THF, 2-Me-THF, dioxolane); 3. Low dielectric constant, high polarizability solvents such as toluene and mesitylene;

350

Chapter 13

TABLE 13.8. Voltage Windows of Some Electrolyte Solutions Solvent electrode AN PC PN PNlEtCI,1I2 (-160°C) EC/SL, (111)

Electrolyte

Anodic limit

Cathodic limit

LiClO4 LiClO4 BU4NPF6 BU4NPF6

2.6 V 2.3 V 3.7V 3V

-3.5 V -2.2V -3.0 -3 V

Et4NPF6

-2.8

2.4 V

Working electrode Pt Pt Pt Pt

Reference Ag-AgCI0 4 Ag-Ag+ Ag-AgCI Ag-AgCI

Glassy carbon SCE

Source: From Barthel and Gores. [3 Notes: PN : propionitrile; Sl : sulfolane; Et: ethyl; Bu : n-butyl.

4. Intermediate dielectric constant aprotics such as acetonitrile, DMF, dimethyl acetamide (DMA) and butyrolactone (BL). From various works it appears that the best nonaqueous electrolyte for higher voltage electrochemical capacitors is Et4NBF4, 1 M, in propylene carbonate, possibly mixed with dimethoxyethane to improve (lower) its viscosity and thus raise ionic mobilities. The above solution has a specific conductance of 13 mS em-I. 13.6.3. Molten Electrolytes

The use of low melting-point organic salts as fused electrolytes for aluminum batteries using AICl4 or AI 2CI? salts of alkylimidazolium cations has been explored in recent years. In an original development, Koch, Nandjundiah, and Godlman l5 have employed these kinds of salts in fused electrolyte capacitors. With suitable anions, good anodic decomposition voltages have been achieved, as was demonstrated comparatively in the anodic half-cycle voltammograms for Li salts of various anions (AsF6, PF6, Im-, Me-) in acetronitrile. The voltammetry behavior of methylethylimidazolium (+) AICl 4 electrolyte in three melts (Lewis basic, Lewis acidic, and neutral mixtures) was studied by Melton and Simpson. 16 Depending on composition, the operating ranges lie between +2.4 and -2.2 V vs. an Al reference electrode in the melt. At a Pt electrode, AsF6 and PF6 imidazolium salts had anodic decomposition limits of ca. 5.1 and 5.2 V vs. Li. A relatively new family of anions (also examined as the basis for Li battery electrolytes and for aqueous solution electrolytes for fuel cells where O 2 reduction catalysis is improved), based on trifluoromethylsulfonic acid, has also been explored for capacitor electrolytes. The family of structures is shown below. The oxidation potential limits, vs. Li, for various complex anions and cations at glassy-carbon (gc), W, and Pt electrodes are listed in Table 13.8 below.

The Electrolyte Factor in Supercapacitor Design

Trifluoromethylsulfonate (triflate)

Perfluoromethylsulfonyl imide

351

Perfluoromethylsulfonyl methide

The oxidation potentials for a series of substituted tetraphenylborate anions also show a clear relation to the energy of highest occupied molecular orbitals (HOMO). Correspondingly, the HOMO energy for the two sulfonic N and C derivatives, and for CF3·SOZ·O-, increases negatively with increasing number of SOz·CF3 groups as in the order: CF3·SOZO- - 6.96 eV, (CF3 ·SOzhN- - 8.19 eV and (CF3·SOZ)3C- - 8.70 eV. The specific double-layer capacitances realized at a glassy-carbon electrode were as follows, in relation to an LiPF6 electrolyte in PCIDMC solvent and in 3.6 M aqueous H zS04 : A number of other low-melting point salts (e.g., of the N-alkylpyridinium type) have been investigated in various works by Robinson and OsteryoungP Related to this class are the alkylimidazolium salts (the general structure is shown below), referred to earlier. 15,16

13.7. PROPERTIES OF SOLVENTS AND SOLUTIONS FOR NONAQUEOUS ELECTROCHEMICAL CAPACITOR ELECTROLYTES

Apart from capacitor devices based on aqueous electrolytes, there are a number of special requirements for electrolyte properties when higher operating voltage capacitors are considered that must be based on nonaqueous solutions as the ionically conducting fluid. The principal purpose of improving the operating voltage range is the quadratic dependence of the energy density, e.d., on the maximum attainable voltage, l1V, on charge [e.d. = 1I2C(11V)z], as was emphasized in Chapter 2 and will be discussed further in Chapter 15, as well as here. The main requirements for suitable nonaqueous solutions are as follows: 1. An adequate voltage window of electrochemical stability (i.e., the decomposition voltage of the desired solution) (Fig. 13.4), which should be somewhat larger than the intended operating range of the capacitor device in order to minimize problems arising from adventitious overcharge. Also, it is not just the overall range of operating voltage that is important for solution stability, but the

352

Chapter 13 1 +3

+2

+'

Aqueous

-,

°

-2

-3

\------111.11 H,SO,(Pt)

I - - - - - - - l i pH 7 Buffer IPII

p, {

! - - - - - - - ; I ' .II N.OHIP'I

\-_ _ _---;1' .II H,SO,lHgl

~{ C{

rl- - - - - - - ( 1 1 . 1 1 KCllHgi

I r - - - - - - - - I I ' .II NaOHIHgl ) - - - - - - - - - ; 1 0 , ' .II Et,NOHIHgl

)--------lI'.M HCIO,ICI 1 ) - - - - - - - - - - 1 1 0,' .II KCIICI

Nonaqueous MoeN p, 1 - - - - - - - - - - - - - - - - - - - ; 1 0,' .II TBABF.

I - - - - - - - - - - - - - - - - - l i ~,~~l TBAP \----------------11

Senzonitrile

O".If TBABF.

\ -_ _\---------------...,1 _ _ _ _ _ _ _ _ _ _ _ _---;1 PC

\--------------;1 1------------11

u

>-

"C

(J')

§

t:

z

W Cl I I-

Z

W

)-------------ll

-,

0,' AI TBAP

CH,CI, O.lA.TEAP 0,',01 TBAP

~,~'.w KI

1 -3

-2

E V "', SCE

I

/

~----------.--~------------I

I

"F(1') I I I

O~--------_+--_+------------Lr--------+_---I------

V-

~

::)

.,

+2

0

~

U

+3

~~'M TBAP

THF

....----- , i.(dl)-

~-----U

"C

o

.c

c;

3

1 :

V+ , / POTENTIAL I

1

..

I

I FARADAIC

:

: DECOMPOSITION

~I·------------~·I

: -FA-R-A-O--i. .... ·I-C-- 1

IDEAL POLARIZABILITY RANGEl :

OECOMPOSITION:

FIGURE 13.4. Estimated potential ranges (decomposition limits) for aqueous and nonaqueous electrolyte solutions. (From A. 1. Bard and L. R. Faulkner, Electrochemical Methods, Wiley, New York (1980). Reprinted by permission of John Wiley & Sons.) The lower curves illustrate decomposition limits, V+ and V-, in a cyclic yoltammetry experiment; i(dl) = double-layer charging currents and iF-Faradaic decomposition currents.

individual positive (anodic) and negative (cathodic) potential limits of stability of the solution relative to some reference electrode potential in the same solution. Furthermore, it is also not just the electrochemical stability of the solvent that is the principal factor but also that of the solute ionic species that must not be discharged or decomposed (e.g., with R4N+ solute salts); this is also a factor

The Electrolyte Factor in Supercapacitor Design

353

of major practical importance. Thus, tetraalkylammonium salts of anions that are difficult to discharge, such as PF(;, BEl, and AsF(;, are preferred solutes. Li+ salts could also be used provided the cathodic limit for Li metal deposition is not reached on charging of the capacitor. 2. Minimum viscosity of the solvent (or solution) in order to maximize ionic mobility and resulting conductance. 3. Maximum solubility of the solute salt to maximize conductance. 4. Minimum ion pairing at given practical solute salt concentrations, again to maximize conductance. 5. Optimum dielectric permittivity or donor number of the solvent to maximize salt solubility and minimize ion pairing. This requirement also determines the solvation of the ions of the solute salt. Other factors are the ionic radii of the solute salt ions and the magnitudes of their charges, often ± Ie, which the determine strengths of solvation and of ion pairing or degrees of dissociation of the solute salt. A summary of the various properties involved is given in Table 13.9. All these factors are involved in determining the two leading quantities that determine specific conductance at a given concentration, namely, (1) the mobility of the dissociated ions and (2) the concentrations of free charge carriers, the cations and anions, at the experimental salt concentration. A further factor that determines optimization of solution properties for electrochemical capacitors is the temperature coefficient of specific conductance, which is determined by the temperature coefficients of factors (2) to (5) in a complex, interactive way. A number of relevant properties of nonaqueous solutions have recently been considered in a comparative way, with tabulations of data, by Barthel and Gores 18 in their useful article on solution chemistry. Various improvements to the properties of a nonaqueous solvent for electrolyte solutions can be achieved by mixing it with one or more other polar aprotic solvents (e.g., dimethoxyethane) since the mixture properties of binary or ternary mixed-solvent systems are rarely additive in proportion to the mole fractions of the components, nor are they usually predictable. 4 Overall, the two main factors for optimization of the electrochemical performance of double-layer-type capacitors are good electrolytic conductance and the widest voltage range of electrochemical stability. Some tabulations of selected data taken from the articles by Barthel and Gores 13 ,18 are reproduced, with permission, in the appendix to this chapter. Readers are also referred to the very large computer-accessible databank (ELDAR) on electrolyte solution properties that has been assembled by Barthel and his associates at the University of Regensburg, Germany. The description of this databank has recently been published. 19

354

Chapter 13

TABLE 13.9. Summary of Factors Determining Electrolyte Solution Properties Ion/salt properties Ionic radius Ionic charge Ionic solvation energy Ion pairing Solubility of the solute salt

Temperature coefficients of ion pairing and solubility Thermodynamics of the solvent

Solvent/solution properties Dielectric permittivity of the solvent Donicity or electron acceptor numbers Solvent viscosity Solvent dielectric relaxation times Mixture properties of solvents Melting point of solvent or solution Temperature coefficients of viscosity and dielectric permittivity Solvent vapor pressure, as a function of temperature Electrochemical stability range Liquid phase temperature range (difference between boiling and freezing points)

In the choice of electrolytes for capacitors, especially those utilizing nonaqueous solutions, the difference in the electromotive series of the elements in such solutions relative to that for water must be noted. Quite large differences of standard electrode potentials arise in such a comparison as a result of the usually substantial differences of standard ion solvation (Gibbs) energy in the various solvents 4 (and of solvent donicities 10) relative to the respective values in water. For the latter, see the tabulations in Refs. 20 and 21. These differences of electrode potentials of the elements can determine the behavior of ionic impurities with regard to self-discharge processes in capacitors (Chapter 18). Parsons 22 and Parker 23 have made some useful diagrammatic scales of electrode potentials in various solvents, as illustrated below. The conventional reference electrode for the comparisons given in Refs. 22 and 23 (Schemes 1 and 2) is the HrH+ electrode in the given solvent. Note that no thermodynamically significant and absolute cross-differences between the potential scales for one solvent and another can be specified since such differences cannot be measured without substantial liquid-junction potentials between two solution interfaces. Parsons 22 has also given a comparative tabulation of so-called "real" solvation energies of some common inorganic ions in various nonaqueous solvents, as shown in Scheme 3. The real solvation energies are the energies associated with taking ions from the gas phase (in some standard state) and transferring them into the solution at some other standard state (or for enthalpies, at infinite dilution) across the vaporlliquid interface of the solution where some surface potential, !:lX, exists. This component can account for some 10-20% of the overall solvation energy for univalent ions. For electrochemical capacitors, as also for various battery systems, the thermodynamic decomposition voltage of the solution (i.e., the solvent and

The Electrolyte Factor in Supercapacitor Design

li K

Zn Cd

355 AgCI Cu

- -+ +- - - - - - - - - - -1- -I -'It- -

Ag

"t ~ - 1" -

_~i~ ___________~~.sj~A~~

__ ~g_

-~~---------- - ~x~--~-~---_ ~i~___________ -~-~_)t-A9SJ.f~ _.!:+~

__________

Li K --t- -t- -

~,n_~~u__~

-f!-

___ _

MCI Ag -------------t---+

AgCI Ag ----------------~-~--+-2

-3

-1

POTENTIAL I V vs H2 /H+

DMSQ

6

SCHEME 1. Potential scales for several solvents, using the hydrogen electrode in the given solvent as the conventional reference electrode. The point of zero charge of mercury in the absence of specific adsorption is marked as a cross. Reprinted from R. Parsons, Electrochim. Acta, 21, 681 (1976), with permission from Elsevier Science.

whatever solute ions are within it) is a quantity of prime importance because it determines the theoretical and often practical voltage range over which the device can be operated. The practical operating range can usually exceed the thermodynamic decomposition limit if substantial overvoltages at the cathode (e.g., Hg) or anode (e.g., Pb0 2) arise in aqueous solutions, or for processes in nonaqueous media.

Reduction potentials

I

-0.5

o

I

0.5

1.0

1.5

POTENTIAL I V vs H 2 /H+ SCHEME 2. Reduction potentials of redox systems in water versus nhe(aqueous) and in acetonitrile vs. normal H2 electrode (acetonitrile). Reprinted from A. 1. Parker, Electrochim. Acta, 21, 671 (1976), with permission from Elsevier Science.

356

Chapter 13 Ag Li

CI Rb K

--+++ - - - ---t--+ H20 CI Rb K Ag Li -;----+ +------+-+- - CH 30H CI RbK AgLi - -+ - -t;- - - - - - - f t - - -HCOOH CI

Rb K

--+--t+ -? CI Rb K Li Ag -t----++ - - - - --++- CI

CI

-r- - - - - - - - - -+

Rb K

----1-1- -

2

CH 3CN

DMF

-1---

CI

HCONH 2

3

-

- -

I

Ag

Ag

?

-1-

DMSO

Li

- I - -1-- -(CH 3 )ZCO I

6

_ -Sd____ _Zt_1~_

H

--+---H

Cd

Zn Cu

Cd

Zn

--+---+-+

---f- H

-1-----

---I----f--

H - --1--

--I- -----I-I-

Cd

Zn Cu

Cd Zn Cu --+----1----1-

H

--1---H

------1-

DMSO

H

----+-I

I

1\

12

a~ leV

18

20

22

al/eV

SCHEME 3. Real solvation energies of some common ions in various solvents. The lengths of the arrows indicate schematically the probable contribution of the oriented solvent dipole at the surface to the real solvation energy of a monovalent cation ex potential effect). Reprinted from R. Parsons, Electrochim. Acta, 21, 681 (1976), with permission from Elsevier Science.

Figure 13.4, which is based on data assembled by Bard and Faulkner, 14 gives a useful idea of the relative stabilities of various aqueous and nonaqueous solutions in terms of their electrochemical decomposition voltage ranges. The accuracies are probably ±50 mV for the aqueous and ca. 100 mV for the nonaqueous solutions, depending on the electrode metals, and practically on the type and state of the anode and cathode materials. For the nonaqueous sol utions,

The Electrolyte Factor in Supercapacitor Design

357

tetraalkylammonium salts, often with PF6 or BF4 anions, are the preferred solutes for capacitors, while Li+ salts are often involved in high-energy rechargeable battery systems where deposited Li is the anode material, as metal or an intercalate in carbon. The donor properties of solvents as classified by Gutmann 10 (Table 13.5) provide a basis for ordering of solvation energies, mainly of cations, among nonaqueous solvents. The donicity scale also provides a measure of the strength of interaction of various solvents with polar molecular solutes, e.g., CF3I where the 19F NMR chemical shift is linearly related24 to the donicity number (DN) (Fig. 13.5). An opposite trend is observed when the chemical shifts of 23Na+ (CI04") in various solvents are plotted against the DN (Fig. 13.6). From an electrochemical direction, there are trends of polarographic halfwave potentials, E 1I2 , for Zn2+, Cd2+, and T1+ with solvent donicity as shown in Fig. 13.7. This plot reflects the dependence of the Gibbs energy of solvation of the cation MZ+ in the solvents on the DN in the redox reaction MZ+ + ze + Hg ~ HgIM. Generally, the E1/2 values shown tend to increase with solvent donicity number. However, in this plot from the literature, it is unclear on what scale the E1I2 values have been referenced. If a reference electrode had been used internally in the respective solvents, then a solvent effect on the reference electrode potential would have been

15

DMF. DMA

12

TMPO

0°

°OMSO

9

E

a. a. "-

u..

6

a>

(,()

3

8

16

FIGURE 13.5. Relation of Gutmann donor numbers lO to of solvents with CF3I. (From Erlich, Roach, and Popov?4)

19 F

NMR chemical shifts in interaction

358

Chapter 13

12

E

9

......

6

8:

0

Z

If)

N

3

flO 0 -3

FIGURE 13.6. Relation of Gutmann donor numbers lO to 23 Na+ (CI04) chemical shifts in various solvents.

included. Alternatively, if an external reference had been used,25 then solventdependent, liquid-junction potentials would be included and their trends would be difficult to predict. In a reverse sense, a scale of acceptor numbers (AN) has also been proposed, as listed in Table 13.10. 26 Clear relationships are also found between ANs and solute-solvent interaction behavior [e.g., with 13C=O chemical shifts of acetone in various solvents and with the Gibbs energies of solvation of the Cl- ion (relative to acetonitrile)] in a series of nonaqueous solvents and water. The AN values are also related to values on other scales for Lewis acidbase interaction strengths, e.g., Kosower's Z-values 27 and the values of Dimroth

DMF -0.3

>

......

5

DMSO

AC

SF

-0.1

~

W

0.9 L-__

o

~

____~__~____-L__~~__- L _

10

ON

20

30

FIGURE 13.7. Trends of polarographic half-wave potentials for Zn2+, Cd2+, and TI+ with solvent donicity values.

The Electrolyte Factor in Supercapacitor Design

359

TABLE 13.10. Selection of Acceptor Numbers for Some Solvents Used in Electrochemistry Solvent

ANa

Tetrahydrofurane (THF) Diglyme Dioxane Dimethylacetamide (DMA) Benzonitrile Dimethy lformamide Acetonitrile (AN) Ethanol Water

8.0 10.2 10.8 13.6

15.5 16.0 19.3

37.1 54.8

Source: From Mayer, Gutmann, and Gerger. 26 numbers are relative to 0 for hexane, adopted as a reference solvent.

a AN

et a1. 28 The AN values, in relation to solvation effects in electrolytes for electrochemical capacitor applications, are relevant more to the states of anions in various solvents while the donor numbers apply more to the states of cations. Generally, of course, a good donor number for cation solvation will be a poor one for anions, and vice versa for acceptor numbers. Herein lies the basis of optimization of solvent properties for maximization of electrolyte solubility and dissociation, leading to good specific conductivity and minimum esr in capacitors: mixtures of two or more solvents having respectively good AN and DN values, can be advantageously employed in developing optimized electrolyte solutions for nonaqueous capacitor systems. Also, good donor and good acceptor mixtures tend to provide widely miscible solutions in binary and ternary solvent mixtures. A suitable choice of mixed solvents can lead to maximized conductance of capacitor electrolytes and hence to desired high-power performance that can be otherwise restricted by the internal resistance of the electrolyte. The charge distribution in several dipolar aprotic solvents that determines in part their DN or AN values has been illustrated by Parker23 as shown in the diagram below, which indicates resonance canonical forms and resulting dipole moments. The resonance aspect can be important in accounting for dipole moment values or magnitudes that are not always obvious from inspection of the molecular formula. Thus, for example, CO might be expected superficially to have a dipole moment similar to the )CO groups in ketones for which the dipole moment is quite large. However, the dipole moment of CO is very small and it is an almost nonpolar gas. This is because two canonical forms, 0+

0-

0-

0+

C =OandC=O,

360

Chapter 13

)N

)N

) N-P

0

)N

++

)N-P-O

HMPA

)N

5.20

+

) S

0

++

)S-O:

-c

N:

++

-c

N: C

+

)~

0

++

)N

)~

S

++

)N

+

OMSO 3.90

AN 3.40

0

C-S

OMF 3.90 SOMF 4.40

are comparably significant in the resonance hybrid, so a very low dipole moment results. In the diagram, the dipole moments are in Debyes, i.e., units of 10!8 esu (see Chapter 4), which is approximately the electronic charge, 4.80 x 10- 10 esu, multiplied by an effective charge separation distance of about 0.0205 nm (or 0.205 A). Similarly, DN and AN values depend on resonance aspects of the electronic structure of the solvent molecule except in simple cases. Examples 23 are illustrated below.

13.8. RELATION OF ELECTROLYTE CONDUCTIVITY TO ELECTROCHEMICALLY AVAILABLE SURFACE AREA AND POWER PERFORMANCE OF POROUS ELECTRODE SUPERCAPACITORS

Here we return to the key practical matter; that is, the complex equivalent circuit by which it is necessary to represent29•30 the electrical response of porous electrode supercapacitors and the associated distribution of Rand C elements. Here the main concern is the distributed electrolytic resistance Re in the liquid electrolyte, as shown in Fig. 13.3, which illustrates the equivalent circuit of a lengthy pore (see Chapter 14) filled with a resistive electrolyte in contact with an electronically conducting surface at which a double-layer capacitance Cdl or a C¢ + Cdl is exhibited. 3 ! As explained in Chapter 14, such a circuit has no single or unique relaxation frequency w" or values of the time constant "ReC;" there is a widely distributed range of Wr values or ReCs which corresponds to a power spectrum on discharge or recharge.

The Electrolyte Factor in Supercapacitor Design

361

This situation occurs because the inner regions of the pore (i.e., its surface elements) are electrically accessed only through a progressively increasing Re down the pore (ERe), i.e., there are progressively increasing IR drops. The double-layer capacitance is completely chargeable, or dischargeable, only under dc or low-frequency time scales. Hence a dispersion of capacitance occurs. A detailed treatment of the electrical behavior of porous electrodes is given in Chapter 14.

13.9. SEPARATION OF CATIONS AND ANIONS ON CHARGE AND ITS EFFECT ON THE ELECTROLYTE'S LOCAL CONDUCTIVITY

An important but little-investigated effect in charging of high-area supercapacitor powder electrode materials occurs on account of the separation of cations and anions by their respective accumulation in the high-area double layers at the cathode and the anode. The extents of separation and accumulation match the charges developed on each side of the capacitor. The net charge accumulation is at least about ± 15 f.iC cm- 2, so that for a 1000 m 2g- 1 carbon matrix, the net (univalent) ion accumulation within about 0.3 nm of the distributed surface will be equivalent to 1000 x 104 x 15 X 1O-6 C g-l, i.e., 1.5 x 102/96,500 mol g-l of carbon, "" 1.5 x 10-3 . This is the quantity of electrolyte ions removed from the bulk electrolyte invading the powder matrix upon completion of a charge. Depending on the volume fraction of solution in the matrix, this extent of removal can significantly diminish the conductance of the electrolyte remaining in the pores of the electrode. For, say, a 2 M electrolyte (i.e., 2 x 10-3 mol cm-3) and a 66% porosity, in the uncharged state there would be ca. 1.3 x 10-3 mol g-l of matrix. Thus the charging process can appreciably deplete the concentration of free ions in the bulk electrolyte in the pores, depending on the porosity ofthe matrix and the initial concentration of electrolyte in the discharged state of the electrodes. This depletion then, of course, has a negative effect on the conductivity of the remaining electrolyte, so power performance is impaired and will depend on the state of charge. In charging electrochemical capacitors, there is not only a migration of cations and anions of the electrolyte to the surfaces of respective cathode and anode matrices, which can deplete the bulk concentration of the internal electrolyte, as explained above, but there can also be a significant electro-osmotic movement of the whole solution on charge (depending on the electrode or cell design), or in an opposite direction on discharge, that should be taken into account in assessing the solution factor in the performance of electrochemical double-layer or other porous capacitor systems. The electro-osmotic movement depends on (1) the sign and density of the surface charge on the porous matrix surface, (2)

362

Chapter 13

any specific adsorption of cations or anions of the electrolyte on that surface, and (3) the sign and density of charge on the separator. Such effects are well known in the literature32 of double-layer and colloid science, and can usually be experimentally measured in separate experiments on flow-through matrix cells. 13.10. THE ION SOLVATION FACTOR

Ion solvation, the interaction of solvent molecules with ions, determines the solubility of salts; the extent of dissociation of acids; the extent, indirectly, of ion pairing; and the mobility of free ions, since their radii in solution are determined by the number of coordinating solvent molecules strongly interacting with the ions determining the so-called Stokes radius. These are all factors that have a major influence on the conductivity of electrolytes. Ion solvation (hydration) energies are defined as the difference between the ion's energy in the solvent and in the gas phase. 14 In the solvent, the ion experiences a large interaction energy with the solvent dielectric or, more specifically, with the electric dipoles of solvent molecules. Experimentally, heats of solvation of the' ions of a salt are derived from the calorimetrically measurable heats of solution, M-Isoln , of the salt; this is usually a relatively small quantity, 0 ± 10 kcal mol-I. The heats of solvation, M-Is, are derived from the following thermodynamic cycle4 involving the dissolution of the salt:

M;

~sub

II

+

Ai

(SUblimatio~t

M+A- (salt crystal) + solvent -----:-:-:---_ /l,.Hsoln

+ A- (mS) (solution)

M+(nS)

With acids, HA, there is a chemical proton transfer reaction with the solvent (here H 20): HA + (m + n) H 20

~

H30+ (m - I H 20) + A- (nH 20)

which determines the concentration of free current-carrying H+ and A-ions, and hence the molar conductance. M-Isub is the sublimation energy of the salt, the negative of the crystal lattice energy, a large negative quantity. The sum of the solvation energies of the two ions of the salt is equal to M-Isoln - M-Isub ' This is therefore a large negative (exothermic) energy quantity, on the order of -320 to -500 kJ mol- I for ordinary monovalent ions. Hence the solvation energy provides a large "driving force" for dissolution of the crystal lattice or ionization (dissociation) of acids, HA.

The Electrolyte Factor in Supercapacitor Design

363

Ways are available for separating the derived salt (M+, A -) values into individual ionic components,4,33 but they can have no fundamental thermodynamic basis. However, various empirical procedures are quite successful. 33 Calculations based on ion-solvent dipole electrostatic energies give values in moderate agreement with the data derived from experiment. Approximately, the ion-dipole energies gained when an ion enters a polar solvent balance the energy that must be provided to break up the crystal lattice of strongly interacting ions in the solid phase and transfer them as free but solvated ions into the solution bulk. The equilibrium solubility of a salt is detennined by the standard Gibbs energy (~GO) of solution of the ions from the salt crystal. Negative ~GO values correspond to good solubility of a salt (i.e., to provision of a high concentration of plus and minus ion charge carriers) and good conductance. Thus the conductance of a salt, among other things, depends greatly on the polarity or dielectric properties of the solvent. The latter also detennine what fraction of the dissolved ions at a given concentration (below the solubility limit) are plus or minus paired (factor 3), usually partially retaining their solvation shells of coordinated solvent molecules [Fig. 13.I(b)]. Table 13.11 gives some idea of ionic hydration energies for some simple monovalent ions relative to the value for the proton. They are all large (negative) values on the absolute scale. Di- or trivalent ions have much larger solvation energies, which are related to the square of the charge number. There have been various attempts to make a priori calculations of individual ionic hydration (or solvation) energies. The first procedure was carried out by Born in 192034 and was based on the difference between the self-energy of charging (see Chapter 4) the ion in a solvent of dielectric constant s and that in a vacuum (B = 1). This difference is a Gibbs energy given by (13.7)

for an ion of charge Zie and radius rio Equation (13.7) gives the correct magnitude for I1G s for ions in water. Better estimates are obtained if calculations 35 ,36 are based on the electrostatic energy of coordination and orientation of solvent dipoles around the ion, with some allowance for solvent structure modification4 in the region around the solvated ion. A critical examination of the various types of ab initio calculations and corresponding models used is found in Ref. 14 and in the relevant original papers. Equation (13.7) relates the I1G s to the dielectric constant of the solvent, B, in the term liB. From the nature of this relation it will be seen that it will not give a good account of the specificity of ion-solvent interaction energies since for B values above about 15-20 (i.e., for a variety of common solvents that are good

364

Chapter 13

TABLE 13.11. Relative Standard Gibbs Energies, Enthalpies, and Entropies of Hydration of Ions Ml? (kcal mol-I) ~? (cal K- I mol-I) tlG? (kcal mol-I) Ion W(ref.) Li+ Na+ K+ Rb+ Cs+ Cu+ Ag+ Tl+ p-

ClBr-

r-

OW CIO] S2-

0.0 137.7 163.8 184.0 189.9 197.8 118.7 147.1 182.8 -366.3 -348.8 -340.7 330.0 -345.7 -427.6 -849.4

0.0 -2.4 5.1 13.6 16.5 17.2 -18.7 3.7 14.6 -63.1 -49.49 -45.79 40.32

0.0 138.4 162.3 179.9 185.0 192.7 124.3 146.0 178.5 -347.5 -334.0 -326.0 318.3

-83.1

824.6

Source: From Benjamin and Gold 21 Copyright 1954 Royal Society of Chemistry. Notes: Absolute value of MIl for H+ aq. is -265 ±5 kcal mol- 1 and for!!S1 it is -5 ±I cal- I K- 1 mol-I. Absolute values for other ions are then obtained by adding -265 to the above values for cations or subtracting it from the values for anions.

media for dissolving salts), lie is a small quantity compared with 1, so that tl.G s remains a relatively large negative quantity. In fact, it can be seen from Eq. (13.7) that except for quite small values of e for nonpolar solvents, the magnitude of tl.G s is determined mainly by the loss of the ion's charging energy in a vacuum (the Zie212ri term with e = 1) as the ion is transferred into the solvent medium. The ion-dipole interaction mode1 34•35 gives a better account of specificity for ion-solvent interactions through the individuality of solvent dipole moments and donicities,1O as well as coordination numbers for solvent molecules around ions. Since the solubility equilibrium constant (the solubility product Ksp) is logarithmically related to AGO (AGo = -RTln Ksp ), relatively small differences of AGO, associated with differences oflattice and solvation energies, make large differences of Ksp; these can vary from 10- 13 to Wi for a variety of salts, and also from one solvent to another. Hence, large differences of conductance can arise, so that for battery or supercapacitor electrolytes either well-dissociated acids or extensively soluble electrolytes must be used. For nonaqueous solvents, solvation energies are usually smaller than for water so solubilities and conductances are often less. Tetraalkylammonium salts or salts of large anions such as Ph4B-, Ph4As-, PF6, AsF6, and BEl are often reasonably soluble in nonaqueous solvents, although they are somewhat ion

The Electrolyte Factor in Supercapacitor Design

365

paired. The differences between solvation energies of ions in nonaqueous solvents and in water are usually not well accounted for by Eq. (13.7) in terms of their dielectric constants. Better comparisons can be obtained by using dipole moments 35 •36 and donor numbers, while taking into account steric factors and the accessibility of the polar center to the ion, as noted earlier in this chapter.

13.11. COMPILATIONS OF SOLUTION PROPERTIES The following are sources of published data on electrolyte solution and solvent properties relevant to capacitor development and testing. 1. Electrochemical Data, B. E. Conway, Elsevier, Amsterdam (1952). 2. Organic Solvents, 1. A. Riddick and W. B. Burger, Techniques of Chemistry II, Wiley Interscience, New York (1970).

3. Physical Chemistry of Organic Solvent Systems, A. K. Covington and T. Dickinson, Plenum, New York (1973). 4. Chemistry Data Series, vol. IX, Activity Coefficients, J. Gmehling et aI., Chemistry Data Series, Dechema e.v., Frankfurt, Parts 1,2,3,4 (1986, 1994). 5. Chemistry Data Series, vol. X, Thermal Conductivity and Viscosity Data of Fluid Mixtures, K. Stephan and T. Heckenberger, Chemistry Data Series, Dechema e.v., Frankfurt (1989). 6. Chemistry Data Series, vol. XII, Electrolyte Data Collection, Parts 1 and 1a, J. Barthel and R. Neueder, Chemistry Data Series, Dechema e. v., Frankfurt (1992, 1993). Parts Ib and Ie, in press, Dechema e.v. (199611997). 7. ELDAR, a Knowledge Base System on Microcomputer for Electrolyte Solutions, J. Barthel and H. Popp, 1. Chem. Information Compo Sci., 31, 107 (1991). 8. Electrochemical Tables, D. Dobos and Muzaki Konyvkaido, Budapest (1965), in English translation [see Chern. Abstracts 65, 11767 (1966)]. 9. Chemistry of Nonaqueous Solutions, G. Mamentov and A. I. Popov, eds., Chapter 1, Solution Chemistry, 1. Barthel and H. J. Gores, VCH Publ., Frankfurt (1994). 10. Nonaqueous electrolyte solutions, H. J. Gores and 1. Barthel, Pure Appl. Chem., 67, 919 (1995). 11. Ionic Hydration in Chemistry and Biophysics, B. E. Conway, Elsevier, Amsterdam (1981). 12. Electrolyte Solutions, R. A. Robinson and R. H. Stokes, Butterworths, London (1955). 13. Comprehensive Treatise of Electrochemistry, vol. 5, Thermodynamic and Transport Properties of Aqueous and Molten Electrolytes, B. E. Conway, J. O'M. Bockris, and E. Yeager, eds., Plenum, New York (1983).

366

Chapter 13

13.12. APPENDIX: SELECTION OF EXPERIMENTAL DATA ON PROPERTIES OF ELECTROLYTE SOLUTIONS IN NONAQUEOUS SOLVENTS AND THEIR MIXTURES

13.12.1. Summary Tables

A summary of significant physical properties of solvents of interest for electrochemistry in nonaqueous solutions is given in Table 13.12. Commonly used abbreviations for such solvents are listed in Table 13.13.

13.12.2. Some Graphically Represented Data from the Literature

A selection of viscosity, conductance, and Walden product data, together with dielectric constant information, is given in Figs. 13.A 1 to 13.A12 as a function of electrolyte concentration or solvent composition, as appropriate. They are based mainly on the work of Matsuda7,8 and ofPlakhotnik. 6 Unfortunately, the electrolyte data are for Li salts; much less plotted data are available for tetraalkylammonium salts in nonaqueous solvents, especially mixtures. Some are in proprietary reports that are not yet in the public domain.

13.12.3. Selected Tabulations

Selected tabulations from the review publications by Barthel and Gores 13 ,18 and other sources are given in Tables 13.7,13.8, and 13.12. TABLE 13.12. Physical Properties of Some Nonaqueous Aprotic Solvents Used in Electrochemistry for Systems Requiring High Decomposition Potentials Solvent

b.p.(°C)

Acetonitrile y-Butyrolactone Dimethoxyethane N,N-Dimethylformarnide Hexamethyl-phosphorotriarnide Propylene carbonate Tetrahydrofuran 2-Methyltetra-hydrofuran Dimethyl sulfoxide Dimethyl sulfite Sulfolane (tetra-methylenesulfone) Nitromethane Dioxolane

81.60 202 84 158 233 241 66 80 189 126 285 101.2 78

f.p.(°C) -45.7 -43 -58 -61 7.2 -49 -108.5 18.55 -141 28.86 -28.6 -95

Source: From Barthel and Gores 13 ,18 and other sources. Notes: d is in g cm-3 at 298 K; D is in debyes, 10-18 esu cm.

d(g cm- 3)

0.771 1.125 0.859 0.944 1.02 1.19 0.88 0.855 1.096 1.207 1.262 1.131 1.06

e

,u(D)

37.5 39 7.20 36.7 29.75 64.4 7.58 7.3 46.6 22.5 43.3 35.94 7.13

3.44 4.12 3.86 5.38 4.94 1.75

367

The Electrolyte Factor in Supercapacitor Design

Cl. u

....

f:"

100%

PC

mOl %

100%

OME

FIGURE 13.A1. Kinematic viscosities 17 of 1 M KBF4 solutions in PC + DME as functions of mixture composition; numbering of the curves corresponds to the following temperatures: x, -30; (0) shows the data for PC + DME mixtures at 25°C. (Data from Plakhotnik et al. 6)

t., -20; D, 0; 0, +25; *, +40 and 0, +50°C; curve

.. I

a

e Ne u

en

.....

0.8

c

:;: 0.6

c

CD

15

e- OA CD

> 0 0 .2

O~------~~-----J--------~

10 -2

10-1

1

101

______~

LOG [Current density I A cm- 2 ) FIGURE 14.12. Overpotential vs. current-density relations for the case of activation and ohmic overpotential at a two-phase porous electrode or at a porous gas-diffusion electrode using the simple_pore 6 model (Tafel approximation calculation). J 5 or even> 3, approximately).

Electrochemical Behavior at Porous Electrodes; Applications to Capacitors

403

0.6 2

0.5

(t)

0.4 0

_0:: ...... 0.3 N I

0.2 0.1 0

0

0.1

0.2

Z'IR o

0.3

0.4

0.5

FIGURE 14.14. Master complex-plane impedance curve for form-factor values, ttl., and Ro = IJ7rKr2. l. = 0.5 (rKlwC)ll2; notation of Ref. 15. Reprinted from Keiser et aI., Electrochim. Acta, 21, 539 (1976), with permission from Elsevier Science.

Note that for pore-form numbers 3 and 4, there is a tendency for an initial (high tf).. values) semicircle to become developed. It is interesting that this has been observed experimentally with a compressed porous Ni electrode. 7

14.3.3. Configuration of Double Layers in Porous Electrodes

In very fine pores of porous electrode structures, the dimensions of double-layer charge distributions can be comparable with the effective widths of pores. This depends on the Debye reciprocal length, K, of the diffuse layer (Chapter 6) and hence on ionic strength. This matter is dealt with in more detail in the chapter on theoretical treatments of the double layer (Chapters 6 and 7). Briefly, since most double-layer capacitors employ relatively concentrated electrolyte solutions having Debye lengths on the order of 0.5-1.0 nm, the problem of extension of the diffuse layer over much of the pore diameter, and overlap from one side to the other, will not arise except in the very finest pores where a special model would then, in any case, be required. The electrostatic problem here of overlapping diffuse layers is similar to that treated by Verwey and Overbeeck in their classic monograph on the Theory of the Stability of Lyophobic Colloids. 16 The situation referred to above regarding overlap of diffuse-layer ion distributions between charged colloidal particles and in thin charged pores was examined theoretically and quantitatively by Farina and Oldham 17 for the case of

Chapter 14

404

CIC

0~-~8~~~-~4~~~0~~~4~~~8~

X FIGURE 14.15. Average relative ionic concentration across the cell for various values of the y parameter. The dashed line represents the ionic concentration profile before charging; Yl = 1.155, Y2 = 10.40, Y3 = 22.49, Y4 = 46.20, and Y5 = 104. Reprinted from C. J. E. Farina and K. B. Oldham, 1. Electroanal. Chern., 81, 21 (1977), with permission from Elsevier Science.

12

10 8

>
C2)« R F ]· At sufficiently low w, Rp(jwRFCd1 + 1) becomes RF and Z is then the impedance of RF in series with Cz, namely, (16.140)

AC Impedance Behavior of Electrochemical Capacitors

523

together with any solution resistance, if significant. This gives a rising vertical plot of Z" in the complex-plane with decreasing W having an intercept of RF on theZ'axis(Fig.16.16). After rationalizing the denominator, the overall impedance, not including an R" is given by (16.141)

(16.142)

from which the real and imaginary components of Z are recognized. The above relation reduces to the limiting cases for W -4 0 or W -4 00 identified above. Note that unlike the equivalent circuit

considered earlier, the capacitance C2 introduces blocking ofthe alternating current at low w, so Z -400 as W -4 O. In the circuit shown on p. 521, when W -4 00, there is admittance to Cp through Rs and RF in series. In the circuit involving C2 , an initial semicircle associated with the frequency response of Cdl and RF with its maximum at We = lIRF Cdb arises with an intercept at Z' = RF (or RF + Rs, if Rs "# 0); beyond the intercept at RF, with continuing scan of W to lower values, the Z is determined by lIjwC2 , i.e., the vertical line of increasing Z' appears with decreasing w, as illustrated in Fig. 16.16. Note that for the above cases and the ones considered earlier, RF will normally be potential dependent according to a Tafel exponential function in Vor '7. It could be represented in the equivalent circuit as an exponentially variable Faradaic resistance or, near the reversible potential, by a linearly variable resistance, which is the basis of treatments of small-amplitude av modulation around the reversible potential of a Faradaic process (Eqns. 16.5 and 16.15). Finally, it should be noted that self-consistency of analyses and modeling of frequency-response of various RC or RCL circuits can be usefully made in terms of the so-called Kramers-Kronig relations,4,22 referred to earlier.

524

Chapter 16

REFERENCES 1. 2. 3. 4.

5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

K. S. Cole and R. H. Cole, J. Chem. Phys., 9, 341 (1941). 1. E. B. Randles, Disc. Faraday Soc., 1 (1947). M. Sluyters-Rehbach and 1. Sluyters, Rec. Trav. Chim., P.B., 83, 217, 581, 967 (1964). For discussion of Kramers-Kronig relations, see, e.g., 1. M. Ziman, Principles of the Theory of Solids, p. 222, Cambridge Univ. Press (1964), and A. H. Morrish, The Physical Principles of Magnetism, pp. 88, 209, Krieger Publ. Co., Huntington, N.Y. (1980). A. N. Frumkin and R. Melik-Gaikazyan, Dokl. Akad. Nauk. S.S.R., 77,855 (1951). R. D. Armstrong and M. Henderson, J. Electroanal. Chem., 39, 81 (1972). M. Keddam, O. R. Mattos, and H. Takenouti, J. Electrochem. Soc., 128, 257 (1981). B. V. Erschler, Disc. Faraday Soc., 1 (1947). B. Proskumin and A. N. Frumkin, Trans. Faraday Soc., 31, 110 (1935). M. Breyer and F. Gutmann, Aust. J. Sci., 11,96 (1948). D. E. Smith, Anal. Chem., 35, 1811 (1963); 36, 962 (1964). D. E. Smith, in Electroanalytical Chemistry, A. 1. Bard, ed., vol. I, pp. 1-155, Marcel Dekker, New York (1966). A. Sevcik, Coil. Czech. Chem. Comm., 13, 349 (1948). F. Will and C. A. Knorr, Zeit. Elektrochem., 64, 258 (1960). A. Eucken and B. Weblus, Zeit. Elektrochem., 55, 114 (1951). B. E. Conway and E. Gileadi, Trans. Faraday Soc., 58, 2493 (1962). E. Gileadi and B. E. Conway, J. Chem. Phys., 31, 716 (1964). E. Gileadi and S. Srinivasan, Electrochim. Acta, 11,321 (1966). 1. Klinger, H. A. Kozlowska, and B. E. Conway, 1. Electroanal. Chem., 75, 45 (1977). H. A. Kozlowska and B. E. Conway, J. Electroanal. Chem., 95, 1 (1979). B. E. Conway, H. A. Kozlowska, and E. Gileadi, 1. Electrochem. Soc., 112,341 (1965). H. A. Kramers, Atti. Congo Fisica (Como), 545 (I927) and R. Kronig, J. Opt. Soc. Am., 12, 547 (1926).

Chapter 17

Treatments of Impedance Behavior of Various Circuits and Modeling of Double-Layer Capacitor Frequency Response*

17.1. INTRODUCTION AND TYPES OF EQUIVALENT CIRCUITS

In earlier chapters it has been emphasized that the high-area, porous carbon structures that provide large specific (per gram) capacitance values (in practice, 5 to 50 F g-l) cannot be represented by a simple capacitance or even by a simple RC circuit. The diagrams in Fig. 17.1 represent a hierarchy of equivalent circuits, from those for a simple capacitor through those involving simple combinations of a capacitor element with either one or two ohmic or equivalent ohmic resistors, to more complex equivalent circuits involving distributed capacitance with ohmic elements in series or parallel coupling with the capacitative elements and, for some conditions, with Faradaic leakage resistances in parallel with the capacitative elements. The latter may also include contributions from pseudocapacitances associated with surface redox functionalities (Chapter 9) on the interfaces or edges of carbon particles or the main pseudocapacitance associated with redox oxide capacitor materials (Chapter 11). In some cases, considered later, the equivalent circuit may also include an inductive element, L. 'This Chapter owes much to the derivations and analyses given by Dr. Miller at the various Deerfield Beach (Florida) seminars on electrochemical capacitors held in recent years.

525

526

Chapter 17

----1 ~

a) Simple capacitor

b)

Capacitor with equivalent or real series resistance A

C

~ RF

c) Capacitor with series resistance and potential - dependent Faradaic leakage resistance, RF

A

o

B

d) parallel combination of n C and RF leakage elements. Overall "RC" constant is RF/n x nC =RFC for the single element.

e) parallel C, RF elements connected B

C

with pore - resistance elements, Rp (related to transmission line, constant phase element but with RF leakage pathways).

FIGURE 17.1. Hierarchy of equivalent circuits, from that for a pure capacitor, with no esr, to that for a porous electrode with distributed C and R components.

This section examines the frequency response of the impedance of various equivalent circuit arrangements that may represent the impedance spectra of some practical capacitor devices,l-5 as measured experimentally by means of a Solartron instrument in two-terminal measurements of electrical response. Figures 16.2, 16.3, and 16.4 in Chapter 16 illustrate the definitions of the time (t)-dependent current let) and voltage Vet) associated with a sinusoidal modulation frequency, ill. let) and Vet) may differ in phase by a phase angle tP.

Treatments of Impedance Behavior of Various Circuits

527

In general, the impedance Z(w) for a given imposed frequency is Z(w) = V(t)I/(t)

(17.1)

analogous to Ohm's law, where Vet) = Vo sin wt and /(t) =10 sin (rot + rp), are the time-variant voltages and currents that are modulated according to the frequency w, as expressed earlier. Z(w) may include ohmic (R), capacitative (C), and sometimes inductive (L) elements; the latter two are dependent on frequency according to Z(w) = l/jwC and Z(w) = jwL, respectively, as displayed below. The elements of equivalent circuits and their respective dependences on frequency are as follows: Z(w)

Element Resistor, R Capacitor, C Inductor. L

R (dR/dw=O) l/jwC=-jwC jwL

where j, as usual, is ~. In a series arrangement of n impedances n

~otal =IZ;

(17.2)

;=1

while for a parallel arrangement n

l/Ztotal = I

liZ;

(17.3)

;=1

so that for this case, the smallest of the Zs principally determines the overall Z. In electrochemical systems the same R, C, and L elements can be recognized as in hardware circuits, except that C is a double-layer electrostatic capacitance or a pseudocapacitance (Ct/J; see Chapter 10) and L is a pseudoinductance, usually of connecting wires the effect of which is usually small except at high frequencies or rapid transients. Another qualitative exception is the existence of a so-called Warburg impedance element (represented by W) that arises when an electrochemical process is or becomes diffusion controlled. The element W contains two components, a diffusional capacitance and a diffusional resistance (see Chapter 3), each of which is dependent on the squareroot of frequency. In the material that follows, the behavior of various elementary RC and RCL equivalent circuits will be examined, including the case of distributed capacitance and resistance elements that represent the behavior of high-area porous electrodes.

528

Chapter 17

17.2. EQUIVALENT SERIES RESISTANCE 17.2.1. Significance of esr Most capacitors, except those of the vacuum or air type, do not exhibit ideal pure capacitative behavior according to the criterion that the real and imaginary components of their impedance should be out of phase by 90° independent of frequency between periodic changes of current and voltage when addressed by a sinusoidally alternating voltage. Most practical capacitors deviate from this requirement because their impedance is not that of a pure capacitance but of a capacitance linked in series with an element exhibiting ohmic behavior that in combination with the capacitance, gives rise to a phase angle that is less than 90° and is usually frequency dependent. This ohmic component is referred to as the equivalent series resistance. In many cases, especially with electrochemical or electrolytic capacitors, the esr is a real series resistance involving components of the capacitor cell, e.g., resistance of the electrolyte and/or contact resistances. Only when the dielectric of the capacitor itself exhibits so-called "lossy" behavior due to kinetically limited dielectric relaxation processes in molecular or oxide dielectric materials is the behavior truly represented by an "equivalent" series resistance, i.e., a value of a supposed series resistance that would imitate the observed phase-angle behavior. This is a fundamental aspect of the behavior of regular dielectric electrostatic capacitors. The kinetic limitation arises at very high frequencies when the polarization induced by the voltage applied to the dielectric cannot cause simultaneous, in-phase, dielectric polarization (Chapter 5) of the dielectric medium (i.e., of its assemblies of molecules); there is a delay. Commonly with packaged capacitor devices, a further and larger real esr arises from "hardware factors" (interparticle and external lead contact resistances, electrolyte resistance in electrolytic and electrochemical capacitors, etc. as referred to in Chapter 16). In the case of double-layer-type electrochemical capacitors, the esr is mainly associated with the latter factors because dielectric loss of the interphasial solvent and ions forming the double layer itself does not normally occur until frequencies greater than hundreds of megahertz are applied. The usual and simplest equivalent-circuit representation of the involvement of an esr is as follows:

which has a frequency-dependent impedance, Z(w) given by

Treatments of Impedance Behavior of Various Circuits

Z(w)

= esr + l/jwC

529

(17.4)

which tends to Z(w) = esr at infinite w. This result usually provides the formal means of evaluation of esr of a capacitor device. Thus, in complex-plane plots of the imaginary (Z") vs. the real (Z') components of the impedance vector (Z) at various frequencies (Chapter 16), the effective esr (including any series electrolytic resistance) is revealed as the intercept on the Z' axis as w ~ 00. The involvement of an esr in electrochemical capacitors is a factor of major importance in their electrical behavior, especially their power performance (Chapter 15). The presence of significant esr restricts the rate at which the capacitance can be charged or discharged upon application of a given voltage difference between the electrodes or the alternating currents that can be generated at a given moment by applying an alternating voltage at a given frequency. The esr limits the power at which the device can be operated from a given direct or alternating voltage, and is thus one of the most important properties of a supercapacitor, in addition to its capacitance density in F g-l or F cm-3 . The impedance of the capacitor device then differs from that of a pure capacitor because a frequency-dependent phase angle (less than 90°) is developed between the alternating voltage and the resulting ac modulation, as indicated earlier on the basis of general principles. Power loss and energy dissipation then ensue. In the case of electrochemical double-layer capacitors, the esr situation is usually substantially more complex than that illustrated in the equivalent circuit here. As explained in Chapter 14, distributed electrolytic, interparticle, and intraparticle resistances arise because of the porous, particulate nature of the electrode matrix. Rather similar behavior can arise with oxide-type redox pseudocapacitors (e.g., Ru02) having high-area microporous matrices. In the above case there is no unique esr except the component of resistance ofthe external solution. The impedance behavior, plotted in the complex plane, exhibits a 45°, almost linear relation at elevated frequencies (transmission-line effect; Chapter 14), with an intercept on the Z' axis as w ~ 00, corresponding to the resistance of the solution and any separator external to the outer surface region of the porous electrode matrix. In state-of-the-art double-layer-type capacitors, effective esr values are in fractions of milliohms up to a few tens of milliohms. The higher values occur when electrochemical double-layer capacitors are designed for higher voltage operations using nonaqueous electrolyte solutions, which have intrinsically lower specific conductances. The minimization of esr is obviously desirable in constructing electrochemical capacitors that are intended for operation at elevated power levels (see Chapter 15).

530

Chapter 17

17.2.2. Impedance Limits for Some Commercial Capacitors Due to esr

Eisenmann6 has evaluated the practical impedance limits for a series of commercially available electrochemical capacitors in relation to the ohmic component of their impedance. The capacitance falloff with decreasing ohmic component (Z') of the impedance is shown in Fig. 17.2 for a series of ten commercial capacitors. The cutoff capacitances shown in Fig. 17.2 as a function of Z' can be identified by the effect of esr. If the capacitor were to be shorted, all its energy would be dissipated at the fastest possible rate with a time constant in seconds given by:

tcutoff = Resr

xC

(17.5)

The initial voltage and current drawn at the start of discharge and hence their product, the power, P, depend on the energy, E, stored in the capacitor device so that6

PIE

= 21tcutoff

(17.6)

S-1

1000 100 10 LL

......

Cl)

u

C

g u

'r

0.1

0

a. 0

U

0.01 0.001 0.01

10

0.1

100

IZI/Ohms FIGURE 17.2. Impedance plots for ten commercial capacitors showing cutoff of capacitance with decreasing esr component. I Soleq 470 F; 2 Soleq 70 F; 3 Maxcap 1 F; 4 NEC 0.47 F; 5 Maxcap 0.47 F; 6 Maxcap 0.22 F; 7 NEC 0.1 F; 8 Maxcap 0.1 F; 9 Maxcap 0.047 F; 10 Sprague 0.022 F. (From Eisenman. 6)

Treatments of Impedance Behavior of Various Circuits

531

The physical construction and materials of the capacitor determine PIE, while the actual capacitance and maximum charging voltage do not enter as variables. 6 The current U.S. Department of Energy goals specify a PIE ratio of 0.0278 s and a cut-off time of 72 s, i.e., twice the 36-s timeline of the tests. Then the power at the time of the short circuit would be (after a revision of the Equation in Ref. 6):

P =

V2IR esr = CV2ltcut-off W

(17.7)

where C is the capacitance and V the voltage at the start of discharge. The addition of an external load resistance, RL , to the transmission-line equivalent circuit causes the cut-off impedance to increase and its approach to become steeper. For measurement evaluation of capacitors, an appropriate load resistance in series (= esr), yields their power ratings according to Eq. (17.7). As an example, Eisenmann 6 considers a I-F capacitor, charged to I V, being discharged through a I-ohm load resistor; then the power delivered will be I W provided that Resr « I; this latter condition determines whether the expected power can be delivered, as may be obvious. The behavior of the porous carbon capacitors actually tested in Eisenmann's work exhibited some interesting features. For material activated in four stages, the slopes of the capacitance vs. frequency curves (capacitance dispersion) were steeper (about -0.8 vs. -0.5) than those (slope about -0.5) of the commercial electrochemical capacitors tested. Also, only the maximally activated material exhibited an initial low-frequency plateau. A Sprague (electrolytic) capacitor typically exhibited almost no dispersion of capacitance with frequencies up to 103 Hz. It was concluded, significantly, that the former behavior was consistent with a resistive layer at the surface of the carbon materials, probably caused by a film blocking entrances (mouths) of pores. This is consistent with the equivalent circuit when low-indexed resistive elements at the outer end of the transmission lines assume high values. This may be identical with the low porosity envelope that has been detected by transmission electron microscopy (TEM) at some porous electrode carbon materials derived by pyrolysis of certain polymers, e.g., polyimides. The observed impedance behavior was for some carbons derived from poly(vinylidene chloride) polymers activated at 850°C in CO 2. This activation procedure is believed to improve accessibility. If the outer resistive film can be removed first, then less activation will be required. These observations provided useful new information about the microscopic resistive behavior of polymer-derived porous carbon materials for double-layer capacitors.

532

Chapter 17

17.3. IMPEDANCE BEHAVIOR OF SELECTED EQUIVALENT CIRCUIT MODELS

For a qualitative interpretation of the frequency response of an equivalent circuit, it is often possible to evaluate "by inspection" the limiting impedance behavior for zero and for infinite frequency (w =0 or w =00 in the equations for Z). For most cases, in circuits where L is insignificant, the frequency dependence of Z arises entirely from that of capacitative elements and their combination with resistive ones. Then for w ~ 00, capacitative elements have zero impedance (Z" lIjwC) while for w ~ 0, they have infinite impedance, i.e., the current is blocked. Resistive components (if purely ohmic) have constant impedance (Z == R), independent offrequency. Thus, for example, for a parallel RC circuit, the R is bypassed by the C component as w ~ 00, i.e., R is short circuited by C. Conversely, C passes no ac current when w ~ o. w ~ 0 is, of course, equivalent to a dc condition. Then current will pass only if there are one or more resistive elements continuously in series between the input and output sides of the equivalent circuits, with no capacitative element in series with them. The simplest equivalent circuits are those where Rand C elements are connected either in series or parallel, as considered in Chapter 16 for the origins of semicircular complex-plane plots of Z" vs. Z'. For the series RC circuit, the impedance is

=

Z=R + lIjwC==R- jlwC

(17.8)

The modulus of Z is given by (17.9) and the phase angle if> = tan- 1(-1/wRC)

(17.10)

The d.f. = Z'IZ" = wRC and dJ. = 1 when w = lIRC or/ = 1I2nRC (2n/ = w). where/is the frequency in radians/s. Similar considerations apply to the parallel RC circuit treated in Chapter 16 in order to demonstrate the origin of semicircularity in the complex-plane plots. For this case, the impedances of the components add up reciprocally:

liZ = lIR + l/(l/jwC) so that after the rationalizing multiplication

(17.11)

Treatments of Impedance Behavior of Various Circuits

Z = _R_(1_---'l:-·w...".R_C:'-) 1 + w2R2CZ

533

(17.12)

and (17.13) As derived in Chapter 16, the real and imaginary components of Z are (17.14) and (17.15)

while Z' and Z" combine, as shown in Chapter 16, to give the complex-plane plot

(Z' - R12)2 + (Z")2 = (RI2)2

(17.16)

the latter being the equation to a circle (manifested as a semicircle in the complex-plane diagram) of radius RI2. The last equation arises from eliminating w from Eqs. (17.14) and (17.15), for Z' and Z". The third degree of complexity (a practically realistic one for electrochemical capacitor devices) is when a series resistance, Rs ' representing solution and/or contact resistance (esr) is combined with the parallel circuit of the previous case. Then Z=Rs+

RF (1 - jwRFC) w2 2CZ 1 + RF

(17.17)

where RF is a Faradaic, parallel leakage resistance that may be potential dependent according to a Tafel exponential factor. In this case, a semicircular plot of Z" vs. Z' is retained, but it is displaced along the Z' axis with an intercept of Rs. The equation for the semicircle is then (Z' - Rs - RFI2)2 + (Z')2 =(RFI2)2

(17.18)

Multiplying out the term in Z containing RF andjwRFC, we find RF

Z =R + --"---s

(1

jwR~C

(17.19)

+ w2R~CZ)

from which Z' and Z" are directly recognized and distinguished. The phase angle tan ifJ = Z"/Z' is found as

534

Chapter 17

+ w2R~CZ) tan t/J = Rs + RF/(l + w2R~CZ) jwR~C/(l

-jwR~C

(17.20)

(17.21)

For this case, note the limiting results that for w --t 0, tan t/J --t 0; and for w --t 00, tan t/J also --t 0, so there is a maximum in tan t/J, as is commonly observed. When Rs --t 0, then tan t/J = -jwRFC, which varies from 0 to 00 with 0< w < 00, i.e., there is no maximum with increasing w. This is the situation in dielectric relaxation plots 7 of the real and imaginary components of the dielectric constant as a function of frequency. When RF --t 00 (no leakage currents), the circuit becomes that for a series RC connection. For tan t/J = 00, t/J = n12, the phase-angle for a pure capacitance, while for tan t/J = 0, t/J = O. Although most circuits involving electrochemical capacitors do not include any deliberatley added inductances, except perhaps for some resonance purposes, adventitious inductance can be introduced that is significant, e.g., when large current pulses are employed in charge or discharge. Then it is the wiring of the circuit that can introduce a significant self-inductance. In some other cases, however, where special types of electrode processes (e.g., metal oxide film formation and passivation) are involved, an intrinsic pseudo inductance may be exhibited in the impedance frequency response. This type of situation can arise when inhibition of an electrode process leads to Tafel behavior that corresponds to a negative resistance (current decreasing with increasing overvoltage), or when surface coverages by some adsorbed intermediates decrease with increasing overvoltage. In the case of corrosion processes involving onset of passivation, very complicated Z' vs. ZIt complex-plane plots appear in two or three quadrants, where reentrant curves arise and ZIt enters the positive quadrant. This is usually due to a situation of decreasing current with increasing potential, i.e., a negative resistance arises. In a series R, C, L circuit Z = R + l/jwC + jwL

=R + j(wL -

lIwC)

(17.22) (17.23)

and (17.24) with

Treatments of Impedance Behavior of Various Circuits A-.

wL - 11wC

tan'f'=---R

535

(17.25)

Note that when wL - lIwC = 0, w 2 = LC or f (= 2nw) = 1I2n (LC)1I2, Z is real. This situation occurs at the so-called "self-resonant frequency," w = wo = 1/(LC)1I2, for which Z is at its minimum (see Figs. 17.6 and 17.8 later). In effect, the opposite variations of ZL and Zc with w tend to cancel each other out, leading to a minimum in IZI. For the above RCL circuit, the impedance as a function of frequency has the form shown in Fig. 17.3, where the components of Z are plotted against frequency, and the minimum in Z is shown for cases where the equivalent series resistance, Rs, is finite or zero. In the latter case Z can go toward zero when complete compensation between Zc and ZL arises at some characteristic frequency. Note that when two parallel RC circuits are in series, and further in series with an Rs, namely.

Rs

two semicircles can arise in the complex plane plot with a high-frequency intercept on the Z' axis of Rs (Fig. 17.4). In another case for a series/parallel combination, namely,

the complex-plane plot has the form shown in Fig. 17.5, where a single semicircle becomes transformed to a vertical line in the -Z" direction with decreasing frequency; the intercepts on the Z' axis are Rs + R1, and Rs at high frequency.

536

Chapter 17

-

IMPEDANCE WITH ESR AS SHOWN

- - - - IMPEDANCE WITH ZERO ESR

N

W

U

ESR

Z 0) has declined from its initial value, Vo. Since I = VIR according to Ohm's law, P = V2IR. P has a maximum value when Rs =RL. The analysis for this case was given in Section 16.4.1. Pmax

P

The current is Vo/(Rs + RL) and the delivered power is 2 (17.28)

For the matching load situation (Rs =RL), the maximum power is found as P max

= VO/ 4Rs

(17.29)

(17.30) i.e., it arises at V = V012 or half the potential for full discharge. Note that when RL = 0, P = 0, and when RL = 00, PL also is zero, while a maximum in P arises between these conditions for the load resistance. This situation is similar to that found in the theoretical analysis for Ragone plots of power density vs. energy density (Chapter 15). For a series RC circuit with a resistive load, the power supplied in relation to the maximum power utilized (P max) is given by P = [4P max r/(1

+ r)2] exp[-2r/(l + r)]

(17.31)

where Pmax = V6/4Rs, as shown earlier. The parameter r =RL/Rs is the ratio of the load resistance to the series or equivalent series resistance. r = 1 is referred to as the "matched load" condition. The time parameter r = tiRsC, where RsC is

540

Chapter 17

the time constant of the Rs' C combination. r is thus a reduced time scale, as arises in other related problems (Chapter 16, Section 16.3.2). For a voltage range of Vo (the initial voltage), down to Val2, analytical results calculated by Miller 1•2 are shown in Fig. 17.7. It is seen that when r = 1 (matched load), PIPmax (ordinate scale of the plot) becomes unity since (l + r)2 is then 4 and exp[-2r/(l + r)], when r = 1 and t =0, is also unity. The supplied power, P, is equal to the maximum power when r = 1, the matched-load condition. The origin ofEq. (17.31) is as follows. Consider a capacitor of capacitance C having an esr of Rs discharging into RL from an initial voltage, Vi, and a voltage during discharge of Vt across the capacitor. Then, at any time, (17.32) where the latter two Vs are the voltages developed across Rs and RL, respectively, when a discharge current I(t) passes. The current for discharge of the capacitor is: 1/ = Ii exp[-tIRC]

(17.33)

also (17.34) and

)(

c

E a.. .....

0.1

..J

a..

r =10

0.01 o/O""'----'---....L2--~--_i--_t- . -=6""""----'--7--~ T=t/RsC

FIGURE 17.7. Behavior of a series (esr) RC circuit with power delivered into a resistive load (RL) for discharge from initial voltage Vo down to VoI2 based on results analytically calculated by Miller. I•2) Parameter r =RL/ Rs and r = 1 is a matched load.

Treatments of Impedance Behavior of Various Circuits

541

(17.35) For an initial charge, qi' on the capacitor, (17.36)

At t = 0, Vi - IiRS - IiRL = 0 or IiR = Vi where R is written for Rs + R L. Then integrating Eq. (17.36),

Vt = Vi - (fiR - IiR exp[-tIRCD

=Vi -

(Vi - Vi exp[ -tIRCD

=Vi exp[-tIRC]

(17.37)

(17.38)

(17.39)

The power delivered by the capacitor is Peap

= ItVt

(17.40)

=Ii exp[-tIRC] . Vi exp[-tIRC]

(17.41)

=

vr Rs+RL

. exp[-2tIRC] = 4RsPmax . exp[-2tIRC]

(17.42)

Rs+RL

where P max, the maximum power developed, is Recalling that RLIRs was defined as r

Vr 14Rs.

4Pmax

(17.43)

4Pmax

(17.44)

P =- - . exp[-2tlC(Rs + R L)] 1+r

= - - . exp[-2r/(1 + r)] l+r

where a time constant r has been defined as r = tiCRs. The power, PL, dissipated across the load RL is I;Rr- Then substituting for II derived earlier in Eq. (17.33),

542

Chapter 17

(17.45)

=

4rPmax (1 + r)

2 .

exp[-2r/0 + r)]

(17.46)

Similarly, the power, Ps' developed across Rs is Ps =

4Pmax (1

+ r)

2 .

exp[-2r/(l + r)]

(17.47)

The power drawn from the capacitance minus that lost in Rs is the power developed across the load, i.e., Peap -

Ps = 4Pmax

O+r) 4Pmax 2· exp[-2r/(1 + r)] 2· exp[-2r/(1 + r)] (1 + r) (1 + r)

(17.48)

=

4Pmax r (1

+ r)

2 .

exp[-2r/(1

(17.49)

+ r)]

which is the required time dependence of power developed in the load from the discharge of the capacitor. The behavior of several RC or RCL systems that may be equivalent to that of any practical capacitor has been simulated in calculations by Miller. 2 A firstorder model for any capacitor is (cf. p. 537), in the simplest analysis:

c L

Rs

where Rs and Rp are series and parallel resistive components coupled with C. L mayor may not be significant. Rp can be a Faradaic (potential-dependent) reaction resistance when C is a double-layer capacitance. Rp is much greater than Rs for practical capacitors except when Faradaic leakage currents arise, leading to self-discharge (see Chapter 18). Z is then (cf. Eqn. 17.23)

Treatments of Impedance Behavior of Various Circuits

543

Z = Rs + j[wL - lIwC]

(17.50)

which exhibits the self-resonant frequency Wo = 2nlo, where 10 = 1I(2n -VLC), as above. The device then stores energy effectively for/

I.

~~):

8

~ H.u

7

6 5

4

.. ..

~ . , ~~ ~::. ~:~:~:~':~'::~-_": •.,,·.... '110·,.·,.·

('J,:.::::>..

R3: 99k Cl·C2 o C3-C.·135

/(d)

3

o

IlA

charge curren! 50 Al -! Oak R2 a 99k

20

40

60

eo

100

120

140

160

IJF

180

200

lime I seconds FIGURE 18.5. Time variation of potentials on open circuit across four capacitors in a transmission-line type of network following charging at 50 /lA. Experimental responses on a hardware R-C network. (From Pell and Conway, in press, 1. Power Sources, 1999).

20 -


1.2

2

(3

4.5

1.0 .::

4.0

0.8

CI)

3.5

Q) (/)

3.0 2.5 2.0

r

0.6 0.4 0.2

OJ

cU (3

>

>..

iii > o u

~

L......J.._~-'---~--'-~--'-~--L-~-'500.0

o

10

20

30

40

time / hours .-.. CI) C)

~

0.75 0.70

(3

0.65

...

0.60

>

CI)

C)

cU

.r:

u

0.55

:0

0.50

(/)

r-r~-.------r~--,--.......~--,--.,.........,.~-,-.,....,--r"1

(b 1.2 Q)

..

~

0.0

...

o >

>..

CI)

> o u

CI)

-0.6 ..=. OJ .2

0.45 0.40 0.35

OJ

0.6

L.....LO~--'-~J........--'-~--'-~...........--'-~-'-~L...........J...-'

5

1 0 15

20

25

30

35

40

-1.2

45

time / hours FIGURE 18.10. Diagnostic test plots for evaluation of mechanism of self-discharge observed at a Panasonic 3.5-F Gold Cap capacitor. Potential recovery relationships after forced discharge are also shown. (From Pel!. Liu, and Conway. 16)

Following galvanostatic discharge, potential recovery on open circuit was also monitored. This recovery is also shown in Fig. 18.10. The potential recovery of this multicell device did not plot as a linear function of...Jt or log (t + r), nor was log(V) linear in t. The increase in open-circuit potential, in this case, is probably the result of a nonuniform charge distribution in the CR matrix of the porous carbon electrodes and between the individual cells that made up this 5.5V capacitor. On open circuit, the charge is redistributed, giving rise to the apparent recovery (see Section 18.11).

586

Chapter 18 en

5.5

>

5.0

.~

Q)

CD

!!

4.5 4.0

CD

3.5

Q)

.~

--

3.0

'C

(j)

en

1.2

•

~

'0 1.0 >

-Q)

'0 > ~ .r:. u

(c) 1.4

2.5 2.0

0.8

(

0.6 0.4 0.2

0

100

200

300

400

Cl

!!

'0 >

>Q; > 0 U

~

0.0

time 1/2 / seconds1/2 ~

'0 > Q)

Cl

!!

'0

5.5

(d) 1.4

5.0

1.2

4.5

1.0 O.B

Cl

3.5

0.6

--.

3.0

0.4

2.5

0.2

Q)

~ .r:. u .~

'C

(j)

en

en

>

Q)

4.0

>

...

'0

2.~ 00

101

102

103

104

105

Cl

,!g

'0 >

>Q; > 0

U

~

0.0

time / seconds FIGURE 18.10. (continued)

18.15. SELF-DISCHARGE AND POTENTIAL RECOVERY BEHAVIOR AT AN RU02 ELECTRODE

18.15.1. Background

It was shown in Chapter 11 how ruthenium oxide (Ru02) has attracted a lot of attention as a possible electrochemical supercapacitor material since its electrochemical behavior was recognized as being like that of a capacitor (e.g., under cyclic voHammetry study) by Trasatti and Buzzanca,17 and Galizzioli, Tantardini, and Trasatti 18 and following our own investigations of that system (cf. Refs. 19 and 20). The large specific capacitance (many farads per gram) of Ru02 prepared either electrolytically as films at Ru metal 19.20 or thermally from RuCl 3 or

Self-Discharge of Electrochemical Capacitors in Relation to that at Batteries

587

(~)RuCI6 at Ti (DSA electrode formation)17 has been variously attributed l9- 21 to redox pseudocapacitance or double-layer capacitance at a large developed surface area. It now appears that both these origins of the capacitative behavior apply (Chapter 11), but with a substantially larger fraction being due to redox pseudocapacitance (i.e., from a series of oxidation states ofRu from +11 to +VII) in the case of the electrolytically generated RU02 film that is formed in a hydrous state capable of freely admitting mobile protons into the bulk oxide material. Contrary to the situation with double-layer capacitors and battery electrodes, very little has been done to study the possible self-discharge behavior of RU02 electrodes except for some effects briefly described by Galizzioli, Tantardini, and Trasatti 18 and later by Arikado, Iwakura, and Tamura,22 together with some recent general studies by Zheng and Jow 21 on RU02 charging. We now describe some new and interesting results on this question recently obtained in our laboratory. The Ru02 films studied 15 were prepared in three different ways. The first was by controlled firing of a Ti wire, multiply painted with RuC1 3 or (NH4)RuCI6 solution, at 350°C (DSA electrode procedure). The second was by anodic and cathodic cycling of a Ru rod in aqueous H 2S04 between 0.05 and 1.50 V under cyclic voltammetry conditions at 50 m V S-I. The third procedure employed cycling at Ru metal electrodeposited at an Au or Ti substrate. The electrochemical behavior of the films was characterized by evaluation of the oxidation and reduction charge in the voltammograms or by dc galvanostatic charge, discharge, and recharge cycles at given currents.

18.15.2. Potential Decay (Self-Discharge) and Recovery in Relation to Charge and Discharge Curves

The observations of principal significance from this work are as follows: 1. Following completion of charging of a thermally formed RU02 film to 1.40 V or to other lower potentials (1.30 V to 1.00 V, RHE) and then interrupting the charging current, the self-discharge associated with a fall in potential takes place over a lengthy period [Fig. 18.1l(a)], eventually reaching a steady potential of about 0.75 V (RHE), which is close to the initial rest potential of the electrode in the solution. Similarly, self-discharge down to 0.80 V was recorded for electrolytic Ru02 on Ru [Fig. 18.11(b)] and to 0.70 V for a thin RU02 film formed on the electrodeposit of Ru on Au. 2. If the thermally formed electrode film had been cathodically discharged to 0.1 V from a previously charged state (potentials 1.40 V to 1.00 V) and then allowed to stand on open circuit, a lengthy recovery of potential took place [Fig. 18.1l(a)], with an asymptotic approach to almost the same potential of 0.75 V

Chapter 18

588

1.5

(a)

w

:c a:

'" > >

1.2

~

0.9

c:

---

Q)

-

'0 c=> ~

- -

-

0.6

.~

c=

Q)

c-

0.3

o

0.0

0

4

2

6 8 Time I hours

1.5 '" > >

14

1.2

~

0.9

-

0.6

"E Q)

12

(b)

w

:c a:

10

--

-- -

(5

c-

'5 ~

'u C:

(l)

c-

0.3

o

0.0

0

2

4

6

8 10 12 14 16 18 20 22 24 Time f hours

FIGURE IS.11. Open-circuit potential decay following anodic charging to 1.40 V (RHE) and open-circuit potential recovery following cathodic discharge to 0.1 V (RHE) for (a) thermally prepared RU02 and (b) electrochemically prepared RU02 electrodes in 0.5 mol dm- 3 H2S04 at 298 K.

Self-Discharge of Electrochemical Capacitors in Relation to that at Batteries

589

as that attained in potential decay from the charged state. Similar recovery behavior was observed at the electrolytically formed films [Fig. 18.11(b)]. Plots of decaying or recovering potential (even after inclusion of the integration constant time r; cf. Ref. 1 and Eq. IS.6) vs. log [time] are not linear, indicating that a Faradaic process of self-discharge, of the kind treated in Refs. 1 and 4 and examined earlier here, is not applicable. 3. The most remarkable observation is that following the recovery of the potential to ca. 0.6 V, a further cathodic discharge down to 0.10 V, involving passage of an appreciable further cathodic charge, can be conducted; then, following that further discharge, the potential again recovers to near the 0.6 V value [Fig. IS.12(a)]. This discharge-potential-recovery cycle can be repeated many times and the successive recovery potentials remain almost the same. A "perpetual" source of charge is, of course, impossible, so presumably there is a sufficient bulk reservoir of oxidized Ru to maintain a surface region, which is evidently potential determining and sufficiently oxidized that (through diffu-

0.8 (a)

0.6 w

:::c a: en

>

>

Cii

0.4

'E Q) '0

c...

0.2

o

2

4 Time I hours

6

8

FIGURE 18.12. Multiple potential recovery transients at RU02 in aq. H2S04 at 298 K following successive galvanostatic discharges, without interim recharging: examples of successive recoveries of potential at (a) a thick, thermally formed RU02 film;

590

Chapter 18

sion processes, see later discussion) an almost unchanged recovery potential is eventually reached. In order to test this point further, similar successive discharge and recovery cycles [Fig. lS.12(b)] were carried out at a much thinner film of oxide formed thermally on Ti. Then the final rest potentials after successive recoveries were found eventually to decrease, discounting the apparently indefinite source of cathodic charge indicated by the results obtained with thicker films. This remarkable succession of recovery-redischarge cycles [Figs. lS.12(a) and lS.12(b)] must arise through a "diffusion of oxidation state", following discharge of the surface region, back to the surface from inside the bulk of the RU02' The following succession of processes can be suggested as indicated below: Ru lV

+e

~

Ru IIl

+e

~

Ru ll

0.8 (b)

0.6 w

:x:

II:

en

>

>

cti

0.4

~ Q)

"0

a..

0.2

o

100

200

300

Time I hours FIGURE 18.12.

(b) as in (a) but for a thin RU02 film.

400

Self-Discharge of Electrochemical Capacitors in Relation to that at Batteries

591

at the electrode surface on discharge, coupled with a redistribution of the oxidation state of Ru in the oxide by electron hopping following such a discharge, according, e.g., to: Ru III bulk + RuII surface ~ Ru II bulk + RuIII surface probably involving electron and proton exchange (cf. Ref. 1); and also RuIV bulk + RuIII surface ~ RuIII bulk + RuIV surface Coupled with such redox processes must be adjustment of the local distribution of 0 2- and OH- ions in the lattice by proton transfer, namely OH- (surface) + 02-(bulk)

~

0 2- (surface) + OH- (bulk)

Proton transfer equilibria with H30+ and H 20 in the surface region eventually will also be involved 0 2- (surface) + H30+ (solution) ~ OH- (surface) + H 20 (solution) Evidently, either from self-discharge or potential recovery, the surface region eventually provides a fairly stable thermodynamic potential that is dependent on the previous cycle history of the oxide film. Similar time-dependent adjustments of potential took place at electrolytically formed Ru02 films, as was shown in Fig. 18.11 (b), but over rather longer time scales; the recovery phenomenon was also similar to that observed at the thermally formed oxide film [Fig. 18.11(a)]. 18.15.3. Model for Potential Recovery

Here we discuss the potential recovery behavior exhibited by electrodes that have been cathodically discharged. The model shown below for the bulk, near-surface, and surface regions of an Ru02 electrode is considered. Three potentials are identified: Vbulk for the bulk, V2 for the near-surface region S2, and VI for the surface region, SI' This is predicated on the requirement that depending on its mode of preparation, although the RU02 is conducting, the conductance is not fully of the metallic type, and the model here, by its nature, involves regions having different states of oxidation. It is supposed that it is VI that is measured during the course of charging or discharging, and in potential decay and recovery. It is determined primarily by the (oxidation) state of region SI' V2 is determined by the state of S2' This can change during charging but apparently not materially during discharging. For a fresh electrode, the role of V2 in determining the potentials measured is small, but for an old, cycled electrode the influence of V2 increases with time of use of the electrode.

592

Chapter 18

rutheniwn oxide film On charge it would be expected (as with nickel oxide electrodes, Ref. 1) that a gradient of state of oxidation would develop outward through S2, toward and involving Sb while on discharge the opposite would tend to occur. When the electrode is in an equilibrium (or virtually steady) state, VI = V2 = Vbu1k, while during polarization (e.g., charging or discharging), internal potential differences (apart from any internallR drop) LlV = V2 - V! and LlV' = Vbu1k - V! become established. After interruption of charging or discharging during which Ll V is finite, a process takes place that leads to Ll V ~ 0 after some lengthy period of time. This process could be proton diffusion (or proton hopping) coupled with internal redox electron transfer between Ru ion centers, which could be rate controlling if that process is slow enough. The possibility of accounting for the observed behavior in terms of a diffusion of oxidation state though proton and electron migration between an outer (potential-determining) surface region and an inner or bulk region has been referred to earlier. Pell, Liu, and Conway15 developed a mathematical representation of the assumed diffusion process which represented the repetitive potential recovery behavior quite well, based on solution of I-dimensional, semi-infinite diffusion equations. A value of the proton diffusion constant in the hydrous oxide of ca. 10- 12 cm2 s-I was derived. 18.15.4. Quasi-Reversible Potentials of Ru02 after Self-discharge

The decline of potential of a charged Ru02 electrode takes place down to a value well below that for the Oz-H20 couple (unlike the situation at nickel oxide!) and the same value is approached in recovery after forced cathodic discharge. This suggests that some quasi-reversible Nernstian potential is

Self-Discharge of Electrochemical Capacitors in Relation to that at Batteries

593

established. It is about 0.7 V, or 0.9 V, RHE, as seen in Fig. 18.11, differing between the thermally and electrolytically formed RU02 films. In order to elucidate what kinds of equilibrium redox processes might be involved in establishing these potentials, the pH dependence of the steady potentials was determinedY The results are shown in Fig. 18.13 from which dE/dpH = -30 m V was reproducibly found for the thermally formed film compared with -60 m V for the electrochemically formed one. These data provide a pH/potential kind of Pourbaix relation for the potential-determining processes at the two oxide materials following self-discharge and potential recovery. Evidently the oxide in its supposedly RU02 state at 1.4 V (or at 1.3, 1.2, 1.1 V) is not stable in aqueous solution at any pH between 0 and ca. 10 (Fig. 18.13), and prefers to establish a substantially less positive steady potential. While a -30m V dependence on pH has not previously been reported, Siviglia, Daghetti, and Trasatti 23 found that the open-circuit potential of a thermal RU02 electrode exhibited a -60-mV dependence on pH. In Fig. 18.13 the potentials asf(pH) have been corrected for the 60-mV pH dependence of the hydrogen reference electrode used, so the pH dependence of the potentials plotted can be interpreted in terms of half-cell processes involving two oxidation states of Ru in the potential-determining process, together with a stoichiometric number of involved protons and a corresponding number of electrons.

0.9 w

::r: :z en >

• •

0.8

>

0.7

:;:; c

'"

0.6

"0 0.

0.5

thick film of electrolytic RU02 on Ru thick film of thermal RU02 on Ru metal

Q)

::>

.~

0.4

'c..=

0.3

u

Q)

0

0.2

a

2

4

6

8

10

12

pH FIGURE 18.13. pH dependence of open-circuit decay and recovery potentials after approach to steady values for _ thermally and. electrochemically prepared thick film of RuOz on electrodes. Reprinted from Pell et aI., Electrochim. Acta, 42,3541 (1997), with permission from Elsevier Science.

594

Chapter 18

Consideration of the two well-defined Nernstian slopes of -30 and -60 m V (Fig. 18.13) requires examination of the numbers of protons and electrons that are involved in the potential-determining redox reactions. Note that none of the many half-cell reactions listed by Pourbaix 24 for processes involving neutral, or cationic, or anionic Ru species corresponds to processes giving the above Nernstian slopes. The only process listed by Pourbaix 24 that has a reversible potential anywhere near the observed stationary potential of ca. 0.75 V in acids is a redox process involving Ru-Ru203 (EO = 0.738 V) with a -60-mV slope. Cyclic voltammetry at RU02 shows (like the situation for Ir0 2 but unlike that for Pt, Rh, Pd, or Au), however, that no redox processes occur where reduction back to Ru(O) takes place, unlike the situation at Pt, Ru, or Au. The observed -60-mV slope for the electrolytically formed Ru02 indicates that the same number of protons as electrons is involved in the potential-determining reaction, so a variety of possibilities could be envisaged. However, none of the reactions listed in Pourbaix24, which fulfill the above condition for protons and electrons, has an EO value near the observed stationary potential (0.75 V). This is probably because the latter potential is one set up at a nonstoichiometric7.8 oxide surface state, whereas the potentials in Pourbaix are for supposedly well-defined phases and oxyanions. The -30-mV slope observed for the thermally or thin electrolytically formed films requires a redox reaction involving an electron-to-proton ratio of 2. Processes in Pourbaix24 in which there is a discharge of protons and electrons are only those for which the opposite H+/e ratio arises, e.g., reactions involving RuO~- [Ru (VI)] or RU04 [Ru (VII)] anions. The observed pH or potential behavior is therefore quite unusual and led US l5 to propose a process such as: Ru+Y - (OH)x - Ru+Y + H+ + 2e ~ Ru+(y-l) - (OH)x_1 . (H20) - Ru+(y-l) involving hydrated oxides (on the lhs and rhs) of the type suggested by Burke et a1. 25,26 These quasi-thermodynamic considerations indicate interestingly that the redox equilibria at the anodically formed oxide film and at the thermally generated one must be quite different. It may be significant that it is the -30-mV slope that arises with the thermal film that still retains some hydrous character provided the thermal treatment is not at very high temperatures. The slope value of -30-mV can be accounted for only by the above process involving a hydrous structure of some kind. Other hydrated Ru02 phases require the same numbers of electrons and protons in the redox equilibria, e.g.

Self-Discharge of Electrochemical Capacitors in Relation to that at Batteries

595

The unambiguous difference between the Nernstian slope for the thermally prepared compared with the electrolytically generated Ru02 film is of considerable interest in relation to the chemical constitution of the surfaces (or bulk as well, in the case of the electrolytic film) of the two materials despite the qualitative similarity between their respective potential decay and recovery behavior (Figs. 18.11 and 18.12). The above thermodynamic and potential decay information indicates that the self-discharge process does not originate (as it does at Ni·0·OH-Ni02 electrode surfaces I) from continuing O 2 evolution since (1) the potential decay process continues down to ca. 0.75 V rather than 1.23 V (the OrH20 reversible potential) and (2) it is also observed from several initial potentials below that reversible potential. Hence the potential decay process must involve readjustment of Ru redox couples in the surface region of the oxide. The same argument also applies to the process involved in potential recovery following cathodic discharge down to 0.1 V, or other potentials less than 0.75 V.

18.16. SELF-DISCHARGE IN A STACK

It is well known that in a series stack an unmatched faulty component capacitor may undergo overcharge or overdischarge if it is adventitiously of lower C than that of the other capacitor units with which it is unmatched. Related problems can arise if one or more components in a stack suffers self-discharge at rates greater than others, so uniformity of self-discharge behavior is desirable, if not essential. High-level manufacturing control conditions are therefore desirable.

REFERENCES 1. B. E. Conway and P. L. Bourgault, Can. J. Chem., 37, 292 (1959); Trans. Faraday Soc., 58, 593 (1962). 2. K. Kinoshita, Carbon: Electrochemical and Physicochemical Properties, Wiley, New York (1988). 3. 1. A. V. Butler and M. Armstrong, Trans. Faraday Soc., 29,1261 (1933). 4. B. V. Tilak and B. E. Conway, Electrochim. Acta, 21, 745 (1976); 22,1167 (1977). 5. B. E. Conway, in Power Sources, vol. 16 (Brighton Power Sources Conference Proceedings) (1997), Inti. Power Sources Symposium Ltd. 6. B. Pillay and 1. Newman, J. Electrochem. Soc., 143, 1806 (1996).

596

Chapter 18

7. O. S. Ksenzhekand V. V. Stender, Dokl. Akad. Nauk. SSSR, 106, 487 (1956); see also Russian 1. Phys. Chem., 27, 1089 (1963). 8. 1. Crank, The Mathematics of Diffusion, The Clarendon Press, Oxford (1956). 9. H. S. Carlslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford Univ. Press, Oxford, (1947). 10. P. Delahay, New Instrumental Methods in Electrochemistry, Chapter 3, Interscience, New York (1954). 11. D. A. Harrington and B. E. Conway, 1. Electroanal. Chem., 221, I (1987). 12. S. Shibata and M. P. Sumino, Electrochim. Acta, 20, 739 (1975). 13. S. Shibata, 1. Electroanal. Chem., 89, 37 (1978). 14. F. G. Will and C. A. Knorr, Zeit. f Elektrochem., 64, 258 (1960). 15. W. G. Pell, T.-C. Liu, and B. E. Conway, Electrochim. Acta, 42, 3541 (1997). 16. W. G. Pell, T.-C. Liu, and B. E. Conway, in Proc. Sixth IntI. Symposium on Electrochemical Capacitors and Similar Energy Storage Devices, S. Wolsky and N. Marincic, eds., Florida Educational Seminars, Boca Raton, Fla. (1996). 17. S. Trasatti and G. Buzzanca, 1. Electroanal. Chem., 29, App. 1 (1971). 18. D. Galizzioli, F. Tantardini, and S. Trasatti, 1. Appl. Electrochem., 4, 57 (1974). 19. S. Hadzi-Jordanov, H. A. Kozlowska, M. Vukovic, and B. E. Conway, 1. Electrochem. Soc., 125, 1471 (1978). 20. S. Hadzi-Jordanov, H. A. Kozlowska, and B. E. Conway, 1. Electroanal. Chem., 60, 359 (1975). 21. J. P. Zheng and T. R. Jow, 1. Electrochem. Soc., 142, L6 (1995); 1. P. Zheng, P. J. Cygan, and T. R. Jow, 1. Electrochem. Soc., 142,2699 (1995). 22. T. Arikado, C. Iwakura, and H. Tamura, Electrochim. Acta, 22, 513 (1977). 23. P. Siviglia, A. Daghetti, and S. Trasatti, Coli. Surfaces, 7, 15 (1983); see also 1. Electroanal. Chem., 122,395 (1981). 24. M. Pourbaix, Atlas of Electrochemical Equilibria in Aqueous Solutions, Nat!. Assoc. Corrosion Engineers, Houston, Tex. (1974 reprint). 25. L. D. Burke and 1. F. Healy, 1. Electroanal. Chem., 124, 327 (1981). 26. L. D. Burke, M. E. Lyons, E. J. M. O'Sullivan, and D. P. Whelan, 1. Electroanal. Chem., 122, 403 (1981).

Chapter 19

Practical Aspects of Preparation and Evaluation of Electrochemical Capacitors

19.1. INTRODUCTION'

The key components in an electrochemical capacitor are the electrode, electrolyte, and the separator materials. Most electrochemical capacitors, except the hybrid type, l are made with identical electrodes, i.e., preparation of only one type of electrode is necessary, unlike in batteries. However, as the specific capacitance (e.g., of a carbon electrode) when positively polarized is usually different from that of the same material when it is negatively polarized, some allowance for this can be made by using a different quantity (weight or volume) of the active material in the two electrodes. Then the resulting device should be polarity labelled. The packaging for capacitors depends on their type, size, and ultimate use. Coin-type cans and electrolytic capacitor-type housings are frequently used for PC-board applications. Larger cells are of prismatic construction, with heavywalled containers and robust terminal connections. The end plates must provide good electrical contact in order to minimize esr, and some compression may be advantageous in construction, but not at the expense of squeezing out significant quantities of electrolyte. The focus of this chapter is on laboratory-scale construction and testing of capacitor cells. An example follows. *This section is based on material contributed by Dr. B. V. Tilak (Occidental Chemical Corp.), Grand Island, N. Y. and Dr. S. Sarangapani, ICET Inc., Norwood, Mass., and edited for this volume by the author.

597

598

Chapter 19

19.2. PREPARATION OF ELECTRODES FOR SMALL AQUEOUS CARBON-BASED CAPACITORS FOR TESTING MATERIALS

A carbon black or activated carbon powder (e.g., Vulcan XC-72, Ketjen black, Black-pearl 2000 or the like), including activated proprietory preparations, is first mixed with a sufficient quantity of aqueous 6 M sulfuric acid to obtain an almost dry paste. The exact quantity of sulfuric acid depends on the type of carbon and has to be determined by prior experiment. Care should be taken to avoid too much sulfuric acid, since the process of pellet making is rendered difficult if the paste contains excess liquid. For initial mixing experiments, water can be substituted for sulfuric acid. The semidry paste is then loaded into a pellet die (an infrared pellet die can be used for this purpose, Fisher Scientific or Spex Corp.) and a pressure of 5000 psi is applied to compact the paste into a pellet. An amount of 200-300 mg of the dry paste should be sufficient for 0.5-0.75 in. diameter pellets, although some trial is again necessary, depending on the size of the die and the type of carbon used. Activated carbon fiber available from Spectracorp, Lawrence, Massachusetts, can be substituted for or used in conjunction with activated carbon in the preparation. A nonaqueous electrolyte such as Li or Na tetrafluoborate or tetraethylammonium perchlorate or tetrafluoroborate dissolved in propylene carbonate or dimethylformamide can be substituted for aqueous sulfuric acid as electrolyte in the fabrication of nonaqueous solution, higher-voltage capacitor units. The use of nonaqueous solvents increases the voltage of capacitors to 2.5-3.5 V, depending on the solvent and the electrolyte, as explained in earlier chapters. The electrode-making procedure is the same as that described above, except that the dry paste is made with the nonaqueous electrolyte. The preparation of the paste has to be carried out inside a glove bag or glove box to maintain dryness, because any water present can lead to gassing on charge and/or self-discharge problems. Usually the nonaqueous solvent or solvents used must be chemically dried and redistilled, in the case of PC, under reduced pressure. Electrodes for carbon-based capacitors can also be made according to the following procedure for the preparation of fuel cell electrodes, given here as an example. A total of 200-300 mg of the activated carbon powder or the chopped activated carbon fiber is made into a slurry with approximately 50 ml of distilled water. Ultrasonication and stirring with a magnetic stirrer facilitates the slurrymaking process. The pH of the slurry is adjusted to 3.0 using sulfuric acid. To this slurry, 5-10 wt.% Teflon 30 emulsion is added (prediluted with about 20 ml of water) and the slurry subjected to ultrasonication for 3 min. The mixture is then filtered through a glass or stainless steel microanalysis holder (Micro Filtration Systems, Dublin, California, or Whatman, available through catalog houses) using a microporous PRFE 511 filter disk available, e.g., from the suppliers of the filtration setup. The filtered cake is removed from the funnel and

Preparation and Evaluation of Electrochemical Capacitors

599

placed over a Toray H060 (Toray Marketing, New York) carbon paper (or the Spectracorp equivalent) and gently rolled with a rolling pin between several layers of filter paper to remove excess water. The electrode is then sandwiched between several layers of dry filter paper and pressed in a hydraulic press at 100-400 psi for 1-2 min. At this stage, the microporous filter membrane can be removed by gently peeling off the layer from the electrode. The electrode is then placed between PEP Teflon films and pressed at 100°C at 100-400 psi for 1-2 min. The electrode is then dried at 100°C for 1 h, baked at 270°C, and sintered at 340°C for 20 min. The hydrophobic nature of such electrodes prevents the full realization of the maximum available capacitance during the initial testing period. However, these electrodes undergo a "break-in" phenomenon, during which period the electrolyte seeps into the electrode and wets the internal finer pores. Vacuum filling of the electrolyte is desirable to speed up the break-in process, and/or a trace of surfactant can be added to the electrolyte.

19.3. PREPARATION OF RuOx CAPACITOR ELECTRODES

A titanium foil is immersed in hot oxalic acid for 2-3 min; it is then ultrasonically cleaned in distilled water. A solution of the following composition is prepared (South African Patent, 680,834): n-butyl alcohol: concentrated aqueous HCI: RuClf

6.2 cc 0.4 cc 1g

This solution is either sprayed or brushed onto the clean titanium foil. The foil is air dried and the process of brushing and drying is repeated until a sufficient loading of the ruthenium is obtained, typically 12 times. Once a sufficient thickness of the coating is built up, the foil is heat treated in air at a temperature of 350-500°C for about 5 min. Care should be exercised to not build up too thick a layer, since thick layers tend to spall on heat treatment. Other transition metal oxide electrodes (e.g., Ir02, C030 4 ) can also be prepared by this procedure that is based on the recipe for preparation of DSA RU02-Ti anodes for the chloralkali industry (Beer patents). For comparative examination of electrochemically formed RU02 films, Ru metal (bulk or electrodeposited on C or Au) is subjected to repetitive anodic and cathodic cycling under cyclic voltarnmetry conditions over a potential range of 0.05 to 1.40 V (RHE) in aqueous H2S04 for a number of hours. This leads to the formation of quite thick films of hydrous ruthenium oxide (up to microns), which are easily visible under an SEM or optical microscope (see Chapter 11). The hydrous oxide exhibits appreciable redox pseudocapacitance, while the thermally formed DSA type has a larger component of double-layer capacitance.

600

Chapter 19

19.4. PREPARATION OF RuOx CAPACITORS WITH A POLYMER ELECTROLYTE MEMBRANE (U.S. Patent 5,136,477)

A paint of ruthenium oxide or ruthenium on a carbon catalyst powder (Aesar) in a solution of 1100 EW Nafion ionomer (Solution Technology, Mendenhall, Pennsylvania or Aldrich) is prepared by suspending and dispersing the catalyst particles in the Nafion solution. An amount of 25-30 wt. % N afion (dry weight basis) may be adequate to prepare a satisfactory paint. An oxidized niobium foil is coated with this paint and dried in an oven to remove the alcohol solvents. A Nation 117 membrane, in the acid form and well equilibrated with water, is placed in between two such coated niobium foils and hot pressed at 110°C for 1-2 min. The membrane electrode assembly is then removed from the press, soaked in distilled water, and subjected to charge and discharge cycling.

19.5. ASSEMBLY OF CAPACITORS

A schematic diagram of an assembly arrangement is shown in Fig. 19.1. Each cell consists of a pair of electrodes prepared as above, interposed with a

. - - - - - - - Electrode 1 . - - - - - Separator Electrode 2

Heater Pad

__ .1 ~ ~ '/4.1r-~1/4' 1/2"

1/2'

FIGURE 19.1. Sectional diagram of laboratory-scale electrochemical cell for testing capacitor electrodes at various temperatures.

Preparation and Evaluation of Electrochemical Capacitors

601

glass fiber paper (Whatman) soaked with 6 M sulfuric acid electrolyte. Suitable end plates made of resin-filled graphite (Carbone of America, JP1345 or equivalent) can be used as current collectors. A simple framework that consists of two stainless-steel end plates, with pedestals machined into them, serves as the cell housing, as shown in Fig. 19.2. This arrangement is useful for laboratory screening of potential capacitor materials in successive series of tests. The cell assembly is placed between the two end plates, which are clamped together with bolts and wingnuts. Care should be taken to apply insulating sleeves on the bolts together with nylon washers to prevent shorting of the two electrodes. Sufficient pressure can be applied to the cell by varying the thickness of the gaskets in the cell andlor tightening the nuts. This design also permits cells with nonaqueous electrolyte (assembled inside a glove box) to be tested under ambient conditions. A spring-loaded cell construction suitable for the present application has been described by Kronenberg, Stein, and Codd. 2 For the assembly and testing of solid polymer electrolyte capacitors, the cell assembly shown in Fig. 19.2 can be used (Sarangapani et a1. 3.4).

Gold Plated Tab

Cell Assembly

~

Teflon Gasket Insulated Bolt

FIGURE 19.2. Cell assembly based on arrangement of electrodes illustrated in Fig. 19.1, but for a solid polymer electrode.

602

Chapter 19

19.6. EXPERIMENTAL EVALUATION OF ELECTROCHEMICAL CAPACITORS

A useful reference manual is available for testing capacitors,4 and Burke and Miller5,6 have published material on the same topic. The following test procedures (see also Chapter 3) are useful as the basis of initial screening tests for capacitors. 19.6.1. Cyclic Voltammetry

Cyclic voltammetry (Chapters 3 and 10) is useful as a quick screening procedure to identify potential capacitor materials. The experimental procedure involves potential cycling between two voltage values preselected for a given electrolyte. The experimental details of conducting a cyclic voltammetric experiment can be found in textbooks, e.g., those by Gileadi 7 or Conway.s A simple electrochemical cell and electrode holder are shown in Figs. 19.3 and 19.4, respectively. Note the inclusion of a reference electrode (Fig. 19.3). 19.6.2. Impedance Measurements

The frequency-response characteristics of a capacitor and its equivalent series resistance are important in an evaluation of a capacitor and are dependent upon: (1) the intrinsic nature of the electrode material; (2) the pore-size distribution of the high-area material used in the fabrication of the electrodes; and (3) the engineering parameters used in the preparation of the electrodes, e.g., the thickness of the active material of the electrode and the nature of the particleparticle contact, as well as the macropore distribution as determined by the applied pressure. Impedance spectroscopy provides a convenient way to assess the frequency -response characteristics of a capacitor material, especially the powerlimiting internal resistance. Impedance measurements as a function of frequency can now be readily made, even down to low frequencies of 0.01 Hz, e.g., by means of a Solartron Frequency Response Analyzer or an EG and G/PAR lock-in amplifier. The analyses of the resulting data are usually not simple and have been treated in Chapters 16 and 17. However, various software packages are available for processing the results and performing equivalent-circuit simulations of recorded impedance spectra. Impedance measurements are more useful for evaluating capacitor behavior than that of batteries because the capacitance component of the impedance is the main quantity of interest and is directly determinable as l/jwC (Chapters 16 and 17). In addition, the combination of C with any series resistance (external and esr) can be easily evaluated and the role of porous-electrode behavior (Chapter 14) readily perceived from the frequency-response behavior of the im-

Preparation and Evaluation of Electrochemical Capacitors ~

603

Electrode Holder

1 1 - - - Counter

Electrode

DHE or HgS04

Reference Electrode Bridge

FIGURE 19.3. Electrochemical three-electrode cell (thermostated) for screening the behavior of capacitor electrodes by means of cyclic voitammetry. DHE signifies dynamic hydrogen (reference) electrode.

pedance. Furthermore, when pseudocapacitance arises (see Chapter 10), it can often be distinguishably evaluated, again from the frequency response-behavior of the real and imaginary components of the impedance or admittance recorded over a wide frequency range.

19.6.3. Constant Current Charge or Discharge Charging at constant current, followed by discharge across a known load, is a traditional method of testing batteries, and this test applies equally well to capacitors. These tests can be done in the following combinations: 1. Charging at constant current followed by immediate discharge across various load resistors. 2. Charging at constant current followed by a holding period (variable, followed by discharge across a preselected load resistance).

604

Chapter 19

Noryl Base Viton Pad Gold Foil

Test E.lectrode ~ Viton Washer

~ ~ / ~;w

I ~

/

/

~

Noryl

Hoi. '''' Lead Wire

FIGURE 19.4. Simple test-electrode holder for mounting in the electrochemical cell shown in Fig. 19.1.

3. Charging at various rates and discharging across a fixed load resistor. The charge, q, and energy, E, input and output to the capacitor can be calculated through integration of the voltage time transients according to: (19.1)

where q is the charge, V is voltage, RL is the load resistor, and E is energy. By computing the input energy and the realized output energy, figures of merit for the charge and discharge efficiency can be derived. The internal resistance (esr) of the capacitor can be determined by applying a square-wave current pulse of proper magnitude (to be determined from the size of the capacitor) and

Preparation and Evaluation of Electrochemical Capacitors

605

measuring the instantaneous voltage decay at the breakpoints. The magnitude of the internal resistance can be used to understand the computed energy efficiency, i.e., large internal resistances lead to poor energy efficiency. The effective or mean RC time constant is determined from the voltage decay curves by measuring the time to reach VJe where Vo is the initial potential before decay. The external resistance of the system (e.g., solution resistance) can be determined by means of the impedance measurements, with data being plotted according to complex-plane analysis (see Chapter 17). Note that a porous electrode does not have a unique RC time constant owing to the distribution of Rand C elements (Chapter 14). Charge and discharge and self-discharge characteristics (Chapter 18) should also be determined as a function of temperature in order to fully understand the characteristics of the device and its operational limits in relation to utilization requirements and ambient conditions. 19.6.4. Constant Potential Charge or Discharge

Here a potentiostated constant potential step is applied to the electrode and the time-dependent current transient is recorded. The integral of this transient in time gives the delivered charge. Usually a series of transients is recorded to various potentials in the operating range of the electrode. In the various electrical procedures, digital recording of either potential or current is advantageous and can be set up through a computer or e.g., a Nicolet digital oscilloscope. 19.6.5. Constant Power Charge or Discharge

This is a common procedure used in battery testing and is advantageous for evaluating electrochemical capacitors, where the output power capability during declining voltage on discharge can be an important practical test parameter. 19.6.6. Leakage Current and Self-Discharge Behavior

Leakage current or self-discharge behavior is conveniently determined by monitoring the residual current flow, the so-called "float current," when a fully charged capacitor is kept under constant voltage control. This measurement can be carried out using a potentiostat in a two-electrode mode; the cell is charged at constant voltage (i.e., corresponding to the open-circuit voltage of a fully charged capacitor) and the residual current flow is monitored when the capacitor is fully charged, as indicated by the constancy of the current observed beyond 5 RC times. Leakage current can also be measured using power supplies with good voltage control, and shunt resistors should be included in the circuit to monitor the current flow accurately. Under these conditions, leakage current is sometimes referred to as the "float current."

606

Chapter 19

Self-discharge behavior can be more directly determined by following the voltage decay of a fully charged capacitor over a 24-h period. There will be a rapid initial drop in voltage from the elimination of any IR drop, but the voltage drop will level off over longer periods of time. The analysis of the time-dependent forms of this potential decay (e.g., in log time or a square root of time) can lead to diagnostic information on the mechanism of the self-discharge process as treated in Chapter 18 in some detail. Sarangapani4 has employed an indirect procedure for following the effects of self-discharge: the energy efficiency on discharge vs. recharge was evaluated after various durations of self-discharge on open circuit; see Chapter 20, Section 20.6. It is important that the potential-measuring system has a high input resistance, ideally about 1010 ohms or more, in order to ensure that the measuring circuit itself does not draw significant charge from the capacitor, leading to an enhanced (anomalous) rate of self-discharge. Again, a digital potential recording system is desirable. Finally, it is best if single electrode measurements can be made against a third, reference, electrode in the potential-measuring system, as in "single electrode" polarization experiments (Chapter 3). Whole-cell measurements are less informative since it is then not known which of the two electrodes of a capacitor cell may be suffering the greater self-discharge rate.

19.7. OTHER TEST PROCEDURES

Other nonelectrochemical test procedures are also important for evaluation and safety tests of devices. Principally these involve: (1) recording internal temperature and changes in temperature (and sometimes pressure) on high-rate discharge and recharge; (2) measuring tendency for gassing; (3) long cycle-life testing, up to 105 cycles or more; (4) maintenance of satisfactory performance over long times, especially with bipolar, stacked cell devices; (5) long-term internal corrosion; (6) evaluation of any deleterious effects from adventitious overcharge or overdischarge; (7) performance evaluation over an appreciable range of temperature; (8) determination of the effective voltage range of operation; and (9) resistance, or otherwise, to overcharge or overdischarge.

REFERENCES 1. T. C. Murphy, G. H. Cole, and P. B. Davis, in Proc. Fifth IntI. Seminar on Electrochemical Capacitors and Similar Energy Storage Devices, S. P. Wolsky and N. Marincic, eds., Florida Educational Seminars, Boca Raton, Fla. (1995). 2. M. L. Kronenberg, B. J. Stein, and B. P. Codd, 1. Electrochem. Soc., 141, 2587 (1994).

Preparation and Evaluation of Electrochemical Capacitors

607

3. S. Sarangapani, P. Lessner, J. Forchione, A. Griffith, and A. B. LaConti, in Proc. 25th lntersociety Energy Conversion Conference, vo!. 3, p. 137 (1990). 4. S. Sarangapani in Handbook of Solid State Batteries and Capacitors, M. Z. A. Munshi, ed., p. 601, World Sci. Pub!. Co., Singapore (1995). 5. A. F. Burke and J. F. Miller, Idaho National Engineering and Environmental Laboratory Report, DOFJED-l 049 1, Pocatello, Idaho October (1994). 6. A. F. Burke and J. F. Miller, in Proc. Electrochem. Soc. Symposium on Electrochemical Capacitors, F. M. Delnick and M. Tomkiewicz, eds., vo!. 95-29, p. 281, Electrochem. Society, Pennington, N.J. (1995). 7. E. Gileadi, Electrode Kinetics for Chemists, Chemical Engineers and Material Scientists, VCH Pub!., New York (1993). 8. B. E. Conway, Theory and Principles of Electrode Processes, Ronald Press, New York (1964).

Chapter 20

Technology Development

20.1. INTRODUCTION In presenting this chapter on technology development, it must be stated that attempts to make an up-to-date technology survey are restricted, unfortunately, by the proprietary nature of recent advances, details of which are often unavailable in the public domain, especially information on the preparation of electrode materials and electrode fabrication. Consequently this chapter is based substantially on material garnered from conferences and seminars, and conference proceedings (see Appendix, Section 20.18) on the technological aspects of the developing field of electrochemical capacitor energy storage. Especially valuable are the conference proceedings volumes of the electrochemical capacitor seminars organized by Florida Educational Seminars, Inc., which are referenced in the text that follows by the abbreviation FES and the date. Some technological information has, however, also been obtained directly from companies currently active in the field, and a summary review of a selection of patents is given in Chapter 21. Nevertheless, limitations on the availability of production and process details, owing to intellectual property and proprietary information restrictions, have naturally led to a less than comprehensive account of technology development in this chapter. However, the main and general lines of development over the past 5 to 7 years are available and are reviewed here. The sources of information from various conferences and their proceedings volumes are listed in the appendix. Around 1990-1991, developers of electrochemical capacitors became aware of a new market for relatively large units required for power trains of electric vehicles based on hybrid battery-capacitor power sources. This devel609

Chapter 20

610

opment occurred on account of the now well-known regulatory actions of the State of California requiring a certain percentage (15) of automobiles to have zero toxic gas emissions by the year 2000. This requirement can be met only by electric-powered vehicles. However, pure battery power for such vehicles is not optimal owing to the unsuitability of most battery power sources to provide intermittent high-load power without deterioration of other properties, such as cycle life and output voltage. Hybridization with large power electrochemical capacitors serves to provide load leveling and leads to much improved and longer life batteries.

20.2. DEVELOPMENT OF THE TECHNOLOGY OF ELECTROCHEMICAL CAPACITORS

Generally, the course of technology development has followed several main and subsidiary directions, as indicated in the following sections. 20.2.1. Classes of Capacitors

The following classes of capacitors are recognized: Aqueous electrolyte, carbon double-layer types; Nonaqueous electrolyte, carbon double-layer types; Aqueous, mixed oxide, redox pseudocapacitance, and double-layer types; Aqueous or nonaqueous, redox conducting-polymer types; Electrolytic hybrid capacitor (high voltage); Electrostatic supercapacitors (high voltage).

In very recent years, the following possibilities for new and expanded utilization of such devices, including substantially scaled-up systems, both in actual size and energy storage capacity, as well as in operating voltage, have been recognized: Hybrid power systems, with batteries, for electric vehicles; Cold, or heavy-load (diesel locomotive) starting assist; Utility load leveling (large units); Power-supply low-frequency smoothing; Large capacitance ballast; Automotive subsystems such as catalytic converter preheating, and possibly power-steering assist; Military and medical (defibrillator) applications; Actuators and motor drives; Possible applications in robotics, probably in hybrid systems;

Technology Development

611

High-rate, short-pulse delivery of large charges. Cooperation with standard-type capacitor manufacturers has been proposed by Murphy, Cole, and Davis (PES 1995) to facilitate the development and production of electrochemical capacitors. Thus, although much contemporary interest centers on the science and technology of electrochemical capacitors, actual production at present is restricted to a relatively small number of companies. More activity in the production (and sales) side will stimulate new research and development, with resulting improvements in design and performance. The current interest in the field is demonstrated by the large attendances by scientists and engineers at the various electrochemical-capacitor seminars and conferences that have been held (e.g., the PES series) over the past 8-10 years or so. More attention, however, is desirable on such aspects as mass-production techniques and "real-world" applications. A brief summary of the types of electrochemical capacitor technology, as viewed in 1995, was tabulated by Murphy and Davis.! This list mainly covered existing commercial or military applications in this field, but there are a variety of other related ongoing research activities that do not appear in their summary. They have, however, been covered in various ways in this and earlier chapters of this volume. Special mention should be made of the developing technology at Los Alamos National Laboratory (by Gottesfeld) of redox-pseudocapacitor electrode materials using electrically conducting and reactive polymers (Chapter 12). Many physical and engineering aspects of the technology of electrochemical capacitor development are similar to those for battery production, e.g., packaging, electrode sealing, especially for bipolar embodiments, use of separators, and accommodation of an electrolyte solution. This enables battery-type production machinery to be employed with economic advantages. On the other hand, a number of different and specialized requirements arise, such as preparation of carbon materials that have pore structures that correspond to maximized areas per gram of material yet are accessible to the electrolyte and hence to charging (Chapter 14). In addition, their intrinsic specific resistivity must be at a minimum to provide the smallest esr values, and the pore-size distribution must be optimal. A further requirement is that of the purity of carbon materials and of electrolytes and solvents, so that self-discharge shuttle reactions (Chapter 18) are minimized, especially since a number of the smaller-scaled capacitor units are designed for use as memory backup for computers or for related purposes in telecommunications technology. Although these requirements have been recognized and addressed in carbon double-layer capacitor development, several apply equally to other types of electrochemical capacitor devices, e.g., the desirability oflow resistance to minimize esr and stability of materials to minimize self-discharge.

612

Chapter 20

20.3. SUMMARIES OF DEVICE DEVELOPMENTS AND TECHNOLOGY ADVANCES

Table 20.1 shows a brief historical outline of the stages of development of electrochemical capacitors from the first traceable patent in 1957 as stated by Arthur D. Little, Inc. (B. Barnett with S. P. Wolsky, PES 1994) and updated by the present author in Chapter 21. The growth of research and development activity has been substantial since 1990, encouraged by the requirements for capacitor-battery hybrid systems of large power and energy capacity for applications in electric vehicle power drives. Related to this table is the sequentiallist of patents catalogued by Sarangapani, Tilak, and Chen,2 and given in Chapter 21. An interesting graph of progression of energy densities achieved in capacitor technology from 1962 to the 1990s is contained in a paper by Trippe (PES 1991). Increases from ca. 20 J kg- 1 (1962) up to ca. 30 MJ kg- 1 (in current weapons systems) were indicated. The large energy density systems have been developed by the Maxwell-Auburn group. Table 20.2 contains a summary of the general status of development of electrochemical capacitors. Other developments are for energy storage from solar photovoltaic sources (projected for the future) and examination of the use of special high-voltage TABLE 20.1. Year

History of the Development of the Electrochemical Capacitor

1957

1969

Development G. E. Patent SOHIO Patents

1975 University of Ottawa Group (funded by Continental Group, Inc.) Mixed oxides (RuOr Ta20s)

Construction Tar-lump blacksulfuric acid electrolyte

Carbon pastesulfuric acid electrolyte

Market

No market for low-voltage capacitors

No market Military for low-voltage capacitors

Principal energy storage

Double layer

Double layer

Source: From Barnett and Wolsky (FES 1994).

1978

1984

NEC/Matsus Pinnacle hita Research

Activated carbon fiber-cloth organic electrolyte Memory backup

Double Redox pseudocapaci layer tance

1990 Matsushita! Isuzu

Mixed Activated carbonoxides (based on aluminum UOI foil organic Continental electrolyte project) Military, Ultra-high high rate current power capacitor charge and dic,(;harge Pseudocapac Double itance plus layer double layer

613

Technology Development

TABLE 20.2.

Summary of the Status of Electrochemical Capacitors (1995)

Developer

Electrode-electrolyte Material

Panasonic Pinnacle Research Institute Maxwell Laboratories Maxwell Laboratories Livermore National Lab

Carbon-organic Mixed metal oxides-aqueous Carbon-organic Carbon-organic Aerogel carbon-Aqueous Sandia National Lab Synthetic carbonaqueous Los Alamos National Lab Conducting polymer-aqueous

Wh/kg a

Whlliter

2.2 0.8

2.9 3

6 7 2 105 times; Long-term stability, related to cycle life; Resistance to electrochemical reduction or oxidation of the electrode surface; High specific surface area, on the order of 1000-2000 m2 g- 1; Maximum operating potential range of cycling within the decomposition limit of the solution (related to resistance to oxidation or reduction);

614

Chapter 20

Optimized pore-size distribution for maximum specific area but minimized internal electrolyte resistance (minimum esr); Good wettability, hence favorable electrode/solution interface contact angle (depends on pore structure); Minimized ohmic resistance of the actual electrode material and contacts; Capability of material being formed into electrode configurations having mechanical integrity (e.g., compressed powders with binder, fiber or woven fiber matrices, aerogel materials, glassy-carbon structures) and minimum self-discharge on open circuit. In relation to the above ideal requirements, the active area, or active weight or volume, is a common concept involved in descriptions or specifications of battery electrodes. It refers to the active mass of electrode materials that can undergo the required electrochemical redox reactions of the battery. A similar term can be applied to electrochemical capacitors; in the case of double-layer-type capacitors, it refers to the real area of electrode matrices that can undergo electrostatic charge acceptance or charge depletion. In the case of redox-pseudocapacitance devices, the terms have the same significance as for battery systems, but the real accessible electrode area is also usually important because this determines power performance. The electrochemically significant active area in an electrochemical capacitor device is usually significantly less than areas of carbon materials determined by BET measurements in the dry state. As referred to earlier, this is due to problems of wetting and invasion by the electrolyte into the finer pores of the pore distribution. A useful summary of the characteristics of various electrode materials is shown in Table 20.3. TABLE

20.3. Summary of the Characteristics of Electrode Materials

Electrode material Carbon-metal composites Aerogel carbon Cellulose-deriv. foamed carbon Doped polymer (polyaniline) Anhydrous ruthenium oxide Hydrous ruthenium oxide

Specific Cap. (F g-l)

Surface area (m2 g-l)

Densi~

(g.cm- )

Resistivity (ohm-em)

100 to 200 120 to 160 70 to ISO

100 to 1500 500 to SOO 500 to SOO

0.5 to 0.7 0.4 to O.S O.S to 1.0

15 WhIliter were reported. Solvents: acetonitrile, succinonitrile, glutaronitrile, propylene carbonate, ethylene carbonate. Salts: alkali metal-tetraalkylammonium cations with CF3 SO;, N(CF3S0 z 2, BF:! , PF6' , AsF6' and CIO:!. Accessible potential range of -4 V achieved with energy densities of 10 WhIliter. (continued)

r

680

Chapter 21 TABLE 21.2.

U.S. Patent No. (Date) 4.

5,303,118 (4112194)

5.

5,121,301 (6/9/92)

6.

5,038,249 (8/6/91)

7.

4,748,542 (5/31/88)

8.

4,725,927 (2116/88)

9.

4,725,926 (2116/88)

10.

4,713,731 (12115/87)

11.

4,622,611 (11111186)

12.

4,363,079 (1217/82)

13.

3,652,902 (3128172)

Continued Electrolyte

Glass fibers or water-absorbing polymer impregnated with gelatinous electrolyte solution. Silica or alumina gels in H2S04 gave capacities of about 168 F with carbon-based electrodes. These capacitors showed negligible reduction in capacitance and escalation in resistance following operation over a long period of time. Paste of activated carbon + electrolyte. Patent describes the method of preparation. Electrode elements are arranged in honeycomb construction. Soap or polymorphic solid solution. Sodium palmitate provided a dc capacitance of 600 flF/cm2 at 10k ohm discharge at 88°C. Conductive polymer such as polyethylene oxide with an inorganic salt consisting of LiCl04, NaCl04, LiAsF6, LiBF4, and finely divided electronic conductor such as carbon for making a solid-state electrochemical capacitor. Electrolyte is sulfolane and derivative thereof. Carbon electrodes in sulfolane mixed with 10 to 80% by volume of propylene carbonate and containing (C4H9)4PBF4 gave 2.2 F over 2.8 V at 70°C. Quaternary phosphonium salt of the formula R JR2R3R4PX where R J to R4 is H atom or alkyl group having 1 to 15 C atoms or aryl group having 6 to 16 C atoms, X = BF4, PF6 , Cl04, AsF6 , SbF6, AlCl4 or RFS0 3 where RF = fluoroalkyl group having 1 to 8 carbon atoms. Capacities as high as 100 Fig and voltage range of 5 V realized with activated C electrodes in propylene carbonate containing tetraethylphosphonium cations and BF4, SbF6 , Cl04 anions. Carbon-based electrodes in propylene carbonate containing tetraethylphosphonium trifluoromethyl sulfonate showed capacities of -100 Fig. Carbon electrodes in sulfuric acid with voltage-regulating agents such as Br-,Cl-, r, and r at a concentration of 0.5 to i.5 moleslliter gave 12 to 30 F over 0.6 V at a charge densities of 9 to 19 C/g. High surface-area carbon electrodes in 100 gIliter K2S0 4 + 3.5 gave 3 to 4 F/cm 2 with