Electrochemical Impedance And Noise 2p3y3t

Corrosion Testing Made Easy

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Electrochemical Impedance ---------and Noise Robert Cottis Stephen Turgoose

Barry C. Syrett Series Editor

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/\II rights reserved.

Printed in the United States or America Library uf Congn:ss Catalog Card Number: CJ9-XOOX I ISI3\' : 1-57590-091-()

Reproduction cf contents in whole or par1 or trano,fer into electroriiJ or photographic str,rage without perm1s,ion of copyright is expressly forbidden. Neither NACE International, its officers, directors, nor thereof accept reo,ponsibility for the use of methods and materials discussed herein. No authorization is implied concerning the use of patented or copyrighted material. The inf9rmation is advisory only and the use of the materials and methods is solely at the risk of the . · ··

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Preface to the· Series The Corrosion Testing Made Easy series of books was conceived in 1983 by of the NACE Publications Committee in recognition of the neBd for easy-to-follow guidelines on specific types of corrosion testing. While many test procedures are defined in NACE and other industry standards and some textbooks. it was noted that nune of these sources was written with the rank beginner in mind ; all assumed some prior experience with corrosion testing and familiarity with corrosion terminology. This situation inevitably leads to a need for the entry level technician to be trained initially by a more experienced experimentalist. The cost of such training can be quite high, especially if a highly paid professional scientist or eng ineer must do the training. Furthermore, because of demands on his time, the more experienced experimentalist may not always be available to give timely advice . Consequently, there seemed to be an urgent need for a series of books that could guide an entry level technician through a corrosion test procedure with little or no assistance. These books would make no assumptions about prior knowledge or expertise. They would describe the possible difficulties in performing the experiment and would indicate how to correct mistakes if they were made. The books would each c.ontain an unusually large number of illustrations on the grounds that "one good picture is worth a thousand words." Hence, the Corrosion Testing Made Easy series of books was born. Each author for the series was chosen with great care. All authors are technically very competent, all have had many years of hands-on experience in performing corrosion testing, and, perhaps even more important, they have a reputation for clear, concise writing. Most of the books in the series can be used as step-by-3tep guides to performing specific types of corrosion test, e.g., stress corrosion tests or electrochemical test. However, the introductory volume. The Basics, describes techniques and equipment that are common to most. if not all , corrosion tests. It provides a glossary of and definitions that relate to corrosion, and it contains some basic principles of corrosion and electrochemical theory. Although the practical aspects of corrosion testing are emphasized in all of the volumes in the series, theoretical aspects are introduced when they can help the experimentalist perform a test with greater ease. More than a dozen volumes are planned for publication in the next few years Whi le they are intended primarily for entry level technicians. it is clear that these books may also be useful for morP. e\::e rienced scientists and engineers. For instance. c:m engineer skilled in thP. ;:ut of performing fatigue tests may use the Corrosion Testing Made Easvseries of books to obtain practical advice on extending his knowledge to corros1on fatigt.;e tests. Or an experienced biolog ist may need help in performing a seawater corrosion test as part of a larger study 2'l marine toxicology. Whoever uses the series. one thing is certain: the volumes are unique and fil l an important g?.:: n the scientific literature. Barr y C. Syrett

Preface ''

This book is an introduction to electrochemical impedance spectroscopy and electrochemical noise for corrosion studies. The primary objective is to provide a basis for the reliable acquisition of data and for data presentation and subsequent analysis. The contents are based on the authors' experience of common difficulties and uncertainties that exist when starting to carry out such measurements. The main text aims to use the minimum of mathematical detail that is consistent with a clear presentation of the subjects. The appendices contain more detailed mathematical treatment of electrochemical kinetics, of the o; 1gin of the signals, and the relevant signal processing techniques for the two methods. These are included for the interested reader, and as a reference source. However, they are not essential for the appreciation of the main text. The book is ed by the accompanying CD-ROM, which contains relevant papers presented at the annual NACE Corrosion conference, listings of relevant papers for the CorrAbs database, spreadsheet and other data files used in the preparation of the book, and programs for the simulation and analysis of electrochemical noise.

Acknowledgments ';

The authors would like to thank other staff and students of the Corrosion and Protection Centre who have contributed to our understanding of the sUbjects. They are too numerous to mention , but for the Impedance Section we would like to recognize the contributions of Catherine Hawkins, Kevin Lawson, Max Pech-Canul , Jose Mora-Mendoza, and Hilal AI-Oumshoey. Early workers at UMIST (notably Karel Hladky working with John Dawson) initiated interest in electrochemical noise; the recent improvement in our understanding has been based on the work of Walter Richter, Juan Mendoza-Flares, Mohamed AI- Ansari, Gillian Bagley, Abdulmajeed Alawadhi , and otl1 ers . Roge r Newman made many valuable comments at various stages of the preparation of the text, and convinced us of the need for such a book. Barry Syrett encouraged us to produce this book and made many helpful suggestions on the draft. Francois Huet read the final draft with great care and expertise, and prevented at least some of our errors making it to the book. The errors that remain are, of course, our responsibility alonel We shall record any that we discover on the UMIST Web site (www..umist.ac.uk). Finally we should like to thank our families, lyfl, Clare and Rhian, and Mary, Luke, Hannah and Toby. for tolerating us during the production of this book. Steve Turgoose, Bob Cottis Manchester 1999.

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Preface to the Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . f

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ix

Preface ..... . .............. ·. . .... . .".. . . . ... . ....... . .... :. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Acknowledgments .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

-1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . · 1 Principles of de Electrochemical Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What Is Electrochemical Impedance Spectroscopy? . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement of Impedance Spectra....... . . . ..... .. .... . .... ... .............. . .. . . ..... . ..... ... What Information Is Given by Impedance Spectroscopy? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What Is EIS Used for? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . · What Is Electrochemical Noise, and What Produces It? .... . ....................... : .... ..·....... . . . How Do We Measure Electrochemical Noise? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What Can It Tell Us? .. . ............ ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Walk-Through of an Electrochemical Noise Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrochemical Noise and the Law! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . Reference........ . ..... . .... .. ........... . ... . ....... . .......... . .... . ...... ........... . ......... .. ......

1 1 2 2 3 3 4 5 6 6 7

- 2. Principles of Electrochemical Impedance Spectroscopy ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

Comparison of Electrochemical Impedance Spectroscopy and Linear Polarization Resistance Methods .. :. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Current-Time RespQnse to a Linear Potential Sweep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Why Is a Small-Amplitude Potential Used? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Current-Time Response to an ac Potential Input . . . . •. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impedance and ittance .. .. . .. . .. .. .. .. .. .. .. . .. .. .. .. .. . .. . . .. .. . .. . .. .. . .. .. .. .. .. . . .. . .. .. .. .. .. Impedance of Simple Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DatCl Presentation .. .. . . . . .. .. .. .. .. .. .. .. .. . .. .. .. . .. .. . .. . .. .. . .. .. .. .. .. . . .. . .. .. .. . . . .. . Nyquist Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bode Plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 10 11 11 13 15 1n 16 17 18

iv

Contents•

3. Measurement of Impedance Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . 19 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Cell Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 · Two- and Three-Electrode Measurements ... .. . . .. '.......... ·. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Circuits with Manual Potentiostats ...... ... ....... . . ... . . ...................................... :. . . 21 Working Electrodes ........ . ............. ..... ~ ..... , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Reference 8ectrcioe·~ ................................. ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Counter (Auxiliary) ·Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 · · Cell Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 ' Connections to the Cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Experimental Setup ....................................... .'. : . .......................... ...... 24 Counter Resistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Frequency Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Number of Data Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . · 26 Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Signal Averaging ..... .t. ·.... ,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1 Area Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . • . . . . . . 27. -~ Common Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Experimental Artifacts in EIS ............................... .... ............ .. ................ :......... 27 · Problems at All Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . 28 High-Frequency Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Midfrequency Problems . ...... .......... . ............................... . ............... .·......... : 30 Low-Frequency Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Less Obvious Experimental Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Problems That Are Due to Changes with Time .................................. :. . . . . . . . . . . . 31 Depressed Semicircles .......................................................... ; • . . . . . . . . . . . . . 32 Effect of Amplitude of Voltage ...................... . ......................... ; ......... , . . . . . . . 32 Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 M

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- 4. Analysis of Electrochemical Impedance Spectroscopy Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equivalent Circuits for Corroding Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical Analysis ..............................._.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equivalent-Circuit Analysis ........... ; ..•.... .. ....; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diffusional or Warburg Impedances . . . . . . . •. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Activation (Charge-Transfer) Control of Both Anode and Cathode Reactions.................. Activation (Charge-Transfer) Control of Anodic Reaction, Cathodic Reaction at the Limiting Current .......................... ·............·.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Activation (Charge-Trar.sfer) Control of Anodic Reaction, Cathodic Reaction under Mixed Control .... . ._.................................................... ,..... . ............... Diffuiionallmpedances of Anodic Processes ......... :~ .. -. .... •............ , .. . . . .. .. .. .. .. .. .. . Selection of Appropriate Equivalent Circuits ............................... . Fitting Dat.: ·,vitr. 8aprJ:;3e~ Semicircles ....... ,.•.. ~ ... .... .... . .... .. ........ ... ..: ... ~. .... ... ....... Other Types of Impedance Response ....... ..... .....! ....... _, .... : . ... •. ..... ....... :. . . . . . . . . . . . . . . . Estimation of Corrosion Rate from EIS Data ...................................... : .. ...... ....... ;... Use of Charge-Transfer Re~tance for Corrosion Rate Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Use of Polarization Resistance for <:;orrosion Rate Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . «

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35 35 36

38 38 40 41 41 42 43 45 46 47 49 49

vi

Contents

Standard deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Root Mean Square. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Skew................................................................................................. 75 • Kurtosis ........................................................ :. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Coefficient of Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . · · 77 Noise Resistance .............................·. . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Frequency-Domain Methods .............................. ~ .. ·. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Spectral Estimation ................................. "... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Power Spectra and Power Spectral Density .......................... , . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Bels and decibels ............................................... .. ................ . ., . . . . . . . . . . . . . . . . 79 Power Spectrum Estimation Methods ................... :.: ........................................... . 79

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Trend Removal ....................................................·....... . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . BO Windowing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Limitations of Spectral Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrochemical Noise Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . •. . . . . . Characteristic ·Features o(Power Spectra.............................................................. Chaos Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . Discri.minant Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications to General Corrosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relationship between Noise Impedance and Conventional Impedance . . . . . . . . . . . . . . . . . . . . . . . . Application to Localized Corrosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coefficient of Variation .................. , ................................... , . . . . . . . . . . . . . . . . . . . . . . Localization Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Charge in Transient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Artificial Neural Network Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~tudy of Painted Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . Thin-Film Corrosion and Low-Conductivity Solutions ..................................... ; . . . . . . . . . . Resistance Noise .......................................................... , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanistic Studies ..........................................................~·. . • . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80 80 82 82

Appendix A. Electrical Circuits . ......................................................................... · Charge................................................................................................... Current......................... . ......................................................................... Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Voltage or Potential Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resistance.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capacitors a net Inductors ...... . ....................·. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constant-Phase Elements .................................. .-.................................. .. . . . . . . Impedance...................... . ........................... . ............... . ............................. Cviiibi;;i;-;g :;-;jpedancas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kirchoff's Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superposition Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear and Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electronic Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95 95 95 95 95 95 96 96 96 96 97 97 97 97

86 86 87 89 89 89 90

90 90 90 91 91 91 93

Contents

v

• 5. Description of Noise Signals ................... ..... ................................................ . 51 Electrochemical Noise Terminology ................. .......... ................. .... .................. . 51 Potential- and Current-Time Records ............. . ............................... ... ................ . 51 Potential and Current Noise ......................... ~ ......... : . ...................................... . 51 Average Amplitude of Noise Signals ................................................................. . . 52 .. Effect of Specimen Area on Measurements : . ...... . ................................................ . 52 Continuous-Time imCJ.Discrete-Time Signals .........•. ..... .............. ... .. . ..... .. .........·.. ; .. . 53

6. Measurement of Electrochemical Noise ........................................................... . Potential Noise Measurement ......................................................................... . Current Noise Measurement ........................ . . : .:. .. ....................... . . . ............. ·.... . Noise Resistance and Noise Impedance Measurement ....................... .. ................... . Cell Design ............................................................................ .. ............... . Electrode Preparation ..............•............................................................... Reference Electrodes ............................................................................. . What Is the Best Samp'ling Frequency? .............................................................. . How Long Should a Time Record Be? ........................................... . ................... . Signal Processing and Recording .................................................................... . Voltage Amplifier.~ ............................. . ................................... .. .................. . Current Amplifier .......................................................................................· . Filtering .................. ... ....... ; .................. . ..................................•..... : ........ . Analog Processing ..........................................................•..................•......... Digital Recording ........................................................... , : . ......................... . Types of ADC ......................................... ... ............................................... . Transferring Data to the Computer ......................................... : .......................... . IEEE-488 Instrument Bus .... . .................................... •................ : .. , ........... . AS 232 ................................................................................. .............. . Parallel Port ........................................ . ................................................ . Universal Serial Bus .............................................................................. . Internal Data Acquisition Cards ................. ... ............................................... . Programs for Control of Data Acquisition ........................................ .. .................. . What Can Go Wrong? .... ; ............•.............. . ................................................. Interference ..... . ............................................................... · · ... · · · · · · · ·. · · · · · · Ground Loops .......................................................... ; .... ·.. . .......... .'.' ....... . Aliasing .... . ...................................... . ... . ............................................. . Drift ....... ; ......... .. ........... . ......................... · · · · · · · · · ·· · · · · · · · · · · · ·· · · · · · · · · · · · · · · · · · · Quantization .....................•.............................................. ... .... .... ....... . .. Instrument Noise ................................. .. ................... ~ .. ' ........ ~· ................ . References ........................................ . ... · ... . ....................... · .. .. · · · · · · · · · · · · · · · .. ·

55 55 .55 56 56 56 58

7. Analysis of Electrochemical Noise .................. , ................................. :, ... 1 .... .. .. ,.l'.,na!ys:s Domains . ............................... ...... ... .. . . ............. : . ... -'· .. · · · · · · · · · · · · · · · · · · · · ·

71

Time Record Analysis .. ............................ ... . .. . ...... ............... .. ..... · ............... . Measurement of Properties of Transients . . .. . .. ....... .. .. ....... . ........ .. ..•. . ................ Statistical Methods ................. .. ........·, ..... .. . . . .... . . .. ..... . . .... ...... .... .. ............ ..... . Mean .. .. ............ ... . ,.: .. , .... . .. i ... .... ............. ..•....•.. . ... . . ...• .•.. i . .. . . .. .. ...... . . . .. . Variance ......................... . ; ........ : .... .. . . . ....... .... •..... ;._,, ........ :·::·... .... ·. · · · · · ·. · · · · · ·

71 71

59 59

60 60 61 61 61

62 63 64 64 64

65 65 65 66 66

66 67 68 68 68

69 69 "'7~

I

I

74 74 75

Contents

Resistors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capacitors ..................... . ..................... ..,...... . ...... .. ....... . ............. . ............. Inductors .. .. .. ... .............. .. .. .. ........ . .. . .... . .......... :.... .. ...... . ...... . ............. .. .... . Specification and Sources of Instrument Noise .......... . ...... .. ....... . .................... :·: . . ::. . Noninverting Buffer ................. . ..... .'.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - ~~ l'f' • C urren t A mp 11er ........... .. ..... . . . ........... .. ..................... . ........ .. ............... . . . Electronic Filters ..................................... . ...... . ....... . ..... : . .......·........... . . . . . . . . . . . Signal Conditioning Circuit for Electrochemical Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. Electrochemical Kinetics ...... . ...... . .. ,_._........... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct-Current Electrochemical Kinetics ... .. .............. . .. .. . ... ... .. ... . ..... . .... . . .. ..... . .... ·: . Activation-Controlled Systems .......... . ............. ~ ......·. . ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass Transfer Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alternating-Current Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Activation-Controlled Prqcesses . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Amplitude of Applied Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Irreversible Cath~ic Process with Mass Transfer Effects ................. . ...... , . . . . . . . . . . . . . . Diffusional or Warburg impedances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relationship of Warburg Coefficient to Physical, Chemical, and Electrochemical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... .

vii

97 98 99 99 100 ~ 100 101 102· 106 107 107 107 108 109 109 109 110 111

112

Appendix C. Walk-Through of a Noise Measurement and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Equipment Required . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 113 Measurement ..... . ................................. . ......... . ..................... . .................. . .. 113 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Appendix D. Spectral Estimation Techniques and Signal Processing ..........·.......... . . . . . . . . 117 Sine-Wave Correlation ................. . ..... . ...................... . ....... . ... . . . .. ·~... . . . . . . . . . . . . . . . 117 Spectral Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 The Fourier ·Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 The Finite Fourier Transform ....... . ... . ....... ... .... . ....... . ....... . .......... ; . . . . . . . . . . . . . . . . . 118 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Estimation of the Power Spectrum from the Fourier Transform ........... . ................ . .... · 119 Maximum Entropy Method ... . .................... . .. . ................... . .................... ~ . . . . . . . . 120 The Autocorrelation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Other Spectral Estimation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Wavelet Transform ..... . ...... . ....... . .... : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Appendix E. Theoretical Basis of Noise Processes ......... .. . . .............. ... .... .. .. .. . . ...... . . 123 Description of Noise Signals .. . .................................... . ....... . ......................... .. 123 The Mean . .. ... . . ...... ... .. ........ .. .. .. .. ... . .. . .... . .-. ..... .... ... . . .. ... . . . .. . ... .... . . ... .. . . · · 123 Statistical Moments . ............. . ....... .. ...... . ....... . .... . ............... .. ..... . ...... . ...... . 123 < ,._ Varianca . . . . . . ......... . .. . .. . . . . ... . . .. . .. . ..... ... ....... . . .. · · .... · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · '"-.,. Standard Deviation ......... . ............. . .... . . .. .... . . . .... ... ....... . .... . ...... .. .. . .... .. ... . . 124 Root Mean Square . ... .... .. .•.. ... . . ..... . ... . .. ... ... ... ... . .... .. ... . . ... . . . ..... .. ..... .. ..... .. 124 Skew . .. . .. ..... ... .. . ... . ... . . ..... . ..... .. ... . . .... .. . .. .. .. .. . .. .. . ..... . .. ... ... .. . .. . .. . . . . . . .. . . 124 Kurtosis . ..... . ·. ........... .... .... .. .... . . . .. .... .. ...... . ...... . .............. . ...... .. .... ... .... . . 124 A

)

viii

Cootents

Coefficient of variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Noise Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Continuous-Time and Discrete-Time Signals ......... , . . ........ ·. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Combining Noise Signals ......................................... ~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Subtraction of Noise Signals ..................... , ...................................................... · 129 Noise Colors .. . ..... : ... .-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Thermal (Johnson) Noise Processes.................................................................. 129 Shot-Noise Processes ......................................................... -. ........................ 129 1/ f Noise Processes ...................................................... _............... .• . . . . . . . . . . . . 120 Shot-noise Analysis of Electrochemical Noise ..........•. 1 • •••••••••••••••••.•••••••• , • • • • • • • • • • • • • • • • 130 •· · Relationship·l:>etween Power Spectra and the Underlying Transient Events ....................... 131 '

.

Appendix F. Transient effects in EIS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Responses with the _Presence of Solution Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of EIS with Potential Step Polarization Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Validation by Kramers-Kronig Transforms ......•...............•........•... -. .... : • • . . . . . . . . . . .

f33 134 135 135

Appendix G. Contents of CD-ROM .......... ~ .....................•. ~.: ... :~ ......... ·:;::~ ................ Papers Presented at Corrosion/96, 97, and 98 ............... ."~ ..... : . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. Bibliography from Corrosion Abstracts ................................ : . ...... , ................._. . . . . . Data for Electrochemical Impedance Spectra ....................................................... ; EnAnalyse Program ...........................................................,.• ~·-·-- · ................. Sample Time Records ..... . ............................................ . ... . ......... '.......... ; .. ; . . . . Spreadsheets .......................................................................... : ...·.... : . . . . . . . .

137 137 137 137 137 138 138

Appendix H. Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Appendix I. List of Symbols : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Index ....-....................................... . ............................................................. 145

..

•

CHAPTER 1

lntroduc.tien · .. This book is about two techf.liques, electrochemical impedance spectroscopy (BIS) and electrochemical ~oi&e measurements, thathave been developed for the examination of the electrochemistry of corroding metal samples. These are both developments of the de techniques that have been discussed in an earlier book in this series. 1 In some ways the techniques are.quite similar; we can liken them to different methods for testing for cracks in railway wheels. • The original de electrochemical methods were com. parable with those used by the original "wheeltapperS: The wheel is struck with a hammer, and the characteristic difference in the ~~nd produced by the cracked and the uncracked wheel is identified. • We may be able to get a more accurate picture of the size and the location of the crack if we separately excite the wheel at different frequencies and analyze the response. This is essentially what the ac impedance method is trying to do. • Alternatively, to save taking,ihe ~ out of service, we may try tO.@D~yze Jllp Sdunds made a;? the wheel rolls along thn8c~Tnis is comparable with what we are trying to achieve with electrochemical noise measurements.

-

Because the natures of the .;_;o techniques are rather different {even !hough they are tryin!:' tn nieasurP. mc:1ny of the same things), we examine them separately; the first part of the book is concerned with EIS and the. second with electrochemical noise measurement. The appendices deal with the more mathematical aspects o1 the two subjects and with other ing material.

In this introductory chapter brief introductions to the two subjects are presented, VJith pointers to where more details can be found, but first a very brief introduction to the common de corrosion rate measurements is given for comparison.

Principles of de Electrochemical Techniques

Corrosion rate measurements by de methods usually involve applying a small amplitude de signaJ, of either ki c): ~1 . voltage or current, to a corroding metal. The resulting .c. urrent orvoltage, respectively, is measu. red. The ~-::.l-_ change in voltage divided by the change in current gives ~ ~· known as the polarization resistance. Thi~ polarization resistance is usu~lly inversely proportional . to the corrosion current density. Using an appropriate , constant (see Appendix B) the corrosion current density can be calculated, and once this is kriown the cor. rosion rate can be expressed in any desired units, via Faraday's laws.

What Is Electrochemical Impedance Spectroscopy? EIS is an electrochemical method in which an ac signal fhenr.e the
Se.. o..~\ 1 c.o. ~

(!)rm,...\-c. a\-\or r-iA.

o.\ e~ddm~

l

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~~.$\or l\~rr-,)~\~. 2

Corrosion Testing Made Easy

to pr:ovidatf:la impedance at 9ift&r&Rl impedance spectrum.

freq••enci~,

the

·

The actual measurements are similar to the common de electrochemical methods in that a current response of a corroding metal tp a voltage is determined. However, the current anct the voita_ge data are processed in such a way that the similarity to the de methods can be difficult to see and the interpretation of data requires different approaches.

•

Measurement of Impedance Spectra With .present technology this is comparatively simple and often requires only s·etting a"iew menu options on a screen. Because the spectra for corrosion systems can be quite complex, and especially because the data are presertted.in the processed form of impedance at different frequencies, it may .be less easy to identify artifacts or errors in the measurements than it wou1d be with de methods. To carry out impedance measurements and to obtain useful data, it is necessary to have an understanding of the types of response that may be expected from corrosion systems and of the types and the origins of experimental artifacts. 1 .

~mpedance is the term used to describe the ac equivalent 9f de resistance. For de the relationship between voltage and current is given by Oh~'s law:

/

/;~

IV

= _I____:./, R,l

(1 .1)

'·····..

where V, in volts, is the voltage across a resistor, R, in ohms, and I, in amps, is the current. For ac signals,

I

angle (pfoportional to the shift in time between peak current and peak voltage). The collection of these parameters for different frequencies is the impedance spectrum. Very often these two parameters are expresse9 in of the real and imaginary parts of the «npedance (see Chapter 2).

(1 .2)

' i

1

I

where Z is the impedance of the circuit. Unlike resistance,I!he impedance of a cir~t may depend on the · frequency of the applied sign~For: an :IS the frequency f. in hery, js thA number of cyctes per sec ond . ...--:--

What Information Is Given by Impedance SpeCtroscopy?

steam'

LThe ittance Y of a circuit is the reciprobal of the impedance, in the same wa{ that conductance is the reciprocal of resistance, so ~~t'(..\O..

_Qirect _cyrre~etg~such as linear polarization, are generally used .to qbt(iliD the total resistance Of the ~ystern..Jhe polarization resistance..&:....

J!

When an ac signal is applied, even a simp!~ corsystem behaves in a more complicated way 1 1 u ., than a single resistor. There is also a rap&Ji!ancs. the Y v ( 1 · 3 ~ double-layer caBacitance G~1 (in farads) that is due fo 3 ~ charge separation close to or at the surface, and this It should be noted that formally the function being mea-~ is present in all corroding systems. In addition there is sured is defined by the output divided by the input. With 't. the resistance of ~.solution Rs to be a!lpwed.,.!9r. The a controlled voltag~ input, we strictly are measuring~ad:i --t ~hi nation of these three companeA&i provides a sjmmittance, not impedanee. Here, the measured data are-1 pte equivalent circuit for a corroding metal. An equivreferred to in the conventional way as impedance, which) alent circuit· is a combination of electrical elements, is the way in which the data are usually presented. Howsuch as resistors and capacitors, which,gives the same ever. the distinctiOn does become important with the response to electrical signals as the actual corroding use of ~orne mathAmatical methods for data validation metal. The simple circu!t described by the three com(see Appendix F). ponents above is shown in Figure 1.1.

=z=

1

~

The imp~ance of a system' at a given frequency is defined by two that relate the output current to the input voltage These are the amplitude of the ac current divided by t e amplitude of the ac voltage and the pt\ase

::{:j rosion

l

"l

Alternating current measurements at different frequencies enable these different components to be identified and separated. ~aoce spectra can be usedJQ ,jQentify and q••antity capsg::it;mces in the syste~ a•u;j to_

,j .J

3

Electrochemical Impedance and NoiSe

The latter can be subsequently used to provide more confident corrosion rate information, and so a basic understanding of this is required. · I

FIGURE 1.1. Simple equivalent circuit for corroding metal.

f

At lower frequencies, different type~ of process occur over different time scales (at different frequencies), and it may be possible to i~ntifY, diffusion effects or adsorption/desorption phenomena: .

What Is EIS Used for? Once data are obtained, there is the question of what to do with them. It js often SJ •fficient to obtaiA AWfflerjcal values of resjstQJ:& .and capaaits;ar in the equ~va­ lent circuit of Figure 1.1; by extrapolation of a plof or plots of the data. Even this simple (but useful for the ~ ~timation of £_orrosion .rate) approach requires some understanding of the possible responses of corroding metals to ensure sensible extraction of useful numbers. However, especially with more complex responses, a fundamental physical understanding is required before useful fitting of the data to equivalent circuits may be considered. At one level, EIS is very easy to use for corrosion testing. Ibe lownl~"&q' 1ency limit of the impedance is obtained, and this is then used in place of the poladzatjan resistance that would have been obtained from de methods. The advantages over de methods are (1) the ability to correc;t the polarization resistance values for high solution resistance, and (2) that having the whole spectrum gives confidence in the value of the low-frequency limit of the impedance. It may further help to valid~te the choice of coefficient used to obtain thP. c:orro~ion rate from polarization resistance. However, any of these benefits over de methods, apart from the simple subtraction of solution resistance from polarization resis,tance, involves some degree of data processing to provide mechanistic information.

•

There are also major benefits of EIS in the establishment of corrosion mechanisms, and these also require significant data processing. In this book, Chapter 2 describes the principles of EIS . and the expected responses of simple corrosion systems, including their equivalent circuits. These provide a basis for Chapter 3, which describes experimental procedures, and also common eXPerimental problems and their solutions. Chapter 4 considers data analysis both by graphical means and by equivalent-circuit analysis and describes some more complex types of process and the corresponding circuit elements and equivalent circuits. It discusses factors such as the selection of appropriate equivalent circuits· and the similarity and the identicality of the responses of different circuits. Finally, the estimation of corrosion rate from EIS is considered. The appendices contain a more formal approach to defining the responses of corroding systems and methods of obtaining impedance parameters from the measured current-time response.

What Is Electrochemical Noise, and What Produces It? The term noise commonly means an unwanted sound, although what is unwanted noise to one observer may be a wanted sound to another. Thus the music inside the disco v.·m be noise in the house next door. Also, just because noise is unwanted doelLlSl- rnee.nJ.,tw;t ~ it does not contain information. The ~c:mf!Bfianu~ can interpret the noise of the engine to identify various fault conditions; the family dog can use almost inaudible noises to detect people arriving at home and to identify ' family . In the study of electrochemical noise we have to make a few conceptual changes, but the general principles are. simil~r to tho~~ of the mP.chanic identifying faults. The main chaf!ge · is that we are. not concerned with audible sounds (fluctuations in air pressure or acoustic noise), but with fiuctuations in electrochemical potential and current. Thus electrochemical potential noise is the fluctuation in the electrochemical potential of an electrode relative to a reference electrode, whereas

tot·

4

Corrosion Testing Made Easy

be analyzed. In this chapter we leek at a simple electrochemical noise measurement and at some examples of noi!ie produced by a range of corrosion processes.

How Do We Measure Electrochemical Noise? ·~~-+-t-Wcllcing

.. .

Elec!rOde I

Tosee whatis involved in measuring electrochemical noise, let us consider the measurement for steel in NaCI solutloh.

WOOing EleclrOde 2 ......,.__.,_ Reference Electrode !mov be sane materic: OS WOOing ele<:trode)

To measure the electrochemical noise of steel, we first need test :elec~odes. Tp allow the measurement of both current and potential noise, we use two steel electrodes (see Figure 1.2). We record the current between the two steel electrodes and the voltage between one of the two steel electrodes and a reference elec· trade. Because we assume an ideal ammeter (one that measures current with no voltage drop), the two steel electrodes used to measure the current effectively behave as a single electrode of twice the area, even though we actually only connect to one of them.

FIGURE 1.2. Measurement configuration for potential and cur· rent noise.

is

electr
'i "electrochemlcat current

We immerse the three electrodes in a salt solution and record how the current and the electrode potential vary with time (see Figure 1.3). The records .s;~l~~ rent or pOtential against time are known as time r~rCIL

In subsequent chapters we look at the

detailed requirements for the accurate measurement of electrochemical noise and the ways in which the data can

0.00005

-().52 0.0(1003

rf

-o.54

0.00001

fl)

: ~ -o.ss

-i 4

-o.00001

ii ;:; c

t

::1

()

~ -o.sa a.

-0.00003

-o.oooos •0.6 -().00007 .0.62 .1...---------..------.-------~--~ .0.00009

0

250

500

750

1000

1250

1500

Time is FIGURE 1.3. Noise time records lor steel in 3.5% NaCI solution. SCE. saturated calomel electrode.

.

5

Electrochemical Impedance and Noise

0.00005

0

., 0

-0.1

-0.00005

w

-0.2

-

uen

.,;

> -0 3

>

~

c ~

-0.0001 -0.00015

.

-0.4

-0.0002

:

•. C1.

-c or(

!!....

::J

u

-0.00025 -0.5 -0.0003 -0.6

-0.00035

-0.7 0

1000

2000

3000

4000

5000

6000

7000

8000

-0.0004 9000

nme/s FIGURE 1.4. Effect of tapping an electrode on the noise time record for steel in 3.5% NaCI solution.

We look at the requirements for a good measurement in Chapter 6.

corrosion reaction in this case is controlled by the transport of oxygen to the metal surface, probable causes of the observed noise will be associated with fluctuations in the oxygen transport. We can see this if we tap one of the electrodes (Figure 1.4). Note that the scales on this plot are very different from those in Figure 1.3, which is why the background electrochemical noise looks rather different.

What Can It Tell Us? In the case of steel in sa_lt solution we can see that there are at least two parts to the behavior: a long-term transient decay as tt)e rust fifm develops and provides an increasing barri~r to oxygen and·fluctuations with a shorter time scale. It is a question of definition as to whether or not the slow transient in potential should be treated as electrochemical noise. It certainly contains information about the corrosion process, but we must be very careful ing many of the stati:;ti9al analyses that assume that the properties are constant over time, D.S this is obviously not the casr for the transient behavior. . The shorter-term fluctuations in current and potential :tre ml""'ra~,h, •it"'\ • · ~"'+,., I "'\ ,...,,""',.. ........ ,..heml·,...~~ . , ,..,;·~-;... 't..., ·• ...... •:...·";_ • ·· - · - - - . · - '""'-•J o.o .......... """'OJ"""''_,...., • V\.4o I IVIV'ti 1 ltJU~ CY'""' ' here we must recognize ~hat the properties of the noise will be changing as the corrosion product film develops. For the case of steel in salt solution there is no obvious structure to the noise. and it is not immediately apparent what physical processes are producing it. As the

As well as demonstrating the possible role of oxygen transport in producing electrochemical noise, this also reminds us that it is very easy for outside influences to produce effects on the measured electrochemical noise. It can be very difficult to separate these effects from true electrochemical noise after a measurement has been made, and we need to take great care to avoid them at the measurement~~ 1 F~ >~ •

.

The electrochemical noise produced by other systems may show a much stronger structure in the time . r~ord. !=Gi example, Figure 1.5 shows the current and potential noise for steel in saturated Ca(OH)2 plus NaCI. In this case the steel tends to be ive because of the alkalinity of the solution, but the chloride causes localized breakdown of the ive film . and pitting corrosion is observed. This results in very clear current and potential transients as the pits initiate, grow, and reivate. ,.

6

Corrosion Testing

-0.266

Ma~e

Easy

100 St&<:l in CaiOHh alter 1 hour

-0.268

80

.' I

w u

Current

!!!. -0.270 >

-

!~ - -Q.272

60 40

0


...... :1

0

0..

20

-0.274

-0.276

'··

2()'.)

0

400

800

600

0 1200

1000

Time/s FIGURE 1.5. Electrochemical noise for Iron in O.OSM Ca(OH)2+0.025M NaCI solution.

-0.5636

-r---------------------,- 30 Steel in 0.27M HCI after 5 hours

·0.5638 ~

w

29

-o.5640

u !!!. -0.5642 > ~0.5644

i

~ -o.5646

28

1

27

-.. i...... :1

26

°

-0.5648 25

-0.5650

-o.5652 +-------1-----+-----t---'----t-----1 24

0

200

400

600

800

1000

1200

Time/s FIGURE .1.6. Electrochemical noise for steel In HCI.

Because of the clear structure in the time record and the well-defined relationship between the features in the tim!!' record and the underlying physical process, it is quite easy to see how information about the physical process can be extracted from the time record, and we look at the methods for doing this in Chapter7. ' ' The ~j~ation, ~owth a~ ~p::~n nt ~~ be POP i~wFG8 ~ iR .sion of metals jn acjds (s,ee Figure 1 .6). However, because there will almost inevitably be many simultaneous sites of bubble formation, the time record does not show clear transient events. and. we need to look at other methods to extract information about the process. These are discussed further in Chapter 7.

~ubble; wil'

.Ro;;;

Walk- Through of an Electrochemical · Noise Measurement In Appendix C we present a walk-through of a simple electrochemical noise measurement. Readers may wish to work through this, although some of the oSed·are not introduced until later chapters in this book.

th;:;;\uo

Electrochemical Noise and the Law! Readers should be aware that aspects of electrochemical noise technology are th£~Ct of patent protection. As we are not patent attorne'lS.)Ne have not •

I

7

Electrochemical Impedance and Noise

attempted to indicate the techniques that may be subject to such protection or to discuss the merits of the inventions claimed. We should not expect individual s in a research environment to experience difficulties with the existing patents, although we specifically disclaim any and all liability for claims that may arise 'in connection with breaches of patent protection. Intending producers of commercial electrochemical noise measurement or monitoring equipment or services and

commercial s would be well advised to clarify their legal position before embarking on a major investment.

Reference 1• Thompson, N.G., J.H. Payer. "DC Electrochemica! Test Methods." Corrosion Testing Made Easy, Vol. 6. Houston, TX: NACE, 1998.

i '

•

CHAPTER 2

Principles of ~lectrochemical ImPedance Spectroscopy. ':

This chapter introduces the principles of the technique, and discusses the expected tesponses for simple corrosion systems, to provide a basis for the identification of experimental errors and artifacts that are considered in Chapter 3. A fuller description of different types of impedance response is provided in Chapter 4.

Comparison of Electrochemical Impedance Spectroscopy and Linear Polarization Resistance Methods Many readers will be familiar with de electrochemical techniques for corrosion rate measurement, such as linear polarization, ·used to obtain the polarization resistance Rp. and it !s useful to compare EIS with this. ( Readers unfamiliar with any electrochemical measurement techniques would be advised to read the previous volume in this series on de electrochemical test methods 1 and the first section of Appendix B of the current volume before continuing). Linear polarization is often carried out by applying a slow potential sweep of low magnitude about the corrosion potential and measuring the resulting net current. The net current (in amps) is defined here as the net anodic current, i.e.,1i._nQL = Ia- Ic.) where l.a and lc are considered positive. We will return shortly to why we use a slow potential sweep and why we use a small magnitude of applied potential. F

II applied overpotentials (overootentiai.G here'Qis the corrosion potential in volts), t e net current is proportional to the overpotential: I]

lnel

= R'

(2.1}

~

-

where Rp (in ohms) is the polarization resistance. The

corrosion rate of·the metal can then be'"l"elated to 1I Rp by a coeffici~nt, which depends on the mechanism of the corrosion process. This® is one component of the simplest equivalent circuit for the corroding metal. In fact, for the experiment described above it is the only component needed and is the equivalent circuit. However, for arternating applied voltages there is another component of the resulting current that must be allowedt.gr. and this is due / to the capacitance@of the surface. ~cause a current can either flow through the interface, causing oxidation or reduction of species, or charge or discharge thecapacitance, these two components of the circuit are in parallel, and a better equivalent circuit for the corroding interface is that in Figure ~At this stage we will not consider the effect of solut1on resistance since this greatly complicates the derivations, but adds little to understanding. For this circuit we can now consider how the current varies with potential or, if we vary potential with time, how the current varies with time. Let us describe the voltage applied to the circuit of Figure 2.1 as V(t), where {t) indicates that the voltage depends on time. • The current flowing through the resistance is given by Eq. (2.1 ), in this case written as "..

V(t) Rp

l~iJ=·-.

\Z.2)

' We generally use E lor electrode potential (often abbreviated to potential) when referring to a real electrochemical measurement and V lor voltage in an equivalent circuit. Readers should. however. be aware that the two ae often used essentially interchangeably.

10

Corrosion Testing Made Easy

such that V(t) Rp = ./(t),

c1a.uit

FIGURE2.1. Slmpttt ....ulvalent for a ~ing metal with the double-layer capllcttance in· patallel with the polariZation re-

(2.5)

and this is what is obtained with the conventional polar· izatioQ resistance methods with slow potential sweeps.

sistance.

Current-Time Response to a Linear Potential This current is often known as the Faradaic current it is the current 'related to oxidation or reduction of species, in which the amount of material oxidized or reduced is related to the current by Faraday's laws.

as

Swe.ep Suppose we apply a voltage sweep given by Figure 2.2. For this, with the direction of the potential sweep changed at t1 and t2 , for t < t,,

The current flowing through the ~tor' depends not on the voltage but on the rate of change of the volt· age: -

/(t)

= ,..

d V(t)

V(t)

= ut,

dV(t)

(2.6)

dt=U,

dt '

(2.3)

+Go dV(t).

(2.4)

The current-time response is shown in Figure 2.3, which shows sudden jumps as d Vldt changes sign.

If the·rate of change of voltage is small, then the second term is small, and ij'le current is given by the first term

Commonly, the current is plotted against the volt· age to give a plot of the form of Figure 2.4, In this

VOl

=

arid the total current is

/(t) = V(t) Rp

where u is the voltage sweep rate (in volts per second) and Vo = ut,. For t, < t < t2, d V(t)jdt "-U, and for t > t~. d V(t)/dt again equals u.

dt

time t,

FIGURE 2.2. Sawtooth applied potential.

• See Appendix A lor a brief intrOduction to the properties of cape.: · tors and other electronic components.

11

Electrochemical Impedance and Noise

l -2...to•-:: L\...,._ t . to\ -: .. .:;~...

I()

time

••

.: FIGURE 2.3. Current-time plot of response of circuit in Figure 2.1 to voltage in Figure 2.2.

figure curves are shown for( a, Voi Rp » G,1d Vldt; b, Vol Rp = G,, d Vldt; and c, Vol Rp « G,, d Vldt. Linear polarization experiments are usually planned such that the capacitive current is negligible, by use of a very low sweep rate, and the response is similar to that in Figure 2.4a.

This is plotted in Figure 2.5. Vo is the peak (maximum) amplitude of the voltage, and w is the angular frequency of the sine wave, in radians u · of s- 1 ; wt ~,s11: is an angle, in radian 2Jr rad = o One complete cycle of the ac signal takes 2]'(1w s [because sin(O) = L~ho sin(2]'() = sin(4]'(), etc]. The angular frequency can be l....s'o ::.1 related to the frequency f cycles per second, in units 1.:~•.-r- l. of hertz, by \..o~-~ 'i. (2.8)

Why Is a Small-Amplitude Potential Used? The period of the ac cycle, in seconds, is then The most valid reason for using a small applied POtential for linear polarization resistance (LPR) is so as not to change the surface being measured by the act of carrying out the measurement. If Rp is defined as the slope of the E-i curve at Ea,,, then an approximately linear response is not required for determining R0 accurately. However, if the average slope of the E-i curve for the whole potential range is used, or if just the currents at + 17 and -17 are used to obtain Rp then the E-i curve must be approximately linear over the whole potential range. We say approximately because it is never precisely linear, however small a value of 11 is used.

Current-Time Response to an ac Potential Input In this book when we talk about an ac signal we mean a sinusoidal signal, and generally the applied signal will be an applied voltage as described by (2.7)

. d 1 2]'( peno =- = - . f (J)

(2.9)

The rate of change of the sinusoidal voltage in Eq. (2. 7) is given. by

d~~t) = Vowc9.~(wt),

(2.10)

and so when we apply this ¥{>1tage to the circuit in Figure 2.1 t~e current is, from f:q. (2.4),

I (t) =

~

sin(wt) + V0 wGl, cos(wt).

(2.11)

p

These two como_9.Q.e~e current can be plotted against time, as i~ The first term is in phase with the applied voltage (zero current occurs at the same time as zero voltage, and the peak current'occurs at the same time as the peak voltage). The second term is 90' '(]'(/2 rad) out of phase (zero current occurs at the same time as the peak voltage, and the peak current

12

Corrosion Testing Made Easy

,---

r. I

-

.

-

D

~ L-----

..

.

·~

• I volbge

..

-

.:

FIGURE 2.4. Current-voltage plot of response of circuit In Figure 2.1 to voltage in Figure 2.2.

time

FIGURE 2.5. Sinusoidal voltage signal.

occurs at the same time as the zero voltage). The magnitude of both is proportional to the magnitude of the applied voltage Vo. We can plot the in-phase and tile out-of-phase components of the current, divided by lfo, as a point in a plane, with the in-phase component on the x axis and the out-of-phase component on the y axis, as in Figure 2.7. We can represent this by saying that the in-phaSe CUrrent iS prOpOrtional tO 1/R, qnd tbp O!Jl:.. of-phase current is proportional to jwGn . where the j operator indicates a 90~ phase shift between the input voltage and the resulting current. When this is done for different frequencies, we obtain a plot such as that in Figure 2.8, which, apart from the labeling of the axes. is identical to the Nyquist plOt of ittance (see below) for the circuit of Figure 2. 1.

scribed above, and it is useful to have a simpler expression for the total current. We could also express Eq. (2.11) as

I (t)

= /0 [sin(wt) cos

or /(t) = /0 sin(wt+

;, ·'

(2.13)

which describes another sine wave, peak amplitude 10 , in amps, but shifted along the time axis (shifted in phase) compared with the voltage by an ang!e ¢. In Eq. (2.12),

/oCOS0 When we carry out a measurement. it is the total current that is measured. not the two components de-

(2.12)

Vo = -. R~.

lo sin<.;) = Vo (,, Co1.

(2.14)

Electrochemical Impedance and Noise

13

1-outofph;lse

FIGURE 2.6.

~omp~nents

~

of current response of circuit In Figure 2.1 to sine-wave voltage.

.

• V0 CllC

t------VJA

In phase

AGURE 2.7. In-phase and out-of-phase current components at one frequency.

and so (2.15)

We can also show these, for different frequencies, on a plot, Figure 2.1 0, giving a result identical to Figure 2.8. In this case both 1Yl and phase angle, ¢, depend on frequency.

(2.16)

Impedance and ittance The term 10 ; \~ is known as the ittance modulus. often written as 1YJ. and q, is the phase angle. When we add the two currents given by Eq. (2.11 ). we get the total current described by Eq. (2.13), as shown in Figure 2.9. which shows the values of 1Y· and t~.

The equations above provide two ways of representing the current response to an applied voltage, as in-phase and out-of-phase components or as a modulus and a phase angle. These also describe the ittance of the circuit. We have ittance here because we have measured the current response to a voltage. i.e.. I f V.

14

Corrosion Testing Made Easy

• •.

~ Q.

0 •

lncreasiug frequency

•

~

.

• ·••

:

in phase

AGURE 2.8. h)-phase and out-of·phase current components at different frequencies.

.

4.

.

Y = lol V0

time

,_...,... -.olll9tl

FIGURE 2.9. Overall current response to a sine-wave voltage .

• :

{ 0

•

'5 0

'~~hase

• ••

angle in phase

FIGURE 2.10. Modulus and phase angle at different frequencies.

15

Electrochemical Impedance and Noise

By using the symbol Y for the ittance, we can write, similar to Eq. (2.14), Y((J))

=~

+ j(J)GJJ

Table 2.1 lmpe9ance and ittance of Simple Circuit Components

(2.17)

p

Component

or

Impedance

ittance

R

1/R jt.JJC

•

(2.18)

where Y' is referred to as the real component and Y" €tb9-l!Il..agmary_cg_mponent. Note that we have de•. scribed the ittance as Y(w) because it depends on frequency. Generally, but not in this case, both Y' and Y" also depend on frequency. Y' is prooortjopal to the in_:Rhase current component and Y" to the out-of-phase ~c•1rrent component. Hece we peed to recognize that j is equal to the square root of -1 , and the expression for the ittance is a complex number and can be manipulated, e.g., added,as such. With a conventional impedance measurement, we provide a signal given by Eq. (2.7) and measure a current response given by Eq. {2.13). Formally, this is an ittance measurement because we look at the· current response to an applied voltage, but in the remainder of this book we refer to the impedance measurement as this is the normal terminology. The analyzer protesses the data to provide either the impedance modulus 1Z] and phase angle, or the real and the imaginary compo'nents of the impedance, Z' and Z". Appendix D provides a description of the sine-wave correlation method commonly used for this.

Resistor R(Q) Capacitor C(F)

- j/rnC

This is exactly the same as calculating the total resistance that is due to two resistors R1 and R2 in series or in parallel. If they are in series we add the impedances, and the total resistance is R1 + R2. If they are in parallel then we add the ittances and

~

Here we have to calculate the ittance and then take the reciprocal to get the impedance. Dropping the subscripts, we obtain

Y= y

From these we could immprli~tel~· h?ve •witten Eq. (2.17) for the ittance of the parallel RC combination. To obtain the impedance of the parallel RC circuit, we cannot add the impedances. For elements in parallel we add the ittances. Only if the elements are in series do we add the impedances.

~+ jwC,

= 1 + jwCR,

(2.20)

R from which the impedance Z is given by

R

(2.21)

z = 1 + jwCR'

Impedance of Simele Circuits ' To build up the expected response of more complex circuits, we do not need to consider the currenttime response, but can simply use the impedances or ittances of the individual elements. Table 2.1 shows the impedance and the ittance expressions for resistors arid capacitors. More complex circuit ~lements are considered in Chapter 4.

(2.19)

To convert this into an expression with real and imaginary parts we need to remove the j term from the bottom, and we can do this by multiplying top and bottom by the complex conjugate of the bottom, (1 - jwCR), because (a+ b)(a- b)= ;i- fi!. This gives

z~

R

jwCR

2

\_~=2C2R2- 1 +w2C2R2 = Z' + j Z".'

(2.22)

To express this in of modulus and phase angle,

IZ tan~~

R

= -;==::;;:::::=;~ J1 + ~.,,2C2R2 = -wCR.

(2.23)

Corrosion Testing Made Easy

16

J~---

1------lc :

These two methods of describing the data are also the basis of the two common methods of data presentation, the Bode and rhe Nyquist plots, respectively. There are several different types of Bode plot used, and both types of plot can be applied to impedance or ittance d~:~.

FIGURE 2.11. Equivalent circuit for a simple corroding system • with a solution resistance.

Nyquist Plots

Note that the magnitude of the phase angle is the same for impedance and ittance, but of opposite sign.

.If we now want to consider the effect on the response of the presence of a solution resistance, we can add this to the circuit in Figure 2.1 to give the equivalent circuit in Figure 2.11 , which is the simplest circuit encountered in corrosion studies. We note that tHe solution resistance Rs is in series with the parallel RC combination, so we simply add the impedance of the solution resistance to the expression in Eq. (2.22) above: jwCR2

.

1 +w2 0R2 .

(2.24)

To calculate the ittance of this circuit we take the reciprocal of the impedance. This gives a rather more complicated expression:

+ w2 C2 R2 ) + R y = (Rs + R)2 + R'fw20 R2 Rs(1

jwCR 2

~ Sometimes known as a complex plane plot, this is a plot of the imaginary part of the impedance Z" against the real part Z'. Because the majority of ~he responses of corroding metals have negative Z", it is conventional, { for corrosion studies, jQ plot Z" against ~·. Consider the response of the resistor capacitor circuit in Rgure 2.1. The expressions for the real and the imaginary parts of the impedance have been given abo e, in Eq. (2.22), and, when_plotted, give Figure 2.12. he corresponc:ling plot of rea] against i · ce is also shown.~3. and can be compared with the plot of the two current componerts given above in Figure 2. 7. Each point on the plots corresponds to the impedance (or ittance) at one frequency, but this frequency information is not included on the plot t.~nless the frequencies corresponding to the individual points are marked. If we now add a solution .resistance to the circuit to give the circuit1n Fi_gure 2~11, we obtain the

(2.25) ·1250.-------..,--..,------,-:1 A= 1000 ohm, C = 0.0001 F

Data Presentation

·1000f-

The data are usually provided by the measurement equipment in one of two forms: • impedance modulus 1Zl and phase angle 4J • real Z' and imaginary Z" parts of the impedance These are related by IZI2

z" ·500

••

• ••

•

r;

= z:2 + 2"'2, 0

tan¢=

z.·

Z'=IZICOSL1,

=

IZI sinL~.

(2.26)

0

•• • •• •• •• •

•

·250 f- .•

Z''

Z"

·750

250

500

z'

750

\

1000

1250

FIGURE 2.12. Nyquist impedance plot for circuit in Figure 2.1; R= 1000n. C = 0.0001 F.

Electrochemical Impedance and Noise

0.075

17

by

i

~

1

,

0.050

_-

The

Wmax

\

= 2rr fma, = Rp~i . )

(2.27)

ittanc~n. Figure:2.1s, -·

anp also traces a semicircle, but in this case the maxi-

y"

mum of Y" is at

~

~

0 . 025~

--- -~

~Jmax = 2:r fmax = -------------

0.000 ---'-----'----'----'--~--'----' 0.000 0.025 0.050 0.075

y' I

FIGURE 2.13. NyqUist ittance plot to·r data In Figure 2.12 of the circuit In Figure 2.1.

corresponding Nyquist impedance plot shown in Figure 2.14. The points trace Q_u t a semicircle, withcenter at Z' As+ Rp/2, Z" = 0, and diqmeter Rp. As apparent from equation (2.24) the solution resistance can be obtained from thebigb_freque~t with the real axis, and the total resistanct! (_~ + R_e;trom the low frequency intercept. ·- -- --·· ·

R,R r ..

..

(2.28)

p y-Ji

---- - ---·--- ---·-·

The on the right of Eqs. (2.27) and (2.28) dictate the time scale of voltage and current responseSci Jbe circl•it.of Figure 2.11 to applied current or voltage / respectively~ese time constants (in seconds) are defined for applied current

9

(2.29)

and for applied voltage

=

One point to note is that the frequency at the top of the semicircle, i.e., when - Z' is at a maximum, is given

(Rs+Rp) \

r,

= (RsR:JGJJ Rs - Rp) ·

(2.30)

The high-frequency limit of Y' is 1I As. and the lowfrequency limit is 1/(Rs + Rp). ··

Bode Plots · These are plots of the impedance parameters against frequency. and different forms ·are possible. The most

-1500.....------..---------,

0.0100.---------------,

0.0075

y"O.C•)50 [ o.:·)25

.

~'-'-~:

0 :•cOOL--[

J.OOOO FIGURE 2.14. Nyquist impedance plot lor circu it in Figure 2.11 for Rs =: 1000, Rp = 1000!!. and ~ 1 0.0001 F.

=

•

_...____.___

0 :•: 25

_.____.\___._

0.00:< 0.0075 y

0.0100

FIGURE 2. 15. Nyquist ittance plot for same circuit as that in Figure 2.1~ .

..

·~

· Corrosion Testlrig Made Easif ,;~

18

4

.

10

. r·

.------------------------,

::t. ~ " '" . :::::. . . . . .,~ 10'

~

2

10.

10.

1

10°

1

10

...J

2

10 Frequency (Hz)

103

4

10

-75.-------------·- ----------,

c;,

lii

3

-50

~

~ -25

... .A...·.

\ :

.J::.

a.

o~--~~~~~~--~----• 1 3 2 1 4 2 3 10' 101010° 10 10 10 Frequency (Hz)

FIGURE 2.16. Bode ure 2.11.

10

'

121 and phase angle plots for circuit of Fig-

FIGURE 2.17. Bode Z' and Z" plots for circuit of Figure 2.11• .

common is to plot loglzt and phase angle against log frequency. For )be.data above, this Bode plot is shown in Figure 2 . 16.~ese plots include all the information obtained, since the frequencies are iqentified. An alternative type of Bode plot.....log ·Z' and log Z" against frequency is shown in Figure 2.17. This pt~ has advantages for the estimation of capacitance ,values, as discussed in Chapter 4.

the stage at which we can consider the effects of exi1eri- .;: mental variables and artifacts on measured impooancil :~ spectra; The response of the circuit iri 2.1' 1 is u~~ .a~-·-~ a baseline for discussing these practical aspectS of thf ::~ technique. · · · .::; ,~ ,:-: .; • ·'!<:::~1

More complicated circuits and more detailed discussion of the information available from trye di.Herent types of plot are covered in Chapter 4, but we are now· at

1. Thompson, N.G., J.H. Payer, "DC ElectrochemiCI.i Test Methods," Corrosion Testing Made Easy, Vol. 6, ,rg ;~ Houston, TX, NACE, 1998; jfJ

Reference :~

·P

'

·- ~

.;~i

.•

.c,~

j

· ~1

·:'~:'

;~

· ~J

..

...

1

'•

~~

~ .t

-~

CHAPTER 3

Measurement of . Impedan-ce~ Spectra\ .:

In this chapter cell and circuit setups for EIS measurements are described, and tM parameters that must be set to measure a spectrum are explained. Common distortions of experimental data and their causes are described. In the case of experimental artifacts, the solutions to these problems are also considered. Most of the features of electrode selection and preparation and cell design are common to ac and de measurements and have been covered in detail in a previous volume in this series. 1 •

Equipment Impedance measurements use an ac genera~ tort ana'!'~ [generally a frecu~ response;;;;-~ly-;;- • (FAA) or a lock-in amplifier} to provide a voltage signal to, and analyze current "and voltage outputs from, a potentiostaf(sometimes more generally termed an electrochemical interface). The setup 0fthe circuit is similar to de potentiodynamic measurements, with the generator being used to provide a signal to the external input of the potentiostat. The analyzer takes the current and the potential outputs and processes the data to provide the impedance. It is possible to provide a small current signal to the cell and measure the resulting voltage (a genuine impedance measurement), but this is not common and is not considered further here. Manv of the or"i~tical aspActs of r.ell ar.rl rirct!it design are the same as those for de methods. and these have been discussed in a previous volume. on de electrochemical test methods. in this series. 1 Readers unfamiliar with any electrochemical corrosion measurements are advised to read this. especially with relation to cell and circuit design . Here the major factors are

emphasized and the experimental features specific to ac measurements discussed. It is assumed that a potentiostat and an ac generator/analyzer are being used. Commonly, the whole experiment is under computer control, but it is also possible to use a manual, not computer-controlled, potentiostat. The generator/an~lyzer can also often be operated manually, but it is convenient to have a computer interface so that the impedance data can be conveniently ~foreabefor~ '> analysis, but even this is not essential. Precise software menu and equipment setup options can be specific to individual suppliers, but the general principles apply to ail equipment It is not intended here to provide system specific advice on setup, but experience indicates that any options not referred to here can be left on their default settings, which are designed for the majority of measurements. If there are menu options that are not clear the operator should refer to the equipment or software manual.

Cell Setup Two- and Three-Electrode Measurements Figure 3.1 shows circuit diagrams for two- and threeelectrode measurements. The diagrams assume a fourterminal potentiostat. which is common, but if a threet~rminal potentiostat is used. '!llith working ~!ectrode. reference electrode. and counter or auxiliary electrode connections, then the two connections to the working electrode both go to the working electrode terminal. It is assumed here that the current and the potential are measu red internally to the potentiostat and provided to the ac analyzer (in fact. it is a potential

'I

20

Corrosion Testing Made Easy

CE

RE

RE

WE

':

Counter electrode

Working electrode

Reference electrode

(a)

•• CE

RE

\. RE

Counter d,·,·troJe

WE

Workin); dcctr<>J<'

(b) FIGURE 3.1. Circuits for (a) three-electrode, (b) two-electrode measurements. CE, counter electrode; RE. reference electrode: WE, working electrode.

21

Electrochemical Impedance and Noise

proportional to the current that is provided to the analyzer). The major difference between thes~wo ty-pes rotcelll arrangement is that a reference electrode is not required for the ~wo-electrode measurement. ~As will be seen later,Jn tbe sectjon on high treoyency problems, the use of refer&Rce electrodes contrjbutes to jnstr!,!mental effects jn measured ·seectra, and operating without a reference electrode can ~e aduaqtage&. However, a two-electrode measurement measures the sum "t of the impedances of the two electrodes and is use...r ~ ful only if the ~o electrodes have essentially identi_ ~~~cal impedances ,(practically,· two "identical" electrodes ~:f·e- never will be identical), and if the corrosion ~tential . ..l \_is required this must be measured separately.~~ .Jt ~advantage at twa-electrode measurements is 1n very ~corrosive or . . s in which · "-V B suitable ref n t be available.

. a

Circuits with Manual Potentiostats We discuss this in of combining a potentiostat with a FAA. The ac generator provides a sine-wave

CE

voltagE! to the external input channel of the potentiastat. The two analyzer channels receive two voltages. One, call4;ld here lf.:h. is the potential, either between 'the working and the reference electrodes or two working electrodes. The other voltage, len. is a potential proportional to the current flowing through the cell. The· simplest way of connecting the analyzer channels is shown in Figure 3.2, in which ·an externaJ·resistor R,, the counter resistor, is put in the counter electrode lead. The voltage across this, IR ,, where I is the current flowiCJg, goes to fch. and the potential for Vcn is measured directly between the working and the reference electrodes. With a two-electrode measurement, it is important to connect from the reference terminal, as shown in Figure 3.2, to avoid the voltage drop across the counter resistor being included in the control voltage. These circuits have some advantages, but also limitations. The advantages are that the signals to the analyzer come directly from the cell and cannot be subject to distortions or phase shifts in the potentiostat. The limitations are associated with the maximum measurable impedance. Typically FRAs have an input impedance

RE

RE

WE

.....

(a) FIGURE 3.2. Possible circuits with external counter resistor and direct connections to the analyzer C'ntinued.

22

Corrosion Testing Made Easy

.

RE

CE

t

.

..

WE

Working electrode

Counter electrode (b) FIGURE 3.2. Continued.

on each channel of ....... 1 MQ. If the impedance of the working electrode is greater than 1 MQ then the majority of the current though the counter resistor will flow, not through the working electrode, but through the resistor in Vcn. For insignificant error, the cell impedance should be no greater thr % of the input impedance of the measuring device, ; case < 10 kQ. Because the cell impedance inc 3 as the corrosion rate decreases. the circuits in e 3.2 can be used only for systems corroding at a ;onably high rate, in which Rp is less than 10 kQ. Most potentiostats provide outputs of potential and of a potential proportional to the current. In this case thj:! measurable impedance is limitetl by the input impedance of the potentiostat, often in the range of 10 10 to 10 12 n. Again the limit of accurate measurements is ..... 1% of this input impedance.

circuit, as discussed later in this chapter, in the section on Experimental Artifacts in EIS.

Working Electrodes

~

These are the electrodes whose impedances are to be measured. Their design has been described in a previous volume in this series. 1 One factor that may be of importance in EIS . measurements is that the technique is often used for high-impedance systep1s. Because tb~ electrode jmpedance is iRllerscl) ~l'fl'iJnr· tignal to the area gt lbi ilisttrosfe thprp ace sometjpes henetjts frprn IJ5jog working plpctcMpg of m••ch l&fiiBf

area man is common for de measyrements. a.gdnjbe1 study of coated nwtals.

...........

There are many different potentiostats available, with different features and specifications. T re are some high-quality potentiostats ' not suite for impedance measurements because they have been deliberately constructed for very stable de cont ol ai!Q~ttHJs cannot control a rapidly changing voltage. this case the potentiostat may introduce phase shifts. especially at'high frequencies. This can be checked with a dumm~

Reference Electrodes

can be one of

~-~a ~!Ice solution resistanc~ How· ever. this is less important in ac measurements than in de because the solution resistance is rnea· sured and can be allowed for. If a two-electrode arrangement is being used there is no reference

..c .......;..

-t-~V

~ ~;

.3?

...1-

23

Electrochemical Impedance and Noise

electrode and the solution resistance will be dictated ~y the spacing of the two electrodes IP practjce many P.~or.>l"'niph-freguency distortions can be associated with the !Jigh impedance a&&g"iated with the Lyggin-refereoce electrode combination, and better data can often be ob~ained with djrecl jmmersjgn of the reference electrode in the ceiL Of coursE(Jrl this case it is important that the leakage of ions from the·reference electrOde does not affect the working electrod~

measured, and the impedance of the counter electrode is unimportant in three-electrode measurements. '

However, it is important that the shape of the electrode js sr1ch as to provide a uniform current distribution ·· on the working etectcoda (see cell design below), and tl'\is may dictate a minimum size.

For two-electrode measurements current between the two iden~ical working electrodes ven though one is plugged into the counter electr e terminal). These are usually the same size and shape but this is not essential. ~~,_w>;~~WW'"""'~the

j_

Counter (Au~~liary) Electrodes

c..vW lr'\. )xd' oF

For three-electrode measurements, these are usually made out of a material that will not corrode, even when polarized, In the test solution · otbQr noble metals s1•ch act gqld or inert matecjel§ SUCh graphite, Can bp !!Spd Jhe impedance Of the electrode is unimportant, as it does not form part of the measured data, and so the surface area is not critical.

(()~ ~lt"d-~o ~b,. Cell Design

as

The mast important factor f.or impedance meast •ce- -'tsl..lll.o.{

+-).. , \

meats js to attempt to nbtaiR '-!RiforFR 6YFF&Pt distrih11rl tion on the working electrode This is more important \l.l~ IC.~ ! than for de measurements because the impedaR~a Qf ~lrec-fGk 1 lbe electrode js low and controlled by its capacitance at .bigR fraqug_ncy. This means that at high frequency the : current will take the shortest path between the counter and the working electrodes, but will be more uniformly spread at lower frequencies.

We can see this by considering the equivalent circuit of the whole cell, Figure 3.3(a). Since we control the voltage between the reference electrode terminal and the working electrode terminal, and measure the current, it is only the highlighted elements that are

_.J_ CE

~ RE

RE

0

•

/

d:Sid -

(a)

FIGURE 3.3. Equivalent circui!!> Qf electrochemical cells Cor: nued.

!

24

Corrosion Testing Made Easy

CE

KE

RE

WE

.. .:

I

I \\'

Working electrode

orking electrode

(b)

FIGURE 3.3. CQfltinued.

_j -1 0 6 o cm 2 , is not uncommon for ive metals, and

Connections to the Cell Often EIS measurement equipment is provided with shielded leads designed to match the equipment. .J.o. aliOid nojse poy other leads jn the cell connections should be as short as possible and also preferably shielded to reduce mains noise. Long trailing un~je!ded leads are a common cause of noisy data.

e~cofo.r l irl--*0. \air I COl'\(\Su iO-( Experimentai.S.etup

'

After the cell is set up there are several experimental parameters, defining the measurements to be taken, to be set before the measurements begin. Some of these can be set automatically with computer-controlled po- ~ tentiostats, but they are still considered here. ~

Counter Resistor

d

~

coated metals can give impedances much higher than this). Many commercial software packages will automatically select the appropriate counter resistor for a given frequency if a computer-controlled potentiostat is used. With a manual potentiostat one counter resistor has to be selected for the whole frequency range, and this may mean that more than one experiment, with different counter resistors, is needed to obtain goodquality data at both high and low frequencies. As a guideline, there is likely to be significant degradation in the quality of the data if the measured impedance differs by more than a factor of 1000, either way, from the counter resistor.

Frequency Range

There are maximum :md minimum limits imposed by the generator/analyzer. but these are often beyond the As with de measurements the optimum counter re- , ,......:range of useful measurements. Instruments will often sistor is one that matches the resistance .(impedance)l ~permit the use of frec'Jencies between 10okHz and ;J being measured. However. EIS measurements often 0 ..0.1 mHz 'or lower. 1Some FRAs can operate in show a wide range of impedances at different frequen-t"..i'> the OliQ---;hArtz IDAQP h11t these frew •encjes are not cies (a range of 6 or more orders of magnitude, e.g}'~ uset~11 for corrosion s·udiesLMeasurements ateigh_. ~

... 25

Electrochemical Impedance and Noise

frequency at the top of the semicircle, as shown in 1..J 0 )UA' Figure 3.4. r~

-1000.-------------~

-*''

..,.....tcN"t.\ •

z"

1.(}

.. . . /. . . 500

z'

.. .

Double".layer capacitances typical for metal surfaces, although much higher vahi~s ,can be found for metals with porous conducting surface ,films_. 1With GJ1 = 100 tJ.F cm- 2 , Table 3.1 shows typical values of the frequency of maximum Z" and .the frequency range required for Z" to approach zero at both high and low frequency for different values of Rp.

0. 1

0.0 I

1000

1500

=

FIGURE 3.4. Nyquist and Bode plots for P.,. 100 n, Rp •· 1000 n, Celt= 0.0001 F, with selected frequencies marked.

~ ~ ~ t1Ut'"Ct r~....,..

t~uJ

'-'-

~'

t.el'- ICc\\ lk~\ra--"'-'

-l>

!.\,.b\h~ Of -4\-.c.

e

=

. I

f~to:k,. ..

<;/frequen-cies l_are often attec;tpd lly eistortions . that are d(le to tbe.cell or cell/instrument interactions. whereas gt low freQIISCCiQi \t;JQ pragti'il&llimi* is often dpten:Ainecl., 1:>).! the stati!iey I d tbe " Atmde IP reality, other factors, such as the patience of the operator and the need to carry out a certain number of tests in a given time, can also ii'TIROSe a lower limit on frequency. One indication of the required range of frequencies comes from considering the Nyquist plot and the need for Z" to approach 0 at high and low frequencies. The frequency of maximum Z" depends on the value of the time constant RpGJ1 for the interface for a simple parallel RC circuit, as in Eq. (2.27). Z" is ..... 1% of its maximum value 2 decades of frequency above and below this frequency. The frequency range \

~> f> 0.01 2rr RC

o~are

'

2rr RC

(3.1)

covers 2 decades of frequency on either side of the

'I

Thus, even for a rapidly corroding system, there is little useful · information above 10 kHz, and the data do not suffer significantly from being restricted to 10kHz. t:_orJ9vv~r. 9.orrosjon rates more tvoical of natural environments there is often little useful informatjoo _shove 1 kHI. Table 3.1 shows that, with slowly corroding systems, it may not be possible to record the full Nyquist plot semicircle with all equipment, because the low-f~equency limit can be below the minimum frequency possible. The choice of minimum frequency, together with the number of data points collected, has a very significant effect on the time taken to acquire the spectrum. The absolute minimum time required for acquiring a data point at a frequency f is 1// ~hich is the time tor eRe ac cy~le. Figure 3.5 shows the minimum possible time to acquire a spectrum with 7 points per decade of frequency for different minimum frequencies.for a starting frequency of 10 kHz. Even going down to 1 mHz takes -1 h, ~orrare required for reaching 0.1 mHz. In practlee, depending on the signal-averaging method used, more than one cycle may be required at each

Table 3.1 FrequQncy of Maximum Z" and Frequency Range Required for Different Corrosion Rates for Cd, = 100 J.LF cm- 2 Rp

(ft em:)

20 500 10,000 1,000,000

Approximate Corrosion Rate (mmiyear)

fat max Z'' (Hz)

14 0.6 0.03 0.0003

80 3.2 0.16 0.0016

Frequency Range (Hz)

0.8-8000 0.032-320 0 .001~16

0.000016-0;16

Exarnpie Mild steel-strong acid Mild steel-natural water Mild steel-inhibited natural water ive metal

26

Corrosion Testing Made Easy

600 j.

500

..

~400

• :; c:

§_300

•E

=200

.:

100 0 -4

-3 ~

.

'

·2 log (minimum frequency)

FIGURE 3.5. Minimum time for EIS spectrum collection with 7 points per decade.

frequency, increasing the time required. For usefui data, it is required that the system be unchanged during the course of the experiment, and many corrosion systems will show slow drift in corrosion potential or corrosion rate over a 1-h period, and it is effects such as this that provide the practical low-frequency limit. The detection of drift in spectra is covered later in this chapter. In tlie last case in Table 3.1 the frequency required for approaching the low-frequency limit is much less than 0.1 mHz. The frequency at the top of the semicircle is 1.6 mHz, and if measurements are made only down to 10 mHz, which is not atypical, it is very difficult to obtain an accurate estimate of Rp from the impedance plots. Similar effects can also occur in more rapidly corroding systems with very high capacitance, as the factor that determines the frequency range of the response is the value of RpGit- In both such cases,jjp ~=~ ~ tain from de methods, especially from ~ 1'\'fr method; which... depending on the solution resistance. can give accurate 8p measurements in a matter of seconds for these conditions. 1t is also true that it is easier to extract an accurate value of Rp from an ittance plot of these data. as discussed in Chapter 4. In practice, with a new system. it is sensible to set a IQlA! rviRir:t:u,trv f~iq~eRe~· fgr tRe 'irst iX(ii'i&irventfi.. If

measurements are made first at the high end of the frequency range. then at successively lower frequencies. a frequency may be reached (above the chosen minimum frequency) below which no useful data are obtained. In

such an event the run can be stopped manually. However, the converse is not true. If too high a minimum frequen~y is set thEm the experiment has lo .be repeated · . to obtain lowerfrequencies.

Number of Data Points The hjgher the number of frequencies in the spectrum, the higher the accurac of an I sis but t e tota time for collection of the data increases. In general 7- !Q...PpJnt§ gee decade js a good compromise between time and accuracy, but for speed of measurement in a new system, fewer data points could be coUected until the other experimental parameters are optimized.

Amplitude The amplitude of the voltage set is always the rootmean-square (rms) voltage, and the peak voltage (the maximum applied voltage) is ...... 1.4 times higher than this. This amplitude is probably the most important experimental variable, especially because the effect of choosing an inappropriate value is not usually apparent. Appendix B describes· the influence of amplitude of applied voltage on measured response in more detail. but the general requirement is that the ·applied vo1ta9e should be small enough for the electrochemi:, _9[r.esponses to be aQproximately linear.. This requires that the applied voltage should be much less than the

27

Electrochemical Impedance and Noise

• exponential Tafel coefficients (~) for the processes ocbeing A, and A! cm 2 for the two electrodes, then curring. With typical values of Tafel coefficients in the . range 13-50 mV (Tafel slopes of ~3o-120 mV per , (Q 2) = Zm A, A! · (3 3) : em easured A,+ A!. · decade), depending o~s of the prijz!>J"~ cesses, an amplityde 5 mV · rarely g1ve sj Uifjcant errors. With a 10-mV rms applied voltage the Some software has a two-electrode setup option, and in this case the software manual should be checked to errors in the measured polarization resistance or other impedance parameters a're likely to be less that 10% e~tablish whether it is the area of one electrode required. for most systems. In principle, the smaller the amplitude or the corrected area as in Eq. (3.3). the better, but as the amplitude of the applied signal decreases so does the amplitude of the curre_!lt response, It should be noted that the solution resistance is deterhlined by the cell geometry and, in general, will not and external noise becomes relatively larger (the signal•. to-noise ratio decreases)__and 5-10 mV rms is often a decrease with increased electrode area in the same good compromise. way as the electrode impedance.

\) · z

-

Common Experimental Setup

Signal Averaging There are generally several choices for signal averaging (often called integration time becau$.8 the impedance calculation involves integrating the current response over a number of cycles). The greater the number of ac cycles averaged for 1 point on the spectrum, the lower the noise (the greater the signal-to-noise ratio), but, as with many parameters, better-quality data are obtained at the expense of increasing the time taken to collect the spectrum.

~P:o':·n:ts~/~d:e:ca:d:e~---7~-----------~· The frequency range given is suggested for the first experiment in a ilew system. The range, especially at low frequency, can be reduced ·if it is established that little useful additional information exists there.

Area Correction This is included here, although much software will automatically correct for the electrode area if it is entered in the setup. Otherwise it is necessary to apply this correction to the data · after they have been collected. For a three-electrode measurement the measured impedance must be multiplied by the area of the electrode to obtain the impedance in ohms times square centimeters. If the data are in. the form of. Z' and ZH these are both multiplied by the area, but if the data are in the forJ11 1Zl and phase angle, only .z1 and not the phasf,'_angle · ied b th ar

0,

The discussion above of the role of the various factors can be summarized by suggesting experimental setup conditions that will rarely cause any problems. These would be . ~"-O --v SCfj~r -..J e~rl~,..JaA. Amplitude 5 mV rms Frequency range 20 kHz-o.001 Hz

For a two-electrode measurement, with equal areas t'"1e two electrode~. A \:r:' 2 <><>ch,

A Z (Q em ) = Zmeasured -. 2 2

(3.2)

because the measured ·impedance is twice the impedance of one electrode. If the areas are notequal,

With much automated equipment .it is not necessary

to set any other parameters, .and the default settings of the equipment will often suffice.·

"<Experimental Artifacts in EIS With an appropriate experimental setup, problems should be minimized, but it is still important to be able to recognize and solve common problems with data ac· quisition. In general, good impedance data, unaffected by measurement artifacts, will show several features: • at high frequencies Z" will tend to zero and Z' will equal the expected solution resistance, • the data will change smoothly with changing frequency. without spikes or sudden jumps, and • !he data should not be widely scattered around the smooth trend.

(

t

28

Corrosion Testing Made Easy

CE

RE

RE

i

WE

1

'··

FIGURE 3.6. Connection of dummy cell to potentlostat.

It is comparatively simple to see if these conditions are met by inspection of the data, as Nyquist and/or Bode plots. The following subsections cort$iqe,r the origin of ·some common problems and suggest approe.ches to their avoidance. However, there. are some spectra that would meet the JibQve conditions and yet not be reliable data, and these are also discussed.Jf jt js believed that UJe data .bejog oblfiPQA are subiarJ to 8ffQf6 ·Q[ exper. iW8£ltal Qr instrumeotat "i&'Mixe 'bw ''? turg iAitiil

rsteps thpt •be· ''d be *eke"'"

1. Check the eguipment with a dummv circuit, prefer· ably one with an ilnpedance similar to the. cell being measured: A convenient dummy circuit is one as in Figure 2.11 , with the values of the circuit elements chosen to be similar to the system being measured. This should be connected to the potentiostat in the manner shown in Figure 3.6. Dummy cells provided by many equipment manufacturers can be used to test hardware but will generally not detect condi· tions where the limits of the equipment have been reached. lf the dummy circuit i~ measured correctly the hardware is performing correctly, and the source of any problems lies in the electrochemical cell. 2. Repeat the ewerifl)eOt to check that the data obtalr\e(j are repeatable. It is not uncommon for ex- "'" tremely strange data to be obtained once and never again. for reasons that never do become apparent.

and much time can be wasted trying to analyze such data. It is convenient to separate experimental problems according to the frequency ranges at which they occur. Many of these problems and distortions do not affect simple data analysis approaches, but can significantly affect the quality of data fitting to equivalent circuits, al· though in most cases they'can be compensated for if they are recognized.

Problems at All Frequencies If the data are nnirw (noi~e is t here in a conven· '"' ~ lP"~. ~.~~\~pc""'"'' tiona! sense to indicate w1 e ~r, not in the sense of electrochemical noise) at all trag• •eocjm, thjs js Jj1cely tQ be dye tO the lack of a GQPMGtipO Of perhaps 8 QOOf qs>nnectjon For example, this can happen with gas bub· bles in or on the tip of the Luggin capillary or with a poor or intermittent electrical connection to the working electrode. It is also possibly an equipment fault that should be checked with a dummy cell.

Hig!}..:.frequency Prob'sr:r:~s

~II/ lc.\\ f"'~ru~V\tf'\+·

Very high apparent solution resistance should not occur with automatic counter resistor selection. If a

29

Electrochemical Impedance and Noise

•

•

·0.7 5 , . - - - - - - - - - - - - - .

manual selection has been made it is likely that the counter resistor is too high compared with the actual impedance at high frequency. This commonly occurs when a high-impedance system, e.g., a coated metal, -is being m.:~ured ag9 J~~ courterresist~! ~been selected to matdi"ffi'e i~pe_~nce of the coat ~etat

-

.

-

.

·0.50

,

.. ..

z" -o.25

.

Scattered data at a high frequency but not at a lower frequency is not too common a problem, but if it occur~ it is likely to be due to the use of too high a counter 'resistor. . This should not ha"'j)'i)e'n with equipment that automatically selects the counter resistor, unless a resistor has failed.'

,I

'-

t

or--~~~.--------~

.:

........

·~

•:.

0.25 ....___.__

4.75

1lnductive behavior at higft frequency, Le~, PoSitive Z" or data below the axis in a conventional Nyquist plot, as in f!gure 3.7lal, is always an ..artifact. 11 g_ enerally arises either from thejinductance of the leads to the ' cell or from <:~!llp_oten!!_ostat interactions. _' The problem 1 is usually much worse w_!len a reference ele~trode is used, rather than a two-electrode system or a dummy . e.en, and inductive effects tend to increase in magnitude as the resistance of the Luggin capillary increases. To establish this high resistance as the origin of the problem, the reference electrode co1,1ld be replaced by a low-impedance pseudoreference electrode, e.g., another piece of the same material as the working elec.trade, or a two-electrode measurement could be carried out. If these changes reduce the problem, then attempts should t?e m~de to minimize the resistance between the reference electrode and the working electrode, including that down the Luggin capillary.

..·....... -:

_._____.__

5.00

~ _._____.._......__.____

5.25

5.50

5.75

Z'

(a)

\<::. c orro':>fo"' a-1

\-.d ... ~

-7500,.-----------,~-\- ~~

a

6clb..lk.

h~~ f rrtr.Ju "':j ~I' l'('t:i.t ~ L\

,,I'"\ -5000

~ ~ O<.h CC(j)

(. i..\- \\A.

..

Z" ·2500

.. ...· .

~1.· 29500

...L

o~---'--~~~~-~--~

27000

•

A much less common problem, but seen with some loop·.o....Jo... potentiostats, has similar origins but results in a §Q.U:. rlous capacitive loop at high freQ••ency In this case, it is more difficult to be sure that it is an artifact, but QD.._

Z'

32000

34500

· 5 .------------------.,

10

·······································

ana~~ ~ value of the ca~aci= :'ili~~; ~:d~ till 'u lpwer than any ( a l j ance)-iow er

i1 js also pos&igjg, iR syste~s with ''Sry high solution :resistance, Jor this to be a g~nuine measurement of the capacftance of the ·solution. Ao__ example is shown in 'r1~e 3. 7(b) for a system with a ve'ty_ high solytion res is, mud sfeel in deionized water containing <1o- 4 M zinc phosoh<m,ate c:orrosion iflhibitor~Jn this case~ qapacitance &tfclrlatpd frv fh&bjgb,f(fQ! !ftDQ¥ sarnjcjr • cJe js --...10- 1 F cm- 2 .and the high-frequency limit of this semicircle, the apparent solution resistance. is very small. The capacitance is as expected for the , 1-mm solution path between t~e reference and the working electrodes.

Q)

c;, c: "'

clc\~ ~

(\ "::

'i . .

10

2

•

I

.•

•••

I!!J

10

••

.·.~;,;.J····•·i ...... ········~Ill! J

1

10c

10

1

10;

3 10

4

10

FrecuMcy (Hz) (b)

FIGURE quency.

t-- ~ ~~ cl rcup'<4 ~~

5

~ Q'

°

\

ro~ I...

-15

"' ·10 ~

~ . b_ \~dOCC(Q''-"'

3.7.

Inductive

and capacitive qehaviors at high fre-

~ ~o..t~~lu,.,

~Y""'"JJc..CL

'a::>f>·

Corrosion Testing Made Easy

30

until at some frequency the current reaches the base of the pores and more normal behavior follows. The origin of these features as artifacts is similar, being .n onuni-

Reference electrode terminal

form current distribution, and can be avoided, or minijmized, by cell design, such that the counter electrode is symmetrical with the working electrode.

Capacitor

~­

fit

Platinum wire

Midfrequency Problems Spjkes in the data close to mains frequency {50 Hz Europe, 60 Hz U.S.), often affecting only 1 data point,

FIGURE 3.8. Reference electrode system to minimize highfrequency distortions. ··

ere commonly seen and are due to noise from other . rl1 :electrical equipm'~nt. . !)hielding dnhe eell .and allieads

If these high-frequency distortions are shown to be due to the reference, i.e., if they di.sappear in two- · "' electrode mode, it is possible to modify the reference system by adding a platinum wire, with only the tip of ~ the wire exposed at the mouth of the Luggin capillary. · This is connected to the reference terminal by a small capacitor, e.g., 0.1 t-tF. as shown in Figure 3.8. The platinum provides a low-impedance path at high frequencies, where the capacitance conducts, but has no effect at low freq~ncies.

-can s61W ttle pro61em, but often a slight modification to th~ setup of th$ . fr~quency sweep will move the . measured frequenei~s. ~uffic!e'1~!Y away from mains frequency th~i:lt the pr~etri appareHtry diSappears. We say •apparently" becatJse the cell will still be subject to the effects of the mains noise. ·

Linear regions in a Nyquist plot at high frequencies, as shown in Figure 3.9 and often apparent only if the plot is expanded, may be real but may also be due to cell geometry. The real cause can be the existence of •porous conducting films ion the surface, with increasing penetration of the current as the frequency is lowered

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

0.25

:• ·' • •

:

d~ +c \~J : ~ t~1' 1 ,j , (.,( \\ ~ t,tfiO~ l

Low-frequency noise is probably the most common problem with EIS measurements and can have different origins.

Too low a en• mter resjstQ!L much less than the cell impedance at low frequencies, will cause noise. If this happens with ~utomatic.counter resistor selection, it is possible that the impedancejs too high to be measured with the equipment being used. If the upper limit of the measurable impedance is not known, it can be checked by use of a dummy circuit that consists simply of a capacitor. The magnitude of the impedance of this is 1/wC, . so a 10-nF capacitor has an impedance o.f ""1600 n at 10 kHz, incr~asing to more than. 1010 Q at 1 mHz, as shownii+Agure3.1Q, Oetria'in'arte uaeer.,ehe.,jmor noise in the Bode P'nt ipdjgate thatthe eqqjpmentjs gp- . /. /erating outside the limi he · u~ , cirC'lfit C":an also be used to check the \ \:, ~ measurable flmits with manual potentiostats. .

,\ 0.25

~

l>

~ C>.."~ ~ It"\..-f\ I t>~l !ttJ

Qof()-l">

o.sot 0.00

There are many types of artifacts with different origins that can affect low-frequency data.

,.

.0.25

z" o.oo

-Low-Frequency·Problems

o.so

o.75

1.00

· z' FIGURE 3.9. Linear response at high frequency.

...

Alternatively the system may be subject to genuine electrochemical noise. In this case, better data may be obtained by increasing the data collection or averaging

31

Electrochemical Impedance and Nois'e

baseline. If used in the ac position for impedance measurements, the effect is to filter out the lower-frequency signals, which are generally those with the corrosion ~nformatio'n .

Less Obvious Experimental Problems . Even if smooth, apparently good data are obt~ined · there are still some types of artifact that may be present.

l~t:_ . ..,. . .:. . ".J 10'3

10' 2

10.

1

1

10°

102

10

10;

PrOblems Tl1at ,\Fe Due to Changes wjth !jme These arise when the time taken for the spectral measurement is large, which allows th f the work' e han in the me ese changes may be reflected in the corrosion potential being different before the measurement from the value after the measurement. However, the absence of a potential change does not prove the electrode has not changed. The two types of drift in the electrode with time result in spectra of the types seen in Figures 3.11 and 3.12. In both figures, measurements were made first at high freqt~encies, then at successively lower frequencies. ii21:1re 3 11 in which Z' .decreases at low frequency, C~}Q be dpe to a GQr- ·: , rosion rate th~t increases durjng the measurement: whereas Figure 3, 12, with increasing Z' at low fre-quency, could be due to a decreasing corrosion rate

4

10

Frequency (Hz) FIGURE 3.10. Ideal impedance bphavior of a 1G-nF capacitor.

..

.

time at each frequency, but this is at the expense of increasing the time taken for the measurement, and may introduce problems that are due to drift in the electrode with time (see below). Some equipment has an ac/dc coupling option. Although we want an ac measurement, the equipment must be set to the de mode. ln.the ac mode, de and lowfrequency signals are filtered out to permit the measurement of small ac signals superimposed on a large de

800

•

.

600

•

•

• • •

• -Z" 400

•

•

• • • • •

•

•· •

200

••

0

I

•

.I I

200

400

800

600

1000

1200

1400

z FIGURE 3.11 . Low-frequency distortion showing the effect of the corrosion rate's increasing during the measurement.

Corrosion Testing Made Easy

32

-400r--------------,

-75

!

-300 ·50

..,.

-200

l" ·25

l. 200

Z'

300

400

... .

"1'•

0

.. 100

...

...

25

500

25

0

75

50

100

Z'

2 ,

10

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

....

E

IZI

10'

'

10° 10.2

10. 1

10°

1

10

2

3

10

10

4

10

Frequency (Hz)

.l~td . -~..J . 2 1 1 3 10· 10· 10. 10° 10

3

frl 10

4

10

5

10

Frequency (Hz) FIGURE 3.12. Low-frequency distortion showing the effect of

the corrosion rate's decreasing during the measurement.

with time. However, increasing Z' at low frequencies can be due to real surface or solution processes, such as adsorption/desorption, film growth, or diffusional effects (see Chapter 4). To distinguish artifacts from real effects it is. necessary to repeat the measurement, and ideally to carry out a measurement immediately, sweeping from low frequency to high. If the effect js real ~ ~me spectrum wjl! be obtajneC

-75 Q)

c;, c:

t1l Q)

-50

f/)

t1l

L:

a.

................. . . ......•·· ........... ;

-25 0 25 ·2 10

10' 1

10°

1

10

2

10

3

10

4

10

Frequency (Hz) FIGURE 3.13. Depressed semicircle.

of this depression have not been clearly identified in all ;cases, but it should not neccessarii"y be cons.idereCi 'a.l 'error or artifact. 'Approaches to analyzing data with depressed semicircles are described in Chapter 4. This spectrum also shows some low-frequency noise, s~ gesting that there are stow fluctuations in the surface condition. .

Effect of Amplitude of Voltage

Compared with the plots produced here. in Chapters 1 and 2 it is usually found that perfect semjcir~lii ~ rare and that "!lepressed" ~miclrcla$ are often ob,lained. An example is shownTn Figure 3.13. The origins

The measured impedance will also be' affected by the amplitude of the applied voltage. It is difficult, if not impossible. to determine if a sufficiently small amplitude has been used from only one spectrum. For simple corrosion systems. any amplitude of an applied

33

Electrochemical Impedance and Noise

• sufficiently small is to repeat the experiment 111itt:l a 8if ferent amgtitude.

-400 -300

.

:\'J. -;t>.§J •

-200

~\O

Z" -100

{,

100f-

lzl

.

•.,

200 1900

4

..

.

.- .

~··

0

10

'

• To complete this section we can show a real data file that shows several of the distortions mentioned abovP.., but also iliustrates another, rather unusual source of useless data. Figure 3.14 shows the impedance ob' from an experiment intended to measure the tained impedance of mild steel in aerated NaCI solution by use of a three-electrode system with a saturated calomel re~erence electrode. lndweti¥a behayiw at fmguencies above J9 kHz is seen the high-frequency semicircle -Ts slightly depressed, and there is a large spike, corresponding to 50.2 Hz, i.e., mains frequency. The solution resistance, > 1500 n, is also much higher than that expected for a NaCI solution. However, apart from these, the data seem sensible, although unexpected, in this system, with a small charge-transfer resistance and a large diffusional impedance (see Chapter 4) at low frequency. The data were not at first repeatable, until the mistake made in the measurement was realized. This mistake was that the reference and tl:le werkiRg &lectred& 99RR&ctions were reversed and Ejgyre 3 14 js jo fact, the impectcmce of a commercial _saturated calomel electrode (SCE).. If this had been an unknown system and the experiment had not been repeated, much time could have been wasted in interpretation.

1

I

2000

2100

2200

z·

2400

2300

2500

.!------------------. ••·•·•··········••··•············•··

103 "" .I . .I 1 3 2 0 10- 10- 10. . W

·,J

.I

1

10

.I

2

10

.I

3

10

4

10

5

10

Frequency (Hz)

.... •

-10r--------------,

•••••••••

o t-----=.I..&.L&-.&&..a.&.L.................~ .....ro=-,.....---1 ~,,c..\~' "" .

J

,,,J ~'lot ...~·~ ~~~

,J

°

1 10· 10

cv.\

1

10

2

10

3

10

4

10

5

10

'*

Finally it should be mentioned that there are mathematical methods for the validation of EIS responses, iuch as Kramers-Kronig transforms. They are beyond the scope of this text, but Appendix E introduces them and some of the requirements for their use.

Frequency (Hz) FIGURE 3.14. Impedance data inadvertently measured for a saturated calomel electrode.

signal will still give a semicircle, but its diameter and the apparent Rp will reduce as the amplitude increases. The only way to check whether the ampUl! 1de ''Sfd was

Reference 1. Thompson, N.G., J.H. Payer. "DC Electrochemical Test Methods," Corrosion Testing Made Easy, Vol. 6. Houston, TX: NACE, 1998.

CHAPTER4

Analysis of ~lectrochemical-·­ lmpec:Jonee; Spectroscopy Data In this chapter we consider methods of data analysis, both graphical and those based on fitting of data to equivalent circuits. The possibilities and the limitations of these methods are discussed in relation to typical corrosion responses, particularly with regard to the estimation of the corrosion rate from EIS. Although much of the benefit of EIS is associated with the mechanistic information it can provide, a detailed discussion of the mechanistic interpretation of EIS data is beyond the scope of this book. This chapter does, however, provide a basis for this. There are many cases in which most of the information required from data can be obt~ined by inspection of plotted data, together with some simple calculations. This type of analysis can be done as the data are being collected and can provide a basis for the early identification of experimental problems. This approach is often viewed as different from an equivalent-circuit analysis, in which circuit parameters are estimated from the best fit of data to a proposed equivalent circuit. However, any analysis by inspection of plots necessarily involves the assumption of an appropriate, if approximate, equivalent circuit for the system being measured.

Equivalent Circuits for Corroding Systems An equivalent circuit is a combination of ive elements (i.e., resistors, capacitors. inductors. and various forms of distributed impedances) that give the same response, at all frequencies. as the actual corrosion cell,

i.e., it is a circuit that gives the same response at all frequencies as the system being studied. There are an almost infinite number of possible equivalent circuits, and they cannot all be considered here. The most important factor to appreciate is that the equivalent circuit, or fitting of data to an equivalent circuit, is not an end in itself. It is a step on the way to data analysis. The values of the circuit components calculated are s• •hsequently used tQ pJ"ov.ide iAfeFR'IetieR about ejther cor~sion rate er serresieA R~teef:laRis~s. A simple example of this already discussed is the use of polarization resistance for corrosion rate estimation. From this comes one essential factor in the selection of an equivalent circuit: The origins of the components in the circuit and the circuit itself must be physically realizable. This is espeeially important as there are usually several different,equivalent circuits that equally well fit the data. This will be discussed later in the chapter, but there are two sets of circumstances that give rise to this range of possible circuits. One is that two circuits, although different in response, give sufficiently similar impedance spectra that the data do not permit distinction between them. The other is that there are often several possible combinations of, e.g., resistors and capacitors. that can give identical impedance spectra, and a decision as to the most appropriate circuit must rely on an underlying understanding of the corrosion system being studied. As an example of this. a simple equivalent circuit for a corroding metal has already been used in

Corrosion Testing Made Easy

Graphical Analysis In Chapter'2, the expressions for the real and the imaginary parts of .the impedance and the ittance were given for a simple circuit of the type in Figure 4.1 a. From the impedance Nyquist plot, the solution resistance can be obtained from the high-frecuency limit of Z', the sum . . of solution and Pc;>larization resistances is equal t? the ,low-frec~ency _limit of Z' and the capacita_nce can be ~

calculated from knowing the frecuency at the top of the FIGURE 4.1. Two equivalent circuits with the same frequency raaponse.

Chapter 2, Figure 2.11, and Chapter 3. This circuit was arrived at because we know there is a capacitance associated with the metal-solution interface, and this capacitance must appear in parallel with the Faradaic impedance, i.e., that due to the electrochemical corrosion processes. In addition there js a solution resistance between the reference electrode, er bl:l§t§tiA tip, aR& tl:le metal In the case considered, the Faradaic impedance is simply a resistance, but in general it can be more complex than this. However, if we simply take the impedance data and try to establish an equivalent circuit that gives the same response, we have two indistinguishable choices. The two circuits in Figure 4.1 have, with appropriate values for the elements, identical frequency responses. To obtain the sarne impedance spectrum from the circuit in Figure 4.1 b as from Figure 4.1 a, the values of the elements in Figure 4.1 b have to be

R,

.semicir~le and Rp, ~~- . -12.27), as shown in Figure 4.2. To ca(culate this capacitance involves inspection of the data because the information is not included on the Nyquist plot. The frequency at the top of the semicircle is not easy to estimate from the phase angle Bode plot commonly used, but can be obtained directly from the Z" vs. frequency Bode plot. L

Many texts will describe an estimation of capacitance from the Bode plot of log 1Zl vs. log frequency. If the solution resistance is very small, the impedance modulus is given by Eq. (2.22), and at high frequency, when w RC is large, log IZI

= -logw -log C.

(4.2)

At high frequencies the plot is a straight line, of slope -1, and if extrapolated to w = 1(f = 0.16 Hz) then log IZI =-log C. However, this approach can give significant errors if the solution resistance is not negligible. 'Figure 4.3 shows the effect of increasing Rs on the Bode plot._

This problem can be avoided if the Bode plot of Z" vs. log frequency is used, as solution resistance does

= Rs+ Rp.

Rs R2 = R(Rs+ Rp).

-1500~--------------------------~

p

C, = Cct{ Rs': Rpr.

(4.1)

Z" Because it is difticult, if not impo·ssible, to provide a physical explanation for the circuit in Figure 4.1 b, for corrosion systems, it is not a useful equivalent circuit. Many systems do give comparatively simple impedance spectra. being essentially a single semicircle, and from these the values of R 5 • Rp. and Cctt can often be obtained quite accurately without a CQmplicated equivalent-circuit fitting procedure.

0~0

"·

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

,: 500

R,,

100C

\

1500

2000

z' FIGURE 4.2. Estimation of circuit parameters from a Nyquist plot.

P-., p

37

Electrochemical Impedance and Noise

·15000..--------------.

d -_lObO

c -:: \ob 0:10 a-:

1. ~~·1

·10000.

z" ·100.----------------.

res

a;

.

-75

·5000- •

:

..

·50

=

.

·25

o~-~----~-~·~-~-~

o~~~~cu~~~~~~~~

10"

10·

2

1o·

1

10'~

to 1

10°

0

5000

Frequency (Hz)

..

(

f

• FIGURE 4.3. Effect of solution resistance on the Bode (modu~· Ius) plot: a, Rs:ot1; b, Rs=10; c, R8 :100; d, Rs=1000 S'l.

3

\iii

i

s-J

not affect Z", as shown in Figure 4.4. Extrapolation of log Z" vs. log w to w 1 gives log Z" -log C. ·

=

2

z'

=

to\10° 3 10"

Either the modulus plot or the Z' Bode plot can ~ be _used t~ estim\te the high- andJhe low-frequency "' \; resistances, Rs and ( Rs + Rp). respectively.

3

10"

1o'

1

----

10°

1

10

2

10

3

10

1~r-------------~ 4

One problem often encountered is the estimation of

-~-~ • j~ _jparameters from data that not reach a low-frecuency , v, o ~ ~limi~ and show onl~ a partial _semicircle. Thi~ happens if

•'1----

2

Frequency (Hz) ·

I.

i~

........ .... ······

10

~ ~

15000

........

4

10 10

·•

10000

z'

z"

RC 1s large, e.g., w1th a 1ve metal, or w1th a porous

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

10 3 10 2 1o 1 10 10°

o·' ~.u.ww~...................a..1.................L.....L.........w~............w.~~..""""""LWMI

1

10'

4

.z

10 . . . - - - - - - - - - - - - - - - - . 3 ----- --:- 10 2 10 101

---

10° 2

10"

10"

1

10°

10

1

2

10

I

1~~:tl ,,,J I ""'"' I .....~ I .....-~ I ,,,J· I I!JO~ 3

10'

2

10'

1

10°

10

1

2

10 Frequency (Hz)

3

10

10°

1

10

2

10

3

10

FIGURE 4,5. Impedance data for a highly capacitive system, mild steel in H2 S containing solution witb inhibitor.

conducting surface film that has a very large surface . area and doubl.e-layer capacitance, such as iron with a sulfide film . .Eor example, it is difficult to have confidence in a low-frequency limit determined by extrapolating the data in Figure 4.5.

3

Frequency (Hz)

10"

1o'

b

10

z:f.t~

2

a

10., ...,._~"-:'-..........................................._._.........._._..........._._.......

10-3

10'

Frequency (Hz)

d

c

3

4

10

FIGURE 4.4. Effect of solution resistance on the Bode (real. imaginary) plot.

lo

such casP.s the jttance plat cen be useful

~Figure 4.6). Even though the data are then same. their different features become apparent with different representations. In this last example. the low-frequency limjt could have also been estimated by fitting the data to an

-

Corrosion Testing Made Easy

38

0 . 2 0 f ·-- - - ·

0.15

y" 0.10 FIGURE 4.7. Equivalent circuit with parallel anodic and cathodic branches.

0.05

..

..

v

0.10

0.15

0.20 ~

y'

-o aL

Lf.

O.OOOr-----....:::..:-~-.-••- - , . . - - - -..-.... '

y"

0.02510.050

·... ....··· .. ... "'....

circuits give identical responses, but in others cases it is simply that the data can be equally well fitted to two different circuits. Eor this reason it is important that !he ysec choose a circuit in which the elements could have physical significance. In this way, elecfrochemtcal parameters can subsequently be extracted from the values ofthe circuit elements as further validation of the circuit model. The simplest physically realizable equivalent circuit · s a s lution resistance, a double-layer capacince, and a£aradic. imlieC!an~ tha! is due to the electrode processes (Figure 4.7). The Faradaic impedance is made up of a parallel combination of the ~mpedcinces of the allodic ..andcathodic reactions. ,'roconsiqer the respons~ that could be expected we carl Consider the FaradaK; impedappe asar.isjng f~om different controlling processes, and we need to introduce at this stage the · concept of diffusional impedances.

,ul

0.075 L...·3...................--L..L..U.W.L..l.........."'--'...LJ.1Jw.L..I...I..I.WIII..--L..L.J.WW 2 1 1 10 10. 10. 10° 10 102 103

Diffu$ional

Frequency (Hz) · FIGURE 4.6. ittance pfots for data in Figure 4.5.

,

~guivalent cjrc1.1U....a~sumiog .t bat oo QJher teatuca ap-, pear jn the SDfiClfum at trecl"encies be!ow UJfjt ·Qlf3a , $UC8d...ln this case, the use Of da:ta in impeda(lce or ittance form is equally acceptable. For systems whose response is more complex than those discussed so far, fitting of data to equivalent circuits is the normal approach to data analysis.

t"-

Equivalent-Circuit Analysis To emphasize an earlier statement, in the use of equivalent circuits to fit or model data, it is important to recog. nize that there are often many circuits that give equally good fits to data. In some cases this is because the

o~~pedances

Appendix B discusses these more formally, but here we simply need to recognize the origin of diffusional impedances and the types of imp~dance response they give. The term Warburg impedance is often used because the original calculation of the expected response was due to Warburg. Actually Warburg did this calculation for redox systems, with two reacting solution species, not for corrosion, and '(iarhttrg did not consid~ h · usion boundary-layer thickness. · tion assumed that the di usion boun aryl::lyP.r i.e. f ti n I' ~centratjons at species are different !rom t e bu1!5_ '-,..coAeeAtr:aticas, cbyld be of infinite thickness.Jn praQ:. tice con~cti.on in the.sohwon will mean that at some distance from the surface the concentrations of solution sQecies are con&lant, a Ad t"is aistaRs& is the 9o• •ndarC

~-

39

Electrochemical Impedance and Noise

A diffusional impedance arises when the surface concentration of an elect,:oc;:l:ler:J~ica"y active species changes during the ac cycle. We consider here the impedance of a cathodic reaction, such as the reduction of oxygen that is common in corrosion systems. The expressions for the' Faradaic impedance of the individual processes are described in Appendix B. The general shape of a finite-length Warburg impedance (i.e., allowing for constant concentrations at some distance from the surface} is shown in Figure 4.8. At high frequencies, the real and the imaginary parts are equal, and the impedance is given by

z = .jW u -

(4 )

. u 1 .jW

.3

where u, ohm cm 2 s- 112 , is the Warburg coefficient. -1000..-------------,

This diffusional impedance is additional to, and in series with, the charge-transfer resistance for the process. This charge-transfer resistance is given by (4.4)

where ic is the cathodic· current density, /Jc is the exponential Tafel coefficient, and be is the Tafel slope (in volts per decade). The diffusional impedance in Figure 4.8 has real and imaginary parts and is defined by two :

1. The frequency range over which it occurs is such that the frequency of maximum Z" is given by Eq. (4.5) in which D is the diffusion coefficient of the species involved and diffusion boundary layer:

~

is the thickness of the

.o

-750

fmax = 0.4 ~ 2 •

z" -500

. ..

2. The magnitudeof the impedance can be expressed

.. ..

500

(4.5)

'

in of various parameters, and diff~rent commercial software uses different parameters. The diffusional resistance Rd is the low-frequency limit of Z'. This can be related to the Warburg coefficient and the above by

1000

750

z'

(4.6) or to the cathodic current and the limiting current

by Rd=.

f3c

(4.7)

..

llim -lc

-so r---,.;;;;::::;::::;::::;::::;::::;::::;::::;:::.

-40~



:

... -

£- -2o l )..: -1~~.:, 3

10"

10"

2

I

. , .... 10.

1

I

,,,.,J.,,.,, . , , ; 10°

10 1

10~

•. !

10J

' .....

J

10"

Frequency (Hz) FIGURE 4.8. Finite-length Warburg impedance.

The Warburg coefficient can be related to physical, chemical, and electrochemical parameters by expressions that depend on the nature of the electrochemical processes. For the oxygen reduction reactiOI1 considered here,

cr=

fJ nFC5 J2B

.

(4 .8)

where C; is the surface concentration of oxygen.

-Corrosion Testing Made Easy

40

log i .

.. .

.

'·· E FIGURE 4.9. E-i curve tor cathodic reaction •

..

These expressions show that the magnitude of the diffusional impedance becomes larger as the current approaches the limiting current, i.e., as the surface concentration falls. We can consider the behavior at different potentials on the current-potential curve shown in F"tgure 4.9. At high potentials, region A, {3

Ret= · -~

i

To see the effect of the diffusional impedance on the overall impedance of the corroding metal, we need to in· corporate the parallel anodic branch and double-layer capacitance; and we can do this for the three cases considered above, to give the overall equivalent circuit in Figure 4.1 0. We assume activation control of the anodic process so that its impedance Za is simply

Rcr.a·

i « i1im. and so

hut: u

{3

Rd=-ium- i'

(4.9)

and little effect of the diffusional impedance is seen. At low potentials, region C, Rd » Ret and also Cs is small close to the limiting current. Thus u is high and the diffusional impedance can be much greater than the charge-transfer resistance, even at high frequencies.

\OcA /o~.

Activation (Charge-Transfer) Control of Both Anode and Cathode Reactions This corresponds to case A in the schematic E-log i diagram of Figure 4.11. The impedances of both the anodic and the cathodic reactions are simply resistances, and so the overall Faradaic impedance is also a resistor,

At intermediate potentials, region 8 , Ra and Ret are comparable, and the diffusional impedance is small at high frequencies but significant at low frequency. log

Cell

j

, Rct.a

l.t.c FIGURE 4.10. Equivalent circuit for corroding metal.

FIGURE .4.11 . Schematic E-log i diagram for different corrosion mechanisms.

41

Electrochemical Impedance and Norse

given by a parallel combination of the two resistors:

1 1 1 -- =- + -. Rct,tot

Rct,a

(4.10)

Rct,c

This is exactly equivalent to the common expression4or the polarization resistance, also defined for activation control, given in Appendix B. The resulting impedance spectrum is of the form shown in Figure 4.12. The Nyquist plot is a semicircle with diameter Rc1•101 • This raises one point that will be considered later in this chapter. the physically realizable equivalent circuit contains two resistors, but they can be combined

-12500

·10000

-7500

z" -5000 f-

. ... .

.

...... .

-2500

..

( 0

\

I

0

5000

2500

10000

7500

z'

12500

~:~I.. ~ . .~ I I

1~

1~

1~ 1~ 1~ 1~ Frequency (Hzl

-

-75

t-

1~

-

·25:...

0~~~~~~~~~~_.~4 2 1 3

10-J

10·

to·'

10°

10

,o=

This corresponds to case C in the schematic Elog i diagram in Figure 4.11. The anodic impedance can still be represented by a resistor. The cathodic impedance is a series combination of a charge-transfer resistance and a diffusional impedance. The magnitude of the diffusional impedance increases as the frequency decreases and also as the cathodic current becomes nearer to the limiting current. If the current is sufficiently close to the limiting current, this diffusional impedance becomes much greater than the anpdic charge-transfer resistance at all measured frequencies, and the overall Faradaic impedance is that due to the anodic branch. The resulting impedance spectra are as shown in Figure 4.13. As for the preceding case, the Nyquist plot is a simple semicircle, but in this case the diameter is Rci.a· It is important to recognize that a corroding system with the corrosion rate controlled by diffusion of the cathodic reactant to the surface will show no evidence of a diffusional impedance, unless there is one associated with · the anodic process. From the impedance spectra it is not possible to distinguish between this case and that for activation control of both processes. but calculation of corrosion rate from Rp requires the use of different coefficients in the two cases, as for de measurements.

1~

..· ../'.......·.

1:. •..;"' ~-

Activation (Charge-Transfer) Control of Anodic Reaction, Cathodic Reaction at the Limiting Current

'

·100-L----------~---,

!9

so that only one is required. Any analysis of data from a circuit such as this will give the value of only the combined resistor, and the second, in the equiv• alent circuit, is redundant. In more complex circuits the existence of redundant elements is not always so clear.

10

t0

Frequency (Hz\ FIGURE 4.12. Impedance spectrum for case A in Figure 4.11.

Activation (Charge- Transfer) Control of Anodic Reaction, Cathodic Reaction under Mixed Control This correspor~ds to case B in the schematic E-log i diagram in Figure 4.11. The anodic impedance can be represented by a resistor. The cathodic impedance is again a series combination of a charge-transfer resistance and a diffusional impedance. but now the magnitude of the diffusional impedance is low at high

42

Corrosion Testing Made Easy

-Soo

·1000r,- - - - - - - - - - - - - - ,

-400

-750-300r

.' .

z"

z"• -soo~

.. . . . . ... . : l ....

-200

..

·100

..

.: .

0 100

-250,.-

.;

•

I

I

I

I

I

200

300

400

500

z'

600

0' 0

l

250

500

~·'='

750

1000

z'

3

10

IZI 2

10 3 10"

2

10"

10.1

10°

1

10

2

10

3

10

4

10

Frequency (Hz) ·50

0 3 10"

-75r--------------,

..• .··"'·... ... . 10.2

10. 1

10°

1

10

.I".... 2

10

3

10

4

10

Frequency (Hz) FIGURE 4.13. Impedance spectrum for case C in Figure 4.11.

FIGURE 4.14. Impedance spectrum for case BIn Figure 4.11.

frequencies, but significant at low frequencies. The resulting impedance spectrum is of the form shown in Figure 4.14 and does show two features. As always for this type of circuit, the lower-frequency diffusional impedance is small compared with the higherfrequency feature.

impedance spectra, because in this case Rd can be large compared with Ret. unlike the case above in which the diffusional impedance was associated with the cathodic reaction. An anodic reaction can show a diffusional impedance if the surface concentration of metal ions is such that the reverse, metal ion to metal reduction, reaction is significant. An example of this has been shown in Figure 2.12 for the mercurous-mercurv svstem of a calomel electrode. In this case the charge-transfer resistance relates to the rate of the reversible metalmetal ion process, whereas the low-frequency limit of Z'( Rp) is what must be used to estimate the corrosion rate of the metal.

Diffusional Impedances of Anodic Processes These are less widely discussed than those of cathodic processes but can be significant features of

Electrochemical Impedance and Norse

43

Selection of Appropriate Equivalent Circuits This is an appropriate time to consider what could be done with data of the type in Figure 4.14 if it were obtained experimentally and to demonstrate that there Are often several equivalent.circuits that give equally good fits to a set of data. The data here were calculated with the circuit of Figure 4.10 with the parameter values in Table 4.1. These correspond, approximately, to a system with a cathodic limiting current density of 50 JJ-A cm- 2 , a corrosic;>n rate of 25 JJ.A cm- 2 , {3 8 = 25 mV, and f3c =50 mV. It is apparent that there are two features in the Nyquist plot, and so many circuits could be proposed to give such a response. Figure 4.'15 st:lows several circuits that will give two features {semicircles) in a Nyquist plot. Four of the circuits contain only resistors and capacitors whereas the fifth has a diffusional impedance (a sixth circuit is of course the one in Figure 4.10 used to create the data). Circuits A-D in Figure 4.15 all give identical responses, and the quality of data fitting to all of these will be the same. A and B are often used for coated metals, where one RC pair is due to the coating and the other to the corrosion process. The physical significance of circuits C and D is less clear. Here just the impedance spectrum corresponding to the best fit to circuit A, superimposed on the original data, is shown in Figure 4.16. The quality of the fit is excellent, and with real data subject to some noise this fit would be as good as is ever found. The fit to circuit E, WittJ only one Faradaic branch, is equally good (Figure 4.17). Table 4.2 shows the parameter values obtained from the fitting to circuits A and E. The average error, as

Table 4.1 Parameter Values for Figure 4.14 Parameter

Value

R$

100 Q 0.0001 F 1000 Q 2000Q 2000Q 10 s

Cat R.:l.a R.:l.c Rct

_,; . D

FIGURE 4.15. Possible equivalent circuits to give two semicircles.

provided by one commercia! program, is sufficiently low that _it would not be possible to distinguish these circuits on real data. Other commerc!al programs have different ways of expressing the quality of fit, but produce similar results. One other program used for fitting the data from Figure 4.14 gave parameter values that varied from these in Table 4.2 by less than 0.1 o1, in all cases. One word of caution is appropriate here. Leastsquares-fitting programs are most effective if the initial estimates of the parameters are close to the actual values. and it can happen that an appropriate circuit

Corrosion Testing Made Easy

44 -1000...--------------,

-1000..---------------,

-750

-750

z " -500

z" -5oo

·250

-250 'I

o~~-~~-~--~~~

0

500

250

1000

750

z•


1~

1~ 1~ 1~ 1~ Frequency (Hz)

1~

J 1~

-75 . . . - - - " ' - - - - - - - - - - - - - - - :

~ ~ ' . ~. i-: C-50

A

to" to·2 to· 1 to0

10

1

2

10

I 3

10

4

10

Frequency (Hz) FIGURE 4.16. Fit of data to equivalent circuit A; symbols are data, the curve Is the fit.

does not give a good fit because of the selection of initial parameters. For example, in the fit to circuit E above we can use, as initial estimates, the values from the fit in Table 4.2 except for the capacitance. We find that with initial estimates of C between 0.00001 and 0.0008 F, all fits converge on the results in Table 4.2. However, if we select an initial estimate of C of 0.0009 F, the fit does not converge. However. this is clearly sensitive to the actual software being used because another commercial·package convergerl on the correct C\!"'~wer eve:-1 if the initial estimate was 0.01 F. This example shows that there can be circuits with very different physical significance that give data suificiently similar that no particular circuit can neces-

~~----f-~.A~ . · .· .- . .- . ~· · . . _.] 1~

1~

1~

· ~~ · · 1~

1~

1~

1~

Frequency (Hz) FIG.URE 4.17. Fit of data to equivalent circuit E; symbols are data, the curve is the fit.

sarily ~ · selected . as the best We can distinguish among possible equivalent circuits perhaps by looking at the required values of the underlying physical and chemical parameters, e.g" surface concentrations or diffusion layer thickness, which may be unrealistic. Alternatively. the influence of solution hydrodynamics or of small applied polarizations may help distinguish possible mechanisms. This is restating the ,preceding comment, that equivalent circuit fitting is not an end in itself, but a step on the way to determining more fundamental parameters. Even taking circuit E. which attributes the low-frequency feature to a diffusional impedance. gives very different parameters for the diffusional impedance from the true values, e.g .. values of 126 ohm for Ra and

Electrochemical Impedance and Noise

45

·250

.Table 4.2 Circuit Parameters from the Fitting of the Data in Figure 4.14 to Different Equivalent Circuits in Figure 4.15 Parameter

Circuit-A ·

100.1 0.000101

Rs

c.,.; c,

Circuit E

Units

100 0.0001

2 F 2 2

R:t.. Ra.c R, R2 ~

z" ·100

·50

..

n

126.1 6.02 .. 0.25

0.6

s %

' ··

~

.

'-·~

~.

0

F

Rd fJ2 / 0 Average error

· 150

n n

673.4

·688.9 109.4 O.Q196

...

., ·200

0

50

100

z'

150

200

250

FIGURE 4.18. Impedance of parallel R·E clrcui~ fot,itterent values of tL n_. • C:

fJ

f rel="nofollow">JrD •

If n = 0, then the E is a r~~~=A. If n=1 __

6 s for s2 1D, come from the fit here and this can significantly affect interpretation. For example, we might want to use Rd to estimate the surface concentration of reactant, using the equations in Appendix B, and from this identify possible species responsible for the diffusional impedance. Using the fit to circuit E, we would estimate a surface oxygen concentration significantly higher than the bulk oxygen concentration in air-saturated solution and we would then be ter;npted, wrongly, to look for an alternative source of the difk.Jsional impedance.

~he E is a capacitor with

,i_lll_P~-~ance (at ~ith

.

The physical origins of depressed semicircles are still not clear, and so the meaning the value ~n,~i~--.......... uncertain, and usually if the value of n is o.~ or more

o!

jthe element can be view_ed as a capacitor and the calpacitance calculated_f..:_om ~0. . _Figure 4.18 shows the

impedance response of a parallel R-E circuit for different values n. A practical example, corresponding to an n value of 0.9, is shown in Figure 4.19. ~ c~o "'~ ~ 1 q-c,---c. s•,. co..,.7 • c 1 r (, ~"'- ~ c~ v ..... t:.a...t- ct I ct·4

of

..-..JA

Fitting Data with Depressed Semicircles

·75.-------------,

Constant-phase Elements (Es) are widely used in data fitting to allow for depressed semicircles.._The capacitor is replaced with a E for better fit quality. The E is, in fact. a general expression for many circuit elements. Formally, the impedance of a E is given by

·50

l

0~----------~

or

25 0

Pur~ c.)c.tcr"n.l~r ~l (J.t -...

l (t't· Rl 'h R.

· j Zo<,,- n Sin

tr\

(

n:-r ) ·. 2

(4 .12)

.. .

..

Z" ·25

(4.11)

= Z0 (o - n cos· ( 2n:-r ) -

1I Zo_:...:The War burg

n=1/2.

,,

Z

C=

high frequency) -is also a special case,

FIGURE

51~-k"W..L':> dri'""J

~ . 19 .

I 25

50

z·

75

100

Practical impedance of R· E circuit, n=0.9.

1PJ1'.J

Corrosion Testing Made Easy

46

• Other Types of Impedance Response There are a variety of processes that can give rise to an equivalent circuit like that in Figure 4.158, but for which the values of R2 and ~ may be either positive or negative. Figure 4.20 shows three Nyquist plots corresponding to R2 and ~ both positive (curve A), R2 and ~ both negative (curve "B) and R2 negative and ~ positive (curve C). In these cases R1 can be attributed to the charge-transfer resistance and the second semicircle arises from a change in the surface that can occur at low frequency, i.e., if sufficient time is·anowed. · Case A in Figure 4.20 could be due to the adsorption of a species that increases with increasing potential -~.-------.-----~-------.

·250

/

c /

I

A

•

and in which the adsorbed species reduces the corrosion rate. At high frequencies the adsorption/desorption process is not fast enough to have any effect, and the surface covecage does not change, whereas at very low frequencies the increased coverage on anodic polarization decreases the measured current reflected by the increased resistance. The capacitance here is of' ten termed a pseudocapacitance as it is not physically a capacitor, but a reflection of the kinetics of the adsorption process. The term adsorption is used loosely here, a~ the surface layer responsible for corrosion rate reduction could equally well be a surface film of oxide, for example. Case B in Figure 4.20 is similar to A except that it would be attributed to adsorption that decreased with rising potential or increased with potential but in which the intermediate increases the corrosion rate. This case could also be modeled by a positive resistance in parallel with a positive inductance, hence the general term inductive behavior, but there does not seem any benefit in introducing pseudoinductances into equivalent circuits, and the presence of a different type of circuit element . can only cause confusion. It is simpler to use negative resistance and negative capacitance to generate these inductive features.

', B

Case C in Figure 4.20 appears very different, as \he Nyquist plot changes direction and the low:.frequency limit is a negative resistance. In fact it is simply an extension of case A, in which the effect of adsorption or film growth is such that the de current decreases with increasing potentiat This type of behavior can be seen on the activ~ive .1ransition for a ive metal, where the current decreases with increasing potential. This can be seen from the ittance plots Figure 4.21 for A and C, in which in both cases the ittance continually decreases with decreasing frequency with no sudden change of direction.

z'

-~~:~~-c'-. ·~

z.

1o

1

·-.. ,..-

100

10·

1

...

A

· ·~~ 3 2 1 2 3

10·

.10·

1o·'

10"

10

10

10

·o'

Freouencv 1Hz\ FIGURE 4.20. Possible low-frequency eHects that are due to adsorption.

This is one of the few real cases in which the distinction between impedance and ittance is significant. With potential control. in which it is actually the ittance that is me~c;ured. both the a~t ive-p~s~i"e tr?,s!tion (in de measurements) and this type of impedance spectrum car. be obtained. With current control, neither can be measured. This is discussed in more detail in Appendix A.

Electrochemical Impedance and Noi~e

47

-3000.-------------

0.10

_-(-

0.05

y"

-

-20001-

z"

\

-1000

0.00

v~

'··

,··

.· ........ ..:,

·~

0~-~-~-~-~~~~

-0.05 -0.05

0.00

y'

0.10

0.05

0

1000

z'

2000

3000

v·:~l=·~c . . . :.::..J 1~

1~

1~ 1i 1~ 1~ Frequency (Hz)

1~

1~

FIGURE 4.21. ittance plots corresponding to Figure 4.20.

These low-frequency features have an origin similar to that of diffusional impedances in that they arise because the surface conditions change as a result of the applied ac signal. With diffusional impedances, it is the concentration of the reactant, whereas here it could be viewed as being the surface area that changes. Both affect the rates of the processes occurring and hence the measured resistance of the system. An example of such a low-frequency response is :>hOwn in Figure 4.22 for a mild steel rotating cylinder in C02"saturated solution. The low-frequency response cannot be attributed to a diffusional impedance in this case because it occurs at too low a frequency, compared with that expected from Eq. (4.5).

FIGURE 4.22. Impedance response of steel in C0 2 -containing solution.

Estimation of Corrosion Rate from EIS Data The principles of this are the same as those for polarization resistance measurements, with the improvement that impedance allows correction for the solution resistance: The corrosion rate is related to the polarization resistance by

B

icorr

= R'p

(4.13).

where B is the Stern-Geary coefficient (in volts); see Appendix B.

Corrosion Testing Made Easy

48

There are some assumptions in the use of this equation, which mostly apply equally to de and ac measurements. The major one is that the resistance measured relates to the corrosion process. There are several aqueous solutions of practicai interest, e.g., chemical cleaning solutions, that contain solution redox couples, and the polarization resistance measured relates to the kinetics of this redox process, not to the corrosion rate. The appropriate value of 8 depends on the corrosion mechanism, but would generally be in the range of 13-

·1500 a

-1000

z"

+--Rp

·500

..··•

.:

...

~ct

..·· ....... ··..

SOmV.

500

Let us assume we have only one resistance extractable from the data, which wr; shall call Rp here, as for example in Figure 4.19. We have· to decide what the appropriate value of the Stern-Geary coefficient is, and this depends on a mechanistic understanding that is not available from one impedance spectrum.

. lcorr

1 = f3af3af3c -. + f3c Rp

(4.14)

For a_corrosion process in .which the cathodic reaction is mass transfer controlled, . lcorr

f3a

=

R' p

(4.15)

··.. 1000

z'

150(

·10000 b

7500

z" -sooo

For activation control of both processes,

...

.. ........ .. ..·.·

-2500

2500

5000

7500

. 10001

z'

-~~--------------------~ c

or if the anodic rate does not vary with ·potential, e.g., for a ive metal, . lcorr

f3c

= R' p

(4.16)

The potential confusion with EIS data is that there may be more than one resistance that can be chosen. For example. with spectra of the type in Figure 4.23 we can obtain the diameter of the high-frequency semicircle Ret C111J the total resistanct: np.

FIGURE 4.:.!3. Schematic Nyquist impecian::e fJiOis.

In most of the cases considered so far the diameter of the high-frequency semicircle (the charge-transfer resistance) is only slightly less than the low-frequency limit (the polarization resistance). so t11e errors are small whichever resistance is taken. Given the un-

certainty in the precise value of the Stern-Geary coefficient the error is not often significant. However. in many cases the resistances are significantly different and a choice must be made.

500

1000

z·

1500

2000

2500

49

Electrochemical Impedance and Noise

• Use of Charge-Transfer Resistance for Corrosion Rate Estimation In principle, for data such as those in Figure 4.14 or Figure 4.23a, we can use the e-transfer resistance with the Tafel coefficients;---. lcorr

=

- .f$~/1c

!Ja + /1

-

1

Figure 4.20 or Figure 4.23c, we can again use Ret and Eq. (4.17) to obtain the corrosion rate. However, in practice 1 it may not be easy to determine whether a spectrum fits in this category, with a capacitive semicircle, or whether it contains a diffusion& impedar.ca, ir• which case Ret should not be used.

(4.17) t

An advantage of this is that the Tafel coefficients (/1) do not vary very widely, and we can generally choose a value of the Stern-Geary coefficient within a factor of 2 •· of the correct vaiue. However, if the spectrum contains a large diffusional impedance, such as in Figure 3.14, then the use of Ret and Eq. (4.17) will not give th~ correct corrosion rate, but will give a rate related to the reversible redox process at the surface. In Figure 3.14, the chargetransfer resistance presumably relates to the mercurous/mercury redox process. If we consider a spectrum in which there is a second capacitive or inductive feature, as for example in

Use of Polarization Resistance for Corrosion Rate Estimation In all cases we can use the polarization resistance if we know the actual slopes of the potential, In i curves at Ecorr. for the individual processes. In this case we replace Ret with Rp in Eq. (4.17) and replace the {3's with the actual slopes. However, we do not usually know these actual slopes, and they can be difficult, if not imposs_ible, to determine. In spite of this the use of the polarization resistance for corrosion rate estimation is unlikely to result in large errors even with an estimated B value, and the use of Rp rather than Rc;1. is safer unless the corrosion mechanism is well understood.

•

CHAPTER 5

·Description of ·Noise-Signals·· I

Electrochemical Noise Terminology t

..

As a relatively young subject and one that crosses the boundaries of several subject areas, electrochemical noise has suffered from a variety of names being used for the same phenomenon and from the same names being used for different phenomena. There are currently efforts within NACE and ASTM to define a consistent terminology for electrochemical noise. The following are used in this book: Electrochemical Potential Noise. This is the fluctuation in potential of an ·electrode relative to a reference electrode. As the second half of this book is exclusively about electrochemical noise, we often omit the word electrochemical and refer simply to potential noise. It remains to be seen what will be permitted in the official designations. The acronym EPN may be used but we have avoided it in this book as we feel that it makes for harder reading. Electrochemical Current Noise. This is the fluctuation in the current to one electrode or between two electrodes. As with potential noise we tend to use the term current noise. The acronym ECN may be used. Electr6chemical Noise Resistance. This is a resistance that is comparable to a linear polarization resistance. but estimated from eiectrochemical noise parameters. The term electrochemical resistance noise is often used, but this has incorrect connotations and is not recommended.

•

but estimated from electrochemical noise parameters. The spectral noise resistance and spectral noise impedance have also been used, but we believe that they are less appropriate.

Potential- and Current-Time Records Several symbols are used to represent noise currents and potentials. We use a mathematically based approach that has the advantage of providing an ~~urate description of the meaning of each symbol w1th1n the symbol itself, although it does result in rather complex symbols. · The fundamental parameter to be measured is the time-varying potential or current, given the symbols E(t) and /(t), respectively. This terminology emphasizes that the potential or the current is being treated in the time domain. The same data may be examined in the frequency domain (see below), in which case the symbols E(f) and /(f) are used. In many cases it is either unimportant or self-evident whether a timeor a frequency-domain representation is being used, in which case the symbols E and I may be used. As with preceding chapters, current and current density are distinguished by I for current and i for current density, although current density is rarely used as it is not appropriate for electrochemical noise studies.

Potential and Current Noise Electrochemical Noise Impedance. This is an impedance (strictly an impedance modulus) that is comparable with a conventional electrochemical impedance.

The raw potential- or current-time record may be considered as consisting of two parts:

52

Corrosion Testing Made Easy

• a de value • a time-varying noise potential with a mean value of zero The de value is simply the mean value of the time record, E(t) or l(t), where the overbar has the conventional statistical meaning ofi'Dean value ot We can omit the (t), indicating the mean over time, as the mean over frequency has no real application; hence we write E and l. The noise value, denoted by En(t) or ln(t), is then the time record minus the de value:

En(t)

= E(t)- E,

ln(t) = l(t)-

i.

(5.1) (5.2)

Note that En(f) is essentially the same as E(f), as the de value will simply be E( f) for f = 0, and all other components will be the same for both parameters.

Average Amplitude of Noise Signals In analyzing noise circuits it is often convenient to use a single number to describe the amplitude of a noise signal. We see below that there is no completely general way of doing this, but for the time being we describe the average amplitude of a noise signal by the variance of that signal. In simple , the variance is the average value of the square of the noise signal. Practically it is a measure of the ac power present in the signal. See Appendix E for a formal· definition of variance as it relates to a time record. As with the mean~e take the time aspect as implied and write E~ and I~ for the variance. Note that the variance has units ofV 2 or A2 ; because the power dissipated in a resistor is equal to V2 / R or 12 R, it is common to refer to E~ as the potential noise power and to 1.~ as the .current noise power. It should be appreciated. however, that these do not strictly have units of power. It is also important to appreciate that the variance will depend on the bandwidth of the data acquisition system. It is somewhat unnatural to work with units of v2. and A2 , and it is common to report noise amplitudes as the positive square root of the variance or the standard devi-

,_

-

ation. We write , · E~ and \ 11~ for the standard deviation of potential and current. These have units of volts and amps. respectively.

It is useful to be able to estimate the standard deviation for a time record. For a Gaussian signal (one for which the distribution of values has a normal distrioution) the. peak-to-peak amplitude will usually correspond to approximately six standard deviations. so we can estimate the standard deviation by dividing the ~ak-to-peak amplitude by six. Note, however, that many electrochemical noise signals are not Gaussian, and this estimate may not be very accurate, especially when there are a few large peaks in the signal.

Effect of Specimen Area on Measurements Readers who are familiar with de electrochemical measurements will be used to the idea that electrochemical potentials are largely independent of specimen area, whereas electrochemical currents are proportional to specimen area. Hence we talk about corrosion potentials without worrying about the area, and we talk about corrosion current densiti~;~s {current. per unit area) to compensate for the effect of area on the measured current. Electrochemical noise is not like this! The exact effect of area on electrochemical noise will depend on the process that is generating it, but as a general rule we should expect current noise power 'to be proportional to the specimen area, and we should expect potential noise power to be inversely proportional to specimen area. In simple , we can see what will happen by considering connecting two equivalent electrodes together, such that the area i~ doubled. The current noise powers add(see Appendix E for a detailed explanation), so the standard deviation of the current is ./2 times as much as that for a single electrode. Note that we are assuming that each of the electrodes involved in the measurement of the current noise has had its area doubled. Consequently, if we are measuring current noise as the current between two electrodes, then each of these will have been doubled in size, and will therefore contribute a current that is "2 times as much as the oric;inal electrode size. The behavior of the potential noise is slightly less obvious, but if we assume that En results from the application of 1, to the polarization resistance Rp. we find that when we double the area. E, reduces by a factor of /2.

53

Electrochemical Impedance and Noise •

Continuous-Time and Discrete-Time Signals The potentials and the currents from the electrochemical cell exist at all times from the beginning to the end of the test, and we can describe them as continuous- • time signals. If we look at S.!JCh a signal from the point of view of the frequencies contained in it, it may, in prin- ciple, contain infinitely high frequenCies. It is not possible to retain all of the information in this signal when recording it. Analog recording methods are rather limited in of t!'le subsequent processing that can ·be performed, and it is therefore usual to sample the continuous-time signal to obtain a discrete-time signal or time record consisting of a number of values that are equally spaced in time. This discrete-time signal can then be recorded on a computer or similar device. The sampling of the signal creates significant limitations on what can be recorded, particularly in of the frequency content of the time record. Appendix E includes an introduction to the theory of discrete-time signals, but a few results are important for an understanding of data collection and analysis. The sampling process creates some limits in what can be represented by the time record. The most important of these limits relates to the frequencies that can be

represented. In simple , for a particular frequency to be properly represented. there must be at least two samples within each cycle of the sine wave. This leads td an upper .frequency limit know as the Nyquist limit or the Nyquist frequency. The Nyquist frequency is thus half the sampling frequency or 1/2t:.t.

•

Unfortunately, signals above the Nyquist .frequency do not simply disappear when they are sampled; but_ reappear as false low-frequency signals or aliases. Aliasi.ng can give rise to serious measurement errors, esp'ecially when high-amplitude power line signals are aliased into the measurement range. For this reason, frequencies above the Nyquist frequency should always be filtered out of the signal before it is sampled, using antialiasing filters. The finite duration of the time record also limits the lowest frequency that can be recorded. To represent a signal reliably it is necessary to include one full cycle of the signal within the time record. It is not quite as obvious, but the limited duration of the time record also leadsto a limit to the resolution with which neighboring frequencies can be resolved. Both of these features are described by the frequency resolution of the measurement, which is equal to 1/(time record duration) or 1/(Nt:.t), where N is the number of samples in the time record.

CHAPTER 6

Measurement of . ··Electrochemical Noise~:, Potential Noise Measurement I

proach, but one that has been little used, is to make both measurements.

Essentially there are two methods of measuring potential noise:

Current Noise Measurement • The potential of one Working electrode is measured relative to a low-noise reference electrode. • The potential between two equivalent working electrodes is measured. In this case the two electrodes will typically be producing equal amounts of noise, so the noise powers add, and the standard deviation of the noise will be ./2 times that of a single electrode. • The method used can have an important impact on the measurement of the noise potential. When a reference electrode is used, the de potential difference is usually relatively large, in the range 100 mV-1 V or more. The measurement of small fluctuations on this large voltage is relatively difficult. On the other hand, the use of two identical electrodes will tend to lose any initial transient change in potential that is due, for example, to corrosion product development, because the two identical electrodes will tend to suffer the same shift in potential. It is an interesting, but largely unanswered, question as to whether or not this initial transient should be incl~ded in the true electrochemical noise. A logical ap• Note that on page 52 the potential noise was shown to be reduced when we increased the electrode afea by (conceptually) coupling two e!ectrodP.S I<'<;P.thP.r ,.,. .... ,P~" here thP. noise is increasPct The difference betwee11 the two .:ases results from tha way in which the two electrodes are connected:- when potential noise is measured between the two electrodes. the two electrodes are effectively con· nected in series. and the ~.:-tentials add: when the area of a single measuring electrode is in.:-reased. the old and the new areas of the electrode are effectivelv in parallel. and the measured J.:'Otential noise is reduced.

As with potential noise, there are two ways of measuring current noise: • The current flowing to a potentiostatically controlled working electrode can be recorded. This has some advantages in interpretation, ~specially when individual transients are being analyzed. Because the electrode potential is constant, little or no current is involved in charging or recharging double-layer capacitances (some such current may be observed because of local potential changes caused by high current densities at pits and similar features). On the other hand, the use of a controlled potential means that the potential noise cannot be measured on the same electrode. Control at a fixed potential also causes problems with the drift of the corrosion properties, which will show up as a changing mean current. One way around this problem is to use a large-area electrode of the same metal in place of the reference electrode. The large electrode area will usually give a lower potential noise amplitude (see page 52), but should have the same mean potential as the smaller test electrode. i • The current flowing between two working electrodes can be measured. This has the advantage in that the potential of the electrodes will find its own level. (We Note that this may not be valid in some situations. For example, stainless steel may suffer from stable pitting for the large electrode. but not for the small electrode. in which case the two corrosion J,:'Otentials may be quite different.

r

..

Corrosion Testing Made Easy

assume that the current measuring method used is near ideal, with zero input impP.dance, and the two working electrodes are ~erefore at the same potential). This also means that the potential noise of the coupled pair of working electrodes may be measured at the same time as the currE!nt noise. Disadvantages of the method·are that the changes in potential that are eaused by the noise current lead to some of the current's being consumed in charging and discharging the double-layer capacitance of the electrodes, and that it is not possible to identify which of the two el~ctrOdes is responsible for the generation Of transient events'··(although it is often possible to deduce this from the polarity of the event). The two electrodes are usually nominally identical, although there may be advantages for some applications in using nonidentical electrodes.

resistance R,. Similarly, if we divide the square root of the potential power spectral density (PSO) (see Appendix D) at each frequency by the square root vf the corresponding current PSO, we obtain the electrochemical noise impedance Zn . At their simplest. noise • resistance and impedance measurements simply require t~e measurement of the current and potential noise at more or less the same time. However, the measurement can be expected to have a greater validity if the current and the potential noise measured relate to the same specimens, especially with relatively unreproducible borrosion processes such as pitting corrosion. For this reason, .noise resistance measurements typically use a coupled pair of working electrodes to measure the current noise, with the potential of this coupled pair being measured against a quiet reference electrode (or at least a reference electrode that is assumed to be quiet). Some workers use a third electrode of the same material as the current measuring electrodes. However, it should be recognized t.hat the potential noise in this case will be dominated by the noise on the pOtential measuring electrode, unless its area is at least equal to, ano preferably . {TIUch greater. tban, Jhe COmbined area of the coupled working electrodes (as the potential noise is expected to be inversely proportional to specimen area). There are also possible problems if the reference electrode suffers from stable pitting corrosion (especially if it has a larger area than the working electrodes).

56

Both of these methods have their pJace. The first is best when it is intended to study individual current transients, such as in studies of the initiation of pitting corrosion, whereas the second is best (indeed only the second can be used) when simultaneous recording of current and potential noise is required. A recent modification of the second method 1 uses two identical electrodes that have their potential separated by a small amount. This has a number of interesting effects:

• The more anodically polarized electrode is subject to more severe conditions than free corrosion and may therefore provide an early warning of localized corrosion problems. • The polarization will tend to force pitting events to occur preferentially on the anodic electrode, and this may assist in interpretation. • The polarization may allow a larger proportion of the current associated with pitting events to be detected (this is not an inevitable result of the technique and depends on the characteristics of the electrodes) .

Noise Resistance and Noise Impedance Measurement If we measure both potential and current noise, we can divide the standard deviation of potential by the standard deviation of current to obtain a value with units of resistance, known as the electrochemical noise

Cell Design For general advice on cell design and specimen preparation, the reader is referred to the earlier book in this series on de techniques.2

Electrode Preparation The requirements for the working electrode(s) are similar to those for impedance measurements, with the avoidance of crevices being particularly important with systems that are susceptible to crevice or pitting corrosion. As noise measurements are generally concerned with small-amplitude signals and low frequencies, the electrical requirements are easily met (at least in of the electrical properties of the electrodes), although shielding to minimize the pickup of extraneous noise is important .

Electrochemical Impedance and Noise

The area of the specimen is particularly important, and not just for the normal reason that the measured current increases as the area increases. Theoretical analyses predict that the current noise power is proportional to the specimen area. whereas the potential noise power is inversely· proportional to the area~ These factors, together with the ooise properties of the measurement system; need to be taken into when deciding on the specimen size. In cases where both the current and the potential noise powers are low, it may be worth considering the use of small electrodes for the measurement of potential noise and ··much larger electrodes for the measurement of current noise. This will allow the amplitude of the currents and potentials to be maximized, although · it will also lose the direct link between the potential and current measurements. A second important factor is the relationship between the area and the physical process occurring. This is perhaps seen most clearly in the case of pitting corrosion in a system such .as stainless steel in chloride solution. Various causes of pit initiation have been studied, but it is clear that an important factor is the presence of sulfide inclusions in the exposed surface. Consequently the pitting behavior of a sample will depend on the most damaging sulfide inclusion (in simple , the largest inclusion, although the inclusion chemistry will also be important) that is exposed. As the sample gets larger, so a larger number of sulfide inclusions will be exposed. and the size of the largest inclusion will tend to increase. Thus pitting becomes more likely as the sample size increases. This is a particular challenge with respect to the use of electrochemical noise for corrosion monitoring in service, as the size of the structure is inevitably larger than th~ si+e of th,e monitoring probe, so it is probable that pits will occur in the structure when they are not observed on the probe. This is a rather subtle problem; as there are many material and environmental factors that influence pit initiation and propagation, the responses to changes in these factors will be ra.ther complex. APproaches that may be considered for the preparation of test probes that will give an early warninQ of the oossibilitv of pitting corrosion include · • the use of less corrosion-resistant materials for the probes • the use of an applied anodic polarization of one or more of the probe electrodes (see page 56)

57

• the application of more susceptible surface finishes to the probes

A further complication is the availability of cathode area to drive the pitting process. If the specimen area is too small, it may not be sufficient to provide enough cathopic current to allow the pit to reach the critical current _ for stable pitting. Again, this is a rather subtle questien; and one that has received little attention. ~')Other important factor in the preparation of the electrodes is the surface finish. In general, localized corrosion processes are facilitated by rough, dirty surfaces, as the roughness and the dirt deposits help to the local changes in chemical conditions that are responsible for the localization of the corrosion. Specific factors that should be considered are

• MechanicaVchemical finish (polished, electropolished, ground, pickled, ivated etc.). Polished or finely ground stainless-steel samples will typically have pitting potentials that are considerably more positive than coarsely ground samples. The influence of mechanical finish on metastable pitting is slightly less well defined, but polishing .will typically remove sites for pit nucleation and thereby reduce the number of metastable pitting transients. This is certainly true for electropolishing and other chemical treatments, for which it appears that the electropolishing process removes active sulfide inclusion sites. Thus one approach to making probes particularly sensitive to metastable pitting is to use a coarsely ground surface. • ·s urface oxidation (natural air-formed. oxide, highertemperature oxides). This is particularly important for samples or equipment containing welds, for which the heat tint around the weld can reduce the critical pitting temperature by 1ooor more. This again leads to a possible method of enhancing the sensitivity of probes to metastable pitting events by oxidizing the probe. • Surfac;e deposits. Simple dirt deposits act to provide a diffusion barrier similar to a crevice, and this helps metastable pits to become larger before reivating or to achieve stable growth. Rust stains (e.g .• from the corrosion of nearby steel) behave in a similar w'Piy, htrt rnay be more active as & consequence 0f the Fe2+ /Fe 3..,. redox process. Biologically active deposits may be more aggressive for a number of reasons, including the formation of H 2 S by sulfate-reducing bacteria and the formation of H2 0 2 by other species (the latter acts as an alternative cathodic reactant and thereby raises the corrosion potential).

Corrosion Testing Made Easy

58

C'\ ~

Ideal (and impossible) - sample suspended in test solution with no

interfaces

Sample posses lhroCJgh surface - gives waterhne ottocl< due to enhancedoxygen lronsport

'· •

Sample mounted in

Table 6.1 Noise Pr9duced by Typical Reference · Electrodes ' PSD/10 8 V 21Hz 0.01 0.1

Reference Electrode

0.001

Hz

Hz

Hz

Ag/AgCI SCE Cu/CuS.Oc

1000 800 30

10 6 0.2

0.5 2 0.008

Note that the actua~ value lor a particular electrode will depend on its construction, and this table should be taken as indicative only.

insulator - gives possibily ot crevice effects where mount meets sdmple...

FIGURE 6.1. Crevice corrosion and waterline problems.

• Crevice and waterline effects. The ideal corrosion sample is a hom~enous sphere exposed to the solution without any other . In practice, this is .essentially impossible, and we must either have a t between · the ·sample and an insulator in the. solution, or we must. · the sampl~ th.rqugh the surface of the solut.lon. These · both lead ·to possible problems (Figure 6.1). A range of solutions to these·problems have been devised, see the book in this series on de techniques for further information .~

Reference Electrodes When a reference ele.ctrode is being used . in noise measurements, it is obvjously important that it contribute as little as possible to the .noise being measured. Whether or .not this is possible, the noise level associated with the reference .electrodes should be determined. For potential noise measurements this is done most simply by measuring the noise between two or more reference electrodes of the same type. Som~ electrode types, inCluding the saturated calomel electrode (SCE) and the Ag/AgCI electrode. are liable to become more noisy if they are maltreated . If this is suspected, comparisons between several electrodes can be performed to identify the good and bad electrodes. Table 6.1 presents the noise levels measured for several electrode types in our laboratory.

These values should 'be taken only as illustrative· the noise level will vary according to the constructi~n of the ele~trode , including ·the des!gn of the salt bridge connecting the electrode to the cell, and the way in which the electrode has been treated bOth before and during use.• You should be aware that reference electrodes can be light sensitive and may generate sig~ific~nt noise at twice the power-line frequency (pnmanly because of fluorescent lighting) and at lower frequencies (e.g., because of clouds ing in front of the sun). The AgiAgCI electrode is well known to be light sensitive and for this reason is probably best avoided for electrochemical noise measurements. When reference electrodes are used to measure electrochemical noise currents with the potentiostatic technique, it is less obvlous how they affect the measurement. In simple the reference electrode noise will be applied across the cell by the action of the potentiastat. This will cause a current to flow through the cell. Consequently the reference electrode potential noise power will generate a current noise power given by

E~retf Zi. where E~ret is the potential noise power of the reference electrode and Zs is the impedance of the cell generating the noise current As Zs is probably not known in this situatipn. it can be difficult to determine what the effect of the reference electrode noise will be.

• The figu ~es given are higher than have been previously reported (Gabnelh, HLJet and Kecjam. Electrochimica Acta. 31. pp. 10251039(198611. The differe.,ce may be associated with the lower frequency used for the re;; ~:lts given here. or the use of a polymer gell-lilled sal: oridge.

Electrochemical Impedance and

Noi~e

What Is the Best Sampling Frequency? The optimum sampling frequency for electrochemical noise depends on the processes that are generating the noise and the analysis that is to be used. Several analysis criteria use the low-frequency limit of the noise PSD, in which case there is no need to sample particularly fast. However, a very.slow sampling rate will require a correspondingly low-frequency antialiasing filter (this is essentially a low- filter to, prevent frequencies that are higher than can be recorded from corrupting the measurements) that may be difficult to obtain or produce, and it"may be easier to sample faster to simplify the filtering requirements. If necessary, subsequent digital filtering may be used be to reduce the amount of data that needs to be stored.

,

There are two schools of thought on the maximum sampling frequency .that is needed. Probably the majority of workers use a sampling frequency of -1 Hz, which is easy to achieve with digital 'JIOttmeters. It also provides a good clearance from power-line frequency, allowing relatively easy power-line frequency rejection. Another group (largely based in Europe) considers it important to sample at a higher frequency (10:-30 Hz). This is much more challenging. It allows little clearance for filtering to remove power-line frequency interference, anQ the short time ava!lable for each sample makes it more difficult to achieve high resolution and accuracy. Electrochemical noise PSD typically falls with increasing frequency, whereas instrument noise is typically close to white at higher frequencies (i.e., the PSD is constant). Cons"3quently it becomes more and more difficult to achieve an instrument noise level below the electrochemical noise level as the-sampling frequency is increased. Thus the fundamental upper limit to the sampling frequency is determined by the instrument noise level that can be achieved. In general , for corrosion research it is desirable to obtain all possible information about a corrosion system being studied. This implies that the sampling frequency used should be as fast as possible in order to gain the maximum possible high-frequency information. To achieve this, it is important to know thP. capabilities of your measuring system and to match these to the observed properties of the electrochemical 'noise in order to optimize the sampling frequency. For corrosion monitoring with electrochemical noise. the required information will generally be known in ad-

59

vance of the measurement, and the data that are output from the electrochemical noise system should be limited to avoid overloading the information analysis sys• tern. However, this should not prevent the use of more rapid sampling than is currently common, with subsequent digital and computational reduction of the data, if t~is gives the most useful information.

How Long Should a Time Record Be? with modern computer systems there is no inherent limit on the duration of a time record, at least for laboratory studies. With multitasking operating systems, data can be written to removable storage media while collection is continuing, and high-capacity fixed and removable storage is available cheaply. For convenience the time record is best stored as reasonably short segments, with 1024 to 4096 samples being common (a power of 2 is convenient if subsequent fast Fourier transform (FFT) processing is anticipated). Sequential 1024-point time records have been recorded for a total of -600,000 points, and these allow interestirig analyses to be performed, such as the estimation of electrochemical no,ise impedance.down to~ x 1Q~~ Hz. 3 However, it is very doubtfUl whether many corrosion systems are stationary• over such a .long period. On the other hand, many analysis are concerned with the low-frequency limit of the noise PSD, and the time record should be long enough to ensure that a reliable low-frequency limit has been ~chieved if one exists. It is therefore recommended that continuoUs- (or, perhaps more accurately, contiguous-) time records should always be taken, as this allows the analysis to be extended to lower frequencies if necessary.

methods

For corrosion monitoring applications, the measurement will inherently continue for long periods. However, in this situation the quantity of data produced by continuous collection of time records is liable to present more of a problem, and it will usually be necessary to reduce the quantity of data before storage. This does not necessarily limit the ability to obtain low-frequency information (simply collecting the mean potential and current every hour wi!l allow t~e low-frequency power spectrum to be estimated, although there will be some aliasing of higher frequencies). As with the question of • We use the term 'stationary' here in its statistical sense. which essentially means that the statistical properties such as the mean and standard deviation are not changing over time.

-

Corrosion Testing Made Easy

60

~-

.

'·•

FIGURE 6.2. Block diagram of data acquisition system. DAC, digital-to-analog converter.

sampling frequency, the optimum corrosion monitoring data reduction and storage procedure will depend on the application and the expected noise response.

uses direct acquisition by a computer system, although · stand-'alone digital recorders can be used. A variety of methods are used to transmit the digital data to the computer for storage and analysis.

Signal Processing and Recording

Voltage Amplifier

In principle electrochemical noise measurements can be made with analog techniques (e.g. using a tape recorder to record the time .record, with analog computing· methods to perform the analysis). However, this limits the accuracy with which the time record can be recorded and the fleXibility of subsequent data analysis. Consequently, almost all modern electrochemical noise measurements use digital techniques.·A block diagram of a suitable data acquisition system is shown in Figure 6.2.

It is quite common to feed the potentiai noise signal directly into a digital voltmeter. This does have some advantages, notably the avoidance of possible additional sources of instrument noise. However, this relies on the voltmeter to remove frequencies above the Nyquist frequency, • and there will ustially be a problem of aliasing of frequencies between the Nyquist frequency and the frequencies that will be removed by the meter. Consequently an amplifier should be used to condition the potential before antialiasing filtering. Often this is a buffer amplifier with a gain of 1, but ·there are advantages in the use of an amplifier with gain, as this reduces the contribution of subsequent stages to the instrument noise. One of the problems' with the measurement of electrochemical potential noise is the large de leve! that must be aecommodated if a reference electrode is used to measur-e the potential of the coupled working electrode pair. This limits the gain that can be used for the potential amplifier to 10 or less. In principle, it is possible to use a stc.:.:.:a. low-notse vcltage offset to bring tth:: mean voltage close to zero so that larger gains can be

The current between the coupled working electrode pair is converted to a voltage by the current amplifier [also known as a zero-resistance ammeter (ZRA)]. The voltage .between the coupled working electrode pair and a reference electrode is conditioned by the voltage amplifier. Following the conversion of the current and potential to voltages. !ow- antialiasing filters should be used on each channel, although this has often been omitted (the significance of this is discussed further on page68). input~

The conditioned signals are then converted to a digi· tal form and recorded. The recording process generally

• The Nvcuist frequency is hal: :;e samp t -g frequency for a sampled signal · ;;.,dis the maximum ' ·equency :;at tan be recorded . See page 12~ onwards lor furt11,; · ;3xplana: :n of this and the problem of aliil5 ' ~ ·;J of higher frequen: ::>s.

61

Electrochemical Impedance and Noise

used without overloading the amplifier. This method has not been used by many workers, as it is difficult to produce a voltage offset without introducing further noise. The voltage amplifier may incorporate low- filtering to prevent saturation of the ampli,fif:lr by high-frequency signals, notably power-line frequency.

Current Amplifier The current amplifier (commonly termed a ZRA in corrosion circles) .works by balancing tha input current •. against a current fed back through the current measuring resistor (Figure 6.3). There are some advantages

in using a modest voltage gain in the amplifier circuit (Figure 6.3b); when small currents are being measured, it makes it easier to obtain the necessary values for the • 'current measuring resistor. A potentiostat can be configured to operate as a current amplifier {Figure 6.3c), but most potentiostats have relatively poor noise charac;teristics and do not make ideal current amplifie.r?. _ As with the voltage amplifier, the current amplifier may incorporate low- filtering, usually in the form of a capacitor in parallel with the measuring resistor. This reduces the amplifier gain for power-line frequency interference and helps to prevent saturation of the amplifier.

Filtering kr4:iinw Input Is a -v ligh ~ICe, 10 ous:u '«
'--"tiCII1I\ 10 / VGA Vw•

~xR

Copoclor c ~~
SCIIuiCitlon ~ ~ bv

mahiiNiquency nolle Q

• Gain is the ratio of the output voltage at a given frequency divided by the input voltage. • Phase shift is.the change in phase of the output voltage relative to the input voltage.

Slcr.doR:I Cl1181'11 ~

In 11111 rnodllk;CIIIon ot !he CINt'll ~h \dage

QR:llld aaoa R II IIIQM:ed bvlte YOIIagll dMder AI and A2.10 1118 ~ 110110g8 II

>--+--.-Vw

lnCieaaed bv a factor ot (Rl +A2~ (1ne WOolcge QC!h.

~ the OI\'¥)II1IIIJ.

v.,..

H41nee

~xRx(Rl+R2}112

1t'e 1oOiage

oan11 lmled by

tne effecll ot Ofl'1l;tller eaocs to a trOJfrT'UTt ot atiout 1000.

CU!!ent 10 be meoued 11 applied be'-! RE and WE.

AE

Potentlostot

>+-.......-.....-

CUllen! acluatf ~hom AE rnoougn R.t.ence \'OIIoge . acooa R, _ _ potaniiCII.

RE L}-1--'--'-- \tillage II actually treOSU9d be,_..., AE unc1W" ·~ O'<('ld ~00k10 lllgll ~· ~ .___ _ _ _ we_Q..;;.__j'-i--- or RE tree.....: andRE ae conllollld ro be ar some potenliet~ Copoctrot rroy be weo ro tlllel w ~: 1ne

I

fi&QUenCY

c

Filters are electronic circuits that allow some frequencies to while blocking others. The properties of a filter are defined by its gain and phase shift, both of which depend on frequency:

Poten~ostar

as o cunent

omptK~r

FIGURE 6.3. Current amplifier operation.

For electrochemical noise measurements we generally use a low- filter that has a g~n of 1 for low frequen. cies, but blocks higher frequencies. A notiQnal ideal low filter will have a gain of 1 up to a specified cutoff frequency and a gain of 0 at higher frequencies. Such behavior is impossible for real electronic devices, and real filters show a falling gain aboVe the cutoff frequency. A very steep fall in gain will be coupled .with a rapidly changing phase shift, which can lead to a distortion of transients, and some care is required in the design and selection of .filter circuits. This is discussed further in Appendix A. A low- antialiasing lUter should always be used ahead of digital data acquisition in order to avoid contamination of the sampled data with allased higherfrequency components.

Analog Processing It is possible to derive some of the simpler electrochemical noise parameters from relatively simple analog circuits (Figure 6.4).

l

Corrosion Testing Made Easy

62

v., --[JC}-r--- Vw Mean Low-pcu Iller CMIIOg8S llequency

I

:::rc

c:on'4)0I MiniS above l/2xRC.

i

RMS· RMf computation Integrated c~cull (ICJ ~

Wlpulldoge. apples lOw- IIIIer. then tokes squae 1001. Core Is needed to EI'1SUie that a sUioble range ot -~

ore inctJded In tne

IT\80S\6ement.

··-

·-

.

'·~~-1 v~ Ri ·

.

1

Eq. (E.8) on page 124 (although it is important that the frequency response of the two measurements is the same), or the mean can be removed ffom the signal with a high- filter and the rms value of the result measured. • Frequency spectra: The PSD at a specific frequency can be estimated with a band- filter to select a narrow range of frequencies and by. measuring the power present in these frequencies with a rms measurement circuit. When several such measurements are made, the full power spectrum can be estimated, althot.igh it becomes expensive to make measurements at a large number of frequencies.

Standard Deviation ~high-

rllfl'l0¥e mean.

Iller belole RMS IC to

• emb&dnlc:ut to get

the required frequency response.

\./ Glcx.nd

vo.J Power Spectral Density ~ bond.poss IIIIer before RMS IC to setect I¥XfOW range ol ftequencies.

FIGURE 6.4. Analog analysis of noise signals.

• Mean: The mean value of the noise signal can be deteimined by use of a low- filter to remove the high-frequency components of the signal. It is difficult to produce low- filters with a very low cutoff frequency, and low-frequency changes in the mean would be observed. • Root-mean-square (rms) value: The rms value can be calcutated with standard analog computing components, and many digital multi meters can measure a true rms value. Note, however, that most cheaper meters estimate a rms value for aG signals by measuring the peak-to-peak value and converting the result to rms-by assuming that the input is a single sine wave. These meters will not calculate a reliable rms value for electrochemical noise signals. For meters that are able to measure a true rms, the lower limit of the frequencies that can be measured is typically --1 Hz, which is far higher than is desirable for electrochemical noise measurements. ·This limitation results .from the filter characteristics us'ed to remove de, and this limitation can be removed in dedicated circuits (although with some difficulty). • Standard deviation: The standard deviation can be calculated from the rms value and the mean with

It should be noted that some analog approaches to the measurement of electrochemical noise parameters are the subject of patent protection. Owing to the relatively limited capabilities of analog · methods and the reducing cost of digital methods, analog techniques are rarely used for modern electrochemical noise measurement systems.

Digital Recording Modern digital recording systems used in the laboratory are typically based on the use of a computer to control the measurement and to record the data obtained, and this approach is assumed here. It is also possible to use dataloggers, transient recorders, or similar devices to perform the same function. They do essentially the same thing as computer- b.ased systems, and the is referred to the manufacturer's instructions for descriptions of how to use the instruments and how to transfer the data to a computer for subsequent analysis. Any digital recording system must perform three functions in the recording process:• • sampling • analog~to-digital conversion • · data storage Samplinq is the process of producing a voltage that is characteristic of the time interval being considered . The most obvious method can be described as ' Note that although it is convenient to consider the first two steps independently. for some data acquisitton methods the sampling and the conversion processes are performed by a single circuit element.

63

Electrochemical Impedance and Noise

• instantaneous sampling and consists of taking the voltage at a specified time (e.g., the start of the time il")terval). This method is typically used by high-speed data acquisition systems (in this context high speed typically means anything with sampling r&tes jn excess of powerline frequency). Instantaneous sampling is particulafly susceptible to aliasing (see page 68 and Appendix E). The other common sampflng method may be described as integrating sampling, and this effectively takes an average value of the input voltage over all or part of the sample interval. The averaging process reduces the susceptibility to aliasing. In particular, when the pe• · riod over which averaging is performed corresponds to one or more power-line cycles, the averaging process effectively eliminates the power~line frequency component of the signal (as the average of sin x over one full cycle is 0). Note, however, that th~re will always be a range of frequencies between the Nyquist . frequency and 1/(the integration period) that will be aliased into the m~asured data. Conseqf,Jently the use of integrating sampling should not be used as a justification for the omission of antialiasing filters. Analog-to-digital conversion is the process of taking the sampled anaput voltage and converting it into a digital number. Analog to digital converters (ADCs) may use the decimal number system, particularly when they are incorporated into digital meters. but they also commonly use the binary system, as the latter is somewhat more natural for the AOC electronics and for subsequent computer processing. The characteristics of AOCs are defined by a number of properties: • Resolution is the ability to resolve small changes relative to the value thatcan be measured. This is usually defined for ADCs in of the number of discrete values that can be measured. Far decimal AOCs this is specified in of the number of digits of resolution, in . which 3 digits implies a range of -999 to +999, 3~2 digits implies -1999 to +1999, 3~4 digits impJies -3999 to +3999 and so on. For ADCs that use the binary number system, the resolution is specified by the number of bits in the binary number. The specified resolution usually includes the sign bit, so an 8-bit ADC will resolve -128 to + 127, a 12-bit ADC -2048 to +2047. and a 16-bit AOC-32,767 to 32,768. For electrochemical noise measurements we generally want a high resolution to minimize quantization errors (see page 95). although this is also linked to the mean value of the signal and the gain of the signal conditioning circuit.

• Sensitivity is the ability to resolve a small value irrespective of the largest value measured and is specified in voltage difference between adjacent • values.: If possible the sensitivity should be sufficient that the quantization noise (see page 68) intra:, duced by the ADC is leSs than other instrument noise ,sources. • Accuracy is a measure of the error in a measurement relative to an absolute standard. It may be considered as having two parts: the precision, which describes ttle scatter of repeat measurements about their mean . value; and the bias, which describes the deviation of the mean of the measurement from the true value. • Reproducibility is the range of output values that may be produced for a single input value. The best that can be expected is ±1 LSB (least-significant bit), as the input value may be on the border between two input values. Values larger than ± 1 LSB constitute a form of instrument noise (see below). • Missing codes may be experienced with some types of ADC when the output skips a given number (or group of numbers) as the input voltage is smoothly increased. Missing codes are most likely when the most-significant bits change (e.g. for an 8-bit converter, between 10000000 and 01111111). Missing codes effectively increase the quantization noise of the ADC. Certain types of ADC are inherently immune from this error. • Nonlinearity occurs when the output number departs from a linear relationship with the input. This is not usually a serious problem for electrochemical noise measurements.

Types ofADC There are several ways in which an analog voltage can be converted to a digital number. As a general rule there are trade-offs between cost, conversion speed, and resolution: • Flash converters provide the fastest conversion. They use ( N-1) comparators to detect N possible input voltages (e.g .. an 8-bit flash converter will use 255 comparators). Flash converters are not appropriate for electrochemical noise measurements, which do not need the very rapid conversion, but which do need a higher resolution than can be obtained economically with this method. Flash converters are typically used for conversion rates above 1 MHz.

Corrosion Testing· Made E...,

64

• Successive appr-oximation converters work by comparing the input voltage with a voltage generated by an internal digital-to-analog converter (DAC). The DAC voltage is successively refined until the required precision has been achieved. A 2N successive approximation ADC requires N steps for performing the conversion; coosequently the conversion can be performed rapidly. ·successive approximation ADCs are the most common devices for conversion rates between ~1 kHz and 1 MHz. The conversion speed is unnecessarily fast and the resolution rather low for electrochemical noise measurements, ~!though the method is commonly used, especially when data acquisition cards are used inside a computer. • Delta-sigma converters determine the difference in voltage between the filtered outJX,Jt of a low-resolution DAC and the input voltage. They are relatively slow, but can proquce very high-resolution conversion (up to 20 bits or better) at a low cost. It is also possible to piOgram some delta..-sigma converters to produce either e, . low sampling rate and high resolution or a faster sampling. rate with a lower resolution. The best of th• devices~re very well swted to the measurement of electrochemical noise, and are used in some commercial ins.truments. ~ Voltage~to-frequencyconverters (VFCs) can be used to measure voltage by counting the number of cycles in a 'given period (i.e., by measuring the frequency). This method is rather susceptible to drift in the VFC, and it is not widely used, although it does have interesting properties fcir the measurement of electrochemical noise. • Dual-slope converters use a modification of the VFC method that corrects for changes in the characteristics of 1he VFC· by comparing the frequency for the test voltage with that for a reference voltage. More sophi~ticaled ~hemes also correct for changes in the VFC output for a zero-input voltage. Dualslop~ converters have a rather limited sampling frequency, but can provide very high resolution. They form the basis of the majority of digital voltmeters and are thereby used for many electrochemical noise measurements.

Transferring Data to the Computer Several methods are available for connecting the ADC system to a computer for data storage.

IEEE-488 Instrument Bus The IEEE 488 bus is based on a design Otiginaily d&· vetof)ed by Hewlett-Packard with ·the specific purposeof connecting instruments such as voitmeters to r.om• puters. It uses an 8-bit parallel bus• plus a number ot1 contra~ signals. ·· · The methods used by IEEE 488 are rather ineffi~ient · by modern standards and thus make it rather expe~sive · to implement and more susceptible to noise than it could be. The maximum transmission speed is 1 Mbyte/s, which is more than sufficient for the data rates required . by electrochemical noise measurements. IEEE 488 allows 15 devices to be connected to the bus,; which' · is also more than sufficient for electrochemical noise applications. Unfortunately, although the low-level p.rOtocols of · IEEE 488 are well defined, there is no standard for the messages to and from the instrument, and these vary from one instrument to another, although individual manufacturers tend to use consistent protocols. Programs to control the IEEE 488 bus tend to be rather tricky to write, as it can be quite easy to get into state in which the voltmeter and the computer' are .waiting for each other, especially when the bus is affected by noise-induced errors. It is generally simplest ·to use·· commercial pr.ograms, ·although these may not always · provide the required capabilities.

a

RS232 The RS 232 protocol (generally called simply the serial port on PCs) was originally designed for tongdistance communication across telephone lines, and it includes a lot of control signals to manage the communications devices. However, it has become widely used for connecting computers to moderate-speed devices such as printers and instruments. Until recently the maximum transmission rate that was commonly used was ·~ 1000 bytes/s, although higher rates (up to i s.'ooo bytes/s or mnre) are now used ovE>r shortciistances. It is • In electronic . a bus ,:;: a set of 51gnal lines that connect to several devices: a parallel b:..s is one ir. which there is an individual data line for each bit of d?.:-" being tr:.~smitted. whereas a serial bus uses just one data line .•·.•th the d:.:a bits being sent one alter another.

65

Electrochemical Impedance and Nois'e

a point-to-point method, so only one voltmeter or other device can be connected to a serial port on a computer. Almost all computers are provided with one or more .AS 232 , interfaces as ·standard, suitable · cables are available cheaply and the low-level communication is ed by tne operating system. Consequently AS 232 is a very convenient method of communicating with voltmeters, and even the lower data rates are adequate for electrochemical noise measurements.

· Parallel Port · · The parallel port on modern computers was originally developed by Centronics for commul!lication between computers and printers. There is np fixed data rate for the interface, and individual characters are transmitted as and when they are available. The original interface was unidirectional, as it was necessary to send characters only from the computer to the printer. The original interface can be used to.transmit data to the computer (i.e., in the wrong direction) by using some of the con. trol lines, and this method has been used to connect instruments to computers. Developments of the interface allow the transmission of data in both directions, and this allows high-speed data transfer. The parallel port has been widely used to connect data storage devices to computers, and some low-cost data acquisition devices are available. However, few of these are appropriate for use for electrochemical noise measurements as they offer rather limited resolutions.

Universal Serial Bus The universal serial bus (USB) is a recently developed serial bus standard that has be~n heavily promoted as the new way of connecting lower-speed devices to computers. It operates at -1 Mbyte/s, although this capacity must be shared among all of the devices<::onnected to the bus. However, multiple USB controlfers can be fitted in a single computer, and this should provide ideal capabilities for the connection of voltmeters and similar instruments to computP.rs ..ThP. wictesoread:Ose of the system should ensure that the necessary components are available cheaply (most new PCs already include USB ports) . Thus it is expected that USB will become an important method of connecting instruments to computers. USB devices are ed in the latest versions of PC operating systems. such as Windows

98 and NT 5.0 (Windows 2000), and this will also help to simplify the task of connecting instruments to the computer..

Internal Data Acquisition Cards A wide range of data acquisition cards is available to. plug into PCs and other computers. Most of these cards are aimed at relatively high-speed low-resolution measurements, and they are not well suited to electrochemicaVnoise measurements. However, some lower-speed devices, based on dual-slope or delta-sigma ADCs, are available. · Communication between internal ADC cards and the computer will generally use derncated input/output (1/0) ports that can be read and written by the central processor unit (U). Higher-speed devices also allow for direct memory access (DMA) transfer directly into the main memory of the computer, although this is not generally necessary for the lower-speed devices that are appropriate for electrochemical noise measurements. It is relatively straightforward to write programs to communicate -with internal ADC cards, especially as the card manufacturers usually provide example codes for a range of programming languages. However, the move to sophisticated multitasking operating systems introduces significant problems. Using internal ADC cards usually implies that the . control .·program :· wiU be responsible for managing the timing of readings. When the program is running as a single task under DOS, this is easy to achieve (providing very highresolution timing is not required, as the PC clock ticks only -18 times a second). Under Windqws 3.1 and 95, multiple programs' can be run. All progra'fns in Windows 3.1 and some in Windows 95/98 use cooperative multitasking, whereby a particular program will run until the program releases control back to the operating systems. As an example of a particularly aggressive program, the Pegasus mail program does not release control while it .is loading new mail, a process that can take several seconds. A data acquisition program would lose readings over this period unless special precautions were taken to continue thP. readings by use of interrupt-driven processes (which are particularly difficult to write). Also. these operating systems are rather susceptible to system crashes as a result of faulty programs. These would obviously stop a data acquisition program. OS/2 and Windows NT provide better protection against rogue programs and use

66

preemptive multitasking to ensure that all programs are run reproducibly. With appropriate tuning of the system operation, OS/2 allows programs written for DOS to be used for data ·acquisition while permitting other programs to be multitasked at the same time. In principle, Windows Nt is an ideal base for data acquisition programs. It is hlghfy stable and prevents rogue programs from crashing the system. However, part of the protection of the system against problems entails the prevention of programs from direct access to 1/0 devices. This rrieans that it is necessary for for flO 'devices to be added to the system as a device driver. Device drivers for the serial port are incorporated into the oper~ting sy~tem, and device drivers for IEEE 488 controllers are starting to become available. Data acquisition card&. tend to present more of a problem unless the manufacturer. has produced a Windows NT device driver for them.

Programs for Control of Data Acquisition The reqUirements for the control' of data acquisition for elec1rochemical noise measurements are relatively · simple. If both current . and potential are being measured, it is just necessary to record time records of pairs of voltages (one· voltage being the conditioned output of the current amplifier and the other the conditioned output of the voltage amplifier). It is convenient for subsequent data analysis to take 2n points in the time record, where n is frequently 10, giving 1024 samples in the time record . Many standard data acquisition programs can record single or multiple time records. If it is required to make continuous measurements (I.e., to take an indefinite number of time records . with no gap between them) and particularly if.it is required to take readings from more than one sample, it may be more difficult to find standard f:>rograms with the required capabilities, and it rnaytherefore be necessary to write aspecial-purpose control program.

What Can Go Wrong? Interference Interference is noise p .:ked up by inductive coupling between the source of :--:e interference and the mea-

Corrosion Testing Made Easy

suring circuit. The most common source of interference is ac power lines, which will give a signal at the power supply frequency (50 or 60 Hz) together with smailerarriplitude signals at multiples of the power supply frequency. This form of interference is easily distinguished from electrochemical noise, but it can still present significant problems. If the interference is present at a higher amplitude than the noise being measured, it can overload amplifier inputs and cause them to stop working properly (known as saturation of the amplifier). Aliasing will trctnsform the 50- or the 60-Hz signal to a lower frequency, and this can be mistaken ·for... a real lowfrequency oscillation; even if it is recognized as interference the contamination of the .signal is difficult to remove. Consequently every effort should be made to minimize the pickup of interference and to prevent it from reaching the data acquisition system by use of suitable filters. Another form of interference that may be more difficult to distinguish from elec1rochemical noise is produced by arcing when the power to an inductive load is disconnected. The voltage across an inductor is proportional to the rate of change of current through it. If the current is interrupted by a switch ·or relay, a very large voltage can be produced, leading to the formation of an arc across the s (in effect the ,current continues to flow, despite the best efforts of the switch). The arcing produces a high-frequency electromagnetic field, which couples with the measuring circuits to produce one or more spikes. A common source 'Of such interference in laboratories is the switching off of the electric motor in a refrigerator by the action of the thermostat, but many other sources are possible, including electric power tools, ignition circuits in gasoline engines, and arc-welding equipment. In many cases the interference can be reduced by adding suppressors to the equipment causing it. This type of interference can usually (although not always) be distinguished from real electrochemical noise by virtue of its short duration (milliseconds rather than seconds). Interference is carried by electromagnetic radiation, and the pickup of interference by the measuring circuits can be minimized by shielding the circuits. In essence this consists of surrounding the measuring circuits with a conductor connected to ground (or another low-impedance point). Then the radiation will not penetrate the conductor. and it does not therefore lead to interierence. The surrounding conductor is known as a shield . screen. or, in the case of a shield in the form of

67

Electrochemical Impedance and Nols'e

• a conductive box, a Faraday cage. Note that a shield does not have to be solid to .be effective; all that is required is for any holes in the shield to be much smaller than the wavelength Of the highest frequency of electromagnetic radiation to be excluded. As the wavelength of power-line frequency radiation is ...... 5000 km, qui1e coarse mesh shields can be used (although switching transients contain much h.igher frequencies, and a little more care is needed).

ditioning module mounted at or close to the cell, with the output being transmitted in a robust analog (e.g., a 420-mA current loop) or digital (e.g., AS 232) form back 'to the main measuring equipment. It is not advisable to transmit tl)e .raw surren~. and potential signals over distances of more than a few meters (although this has a~parently been done successfully).

Ground Loops At least for lower-frequency interference, the cables in the measuring circuit may be thought of as forming •· a single-turn coil that interacts with the interfering magnetic field. Then the amplttiJde of the interference will be proportional to the area inside the coil. 1This area can be minimized by ensuring that the cables in critical parts of the measuring· circuet are.. kept together, ideally by using a multicore cable with a grounded shield. For the same reason cables should be kept as short as possible. A good general solution to interference reduction in the laboratory is to place the cell and measuring equipment inside a Faraday cage, and then to use a shielded · multicore cable· to connect from .the. ..cell · to the measuring equipment. For very sensitive measurements it may be beneficial to use battery-powered instruments to keep all power-line connections outside the Faraday cage. In the field, the measurement problems are somewhat more complex, as the main measuring equipment will typically be a considerable distance from the electrochemical cell. The best solution is to use a signal con-

'J.

particular interference problem occurs when a continuous circuit is formed in the signal ground. This happens, for example, when two instruments have their ground circuits connected togetherboth through direct connection (e.g., by means of a connection to an electrochemical cell) and through their connections to power-line ground (Figure 6.5). The resulting loop of conductor acts as a shorted single-turn coil, and large currents can flow aro~nd this loop as a result of power-line freque11cy interference. This can result in significant potentia:! differences between nominally identical ground connections. To avoid.ground loops, ensure that_circuits are connected to ground by means of only a single conductor. In some cases, this may nec~ssitate discqnn~cting the ground connection from the . power supply to "one or more instruments. This has safety implications, and you should take professional advice if you ne~ to do this. A better solution may be to use battery-powered instruments; not only does this avoid problems of ground loops, but it also removes some of the sources of interference.

FIGURE 6.5. Ground loop formed by connection ot the electrode to the grounded working electrode terminal of the potentiostat, to the negative input of the oscilloscope, and through the mains power supply.

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68

Corrosion Testing Made Ea${

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Although the Nyquist frequency defines the maximum frequency about which information can be obtained from the sampled data, this does not prevent higher frequencies from appearing in the sampled data if they are present in the input signaL :this process is knows as aliasing, and it is important to appreciate that alias~ ing cannot be treated after the sampling has occurred, as the samples obtained from input frequencies above and below the Nyquist frequency are identicaL

"ro avoid aliasing problems, it is necessary to remove

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frequencies above the Nyquist frequency by use of a low- antialiasing filter. Many electrochemical noise studies pave been performed without using antialiasing filters, usually relying on. integrating voltmeters to remove power~line frequ~ncy interference. Electrochemical noise power sp~ctr~ typically fall with increasing frequency, and this gener8Jiy means that the effect of aliasing is relatively minor, con:sisting o.t a flattening of the high-frequency end of the spectrum. ·

FIGURE 6.6. Initial drift.

operation well beyond this period. On the other hand, fluctuations in process conditions may frequently occur, such as start-up and shut-down, changes in process temperature, flow rate, and composition of the !SfOCess fluid. Such changes will all give rise to apparent electrochemical noise signals, and these must be allowed for if electrochemical noise monitoring is to provide reUable performance for corrosion monitoring.

On the other harid, some workers have used ADCs that.do not inherently remove power-line frequency, and they have suffered from very.s~vere contamination of their slgnal by aliased power line frequency.

As well as drift that forms a genuine part of the measured electrochemical noise, other sources of drift may also be present. These include

Drift

• drift in reference electrode potential • drift in amplifier error voltages •· • temperature effects on reference electrodes, instruments, and the corrosion process

Corrosion potentials typically vary in a systematic way over time, especially in the first hours and days of exposure. This drift is particularly pronounced when protective or partially protective corrosion product films are produced. Figure 6.6 shows the initial drift for stainless steel in 3.5% NaCI solution, with the potential rising as t~e ive film becomes established · and the initial couRiing current disappearing. It is questionable whether such slow changes in corrosion potential should be regarded as part of the electrochemical noise signal or not. They certainly contain information about the corrosion process. and as such they should be recorded, but it may be ..appropriate to remove them by driit removal procedures at the analysis stage. In plant corrosion monitoring situations. the initial·drift is not usually ternbly relevant. as the plant will be in

These sources of drift will usually be small compared with the inherentdrift of the electrochemical system.

Quantization · Whatever analog"to-digital conversion technique is used, the converted numbers will have a minimum step size or resolution that can be represented. Ttlus a 3-digit voltmeter reading from -0.999 to +0.999 V will have a resolution of 1 mV. When this is used to convert a smoothly changing voltage. each measurement will have a quantization error that will have a uniform distribution in the range -0.5 to +0.5 mV (for an ideal ADC). This \',ill give rise to white noise with a PSD of 8' 10 5 V2 ;Hz. In general. quantization noise will give

69

Electrochemical Impedance and Noise

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a PSD of 6. V2 j(12 x Nyquist frequency) V2/Hz, where 6. V is the voltage resolution. Note that this is not a fault of the ADC, but an inherent limitation of the use of digitally sampled data (the quantization noise can be reduced by increasing the resolution of the ADC, but it can never be eliminated). Quantization noise is characterized by clear reproducible levels in the time record (Figure 6.7).

Instrument Noise Any electronic device will tend to introduce noise in the measurement. At a very fundamental level, two forms of noise. shot noise and thermal noise, derive from fundamental physical processes and are essentially unavoidable (see Appendix E) .

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FIGURE 6.8. Dummy cell for testing Instrument noise. R1, R2 and C1, C2 should have similar values to those expected for Rp and cdl for the system to be studied.

replacing the cell with a dummy cell. Some components of the instrumentation noise will be affected by the cell impedance, so the dummy cell should have similar characteristics to the real cell. A typical dummy cell is shown in Figure 6.8.

Almost all amplifiers will also suffer from 1If noise at low frequencies .

References

The instrumentation noise should always be determined as part of the electrochemical noise measurement procedure . This can be done quite simply by

1. Benish. M.l. , J . Sikora , B. Shaw, E. Sikora, M. Yaffe, A. Krebs. G. Martinchek. Corrosion/98 , paper 370: see the CD-R01v1 for the full copy of this paper.

Corrosion Testing Made Easy

70

2. Thompson, N.G., J.H. Payer, "DC Electrochemical Test Methods." Corrosion Testing Made Easy, Vol. 6. Houston, TX: NACE, 1998. 3. AI Ansari, M.S., A.A. Cottis. "Long term monitoring

of electrochemical noise." Proceedings of the 13th

International Congress on Metallic Corrosion, paper 212. Austra)asian Corrosion Association, Clayton, ALstralia, 1996.

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CHAPTER 7

Analysis .of ~lectrochemical Noise~· .. Analysis Domains

Time Record Analysis (

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In the analysis of electrical circuits and their electrochemical equivalents, it is convenient to examine various domains. These are equivalent views of the same information, with the different views offering various advantages in data analysis: • The time domain considers the fluctuation of the instantaneous potential or current as a function of time. It is the normal way in which signals are recorded and is probably the most natural way of viewing the data. • The frequency domain examines signals in of the power present at various frequencies. Although a potential in the time domain has a value that varies as a function of time, the same signal in the frequency domain will vary as a function of frequency and will also have both phase and amplitude at each frequency. In principle, data can be transformed from the time domain to the frequency domain by the Fourier transform. Analysis of electrical circuits is relatively easy in the frequency domain. • The Laplace or s domain can be regarded as an extension of the frequency domain. It tends to be most widely used in the analysis of electrical circuits, largely because it allows the simultaneous treatment of steady-state and transient behavior. However, it has not been widely used in the interpretation of electrochemical noise, and is not be considered further here.•

• See Damper (full reference on page 1251 for an intrcauction to the Laplace s domain for electrical circuit analysis.

Measurement of Properties of Transients The simplest approach to data analysis, and proba· bly one of the most powerful, is to examine the time record(s) for features that are characteristic of particular types of corrosion.t In understanding how transients arise it is helpful to determine which is the controlling parameter, current or potential. In most cases we find that current is the controlling parameter and' that the potential is simply the response of the rest of the specimen to the local corrosion currents. Thus pitting corrosion is frequently associated with current transients that occur as pits nucleate, propagate, and finally die. For steel, the current transient exhibits a relatively rapid rise, followed by a slower decay as the pit reivates. The current from the pit is largely drawn from the capacitance of the ive film, causing the potential to fall over the period of the current transient and then rise rather more slowly as the ive .film recharges as a result of the cathodic reaction. This can be seen in Figure 7.1 . Note that the pitting events in this case are particularly clear and regular (indeed it could be argued that they are not normal metastable pitting events, but stable pits that are switched off when the potential falls too low as a result of polarization of the cathode reaction). t Note that because much early work was concerned with only potential noise. there was a tendency to talk simply of a particular characteristic of the electrochemical noise. This can lead to very ambiguous statements. and it is important always to specify elec!rochemical potential noise or electrochemical current noise.

Corrosion Testing Made Easy

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For stainless steel, the current transient is somewhat d.ifferent and typically shows a steadily increasing current during the growth phase, which then falls rapidly as the pit reivates quickly, with little d~tectable current decay. As with carbon steel, the potential tra'nsient consists of the response of the ive surface (that must provide the cathodic current) to the current produced by the pit. Figure 7.2 shows ti.me records for stainless steel in 3.5% NaCI solution. Note that as well as having a different transient shape, the meta~table pitting events are much less predictable in both amplitude and time of occurrence than for the carbon steel. In this case both of the current measuring electrodes are pitting, so both positive·and negative current transients are observed, depending on which electrode is giving rise to the transient. As the potential noise is being measured with a reference electrode, all of the potential transients are negative (if a third stainless-steel electrode were used as a pseudoreferenr.P. e!ectrode, then pits occurring on this electrode would give rise to positive transients) . If clear isolated transients are observed. it is possible to derive considerable information about the pitting process. The area under a transient in a current noise time

+ 0.025M NaCI solution, showing pitting events.

record indicates the charge involved in the transient and hence the volume of pit formed. More comprehensive analyses may examine the statistics of the birth, growth, and death of the transient events. Pitting processes have also been known to give rise to regular oscillations (Figure 7.1 is arguably an example of this), although these are more commonly associated with crevice corrosion or corrosion fatigue crack growth (in the latter case, the cycUc frequency is obvioJJsly that ot the mechanical fatigue cycle). Long-term changes in potential may also be .associated with the commencement of localized corrosion. Thus the initiation of crevice corrosion will usually cause the potential to fall (Figure 7.3), as the active crevice pulls the potential of the cathode down. This is an example of a time record that would have been interpreted in the past in of changes in the corrosion potential · with time. This remains the most obvious interpretation, and serves to iemind us that we should not forget older approacl1es in our efforts to get the best information out of electrochemical time records. Observations of individual transients in the time record may also be useful if they can be correlated with

Electrochemical Impedance and Noise'

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74

Corrosion Testing Made Easy

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. F.JGURE 7.3. Long-term potential time record for an austenitic stainless-steel sample undergoing crevice corrosion ·in a chloride solution.

other events that are also associated with the corrosion Thus the correlation of electrochemical transients with acoustic emissions during stress-corrosion crack (SCC) growth has contributed to the study of crack growth mechanisms, and may aiso be of value in monitoring for SCC (Figure 7A).

Statistical Methods

proces~.

AE TRIGGER

4.4 <S:

::I. ..: 4.0

z

The simpler statistical methods treat the time record as a collection of individual potentials or currents (or, in statistical , as a sample from the population of all values that will occur over infinite time) and ignore the relationship between one value and the next. On this basis, the time record is defined completely by its distribution of values, usually expressed as a cumulative probability function or, equivalently, as a probability density function. Perhaps surprisingly, it is rare to see the actual distribution plotted for an electrochemical noise signal, and instead we tend to use statistical parame· ters that describe the properties of the distribution.

~

0.::

The formal definitions of the various statistical parameters are presented in Appendix E.. Here an indication of their physical meaning and the information that is contained with!n t~em 2.~e g!•:e~ .

~ 3.6 u

0.2

0.4 0.6 TIME, S

FIGURE 7.4. Correlation of electrochemical noise with SCC. AE. acoustic emission.

Mean The mean. or average. value is well understood. It can be calculated as the average of all of the values in the

75

Electrochemical Impedance and Norse

• time record. It can also be estimated by applying a low filter and measuring the de voltage. However, this is difficult in practice for the very low frequencies that are of interest for electrochemical noise measurements. The mean potential is obviously the average corFOsion potential and can .be interpreted in conventional ways. It is not generally considered to form a part of an electrochemical noise measurement, although it forms a part of some parameters used to test for localized corrosion (see page 89). The fluctuations in the mean over the longer term may also be directly related to •· changes in the corrosion processes occurring (e.g. as in Figure 7.3). The mean current is usually expected to be zero when we are measuring the current be~een two nominally identical electrodes. In practice it rarely is zero because of small differences in the corrosion behavior of the two electrodes. When we test dissimilar electrodes, then it is no longer expected that the mean current will be zero, and the mean may therefore carry more useful information.

Variance . The va~iance is a measure of the average ac power in the signal (see page.52). Note that the variance of a signal depends on the range of frequencies included in the signal, so the variance is not a uniquely defined description of an electrochemical noise signal. Variance has units of V2 or A2 , and is often referred to as the noise power. In general , it is reasonable to expect that the variance of the current will increase as the rate of the corrosion process increases and as the corrosion becomes more localized. In contrast, the variance of the potential is expected to decrease as the rate of the corros~on process increases, but to increase as the corrosion becomes more localized. These results can be shown formally for certain classes of noise by use of a shot-noise analysis (see page 130), but they are expected to be generally true for most types of corrosion.

standard deviation is slightly more intuitive than the variance and is related to the width of the trace on the time record.

Root Mean Square The rms value is the square root of the average value of the square of the potential or current (without subtraction of the mean). In practical it is a measure of the power available from the signal, including the effect of any de or mean potential or current. The rms, variance, and mean are related, and it is not necessary to measure all three.

Skew Skew (or skewness) is a measure of the symmetry of the distribution (Figure 7.5). It is nondimensional. A value of zero implies that the distribution is symmetrical about the mean, a positive skew implies that there is a tail in the positive direction, arid a negative skew implies · that there is a tail in the negative direction. A time record consisting of unidirectional transients will typically be heavily skewed, and this may be a useful measure for detecting such transients associated with metastable pitting. Of· course, if the current noise is measured between two identical electrodes, it is possible for the transients to be unidirectional (if only one electrode happens to be active) or bidirectional (if both electrodes are active), and in this case the skew will be a rather unpredictable indicator. The value of skew calculated from a sample has a relatively large uncertainty. For samples taken from a normal distribution, in which the expected skew is zero, the expected standard deviation of the measured skew will be approximately J6f N (where N is the number of points in the time record). Thus for a 1024-point time record we should not regard skews as being significantly different unless they .di!fer by more than, say. 3 standard dev1a11ons or "'0.23. ' · · · · ·

Standard deviation Kurtosis The standard deviation is simply the square root of the variance. and the calculation and the interpretation are essentially equivalent to those of the variance. The

Kurtosis is a measure of the shape of the distribution compared with the normal distribution (Figure 7.6).

76

Corrosion Testing Made Easy

• 0.45 0.4

0 =0.35 u c: :I ""'?: 0.3

~o.25 4D

0

~ 0.2

:0

~0.15

..

.

•

..

...0

Q.

0.1 0.05 0 ·5

-4

·3

·2

·1

0

2

3

4

5

Value FIGURE 7.5. Skew of a distribution.

0.45 ,.--------------.r----------'"---,

Positive kurtosis

.0.4 . c:

0

ti

0.35

at

0.3

c:

?: 1i

! 0.25 ~ 0.2

:a~ 0.15 2 Q.

0.1 0.05

·3

·1

0

2

3

4

5.

Value FIG !mE 7.6. Kurtosis of a distribution.

As with skew, kurtosis is nondimensional. A kurtosis of zero implies that the distribution has a shape similar to that of the normal distribution (although it does not prove that the distribution is a normal distribution). A positiv~ kurtosis irr.plies a more s~il\y dis· tribution, whereas a negative kurtosis implies a flatter distribution. Kurtosis has a standard deviation approximately twice that for the skew (i.e. , 24. N ). so quite large dif-

ferences are necessary (of the order of 0.5 for a 1024point time recori) before we can consider them to be significant. As ;;.;.::: a6 :he inhei.:nt sto.tistica: u;·.~..ertainty in the values of skew and kurtosis, there will also be significant problems introduced by the effects of long-term drift of . the mean potential (Figure 7.7) . The current may also drift. but the extent is usually less significant than for the potential.

n

Electrochemical Impedance and Noise

0.4 5 . - - - - - - - - - - - - - - - - - , 0.4 c 00.35

I

I

,. ~

•

0.3

~~0.25

L-Yf .

I.

0.2 0.15 0.1

0.05

nrne

Potential

Rising exponential decay produces posiHve skew and negative kurtosis FIGURE 7.7. Skew and kurtosis produced by drift.

Despite these limitations, skew and kurtosis and other higher-order statistical measures, such as the bispectrum, seem to offer interesting possibilities for extracting information about localized corrosion from electrochemical noise signals.

Coefficient of Variation The coefficient of variation is the standard deviation divided by the mean and is a measure of the.noisiness of the signal compared with its mean value. For some stochastic proce_sses it is expected that the coefficient of variation wUt bE!.aeonstant.Thus it is n6t, at first sight, unreasq11~e tqexpeCt that the coefficient of variation should beJ·gOOd indicator of the type of process occurring. However, if we are concerned about the coefficient of variatio'n of corrosion currents we should treat the anodic and cathodic reactions separately, and in consequence the mean current that we divide by should be the sum of the magnitudes of these two currents (i.e., 2/c:orr), not the difference between them that we actually measure as the galvanic coupling current.

presented in detail in Appendix E, s this view, although some of the necessary assumptions are not always valid. Practical measurements also tend to the analysis, and the noise resistance is probably the best-established parameter for the interpretation of electrochemical noise, although it obviously gives useful information for only uniform corrosion processes. It has often been assumed that, because electrochemical noise measurements do not involve the application of current to the cell, they are not susceptible to solution resistance effects. However, it appears that 11, incorporates the solution resistance in just the same way that Rp does. Noise resistance can be easier to measure than LPR in solutions of high resistance, but the result is no more useful than Rp would be! Some authors use the term electrochemical resistance noise for Rn. This is both inaccurate (it implies a noise in a resistance rather than a resistance calculated from noise), and misleading (it is possible to measure noise in the solution resistance, for which this term is also used, in this case correctly; see page 69).

Frequency-Domain Methods Noise Resistance Spectral E;stima_tion The electrochemical noise resistance Rn is calculated as the standard deviation of potential divided by the standard deviation of the current. It has been argued that the noise resistance is the same as the polarization resistance . A simple. analysis,

Spectral estimation is the process of estimating the power p~asent at the various frequencies. The term estimation is used because the objective is assumed to be to es: mate the power present for all frequencies for an infini:ely long time record.

Corrosion Testing Made Easy

78

0.8 0.5 0.4 0.3

>

~~~

0.2

..

ii 0.1

c .! 0 a.

..

0

~

-G.1

~

-0.2

I

I

.~

-0.3

~

\~

-0.4 -0.5 0

..

200

800

800

1000

1200

Time/sec (a)

~

... 10

~

l

I

1 0.1 ~--------------~----~------~--~--------~~--~ 0.001 0.01 0.1

Frequency I Hz

(b) FIGURE- 7.8.• Typical power spectrum: (a) time record, (b) power spectrum.

Power Spectra and Power Spectral Density When we combine uncor'related signals. we add the powers in the two signals. Similarly, when we ana- · lyze the frequencies present in a complex signal we divide the power between the various frequencies . A plot of the power present as a function of frequency is

known as a power spectrum . We plot frequency on the x axis and a measure or the power on the y axis (see Figure 7.8). We could separate the signals into a number of narrow frequency bands and plot the power present in that range of frequencies. However, if we did this the y-axis scale would depend on the width of the frequency bands used. We therefore divide the power

79

Electrochemical Impedance and Noise

by the width of the frequency band to obtain the PSD. This will have units of power per hertz, i.e., V2 /Hz or A2 /Hz. It is also common to plot the square root of the PSD, giving units of V1.JRZ or A/ .JRZ. It is obviously _i~po'­ tant to specify which unit is being used, although 1t IS not uncommon for the y axis ifl a powe~ spectrum simply to be labeled potential noise or amplitude. without giving any units, thereby leaving the reader to guess what is intended.

Bels and Decibels

:.k

::

Power spectra were initially developed by communications engineers. As is common tOSfay, they plotted log PSD against log frequency. However, for reasons that are somewhat obscure, they chose to treat log PSD as the parameter being plotted, and gave it the unit of the bel. One bel corresponds to a change of one unit in log 10 (PSD), i.e., a change in the PSD by a factor of 10. For even more obscure reasons, bels are never used · directly, but only the derived unit of the decibel, abbreviated dB and pronounced deebee (10dB = 1 Bel). Thus 1 dB corresponds to a change in power by a factor of 1oo.1 or 1.3, whereas 10 dB correspond to a change in power of 10 times. One common use of decibel units is in describing slopes of regions of a power spectrum. Here It is common to quote slopes in units of decibels per octave (which must be a good candidate for the most confusing unit ever). One octave corresponds to a doubling of frequency (from musical terminology). Thus a slope that is commonly observed is 6 dB/octave. 6 dB corresponds to a four-time increase in power (because 10°·6 ::::: 4), or, equivalently, a two-time increase in potential. Hence a slope of 6 dB/octave is the same as saying that potential is proportional to frequency. Similarly a slope of 12 dB/octave will correspond to the potential's being proportional to the square of frequency. Units of decibels per decade may also be used to specify slopes. in which the decade implies a factor of 10 change in frequency. Table 7. 1 lists some common slopes and the relevant values in the different units. Decibels are inherently relative units. but 0 dB may be defined as an arbitrary value. such as 1 V2 /Hz. in which case decibels may be used as absolute units. It

Table 7.1 Slopes for Power Spectra Function

dB/octave

V'! -x f

3

\1"-x f

6 12 -6

v 'X

(2

V -x 1/f V 'X fn

6n

dB/decade

10 20 40 -20 20n

;

log (V)/Iog (f)

0.5 . 1 2 -14 n

is not unknown for decibels to be used as absolute units without specifying what 0 dB represents - this is bad practice and should be avoided. In general, units based on decibels are prone to confusion, and you are advised to avoid them. Quote PSDs as V2 /Hz or V1.JRZ; being sure to specify which you are using (V;.JRZ is not strictly a PSD, but .JPSD). Slopes require particular care; they are best reported as (log PSD)/(Iog frequency), but (log .JPSD)/(Iog frequency) is also used. The units of both are numbers, but it is very important to specify which of the two is being used (as they differ by a factor of 2).

Power Spectrum Estimation Methods · Two methods are commonly used to estimate power spectra in electrochemical noise. studies: th~ Fourier transform in the. form of the FFFahd the max1mum entropy method (MEM). The underlying theory of ~hese methods is presented in Appendix D. Here we Simply consider them as alternative methods of estimating the power spectrum.* As the methods are commonly used, the FFT produces noisy spectra, whereas the MEM produces smoother spectra. However, the FFT can be smoothed and the MEM will produce noisier spectra if a higher order is used (in essence, the order is the number of used to define the spectrum; a higher order allows more wiggles in the spectrum). The question of the optimum order to use for the MEM has not been given • We talk about spectral estimation t:-ecause we assume that the ob· jective is to derive the general power spectrum for the process rather .than simply tne power present in :-,e of time record that happens to have beer collected. This is mu:n the same distinction as when we try to es: ~ate the mean and :· e standard deviation of a large population !tom a more limited sa:-,ple.

•._..... ,.

Corrosion Testing Made Easy

80

• much attention by electrochemical noise researchers. It is suggested that both MEM and FFT spectra should always be plotted and that the optimum order for the MEM calculation is the smalle$t value that gives a good fit between the MEM and the FFT spectra.

Trend Removal Spectral estimation methods require that the signal be stationary (the statistical properties do not change over time). Corrosion potentials typically drift over time. .especially in the ear~y stages of exposure, and this leads to problems with spectral estimation. It is therefore common to remove the drift, usually by subtracting a linear regression line from the data (Figure 7.9). The detailed implications of trend removal procedures are not entirely clear at present. However, it should be appreciated that trend removal tends to produce a low-frequency plateau, particularly in the MEM spectrum, that may not be 'real'.

Time

~

Fit trend line (e.g. using regression analysis)

~

SUbtracttrendnne

~

Windowing Windowing refers to the process of selecting a section of the time series (assumed to be infinite) fot examination. In effect, the infinite time series is multiplied by a windowfng function that is zero everywhere Out$JQe the window being examined and nonzero within that window (Figure 7.10). Implicitly, we use a square window (one for which the windowing function is 1 within all of the window and zero everywhere else) to select a time ·record for examination. We therefore tend to assume that windowing refers to the use of a windowing function that is not constant over the window, and in particular to functions that taper to zero at each end of the window (Figure 7.11 ). Windowing of the time record before a11alysis reduces the spreading of frequencies in the FFT spectrum (the.process known as leakage). As electrochemical noise ·usually contains rather smoothly changing.power as a function . of frequency, the problem of leakage is relatively minor. However. a windowing Junction shoUld probably aiways oe used (and the function usedreported). It is less clear whether windowing is necessary or desirable for MEM analysis. and it can probabli be omitted (although this does lead to questions about the comparison between the FFT and the MEM spectra) . Note that windowing

Time

FIGURE 7.9.

R~ng

a trend from a time record.

should always be applied after trend removal (if the latter is used); otherwise the trend removal will usually recreate nonzero values at the end of the time record.

limitations of Spectral Estimates R<;~.w

FFT power spectra nave clear hmits:

• The maximum frequency that can be analyzed is the Nyquist frequency. • The frequency resolution is limited by the duration of the time record .

81

Electrochemical Impedance and Noise

• Infinite (or quasi-infinite) signal

.:

lime

Time

Select segment wtth window

0------------------~------

~

Windowed signal

Tlme

FIGURE ·7.11. Windowing a time record (after trend removal).

lime

FIGURE 7.10. Windowing of a time series.

When an FFT spectrum is smoothed by averaging multiple spectra. these limits are unchanged. When a single FFT spectrum is smqott:led_by averaging groups of frequencies. the frequency resolution is increased in · proportion to the number of frequencies in each group. The limits of MEM estimates of power spectra are less clear. At the high-frequency end the Nyquist frequency remains the limit. Arguably the frequency reso-

lution of the estimate is determined by the Nyquist frequency divided by the MEM order, because the order determines the number of poles that define the spectrum. For spectra containing sharp peaks, this may be slightly pessimistic. ~ut for noise speCtra it is probably quite a good estima::At has been argued that the MEM estimate can be ex1;apolated to frequencies below the frequency resolutior of the FFT estimate, but it seems clear that this is not ·:alid under normal circumstances. It is rare for the ,: ·jer used in the MEM calculation to be reported in pu: ished work. In part. this may have

82

Corrosion Testing Made Easy f

been because one well-used program used a technique of automatically determining the best order, such that even s did not usually know what order was being used. However, it is now clear that the order can have a marked influence on the abilitY of the MEM to model a spectrum, and the order should always be re- • ported. There is a tendency{o use a small order to give a smoother spectrum, but it ·must be appreciated that this also leads to a loss of detail at low frequencies. It is recommended to start with an order of --.. 1%-2% of the number of points in the time record and then compare the MEM with the FFT to ensure that they fiive a good match·. particularly in the low-frequency region. Some workers comparing the FFT and the MEM (largely by using a practical approa9h) have implied that there is a fundamental difference between the two methods and that one may be right and the other wrong. However, once the background to the two methods is understood, it is clear that the MEM is better seen as a method of smoothing the FFT, with an unfortunate, but unsurprising, tendency to distribute errors uniformly over the linear frequencyIPSO plane, thereby giving errors in the low-frequency region that appear large when viewed on log-log scales.

Electrochemical. Nolsa Impedance If we have derived both the potential and the current noise power spectra, we can divide the potential PSD at each frequency by the current PSD at each frequency. We need to use smoothed spectra to obtain a reasonably smooth impedance spectrum, and the MEM is commonly used for this (although, in principle, other methods of obtaining smooth spectra could also be used). The result has units of square ohms, and, by taking the square root, we derive the electroChemical noise impedance Zn. Other terminology has been used for this ·parameter, notably spectral noise impedance. However, the parameter is directly comparable with the electrochemical noise resistance, and the spectral in -the alternative definition is tautological. as impedance is necessarily rt function of freauency. The analysis of the relationship between Zn and the conventional electrochemical impedance is exactly ~nalogous to that for Rn and Rp. Hence, providing the two electrodes have equivalent behavior. Zn should measure the modulus of the electrochemical

impedance of the system. It is more limited than the electrochemical impedance, both because the power present in electrochemical noise signals is not sufficie5lt to obtain useful measurements above 1-1 0 riz and because "it is not feasible to estimate the phase angle reliably. In some circumstances, the ffiherent noist:' of the instrumentation may aUciw valid impedance measurerflents to be made at higher frequencies but this depends on rather particular experimental conditions. It is not, in any case, significantly different from the deliberate use of noise signals for conventional impedance measclrements. One possible advantage of electrochemical noise impedance measurements is the ability to make measurements at very low frequencies. Figure 7.12 shows an impedance spectrum derived from noise measurements that measures down to frequencies of 2 x 1 5 Hz (this was obtained from the time record of Figure 7.3). While this is an interesting ability, it is very doubtful whether the result has any real validity, as the corrosion system was almost certainly non-stationary over the period of the measurement. This is liable to be true fer almost all corroding systems.

o-

Characteristic Features of Power Spectra Noise power spectra can typically be interpreted as line segments on the log-log plot. Commonly, a lowfrequency plateau is observed, followed by a falling power at higher frequencies and possibly a further plateau at high frequencies. Various attempts have been made to characterize noise generation processes in of the slopes of the various parts of the spectrum and the transition frequencies between the various sections of the plot. In interpreting these analyses, it is very important to distinguish between potential and current noise spectra. If we regard the current noise as the fundamental process, then the potential noise will be produced by the current noise applied to the electrochemical impedance of the metal-solution impedance . {Figure 7.13). Instrument noisd often becomes dominant at higher frequencies, and may · therefore produce a highfrequency plateau that has nothing to do with the electrochemical system. Similarly, aliasing that has not been removed by antialiasing filters may produce a small false high-frequency plateau (Figure 7. 14).

Electrochemical Impedance and Noise

N

X

N< ....

0

N

X

~ ?: ·;; r:: Gl

c

'i .... ti

Gl Q. (/)

~

. ~

0

Q.

1.00E+02 1.00E+01 1.00E+OO 1.00E.{)1 1.00E-o2 1.00E.{)3 1.00E-o4 1.00E-o5 1.00E-oe 1.00E.{)7 1.00E.{)8 1.00E-o9 1.00E-10 t.OOE-11 1.00E·12 1.00E-13 1.00E·14 1.00E·15 1.00E·16 1.00E-17 1.00E·18 1.00E-oe

1.00E-o5

83

1.00E-{)4

1.00E.OO

1.00E-{)2

1.00E.{)1

1.00E+OO

Frequency I Hz (a)

E .c 0 Gl

u 1.00E+05 r::

= 8. .5 "C

1.00E~+---------~--------~----------~--------~--------~--------~

1.00E..()6

1.00E-o5

t .OOE-{)4

t .OOE.OO

t.OOE-{)2

t .OOE-{)1

1.00E+OO

Frequency I Hz (b)

AGURE 7.12. Lcw-tr::;~ency impedar.ce sf)
M

84

Corrosion Testing Made Easy

1.00E+07 . . - - - - - - - - - - - - - - - - - - - - - - - - - - , 1.00E+06 1.00£+05

t-------------

1.00E+04 1.00E+03 1.00E+02

)

i

lE

1.00E+01 1.00E+OO 1.00E-01 1.110E-d2

Current PSO results from Potential PSD of one electrode applied to lmpedance2 of the other

+---~

1.00E-«< ';: 1.OOE-04 ~ t.OOE.05 ~ 1.00E-o6

1.00E-07 t.ooe-oa t----~ I.OOE-09 t.OOE·10 1.00E·11 1.00E·12 1.00E·13

+----..-----..-----.-----...----....-----!100

o.ooof

O.ot

•• 0.001

0.1

10

Frequency I Hz FIGURE 7.13. Relationship between current and potential power spectra and the electrochemical impedance.

0.1

0.01

' ' ...

''

' ' ...

0.001+--------~------~------~--------'~ 0.001

0.01

0.1

10

Frequency I Hz FIGURE 7.14. High-frequency plateau produced by aliasing of 1/ f noise.

FFT power spectra for noise signals are inherently noisy. An unusually smooth FFT power spectrum or one showing per!0d!c- beh?.•.'iiJr t~'~;'=::!"~' indicates that the spectrum is dominated by one or two features in the time record, such as a single' large-amplitude transient . An example of this can be seen in Figure 7.15, in which the six simulated transients lead to the characteristic ripple in the spectrum, particularly in the FFT spectrum

at--10- 1 Hz. This does not mean that the spectrum is wrong, but it must be questioned what advantage there ;~ in plotting the spectrum, when it is likely that the time record has a very clear pattern to it and one that may be easier to interpret. If a time record is produced as the sum of a randomly occurring sequence of identical transient events,

-

Electrochemical Impedance and Noise

85

the spectrum of the time record will, on average, be the same shape as that for the individual events, but with the PSD increased in proportion to the number of events. If the events differ in amplitude but not duration, the overall spectrum will still have the same shape. However, if the events have varying duration the spectrum will be

smeared out and its shapE' will no longer be the same as that of an individual event. Consequently the slope of the po~er spectrum rr -=..y be related to the shape of the elementary transients. giving rise to the noise, but it may also be influencec ::JY the distribution of the duration of the transients (sc-= Figure 7.15).

2~--------------------------------------------------~

1.5

., CD

~

Q.

E

<

0.5

~.5

0

600

200

800

1000

1200

nme (a) 10000

1000

100

Q

en

a.

10

0.1

0.01 -'"·------------------------....---------------~ O.CJI

().01

0.1

Frequency I

·.(· ,_ .

Hz

(b)

FIGURE 7.15. Relationship between spectra for individual transients and overall time record: (a , artificial time record with six tran. sients: (b) power spectra lor individual transients (the flattening at high frequency is due to ali asing effects; also note the smooth spectrum with a regular ripple that is due to the single events); (c) power spectra for total signal (r-oisy curve), compared with the sum of the spectra for "the individual events (smooth curve) Continued.

Corrosion Testing Made E~$Y

86

• 10000

1000

100

a

U)

a.

•.,

.

10

0.1

.f---------.--------....--------1

0.01 0.001

0.01

0.1

Frequency I

Hz

(C)

FIGURE 7.15. Continued.

As a consequence of this relationship between the spectra for the individual transients and the overall time record, we may expect that the slope(s)• and changes in slope of the power spectra will contain information about the type of corrosion process occurring. This was suggested by early workers, and many papers quote specific slopes and/or transition frequencies as being characteristic of a particular type of corrosion. This does seem an attractive argument and offers the prospect of a simple method of discriminating between types of corrosion. However, we doubt whether the work to date has really demonstrated the generality of the method, and much of the analysis used is based on rather flimsy evidence and questionable d:lta. Thus, we don't feel that it is valid to provide a universal table of slopes for specific corrosion types. However, this is an approach that is worth exploring for specific corrosion monitoring applications for which the range of materialenvironmt;!nt systems under consideration is limited and for which the relevant slopes can be established by experiment.

it is vital to avoid the pitfall of comparing the slope of a potential noise power spectrum with that of a current noise spectrum.

Chao.s Methods Several workers have applied the techniques of chaos analysis to electrochemical noise data, and evidence of chaotic behavior has been reported. These observations are interesting from a theoretical point of view (as they imply that the noise generation processes may cause fluctuations by virtue of. Jhe . Jnstability of the governing equations, rather than purely random processes), but it is not yet clear whether they have practical applications. As an example, Figure 7.1 may reasonably be expected to have chaotic character, as is indicated by the rather regular occurrence and amplitude of the transients.

Discriminant Analysis As with all electrochemical noise analysis methods, it is important to distinquish between potentia! and current noise. Tllese are expected to have different slopes, and • The stope ef the power spectrum o~er its cemral range is often referred to as the roll-off slope, from terminolo9·cal used tor electromc filters.

In many corrosior: measurements, we are concerned with discriminating between one of two or more states or conditions. For example, we might wish to discrimin9-te between a pitting and a nonpitting state. This is generally achieved by making a measurement that is known to give different values for the two states and

Electrochemical Impedance and Noise

87

comparing the value observed in service with the values that have been measured previously for the known states. However, electrochemical noise measurements give us the possibility of deriving many parameters, such as the potential and the current power spectra and the noise impedance spectrum. In this situation, it becomes a Iitt~ more. difficult to decide on the best method of discriminating between the two states. A statistical approach to the problem will determine the separation between the mean values of the given parameter for t~e two states relative to the standard deviation for that parameter. If we are considering using power spectra computed with unsmoothed Fourier transforms, we can make ' use of the relationship between the standard deviation and the observed power to derive a single discri!Tlination function: PSD,

d( f)

= ( PSD2 -

PSD2 ) PSD1 - 2 O.S,

(7.1)

where PSD1 and PSD2 are the PSDs for the two states at frequency f. Then the larger the value of d( f), the more reliably the PSD at that frequency can discriminate between the two states. If we plot d( f) as a function of frequency, it is usually easy to detect those frequencies at which a reliable discrimination can be achieved. Figure 7.16 shows a plot ofthe discriminant function for steel in carbonate solution of two different concentrations, and good discrimination can be expected for frequencies centered on-1Hz, whereas there is much poorer discrimination at higher frequencies.

Applications fo General Corrosion General, uniform corrosion.is arguably the one case for ' which there exists a well-characterized method for the use of electrochemical noise. Unfortunately, it is also the case for which it is questionable whether electroq"Jemical noise offers significant advantagel? O'(er conventional methods. The well-characterized analysis method for electro~~emical noise is the i~erpretation of the noise resistance by use of the Stern-Geary equation. This ·has been shown by several workers to give a good estimation of the corrosion rate, and it has a simple t~eoretical justification. To perform a theoretical analysis of the noise resistance method, we need to make a few simplifying assumptions: • We assume that the metal-solution interface can be treated both as a noise source and as a linear equivalent circuit. • We assume that the two electrodes used for the current measurement are similar in their nois& _generation characteristics and their polarization behavior. This assumption is not essential, but the r~sults of assuming differing properties are dependent on several variables and consequently difficult to Interpret. In general, major differences between the two electrodes present a problem-see page 68 for further discussion of this. 10 ~------~----~---r--~----~,

1!-11

· - - - ··· ··.,

11!-12

10

~11!-U

--

'\:: 1!-14

....,

8

..,......

~11!-U

j-~·~

~

20

IE-11 IE· II

_....__ _ _-!

11!019

0 10

10

0

IS

Freq. (Hz) (bl

FIGURE 7.16. Discriminant analysis of power spectrum for steel in carbonate solution with a concentration of 7 or 96 mgfl: (a) power spectra, (b) discriminant function. .

'·'

/

88

Corrosion Testing Made Easy

• To simplify the analysis, we also assume that the potential noise is measured with a noise-free reference electrode and we ignore the effects of the sotution resistance. Neither of these assumptions are necessary, but the added complexity of the analysis obscures its fundamental simplicity. 1

the noise source on electrode 2, we find that two current noises subtract from each other, whereas the potential noises add. Hence we cannot estimate Rp from the instar'ltaneous potential and current noise. However, if we consider the potential and the current noise powers, we can write

. There are various ways in which the electrochemical noise measurement system can be described in of an electrical equivalent circuit. We assume that the corroding metal electroaes provide a source of current noise as a result of fluctuations in the corrosion currents. These fluctuating c~·rrents return either to the local electrode through the metal-solution impedance or to the other measuring electrode through its metal-solution impedance. For simplicity, we treat the me.tal-solution impedance as a resistance Rp. although the full ac analysis is essentially the same as this simplified method. We use the superp9sition theorem (page 97) to allow us to analyze the current sources from the two electrod~s independently. Considering the current noise from electrode 1 (Figure 7.17) we can see that this will produce a potential noise 5n between the reference electrode and th~ coupled working el~ctrode pair. This will then produce a current /~ to working electrode 2, given (from Ohms' law) by (7.2)

t;

This current noise will be measured as the electrochemical noise current (or that part associated with the noise source on working electrode 1). Note that Eq. (7.2) relates to the instantaneous current and potential. However, when we come to add the effects of

Current

FIGURE 7.17. Equivalent circuit for noise measurement.

-. ·2 /

n

E2

n = R2' p

(7.3)

Now,· when we combine the effects from the noise sources on the two working electrodes there is no difference between the addition and subtraction of the noise powers (page 101). Consequently, if we assume that the two working electrodes are equivalent, this simply doubles both the potential and the current noise power, and we see that the measured electrochemical noise resistance Rn is identical to the polarization resistance Rp:

(7.4)

Note that this analysis makes no ~ssumptions about the source of the current noise or about its nature in of amplitude, frequency content, and so forth. Hence we expect the resu4t to be reasonably reliable, providing the two electrodes have similar values of Rp. So far, we have not taken of the frequency at which Rn is being measured. For the most common measurement technique, R;, is measured by taking the standard deviations of En and In for a time record of arbitrary duration and sampling frequency (1 024 points sampled at 1 sample/s being common). If the time records are properly filtered to avoid aliasing, the measured standard deviations will contain frequencies between 1/(time record duration) and the Nyquist frequency, with the high-frequency end being modified by the antialiasing filters. The distribution of power over this frequency range will depend on the characteristics of the noise generation process and the impedance of themetai-solution interface. The calculated value of Rn is effectively given by the square root of the average PSD of the potential noise divided by the average PSD of the current noise. These averages will often be dominated by higher-frequency components, and for typical time record characteristics, Rn is effectively measured at a rather higher frequency than would normally be

89

Electrochemical Impedance and Noise

used to measurP. Rp. For this reason we would suggest that Rn is better measured either by explicitly filtering the potential and the current noise time records to remove higher frequencies (e.g .• frequencies above 0.01 Hz). or by taking the noise impeda'nce at a low, specified frequency. Arguably, the low-frequency limit of the nois~ impedance should be use.d·to determine Rn. but there can be some uncertainty about the low-frequency limit of the noise spectra, and it may be necessary to use a higher frequency to ensure reproducible measurement.

...

['

Relationship between Noise Impedance and Conventional Impedance

-[

It can be shown that th~ noise impedance Zn should measure the same values ·as conventional ac impedance measurements. The argument is essentially identical to that given above for Rn. with the standard deviations of potential and current being replaced with the PSO for a given frequency and Rn and Rp with Zn n .. and the conventional ac impedance for the correspond·•· ing frequency.

"'· b :. 'L:

., ..

+ .. i•!

n: .. r ·.· · ,

The noise impedance method is a somewhat limited way of estimating the ac impedance spectrum. The power in electrochemical noise signals .typicaUy falls to very low levels . above 1 Hz. and instrument noise -will often swamp the electrochemical noise measurement. (Note that the theoretical analysis above does not apply to noise originating from the instrumentation). Consequently, it is not usually possible to measure the higher-frequency parts of the spectrum. On the other hand electrochemical noise spectra can be measured .down to very low frE;!quencies, 2 and jtjs possible that such measurements may offer advantages compared with conventional ac impedance measurements. There are serious questions about whether corroding systems are sufficiently stationary for measurements at very low "frequencies to. be valid. but similar questions apply to conventional ac impedance measurements.

" · Application to Localized · ·'· _..:;, ~,

.. -r

~orrosion

Electrochemical noise is the electrochemical method with the best prospect of detecting and monitoring localized corrosion processes such as pitting. crevice cerro. sion. and sec (although this is more a statement that other methods do not work than that eiectrochemical

noise does). A considerable amount of work has been done to evaluate electrochemical noise for this purpose, but the be~t analysis methods remain somewhat uncertain. Localized corrosion is generally associated with larger electrochemical events .than uniform corrosion. For example, an electrode undergoing pitting corrosion ' fr~uently displays characteristic current transients-that-are associated with the initiation, growth, and reivation of metastable pits (see Figures 7.1 and 7.2). Jience methods used to detect localized corrosion are generally concerned with the identification of largeamplitude events. One obvious way of doing this is to look at the time record; the human visual cortex provides ·an ·extremely powerful pattern recognition toot. and one that we have difficulty in emulating with mere mathematics! However, visual examination suffers from a severe boredom factor, and analytical methods of identifying localized corrosion are needed to allow the technique to be used for routine corrosion monitoring.

Coefficient of Variation For a noise process that consists of a few large events, we expect the standard deviation of the current to be large, whereas we expect the mean current to be small. Hence we expect the caefficient of variation (defined for this purpose as the standard deviation of the measured coupling curren~ divided by the mean) to be large, and the coefficient of variation of the current has been used as an indicator of localized corrosion. Unfortunately it suffers from the fundamental problem that the expected value (in the statistical sense of the best guess for the value) of the mean is zero. If the rnean is indeed zero, the coefficient of variation will be infinite, whatever the standard deviation. Thus there is a possibility thatthe coefficient of variation wilt give a false-positive indication of localized corrosion if the mean happens to be close to zero. An approach that may overcome this limitation is to use nonidentical electrodes to measure the current or to f')()larize them to slightly different potentials (see page 56), such that the expected value of the mean current is nonzero and the actual value is rn-::-re '"'::'•"Jdt•-:-if:-!c A.nother ~ossib 1 ~ way :~f -:-•.•ercom· ing this limitation is to use the sum of the magnitudes of the individual anodic and cathodic currents {as noted on page 77) rather than the measured difference, but as far as we are aware, this has not been examined in any detail (and these currents cannot, of course. be measured directly) .

90

Corrosion Testing Made Easy

Localization Parameter As a way of avoiding the problem of the Qf:!pendence on the nonzero value of the mean current, the localization parameter has been proposed. This is the standard deviation of the current noise divided by the rms current. In this case the actual value will always be between 0 and 1, with a value approaching 1 implying a large standard deviation compared with the mean, and a value approaching 0 corresponding to a small standard deviation compared with the mean. Thus the IQ.calization p~rameter avoids the problem of the coefficient' of variation of tending to infinity as the niean tends to zero. However, it can be shown that the localization parameter is a simple mathematical transformation of the coefficient of variation, and it re. ally suffers from the . . I same dependence on the chance valtJe of the mean current.

Charge in Transient If we make a number of assumptions, we can apply a shot-noise analysis to localized corrosion (see Appendix E). This leads to a method of calculating the average charge in the individual transients that make up the electrochemical noise. The charge is given by

on the basis of several statistical parameters, including mean, standard deviation, skew, and kurtosis of potential and curre~t. Preliminary results appear promising, althOugh the producti,on of good-quality training data presents a major challenge.

•

Study of Painted Metals Painted metals have proved difficult to analyze with conventiqn,al electrochemical methods, and it is perhaps natural that electrochemical noise should be tried with this system. Unfortunately, painted metals present .significant problems for electrochemical noise measurements. Most importantly, the electrical resistance of intact paint films is very high (in the region of 109 0 or more); consequently the electrochemical current noise is very low and extremely difficult to measure. The. electrochemical potential noise is similarly susceptible to errors introduced by the amplifier input current noise. Many earlier results for measurements on painted metals were probably seriously affected by instrument noise and are best .discarded.' More recent work has tended to use the electrochemical noise resistance and has shown reasonable results from this method (although one would expect the resultant value to be dominated by the resistance of the paint film for many systems). Reference 4 provides a good indication of recent work on coated systems.

(7.5)

where b is the bandwidth of measurement and 8 is the Stern-Geary coefficient. Note that this analysis applies to only frequencies that are sufficiently low that the individual transients _can~ treated as instantaneous pulses of charge. Hence

j/#

J~

and are best replaced with the square root of the low-frequency limit of the corresponding PSD, in which case the bandwidth term becomes unnecessary. This method of analysis has been relatively little used, · but it has tieen shown to provide a realistic estimate of ~~c ::1E:!:stable p!t size f,y ~li.tmin11m a!loys.3

Artificial Neural Network Methods An interesting approach to the use of electrochemical noise for detecting localized corrosion uses an artificial neural network trained to identify localized corrosion

Painted systems are particularly liable to complications associated with differences in the properties of the two, nominally identical, electrodes used to measure current noise. The resistance .of.the. paint film will fall by several orders of magnitude as it starts to break down, and it is very unlikely that both electrodes will start to break down at the same time. We are therefore concerned with two very different electrodes. An important question is then which electrode will dominate the measurement. We have to make some assumptions about the relationship between potential and current noise in order to see how this will work out. For the high-resistance intact paint film we can assume that the propensity to drive current into an external circuit will be very small. both because the resistance is high and because the corrosion rate is very small. Consequently, when the two electrodes are coupled. this electrode will provide a negligible contribution to the current or potential noise. In contrast, the actively corroding low-resistance electrode will be able to supply much more current. and this will dominate the fluctuations in potential of the coupled electrodes.

Electrochemical Impedance and Noise

91

Furthermore the fluctuation in the measured current that results will consist of that current produced by the fluctuation in potential that is due to the low-resistance electrode ing through the resistance of the paint film on the high-resistance electrode. Consequentfy, if we measure Rp. we shall measure the resistance of the good high-resistance e~ctrode . This is not usually what · we want to know!

best analysis methods depend on measuring the current noise, and this will be affected by the high solution resistance in just the same way as the externally applied currents used to measure ac impedance or polarization resistance. Thl!S electrochemical noise measurements are not a j)anacea for high-solution-resistance systems, although they are a useful tool to add to other. electro-. chemical techniques.

This is a general problem with electrochemical noise measurements on electrodes that happen to be dissimilar (although the probiem is accentuated for painUilms by the very large differe.nces that are liable to occur) •.As a general rule (although not one that can be guaranteed to be valid in all cases), we expect the electrochemical · noise resistance measUred to be predominantly that of the electrode that is corroding more slowly.

As an illustration of these effects, Rgure 7.18 shows p :comparison of icorr obtained from Rp measured by LPR and by electrochemical impedance and Rn measured from the standard deviation of the potential and the current noise (in the frequel")cy band 0.001-o.s Hz, as it was based on 1024-point time records sampled at 1 Hz).

Thin-Film Corrosion and Low-Conductivity Solutions Atmospheric corrosion and other aqueous corrosion processes in nominally gaseous environments, such as · flue gas, ·occur in thin layers of liquid on the surface of the metal. This presents major problems to the investigator concerned with studying the corrosion mech8~ . nisms involved. Electrochemical methods can·be used by constructing two-dimensional probes with narrow electrodes separated from one another by narrow insulating gaps. Even with such electrode designs, the solution resistance tends to be high compared with the impedance of the charge-transfer processes on the small-area working ei!J.Ctrodes. · Similarly, very high solution resistar\ces ·are experienced in low-conductivity environments such as crude and refined oil products. In both cases, electrochemical noise measurements were expected to have an advantage over conventional electrochemical methods, in that there is no applied current; hence the effect of solution resistance was expected to be unimportant. Unfortunately, it is not quite that simple. It is true that the electrochemical potential noise will not be affected by the solution resi~tanr.e.• Howeve~. as we r.2·:e seen. mc:Jt of the ' This is correct in of the measurement's providing a true representation of the actual average ·fluctuation in the· potential of the sample being monitored. However. the high solution resistance will also influence :he local currents on the wo;i\ing electrode. causing them to be m.:-re localized. As the potentia: measured will usually be a far-field average. this ·may ·mean that smalllcc.~l changes associated with '"calized corrosion will be los:.

Within the experimental error, it can be seen that the actual corrosion rate (as determined by the weight loss measurement) is relatively constant, as would be expected, as it is limited by oxygen transport. Furthermore, Rp from LPR and Rn are affected by the solution resistance in the same way.

Resistance Noise Fluctuations in the solution resistance can be measured with a high-frequency (100-kHz) signal applied to the cell. 6 T'- solution resistance will be modified by gas bubbi$Sca,f\d nonconductive particles or second-phase fluid globUles in the solution, and the detailed examination of resistance noise may provide useful information on the generation of gas bubbles or the impact of particles on the surface. As an example, Figure 7.19 shows changes in solution resistance that are attributed to the evolution of hydrogen bubbles on two UNS 92024 aluminum alloy electrodes in EXCO solution.

Mechanistic Studies To date, the main application of electrochemical noise to m~ch~!"iS!!t: ~~u~i95 has been in ~elatic:~ ~::: pitting ' corro'sion, and the analysis has been performed in the time domain by use of analyses of the current and the potential transients associated with pit initiation and metastable pit growth. Thus several groups have looked at the statistics of the properties of metastable pitting events (such as frequency of occurrence, lifetime. and growth law).

Corrosion Testing Made Ea'y

92

1~~------------------------------------------, Unear Polarization Resistance

O

81) Electroc~mical IITlJlEld
120

•

Electrochemical Noise Resistance

•

Weight Loss

..E 0

'1

60

j

.. .

~

'·· 20 0

t

0.1

0.01 %~504

(a)

O

Unear Polarization Resistance

[ill Eledrochemic:al Impedance Spectroscopy •

Electrochemical Noise Resistance

•

Weight Loss

0.1

% N
Although the power spectra are derived from the individual metastable pitting events and can give some information about the properties of the events. it is difficult to derive the detaiied statistics of the events from the power spectrum. As a simple illustration. identical power spectra will be produced for potential time records with transients of peak amplitude Ema, and for

time records with 4 times as many transients with the same duration but peak amplitude 0.5 L.nax · Thus the power spectrum . does not allow us to ·distinguish between the number of events and their amplitude. If a simultaneous recording of potential and current noise is used, the additional information may allow more detailed analysis. If a shot-noise analysis is applicable,

93

Electrochemical Impedance and Noise

• variance. Its application to electrochemical noise is still the subject of ongoing research, and it is not discussed further here . See Ref. 7 to~ further-information.

50

0

,..

References

c:-= -50

•1.

e

See Bertocci, U., C. Gabrielli, F. Huet, and M. Keddam (plus P. Roussean for part II), "Noise resistance applied to corrosion measurements: I thea. . retical analysis and II experimental tests." J. Electrochem. Soc. 144, 31-36 and 37-43 (1997), for a full analysis.



·100

0

50

150

100

liJDe I

200

I

FIGURE 7.1 9. Variation of solution resistance that is due to hy· drogen bubble evolution.

this allows the estimation of the average charge in the transients {see page 90) and the corrosion rate (from Rn) . and hence the number of transients can be estimated. The shape of the current and the potential power spectra will also provide an indication of the shape of the individual current and potential transients {see page 84), and the low-frequency part of . the metal-solution impedance (from the electrochemi·cal noise impedance). It appears that the higher-order statistical properties of the time record, such as skew and kurtosis, and higher-order spectra, ·such as the bispectrum, may be particularly useful in extracting information from the noise. The bispectrum has roughly the same relationship to the power spectrum as tlie skew has to the

2. AI Ansari, M.S., A.A. Cottis, "Long term monitoring of electrochemical noise." Proceedings of the 13th International Congress on Metallic Corrosion, paper, 212, Australasian Corrosion Association, Clayton, Australia, 1996. 3. Cottis, A.A., S. Turgoose. "Electrochemical noise measurements: a theoretical basis." Mater. Sci. Forum 192-194, 663-672 (1995). 4. Bierwagen et al. Corrosion/98, paper 380: see the CD-ROM for the fullcopy of this paper. 5. Mendoza Flores, J. "Electrochemical techniques applied to the study of mild steel corrosion in 1%, 0.1 %, 0.01% and 0.001% Na2so• ." M.Sc. thesis. UMIST, Manchester 1993. 6. Huet et al. Corrosion/98, paper 378: see the CDROM for the fullcopy of this paper. 7. Cottis, A.A., M.A. AI-Ansari, G. Bagley, A. Pettiti. "Electromechanical noise measurements for corrosion studies." Maier. Sci. Forum 289-292(2), 741754 (1998).

:I

I

I

.~

Appendix A

Electrical Circuits:·· .:

~-~

Before we look at the analysis of electrical circuits, it is . useful to review some of the besic p_l"lysical phenomena and relationships of electrical theory.

Charge Charge is the fundamental electrical property. It derives from th~ properties of the charged elementary particles, such as the electron and the proton. The units of charge are coulombs, given the symbol C. The fixed charge on elementary particles results in the charge being quantized in units of the charge on one electron (1.6 x 10- 19 q;

Current Current is the motion of charge through a conductor. A flow of 1 C/s gives a current of 1 A. We generally use the symbol I for current variables.

Potential Electrical potential is a measure of the potential energy of a charge at a specific point. It is measured (hypothetically) as the work done in taking 1 C of positive charge from infinity to the specified point. The potential is 1 V when the work done is 1 J.

Voltage or Potential Difference The potential difference or voltage between ":'.'10 points is simply the difference in electrical p9tentia 1 between

the twa points. A voltage of 1 V exists between two points if 1 J is required for moving 1 C of charge from one point to the other. A consequence of the definitions of voltage and current is that the power dissipated when a current flows is given by the -voltage times the current: power (W)

= voltage (V) x current (A).

(A.1)

Resistance When a voltage is applied across a conductor, it is often found that the resultant current is directly proportional to the voltage. This leads to Ohm's law: ·

= current (A) x V = IR

voltage (V)

resistance (0), (A.2)

where R is known as the . resistance qf the conductor (with units of ohms, n =VIA). Like many laws, Ohm's law is an approximation. Metals generally obey Ohm's law very closely, but semiconductors and metalsolution interfaces do not. When the relationship between the current and the voltage is nonlinear, the term resistance is often used to refer to the slope of the V-1 curve, in which case the resistance is a function of the current. Ohm's law applies to both the time and the frequency domains. In the time domain. V and I are the instantaneous values of voltage and current. In the frequency domain, both V and I are complex functions of frequency. having both aiT'plitude and phase. Howe\'er. R is a real constant. Consequently the current

96

Corrosion Testing Made Easy

through a resistor and the voltage across it have the same phase.

angular frequency of the sine wave being considered 2rr f, where f is the frequency in cycles per second, or hertz) .

(=

•

Capacitors and Inductors

Constant-Phase Elements

Capacitors are devices that ~o·re charge witoout allow~ ; ing continuous currents to flow. A typical capacitor construction consists of two parallel plates separated by an insulator. Charge will be stored by the electrostatic attraction between the two plates when the potential
•

In electrochemistry, we frequently encounter elements in equivalent circuits that have a character somewhere between a capacitor and a resistor. These are known as CJ;~s. and the best-known example is the Warburg or diffusional impedance (see page84).

Impedance The relationships between V (in volts) and I (in amps) in the frequency domain may be generalized as a form of Ohm's law:

V= IZ, .

capac1tor

. d uctor 1n

I

= CdV Cit,

v=

dl L-,

dt

(A.3)

where C and L are, respectively, known CiS the capacitanc~ and the inductance of the device or ~ircuit being

considered. The units of capacitance are farads (abbreviated F); 1 F will give a current of 1 A when the rate of change of voltage across it is 1 V/s. It is unfortunate, but unavoidable, that the symbol C is conventionally used for capacitance as it conflicts with the use of C for coulomb, the unit of charge (although the C for capacitance is usually italicized , whereas the C for coulombs is not). The uni~s of inductance are henrys (abbreviated H); an 1inductance of 1 H will produce a voltage of 1 V when the current through it is changing at 1 A/s. The relationships in the frequency domain are somewhat easier to handle:

.

capac1tor

V

I = ;,..r.· J

inductor V

~-

-

= j w L I.

(A.4)

In these equations j is ,; =1 (implying a 90 · difference in phase between the current and the voltage). and w is the

(A.S)

where Z (in ohms) is known as the impedance of the device or circuit in question, and V, 1, and Zare all complex quantities. From the discussion above, we can see that the impedances of the three simple components are resistor capacitor inductor

Z= R,

z=

1

jwC'

Z= jwL.

(A.6)

As 11j = - j, the impedance of a capacitor can also be written as - j 1wC.

Combining Impedances Once we start to analyze circuits containing more than one element, we need to work out the combined properties of series and parallel combinations. In the time domain, it is relatively complicated to analyze combinations nf impedances. a!=\ we usually need to solve Simultaneous differential equations. In the frequency domain, the analysis is relatively easy, and we simply need to apply the rules shown in Figure A.1 . using complex arithmetic (i.e .. using complex numbers) for the calculations.

97

Elec:;trochemical Impedance and Nois'e

~' ~--------------------------------~----------------------------------------------

,..,.., _.

,. ! -:~~- ~- . --: _?!_I· . 1

-

z

lmpcll:mcc~

~

1

1

-+ --Z1 Z2

Z1

Z2

·. · Impedances in series - add impedances

FIGURE A.1. Combining impedances.

·'

, To-analyze electrical circuits we use Kirchoff's Law: "The sum of all of the currents into any node in an electrical circuit is zero." ', . (A node is simply a region of an electrical conductor

§\•. ' connected to two or more circuit elements.)

f

Then for any circuit we can write a set of simultaneous equations relating the currents and voltages at the various parts of the circuit. Solving these equations allows !: us to determine the behavior of the circuit. Note that the f: analysis of time-varying signals in the time domain for ~ circuits i~volving ~pacitorsan~or ind~ctors will_ involve rfJ : the solUtion of Simultaneous dtffer.enttal equattons. In ~ the frequency or Laplace domains, the equations are ~urely alg~braic and hence far simpler to solve. ~--

p.\

lr. , lt\•r 'i.. ...

iL '

i.w . Superposition Theorem ~~ .

ti-

~-

tr

~

" 1

J .

Linear and Nonlinear.Systems

i.n fl:•ralicl -add If/. fittancc)

z

t
cat systems). Note that this theorem (and most electrical circuit theory) depends on the system's being linear.

In circuits involving multiple voltage or current sources, it is not oecessary to analyze for the effects of all of the sources simultaneously. The behavior'resulting from individual voltage or current sources can be analyzed individually, with other voltage sources replaced with open circuitl" ~nd current sourc:P.~ with short circuits. Then the overall currents and voltages wiil be the sum of those obtained for the individual sources. The one exception to this rule is when one source is dependent on another (this is rare :or eiectrochemi-

Ttle above analyses assume that the_systemis Unear (i.e., the relationship between current and voltage does not depend on their magnitude). The analysis. of nonlinear systems is rather more complicated, as the behMior becomes amplitude dependent, and the distortion of sine. waves produces harmonics of the input signals. Unfortunately etectrochemicat systems are very definitely nonlinear, but to make use of simple analyses, it is generally assumed that they can be treated as linear, providing the amplitude of any fluctuations is small (less that 10 mV should give reasonably linear behavior). Full analysis of nonlinear circuits is beyond the scope of this book.

Electronic Components It is frequently necessary in electrochemical measurements to use electronic ·components. They are used to construct dummy cells, to act as references for impedance measurements, and to convert current to voltage before measurement. Although modern electronic components generally offer a very high quality at a lowprice, it is important to be aware of the types of device available, especially when it is desired to make very low noise measurements. For this reason, this section summarizes the characteristics of standard electronic components.

Resistors The types of resistor available are summarized in Table A.1. When selecting a resistor, the most important requirement is to ensure that the device has a suitable power rating. The power dissipated in a resistor is given by V2 R or / 2 R. wh~re V il" the voltaae drop acrns~ the resistor, R is the resistance of the resistor, and I is the current through the resistor. The power rating of a resistor is the maximum continuous power dissipation that can be tolerated without permanent damage occurring. However, a resistor working close to its power rating will

98

Corrosion Testing Made Easy t

Table A.1 Resistor Types· Resistor Type

Typical Power Rating(W)

Carbon composition Carbon film Metal oxide

0.25-1

Metal film

Wirewound

0.25-2 0.25-3 0.1-0.5 1-1000

Noise, Power:

Range of Values Available (0)

1-107 1-106 0.1-106 0.001-104

Standard Precision (%)

' High

5

Moderate Low Low Low

5 5

Temperature coefficient!i (ppmf>C)

Costs~ Low

15Q-1000 300 15-100 5-200

1

5

Low Low Low High

• I

~· '

The values quoted are typical ranges, taken from a 1996 electronic components catalog. Values and properties outside the ranges given will be available from specialist suppliers, usually at a price.

·

t Carbon composition resistors are no longer ~y available, largely because they have been superseded by better types. They are included

largely as a warning that they tl!lnd to ha~ high noise.levels. t "TYPes shown as having tow noise powers will give noise levels close to the theoretical thermal noise.

SThe temperature coeffiCient is the artlount gy which the resistance will change as the temperature changes. Note that the temperature may change because of the power dissipated in the resistor, as well as fluctuations in the ambient temperature. '{ Low-cost devices will usually only cost a few pennies each. High-power and especially high-precision devices may cost much more. For example, 0.1% metal film resistors cost-20 times as much as 1% devices.

get rat~er hot, and this will tend to change the resistance. Consequently, you should aim to work well below the maximum power rating (10% of the maximum rating will usually be acceptable, except in high-precision applications).

.

.

The next important property of a resistor is its pre· cision. Low-cost standard devices typically have a precision in the range ±1% to ±5'}'o. ln a9(iition. these devices will typically have a rather.high temperature coefficient. Higher-precision devices can be obtained, but the costs become very high. .,

A third

importapt property of a resistor is the noise level. An ideal resistor will exhibit a thermal noise given by ~ 4kTR, where ~is the voltage noise PSD, · k is Boltzmann's constant (1.38 x 10- 23 J/K), Tis the absolute temperature, and R is the resistance. Thermal noise is white (i.e,, the PSD is independent of fre· quency). Most modern resistors approach this ideal. but carbon composition resistors have a higher noise level and should be avoided.

=

Example calculatiohs We require a set of resistors to allow current to be measured with a voltmeter. The voltmeter measures ±200 mV full scale, and we need the following ranges:

±2J.LA full scale ±200J,LA fuH scale ±20 mA full scale ±2 A full scale As an example we shall calculate the resistance value and power rating for the ±20-mA range. ·

=

=

We want V 200 mVfor/ 20 rnA. From Ohm's law ( V = IR), this implies a resistance of v; I 10 Q.

=

We also need to check the powenfissipated in the resistor. At first sight, we should calculate the power with 200 mV across the resistor: Power is given by R = 0.004 W, so -a standard 0.1-W resistor would probably be adequate, although we might use a 0.25-W device to minimize heating effects and keep well within our guideline of dissipating only 1Oo/o ofthe rated power. However, this assumes that ~ never apply more than 20 rnA to the circuit. As we cannot guarantee this, we need to allow for overload conditions. There are two ways in which we can approach this:

\12;

• We can put a fuse in series with the resistor to limit the maximum current. In this case we could use a fuse rated to fail at 100 mA, in which case the maximum power will be given by / 2 R = 0.1 W [= ( 100 mA) 2 x 10 Q]. In this case a 0.25-W resisior

99

Electrochemlcai Impedance and NoiS'e

will still be adequate (as it will never be required to dissipate more than 0.1 W for long). • We can design to accommodate the maximum probable voltage drop across the resistor. If this resistor were to be used with a potentiostat with a ±15-V output capability, it would be reasonable to design f~r 15 V across the resistor, ·leading to a design power of 22.5 W [= (15V) 2 j10 n]. This power rating is obtainable, although it will be rather large and expensive, and it may be difficult to get the required precision. An alternative that may be attractive is to use a number of equal resistors of lesser power rating. For example, 10 100-n 3-W·resistors could be wired in parallel to give an overall rating of 30 W. As this power will be dissipated only under fault conditions, we can reasonably design on the basis of the full power rating of the resistors. In this situation it may.. also be necessary to protectthe voltage measuring circuit, as it may be damaged by having 15 V applied to it. The calculation of the resistor values for the other ranges is left as an exercise for the reader.

~·

8

, '·

:::. !

Circuits such as this will be found in almost all current measuring devices. You should..be aware that not all commercial instruments will provide adequate resistor power ratings, and it is common for current measuring circuits to be damaged by current overloads. It is also common for the fuses used to protect the highereurrent ranges against overload to have blown, and this is often a cause for a multimeter to appear to have broken. It is always worth checking the fuse-a new fuse is much cheaper than a service call. A second factor that needs to be taken into , · - e~pecially when high currents ar~ being .measured, is the influence of the resistance of connecting wiring and switches. This will often be a significant fraction of an ohm and could give large errors. Consequently we should use a four-terminal connection to each resistor (Figure A.2). By separating the currentand voltage connections, we ensure that the resistance across

\,:-:- -~,;le /

Con:-~tions

"-...

!1. --.' C' ... -2-~~t Con~'~·..:·t:,:.rs

~--

FIGURE A.2. Four-terminal connection to a resistor.

which the voltage is measured is unaffected by resistances .

Capacitors Wpereas it is relatively easy to produce resistors with a wide range of values, larger values of capacitance require quite large volumes to store the charge. Consequently, there are several methods used to produce caRacitors, and these have different characteristics. These are summarized in Table A.2.

j

:! .,I

The two . major categories of capacitor are those based on a classical paranel plate capacitor With a solid insulator ·(dielectric) .between metal plates, .and those based art the capacitance of an electrochemical doublelayer or ive oxide film (known as electrolytic capacitors). At the relatively low frequencies of interest in EIS and electrochemical noise measurements, nonelectrolYtic capacitors can usually be assumed to be near ideal, although the precision of low-cost devices is generally rather poor. In contrast, electrolytic capacitors have important limitations: They are usually polarized (i.e., one terminal must always be more positive than the other), they exhibit a leakage current (see Table A.2 for typical values), and at higher frequencies they do not have a purely capacitive behavior. Thus electrolytic capacitors should be avoided wherever possible, except for noncritical applications such as filtering power supply noise or acting as charge--storage devices on power supplies, for which their high capacitance values are essential. · · ·. · · · Costs have not been indicated in Table A.2; the cost tends to depend on the size, voltage rating, and precision of the capacitor, ranging from a few cents up to several dollars.

Inductors Inductors are the most expensive to produce of the three major electronic component types, and consequently they are avoided as far as oossible. Their main applications are as power storage and conversion de· vices in power supplies and as components of highfrequency-tuned circuits. Inductors are invariably fabricated as coils, usually wound on a former of magnetic material that serves to increase the inductance (but also makes it nonlinear). There is a trade-off between high

:I

,._

100

Corrosion Testing Made Easy

Table A.2 Typical Capacitor Properties Capacitor Type•

Range of Values Avallable'(F)

Electolytic Tantalum Double layer Polypropylene Polycarbonate Polyester Polystyrene Mica Ceramic

Precision (%)

10 7-0.1 10 7-10 4 0.1-10 10··10_10 7 10' 10-10 5 10 "-10 5 10- 11 -10 8 ... 10-'2-10 a 1Q-11-1Q -S

20 20 50

5-lO 5-10 5-10 1-5

Temperature Coef:ficient · (ppm!" C)

6-450 6-35 2.5-5 63-1500 63-400 63-630 ' 5()-,.630 500 5Q-15000

·200 100 200 so!-2oo

1

1Q-20

Voltage Ratingt(V)

3Q-100

Leakage Current (J£A)

5 0.1

• The fwst two capaeitor types rely on the capacitance of the ive oxide on a metal in an electrolyte (aluminum for standard electrolytic capacitors); Double-layer capacitors rely on the eiectrochemical double layer between an aqueous solution and finely divided carbon (hence the vwy low-voltage rating). The remainder(Jse the material specifying the type as the dielectric in a parallel plate capacitor. I Values in..liM ranges quoted will typicaUy not all be available simultaneously. Thus the maximum capacitance available will go down as the voltage ratinggoee .up. The size and the cost will usually also increase as the capacitance and/or voltage rating increases.

inductance (which requires many turns in the coil) and high current rating (which requires thick low-resistance wire In the coil). Because of the wide range of possible current ratings and _inductances and the low usage of inductors in electronic circuits, relatively few standard devices are available.

Spe~ification and Sources of Instrument NQise

We can examine the general principles of the specification of instrument noise -by considering the noise characteristics of a standard operationa!. amplifier~ This is a device with two inputs, the inverting and noniiwerting input, and one output (see Figure A.3): The transfer function (the way in which the output voltage is rel~~ed to the input voltages) of the amplifier is

.

+Ve e.=l?-e·..,r e . power supplv

+

"'

-ve pow~r suppl\ . FIGURE A.3. Operational amplifier.

given by

eo =

(~

-' a.)A

(A.7)

where A, the gain, is large (105-108 ), e 0 is the output voltage from the amplifier, and ~ and a. are the voltages at the noninverting and the inverting inputs of the amplifier, respectively. There will be many sites witt)in the circuit of the amplifier at which noise will be generated, but these can all be converted fc5 an equivalent noise at the input of the amplifier. Thus an amplifier is specified in of its inpt.it noise. However, it is not possible to combine the effects of the voltage noise and the current noise at the input. Thus an amplifier will have two noise values specified, the input voltage noise and the input current noise. There is a tendency to assume th~t the input voltage noise controls the behavior when the amplifier is used to amplify voltage, whereas the input current noise is relevant when amplifying current. However, this is not necessarily the case, as we can see by considering simple operational amplifier circuits.

Noninverting Buffer This circuit (see Figure A.4) provides a lowimpedance output voltage that should be the same as

.

Electrochemical Impedance and Noise

101 .

.

'··

v....

FIGURE A.4. Noninverting buffer amplifier. FIGURE A.S. Current amplifier"1:ircuit.

the input voltage. The input voltage noise Vn (in volts), clearly just adds to the input signal and is reproduced at the output. The effect of input noise current In (in amps), is slightly less obvious, but if we recognize that this current must flow th.rough the impedance of the voltage •· source, we can see that it will result in voltage noise given by (A.8) t •. where Zs (in ohms) is the source impedance at the relevant frequency. To see which of these noise sources is the more significant, we need to consider typical values. As an example we can consider the OP27, which is an inexpensive but high-quality modern device. One grade of this amplifier has specified maximum noise figures, as given in Table A.3. Note that the noise is close to being white at higher frequencies· and can the.refore be specified by its PSD (or .JPSO as is normafty used). At frequencies below 10 Hz, the noise tends toward 1/f characteristics, so the total noise power over a range of frequencies is specified. For the OP-27 these figures imply that the inp•1t current noise will typically be more significant than the input voltage noise when Zs is greater than -10 kQ.

Table A.3 Nqise Characteristics for the OP27 Frequency

.' 0.1-10 Hz

Input Voltage Noise

Input Current Noise

0.08 ttV

10Hz

3.5nVI./Hz

30Hz 1kHz

3.1 nV/JHz 3.0 nV/JHz

1.5 1.0 0.4

Current Amplifier 'I

The current nmplifier (or ZRA) converts an input current to an output voltage by ing it through a measuring resistor (see Figure A.S). The amplifier keeps the two input terminals at essentially the same potential. In this circuit, the amplifier input noise current will essentially just add to the current being measured, whereas the input voltage noise Vn (in volts), will produce a current through the amplifier resistor that is given by Ohm's law:

(A.9) where Rib is the resistance (in ohms) of the resistor. As with the voltage amplifier, the input cur~ent noise will dominate the amplifier noisewoen lnRn; > Vn. If we look at these two analyses, we can see that we can specify a critical impedance for the operational· amplifier, below which ·tfte·vottage·noise is the more important, and above which the current noise dominates. Note that the ·critical impedance is highly dependent on the amplifier technology. The OP27 is bipolar; field-effect transistor (FET) technologies typically give a somewhat higher voltage noise. but a much lower current noise, so the critical impedance is much higher. For exam!')le the TLC 2201 C has a critical impedance in the region of 106 n,.so the behavior is dominated by the voltage noise under most circumstances. This type of device should therefore be used for high-impedance systems, such as painted metals, whereas low-voltage noise bipolar devices such as the OP27 should

Corrosion Testing Made Easy

102

0.1

iCJ ...

K.gh lniquency • gain falls rapidly, so signal is blocked

Low frequencies • gain is nearly one, so signal is ed

O.o1

.:

0.001

0.0001 ~--.-----..--__,..--....-------.,......-1000 10000 100000 1000000 10000000 10 100

l .

Frequency l Hz

FIGURE A.6. Typical low--filter transfer function.

be used for low-impedance systems such as active metals.

Electronic Filters EleCtronic filters are circuits that allow some frequencies .to while removing others. The transfer function of a filter can be displayed on a Bode plot, which presents the gain of the circuit as a function of frequency, togethef with the input-to-output phase shift ·c· (Figure A.6), · The simplest filter circuit consists of one resistor and one capacitor. The input voltage ~s applied across the two components in series, while the output voltage is taken across one component. If the output is taken across the capacitor (Figure A.6) the gain at low frequencies is 1, as the impedance of the capacitor will be high relative to that of the resistor, and the phase shift will be zero. At the frequency at which the impedance of the capacitor is equal to that of the resistor (f = 1/2JT RC). the gain will be 11 ../2 and the phase shift will be -45'. At higher frequencies the gain will decrease continuously, wtth a slope approaching - 1 on the log-log scales, and,the phase shift will tend to -90". This circuit is known as a low- filter because it allows low frequencies to unchanged while reducing the amplitude of higher frequencies .

If the resistor and the capacitor are exchanged, so is the behavior of the filter (Figure A.7). This is known as a h_ igh- filter. ·

Filters that use just one resistor~apacitor pair are known as first-order filters. In principle, additional resistor~pacitor stages can be added to give a sharper drop in gain versus frequency ('known as the roll-off slope). Roll-off slopes are often described in units of decibels per octave or decibels per decade; see page 79 for further explanation. Thus adding a second resistor~apacitor stage will give a slope of -2 or 2 on the log-log scales, and this is known as a secondorder filter. In practice, problems occur when resistorcapacitor stages are coupled together . because the current drawn by the later stages perturbs the earlier stages, and intermediate amplifiers are needed to overcome this problem. The resultant devices are known as active filters. The most common active filter circuits are based on the work of Sallen and Key, and these use one amplifier for every two resistor~apacitor pairs. Figure A.8 shows second-order low- and high- active filters. The resistor and the capacitor values can be adjusted to give different filter characteristics. Three common filter characteristics (Figure A.9) are • Bessel: This has q relatively shallow transition from the band (the region where the gain is 1) to

103

Electrochemical Impedance and Ndise

Low frequencies • gain falls rapidly, so signal is blocked

High frequency • gain is nearly 1, so signal is uricnanged

0.1

·c iii Cl

0.01

0.001

.:

0.0001

..._--.-------,----.----,.--.....,...---.----1 10

100

1000

10000

100000

1000000

10000000

Frequency I Hz FIGURE A.7. High--filter transfer function.

the stop band (the region where the gain is faning). This leads to a gradual change in the phase shift, which gives the filter a good response to transient signals. • Butterworth: This has intermediate characteristics, With no overshoot in the band but a reasonably sharp roll-off.

Cl 47 nF

Output Input TC222nf

• Chebyshev: This has a sharp transition from the band to the stop band, but at the expense of fluctuations in gain in the band and a sharp change in phase shift, which tends to cause ringing in transient signals. Higher-order filters give a steeper roll-off slope (Figure A.10). The circuits shown in Figure A.B give a filter with a transition frequency of 1 kHz; for higher-order filters the second-order circuit is repeated the required number of times, but ideally the component values need to be modified to obtain the optimum response. To adjust for an alternative frequency, scale the values of the resistors and capacitors appropriately (the frequency

'\;'Ground

Second-order low- active filter Rl S. l k!l

0 Output

••••

_10 u \ ··y&.fi3r..M:>rth ....... \

Second-order high- active filter FIGURE A.S. Active filter circuits.

.

\ \

• .• ' ') k!l

~Gr0Und

CD

~··...-Bessel

Cheybeshev

_,A\ .

, \

-~

-20

\ \.\. ~--~----~-~~~~ 100) 100 lOOXl

Frequency /Hz

FIGURE A.9. Filter characteristics.

0

<.!>

104

Corrosion Testing Made Easy

• 0.1 0.01 0.001

..

c:

"

•.,

0.0001 1e.os 1E-06 1E-07 1E.OS 1E..Q9 1E·10 10

100

1000

10000

100000

1000000

Frequency FIGURE A.10. Response of active filter circuits of orders 1-8.

·15V ' +ISV

CO'INl

"

·ISV

FIGURE A.11. Signal conditioning circuit for electrochemical noise. Note: All operational amplifiers are shown with noninverting input at the top.

Electrochemical Impedance and Ndise

105

""'

CO*Q

Silk Screen CCHn

.....

e~~~ I "' il ca

ct1jf~

C5

co

iY"~

rr· rtff i:J''S

o. . .

·o.c t\.

.. ,

C1

cz

I I

~··

[f

-cJC'9 ..,

IT

C1li(J C3 C7

IT

-QC11

Component

S~PCB··

Solder Side PCB

FIGURE A.12. Printed circuit layout for signal conditioning circuit (not to scale). PCB, printed circuit board.

is proportional to 1/ RG) . The capacitor values listed are drawn from the E6 series, • and it will generally be * In order to provide a measure of standardization. standard senes

have been defined for component values.The values have an ao· proximately constant multiplier between adjacent values.The valu~s defined for the E6 series are ...• 1. 1.5. 2.2. 3.3. 4.7. 6.8. 10 .. .. giving six values per decade.

simplest to change these values only by factors of 10, with the resistor values being used to fine tune the freq'Jency. ld~ally, the resistor values should be ,.,_ 10 kQ io minimize the noise intro9uced by the resistor, although larger values may be necessary to achieve low cutoff frequencies with realistic capacitor values (it is difficult to obtain values of more than 10 11 F for nonelectrolytic capacitors).

106

Signal Conditioning Circuit for Electrochemical Noise

Corrosion Testing Made Easy

•

A complete signal conditioning circuit for electrochemical noise measurements is presented in Figure A.11. For the potential measuring circuit this uses a voltage amplifier (U4) with a gain of 10, followed by a fourthorder low- antialiasing filter (US and U6). The current measuring circuit uses a conventional current amplifier (U1) with a choice of one of three feedbact< resistors selected by switch SW1. This is follriwed by a fourth-order low- antialiasing filter (U2 and U3, identical to that for the voltage circuit). To allow operation of several copies of ,the circuit from a single power supply without leading''to ground-loop problems and to allow the use of low-cost plug-mount~d de supplies a de-de converter (VR2) is used to provide isolated ±15-V power supplies. This is preceded by a voltage regulator (VR1) to ensure that the power supply voltage is stable over a range of input voltages. In addition, the input is ed through a bridge rectifier (SA-labelled 01 in Figure A.11) so that the circuit will function correctly with either polarity. of Input voltage. The .de-de convertor may introduce a certain amount of high-frequency noise, and additional decoupling

capacitors are used to reduce this noise and to ensure stability of the operational amplifiers. A printed circuit layout for this amplifier is shown in Figure A.12; f7igure A.13 shows noise power spectra for typical version of the amplifier. The version shown used OP27 amplifiers-somewhat better low-frequency • performance could probably be obtained with chopperstabili~ed amplifiers such as the MAX 320. Owing to reproduction limitations, the layout is not reproduced to scale, and interested readers should the author for a better-quality original.

a

.:

~ 0.01

">

0

~

1E·7

Potential + Reference Electrode NoiSe

"~

......

0 1E-12

~ 1E-17

Current Noise

1E-5

1E-4

1E·3

0,01

0.1

Frequency I Hz FIGURE A.13. Instrument noise spectra for signal conditioning circuit.

•

Appendix 8 .

I;

It is not the intention here to review the whole area of electrochemical kineti~ but .to provide a basis for the used in impedance spectroscopy and a formal approach to .the origin of some common types of response. There are many texts that will provide the derivation of the basic equations used here.

If the potential is changed, then (8.3) which defines the charge-transfer resistance the anodic process:

dE -d.

Direct-Current Electroc.h.emical Kinetics Activation~Controlled

·

Systems

= Rct,a·

For a simple activation-controlled metal dissolution (anodic) process,

+ ne,

(8.1)

with the rate of the metal ion reduction reaction considered to be small, the current density ia is given by

OX+ ne-+ Red

I

.

(. E- E~\

la=lo,aexp.

.

lc

. C = fo,c

$

( (E- Eg)) exp f3c

u

I·

..

'

I

(8.6)

where Cs is the surface concentration of the species being reduced, often oxygen or hydrogen ions in corrosion systems. The other are as for the anodic reaction in Eq. (8.2), with the subscript chere indicating thecathodic reaction. From Eq. (8.6), if Cs does not change with potential, i.e., for activation control, we obtain (8.7)

(8.2)

fJa

··.·:

(8.5)

the cathodic current density is given by

1.;:: _; , .

(8.4)

For an irreversible cathodic reaction,

These are systems in which the rates of the electrochemical processes are controlled by the charge transfer across the metal solution interface (hence the alternative term, charge-transfer control).

M-+ Mn+

Ia

Rct.a for

io." is the exchange current density for the anodic reaction, E is the electrode potential, E~ is the standard electrode potential. and ~.; is the (exponential) Tafel coefficient.

where we can define, as above,

dE --d. lc

= Rct,c•

(8.8)

with the minus sign because the cathodic current

( ; r..

Corrosion Testing Made Easy

108

(8.9) • and so (8.1 0)

Above it was assumed that the surface concentration'of the cathodic reactant did not vary with potential. This is approximately true if the cathodic current density is much less than the limiting mass-transfer-controlled curref}t density. For cathodic reaction, neglecting any participation of the reve'i'se (oxidation) reaction, the limiting diffusion current itim is given by Fick's first law:

a

.: 1

1

1

Rct,tot

Rct,a

Rct,c

--=--+--.

(8.11)

Thus the charge-transfer resistances for the two reactions are added in par~llel to get the overall resistance of the interface.

nFOCbutk

1 = . (1 + 1) lcorr

f3a

f3c

(8.13)

ilim = -----:--()

where D (in square centimeters per second) is the diffusion coefficient, 8 (in centimeters) is the diffusion layer thickness, and C bulk is the bulk concentration of reactant (in moles times inverse cubic centimeters for consistency of length units). n is the number of electrons consumed in the reduction of 1 molecule of oxidant and F is Faraday's constant (96,485 C mol- 1 ). The dependence of surface concentration on current is gi_ven by

For a system at its corrosion potential, ia = ic = icorr and so, by combining Eqs. (8.4), (8.8), and (8.11) we find

Rct,tot

~···

Mass Transfer Effects

increases as the potential decreases. For the corroding interface, the net current is given by

ic<'l Cootk- Cs = nFD'

(8.14)

A combination of Eqs. (8.6), (8.13), and (8.14) give an alternative expression for the de current: (8.12)

'

ic= io,c .( 1- -..ic) lhm

where Rct,tot is the slope of the E-i curve at Ecorr· This is the standard Stern-Geary equation used for polarization resistance m~asurements, with two differences. Here the term charge-transfer resistance Ret is used instead of polarization re~istance Rp (as seen later in this case, with the assumptions above, Rp Ret). Also, the derivation here does not require any assumptions to be mad~ about the · linearity of the current-voltage curve. The corrosion rate is obtained from the slope of the curve at Ecorr. and the results obtained will be insensitive to. the amplitude of the applied voltage. However, the amplitude should be kept as low as possible to avoid causing changes to the corroding metal under test.

cbulk

exp

[-(E-Eo)] . .

fJc

c

If ic « ium then this is the same as Eq. (8.6), but this form of the equation allows for the change in surface concentration with potential (or current)~ Rom this,

=

When ac methods are used to ohtain corrosion rates. the same equation is used, with the resistance determined from the ac response of the system. In this case, nonlinearity of the current-voltage curve is important. This is because the current throughout the ac cycle is used in the calculation of the resistance. rather than the slope at Ecorr·

. (8.15)

(8.16) Now, at the corrosion potential,

dine! dE

ic0rr f3a

icorr f3c

i~r

1 Rp

- - = - ·+ - - - . - =·- ' f3c11im

(8.17)

and this defines the polarization resistance of the sys· tern. As icorr approaches itim then (8.18) The charge-transfer resistance is the slope of the potential-current curve with constant surface concentrations. whereas the polarization resistance is the slope of the steady-state potential-current curve.

Electrochemi.c al Impedance and Noise

109

•

).

E FIGURE 8.1. Current versus potential for reaction with a cathodic limiting current.

Equation (8.17) could be written as

(8.19) ~;

. . D

such as a second cathodic reaction. To obtain the overall impedance of an electrode we have to consider only the impedances of the individual processes.

Activation-Controlled Processes and Rp is greater than Ret. .

··. . · It is often convenient to produce schematic potential::: current curves for corroding systems, and because ~· · of the exponential nature of the relationships these ' are often drawn as E-log i curves. Figure 8.1 shows the current potential curves for an activation-controlled anodic reaction, Eq. (8.2), and a cathodic reaction with a limiting current, E~. (8.15).

Alternating-Current Responses From de measurements, we obtain only the one value of resistance. Definitions of polarization and chargetransfer resistances were given above because, with ac measurements, it may be possible to separate the charge-transfer resistance from mass transfer effects, as the responses occur at different ·frequencies. The 0verall i&;;istan~.;~::: ur iiiiJJt>udnce of the electrode comes from a parallel combination of the impedances of the individual processes, or we could say that the overall ittance is the sum of the ittances of the electrode processes. The derivations above were for one anodic and one cathodic process but can easily be extended to include other electrochemical processes,

The impedance of these is simply the charge-transfer resistance of the process, and the equivalent-circuit component for this reaction is a resistor:

Ret ::;

~. I .

(8.20)

This assumes a small-amplitude applied voltage, and what constitutes a small amplitude needs to be considered . .

Effect of Amplitude of Applied Voltage Considering just one activation-controlled anodic process, e.g., for a system with a mass-transfer-controlled cathodic reaction whose rate does not vary with potential, we find

. = .• .. , f") \f3a t~o·•

ta

where 'I

= E-

Ecorr

The net current is i a

exo -

(8.21)

= Vo sin(wt). -

icorr or

inet = icorr [ exp (

;a ) - 1] ·

(8.22)

1 Corrosion Testing Made Easy

110

conditions above, the measured resistance depends on amplitude according to

Using the Taylor expansion, we obtain exp(x)

x2

x3

= 1 + x + -2! + -3! · -·

(8.23)

.' Rrneas

then

.

.

1 '7 = lcorr [ - + - ( - ) fia 2! fia .

/net

'7

2+.

)3 · · ·] ,

'7 . -3!1 ( -fia

(8.24)

eix + e-ix 2

,

(8.27)

'"), v;;

1 + 8 02

where, as above, fial icorr is the resistance at small amplitude. A plot of the· error in measured resistance against amplitude is shown in Figure 8.2. The error is only few percent if Vo!f3a is <:112,-and for typical anodic reactions th is corresponds to between ~ < 12 mV (ba 60 mV per decade) and ~ < 6 mV (ba 30 mV per decade). The amplitude set for the measurements is Vrms. which is 1.4 times less than the peak voltage '4> used here, and so an amplitude of 5 mV rms will rarely give significant errors.

a

=

= eix_e-ix · 2j.t

cos(x) =

= -lcorr . (

f3a

and the in 17 2 and higher powers are responsible for ~he nonlinearity Q( the potential current plot. Replacing 71 with ~ sin(wt) and using

sin(x)

1

fia

(8.25)

we then obtain

=

Irreversible Cathodic Process with Mass Transfer Effects

. =.lcorr [Sin· (wt>(Vo 1\{? - +- · · ·) · · ·]

lnet

B /3~

f3a

plus sin(wt) in higher powers of in cos 2wt and sin 3t.>t and higher.

~

(8.26) . .

1 f3a and

Thus the nonlinearity of the response affects the amplitude of the current response at the applied frequency, and this affects the apparent resistance. For the

For this we can use the de calculations to define two resistances. The charge-transfer resistance is that measured if the surface concentration of reactant is constant, and this happens at high frequencies. At lower frequencies, the surface concentration changes during the ac cycle, and this gives rise to a diffusional impedance. At very low frequencies the measured resistance is the polarization resis~nce.

0.9

1 a: ....

I

a:

0.8 .

o.i ..._ _ __ - -··· -·- - . ---·- ··---- --·--

o

1.5

0.5

·v•• beta FIGURE 8 .2. Error in measured resistance as a function of amplitude.

111

Electrochemical Impedance and Noll;e

Diffusional or Warburg impedances

C(1l, t) = Cb

Equation (B. 7) defines the cathodic current density as a function of potential, but to calculate the impedance we need to know how the potential-varies with current. In general, the potential is a function of the current and surface concentration of .metal ions,

E = f(i ,C5 ),

(8.28)

from which the response of the electrode to a timedependent pe~turbation is given by

...

dE

(a£)

di

( oE) dC,

Cit= · a; · c..-dt+ · oc, ; d"t

iiC(x,t)J [

ax

x=O

(for all t), ·

= i(t) , nFD .

(8.35)

where Cs is the steady-state value of the surface concentration of the cathodic reactant and Ct~ is the bulk ctmcentratiOn, and 8 is the diffusion boundary-lc;tyer thickness. Solution of these gives ... (8.36)

.: where

(8.29)

For a sinusoidal current of a'ngular frequency w (in ra-

t · dians per second), applied in addition to the de current, the dependence of current on time is given by

i(t) =ide+ i0 sinwt,

(8.30)

l.

where ; 0 (in amps per square centimeter) is the peak current density and ;de is the steady-state cathodic current under the de conditions qefined by Eq. (8.15). Three of these are simple to calculate:

di

·0

t

dt

( a~) C. = -Rct.c = -~.

(8.32)

(:~\ = 6,-

(8.33)

This can also be written· as

,

Zd{w)

, h•(.Jw 82)2 0

= aw2 tan

(1 - 1') •

(8.38)

with

lc.

The other term describes the time dependence of surface concentration and is obtained by solving Fick's second law, Eq. (8.34), with appropriate boundary conditions:

(8.34) where C(x,t) is the concentration of metal ions at distance x from the surface at time t. This can be solved by r_apla~P. transform~tion . For finite planar diffusior!, for which the concentration of the reacting species is constant at greater than ·some distance from the surface, the initial and the boundary conditions are C(O. 0)

Zci

(8.31)

- = I wCOSw,

ole

Rd is the diffusional resistance, or the limit of the real part of zd at low frequency, and 0/8 2 is the parameter that defines the frequency response, such that the maximum of occurs at f - 0.40/8 2 .

= Cs.

(8.39) where the term a is known as the Warburg coefficient. Equation (8.38) can more usefully be expanded as

Zd(w)

, [sinh(u) + sin(u)]- j[sinh(u)- sin(u)] cosh(u) + cos(u) '

= aw'

,

~= (2~2r

(8.40)

which give the real and the imaginary parts of the diffusional impedance. At high frequencies both Z' and Z '' tend io a jw;, anu the low-frequency limi~~ are

Z' =aS Z" =0.

D . ( 2): (8.41)

..

112

.

I

Corrosion Testing Made Easy ""

I Relationship of Warburg Coefficient to Physical, Chemical, and Electrochemical Parameters

For the irreversible reduction of a cathodic species, e.g., oxygen, fj

a= The Warburg coefficient a car. be related to electrochemical, physical, and chemical parameters. The ex- • pression usually given is rerevant to only a redox process, Eq. (6.42), witn rapid kinetics:

· 0+ ne-

=R

nFC5 .J2D

•

(6.44)

where C5 is the surface concentration of the reactant befoPe the application of the ac signal. Inserting this into (6.39) gives an alternative expression for Rd:

(6.42)

(8.45) I

-Equation (6.43) was originally derived by Warburg:

!

I

For an anodic process for which the reverse (metal ion to metal reduction) reaction cannot be ignored, it

(6.43)

M ~M'*+ne,

(6.46)

'" then C0 and CR are the equilibrium surface concentrations of 0 and R, respectively. However, this expression has limited relevance in corrosion studies.

cro~rEo

(8.47)

,

t

Appendix C

Wolf<- Through of a Noise MeOSutetnent and AnalysiS: '

·~

'· ..~~

..

In this appendix, we go through an electrochemical noise measurement in detaik It is ~uggested that readers without previous experience of similar measurements might like to try to repeat the experiments. A system of steel in 3.5% NaCI solution is used.

Equipment Required It is possible to record both the potential and the current noise by using a chart recorder, although this will restrict the analyses that can be performed on the time records. Consequently, it is recommended that the potential and current noise should be sampled and recorded on a computer. The sampling can be done either with external voltmeters or with a data acquisition card mounted inside the computer. The relatiVe merits of the two methods and the details of the connections required are considered in Chapter 6. We simply assume that two voltage measuring devices are available, together with software to control the measurement. To convert the current to a voltage, it is preferred · to use a current amplifier (also known as a ZRA; see page 61 for more information), and to avoid aliasing (see page 68), both the potential and the current signals shouid be filtered to remove higher frequencies than can be recorded (see page 61 ). Thus the measuring circuit will be as shown in Figure C.1 . For our practice measurement we use simple epoxymounted steel disk electrodes (Figure C.2) with a diameter of ...., 10 mm. The electrode should be abraded to provide a reproducible surface. The system we are

.. going to measure is not subject to crevice-corrosion problems, so this relatively crude specimen configuration will be acceptable. We use two steel working electrodes to measure the electrochemical current noise and a third reference electrode to measure the potential noise. For these measurements we use a commercial saturated calomel reference electrode (see page 58 for further discussion of the selection of suitable reference electrodes), but you can also use a third steel electrode.

Measurement Having arranged the measuring instrumentation and

the electrodes, all that is reqtJired is to insert the electrodes into the solution · arid start to take readings.

For these measurements, we -,Ut$e .a $ll~n of 3.5% NaCI. To obtain reproducibl8 ,..ufts tfl&.Sofution should be made with reagent-gra~>watef1 and reagent- or analytical-grade chemicals, ·a®ough for a simple trial run of an electrochemicaii"'IOse measurement, the purity of the solution is obViously, not critical. For simplicity the solution is used in the naturally aerated state (i.e. exposed to the air, but not explicitly aerated). This means that oxygen transport to the steel surface plays an important part in the corrosion process, and care should be taken to avoid mechanical disturbance of the system-the electrodes should be held firmly in place, and the cell should be placed away from sources of mechanical vibration. Otherwise the cell design is not critical. Figure C.3 shows a simple cell that uses a small ' See ASH.l Standard 01193-91 · s :andard s.-ecification for Reagent Water:

114

Corrosion Testing Made Easy

•

FIGURE C.1. Block diagram for electrochemical noise measurement.

FIGURE C.4. Instruments used to measure electrochemical noise.

c.a.

FIGURE Meuurlng electrodes tor simple electrochemical noise measurement .

electrochemical noise· properties change considerably, several time records should normally be taken, both to capture the initial transient and to record the later behavior as the system approaches steady-state conditions. For this trial measurement, we suggest recording time records of 1024 samples at 1-s intervals, with the time records being taken continuously for a period of 24 h (although this will probably not be long enough to achieve true steady state). Some recording systems will be unable to take truly continuous-time records, in that a few seconds will be required for saving the data between time records, but this will not usually present · a significant problem. Figure C.4 shows the full experimental system used to measure the ~lectr~hemical noise for this system. Note that the voltmeters, the ZRA, and the cell are mounted inside a Faraday cage (in this case a galvanized steel bin) to minimize the pickup of radiated interference.

FIGURE C.3. Simple electrochemical noise measurement cell.

Analysis

glass jar with the electrodes mounted in holes in the lid of the jar.

The selection of the best method for analysis of electrochemical noise data remains one of the major challenges. For this simple trial experiment we

Once the electrodes are ·inserted into the solution. the recording of measurements may start immediately. However, as many corrosion systems will give an initial period of transient behavior in which the measured

• look at the potential and the current time records, • calculate the standard deviation of the potential and current noise. and • look at the power spectra.

Electrochemical Impedance and Noi~e

115

.: FIGURE C.6. Dragging down ..-Kf right Over an area of a graph to expand the vhlw: 0{89 left and Up JO .....,.. the full scale view.

FIGURE C.S. File Load Dialog.

We perform the analysis by using the program ENAnalyse.exe that is included on the CD-ROM enclosed with this book. Note .t hat this program requires a 32-bit version of Windows (Windows 95198 or NT). • The current version of the ENAnalyse program handles only a single time record, although it can extract arbitrary columns from the input data file. Consequently, we need to analyze the electrochemical current noise and potential noise separately. . To use the ENAnalyse program, we first load the time record. Press the load time record button and the load dialog opens. Select the appropriate parameters (see Figure C.S) and press OK to load the ftle. -Note that it is possible to load fewer points than are present in the data file. If you attempt to load more points than are present you will get a run-time error, and the program will end with an error message. The number of points is restricted to an integral power of 2 for compatibility with the "FFT. Once the file has been loaded. you can inspect graphical views of the time record and power spectra by sele~tir') the relevant notebook pages. Note that thP. !rdividual graphs may be disabled by unchecking the relevant Graph Enable box. For input time records of more than 65,768 (2 16 ) points, all graphs are disabled when the time record is loaded. This is done to avoid slow operation on low-performance PCs. The graphs can be reenabled if required, and the program will handle time records of up to 10 6 points, providing sufficient real and

virtual memory is available. Note, however, that there will be a marked degradation in performance when the program starts to use virtual memory. To provide data that you can try with the program, the file Gnoise.Txt provides a simulated potential and current noise time record with 1 M (220 ) samples. Owing to the very large number of samples, a plot of the full time record and most of the FfT pow~J spectrum have the appearance of a ~ band.·Jt yO..,.wish, you can zoom in on regions oUhe graphs by :selecting an area with the mouse (Ftgure C.6). · TA2.TXT contains real·· ¢urre~t · and pqtential time for stee.l jn 3.5%. NaCI,..with 1924 records . ... . . . ·. . . ' data points taken with a sample interval .of 1, s. We shall work through the analysis of thestJme frecords in some detaU.. _Yqu can a1so u$e the .pfocedure to analyze . your own ·time recort:U~."'" '' -. ~~:: . ,''J.' ,. . ~· ·

m,e

1. Load the file TR2.TXT ·With the parameters of FigureC.7. This will load the potential time record. 2. Record the statistical parameters of the potential time record. You should obtain the results shown in Figure C.8. 3. Look at the MEM and FFT potential power spectra. You should see the display shown in Figure C.9 .(the rather rnessec:t-:uJ9axes · for thi~ log-16g .plot are a known problem with the program). 4. Save the MEM spectrum as TR2E.MEM in a suitable directory, such as C:\WORK. 5. Repeat the above process, but for the current time record (you will just need to change the input column from 1 to 2). Save the MEM spectrum as TR21.MEM, and note its statistical properties.

I~

116

Corrosion Testing Made Easy

6. Calculate Rn (standard deviation of potentiaV staJ1dard deviation of current). Use the value for the standard ~tton after trend removal to avoid beir•y ·confused.by the trend in the potential. You should get a value of 304 7. Calculate the localization parameter (standard deviaJion .of currentlrms current). ENAnalyse does not determine the rms current, so yoU will have to use .Eq. (E.8) to calculate this from the mean and standard deviation ~ In this case, you will have to use the m~~n and standard deviation without trend removal (because by definition, trend removal will reduce the mean to zero). You should get a value of 0.100. 8. Calculate the characteristic Charge {(standard deviation of current x standard devi~tion of potential)/ (Stern-Geary coefficient x bandwidth)]. Use 0.026 V for the Stern-Geary coeffiCient and 0.5 for the bandwidth. In this case use the values after trend removal. You should get a value of 2.33 J.LC. 9. If you are experienced in the use of a spreadsheet package, import TR2E.MEM and TR2J.MEM into the spreadsheet and plot the electrochemical noise impedance. In essence this will involve the following steps:

o. ··.

. FIGURE C.7. Parameters for loading TR2.TXT.

• Import TR2E.MEM into a blank spreadsheet page. • Import TR21.MEM into new columns of the same spreadsheet page, making sure that the two sets of numbers are aligned. • Create a formula in a new column that calculates the noise impedance as J(potential PSO/current PSO). • Plot the noise impedance against one of the frequency columns that will have been imported from the two MEM files. • Figure C. 10 shows a graph that has been plotted. ~URE' c.t. Statistical properties for TR2.TXT.

i li.00€+04

Ul0€+03

1---------

~J

--,.---~~

i . . . . - - - - 1

l .OOE.o4

1.00€-03

1.00E.Q2

l.OOE.01

1.00€+00

FAtqUency/Hz

FIGURE C.9. Plot of FFT and MEM spectra for potential.

FIGURE C.10. Electrochemical noise impedance spectrum for TR2.TXT.

r

...

.

.

Appendix 0

Sp~ctrcil

Estimation Techniques· andSignal Processing

In this appendix, the aim is to give a general appreciation of spectral estimation and other signal processing methods. It is not intended to be a complete or rigorous treatment of these ~ubjects. Readers requiring further information, especially in respect to computer codes for the performance of these methods, are recommended to read one of the various version of the book, Numerical Recipes inC (or FORTRAN or Pascal}, by Press et al., published by Oxford University Press.

and so

i(t) sin(wt) = io[sin2 (wt) cost; + sin(wt) cos(wt) si~ t; ].

The integral with resPE!ct to time, or sum, of this over one cycle of the ac signal is · · 2,,

i 0 fo~ sin2 (wt)cost;dt,

because the integral of sin(wt) cos(wt) cycle, and this equals

Sine·Wave Correlation

1C .

For the analysis of the transfer function at a single frequency, a sine-wave correlation method is commonly used. For an applied voltage V = V0 sin(wt}, the current response is multiplied, separately, by both sin(wi) and by cos(wt), and the two results are integrated over a complete number of cycles. These integrals give the inphase (real) and out-of..,phase (imaginary) components of the current response. If fhe response of a system to a sine-wave voltage

Vo sin(wt) is a sine-wave current, i(t\

= i0 sin(,<>t + ¢) .

(0.1)

this can be expressed as i(t) = io sin(wtl cos¢+ i0 cos((ut) sin 1/>.

(0.2)

(0.3)

2 /oCOSt/J.

(0.4)

= 0 over one (0.5)

Similarly, the integral of i(t) cOs(wt) is (0.6)

Thus the moduli of the ittance and the phase angle are given by

y2 =

(:r

118

Corrosion Testing Made Easy f

Contributions from responses at other frequencies are rejeeted by this analysis, because the int~ls are zero.

this will be proportional to the cosine component present in the signal.

Spectral Estimation

We can cOmbine the two calculations by using the relationship . ~ :%""" -

SpeQtral estimattpn Is-~~ pr,oce~ of ~stimating_the amplitude of the various frequ~ies present in a signal (the power spectrum). The word estimation is used because it is newr possible to·rnelOde au possibfe'iritormation &bout a signal in a.sampled time record, so we can never.determine a true power spectrum. There are many w~s in wh'lch a power spectrum can be estimated, but two are commonly used with electrochemical noise signals, the Fourier transform in the form of the FFT and the MEM.

The Fourier Transform The Fourier transform converts from the time to the frequency dQmain by using the same basic approach as . that used· in sine-wave c9rrelation. It depends on the properties of the integrat:

..·.. J:

sin(at)sin(,Bt)dt.

(0.9)

When a 1= ,8 (i.e., when the two sine waves have different frequencies), the product sin(
h(t)sin(at)dt,

(0.10)

we obtarn a value that is proportional to the amplitude of the sine wave, sin(at), present in the signal h(t). If we consider the related cosine wave for signals of different .frequency the integra1 is zero. as above. for a -::;: ,8. ·However, for a ,8 the product becomes s!!'! ::t t cos at, the integral of which is alsc zero eve~ c~e complete cycle.

=

However, if we take the integral / : h(t)cos(at}dt.

~ading

H(f•)l=

= sin wt + j cos wt,

(0.11\

(0.12)

to the full definition of the Fourier integral:

£:

h(t)ei"'' dt

£:

=..

h(t)(sinwt + j coswt)dt

. (0.13) It is, of course, impossible to determine the Fourier integral for anything other ·than a simple, well-behaved analytical function, as we can never have a continuoustime record from -oo to oo. Consequently Fourier analysis on reiil sampled time records is always required to ~ke two approximation~: that the time record is of finite duration and that the time recOrd consists of discrete samples. The effects of these two approximations are essentially independent of each other, and we con· sider tt)em separately.

The Finite Fourier Transform If we take a time record of finite duration, problems arise for frequencies that do not have a'ri integral number of cycles in the time record. If we consider the Fourier integral over a ftnite period, ("- h(t)eiwl dt,

lo

If we take the integral

j_:

ei"'1

·

(0.14)

the value is no longer necessarily zero for frequencies other than · these for which tnax r¥27rctJ for integral values of n.

=

We can force the eiwt term always to have an integral number of cycles within the integration period by choosing to analyze for only 2mu nltmax. where n is a positive integer. However, there will still be components in the signal being analyzed that do not have an integral number of cycles in the period of the time record and that therefore make a positive contribution to the integral even though they are not at the analysis frequency. As the,s e errors are associated with the incomplete cycles at the end of the time record. their extent can be limited by artificially reducing the amplitude of the time record at the ends, the process known as w.indowing.

=

119

Electrochemical Impedance and Noise

j

~

l

j J j j

j

~ j

~

lj

1

~ 1

. . .·

~ ~ ~

-3

-z

-1

o

1

z

offset in units of mqucncy bins FIGURE 0.1. Leakage envelopes for a number of common windowing functions (after Press et al.).

Note that windowing does not eliminate the spreading of the measured frequencies around the analysis frequency (known as leakage); rather it controls the shape of the envelope of frequencies included in the measurement. Figare 0.1 shows examples of the leakage envelope for a number of common windowing functions. Other than the square window (i.e. no modification to the time record), which has a rather extensive pattern of leakage extending away from the analysis frequency, it can be seen that there is relatively little difference among the various windowing functions.

Discrete Fourier Transform If the time record is sampled with a sample interval . Ll.t, the Fourier integral is replaced with a summation, X

L

h(kM)eiwkt..t,

(0.15)

allows us to analyze only for frequencies below 1/(2Llt) (known as the Nyquist frequency or the Nyquist limit). When we combine.the effects of the finite time record length and the sampling, we can conclude that the Fourier transform of a sampled time record will give the following output. For time record of N samples with Ll.t, we have the fo11ewing.

a sample period

• The spacing of frequencies that we can analyze for is 1/(NM). This can be defined~ the frequency resolution of the analysis, although the range of frequencies actually included will depend on the windowing function used. • The maximum frequency that we can analyze for is 1/(26.t), the Nyquist limit. · • We can analyze for ( N /2) + 1 frequencies: 0, 11 (NM), ... ,kf(ND.t), ... ,(N/2- 1)/(ND.t),1/(2M).

k=-:-.:

where k is an integer .and Ll t is the interval between samples. This summation is valid only for values of 6.t such that there is at least one sample in half a cycle of.the sine wave. Hence the discrete Fourier transform (OFT)

(

Estimation nf the Power Spec:trum from the Fourier Transform The OFT computes the amplitude and the phase of the ( N /2) + 1 frequency components present in the time record. Fortuitously. the effect of leakage is to ensure that all of the power in the time record is

-·

- - --

120

included in one component or another. Hence the power presentin a frequency band of width of 11N M can ba estimated as the amplitude squared from the Fourier transform~

The power spectfum of a noise signal as estimated with the raw Fourier transfo~m .is inherently noisy (the expected PSD has a standard deviation equal to the mean). There are various ways in which a smoother estimate may be obtained:

1. Averaging several time records. If several time • .records are avail~ple, the power spectra can be obtained ,individually, and these can then be averaged to produce a smoother spectrum (in accordance with normal standard error behavior, the standard deviation of the average falls as 11../M. where M is the number of time record~). For a single time record, a similar process can be undertaken by an. alyzing segments of the time record and averaging these. 2. Averaging in the frequency domain. Groups of adjacent points in the power spectrum may be combined by averaging the frequency and the PSD. As with option 1 the standard deviation falls as 11 ./M, Where M is the number of points in each group. It may not be immediately apparent, but for a given set of samples in the time domain, methods 1 and 2 are exactly eq~lvalent. Note that they make the frequency resolution larger (and hence the lowest frequency determined) by aJactor of M compared with the analysis of the same data as a single time record. 3. Smoothing the power spectrum: The raw power spectrum may be smoothed ~Y fitting an appropriate function to it. Method 2 is arguably a special case of this approach. In that a straight line is fitted to groups of M points. More generally, if the spectrum is expected to consist of a set of line segments, as would be obtained for the impedance spectn.Jm of an equivalent-circuit, the appropriate functions can be fitted to the Fourier transform. This will ·produce results that are very comparable with those of MEM spectra.

Maximum Entropy Method The Autocorrelation Function To understand how the MEM works. i~ is probably easiest to look at an alternative method of calculating tne power spectrum. This uses two steps. In the first step

Corrosion Testing Made Easy

the random element of the power spectrum is averaged out by calculating the autocorrelation function (ACF). In simple , the ACF describes the average value of the product ot' the voltage (or current) at one time and the voltage (or current) a given time later. For a discrete • time signal h( t) this is defined formally as

•

1 N ACF(r) = jh(kM)IIh(kM + r)l,

·

N

L

(0.16)

k=1

where r is the time difference between the pairs of voltages (known as the lag). · NcitEf'that there are some complications when a time record of finite duration is concerned, as the number of that can be averaged to estimate the ACF decreases as the lag increases. Thisfunctionisdefinedforr =0,t.t,2t.t, ... ,(N-1). M. Where nt.t+r is beyond the end of the time record, the calculation "rolls round" to nt:.t + r- Nt.t. As aresult, the values from (NI2)t.t to (N- 1)t.t duplicate the values from t.t to [(NI2) -1] t.t, and we need to consider only the ACF from 0 to.(N/2)t..t. Figure 0.2 shows some examples of the autocorrelation function for a selection of noise power spectra. To translate from the ACF to the power spectrum, we use the result of the Wiener-Khinctlin theorem, namely tbat the power spectrum is the Fourier transform of the ACF. This is one way to calrulate the power spectrum with the FFT, but it is not nortnally used as it is slower than the direct calculation from the time record. The MEM, in common with several other methods of estimating power spectra, works by fitting a function to the ACF and then computing the power spectrum that corresponds to the fitted function. The function used by the MEMisoftheform ao+B1r+B2r 2+a3r 3 +·. +Bmrm. where in, the number of coeffici~nts used, ·is known as the order of the MEM. When this is converted to a power spectrum, it results in a function of the form

P(f)

=· 1

aa m

12

(0.17)

11 + {;~zkl where P( f) is the PSO. ~ to ak are the MEM coefficients, m is the order of the MEM and z is a complex frequency term (similar to the Laplace s term, but

Electrochemical Impedance and Notse

121

Methods for the Calculation of Power Spectra 1.5

These ligures illuslrate the methods that can oe used to c.lculate power spectra. They have bftn produced using 'brulalolu'l'ft8ttl!!ds In 811 Excel spreadsheet, available on itlll CO-ROM as AulcxorNialion.rla in the~· dftcloly. In patlicular,lhe t.IEM hal not been c.lc:ulaMd 1¥ lhe norTMI nWihod, bu1 1¥UIIng lhe Excel Solver to fit the relevant polynomial .

.... ·1

- ·1.0

•

ca1o.J'I1A Fourier

l

Trar.!onn ol bme rec.ord

,.

You may lind l interesting to look at the spraadsMet to sea how the c.lculatlons

'-----_.,:...------J 0

••

Tlmo

I Vn

!lave bean par1onned.

20

n

V

.

ca~cutw

-......,, illlldion

ao a-.qe VU. ol x(1J x x(l•tl

.. ,...--------------,

,...-r..-omo

Fltlecl ACF

r------------, • I• .• 10

li •.•

. i

i!

:I ·\J ·.·.· \. j/ J

2

t:

0.1

..,

i :.

-10

_,,_... ·5

0

-tO

tS

·10

•

Poww Speclrum " - FFT

\ .- ..-.-'

..

10

••

20

n

'ICalcuialol Fourier nnolorm <.j.ol tlaed ..-lion

u

'

fu

fu

~

2

~ a

2

-

fu

~u I

f

I

o.s 0.4

..

.!

Otloyl.....,...

u

3

0.3

# ...

Po_, Speclnlm trom ACF

u

.._,...

·10

-~S

.Q;I=:.:::-~"'1

JJ.I=:==I

G.2

·· .•

'

•

oL----~~---~:.._·---~

·tS

~/Ill

O.t

.·

l ~~

...

-~~-------------~--~ 0.1 0.2 O..J 0.4 <1.1

~

0.7

0.5

0.1

0.1

0.2

0.)

0.4

0.5

t.S 1

...

0.1

0.1

o..i

0.3

0.4

Q.S

u

F~cytHz

'-Y/111

RGURE 0.2. Autocorrelation function.

modified to allow for the discrete nature of the original time record). This resulls in a spectrum with particular characteristics. The polynomial in the denominator will have m roots (i.e., the value of the polynomial will be zero) in the complex frequency plane, and the function will go to infinity at these points. These points are known as poles because the amplitude rises sharply, giving the appearance .of a flagpole. Because all of the roots are in the denominator. the MEM is known as an all-poles model. It is also known as an autoregressive or AR model. as this is a statistical description of the model used. The sharp poles make it particularly good at modeling sharp peaks in the spectrum. The details of the mathematical processes involved in converting from the ACF to the power spectrum are beyond the scope of this book (see Press et al. for details) .

Other Spectral Estimation Methods 5

It is also possible to model th~ power spectrum by a · function of the form

P( f)

=co+

It. c,z'l'

where c0 to ck are coefficients. This is known as a moving average or MA model. A combination of the MA and AR models leads to the ARMA (autoregressive moving average) model. The MA model is an au-zeros model-there will be zeros in the complex frequency plane, but no poles. The ARMA model has both poles and zeros.

It could be argued that the ARMA model. by virtue of its greater generality, should give a better power

122

spectral estimate, but this has not been examined for electrochemical noise power spectra.

Corrosion Testing Made Easy

0.1

c

0 .01

~o•

Wavelet Transform .A wavelet is a signal that ha-a a nonzero value over only a finite time. Just as a signal can be reconstructed as an infinite seri~ of sine waves of varying frequency, amplitude, and phase, so it can also be reconstructed as an infinite sum of wavelets of varying frequency, am.,plitude, and locatio.n in time. As wavetets outside the time interval being considered are ignored, and there is no reason why wavelets at one time should have the same amplitude as wavelets of the same frequency at another time, the wavelet transform is, at least iri principle, better at fitting nonstationarf signals. Unfortunately it ia di(tacult to make use of this property, as It is generally necessary to average the properties of all wavelets at a given frequency in order to produce a manageable output. Thus, Figure 0.3 shows a wavelet transform for the potential fluctuations occurring during a fatigue test. As they are generally applied, wavelet transforms use frequencies that increase ·by factors of 2 at each step, leading to a rather coarse frequency resolution.

0.001

(.)

Q) Qi

'

0.0001

>

10-s

Ill

3:

~

10 10.7

c:

Ill Q)

:E

·8

10 I

f

10-11 0.0001

0.001 .. . . O.Ql

· 0.1

Frequency I Hz FIGURE 0.3. Wavelet transform.

In practice the wavelet transform is almost infinitely flexible-the wavelet may have a wide variety of shapes, and any set of frequencies may be used. Witl'l further work, this may lead to improved analysis procedures, but at present it just leads to an extremely confused situation, with few obvious advantages compared with those of conventional spectral analysis.

•

Appendix E

-Theoretical Basis of Noise· Pro.cesses·: •

Description of Noise Sianals

For a sampled time record, this converts to the equivalent summation

t

Almost by definition, the exact value of a noise signal at any particular time is unpredictable, and, other than reproducing the full time record, noise signals can be described by only their statistical properties or power spectra. To use the more common st~tistical descriptors it is necessary for the signal to be stationary. This implies that its mean and standard deviation do not change with time (strictly stationarity reql!ires that all statistical moments be independent of time, but in view of the inherent uncertainty of the higher moments, it .is usually considered to be sufficient for the mean and standard deviation to be independent of time). The simpler statistical properties of the time record treat it simply as a collection of values, with no significance being given to the order in which the values occur in time. In consequence, these properties lose a lot of the information that is present in the time record, and it is perhaps not surprising that they have some limitations. Note that the formulas are given for the potential noise; the current noise formulas are obviously identical, b~t with I replacing E.

The Mean The mean of a continuous-time record is given by

mean= E(t)

=E = 1- j'lnw. fmax · 0

E,dt.

(E .1)

1 N

mean

= E = N L E[k],

(E.2)

k::1

where E[k] is the value of sample corresponding to time kA.t, i.e., the kth sample in the time record.

Statistical Moments In statistical terminology the mth statistical moment is given by•

mth moment

=.

1 N N

L E[k]m

(E.3)

Jc-1

Frequently, the second and higher moments are taken about thl3 mean, in which case the E[k]m in Eq. (E.3) is replaced with (E[k] -E)m. The first moment is equivalent to the mean and the second moment (taken about the mean) to the variance. The third and higher moments are not commonly used as they have expected values (for a normal distribution) that are rather nonintuitive. Consequently, we tend to use derived parameters (such as the skew and kurtosis- see below) that have a value of 0 for a normal distribution. • Note that throughout these appendi.:es we have essentially ignored the standard statistical problem of whether to divide by N (which is correct if we have the whole population available) or by (N- 1) (which applies when we are using an estimated value of the mean). For electrochemical noise time records. which typically have 1024 or more points. the distinction is nvt usually significant.

Corrosion Testing Made Easy

124

Variance

Root Mean Square

The variance of a contirioou_s.signal is given by varlan~e

,'

= En( t) 2 . 1 ~ .Ia- ,_, = En(t) 2 dt.

enax

The rms value is the square root of the average value of the square (without subtraction of the mean):

Erms =

- 1-

Erms(sampled signal)=

2_

(E.4)

0

As the variance is meaningful for signals only in the time .Qomain, we normally take the time element as read and write E~.

I

N

{

Jo

t

E{k]2 .

2

k=l

The frequencies included in the definition of the variance for a continuous function have no upper limit, but frequencies with a period longer than tmax will not be captured reliably. In any real measurement, there will also be an upper frequency limit because of the limited frequency response of the measuring instrument. When the sampled data are examined, the variance should include frequencies only below the Nyquist frequency, although .if aliasing of higher frequencies has been allowed to occur this power will also be included in the variance. Providing aliasing has not occurred, we can say that the variance will include frequencies between 1/(th~ duration of the time record) and the Nyquist frequency. Which parts of this frequency range dominate the measured value is a more complex question, as ·this will depend on the power spectrum and can be treated on only an individual basis.

Standard .Devia!far: The standard deviation is simply .the square root of the variance. This gives it the advantage of having the same units as the mean and therefore being somewhat more natural to work with.

(E.?)

k=l

-2

-

Erms =E + E~.

(E.5)

(E.6)

The rms, variance, and mean are related as follows•:

For a sampled signal, the variance is given by

1 N 2 variance= N z)En[k]) .

ft.... E(t)2 dt,

fmax

(E.8)

Skew Skew (or skewness) is a measure of the syr:nmetry of the distribution. It is calculated as

1 skew=-

N

N

-E)

_, 3

L

· · En[kJ

k=l

(

J En[k]2

(E.9)

Skew is derived from the third moment by dividing it by the standard deviation cubed, which makes it nondimensional. A value of zero implies that the distribution is symmetrical about the mean, whereas a positive skew implies that there is a tail in the positive direction, and a negative skew implies that there is a tail in the negative direction.

Kurtosis Kurtosis is a measure of the shape of the distribution compared with the normal distribution. It is calculated as

1 ku•tosis = N

L

N (

k=l

__-)4 - 3.

En[k]-E

JEn[k]2

(E.10)

.

-;-; . ~ r.urtosis is derived from t:·,.; :vUI th momant, whictl is divided by the standard deviation to the fourth power. • Readers are reminded that

En has been defined as the potential E

after subtraction of the mean E. Also. where we write E~ . we imply an average over the N points of the time record .

125

Electrochemical Impedance and No1se

This makes it nondimensional. Three is then subtracted to bring the kurtosis for a normal distribution to zero. Hence a kurtosis of zero implies that the distribution has a shape similar to that of the normal distnbution. A positive kurtosis implies a more spiky distribution, whereas a negative kurtosis implies a flatter distribution.

Coefficient of variation The coefficient of variation is defined as the standard deviation divided by the mean:

coefficient of variation

m

= · En. ; ·

(E.11)

Noise Resistance The electrochemical noise resistance Rn is calculated as the standard deviation of potential divided by the standard deviation of current:

to use a chart recorder for a first look at a new system. However, most of the analysis methods that we are discussing require processing that is difficult or ' impossible with continuous-time inputs. Consequently, one of the early steps in data acquisition for electrochemical noise measurement is the taking of samples tram the continuous-time system. The sampling process converts a continuous time inputinto a discretetime output. • The discrete-time output will consist of a finite number of values for the signal corresponding to ppecific times. We generally assume that the sampling process occurs at regular intervals. "this gives rise to a time record with the characteristics shOwn in Figure.E.1. There are a number of ways in which we can convert from the continuous-time input to the discrete-time output. Whichever method is used it is clear that information is lost in the conversion; once we have sampled the input we cannotreverse the process and reconstruct the input from the sampled output. In information theory

(E.12)

. Gontlnuous-Time and Discrete-Time Signals The majority of modern methods for the collection of electrochemical noise data rely on computer-based data acquisition. Computers necessarily deal with a sequence of sampled voltage or current values, and it is important to be aware of the limitations and possible problems introduced by the sampling process.

0.6

0.5 0. ~

0.3

It is possible to record an electrochemical noise signal and apparently to retain its continuous-time nature. For example, this can be done with a chart recorder or an analog tape recorder. However, in the process of making such a recording. information at some frequencies will inevitably be lost. Thus most chart. recorders will have a rapidly falling sensitivity at frequencies above a few tens of hertz. T<=tpe recorders will h!'lvP. a higher frequency limit that will depend on the tape speed and the recorder design. but will typically be in the kilohertz region. Chart recorders and ·analog tape recorders have some useful features : in particular it is often useful

:::

:!

c: ~

... 0

0.2 0. ~

0 ~1

-<1.2 -<1.3 -:'.! ~5

0

10

15

20

nme t s

FIGURE E.1 . Time record before and after sampling by instantaneous sampling.

• For a gcod discussion of the principles underlying discrete-time signals. sCJe Damper. R. I. Introduction to discrete-time signals and system$. London: Chapman & Hall. 1995.

126

Corrosion Testing Made Easy

0.8 r-:~-:--:--:~-:--:--:~;--:--:~;-~~~ 0.8

lt1

0.4

?::

0.2

Q,

.0.2

I

a

.«!~

0 .0.4 .0.6 .0.8 . 15

0.6. 0.5 0.4

-

. f. .

0.3

> 0.2 ]

'E

0.1 0

Q,

.0.1

l

20

: :

: : :X

Chapter 6, however, the problem is that higher frequencies do not simply disappear when the continuous-time signal is converted to discrete time but reappear as false indi~tions at lower frequency. This process is known as aliasing (because the low frequency is an alias. or false name, for the real high frequency). This effect can produ<;e very serious errors in electrochemical noise measurements, especially when power-line frequency noise, which is almost inevitably present in the measured signal, is aliased to a lower frequency. It is importa~t :to note that instantaneous and integrating samplings can have very different responses to frequencies above the Nyquist frequency, especially when the sample period corresponds to an integral number of cycles of the high frequency. In this case, the averaging effect of the integrating sampling process eliminates the high frequency, whereas instantaneous sampling converts it to an unpredictable de error (or, if the sampling fre- . quency is not exactly the same as the high frequency, to a strong signal at a low frequency) .

).(

:X: : X

:- : x

: :X :

.0.2 : )i(

.0.3 .0.4

:

.0.5 0

5

10

15

20

Time/a

FIGURE E.2. Integrating sampling.

: the information in a continuous-time input is infinite, and we inevitably lose information in the sampling process. Consequ~mtly it is very important that the sampling process extracts the right information. Figure E.1 shows the simplest method of instantaneous sampling. Figure E.2 shows the slightly leSs obvious method of integrating sampling.

Whichever conversion method is used (see Figure E.4), it is important to remove frequencies above the Nyquist frequency before sampling, as it is impossible to distinguish an aliased high frequency from a real low frequency once sampling has occurred, and even the integrating converter allows some aliasing to occur. The high frequencies are removed by use of a low- filter (so called because the low frequencies through the filter unaltered, whereas high frequencies are blocked) also known as an anti-aliasing filter. See Appendix A for a discussion of filter properties and designs for simple filter circuits.

Combining Noise Signals

An important aspect of the ~mpling process is what happens to frequencies close to and above the sampling frequency. Rgure E.3 shows the effect of sampling two sine waves by instantaneous and integrating sampling methods.

If we add two noise voltages together, the instantaneous voltage obeys normal rules, and we can write

These figures were produced with a modification of the Excel spreadsheet NYOUIST.XLS, and this is included on the CD-ROM if you want to explore this behavior f:..;rth:~.

however, when we come to work out the amplitude of the new noise signal, we find a somewhatcounterintuitive result:

A conclusion of the matheinatical analysis of the sampling process is that a discr9te-time signal can contain information about frequencies only up to half the sampling frequency. This limit is known as the Nyquist frequency or the Nyquist limit. As we have seen in

Esum (t)

= Et (t) + E2(t).

(E.13)

Electrochemical Impedance and Noise

127

1.4r----------------------------------------------------, -Original Sine Wave • Instantaneous Sampling A Integrating ~mpling

1.2

Sampling F~equency 1 Hz Signal Frequency 0.3 Hz

0.8 0.6 0.4 G)

;-g Cl

IV

0.2 0 -0.2 -0.4 -Q.6

~-·

.·

-0.8 -1 -1.2 4

2

0

10

8

6

12

Time/s (a)

1.4 -Original Sine Wave • Instantaneous Sampling A Integrating Sampling

,

1.2 ~

I!

n

0.3

, , , , ,

~

~

0.6 0.4

I

G)

;-g Cl

IV

0.2 0

.....

A

-0.2

~

A

A

...

...

.t

'

-0.4 -0.6 -0.8

v

-1

v

v

v

v

-1.2 0

5

"

~ 15

10

\j

20

v

v ~5 ::."

Time (b) FIGURE E.3. Sampling of sigri'als with frequencies below and above the Nyquist frequency: (a) Signal frequency below Nyquist frequency; both sampling methods give good results, although integrating sampling reduces the amplitude somewhat. (b) Signal frequency well above Nyquist f.requency, severe aliasing with instantaneous sampling,afiasin~. reduced with integrating sampling.

128 Corrosion Testing Made Easy f

-Original Sine Wave • Instantaneous Sampling A Integrating Sampling Sampling Frequency 60Hz Signal Ffequency 60 Hz I

0.8 0.6

'

0.-4 CD

Cl C'G

;g

0.2

.

0

'

-Q.2

'··-Q.4

Jl

II II

II

II II II

-0.6

-o.a -1 -1.2 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

nme/s (a)

1.4 1.2

,

-Original Sine Wave • Instantaneous Sampling A Integrating Sampling Sampling Frequency 60 Hz Signal Frequency 59 Hz

~

0.8

•

0.6

~

0.4

-~f

0.2

a ·0;2 -0.4 -0.6 I

-0.8

-1 -1.2 0

0.05

0.1

0.15 .

0.2

0.25

0.3

0.35

0.4

0.45 -

0.5

Time/s (b) AGURE E.4. Instantaneous and integrating converters: aliasing of po~r-line frequency. (a) Sampling at exactly mains frequency; instantaneous sampling gives ad¢ error that depends on the relative phase of the two signals. (b) Sampling close to mains frequency: instantaneous sampling gives a low-frequency output at the same amplitude as that of the interfering mains frequency signal.

129 I

Electrochemical Impedance and Noise

If the two noise signals are uncorrelated (there is no relation between the instantaneous values of the two signals), the average value of the product of the two signals will be zero. Consequently, the middle term of the integral disappears, and the voltage noise power is simply the sum of the powers of the two signals bein~ added:

This relationship is also explored in ADDITION.XLS. Note that although the expected values of the powers 9f the sum and the differe~e signals are the same, the actual 'Values measured for a.flllite time record are usually slightly different. '

Noise Colors (E.15)

Note that this means that the standard deviation of the sum of the two ~!gnals is not the. same as the sum of the standard deviations of the tWo signals being added. This is why we tend to define the amplitude of the noise in of its power or variance rather than its standard deviation. I

By analogy with the colors of Ught, it is common to describe noise as having a color: • White noise has the same PSD ~tall frequencies. • Pink noise has a PSD that f~ with increasing frequency.

~

If you have a copy of Excel, you can explore the addition of two noise signals by using the . file ADDITION.XLS on the accompanying CD-ROM. You will note that the rule for the addition of the variances is not exact for individual time records of finite length, but as you take longer time records, the result tends to the value expected. Strictly, we say that the expected value (in the statistical sense of our best guess of the value) of the variance of a summed signal is the sum of the individual variances.

Thermal (Johnson) Noise Processes Thermally activated random movement of electrons will cause charge separation to occur across resistors. This results in white noise with a voltage noise PSD of 4kTR, where k is Boltzmann's constant, T is the absolute temperature, and R is the resistance. Thermal noise is generally very small compared with electrochemical noise levels, but may become significant in very high-

resistance circuits.

·

Subtraction of Noise Signals Shot-Noise Processes What happens when we subtract one noise voltage from another is perhaps even more unexpected. A noise voltCurrents in electrical circuits consist of drift of charge age has a mean value of zero and is as likely to be carriers (electrons in metals and ions in aqueous sopositive as negative. Taking minus the voltage simply lution) through conducting materials. For a given curswaps over the positive and negative periods, and pro.. rent the aVerage number of,charge carriers (electrons duces another noise signal. Consequently subtracting in a metal or ions in solution) will be defined by the one noise voltage from another has exactly the same . average current divided by the charge on the charge effect as adding them; the voltage powers add! We can carrier. However, the actual number of charge carriers prove this mathematically as above: ing a particular point in a circuit in a given time interval will nave a statistical distribution about this average value, and this will give rise to noise in the observed current. p'roviding the motion of individual charge carriers is independent of other charge carriers, a simple stati~tical analysis (see .below fer a partial derivation) leads to the shot-noise formula: as above, j~tm..[2 E1 (t) E2 (t) dt] has an expected value of zero; hence (E.17)

Current. noise power= 2elb,

(E.18)

where e is the charge on the carge carrier, I is the mean current. and b is the bandwith.

.•

130

.

Corrosion Testing Made Easy .

In metallic condu~ors, the drift of electrons is correlated, and the observed noise is less than that predicted by the shot-noise formula. However, conduction in ionic solutions and .semiconduCtors should be assumed to obey the shot-noise formula. Note that the relevant charge in ionic solutions is the average charge • on ions that are carrying the £urrent and may be more than one electron. · •

1/f Noise Processes

current will be the variance of mq divided by t:.t2 to convert from charge to current: ·'

(E.19) If we analyze over a reasonably large number of sampies,• we .inclu(fe frequencies from essentially zero to the Nyquist frequency 11(2t:. t) in our measurement. We may therefore replace 1/(2!i. t) with b, the measurement bandwidth, leading to the shot-noise formula: I

-BQth thermal and shot noise are white. However, many practical . noise sources exhibit a form of pink noise, generally known as 1/f or flicker noise. The source of this noise remains somewhat unclear, although theoretical explanations have been develgped (see the paper by Eden* for further information). This type of noise has also been reported for electrochemical systems, although -the term is often used rather loosely to describe any noise that has a power Spectrum that falls with increasing frequency.

Shot-noise.Analysis of Electrochemical Noise · Electrochemical noise in .corrosion often arises from discrete events, such as the generation of ~urrent transients associated with metastable pit grOWth arid death. It, for ·the moment, we assume that we sample slowly, using an integrating converter such that we capture the power in the individual transients, in each sample period we capture a random number of transients. Considering the current signal, each transient will contribute a certain amount of charge q and, within the sample interval, the charge ed will be q times the number of transients m To know how much noise this will produce, we need to know the standard deviation of m, as this will be the standard deviation of the time record. tf the transients are independent random events, this forms a Poisson process, for which it can easily ·be shown that the variance of m is equal to the mean value of m. If the average current is f. then the average number of transients in a sample Interval of t:. t is clearly the average charge in the sample interval I t:. t divided by the charge on a transient q. Hence the variance of the

I

IN= 2/qb.

For shot noise in electronics, the charge is simply the charge on the electron, but for our transients it is the much larger charge in each transient. An important point to note is that I in the shot noise formula is the average current corresponding to the age of the transients. If we are concerned with anodic metastable pitting transients, the current is therefore the average anodic current. In free-corrosion conditions this will, of course, 0. the corrosion current lcorr. If we have measured only the current noise, we shall have two unknowns ( lcorr and q) and only one measured value, so we cannot solve for lcorr and q. However, we can also measure the potential noise to provide asecond measured value. We assume that half of the current produced by the transients on one electrode-es to the second electrode. ~Halving the current will also halve the standard deviation, hence dividing the variance by 4. (In this case, because the two currents are correlated, the normal rule that noise current powers add do not apply). This current will actually be measured as the .electrochemical cur(ent noise and it will also polarize the second electrode, thereby producing a potential noise given by ~ _ c:;n -

2

IN ..-.2 _

4 "P -

lcorrQb ..-.2

2

"P·

(E.21)

At first sight, this introduces a third unknown Rp. but we can eliminate this by assuming that the Stern-Geary equation applies:

a

fcorr

a

• Eden, D. A. Corrosion/98 . paper 386. See the CD·AOM for a copy.

. 1E.2Q)

= R'p

(E.22)

where is the Stern-Geary coefficient (see page 82). Substituting for Rp from Eq. (E.22) in Eq. (E.21 ), we

131

Electrochemical Impedance and Noise

I

obtain 2 qbfi --En2-!2R - n P - 2/corr.

-

(E.23)

If we assume that both working electrodes are produc-. ing transients, this will incr~as!3 the current and potential noise power by a factor of 2 (assuming that the noise currents associated with the two electrodes are uncorrelated). Hence we have (for the cell as a whole)

..

I~ = lcorrqb, .

E5=

(E.24)

qbfi .

(E.25)

logic of the argument is the sarrie as that of other such proofs. However, in our derivation we have assumed that it is vali,d to use Rp to analyze electrochemical noise Signals, and the validity of this assumption has not been proved. Note that a basic assumption is that we consider low frequencies (as we need to include a significant number of transients in each sample). If we use the lowfrequency limit of the potential and the current PSDs in plap~ of the corresponding variances, we ensure that we meet this requirement, and the b in Eq. (E.26) disappears.

lcorr

Solving Eqs. (E.20) and (E.23) and for lcorr and q we get•: t

v~ t:(j l(j q= lcorr

=

Bb

a/fi

m

'

(E.26)

(E.27)

Equation (E.27) might be taken as proof that Rn and Rp are the same. In some senses this is correct, and the

*Note that in earlier work (Collis, R. A., S. Turgoose. "Electrochemical, noise measurements: a theoretical basis." presented to EMCR'94; · Sesimbra, Portugal, September~ 1994, Mater. Sci. Forum 192194, 663-672 (1995)) the denominator in Eq. (E.26) is 2bB. This was obtained by dividing the current noise power between the two

Relationship between .Power Spectra and the Underlying Transl~nt Events If we consider a single transient, we can determine the power that it adds to the overall noise signal by calculating the variance for that transient .alone -over the period of our time record. Furthermore, we can calculate the PSD that is due to that transient by determining the power spectrum for that transient alone. If we have additional transients in the time interval, -their variance and PSD will (on average) add to produce the overall variance or PSD.

electrodes, rather than simply dividing the noise currents by 2 (and hence dividing the power by 4) . We believe the result here to be correct, although an error of a factor of 2 is rather 111odest by the standards of electrochemical noise measurements!

.

.

Appendix'F

Transient effe.cts··in.-'EI$ .: The example of current response to an applied voltage given in Chapter 2 is one of the feY
V dV I =-+C-

R

dt'

(F.1)

and so if we apply a sine-wave current signal [I = 1° sin(wt)], the equation we have to solve to obtain the voltage is

dV

dt =

V 1° . - RC + C stn(wt),

(F.2)

e<- ~) + 1° 1 +w2C1-R2 R sin(wt)

wCR2 0 -I 1 + ccl-Cl R2 cos(wt).

R

jwCR2

z = 1 + w2Cl R2 - 1 + ccl-Cl R2' .

(F.S)

and the relationship between the impedance and the in sin(wt) and cos(41l) can be seen. However, Eq. (F.3) has another term that contains the term f!T'IPC . This is a transient response to the applied signal and decays to 0 at long times if RC is positive. This transient response is additional to the other two that describe the cyclic steady-state response. As previously described [Eq. (2.29)], the term RC is the time constant for the circuit under current control.

A similar transient response occurs with other types of applied current. For example for the expression for the voltage that is due to a current step from 0 to 1° we need to solve

which on integration gives

V[t] = C,

The cos(wt) term is an out-of-phase voltage and defines the imaginary impedance. Equation (2.22) for the impedance of this circuit was derived from the ittances or impedances of the two circuit components,

dV

(F.3)

The integration constant C, depends on the initial conditions but with V(O) = 0, (F.4) Equation (F.3) contains three . The sin(wt) term is a voltage in phase with the applied current. and its magnitude defines the real part of the impedance.

V

1°

- - -RC+ dt C'

(F.6)

V(t)= / 0 R(1-e-~).

(F.7)

which gives

The question occurs of what happens if the time constant RCis negative. In this case the exponential term in both Eqs. (F.3) and (F.7) increases with time and eventually tends to infinity, i.e., the circuit does not have a stable response to an applied current. This is different

134

Corrosion Testing Made Easy

from the response to an applied voltage, Eq. (2.11 ), which is stable whatever the values of Rand C.

• that a given circuit may be stable with potential control 1 but not with current controt(later we will see that the opposite may apply for some Circuits} • that the expressions developed for the impedance of a circuit by considering the impedances or ittances of the individual elements, as in Table 2.1, gives only the cyclic (ac) steady state and does not include tran- .sient effects, and. • that the criterion for stable response of a circuit to an ac signal is the same as for a more simple applied de signal.

Responses with the Presence of Solution Resistance

The ·expression relating voltage and cu·rrent is

v- \11 ·= cdV1 + ft

dt

+ Rs

dt

_V_

+ RsRC.

(F.11)

The .first term in each of these expressions will give a transient response to any applied waveform. Considering the expressions for current or potentiai steps, Eqs. (F.1 0) and (F.11) can be written as ' I

.dV = _ dt

V+I( Rs+RC R). I

d/

dt

(F. 12)

TJ

V

(F. 13)

=-Tv+ R,RC'

where '' and 'fv are the time constants for current and potential control, respectively, as defined in Eqs. (2.29) and (2.30).

From this we can go on to consider the response of a circuit including a solution resistance, as in Figure F.1 .

=

..!.. dV

d/ =_;I( Rs+ R) dt RsRC

This 'example.shows

I

or for the response to an applied voltage,

\11

(F.S)

R'

which, when \11 is eliminated, can be written as

From these, for a current step to 1° we get Eq. (F.14) and for a voltage step to 'V 0 we get Eq. (F:15):

V(t)

=1°ft+ 1°~1- exp (- :, ) ].

(F.14)

~[1 +.!! exp (- .!..)]· Rs+ R ft

(F.15)

l(t) =

'fv

I(

ft+

' R

R)

0

+ 'S I

C d/ dt

= Cd V dt

+

V R"

(F.9)

For the response to an applied current, we need to solve dV dt

=-~ 1(ft+R) 0 .dI RC + RC + r'Sdt'

(F.1 0)

v

These equations show that stable responses are obtained only if the value of the relevant time constant is positive, and this also applies to ~pplied ac signals. To summarize: • For current control (impedance measurement) a stable response requires 'ft

= RC> 0.

(F.16)

• For potential control (ittance measurement) a stable response requires

rv FIGURE F.1. Circuit with solution resistance R., polarization resistance R, and double-layer capacitance C. V is the total voltage, and V, is the voltage across the interface.

=

RsRC Rs+ A> 0.

(F.17)

The conditions in Eqs. (F.16) and (F.17) explain why the data in Figure 4.20c, where RC < 0, cannot be

.:

]

.·~ .·~~

.",-·-<")

~ 1~'

135

Electrochemical Impedance and Noise

0~~-----------------------------,

I

frequencies it is possible to calculate Z' at all frequencies, and vice versa. • The conditions for the transforms to work are several:

0.001

ieo.ooo~----~--~~+------+--------_, ~.

~.001

• Causality: The response should be caused b~.the input. • Stability: the system should return to its original con· 'dition after the signal is removed. • Linearity: the response and the input should ·be lin-· early related. • Finiteness: the response should be finite at all fre-

·~uencies.

~.OQ2L---'--.,....-'----'------J'-----'-----'-----'--........J

0

1000

2000

3000

Tune(Sec) FIGURE F.2. Current response to a potential step.

obtained with current control, but can be with potential control because rv is positive.

Comparison of EIS with Potential Step Polarization Resistance Equation (F.15) is the basis for the potential step method . of obtaining polarization resistance. Figure F.2 shows the current response to repetitive -V and +V steps, in this case 10 mV, appiied to a highly capacitive system (mild steel in H2 S containing solution), where ~ = 11 Q, R = 28 Q, and C = 1 F for the 7-cm2 electrode. The frequency of maximum Z' in the impedance Nyquist plot is thus -
Data Validation by Kramers-Kronig Transforms The Kramers-Kronig transforms are mathematical relationships between the real and the imaginary parts of the impedance that must be obeyed by valid impedance or ittance data. If, for example. Z" is known for all

In of validatingt real data, detection of violation of the stability criterion is probably the most useful. For example the data in Figures 3.11 and 3.12, showing low-frequency drift, do not transform correctly. The linearity condition is generally met by impedance data. It is not the same as requiring a linear current potential relationship, as the nonlinearity of this shows up in higher harmonics (Appendix B), which are rejected by the data processing. There has, however, been much discussion over the past 10 years about the apparent invalidation of certain types of impedance spectrum. The reasons for this are clear and are briefly mentioned here. Using the general term lm (immitance) for either impedance or ittance, the basic equations are 1. for calculation of real from imaginary, 00

'() 2/xlm"(x)-wlm"(w)d Im'() w- 1m oo = - ; x2 _ w2 x, 0

{F.18) or 2. for calculation of imaginary from real 00

Im"(w )-2w/lm'(x)-lm'(w)d ..? x. 2

(F.19)

A - - (l)

T(

0

One apparent problem becomes evident on inspection of Eq. (2.22) for the impedance of a parallel RC circuit. The term for Z .. contains in R2 but not in R Any set of !" data is consister.! •.•tith two spectra, • See. for example, "Electrochemical Impedance: Analysis and In· terpretation," ASTM Sp. Tech. Publ. 1188, eds. Scully J.R., D.C. Silverman (1993). I Strictly speaking, the transforms can only invalidate data. not validate them.

Corrosion Testing Made Easy

136

corresponding to the positive and the negative roots of Ffl. Similarly, any set of Z' data is consistent ·with two spectra, corresponding to the positive and the negative roots of (j. However, the integrals above will return only one of these two possibilities. In light of the sections above, it should not be surprising thlit the one. data • set calculated is that con~lstent with RC > o. With the slightly more complicated circuit of Figure F.1 the transforms will return a data set consistent with

a positive value of the relevant time constant, as described in Eqs. (F.16) and (F.17). This is one case in which it is important to distinguish between impedance an'd ittance, and if a conventional impedance measurement is made (i.e., with controlled poten1ial. so really one is measuring ittance), these transforms mus\ be applied to ittance data. For example, for data such as in Figure 4.20c the transforms applied to impedance data will suggest that the data are invalid, but will work if applied to ittance data. ' t

.....

,.

I

.

.

Appendix G

Conterits of CD-IlOifll::· .

:

The CO-ROM that accompanies this volume contains a number of resources to assist'the reader in following the text and obtaining further information about impedance and noise methods applied to corrosion.

Papers Presented at Corrosion/96, 97, and 98 · Thanks to the conference department at NACE we have been allowed to include the full text of ali of the papers related to the subject of this volume that have been presented at the annual Corrosion conference in the period 199&-1998 (since the Papers were available in electronic form). These are stored in the subdirectory 0:\Papers, where 0 is the drive letter of your CO-ROM. All of the papers are provided in Adobe Acrobat format. The Adobe Acrobat reader can be obtained from the Adobe Web site (http://www.adobe.com) or many other sources. A text index to the paper is provided in the file INOEX.TXT, while an HTML version is provided in INDEX.HTM. Providing your web browser is configured for the use of Acrobat as a 'plug-in', you can open this file. with your web browser, and click on the paper number to open a specific paper.

.

It you find these papars useful, you may wish to consider obtaining the full sets of conference papers on CD-ROM from NACE. See the NACE Web site at http://www.nace.org for further details.

Bibliography from Corrosion Abstracts Thanks to Cambridge Scientific Abstracts we have repr~duced listings of papers relating to impedance and no1se that have been produced from Corrosion Abs-

tracts. This is available on-line in the CSA Internet Datcwase Service, as well as being available on CDROM (see the NACE web site on http://www.nace.org for further information and special offers .for NACE ). These listings are found in the D:\Papers\CorrAbs subdirectory, in files Noise.txt and and lmped.txt.

Data for Electrochemical lmpedapct Spectra The original data for all of the electrochemical impedance spectra presented here are Included in directory 0:\lmpedance. They are provided in text file format, exported from "Zview." The file format is as shown in figure G.1. The file name is as this example: fig(figure number).txt (fig313.txt, corresponding to Figure 3.13).

EnAnalyse.Program This program provides a starting point for the analysis of electrochemical noise time records. The program (EnAnafyse.exe) and two ing files (RAKG 1.BPL and VCL40.BPL) are stored in the directory D:\ENAnalyse/Program on the CD-ROM. You can run the program directly from the CD-ROM (the simplest way of. doing this is to double click on the program from within the Windows 95 or NT Explorer). If you wish to install the program on your computer, then run D:\ENAnalyse\Setup.exe and this will run a conventional installation program. This program requires a 32-bit version of Windows to run (Windows 95 or Windows NT). The

! ,j I

:!

I

·!

·'

Corrosion Testing Made Easy

138

•zview Export Data File: Version 1.1" •Export From File:" •Date: 2·24·1999 •Time: 7:34:45•

""

"fiq313. ~t" •Pt. Freq _ . 1, t~OOOOOOE+0004, 2. 7.943SOOE+0003.

z•••

Z' 4.048599E+OOOO, 4.014400E+OOOO.

1. 065248E · 0001

1.168238E-0001

FIGURE G.1. Ale format. I

'··

f

=

program is supplied subject to the license included in 0:\ENAnalyse\License.txt. Updates to the program may become available in due course-look at the UMIST Corrosion. and Protection Centre Web on . I . .· site . http://www..um1st.ac.uk.

consists of calculating Ym 0.9ym-1 + 0.1xm. where Xm is the mth element of the original time record and Ym is the mth element of the filtered time record (taking Jt 0). This reduces the PSO at high frequencies, whereas the low frequency limit will still be At V2/Hz.

Sample Time Records

The program.used to produce these time records is also provided in 0:\AandNum\, with the same layout as for ENAnalyse. ·

A number of sample time records are provided in di· rectory 0:\ nmeRecs. These are all ASCII text files and can be examined with a standard editor such as Word· pad (except for the Windows NT version. Notebook will nOt handle most of these files as they are too large). GNoise.txt: This is· an artificial white-noise time record. The file consists simply of around 1.1 million (more than~) numbers arranged one to a line. The numbers are drawn from the normal distribution with a mean of zero and a·variance of 0.5. If these are used to comp.\.lte a power spectrum, the spectrum should be flat, with a PSO of At V2/Hz (where At is the asst.~med time between samples, and taking the num~rs as ·· voltages). FNoise.txt: This contains -1.1 million points that · have ~n obtained by applying-a simple'digital filter to data equivalent to Gnoise.txt. The filtering process

=

. TR2.TXT: This is a real time rerd of 4096 samples obtained for steer exposed to 3;5% NaCI solution, with air bubbling over the surface of the steel. See Appendix C for a description of the analysis of this time record. It consists of three header lines followed by the noise po~ential and current: . Line 1 Title Line 2 Start time Line 31nterval between samples (in seconds) Line 4 Potential 1, Current 1 · Line 5 Potential 2, . . . .

. Spreadsheets A number of the spreadsheets used in the creation of diagrams for this book are included in directory 0:\Spreadsheets\.

f

.

Appendix H

.: ~-

.

The following definitions may be of assistance to readers. There is also a much m
ARMA: Autoregressive moving average.

'

Artificial neural network: An electronic device or computer algorithm that performs a ·mapping (production of output values from a set of input values) on the basis of a set of training points. Bode plots: Plots of either modulus and phase, or real and imaginary parts, against 1og frequency. Capacitive current: That part -of the total current that charges or discharges the double-layer capacitance. Conductivity: The ability of a material or fluid to conduct electricity. Defined as the conductance of a unit cube of the material or fluid.

Aliasing: The conversion by a sampling process of frequencies above the Nyquist frequency to false frequencies below the Nyquist frequency.

Corrosion potential: The · potential that a corroding metal adopts with no externally applied current. Measured relative to a reference electrode..

Alternating current (ac): A periodically varying current, used here to mean a sinusoidally varying current.

Counter electrode: An electrode used to provide current to an electrochemical cell. Usually made of a noncorrodible metal.

Amplifier: An electronic device that produces an output (either voltage or current) that is proportional to an input (also either current or voltage) . Analog: A signal that has a continuous range of possible values, such as a voltage or current. Angular frequency: .The frequency exP.ressed in radians per seconds = 2.Jr times frequency. • Verinck. E.D. "The basics: Corrosion Testing fo.lada Easy, Vol. 3. Houston. TX, NACE, 1994.

Counter resistor: A resistor used to provide a voltage . proportional to the current flowing, by means of Ohm's law, V= IR. E: Constant-phase element. U: Central processor unit. Crevice corrosion: Corrosion associated with a narrow crevice between two metals or between a metal and an insulator. It normally occurs with ive

.•

.

Corrosion Testing Made Easy

metals and is associated with chemical changes (usually acidification) in the crevice that lead to the breakdown of the ive film in the crevice.

Faradaic current: That part of the total current that is related to amount of material consumed by Faraday's laws (see capacitive current).

DAC: Digital-to-analog converter, a device that converts a digital input to an analog output.

FET: Field-effect transistor.

140

.

OFT: DIScrete Fourier transform. Diffusion: The transport of species in solution that is due to a concentration gradient. Diffusion control: Control of the rate of an electrode process by slow diffusion of a reacting species to the surface. t

Diffusional Impedance: An impedance associated with the change in surface concentration of an electroactive species during the ac measurement. D.lgital: A representation of a parameter by a number with a limited range of possible values. Digital electronic devices (such as digital compUters) are easier to produce and more reliable and repeatable than analog devices and hence are very widely used.

.

.

FFT: Fast Fourier transform. Filter: An electronic device that produces an output that is derived from an input, but with a proportionality that varies as a function of frequency. : Fourier transform: A mathematical operation that calculates the phase and the amplitude of the frequencies present in a time record.

.

FRA: Frequency response analyzer. Frequency: The nurnber of ac cycles per second, in units of hertz or inverse seconds. Frequency domain: This relates to the representation of a signal in of the phase and the amplitude of the various frequencies that are pre$8nt in it, rather than the variation over time (the time domain). Gain: The ratio of the output from an amplifier to its input.

DMA: Direct memory acce~.

IEEE: Institute of Electrical and Electronics Engineers.

Double-layer capacitance: The capacitance associated with the· metal solution interface that is due to charge separation across the ln~erface.

Impedance: The ac equivalent of de resistance. Z VII, where V and I are ac signals.

E~S:

Electrochemical impedance spectroscopy.

Electrochemical noise impedance: The impedance that is obtained by taking the square root of the power spectral density of potential divided by the square root of the power spectral density of current. Electrochemical noise resistance: The resistance that is obtained by dividing the standard deviation of potential noise by the standard deviation of current noise. Electropolishing: The polishing of a metal surface by use of a controlled electrochemical reaction. Equivalent circuit: A combination of electrical elements that gives the same impedance response as the corrosion metal being studied.

=

Impedance modulus: The ratio of the magnitude of ac voltage to ac current. VO: Input/Output. Laplace domain: This relates to the representation of a signal in of the real and the imaginary frequencies thaf are present in it. Imaginary frequencies are effectively exponentially increasing or decreasing transients, so the laplace domain allows for transient processes as well as steady-state processes. Linear polarization: A method for obtaining the polarization resistance by.applying a low-amplitude, slow potential sweep about the corrosion potential. LPR: linear polarization resistance. LSB: Least-significant bit.

Electrochemical Impedance and Noise

.

141 .

Maximum entropy method: A mathematical operation that estimates the power spectrum corresponding to a time record.

Quantization: An error in a digitized value that results from the limited resolution of the digitizing device. •Referenc~

MEM: Maximum entropy method.

electrode: An electrode of known potential against which the potential of a corroding metal is measured.

Nyquist frequency or limit: The maximum frequency that can be represented in a sampled time series. Any frequencies above the Nyquist limit that are present in the signal before sampling will lead to aliasing.

Rt!sistivlty: The resistance of a material to !be age of current. Defined as the resistance across opposite faces of a unit cube of the material.

Nyquist plot: .A plot of the imaginary part of the •· · impedance or the ittance against the real part. Overpotentlal: The difference between the actual potential of a metal and some baseline potential, e.g.,

E-

Ecorr.

rms: Root mean square. Stern-Geary coefficient: 8 volts, used -to calculate the corrosion current from polarization resistance lcorr = 81 Rp. The value depends on the mechanism of the corrosion process.

PC: Personal computer.

Stress-corrosion cracking (SCC): The cracking of an alloy as a result of the combined action of applied stress and a corrosive environment.

Phase angle: A measure of the time lag between peak current and peak potential for an ac system. One cycle of ac is equivalent to 360°.

SCE: Saturated calomel electrode.

Pink noise: Noise with a power spectral density that is inversely proportional to the frequency. Pitting corrosion: Corrosion occurring in localized spots, leading to pits (depressions in the surface), with little or no corrosion outside the pits. It normally occurs on ive metals, and is associated with local changes in chemistry (usually acidification) in the pit.

Time domain: This relates to the representation of a signal in of the value as a function of time. Time record: A set of samples of a given parameter, obtained at successive, usually equal, time intervals. USB: Universal serial bus. VFC: Voltage-to-frequency converter.

Polarization resistance: The slope of the steadystate potential-net current curve at the corrosion potential.

Waterline attack: Enhanced corrosion at the air-water interface, caused by the easier :transport of oxygen in this region.

Power spectral density (PSO): The power present in a signal at a given frequency, expressed as the power per unit of frequency.

White noise: Noise with a power spectral density that is the same at all frequencies. Working electrode: The electrode being studied.

Power. spectrum: A graph of the power present in a signal as a function of frequency.

ZRA: Zero-resistance ammeter.

,.

.

.

Appendix I

List of Symbols> .: Symbol f3a f3c

Ll.t 8 '1 a t"

(J)

t/J A A b

B

ba be

c c

cburk Glr

Cs 0 riff)

E

E

m

Definition Exponential anodic 1 · Tafel coefficient Exponential cathodic Tafel coefficient lntersample period Diffusion layer thickness Overpotential Warburg coefficient Time lag or time constant Angular frequency Phase angle Area Amplifier gain Bandwidth Stern-Geary coefficient Anodic Tafel slope Cathodic Tafel slope Capacitance Concentration Bulk solution · concentration Double-layer capacitance Surface concentration Diffusion coefficient Frequ.,.ncy-•:11?;' 0 "dent discrimination function Electrode potential Mean potential Standard deviation of potential

Units

..v v

Symbol E2 n

e...

E(f) s em

v

n cm2 s-1/2

E(t)

E[k]

s radls rad cm 2

14

Ecorr En(t)

Hz

v

En, ref V or Vldecade V or V/decade F mole cm- 3 molecm-3

eo f

F F

H(f)

mole cm- 3 crr;2 5 -1

v v v

h(t)

i

i

r=:

\1 f2n

Definition Variance of potential Operational amplifier inverting input voltage Frequency domain representation of a potential signal Time-varyi~g

electrode potential kth element of time record Operational amplifier noninverting input voltage Corrosion potential Time-varying noise potential (with zero mean) Potential noise of ·reference electrode Operational amplifier output voltage Frequency

e

Faraday's constant Representation of h(t) in the frequency domain General time-varying !unction Current Current density Mean current Standard deviation of current

Units

V2

v

v v v v v v v v Hz (=cycles per second) 96,500 C mole- 1

A A cm- 2 A A

•of

:~

:~~

-~

~~

.'

''·!

Corrosion Testing Made EQy

144

.. .,. I '

Symbol

,.

72n n

., l(f)

/(t) /[k]

lo io

'· ;. lc ic icorr ilim

In In1. t)

Inet j k

k L

m m

Definition Variance of current Current noise due to potential noise from one electrode acting on another Freque~-domain . representation of a current signal Time-varying current kth element of time record I Current amplitude (of sine .wave) Exchange current density · Anodic current Anodic current density Cathodic current Cathodic current density Corrosion current . density . Umiting current density Operational amplifier input current noise Time-varying noise current (with zero mean) Net current Square root of -1 General-purpose index Boltzmann's constant Inductance MEM order Number of transients

Units A2 A

Symbol

n N

q •

A

R

R:t A A

Res.:

Rib A

Rn

Acm- 2

Rp

Rs t

~A

Acm- 2 A Acm- 2 Acm- 2

T

u

v

\.1> Vn

Acm- 2 A

X X'

A

y

Definition Number of electrons ,(also used as a generic · subscript for noise) Number of points in a time record Charge Resistance Charge-transfer resistance Diffusional resistance resistance Electrochemical noise resistance Polarization resistance Solution resistance Time Absolute temperature Voltage sweep rate Voltage Voltage amplitude (of sine wave) Operational amplifier input voltage noise Real component of X Imaginary component of X ittance

Units

c

n n n n n n n s K V/s

v v v n n

1/n (=Siemens) complex

Y' Y"

A

1.38 H

X

10-23 J K- 1

z k 7

Ln

Zs

Real component of Y Imaginary component of Y Impedance Diffusional impedance Electrochemical noise impedance Source impedance

1/n 1/n

n, complex n n n

f

•

.: A l

AC impedance method. See Electrochemical impedance spectroscopy ACF. See Autocorrelation function Activation control, 40--43, 109 Active-ive transition, 4~7 ADCs. See Analog-to-digital converters ittance calculation of, 15 conductance and, 2 eleqrodes and, 109 impedance and, 13-15, 46-47 plots of, 12, 38 Adsorption/desorption, 32, 46-47 Ag/AgCI electrode, 58 Aliasing, 53, 60, 66, 68, 88, 106, 124, 126 All-poles model, 121 . Alternating-current responses, 109-110 Aluminum alloys, 90 Amplitude, average, 52 Amplitude, of voltage set, 26 Analog techniques, 60-63 Analog-to-digital converters, 63, 139 Anodic process, 40--43, 48, 107 Antialiasing, 61, 106 Aqueous corrosion, 91 AR model. See Autoregressive model Arcing, 66 Area correction, 27 ARMA mooet. ::>ee Autoregressive moving average model Artificial neural network, 139 Atmospheric corrosion. 91 Autocorrelation function (ACF). 12G-121, 139

Autoregressive (AR) model, 121 Autoregressive moving average model (ARMA), 121-122

8 Bel units, 78 Bessel filters, 102-103 Binary number system, 63 Bispectral analysis, 93 Bode plots, 17-18,25, 3G-31, 36 Boltzmann constant, 98 Buffers, noninverting, 100 Bus, IEEE 488 type, 64 Butterworth filters, 103

c Calomel electrode, 42 Capacitance, 10, 140 double-layer, 2 frequency range, 26 potentiostats and, 29 pseudocapacitarice, 46-47 Capacitors, 96, 99-100 Carbonate solution, 87 Cathodic reaction, 41-43,71, 107 Causality conditions, 135 Cell design, 23, 56-58 Chaos methods, 86 Charge, defined, 95 Charge-transfer control. See Activataon control Chart recorders, 125 Chebyshev filters, 103 Chemical cleaning solutions. 48 Chopper-stabilized amplifiers. 106

Corrosion Testing Made Easy

146

Cleaning solutions, 48 Coated metals, 29, 9Q-91 Colors, of noise, 129 Complex numbers, 15 Constant-phase elements, 45-46, 96 Contlnuous·tin)~ signals, 53, 125 Convection, 38 Corrosion rates, 56-58 charge-transfer control, 49 low-frequency distortions, 32 polarization resistance, 47 potential, defined, 139 See also specinc types, tests Counter resistors, 2-4 Coupling options, 3Q-31 E. See Constant-phase elements Crevice effects, 58, 72, 139-140 Current amplifier, 61, 101 Current control, 46-47 Current, defined, 95 Current noise measurement, 55-56 Current time records, 51-53 Current time response, 1o-11

D DAC. See Digital-to-analog converters Data transfer, 64-66 Decibel units, 78 Delta-sigma converters, 64 Depressed semicircles, 32 Desorption/adsorption, 32, 46-47 Dielectrics, 99 Diffusion boundary-layer, 38 Diffusional impedance. See Warburg impedances Digital recording, 62 Digital-to-analog converters (DAC), 64,

139-140 Direct-curFent kinetics, 107-112 Direct memory access (DMA), 65 Dirt deposits, 57 Discrete-time signals, 53, 125 Discriminant analysis, 8&-87 DMA. See Direct memory access Double-layer capacitance, 2, 10, 25,

140 Drift, 26, 30-31, 68 Dual-slope converters. 64 Dummy cells, 28, 68

'

E ECN. See Electochemical current noise EIS. See EIEfctrochemical impedance spectroscopy Electrochemicai current noise (ECN), 51-53 Electrochemical impedance spectroscopy {EIS) · c1nalysis of, 35-49 linear polarization resistance methods,

9-13 mathematical methods for, 32 measurement of, 19-33 noise and, 9-18 principles of, 1 uses for, 3 Electrochemical interface. See Potentiostats Electrochemical kinetics, 107-112 Electrochemical noise, 55-75 . analysis of, 71-93 defined, 3-4 impedance, 82 measurement of, 4-5 oxygen transport, 5-6 potential noise, 51-53 resistance of, 51-53, n sample measurement process, 113-116 steel, 4-5 two parts of, 5-6 Electrodes, 55-58 Electrolytic capacitors, 99 Electronic filters, 102-1 03 EnAnalyse program, 115, 137-138 EPN. See Electrochemical pot~ntial noise Equivalent circuits, 35-36, 43-45

F Far-field averaging, 91 Faradaic current, 10 Faradaic impedance, 40 Faraday cage, 67 Faraday constant, 108 Faraday's laws, 1 Fast Fourier transforms {FFT), 58, 78,

8o-B4 FET. See Field-effect transis!cr FFT. See Fast Fourier transform Fick's law, 111 Field-effect transistor (FET), 101 Field problems, 67 Film growth, 32

147

Electrochemical Impedance and Noise·

Filtering, 61, 102-103 Finish, of surface, 57 Finiteness conditions, 135 Flash converters, 63 Flicker noise, 130 Fourier transform, 118-120. See also Fast Fourier transform . · Frequency-domain methods, 71, n-79 Frequency range, 24-26 · Frequency response analyzer, 19 ~ Frequency spectra, 62

Gaseous environments, 91 Generator/analyzer, 19 Gold electrodes, 23 , Graphical analysis. See Bode plots;'Nyquist plots Ground loops, 67

H High-frequency problems, 28 High impedance systems, 29 Higher-order fifters, 103 Hydrogen bubbles, 6, 93

IEEE 488 bus, 64 IM. See lmmitance lmmitance (IM), 135 Impedance, 140 ittance and, 13-15,46-47 calculation of, 15-16 combining, 96 diffusional, 39 frequency and, 2 highly capacitive systems, 37 spectral, 2 Warburg impedances, 38, 39 Impedance spectroscopy. See EIS Inductors, 29, 96, 99-100 Inhibitors, 37 Instrument bus, IEEE 488, 64 ir.5~iumsr.~ :,,jise, 68, 82-83, 10o-101 Interfacial effects, 87 Irreversible processes, 111

J Johnson noise. 129

K Kirchoff's law, 97 ~amers~K~nig transforms, 32, 135 Kurtosis, 75-76, 124

L I

Laplace domain, 71 Laplace transformation, 111 Leakage current, 99, 119 Lea~t-squares fitting, 43-44 Limiting current, 41-43 Linear polarization methods, 2 Linear systems, 97 Linearity condition, 135 Localization parameter, 9Q-91 Lock-in amplifier, 19 ~­ Low-conductivity solutions, 91 Low frequency limit, 37, 44-47 Low-frequency problems, 30 Low- filter, 126 Luggin capillary, 22, 28-29 Luggin tip, 36

M MA model. See Moving average model Mass transfer effects, 48, 108, 11 0 Maximum entropy method (MEM), 78, 8o-81,

12Q-121 Mean, defined, 62, 74-75, 123 Mean potential, 74-75 Mechanistic studies, 91-92 MEM. See Maximum entropy method Mercurous-mercury system, 42 Metal-solution interface, 87 Midfrequency problems, 30 Mixed control, 41-43 Moving average (MA) model, 121

N Neural networks, 139 Noise. See Electrochemical noise Noise impedance method, 89 Non inverting buffers, 100-101 Nonlinear systems, 97 Nyquist frequency, 53, 60, 63, 119, 126,

141 Nyquist limit. See Nyquist frequency Nyquist plots, 25, 36 ittance, 12

..

Corrosion Testing·Made Easy

148

Nyquist plots (cont.) impedance, 16-17 lin.ear regions in, 30

0 Ohm's law, 95 Overpotentials, 9-13, 141 Oxygen transport, ~. 39

p Painted metals, 9o-:-91, 101 Parallel port, 64-66 Patent protection, 6-7 Phase angle, 13, 141 Pink noise, 130 Pitting, ~. 57, 72, 89-90, 91 Platinum electrodes, 23 Poisson process, 130 Polarization resistance, 1, 48 activation and, 41 charge-transfer and, 108 corrosion rates and, 47-49 double-layer capacitance, 10 EIS and, 9-13, 135 noise and, 88 . See also Low-frequency limit Polymer gels, 58 · Porous conducting films, 30 Potential control, 46 Potential, defined, 95 Potential noise. See Electrochemical potential noise Potential step method, 26 Potential time records, 51-53 Potentiodynamic measurements, 19 Potentlostats, 19 capacitance and, 29 cell setup and, 19 computer-controlled, 19, 24 dummy cells, 28 manual, 19, 24 phase shifts, 22 Power spectral density (PSD), 78, 141 !-'ower spectrum, 78 PSD. See Power spectral density

a Quantization, 68, 141

R Redox systems, 38, 48 Redundant~l~ments, 41 Reference electrodes, 22-23,29, 36, 58 Reproducibility, 63 Resistance, defined, 95 Resistance noise, 91 Resistivity, 141 Resistors, 97-99 Root-triean-square value, 62, 75, 124 AS 232 protocol, 64

s Sallen-Key theory, 102 Sampling methods, 53, 58, 62-63 Saturated calomel electrode (SCE), 32, 58 Sawtooth potential, 10 Scattered data, 29 SCC. See Stress-corrosion crack growth SCE. See Saturated calomel electrode Semiconductors, 95 Sensitivity, 63 Shielding, 30, 67 Shotnoise,68,90-91,92-93, 129,130 Signal averaging, 27 Signal conditioning circuit, 104, 106 Signal processing, 60, 117-122 Sine-wave correlation, 117-122 Sinusoidal signals, 11-12 Size, of electrodes, 55 Skewness, 75, 124 Small-amplitude potential, 11 Solution resistance, 16, 22, 92-94, 134 Specimen area, 52, 57 Spectral estimation, 77-78, 117-122 iimits on, 8Q-81 Spikes, 27 Standard deviation, 62, 75, 124 Stationary systems, 59 Statistical methods, 74-77 Statistical moments, 123 Statistical properties, 59 Siearn-Geary ~,;oefficient, 47-49 Stearn-Geary equation, 87, 108 Stress-corrosion crack (SCC) growth, 72. 141 Sulfide films. 37 Superposition theorem, 88, 97

Electrochemical Impedance and Noise

.

149 .

T Tafel coefficients, .27, 39 Thermal noise, 68, 129 Thin-film corrosion, 91 Three-electrode measurements, 23, 27 Time domain, 71 Time records, 51-53, 71~2. 82-84 Transfer function, 100 Transient analysis, 55-56, 71-72, 90-91, 131-136 Trend removal, 80 Two-electrode measurements, 23, 27,29

..

·~

Universal serial bus, 64-66 USB. See UniVersal serial bus

v Variance, defined, 75, 124 Variation, coefficient of, n. 89-90, 125

VFCs. See Voltage-to-frequency converters Voltage amplifier, 60 Voltage, defined, 95 Voltage-to~frequency converters (VFCs), 64

w W~rburg coefficient, 39, 112 Warburg impedances, 38-42, 96, 111 Waterline effects, 58, 141 Wavelet transform, 122 W,eJds, 57 Wheeltappers, 1 White noise; 129 Windowing, 80 Working electrodes, 22

z Zero-resistance ammeter (ZRA), 60, 101 ZinC'phosphate, 29 ZRA. See Zero-resistance ammeter

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