Electrochemical Impedance Spectroscopy and its Applications Andrzej Lasia Département de chimie, Université de Sherbrooke, Sherbrooke Québec, J1K 2R1 [email protected] A. Lasia, Electrochemical Impedance Spectroscopy and Its Applications, Modern Aspects of Electrochemistry, B. E. Conway, J. Bockris, and R.E. White, Edts., Kluwer Academic/Plenum Publishers, New York, 1999, Vol. 32, p. 143-248. Table of content I. Introduction..............................................................................................................................4 1. Response of Electrical Circuits............................................................................................4 (i) Arbitrary Input Signal......................................................................................................4 (ii) Alternating Voltage (av) Input Signal..........................................................................6 2. Impedance of Electrical Circuits..........................................................................................7 (i) Series R-C Circuit............................................................................................................8 (ii) Parallel R-C Circuit......................................................................................................8 (iii) Series: Rs + Parallel R-C Circuit..................................................................................9 3. Interpretation of the Complex Plane and Bode Plots...........................................................9 II. Impedance measurements ......................................................................................................10 1. Ac Bridges.........................................................................................................................10 2. Lissajous Curves ................................................................................................................11 3. Phase Sensitive Detection (PSD).......................................................................................11 4. Frequency Response Analyzers.........................................................................................12 5. Fast Fourier Transform (FFT)............................................................................................13 (i) Pulse Perturbation..........................................................................................................14 (ii) Noise Perturbation.....................................................................................................14 (iii) Sum of Sine Waves....................................................................................................14 III. Impedance of faradaic reactions in the presence of diffusion................................................15 1. The Ideally Polarizable Electrode......................................................................................15 2. Semi-Infinite Linear Diffusion ..........................................................................................15 3. Spherical Diffusion............................................................................................................20 4. Cylindrical Electrodes........................................................................................................22 5. Disk Electrodes..................................................................................................................23 6. Finite-Length Diffusion.....................................................................................................23 (i) Transmissive Boundary.................................................................................................24 (ii) Reflective Boundary ..................................................................................................25 7. Analysis of Impedance Data in the Case of Semi-Infinite Diffusion: Determination of the Kinetic Parameters.....................................................................................................................26 (i) Randles’ Analysis64,65,67 .................................................................................................26 (ii) De Levie-Husovsky Analysis ....................................................................................27 (iii) Analysis of cot ϕ........................................................................................................27 (iv) Sluyter's Analysis.......................................................................................................28 IV. Impedance of A faradaic reaction involving adsorption of reacting species .........................29 1. Faradaic Reaction Involving One Adsorbed Species.........................................................29 2. Impedance Plots in the Case of One Adsorbed Species ....................................................31 3. Faradaic Impedance in the Case Involving Two Adsorbed Species..................................34 4. Impedance Plots in the Case of Two Adsorbed Species....................................................36 5. Faradaic Impedance for the Process Involving Three or More Adsorbed Species............36 V. Impedance of solid electrodes................................................................................................37 3 1. Frequency Dispersion and Electrode Roughness...............................................................37 2. Constant Phase Element.....................................................................................................37 3. Fractal Model.....................................................................................................................40 4. Porous Electrode Model.....................................................................................................42 (i) Porous Electrodes in the Absence of Internal Diffusion................................................42 (a) De Levie’s treatment..............................................................................................43 (b) Rigorous treatment.................................................................................................45 (ii) Porous Electrodes in the Presence of Axial Diffusion...............................................46 (iii) Other Pore Geometries...............................................................................................49 5. Generalized Warburg Element...........................................................................................49 VI. Conditions for "good" impedances ........................................................................................50 1. Linearity, Causality, Stability, Finiteness..........................................................................50 2. Kramers-Kronig Transforms..............................................................................................51 3. Non-Stationary Impedances...............................................................................................53 VII. Modeling of experimental data ..............................................................................................54 1. Selection of the Model.......................................................................................................54 2. CNLS Approximations ......................................................................................................56 (i) CNLS Method................................................................................................................56 (ii) Statistical Weights.....................................................................................................57 (iii) AC Modeling Programs .............................................................................................58 VIII. Instrumental limitations .........................................................................................................58 IX. Conclusion .............................................................................................................................60 4 I. INTRODUCTION Electrochemical Impedance Spectroscopy (EIS) or ac impedance methods have seen tremendous increase in popularity in recent years. Initially applied to the determination of the double-layer capacitance1-4 and in ac polarography,5-7 they are now applied to the characterization of electrode processes and complex interfaces. EIS studies the system response to the application of a periodic small amplitude ac signal. These measurements are carried out at different ac frequencies and, thus, the name impedance spectroscopy was later adopted. Analysis of the system response contains information about the interface, its structure and reactions taking place there. EIS is now described in the general books on electrochemistry,8-17 specific books18,19 on EIS, and there are also numerous articles and reviews.6,20-31 It became very popular in the research and applied chemistry. The Chemical Abstract database shows ~1,500 citations per year of the term "impedance" since 1993 and ~1,200 in earlier years and ~500 citations per year of "electrochemical impedance". Although the term "impedance" may include also non- electrochemical measurements and "electrochemical impedance" may not include all the electrochemical studies, the popularity of this technique cannot be denied. However, EIS is a very sensitive technique and it must be used with great care. Besides, it is not always well understood. This may be connected with the fact that