Lessons / Tools
EIS playground
Pick an equivalent circuit, change Rs, Rct, the constant phase element and the Warburg element, and watch the Nyquist and Bode plots form from 100 kHz down to 10 mHz.
How to read the plots
Impedance Z = E/I is measured with a small sine wave, usually 5 to 10 mV, at one frequency after another. Each frequency gives one point with a real part Z′ and an imaginary part Z″.
- Nyquist plot. −Z″ against Z′. High frequencies sit on the left. The first crossing of the real axis is Rs, the resistance of the electrolyte between the working and reference electrodes. This is the value most often used for iR correction.
- The arc. A resistor in parallel with a capacitor gives a semicircle of diameter Rct. A smaller arc means faster charge transfer at that potential.
- The Warburg tail. At low frequency, diffusion of the redox species adds a straight line at 45°. Its size is set by σ, which falls as concentration and diffusion coefficient rise.
- Bode plots. |Z| and phase against log frequency. They show frequency directly, so they make it easier to see how many time constants a spectrum holds and where each one sits.
Why a CPE and not a capacitor
Real electrodes are rough, porous and uneven, so the measured arc is usually flattened. A constant phase element, Z = 1/[Q(jω)n], describes this with an exponent n between 0.5 and 1. When n = 1 it is an ideal capacitor. Q is not a capacitance when n is below 1, so the playground also shows the effective capacitance from the Brug formula. Report n with Q, and treat a value of n far below 0.8 as a sign that the model may not describe the electrode well.
Good practice
- Record a spectrum only at a steady state, and check linearity, stability and causality. Kramers–Kronig tests are built into most analysis software.
- Choose the simplest circuit that each element can be explained physically. More elements always fit better, but that alone does not make the model right.
- Give the DC potential, amplitude, frequency range and points per decade with every spectrum.
- Download the CSV here and fit it in your own software to practise. You should get the values back.
References
- A. Ch. Lazanas, M. I. Prodromidis, Electrochemical impedance spectroscopy, a tutorial, ACS Meas. Sci. Au 3 (2023) 162–193. doi:10.1021/acsmeasuresciau.2c00070
- G. J. Brug, A. L. G. van den Eeden, M. Sluyters-Rehbach, J. H. Sluyters, The analysis of electrode impedances complicated by the presence of a constant phase element, J. Electroanal. Chem. 176 (1984) 275–295.
- M. E. Orazem, B. Tribollet, Electrochemical Impedance Spectroscopy, 2nd ed., Wiley, 2017.
- A. J. Bard, L. R. Faulkner, Electrochemical Methods: Fundamentals and Applications, 2nd ed., Wiley, 2001, chapter 10.