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Chapters / Chapter 1, General electrochemistry / Lesson 1.3

1.3 Electrochemical impedance spectroscopy

What impedance measures, how to read Nyquist and Bode plots, and how to build, fit and check an equivalent circuit, with a playground to try it.

About 30 minutesOpen to everyoneQuiz at the end

What impedance measures

In a CV the potential is swept over hundreds of millivolts. Impedance spectroscopy does the opposite. It holds the electrode at one potential and adds a small sine wave, usually 5 to 10 mV, at one frequency after another, typically from 100 kHz down to 10 mHz. The current answers with a sine wave of the same frequency, smaller or larger, and shifted in time.

Z(ω) = E(ω) / I(ω) = Z′ + jZ″|Z| is the ratio of the amplitudes and the phase angle θ is the time shift. ω = 2πf. Z′ is the real part and Z″ the imaginary part.

Because each process at the electrode has its own speed, each shows up in its own frequency range. Electrolyte resistance answers at every frequency, charging of the double layer and charge transfer at middle frequencies, and slow diffusion only at low frequencies. That is what lets impedance separate processes that overlap in a CV.

Three conditions behind every spectrumThe system must be linear (small amplitude), stable (not drifting while it is measured) and causal (the current is a response to the applied signal only). Most errors in EIS come from breaking one of them.

The building blocks

ElementImpedancePhaseWhere it comes from
Resistor, RR0°Electrolyte (Rs), charge transfer (Rct), films
Capacitor, C1/(jωC)−90°Double layer, an ideal blocking electrode
Constant phase element, CPE1/[Q(jω)n]−90n°A double layer on a rough or uneven surface, 0.5 < n ≤ 1
Warburg, Wσω−1/2(1 − j)−45°Semi-infinite diffusion of a redox species
Inductor, LjωL+90°Usually cables and connections at high frequency

A resistor and a capacitor in parallel have a time constant τ = RC. On a Nyquist plot they give a semicircle whose top sits at f = 1/(2πRC). Processes whose time constants differ by a factor of about 100 or more give separate arcs. Closer than that, they merge into one distorted arc.

Reading Nyquist and Bode plots

Try it in the playground below. Start with Simple interface, then switch to Randles, with diffusion and Two time constants.

  • Nyquist plot, −Z″ against Z′, with equal axes. The high-frequency intercept is Rs, the width of the arc is Rct, and a 45° tail at low frequency points to diffusion. Frequency is hidden on this plot, so label a few points.
  • Bode plots, |Z| and phase against log f. Frequency is visible, flat parts of |Z| are resistances, sloped parts are capacitive, and each peak in the phase marks a time constant. They are often better for counting processes.

The full EIS playground also lets you download the spectrum as a CSV file to practise fitting.

Building and fitting an equivalent circuit

  1. Look before fitting. Count the arcs and tails in the Nyquist and Bode plots, and estimate Rs, Rct and the arc frequency by eye.
  2. Choose the simplest circuit in which each element has a physical meaning, for example Rs + (Rct ∥ CPE) for a single interface, or the Randles circuit when a redox species diffuses.
  3. Fit with complex nonlinear least squares, usually with modulus or proportional weighting, using the real and imaginary parts together.
  4. Check the result. The residuals should be small and random over the whole frequency range, the parameter errors should be much smaller than the values, and the values should make sense, for example a double-layer capacitance of tens of µF per cm² of real area.
  5. Report the circuit, every fitted value with its error, n for each CPE and the effective capacitance from the Brug formula if you compare capacitances.
More elements always fit better. A good fit alone does not show that a circuit is right, so a new element should only be added when the data clearly need it and it can be explained physically.

Good data first

  • Linearity. Measure at two amplitudes, for example 5 and 10 mV. The spectra should overlap.
  • Stability. Wait for the open-circuit potential or the current to settle before you start, and compare a spectrum recorded from high to low frequency with one recorded in the other direction, or repeat it. Large changes mean the system is drifting.
  • Kramers–Kronig test. Valid data satisfy these relations. Most analysis software offers a quick linear Kramers–Kronig test, and points that fail it, often at the lowest frequencies, should be removed before fitting.
  • Artefacts. A loop or inductive tail at the highest frequencies usually comes from cables or the reference electrode. Keep leads short, use a reference with a low impedance, and treat the first points with care.

Where we use impedance

FieldWhat impedance givesChapter
ElectrocatalysisRu for iR correction, and Rct to compare catalysts at the same potential2
SupercapacitorsEquivalent series resistance and the frequency at which the device stops behaving as a capacitor3
SensingChange of Rct of a redox probe when a target binds to the electrode4
CorrosionPolarization resistance and coating performance5
BatteriesResistance of the electrolyte, the SEI and charge transfer, and how they grow with ageing6

Check your understanding

Quiz for lesson 1.3

Pick one answer per question. You see the explanation straight away.

References and further reading

  1. A. Ch. Lazanas, M. I. Prodromidis, Electrochemical impedance spectroscopy, a tutorial, ACS Meas. Sci. Au 3 (2023) 162–193. doi:10.1021/acsmeasuresciau.2c00070
  2. M. E. Orazem, B. Tribollet, Electrochemical Impedance Spectroscopy, 2nd ed., Wiley, 2017.
  3. B. A. Boukamp, A linear Kronig–Kramers transform test for immittance data validation, J. Electrochem. Soc. 142 (1995) 1885–1894.
  4. 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.
  5. A. J. Bard, L. R. Faulkner, Electrochemical Methods: Fundamentals and Applications, 2nd ed., Wiley, 2001, chapter 10.