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Electrochemical Impedance Spectroscopy (EIS)

date2026-07-24tags:chem:

Electrochemical impedance spectroscopy (EIS) is the AC counterpart to cyclic voltammetry. It uses a tiny sinusoidal perturbation (a few millivolts) to keep response linear. It measures how current differs in amplitude and phase from applied voltage. A frequency sweep separates processes by speed. DC techniques cannot do this alone. A DC measurement sums all processes into one number. EIS pulls them apart along the frequency axis. This page complements Electrochemistry and Corrosion. It shares the linearized-kinetics assumption with LPR.

What EIS actually measures

The instrument applies $E(t) = E_0 \sin(\omega t)$ on top of a DC potential. The angular frequency is $\omega = 2\pi f$. Amplitude $E_0$ is typically 5-20 mV. Typically the DC potential is the open-circuit potential. The instrument records $I(t) = I_0 \sin(\omega t + \phi)$. Current shifts by phase $\phi$. Amplitude scales. The complex impedance is:

$$ Z(j\omega) = \frac{E(j\omega)}{I(j\omega)} = Z' + jZ'' = |Z|\,e^{j\phi} $$

$Z'$ is real (resistive). $Z''$ is imaginary (capacitive or inductive). Sinusoids work because small $E_0$ keeps the local current-voltage curve linear. A Taylor expansion of any nonlinear kinetics gives a sinusoidal response. The current stays at the same frequency. The system behaves linearly during the measurement. Large $E_0$ produces harmonics at $2\omega, 3\omega, \dots$. This is a testable sign of leaving the linear regime. A full spectrum sweeps ~$10^5$ to $10^{-2}$ Hz or lower.

Nyquist and Bode representations

The spectrum $Z(\omega)$ plots two ways:

Equivalent circuit elements, by frequency regime

Fit EIS data to an equivalent circuit. It uses resistors, capacitors, and diffusion elements. The fit reproduces the measured $Z(\omega)$. The physical limits of this approach are in the model-ambiguity section below.

Solution/ohmic resistance, $R_s$

$R_s$ is the electrolyte resistance between working and reference electrodes. It includes contacts and leads. It is frequency-independent. It appears at the real-axis intercept at the highest frequency. At high $\omega$, all capacitors look like short circuits. This is the same ohmic drop as in Potentiostat (iR compensation).

The double layer: $C_{dl}$ and the constant phase element

Every electrode/electrolyte interface acts like a capacitor. This is the electrical double layer. It exists regardless of redox reactions. It is the same $C_{dl}$ that appears as capacitive background in cyclic voltammetry. An ideal capacitor gives impedance $Z_C = 1/(j\omega C_{dl})$. In parallel with charge-transfer resistance, it traces a perfect semicircle in the Nyquist plot.

Real electrodes rarely show perfect semicircles. Surface roughness and porosity create a distribution of local time constants. Replace the ideal capacitor with a constant phase element (CPE):

$$ Z_{CPE} = \frac{1}{Q\,(j\omega)^n} $$

$n \in [0, 1]$. At $n=1$, the CPE is an ideal capacitor ($Q=C$). At $n=0$, it is a resistor. At $n=0.5$, it behaves like a Warburg element.

A parallel $R$–CPE gives a depressed semicircle. Its center sits below the real axis. When the depression is modest ($n \gtrsim 0.75$), estimate an effective capacitance:

$$ C_{eff} = \frac{(RQ)^{1/n}}{R} $$

This comes from Matt Lacey's write-up on constant phase elements (Lithium Inventory ─ Constant Phase Elements). It approximates, not recovers, a physical capacitance. It gets worse as $n$ drops from 1.

Charge-transfer resistance, $R_{ct}$

$R_{ct}$ is the resistance to the faradaic reaction. For a one-electron couple at equilibrium, linearizing Butler-Volmer for small overpotential $\eta$ gives $i \approx i_0\,nF\eta/RT$:

$$ R_{ct} = \left.\frac{d\eta}{di}\right|_{\eta=0} = \frac{RT}{nFi_0} $$

This parallels LPR, which linearizes Wagner-Traud kinetics of a corroding electrode. $R_{ct}$ is the charge-transfer resistance at the corrosion potential. $R_p$ from LPR is the combined resistance of anodic and cathodic partial reactions. Their relationship depends on Tafel slopes:

$$ R_p = \frac{B}{i_{corr}} = \frac{R_{ct}}{1 + \beta_a/\beta_c} $$

When $\beta_a \approx \beta_c$ (common for symmetric corrosion), $R_p \approx R_{ct}/2$. When slopes differ significantly, they can differ by a factor of 3 or more. For passive systems (anodic branch suppressed, $\beta_a \to \infty$), $R_p \to 0$. Do not treat $R_{ct}$ from a single-time-constant EIS fit as equal to $R_p$ from LPR without evidence that Tafel slopes are symmetric.

Warburg impedance, $W$

Warburg impedance models diffusion limitations. A reactant must diffuse to the electrode. For semi-infinite linear diffusion in an unstirred solution:

$$ Z_W = \sigma\,\omega^{-1/2}(1 - j) $$

$\sigma$ is the Warburg coefficient (set by diffusion coefficients and concentrations). The real and imaginary parts have equal magnitude at every frequency. $Z_W$ traces a 45° line on the Nyquist plot. Diffusion is "the 45° line", parallel to how a semicircle means an $R$–$C$ pair. Diffusion dominates at low frequency. It is slow. At high frequency, the perturbation reverses before the diffusion layer develops.

The Randles cell

The Randles circuit is the most common arrangement in electrochemistry. It has solution resistance in series with a double-layer capacitance in parallel with a charge-transfer-resistance-plus-Warburg branch:

$$ Z(\omega) = R_s + \frac{1}{j\omega C_{dl} + \dfrac{1}{R_{ct} + Z_W}} $$

Faradaic current (through $R_{ct}$ and $Z_W$) and capacitive current (through $C_{dl}$) are parallel paths. The series placement of $Z_W$ after $R_{ct}$ reflects the fact that reaction cannot outrun reactant supply. Nyquist: a semicircle at high-to-mid frequency ($R_{ct}$–$C_{dl}$ arc, intercepts at $R_s$ and $R_s+R_{ct}$). A 45° Warburg tail at low frequency. Real interfaces replace $C_{dl}$ with a CPE. This gives a depressed semicircle. See Lithium Inventory ─ The Randles Circuit. All corrosion circuits are variations on this.

Porous electrodes: the de Levie transmission line

Flat electrodes are unusual. Real electrodes (porous coatings, corroded pits, battery composites) have pores. De Levie treats a single cylindrical pore as a distributed network. Ionic resistance runs down the electrolyte "rail." Double-layer capacitance distributes along the pore wall. Charge must fight resistance before it reaches capacitance deep inside. This diffusion-like math produces the same 45° line as Warburg (even without physical diffusion). An infinitely long pore looks like Warburg. A finite pore shows 45° behavior only down to the frequency where the full pore reaches. Below that, the response saturates into a vertical capacitive line. See Lithium Inventory ─ Diffusion Impedance. A continuous distribution of pore depths creates a distribution of $RC$ time constants. A single CPE exponent $n$ summarizes this compactly.

Model ambiguity: when a good fit isn't proof

$\chi^2$ minimization is necessary, not sufficient

Fitting an equivalent circuit is a nonlinear least-squares problem. Adjust element values to minimize $\chi^2 = \sum_i \left[ (Z'_{i,\text{exp}} - Z'_{i,\text{model}})^2 + (Z''_{i,\text{exp}} - Z''_{i,\text{model}})^2 \right] / \sigma_i^2$. This gives a best-fit answer for a fixed circuit topology. It does not prove that topology is correct. Different circuits can reproduce nearly identical impedance when time constants overlap or frequencies are incomplete. A low $\chi^2$ means the circuit can describe the data. It does not mean the circuit is physically real.

Bayesian inference and AutoEIS

A 2024 study applied Bayesian inference to corrosion equivalent-circuit models (/npj Materials Degradation/ 2024, arXiv:2407.20297). It computed the full posterior probability distribution over parameters. It found that three commonly used corrosion ECMs are statistically indistinguishable. A good least-squares fit to one is not evidence that it is physically real.

AutoEIS automates model comparison (/J. Electrochem. Soc./ 170, 086502 (2023), arXiv:2305.04841). It validates data against Kramers-Kronig first. It generates candidate circuits with an evolutionary algorithm. It fits each with Bayesian inference. It ranks survivors with WAIC, which penalizes unnecessary complexity.

What wrecks a spectrum

Drift and non-stationarity

EIS assumes the system does not change during the sweep. A low-frequency sweep can take many minutes. If $E_{corr}$, temperature, or the surface drifts, the low-frequency data becomes unreliable. Matt Lacey's practical check: sweep high-to-low, then repeat low-to-high (Lithium Inventory ─ Kramers-Kronig Transform). A stationary system gives identical spectra in both directions. This applies the same OCP stability gate from the Electrochemistry hub.

The Kramers-Kronig validity test

The Kramers-Kronig relations are mathematical conditions. Any impedance from a linear, causal, stable, time-invariant system must satisfy them. They relate real and imaginary parts by transform, independent of any circuit. KK checks four assumptions: causality, linearity, stability, and finiteness. Data that fails KK "probably cannot be fitted to an equivalent circuit" ([[https://lithiuminventory.com/experimental-electrochemistry/eis/kramers-kronig/][Lithium Inventory]) (see AutoEIS pipeline and Clarkson's corrosion-EIS notes). Always KK-validate before fitting.

A related computational tool: ElectroKitty

Research Contacts notes ElectroKitty, a Python package for simulating and fitting electrochemical mechanisms (Talić, /ACS Electrochem./ 2024). It uses non-Langmuir adsorption isotherms and Bayesian inference for uncertainty. As published, it handles voltammetric transients (cyclic voltammetry, chronoamperometry), not EIS. It shares AutoEIS's philosophy: report how well data constrains the mechanism.

A note on this page's sources

Corrected sources from the original research brief:

Sources and further reading

See also