Warburg Impedance

The low-frequency tail on a lithium-ion cell’s Nyquist plot is not just the last stretch of the same curve as the semicircles before it, it is the signature of a different physical process. It represents lithium ions diffusing through the solid electrode material, and it is normally described using a Warburg element.

The Classical Warburg Element

Warburg’s original solution describes an electroactive species diffusing into a semi-infinite medium under a small sinusoidal perturbation. For one-dimensional diffusion this gives an impedance of the form:

ZW(ω) = σ(1 − j)/√ω = σ/√ω − jσ/√ω

where σ is the Warburg coefficient (Ω·s-1/2) and ω is the angular frequency. The real and imaginary parts are equal at every frequency, which is why an ideal Warburg element traces a straight line at 45° to the real axis on a Nyquist plot. See the low-frequency tail described in our Idealised Nyquist Plot of a Lithium Ion Battery.

σ carries the diffusion coefficient: for a species with bulk concentration C* and diffusion coefficient D diffusing at an electrode of area A, σ is proportional to 1/(A·C*·√D). In principle that lets you extract D from a measured EIS spectrum. In practice, this is where the semi-infinite assumption starts to cause problems for a battery electrode.

Why a Battery Particle Is Not Semi-Infinite

The classical Warburg element assumes the diffusing species never reaches a boundary. That holds up reasonably well at high and mid frequencies, where the AC perturbation only pulls lithium a short distance into the active material before the signal reverses direction. At low enough frequency, though, the diffusion length grows until it reaches the edge of the particle, and the semi-infinite assumption stops being valid.

What happens next depends on the boundary condition. Most treatments of an intercalation particle assume a reflecting, zero-flux boundary, since lithium cannot diffuse out through the current collector or the particle core. Once the diffusion length reaches that boundary, the impedance peels away from the 45° line and bends toward vertical: the particle behaves like a finite reservoir filling up, which reads as capacitive rather than diffusive. This is usually called a finite-length, or reflective, Warburg element, and its impedance uses a hyperbolic cotangent term rather than the simple square-root form above:

Zfinite(ω) = Rd · coth(√(jωτd)) / √(jωτd)

Here τd is the diffusion time constant, related to the diffusion length L (typically the particle radius, or electrode thickness if that is the limiting dimension) by τd = L²/D. At high frequency (ωτd ≫ 1) this expression reduces back to the ordinary Warburg line; at low frequency (ωτd ≪ 1) it flattens into a resistance in series with a capacitance. The frequency at which the curve bends is set by τd, which is often the more useful number for a battery engineer, since it only needs an estimate of the diffusion length rather than the electrode area and bulk concentration the classical Warburg coefficient requires.

Fitting Real Data

Measured low-frequency data on commercial cells rarely gives a clean 45° line or a clean finite-Warburg bend. Porous electrode structures, a spread of particle sizes, and overlapping contributions from both electrodes blur the picture. In practice, a constant phase element or a generalised finite-diffusion element with an adjustable exponent is often fitted instead of the textbook Warburg, because the fitted exponent frequently lands away from the ideal 0.5. BioLogic’s application note on low-frequency battery impedance works through fitting several of these variants against real 18650 cell data and is a useful illustration of how far measured spectra can drift from the idealised picture [2].

The practical takeaway: a 45° line in a measured spectrum is a reasonable indicator that solid-state diffusion is rate-limiting in that frequency range, but the exact angle, and any diffusion coefficient fitted from it, should be treated as approximate. See our Electrochemical Impedance Spectroscopy page for how this tail fits into the rest of the spectrum, and DCIR of a Cell for the time-domain view of the same underlying resistance and diffusion processes.

References

  1. Barsoukov, E. and Macdonald, J.R. (eds), Impedance Spectroscopy: Theory, Experiment, and Applications, Wiley, 2nd ed., 2005 – standard reference for the Warburg and finite-length diffusion impedance derivations.
  2. BioLogic, How to interpret lower frequencies impedance in batteries (Application Note 61) – worked fits of Warburg and finite-diffusion elements against commercial 18650 cell data.
  3. Alexandros Ch. Lazanas and Mamas I. Prodromidis, Electrochemical Impedance Spectroscopy – A Tutorial, ACS Measurement Science Au, 2023.
  4. Nina Meddings, Marco Heinrich, Frédéric Overney, Jong-Sook Lee, Vanesa Ruiz, Emilio Napolitano, Steffen Seitz, Gareth Hinds, Rinaldo Raccichini, Miran Gaberšček, Juyeon Park, Application of electrochemical impedance spectroscopy to commercial Li-ion cells: A review, Journal of Power Sources, Volume 480, 2020

    See also: Randles Circuit