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Popular topics · Modelling and fitting (ZFit, ZSim)

What is the M element? How is the impedance derived for restricted linear diffusion (insertion reaction)?

A step-by-step derivation from the kinetic model of cation insertion into a thin host film (battery electrodes, electrochromic and grafted polymer films) to the impedance expression of the M element in ZSim and ZFit.

Key points

  • Model: electronically conducting host film of constant thickness; no phase change, substrate impermeable to the inserted species (zero flux at x = L), no interactions between species (Langmuir isotherm).
  • At steady state the concentration profile in the film is flat and the current is zero; every electrode potential is an equilibrium potential. The filling fraction varies sigmoidally (Langmuir) with potential.
  • Faradaic impedance: Zf = Rct + Rd·coth√(τd·s)/√(τd·s), τd = L²/D. The electrode impedance is obtained by connecting it in parallel with Cdl; this is the Randles circuit with restricted diffusion instead of Warburg.
  • It behaves like Warburg (45°) at high frequency and like a series RC at low frequency: at the low-frequency limit Re Z = Rct + Rd/3 and Clf = τd/Rd.
  • The 45° line is visible only when Cdl is small enough and Rd/3 is not negligible compared with Rct; the shape of the diagram depends on the Rct/Rd ratio and the time constants.
  • Extensions: spherical or cylindrical particle geometry, particle size distribution in composite electrodes, diffusion with trapping and two-step (indirect) insertion. The Ma and Mg elements are also available in ZSim/ZFit.

Figures

Insertion of M⁺ into a thin host material: electrons come from the current collector, cations from the electrolyte; the flux is zero at x = l.
Figure 1. Insertion of M⁺ into a thin host material: electrons come from the current collector, cations from the electrolyte; the flux is zero at x = l.
Scheme for calculating electrode impedance: linearisation, Laplace transform, steady-state solution and addition of the double-layer capacitance.
Figure 2. Scheme for calculating electrode impedance: linearisation, Laplace transform, steady-state solution and addition of the double-layer capacitance.
Langmuir insertion isotherm: the filling fraction y⟨M⟩ varies sigmoidally with electrode potential.
Figure 3. Langmuir insertion isotherm: the filling fraction y⟨M⟩ varies sigmoidally with electrode potential.
Randles circuit with restricted linear diffusion: C1 / (R1 + M1).
Figure 4. Randles circuit with restricted linear diffusion: C1 / (R1 + M1).
Possible Nyquist shapes as a function of Rct/Rd and time constants; the Warburg line is visible only under certain conditions.
Figure 5. Possible Nyquist shapes as a function of Rct/Rd and time constants; the Warburg line is visible only under certain conditions.
As frequency decreases the concentration wave reaches the full film thickness; impedance moves from the Warburg line to a vertical capacitive branch, with Re Z limit Rct + Rd/3.
Figure 6. As frequency decreases the concentration wave reaches the full film thickness; impedance moves from the Warburg line to a vertical capacitive branch, with Re Z limit Rct + Rd/3.
Equivalent circuit for a two-step (indirect) insertion reaction: C1 / (R1 + C2 / (R2 + M3)).
Figure 7. Equivalent circuit for a two-step (indirect) insertion reaction: C1 / (R1 + C2 / (R2 + M3)).

Images: Bio-Logic Science Instruments; taken from the source article.

Related application notes

Related curriculum module: Electrochemical Impedance Spectroscopy (EIS)

Related products:EC-Lab®MPG-200 Series (MPG-2 / MPG-205)VMP-3e
Source: Bio-Logic Learning Center, “The M element in EIS: Impedance under restricted linear diffusion” · Source updated: February 6, 2025
Copyright of the images and the original text belongs to Bio-Logic SAS.

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