A redox electrode's impedance: the Randles circuit

Platinum in 1 mM Fe³⁺ and Fe²⁺ with KCl (and 10 mM HCl), at rest, behind a stagnant layer of solution: its impedance spectrum, simulated from the ions and the reaction, against the Randles equivalent circuit with every element worked out from the same physics. A semicircle for the reaction and its double layer, then diffusion's arc: a 45° line while the diffusion is short of the stirred bulk (a thick stagnant layer shows it best), bending back once it reaches it. Vary the rate constant, the salt and the stagnant layer. (More about the model below.)

k⁰ KCl stagnant layer

About this simulation

A small sine wave of voltage on the electrode (small next to RT/F, 26 mV, so the response is linear) drives three things at once. Charge piles onto the double layer, like a capacitor C_dl. Fe³⁺ is reduced and Fe²⁺ oxidised a little faster and slower in turn, which near rest is a resistor, the charge-transfer resistance R_ct = RT/(Fi₀), with i₀ = Fk⁰c the exchange current (k⁰ in m/s; so k⁰ = RT/(F²R_ctc) gives it back from a fit). And the reaction's own small changes in concentration have to diffuse to and from the bulk, which lags: the Warburg impedance Z_W, at first σ(1 − i)ω^−½. The current to all of them crosses the solution, R_s. So Z = R_s + 1/(iωC_dl + 1/(R_ct + Z_W)), Randles' circuit. Its corner at ω = 1/(R_ctC_dl) is the top of the semicircle while the Warburg there is small; then the 45° line takes over; and once the diffusion reaches through the stagnant layer to the stirred bulk (about D/(2πL²) in frequency), Z_W stops growing and the spectrum bends back to the real axis at R_s + R_ct + R_W, with R_W = (RTL/F²c)(1/D_Fe³⁺ + 1/D_Fe²⁺). Reading those off the plots takes care where the features overlap. The dip between the arcs sits about 11% beyond R_s + R_ct (both ends are marked on the axis), as the Warburg's start bends the first arc, so k⁰ from the dip comes out low. The phase plot's tall peak isn't the R_ctC_dl corner (marked, 37 Hz at the start) but sits at 1/(2πC_dl√(R_s(R_s + R_ct))), near 1 kHz, and is tall only because R_s here is so small; the corner is the knee in |Z|. And the Warburg's σ, from Z′ against ω^−½, comes cleanly only from a thick layer. That's the transmissive finite Warburg of a Nernst diffusion layer (fitting programs' "Ws"); a rotating disk is the usual way to make one.

driftlet knows nothing of the circuit: it solves the ions, the potential and the reaction for their small oscillation at each frequency. The circuit's elements here come from the same physics, by hand: R_s = L/κ from the solution's ideal conductivity; C_dl, an inner (Stern) layer of 20 µF/cm² in series with the diffuse layer's εκ_D (Gouy–Chapman at zero charge); R_ct from k⁰; and Z_W for a layer of thickness L ending at a fixed bulk, (RT/F²c)·Σ tanh(L√(iω/D))/√(iωD) over Fe³⁺ and Fe²⁺. The readout gives how far the two disagree, under 1% across these controls. Most of that shrinks with less iron: the iron's own migration, which the circuit leaves out (in the most dilute salt Fe³⁺ and Fe²⁺ carry a few percent of the current, but both are cations pushed the same way, and their corrections to R_W mostly cancel: a couple with one neutral form would be off by more). The rest, a tenth of a percent or two, isn't traced here. (Not the Frumkin effect: at rest a change in ψ moves the forward and back reactions alike, since z_Fe³⁺ − z_Fe²⁺ is the electrons' charge, as for any couple, so R_ct sees the whole of the overpotential.)

Three choices make this cleaner than a measurement. R_s here is only the stagnant layer's, to a reference at its edge; in a real cell it's set by the cell's shape and where the reference sits, and is often tens of times more. The double layer (driftlet's diffuse: { epsr }, below the grid) is set to have no charge at rest; real platinum at this potential sits well positive of its zero charge, with chloride adsorbed, and off zero charge ψ would shift the exchange current (the Frumkin effect, on its own page) and C_dl. And it's an ideal capacitor, where real solid electrodes give depressed arcs, which fits stand in for with a constant-phase element (CPE). The solution has 10 mM HCl so that Fe³⁺ doesn't hydrolyse and precipitate; in reality the chloride also complexes Fe³⁺ and speeds its reaction, which this model leaves out. k⁰ is illustrative, in the range reported for this couple at platinum.