The Frumkin effect: a reaction its double layer repels

Peroxodisulfate (S₂O₈²⁻) reduced to sulfate at a mercury electrode in sodium fluoride, stirred as on a rotating disk: the steady current against the electrode's potential. Negative of the potential of zero charge (pzc) the electrode's charge turns negative and its double layer pushes the anion away from where it reacts, so in dilute salt the current falls as the drive grows. Vary the salt, the potential and the ions' size. (More about the model below.)

NaF ion size E log current

About this simulation

Kryukova found this inhibition in 1949 at a dropping mercury electrode, and Frumkin and Florianovich explained it soon after. The electron jumps to the reactant at the plane of closest approach, a few ångströms from the metal, so only the drop across that inner (Stern) layer drives it. The diffuse layer outside sets how much reactant is at the plane: Boltzmann's c e^−zFψ/RT, with z = −2 and ψ the diffuse layer's drop. Past the pzc, ψ turns negative, and the plane loses its reactant faster than the inner layer's drop speeds the reaction up. More salt thins the diffuse layer and shrinks ψ, so the fall starts later and goes less deep, while the minimum moves the other way, to less negative potentials (with small ions, −0.79 V in 1 mM and −0.61 V in 0.1 M, where it's a dip of under 10%): in the classic correction it sits where the diffuse layer's capacitance is eight times the inner layer's (dψ/dE = 1/9, for a transfer coefficient of 0.25 and z = −2), which more salt reaches sooner. (Positive of the pzc it works the other way: the layer draws the anion in, and the wave sits at more positive potentials than the bare kinetics would put it.)

The dashed line is that classic correction, which takes the reactant at the plane to be in equilibrium with the solution just outside the double layer. The full line adds what that leaves out, Levich's non-equilibrium double layer (1949): the reactant has to cross the diffuse layer against its field to reach the plane, and the layer resists it, by a length K = ∫(c/c(x) − 1) dx that grows as the plane empties. However fast the reaction, that caps the current at the limiting current times L/(L + K) (L the stagnant layer's 20 µm), and the cap falls as the electrode goes negative.

How much of the fall is the cap and how much the classic correction depends on how fast the reaction at the plane is. With this page's kinetics its rate constant k_f (in the readout) climbs from a fraction of a centimetre a second near the pzc to about 5 m/s at −1.2 V in dilute salt, 14 to 21 in the strongest (below the collision ceiling, but far above the 10⁻³ to 1 cm/s of everyday couples), and the cap trims the far end, by up to a third at −1.2 V; with faster kinetics the cap would be most of it. A 2023 study of this experiment (Pendergast and White), with transport through the double layer and electron transfer at a distance, compared against measurements, finds the anion's migration within the double layer the main cause of the fall. Most of K builds up within an ångström or two of the plane, so its size also rests on the reaction happening at a sharp plane, which electron transfer at a distance softens. The simulation (points) follows the full line, to 0.1% where the current is down: that checks the solver against this model, not the model against mercury.

On real mercury the current passes a minimum and rises again at more negative potentials. Here, with small ions (0.1 nm is all but the plain Gouy–Chapman layer of the textbook correction), it about doubles from the minimum to −1.2 V in dilute salt (the classic correction's triples), but it can't keep rising: whatever the kinetics, the cap L/(L + K) keeps falling. On sites 0.7 nm across it doesn't turn at all within the window: the crowded layer holds ψ more negative, and the crowding cations push the reactant off the plane. The illustrative kinetics move all this as much as the size does. What real mercury has that this model doesn't: molecular models of the reaction (Nazmutdinov, Glukhov, Tsirlina and Petrii, 2002) put the rise down to a reaction layer that thickens, with ions farther out in the diffuse layer taking electrons too, and the rate depends on the cation, which can pair with the anion. Here the reactant reacts only at the plane, and the ions are point charges on sites.

The double layer is thousands of times thinner than the stagnant solution (0.5 to 10 nm against 20 µm), so driftlet holds it below the grid: a Gouy–Chapman layer beside the face (diffuse: { epsr }), its ions on sites of the chosen size (Bikerman), in series with an inner layer of 17 µF/cm² (constant here; mercury's is near that at negative charge). The pzc is mercury's in NaF, −0.19 V vs SHE. The reactant is a trace, 10 µM, where the experiments used millimolar peroxodisulfate in dilute salt, enough to shape the layer itself. The kinetics are illustrative: the accepted mechanism's rate-determining first electron (S₂O₈²⁻ + e⁻ → SO₄²⁻ + SO₄•⁻, the O–O bond breaking) with a transfer coefficient of 0.25, a Tafel slope of 236 mV per decade, written here as the two-electron step with driftlet's α = 0.125; chosen so the rate constant at the plane stays below 25 m/s in this window, where the heterogeneous collision factor √(RT/2πM), an order-of-magnitude ceiling, is 45 m/s for S₂O₈²⁻. A Butler–Volmer rate allowed to grow without limit over 3 V would change both lines' shape past the minimum. The theory lines are worked out apart from the solver: the Frumkin correction with the same layer, the layer's resistance by quadrature, and diffusion across the stagnant layer. They leave out migration there (the electrode's field in the solution holds the anion back too), which lowers the plateau by about 1% in the most dilute salt.

The level diagram has the mercury's Fermi level, the S₂O₈²⁻/SO₄²⁻ redox level and its standard level, E° = 1.94 V from the μ° used here (tables of E° give 2.01 V), and φ, which is the SHE's level here. It's drawn in volts, so electron energy is down the page: reducing the oxidant, electrons go from the Fermi level up to the redox level, toward higher voltage. The gap between the Fermi level and the redox level, about 2.4 V at the pzc, is set partly by the oxidant's strength and partly by the trace of sulfate (20 nM) in the bulk. The inner layer's drop and ψ sit between the mercury and the solution's first node, far too thin to see at this scale; the last two panels open the diffuse part up, from the plane outward: φ across it, and each ion's concentration there (log scale). Negative of the pzc the reactant is drawn down below Boltzmann's only within a fraction of a nanometre of the plane, too close to see here; the readout gives both at the plane.