Interface kinetics

Throughout this book, interfaces have been where the interesting steps happen: Donnan steps, junction steps, half-cell steps. So far those steps were all equilibrium facts. In the Electrode potential topic we saw what a current does to the picture: the metal's Ve−V_{\mathrm{e}^-} peels away from the reaction level it was pinned to, by the surface overpotential

η=Ve−(electrode)−Ve−(Ox/Red).\eta = V_{\mathrm{e}^-}(\text{electrode}) - V_{\mathrm{e}^-}(\mathrm{Ox}/\mathrm{Red}).

What we deferred is the question a device designer actually cares about: how much current does a given η\eta buy? Within a material, we answered this with a conductivity, Ji=−σi∇ViJ_i = -\sigma_i \nabla V_i (Basic transport). At an interface, the answer is not a slope but a step, and the current is generally exponential in the step. That exponential element is the last circuit ingredient the ViV_i picture needs.

The Butler–Volmer element

For an electrode reaction transferring zz electrons, the standard current–overpotential relation is the Butler–Volmer equation, which in our terms reads

J=J0[eαzfη−e−(1−α)zfη],f=F/RT,J = J_0 \left[ e^{\alpha z f \eta} - e^{-(1-\alpha) z f \eta} \right], \qquad f = F/RT,

with η\eta the Ve−V_{\mathrm{e}^-} step defined above. Two parameters characterize the interface:

  • The exchange current J0J_0 is the two-way traffic at equilibrium. At η=0\eta = 0 the anodic and cathodic partial currents each run at J0J_0 and cancel; equilibrium is busy, not idle. A large J0J_0 means the interface is hard to pull away from equilibrium (a good electrocatalyst, or a fast reference-electrode couple); a tiny J0J_0 means the level can be dragged far with little consequence.
  • The charge-transfer coefficient α\alpha splits the influence of η\eta between the two directions: a fraction α\alpha of the step lowers the anodic barrier, and the remaining 1−α1-\alpha raises the cathodic one.[1]
Voltage Ve−V_{\mathrm{e}^-}Reaction coordinateVe−V_{\mathrm{e}^-}Ve−(Ox/Red)V_{\mathrm{e}^-}(\mathrm{Ox}/\mathrm{Red})
−0.10−0.050.000.050.10−15−10−5051015Current J/J0J/J_0Overpotential η\eta (V)JaJ_{\mathrm{a}}−Jc-J_{\mathrm{c}}



The interface as a nonlinear circuit element. Left: the two levels, η\eta apart, with the transition state between them — high electron energy is low voltage, so the activation pass hangs below the levels, and it rides with the metal level by the fraction α\alpha (the grey dash marks its equilibrium position). Right: the resulting current, with the anodic and cathodic partial currents drawn faint; they cancel at η=0\eta = 0 where both equal J0J_0. Slide α\alpha to 11 and the cathodic branch saturates: this is the Shockley diode law. Raise zz and the electrons cross as a convoy — each one multiplies the leverage of η\eta, steepening the response by that factor.

For small η\eta the exponentials linearize and the interface is just a resistor, J≈J0zfηJ \approx J_0 z f \eta, the charge-transfer resistance Rct=RT/(zFJ0)R_{\mathrm{ct}} = RT/(z F J_0) per unit area. For large η\eta one exponential dominates and η\eta grows only logarithmically with current — the Tafel equation, a straight line of (ln⁡10)RT/(αzF)≈59 mV/(αz)(\ln 10) RT/(\alpha z F) \approx 59\,\mathrm{mV}/(\alpha z) per decade on a log-current plot. Toggle the figure above into its Tafel view: the two partial currents become the straight asymptotes, crossing at the exchange current J0J_0; the net current, meanwhile, notches to zero at η=0\eta = 0.

The diode connection

Electronics has its own famous exponential interface law, the Shockley diode equation

J=J0[efV−1],J = J_0\left[e^{fV} - 1\right],

and it is exactly the Butler–Volmer equation with α=1\alpha = 1: forward bias lowers the barrier one-for-one, while reverse bias raises no barrier at all — it only shuts off the forward traffic, leaving the fixed −J0-J_0 of carriers that fall down the junction regardless. The clean semiconductor twin of the electrode is the Schottky diode: a metal's Ve−V_{\mathrm{e}^-} meets a semiconductor's carrier levels across one sharp interface, and thermionic emission over the barrier plays the role of the electrode reaction — watch it in the figure below. Here the electrochemists' α\alpha is bolted at 11 for a reason you can see: the barrier as approached from the metal side is pinned, while the semiconductor side follows the bias in full.[2]

Metaln-type semiconductor Ve−V_{\mathrm{e}^-}Ve−∘V^\circ_{\mathrm{e}^-}−0.6−0.4−0.20.00.20.40.60.81.0Species Voltage (V)

A Schottky diode: the sharp interface with a pinned barrier. The conduction edge meets the contact a fixed ϕB\phi_B below the metal's Ve−V_{\mathrm{e}^-} at every bias, and the applied bias is taken up entirely by band bending on the semiconductor side (the depletion zone widening in reverse). The metal-side barrier never changes — that branch of the current saturates — while the semiconductor-side barrier follows the bias one-for-one: α=1\alpha = 1, drawn in space.

Where the electrons jump: Marcus–Gerischer

Why an exponential, and what sets J0J_0? The classic microscopic picture is due to Marcus and Gerischer. Each Ox\mathrm{Ox} ion offers a vacant electronic state and each Red\mathrm{Red} ion an occupied one, but solvent fluctuations smear these into two broad distributions — a filled band DredD_{\mathrm{red}} and an empty band DoxD_{\mathrm{ox}}, Gaussian bumps offset to either side of the couple's standard level Ve−∘V^\circ_{\mathrm{e}^-} by the reorganization energy λ\lambda. Electrons tunnel between the electrode and whatever portion of these bumps lines up with the electrode's own filled or empty states; bias slides the electrode's levels across the bumps, and the overlap integral sets the current.

This is the one place in the book where the traditional electron-energy diagram and ours meet almost verbatim: a Gerischer diagram is a redox-level band diagram, and our Ve−V_{\mathrm{e}^-} diagrams are the same picture flipped upside down (energy up = voltage down), with the DredD_{\mathrm{red}} bump sitting above the couple's Ve−V_{\mathrm{e}^-} and DoxD_{\mathrm{ox}} below. The quantitative machinery is well covered in the literature,[3] so I'll restrict myself to two points the standard cartoons tend to blur:

  • The bump amplitudes scale with the concentrations of Ox\mathrm{Ox} and Red\mathrm{Red} — they are densities of actual ions, not of abstract states. A couple that is all Red\mathrm{Red} has no DoxD_{\mathrm{ox}} bump to speak of.
  • A disequilibrated solution simply has several pairs of bumps, one per couple, each pinned to its own standard level — the multi-level picture from Half-reactions carries straight over.
−0.4−0.20.00.20.40.60.81.01.21.4Voltage Ve−V_{\mathrm{e}^-} (V)Density of statesVe−∘(Ox/Red)V^\circ_{\mathrm{e}^-}(\mathrm{Ox}/\mathrm{Red})Ve−(Ox/Red)V_{\mathrm{e}^-}(\mathrm{Ox}/\mathrm{Red})Dred  (filled)D_{\mathrm{red}}\;\text{(filled)}Dox  (empty)D_{\mathrm{ox}}\;\text{(empty)}

The Marcus–Gerischer picture on a Ve−V_{\mathrm{e}^-} axis — upside-down relative to the usual energy plot, so the filled DredD_{\mathrm{red}} sits above and the empty DoxD_{\mathrm{ox}} below. The bumps sit ±λ\pm\lambda about the standard level Ve−∘V^\circ_{\mathrm{e}^-} (Gerischer's E∘E^\circ, always their midpoint), and their amplitudes follow the actual ion populations. The couple's actual level Ve−V_{\mathrm{e}^-} stays pinned as the ratio slides — the electron reservoir holds still while the whole density-of-states structure shifts beneath it by the Nernst term.

Mixed potentials, quantitatively

The Electrode potential topic introduced mixed potentials as a fact about levels: an electrode coupled to several half-reactions settles at a Ve−V_{\mathrm{e}^-} matching none of them. Kinetics says where: each couple kk contributes a Butler–Volmer current driven by its own ηk\eta_k, and the electrode floats to the level where the currents cancel, ∑kJk=0\sum_k J_k = 0. A corroding metal is the canonical case — the metal-dissolution couple runs anodically, the oxygen couple cathodically, and the balance point (the corrosion potential) sits between the two equilibrium levels, with a steady corrosion current circulating even though the electrode as a whole draws nothing.

Ve−V_{\mathrm{e}^-}Ve−(O2/OH−)V_{\mathrm{e}^-}(\mathrm{O_2}/\mathrm{OH}^-)Ve−(Fe2+ ⁣/Fe)V_{\mathrm{e}^-}(\mathrm{Fe}^{2+}\!/\mathrm{Fe})ηO2\eta_{\mathrm{O_2}}ηFe\eta_{\mathrm{Fe}}ElectrodeSolutionVoltage Ve−V_{\mathrm{e}^-}

A corroding metal, coupled to two half-reactions at once. The electrode floats to the level where the two Butler–Volmer currents cancel, leaving both overpotentials nonzero: the oxygen couple runs cathodically and the iron couple anodically, a steady corrosion loop with zero net electrode current. The slider sets the ratio of exchange currents — the mixed potential slides toward the kinetically faster couple.

Takeaways

An interface passing current carries a step in ViV_i, and the current is exponential in that step: Butler–Volmer for electrodes, Shockley for diodes. Exchange current sets how stiff the interface is, α\alpha how the step splits between the two barriers, and Marcus–Gerischer supplies the microscopic picture — one that lives natively on these diagrams. With transport (slopes) and kinetics (steps) both priced in ViV_i, a driven electrochemical device really can be read end to end like a circuit.

That closes the main sequence. The appendices dig into the foundations underneath, starting with the one this whole framework rests on:

NEXT TOPIC: Understanding electrochemical potential

  1. The charge-transfer coefficients could be better named "barrier-lowering coefficients," reflecting the degree to which the overpotential controls the reaction barrier in each direction. Practically they end up as empirical fitting parameters, much like a diode's ideality factor. ↩︎

  2. The everyday pn junction obeys the same exponential law but arrives at it by a different route — minority-carrier injection and diffusion, with the exponential coming from equilibrium carrier statistics rather than a rate-limiting barrier crossing — so the barrier-lowering reading applies to it only loosely. ↩︎

  3. Gerischer, H. (1960–61). Über den Ablauf von Redoxreaktionen an Metallen und an Halbleitern, I–III. Z. Phys. Chem. NF, 26, 223–247 & 325–338; 27, 48–79 (in English, his monograph Semiconductor Electrochemistry, Lawrence Radiation Laboratory preprint UCRL-18145, 1968). Marcus, R. A. (1965). On the Theory of Electron-Transfer Reactions. VI. Unified Treatment for Homogeneous and Electrode Reactions. J. Chem. Phys. 43, 679–701. Modern treatments: Schmickler, W., & Santos, E. (2010). Interfacial Electrochemistry (2nd ed.). Springer; Bard, A. J., & Faulkner, L. R. (2022). Electrochemical Methods (3rd ed.), ch. 3. ↩︎