The Curve We Cannot Read:
On the Structural Indistinguishability of Interfacial Kinetics

Plot current against overpotential for almost any electrode reaction and you get a shape that looks reassuringly familiar: flat near the origin, rising into a curve, eventually bending over into something like a plateau. Every kinetics textbook shows a version of this curve. The reassuring part is the problem.

Let me put the claim precisely. Over the overpotential range a real cell gives us, several genuinely different hypotheses compete to describe the interface: symmetric Butler–Volmer, asymmetric Butler–Volmer, resistance-corrected ("modified") Butler–Volmer, Marcus–Hush–Chidsey, and CIET. Their agreement runs deeper than a shared ability to fit the same data. To leading order, they are the same equation. The differences meant to distinguish them surface only in a regime where two unrelated artifacts produce the identical signature. Better instruments would not settle it; the obstruction is in the algebra.

The Family We're Actually Choosing Among

It is easy to talk about "BV versus CIET" as if there were two candidates. What we are choosing among is a family, all of it built from the same move: assume a shape for the free-energy landscape of the elementary charge-transfer step, then read off a current from it.

• Symmetric Butler–Volmer: a fixed 50/50 split of the driving force between the anodic and cathodic activation barriers.
• Asymmetric Butler–Volmer: the same split, but at a fitted ratio $\alpha : (1-\alpha)$ instead of $1/2 : 1/2$.
• Modified (resistance-corrected) Butler–Volmer: either of the above, with an assumed series resistance subtracted from the applied bias before fitting.
• Marcus–Hush–Chidsey (MHC) / Gerischer: a parabolic free-energy landscape integrated over a continuum of electronic states, appropriate for a metal or, with a weighting density of states, a semiconducting oxide like NMC or LCO.
• CIET: the MHC formalism extended so that the transferring electron and the inserting ion move together, still inheriting MHC's underlying activation picture.

Every member of this family ends in the same place: a current–overpotential relation $i(\eta)$. The question I want to answer is: how much of that function do we actually need to see before the family members stop agreeing with each other?

Where the Whole Family Agrees

Start with the general (asymmetric) Butler–Volmer equation, since it is the one member of the family we can expand by hand without special functions:

Let $x \equiv F\eta / RT$, so that $i/i_0 = e^{\alpha x} - e^{-(1-\alpha)x}$. Expand each exponential to third order:

Subtracting term by term, the constant "1"s cancel, the linear terms add ($\alpha x + (1-\alpha)x = x$), and the quadratic terms combine as a difference of squares, $(\alpha x)^2 - [(1-\alpha)x]^2 = (2\alpha - 1)x^2$. What survives is:

Look at where $\alpha$ first enters: not in the leading term, only in the second-order one. To leading order,

The same conclusion holds, model for model, if the exponentials are replaced with the Marcus–Hush–Chidsey rate integral that CIET is built on. Expanded near equilibrium, the MHC current reduces to the identical linear form, $i \approx i_0 F\eta/RT$, with an effective transfer coefficient of exactly $1/2$ (Chidsey, 1991; Zeng et al., 2014). A fixed asymmetric split and a full Marcus parabola integrated over an electrode's density of states are, in this regime, the same number wearing two different derivations. Neither leaves a trace in the one quantity a linear $i\text{-}\eta$ or small-amplitude EIS measurement can deliver.

Where the Family Is Supposed to Split

The disagreement is real, but it only shows up far from equilibrium. Butler–Volmer has no ceiling: each exponential term grows without bound as $\eta$ grows, so the current keeps climbing on a Tafel plot forever. Marcus theory behaves differently by construction. The activation energy for a single barrier is a parabola in the driving force,

where $\lambda$ is the reorganization energy. As $\eta$ grows from zero, this barrier first shrinks, reproducing something that looks Tafel-like, until $F\eta = \lambda$, where the reaction becomes barrierless. Push $\eta$ past that point for an isolated molecular reaction and the barrier grows again: the famous Marcus inverted region, where the rate turns over and falls.

An electrode does not behave this way, which is Chidsey's (1991) central point. A metal or a semiconducting oxide offers a whole continuum of electronic states, not one. Past the barrierless point, there is always some subset of states with the right energy to react without paying the inverted-region penalty, so the net current saturates instead of turning over. CIET inherits this saturating, S-shaped curve because it is built on the same MHC formalism. The one place the family is supposed to disagree with Butler–Volmer is exactly this large-$\eta$ plateau, a feature set by $\lambda$, and invisible to any EIS spectrum taken at low bias.

Two Impostors Wearing the Same Bend

A saturating curve is exactly what an intrinsic Marcus-type mechanism predicts. It is unfortunately also what two purely extrinsic effects predict, at similar current densities, in almost any real cell.

The first is mass transport. As $\eta$ grows, the reactant at the surface depletes, and the current approaches a diffusion-limited value that, for a film of thickness $L$, scales roughly as

This plateau has nothing to do with the reorganization energy. But taken alone, an $i\text{-}\eta$ curve that flattens at high $\eta$ cannot tell you whether it flattened because of $\lambda$ or because of $L$.

The second is uncompensated resistance. Any series resistance between the reference point and the interface shifts the true local overpotential away from the applied one, $\eta_{\text{true}} = \eta_{\text{applied}} - iR_s$. At high current, this steals back exactly the kind of curvature that would otherwise look like unbounded Tafel growth. This is precisely why "modified Butler–Volmer" schemes exist: fit $R_s$ alongside $i_0$ and $\alpha$, and subtract its effect before reporting kinetics.

I used to think of resistance correction as a fix, a way of cleaning the data before the real kinetic question gets asked. I don't think that anymore. Adding $R_s$ as a free parameter does not remove the degeneracy between an intrinsic saturation and an artifact; it relocates it. A curve that bends over at high $\eta$ can now be explained by any combination of a real $\lambda$, a real diffusion length $L$, and an assumed $R_s$ that trades off against both. "Modified BV" sits inside this family alongside CIET — a third curve that bends like the others, for a reason of its own.

Counting What We Can Actually Solve For

Put the pieces together and the problem is one of counting. A real, finite-range $i\text{-}\eta$ curve, measured with real noise, realistically yields perhaps three independent features: an initial slope ($R_{ct}$), the overpotential at which curvature becomes visible, and a limiting or asymptotic current. Any model in the family with three or more free parameters can be tuned to reproduce those three features from more than one underlying mechanism, and every model past plain symmetric BV clears that bar the moment you add $\alpha$, or $\lambda$, or $R_s$, or $L$. The map from "which physical process is happening" to "what the curve looks like" is not one-to-one. Fitting harder does not invert a map that isn't injective.

I don't think this is a coincidence, and I don't think it is unique to kinetics. It is the same shape of problem I keep running into when people claim a DRT deconvolution has cleanly separated charge-transfer resistance from diffusion admittance in an impedance spectrum: an underdetermined inverse problem, dressed up differently depending on which domain (time, frequency, or overpotential) you happen to be squinting at. That comparison deserves its own essay. For now, the point is narrower: no amount of clever regression rescues an argument of the form "the fit is good, therefore the mechanism is right," when several mechanisms are guaranteed to produce equally good fits by construction.

What Would Actually Break the Degeneracy

If the curve itself can't decide, the only way out is architectural: build an experiment where the competing explanations are forced to make different predictions about something other than the shape of one curve.

The diffusion-versus-intrinsic-saturation confound has a clean handle, because the two depend on geometry differently. An intrinsic, $\lambda$-controlled plateau does not care how thick the electrode is. A diffusion-limited plateau scales predictably with the transport length. Vary that length scale independently, rather than inferring it after the fact from the same curve you're trying to interpret, and the two hypotheses stop agreeing.

The resistance-versus-intrinsic-saturation confound needs the same kind of move: a way of reading the true interfacial $\eta$ directly, close enough to the reaction plane that $R_s$ stops being a free parameter to fit around and becomes something the architecture pins down on its own. And underneath both sits the confound my last two essays were circling: none of this length-scale or resistance bookkeeping survives a reference potential that drifts as the reaction proceeds, which is exactly what an intercalation electrode's $U_{eq}(x)$ does and a catalytic electrode's does not. That has to be settled first; the other two become answerable only once the baseline holds still.

Working out what such an architecture looks like for a lithium-intercalation electrode, and being honest this time about the assumptions it would quietly carry, is where I want to go next.