Incoprea research · leads batch J · 3 November 2026

Electrochemistry and interfaces: 200 leads and five computations

0 · The structure being generalised

An interfacial measurement reports a quantity against four separate conventions, and each can carry its own bias:

• A reference potential — an additive offset plus a liquid-junction term that depends on the analyte.
• A circuit model — a rational function whose parameters are identified only up to the correlations the frequency window allows.
• A double-layer model — a structure inferred from a capacitance that is usually dominated by the layer the model does not describe.
• An area convention — a denominator that is itself a measurement divided by an assumption.

Each has a bias factor and an elasticity in the sense of the batch-B page: how much the reported number moves when the convention moves. The five tests below measure that for the four sections, and the leads ask where else the same structure appears.

E · Reference electrodes, liquid junctions and potential conventions

A reference enters a measurement as an additive offset, and the junction in front of it as a second offset that depends on the analyte's composition. The two are usually treated as one "reference error", which hides that the first is constant within a session and the second moves whenever the electrolyte does. The leads below separate them.

F · Impedance: what an equivalent circuit can and cannot identify

An impedance spectrum is a complex function measured over a finite band with finite precision. An equivalent circuit is a rational function fitted to it. The question is not whether the circuit fits — some circuit always fits — but how many of its parameters the data determine, which is a property of the band, the noise and the separation of the time constants, not of the software.

G · Double-layer structure in concentrated electrolytes

At 1 M the Debye length is smaller than a hydrated ion, so the continuum model is being used outside its validity exactly where electrocatalysis is done. The corrections that replace it — steric saturation, dielectric decrement, ion correlation — change the capacitance in different places: some at the point of zero charge, some only at finite potential. The leads below ask which observable each correction actually moves.

H · Electrocatalysis: surface area, turnover, and the circularity of per-area activity

A per-area current is a geometric current divided by an area, and the area is a capacitance divided by an assumed specific capacitance. A turnover frequency divides again by a site count. Each division introduces an assumption that the next quantity inherits, and the leads below ask which of those assumptions cancel and which do not.

Tests run

Five computations, all in incoprea_nov3/code/electrochem_tests.py, writing incoprea_nov3/results/electrochemistry_interfaces.json. Tests 1 and 4 are closed-form evaluations; tests 2, 3 and 5 are numerical, with the Monte Carlo seeded at 20261103. No measured data are used anywhere on this page.

1 · Liquid junctions: when the bridge matters and when it does not

The Henderson equation with limiting ionic mobilities, evaluated at 298.15 K (RT/F = 25.69 mV) for each analyte against a 3.5 M KCl bridge. The sign convention is E_j = φ(analyte) − φ(reference). Activities are not included: the model uses concentrations, so the absolute values are indicative and the comparisons are what the table is for.

What the table says. Against a 3.5 M bridge every junction is under 15.4 mV, and the largest is 1 M HCl at 15.4 mV. The alkali and sodium cases sit at one or two millivolts. On a potential axis these look like bookkeeping. On a current axis they are not: a 5 mV offset is a factor of 1.10 in apparent current at 120 mV/dec and 1.47 at 30 mV/dec, and a 20 mV offset is 4.6 at 30 mV/dec.

The bridge is a concentration device. Holding the analyte fixed at 0.1 M HCl and changing only the bridge:

The HCl junction falls from 26.9 mV against a 0.1 M bridge to 4.3 mV against a 5 M one, a factor of 6.2, while the mobility mismatch between H⁺ and Cl⁻ is identical in the two cases. What the bridge buys is that its own ions carry most of the current, so the analyte's transport numbers stop mattering. The KOH column changes sign, which is the same mechanism with the hydroxyl mobility in place of the proton's.

Propagation to an apparent current. i_app/i_true = 10^ΔE/b:

Internal checks. Swapping the two sides changes the sign and nothing else: the largest antisymmetry residual over the ten junctions is 0.0e+00 mV. For a single electrolyte across two concentrations the Henderson expression collapses to (u₊ − u₋)/(u₊ + u₋) · ln(c₁/c₂); the 0.1 to 1 M HCl cell reproduces that closed form to 3.7e-14%, and its value is −38.0 mV. The 0.1 to 1 M KCl cell gives 1.11 mV, which is small because the two mobilities nearly match, not because the concentrations do.

The expectation this contradicted. I expected the junction potential to track the mobility contrast between the analyte's ions, which would make HCl the dangerous case at every bridge strength. It tracks the concentration ratio instead: 0.1 M HCl against a 0.1 M bridge gives 26.9 mV, 6.2× what the same acid gives against a 5 M bridge, with the same mobilities throughout. The consequence for reporting is that the bridge concentration belongs in the methods section and the analyte concentration belongs in the result, because it is the analyte that moves the offset.

2 · How many circuit elements a spectrum can identify

Model: Z = R_s + R₁/(1 + jωτ₁) + R₂/(1 + jωτ₂), five parameters in log form, with R_s = 5 Ω, R₁ = 100 Ω, R₂ = 40 Ω, τ₁ = 10⁻⁴ s. Noise: independent Gaussian errors on the real and imaginary parts with σ = ε|Z|/√2, so the magnitude error is ε|Z|. The Fisher information F = J^TJ/σ² gives the Cramér–Rao covariance; a parameter counts as identifiable if its relative standard error is under 10% and its largest correlation with another parameter is under 0.9. Both thresholds are stipulated, and where a table carries a second count it is the count at a 0.95 correlation threshold, so that the arbitrariness is visible.

Six decades, 1% noise, and the count depends on separation. At τ₂/τ₁ = 10 the count is 3 of 5; at 3 it is 1 of 5, and the surviving parameter is R_s. The failure at small separation is not a large marginal error — R₁ has a 2.8% marginal error at ratio 3 — but a correlation of 0.999 between R₁ and R₂, which means the two are determined only through their sum.

Precision does not help. At ratio 10, changing the noise by two decades:

From 0.1% to 2.0% the count does not move at all: the marginal errors fall by an order of magnitude and the correlation matrix, which is noise-independent in this model, does not. Only at 5.0% does a marginal error become the binding constraint. A correlation that comes from the model's geometry cannot be bought down with a better potentiostat.

Bandwidth is not monotone either. Windows centred on the geometric mean of the two corner frequencies, 10 points per decade, 1% noise:

Five decades give 5 and six give 3. The extra decade lies below both corners, where the two resistances are seen only as a sum, and it raises corr(R₁,R₂) from 0.879 to 0.903 — across the 0.9 threshold. The count at a 0.95 threshold is 5 in both cases. More data of the wrong kind is not neutral: it can make a parameter set less identifiable under a correlation criterion even as every marginal error falls.

Point count saturates as √N. At ratio 10 and 1% noise over six decades:

From 10 to 40 points per decade, a factor of 4 in point count, σ(τ₂) falls by 2.00 — the square root, as it must be when the information is additive. The information itself was fixed by the bandwidth, not by the sampling.

A check that the covariance is the right one. 150 spectra were simulated at the 1% level and fitted by Levenberg–Marquardt; the realised scatter is compared with the Fisher prediction:

The largest deviation from unity is 0.12, and the biases are at the third decimal. All 150 fits converged. As a contrast, a single RC with R_s in series has three parameters and all 3 are identifiable in the same band, with a condition number of 6.9 rather than 535. The difficulty is not impedance spectroscopy; it is two overlapping time constants.

The expectation this contradicted. I expected identifiability to be a monotone function of information: more decades, more points, less noise, more parameters. The noise sweep contradicts it first — from 0.1% to 2% noise the count is fixed at 3, because what fails is a correlation, not a variance. The bandwidth sweep contradicts it second, in the other direction: the sixth decade makes the count worse under a stated criterion, since the new points constrain only the sum of the two resistances. Any statement that a circuit "was identified from the spectrum" therefore carries three unreported numbers — band, noise and correlation threshold — without which it cannot be checked.

3 · Double-layer structure in a concentrated electrolyte

The Bikerman lattice-gas modification of Poisson–Boltzmann for a 1:1 electrolyte, c_± = c₀e^∓u/[1 + 2φ₀(cosh u − 1)] with u = eψ/kT and φ₀ = c₀a³ the bulk site occupancy, ion diameter a = 0.4 nm, ε_r = 78.5. The surface charge follows from the first integral, σ² = 2εε₀kT ∫(c₊ − c₋)du, and the differential capacitance from its derivative. A compact layer of 20 µF/cm² is placed in series — stipulated, not fitted — because that is what a measured capacitance actually contains.

Capacitances in µF/cm². The attenuation column is the fractional change in the series total divided by the fractional change in the diffuse layer. It sits near 12% at every concentration: the compact layer absorbs about 8× of whatever the diffuse layer does.

The diffuse-layer capacitance at the point of zero charge is identical to Gouy–Chapman at every concentration. The steric term multiplies both ion densities by the same denominator at u = 0, so it cancels in c₊ − c₋; the numbers in the C_diff PZC column therefore depend only on √c. What the lattice gas changes is the curvature away from the point of zero charge: at 5 M the diffuse-layer capacitance falls from 511 to 138 µF/cm² between 0 and 0.5 V, a factor of 3.71, while Gouy–Chapman would have risen by cosh(u/2) = 8,411.

Shape. Scanning the potential for eight site occupancies at fixed concentration, where the capacitance maximum sits:

At φ₀ = 0 the curve is monotone — the Gouy–Chapman limit, where the capacitance rises without a maximum. The maximum then moves from 0.120 V at φ₀ = 0.02 to zero at φ₀ = 0.20, which for this ion size is 5.2 M. Two different transitions are visible and they are not the same one: the hump position moves continuously with concentration, while the hump disappears (the maximum reaches the point of zero charge) only at the highest occupancies.

Internal checks. Setting φ₀ = 0 must recover Gouy–Chapman. The surface charge at u = 3 agrees to 8.7e-13% and the differential capacitance to 8.9e-13%; the small-potential limit agrees with εε₀κ to 3.7e-06%. The Debye lengths in the first table are the dilute-limit values, not fitted ones.

The expectation this contradicted. I expected a concentrated electrolyte to be a double layer with a shorter screening length, so the lattice-gas correction would show up in the screening length and in the capacitance at the point of zero charge. It shows up in neither: the linear-response length is the Debye length at every φ₀, and C at the point of zero charge is exactly the Gouy–Chapman value because the steric factor cancels there. The only signature is the potential dependence — which is then attenuated about 8× by the compact layer that any real measurement includes. A paper that reports a concentration-dependent capacitance and calls it double-layer structure is reporting a 12% effect riding on an 8× larger assumption.

4 · The Tafel slope that uncompensated resistance invents

Model: E = b log₁₀(i/i₀) + iR_u with b = 120 mV/dec, fitted by least squares over one decade. The units are Ω cm² for R_u and A/cm² for i, so iR_u is in volts.

The 5% threshold is an iR drop, not a resistance. Solving for the R_u that produces a 5% slope error at each window position gives 7.23, 0.723 and 0.072 Ω cm² for windows starting at 0.1, 1 and 10 mA/cm². Multiplied by the top current of each window these are all the same number: 7.2 mV. The error is a function of the iR drop at the top of the fitted window and of the window width, and of nothing else.

That number is stricter than the usual rule of thumb. At R_u = 10 Ω cm² and a 10–100 mA window, the iR drop at the top is 1,000 mV, the apparent slope is 950 mV/dec, and the transfer coefficient inferred from it is a factor of 7.9 too small. At the same resistance and a 1–10 mA window the drop is 100 mV and the slope is 203 mV/dec.

Under-correction and over-correction. With R_u = 10 Ω cm² over 10–100 mA, applying a fraction of the correction:

A 20% error in the measured R_u leaves a fifth of the distortion, which is an error of 138% — not five. Partial compensation is the worst of the three cases, because it looks like a correction. The closed form for the fitted slope, b + R_u·cov(x, 10^x)/var(x) over the window, reproduces the numerical fit to 2.8e-14% — an exact check, since both are the same regression.

The expectation this contradicted. I expected the fitted slope over a decade to pick up the local distortion at the top of the window, which for a 100 mV iR drop would be b + ln(10)·iR = 350 mV/dec — an apparent tripling. The regression picks up 83 mV/dec of the available 230, about a third, because the least-squares line is dominated by the low-current end of the window where the distortion is small. The bias is real but the naive estimate overstates it by roughly a factor of three, and the practical rule that falls out is an iR budget in millivolts at the window top rather than a resistance in ohms.

5 · The assumed specific capacitance, and what it does to a per-area current

Two stipulated catalysts: A with C_dl = 1.0 mF/cm²_geo and j_geo = 2.0 mA/cm²; B with 3.0 mF and 4.0 mA. The roughness factor is C_dl/C_s and the specific activity is j_geo divided by it.

Areas in cm²_real per cm²_geo, currents in mA/cm²_real. From 20 to 80 µF/cm² the area changes by 4.0× and the specific activity by the same factor in the other direction. The last column does not move: the within-paper ranking is invariant to the assumption, because both catalysts are divided by the same number.

Across papers it is not invariant. Paper 1 reports A at C_s = 20 with a specific activity of 1.0 mA/cm²_real; paper 2 reports B at C_s = 60 with 0.5 in the same units. As reported, A wins; harmonised to any common convention, B wins:

The harmonisation target cancels out of the comparison — the same winner at 20, 40 and 60 — so the choice of convention does not matter once there is one. What matters is that comparing the reported numbers compares 3× different units and reverses the answer.

Turnover frequency: which normalisation cancels. With a bulk site density of 1.5×10¹⁵ cm⁻², or with a site density from redox integration (Q = 1.2 mC/cm²_geo) normalised to the same area:

TOF in s⁻¹. The bulk-route TOF runs from 0.111 to 0.444 s⁻¹, a factor of 4.00, while the coulometric route gives 3.33 s⁻¹ at every C_s (ratio 1.0000). The invariance is exact, not approximate: the assumed capacitance appears once in j_spec and once in Γ, and cancels in the quotient.

The uncertainty that does not cancel. Three independent 5% errors in j, C_dl and C_s combine to 8.66% in quadrature, and a Monte Carlo with 200,000 draws gives 8.71%. The assumed specific capacitance is one of three equal contributions, so it cannot be treated as a correction applied after the error budget is written.

The expectation this contradicted. I expected the turnover frequency to be the robust quantity — the one that survives the area convention because it divides the area out. It survives only if the site density is measured on the same area basis as the current. With a bulk site density the TOF moves by the full factor of 4.00 across the same C_s range, and at the stipulated values the two routes disagree by a factor of 30 at the same electrode. The circularity is not that a per-area number contains an assumption; it is that the two ways of removing the assumption give numbers that differ by more than the effect being claimed.

Counts

Tags are editorial: test marks a lead that can be attacked with a computation, a dataset, or a re-analysis already to hand; note marks one that needs a literature sample, a new measurement, or a change in reporting practice. A "test" tag is not evidence that the lead is true.

Limits

• The junction model uses concentrations, not activities. Limiting mobilities at infinite dilution are applied at 3.5 M, where they are certainly wrong; ion pairing and activity coefficients are ignored. The ten junction values are therefore indicative of magnitude and robust only in their ordering; the internal checks (antisymmetry, the single-electrolyte closed form, the KCl near-null) test the implementation, not the physical model.
• Identifiability is a local, Gaussian, known-model calculation. The Cramér–Rao bound assumes the model is correct, the noise is Gaussian with known variance, and the parameters are far from a boundary. It says nothing about a wrong model, and the counts are sensitive to the two stipulated thresholds — which is why two of the tables carry the count at both a 0.9 and a 0.95 correlation threshold.
• The double-layer model is a lattice gas with one ion size. a = 0.4 nm, ε_r = 78.5 fixed, no dielectric decrement, no ion correlation, no overscreening, and a compact-layer capacitance of 20 µF/cm² that is stipulated rather than measured. The model cannot produce underscreening by construction; its inability to do so is a property of the model, not evidence about any real electrolyte.
• The Tafel test is an ideal one-step law. No mass transport, no double-layer current, no potential-dependent transfer coefficient, no distribution of site energies. The 7.2 mV rule is a property of a one-decade least-squares fit to that law and will move with the window width and the fitting weights.
• The catalysis numbers are stipulated, not measured. C_dl values of 1.0 and 3.0 mF/cm², currents of 2.0 and 4.0 mA/cm², a bulk site density of 1.5×10¹⁵ cm⁻² and a redox charge of 1.2 mC/cm² were chosen to be plausible, not taken from any sample. The conclusions are about the algebra of normalisation and hold for any values; the numbers illustrate them.
• Monte Carlo. 150 impedance fits and 200,000 normalisation draws, seeded at 20261103. The impedance check compares realised scatter with predicted scatter and agrees to 11.6% at worst, which is the resolution of 150 draws, not a disagreement.
• No literature search. Several leads are certainly known under other names — liquid-junction theory, Cramér–Rao identifiability, the Bikerman lattice gas, Tafel distortion by ohmic drop, ECSA normalisation — and are stated here as falsifiable claims rather than as discoveries.
• All 200 leads are untested. The tags describe intent, not status. Nothing on this page is a measurement; the only numbers are closed-form evaluations and seeded simulations.

If one sentence survives: every interfacial number is a ratio whose denominator was chosen — a reference potential, a circuit, a layer, an area — and the denominator is testable in the same way as the numerator: change it and see whether the reported quantity moves. Where it does not move the choice was harmless; where it moves, the paper has reported a convention and called it a property.

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