Why use asinh instead of the Tafel equation to fit activation overpotential?

The Tafel equation, η = (RT/αF)·ln(j/j0), is undefined at j=0 and diverges as current approaches zero, so it discards or distorts exactly the low-current data points that best constrain the exchange current density. The full Butler-Volmer form using asinh, η = (RT/αF)·asinh(j/2j0), stays finite through j=0 and converges to the same Tafel behavior at high current — it is a strict upgrade, not a different model.

Why does the fit warn that j0 is not reliable?

Exchange current density (j0) and the charge-transfer coefficient (α) are only separately identifiable from data collected at low current density, typically below about 0.05 A/cm². If a dataset starts at higher current, the fit can still find a Tafel slope, but j0 and α individually become highly correlated — many different (j0, α) pairs fit the data almost equally well — so the tool flags this rather than reporting a falsely precise number.

Free · Tool 02

Your test bench made a curve.
This is what's actually in it.

Paste the I-V data off your potentiostat. Get exchange current density, charge-transfer coefficient, ohmic resistance and mass-transport limit — each with a real 95% confidence interval, not just a solver's best guess.

Test data

Paste two tab/comma-separated columns from Excel or your test bench export. A header row is fine — it's auto-detected.

Column 1: current density j (A/cm²) — Column 2: cell voltage V (V)

Cells in series

Active area (cm²)

Temperature (°C)

Technology

Cathode pressure (bar)

Anode pressure (bar)

Fit mass-transport limit (j_lim)

Optional: second curve for degradation rate

Paste a later-life curve (same format) to get a µV/h degradation rate you can carry straight into the LCOH and guarantee-correction tools.

Hours between the two tests

Second curve data (optional)

Paste your data and click Fit curve.

One curve, once. HYDRA OS refits this continuously against live stack data, catches drift before it shows up in production, and feeds it straight into degradation-adjusted LCOH.

See the Founding Pilot Cohort

Worked example

The dataset this page loads with — fourteen I-V points from a 60-cell, 300 cm² PEM stack at 80 °C and ambient pressure on both electrodes. Press Fit curve and the tool returns exactly this.

Fitted parameters

j₀ (anode): 3.36 × 10⁻⁵ A/cm²
α (anode): 0.492
ASR: 177.4 mΩ·cm²
j_lim: 70 A/cm² (at bound)
Tafel slope: 142.4 mV/decade

95% confidence intervals

j₀ (anode): −4.2 × 10⁻⁶ … 7.1 × 10⁻⁵
α (anode): 0.426 … 0.558
ASR: 165.8 … 189.0 mΩ·cm²
j_lim: not identifiable
RMSE: 10.9 mV
R²: 0.9990

R² is 0.9990 and the fit is still not telling you what you think it is: the confidence interval on j₀ crosses zero, and j_lim sits at its bound with zero standard error. Only ASR is actually constrained by this data.

This is the failure mode the tool exists to catch. j₀ and α are correlated at −0.99 in this fit, meaning the data cannot separate them — a lower exchange current density and a higher transfer coefficient describe these fourteen points equally well. And because the measurements stop at 3.5 A/cm², nowhere near mass-transport limitation, j_lim is unconstrained and the optimizer parks it at the bound. Quoting a j₀ from a dataset like this in a vendor comparison is quoting noise with four significant figures. To constrain j₀ and α separately you need low-current-density points below roughly 0.01 A/cm²; to constrain j_lim you need to drive the stack into the mass-transport regime.

Method & sources

This tool fits a two-electrode, lumped Butler-Volmer model to your data using Levenberg-Marquardt nonlinear least squares — the same class of algorithm behind commercial electrochemistry fitting software, implemented here in about 150 lines of dependency-free JavaScript so it runs entirely in your browser. Nothing you paste in is uploaded anywhere.

The model

V_cell(j) = E_rev(T,p) + η_act,anode(j) + η_act,cathode(j) + j·ASR + η_conc(j) E_rev(T,p) = 1.2291 − 8.5×10⁻⁴·(T−298.15) + (RT/2F)·ln(p_H2·√p_O2) η_act(j) = (RT/αF)·asinh( j / (2·j0) ) η_conc(j) = (RT/nF)·ln( 1 / (1 − j/j_lim) )

Cathode kinetics (hydrogen evolution) are fixed at a fast default (j0,cat = 0.1 A/cm², α = 0.5) rather than fit — HER is fast enough on typical cathode catalysts that the anode (oxygen evolution) dominates the activation loss, and fitting both simultaneously with only a total-voltage measurement is not identifiable from most test datasets. If your cathode is known to be limiting, treat the fitted ASR as absorbing some of that loss.

Why asinh, not the textbook Tafel equation

The Tafel form, η = (RT/αF)·ln(j/j0), is undefined at j=0 and diverges as current approaches zero — exactly where your test bench's minimum-stable-load points live. The full Butler-Volmer asinh form used here stays finite through j=0 and converges to identical Tafel behavior at high current. It isn't a different model, it's the same model without the part that breaks on the data you actually have.

Confidence intervals and identifiability

Parameters are optimized in a transformed space (log for j0/ASR, logit for α, and j_lim is parameterized as "above your highest tested current density plus a free positive offset" so the solver can never collapse it inside your data range) — this is what keeps the fit from wandering into a false local minimum on noisy data. The covariance matrix (and from it, the 95% confidence intervals shown) comes from σ²·(JᵀJ)⁻¹ at the converged solution, propagated from the fit's own residual scatter — not assumed.

j0 and α are only separately identifiable when your data includes low-current points (below roughly 0.05 A/cm²). Without them, the fit still finds a good Tafel slope, but reports a warning rather than a falsely precise j0.

Sources

E_rev0 = 1.2291 V — standard reversible cell voltage at 25°C, 1 atm, liquid water reference; equals ΔG°_f(H₂O,l)/(n·F) using the standard Gibbs free energy of formation of liquid water.

−8.5×10⁻⁴ V/K temperature coefficient — dE_rev/dT = −ΔS°_f(H₂O,l)/(n·F), derived from standard molar entropies (S°: H₂O(l)=70.0, H₂(g)=130.7, O₂(g)=205.2 J/(mol·K)), which gives −8.46×10⁻⁴ V/K — reproducible from any standard thermochemistry reference, not one paper.

Cathode defaults (j0,cat = 0.1 A/cm², α = 0.5) — a representative mid-range geometric exchange current density reflecting hydrogen evolution's well-known kinetic advantage over oxygen evolution on any practical PEM anode catalyst, not a specific measurement; see "The model" above for why this isn't fit.

What this tool deliberately does not do

It does not distinguish anode vs. cathode contribution to ASR — the fitted ASR is the total ohmic + any lumped cathode loss.

It does not fit crossover or hydrogen-in-oxygen behavior — use the turndown tool for that, with this fit's parameters as an input if you want them.

Degradation rate from two curves assumes the whole curve shifted uniformly in voltage — a standard simplification for a screening tool, not a mechanism-resolved degradation model.