2D companion to the 3D nanocatalyst hydrogen electrolyzer: the same nanoparticle-roughness + Butler-Volmer/Tafel + bubble-shielding pipeline, read off four live panels instead of a rendered electrode.
Electrode roughness from a fixed catalyst loading split into smaller nanoparticles:
R_f = 1 + k · (coverage/100) · (d_ref / d)
Tafel kinetics for the hydrogen evolution reaction 2H⁺ + 2e⁻ → H₂ (cathodic Butler-Volmer branch):
η = max(0, V_cell − 1.23 V)
j₀(T) = j₀,ref · exp[ −(Eₐ/R)·(1/T − 1/T_ref) ]
j_kinetic = R_f · j₀(T) · exp(α F η / R T)
with α = 0.5, F = 96,485 C/mol, R = 8.314 J/(mol·K). This gives the real ~120 mV/decade Tafel slope, so current rises exponentially with overpotential rather than staying flat.
Mass-transport limit + bubble shielding cap the kinetic current the way a real cell does at high voltage: a fixed limiting current jlim combines with jkinetic Koutecky-Levich style, then growing bubble coverage θ throttles the surface further:
1/j_pre = 1/j_kinetic + 1/j_lim
θ = min(0.7, j_pre / (j_pre + 120))
j = j_pre · (1 − θ)
Hydrogen production follows Faraday's law, n(H₂) = j·A / (2F) mol/s, converted to mL/min at STP (22.4 L/mol).
- Electrode cross-section (top-left) — nanoparticle sites on the cathode nucleate bubbles at a rate proportional to the live current; bubbles rise and crowd the surface at high current.
- Tafel plot (top-right) — log₁₀(j) vs. η, the diagnostic electrochemists actually use to read off the exchange current and Tafel slope; the live point is marked.
- Current vs. voltage (bottom-left) — j across the full 1.00-2.20 V sweep at the current coverage/diameter/temperature, showing the exponential onset and the mass-transport plateau.
- Current vs. temperature (bottom-right) — j across the full 10-90 °C sweep at the current voltage, showing the Arrhenius boost to j₀ as the electrolyte warms.
Fix vs. the 3D version: the 3D engine's kinetic exponent carried a stray ×0.001 scale factor that nearly cancelled the Tafel exponential, making current density almost independent of voltage — the opposite of what "Tafel kinetics" and the 3D tooltip's own text describe. This 2D build removes that factor and instead caps the exponential with a physically motivated mass-transport limiting current, which keeps the numbers on-screen sane while preserving the real exponential onset.