Protons cross a PEM fuel cell's Nafion membrane by the Grotthuss mechanism — hopping between water molecules bound to sulfonic-acid (SO₃⁻) sites — so conductivity depends entirely on how much water (λ, mol H₂O per mol SO₃⁻) is sitting inside the membrane at each point.
Two competing flows set that water profile: electro-osmotic drag physically carries water from anode to cathode along with every proton (nd ≈ 2.5 H₂O per H⁺ when saturated), while back-diffusion pushes water the other way down its own concentration gradient. The simulation integrates this 1-D diffusion–drift PDE across the membrane thickness every frame (time is accelerated for visualization):
∂λ/∂t = D_w(λ,T)·∂²λ/∂x² − v_drag(λ,i)·∂λ/∂x
D_w(λ,T) = D₀(λ)·exp[2416·(1/303 − 1/T)] (Motupally et al.)
n_d(λ) = 2.5·λ/22 (Springer et al.)
σ(λ,T) = (0.005139λ − 0.00326)·exp[1268·(1/303 − 1/T)] S/cm
At the anode boundary, humidified feed gas clamps λ to the Springer sorption isotherm λ_eq(a) = 0.043 + 17.81a − 39.85a² + 36.0a³; switching to Dry removes that external supply, so the anode side survives only on whatever water back-diffuses in from the cathode. At the cathode, oxygen-reduction product water raises the effective humidity directly with current.
- Current density i — more current means more electro-osmotic drag pulling water toward the cathode, and more product water generated there.
- Temperature — both D_w and σ follow Arrhenius terms; higher T speeds diffusion but also the saturation vapor pressure, indirectly drying the feed.
- Inlet RH — sets the equilibrium hydration the humidified boundary is clamped to.
- Anode feed toggle — Dry + high current is the classic failure mode: drag outruns back-diffusion and the anode side desiccates, conductivity collapses locally, and ohmic loss (i·R) climbs even though the cathode stays wet.
The bright drifting dots are protons; their hop speed is scaled by the local conductivity, so they visibly stall wherever the membrane runs dry.