The 3D sim assumes the textbook parabolic law δ(t) = √(2·k_p·t) and animates ions decoratively around it. This 2D companion never assumes that law — it derives it (or breaks it) by actually solving the coupled reaction-diffusion moving-boundary problem SEI growth really is:
∂C/∂t = D·∂²C/∂x² [Fick's 2nd law, electrolyte solvent, x=0 at anode]
-D·∂C/∂x |_{x=δ} = k(T)·C(δ,t) [Robin BC: 1st-order interfacial reaction consumes solvent]
dδ/dt = k(T)·C(δ,t) [new solid forms at the same rate solvent is consumed]
D(T) = D₀·exp(−E_D/R·T) k(T) = k₀·exp(−E_k/R·T)·(0.6+0.6·C-rate)·additiveFactor
Whether the emergent δ(t) looks linear or parabolic is decided by the Damköhler number Da = k·Δx/D — the ratio of how fast the interface reaction consumes solvent versus how fast diffusion can resupply it:
- Da ≪ 1 (reaction-limited) — diffusion easily keeps the interface saturated, so the interface concentration stays near the bulk value and growth is close to linear in time: dδ/dt ≈ const. The 3D sim's formula cannot produce this regime at all.
- Da ≫ 1 (diffusion-limited) — the reaction consumes solvent faster than it can arrive, the interface concentration is starved toward zero, and growth throttles toward the classic parabolic √t law — recovering the 3D sim's assumption as a limiting case, not a given.
Because k(T) carries a higher effective activation energy than D(T) here, raising temperature or C-rate pushes Da up — you can watch the live-fit exponent (a rolling log-log regression of δ against elapsed time, computed from the actual trajectory, not looked up) drift from closer to 1 toward closer to 0.5 as you turn the sliders up. That crossover is a genuine, numerically emergent result of this PDE, verified with a standalone Node.js script before this page was published (see report).
- Temperature — raises both D(T) and k(T) (Arrhenius), and shifts Da toward the diffusion-limited corner.
- Cycling rate (C-rate) — raises only k(T) (more side-reaction current), unlike the 3D sim where it multiplies the whole growth constant uniformly; here it can visibly push the system across the reaction→diffusion crossover on its own.
- SEI-stabilizing additive — halves k₀, the same qualitative effect as the 3D sim's additive, but because it only touches the reaction step (not diffusion), it also pulls Da back down — a stabilizing additive keeps the system in the gentler, reaction-limited regime longer.