The 3D version of this lab grows a height field: every grid column has one number (its height) and a new atom is assigned to whichever column already sticks out the most. That is a fast approximation of dendrite growth, but it can never show a branch curving back over its neighbor.
This 2D companion instead simulates a real cross-section of the cell (x = lateral position, y = distance from the anode) and grows it the way diffusion-limited aggregation actually works: individual Li⁺ ions are launched from just above the growth front and take a random walk — some steps down toward the deposit, some sideways — until they touch a solid cell. Whether they actually stick is itself probabilistic:
p_stick(site) = (empty neighbors / 4)^η
η = η₀ · (J / J_ref) · (1 − 0.85 · additive/100)
A site with more empty neighbors is more "exposed" — sticking out further into the electrolyte, which is exactly where the electric field concentrates in the real cell. Raising η (via higher current density J, or lower additive) makes the ion strongly prefer those exposed tips, so mass gets funneled into a few thin fast-growing fingers — a genuine branching fractal, not a smoothed height field. At η ≈ 0 (high additive) almost every contact sticks, so growth fills in evenly and stays low and compact, matching how film-forming additives are known to produce smoother, mossy deposits instead of needles.
Applied stack pressure acts on the ion's walk itself: while diffusing sideways, a pressurized ion is biased toward whichever neighboring column is currently shorter, mechanically redirecting flux away from tips and into valleys — the 2D analogue of physically flattening the front.
The predicted time-to-short is Chazalviel's Sand's-time expression for a binary electrolyte, evaluated with literature-typical constants (D ≈ 4×10⁻¹⁰ m²/s, C₀ = 1 M, t₊ = 0.3) instead of the 3D lab's illustrative-only scaling:
τ_Sand = π D · ( F·C₀ / (2 t₊ J) )² ⇒ τ ∝ 1 / J²
With those real constants the baseline (J = 3 mA/cm², no additive, no pressure) works out to roughly 10 hours — much longer than the 3D lab's illustrative "22 minutes," because that number was explicitly scaled for a snappy demo rather than derived from a diffusion coefficient. Both panels keep the same physically-required J⁻² scaling; only the absolute baseline differs. Additive and pressure apply the same directional multipliers used in the 3D lab. The visual growth rate on screen is deliberately accelerated so a dendrite bridges the gap in seconds rather than hours — the readout is the calibrated number, the animation is sped up for legibility.