The 3D version scales instanced spheres by a single closed-form mean radius r̄(t)³ = r̄₀³ + K·t applied identically to every particle's own fixed size ratio. This 2D companion instead integrates each of 140 nickel grains as its own ordinary differential equation — a discrete Lifshitz–Slyozov–Wagner (LSW) population balance, the actual mean-field theory the closed form is derived from:
Per-grain growth: dr_i/dt = (K(T,j)/r_i) · (1/r_c − 1/r_i)
Self-consistent r_c: r_c = mean(active r_i) [enforces Σ r_i²·dr_i/dt ≈ 0,
i.e. total Ni volume ≈ conserved]
Grains below a critical size vanish (Ni fully redissolves);
survivors above r_c grow at their expense — the actual
"small ones shrink, large ones grow" mechanism, not a
uniform rescale of a fixed shape.
The left panel is a packed 2D cross-section (a polished-section micrograph analogue) — genuinely distinct from a 3D camera view, since it is the direct 2D rendering of the population-balance state itself. The right panel is a live histogram of the grain-size distribution: watch it broaden and skew right as ripening proceeds — the classic LSW fingerprint that a single mean-radius number cannot show.
Bug found while building this companion: the 3D sim.html's own inline comment claims its rate constant K0 = 2.4e-21 was "chosen so ~800 °C/40,000 h gives ~50% radius growth" — but that K0 actually predicts under 0.2% growth at those conditions (verified numerically), roughly 800× too slow to match its own narrative and the 50,000+ hour time-acceleration slider it ships with. This 2D companion uses a corrected rate constant (K0 = 1.86e-18 m³/s) reverse-solved from the same Arrhenius/current-coupling formula so the visualization actually shows the coarsening the UI promises; temperature and current density still move the rate exactly the same way (Arrhenius + linear current coupling).
- Temperature — sets the Arrhenius rate; +50 °C is roughly a 3× rate jump here (100 °C span 650→850 °C is a ~100× jump, matching the exponential Eₐ ≈ 200 kJ/mol activation barrier).
- Current density — adds the same illustrative electromigration coupling term as the 3D model.
- Surviving grains — count of the original 140 that haven't fully redissolved; falling faster at high T/current is itself part of the coarsening signature.