This is a top-down, 2D-native counterpart of the 3D anodic-alumina simulation. Instead of rendering the pore lattice as an oblique 3D stack of cylinders, it treats each pore's position as a particle undergoing an Ornstein–Uhlenbeck relaxation: a spring-like restoring force (mechanical stress between neighboring growing pores, the accepted physical driver of self-ordering) continuously pulls each pore back toward its ideal hexagonal site, while a stochastic forcing term — scaled by how far the applied voltage sits from the electrolyte's self-ordering window — pushes it away. The two compete every animation frame, so order is an emergent, live equilibrium rather than a one-off jitter.
Per-pore relaxation (unit-cell fractional coordinates u):
du/dt = -k·u + σ(U)·ξ(t) (Euler–Maruyama, ξ = unit Gaussian white noise)
stationary variance Var(u) = σ(U)²/(2k) → tighter lattice as σ→0 (in-window U)
Interpore distance: D_int = ξ·U (ξ ≈ 2.5 nm/V, Nagayama et al. 2001)
Cell porosity: P = (π / 2√3)·(D_p / D_int)² ≈ 10% at self-ordering
Barrier thickness: t_b ≈ λ_b·U (λ_b ≈ 1.0-1.3 nm/V)
Layer growth: dh/dt ∝ J (Faraday's law of electrolysis)
- Voltage U — sets the hexagonal cell size (D_int) and barrier thickness, and how far the noise term σ(U) is from zero. Each electrolyte has a narrow "self-ordering" voltage window (≈25 V sulfuric, ≈40 V oxalic, ≈195 V phosphoric); inside it σ→0 and the lattice relaxes to near-perfect hexagonal order, outside it σ grows and the live relaxation settles into a wider, visibly disordered steady state.
- Electrolyte — changes which voltage window gives ordering, and the practical current-density range.
- Current density J — sets the anodizing rate (charge passed per second, shown in the growth column on the right) and slightly widens pores at higher J through enhanced field-assisted dissolution.
- Time t — the oxide column on the right thickens as anodizing proceeds; the pore lattice geometry itself (spacing, diameter, barrier) is steady-state and time-independent once the process self-stabilizes, exactly as in the 3D model.
This self-ordered array is the basis for real nanotechnology: AAO templates are used to grow nanowires, as anti-reflective coatings, and as the etched pattern in hard-anodized cookware and aerospace aluminum finishing.