Each carbon-nanotube-like filler is modeled as a thin rod (a 2D "stick") of length L and width w, scattered at a uniform-random position and orientation across a square cross-section of the polymer film — the 2D analogue of the 3D CNT nanocomposite this simulator is paired with. Two rods are electrically connected the instant their centerlines pass within a tunneling gap, approximated as one rod-width. Connectivity is tracked with a union–find (disjoint-set) structure rebuilt every time φ, the aspect ratio, or the alignment changes.
2D continuum stick percolation (Widom 1974 / Balberg 1984 / Mertens 2012):
N_c · ⟨A_ex⟩ / A ≈ 5.637 (isotropic 2D rods — a DIFFERENT universal
constant from the 3D sibling sim's 1.4;
2D percolation is a distinct universality
class from 3D, not a flattened copy of it)
⟨A_ex⟩ ≈ (4/π) L² (excluded area of two randomly oriented
zero-width sticks, angle-averaged: the
exact excluded area between two segments
at angle γ is 2L²|sinγ|, and ⟨|sinγ|⟩=2/π
over an isotropic 2D angle distribution)
φ_c ≈ 5.637 · (π/4) · (w/L) ≈ 4.43 (w/L) → thinner, longer rods still
percolate at lower loading, exactly as in 3D
Note: the paired 3D sim's on-page derivation carries a stray factor — it states ⟨v_ex⟩≈2L²d but that value plugged into N_c⟨v_ex⟩/V=1.4 gives φ_c≈0.55(d/L), not the 0.7(d/L) the 3D sim actually displays. The self-consistent 3D excluded volume is ⟨v_ex⟩=(π/2)L²d (verified numerically); this 2D engine's ⟨A_ex⟩ and φ_c above were re-derived independently and checked for internal consistency the same way.
- Filler area fraction φ — how much of the film cross-section is nanotube. Raising it adds more rods until the excluded-area sum crosses the Widom/Balberg threshold and a giant conductive cluster spans the sheet.
- Aspect ratio L/w — matches the article's real fillers: high-aspect CNT/graphene ribbons percolate at a tiny loading, while low-aspect fillers need many times more — the same trend the 3D sibling shows, now visible edge-on in a single plane.
- Alignment — shear or field alignment orients rods along one axis. Parallel rods rarely cross, shrinking their mutual excluded area, so aligned sheets need a higher φ_c than a random dispersion — reproduced here by shrinking the touching gap as alignment rises.
- Largest cluster — the fraction of all placed rods in the single biggest connected component. The badge turns "conducting" once that cluster touches both the left and right edges of the square — a real spanning path for current.