Each carbon-nanotube filler is modeled as a hard capped cylinder (a "stick") of length L and diameter d, placed at a uniform-random position and orientation inside a cube of polymer matrix. Two rods are electrically connected the instant their surfaces come within a tunneling gap — approximated here as their centerlines passing within one rod diameter of each other. Connectivity across the whole set is tracked with a union–find (disjoint-set) structure rebuilt every time you change φ, the aspect ratio, or the alignment.
Continuum stick percolation (Balberg 1984):
N_c · ⟨v_ex⟩ / V ≈ 1.4 (isotropic 3D rods, N_c = critical rod count)
⟨v_ex⟩ ≈ 2 L² d (excluded volume of two randomly oriented rods, L ≫ d)
φ_c ≈ 1.4 · d / (2L) = 0.7 (d/L) → thinner, longer rods percolate at LOWER loading
- Filler volume fraction φ — how much of the matrix volume is nanotube. Raising it adds more rods until the excluded-volume sum crosses the Balberg threshold and a giant conductive cluster spans the box.
- Aspect ratio L/d — matches the article's real fillers: SWCNT/MWCNT (L/d up to ~2,500) percolate at φ_c as low as 0.01–0.1 wt%, while low-aspect fillers like spherical nano-SiO₂ (L/d → 1) need many times more loading — exactly why CNT and graphene are prized as "efficient" conductive fillers.
- Alignment — real processing (extrusion shear, electric-field alignment) orients rods along one axis. Aligning rods shrinks their mutual excluded volume in the perpendicular directions, which is why aligned nanocomposites need a higher φ_c to percolate than a random (isotropic) dispersion — the model reproduces this by shrinking the effective excluded volume as alignment rises.
- Largest cluster — the fraction of all placed rods that belong to the single biggest connected component. The badge turns "conducting" once that cluster touches both the left (−x) and right (+x) faces of the cube — a real spanning path for current.
This is the same physics behind the article's "critical concentration ~1–2% for CNT electrical conductivity" figure and the broader electrical-percolation section of nanocomposite reinforcement — mechanical reinforcement (stiffness, strength) is dominated by interfacial load transfer, but electrical/thermal conductivity in a filled polymer is a purely geometric percolation problem, which is what this simulator isolates.