Polymer Reptation 2D: Brownian Bead-Spring Chain in a Fixed Obstacle Lattice

Chain & Lattice

Chain length N 48 beads

Obstacle density 6 / 10

Temperature (k⁫T, relative) 1.00

A real Brownian-dynamics bead-spring chain is integrated live, wiggling through a fixed square lattice of point obstacles it cannot easily cross — this is genuinely simulated, not a pre-baked animation.

Live Measurement (from the running simulation)

Entanglements Z = N/Nᵇ

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Sim time elapsed

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Measured D (from MSD fit)

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Rouse D₀ (theory, N only)

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Measured ratio D/D₀

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Theory ratio 1/(3Z)

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How it works

This is a direct overdamped Langevin (Brownian-dynamics) integration of a bead-spring chain — the same Rouse model used in polymer physics — stepped forward every frame with real spring forces plus real Gaussian thermal noise. In "Entangled" mode, every bead also feels a soft repulsive force from the nearest node of a fixed square lattice of point obstacles (the classic Evans & Edwards 1981 fixed-obstacle model of a 2D entangled melt): the chain must wiggle around these pinned points rather than crossing them freely, which is exactly what a real chain does when caged by its topologically-fixed neighbours.

Langevin step (per bead, per dt):
  γ·dr = Fᵢᵢᵢᵢᵢ·dt + √(2γk⁫T·dt)·ξ(t)
  F_spring = k·(rᵢ₋₁−2rᵢ+rᵢ₊₁)      (harmonic Rouse springs)
  F_obstacle = repel(rᵢ − nearest lattice node)

Entanglement length (derived, not guessed):
  Nᵇ = (lattice spacing / equilibrium bond length)²
  Z = N / Nᵇ

de Gennes prediction being tested live:
  D_reptation / D_Rouse ≈ 1 / (3Z)   for Z ≥ 1

Chain length N — more beads means the same lattice traps more entanglements (Z grows), which is the regime where reptation confinement should visibly suppress the measured diffusion coefficient.
Obstacle density — sets the lattice spacing; a denser lattice shortens N_e so the same chain becomes more entangled.
Temperature — scales the thermal noise driving both the spring fluctuations and the obstacle-lattice hopping; it also sets the equilibrium bond length used to derive N_e.
Measured D — the simulator tracks its own chain's center-of-mass position every frame, accumulates a mean-squared-displacement curve since the last reset, and fits a live diffusion coefficient from its slope (MSD = 4Dt in 2D) — a genuine measurement of the running trajectory, not a formula.

Real-world relevance: this is the same fixed-obstacle Brownian-dynamics technique (Evans & Edwards, J. Chem. Soc. Faraday Trans. 1981) used to numerically test de Gennes' reptation hypothesis before more elaborate melt simulations existed — watching a single chain's center of mass diffuse more slowly than free Rouse motion, by roughly the 1/(3Z) factor, as it gets more entangled.

Entangled

Chain beads Springs (bonds) Fixed obstacle lattice