Polymer Chain 2D: Worm-Like-Chain Langevin Dynamics & Live Flory-Exponent Fit

A freely-jointed chain has bond directions that are completely uncorrelated from one segment to the next. A worm-like chain (WLC) instead has a bending energy that resists sharp turns, so nearby bond directions stay correlated over a characteristic length called the persistence length ℓ_p. Over distances much longer than ℓ_p, a worm-like chain behaves like a random walk again; over distances shorter than ℓ_p, it looks like a stiff rod.

Every animation frame runs several Langevin (Brownian-dynamics) sub-steps: each bead feels a harmonic spring force from its neighbours (keeping bond lengths near their rest value), a discretized Kratky–Porod bending force proportional to the local curvature (rᵢ₋₁−2rᵢ+rᵢ₊₁) scaled by the bending modulus, and independent Gaussian random kicks scaled by the temperature. This is numerically integrated, not pre-baked.

Five chains of different lengths (8 to 128 beads) run under identical stiffness and temperature. As each accumulates samples of its own radius of gyration, the simulator plots ln(mean Rg) against ln(N) for all five and fits a straight line through them by least squares — the slope of that line is the effective Flory exponent ν in R_g ~ N^ν, measured directly from the running simulation rather than assumed.

When the persistence length ℓ_p is small compared to the chain length, bending stiffness only matters locally and the chain scales like an ideal random coil with ν≈0.5. When ℓ_p is comparable to or larger than the number of bonds, even the longest chain in the ensemble still looks mostly straight, pulling the fitted exponent toward the rod-like limit ν≈1 until enough beads are added for the coil regime to reappear.

The worm-like chain model is the standard description of double-stranded DNA (persistence length ≈ 50 nm), unfolded or partially folded protein backbones, and semiflexible cytoskeletal filaments like actin and microtubules. The same rod-to-coil crossover measured here by the live exponent fit governs how DNA is packaged in chromatin, how stiff biopolymers respond to stretching in single-molecule experiments, and why short synthetic polymers can behave very differently from long ones of the same chemistry.