Polymer Reptation & Tube Model

de Gennes reptation, τ_rep ~ N³, Rouse vs reptation MSD regimes

Reptation Animation — Chain Diffusing in Tube

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Tube model (de Gennes 1971, Doi-Edwards): chain confined to "tube" defined by entanglements. Reptation = snake-like diffusion along tube contour. τ_rep ~ N³, D_rep ~ N⁻². Tube length L ~ N·b (b = Kuhn length).

MSD Regimes

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Four MSD regimes:
t < τ₀: ⟨r²⟩ ~ t^(1/2) (Rouse in tube, free modes)
τ₀ < t < τ_e: ~ t^(1/4) (confined Rouse)
τ_e < t < τ_rep: ~ t^(1/2) (reptation along tube)
t > τ_rep: ~ t^1 (Fickian diffusion)

Relaxation Time τ_rep ~ N³

Rouse (unentangled): τ_R ~ N², D ~ N⁻¹
Reptation (entangled): τ_rep ~ N³, D_rep ~ N⁻²
Experimental: τ ~ N^(3.4) (due to contour length fluctuations, constraint release). Crossover at N ≈ Ne (entanglement length, typically 50-300 monomers).

Viscosity η ~ N^α

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Unentangled (N < Ne): η ~ N^1 (Rouse). Entangled (N > Ne): η ~ N^(3.4) (reptation predicts 3, experiment gives 3.4 due to fluctuation corrections). Crossover at Ne.

Rouse Modes — Normal Coordinates

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Rouse modes: X_p(t) = (1/N)Σ_n r_n cos(pπn/N). Mode p has relaxation time τ_p = τ_R/p². Longest mode (p=1) governs overall chain diffusion.