Polymer Reptation: Entanglement & Melt Viscosity

Chain & Melt

Chain length N 150 segments

Entanglement density 6 / 10

Melt temperature (relative) 1.00 T₀

The highlighted chain is trapped in a tube formed by neighboring chains (grey rods) and can only diffuse back and forth along its own contour — de Gennes' "reptation".

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Entanglements Z = N/Nᵇ

3.75

Relative viscosity η/ηₑ

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Disengagement time τ_d/τ₀

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Diffusion D/D₀

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How it works

Above a critical length N_e (the entanglement length), a polymer chain can no longer move freely — it is topologically caged by the chains around it. de Gennes' reptation model treats this cage as a tube of diameter roughly N_e: the chain can only crawl back and forth along its own contour, one-dimensional "curvilinear" diffusion.

Entanglements per chain:  Z = N / Nᵇ

Rouse time (unentangled):  τᵣ ∝ N²
Disengagement time:        τᵈ ≈ 3·Z·τᵣ   ⇒  τᵈ ∝ N³  (Z ≥ 1)

Center-of-mass diffusion:  D₀ ∝ 1/N          (Rouse, always)
                            Dᵣᵉᵣ = D₀ / (3Z)  ⇒  D ∝ 1/N²  (Z ≥ 1)

Zero-shear viscosity:      η ∝ N        for N < Nᵇ  (Rouse / unentangled)
                            η ∝ N³·⁴   for N ≥ Nᵇ  (reptation, empirical exponent)

Chain length N — more statistical segments means more entanglements once N exceeds N_e, which is exactly the regime where the famous N^3.4 viscosity scaling takes over from linear Rouse behaviour.
Entanglement density — a denser melt has a shorter N_e (obstacles are closer together), so the same chain gets trapped by more entanglements and the tube narrows.
Temperature — raises monomer mobility, speeding up curvilinear diffusion along the tube without changing the scaling exponents themselves.
Mode toggle — switches the same chain between confined reptation and a hypothetical unentangled Rouse chain, so you can compare the τ, D and η each regime predicts for the identical N.

Real-world relevance: this N^3.4 law is why doubling a polymer's molecular weight can raise a plastic melt's viscosity more than tenfold — it governs how polymers are processed (extrusion, injection molding) and why entangled melts flow so much more sluggishly than short, unentangled oligomers.

Reptation

Test chain (beads) Confining tube Neighboring chains

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