Polymer Reptation: Entangled Chain in a Tube

Polymer Dynamics

Chain length N 40

Obstacle (mesh) density 1.0

Thermal wiggle 1.0

MSD slope (log-log)

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Entanglements Z

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Centre-of-mass MSD

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Reptation time τ_d (est.)

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How it works

An entangled polymer melt behaves like de Gennes' tube model: neighbouring chains act as fixed topological obstacles a given chain cannot cross, confining it to a curvilinear tube along its own contour. The chain can only move by sliding back and forth along the tube's axis — snake-like reptation — occasionally poking a new segment out of one end while the old tube memory at the other end relaxes away. An unentangled (short, or low-density) chain has no such cage and instead diffuses by ordinary Rouse dynamics, with beads dragging independently through the solvent.

MSD(t) ~ t^0.5   Rouse, short/unconfined chains
MSD(t) ~ t^0.25  reptation, chain still exploring old tube
MSD(t) ~ t^0.5   reptation, tube renewal regime
MSD(t) ~ t^1.0   free diffusion, t ≫ τ_d
τ_d ~ N^3        reptation time scales steeply with length

Chain length N — more beads means more entanglements Z and a much slower, steeper τ_d ~ N³ reptation time.
Obstacle (mesh) density — denser surrounding chains narrow the tube, sharpening confinement and lowering the short-time MSD slope.
Thermal wiggle — the effective temperature driving bead fluctuations along (and, if unentangled, across) the chain.
Entangled vs Unentangled — switches whether lateral obstacle points constrain the chain to its tube, or whether every bead random-walks freely (Rouse).
The live log-log MSD slope readout is the exponent α in MSD ~ t^α, measured from the centre-of-mass trajectory of the running simulation — watch it drop toward 0.25 once entanglement kicks in, and climb back toward 1 as the chain reptates out of its original tube.

Real-world relevance: reptation theory explains why melt viscosity of plastics rises so steeply with molecular weight (η ~ N^3.4), why longer polymers process more slowly, and why extruded/injection-moulded parts retain molecular orientation that a manufacturer must anneal away.

Chain (bead-spring) Chain ends Obstacle mesh Tube envelope

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