Tube Dynamics Works for Randomly Entangled Rings

The tube model is the cornerstone of molecular theory for polymer rheology. We test its microscopic assumptions by simulating topologically equilibrated ring polymers, whose dynamics is free from end segment relaxation. We show that a closed-form expression derived from the tube model adapted to ring polymers quantitatively predicts the segmental mean squared displacements over the entire range of time scales from local motion to complete equilibration, with a time-independent local friction factor.

Figure 1

Left: A ring polymer entangled with its periodic images. Right: Ring rebridging moves, which equilibrate the entanglement topology.

Figure 2

Bead mean squared displacements, for three rings with $N=400$, 800, and 1600; MD results (solid) and tube theory predictions (dashed). Inset: Friction factors $\zeta$ from linear chain simulations, with fit of form $\zeta (N)=\zeta (\infty )-c/N$.

Figure 3

Expected regimes of bead MSD regimes for the tube model. I: Beads do not feel the tube, and move by 3D Rouse dynamics. II: Beads move by 1D Rouse dynamics along the tube. III: Beads move by ring reptation. IV: Beads have traversed the tube many times; MSD reaches a plateau.

Figure 4

Tube contours (colored red to blue by bead index) from the isoconfigurational average over time ${\tau }_a=2\times {10}^4\tau$. Different contours correspond to elapsed times ${\tau }_e$, ${\tau }_R$, and ${\tau }_d$.