Translationally Invariant Slip-Spring Model for Entangled Polymer Dynamics

The topological effect of noncrossability of long flexible macromolecules is effectively described by a slip-spring model, which represents entanglements by local, pairwise, translationally invariant interactions that do not alter any equilibrium properties. We demonstrate that the model correctly describes many aspects of the dynamical and rheological behavior of entangled polymer liquids, such as segmental mean-square displacements and shear thinning, in a computationally efficient manner. Furthermore, the model can account for the reduction of entanglements under shear.

Figure 1

Monte Carlo updates of slip-spring conformations: (a) Shuffling of a slip spring along the backbone of a chain molecule (reptation), (b) grand canonically creating or deleting a slip spring at a chain end, and (c) transferring one end of a slip spring from one chain to another bead (constraint release and re-entanglement).

Figure 2

Chain dimensions $R_e$ as a function of the activity of slip springs at density $\overline{\mathcal{N}}=4$, without and with compensating potential $-\Delta \Phi$. Inset: Average number $⟨n_{ss}⟩/(2nN)$ of slip springs per monomer as a function of the slip-spring activity $z$.

Figure 3

Mean-square displacement $g_1(t)$ of all segments and of the molecule’s center of mass $g_3$, with and without slip springs for $N=128$. The activity of slip springs is $z=35.87$, resulting in an average distance between the slip springs of four segments; $⟨R_e^2⟩=114\pm 1{\sigma }^2$. The simulation cell size is $L=20\sigma$. The lines show the predictions of the tube model, where $N_e$ is extracted from the behavior $g_1\propto t^{1/4}$ of the entangled system and the measured self-diffusion coefficient $D$ of the unentangled system.

Figure 4

Main panel: Shear viscosity $\eta$ as a function of shear rate $\dot{\gamma }$ for the system with and without slip springs. The slip-spring activity results in approximately four monomers between slip springs in equilibrium. Inset: Decrease of the number of entanglements per chain with shear rate $\dot{\gamma }$ for $N=32$.