Persistence length and plateau modulus of semiflexible entangled polymers in primitive chain network simulations

The relationship between the plateau modulus (G_0) and the persistence length (L_p) of entangled semiflexible polymers is still uncertain. Some previous theoretical models have predicted that G_0 decreases with increasing L_p, while experiments indicated the opposite. In this study, we extend the primitive chain network (PCN) model to incorporate bending rigidity at the scale of the entanglement length, consistently with the coarse graining of the model. Simulations investigate the effects of rigidity (and of molecular weight) on chain conformation and relaxation modulus. Our results reveal a relationship described as G_0~L_p^(2/3), indicating that G_0 increases with increasing L_p, consistently with experiments. It should be considered, however, that implicit in our simulations is the condition that the entanglement length is kept fixed with changing L_p.

💡 Research Summary

The paper addresses a long‑standing controversy in polymer physics: how the plateau shear modulus (G₀) of entangled semiflexible polymer solutions depends on the chain persistence length (Lₚ). Classical tube‑based theories (Morse’s binary‑collision approximation, effective‑medium approaches) predict that G₀ should decrease as Lₚ increases, whereas experimental studies on DNA and F‑actin solutions report the opposite trend. To resolve this discrepancy, the authors extend the Primitive Chain Network (PCN) model—a coarse‑grained representation of entangled polymers where each polymer is mapped onto a network of nodes (slip‑links), strands, and dangling ends—by explicitly incorporating bending rigidity at the scale of the entanglement length.

In the extended PCN model, each strand between two consecutive nodes is treated as a Gaussian subchain consisting of four virtual Kuhn segments (length 5 in reduced units). Bending rigidity is introduced through an angular potential U_bend = κ(1 – cos θ) acting on the angle θ between adjacent strand vectors; κ is the tunable bending parameter denoted by “>”. The equations of motion for node positions and dangling ends include drag, tension, osmotic, and stochastic forces, while the sliding of chains through slip‑links is governed by a one‑dimensional force balance that does not contain a bending term (because bending does not generate a longitudinal sliding force). Network topology changes (creation/destruction of entanglements) are handled via a Monte‑Carlo acceptance rule based on the Boltzmann factor.

Simulations were performed in a cubic box (size 10³) with a fixed strand density ν = 10. The authors varied the bending parameter (> = 0, 0.1, 0.2, 0.3, 0.4, 0.6, 1.0) and the average number of entangled strands per chain k (≈ 20, corresponding to different molecular weights). For each condition eight independent runs were carried out to obtain reliable statistics. After equilibration, the stress autocorrelation function was recorded and the linear relaxation modulus G(t) was computed via the Green‑Kubo relation. The plateau modulus G₀ was defined as the time‑average of G(t) over the flat region preceding terminal relaxation.

Structural analysis shows that increasing > leads to pronounced chain swelling: the mean square end‑to‑end distance Rₑ grows with both k and >. Using the worm‑like chain expression Rₑ² = 2LₚL_c