Entanglement of Semiflexiible Polymers: A Brownian Dynamics Study

We report extensive Brownian dynamics simulations of very tightly entangled solutions of semiflexible rods, of length $L$ comparable to their persistence length $L_{p}$, at concentrations comparable to those in recent experiments on Fd-virus and filamentous actin. We find a clear crossover with increasing number concentration $c$ from a regime of loosely entangled rods, in which rotational diffusion is hindered by topological constraints but transverse bending fluctuations are not, to a tightly entangled regime in which bending fluctuations are also restricted, and can relax only by reptation along a wormlike tube. This crossover occurs at a dimensionless concentration $c^{**}L^{3} \sim 500$ for chains with $L = L_{p}$. The tube radius $R_{e}$ is found to depend upon $c$ and $L_{p}$ with the predicted scaling relation $R_{e}\propto c^{-3/5} L_{p}^{-1/5}$ for $c > c^{**}$. The dynamic modulus $G(t)$ has been obtained from simulations of the relaxation of stress after a small amplitude step extension of the simulation unit cell. An elastic plateau in $G(t)$ that is absent at lower concentrations also appears for $c \geq c^{**}$.