Entangled dynamics of a stiff polymer

Entangled networks of stiff biopolymers exhibit complex dynamic response, emerging from the topological constraints that neighboring filaments impose upon each other. We propose a class of reference models for entanglement dynamics of stiff polymers and provide a quantitative foundation of the tube concept for stiff polymers. For an infinitely thin needle exploring a planar course of point obstacles, we have performed large-scale computer simulations proving the conjectured scaling relations from the fast transverse equilibration to the slowest process of orientational relaxation. We determine the rotational diffusion coefficient of the tracer, its angular confinement, the tube diameter, and the orientational correlation functions.

Figure 1

(Color online) Illustration of an entangled needle in a plane. The relevant length scales are the length of the needle $L$ and the mesh size of the network $\xi$. The surrounding point obstacles confine the needle to a tube (shaded areas) of width $d$, a renewed tube (green online) is tilted against the old one (red online) by an angle $ϵ=d∕L$.

Figure 2

(Color online) Simulation results for the rotational diffusion coefficient. (a) Dashed lines are asymptotic fits to the predicted Doi-Edwards scaling, $D_{\mathrm{rot}}^{\infty }=A(n^{*}{)}^{-2}$; the top (red) solid line shows the result from a Boltzmann theory for ballistic dynamics, $D_{\mathrm{rot}}=10.5∕n^{*}{\tau }_0$ [24]. (b) Deviation from the asymptotic behavior; note the different prefactors $A$ in $D_{\mathrm{rot}}^{\infty }$ for ballistic and overdamped dynamics.

Figure 3

(Color online) Typical trajectories of the center of the needle for overdamped dynamics. With increasing entanglement, the needle is confined to a narrowing tube, and reptation dynamics emerges; see also the supplementary movie [25].

Figure 4

(Color online) Time dependence of the rotational motion. (a) The mean-square angular displacement of the needle develops a plateau at high densities, which scales as ${ϵ}^2\sim (n^{*}{)}^{-2}$ (cf. inset). The arrow indicates the disengagement time ${\tau }_{\mathrm{d}}$. (b) The exponential relaxation of the orientation correlation $C_1\left(t\right)$ is preceded by a plateau very close to unity (see inset for a logarithmic scale). Symbols indicate simulation results and solid lines fits to $f\exp (-t∕{\tau }_{\mathrm{rot}})$; the fitted plateau value is consistent with the measured angular confinement, $f=1-{ϵ}^2∕2$.