Propagation and Relaxation of Tension in Stiff Polymers

We present a unified theory for the longitudinal dynamic response of a stiff polymer in solution to various external perturbations (mechanical excitations, hydrodynamic flows, electrical fields, temperature quenches, etc.) that can be represented as sudden changes of ambient/boundary conditions. The theory relies on a comprehensive analysis of the nonequilibrium propagation and relaxation of backbone stresses in a wormlike chain. We recover and substantially extend previous results based on heuristic arguments. New experimental implications are pointed out.

Figure 1

Pulling (schematic): In response to an external force $\mathfrak{f}$, the thermally undulated contour $\mathbf{r}(s)=({\mathbf{r}}_{\perp },s-r_{\parallel }{)}^T$ is straightened within boundary layers of growing width ${\ell }_{\parallel }(t)$.

Figure 2

Double-logarithmic sketch of the tension propagation laws ${\ell }_{\parallel }(t)\propto t^z$. At $t_{\mathfrak{f}}={\mathfrak{f}}^{-2}$ they cross over from a universal short-time regime to (problem-specific) tension-dominated intermediate asymptotics, except for weak forces, $\mathfrak{f}<{\ell }_p^2/L^4$, and for ${\ell }_p$ quenches. The propagation ends when ${\ell }_{\parallel }(t)\approx L$.

Figure 3

Characteristic times (logarithmic scale) for pulling and release against the applied external force $\mathfrak{f}$ (linear scale). The time $t_{\star }$ (stars) separates regions where ordinary perturbation theory (OPT) applies (dark shaded) from regions (light shaded) of linear (hatched) and nonlinear tension propagation and from homogeneous tension relaxation (white). Whereas longitudinal friction is negligible for $t>t_{\star }$, it limits the dynamics for $t<t_{\star }$.