Anisotropic Random Networks of Semiflexible Polymers

Motivated by the organization of cross-linked cytoskeletal biopolymers, we present a semimicroscopic replica field theory for the formation of anisotropic random networks of semiflexible polymers. The networks are formed by introducing random permanent cross-links which fix the orientations of the corresponding polymer segments to align with one another. Upon increasing the cross-link density, we obtain a continuous gelation transition from a fluid phase to a gel where a finite fraction of the system gets localized at random positions. For sufficiently stiff polymers, this positional localization is accompanied by a continuous isotropic-to-nematic ($IN$) transition occurring at the same cross-link density. As the polymer stiffness decreases, the $IN$ transition becomes first order, shifts to a higher cross-link density, and is preceded by an amorphous solid where the average polymer orientations freeze in random directions.

Figure 1

Cross-linker aligning two wormlike chains at arclengths $s$ and $s^{\prime }$.

Figure 2

The first panel shows the schematic phase diagram. The other three panels show the free energy per WLC (in units of $k_{B}T$) as a function of $\eta$ for $l=0.001$ (second), $l=0.15$ (third), and $l=0.4$ (fourth) at ${\mu }^2={\mu }_c^2=1$.