Orientational order and glassy states in networks of semiflexible polymers

Motivated by the structure of networks of cross-linked cytoskeletal biopolymers, we study orientationally ordered phases in two-dimensional networks of randomly cross-linked semiflexible polymers. We consider permanent cross-links which prescribe a finite angle and treat them as quenched disorder in a semimicroscopic replica field theory. Starting from a fluid of un-cross-linked polymers and small polymer clusters (sol) and increasing the cross-link density, a continuous gelation transition occurs. In the resulting gel, the semiflexible chains either display long-range orientational order or are frozen in random directions depending on the value of the crossing angle, the cross-link concentration, and the stiffness of the polymers. A crossing angle θ̃2π/M leads to long-range M-fold orientational order, for example, "hexatic" or "tetratic" for θ=60° or 90°, respectively. The transition to the orientationally ordered state is discontinuous and the critical cross-link density, which is higher than that of the gelation transition, depends on the bending stiffness of the polymers and the cross-link angle: The higher the stiffness and the lower the M, the lower is the critical number of cross-links. In between the sol and the long-range ordered state, we always expect a gel which is a statistically isotropic amorphous solid with random positional and random orientational localization of the participating polymers.