Theory of Chiral Smectic A Twisted Ribbons @ Brandeis University

Author(s):

C. N. Kaplan¹, H. Tu², R. A. Pelcovits² and R. B. Meyer¹

¹ The Martin Fisher School of Physics, Brandeis University
² Department of Physics, Brown University

 Rodlike fd viruses in an aqueous solution, along with polymers that produce an attractive force between the virus particles, have been observed in laboratory experiments to self-assemble into a variety of geometric structures including twisted ribbons (see the schematic illustration and microscope picture to the right). We have developed a theoretical model which explains the properties of these ribbons on the basis of very general features of the fd rods. The theory yields predictions in good agreement with experiment, namely, (a) a phase diagram with a first-order transition from flat membranes to twisted ribbons, (b) the ratio of the ribbon's pitch to width, (c) the tilt angle of the rods at the edge of the ribbon.  The theory has also demonstrated the importance of molecular chirality ("twisting-handedness") in the formation of the ribbons, as well as the tendency of fd rods to assemble into structures with negative Gaussian curvature (as in a saddle shape).

We have studied a theoretical model for the chiral smectic A twisted ribbons observed in assemblies of fd viruses condensed by depletion forces. The depletion interaction is modeled by an edge energy assumed to be proportional to the depletant polymer concentration in solution. Our model is based on the Helfrich energy for surface bending and the de Gennes model of chiral smectic A liquid crystals with twist penetration at the edge.  We have considered two variants of this model, one with the conventional Helfrich Gaussian curvature term, and a second with saddle-splay energy. A mean field analysis of both models yields a first-order phase transition between ribbons and semi-infinite flat membranes as the edge energy is varied. The phase transition line and tilt angle profile are found to be nearly identical for the two models; the pitch of the ribbon, however, does show some differences. Our model yields good qualitative agreement with experimental observations, namely, (1) the existence of a first-order phase transition between ribbons and semi-infinite membranes, preempting a transition to finite-sized disks; (2) the dependence of the tilt angle at the edge of the ribbon on the curvature of the edge; (3) the decrease in the tilt angle at the edge of the membrane as the system undergoes the first-order transition from the semi-infinite membrane to the ribbon; (4) the order of magnitude of the pitch to width of the ribbon.