Instability of flat disks with respect to the formation of twisted ribbons in smectic-A*monolayers

Abstract

Smectic-A* monolayers self-assembled from aqueous solutions of chiral fd viruses and a polymer depletant can assume a variety of shapes such as flat disks and twisted ribbons. A first order phase transition from a flat disk to a ribbon occurs upon lowering the concentration of polymer depletant or the temperature. A theoretical model based on the de Gennes model for the smectic-A phase, the Helfrich model of membrane elasticity, and a simple edge energy has been previously used to calculate the disk-ribbon phase diagram. In this paper we apply this model to the nucleation process of ribbons. First, we study the "rippled disks" that have been observed as precursors of ribbons. Using a model shape proposed by Meyer which includes rippling in both the in-plane and out-of-plane directions, we study the energetics of the disks as functions of the edge energy modulus (a measure of the polymer concentration) and the mean curvature modulus k. We find that as the edge energy modulus is reduced the radial size of the ripples grows rapidly in agreement with experimental observations. For small enough k we find that the out-of-plane size of the ripples grows but its value saturates at a fraction of the twist penetration depth, too small to be experimentally observable. For large k the membrane remains flat though rippled in the radial direction. Such membranes do not have negative Gaussian curvature and thus will not likely spawn twisted ribbons. We also study the creation of twisted ribbons produced by stretching the edge of a flat membrane in a localized region. In experiments using a pair of optical traps it has been observed that once the membrane has been sufficiently stretched a ribbon forms on the stretched edge. We study this process theoretically using a free energy consisting of the Helfrich and edge energies alone. We add a small ribbonlike perturbation to the protrusion produced by stretching and determine whether it is energetically favorable as a function of the size of the protrusion. In qualitative agreement with experiment we find a nonzero value for the critical size of the protrusion needed to make a ribbon energetically favorable, though the value we find is an order of magnitude lower than the experimental value possibly due to our neglect of the director field. As in the case of the rippled disks we find that the mean curvature energy acts as a barrier between the disk and twisted ribbon structures.

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“…Many theoretical approaches have been proposed. A de Gennes framework accompanied by appropriate surface energy terms was used to characterize colloidal membranes, twisted ribbons and π-wall [160,197,198,[223][224][225][226]. Sakhadande et al constructed on the free volume theory for depletant-rod interactions, and a third order virial expansion for rodrod interactions, with the equation of state for a hard disk system to constrain the areal rod density to study 2D colloidal membranes composed of binary mixture of rods with opposing chiralities [228].…”

“…Many theoretical approaches have been proposed. A de Gennes framework accompanied by appropriate surface energy terms was used to characterize the colloidal membranes, twisted ribbons and π-wall [159,196,197,[222][223][224][225]. Sakhadande et al adopted a continuum Ginzburg-Landau theory to study raft stability [226].…”