  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 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 ribbon-like perturbation to
the protrusion producd by stretching and determine whether it is energetically
favorable as a function of the size of the protrusion.
