  We consider 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 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 consider 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 if the sign of the Gaussian curvature or saddle-splay
modulus is chosen to favor negative Gaussian curvature.
