  Smectic-A monolayers self-assembled in aqueous solutions of chiral fd viruses
and a polymer depletant have been shown to exhibit a variety of structures
including large, flat disks and twisted ribbons. The virus particles twist near
the edge of the structure in a direction determined by the chirality of the
viruses. When fd viruses and their mutants of opposite chirality are mixed
together in nearly equal amounts unusual structures referred to as "crenellated
disks" can appear. These disks are achiral overall but the twist at the edge
alternates between left- and right-handedness. To minimize the mismatch where
the two regions of opposing twist meet, the "crenellated" structure exhibits
cusps rising out of the plane of the monolayer. We use a phenomenological
elastic theory previously applied to flat disks and twisted ribbons to analyze
an analytic model proposed to describe the "crenellated" structure . When
compared with flat, circular disks, we find that the model "crenellated disks"
are stable or at least metastable in a wide region of the phase diagram spanned
by the Gaussian curvature modulus and the edge energy modulus, with a large
energy barrier separating the two structures. The director pattern and
geometric parameters of the "crenellated disks" are found to be in qualitative
agreement with experimental observations.
