  Motivated by experiments on colloidal membranes composed of chiral rod-like
viruses, we use Monte Carlo methods to determine the phase diagram for the
liquid crystalline order of the rods and the membrane shape. We generalize the
Lebwohl-Lasher model for a nematic with a chiral coupling to a curved surface
with edge tension and a resistance to bending, and include an energy cost for
tilting of the rods relative to the local membrane normal. The membrane is
represented by a triangular mesh of hard beads joined by bonds, where each bead
is decorated by a director. The beads can move, the bonds can reconnect and the
directors can rotate at each Monte Carlo step. When the cost of tilt is small,
the membrane tends to be flat, with the rods only twisting near the edge for
low chiral coupling, and remaining parallel to the normal in the interior of
the membrane. At high chiral coupling, the rods twist everywhere, forming a
cholesteric state. When the cost of tilt is large, the emergence of the
cholesteric state at high values of the chiral coupling is accompanied by the
bending of the membrane into a saddle shape. Increasing the edge tension tends
to flatten the membrane. These results illustrate the geometric frustration
arising from the inability of a surface normal to have twist.
