  Chiral rodlike colloids exposed to strong depletion attraction may
self-assemble into chiral membranes whose twisted director field differs from
that of a 3D bulk chiral nematic. We formulate a simple microscopic variational
theory to determine the elastic moduli of rods assembled into a bi-dimensional
smectic membrane. The approach is based on a simple Onsager-Straley theory for
a non-uniform director field that we apply to describe rod twist within the
membrane. A microscopic approach enables a detailed estimate of the individual
Frank elastic moduli (splay, twist and bend) as well as the twist penetration
depth of the smectic membrane in relation to the rod density and shape. We find
that the elastic moduli are distinctly different from those of a bulk nematic
fluid, with the splay elasticity being much larger and the curvature elasticity
much smaller than for rods assembled in a three-dimensional nematic fluid. We
argue that the use of the simplistic one-constant approximation in which all
moduli are assumed to be of equal magnitude is not appropriate for modelling
the structure-property relation of smectic membranes.
