  Colloidal membranes are monolayer assemblies of rodlike particles that
capture the long-wavelength properties of lipid bilayer membranes on the
colloidal scale. Recent experiments on colloidal membranes formed by chiral
rodlike viruses showed that introducing a second species of virus with
different length and opposite chirality leads to the formation of rafts ---
micron-sized domains of one virus species floating in a background of the other
viruses [Sharma et al., Nature 513, 77 (2014)]. In this article we study the
interaction of such rafts using liquid crystal elasticity theory. By
numerically minimizing the director elastic free energy, we predict the tilt
angle profile for both a single raft and two rafts in a background membrane,
and the interaction between two rafts as a function of their separation. We
find that the chiral penetration depth in the background membrane sets the
scale for the range of the interaction. We compare our results with the
experimental data and find good agreement for the strength and range of the
interaction. Unlike the experiments, however, we do not observe a complete
collapse of the data when rescaled by the tilt angle at the raft edge.
