Abstract
Using a combination of theory and experiments we study the interface between
two immiscible domains in a colloidal membrane composed of rigid rods of
different lengths. Geometric considerations of rigid rod packing imply that a
domain of sufficiently short rods in a background membrane of long rods is more
susceptible to twist than the inverse structure, a long-rod domain in a
short-rod membrane background. The tilt at the inter-domain edge forces splay,
which in turn manifests as a spontaneous edge curvature whose energetics are
controlled by the length asymmetry of constituent rods. A thermodynamic model
of such tilt-curvature coupling at inter-domain edges explains a number of
experimental observations, including a non-monotonic dependence of the edge
twist on the domain radius, and annularly shaped domains of long rods. Our work
shows how coupling between orientational and compositional degrees of freedom
in two-dimensional fluids give rise to complex shapes and thermodynamics of
domains, analogous to shape transitions in 3D fluid vesicles.