  Experiments on supported lipid bilayers featuring liquid ordered/disordered
domains have shown that the spatial arrangement of the lipid domains and their
chemical composition are strongly affected by the curvature of the substrate.
Furthermore, theoretical predictions suggest that both these effects are
intimately related with the closed topology of the bilayer. In this work, we
test this hypothesis by fabricating supported membranes consisting of colloidal
particles of various shapes lying on a flat substrate. A single lipid bilayer
coats both colloids and substrate, allowing local lipid exchange between them,
thus rendering the system thermodynamically open, i.e. able to exchange heat
and molecules with an external reservoir in the neighborhood of the colloid. By
reconstructing the Gibbs phase diagram for this system, we demonstrate that the
free-energy landscape is directly influenced by the geometry of the colloid. In
addition, we find that local lipid exchange enhances the pinning of the liquid
disordered phase in highly curved regions. This allows us to provide estimates
of the bending moduli difference of the domains. Finally, by combining
experimental and numerical data, we forecast the outcome of possible
experiments on catenoidal and conical necks and show that these geometries
could greatly improve the precision of the current estimates of the bending
moduli.
