  A phenomenological Landau elasticity for the shape, dilation, and lipid-tilt
of bilayer membranes is developed. The shape mode couples with the sum of the
monolayers' tilt, while the dilation mode couples with the difference of the
monolayers' tilts. Interactions among membrane inclusions within regular arrays
are discussed. Inclusions modifying the membrane thickness and/or inducing a
tilt-difference due to their convex or concave shape yield a dilation-induced
attraction and a tilt-difference-induced repulsion. The resulting interaction
can stabilize 2D crystal phases, with the possible coexistence of different
lattice spacings when the dilation-tilt-difference coupling is large.
Inclusions favoring crystals are those with either a long-convex or a
short-concave hydrophobic core. Inclusions inducing a local membrane curvature
due to their conical shape repel one another. At short inclusions separations,
a tilt comparable with the inclusion's cone angle develops: it relaxes the
membrane curvature and reduces the repulsion. At large separations the tilt
vanishes, whatever the value of the shape-tilt coupling.
