Defects in flexible membranes with crystalline order

We study isolated dislocations and disclinations in flexible membranes with internal crystalline order, using continuum elasticity theory and zero-temperature numerical simulation. These defects are relevant, for instance, to lipid bilayers in vesicles or in the ${L}_{\ensuremath{\beta}}$ phase of lyotropic smectic liquid crystals. We first simulate defects in flat membranes, obtaining numerical results in good agreement with plane elasticity theory. Disclinations and dislocations eventually exhibit a buckling transition with increasing membrane radius. We generalize the continuum theory to include such buckled defects, and solve the disclination equations in the inextensional limit. The critical radius at which buckling starts to screen out internal elastic stresses is determined numerically. Computer simulation of buckled defects confirms predictions of the disclination energies and gives evidence for a finite dislocation energy.