Predicting the onset of disclination defects on curved open surfaces

Ordered structures on surfaces with Gaussian curvature are topologically constrained to contain a finite defect charge, while the positioning of these defects is determined by optimization of the surface's elastic energy. Open surfaces of sufficiently small curvature minimize energy without defects in the bulk, while stronger curvature favors the appearance of a disclination in the ground state; the presence of a boundary adds subtlety to the local screening of Gaussian curvature by defects. We find that, in contrast to previous heuristic arguments, the onset of transition is governed neither by local values of Gaussian curvature nor by its global integral. Starting from stringent energy minimization, we instead propose a weighted integral Gaussian curvature as an improved predictor for the transition - one that is universally valid for a large class of bounded rotationally symmetric surfaces. Our formalism also allows for analysis of the first or second order character of the transition, and how it is modified by boundary stresses and/or breaking symmetries of shape or material properties. These findings are of practical and fundamental importance in both engineering and biological cellular systems, such as the pattern of visual processing in invertebrate eyes.