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Neutral versus charged defect patterns in curved crystals

Characterizing the complex spectrum of topological defects in ground states of curved crystals is a long-standing problem with wide implications, from the mathematical Thomson problem to diverse physical realizations, including fullerenes and particle-coated droplets. While the excess number of "topologically-charged" 5-fold disclinations in a closed, spherical crystal is fixed, here, we study the elementary transition from defect-free, flat crystals to curved-crystals possessing an excess of "charged" disclinations in their bulk. Specifically, we consider the impact of topologically-neutral patterns of defects -- in the form of multi-dislocation chains or "scars" stable for small lattice spacing -- on the transition from neutral to charged ground-state patterns of a crystalline cap bound to a spherical surface. Based on the asymptotic theory of caps in continuum limit of vanishing lattice spacing, we derive the morphological phase diagram of ground state defect patterns, spanned by surface coverage of the sphere and forces at the cap edge. For the singular limit of zero edge forces, we find that scars reduce (by half) the threshold surface coverage for excess disclinations. Even more significant, scars flatten the geometric dependence of excess disinclination number on Gaussian curvature, leading to a transition between stable "charged" and "neutral" patterns that is, instead, critically sensitive to the compressive vs. tensile nature of boundary forces on the cap.