Defective ground states of toroidal crystals

Crystalline assemblages of identical subunits packed together and elastically bent in the form of a torus have been found in the past ten years in a variety of systems of surprisingly different nature, such as viral capsids, self-assembled monolayers, and carbon nanomaterials. In this Rapid Communication we analyze the structural properties of toroidal crystals and provide a unified description based on the elastic theory of defects in curved geometries. We find ground states characterized by the presence of fivefold disclinations on the exterior of the torus and sevenfold disclinations in the interior. The number of excess disclinations is controlled primarily by the aspect ratio of the torus, suggesting a mechanism for creating toroidal templates with precisely controlled valency via functionalization of the defect sites.