Universal collapse of stress and wrinkle-to-scar transition in spherically confined crystalline sheets

Imposing curvature on crystalline sheets, such as 2D packings of colloids or proteins, or covalently bonded graphene leads to distinct types of structural instabilities. The first type involves the proliferation of localized defects that disrupt the crystalline order without affecting the imposed shape, whereas the second type consists of elastic modes, such as wrinkles and crumples, which deform the shape and also are common in amorphous polymer sheets. Here, we propose a profound link between these types of patterns, encapsulated in a universal, compression-free stress field, which is determined solely by the macroscale confining conditions. This stress universality principle and a few of its immediate consequences are borne out by studying a circular crystalline patch bound to a deformable spherical substrate, in which the two distinct patterns become, respectively, radial chains of dislocations (called scars) and radial wrinkles. The simplicity of this set-up allows us to characterize the morphologies and evaluate the energies of both patterns, from which we construct a phase diagram that predicts a wrinkle-scar transition in confined crystalline sheets at a critical value of the substrate stiffness. The construction of a unified theoretical framework that bridges inelastic crystalline defects and elastic deformations opens unique research directions. Beyond the potential use of this concept for finding energy-optimizing packings in curved topographies, the possibility of transforming defects into shape deformations that retain the crystalline structure may be valuable for a broad range of material applications, such as manipulations of graphene's electronic structure.