Wrinkling crystallography on spherical surfaces

Significance Curved crystals cannot comprise hexagons alone; additional defects are required by both topology and energetics that depend on the system size. These constraints are present in systems as diverse as virus capsules, soccer balls, and geodesic domes. In this paper, we study the structure of defects of the crystalline dimpled patterns that self-organize through curved wrinkling on a thin elastic shell bound to a compliant substrate. The dimples are treated as point-like packing units, even if the shell is a continuum. Our results provide quantitative evidence that our macroscopic wrinkling system can be mapped into and described within the framework of curved crystallography, albeit with some important differences attributed to the far-from-equilibrium nature of our patterns. We present the results of an experimental investigation on the crystallography of the dimpled patterns obtained through wrinkling of a curved elastic system. Our macroscopic samples comprise a thin hemispherical shell bound to an equally curved compliant substrate. Under compression, a crystalline pattern of dimples self-organizes on the surface of the shell. Stresses are relaxed by both out-of-surface buckling and the emergence of defects in the quasi-hexagonal pattern. Three-dimensional scanning is used to digitize the topography. Regarding the dimples as point-like packing units produces spherical Voronoi tessellations with cells that are polydisperse and distorted, away from their regular shapes. We analyze the structure of crystalline defects, as a function of system size. Disclinations are observed and, above a threshold value, dislocations proliferate rapidly with system size. Our samples exhibit striking similarities with other curved crystals of charged particles and colloids. Differences are also found and attributed to the far-from-equilibrium nature of our patterns due to the random and initially frozen material imperfections which act as nucleation points, the presence of a physical boundary which represents an additional source of stress, and the inability of dimples to rearrange during crystallization. Even if we do not have access to the exact form of the interdimple interaction, our experiments suggest a broader generality of previous results of curved crystallography and their robustness on the details of the interaction potential. Furthermore, our findings open the door to future studies on curved crystals far from equilibrium.