Frustrated Self-Assembly of Hyperbolic Crystals from Icosahedral Nanoparticles

Simple building blocks such as polyhedral nanoparticles (NPs) can self-organize into complex assemblies that resemble structures observed in biology. Previous study showed that the complex order of tetrahedral NP assemblies can be understood based on their crystal structures on the 3-sphere. Icosahedra, however, cannot tile the 3-sphere and is beyond this theory. In this talk, we present an analytical theory of icosahedral NP self-assembly based on the non-Euclidean crystal {3,5,3} in 3-hyperbolic space. Following the assumption that geometric frustration will favor low dimensional morphology of assembled clusters, we show that the local order of the self-assembly is determined by the crystal order on a flat surface in 3-hyperbolic space and icosahedral NPs self-organized into part of cylindrical or spherical shells.