Hard particle self-assembly from the perspective of geometric frustration

It is well established that the appearance and properties of self-assembled structures are affected by the geometry of their constituents. This is especially true for hard polyhedrally shaped particles, which interact solely via excluded volume to form a plethora of entropically stabilized crystal structures. Yet, a priori prediction of these structures is non-trivial for anything but the simplest of space-filling shapes, such as cubes, especially when the thermodynamically preferred structure differs from the densest packing structure. By sufficiently curving space, however, we can eliminate the geometric constraints that prevent polyhedra from forming locally dense packings and theoretically create tessellations for all regular polyhedra. Using Monte Carlo simulations, we show that most hard polyhedra belonging to the family of Platonic solids can self-assemble into space-filling crystal structures when constrained to the surface of a hypersphere. By increasing the hypersphere radius to gradually flatten space, we introduce geometric frustration that prevents the particles from tessellating the hypersphere, and inevitably introduces defects.  Lastly, we compare systems assembled in curved and flat space by applying different local environment metrics and show that all the observed assemblies of Platonic shapes in Euclidean space can be interpreted as shadows of tessellations and defects on the hypersphere.