Hierarchical self-assembly of simple hard polyhedra into complex mesophases.

Nature offers many intriguing examples of hierarchically self-assembled mesophases, such as lamellar, gyroid, hexagonal, and cholesteric phases. These structures are typically believed to emerge from complex, competing enthalpic interactions, as observed in block copolymers and amphiphilic surfactants. Here, using extensive Monte Carlo simulations, we demonstrate that even simple achiral hard particles with distorted tetrahedral shapes and purely excluded-volume interactions can spontaneously self-assemble into a diverse range of mesophases and liquid crystal phases, including the unexpected emergence of chiral structures. We attribute the formation of these phases to geometric frustration in the orientational ordering of neighboring particles, induced by their particle shape. The system resolves this frustration by coupling it with an energetically less favorable elastic deformation mode in the orientational ordering, such as twist or splay. We show that simple shape descriptors, such as anisotropy or biaxiality, predict the self-assembly behavior: rod-like particles stabilize cholesteric and twisted lamellar phases, plate-like particles form biaxial and splay nematic phases with randomly distributed splay domains as well as hexagonal cylindrical phases, while moderately anisotropic particles favor gyroid phases. This framework provides valuable insights for designing mesophases in supramolecular chemistry, liquid crystals, colloid science, and nanoparticle assembly.