Spherical crystals are elementary models of geometric frustration in materials, with important realizations in a range of systems from viral shells and fullerenes to particle- and molecular-coated droplets. Using continuum elasticity theory, we study the structure and elastic energy of ground states of crystalline caps conforming to a spherical surface. We find that the ground states consist of positive disclination defects and that the ground states with icosahedral subgroup symmetries in caps arise across a range of curvatures, even far from the closure point of complete shells. Furthermore, we use Monte Carlo simulations to investigate the kinetic pathway of the formation of viral shells (capsids) and find that the key to the formation of perfect icosahedral capsids is in the strength of elastic energy compared to the protein-protein interactions and the chemical potential of free subunits.
At the end of the talk, I discuss our efforts to understand the formation of SARS-CoV-2 particles in their host cells. In contrast to icosahedral viruses, the structures of coronaviruses are heterogeneous both in morphology and size, significantly complicating any theory of their formation.