ABSTRACT

Unlike periodic patterns and crystals, which have rotation and translation symmetries, quasiperiodic patterns observed in Faraday waves and quasicrystals in solid-state and soft-matter systems lack translation symmetry while possessing long range order. I investigate the formation and stability of icosahedral quasicrystalline structures using a dynamic phase field crystal model. Nonlinear interactions between density waves at two length scales stabilize three-dimensional quasicrystals. I determine the phase diagram and parameter values required for the quasicrystal to be the global minimum free energy state. I demonstrate that traits that promote the formation of two-dimensional quasicrystals are extant in three dimensions, and highlight the characteristics required for three-dimensional soft matter quasicrystal formation.