Since the discovery of quasicrystals (QCs) in metallic systems in 1982, these ordered yet aperiodic phases have been observed in a variety of soft matter systems, including block copolymers, giant surfactants, and star copolymers. Due to their unique properties, QCs have potential applications in photonic materials, surface coatings, and membrane design. Despite the realization of QCs in soft matter, the underlying thermodynamic mechanisms remain poorly understood, and the controlled assembly of quasicrystals continues to be a challenge. Here, we adapt the vertex model to study QC formation in the 2D assembly of bidisperse deformable particles. In our model, the network of polygons represents the arrangement of particles, and the total energy of the configuration is determined by the deviations in the polygon's area and perimeter from their preferred values. First, we determine the phase boundary between the dodecagonal QC approximant sigma phase and the disordered phase in our parameter space. Near the phase boundary, we observe particles rearranging into distinct local environments, whose coexistence has been perceived as a precursor for the formation of QCs in many cases. Furthermore, we are developing quenching strategies to tune the relative fractions of these environments. Our goal is to understand the role of these local environments in the nucleation and growth of QCs, as well as in the defects present in the equilibrium assembly.
