Quasicrystalline long-range order in an ABC star block copolymer

We report the formation of a dodecagonal quasicrystal (DDQC) in a lattice Monte Carlo simulation of a star-shaped three component polymeric alloy. We have observed a series of Archimedean and quasicrystalline phases $(4.8^2) \rightarrow (3^2.4.3.4) \rightarrow {\rm DDQC} \rightarrow (4.6.12)$ with increase of one component of ABC star polymers. This phase behavior can be regarded as a transition from square tiling to triangle tiling via square-triangle tiling. The simulation is associated with the recent striking experimental manifestation of quasicrystalline order: A mesoscopic tiling pattern with twelvefold symmetry in a three-component star polymer system composed of polyisoprene, polystyrene, and poly (2-vinylpyridine). Since, the same kind of quasicrystalline structures have been found for metal alloys, chalcogenides, and liquid crystals, the present result confirms the universal nature of quasicrystalline long-range order over several hierarchical length scales. \\[4pt] T. Dotera and T. Gemma, Philos. Mag. {\bf 86}, 1085 (2006).\\[0pt] T. Dotera, Phil. Mag. {\bf 88}, 2245 (2008). \\[0pt] K. Hayashida, et al., Phys. Rev. Lett. {\bf 98}, 195502 (2007).