A mesoscopic tiling pattern with 12-fold symmetry has been observed in a three-component polymer system composed of polyisoprene, polystyrene, and poly(2-vinylpyridine) which forms a star-shaped terpolymer, and a polystyrene homopolymer blend. Transmission electron microscopy images reveal a nonperiodic tiling pattern covered with equilateral triangles and squares, their triangle/square number ratio of 2.3 (approximately equal to 4/sqrt[3]), and a microbeam x-ray diffraction pattern shows dodecagonal symmetry. The same kind of quasicrystalline structures have been found for metal alloys (approximately 0.5 nm), chalcogenides (approximately 2 nm), and liquid crystals (approximately 10 nm). The present result (approximately 50 nm) confirms the universal nature of dodecagonal quasicrystals over several hierarchical length scales.

New tiling patterns with complex arrangements have been identified for ABC star-shaped terpolymers and ABC/A/B copolymer/homopolymer blends characterized by transmission electron microscopy and microbeam small-angle X-ray scattering. The samples are composed of polyisoprene (I), polystyrene (S), and poly(2-vinylpyridine) (P), whose volume ratios of I:S:P are 1.0:1.8:X, where X covers the range 0.8 ≤ X ≤ 3.2. The other series of the type I1.0S Y P2.0 with Y in the range of 1.1 ≤ Y ≤ 2.7 was also prepared by adding I or S homopolymers to the terpolymers of the type I1.0S1.8P X . The systematic phase transition takes place in both series when the X or Y variables are increased and four new tiling patterns have been observed:  [5.3, 5.3, 8], [4.5, 6, 9], [5, 5, 10], and [4, 6.7, 10], where the three numbers in the brackets denote the average coordination number (ACN) for the I, S, and P components, respectively. A very simple transition rule has been established:  the ACN of the ascendant component increases monotonically with an increase of the variable X or Y, accompanied by the monotonic decrease in the ACN for the descendant component, which is polyisoprene in these cases.

Characteristic cylindrical structures formed by ABC star-shaped terpolymers were investigated by microbeam small-angle X-ray scattering (SAXS) in addition to transmission electron microscopy (TEM). The polymer samples are composed of polyisoprene (I), polystyrene (S), and poly(2-vinylpyridine) (P); their volume ratios for I:S:P are 1:1:X, where X equals 0.7, 1.2, 1.3, and 1.9. The spotlike diffraction patterns were observed by microbeam SAXS due to scattering from a small number of ordered grains in the polymer samples, where the exact packing manners of the cylinders, or their lattice constants, have been clarified. In short, it has been found that the cross-sectional patterns of these tilings have the features of four Archimedean tiling patterns, i.e., (6.6.6), (4.8.8), (3.3.4.3.4), and (4.6.12). All the four SAXS patterns are quite consistent with the structural observation by TEM with regard to the crystallographic data.