Formation of tiling patterns in hierarchical structures through self-assembly of ABCD miktoarm star quaterpolymers

Block copolymers are a key class of soft matter, and their ability to form complex nanostructures is dictated by molecular architecture and block composition. In this study, we investigate the self-assembled structures of ABCD miktoarm star quaterpolymers using simulated annealing. By systematically tuning the volume fraction of the longest arm (D-arm), f_D, while keeping the ratio f_A : f_B : f_C fixed at six representative values, we construct phase diagrams revealing 15 distinct three-dimensional hierarchical ordered microstructures, 14 of which are reported for the first time. At high f_D (∼0.6), we identify six LS₃ structures (an alternating packing of monolayers of "three-color" spheres arranged in tiling patterns and D-lamellae). As f_C/(f_A + f_B) increases, the coordination number of each C-domain increases from 4 to 6 to 8 to 10 and further to 12 in the monolayer. At lower f_D, six CS₃ and three C₂S₂ structures (composed of "three-color" or "two-color" spherical and "one-color" or "two-color" cylindrical domains) form, exhibiting intricate tiling patterns in the cross-sections of the cylinders and on the curved surfaces of each cylinder in the CS₃ structure. At f_A : f_B : f_C = 1 : 1 : 1, 1 : 1 : 0.5 and 1 : 1 : 2, the patterns undergo sequential transitions of [10.6.4; 10.6.4; 10.6.6] → [8.6.4; 8.6.6] → [6.6.6], [10.6.4; 10.6.4; 10.6.6] → [6.6.6] + [12.6.4] → [8.8.4] and [12.6.4]→ [3.3.4.3.4] → [8.8.4], respectively, with decreasing f_D, where the notation [k.l.m] has the same meaning as that used in the miktoarm star terpolymer systems, except that some cylinders are composed of an alternating arrangement of A-, B- and C-spheres or A- and B-spheres. We elucidate the formation mechanisms of the observed structures based on composition-geometry relationships. This work advances the understanding of self-assembly in multi-arm star copolymers and provides foundational design principles for engineering hierarchical nanomaterials with programmable tiling geometries.