Formation of Tiling Patterns in Hierarchical Structures through Self-Assembly of ABCD Miktoarm Star Quaterpolymers

Abstract

Block copolymers are a key class of soft matter, where phase behavior and the ability to form complex nanostructures are 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), fD, while keeping the ratio fA:fB:fC 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 fD (~0.6), we identify six LS3 structures ("three-color" spheres arranged in tiling patterns sandwiched between D-lamellae). The patterns evolve as [8.8.4]→ [6.6.6]→ [8.6.4;8.6.6]→ [10.6.4;10.6.4;10.6.6]→ [3.3.4.3.4]→ [12.6.4] with increasing fC/(fA+fB). As fD decreases, six CS3 and three C2S2 structures form. These structures are composed of "three-color" or "two-color" spherical and "one-color" or "two-color" cylindrical domains, exhibiting intricate tiling patterns both in the cross-sections and on the curved surfaces of the cylinders. In CS3 structures formed at fA:fB:fC =1:1:1 and 1:1:0.5, the pattern undergoes sequential transitions of [10.6.4;10.6.4;10.6.6]→ [8.6.4;8.6.6]→ [6.6.6] and [10.6.4;10.6.4;10.6.6]→ [6.6.6]+[12.6.4]→ [8.8.4], respectively with decreasing fD. In the representative case with fA:fB:fC =1:1:2, patterns in C2S2 structures transition as [12.6.4]→ [3.3.4.3.4]→ [8.8.4] with decreasing fD. We elucidate the formation mechanisms of the observed structures based on composition-geometry relations. This work advances the understanding of self-assembly in multi-arm star block copolymers and provides foundational design principles for engineering hierarchical nanomaterials with programmable tiling geometries.