Numerical simulations reveal a family of hierarchical and chiral multicontinuous network structures self-assembled from a melt blend of Y-shaped ABC and ABD three-miktoarm star terpolymers, constrained to have equal-sized A/B and C/D chains, respectively. The C and D majority domains within these patterns form a pair of chiral enantiomeric gyroid labyrinths (srs nets) over a broad range of compositions. The minority A and B components together define a hyperbolic film whose midsurface follows the gyroid minimal surface. A second level of assembly is found within the film, with the minority components also forming labyrinthine domains whose geometry and topology changes systematically as a function of composition. These smaller labyrinths are well described by a family of patterns that tile the hyperbolic plane by regular degree-three trees mapped onto the gyroid. The labyrinths within the gyroid film are densely packed and contain either graphitic hcb nets (chicken wire) or srs nets, forming convoluted intergrowths of multiple nets. Furthermore, each net is ideally a single chiral enantiomer, induced by the gyroid architecture. However, the numerical simulations result in defect-ridden achiral patterns, containing domains of either hand, due to the achiral terpolymeric starting molecules. These mesostructures are among the most topologically complex morphologies identified to date and represent an example of hierarchical ordering within a hyperbolic pattern, a unique mode of soft-matter self-assembly. chirality | liquid crystals | entanglement | hyperbolic tilings | miktoarm copolymers L iquid crystals formed by molecular self-assembly provide fascinating examples of complicated space partitions in softmaterial science. Relatively complex examples are the bicontinuous mesostructures found ubiquitously in both natural and synthetic soft matter, including lipid-water systems and block copolymer melts, namely the double diamond (symmetry Pn3m), the primitive ðIm3mÞ, and, particularly, the gyroid ðIa3dÞ mesophases. The structure of these mesophases can be described by a molecular membrane folded onto one of the three simplest triply periodic minimal surfaces (TPMS), namely the D, P, and G(yroid) surfaces, named by Schoen in the 1960s (1). From a 3D perspective, these structures are characterized by the nets describing the pair of mutually threaded labyrinths carved out of space by the convoluted hyperbolic architecture of the TPMS. For the gyroid, this is a racemic mixture of two chiral srs nets, one left-and the other right-handed [the three-letter nomenclature follows the Reticular Chemistry Structure Resource naming convention for 3D nets (2)]. This leads to an overall achiral structure when the two nets are chemically identical, which is the case in most experimentally identified gyroid liquidcrystal structures. One such structure recently reported is a gyroid assembly found in an ABC three-miktoarm star terpolymer melt (3). In this structure, the majority C component constitutes the two labyrinth nets while the...
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