Hierarchical self-assembly of a striped gyroid formed by threaded chiral mesoscale networks

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.Liquid crystals formed by molecular self-assembly provide fascinating examples of complicated space partitions in soft-material 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 ), the primitive , and, particularly, the gyroid 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 liquid-crystal 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 A and B minority components together form the dividing membrane. Because of the connectivity of the star molecular architecture and because all components microphase separate, the A and B components segregate on the dividing hyperbolic interface. This structure is an experimental indication of a unique mode of self-assembly, namely “hierarchical assembly of a hyperbolic pattern.” Complementing this finding and further motivating our work reported here, a recent simulation study by one of us (J.J.K.K.) explored self-assembly of blends of equal amounts of two distinct three-miktoarm stars, namely ABC and ABD three-miktoarm star terpolymers (Fig. 1). Both molecules were assigned equal molecular weights, and the proportions of the equal volume C (green) and D (yellow) chains relative to the equal A (red) and B (blue) chains were varied (4). Despite these severe compositional constraints, a number of unique four-colored mesophases were revealed. The most striking feature of the predicted phase behavior in this system was the presence of interesting patterns whose general features are reminiscent of the gyroid, albeit far more complex in both geometric and topological aspects. In the system reported here, two ordering regimes form. At the larger length scale, ordering induces a gyroid-like membrane, which is itself also spontaneously ordered at a smaller length scale, giving unique microdomain patterning due to the membrane confinement to a hyperbolic curved interface. Each of these patterns contain distinct numbers and types of interwoven 2D and 3D A and B domains forming nets of equal hand, immersed within the hyperbolic interface between an enantiomeric pair of C and D srs nets. These structures are spectacularly convoluted in 3D space and correspond to special members of a sequence of chiral cubic patterns that emerge by local striping of the gyroid membrane. We demonstrate how this is performed systematically by mapping a particular family of tilings in the hyperbolic plane onto the gyroid in 3D euclidean space. Careful analysis of the morphologies formed in the simulations, described below, reveals the presence of up to three distinct chiral cubic mesophases within this striped gyroid region of the phase diagram. We explore the geometric and topological variety of these self-assemblies in detail and discuss how they emerge as a response to a hierarchy of frustrations imposed by the three-arm star molecular architecture, acting in both two and three dimensions.Open in a separate windowFig. 1.(A) Model ABC and ABD three-miktoarm star terpolymer molecules. All molecules contain equal-sized A (red) and B (blue) arms, and longer C (green) and D (yellow) arms, also of equal size. The parameter x (equal to in this image), corresponds to the number ratio of C to A beads. (B) C and D domain geometry, a pair of intertwined srs nets. (C–G) Single-unit cell snapshots illustrating the curved striped pattern formed by the minority components A and B for varying x. (C) x = 2, (D) x = 3.33, (E) x = 3.67, (F) x = 5, and (G) x = 6. Note the threefold branching of the stripes for all values of x.