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

J. J. K. Kirkensgaard, M. E. Evans, L. de Campo, S. T. Hyde
2014 Proceedings of the National Academy of Sciences of the United States of America  
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
more » ... e 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 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 Significance Chirality and hierarchical ordering are two fundamental properties found in many of nature's most complex self-assembled structures such as living cells. Simultaneous control over these properties in synthetic systems is vital to mimic or even surpass nature's designs. Via numerical simulations, we describe a class of complex morphologies that afford radically new architectures for self-assembled shapes. Specifically, a mixture of two star block-copolymers are shown to form multiple interwoven 2D and 3D labyrinths-all chiral-and hierarchically ordered on two different length scales. Furthermore, we show that such intricate network morphologies forming at a confined, hyperbolic interface can be classified and modeled in terms of a much simpler isotropic model of packing based on tilings of the hyperbolic plane.
doi:10.1073/pnas.1316348111 pmid:24474747 pmcid:PMC3910609 fatcat:ylxcpjv7lje6bnmew4eexuvff4