Breaking symmetry in self-assembled, triply-periodic, supramolecular networks

Amphiphilic macromolecules, such as block copolymers (BCPs), have long been known to self-assemble into bicontinuous, triply-periodic network phases. Of these, the canonical cubic phases of the double gyroid (DG), double diamond (DD), and double primitive (DP) have received the most attention. However, the responses of these networks to broken crystal symmetries, such as deformations of the unit cell and interruptions in translational symmetry, are poorly understood, yet known to result in significant non-affine deformations and even transformations in the underlying skeletal graphs. We turn to a recently-developed, pseudo-mechanical model of BCP network deformations, "liquid network theory," which is able to capture key elements of network response, including non-affinity, strut length variations, and mass transport. Using comparisons with molecular theory and experiments, we show how this relatively simple construction captures the complex, coordinated motion of macromolecules under different deformation protocols. We additionally explore consequences for the allowed symmetries of "mesoatomic" nodal motifs, leading to heuristics for the assembly of BCP network structures beyond the canonical cubic phases.