Packing Frustration and the Hidden Geometry of Bicontinuous Network Phases of Block Copolymers

Block copolymers can self-assemble into a variety of ordered nanostructures, including the triply-periodic, bicontinuous double-gyroid (DG), double-diamond (DD), and double-primitive (DP) network phases. These phases consist of complex, intercatenated, labyrinthine domains, yet are supramolecular soft crystals with long-range order. Despite a history of experimental observations, as well as theoretical and computational studies, the formation and stability of these phases remains poorly understood and reliant on the vague notion of packing frustration. Here, we paint a picture of packing frustration -- an interplay between the thermodynamics governing interface geometry and chain stretching that is required by packing constraints of the polymer melt -- by turning to the strong segregation theory of block copolymers. The thermodynamic costs of satisfying this packing constraint is closely connected to the geometry of the terminal boundaries of each domain, i.e. the set of points defining the extent of chain stretching from the intermaterial dividing surfaces. We show that these terminal boundaries are well-approximated by a suitable set of medial surfaces, and that such medial geometry is essential to the stability of the network phases, resolving a long-standing puzzle of the stability of network phases in the strong segregation theory. Furthermore, we uncover new heuristics for polymer architectures that promote stability of the DG phase and explore how the medial geometry of the DG phase makes it more optimal than the competitor DD and DP phases. Finally, we introduce a map of packing frustration "hot spots" in each phase and illustrate how targeted infill of these hot spots can stabilize the DD and DP phases over the DG phase.