Medial packing and elastic asymmetry stabilize the double-gyroid in block copolymers

Abstract

Triply-periodic networks are among the most complex and functionally valuable self-assembled morphologies, yet they form in nearly every class of biological and synthetic soft matter building blocks. In contrast to simpler assembly motifs -spheres, cylinders, layers -TPN assemblies require molecules to occupy variable local domain shapes, confounding attempts to understand their formation. Here, we examine the double-gyroid (DG) network phase of block copolymer (BCP) melts, a prototypical soft self-assembly system, by using a geometric formulation of the strong stretching theory (SST) of BCP melts. The theory establishes the direct link between molecular BCP packing, thermodynamics of melt assembly and the medial map, a generic geometric measure of the center of complex shapes. We show that "medial packing" is essential for thermodynamic stability of DG in strongly-segregated melts, reconciling a long-standing contradiction between infinite-and finitesegregation theories, corroborating our SST predictions at finite-segregation via self-consistent field calculations. Additionally, we find a previously unrecognized non-monotonic dependence of DG stability on the elastic asymmetry, the comparative entropic stiffness of matrix-forming to tubularnetwork forming blocks. The composition window of stable DG -intermediate to competitor lamellar and columnar phases -widens both for large and small elastic asymmetry, seemingly overturning the heuristic view that packing frustration is localized to the tubular domains. This study demonstrates utility of geometric optimization of medial tesselations for understanding in soft-molecular assembly. As such, the particular medial-based approach deployed here is readily generalizable to study packing frustration far beyond the case of DG morphologies in neat BCP assemblies.