Modeling chain packing in triply periodic networks of self-assembled block copolymers

Triply periodic network (TPN) phases belong to the natural forms of soft matter self-assembly and underlie a range of attractive function properties. Understanding how a TPN phase is equilibrium among its competitor phases is intricately tied to packing frustration or extension of molecules into centers of the domains, a notion which is ambiguous for TPN phases and confounds attempts to predict thermodynamically optimal packing of molecules. Here, we examine the double gyroid (DG) network phase of block copolymer (BCP) melts, a prototypical amphiphilic system and test the link between terminal boundaries of individual blocks to lie along medial sets, a generic and purely geometric definition of the center of complex domain shapes and the thermodynamics of BCP assembly using geometric formulation of the strong stretching theory of BCP melts. We show that medial packing is essential for equilibrium DG in strongly segregated BCP melts and importantly revise long held notions of packing frustration which assume that terminal boundaries of tubular blocks are constrained to lie along  one-dimensional skeletal graphs. Additionally, we find a previously unrecognized dependence of DG stability on the entropic stiffness asymmetry between tubular and matrix blocks.