Anatomy of triply-periodic network assemblies: characterizing skeletal and inter-domain surface geometry of block copolymer gyroids

Triply-periodic networks (TPNs), like the well-known gyroid and diamond network phases, abound in soft matter assemblies, from block copolymers (BCPs), lyotropic liquid crystals and surfactants to functional architectures in biology. While TPNs are, in reality, volume-filling patterns of spatially-varying molecular composition, physical and structural models most often reduce their structure to lower-dimensional geometric objects: the 2D interfaces between chemical domains; and the 1D skeletons that thread through inter-connected, tubular domains. These lower-dimensional structures provide a useful basis of comparison to idealized geometries based on triply-periodic minimal, or constant-mean curvature surfaces, and shed important light on the spatially heterogeneous packing of molecular constituents that form the networks. Here, we propose a simple, efficient and flexible method to extract a 1D skeleton from 3D volume composition data of self-assembled networks. We apply this method to both self-consistent field theory predictions as well as experimental electron microtomography reconstructions of the double-gyroid phase of an ABA triblock copolymer. We further demonstrate how the analysis of 1D skeleton, 2D inter-domain surfaces, and combinations therefore, provide physical and structural insight into TPNs, across multiple length scales. Specifically, we propose and compare simple measures of network chirality as well as domain thickness, and analyze their spatial and statistical distributions in both ideal (theoretical) and non-ideal (experimental) double gyroid assemblies.