Local v/a variations as a measure of structural packing frustration in bicontinuous mesophases, and geometric arguments for an alternating ${\rm Im}\overline{{\mathsf3}}{\rm m}$ (I-WP) phase in block-copolymers with polydispersity

This article explores global geometric features of bicontinuous space-partitions and their relevance to self-assembly of block-copolymers. Using a robust definition of `local channel radius', based on the concept of a medial surface [Schröder et al., Eur. Phys. J. B 35, 551 (2003)], we relate radius variations of the space-partition to polymolecular chain stretching in bicontinuous diblock- and terblock copolymer assemblies. We associate local surface patches with corresponding cellular volume elements, to define local volume-to-surface ratios. The distribution of these v/a ratios and of the channel radii are used to quantify the degree of packing frustration of molecular chains as a function of the specific bicontinuous geometry, modelled by triply-periodic minimal surfaces and related parallel interfaces. The Gyroid geometry emerges as the most nearly homogeneous bicontinuous form, with the smallest heterogeneity of channel radii, compared to the cubic Primitive and Diamond surfaces. We clarify a geometric feature of the Gyroid geometry: the three-coordinated nodes of the graph are not the widest points of the labyrinths; the widest points are at the midpoints of the edges. We also explore a more complex cubic triply-periodic surface, the I-WP surface, containing two geometrically distinct channel subdomains. One of the two channel systems is nearly as homogeneous in local channel diameters as the Gyroid, the other is more heterogeneous than the Primitive surface. Its hybrid nature suggests the possibility of an “alternating I-WP” phase in polydisperse linear ABC-terpolymer blends, with monodisperse molecular weight distributions (MWD) in the A and B blocks and a more polydisperse C block. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007