Local v/a variations as a measure of structural packing frustration in bicontinuous mesophases, and geometric arguments for an alternating Im3‾m{\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 200702.40.-k Geometry, differential geometry, and topology, 61.30.St Lyotropic phases, 81.16.Dn Self-assembly, 82.35.Jk Copolymers, phase transitions, structure,

Bicontinuous geometries and molecular self-assembly: comparison of local curvature and global packing variations in genus-three cubic, tetragonal and rhombohedral surfaces

Balanced infinite periodic minimal surface families that contain the cubic Gyroid (G), Diamond (D) and Primitive (P) surfaces are studied in terms of their global packing and local curvature properties. These properties are central to understanding the formation of mesophases in amphiphile and copolymer molecular systems. The surfaces investigated are the tetragonal, rhombohedral and hexagonal tD, tP, tG, rG, rPD and H surfaces. These non-cubic minimal surfaces furnish topology-preserving transformation pathways between the three cubic surfaces. We introduce `packing (or global) homogeneity', defined as the standard deviation Δd of the distribution of the channel diameter throughout the labyrinth, where the channel diameter d is determined from the medial surface skeleton centered within the labyrinthine domains. Curvature homogeneity is defined similarly as the standard deviation ΔK of the distribution of Gaussian curvature. All data are presented for distinct length normalisations: constant surface-to-volume ratio, constant average Gaussian curvature and constant average channel diameter. We provide first and second moments of the distribution of channel diameter for all members of these surfaces complementing curvature data from [A. Fogden, S. Hyde, Eur. Phys. J. B 7, 91 (1999)]. The cubic G and D surfaces are deep local minima of Δd along the surface families (with G more homogeneous than D), whereas the cubic P surface is an inflection point of Δd with adjacent, more homogeneous surface members. Both curvature and packing homogeneity favour the tetragonal route between G and D (via tG and tD surfaces) in preference to the rhombohedral route (via rG and rPD). Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 200602.40.-k Geometry, differential geometry, and topology, 61.30.St Lyotropic phases, 81.16.Dn Self-assembly, 82.35.Jk Copolymers, phase transitions, structure,