Single-Gyroid

Since analytical expressions for only a few continuous triply periodic CMC surfaces are known (e.g. the Enneper-Weierstrass parameterization of the single-gyroid minimal surface with H = 0 and a volume fraction of 50 % [9]), these surfaces are typically modeled with the help of level surfaces. 

The single-gyroid (SG) IMDS with 74i32 (No. 214) symmetry was first discovered in 1967 by Luzzati et al. as a cubic phase occurring in strontium soap surfactants and in pure lipid-water systems [12, 13]. In 1970, Schoen identified the minimal... 

Fig. 2.2 Single-gyroid surfaces calculated using Eq. (2.5). a Minimal gyroid surface for 1=0. b Two helical interpenetrating single-gyroid networks separated by the minimal gyroid surface, c Single-gyroid for 1 = 1.3. d Pinch-off surface for t = 1.413...

Fig.20 Top row single unit-cell models of core-shell double gyroid (Q230), orthorhombic (O70), and alternating gyroid (Q214) cross-sectioned to reveal interfacial configuration. Bottom row. direct projections of cross-sectioned interfaces. Sketches of PI-fi-PS-fi-PEO chains show how each morphology is assembled. Projections appear to scale that is, the core-shell double gyroid unit cell is roughly twice the thickness of the other two. From [75], Copyright 2004 American Chemical Society...

Figure 4.25. High magnification image.s obtained by transmission electron microscopy of stained thin sections of a hyperbolic mesopha.se of a linear diblock copolymer, polystyrene-polyisoprene, whose morphology follows the D-surface (single node circled in middle picture) and possibly the gyroid. Staining produces high contrast between the two block domains. Note the very different magnifications.

However, there is a structure consistent with both the required space group and the optical properties. The gyroid surface, which occurs frequently in lipid-water systems, provides such a possibility. If we assume that cholesterol skeletons form rod-like infinite helices, this structure represents an effective three-dimensional packing of such helices. Thus, the rods form a body-centered arrangement as shown in Fig. 5.5. In this structure, there is a helical twist between the rods, in addition to the cholesteric twist within each rod. The h)rperbolic structure is a consequence of the chirality of the esters, which induces torsion into the packing arrangement. A racemic mixture does not exhibit this phase natural cholesteric esters contain a single enantiomer only. 

Figure 7.5 A multicontinuous G-PCS (the gyroid) differei tiating sieve-elements (see cdso Fig. 7.6). (a) A [211] projection of a double bilayer G-PCS. The match of the theoretical projection is excellent (as well as the correlation between the Fourier transforms (a (experiment) and a (theory)) and it is eeisily seen that a single bilayer G-PCS does not account for the experimental projection (see (c)). (b) Serial section of (a). Note the apparent pleomorphic behaviour of the cubic membrane in (b), which shows co-existing D- and G-morphologies, related by an intersection-free, and thus, topologically constrained transformation, (c bottom) Computer generated projections of the single G [211] and the double bilayer G2 [211] projections. Figures (a) and (b) are modified from [32], with permission.

Adv. Mater. 2009 Nanostructured caldte single crystals with gyroid morphologies, A. S. 

The three dimensional (3D) cubic V2 phases are arranged as single continuous lipid curved bilayers forming a eomplex network containing two non-intersecting water channels [90]. Three different bicontinuous cubic nanostructures (a family of closely related phases) have been identified in the literature. They have a primitive (P), a gyroid (G), or a diamond (D) infinite periodic minimal surface (IPMS) [88, 89]. 

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