The hyperbolic realm cubic and intermediate phases

A central issue in the field of surfactant self-assembly is the structure of the liquid crystalline mesophases denoted bicontinuous cubic, and intermediate phases (i.e. rhombohedral, monoclinic and tetragonal phases). Cubic phases were detected by Luzzati et al. and Fontell in the 1960 s, although they were believed to be rare in comparison with the classical lamellar, hexagonal and micellar mesophases. It is now clear that these phases are ubiquitous in surfactant and Upid systems. Further, a number of cubic phases can occur within the same system, as the temperature or concentration is varied. Luzzati s group also discovered a number of crystalline mesophases in soaps and lipids, of tetragonal and rhombohedral symmetries (the so-called T and R phases). More recently, Tiddy et al. have detected systematic replacement of cubic mesophases by intermediate T and R phases as the surfactant architecture is varied [22-24]. The most detailed mesophase study to date has revealed the presence of monoclinic. 

Detailed data are available for some cubic phases, which offer a good test of the model of hyperbolic bilayers wrapped onto IPMS. In these cases, the results suggest that these phases consist of bilayers of cubic S3onmetry, whose mid-surfaces closely follow IPMS [25,26]. 

The equations developed earlier in this chapter offer a useful route to decoding the mysteries of these cubic phases. For example, in a cubic mesophase, the surface averaged value of the Gaussian curvature at the midsurface of the bilayer, K , is related to the lattice parameter of the cubic unit cell (a) via the equations  

This progression fits well experimental data collected so far. In the binary glycerol monooleate-water system there is conclusive experimental evidence for the presence of a body-centred structure of symmetry laSd at lower water contents and a primitive cubic bilayer (symmetry Pn m) at higher water contents [28] (Fig. 4.15). 

This finding fits with the expected relative phase locations well the bilayer transforms from the Cq phase to the Cp) pha upon dilution [29]. The details of this transformation are discussed in more detail in section 5.1.3. 

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