Mesophases meshes

It is impossible to offer estimates for relative locations of mesh mesophases within a phase diagram, due to the extreme variability of shape parameters ( ) with the composition of meshes. Note that Type 1 meshes (stacked layers of 2D apolar labyrinths) can result provided that /2 < s 2/3, and Type 2 meshes (with water labyrinths) can form if s > 2/3, over a range of compositions (Figure 16.8). That means that Type 2 mesh mesophases can be formed in lyotropes containing single-chain amphiphiles in contrast to the other Type 2 mesophases, which form only if s > 1. 

Figure 16.23. (a) View of a single monolayer in a (T)etragonal mesh mesophase. (b) The bilayers are stacked in a staggered arrangement, giving a body-centred tetragonal symmetry (1422), with lattice parameters a and c... 

In addition to 3D columnar packings, these branched polycontinuous forms include novel mesh structures, with 3D arrangements of meshes, rather than the smectic array common to the R and T mesh mesophases. One example, a rhombohedral arrangement of the hexagonal mesh (found also in the R phase), is shown in Figure 16.26. 

Figure 16.26. A theoretical 3D mesh mesophase, consisting of three interwoven graphite nets (the R mesh mesophase consists of smectic stacks of these nets)...

Thus, these mesostructures are predominantly lamellar, and identified as conventional (parabolic) lamellar phases, although they may in fact be hyperbolic. Indeed, unless v/al is exactly unity, a planar interface (lamellar mesophase) incurs a bending energy cost hyperbolic sponge monolayers or bilayers or mesh monolayer mesophases are favoured if v/al differs from unity. It is likely then that many "lamellar"" phases in fact adopt a hyperbolic geometry. Careful neutron-scattering studies of a lamellar phase have revealed the presence of a large number of hyperbolic "defects" (pores within the bilayers) in one case [16]. (An example of this mis-identification of hyperbolic phases in block copolymers is discussed in section 4.10.)... 

The relative stability of mesh and IPMS structures is still unclear. For example, the Ri mesophase (of rhombohedral symmetry) in the SDS-water system transforms continuously into the neighbouring bicontinuous cubic phase (Fig. 4.14) [20]. This suggests that this mesophase is a hyperbolic (reversed) bilayer Ijring on a rhombohedral IPMS. Indeed, the rhombohedral rPD surface is only marginally less homogeneous than its cubic counterparts, the P- and D-svu-faces. 

Between 5=1/2 and 2/3 are mesh phases that include regular tetragonal and hexagonal mesophases and also the defective lamellar phase [295]. For a surfactant parameter from 2/3 to 1, continuous mesh phases or cubic phases are found. Finally, there is the L phase at a surfactant parameter of 1. Indeed, in the system Ci6(EO)6-water, the surfactant parameter is calculated [142] to be 0.65 (Fig. 29), as was found in the L phase of the cesium penta-decafluorooctanoate-water system [265]. In the system C3o(EO)9-water, the surfactant parameter is calculated [262] to be 0.592. 

Other more complex morphologies also arise for A-B mixtures. In particular, domains A and B may enclose each other, forming entangled networks, separated by a hyperbolic interface. Those cases include mesh , bicontinuous microemulsions, bicontinuous cubic phases and their disordered counterparts, sponge phases, which are discussed below. In these cases too, the sign (convex/concave) of the interfacial mean curvature sets the Type . A representation of the disordered mesostructure in a Type 2 bicontinuous microemulsion is shown in Figure 16.3. A hyperbolic interface may be equally concave and convex (a minimal surface, e.g. see Figure 16.2(c)) so that the mesophase is neither Type 1 nor Type 2. Lamellar mesophases ( smectics or neat phases) are the simplest examples. Bicontinuous balanced microemulsions, with equal polar and apolar volume fractions are further examples. 

In this section, we will undertake the more speculative task of introducing novel interfaces, in the belief that they too may be found among lyotropic liquid crystals. A variety of other mesophases have been proposed in the scientific literature over the years. Here, we will focus on the more topologically complex examples, including novel arrays of meshes, sponges and hybrids. 

Similarly, meshes can be viewed as punctured bilayers, where ordered square and hexagonally patterned arrays of punctures result in the T and R mesophases, respectively. Inverting that argument leads to the conclusion that taking account of the diffraction peaks in the scattering pattern only allows one to reconstruct those spatially correlated domains in the mesostructure a bicontinuous membrane could diffract as a smectic or hexagonal lattice, and yet its mesostructure is far from that of the classical lamellar or hexagonal mesophases. 

The biological importance of mesh and other intermediate mesophase structures goes far beyond the finer details of phase studies. Most studies of biological membranes have historically focused on membrane proteins and their influence on biological function. Until relatively recently, that emphasis has been justified by the... 

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