Hexatic phases

In this section we will discuss in some detail the application of X-ray diffraction and IR dichroism for the structure determination and identification of diverse LC phases. The general feature, revealed by X-ray diffraction (XRD), of all smectic phases is the set of sharp (OOn) Bragg peaks due to the periodicity of the layers [43]. The in-plane order is determined from the half-width of the inplane (hkO) peaks and varies from 2 to 3 intermolecular distances in smectics A and C to 6-30 intermolecular distances in the hexatic phase, which is characterized by six-fold symmetry in location of the in-plane diffuse maxima. The lamellar crystalline phases (smectics B, E, G, I) possess sharp in-plane diffraction peaks, indicating long-range periodicity within the layers. 

Another approach to explain tubule formation was taken by Lubensky and Prost as part of a general theoretical study of the relationship between orientational order and vesicle shape.173 These authors note that a membrane in an Lp/ phase has orientational order within the membrane which is lacking in the La phase. The clearest source of orientational order is the tilt of the molecules with respect to the local membrane normal The molecules select a particular tilt direction, and hence the local elastic properties of the membrane become anisotropic. A membrane might also have other types of orientational order. For example, if it is in a hexatic phase, it has order in the orientations of the intermolecular bonds (not chemical bonds but lines indicating the directions from one molecule to its nearest neighbors in the membrane). 

To discuss the models in this section, we should mention two issues. First, the models assume the membrane is sufficiently soft that the tilt direction can vary with an energy cost that scales as (Vc(j)2. This is appropriate if the membrane is in a fluid phase like a smectic-C liquid crystal, with order in the tilt direction but not in the positions of the molecules. It is also appropriate if the membrane is in a tilted hexatic phase, with order in the orientations of the intermolecular bonds as well as the tilt. However, this assumption is not appropriate if the membrane is in a crystalline phase, because the tilt direction would be locked to the crystalline axes, and varying it would cost more energy than (V(f>)2. 

It is possible for chiral mesogens to produce essentially achiral mesophases. For instance, in certain ranges of concentration and molecular weight, DNA will form an achiral line hexatic phase. A curious recent observation is of the formation of chiral mesophases from achiral mesogens. Specifically, bent-core molecules (sometimes called banana LCs) have been shown to form liquid crystal phases that are chiral. In any particular sample, various domains will have opposite handedness, but within any given domain, strong chiral ordering will be present. 

The lifetime of BR generated from 97 (n = 19) was found to be 64 5 and 70+ 5 ns in the isotropic and hexatic B phases of BS [319], The lack of influence on the lifetime of the biradical by the hexatic phase when the E/C ratios are clearly affected is at first puzzling. However, it can be cited as evidence that the T - S rate is independent of the conformation in which a BR is held [263]. Note that the BR from 97 (n = 21) as shown in Figure 60 has its hydroxyl group far removed from the cross-sectional segment of the BS-provided reaction cavity cylinder which is quite polar. In any case, the long lifetime of the BR found in hexatic BS and its near equivalence to that in the isotropic phase indicate that the various biradical conformers have equilibrated in the cylindrical reaction cavity prior to collapsing to products. 

In order to discuss the hexatic phase it is necessary to introduce the idea of a disclination. Imagine a two-dimensional close packed hexagonal lattice drawn on a deformable sheet. If one chooses a particular lattice site as the centre of coordinates, the lattice will consist of six 60° sectors centred on this point. One now has two alternatives. 

If one makes use of the rather limited information available and given above one may infer that a tilt of between 20° and 30° is normal for straight chain azobenzene derivatives when deposited as LB films, even when a homeotropic phase exists. Such a structure can only be produced in a rather loosely packed film. At the moment it is an open question whether monolayers of these materials exist in the hexatic phase as is the case for fatty acids or whether the structure more nearly corresponds to the smectic-A phase. In the case of the birefringent phase described by Jones et al. [151] it was shown that, once this phase was established, further layers deposited by the LB technique go down in an epitaxial manner. 

Experimental evidence was reported for the existence of various additional phases a pre-cholesteric order in the form of a network of double-twisted cylinders, analogous to the thermotropic blue phases [27], a hexatic phase that replaces the hexagonal columnar in very long DNA fragments [31], and a structure with orthorhombic symmetry appearing in the transition to crystalline order [27]. 

The proposition that the quasi-two-dimensional contact layer may exist as a hexatic phase... 

R. Radhakrishnan, K.E. Gubbins and M. Sliwinska-Bartkowiak, On the Existence of a Hexatic Phase in Confined Systems, Phys. Rev. Lett. 89 (2002) art. 076101... 

Halperin and Nelson [53,54] and Young [55] recognized that the vector character of dislocations must be taken into account in calculating the melting temperature, and also recognized that the dislocation-unbinding transition results in a sixfold bond orientationally ordered fluid phase, the hexatic phase, and that a second, discliriatiori- mbm mg transition is required to obtain an isotropic fluid. 

The quasi-long-range bond orientational order present in the hexatic phase is destroyed at a second, disclination-unbinding, transition, which occurs at a temperature T >T. In this transition the tightly bound... 

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