The hexatic phase

The degree of hexatic ordering can be defined quantitatively by fitting the /-scans to a Fourier series of the form 

Orientational long-range order, positional short-range order 

Cg measure the degree of 6n-fold ordering. An application of (5.7.1) to study the bond-orientational order across the S,-Sc transition will be described in 5.8.1. 

The hexatic B-S transition is expected to belong to the XY universality class/ but high resolution calorimetric investigations appear to indicate that this may not be the case. Tilted analogues of hexatic B, namely Sp and Sj, have also been identified.  

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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. 

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. 

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... 

Fig. 5.7.1. (a) The structure of the hexatic phase of a two-dimensional lattice. The orientation of the lattice vectors a and b is preserved over a long range, but the molecular positions are correlated only over a short distance p. b) The orientational order as well as the positional order are short range. (After Litster... 

Recently, two molecular dynamics simulations were published in which hexatic phases were observed for particular systems. In one [69], the phase was observed for a particular concentration in a two-component mixture of Lennard-Jones atoms, and in the other [70], the hexatic phase appeared for an inverse 12th-power potential at a particular value of the pressure. It is unclear why these special conditions should produce the hexatic phase, so extensions of these studies would appear to be necessary. 

SmF C2h X T(l) Tilted analogy of the hexatic phase. A stack of interacting hexatic layers with three-dimensional, long-range, sixfold, bond orientational order and liquid-like positional correlations within the layers... 

More recently, Strobl [18] has stimulated a discussion by arguing that in all processes of polymer crystallization, a mesomorphic precursor phase is first formed before the crystalline phase. Blocks of this mesomorphic state then attach to the growth front, as sketched in Fig. 1.5. This model, inspired by the observation of the hexatic phase in short n-alkanes [19, 20], is actively contested by the polymer community. Whether such mesomorphic phases are stable intermediates before the formation of the crystalline phase and whether a critical chain stiffness is required for such mesomorphic phases are being discussed. Several laboratories worldwide are pursuing experiments to explore this aspect of poljmer crj talUzation. 

As has aheady been indicated a large number of computer simulations have been performed to study phase transitions. In particular, some simulations have predicted the existence of the hexatic phase [256], whereas others have not [132]. Moreover, as Toxvaerd [264] has indicated simulations with a very large number of particles are needed to give a definitive conclusion about the melting order. More complex simulations, including the calculation of the angular correlation fimction [262,265] have indicated a first-order transition. 

The hexatic phase SmB, and the tilted versions SmF and SmI, are characterized by the onset of increased order within the layers, with the positional coupling between layers remaining weak. In general, one cannot distinguish these phases, nor deduce the tilt angles, from the powder patterns, shown for the SmB phase in Fig. 8. The lateral packing of the molecules is locally hexago-... 

In the hexatic phase, the molecules are distributed on a hexagonal lattice, but the positional order does not extend over distances larger than a few hundred angstroms, whereas the bond orientational order extends over very large distances (long range... 

Because of the existence of the quasi-hex-agonal lattice in the tilted hexatic phase, there are three possible molecular tilt directions, The molecular tilt points towards the nearest neighbor for the SmI phase, towards the next nearest neighbor for the SmF phase, and towards an intermediate site for the novel SmL phase (see Fig. 1). Thus the former two have a higher symmetry than the SmL phase, which was first discovered in a lyotropic liquid crystal system [87]. The most common tilted hexatic phases found in thermotropic liquid crystal compounds are SmI and SmF. As mentioned previously, the pseudo-hexagonal molecular arrangement which is the characteristic feature of the hexatic phase, was first identified for the SmI phase. 

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