Tilted hexatic phases

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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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. 

As in the S phase, Sg lacks true long-range translational order because of the Peierls-Landau instability. Theoretically, the existence of molecular tilt implies that there must be a certain degree of bond-orientational order in the Sg phase. This has been verified experimentally by a high resolution synchrotron X-ray study of the transition from Sg to S, in a monodomain freely suspended film. It will be recalled that S, is a tilted hexatic phase. The bond-orientational order parameter Cg (as defined in (5.7.1), but appropriately modified to allow for the fact that the molecules are tilted) is plotted as a function of temperature in fig. 5.8.2. The weak bond-orientational order in evolves continuously into S, showing... 

Figure 23. Structures of the tilted hexatic phases SmF and SmI, and their diffraction patterns with the incident beam perpendicular to the layer normals. (Adapted from [238]).

The smectic F phase can also be considered as a tilted hexatic phase similar to the smectic I phase, the difference between the two phases lying in the direction of the long molecular axes with respect to the two-dimensional arrangement within the smectic layers [151]. Calorimetric and X-ray diffraction studies of several compounds of the homologous TBnA series carried out around the SmC-SmF transition clearly indicate the first order nature of the corresponding transition, with, in particular, a jump in the... 

Due to the symmetry of the order parameters associated with the molecular tilt ("Fc = 00 exp (i 0c)) aiid bond-orientational order ( fH = fnc exp(i 60o)), the model Hamiltonian describing the SmC-tilted hexatic phase can be written as [88]... 

In the SmC phase, and Kf are the Frank elastic constants for bend and splay distortion of the C-director. In the tilted hexatic phase, they are the sums of the director and bond-orientational elasticities. The last... 

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