Deformations columnar phases

Due to the rigidity of Rp-chains, in fluorinated LCs the transition from smectic to columnar organization often takes place via modulated smectic phases (ribbon phases, c2mm, p2gg, and Colob see Fig. 20, left) which completely or partly replace the bicontinuous cubic phases at the Sm-to-Col cross-over. Similarly, in columnar phases the circular columns can be deformed to an elliptic, rectangular, or square... 

We now discuss the fundamental question of fluctuations in the columnar phase. Let us suppose that the liquid-like columns are along the z axis and that the two-dimensional lattice (assumed to be hexagonal) is parallel to the xy plane. The two basic deformations in such a structure are (i) the curvature deformation (or bending) of the columns without distortion of the lattice and (ii) lattice dilatation (or compression) without columnar curvature. There can also be coupling between the two types of distortion but, as shown by Kleman and Oswald, " the coupling term merely rescales the bend elastic constant of the columns. We shall consider only the vibrations of the lattice in its own plane. The free energy may be written... 

In discotic systems, the roles of ATu and K22 are reversed, because in such phases the bend deformations require the lowest energy [181]. Measurements of splay and bend constants in a homologuous series of discotic n-hexa(alkanoylox)truxenes [76] revealed that K22 is always smaller than A, . The splay/bend ratio approached unity at the high temperature transition to the columnar phase. Qualitatively different results have been obtained, however, by Raghunathan et al. [75], who found K22>Kn in a disco-tic nematic phase enclosed between two columnar phases. The authors interpreted this unexpected result as being a consequence of short-range columnar order. 

Fig. 9.32 (a) Phase transition behaviors of N (1,7)-Sn including hexagonal columnar phase (Colh) and optical isotropic cubic (Cub) as a function of carbon number of alkyl tail, (b-d) Schematic illustration of the column and cubic formation by the winding of the layer, (b) The asymmetric molecules are packed with the same directionality. Then, two rigid side arms of the banana molecule take a different orientation to the layer normal, producing a significant difference in the density if the layer is flat. Hence a dilatirai occurs in one half-layer (upper part divided by the dotted line in (a)) while a compression occurs in the other half (lower part) as in (c). This frustration can cause cylindrically and spherically enclosed deformations of the layer as in (d)... 

The molecular dynamics of the co-dendrimer G3-(Li)i6 - (L2)i6 in its two phases, i.e. a low temperature Cok phase and a high temperature SmA phase, was studied by NMR relaxation [221]. In the high frequency range, the motions appear similar in both phases and are ascribed to reorientations of the dendritic segments, whereas in the low frequency ranges, notable differences in the dynamics between the columnar and layered phases were observed, and were discussed in terms of elastic deformations of the columns and layer undulations. 

The simplest phase corresponds to a compact triangular packing (still called hexagonal) of isotropic columns (see Fig. 9.3). The equivalent phases in smectic C liquid crystals are such that molecules are tilted relative to columns. A priori molecules may tilt towards the nearest or second nearest neighbour columns, or in any other direction. These three possibilities correspond to three distinct phases. Columns may also deform themselves into ellipses without molecules tilting relative to the columnar axis this is yet another phase. Columnar symmetry may also occur with local smectic order and an example is shown in Fig. 9.19 for long thin molecules with polar heads. All these phases have common elastic, hydrodynamic and topological properties. 

Just as in the smectic phases and in solids, both edge and screw dislocations can occur. Three typical examples are illustrated in Fig. 9.20. In fact, big deformations do not involve dislocations (or only slightly), but simply the column curvature, which costs far less in energy terms. This feature is easy to understand by considering an example. Let us try to construct a columnar system from a cylinder of revolution of radius Tq with the boundary condition that the columns should be orthogonal to the surface of the cylinder where they meet it. Requiring the columns to be straight, either the space will not... 

First, simple less homogeneous - and therefore likely to be more prevalent among chemically disperse membranes - lower symmetry variations on the known mesostructures deserve consideration. Consistent reports of other mesophases by experienced researchers of the calibre of Luzzati and Fontell in Sweden cannot be discounted (17). Most of the proposed mesostructures involve simple symmetry-breaking deformations of the more common phases, including hexagonal columnar and discrete micellar mesophases. 

It is important to note that, just as there is a major difference between the fluctuations of the columnar and smectic phases, the distortions relax at different lengths scales. In smectics the distortion relaxes in much larger scales than that of the distortions, whereas in columnars they relax in length scales comparable to that of the deformations. ... 

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