Bending, columnar phases

Fig. 9 (a) Sketch of the structure of the hexagonal columnar phase of DNA, showing parallel molecules hexagonally packed in the plane perpendicular to their axis, a and 4 are the lattice parameters, (b) COL developable domains observed in polarized microscopy, w indicates defect walls between differently oriented domains, while 7t stands for point defect around which DNA molecules continuously bend (size bar is 10 pm). Adapted with permission from [27]... 

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. 

The mechanical properties of a columnar phase are, in a sense, similar to those of a sheet of paper - which can easily be curved in one dimension, but which is much more difficult to then curve in another direction (because this would involve stretching as well as bending the sheet). For material with a low birefi"ingence. 

The principal elastic constants for a nematic liquid crystal have already been defined in Sec. 5.1 as splay (A , j), twist(/ 22) and bend(fc33). In this section we shall outline the statistical theory of elastic constants, and show how they depend on molecular properties. The approach follows that of the generalised van der Waals theory developed by Gelbart and Ben-Shaul [40], which itself embraces a number of earlier models for the elasticity of nematic liquid crystals. Corresponding theories for smectic, columnar and biaxial phases have yet to be developed. 

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