Distortions columnar phases

As for crystals, tire elasticity of smectic and columnar phases is analysed in tenns of displacements of tire lattice witli respect to the undistorted state, described by tire field u(r). This represents tire distortion of tire layers in a smectic phase and, tluis, u(r) is a one-dimensional vector (conventionally defined along z), whereas tire columnar phase is two dimensional, so tliat u(r) is also. The symmetry of a smectic A phase leads to an elastic free energy density of tire fonn [86]... 

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 both cases, the loss of mirror symmetry may be caused by a helix either the orientation of the molecular director or the position of the molecules may spiral within the column, or the column lattice is helically distorted. Within the column, mirror symmetry may also be lost by the introduction of tilt and polarization. Similarly to a surface-stabilized (thus nonhelical) smectic C phase of a chiral material, a nonhelical columnar phase of chiral molecules may itself be termed chiral if the optical axis of the molecules is inclined with respect to the column axis the vector of the tilt-induced po-... 

No thermotropic columnar phase with a helical lattice distortion of either of the two above types has yet been reported, but concentrated solutions of DNA and poly-y-benzyl-L-glutamate show undulations and defects that indicate that such helical defect structures exist in these lyotropic systems (see section 11.5). 

For (b), two models are plausible One may assume either an antiferro-electric pattern of tilted columns for which small fields cause a deviation of the tilt directions and for which a larger threshold field changes the pattern of tilt directions into a ferroelectric structure. Alternatively, a helical tilt may be distorted at lower fields and unwound at higher fields. As the helix pitch in columnar phases is usually much shorter than the wavelength of visible light, no selective reflection of light is expected and one may best distinguish between the two cases by X-ray or circular dichroism measurements. Only... 

Results for biaxial smectics and columnar phases have additional compressional terms, but tilted smectic phases can support additional torsional distortions. Such phases are conveniently described in terms of two directors, one along the tilt direction, corresponding to the nematic director n, and the projection of n on the layer plane, known as the c-director (see Fig. 6). 

In a related series of compounds with 5-heptyl-2-phenylpyrimidines replacing the 4 -cyanbiphenyls of compounds 208-210, the sequence SmC-SmA-Colrec was observed on increasing the number of phenylpyrimidine units [367]. This indicates an even stronger distortion of layer curvature by these rod-like units as hexagonal columnar and cubic phases are removed and replaced by a rectangular columnar ribbon phase. 

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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