Fluctuations columnar phases

Understanding the organization of disc-shaped mesogens in columnar phases is of practical interest because of their useful optoelectronic applications [168-170]. It will be instructive to explore the role of fluctuations in influencing transport properties of thermotropic discotic liquid crystals—in particular, in the columnar phase. 

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... 

Dynamic processes that can be investigated by NMR include both the motions of individual molecules, e.g., conformational dynamics and molecular rotations, and collective motions, e.g., director fluctuations in nematic systems, layer undulations in smectic systems, or density modulations in columnar phases of discotic systems. Self-diffusion can be measured by NMR relaxation or field gradient methods, as discussed in Sec. 13 of Chap. VII of this Volume. Table 1 gives an overview of the time scales accessible by the most common experimental techniques and examples of the type of motion that can be studied. 

The apparent disagreement between simulation results and experimental observations on the the existence of power law relaxation near the isotropic-columnar (I-C) phase boundary needs further attention. This appears to be associated with the existence of nematic-like fluctuations in the experimental system. This can be checked, on the experimental side, by studying the relaxation near the I-C phase boundary located away from the discotic-nematic phase. For simulation, a careful analysis should be done by changing the temperature and density along the phase coexistence line and studying the existence or lack of nematic-like orientational fluctuations. 

The invariance of fi with temperature across the entire columnar liquid crystalline phase is quite dramatic. It is a signature of nearest neighbor, random phase diffusion. The sudden change in the magnitude of the mobility at the clearing temperature observed by Adam and coworkers [11] signifies that thermal fluctuations in the positions... 

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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