Hierarchical Discommensuration Pattern and Phase Transitions in the Electrohydrodynamic Convection of Liquid Crystals with a Periodic Substrate Potential
In recent years there has been considerable interest in the statistical mechanics of low-dimensional systems with competing interactions. A typical model of such systems is the Frenkel-Kontorova (FK) model or its extension, in which a one-dimensional array of particles interact with each other in the presence of a spatially periodic external potential . Various concepts, such as discommensuration (DC), commensurate (C) lock-in, and fractal (or hierarchical) nature, have been introduced by the study of the models. In spite of these developments on the theoretical side, the experimental investigation of atomic systems with competitive interactions near the ground state has not been improved. The reasons are that: i) due to the microscopic size of atomic structure, we must see the system in the reciprocal space, ii) the temperature in the experiments is high enough to break the complex commensurate structures, iii) there may exist defects also in the substrate crystal within the commensurate periodicities. On the other hand, convective systems of liquid crystals with a periodic substrate voltage  are considered to be most adequate systems to observe the competitive phenomena, because the rolls which play the role of “atoms” have a semimicroscopic size, and thus the subtle structure is visible in real space, even if it does not have long-range order. Besides, its energy scale is larger than that corresponding to the possible temperature, so we can easily observe the structure at or near the ground state. In this paper, we interpret the experimental results of the liquid crystal system with a one-dimensional roll pattern, regarding it as a particle system with competitive interactions.
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