Smectic, cholesteric, and Rayleigh-Benard order in two dimensions

Order in layered smectic and cholesteric liquid-crystal films is investigated. Although long-wavelength fluctuations in the layer displacements destroy translational order at any finite tem-perature, one must also account for effects of free dislocations. The combined action of layer fluctuations and dislocations is to produce a phase at finite temperature with persistent orientational order in the local normal to the layers. At length scales greater than ${\xi }_D\sim \exp \left(\frac{E_D}{2k_{B}T}\right)$ where $E_D$ is the energy of an isolated dislocation, the properties are those of a two-dimensional nematic, with the local layer normal playing the role of a director field. An intermediate phase of this kind could conceivably exist in bulk cholesteric liquid crystals as well. In two dimensions, an unbinding of disclination pairs eventually produces an isotropic phase. The low-temperature nematic Frank constants $K_1\mathrm{}\left(T\right)\mathrm{}$ and $K_3\mathrm{}\left(T\right)\mathrm{}$ are worked out, and compared with the diamagnetic susceptibility expected in a two-dimensional superconductor. We determine the characteristic nematic frequencies at low temperatures, using a simple model of dislocations interacting with a layer displacement field. Our analysis may also be relevant to the behavior of Rayleigh-Benard convective rolls in the presence of thermal noise fluctuations.