Nematic order director fluctuations and nuclear magnetic relaxation of a confined liquid crystal

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Abstract
The dynamics of order director fluctuations of a nematic liquid crystal confined to microcavities is studied within the Frank elastic theory. For cylindrical capillary and rectangular model cavities, constrained in one, two, and three spatial dimensions, mutually perpendicular components of fluctuations are expanded in terms of eigenfunctions of the Landau-Khalatnikov type relaxation equation. The calculated dispersion of the director fluctuations’ spin-lattice relaxation rates exhibits weak dependence on the nematic structure and the confining geometry, unless the size of the cavity is close to the critical value at which the structural transition is expected. In the latter case, the low-frequency behavior of the relaxation rate is dominated by the slowest fluctuations’ modes. On the other hand, the angular dependence of the relaxation rate is sensitive to details of the nematic structure in the cavity in the whole frequency range.