Director fluctuations in nematic liquid crystals

Abstract

The literature of the past two decades suggests that confusion persists concerning the role of director fluctuations in the theory of nematics. Are these fluctuations necessarily small? If so, the fluctuation spectrum must terminate at a wavevector q _c which is of the order of 10⁹ m⁻¹, yet fluctuation modes with wavevectors in excess of 10⁹ m⁻¹ are commonly invoked to explain, for example, the frequency dependence of NMR relaxation rates. If the cut-off wavevector exceeds 10⁹ m⁻¹ and the fluctuations are not small, are their effects included in standard theories of nematic disorder, or should those theories be renormalized in some way to allow for them? Can we turn to experimental data for nematics, or to the results of computer simulations, in order to learn where q _c lies? This paper suggests answers to these questions, based upon an elementary definition of the director which can be applied to groups of molecules of any size, not necessarily macroscopic. In the course of the argument some widely accepted ideas about nematics are challenged, in particular: (i) the idea that mean field theories of the Maier-Saupe type provide an adequate description of them; (ii) the idea that, in so far as mean field theories are not exact, their deficiencies may be rectified by the cluster-expansion approach; (iii) the idea that the processes whereby molecules in a nematic undergo large changes in orientation, and from time to time experience end-to-end inversion, are essentially distinct from director fluctuations, which do no more than gently rock the cage within which the molecule is constrained.