Continuum theory of thermotropic systems. Hydrodynamics of liquid crystals

Abstract

The hydrodynamical parameters, i.e., the long-wave length and low-frequency properties of any homogeneous system are completely determined by the conservation laws and the static properties of the system, as described by various “elastic constants”. In this paper we present a unified formulation of hydrodynamics equally suitable for normal liquids, liquid crystals and solids, and apply it in detail to the nematic phase. The generality of this approach is of particular importance for the description of the dynamics near phase transitions. As variables we use quantities which have a meaning in any phase, namely, the densities of conserved quantities (particle number, momentum, angular momentum, and energy) and the entropy density, together with the corresponding current densities. In a continuum theory the various phases are characterized by their symmetry and by the static stresses, that they can sustain.The hydrodynamic equations including dissipation are then derived making use only of the familiar local conservation laws and general thermodynamic relations. We thus avoid introduction of hydrodynamic variables related to order parameters which are specific to particular phases. Consequently, we do not need any new conservation laws. This is accomplished by utilizing the conservation of angular momentum which is no longer a consequence of the conservation of linear momentum, if the particles have internal (rotational) degrees of freedom, as is essential for liquid crystals. We discuss for the nematic phase how the number of elastic and dissipative constants is reduced by restricting the most general set of equations by means of approximations which have been made in previous theories.Response functions and fluctuations of hydrodynamic variables are calculated in the hydrodynamic regime, particularly for the nematic phase. The hydrodynamic modes are discussed in detail for nematic liquid crystals. Apart from the compressional waves, one also gets the modes describing mainly the motion of the molecular axes. They are of diffusion type, as has been discussed previously. The coupling of compressional and orientational modes yields a characteristic anisotropy of sound attenuation and would thus supplement the experiments on orientational modes.