Hydrodynamics and Electrohydrodynamics of Liquid Crystals

Abstract

We present the hydrodynamic and electrohydrodynamic equations for uniaxial nematic liquid crystals and explain their derivation in detail. To derive hydrodynamic equations, which are valid for sufficiently small frequencies in the limit of long wavelengths, one identifies first the hydrodynamic variables, which come in two groups: quantities obeying conservation laws and variables associated with spontaneously broken continuous symmetries. As variables that characterize the spontaneously broken continuous rotational symmetries of a nematic liquid crystal we have the deviations from the preferred direction, which is characterized by the director, a unit vector that does not distinguish between head and tail. To derive the hydrodynamic equations we make use of symmetry arguments and irreversible thermodynamics. Among the symmetry properties used are the behavior under time reversal and spatial parity, Galilean covariance, and the invariance under rotations and translations. In a first step one writes down the Gibbs-Duhem relation and expands the thermodynamic forces, which are defined via the Gibbs-Duhem relation, into the hydrodynamic variables. In the second and final step to close the system of hydrodynamic equations, one expresses the currents (and quasi-currents) appearing in the conservation laws (and in the balance equations for the variables associated with the broken symmetries) by the thermodynamic forces. The currents and quasi-currents are split into two contributions, reversible ones that lead to vanishing entropy production and into dissipative ones that are associated with positive entropy production. We discuss how the effect of static and dynamic electric fields (as well as static magnetic fields) can be combined with hydrodynamics to get the electrohydrodynamic equations for uniaxial nematic liquid crystals. We will critically examine which part of the Maxwell equations must be combined with the hydrodynamic equations to get a consistent description at low frequencies and long wavelengths. We consider a number of additions to nematodynamics. First we investi- gate how the electrohydrodynamic equations are modified when thermodynamic fluctuations are taken into account and we analyze which additional terms have to be incorporated if highly nonlinear effects are present or if one deals with spatially strongly inhomogeneous situations (in which case higher order gradient terms enter the picture). In many situations, for example close to phase transitions, when defects are present or for polymeric systems, one must take into account additional variables in a macroscopic description, that are not strictly hydrodynamic but relax sufficiently slowly in the long wavelength limit. One such variable is the modulus of the order parameter, whose spatio-temporal behavior becomes of macroscopic importance close to phase transitions (e.g. to the isotropic phase) or for lyotropic nematic liquid crystals (multi-component systems, whose properties vary predominantly with composition), for which the modulus can vary spatially, since there are spatial variations in the concentration of the constituents. Another example is the strain associated with the transient network in liquid crystalline side-chain polymers for which the mesogenic units are attached to the polymeric backbone via a flexible spacer. Finally we discuss biaxial nematic liquid crystals, which are characterized by two (and thus three) preferred directions. In contrast to uniaxial nematics, which are found for rod-shaped and disk-shaped (discotic) molecules in thermotropic (properties change predominantly as a function of temperature) low molecular weight materials, biaxial nematic phases have been shown to exist mainly for lyotropic and polymeric systems so far. We summarize briefly how the hydrodynamics of other liquid crystalline phases with spontaneously broken continuous rotational symmetries is influenced by director type degrees of freedom. Among these systems are cholesteric liquid crystals, which are characterized by a helical superstructure, and various tilted smectic liquid crystalline phases that have anisotropic in-plane fluidity: smectic C, CM, F, I and L phases and the appropriate chiral phases. In the Appendix we give the complete set of electrohydrodynamic equations for uniaxial nematics in compact form and we show how the present description is related to the frequently used continuum-type approach of Ericksen and Leslie discussing critically the incompressibility approximation inherent to this approach.