Hydrodynamics and Electrohydrodynamics of Liquid Crystals

We present the hydrodynamic and electrohydrodynamic equations for uni-axial nematic liquid crystals and explain their derivation in detail. To derive hydrodynamic equations, which are valid for suuciently small frequencies in the limit of long wavelengths, one identiies rst 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 behaviour under time reversal and spatial parity, Galilean co-variance, and the invariance under rotations and translations. In a rst step one writes down the Gibbs-Duhem relation and expands the thermo-dynamic forces, which are deened via the Gibbs-Duhem relation, into the hydrodynamic variables. In the second and nal 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 eeect of static and dynamic electric elds (as well as static magnetic elds) 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 investigate how the electrohydrodynamic equations are modiied when thermody-namic uctuations are taken into account and we analyze which additional terms have to be incorporated if highly nonlinear eeects 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 suuciently slowly in the long wavelength limit. One such variable is the modulus of the …