Hydrodynamics of Cholesteric Liquid Crystals

The equations governing the linearized hydrodynamics of cholesteric liquid crystals are systematically deduced. They are valid for compressible as well as incompressible cholesterics and for arbitrary direction of mode propagation. The variables which contribute to the hydrodynamics are the conserved variables, mass density, energy density, and momentum density, and one additional broken-symmetry variable whose auto correlation function diverges at zero wave vector $\overrightarrow{\mathrm{k}}$. This divergent auto correlation function is determined from the Frank free energy for cholesterics and is found to diverge as ${({k_3}^2+c{k_{\perp }}^4)}^{-1\mathrm{}}$, where $c$ is a constant, $k_3$ is the component of $\overrightarrow{\mathrm{k}}$ parallel to the pitch axis, ${\overrightarrow{\mathrm{p}}}^0$, and $k_{\perp }$ is the component perpendicular to ${\overrightarrow{\mathrm{p}}}^0$. The form of this divergence implies that an infinite cholesteric is unstable with respect to fluctuations. The dephasing distance is, however, astronomical; and any finite sample is stabilized by its boundaries. The mode structure of the hydrodynamical equations is analyzed for an incompressible choleteric and for a compressible cholesteric for $\overrightarrow{\mathrm{k}}$ along the two symmetry directions. The spectrum for $\overrightarrow{\mathrm{k}}$ parallel to ${\overrightarrow{\mathrm{p}}}^0$ includes a diffusive velocity and a diffusive director mode in agreement with the work of Fan, Kramer, and Stephen. The spectrum for $\overrightarrow{\mathrm{k}}$ perpendicular to ${\overrightarrow{\mathrm{p}}}^0$ has a similar structure. For $\overrightarrow{\mathrm{k}}$ at an angle of 45° to ${\overrightarrow{\mathrm{p}}}^0$, there is a propagating shear wave for sufficiently small $k$. The velocity of longitudinal sound is very slightly anisotropic. Hydrodynamical forms of dynamic response functions are derived, and flow of a cholesteric in a cylindrical capillary is discussed.