Theory of translational diffusion in nematic liquid crystals

The theory of translational diffusion in nematic liquid crystals is developed from the standpoint of hydrodynamics and molecular-hydrodynamic interactions. The anisotropy of the diffusivity is obtained and related to the orientational order parameter, viscosity coefficients, and the detailed molecular geometry. Liquid-crystal hydrodynamics, Riemannian geometry, and molecular chain space are utilized in the theoretical development. The results for the diffusivity parallel and perpendicular to the director in $p$-azoxyanisole (PAA) at 125°C are $D^{\parallel }=4.3\mathrm{}\times {10}^{-6\mathrm{}}$ and $D^{\perp }=3.1\mathrm{}\times {10}^{-6\mathrm{}}$ ${\mathrm{cm}}^2$/sec, respectively, giving an anisotropy ratio $\frac{D^{\parallel }}{D^{\perp }}=1.4\mathrm{}$. In addition, the temperature dependence of $D^{\perp }$ is obtained, using available viscosity data. $D^{\perp }$ ($D^{\parallel }$) shows an increase (decrease)—corresponding to the decrease in order parameter—as the isotropic transition temperature is approached.