Theory of the Frank Elastic Constants of Nematic Liquid Crystals

Abstract

It is shown that the Frank elastic constants may be expressed in terms of even-order Legendre polynomials averaged over the one-molecule orientational distribution function. In particular, it is found that $\frac{({K}_{11}\ensuremath{-}\overline{K})}{\overline{K}}=C\ensuremath{-}3{C}^{\ensuremath{'}}\frac{{\overline{P}}_{4}}{{\overline{P}}_{2}}+\ensuremath{\cdots},\frac{({K}_{22}\ensuremath{-}\overline{K})}{\overline{K}}=\ensuremath{-}2C\ensuremath{-}{C}^{\ensuremath{'}}\frac{{\overline{P}}_{4}}{{\overline{P}}_{2}}+\ensuremath{\cdots},\frac{({K}_{33}\ensuremath{-}\overline{K})}{\overline{K}}=C+4{C}^{\ensuremath{'}}\frac{{\overline{P}}_{4}}{{\overline{P}}_{2}}+\ensuremath{\cdots}$, where $\overline{K}=(\frac{1}{3})({K}_{11}+{K}_{22}+{K}_{33})$, $C$ and ${C}^{\ensuremath{'}}$ are constants, which depend on the details of the system, and ${\overline{P}}_{m}$ is the weighted average of the $m$th Legendre polynomial. Higher-order terms in these series involve ${\overline{P}}_{6}$, etc. The constants $C$ and ${C}^{\ensuremath{'}}$ are calculated for the case of rodlike molecules interacting via a hard-core repulsion. The results are in good agreement with experiments on the substance $p$-azoxyanisole.