Theory of the Frank Elastic Constants of Nematic Liquid Crystals

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 $\frac{(K_{11}-\overline{K})}{\overline{K}}=C-3\mathrm{}{C}^{\prime }\frac{{\overline{P}}_4}{{\overline{P}}_2}+\cdots ,\frac{(K_{22}-\overline{K})}{\overline{K}}=-2C-C^{\prime }\frac{{\overline{P}}_4}{{\overline{P}}_2}+\cdots ,\frac{(K_{33}-\overline{K})}{\overline{K}}=C+4\mathrm{}{C}^{\prime }\frac{{\overline{P}}_4}{{\overline{P}}_2}+\cdots$, where $\overline{K}=\left(\frac{1}{3}\right)(K_{11}+K_{22}+K_{33})$, $C$ and $C^{\prime }$ 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 $C^{\prime }$ 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.