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 ${({K}₁₁{-}K̄)}{K̄}=C{-}3{C}^{{'}}{{P̄}₄}{{P̄}₂}+{⋯},{({K}₂₂{-}K̄)}{K̄}={-}2C{-}{C}^{{'}}{{P̄}₄}{{P̄}₂}+{⋯},{({K}₃₃{-}K̄)}{K̄}=C+4{C}^{{'}}{{P̄}₄}{{P̄}₂}+{⋯}$, where $K̄=(1/3)({K}₁₁+{K}₂₂+{K}₃₃)$, $C$ and ${C}^{{'}}$ are constants, which depend on the details of the system, and ${P̄}ₘ$ is the weighted average of the $m$th Legendre polynomial. Higher-order terms in these series involve ${P̄}₆$, etc. The constants $C$ and ${C}^{{'}}$ 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.