Computer simulations of the elastic properties of liquid crystals

We have carried out extensive computer simulations of the Lebwohl-Lasher model of a nematic liquid crystal to determine the Frank elastic constant K for this system as a function of temperature. We use equilibrium, k-dependent, fluctuation expressions, valid at small wave vector k, and also measure the response to applied, spatially varying fields that couple to the k-dependent molecular orientation tensor. We find good agreement between these two methods. The ratio K/P¯ ${}_2{}^2{}_{}{}^{}$, where P${\mathrm{¯}}_2$ is the nematic order parameter, adopts its molecular-field value at low temperatures, but rises by over 20% on approaching the nematic-isotropic transition temperature ${\mathit{T}}_{\mathit{N}\mathrm{-}\mathit{I}}$. We note that this temperature dependence cannot be accounted for by molecular-field arguments. However, spin-wave theory correctly relates the k-dependent elastic constant with the order parameter at low temperatures, and is quite accurate up to T/${\mathit{T}}_{\mathit{N}\mathrm{-}\mathit{I}}$≊0.8. Finally, direct simulations of the Fréedericksz transition give results in reasonable agreement with the value of K as calculated by the other methods; however, we find that this is not a very efficient route to K.