Computer simulation studies of anisotropic systems. XXVII. The direct pair correlation function of the Gay–Berne discotic nematic and estimates of its elastic constants

Monte Carlo simulations have been performed for a discotic liquid crystal composed of Gay–Berne particles. On the basis of these simulations for the nematic phase, a subset of the spherical harmonic expansion coefficients of the direct pair correlation function (DPCF) were determined from the pair distribution function (PDF) by solving the Ornstein–Zernike (OZ) equation. This was achieved by generalizing the Wiener–Hopf factorization scheme for the numerical solution of the OZ equation. Only the expansion coefficients $g_{l_1{\mathrm{,l}}_2\mathrm{,l}}\mathrm{(r)}$ ${\mathrm{(l}}_{\alpha }\text{⩽4)}$ of the PDF in the laboratory frame were used when solving the OZ equation; this means that the DPCF so obtained is equivalent to that for a nematic in which the director is randomly distributed. From the DPCF, the scaled Oseen–Zöcher–Frank elastic constants $K_{11}^{*},$ $K_{22}^{*},$ and $K_{33}^{*},$ as well as the surface constant $K_{13}^{*},$ have been calculated from the subset of expansion coefficients. Generally, we find that $K_{33}^{*}{\mathrm{<K}}_{11}^{*}{\mathrm{<K}}_{22}^{*},$ in agreement with what is expected and found for discotic nematics. These results are quantitatively but not qualitatively different from those calculated with the help of analytical approximations for the same spherical harmonic expansion coefficients of the direct pair correlation function. For example, the values of the bulk elastic constants determined via the OZ equation are about three times larger than the bulk elasticity obtained with the low density approximation.