Enhanced Landau-de Gennes potential for nematic liquid crystals from a systematic coarse-graining procedure

The macroscopic theory of nematics is conveniently described in terms of the phenomenological Landau-de Gennes free energy. Here we show how such an effective free energy can be obtained explicitly from a microscopic model via the help of a systematic coarse-graining procedure. We test our approach for the two- and three-dimensional Lebwohl-Lasher model of nematics. The effective free energy that we obtain is consistent with the phenomenological Landau-de Gennes form for weak orientational ordering and the Maier-Saupe theory of the isotropic-nematic transition. For strong orientational ordering, however, the effective free energy increases rapidly and diverges logarithmically near the fully oriented state. The explicit form for the regularized Landau-de Gennes potential proposed here restricts the order parameter to physical admissible values and reproduces our numerical data accurately.