Elastic effects of long-range quadrupolar interactions in nematic liquid crystals

We discuss the role of the quadrupolar interaction in nematic liquid crysta l samples in the shape of a slab limiting the study to planar deformations. Our analysis shows that this interaction gives rise to a bulk energy densi ty that, in the elastic approximation, depends linearly on the second spati al derivative and quadratically on the first spatial derivative of the nema tic orientation. We show that this bulk energy density can be separated in a surfacelike term, which gives rise just to a surface contribution, plus a term having the usual form. Both terms depend on the first derivative of t he tilt angle and are proportional to the square of the electrical quadrupo lar density. The bulk term, quadratic in the first derivative of the tilt a ngle, renormalizes the usual elastic energy density connected to the shea-r ange forces. The bulk elastic constant of quadrupolar origin can be negativ e and one order of magnitude smaller than the effective elastic constants f or typical nematic liquid crystals. According to our analysis this interact ion is responsible for an elastic anisotropy proportional to the square of the electrical quadrupolar density, which depends on the nematic orientatio n. The surfacelike term is proportional to the first derivative of the tilt angle. It calls mind to the splay-bend elastic term, although the tilt ang le dependence is more complicated. The relevant elastic constant is of the same order of magnitude as the bulk one, due to the same interaction. We ev aluate also the energy density in the surface layers, where the quadrupolar interaction is restricted by the surface. In this case we show that the fr ee energy contribution due to the surface layers is reduced to a classical anchoring energy. The solution of the variational problem by means of a sim ple version of the density functional theory is presented. [S1063-651X(98)1 1612-4].