Local affine-connection approach to the elastic constants of nematic liquid crystals

In this paper, the affine connection approach [D. Baalss and S. Hess, Phys. Rev. Lett. 57, 86 (1986)] will be defined in terms of a local transformation. Accordingly, the macroscopic anisotropy of the nematic medium will be conceived as resulting from a local transformation where, at each point of the sample, the spherical molecules of an idealized hypothetic isotropic liquid have their shape changed to the ellipsoidal form of the nematic liquid crystals molecules. When such local character is imposed to this transformation, the patterns determined by the director configuration of the nematic medium acquire an intrinsic curvature whose correct treatment requires the replacement of the techniques and methods of the usual calculus by those of the differential geometry of nonflat surfaces. Such an approach will be used in the calculus of the nematic elastic constants. As a result, the dependence of the elastic constants on the scalar order parameter, on the eccentricity of the nematic molecules, and on the interaction between them will be determined and compared with the experimental data of the 4-methoxybenzylidene-4-n-butylaniline.