Física dos cristais líquidos e gravitação : pontos de encontro - Pereira, Erms Rodrigues (2011)
Aspects of the physics of nematic liquid crystals are studied in this thesis from the viewpoint of riemannian geometry through analogue models of gravitation. The topics chosen for study were: geometric and wave optics, elastic waves, hydrodynamics and heat conduction. The main analogue model used is based on the interpretation of Fermat’s principle as a process to obtain null geodesics, where the liquid crystalline material is seen as a riemannian manifold. This approach predicts that the metric effectively felt by the light ray depends on the configuration of molecules in the liquid crystal and on the parallel and perpendicular refractive indexes to the axis of symmetry of the liquid-cristal molecule. It is known that, for the particular case of the existence of topological defects within the material, effective metric similar to cosmological defects (like global monopoles and cosmic strings) are obtained. This thesis develops itself on the situation where there are topological defects of hedgehog type and (k = 1, c = 0) disclination type in the nematic phase of the liquid crystalline material. The first problem studied, as a review, deals with the wave optics, with respect to the light diffracted by the cited defects. Since plane waves of small wavelength have identical trajectories to light rays, the use of analog model is therefore justified. Thus, we show that light scattered by these defects generates a characteristic diffraction pattern, being the location given by an algebraic expression dependent on the parallel and perpendicular refractive indexes to the axis of symmetry of the molecule. We also show how theses patterns depend on the temperature of the material. The second studied problem deals with the geometrical optics and hydrodynamics of the nematic liquid crystals. From a molecular configuration similar to a (k = 1, c = 0) disclination, we let the material flow radially towards the axis of the defect. Then, using the hydrodynamic fact that velocity gradients in the material locally change the refractive index of the molecule, we find the velocity profile that must exist around the defect so that the metric actually experienced by light traveling in the plane perpendicular to the axis the defect is the Schwarzschild one in the equatorial plane, with the Schwarzschild radius interior to the object. We found that the absolute values of the velocity of liquid crystalline fluid can be order of a few meters per second, differing greatly from the values obtained by Gordon metric for an isotropic fluid under identical conditions. The third studied problem deals with the elastic oscillations in the presence of topological defects. Similarly to the first problem, the trajectory of the sound is obtained by an elastic version of Fermat’s principle and, then, compared with a null geodesic. We show how topological defects influence the sound trajectories and the sound diffracted by them. The fourth problem deals with the heat conduction in the vicinity of defects. Considering that the defects come from an addition or removal of portion of the material, letting the medium relaxes elastically, effective metric of the space disturbed by the defect are found, with expressions similar to those obtained by the analogous model based on Fermat’s principle. These metrics generate a modified thermal conductivity tensor, allowing the study of the temperature field in this situation. We show that, depending on the values of parallel and perpendicular thermal conductivity to the axis of symmetry of the molecule and on the defect in question, the temperature gradient can be accentuated or attenuated on the defect, allowing control of the response thermal temperature of the material, according to the presence of defects. Encouraging a greater understanding of the physics of liquid crystals and its use as a background in analogue models of gravity is the main theme of each analyzed problem.