In this thesis we want to study two phenomena that can be interpreted as a consequence of the non-uniform molecular configuration in a nematic liquid crystal sample. It is organized as follows. Chapter 1 gives a brief introduction to nematic liquid crystals. It is meant to provide a neat summary of the mathematical and physical tools that will be used in the remaining Chapters. The equilibrium configuration of a nematic liquid crystal bounded by a rough surface is studied in Chapter 2. The surface wrinkling induces a partial melting in the degree of orientation. This softened region penetrates the bulk up to a length scale which turns out to coincide with the characteristic wavelength of the corrugation. Within the boundary layer where the nematic degree of orientation decreases, the tilt angle steepens and gives rise to a nontrivial structure, which may be interpreted in terms of an effective weak anchoring potential. It is then possible to relate the effective surface extrapolation length to the microscopic anchoring parameters. We also analyze the crucial role played by the boundary conditions assumed on the degree of orientation. Quite different features emerge depending on whether they are Neumann- or Dirichlet-like. These features may be useful to ascertain experimentally how the degree of orientation interacts with an external boundary. Chapter 3 is devoted to the study of biaxiality. Nematic liquid crystals possess three different phases: isotropic, uniaxial, and biaxial. The ground state of most nematics is either isotropic or uniaxial, depending on the external temperature. Nevertheless, biaxial domains have been frequently identified, especially close to defects or external surfaces. We show that any spatially varying director pattern may be a source of biaxiality. Indeed, in the Chapter we introduce a symmetric tensor whose eigensystem may be used to easily determine directions and extent of naturally induced biaxiality. Finally, these general considerations are applied to some examples. In particular, when homeotropic anchoring is enforced on a curved surface, the order tensor becomes biaxial along the principal directions of the surface. The effect is triggered by the difference in surface principal curvatures.

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