Fluctuations and stability of liquid crystal order in the presence of a heterogeneous substrate

Substrates are essential in liquid crystal applications. The imperfection intrinsic to a substrate imposes heterogeneous surface perturbations on the system, the response to which is of both fundamental and applied interest. In this thesis we present a systematic theoretical study of liquid crystal cells with a heterogeneous substrate and study the stability of liquid crystal order under these random surface perturbations as well as induced random textures.;We find that surface heterogeneity can have a substantial influence on the bulk liquid crystal order. For a substrate imposing a strong surface random pinning, topological defects are expected to be induced. Motivated by this we studied as an example a vortex with winding 2pip imposed on the surface. We present an exact three-dimensional solitonic solution to the sine-Gordon-type Euler-Lagrange equation describing the induced bulk texture with a prescribed polar tilt angle on a planar substrate and escaping into the third dimension in the bulk. The solution is relevant to characterization of a Schlieren texture in nematic liquid-crystal films with tangential (in-plane) substrate alignment.;On the other hand we expect that for weak surface heterogeneities no topological defects are induced by the surface pinning, leading to only elastic distortions. To understand this regime, we model a thick nematic liquid crystal cell with a random heterogeneous substrate as a bulk xy model with quenched disorder confined to a surface, and study its statistical properties. We find that at long scales the nematic order is marginally unstable to such surface pinning. We compute short scale correlations and the characteristic length scale that separates the short and long distance behaviors using the random torque approximation. To characterize the induced random texture at long scales, we study the system by the functional renormalization group and matching methods, and find universal logarithmic and double-logarithmic distortions in two and three dimensions, respectively. We also study finite-thickness cells with a second homogeneous substrate, obtaining crossover behaviors in the nematic texture as a function of pinning strength and cell thickness. We derive the corresponding polarized light microscopy signal for future comparison with experiments.;We extend the theory to a smectic cell with weak surface heterogeneities. We model this system as a harmonic elastic model with both surface random positional and surface orientational random pinning. Similar to the surface nematic problem, we studied the short and long scale behaviors. We calculate the two length scales along and perpendicular to the smectic layers on the heterogeneous substrate characterizing anisotropic ordered domains (beyond which smectic order is absent). We then apply the functional renormalization group to characterize elastic smectic distortions on long scales. We find a three dimensional Cardy-Ostlund-like phase transition between a thermal smectic state at high temperature and a glass-like pinned state at lower temperature. We compute the corresponding polarized light microscopy signal and based on it argue that this glass transition provides a reasonable explanation for recent experimental observations in smectic cells.