Equilibrium and Nonequilibrium Statistical Mechanics of Membranes, Liquid Crystal Films, and Other Layered Structures.

In this thesis, we develop and analyze a continuum Landau theory for chiral and achiral lipid bilayers. This theory contains couplings between tangent-plane orientational order and curvature that lead to "rippled" phases with one-dimensional height modulations and to phases with two -dimensional height modulations. We calculate the mean -field phase diagrams by using both analytical and numerical methods. We generalize our theory to study the equilibrium phase diagrams of liquid crystal films. Both bulk smectics and freely suspended films are considered. For flexoelectric systems, continuous structural phase transitions are predicted among square-lattice, hexagonal, and distorted square-lattice phases as a function of the applied electric field. It is also shown that only uniform flat phases are predicted for thin films. One-dimensional ripple phases and two -dimensional square lattice phases can occur with increasing film thickness. Secondly we study the growth and instability of Myelin figures. For quasi-equilibrium growth, we predict a growth rate proportional to t^{-1/2 }, where t is the growth time. The proportional constant is inversely proportional to the viscosity of the fluid. Myelin figures are unstable under dehydration. The initial instability of myelin figures develops periodic arrays of bumps at the surface with a wave length of about 1 mum. This morphological change is induced by increasing the ratio of surface area to volume of myelin figures due to dehydration. We interpret this initial instability from energetical considerations and calculate the preferred wave length. Finally, we study theoretically the swelling kinetics of layered structures, particularly triblock copolymer mesogels. The gels are swollen by a solvent good for the bridging block but poor for the nonbridging block. At late stages the penetration front moves as in ordinary diffusion. However, the bending elasticity of the non -bridging layers leads to an initial t^{1/6 } relaxation of the penetration front. The crossover length between these two regimes is approximately the width of a single layer. However, for a large number of lamellae there is a cooperative effect which leads to a large enhancement of this crossover length. These results apply to the swelling of other layered structures, such as clays.