Smectic membranes are perfect model systems for studying low-dimensional phase transitions and the associated fluctuations. During the last two decades we have seen important progress in the understanding of the structure and fluctuation behavior of these systems, driven by both new experimental techniques and theoretical developments. Phase transitions are reviewed involving liquid, hexatic, and crystalline layers, which provide several types of model system for low-dimensional melting. The authors discuss the influence of the surfaces on the physical properties of the membranes as well as the crossover from three-to two-dimensional behavior. The layer-displacement fluctuations in smectic membranes have been investigated by specular and diffuse x-ray reflectivity. Theoretical and experimental aspects of the displacement-displacement correlation function are discussed. Of special interest is the quenching or enhancement of fluctuations at surfaces, which is directly related to the phenomenon of surface ordering. The authors consider the conditions under which fluctuations are conformal throughout a membrane, and then the dynamic aspects of the layer-displacement correlation function, which include the effects of finite size, surface tension, and viscous dissipation. This leads in smectic membranes to a discrete spectrum of elastic and viscous relaxation modes, which have been studied experimentally with coherent x rays at third-generation synchrotron sources. The fluctuating character of crystalline-B membranes is also considered. Finally, the article looks briefly at thinning transitions, smectic membranes of chiral molecules, smectic films on substrates, and applications to biologically relevant systems. Open questions and future trends in the field are discussed. CONTENTS I. Motivation and Scope 182 II. Smectic Membranes 183 A. Smectic liquid-crystal phases 183 B. Main features of x-ray patterns 185

The natural structure for smectic-A liquid crystals arranged in a sample with homogeneous boundary conditions is the so-called bookshelf structure with uniform layers perpendicular to the sample cell plane. However, this structure often deforms into the so-called chevron structure when the sample is cooled. This deformation is usually thought to result from the mismatch between bulk and surface layer thicknesses. In this paper we study the dynamics of chevron formation. Two possible scenarios are envisaged. In one of these there is strong coupling between layer deformation and fluid flow, and in the other the fluid essentially does not move. In this paper we examine the first scenario, leaving the second, slower relaxation mode for another paper. Analytic solutions are found for near-critical deformations, and numerical solutions are found beyond the critical regime. ͓S1063-651X͑98͒06912-8͔

Coherent dynamic x-ray scattering has been used to study the thermally excited layer fluctuations in freely suspended smectic films of the compound 4O.8. Using 8-keV x rays and films with a thickness around 0.3 &mgr;m we resolve relaxation times down to a few &mgr;s. A combination of damped and oscillatory behavior is observed for the layer undulations, which can be attributed to inertial effects. These are due to the surface contribution to the free energy which cannot be disregarded for thin films.

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We present a theoretical study of the dynamic displacement-displacement and intensity-intensity (for coherent soft x-ray scattering) correlations in Sm-A as well as Cr-B free-standing films. The work is based on a continuous hydrodynamic model that allows one to calculate efficiently the dynamic correlation functions and considerably simplifies earlier analyses of finite-size and surface effects in Sm-A films. The model is extended to Cr-B films. We show that despite the crystalline order, the Cr-B film is a strongly fluctuating system, which is due to an abnormally small shear elastic constant. An easy-shear approximation is developed to describe the fluctuations in the Cr-B phase. We predict nonmonotonic behavior of the intensity-intensity correlation function in both Sm-A and Cr-B films. The analysis can be applied to either coherent x-ray or conventional laser dynamic light scattering experiments.