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

Scanning force microscopy on monomolecular films of eicosylperfluorotetradecane, F(CF(2))(14)(CH(2))(20)H, on mica, silicon oxide, or water revealed spontaneous organization to well-defined nanoscopic ribbon and spiral or toroidal superstructures. Whether ribbons or nanospirals were formed depended on the solvent from which the molecular monofilm was cast. Ribbons were observed when a hydrocarbon or a perfluorocarbon solvent was used, e.g., decalin or perfluorodecalin. When the compound, however, was deposited from nonselective hexafluoroxylene, the molecules assembled into spirals of defined size. The spirals/toroids transformed to ribbons when exposed either to decalin or perfluorodecalin vapor, and the ribbons transformed to toroids when exposed to hexafluoroxylene vapor. These changes could be observed in situ. Scanning force microscopy yielded an identical height and width for the bands forming the spirals and for the parallel flat ribbons. X-ray reflectivity yielded a height of 3.61 +/- 0.05 nm, again identical for both morphologies. Yet, the length of the extended F(CF(2))(14)(CH(2))(20)H molecule, i.e., 4.65 nm, exceeds the layer thickness obtained from X-ray reflectometry. It is, however, consistent with an arrangement where the fluorinated chains are oriented normal to the surface layer and where the alkyl segments are tilted with a 122 degrees angle between the two segments. Within the plane defined by the tilt, this angle allows a dense packing of the alkyl segments compensating for the larger cross-section of the fluorocarbon segment. The tilt plane defines an "easy" direction along which the monolayer structure can preserve order. In the plane perpendicular to this axis, long-range ordered dense packing of the alkyl chains is not possible. Incommensurable packing can in principle explain the finite and regular width of the ribbons and the stepwise turn in the spirals.

We present an x-ray study of freely suspended hexatic films of the liquid crystal 3(10)OBC.Our results reveal spatial inhomogeneities of the bond-orientational (BO) order in the vicinity of the hexatic-smectic phase transition and the formation of large scale hexatic domains at lower temperatures. Deep in the hexatic phase up to 25 successive sixfold BO order parameters have been directly determined by means of angular x-ray cross-correlation analysis (XCCA). Such strongly developed hexatic order allowed us to determine higher order correction terms in the scaling relation predicted by the multicritical scaling theory over a full temperature range of the hexatic phase existence. 1The influence of angular correlations on structural and physical properties of complex fluids, colloidal suspensions and liquid crystals (LCs) remains one of fundamental and unresolved problems in modern condensed matter physics [1]. A prominent example of a system with angular correlations is the hexatic phase that combines the properties of both crystals and liquids [2]. The two-dimensional (2D) hexatic phase shows a sixfold quasi-long range bond-orientational (BO) order, while the positional order is short range [3]. The hexatic phase is a general phenomenon that was observed in a number of systems of various physical nature, such as 2D colloids [4][5][6], electrons at the surface of helium [7], 2D superconducting vortexes [8,9] and, particularly, in liquid crystals [10][11][12][13].The hexatic phase was predicted by Halperin and Nelson [14] as an intermediate state in 2D crystal melting. According to their theory the hexatic phase arises as a consequence of the broken translational symmetry of a 2D crystal induced by dissociation of dislocation pairs.This mechanism does not work in 3D crystals, however, the 3D hexatic phase was observed experimentally in LCs [10]. The multicritical scaling theory (MCST) developed by Aharony and coworkers [15] based on renormalization group approach to critical phenomena enabled quantitative characterization of the BO order in the hexatic phase and, particularly, allowed to study a crossover from 2D to 3D behavior [11,16]. In spite of the extensive experimental and theoretical work the origin of the hexatic phase in LCs and the features of the hexatic -smectic phase transition remain puzzling and controversial.The structure of hexatics is traditionally studied by means of x-ray or electron diffraction in a single-domain area of a hexatic film (see for reviews [17][18][19]). The quantitative characteristics of the BO order, the so-called BO order parameters [12], are typically determined by fitting the measured azimuthal intensity distribution by the Fourier cosine series. In contrast to this approach in the present work we performed spatially resolved x-ray diffraction studies of free standing LC films. Measured x-ray data were analyzed by means of direct Fourier transformation and by using angular x-ray cross-correlation analysis (XCCA) [20][21][22][23]. The latter method enabled a direct d...