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...

The dynamics of the layer-displacement fluctuations in smectic membranes have been studied by x-ray photon correlation spectroscopy (XPCS). We report transitions from an oscillatory damping regime to simple exponential decay of the fluctuations, both as a function of membrane thickness and upon changing from specular to off-specular scattering. This behavior is in agreement with recent theories. Employing avalanche photodiode detectors and the uniform filling mode of the synchrotron storage ring, the fast limits of XPCS have been explored down to 50 ns.

We present an x-ray study of liquid crystal membranes in the vicinity of the hexatic-smectic phase transition by means of angular x-ray cross-correlation analysis. By applying two-point angular-intensity cross-correlation functions to the measured series of diffraction patterns the parameters of bond-orientational (BO) order in hexatic phase were directly determined. The temperature dependence of the positional correlation lengths was analyzed as well. The obtained correlation lengths show larger values for the higher-order Fourier components of BO order. These findings indicate a strong coupling between BO and positional order. About 30 years ago it was realized that the phase transition between a two-dimensional (2D) crystal and the liquid phase can proceed through an intermediate hexatic phase [1]. The corresponding mechanism involves dissociation of dislocation pairs. The 2D hexatic phase is characterized by a sixfold quasilong-range bond-orientational (BO) order, while the positional order is short range and the shear modulus is zero. Phases with hexatic order have been found in several systems, such as electrons at the surface of helium [2], charged polymer colloids [3,4], and smectic liquid crystals (LCs) [5][6][7]. In the last case a hexatic phase was experimentally observed quite unexpectedly in three-dimensional (3D) stacked molecular systems [8]. In the 3D hexatic phase the positional order is short range, while the BO order persists over a long range. However, the mere existence of a hexatic phase does not imply a specific melting mechanism, and the origin of hexatic order and its relation to the defect-mediated melting transition is still controversial. Smectic liquid crystals are particularly suitable to investigate these problems, as they can be suspended over an opening in a solid frame. Such smectic membranes are substrate-free and have a controlled thickness ranging from two to more than thousands of layers [9].Smectic-A membranes can be described as stacks of liquid layers. The in-plane structure is liquidlike with positional correlations between the molecules decaying exponentially with a correlation length ξ 0 . While cooling a hexatic phase may occur, which shows a long-range BO order. This leads to a sixfold rotational symmetry and the BO correlations are characterized by the local ordering field ψ(r) ∝ exp[i6θ (r)], where θ (r) is the angle between the "bonds" and some reference axis. Upon decreasing the temperature, the width of the radial-intensity peak decreases simultaneously with a further development of the BO order. This indicates a coupling * Present address: The University of California, San Diego, La Jolla, CA 92093, USA.† Corresponding author: ivan.vartaniants@desy.de between the positional correlations and the BO order [10]. Additionally, at even lower temperatures a 3D crystalline phase appears with a hexagonal in-plane lattice or a rectangular lattice with a so-called herringbone order [6,7]. The common way to study the BO order is to perform x-ray or electron diffraction me...

We report on the X-ray studies of freely suspended hexatic films of three different liquid crystal compounds. By applying angular X-ray cross-correlation analysis (XCCA) to the measured diffraction patterns the parameters of the bond-orientational (BO) order in the hexatic phase were directly determined. The temperature evolution of the BO order parameters was analyzed on the basis of the multicritical scaling theory (MCST). Our results confirmed the validity of the MCST in the whole temperature range of the existence of the hexatic phase for all three compounds. The temperature dependence of the BO order parameters in the vicinity of the hexatic-smectic transition was fitted by a conventional power law with a critical exponent β ≈ 0.1 of extremely small value. We found that the temperature dependence of higher order harmonics of the BO order scales as the powers of the first harmonic, with an exponent equal to the harmonic number. This indicates a nonlinear coupling of the BO order parameters of different order. We demonstrate that compounds of various compositions, possessing different phase sequences at low temperatures, display the same thermodynamic behavior in the hexatic phase and in the vicinity of the smectic-hexatic phase transition.

An analysis is given of the various contributions to the x-ray diffraction in smectic-A liquid crystals, where several effects due to the real structure are superimposed upon the algebraic decay of the correlations due to thermal fluctuations. The finite size of either the sample or intrinsic domains is shown not only to determine the half-width of the Bragg peak but also to contribute to the wings of the diffraction curve, thus overlapping with the algebraic decay. The asymmetry of the peak-intensity distribution observed for nonoriented (powder) samples is also explained. For samples with a small mosaicity, the transition from the limit of powder diffraction to that of a perfectly oriented sample is calculated as a function of the wave-vector magnitude. Finally the possible broadening of the x-ray peak due to dislocations is analyzed. Results of various high-resolution experiments on monomeric, polymeric, and lyotropic smectic liquid crystals are analyzed and the possible sources of observed real-structure effects are discussed. PACS number(s): 61.30.v, 61.10.Dp

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