X-ray cross-correlation analysis of liquid crystal membranes in the vicinity of the hexatic-smectic phase transition

Abstract

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...

Structural studies of the bond-orientational order and hexatic–smectic transition in liquid crystals of various compositions

“…Within such an approach the BO order parameters are determined by fitting the measured azimuthal intensity distribution with the Fourier cosine series. In our previous X-ray studies [22,23] we have applied the angular X-ray crosscorrelation analysis (XCCA) to study the BO order in single-domain Hex-B films. Angular XCCA has been developed to investigate the structure of partially ordered systems to detect hidden symmetries and weak angular correlations [24][25][26][27][28][29].…”

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