Spatially resolved x-ray studies of liquid crystals with strongly developed bond-orientational order

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.

Figure 1

The scheme of the diffraction experiment showing the focusing system, glass sample stage with drawn LC membrane, and 2D detector.

Figure 2

(a–c) Diffraction patterns from a LC film in the smectic (a) and hexatic phase (b, c). (d) Temperature dependence of the positional correlation length $\xi$ determined from the radial profile of the scattering peak marked by an arrow in (b). The temperature of the hexatic-smectic phase transition is specified by an arrow. In the inset the temperature dependence of the peak maximum position $q_0$ is shown.

Figure 3

Spatially resolved 2D maps of the LC film structure at $T=66.3{}^{\circ }\mathrm{C}$ (a) and $T=66.1{}^{\circ }\mathrm{C}$ (b). Each vector in the plot corresponds to a certain position in the sample. Direction of a vector is associated with the angular position of the center of the peak marked with an arrow in Fig. 2, and the length of a vector is proportional to the magnitude $|I_6\left(q_0\right)|$. The length of the vectors in (a) is magnified five times as compared to the vectors in (b).

Figure 4

(a) Magnitudes of the FCs of intensity $|I_n\left(q\right)|$ with $n=6,12,...,78$ as a function of $q$ at the temperature $T=61.0{}^{\circ }\mathrm{C}$. Solid lines are SRL and Lorentzian fits to the experimental data (points) (see text for details). (inset) Magnitudes of the FCs $|I_n\left(q_0\right)|$ at $q_0=14.16 {\mathrm{nm}}^{-1}$ as a function of the order $n$. Red line represents the fitting using the MCST. (b) Temperature dependence of the HWHM ${\gamma }_n$ of the FCs of intensity.

Figure 5

(a) Temperature dependence of the normalized BO order parameters $C_{6m}$. In the inset the nonmonotonic behavior of the BO order parameters of the orders $m=4,...,10$ is shown. (b) Temperature dependence of the total number of FCs $M$ and positional correlation length $\xi$ [see Fig. 2].

Figure 6

(a) Temperature dependence of the parameter $\lambda \left(T\right)$ (red circles) in the scaling relation (6) from the MCST. Compare with the temperature dependence of the same parameter (black squares) when the linear correction term in the scaling relation (6) is neglected. (b) Temperature-averaged values of ${\langle {\sigma }_m\rangle }_T$ (red circles) and their fit with the scaling relation (6) with the first order correction term (black line). Results of fitting without the correction terms (green dash-dot line) and with the 2nd order correction term (blue dash line) are also shown.

Figure 7

The scattered intensity $I(q_0,\varphi )$ as a function of azimuthal angle at the temperatures $T=66.5{}^{\circ }\mathrm{C},T=66.1{}^{\circ }\mathrm{C},T=61.0{}^{\circ }\mathrm{C}$, and $T=55.0{}^{\circ }\mathrm{C}$.

Figure 8

The fitting of $C_{6m}$ with function (B2) at the temperatures $T=66.1{}^{\circ }\mathrm{C},T=64.0{}^{\circ }\mathrm{C},T=61.0{}^{\circ }\mathrm{C}$, and $T=55.0{}^{\circ }\mathrm{C}$.

Figure 9

Temperature dependence of the exponents ${\sigma }_m$ for all harmonics.