Polar structure of disclination loops in nematic liquid crystals probed by second-harmonic-light scattering

Angle-resolved, second-harmonic-light scattering (SHLS) measurements are reported for three different classes of thermotropic nematic liquid crystals (NLCs): polar and nonpolar rodlike compounds and a bent-core compound. Results revealing well-defined scattering peaks are interpreted in terms of the electric polarization induced by distortions of the nematic orientational field (“flexopolarity”) associated with inversion wall defects, nonsingular disclinations, analogous to Neel walls in ferromagnets, that often exhibit a closed loop morphology in NLCs. Analysis of the SHLS patterns based on this model provides a “proof-of-concept” for a potentially useful method to probe the flexopolar properties of NLCs.

Figure 1

Chemical structures and transition temperatures for the liquid crystal molecules studied.

Figure 2

Schematic diagram of the experimental setup.

Figure 3

Representative CCD images of the SH scattering patterns for the $V\left(\omega \right)\to V\left(2\omega \right)$ process, recorded at $T_{NI}-T=2.2{}^{\circ }\mathrm{C}$ from (left to right) the BCN, 5CB, and C7 nematic samples. The direction of the applied magnetic field is horizontal, as indicated. Bottom: The SH power plotted as a function of scattering angle (in the scattering plane parallel to $\mathbf{H})$, after integrating over vertical columns of $\pm 15$ pixels from the horizontal axis through the center. The region used for this average is shown in white outline in the center image, top panel. (As discussed in the text, the peaks are artificially narrowed due to focusing by the telescope optics.)

Figure 4

SH intensity for the BCN sample, integrated over the scattering pattern recorded on the CCD and plotted versus square of the incident IR pulse energy. The solid line is a fit that confirms the expected quadratic relation between the incident and SH powers.

Figure 5

SH scattering pattern versus temperature in the nematic phase of the BCN for the $V\left(\omega \right)\to V\left(2\omega \right)$ process. Upper left, upper right, and lower left: $T_{NI}-T=0.2$, $5.2$, and $10.2{}^{\circ }\mathrm{C},$ respectively. Lower right: SH power (averaged as described in Fig. 4) versus scattering angle, showing the shift of the peak power to larger scattering angles with decreasing $T$ in the nematic phase $(T_{NI}-T=0.2$, $2.2$, $5.2$, $10.2{}^{\circ }\mathrm{C},$ left to right).

Figure 6

Peak scattering angle $\left({\theta }_0\right)$ versus temperature for the two processes $V\left(\omega \right)\to V\left(2\omega \right)$ (points in red) and $H\left(\omega \right)\to V\left(2\omega \right)$ (blue points) for each of the materials studied. Top row: Raw data. Bottom row: Results after rescaling the data for the $V\left(\omega \right)\to V\left(2\omega \right)$ process to match the $H\left(\omega \right)\to V\left(2\omega \right)$ data. The solid lines are power law fits described in the text, and the dashed vertical lines indicate the first order $NI$ transition temperature. The disappearance of SH signal at $T_{NI}$ is indicated by a single data point in each case.

Figure 7

Top: Integrated SH intensity versus applied field squared for the 5CB sample. Bottom: Linear light transmission (forward direction) versus $H^2$ for a 543 nm laser beam.

Figure 8

Dependence of the SH scattering on the fundamental $\left(\omega \right)$ and SH $\left(2\omega \right)$ polarizations for $T_{NI}-T=2.2{}^{\circ }\mathrm{C}$.

Figure 9

Polarizing microscope images of metastable inversion wall loops in the three different nematic liquid crystals studied: (a) 1 mm thick 5CB, (b) 1 mm thick C7, (c) 1 mm thick BCN, (d) 0.1 mm thick 5CB. The images were recorded after cooling the samples from the isotropic state to a temperature $2.2{}^{\circ }\mathrm{C}$ below the isotropic-nematic transition under a constant 1.13 T magnetic field, directed approximately left to right in the figure. The microscope was focused on a plane just below the sample-substrate interface where the features of the loops are sharpest; the resolution is limited due to the thickness of the samples. The images are rendered in gray scale to facilitate comparison.

Figure 10

Left panel (a): Schematic illustration of an inversion wall loop in an aligned nematic (after Ref. [27]). The dashed oval traces the center of the wall (points where the angle $\varphi$ of the director $\mathbf{n}$ reaches $\pi /2$ with respect to the equilibrium orientations $\varphi =0,\pi$ along the aligning field $\mathbf{H})$. The left and right sides of the loop along the $z$ direction contain a splay-dominated mixture of splay-bend director distortions (labeled $sb$ in the text), while the top and bottom sides along the $x$ direction are richer in bend (and are labeled $bs$ in the text). Arrows indicate the rotation of the $\mathbf{n}$ through the wall. Middle panel (b): Qualitative map of flexoelectric polarization in the $sb$ and $bs$ regions of the loop for the specific case of equal and opposite splay and bend flexoelectric coefficients, $e_1=-e_3.$ (The results of a detailed calculation are presented in Fig. 11.) Right panel (c): Map of the average induced nonlinear polarization resulting from the distribution of flexoelectric polarization in (b). Note the opposite orientations of ${\mathbf{P}}_{2\omega }$ across the diameter $D$.

Figure 11

Top row: Angle of the director $\varphi$ (relative to the applied field $\mathbf{H})$ plotted as a function of $z/\xi$ through the left and right sides of the inversion loop $(sb$ regions), and as a function of $x/\xi$ through the top and bottom sides $(bs$ regions). The plots are based on Eq. (6). The $x$ and $z$ directions are defined in Fig. 9; the origin corresponds to the center of the inversion wall in each case. Bottom row: Corresponding components of the flexoelectric polarization plotted as a function of position for the case $e_1=-e_3$, according to Eq. (6). $P_x^f\left(P_z^f\right)$ is shown as a blue (red) curve.