Fluctuation Modes of a Twist-Bend Nematic Liquid Crystal

We report a dynamic light-scattering study of the fluctuation modes in a thermotropic liquid crystalline mixture of monomer and dimer compounds that exhibits the twist-bend nematic (${\mathrm{N}}_{\mathrm{TB}}$) phase. The results reveal a spectrum of overdamped fluctuations that includes two nonhydrodynamic modes and one hydrodynamic mode in the ${\mathrm{N}}_{\mathrm{TB}}$ phase, and a single nonhydrodynamic mode plus two hydrodynamic modes (the usual nematic optic axis or director fluctuations) in the higher temperature, uniaxial nematic phase. The properties of these fluctuations and the conditions for their observation are comprehensively explained by a Landau-de Gennes expansion of the free-energy density in terms of heliconical director and helical polarization fields that characterize the ${\mathrm{N}}_{\mathrm{TB}}$ structure, with the latter serving as the primary order parameter. A “coarse-graining” approximation simplifies the theoretical analysis and enables us to demonstrate quantitative agreement between the calculated and experimentally determined temperature dependence of the mode relaxation rates.

Popular Summary

The discovery of new phases of matter is a major event in materials’ research. The twist-bend phase of liquid crystals is a recent and significant example: It represents the first fundamentally new, orientationally ordered (i.e., nematic) mesophase found in small-molecule systems in several decades. Despite a flurry of experimental efforts to investigate the structure and static properties of this phase and to explain its formation by theoretical models, the dynamics of the twist-bend phase—specifically the nature of the fluctuations in the relevant parameters that describe its structure—have remained largely unexplored. Here, we describe a dynamic light-scattering study to investigate the nature of these fluctuations; we also present a theoretical model that quantitatively explains their character and behavior.

Our experimental and theoretical investigation focuses on a liquid-crystalline mixture of monomer and dimer compounds. We investigate fluctuation modes, one of which is due to the molecular long axis bending and twisting in space. To simplify our analysis, we adopt a so-called “coarse-graining” approximation. We study the temperature dependence of the fluctuation modes, and our results highlight the important role of a polarization field in the formation of the twist-bend state. We expect that our findings can be extended to other compounds and mixtures forming the twist-bend phase.

Our results may help guide efforts to exploit the twist-bend phase for various optical or electro-optical applications.

Figure 1

Left diagram: Schematic representation of the ${\mathrm{N}}_{\mathrm{TB}}$ phase structure, showing heliconical director $\widehat{\mathbf{n}}$ (with cone angle $\beta$ and helical pitch $t_0$) and helical polarization field $\mathbf{P}$. Right diagram: Frame of reference used to describe spatial variations of the average director or pitch axis, $\widehat{\mathbf{t}}$, on length scales much longer than the pitch (see Sec. 4). The orthogonal unit vectors ${\widehat{\mathbf{e}}}_1$ and ${\widehat{\mathbf{e}}}_2$ form a right-handed system with $\widehat{\mathbf{t}}$. The $xyz$ axes are fixed in the laboratory frame.

Figure 2

Top diagram: Light-scattering geometries G1 (left) and G2 (right) described in the text, with the average director (optic axis) in the sample cell indicated by the arrow pointing out of the page for G1 (homogeneous planar alignment with average $\widehat{\mathbf{n}}$ normal to the scattering plane) or the downward arrow for G2 (homeotropic alignment with average $\widehat{\mathbf{n}}$ in the plane). The orientations of polarizer and analyzer are similarly indicated. Bottom diagram: Chemical structure of the monomer and dimer compounds utilized for the present study. The $30/70\mathrm{wt}\%$ mixture exhibits a $\mathrm{N}\text{−}{\mathrm{N}}_{\mathrm{TB}}$ phase transition at $94.2°\mathrm{C}$.

Figure 3

Polarizing microscope textures for a 5-$\mu \mathrm{m}$-thick homeotropically aligned sample of the studied mixture. The optic axis is normal to the image plane, and the sample is placed between crossed polarizers. (a) Separate regions of N and ${\mathrm{N}}_{\mathrm{TB}}$ phases observed at the transition between the two; the boundary is marked by the dashed line. Both regions are uniform and dark, indicating high-quality homeotropic alignment of the director $\widehat{\mathbf{n}}$ in the nematic and pitch axis $\widehat{\mathbf{t}}$ in the ${\mathrm{N}}_{\mathrm{TB}}$ phase. (b) Under an applied ac voltage (5 V @ 10 KHz), a second-order Freedericsz transition (reorientation of $\widehat{\mathbf{n}}$ in the center of the sample) is observed in the nematic region, while the ${\mathrm{N}}_{\mathrm{TB}}$ region is unchanged. (c) Under higher voltage (7 V @ 10 KHz), the ${\mathrm{N}}_{\mathrm{TB}}$ region undergoes a first-order reorientation of $\widehat{\mathbf{t}}$ in the form of nucleating toroidal focal conic domains (FCDs) and expanding stripes of splay and saddle-splay deformations of $\widehat{\mathbf{t}}$. (d) Several seconds after the voltage has been switched off, the nematic region relaxes back to the homeotropic state, whereas the ${\mathrm{N}}_{\mathrm{TB}}$ region relaxes considerably slower.

Figure 4

Top panel: Normalized homodyne DLS correlation functions taken in the nematic phase of the studied LC mixture for (a) geometry G2 with $T-T_{\mathrm{TB}}=16.2°\mathrm{C}$ and angles ${\theta }_i=15°$, ${\theta }_s=40°$, (b) G1 with $T-T_{\mathrm{TB}}=6.0°\mathrm{C}$ and ${\theta }_i=0°$, ${\theta }_s=60°$, and (c) G2 with $T-T_{\mathrm{TB}}=16.2°\mathrm{C}$ and ${\theta }_i=15°$, ${\theta }_s=0°$ (“dark” director geometry). Solid lines represent fits to a single exponential decay, except for (c), which is fit to a double exponential with the slower component stretched. Bottom panel: Normalized correlation data taken in the ${\mathrm{N}}_{\mathrm{TB}}$ phase for (a) geometry G2 with $T-T_{\mathrm{TB}}=-0.93°\mathrm{C}$ and ${\theta }_i=15°$, ${\theta }_s=40°$, (b) G1 with $T-T_{\mathrm{TB}}=-1.1°\mathrm{C}$ and ${\theta }_i=0°$, ${\theta }_s=60°$, and (c) G2 with $T-T_{\mathrm{TB}}=-2.5°\mathrm{C}$ and ${\theta }_i=35°$, ${\theta }_s=0°$ (dark director geometry). Solid lines are single exponential fits, except for a double exponential in (c).

Figure 5

Dependence of the relaxation rates of the fluctuation modes detected in geometry G2 on the magnitude of the scattering vector $q$. Circles and squares correspond to relaxation rates ${\mathrm{\Gamma }}_2^n$ and ${\mathrm{\Gamma }}_2^p$ of the hydrodynamic director and nonhydrodynamic polarization modes detected in scattering geometry G2 in the middle of the nematic phase ($T-T_{\mathrm{TB}}=25°\mathrm{C}$). The scattering angles ${\theta }_s$ range from $-5°$ to 45° for fixed incident angle ${\theta }_i=15°$ (see definitions in Fig. 2, top right). The slope of the line through the data on the log-log plot for ${\mathrm{\Gamma }}_2^n$ is 2, indicating ${\mathrm{\Gamma }}_2^n\sim q^2$. Diamonds and triangles correspond to relaxation rate ${\mathrm{\Gamma }}_2^t$ of the nonhydrodynamic pitch axis fluctuations at temperatures $T-T_{\mathrm{TB}}=-0.85°\mathrm{C}$ and $-8.0°\mathrm{C}$, respectively, in the ${\mathrm{N}}_{\mathrm{TB}}$ phase. These data are limited to higher $q$ (or ${\theta }_s$ in the range 25° or 35° to 65°) due to a large component of background scattering at lower $q$, whose effect is exacerbated because of the low scattering intensity from fluctuations in the ${\mathrm{N}}_{\mathrm{TB}}$ phase in the G2 geometry.

Figure 6

Top panel: Temperature dependence of relaxation rates associated with director fluctuations detected in scattering geometry G2 in the uniaxial nematic phase (${\mathrm{\Gamma }}^n$ for $T>T_{\mathrm{TB}}$, circles in main figure and inset) and twist-bend phase (${\mathrm{\Gamma }}^t$ for $T<T_{\mathrm{TB}}$, squares in main figure) with fixed ${\theta }_i=15°$ and ${\theta }_s=40°$ (see Fig. 2, top right). Bottom panel: Temperature dependence of the relaxation rate of polarization fluctuations in the nematic (diamonds in main figure and inset for ${\theta }_s=40°$ and ${\theta }_i=15°$) and ${\mathrm{N}}_{\mathrm{TB}}$ (triangles in main figure for ${\theta }_s=0°$, ${\theta }_i=35°$) phases. The solid lines in both panels are fits of ${\mathrm{\Gamma }}^t$ and ${\mathrm{\Gamma }}^p$ to calculated results from the coarse-grained free-energy density model presented in Sec. 4 of the text.

Figure 7

Temperature dependence of the inverse of the total scattering intensity $I_2^{-1}$ recorded in geometry G2 for ${\theta }_i=15°$, ${\theta }_s=40°$. The solid line is a linear fit of the data for $T<T_{\mathrm{TB}}$.

Figure 8

Visualization of certain fluctuation modes in the ${\mathrm{N}}_{\mathrm{TB}}$ phase [29]. Small cylinders represent the heliconical director field $\widehat{\mathbf{n}}(\mathbf{r})$, and surfaces represent the pseudolayers. (a) Ground state. (b) Hydrodynamic mode with wave vector $\mathbf{q}=0$, with uniform rotation of $\widehat{\mathbf{n}}(\mathbf{r})$ and hence uniform displacement of pseudolayers; this mode has no energy cost with respect to the ground state. (c) Nonhydrodynamic tilt mode, with the coarse-grained director $\widehat{\mathbf{t}}$ [average of $\widehat{\mathbf{n}}(\mathbf{r})$] tilted with respect to pseudolayer normal. (d) Hydrodynamic mode with $\mathbf{q}=q\widehat{\mathbf{z}}$, with $z$-dependent rotation of $\widehat{\mathbf{n}}(\mathbf{r})$ and $z$-dependent displacement of pseudolayers (leading to compression and dilation). (e) Hydrodynamic mode with $\mathbf{q}=q\widehat{\mathbf{x}}$, with $x$-dependent rotation of $\widehat{\mathbf{n}}(\mathbf{r})$ and $x$-dependent displacement of pseudo-layers (leading to curvature), accompanied by tilt so that $\widehat{\mathbf{t}}$ remains normal to pseudolayers.