Fréedericksz-Like Transition in a Biaxial Smectic-$A$ Phase

The two main classes of liquid-crystal (LC) phases of rodlike molecules are nematics, where the rods align in the same direction (the nematic director $\mathbf{n}$), and smectics, where the rods not only are aligned but also form layers. The electro-optic effects in LC devices that are a backbone in today’s display industry mainly use the Fréedericksz transition, which is the bulk reorientation of a surface-anchored nematic by an electric field. Conventional (uniaxial) smectics do not present a Fréedericksz transition, because, due to their layered structure, the director reorientation would distort the layers, which would cost too much energy. In a worldwide ongoing effort to extend the variety of LC compounds suitable for applications in the display industry, bent-shaped molecules have recently raised much attention, since they present multiple new LC phases with unusual properties. In this paper, we report on a structural and electro-optic study of the LC phases of a bent-shaped dimer. On cooling from the isotropic liquid, this compound shows a usual nematic ($N$), a twist-bend nematic ($N_{\mathrm{TB}}$), and a biaxial smectic-$A$ phase ($\mathrm{Sm}{A}_b$). Quite surprisingly, contrary to usual smectics, $\mathrm{Sm}{A}_b$ presents a remarkable electro-optic response, with low ($<4\mathrm{V}$) voltage threshold, no reorganization of the smectic layers, and low ($<1\mathrm{ms}$) response time (i.e., 30 times faster than the $N$ phase at higher temperature). We interpret this unexpected electro-optic effect as a Fréedericksz transition affecting the secondary director $\mathbf{m}$ of the $\mathrm{Sm}{A}_b$, and we model it by analogy with the usual Fréedericksz transition of the $\mathbf{n}$ director of the uniaxial $N$ phase. Indeed, a Fréedericksz transition affecting only $\mathbf{m}$ in this biaxial fluid smectic does not alter its layered structure and costs little energy. From the point of view of applications, thanks to its low relaxation time, this “biaxial” Fréedericksz transition could be exploited in electro-optic devices that require fast switching.

Popular Summary

Most liquid-crystal displays rely on nematics, which are fluids made of anisotropic rodlike molecules aligned in the same direction (called the $n$ director). These displays exploit a phenomenon known as the Fréedericksz transition, a reorientation of the director by an electric field. However, there is another class of liquid crystals called smectics, where the molecules are not only aligned in the same direction but also form a 1D stack of equidistant fluid layers. Smectics do not show the Fréedericksz transition, because the reorientation of the director would disturb the layered structure and would therefore cost too much energy. Here, we report on a smectic that displays a fast electro-optic effect that is similar to the Fréedericksz transition.

In the search for new display mechanisms, researchers have recently considered molecules not shaped like rods. These molecules are biaxial, meaning that they lack revolution symmetry. Thus, they can form biaxial smectics, labeled “SmAb,” where their longest axis is aligned along the $n$ director and their medium axis along the $m$ director, parallel to the layers.

During electro-optic studies of smectics with “bent-shaped” molecules, we find that SmAb shows an effect highly reminiscent of the Fréedericksz transition of nematics. Using polarized-light microscopy, birefringence, and dielectric measurements, we show that it is indeed a Fréedericksz transition, but affecting the $m$ director. This in-plane transition is allowed in a biaxial smectic because it does not alter the $n$ director field and therefore the layered structure remains unperturbed.

We also find that the relaxation time of this effect is 30 times shorter than that of the usual Fréedericksz transition of nematics, which means that SmAb could be exploited in electro-optic devices that require fast switching.

Figure 1

Molecular structure and phase sequence of the LC compounds. For the BNA-76 bent-shaped dimer, a simplified sketch of the average conformation of the molecule is also shown, with the electron-rich conjugated parts in red and the alkyl and alkoxy chains in blue.

Figure 2

X-ray scattering patterns of a BP12 sample in the $M_X$ phase ($T=97°\mathrm{C}$) aligned by a 1.7 T magnetic field $H$ (double-headed arrow). (a) With a 60 mm sample-to-detection distance. The solid arrow points to one of the smectic reflections at small angles, whereas the dashed arrow points at the wide-angle diffuse ring. (The white disk at the center represents the beam stop.) (b) With a 120 mm sample-to-detection distance. The solid arrow points to one of the smectic reflections, the dashed arrow to the barely observable second-order smectic reflection, and the dotted arrow to the absence of any scattering at wave vector $q_0=2\pi /L$, where $L$ is the BNA-76 molecular length. (The white arrow points to parasitic scattering around the beam stop.)

Figure 3

Sketch of the structure of the $M_X$ phase. The monomer units form smectic liquid layers with thickness $d_0\approx L/2$. In each layer, the main axes $\mathbf{p}$ of the monomers are tilted in the same direction, and the sign of the tilt alternates from one layer to the next. The dimers span two adjacent layers and form an orthogonal intercalated smectic phase. The orientational order tensor $\mathbf{Q}$ of the dimers (shown in orange) is biaxial, with primary director $\mathbf{n}$ oriented along the layers normal and secondary director $\mathbf{m}$ parallel to the projection of $\mathbf{p}$ on the layer plane. ($\mathbf{k}$ is the third director, which is perpendicular to both $\mathbf{n}$ and $\mathbf{m}$.)

Figure 4

Coexistence of uniform single domains of the $M_X$ and $N_{\mathrm{TB}}$ phases of the BP12 mixture (planar cell, $d=9.8\mu \mathrm{m}$, scale bar $100\mu \mathrm{m}$). The optic axes of both phases are approximately parallel to the rubbing direction $\mathbf{r}$. The cell is viewed between crossed polarizers, using a Berek compensator with slow axis $\gamma \perp \mathbf{r}$. (a) Growth of uniform $M_X$ domains within the $N_{\mathrm{TB}}$ at $T=102.4°\mathrm{C}$. (b),(c) $M_X$ and $N_{\mathrm{TB}}$ domains at $T=100.7°\mathrm{C}$, without a field (b) and when a biaxial Fréedericksz transition (c) is induced by an ac field ($U_{\mathrm{rms}}=9\mathrm{V}$, $f=6\mathrm{kHz}$) applied along the cell normal. The birefringence of the $N_{\mathrm{TB}}$ [yellow regions in (b) and (c)] is independent of the field. The birefringence of the $M_X$ phase varies with the field, from its low zero-field value [brown-red domains in (b)] to a high value above the transition [green domains in (c)]. The defect walls that separate the twin domains in (c) appear as thick dark lines.

Figure 5

Growth of a homeotropic $M_X$ domain within the $N_{\mathrm{TB}}$ under an electric field ($U_{\mathrm{rms}}=25\mathrm{V}$, $d=1.4\mu \mathrm{m}$, $T=100°\mathrm{C}$). The observation of birefringence while rotating the crossed polarizers (white double-headed arrows) reveals that the domain is biaxial, with $\mathbf{m}$ parallel to $\mathbf{r}$. (Scale bar $100\mu \mathrm{m}$.)

Figure 6

Temperature and field dependence of the birefringence of the BP12 mixture. (a) $\mathrm{\Delta }n(T)$ measured in different phases and for different sample alignments. The black symbols show the data obtained using PMT detection, whereas the red symbols show the data measured with the image-mapping technique. The circles show the data $\mathrm{\Delta }n=n_{\parallel }-n_{\perp }$, obtained in the uniaxial $N$ and $N_{\mathrm{TB}}$ phases or regions, whereas the open squares show the data $\mathrm{\Delta }n=n_{nn}-n_{mm}$, obtained in the $M_X$ phase without a field. The full squares show the data $\mathrm{\Delta }n=n_{nn}-n_{kk}$ in the $M_X$ phase, obtained by extrapolation of the results measured under a field. The red line shows the birefringence expected by extrapolation of the Haller fit [53] of the data in the range $140°\mathrm{C}–160°\mathrm{C}$. (b) $\mathrm{\Delta }n(U)$ in the $N$ (black symbols) and $M_X$ (red symbols) phases.

Figure 7

Time evolution of the optical response during the Fréedericksz transition in the $N$ (FrTr, black lines and symbols) and $M_X$ (BFrTr, red lines and symbols) phases of the BP12 mixture (cell thickness $d=1.4\mu \mathrm{m}$). (a) Optical response to dc pulses ($U>U_c$; the dashed straight lines mark the start and the end of the pulse). The relaxation time ${\tau }_{\mathrm{off}}$, which depends only on the cell thickness, is much faster in the $M_X$ phase. Fits of the data with an exponential relaxation law are shown in blue. (b) Temperature dependence of ${\tau }_{\mathrm{off}}$ in the $N$ (open black symbols) and $M_X$ (full red symbols) phases.

Figure 8

Dielectric data of the BP12 mixture (cell thickness $9.8\mu \mathrm{m}$). (a) Cell capacitance versus applied voltage in the $N$ phase (black symbols) and in the $M_X$ phase (red symbols). The solid lines show the best fits of the data with the theoretical model (see Appendix pp5). (b) Temperature dependence of the dielectric tensor eigenvalues measured in the uniaxial $N$ and $N_{\mathrm{TB}}$ phases and in the biaxial $M_X$ phase.

Figure 9

Schematic representation of the nematic directors ($\mathbf{n}$,$\mathbf{m}$, and $\mathbf{k}$), the order parameter tensor ($\mathbf{Q}$), and the main distortion modes in the $N$ and $\mathrm{Sm}{A}_b$ phases. In the uniaxial $N$ phase (top row), the only relevant distortions are those of the main director $\mathbf{n}$. In the $\mathrm{Sm}{A}_b$ phase (bottom row) and at fixed structure of the smectic layers (shown in red), the only relevant distortions are those of the secondary director $\mathbf{m}$.

Figure 10

Geometry of the distortions of $\mathbf{n}$ in the $N$ phase (top row) and $\mathbf{m}$ in the $\mathrm{Sm}{A}_b$ phase (bottom row); in the left and middle columns, the applied voltage is, respectively, below and above the Fréedericksz transition threshold. $\mathbf{r}$ is the rubbing direction, $(\mathbf{x},\mathbf{y},\mathbf{z})$ is a reference frame, ${\theta }^n$ (respectively, ${\theta }^m$) is the angle of the $\mathbf{n}$ (respectively, $\mathbf{m}$) director with the normal to the cell ($\mathbf{z}$ axis). The smectic layers are shown in red.

Figure 11

Temperature dependence of the Fréedericksz transition threshold in the $N$ and $\mathrm{Sm}{A}_b$ phases. (There is no Fréedericksz transition in the $N_{\mathrm{TB}}$ phase, and no data are available for the biphasic $N_{\mathrm{TB}}/\mathrm{Sm}{A}_b$ range due to the nonuniform cell texture.)

Figure 12

Temperature dependence of the splay and bend elastic constants in the $N$ (a) and $\mathrm{Sm}{A}_b$ (b) phases of the BP12 mixture. [The inset in (a) shows a magnification of the $K_{33}^n$ curve.]

Figure 13

Rotational viscosities of the BP12 mixture for reorientation of $\mathbf{n}$ in the $N$ phase (open black symbols) and $\mathbf{m}$ in the $\mathrm{Sm}{A}_b$ phase (full red symbols). The solid straight lines show exponential decay fits of the viscosities (Arrhenius-law behavior).

Figure 14

DSC scan on cooling of the BP12 mixture.

Figure 15

X-ray scattering patterns of a BP12 sample aligned by a 1.7 T magnetic field $H$ [double-headed arrow in (a)] recorded with a 60 mm sample-to-detection distance. (a) In the $N$ phase ($T=117°\mathrm{C}$); (b) in the $N_{\mathrm{TB}}$ phase ($T=107°\mathrm{C}$); (c) in the $M_X$ phase ($T=97°\mathrm{C}$); and (d) in the crystalline phase (room temperature). White disks at the pattern centers represent the beam stop. In (a)–(c), the dashed arrow points to the wide-angle diffuse ring. In (a) and (b), the solid arrow points to small-angle diffuse streaks. In (c), the solid arrow points to one of the two first-order smectic reflections, while the dotted arrow points to one of the two second-order smectic reflections. In (d), the pattern shows only sharp Bragg reflections arising from several large crystallites in reflection position in the x-ray beam.

Figure 16

Powder x-ray diffractograms of a BP12 sample in the crystalline, $M_X$, $N_{\mathrm{TB}}$, and $N$ phases. The inset shows a magnification of the low-angle region.

Figure 17

Azimuthal profiles of the scattered x-ray intensity of a BP12 sample aligned in a magnetic field in the $N$, $N_{\mathrm{TB}}$, and $M_X$ mesophases.

Figure 18

Growth of $N_{\mathrm{TB}}$ monochiral domains (on the left side) in the $N$ phase (on the right side) at the $N\text{−}{N}_{\mathrm{TB}}$ transition of BP12 ($T=108.9°\mathrm{C}$, $d=9.8\mu \mathrm{m}$). To evidence the alternating sign of the chirality of the domains, they are observed in monochromatic light, $\lambda =546\mathrm{nm}$, between crossed polarizers (white double-headed arrows) rotated at 3° with respect to the rubbing direction $\mathbf{r}$. (Scale bar $100\mu \mathrm{m}$.)

Figure 19

Alternating monochiral $N_{\mathrm{TB}}$ domains of BP12 observed between crossed polarizers ($T=108.8°\mathrm{C}$, $d=9.8\mu \mathrm{m}$). In both kinds of domains, the optic axis $\mathbf{N}$ is parallel to the rubbing direction $\mathbf{r}$. When $\mathbf{r}$ is parallel to the input polarizer (b), both kinds of domains appear dark. When the sample is rotated by $\pm 4°$ [(a) and (c)], a strong transmittance contrast appears between the two kinds of domains. The inversion of this contrast with the sign of the rotation indicates the different signs of the chirality of adjacent domains. (Scale bar $100\mu \mathrm{m}$.)

Figure 20

Stripe instabilities in the monochiral $N_{\mathrm{TB}}$ domains of BP12 (a) and their annealing after thermal cycling and repeated application of a weak electric field ($T=108.0°\mathrm{C}$, $d=9.8\mu \mathrm{m}$). After annealing (b)–(d), the domains are again uniform and their optic axis $\mathbf{N}$ is parallel to the rubbing direction $\mathbf{r}$. The weak residual heterogeneities after annealing (d) are due to the surface memory of the alignment layers. (Scale bar $100\mu \mathrm{m}$.)

Figure 21

Biphasic coexistence of the $M_X$ (bright) and $N_{\mathrm{TB}}$ (dark) phases in a $d=1.4\mu \mathrm{m}$ planar-alignment cell at different temperatures: $100°\mathrm{C}$ (a), $98°\mathrm{C}$ (b), and $97°\mathrm{C}$ (c). To visualize the different birefringence of the two phases, the cell is observed under polarized monochromatic light ($\lambda =546\mathrm{nm}$) with a Sénarmont compensator and with an analyzer rotated by an angle optimizing the contrast between the two phases. Each image is recorded after 15 min of relaxation at fixed temperature to reach equilibrium between the two phases. (Scale bar $100\mu \mathrm{m}$.)

Figure 22

Azimuthal alignment of the $M_X$ domains coexisting with the $N_{\mathrm{TB}}$ phase at $101.8°\mathrm{C}$. The sample is observed between crossed polarizers and rotated at angle $\psi =-2°$, 0°, $+2°$, and 0° in (a), (b), (c), and (d), respectively. Here, $\psi$ is the angle between the rubbing direction $\mathbf{r}$ and the input polarizer. The slow axis $\mathbf{N}$ of the $M_X$ regions is approximately parallel to $\mathbf{r}$. The azimuthal deviations of $\mathbf{N}$ from $\mathbf{r}$, at angles typically smaller than $\pm 3°$, are due to slight disorientations of the smectic single domains of the $M_X$ phase. The analysis of the transmitted light and the smectic textures shows that $\mathbf{N}$ is vertical in the pictures, perpendicular to the smectic layers that are horizontal, which confirms the $\mathrm{Sm}A$ (and not $\mathrm{Sm}C$) nature of the $M_X$ phase. In (d), an electric field, larger than the threshold of the Freédericksz-like transition, is applied to the sample ($f=6\mathrm{kHz}$,$U_{\mathrm{rms}}=7\mathrm{V}$).

Figure 23

Homeotropically aligned $M_X$ phase after field removal ($d=1.4\mu \mathrm{m}$, $T=85°\mathrm{C}$). The smectic layers are parallel to the image plane, $\mathbf{n}$ is perpendicular to it, and $\mathbf{m}$ is parallel to the layers, in the image plane. The sample is birefringent, with $\mathrm{\Delta }n\sim 0.02$, and the slow axis is parallel to $\mathbf{m}$. In the striped areas, $\mathbf{m}$ rotates in the plane, and each band corresponds to a 90° rotation. (Scale bar $50\mu \mathrm{m}$.)

Figure 24

Estimations of the pretilt angle ${\psi }^m$ (left) and the anchoring extrapolation length $L_e^m$ (right) for the secondary director in the $M_X$ phase ($T=95°\mathrm{C}$).

Figure 25

Sketch of an $M_X$ phase with a $\mathrm{Sm}{A}_b(\mathrm{Sm}{A}_b)$ structure. As for the $\mathrm{Sm}{A}_b(\mathrm{Sm}{C}_A)$ structure presented in Fig. 3, the monomer units form smectic liquid layers with thickness $d_0\approx L/2$. The dimers span again two adjacent layers and form an orthogonal intercalated smectic phase with biaxial orientational order parameter tensor $\mathbf{Q}$. However, in each layer, the average orientation of the monomer axis $\mathbf{p}$ is parallel to the layers normal, because the tilt of the monomers is doubly degenerated, which is quite unlikely due to a strong energetic penalty.