Transformation of polar nematic phases in the presence of an electric field

Only a few years have passed since the discovery of polar nematics, and now they are becoming the most actively studied liquid-crystal materials. Despite numerous breakthrough findings made recently, a theoretical systematization is still lacking. In the present paper, we take a step toward systematization. The powerful technique of molecular-statistical physics has been applied to an assembly of polar molecules influenced by electric field. Three polar nematic phases were found to be stable at various conditions: the double-splay ferroelectric nematic $N_F^{2D}$ (observed in the lower-temperature range in the absence of or at low electric field), the double-splay antiferroelectric nematic $N_{AF}$ (observed at intermediate temperature in the absence of or at low electric field), and the single-splay ferroelectric nematic $N_F^{1D}$ (observed at moderate electric field at any temperature below transition into paraelectric nematic $N$ and in the higher-temperature range (also below $N$) at low electric field or without it. A paradoxical transition from $N_F^{1D}$ to $N$ induced by application of higher electric field has been found and explained. A transformation of the structure of polar nematic phases at the application of electric field has also been investigated by Monte Carlo simulations and experimentally by observation of polarizing optical microscope images. In particular, it has been realized that, at planar anchoring, $N_{AF}$ in the presence of a moderate out-of-plane electric field exhibits twofold splay modulation: antiferroelectric in the plane of the substrate and ferroelectric in the plane normal to the substrate. Several additional subtransitions related to fitting the confined geometry of the cell by the structure of polar phases were detected.

Figure 1

A pair of interacting polar molecules. Adapted from Ref. [35].

Figure 2

Director distribution in $N_{AF}$ (a), $N_F^{2D}$ (b), and $N_F^{1D}$ (c). Green corresponds to the positive splay and polarization, red corresponds to the negative splay and polarization; the $\mathbf{x}$-axis is either along the domain symmetry axis in (a) and (b) or along the symmetry plane in (c); the $\mathbf{r}$-axis is perpendicular to the $\mathbf{x}$-axis; $\theta$ is the angle between the local director $\mathbf{n}$ and the $\mathbf{x}$-axis; $\mathbf{P}$ is the local polarization.

Figure 3

Electric field–temperature phase diagram (a); Temperature dependencies of the domain radius (b) and characteristic polar order parameter (c) at $E=0$. Green arrow in (a) tentatively corresponds to the phase sequence with temperature variation at fixed $E\ne 0$. Red arrows with numbers in (a) and (b) correspond to the temperatures of specific phase transitions observed experimentally (Sec. III B). Dashed blue line in (b) corresponds to the half-thickness of the cell. Molecular parameters are the same as in the caption of Fig. 12.

Figure 4

POM images of quasicylindrical domains in $N_F^{2D}$ (a) and $N_{AF}$ (b); orientation of planar domains (schematic illustration) in $N_F^{1D}$ (c) and $N_{AF}^{1D}$ (d) at $E=0$; orientation of planar domains in $N_F^{1D}$ at $E\ne 0$. In (a) and (b) DIO material is used; cell thickness is $10\mu \mathrm{m}$. Images of (a) and (b) are reproduced with permission from Ref. [16]. Copyright WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2017.

Figure 5

POM images of the DIO planar cell (cell thickness is $5\mu \text{m}$) at several voltages (sinusoidal signal, $1\text{kHz}$) and temperature $T=74{}^{\circ }\text{C}$ (Row 1), $70{}^{\circ }\text{C}$ (Row 2), and $81{}^{\circ }\text{C}$ (Row 3). P and A are the directions of polarizer and analyzer, R is the rubbing direction.

Figure 6

POM images of the two parts of the cell with DIO at $67{}^{\circ }\text{C}$ under $0.4\text{V}$, $1\text{kHz}$ electric field: (a) mostly $N_F^{2D}$; (b) transformation from $N_F^{2D}$ to $N_F^{1D}$.

Figure 7

POM images of the DIO planar cell (cell thickness is $5\mu \text{m}$) at applied fixed electric field ($0.9\text{V}$, sinusoidal signal, $1\text{kHz}$) during the cooling cycle (temperatures are indicated in the bottom right corners). P and A are directions of polarizer and analyzer, while R is the rubbing direction.

Figure 8

POM images of the DIO planar cell (cell thickness is $5\mu \text{m}$) at applied fixed electric field ($0.9\text{V}$, sinusoidal signal, $1\text{kHz}$) during the cooling cycle (continuation of Fig. 7).

Figure 9

A trial (blue) polar molecule in a combination of the mean molecular field and electric field $\mathbf{E}$. Here $\mathbf{n}$ is the local director at a point where the trial molecule is located.

Figure 10

Tilt of director (a) and splay deformation (b) distributions within the domain in $N_F^{2D}$ [blue curves (1) and (2)] and $N_F^{1D}$ [red curves (3), (4), and (5)]. Here $\tilde{\tau }=0.001$ (1), 0.22 (2), 0.225 (3), 0.8 (4), and 0.843 (5).

Figure 11

Electric field dependencies of the characteristic polar order parameter (a) and domain radius (b) in $N_F^{1D}$ at $J_{202}^{\left(0\right)}/k_B=2032\text{K}$, $J_{101}^{\left(0\right)}/k_B=362\text{K}$, $\lambda =2\mu {\text{m}}^{-1}$, $K{V}_0=5\times {10}^{-35}\text{N}{\text{m}}^3$, and $T=70{}^{\circ }\text{C}$ [curves (1)] and $81{}^{\circ }\text{C}$ [curves (2)].

Figure 12

Distribution of the $S$ [(a),(c),(e)] and $P$ [(b),(d),(f)] orientational order parameters within the domains in $N_F^{2D}$ [(a),(b)], $N_{AF}$ [(c),(d)], and $N_F^{1D}$ [(e),(f)] at $T=57{}^{\circ }\text{C}$ (1); $62{}^{\circ }\text{C}$ (2); $67{}^{\circ }\text{C}$ (3); $69{}^{\circ }\text{C}$ (4); $76{}^{\circ }\text{C}$ (5); $81{}^{\circ }\text{C}$ (6); $83{}^{\circ }\text{C}$ (7); $85{}^{\circ }\text{C}$ (8); $88{}^{\circ }\text{C}$ (9); $90{}^{\circ }\text{C}$ (10); and $92{}^{\circ }\text{C}$ (11) at $E=0$, $J_{202}^{\left(0\right)}/k_B=2032\text{K}$, $J_{101}^{\left(0\right)}/k_B=362\text{K}$, $\lambda =2\mu {\text{m}}^{-1}$, $J_A/k_B=113\text{K}\mu \text{m}$, and $K{V}_0=5\times {10}^{-35}\text{N}{\text{m}}^3$. Radius $r$ is defined in Fig. 2 for all the polar phases. Here $J_A$ is the strength of antiferroelectric decoupling [35, 56].

Figure 13

Distribution of the $S$ (a) and $P$ (b) orientational order parameters within the domains in $N_F^{1D}$ at ${\varepsilon }_{a}E/\left(K{\lambda }^2\right)=0$ (1); 0.01 (2); 0.028 (3) at $T=87{}^{\circ }\text{C}$, $J_{202}^{\left(0\right)}/k_B=2032\text{K}$, $J_{101}^{\left(0\right)}/k_B=362\text{K}$, $\lambda =2\mu {\text{m}}^{-1}$, and $K{V}_0=5\times {10}^{-35}\text{N}{\text{m}}^3$. Radius $r$ is defined in Fig. 2.

Figure 14

Principal geometry of the simulations box and the polar nematic film. Black arrows show the rubbing direction of planar alignment of the film. Orange arrow shows the electric field direction. Yellow dashed arrows show the periodic boundary condition directions of the simulation box.

Figure 15

Dependence of the total free energy on the value of dimensionless electric field $e$ and rubbing direction orientation $\phi$. Violet dashed line traces energy minimum over $\phi$.

Figure 16

Director $\mathbf{n}$ and polarizability $(\mathbf{n}·\mathbf{p})$ distributions at various dimensionless electric field $e$ values in the central cross-cuts perpendicular to the substrate plane (top) and along the substrate plane (bottom). Director distributions are shown in color, corresponding to the direction of $\mathbf{n}$ ($x$, red; $y$, green; $z$, blue). Polarizability is shown in orange [$(\mathbf{n}·\mathbf{p})=1$] and violet [$(\mathbf{n}·\mathbf{p})=-1$].