Spatially periodic deformations in planar and twisted flexoelectric nematic layers

Electric field induced deformations in twisted and untwisted planar layers of nematic liquid crystals possessing flexoelectric properties were investigated numerically. The spatially periodic deformations, taking the form of parallel stripes, were found to have smaller free energy than the homogeneous deformations. The structures of distorted layers as well as the evolution of deformations during changes of bias voltage were recognized. The role of flexoelectric properties was analyzed. Calculations taking into account the peculiar elastic properties of the bent-core nematics were also performed. In the untwisted planar layers the stripes were parallel to the initial director orientation. In the twisted layers, two different kinds of periodic deformation were distinguished. They had different structures and different directions with respect to the initial director distribution. The results were consistent with existing experimental data.

Figure 1

Schematic distributions of signs of ϕ($y,z$) in the cross sections of stripes illustrating the influence of flexoelectricity on the structure of deformed layers. Dark and bright regions correspond to negative and positive angles ϕ, respectively. $r=0.2$. Values of $e_{33}$ (in pC/m) are given.

Figure 2

Director field in the cross section of a single stripe arising in planar layer; $e_{33}=20\mathrm{pC}/\mathrm{m}$; $U=3\mathrm{V}$.

Figure 3

Amplitude Θ of deformations arising in planar layers as a function of bias voltage $U$. Solid lines: periodic patterns; dashed lines: homogeneous deformations. Values of $e_{33}$ (in pC/m) are indicated at the curves.

Figure 4

Spatial period λ of deformations arising in planar layers as a function of bias voltage $U$. Values of $e_{33}$ (in pC/m) are indicated at the curves.

Figure 5

Amplitude Θ of deformations arising in twisted layers as a function of bias voltage $U$. Thick lines: type 1 patterns; thin lines: type 2 patterns; dashed lines: homogeneous deformations. Dashed vertical lines indicate rapid changes of the type of deformations. Dotted line denotes the energetically unfavorable deformation of type 2. Values of $e_{33}$ (in pC/m) are indicated at the curves.

Figure 6

Direction of stripes arising in twisted layers determined by the angle ψ as a function of bias voltage $U$. Thick lines: type 1 patterns; thin line at $\psi ={45}^{\circ }$ corresponds to the type 2 patterns for $e_{33}=20$ and 50 pC/m. Dashed lines indicate rapid change of the type of deformations at $U=U_3$. Values of $e_{33}$ (in pC/m) are indicated at the curves.

Figure 7

Spatial period λ of deformations arising in twisted layers as a function of bias voltage $U$. Thick lines: type 1; thin lines: type 2. Dashed vertical lines indicate rapid change of the type of deformations at $U=U_3$. Dotted line denotes the energetically unfavorable deformation of type 2. Values of $e_{33}$ (in pC/m) are indicated at the curves.

Figure 8

Director field in the cross section of a single stripe of type 1 occurring in the twisted layer. The star indicates position of the escaped disclination line. $e_{33}=20\mathrm{pC}/\mathrm{m}$; $U=3\mathrm{V}$.

Figure 9

Director field in the cross section of a single stripe of type 2 occurring in the twisted layer. $e_{33}=20\mathrm{pC}/\mathrm{m}$; $U=3\mathrm{V}$.

Figure 10

Amplitude Θ of periodic patterns arising in twisted layers containing mixture A as a function of bias voltage $U$. Thick lines: type 1 patterns; thin line: type 2 patterns. Values of $e_{33}$ (in pC/m) are indicated at the curves.

Figure 11

Amplitude Θ of periodic patterns arising in twisted layers containing mixture B as a function of bias voltage $U$. Thick lines: type 1 patterns, thin line: type 2 patterns. Values of $e_{33}$ (in pC/m) are indicated at the curves.

Figure 12

Spatial period of deformations arising in twisted layers containing mixtures A and B as a function of bias voltage $U$. Thick lines: type 1 patterns; thin lines: type 2 patterns. Values of $e_{33}$ (in pC/m) are indicated at the curves.

Figure 13

Direction of the type 1 stripes of deformations arising in twisted layers containing mixtures A and B of calamitic and bent-core nematics as a function of bias voltage $U$. Thick lines: type 1 patterns; thin line: type 2 patterns. Dashed vertical lines indicate rapid change of the type of deformations at $U=U_3.e_{33}=20\mathrm{pC}/\mathrm{m}$.

Figure 14

Schematic presentation of the free energies per unit area of the layer related to different types of deformations plotted as functions of voltage. Left part: twisted layers; right part: planar layers. Dotted lines: undistorted state; dashed lines: one-dimensional deformation; thin solid line: type 2 patterns; thick lines: type 1 patterns in twisted case and periodic pattern in planar layer. Voltages $U_1\cdots U_4$ correspond to changes of the kind of deformations.