# Peculiarities in the Director Reorientation and the NMR Spectra Evolution in a Nematic Liquid Crystals under the Effect of Crossed Electric and Magnetic Fields

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## Abstract

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## 1. Introduction

## 2. Theoretical Treatment of the Orientational Dynamics of Nematics under the Effect of the Crossed Electric and Magnetic Fields

#### 2.1. Deuterium Nuclear Magnetic Resonance Study

#### 2.2. Peculiarities in the Director Reorientation under the Effect of Crossed Electric and Magnetic Fields

#### 2.2.1. Theoretical Analysis in Case of the Linear Balance Equations

#### 2.2.2. Theoretical Analysis in Case of the Nonlinear Balance Equations

#### Switched-on Process in the Positive Sense

#### Switched-off Process

#### Switched-on Process in the Negative Sense

#### 2.3. Simulation of the Time-Resolved ${}^{2}$H Spectra

## 3. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The coordinate system used for analyzing the time-resolved NMR experiments and theoretical analysis. The $\mathrm{x}$-axis is taken as being parallel to the magnetic field $\mathbf{B}$ and both electrodes. The electric field $\mathbf{E}$ and director $\widehat{\mathbf{n}}$ make the angles $\alpha $ and $\theta $, respectively, with the magnetic field $\mathbf{B}$. (

**b**) Three regimes of a voltage pulses.

**Figure 2.**Two sets of time-resolved deuterium NMR spectra for the turned-on (

**a**) and turned-off (

**b**) experiments recorded for 5CB-${\mathbf{d}}_{2}$ at 15 ${}^{\circ}$C and $\alpha =79.{7}^{\circ}$ [11].

**Figure 3.**Plot of the angle $\theta \left(t\right)$ vs. time t for the turned-on (

**a**) and the turned-off (

**b**) processes obtained for a number of values of $\alpha $ [11]. The symbols were obtained from the time dependence of $\Delta \nu \left(\theta \right)/\Delta {\nu}_{0}(\equiv {P}_{2}(cos\theta ))$, whereas the solid lines were obtained from the torque balance equation for a monodomain nematic.

**Figure 4.**Plot of the functions ${\tau}_{\mathrm{ON}}\left(\alpha \right)$ and ${\tau}_{\mathrm{OFF}}\left(\alpha \right)$ (solid lines) vs. angle $\alpha $, obtained from the torque balance equation for a monodomain nematic, whereas the symbols indicate the relaxation times obtained from the best-fits in Figure 3a,b, respectively.

**Figure 5.**Plot of the dimensionless growth rate $s\left({q}_{x}d/\pi \right)/s\left(0\right)$ vs. dimensionless wavelength ${q}_{x}d/\pi $ for a number of values of the voltage (

**a**) $U=20\phantom{\rule{3.33333pt}{0ex}}$V and (

**b**) $U=50\phantom{\rule{3.33333pt}{0ex}}$V [18], and for a number of values of the angle $\alpha $: $0.157$ (∼${9}^{\circ}$) (curve (1)); $0.471$ (∼${27}^{\circ}$) (curve (2)); $0.785$ (∼${45}^{\circ}$) (curve (3)); $1.257$ (∼${72}^{\circ}$) (curve (4)), and $1.57$ ($\sim {90}^{\circ}$) (curve (5)), respectively.

**Figure 7.**(

**a**) Plot of the growth rate $s\left({q}_{x}d/\pi \right)$ vs. dimensionless wavelength ${q}_{x}d/\pi $ obtained for a number of values of the anchoring strength $\mathcal{W}$ [11,18]: cases I (curve 1) and II (curve 2), as well as for the case of the strong anchoring (curve 3), respectively. (

**b**) Same as Figure 6, but for a number of values of the angle $\alpha $: $1.22$ (∼${70}^{\circ}$)(curve (1)); $1.29$ (∼${74}^{\circ}$)(curve (2)); $1.36$ (∼${78}^{\circ}$)(curve (3)); $1.43$ (∼${82}^{\circ}$)(curve (4)), and $1.50$ (∼${86}^{\circ}$)(curve (5)), respectively.

**Figure 8.**Plot of the dimensionless ERV coefficient ${\gamma}_{\mathrm{eff}}\left(\alpha ,\mathcal{W}\right)/{\gamma}_{1}$ vs. the angle $\alpha $ for a number of values of the anchoring strength $\mathcal{W}$ [18].

**Figure 9.**Plot of the evolution of the polar angle $\theta \left(\mathrm{x},\mathrm{z}=0,\tau \right)$ (

**a**) and the velocity field $\mathbf{v}=u\left(\mathrm{x},\mathrm{z}=0,\tau \right)\widehat{\mathbf{i}}+w\left(\mathrm{x},\mathrm{z}=0,\tau \right)\widehat{\mathbf{k}}$ (

**b**) to their equilibrium distributions along the length of the dimensionless nematic cell $(-10\le \mathrm{x}\le 10)$ [19], for a number of dimensionless times $\tau =$2, 4, 6, 8, and 10, respectively. In both cases (

**a**) and (

**b**) the evolutions are shown during the turned-on process ($\mathbf{E}>0$), whereas the solid (curves 1) and the dash dotted (curves 2) lines are the calculated results for the vertical $w\left(\mathrm{x},\mathrm{z}=0,\tau \right)$ and horizontal $u\left(\mathrm{x},\mathrm{z}=0,\tau \right)$ components of $\mathbf{v}$, respectively.

**Figure 10.**Plot of the evolution of the polar angle $\theta \left(\mathrm{x},\mathrm{z}=0,\tau \right)$ to its equilibrium distribution along the length of the dimensionless nematic film $(-10\le \mathrm{x}\le 10)$, for a number of dimensionless times $\tau =$ 2 (∼12 ms), 6 (∼36 ms), 8 (∼48 ms), 12 (∼72 ms), and 20 (∼0.12 s) [19], respectively. (

**a**) shows the case B, while (

**b**) shows the case A during the turned-on process ($\mathrm{E}>0$. In all these cases ${\theta}_{0}=0.01$ (∼$1.{1}^{\circ}$) and $\alpha =1.57\phantom{\rule{3.33333pt}{0ex}}(\sim $$89.{96}^{\circ})$).

**Figure 11.**Plot of the evolution of the polar angle $\theta \left(\mathrm{x},\mathrm{z}=0,\tau \right)$ to its equilibrium distribution along the length of the dimensionless nematic film $(-10\le \mathrm{x}\le 10)$ and for a number of dimensionless times $\tau =$2 (22), 6 (26), 8 (28), 12 (32) and 20 (40) [17], respectively. (

**a**) shows the case B, while (

**b**) shows the case A, respectively, during the turned-off process ($\mathrm{E}=0$).

**Figure 12.**Plot of the evolution of the polar angle $\theta \left(\mathrm{x},\mathrm{z}=0,\tau \right)$ to its equilibrium distribution along the length of the dimensionless nematic film $(-10\le \mathrm{x}\le 10)$ for a number of dimensionless times $\tau =$ 0(248), 2 (250), 4 (252), 6 (254), and 8 (256), respectively. (

**a**) shows the case B, while (

**b**) shows the case A during the turned-on process in the negative sense ($\mathrm{E}<0$) [19]. In all these cases ${\theta}_{0}=0.01$ (∼$1.{1}^{\circ}$) and $\alpha =-1.57\phantom{\rule{3.33333pt}{0ex}}(\sim $$-89.{96}^{\circ})$).

**Figure 13.**The same as in Figure 12, but plot of the evolution of the polar angle $\theta \left(\mathrm{x},\mathrm{z}=0,\tau \right)$ to its equilibrium distribution along the length of the dimensionless nematic film $(-10\le \mathrm{x}\le 10)$, for a number of dimensionless times $\tau =$ 0 (250), 2 (252), 4 (254), 6 (256), and 8 (258) [19], respectively.

**Figure 14.**The deuterium NMR spectra $I\left(\nu \right)$ of 5CB-d${}_{2}$ calculated for a number of times $\tau =0,4,12,16$, and 20, both for the case of strong (

**a**) and weak (

**b**) anchoring, respectively. In both sets of spectra the solid lines show the turned-off process ($E=0$) [17].

**Figure 15.**The deuterium NMR spectra $I\left(\nu \right)$ of 5CB-d${}_{2}$ calculated (dotted curves) and measured (solid curves) for the turned-off process at 14.8 ${}^{\circ}$C.

**Table 1.**The optimal values of the dimensionless wavelength ${q}_{x}^{\mathrm{max}}\left(\alpha \right)$ as a function of both the angle $\alpha $ and the anchoring strength $\mathcal{W}$ [18].

$\mathit{\alpha}\left({}^{\circ}\right)$ | ${\mathit{q}}^{\mathbf{max}}\left(\mathit{\alpha}\right)\left(\mathbf{Strong}\right)$ | ${\mathit{q}}^{\mathbf{max}}\left(\mathit{\alpha}\right)\left(\mathbf{Case}\phantom{\rule{3.33333pt}{0ex}}\mathbf{I}\right)$ | ${\mathit{q}}^{\mathbf{max}}\left(\mathit{\alpha}\right)\left(\mathbf{Case}\phantom{\rule{3.33333pt}{0ex}}\mathbf{II}\right)$ |
---|---|---|---|

74 | $20.5$ | $20.6$ | $21.3$ |

78 | $20.96$ | $21.3$ | $21.7$ |

82 | $21.8$ | $22.0$ | $22.6$ |

86 | $22.1$ | $22.2$ | $22.9$ |

90 | $23.0$ | $23.2$ | $23.6$ |

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**MDPI and ACS Style**

Zakharov, A.V.; Sliwa, I.
Peculiarities in the Director Reorientation and the NMR Spectra Evolution in a Nematic Liquid Crystals under the Effect of Crossed Electric and Magnetic Fields. *Crystals* **2019**, *9*, 262.
https://doi.org/10.3390/cryst9050262

**AMA Style**

Zakharov AV, Sliwa I.
Peculiarities in the Director Reorientation and the NMR Spectra Evolution in a Nematic Liquid Crystals under the Effect of Crossed Electric and Magnetic Fields. *Crystals*. 2019; 9(5):262.
https://doi.org/10.3390/cryst9050262

**Chicago/Turabian Style**

Zakharov, Alex V., and Izabela Sliwa.
2019. "Peculiarities in the Director Reorientation and the NMR Spectra Evolution in a Nematic Liquid Crystals under the Effect of Crossed Electric and Magnetic Fields" *Crystals* 9, no. 5: 262.
https://doi.org/10.3390/cryst9050262