Electrically Driven Torsional Distortions in Twisted Nematic Volumes

Abstract

The purpose of this review is to describe the physical mechanism responsible for the appearances of both traveling and non-traveling distortions in a twisted microsized nematic volume under the effect of a large electric field. Both experimental and theoretical works devoted to the excitation of structured periodic domains in initially homogeneously aligned liquid crystal systems under the effects of strong crossed electric and magnetic fields were analyzed. Electrically driven distortions in the microfluidic nematic capillary in the presence of a temperature gradient in it, based on the number of numerical results, were analyzed. We also focus on the description and explanation of the novel mechanism of excitation of the kink- and $\pi$-like distortion waves of the director field $\widehat{\mathbf{n}}$ in a cylindrical nematic micro-volume under the effect of voltage U and temperature gradient $\nabla T$, set up between the cooler inner and hotter outer cylinders. Electrically driven torsional distortions in twisted nematic micro-volumes in the form of the kink-like running front based on the classical Ericksen–Leslie approach were considered.

1. Introduction

This review is motivated by the large number of studies of liquid crystals (LC), both experimental and theoretical techniques, aimed at showing that deformations of these LC materials under the effects of strong electric or magnetic (or both) fields, which are characterized by some features [1,2,3]. The field-induced effects, specifically, whether the applied voltage induces a homogeneous state or a structured one, are essential for research and technology. Some researchers have studied the processes of excitation of structured periodic domains in initially homogeneous LC systems under the effects of strong electric or magnetic fields. Other researchers, on the contrary, excluded the possibility of the formation of periodic domains in initially homogeneous systems and considered the further evolution of these systems as a whole without structural changes. Whatever the actual purpose, the condition for the initiation of excitation of periodic structures under the influence of strong external fields is important information for this LC system. It is necessary to understand the conditions that define the boundary between the excitation of a periodic pattern and the state when the excitement of this periodic structure is impossible. Knowing this, it is possible to predict the further behavior of this LC system under the influence of a strong electric or magnetic field.

Among all LC compounds, nematic liquid crystals are currently the most popular LC materials used in information technology, various LC sensors, and LC actuators [1,2]. For instance, the textures of twisted nematics (TNs) are obtained by orienting a drop of bulk LC material between two plates oriented perpendicular to each other with convenient treating. In turn, in the case of a uniformly aligned nematic drop placed between two plates oriented parallel to each other, the TN textures can be obtained by applying an electric field $\mathbf{E}$ directed orthogonally to a uniformly aligned LC material. It is shown [3] that there is a critical value of the electric field $E_{\mathrm{th}}$, when exceeded, a distortion of the homogeneous texture of the director field $\widehat{\mathbf{n}}$ occurs between the bounding surfaces. This form for critical field $E_{\mathrm{th}}$ is based on assumptions that the director remains strongly, e.g., homogeneously, anchored at these bounding surfaces, and the director $\widehat{\mathbf{n}}$ is uniformly aligned across the nematic sample for $E<E_{\mathrm{th}}$. On the other hand, with an increase in the electric field $E>E_{\mathrm{th}}$, several relaxation regimes of the director field arise in the microsized volume of the TN film. These regimes, in which the director $\widehat{\mathbf{n}}$ rotates in the plane parallel to both bounding surfaces, exert a torque directed to the normal of these boundaries. This applied electric field $\mathbf{E}$ may alter the molecular configuration of the LC layer and, thus, alter the optical characteristics of the LC cell [1].

The application of an electric field $\mathbf{E}$ along the x, y, or z axes to the TN micro-volume, initially uniformly aligned in the $x-z$ or $x-y$ planes (see Figure 1) can lead to three types of responses.

Firstly, in a uniformly aligned TN micro-volume, the structure does not change, and the further evolution of this LC system as a whole proceeds without structural changes. Secondly, in a uniformly aligned TN micro-volume, the structured periodic domains may be excited. Thirdly, the torsional distortions in the form of a kink- and $\pi$-like running fronts may occur in a uniformly aligned TN micro-volume.

The layout of this review is as follows. In the next section, both experimental and theoretical works devoted to the excitation of structured periodic domains in initially uniformly aligned LC systems under the effects of strong electric or magnetic fields will be analyzed. At the same time, the reorientation of the director field in a microsized nematic volume imposed by crossed electric and magnetic fields will be considered a simple reorientation of the monodomain without structural changes. Electrically driven distortions in a microfluidic nematic capillary in the presence of a temperature gradient in it will be analyzed in Section 3. We will focus on the description of a novel mechanism of the kink- and $\pi$-like distortion waves of the director field $\widehat{\mathbf{n}}$ in the microsized nematic volume under the effect of voltage U applied between two cylinders and a temperature gradient $\nabla T$, which is set up between cooler inner and hotter outer cylinders. Electrically driven torsional distortions in twisted nematic micro-volumes in the form of the kink-like running front will be considered in Section 4. Conclusions will be given in Section 5.

2. Spatially Periodic Distortions in Microsized Nematic Volumes

We are primarily interested in describing the physical mechanism responsible for the excitation of spatial distortions in an initially homogeneously aligned microsized nematic volume under the effect of crossed electric $\mathbf{E}$ and magnetic $\mathbf{B}$ fields. This process almost has half a century of history. For the first time, the dynamics of distortion of a well-aligned nematic liquid crystal film in a perpendicular magnetic field around the critical value of the Freedericksz field was studied both theoretically and experimentally around the early 1970s [4,5]. Experimental methods of thermal and optical birefringence techniques were used to describe the dynamics of a periodic roll structure in a homogeneously aligned methoxybenzylidenebutylaniline (MBBA) compound. In these works, a hydrodynamic model was developed based on the effect of reducing the effective viscosity in the presence of a backflow caused by rotation of $\widehat{\mathbf{n}}$, which occurs in a convective roll pattern. In particular, in [5], it was shown how the coupling between flow and distortion in the dynamics of the Freedericksz problem can lead to the excitation of the transient periodic domains. It was shown in [6] that the striped texture is often observed as the initial reaction of a uniformly aligned nematic liquid crystal to a suddenly applied reorienting electric field. It was also shown that the instability structure depends on the elastic and viscous anisotropy of the liquid crystal, the electric field strength, and, importantly, on the boundary conditions imposed by the sample cell. Moreover, within the framework of the classical Ericksen–Leslie theory [7,8], the nature of the twist-Freedericksz transition in LC samples under the effect of a suddenly applied large magnetic field was studied [9]. It has been shown that when the large magnetic field $\mathbf{B}$ is applied across the LC sample, competition arises between the field, on the one hand, and the surface anchoring forces mediated by the elasticity of the liquid crystal, on the other [4]. Above the threshold field $B_{\mathrm{th}}$, the interior of the LC sample begins to reorient toward $\mathbf{B}$, with twist elastic distortion produced between the interior and the aligning walls, at which $\widehat{\mathbf{n}}$ remains parallel to ${\widehat{\mathbf{n}}}_0$ (the strong anchoring condition). It has been shown that when a field above the threshold is suddenly applied to the LC sample, any small perturbation in the initially uniform alignment will begin to grow exponentially with a rate that is inversely proportional to some effective viscosity ${\gamma }_1\left(\mathrm{eff}\right)$ for that particular reorientation process.

To understand why and how a periodic domain, excited in the homogeneously aligned LC sample, can give a faster response at high fields, several considerations are involved [9]. First of all, a non-uniform rotation domain, e.g., oppositely rotating zones, produces a backflow reaction in addition to the molecular rotation. Second, this backflow reinforces the opposite rotations of neighboring regions of the LC sample. Third, these oppositely rotating regions effectively replace ${\gamma }_1$ with a lower shear viscosity. Thus, the decrease in viscosity can be huge, and as a result, many periodic responses occur. At almost the same time, a new form of the Freedericksz transition was described, in which the uniform splay distortion is replaced by a complex periodic twist-splay pattern having a lower critical field than that for uniform splay [10]. In this paper, it was noted that the apparent periodic splay domain may actually be a pure twist, which was a significant novelty of this structure. It has been shown that in the LC material with a splay elastic constant $K_1$, which is much larger than the twist elastic constant $K_2$, the periodic distortion will have a lower critical field than the uniform distortion since it avoids splay. On the other hand, because of its more complex structure, the periodic distortion requires more elastic energy to relax a given amount of field energy than does the uniform distortion. Based on these observations, it can be concluded that in materials in which $K_1$ is not much larger than $K_2$, uniform distortion will have a lower critical field.

Thus, there is a lot of experimental evidence on the emergence of transitional spatial structures in the early stages of the Freedericksz transition (see Reference [10] and the references in it). All of them require detailed research within the framework of the appropriate hydrodynamic models. In this situation, hydrodynamic aspects, as well as detailed numerical studies that should be carried out on the basis of these approaches, play a crucial role, since the director field dynamically correlates with the backflow.

A nonlinear deterministic analysis of nematodynamic equations has been broadly used to identify the most unstable mode whose characteristic wavenumber is then associated with the observed periodicity of the transient pattern [11]. In turn, the linear analysis assumes that all modes of distortion are independent and that the only mode, which is macroscopically observed, is the one with the fastest growth rate. The subsequent application of the early proposed approach [5], where the nematodynamic equations linearized with respect to the distortion angle were used, has been successful in a qualitative sense. In the framework of this approach, a reasonable estimation for the number of material parameters, such as a wavelength of the distortion as a function of the applied magnetic field, has been achieved. However, the important question of whether the simple analysis of periodic transition can be refined to a procedure for the specification of the condition under which the periodic domain can form in an initially uniformly aligned LC sample has not been resolved. In the general context of the problems of the formation of domains with periodic structures, another interesting question concerns the description of the appearance and disappearance of transitional spatial structures. This situation occurs, for example, in the Freedericksz transition, when a magnetic field whose intensity exceeds a certain threshold is applied to a uniformly aligned nematic liquid crystal [12]. The molecules are locally reoriented in opposite directions, which leads to a transitional pattern consisting of, for a certain geometry, a parallel stripe with a well-defined periodicity. It was shown that at the final stage of evolution, a domain with a periodic structure decays, which leads to uniform reorientation over the entire plane of the LC sample. The appearance of this transient structure is due to backflow effects that couple the director and velocity fields and lead to finite-wavelength instability.

Another direction in the study of the possibilities of the occurrence and disappearance of transient periodic structures in an initially uniformly aligned LC sample under the influence of a strong electric field involves the use of NMR spectroscopy [13].

Initially, our research interests are focused on descriptions of the physical mechanisms responsible for the excitation of spatially periodic distortions in the microsized homogeneously aligned nematic (HAN) volume under the effect of crossed electric $\mathbf{E}$ and magnetic $\mathbf{B}$ fields [13,14,15,16,17]. In the case of the HAN phase in which the director is initially aligned by the external magnetic field $\mathbf{B}$, the deuterium NMR spectrum originated from a group of equivalent deuterons consisting of a doublet whose quadrupolar splitting is denoted by $\Delta {\tilde{\nu }}_0$. When a sufficiently strong electric field $\mathbf{E}$ is applied with respect to $\mathbf{B}$, the director tends to be aligned parallel to $\mathbf{E}$, and the quadrupolar splitting $\Delta \tilde{\nu }\left({\theta }^{\prime }\right)$ is connected with $\Delta {\tilde{\nu }}_0$ as [13]

$$\begin{array}{c}\hfill \Delta \tilde{\nu }\left({\theta }^{\prime }\right)=\Delta {\tilde{\nu }}_0{P}_2\left(cos{\theta }^{\prime }\right),\end{array}$$

(1)

where $P_2\left(cos{\theta }^{\prime }\right)$ is the second Legendre polynomial, and ${\theta }^{\prime }$ is the angle made by $\widehat{\mathbf{n}}$ with $\mathbf{B}$ (see Figure 2).

As the electric field $\mathbf{E}$ is turned on, the director moves from being parallel to $\mathbf{B}$ to parallel the field $\mathbf{E}$. At the same time, the splitting $\Delta \tilde{\nu }\left({\theta }^{\prime }\right)$ will decrease, pass through zero, and then increase to one-half of $\Delta {\tilde{\nu }}_0$ when $\widehat{\mathbf{n}}$ is parallel to $\mathbf{E}$. Thus, it is established that deuterium NMR spectroscopy is a powerful method for studying the dynamic behavior of the director field in the microsized nematic volume.

In turn, when the electric field $E\gg E_{\mathrm{th}}$ (∼$1\mathrm{V}/\mathsf{\mu }\mathrm{m}$) is applied, the state of the nematic system becomes unstable, and the misalignment of the director relative to the direction imposed by the aligning magnetic field increases so much that the reorientation caused by the strong $\mathbf{E}$ manifests itself in the growth of one particular Fourier mode. In this case, the spectral line shape characterizing the initially aligned nematic sample expands with time-dependent splitting, while the initial steady doublet with constant splitting gradually vanishes [13,14,15,16]. Thus, the application of the large $\mathbf{E}$ leads to the appearance of new doublet with vanishing amplitude, which gradually increases with constant splitting, so that the total spectral intensity is transferred from the initial doublet to the new one, with half the quadrupolar splitting. The analysis of these results strongly suggests that the intermediate state is inhomogeneous and perturbed by thermal fluctuations, so that in response to the suddenly applied electric field, spatially periodic patterns can appear in initially uniformly aligned nematic domains [17,18,19,20,21]. These nonuniform rotational modes involve additional internal elastic distortions of the conservative nematic system and, as a consequence, these deformations reduce the viscous contribution to the total energy of the nematic phase [21].

In turn, the decrease in the viscous contribution leads to a decrease in the effective rotational viscosity coefficient and gives a faster response to the director rotation of the director than for the uniform mode, as observed in the time-resolved deuterium NMR spectroscopic measurements [13,14]. Therefore, it is necessary to analyze the nematic reaction to the initial state, which exhibits some thermal fluctuations of the director under the influence of the strong electric field. Such a reaction of the initial HAN domain to the suddenly applied electric field with the appearance of spatially periodic patterns will be analyzed on the basis of the Ericksen–Leslie theory [7,8], supplemented by the charge balance equation. The situation will also be analyzed when, under the effect of externally applied electric field, the kink- or double $\pi$-forms of distortion wave propagating along the normal to the bounding surfaces can occur in the microsized nematic volume [22]. Since anomalous changes in the shape of spectral lines do not provide any information about the average orientation of the director, additional numerical studies of these LC systems have been carried out, which include both director reorientation and fluid flow [17,18,19,20,21].

2.1. Reorientation of the Director Field in a Microsized Nematic Volume Imposed by Crossed Electric and Magnetic Fields as a Simple Monodomain

First of all, the process of reorientation of the director field as a single monodomain was investigated, both by the NMR technique [13,14,15,16,17] and numerically [23], within the framework of the classical Ericksen–Leslie theory [7,8], supplemented by the charge balance equation. For this aim, deuterium NMR spectroscopy was used to investigate the director dynamics of specifically deuteriated $4-\alpha ,\alpha -d_2-pentyl-4^{\prime }-cyanobiphenyl$ (5CB-d₂) subject to both magnetic and electric fields in the nematic phase. The spectra were recorded using a JEOL Lambda 300 spectrometer, which has a magnetic flux density, $\mathbf{B}$, of $7.05\mathrm{T}$. The nematic cell was held in the NMR probe head so that the electric field, whose direction is normal to the substrate surface, makes an angle, $\alpha$, with the magnetic field (see Figure 2). For 5CB-d₂ between the two electrodes with a voltage across the sample of 50 V, the electric potential was applied at an angle, $\alpha$, to the magnetic field of the spectrometer (see Figure 2). This provides a unique reorientation pathway. A sequence of time-dependent deuterium NMR spectra was obtained, which can be used to study the dynamic director orientation. When the electric field is applied to the nematic film, the director moves from being parallel to the magnetic field to being at an angle to the magnetic field (the turn-on process) because the dielectric and magnetic anisotropy are both positive for $5CB-d_2$. After the electric field is turned off, the director relaxes back to being parallel to the magnetic field (the process of turning off). The deuterium NMR spectra were recorded during the turn-on and the turn-off alignment processes as a function of time. In that case, the quadrupolar splitting is given by Equation (1). It is expected that when the electric field is applied or removed, the monodomain director orientation will move in the plane defined by $\mathbf{B}$ and $\mathbf{E}$. The deuterium NMR spectra were recorded during the turn-on and turn-off processes [13,23]. In the turn-on process, the quadrupolar splitting was observed to decrease and then saturate with time corresponding to the equilibrium alignment of the director $\widehat{\mathbf{n}}$. Figure 3 shows the temporal changes in the ratio of the quadrupolar splittings, determined from the time-resolved deuterium NMR spectra recorded at $299\mathrm{K}$ for the turn-on and turn-off processes.

In this case, for the turn-on process, the director rotates from its initial orientation, ${\theta }_0^{\prime }=0^{°}$, at which $\Delta {\tilde{\nu }}_0=49.6\mathrm{kHz}$, and then aligns at the limiting angle, ${\theta }_{\infty }^{\prime }$, of 29.5${}^{°}$, determined from Equation (1) with $\Delta {\tilde{\nu }}_{\infty }=31.5\mathrm{kHz}$, the limiting value of $\Delta \tilde{\nu }\left(t\right)$ as t tends to infinity. In the turn-off process, the time dependence of the director orientation was obtained in the same way and is also shown in Figure 3. The time dependence of the director reorientation, described by the angle ${\theta }^{\prime }\left(t\right)$, has been obtained analytically [13], as

$$\begin{array}{c}\hfill tan\left({\theta }^{\prime }\left(t\right)-{\theta }_{\infty }^{\prime }\left(\alpha \right)\right)=tan\left({\theta }_0^{\prime }-{\theta }_{\infty }^{\prime }\left(\alpha \right)\right)exp\left(-\frac{t}{t_{\mathrm{ON}\left(\mathrm{OFF}\right)}}\right).\end{array}$$

(2)

Here, ${\theta }_{\infty }^{\prime }\left(\alpha \right)$ is the limiting value of ${\theta }^{\prime }\left(t\right)$ when t tends to infinity, ${\theta }_0^{\prime }$ is the initial director orientation, and times $t_{\mathrm{ON}\left(\mathrm{OFF}\right)}$ are related to the material parameters by

$$\begin{array}{c}\hfill t_{\mathrm{ON}\left(\mathrm{OFF}\right)}=\frac{{\gamma }_1}{\sigma {ϵ}_0{ϵ}_a{E}^2},\end{array}$$

(3)

and

$$\begin{array}{c}\hfill \sigma =\sqrt{1+2{\sigma }_{1}cos2\alpha +{\sigma }_1^2},\end{array}$$

(4)

where ${\sigma }_1={\mu }_0{ϵ}_0{\left(\frac{E}{B}\right)}^2\frac{{ϵ}_a}{{\chi }_a}$, ${ϵ}_a$ denotes the nematic dielectric anisotropy, ${\chi }_a$ is the anisotropy of the diamagnetic susceptibility of the nematic, and the limiting value ${\theta }_{\infty }^{\prime }$ is given by

$$\begin{array}{c}\hfill cos2{\theta }_{\infty }^{\prime }=-\frac{1+{\sigma }_{1}cos2\alpha }{\sigma }.\end{array}$$

(5)

The values of the two times, $t_{\mathrm{OFF}}$ and $t_{\mathrm{ON}}$, were obtained by fitting the ratio of the quadrupolar splittings obtained as a function of time for the turn-on and turn-off processes. The solid lines in Figure 3 show the best fits giving the values for $t_{\mathrm{ON}}$ and $t_{\mathrm{OFF}}$ as 0.594 ms and 1.18 ms, respectively.

In order to investigate the monodomain reorientation of the director exposed to both electric and magnetic fields, taking into account the backflow, induced by the electric field and the director’s reorientation, a numerical study of the complete system of hydrodynamic equations was carried out, which include both the director’s reorientation and the fluid flow [23]. For this aim, the theoretical treatment and numerical results were the same as in the NMR experiment sample of $5CB-d_2$, at the temperature $300\mathrm{K}$ and density ${10}^3{\mathrm{kg}/\mathrm{m}}^3$. It was considered a nematic system enclosed in a microsized volume bounded by two horizontal and two vertical surfaces at mutual distances L and $d(L\gg d)$ on a scale of the order of tens of micrometers. The coordinate system defined by this task assumed that the electric field $\mathbf{E}$ is applied normally (or close to normal) to the magnetic field $\mathbf{B}$, which is directed parallel to the horizontal surfaces (see Figure 2). This problem was considered within the framework of the extended Ericksen–Leslie theory [7,8], supplemented by the charge balance equation, and the geometry of this system can be considered two-dimensional since the director $\widehat{\mathbf{n}}$ lies in the plane $xz$ (or $yz$), determined by electric $\mathbf{E}$ and magnetic $\mathbf{B}$ fields. In this geometry, the unit vector $\widehat{\mathbf{i}}$ is directed parallel to the horizontal bounding surfaces and coincides with the direction of the vector $\mathbf{B}$, while the unit vector $\widehat{\mathbf{k}}$ coincides with the vector $\mathbf{E}$ and $\widehat{\mathbf{j}}=\widehat{\mathbf{k}}\times \widehat{\mathbf{i}}$, respectively. It was supposed that the components of the director $\widehat{\mathbf{n}}=n_x\widehat{\mathbf{i}}+n_z\widehat{\mathbf{k}}=cos\theta \left(z,t\right)\widehat{\mathbf{i}}+sin\theta \left(z,t\right)\widehat{\mathbf{k}}$ (see Figure 2), depend only on x, z variables and time t. Further calculations were carried out with the complex flow $\mathbf{v}(z,t)$ induced by the electric field and the director reorientation, together with the incompressibility condition and no-slip boundary conditions on the restricted surfaces $\mathbf{v}(z=0)=\mathbf{v}(z=d)=0$, which implies that only one non-zero component of the vector $\mathbf{v}$ exists, viz., $\mathbf{v}(t,z)=v_x(t,z)\widehat{\mathbf{i}}=u(t,z)\widehat{\mathbf{i}}$. In that case, the Navier–Stokes equation reduces to [7,23,24]

$$\begin{array}{c}\hfill \rho u_{,t}(t,z)={\sigma }_{zx,z},\end{array}$$

(6)

$$\begin{array}{c}\hfill P_{,z}(t,z)+\frac{\partial \mathcal{R}}{\partial {\theta }_{,t}}{\theta }_{,z}=0,\end{array}$$

(7)

where $\rho$ is the mass density, ${\partial }_{,t}=\partial /\partial t$, ${\partial }_{,z}=\partial /\partial z$, $P_{,z}=\frac{\partial P(t,z)}{\partial z}$, and $P(t,z)$ is the hydrostatic pressure in the cell, $\mathcal{R}\equiv \mathcal{R}(\theta ,u)=\frac{{\gamma }_1}{2}\left(h\left(\theta \right)u_{,z}^2+2\mathcal{A}\left(\theta \right){\theta }_{,t}{u}_{,z}+{\theta }_{,t}^2\right)$ is the Rayleigh dissipation function, $\mathcal{A}\left(\theta \right)=\frac{1}{2}\left(1+{\gamma }_{21}cos2\theta \right)$, $h\left(\theta \right)=$$\frac{1}{{\gamma }_1}\left(\frac{{\alpha }_1}{2}{sin}^{2}2\theta +\left({\alpha }_5-{\alpha }_2\right){sin}^2\theta +\left({\alpha }_3+{\alpha }_6\right){cos}^2\theta +{\alpha }_4\right)$, $u_{,z}=\frac{\partial u(t,z)}{\partial z}$ is the gradient of the velocity $u(t,z)$, ${\alpha }_i(i=1,\dots ,6)$ are the six Leslie coefficients, ${\gamma }_{21}=\frac{{\gamma }_2}{{\gamma }_1}$, and ${\gamma }_1$ and ${\gamma }_2$ are the rotational viscosity coefficients. The stress tensor (ST) component, ${\sigma }_{zx}$, is given by ${\sigma }_{zx}=\frac{\delta \mathcal{R}}{\delta u_{,z}}=h\left(\theta \right)u_{,z}+\mathcal{A}\left(\theta \right){\theta }_{,t}$, which involves algebraic expressions of the director components and the velocity gradients [25].

For the quasi-two-dimensional geometry, the torque-balance equation describing the reorientation of the nematic film confined between two electrodes can be derived from the balance of elastic, viscous, magnetic, and electric torques ${\mathbf{T}}_{\mathrm{elast}}+{\mathbf{T}}_{\mathrm{vis}}+{\mathbf{T}}_{\mathrm{mag}}+{\mathbf{T}}_{\mathrm{el}}=0$, and takes the dimensionless or scaled form [23]

$$\begin{array}{c}\hfill {\theta }_{,\tau }=-\mathcal{A}\left(\theta \right)u_{,z}+{\delta }_1\left(\frac{1}{2}{G}_{\theta }\left(\theta \right){\theta }_{,z}^2+G\left(\theta \right){\theta }_{,zz}\right)+\frac{{\overline{E}}^2\left(z\right)}{2}sin2\theta +{\delta }_{2}sin2\left(\theta +\alpha \right),\end{array}$$

(8)

where $G\left(\theta \right)={cos}^2\theta +K_{31}{sin}^2\theta$, ${\theta }_{,zz}={\partial }^2/\partial z^2$, $K_{31}=K_3/K_1$, and $K_1$ and $K_3$ are the splay and bend elastic constants. Here, $\tau =\left(\frac{{ϵ}_a{ϵ}_0{E}^2}{{\gamma }_1}\right)t$ and $\overline{z}=\frac{z}{d}$ are the scaled time and scaled distance from the bottom electrode, d is the thickness of the cell, ${ϵ}_a={ϵ}_{\Vert }-{ϵ}_{\perp }$, ${ϵ}_{\Vert }$, and ${ϵ}_{\perp }$ are the dielectric constants parallel and perpendicular to the director, while ${\delta }_1=\frac{K_1}{{ϵ}_a{ϵ}_0{E}^2{d}^2}$ and ${\delta }_2=\frac{{\chi }_a{B}^2}{2{ϵ}_a{ϵ}_0{E}^2{\mu }_0}$ are two parameters for the system.

The application of the voltage across the nematic film results in a variation of $E\left(z\right)$ through the film [23], which is obtained from

$$\begin{array}{c}\hfill \frac{\partial }{\partial z}\left[\left(\frac{{ϵ}_{\perp }}{{ϵ}_a}+{sin}^2\theta \left(\tau ,z\right)\right)\overline{E}\left(z\right)\right]=0,\\ \hfill 1={\int }_0^1\overline{E}\left(z\right)dz,\end{array}$$

(9)

where $\overline{E}\left(z\right)=\frac{E\left(z\right)}{E}$, $E=\frac{U}{d}$, and U is the voltage applied across the cell.

In order to be able to observe the evolution of the angle $\theta (\tau ,z)$ to its stationary value ${\theta }_{\mathrm{st}}\left(z\right)$, as well as the evolution of the velocity field $u(\tau ,z)$ caused both by the electric field and the director reorientation to its stationary orientation, the scaled analog of the Navier–Stokes equation was considered (see Equation (6))

$$\begin{array}{c}\hfill {\delta }_3{u}_{,\tau }(\tau ,z)={\sigma }_{zx,z},\end{array}$$

(10)

where ${\delta }_3=\rho {ϵ}_0{ϵ}_a{\left(Ed/{\gamma }_1\right)}^2$ is an additional parameter for the LC system. The torque balance transmitted to the surfaces assumes that the director angle has to satisfy the boundary conditions [23]

$$\begin{array}{c}\hfill G\left(\theta \right){\left(\partial \theta \left(z\right)/\partial z\right)}_{z=0}=\frac{Ad}{2K_1}sin2\Delta {\theta }^{-},\\ \hfill G\left(\theta \right){\left(\partial \theta \left(z\right)/\partial z\right)}_{z=1}=\frac{Ad}{2K_1}sin2\Delta {\theta }^{+},\end{array}$$

(11)

where A is the anchoring strength [26], $\Delta {\theta }^{\pm }={\theta }_s^{\pm }-{\theta }_0^{\pm }$, whereas the initial orientation of the director is directed parallel to both surfaces, with $\theta (\tau =0,z)=0$, and then allowed to relax to its equilibrium value ${\theta }_{\mathrm{eq}}\left(z\right)$. Here, ${\theta }_s^{\pm }$ and ${\theta }_0^{\pm }$ are the pre-tilt angles of the surface director and the easy axes at $z=0$ and $z=1$, respectively. The no-slip condition on $\mathbf{v}$ at both solid surfaces assumes that the velocity has to satisfy the boundary conditions [23]

$$\begin{array}{c}\hfill u{\left(z\right)}_{z=0}\equiv v_x{\left(z\right)}_{z=0}=0,\\ \hfill u{\left(z\right)}_{z=1}\equiv v_x{\left(z\right)}_{z=1}=0.\end{array}$$

(12)

The reorientation of the director in the nematic film between the two solid surfaces, when the relaxation regime is governed by the viscous, elastic, magnetic, and electric torques, and including backflow, can be obtained by solving the system of non-linear partial differential Equations (8)–(10), with appropriate boundary (see Equations (11) and (12)) and initial $\theta (\tau =0,z)=0$ conditions. For $5CB-d_2$ between the two electrodes with a voltage across the sample of 50 V, both relaxation regimes, with and without taking the backflow into account, are characterized by a monotonic increase of the angle $\theta (\tau ,z)$ up to the stationary value ${\theta }_{\mathrm{st}}\left(z\right)$, with practically, the same relaxation time (see Figure 4).

Physically, this means that the electric field produces an alignment of the director away from the magnetic field, caused by the applied voltage and that the field has a strong influence compared with the other torques, on the relaxation process. These calculations reveal the weak effect of backflow on the relaxation process in the nematic cell under the influence of the strong (∼$1\mathrm{V}/\mathsf{\mu }\mathrm{m}$) electric field. The calculations also show that the electric, magnetic, elastic, and viscous torques exerted are vanishingly small for the above-mentioned case when the electric field is almost orthogonal to the magnetic field after a scaled relaxation time of 25 which corresponds to a relaxation time of $31.8\mathrm{ms}$. Calculations also show that the angle between the two fields, $\alpha$, has a strong influence on the relaxation process for the turn-on regime (see Figure 5).

Thus, the main conclusion that can be done for the case of the $5CB-d_2$ sample with the thickness of $50\mathsf{\mu }\mathrm{m}$, when the voltage of $50\mathrm{V}$ is applied at an angle $\alpha$ to the magnetic field $\mathbf{B}$ of $7.05$ T, is that the electric field causes the reorientation of the LC sample as a monodomain.

2.2. Spatially Periodic Patterns in a Microsized Nematic Volume Imposed by Crossed Electric and Magnetic Fields

The peculiarities in the dynamics of the director reorientation in the LC film under the influence of the electric $\mathbf{E}$ field directed perpendicular to the magnetic $\mathbf{B}$ field begin when the larger electric field is applied across the LC sample [13,21,25]. For this aim, the time-resolved deuterium NMR spectroscopic measurements of field-induced director reorientations were performed [13,14,25]. Taking into account the fact that the quadrupolar splitting is related to the angle $\theta$ made by the director $\widehat{\mathbf{n}}$ with the magnetic field $\mathbf{B}$ (see Figure 2), deuterium NMR spectroscopy was found to be a powerful method to investigate the dynamic director reorientation in nematic films. The deuterium NMR spectroscopy was used to investigate the director dynamics of deuteriated $5CB-d_2$ subject to both magnetic and electric fields in the nematic phase. A sequence of deuterium NMR spectra was acquired as a function of time, which can be used to explore the dynamic director reorientation. Analysis of these results shows that the value of the relaxation time for the turn-on process ${\tau }_{\mathrm{ON}}\left(\alpha \right)$ monotonically grows with the increase of angle $\alpha$, up to the maximum value ${\tau }_{\mathrm{ON}}\left(\max \right)$. With further growth of $\alpha$, up to the right angle ($\alpha \sim \frac{\pi }{2}$), ${\tau }_{\mathrm{ON}}(\alpha \sim \frac{\pi }{2})$ rapidly decreases with a few milliseconds, with respect to ${\tau }_{\mathrm{ON}}\left(\max \right)$ [13,14,25].

This behavior of the relaxation time can be explained by the spontaneous excitation of periodic structures in an initially uniformly aligned LC sample. Physically, this means that a periodic distortion emerges spontaneously from the homogeneous state. It may induce a faster response than in the uniform mode because it has a lower effective viscosity. To support this point of view, a complete analysis was carried out, which included the system of hydrodynamic equation for describing the director’s reorientation and for the velocity field [17,21,25]. This analysis of the effect of the strong electric field on the reorientation of the director field suggests that in order to correctly describe the dynamic evolution of $\widehat{\mathbf{n}}$, we do not need to include proper processing of the backflow [23]. This means that the main role is played by electric and magnetic forces and the viscous forces become negligible compared to the aforementioned contributions to the dimensionless torque balance equation. In this case, the torque balance equation takes the form [18,19,20]

$$\begin{array}{c}\hfill {\gamma }_1{\theta }_{,t}=K_1\left[\left(2-\mathcal{M}\right){\theta }_{,xx}+\mathcal{G}\left(\theta \right)\left({\theta }_{,zz}-{\theta }_{,xx}\right)\right]+\\ \hfill \frac{1}{2}{K}_1\mathcal{M}\left[\left({\theta }_{,x}^2-{\theta }_{,z}^2-2{\theta }_{,xz}\right)sin2\theta -{\theta }_{,x}{\theta }_{,z}cos2\theta \right]+\\ \hfill \frac{1}{2}{ϵ}_0{ϵ}_a{E}^{2}sin2\left(\alpha -\theta \right)-\frac{1}{2}\frac{{\chi }_a}{{\mu }_0}sin2\theta B^2-\frac{1}{2}(e_1+e_3)E_{,z}sin\alpha sin2\theta ,\end{array}$$

(13)

where ${\theta }_{,t}=\partial \theta /\partial t$ and ${\theta }_{,x}=\partial \theta /\partial x$ denote the partial derivative of the angle $\theta$ with respect to time t and space coordinate x, respectively, while $\mathcal{G}\left(\theta \right)={cos}^2\theta +K_{31}{sin}^2\theta$, $\mathcal{M}=1-K_{31}$, and $E_{,z}=\partial E/\partial z$.

Specific calculations were carried out for the microsized nematic volume bounded between two electrodes, when the director is weakly anchored to the horizontal bounding surfaces with anchoring energy [26]

$$\begin{array}{c}\hfill W_{an}=\frac{1}{2}A{sin}^2\left({\theta }_s-{\theta }_0\right),\end{array}$$

(14)

where A is the anchoring strength, ${\theta }_s$ and ${\theta }_0$ are the angles corresponding to the director orientation on the bounding surface, ${\widehat{\mathbf{n}}}_s$, and easy axis, $\widehat{\mathbf{e}}$, respectively (see Figure 2). In this case, the torque balance transmitted to the bounding surfaces assumes that the direction angle must satisfy the boundary conditions [18]

$$\begin{array}{c}\hfill \mathcal{G}\left(\theta \right){\left({\theta }_{,z}\left(z\right)\right)}_{z=\mp d}=\frac{A}{2K_1}sin2\Delta {\theta }^{\mp },\end{array}$$

(15)

where $\Delta {\theta }^{\mp }={\theta }_s^{\mp }-{\theta }_0^{\mp }$, while ${\theta }_s^{\mp }$ and ${\theta }_0^{\mp }$ are the pre-tilt angles of the surface director and the easy axes at $z=-d$ and $z=d$, respectively. In turn, it is assumed that regarding the two lateral surfaces, the direction angle must satisfy the strong anchoring conditions,

$$\begin{array}{c}\hfill \theta (x=\mp L,-d<z<d)=0,\end{array}$$

(16)

while the initial orientation of the director is directed parallel to both horizontal surfaces, with $\theta (x,z,\tau =0)=0$, and then allowed to relax to its stationary value ${\theta }_{st}(x,z)$.

The application of the voltage across the microsized nematic volume results in a variation of $E\left(z\right)$ through the film [18], which is obtained from

$$\begin{array}{c}\hfill {\left[\mathcal{B}\left(\theta \right)\overline{E}\left(z\right)\right]}_{,z}=0,{\int }_{-d}^d\overline{E}\left(z\right)dz=1,\end{array}$$

(17)

where the function $\mathcal{B}\left(\theta \right)$ is equal to $\frac{{ϵ}_{\perp }}{{ϵ}_a}+{sin}^2\theta$, $\overline{E}\left(z\right)=\frac{E\left(z\right)}{E}$, $E=\frac{U}{d}$, and U is the voltage applied across the microsized film.

The reorientation of the director in the microsized nematic volume under the effect of the external forces was obtained by solving the nonlinear differential Equations (13) and (17) with appropriate boundaries ((15) and (16)) and initial

$$\begin{array}{c}\hfill \theta \left(x,z,\tau =0\right)=0,\end{array}$$

(18)

conditions [18,19,20,21].

In this case, when the dielectric and magnetic anisotropies of the LC phase are positive (the case of deuterated $5CB-d_2$), and when a strong electric field $\mathbf{E}$ is applied at an angle $\alpha$ close to the right angle to the magnetic field $\mathbf{B}$, the reorientation process of the director field in the microsized LC volume proceeds as a uniform reorientation of a simple LC monodomain [21,23]. In turn, the analysis of the NMR results strongly suggests that the intermediate state is inhomogeneous and disturbed by thermal fluctuations, so that in response to the suddenly applied electric field, spatially periodic structures may appear in initially homogeneously aligned nematic (HAN) domains [17,18,19,20,21]. These nonuniform rotational modes involve additional internal elastic distortions of the conservative nematic system and, as a consequence, these deformations reduce the viscous contribution to the total energy of the nematic phase [25]. This decrease in the viscous contribution leads to a decrease in the effective coefficient of rotational viscosity and provides a faster rotation response to the director than for the uniform mode, as observed in measurements of deuterium NMR spectroscopy [13,25]. Therefore, it is necessary to analyze the nematic response to the initial state, which exhibits some thermal fluctuations of the director under the effect of the strong electric field. To find out the role of directorial fluctuations in maintaining spatially periodic structures in a microsized nematic volume under the effect of the strong orthogonal electric field, numerical studies were carried out [17,18,19,20,21]. In order to observe the formation of spatially periodic structures in nematic volume excited by the strong orthogonal electric field, the initial condition for the polar angle $\theta (x,z,\tau =0)$ was chosen in the form [18,19,25,27]

$$\begin{array}{c}\hfill \theta (x,z,0)={\theta }_{\mathrm{fl}}cos\left(q_{x}x\right)cos\left(q_{z}z\right),\end{array}$$

(19)

which determines the fluctuations of the director field over the nematic volume with amplitude ${\theta }_{fl}$ and the wavelengths $q_x$ and $q_z$ of a separate Fourier modulation component.

In a series of computational works [18,19,20,21,27,27] performed within the framework of the classical Ericksen–Leslie theory [7,8], supplemented by the charge balance equation, the processes of excitation and evolution of the director field in the form of a well-developed periodic structure, under the effects of the crossed electric and magnetic fields, have been described. It has been shown that the reorientation of the director field to its stationary orientation, described by the polar angle $\theta (x,z=0,\tau )$ to its stationary distribution ${\theta }_{\mathrm{st}}(x,z=0,\tau ={\tau }_R)$, under the effect of the electric field (∼1.03 V/$\mathsf{\mu }\mathrm{m}$) applied at the angle $\alpha =1.57$ (∼89.96${}^{°}$) with respect to the magnetic (∼7.05 T) field, with the initial condition presented as the Equation (19), is characterized by a distinct periodic structure with lattice points in $x=-10.0,-7.74,-5.74,-3.30,-1,27,1.21,3.21,5,65,7,68,10,0$, and the relaxation time ${\tau }_1\left(A\right)$ is equal to 24.5 (∼31.1 ms), (see Figure 6a).

In this case, the value of amplitude ${\theta }_{fl}$ is equal to 0.01 (∼1.1${}^{°}$) (case 1). In turn, when the amplitude value ${\theta }_{fl}$ is equal to 0.001 (∼0.11${}^{°}$) (case 2), the dependence of the angle profile $\theta (x,z=0,\tau )$ on time has a wavelike profile along the variable axis x growing in the positive direction (see Figure 6b). Only at the later stage of the reorientation process does the completely convex profile grow in the positive direction, and the relaxation time ${\tau }_2$ = 25 (∼31.75 ms) is developed.

Physically, this means that in case 2, the value of the electric field component $E_z$-directed perpendicular to the magnetic field is not enough to maintain spatially periodic structures, but only enough to form wave-like deformations in the microsized nematic volume. With the further decrease of the angle $\alpha$ up to 1.565 (∼88.81${}^{°}$), only for the values of $q_x=0.785$ and $q_z=64.30$, and the amplitudes ${\theta }_{fl}$ equal or greater than $0.02$, the nonuniform rotation mode rather than the uniform one (see Figure 7a,b) is provided.

In this case, two scenarios of relaxation of the angle $\theta (x,z=0,\tau )$ to its stationary value ${\theta }_{\mathrm{st}}(x,z=0,\tau ={\tau }_R)$ over the microsized nematic volume with the strong anchoring condition for the angle $\theta$ and under the effect of the electric field (∼1.03 $\mathrm{V}/\mathsf{\mu }\mathrm{m}$) applied at the angle $\alpha$ = 1.565 (∼88.81${}^{°}$), to the magnetic field (∼7.05 T) are shown. In case (a) the value of amplitude ${\theta }_{fl}$ is equal to 0.02 (∼2.2${}^{°}$) (case 3), while in case (b) is equal to 0.01 (∼1.1${}^{°}$) (case 4), respectively. In case 3, the process of director reorientation is characterized by the non-perfect periodic structures with lattice points in $x=-10.0,-7.42,-6.0,-2.94,-1,52,1.48,2.88,5,90,7,43,10,0$, and the relaxation time ${\tau }_3$ is equal to 20.0 (∼25.4 ms), while in case 4, the time dependence of the angle profile $\theta (x,z=0,\tau )$ has a wavelike profile along the variable axis x growing in the positive direction. Only at the later stage of the reorientation process does one deal with the completely convex profile growing in the positive direction, with the relaxation time being ${\tau }_4$ = 21.0 (∼26.7 ms).

It is obvious from Figure 6 and Figure 7 that for certain values of the electric (∼1.03 $\mathrm{V}/\mathsf{\mu }\mathrm{m}$) and magnetic (∼7.05 T) fields applied across the nematic film of thickness ∼194.7 $\mathsf{\mu }\mathrm{m}$, for the values of the angle $\alpha$ greater or equal to 1.565 (∼88.81${}^{°}$) there is a threshold value of the amplitude ${\theta }_{fl}$, which provides the nonuniform rotation mode rather than the uniform one, while for the lower values both of the amplitude ${\theta }_{fl}$ and the angle $\alpha$ the uniform mode dominate [18,27].

In turn, the time-resolved deuterium NMR spectra give the temporal variation in the ratio of the quadrupolar splitting frequency, $\Delta \tilde{\nu }/\Delta {\tilde{\nu }}_0$. These ratios and Equation (1) give the director orientation as a function of time during the turn-on and turn-off processes at each angle $\alpha$ as shown by the symbols in Figure 8a,b, respectively.

The results of comparing the experimentally obtained values of the relaxation times of the director field under the effect of crossed electric (∼1.03 V/$\mathsf{\mu }\mathrm{m}$) and magnetic (∼7.05 T) fields and the calculated values under different anchoring conditions are presented in the

Table 1. The calculated values of relaxation times ${\tau }_{\delta }(\Delta )$, where $\delta =1,\dots ,8$ and $\Delta =B,C$.

$\alpha =1.57$	${\tau }_1\left(\mathrm{B}\right)$	${\tau }_2\left(\mathrm{B}\right)$	${\tau }_5\left(\mathrm{C}\right)$	${\tau }_6\left(\mathrm{C}\right)$
	$24.5(\sim$$31.1\mathrm{ms})$	$25.0(\sim$$31.75\mathrm{ms})$	$24.5(\sim$$31.1\mathrm{ms})$	$26.0(\sim$$33.0\mathrm{ms})$
$\alpha =1.565$	${\tau }_3\left(\mathrm{B}\right)$	${\tau }_4\left(\mathrm{B}\right)$	${\tau }_7\left(\mathrm{C}\right)$	${\tau }_8\left(\mathrm{C}\right)$
	$20.0(\sim$$25.4\mathrm{ms})$	$21.0(\sim$$26.7\mathrm{ms})$	$20.5(\sim$$26.0\mathrm{ms})$	$22.0(\sim$$28.0\mathrm{ms})$

[27].

This LC system under the effects of the above-mentioned crossed electric and magnetic fields applied across the nematic film of thickness ∼194.7 $\mathsf{\mu }\mathrm{m}$, at two values of the angle $\alpha =1.57$ (∼89.96${}^{°}$) and 1.565 (∼88.81${}^{°}$) show a slight increase in the values of ${\tau }_i\left(C\right)(i=5,6,7,8)$ compared to ${\tau }_i\left(B\right)(i=1,2,3,4)$. Here, ${\tau }_i\left(C\right)(i=5,6,7,8)$ correspond to the case of the weak anchoring ($A={10}^{-6}\mathrm{J}/{\mathrm{m}}^2$) conditions on the horizontal surfaces and two values of the angle $\alpha =1.57$ (∼89.96${}^{°}$) and 1.565 (∼88.81${}^{°}$). In case (5) and the value of the angle $\alpha =1.57$ (∼89.96${}^{°}$), the value of the amplitude ${\theta }_{fl}$ is equal to 0.01 (∼1.1${}^{°}$), while in case (6) and the value of the angle $\alpha =1.57$ (∼89.96${}^{°}$), the value of the amplitude ${\theta }_{fl}$ is equal to 0.001 (∼0.11${}^{°}$), respectively.

In turn, in case (7) and the value of the angle $\alpha =1.565$ (∼88.81${}^{°}$), the value of the amplitude ${\theta }_{fl}$ is equal to 0.02 (∼2.2${}^{°}$), while in case (8) and the value of the angle $\alpha =1.565$ (∼88.81${}^{°}$), the value of the amplitude ${\theta }_{fl}$ is equal to 0.01 (∼1.1${}^{°}$), respectively. The values of the relaxation times ${\tau }_i\left(B\right)(i=1,2,3,4)$ correspond to the case of the strong anchoring condition on all bounding surfaces. These calculations show the weak effect of anchoring conditions on the reorientation of the director’s field under the effect of the above-mentioned external fields.

In turn, a direct comparison (calculated and obtained using the NMR spectroscopic technique of relaxation time data ${\tau }_R(\alpha \sim 88.7^{°})\sim 20\mathrm{ms}$ (see Figure 8)) shows a good correspondence between these values. An analysis of these results shows that in deuterated $5CB-d_2$ at the temperature of $300\mathrm{K}$ and the density of ${10}^3{\mathrm{kg}/\mathrm{m}}^3$, the application of the large electric (∼1.03 $\mathrm{V}/\mathsf{\mu }\mathrm{m}$) field applied at the angle $\alpha$ close to the right angle to the magnetic field (∼7.05 T) leads to the values ${\tau }_R\left(\alpha \right)\sim 20÷25\mathrm{ms}$, for angle $\alpha \sim 88.7^{°}$ and higher. Calculations of the reorientation of the director field in a microsized nematic volume imposed by crossed electric and magnetic fields in the form of a simple monodomain show that the scaled relaxation time is 30, which corresponds to a relaxation time of $38.1\mathrm{ms}$ [27].

Thus, this analysis of numerical results based on the predictions of the hydrodynamic theory, including both the director reorientation and the charge balance equation, provides evidence for the appearance of spatially periodic structures in response to the large electric field applied at an angle to the magnetic field. In turn, periodic distortions of large amplitude modulated in the microsized nematic volume parallel to horizontal bounded surfaces lead to the increase in the elastic energy of the conservative LC system, and, as a consequence, this causes the decrease in the viscous contribution of $W_{\mathrm{vis}}$ to the total energy of the LC system. In turn, the decrease in $W_{\mathrm{sys}}$ leads to lower values of the rotational viscosity coefficient $\${\gamma }_1\left(\mathrm{def}\right)$, and should lead to the faster relaxation time ${\tau }_R\left(\alpha \right)$, as observed experimentally [13,25].

3. Electrically Driven Distortions in Microfluidic Nematic Capillary

In this section, we will focus, on the one hand, on the description of a new mechanism of the kink-like distortion wave of the director field $\widehat{\mathbf{n}}$ in the microsized nematic volume under the effect of a voltage U applied between two cylinders and a temperature gradient $\nabla T$, which is set up between cooler inner and hotter outer cylinders [28,29,30] (see Figure 9).

It was found that, under certain conditions, in terms of the curvatures of cylinders $\kappa$ and the voltage U applied between cylinders, the torques and forces acting on the director $\widehat{\mathbf{n}}$ may excite both the kink-like distortion wave spreading along the normal to cylindrical boundaries, whose resemblance to a kink-like distortion wave depends on the value of U and the curvature of the inner cylinder. On the other hand, conditions were developed in terms of $\kappa$ and U, creating the distortion mechanism of $\widehat{\mathbf{n}}$ in the double $\pi$-form with the intermediate relaxation wall. The interest in studying fractures and deformations similar to $\pi$ in pneumatic systems under the influence of an electric field is due to the fact that $\pi$-cells can be used in various flat-screen displays. A lot of interest was focused on $\pi$-cells in which the director has parallel surface tilts, with LCs of positive dielectric anisotropy. In such cells, there are three different states of the director: the horizontal state, or the H state, the twisted vertical state, or the T state, and the planar vertical state, or the V state, depending on the surface tilts and the elastic constants of the material. When an electric field is applied, there are possible transitions between the different states which have been considered in device applications [31]. These two-stage switch-on dynamics of a nematic $\pi$-cell are explored in detail using a convergent beam fully-leaky guided mode technique [31]. It has been shown that there is a rapid process of switching from the initial symmetric state H to another symmetric but semi-stable state H when an electric field is applied. This state may be useful in device applications. Later, the alignment properties and the flow velocity distribution during the Poiseuille flow through the nematic microchannel with homeotropic surface alignment were measured using a combination of conoscopy, fluorescent confocal polarization microscopy, and time-lapse imaging [32]. It was found that there are two topologically different profiles of the director with corresponding velocity fields of the fluid, and the preferred state is dictated by the volumetric flow rate of the liquid crystal. Almost at the same time, the dynamic field pumping was investigated, i.e., the interaction between director, velocity, electric fields, as well as, a radially applied temperature gradient, where the inner cylinder is kept at a lower temperature than the outer one, was studied [28]. Calculations show that there exists a range of parameter values (voltage and curvature of the inner cylinder) producing a kink-like orientation process in the LC system, as well as a nonstandard pumping regime with maximum flow near the hot cylinder. Subsequently, the problem of the formation of kink-like distortions in nematic samples under the effect of the electric field was further developed [29,30]. The effect of the curvature $\kappa$ of a homogeneously aligned nematic cavity confined between two infinitely long horizontal cylinders was investigated, and the combined effect of the flexoelectric polarization $\mathbf{P}$ [33] and thermomechanical flow [34] on the director’s reorientation was considered. In this study, the effect of the curvature $\kappa$ of cylindrical homogeneously aligned nematic cavity (HANC) with radii $R_1$ and $R_2$ was investigated, where $R_1<R_2$ and $d=R_2-R_1$ is the capillary thickness on a scale of the order of tens of micrometers. Taking into account the fact that both the temperature gradient $\nabla T=T_{,r}\left(r\right){\widehat{\mathbf{e}}}_r$ and the electric field $\mathbf{E}=E\left(r\right){\widehat{\mathbf{e}}}_r$ are directed along the unit vector ${\widehat{\mathbf{e}}}_r$, it can be assumed that the components of the director $\widehat{\mathbf{n}}=n_r{\widehat{\mathbf{e}}}_r+n_z{\widehat{\mathbf{e}}}_z$ and the velocity flow $\mathbf{v}(r,t)=v_x(r,t){\widehat{\mathbf{e}}}_r=u(r,t){\widehat{\mathbf{e}}}_r$, as well as the rest of the physical quantities depend only on the coordinate r and time t, and the coordinate system defined by this task entails that the director $\widehat{\mathbf{n}}$ lies in the $r-z$ plane. Here, $T_{,r}=\partial T\left(r\right)/\partial r$ denotes the partial derivative of temperature $T\left(r\right)$ with respect to space coordinate r, ${\widehat{\mathbf{e}}}_r$ is the unit vector along the dimensionless radius r (i.e., scaled by d), whereas the other unit vectors of the cylindrical coordinate system to be used here are ${\widehat{\mathbf{e}}}_z$, defined by the common axis of the two cylinders, and the tangential one ${\widehat{\mathbf{e}}}_{\alpha }={\widehat{\mathbf{e}}}_z\times {\widehat{\mathbf{e}}}_r$ (see Figure 9). These studies are primarily concerned with the description of the physical mechanism responsible for the electrically driven nematic flow $\mathbf{v}(r,t)=u(r,t){\widehat{\mathbf{e}}}_r$ in microfluidic HANC containing a temperature gradient $\nabla T$. This gradient was fixed between two infinitely long horizontal coaxial cylinders with the planar preferred of the average molecular direction $\widehat{\mathbf{n}}(r=R_1)\Vert {\widehat{\mathbf{e}}}_z$ ($\widehat{\mathbf{n}}(r=R_2)\Vert {\widehat{\mathbf{e}}}_z$) on the bounding surfaces, which are kept at different temperatures, with the outer one $T_{r=R_2}=T_{\mathrm{out}}=T_2$ hotter than the inner one $T_{r=R_1}=T_{\mathrm{in}}=T_1$ ($T_2>T_1$). In the above-mentioned works [29,30], the response of HANC composed of asymmetric polar molecules, such as cyanobiphenyl, at the density $\rho$, and confined between two horizontal coaxial cylinders which are kept at different temperatures ${\chi }_{\mathrm{in}}={\chi }_1$ and ${\chi }_{\mathrm{out}}={\chi }_2$ (${\chi }_1<{\chi }_2$) on the effect of both the radially applied temperature gradient $\nabla \chi ={\chi }_{,r}{\widehat{\mathbf{e}}}_r$ and electric field $\mathbf{E}=E\left(r\right){\widehat{\mathbf{e}}}_r$ was analyzed. Here, both dimensionless temperatures are scaled by the nematic-isotropic transition value, i.e., ${\chi }_i=T_i/T_{\mathrm{NI}}$ ($i=1,2$), and ${\chi }_{,r}=\partial \chi \left(r\right)/\partial r$ denotes the partial derivative of temperature $\chi$ with respect to space coordinate r. The voltage U applied between two horizontal coaxial cylinders can be scaled by the threshold value for producing the distortion (Freederickzs voltage), i.e., $U_0=U/U_{\mathrm{th}}$, with $U_{\mathrm{th}}$ defined in Reference [3].

It was shown that the condition for producing a distortion of the homogeneous aligned nematic micro-volume is $U_0>1$, which inputs the electric field component $E\left(r\right)$ self-consistently with the nematic distortion $\nabla \widehat{\mathbf{n}}$ [29,30]. Taking into account the geometric features of the cylindrical LC system (see Figure 9), the dimensionless torque balance and linear momentum balance equations take the forms

$$\begin{array}{c}\hfill {\mathcal{T}}_{el+P}(r,\tau )+{\mathcal{T}}_{elast}(r,\tau )+{\mathcal{T}}_{vis}(r,\tau )+{\mathcal{T}}_{tm}(r,\tau )=\\ \hfill E^2(r,\tau )n_r{n}_z+E(r,\tau ){\mathcal{P}}_z(r,\tau )+\\ \hfill \frac{1}{\Delta }\left[{\mathcal{K}}_{,r}+\frac{1}{r}\mathcal{K}-\left(1-K_{31}\right)n_{z,r}^2-\frac{1}{r^2}{n}_r{n}_z\right]+\\ \hfill n_r{\dot{n}}_z-n_z{\dot{n}}_r-\frac{1}{2}{u}_{,r}\left[1-{\gamma }_{21}\left(n_z^2-n_r^2\right)\right]+\\ \hfill \frac{1}{2}\frac{{\delta }_2}{\Delta }{\chi }_{,r}\left[n_z{n}_{r,r}(3+n_r^2)-n_r{n}_{z,r}(1+n_r^2)\right]=0,\end{array}$$

(20)

$$\begin{array}{c}\hfill {\delta }_4\Delta \dot{u}-{\nabla }_{,r}\left[{\sigma }_{rz}^{\mathrm{vis}}+{\sigma }_{rz}^{\mathrm{tm}}\right]=0,\end{array}$$

(21)

$$\begin{array}{c}\hfill {\nabla }_{,r}{\sigma }_{rr}^{\mathrm{elast}}=\frac{{\sigma }_{\alpha \alpha }^{\mathrm{elast}}}{r}-{\psi }_{,r}^{\mathrm{el}+\mathbf{P}}.\end{array}$$

(22)

Here, $\Delta ={\left(\pi U_0\right)}^2$ is the electric energy parameter, $\mathcal{K}=n_z{n}_{r,r}-K_{31}{n}_r{n}_{z,r}$, and the Ericksen–Leslie form for the viscous ${\mathbf{T}}_{vis}={\mathcal{T}}_{vis}(r,\tau )\widehat{\mathbf{j}}=\frac{\delta {\mathcal{R}}_{\mathrm{vis}}}{\delta \dot{\widehat{\mathbf{n}}}}$ and thermomechanical ${\mathbf{T}}_{tm}={\mathcal{T}}_{tm}(r,\tau )\widehat{\mathbf{j}}=\frac{\delta {\mathcal{R}}_{\mathrm{tm}}}{\delta \dot{\widehat{\mathbf{n}}}}$ torques, as well as the shear stress (SS) components ${\sigma }_{rz}^{\mathrm{vis}}$, ${\sigma }_{rz}^{\mathrm{tm}}$, ${\sigma }_{rr}^{\mathrm{elast}}$ and ${\sigma }_{\alpha \alpha }^{\mathrm{elast}}$ for the cylindrical geometry are collected in References [29,30]. In turn, the dimensionless charge balance equation, accounting for the flexoelectric radial component $\mathbf{P}={\mathcal{P}}_r{\widehat{\mathbf{e}}}_r+{\mathcal{P}}_z{\widehat{\mathbf{e}}}_z$, takes the form [29,30]

$$\begin{array}{c}\hfill {\nabla }_{,r}\left[E\left(r\right)\left(\frac{{ϵ}_{\perp }}{{ϵ}_a}+n_r^2\right)+{\mathcal{P}}_r\right]=0,\end{array}$$

(23)

which has the solution

$$\begin{array}{c}\hfill E=\frac{\mathcal{B}-r{\mathcal{P}}_r}{r\left(\frac{{\epsilon }_{\perp }}{{\epsilon }_a}+n_r^2\right)}.\end{array}$$

(24)

Here, $\mathcal{B}=\frac{{\mathcal{B}}_P}{{\mathcal{B}}_E}$, ${\mathcal{B}}_P=1+{\int }_a^{a+1}\frac{{\mathcal{P}}_{r}dr}{r(\frac{{\epsilon }_{\perp }}{{\epsilon }_a}+n_r^2)}$, ${\mathcal{B}}_E={\int }_a^{a+1}\frac{dr}{r(\frac{{\epsilon }_{\perp }}{{\epsilon }_a}+n_r^2)}$ following from the condition of electric field normalization ${\int }_a^{a+1}E\left(r\right)dr=1$.

The vector components of $\mathbf{P}$ are given by the classical Meyer model [33] as

$$\begin{array}{c}\hfill {\mathcal{P}}_r={\delta }_1\left[\left(1+e_{31}\right)n_r{n}_{r,r}+\frac{n_r^2}{r}\right],\end{array}$$

(25)

and

$$\begin{array}{c}\hfill {\mathcal{P}}_z={\delta }_1\left[n_z{\nabla }_{,r}{n}_r+e_{31}{n}_r{n}_{z,r}\right].\end{array}$$

(26)

To describe the dynamics of the director field $\widehat{\mathbf{n}}$ relaxation in the microsized homogeneously aligned nematic volume confined between two infinitely long horizontal cylinders with a radial temperature gradient $\nabla \chi$, under the effect of a voltage U applied between the cylinders, it is necessary to take into account the entropy balance equation [29,30]

$$\begin{array}{c}\hfill {\delta }_5\Delta \dot{\chi }-\frac{1}{r}\frac{\partial }{\partial r}\left[r{\chi }_{,r}\left(\lambda n_r^2+n_z^2\right)\right]=0,\end{array}$$

(27)

where ${\delta }_5=\frac{\rho C_p{K}_1}{{\lambda }_{\perp }{\gamma }_1}$ is another parameter of the LC system, $\lambda ={\lambda }_{\Vert }/{\lambda }_{\perp }$ is the ratio of heat conductivity coefficients along and perpendicular to the director, respectively, and $C_p$ denotes the heat capacity [34]. In turn, the set of parameters of the LC system ${\delta }_i(i=1,\dots ,5)$ is collected in the References [29,30].

The system of nonlinear partial differential Equations (21), (22), (24) and (27) must be supplemented by boundary conditions for director, velocity, and temperature, respectively. For the case of the homogeneous anchoring director on both bounding cylindrical surfaces, the boundary condition for the director field takes the form

$$\begin{array}{c}\hfill n_r{\left(r\right)}_{r=a}=n_r{\left(r\right)}_{r=a+1}=0,\end{array}$$

(28)

while the no-slip conditions on these surfaces must be written as

$$\begin{array}{c}\hfill u{\left(r\right)}_{r=a}=u{\left(r\right)}_{r=a+1}=0.\end{array}$$

(29)

In turn, the boundary conditions for the temperature field are reduced to

$$\begin{array}{c}\hfill \chi {\left(r\right)}_{r=a}={\chi }_1,\chi {\left(r\right)}_{r=a+1}={\chi }_2.\end{array}$$

(30)

The above-mentioned system of nonlinear partial differential equations Equations (21), (22), (24) and (27), together with the boundary conditions Equations (28)–(30), and the initial condition $n_r=0.001$, $(a<r<a+1)$ has been solved by the numerical relaxation method [35].

Figure 10 and Figure 11 show the evolution of the director’s field component $n_r(r,\tau )$ to its stationary distribution $n_r^{st}(r,{\tau }_R)\equiv n_r^{st}\left(r\right)$ across the microsized HANC $a\le r\le a+1$, calculated for two voltages $U_0=6$ (see Figure 10) and 14 (see Figure 11), and fixed curvature $\kappa =5(a=0.2)$ [29,30].

In this case, the temperature difference in $\Delta \chi =0.0162$ corresponds to $\Delta T=T_{\mathrm{out}}-T_{\mathrm{in}}\sim 5K$. Figure 10b and Figure 11b show the most pronounced influence of flexoelectricity on orientation dynamics. The torques ${\mathbf{T}}_P$, ${\mathbf{T}}_{\mathrm{tm}}$ produce a kink-like reorientation of the director (curves from 4 to 6 in Figure 10b). This process shows the initial stage at which the electric torque near the inner cylinder $r=a$, during the period $\tau <{\tau }_1$ initiates the distortion of the director. At that time, the sum of torques ${\mathbf{T}}_P+{\mathbf{T}}_{\mathrm{tm}}$ acts in opposite verse near the outer cylinder $r=a+1$. The values of the radial director component have become negative and the kink-like reorientation is building up (Figure 10b solid curves 2 and 3), during the time interval ${\tau }_1<\tau <{\tau }_3$, under the increasing flexoelectric torque. The field $E(r,\tau )$ is strong and the sum of torques ${\mathbf{T}}_P+{\mathbf{T}}_{\mathrm{tm}}$ continues the process of kink motion towards the inner cylinder (Figure 10b, curves 4, 5, and 6). Due to the nematic symmetry, the stationary distributions of $n_r^{eq}\left(r\right)$ (curve 7 in Figure 10a) and $n_r^{eq}\left(r\right)$ (curve 7 in Figure 10b) are equivalent, the relaxation times of the two models are close to each other, but their relaxation ways to the stationary states are different. Results for the model of $\mathbf{P}=0$ show that the torque ${\mathbf{T}}_{\mathrm{tm}}$ tends to perturb the director component $n_r$, but it is not strong enough to produce the kink-like distortion of the director profile (Figure 10a, curves 3 to 6). This slow reconstruction of the orientation profile by the viscous torque is similar to the slow orientation dynamics reported in Reference [31]. The kink-like evolution, shown in Figure 10b (curves 4, 5, and 6) correlates with the electric field function $E(r,\tau )$ (see Figure 4, Reference [30]), $U_0=14$. This value of voltage causes a larger deformation near the inner cylinder than in the other case $U_0=6$ (curves 1 to 5 in Figure 11b). The torque balance keeps the main role of ${\mathbf{T}}_{\mathrm{elast}}$ and ${\mathbf{T}}_{\mathrm{el}}$ over the time interval $\tau <{\tau }_5$, near the inner cylinder. After the time period ${\tau }_5$, quick growth of the negative profile occurs in the relaxation way (curves 6 to 8, Figure 11b) and the negative part of the reorientation profile of the director takes $\pi$-form (curve 10, in Figure 11b). Similar behavior is shown by the director reorientation due to the model of $\mathbf{P}=0$ (Figure 10a). The new voltage is in 2.5 times bigger than the previous one; all processes are significantly sped up, and the inflection points of the electric field (see Figure 5a,b, Reference [30]) have not evolved. Director orientation in the nematic cavity, as plotted in Figure 10a,b, exhibits the inner planar orientation $n_r\left(r\right)=0$, for $r=r_w$. The space position of the planar director (Figure 11a,b) can be classified as a steady wall position, versus the situation shown in Figure 10a,b, showing a moving wall, which in turn disappears at the ${\tau }_R$ (Figure 10a,b).

Thus, numerical methods within the framework of the generalized Ericksen–Leslie theory show which mechanisms are responsible for the formation of both kink- and $\pi$-like deformation modes in cylindrical LC cavities under the effect of the radially applied electric field. It has been shown that in the case when the voltage $U_0\to \infty$, one has that ${lim}_{U_0\to \infty }u(\tau ,r)\to 0$, and any horizontal steady flow $\mathbf{v}(r,t)=u(r,t){\widehat{\mathbf{e}}}_r$ of the LC phase stops in the microsized nematic capillary, since under the influence of strong external electric field $\mathbf{E}$ the dipoles of molecules forming the LC phase are oriented along this field. This once again shows that the macroscopic description of the nature of the hydrodynamic flow of an anisotropic liquid subtly senses the microscopic structure of the LC material.

Notes that the relaxation behavior of the director field $\widehat{\mathbf{n}}$ probably can be observed in the form of the kink-like wave $n_r(r,t)$ spreading across the nematic cavity in polarized white light. Taking into account that the director reorientation takes place in the narrow area of the nematic $5CB$ sample (the width of the kink-like wave) under the influence of the voltage $U_0=6$, applied between two horizontal coaxial cylinders, the kink-like wave can be visualized in polarized white light as a dark strip running across the microsized nematic cavity with the velocity in a few tens of μm/s. This dark running-strip can be recorded using a charge-coupled device camera and video cassette recorder, while the temperature difference, for the experimentally well-studied and technologically interesting case of $5CB$, can be achieved by pumping the cooling material (with a temperature $T_{\mathrm{in}}$ less than 5 degrees below room temperature $T_{\mathrm{out}}$) through the inner cylinder. It should be noted that the planar anchoring of nematic 5CB can be achieved on self-assembled monolayers formed from alkanethiols on gold [36].

4. Electrically Driven Torsional Distortions in Twisted Nematic Micro-Volumes

Among all LC compounds, nematic LCs are currently the most popular LC materials used in information technology, various LC sensors, and LC actuators [1,2]. The main focus is on recent progress in the area of sensors based on the effect resulting from the ability to convert the electric signal to microsized mechanical displacements. This area of research became very attractive recently since the possibilities for applications of LC sensors are growing in many areas, ranging from the detection of mechanical microsized displacements to the detection of chemical and biological agents. Recently, the description of a new mechanism for converting the electrical field $\mathbf{E}$ into additional pressure $\mathbf{P}$, which acts on the boundaries of a microsized TN volume for a very short time [37,38], has been proposed. It was shown that the torques acting on the director $\widehat{\mathbf{n}}$ may excite the kink-like pressure $P(z-vt)$ traveling wave spreading along the normal to both boundaries, whose resemblance to a kink-like wave increases with an increase of the applied electric field. In this case, the texture of TNs was obtained by orienting a drop of bulk LC material between two plates oriented perpendicular to each other with convenient treating (see Figure 1a). It has been shown that the kink-like pressure $P(z-vt)$ traveling wave spreading along the normal to both boundaries under the effect of the external electric field $\mathbf{E}$ is responsible for the kink-like deformation wave $\Phi (z-vt)$ moving from unstable to the stable state in the LC cell [37,38]. Thus, the main goal of this Section is to show how the externally applied electric field can be converted to the running kink-like distortion wave $\Phi (z-vt)$.

It should be noted that the solitons in LCs are important in the science of nonlinear physics and materials science [39,40]. The fact that the solitary waves might be important in ferroelectric LCs (FLCs) was for the first time suggested by Cladis et a1. [40] based on observations of electrically driven director reorientation in an initially helix FLC cell. However, our attention will be focused on describing the possibility of the formation of the kink-like distortion wave in TN cells by the action of the external electric field $\mathbf{E}$, directed both along the x or z axes to the TN micro-volume (see Figure 1a,b). In order to study the evolution of pressure $P(z,t)$ in the form of running kink-like waves, the behavior of the angle $\Phi (z,t)$ in the form of the running wave $\Phi (z-vt)$, where v is the velocity of the front, were studied [37,38]. Because in the case described above, the field $\mathbf{E}$ is aligned parallel to the x-direction, the state ${\Phi }_{z=d}\left(z\right)=\pi /2$ is now unstable, and the front $\Phi (z-vt)$ starts to move away from the one edge ($z=d$) of the cell to their second one ($z=0$). However, this raises a number of questions. Under what conditions are the formation of the running kink-like waves in the microsized TN volume under the effects of the externally applied electric field and the initially disturbed director field possible? How fast will such a front move and how much influence will the external electric field and the boundary conditions have on the resemblance of a traveling wave to a kink-like wave?

The answers to these questions are given on the basis of the analysis of the torque balance and the linear momentum balance equations together with the appropriate initial and boundary conditions [37,38]. The torque balance equation for the above-mentioned geometry, without taking into account the backflow, can be written in the form [37,38]

$$\begin{array}{c}\hfill {\left(\frac{\partial {\mathcal{F}}_{\mathrm{elast}}}{\partial {\Phi }_{,z}}\right)}_{,z}-\frac{\partial \mathcal{F}}{\partial \Phi }-\frac{\partial \mathcal{D}}{\partial {\Phi }_{,t}}=0,\end{array}$$

(31)

while the linear momentum balance takes the form

$$\begin{array}{c}\hfill P_{,z}=-\frac{\partial \mathcal{D}}{\partial {\Phi }_{,t}}{\Phi }_{,z}.\end{array}$$

(32)

Here, $\mathcal{P}=P-\mathcal{F}$ is the arbitrary pressure in the TN cell, $\mathcal{F}={\mathcal{F}}_{\mathrm{elast}}-{\mathcal{F}}_{\mathrm{el}}$, ${\mathcal{F}}_{\mathrm{elast}}=\frac{1}{2}{K}_2{\Phi }_{,z}^2$ is the elastic energy density, ${\mathcal{F}}_{\mathrm{el}}=-\frac{1}{2}{ϵ}_a{ϵ}_0{\left(\mathbf{E}·\widehat{\mathbf{n}}\right)}^2$ is the electric potential, $\mathcal{D}=\frac{1}{2}{\gamma }_1{\Phi }_{,t}^2$ is the dissipation function, $K_2$ is the twist elastic coefficient, ${\Phi }_{,t}=\frac{\partial \Phi }{\partial t}$, ${\Phi }_{,z}=\frac{\partial \Phi }{\partial z}$, and $\Phi \equiv \Phi (z,t)$ is the azimuthal angle, i.e., the angle between the director $\widehat{\mathbf{n}}$ and the unit vector $\widehat{\mathbf{i}}$, respectively.

It has been shown that in the case of a strong attachment, the azimuth angle must satisfy the boundary conditions

$$\begin{array}{c}\hfill \Phi {\left(z\right)}_{z=0}=0,\Phi {\left(z\right)}_{z=d}=\frac{\pi }{2}.\end{array}$$

(33)

In this case, the director field $\widehat{\mathbf{n}}$ is rotated within the plane $\mathrm{XY}$ without any translational motion, so the dynamics of the director field are described in the case of the absence of the fluid flow and the arbitrary pressure $\mathcal{P}$ takes the form

$$\begin{array}{c}\hfill \mathcal{P}(z,t)=\frac{1}{2}{K}_2{\Phi }_{,z}^2(z,t)-\frac{1}{2}{ϵ}_a{ϵ}_0{E}^2{sin}^2\Phi (z,t)-{\gamma }_1{\int }_0^z{\Phi }_{,t}(\eta ,t){\Phi }_{,z}(\eta ,t)d\eta .\end{array}$$

(34)

Here, the first two terms are both contributions to the arbitrary pressure due to the elastic and electric forces, whereas the third contribution is due to the viscous force, respectively. To obtain the evolution of the pressure $\mathcal{P}(z,t)$ in the form of the running kink-like waves we must obtain the values of the angle $\Phi (z,t)$ in the form of the running wave $\Phi (z-vt)$, where v is the front velocity.

In the case of twisting geometry and the absence of flow, the dimensionless torque balance equation takes the form [37,38]

$$\begin{array}{c}\hfill {\Phi }_{,\tau }(z,\tau )={\Phi }_{,zz}(z,\tau )+\frac{1}{2}sin2\Phi (z,\tau ),\end{array}$$

(35)

where ${\Phi }_{,\tau }(z,\tau )=\frac{\partial \Phi (z,\tau )}{\partial \tau }$ is the derivative of the azimuthal angle $\Phi$ with respect to the dimensionless time $\tau =\frac{{ϵ}_0{ϵ}_a{E}_0^2}{{\gamma }_1}t$, ${\Phi }_{,zz}(z,\tau )=\frac{{\partial }^2\Phi (z,\tau )}{\partial z^2}$ is the second derivative of the angle $\Phi$ with respect to the dimensionless space variable z (i.e., scaled by the film thickness d), respectively.

In the case of twisted nematics (see Figure 1a) and strong anchoring, the torque balance equation transmitted to the bounding surfaces assumed that the azimuthal angle $\Phi$ has to satisfy the boundary condition (case A)

$$\begin{array}{c}\hfill \Phi {\left(z\right)}_{z=0}=0,\Phi {\left(z\right)}_{z=1}=\frac{\pi }{2},\end{array}$$

(36)

while in the case of the weak $W_{\mathrm{an}}=\frac{1}{2}A{sin}^2\left({\Phi }_s-{\Phi }_0\right)$ anchoring [26], the torque balance equation transmitted to the bounding surfaces assumed that the azimuthal angle $\Phi$ has to satisfy the boundary condition (case B)

$$\begin{array}{c}\hfill {\left(\frac{\partial \Phi \left(z\right)}{\partial z}\right)}_{z=0,1}=\mathcal{W}\Delta \Phi ,\end{array}$$

(37)

where $\mathcal{W}=\frac{Ad}{K_2}$ is the dimensionless anchoring strength, $\Delta \Phi ={\Phi }_s-{\Phi }_0$, ${\Phi }_s$ and ${\Phi }_0$ are the azimuthal angles corresponding to the director orientation on the bounding surface and easy axis $\widehat{\mathbf{e}}$, respectively. In this case, the elastic torque $T_{\mathrm{elast}}=\frac{K_2}{d}{\left(\frac{\partial \Phi \left(z\right)}{\partial z}\right)}_{z=0,1}$ tends to align ${\widehat{\mathbf{n}}}_s$ along $\mathbf{E}$, and the opposed anchoring torque $T_{\mathrm{anch}}=-\frac{\partial W_{\mathrm{an}}}{\partial {\Phi }_s}$ rotates ${\widehat{\mathbf{n}}}_s$ toward $\widehat{\mathbf{E}}$.

In turn, in the case of the homogeneous nematics (see Figure 1b) and strong anchoring, the torque balance equation assumed that the azimuthal angle $\Phi$ has to satisfy the boundary condition (case C)

$$\begin{array}{c}\hfill \Phi {\left(z\right)}_{z=0}=0,\Phi {\left(z\right)}_{z=1}=0.\end{array}$$

(38)

In order to observe the evolution of the traveling disturbance in time with velocity v, the dimensionless torque balance Equation (35), in the coordinate system $\xi =z/\kappa -v\tau$ moving with the dimensionless wave speed v, takes the form [37,38]

$$\begin{array}{c}\hfill {\Phi }_{,\tau }\left(\xi \right)=v{\Phi }_{,\xi }\left(\xi \right)+{\Phi }_{,\xi \xi }\left(\xi \right)+\frac{1}{2}sin2\Phi \left(\xi \right).\end{array}$$

(39)

Now the relaxation of the director $\widehat{\mathbf{n}}$ to its stationary orientation, which is described by the angle $\Phi \left(\xi \right)$, being initially disturbed perpendicular to the external field $\mathbf{E}$, with

$$\begin{array}{c}\hfill \Phi (\xi ,\tau =0)=\frac{1}{\sigma }exp\left(-\frac{\xi -{\xi }_3}{\sigma }\right),\end{array}$$

(40)

was investigated by the standard numerical relaxation method [35], with the strong boundary conditions on both bounding surfaces (case A):

$$\begin{array}{c}\hfill \Phi {\left(\xi \right)}_{\xi ={\xi }_1}=0,\Phi {\left(\xi \right)}_{\xi ={\xi }_2}=\frac{\pi }{2}.\end{array}$$

(41)

Here, ${\xi }_1=d/\left(2\kappa \right)-v\tau$ and ${\xi }_2=d/\left(2\kappa \right)$ are the dimensionless positions for lower and upper boundaries, and ${\xi }_3$ $({\xi }_1+{\xi }_2)/2<{\xi }_3\le {\xi }_2$ is chosen close to the upper restricted surface.

In the case when the director $\widehat{\mathbf{n}}$ is strongly anchored to the upper and weakly anchored to the lower boundaries, when the anchoring energy takes the form [26] $W_{\mathrm{an}}=\frac{1}{2}A{sin}^2\left({\Phi }_s-{\Phi }_0\right)$, the torque balance transmitted to the surface assumed that the director angle $\Phi$ has to satisfy the boundary conditions (case B):

$$\begin{array}{c}\hfill {\left(\partial \Phi \left(\xi \right)/\partial \xi \right)}_{\xi ={\xi }_1}=\mathcal{W},\\ \hfill {\left(\Phi \left(\xi \right)\right)}_{\xi ={\xi }_2}=\pi /2,\end{array}$$

(42)

where $\mathcal{W}$ is the dimensionless anchoring strength.

For twisted geometry, the dimensionless pressure takes the form [37,38]

$$\begin{array}{c}\hfill \mathcal{P}\left(\xi \right)={\mathcal{P}}_{\mathrm{elast}}\left(\xi \right)-{\mathcal{P}}_{\mathrm{el}}\left(\xi \right)-\int \left(\partial \mathcal{D}/\partial {\Phi }_{,\tau }\right){\Phi }_{,\chi }\left(\chi \right)d\chi =-{\Phi }_{,\xi }^2\left(\xi \right),\end{array}$$

(43)

where ${\mathcal{P}}_{\mathrm{el}}\left(\xi \right)=\frac{1}{2}{sin}^2\Phi \left(\xi \right)$ and ${\mathcal{P}}_{\mathrm{elast}}\left(\xi \right)=-\frac{1}{2}{\Phi }_{,\xi }^2\left(\xi \right)$ are both dimensionless contributions to the total arbitrary pressure $\mathcal{P}$ due to the electric and elastic forces, respectively. It should be noted that the dimensionless $\mathcal{P}$ and dimensional P pressure values are related to the ratio $\mathcal{P}\left(\xi \right)=(K_2/{\kappa }^2)P\left(\xi \right)$.

The distortion of the director field in the microsized TN film under the externally applied electric field $\mathbf{E}$ from the initial state to its stationary orientation, being initially disturbed orthogonally to $\mathbf{E}$, was investigated by the standard numerical relaxation method [35], with the boundary conditions in the form of cases A (Equation (41)) and B (Equation (42)), respectively. In turn, the initial condition $\Phi (\xi ,\tau =0)$ was given in the Gaussian form, (see Equation (40)), being initially disturbed orthogonally to $\mathbf{E}$.

The dynamics of the response in the form of the running kink-like distortion wave in the microsized TN volume, sandwiched between two parallel surfaces, under the effect of the externally applied electric field $\mathbf{E}$ directed parallel to the director on the lower surface in details has been presented in References [37,38]. Here, we demonstrate only one of the variants for the formation of the kink-like distortion wave under the effect of the external electric field.

The results of calculations for the distortion of the director field in the microsized TN film, which is described by the azimuthal angle $\Phi \left(\xi \right)$, when the external electric field is equal to $E_0=1.2\times {10}^{-5}{\mathrm{C}/\mathrm{m}}^2$, and the director is strongly (case A) and weakly (case B, with the value of $\mathcal{W}$ being equal to $1.0$ or $A\sim {10}^{-5}{\mathrm{J}/\mathrm{m}}^2$) anchored to the bounding surfaces, are shown in Figure 12a,b, respectively.

Calculations show that the initial perturbation of the director $\widehat{\mathbf{n}}(\tau =0)$, which is described by the azimuthal angle $\Phi (\xi ,\tau =0)$, with ${\xi }_3=4.75$, under the effect of the externally applied electric field $E_0=1.2\times {10}^{-5}{\mathrm{C}/\mathrm{m}}^2$, can be transformed into the running kink-like distortion wavefront, with the constant shape and speed v (see Figure 12a,b). The motion of this running front along the z-axis starts at the point ${\xi }_1=5.0$, located on the upper bounding surface of TN film, and is directed to the lower bounding surface, located at the point ${\xi }_2=-5.0$. Eventually, this running wave reaches the stationary state after the time ${\tau }_R\left(\alpha \right)$, where $\alpha =A$ and B, corresponding to the cases A and B. A comparison of these two relaxation processes presented in Figure 12a,b show that the evolution of the initial disturbance $\Phi (\xi ,\tau =0)$ to the running kink-like distortion wavefront is faster in case A than in case B.

Such the kink-like wave $\Phi \left(\xi \right)$ is responsible for the excitation of the kink-like pressure wave $P\left(\xi \right)$ (see Equation (43)) along the normal to both boundaries. When the initial orientation of the director is disturbed in the vicinity of the upper edge of the TN cell, orthogonally to the electric field $\mathbf{E}$, the pressure kink-like wave $P\left(\xi \right)$ starts to move away from the upper edge to the lower one. The dimensionless values of $P\left(\xi \right)$, in the case of strong (case A, see Figure 13a) and weak (case B, see Figure 13b) boundary conditions, under the influence of the external electric field $E_0=1.2\times {10}^{-5}{\mathrm{C}/\mathrm{m}}^2$, as a function of the dimensionless coordinate $\xi$, are shown in Figure 13.

Calculations show that the effect of $E_0=1.2\times {10}^{-5}{\mathrm{C}/\mathrm{m}}^2$ on the absolute value of $P\left(\xi \right)\left[{\mathrm{N}/\mathrm{m}}^2\right]$, on the first stage of evolution, is characterized by a sharp decrease of the absolute magnitude of the pressure up to, (practically) a zero value, with the following monotonic increase of the absolute value in the final stage of the evolution, in case A up to ∼32 ${\mathrm{N}/\mathrm{m}}^2$ (see Figure 13a), and in case B up to ∼$26.5$ ${\mathrm{N}/\mathrm{m}}^2$ (see Figure 13b), respectively. Notes that the anchoring effect decreases the magnitude of $\left|P\right(\xi \left)\right|$ up to $21\%$. Calculations also show that the pressure profile $P\left(\xi \right)$ will be determined by the balance of the rate of change in the elastic and electric energy with viscous dissipation; the front $P\left(\xi \right)$ starts to move from the one edge of the cell to the second edge, and, finally, the pressure profile can obtain the lower plate after time ${\tau }_R$. So, by fixing the extra pressure, which will be acted on the lower boundary, one can measure, by using an appropriate setup, the relaxation time ${\tau }_R$ corresponding to the evolution of the disturbance state to the equilibrium orientation in the form of the kink-like running wave. It has been shown that the relaxation behavior of $\Phi (z-vt)$ in the form of the running wave probably can be observed in polarized white light [38]. Taking into account that the director reorientation takes place in the narrow area of the LC sample (the width of the traveling wave) under the influence of the electric field $\mathbf{E}$, for instance, $E>1.2\times {10}^{-5}{\mathrm{C}/\mathrm{m}}^2$ for the $10\mathsf{\mu }\mathrm{m}$ nematic $5CB$ cell, the running wave can be visualized in polarized white light as a dark strip running along normal to both glass plates, with the velocity ∼${10}^{-4}\mathrm{m}/\mathrm{s}$.

Another relaxation regime, when the director $\widehat{\mathbf{n}}$ on both bounding surfaces is aligned parallel to the unit vector $\widehat{\mathbf{i}}$ (see Figure 1b), has been investigated. In this case, the evolution of distortion of the director field in the microsized homogeneously aligned nematic film, with the strong anchoring conditions on both bounding surfaces as in Equation (38) (case C), and with the initial condition in the form $\Phi (\xi ,\tau =0)=0$, under the externally applied electric field $E_0=4.6\times {10}^{-5}{\mathrm{C}/\mathrm{m}}^2$, is shown in Figure 14.

Calculations show (see Figure 14) that under the effect of the externally applied electric field $E_0=4.6\times {10}^{-5}{\mathrm{C}/\mathrm{m}}^2$, two traveling distortion waves running in different directions along the z-axis to both bounding surfaces can be excited. The center of the formation of these waves is located in the vicinity of the point ${\xi }_3=0.0$, while the bounding surfaces are located at the points ${\xi }_{1,2}=\pm 1.3$, respectively. The relaxation time ${\tau }_R$ for these separated processes is the same and equal to $0.0375$, or ∼$290\mathsf{\mu }\mathrm{s}$.

5. Conclusions

This review describes some new numerical advances in predicting the structural and dynamic behaviors of the director field $\widehat{\mathbf{n}}$ in a microsized twisted nematic volume subjected to the crossed electric $\mathbf{E}$ and magnetic $\mathbf{B}$ fields.

First of all, we were interested in describing the physical mechanism responsible for the excitation of spatial distortions of the director field in an initially uniformly aligned microsized nematic volume under the effect of crossed electric $\mathbf{E}$ and magnetic $\mathbf{B}$ fields.

Secondly, we focused on the description of a new mechanism of both kink- and $\pi$-like distortion waves of the director field $\widehat{\mathbf{n}}$, which are excited in the cylindrical micro-volume of the nematic phase under the effects of the voltage U and temperature gradient $\nabla T$, formed across this LC volume. It was found that, under certain conditions, in terms of the curvature of cylinders $\kappa$ and the voltage U applied between cylinders, the torques and forces acting on the director $\widehat{\mathbf{n}}$ may excite the kink-like distortion wave spreading along the normal to cylindrical boundaries, whose resemblance to a kink-like distortion wave depends on the value of U and the curvature of the inner cylinder. On the other hand, the conditions were worked, in terms of $\kappa$ and U, producing the distortion mechanism of the $\widehat{\mathbf{n}}$ in the double $\pi$-form, with the intermediate relaxation wall.

Thirdly, a description has recently been proposed of a new mechanism for converting the electric field $\mathbf{E}$ into additional pressure $\mathbf{P}$, which acts on the boundaries of the microsized volume of a twisted nematic (TN) for a very short time, similar to the kink-like pressure $P(z-vt)$ running wave propagating along the normal to both boundaries. It is shown that for excitation of the kink-like pressure $P(z-vt)$ wave spreading along the normal to both boundaries under the effect of the external electric field $\mathbf{E}$ is responsible for the kink-like deformation wave $\Phi (z-vt)$ moving from an unstable to a stable state in the LC cell. Calculations show that the physical mechanism that is responsible for the electric field-induced distortion of the director field $\widehat{\mathbf{n}}$ in the form of the kink-like wave provides much faster relaxation than in the non-traveling mode.

Notes that the reorientation of the director field $\widehat{\mathbf{n}}$ in the form of the kink-like distortion wave spreading along the normal to both boundaries can probably be observed in polarized white light. Taking into account that the director reorientation takes place in the narrow area of the LC sample, under the large electric field $\mathbf{E}$ (∼$1\mathrm{V}/\mathsf{\mu }\mathrm{m}$) directed parallel to the bounding surfaces, the running kink-like distortion wave can be visualized in polarized white light as a dark strip running along the normal to both LC boundaries, with the velocity $v\sim 1.0\mathsf{\mu }\mathrm{m}/\mathsf{\mu }\mathrm{s}$.

Therefore, further research should be conducted on a wider range of problems on how elastic soft matter, such as liquid crystals confined in a microsized volume, begins to deform under the influence of a strong electric field; this will eventually increase our knowledge in the field of materials science.