In the case of the linearized 2D nematic system, the dynamics of the thermal fluctuation

$\delta {n}_{z}$ under the effect of crossed

$\mathbf{E}$ and

$\mathbf{B}$ fields can be obtained by solving the system of the linear differential Equations (

5) and (

6) with the appropriate boundary and initial conditions. By means of transmittance of the torque balance to the bounding surfaces, one can obtain the boundary conditions for the thermal fluctuation

$\delta {n}_{z}$. In our case it can be written as [

11,

18]

Here

$\mathcal{W}$ is the anchoring strength,

$\Delta n={n}_{s}-e$,

${n}_{s}$ is the director orientation on the bounding surfaces,

e is the easy axis orientation, and

d is the film thickness, respectively. Note that the velocities

u and

w on both electrodes have to satisfy the boundary conditions (case of free boundaries)

In turn, the initial conditions both for

$\delta {n}_{z}$ and

$\mathbf{v}$ can be written in the form

respectively.

In our case the components of the velocity

$\mathbf{v}=u\widehat{\mathbf{i}}+w\widehat{\mathbf{k}}$ have to satisfy the boundary conditions

whereas the boundary conditions on both electrodes for director fluctuation

$\delta \widehat{\mathbf{n}}$ take the form [

11,

18]

Substituting Equation (

12) into Equation (

11) and solving them yields

and one can calculate the ERV coefficient as

where

${K}_{31}=\frac{{K}_{3}}{{K}_{1}}$,

${\overline{\eta}}_{i}={\eta}_{i}/{\gamma}_{1}\phantom{\rule{3.33333pt}{0ex}}(i=1,...,8)$ and

${\overline{\alpha}}_{i}={\alpha}_{i}/{\gamma}_{1}\phantom{\rule{3.33333pt}{0ex}}(i=2,3)$ are the dimensionless elastic and viscous coefficients, respectively. Notice that the linearized analysis is valid as long as the amplitude of the response

$\delta {n}_{z}$ is sufficiently small. Having obtained from Equation (

15) the growth rate

$s\left({q}_{x},{q}_{z}\right)$, one can determine the value of the angle

$\alpha $ which provides a periodic response for the nematic phase at applied electric field

E above threshold value

${E}_{\mathrm{th}}=\frac{\pi}{d}\sqrt{\frac{{K}_{1}}{{\u03f5}_{0}{\u03f5}_{a}}}$.

It should be noted that the NMR measurements were made with deuterated 5CB-

${\mathrm{d}}_{2}$ nematic sample of the thickness of 194.7

$\mathsf{\mu}$m using a JEOL Lambda 300 spectrometer [

4], which has a magnetic flux density

$\mathbf{B}$ of

$7.05\phantom{\rule{3.33333pt}{0ex}}\mathrm{T}$. The values of voltages in our calculations is varied between 20 and 200

$\mathrm{V}$. The last value of

U was chosen to be equal to the value used in the NMR spectroscopy experiment [

4]. So, the values of the dimensionless parameters

${\lambda}_{1}$ and

${\lambda}_{2}$ were estimated to be equal to 47,418 and 5486, respectively. The values of the anchoring strengths

$\mathcal{W}$ for 5CB is varied in the range from

${10}^{-5}$ to

${10}^{-6}$ J/m

${}^{2}$ [

11]. The calculation of the root of Equation (

14) yields the minimal

${q}_{z}^{\mathrm{min}}=0$, at

$z=0$, whereas at

$z=1$ the minimal

${q}_{z}^{\mathrm{min}}$ has to satisfy the following equation:

where

$\kappa ={K}_{1}/\left(\mathcal{W}d\right)$. The calculated values of the dimensionless coefficient

$\kappa $ are equal to 0.044 for the value of

$\mathcal{W}={10}^{-6}\phantom{\rule{3.33333pt}{0ex}}$J/m

${}^{2}$ (case I), and 0.0044 for the value of

$\mathcal{W}={10}^{-5}\phantom{\rule{3.33333pt}{0ex}}$J/m

${}^{2}$ (case II), respectively. These values of

${K}_{1}/\left(\mathcal{W}d\right)$ provide the minimal values of the dimensionless wavelengths

${q}^{\mathrm{min}}\equiv {q}_{z}^{\mathrm{min}}$, which are equal to

$\Delta =\pi +\delta $, where

$\delta =0.1434$ in case I and 0.01434 in case II, respectively. Note that in the limiting case of strong anchoring, when

$\mathcal{W}\to \infty $,

$lim{\Delta}_{\mathcal{W}\to \infty}=\pi $.

“Having obtained the values of

${q}^{\mathrm{min}}=\Delta $ (case I), one can calculate, by using of Equation (

15), the dimensionless growth rate

$s\left({q}_{x}d/\pi \right)/s\left(0\right)$ vs. the dimensionless wavelength

${q}_{x}d/\pi $ under the effect of the electric field

$\mathbf{E}$. Calculations of the dimensionless growth rate

$s\left({q}_{x}d/\pi \right)/s\left(0\right)$ vs. dimensionless wavelength

${q}_{x}d/\pi $, for two voltages

$U=20\phantom{\rule{3.33333pt}{0ex}}$V (

Figure 5a) and

$U=50\phantom{\rule{3.33333pt}{0ex}}$V (

Figure 5b) applied across the nematic film, for a number of values of the angle

$\alpha $:

$0.157$ (

$\sim {9}^{\circ}$)(curve (1));

$0.471$ (

$\sim {27}^{\circ}$)(curve (2));

$0.785$ (

$\sim {45}^{\circ}$)(curve (3));

$1.257$ (

$\sim {72}^{\circ}$)(curve (4)), and

$1.57$ (

$\sim {90}^{\circ}$)(curve (5)), respectively, are shown in

Figure 5a,b [

11,

18]. The main result of this calculations is that the periodic response appear for the high value of the voltage

$U\ge 50\phantom{\rule{3.33333pt}{0ex}}$V, when for each value of the angle

$\alpha >{70}^{\circ}$ there is an optimal dimensionless wavelength

${q}_{x}^{\mathrm{max}}$ corresponding to the fastest growth of the distortion. For instance, in case I, and when the voltage

$U=50$ V, the strong periodic distortion emerging spontaneously from the homogeneous state by the strong

$\mathbf{E}$ may induce faster than in the uniform mode only for the high values of the angle

$\alpha \sim {65}^{\circ}$ and higher. Calculations of the dimensional growth rate

$s\left({q}_{x}d/\pi \right)$ (in s

${}^{-1}$) vs.

${q}_{x}d/\pi $, under the effect of both the voltage

$U=200$ V and the magnetic field

$B\sim 7.05\phantom{\rule{3.33333pt}{0ex}}T$, for a number of values of the angle

$\alpha $ (same as in

Figure 5), are shown in

Figure 6. The main result of this calculation is that the periodic response appears only for the values of

$\alpha \equiv {\alpha}_{\mathrm{th}}\sim {45}^{\circ}$ and higher, and for each value of the angle

$\alpha >{\alpha}_{\mathrm{th}}$ there exist an optimal dimensionless wavelength

${q}_{x}^{\mathrm{max}}$ corresponding to the fastest growth of the distortion. This means that the threshold value of the angle

${\alpha}_{\mathrm{th}}$, at which a periodic structure in the homogeneously aligned nematic film under the effect of crossed electric and magnetic fields begins to form, decreases with increasing the voltage applied across the LC film. Calculations show that for such nematic film, with thickness ∼200

$\mathsf{\mu}$m, the character of the anchoring conditions (strong or weak) has a weak effect on the growth rate

s and the dimensionless wavelength

${q}^{\mathrm{max}}\equiv {q}_{x}^{\mathrm{max}}$ (see

Figure 7a). Note that when

$\alpha $ tends to

${90}^{\circ}$, then the value of the optimal

${q}^{\mathrm{max}}\equiv {q}_{x}^{\mathrm{max}}$, corresponding to the fastest growth of the distortion (see

Figure 7b), increases slightly (see

Table 1). Having obtained the values of

${q}^{\mathrm{max}}\left(\alpha \right)$, corresponding to each value of

$\alpha $, one can calculate the effective viscosity [

18]

as a function of both the angle

$\alpha $ and the anchoring strength

$\mathcal{W}$. In the limiting case of

${q}^{\mathrm{max}}=0$ the ERV coefficient

${\gamma}_{\mathrm{eff}}\left(0\right)$ is equal to

which agrees with the result reported in the literature [

22]. Calculations of the dimensionless ERV coefficient

${\gamma}_{\mathrm{eff}}\left(\alpha ,\mathcal{W}\right)/{\gamma}_{1}$ as the function of both the angle

$\alpha $ and the anchoring strength

$\mathcal{W}$ are shown in

Figure 8. It is evident that when

$\alpha $ tends to be

${90}^{\circ}$, then the value of the optimal

${q}^{\mathrm{max}}\equiv {q}_{x}^{\mathrm{max}}$, corresponding to the fastest growth of the distortion, providing the lower ERV coefficient

${\gamma}_{\mathrm{eff}}\left(\alpha \right)$ being less than one in the bulk nematic phase. Physically, this means that the periodic distortion emerging spontaneously from a homogeneous state may induce a faster response than in the uniform mode. The nonuniform rotation modes involve additional internal elastic distortions that are absent in the uniform rotation mode. This leads to a compromise that determines the wavelength of the fastest-growing periodic structure in the nematic film. In turn, the large-amplitude distortions modulated in the

x direction lead to the increase of the elastic energy of the conservative nematic system and, as a result, it causes the decrease of the viscous contribution to the total energy of the LC system. Both

Figure 5 and

Figure 6 show that there is a certain threshold value of the angle

${\alpha}_{\mathrm{th}}$ which provides the periodic distortion emerging spontaneously. That is, when

$\alpha $ is close to a right angle (see curves (4) and (5) in

Figure 6), the value of

${\gamma}_{1}$ in Equations (

5) and (

6) should be replaced by

${\gamma}_{\mathrm{eff}}$. That is, the lower value of

${\gamma}_{\mathrm{eff}}\left(\alpha ,\mathcal{W}\right)$ initiates another dynamics of the director reorientation in the nematic film under the effect of crossed electric and magnetic fields. In turn, since

${\tau}_{\mathrm{on}}\left(\alpha ,\mathcal{W}\right)\sim {\gamma}_{\mathrm{eff}}\left(\alpha ,\mathcal{W}\right)$ [

22], the lower value of

${\gamma}_{\mathrm{eff}}$ will reduce the relaxation time, as observed experimentally by using the time-resolved deuterium NMR spectroscopy [

4].” [

18].