Thermally excited vortical flow in a microsized nematic cell

Abstract

The theoretical description of the reorientational dynamics in a microized liquid crystal cell, where the nematic sample is confined by two horizontal and two lateral surfaces, under the influence of a temperature gradient ∇T is presented. We have carried out a numerical study of the system of hydrodynamic equations including director reorientation, fluid flow v, and the temperature redistribution across the cell under the influence of ∇T, when the sample is heated both from below and from above. Calculations show that, due to interaction between the gradient of the director field ∇n̂ and ∇T, the bidirectionally aligned liquid crystal (BALC) sample settles down to a stationary bi-vortical flow regime. As for a nematogenic material, we have considered the BALC cell to be occupied by 4-n-pentyl-4′-cyanobiphenyl, and investigated the effect of both ∇n̂ and ∇T on the magnitude and direction of v, for a number of hydrodynamic regimes.

---

I. Introduction

The widely used flat-panel liquid-crystal displays (LCDs) consist of a liquid-crystal (LC) film sandwiched between two glass or plastic surfaces on the scale of the order of micrometers, across which a voltage may be applied, independently to each pixel of the LCD. In this structure, the transmission of light through individual pixels is controlled by a potential difference applied between electrodes on the back plate of the LCD.¹ This applied electric field may alter the molecular configuration of the LC layer and thus alter the optical characteristics of the LCD.

In the field of LC phases, a great deal is known about their deformations under the influence of electric and magnetic fields,² whereas, on the other hand, comparatively little is known about the effect of a temperature gradient on their structure properties.^3–6

Recently, a new method to control LC flow dynamics using a bidirectionally aligned liquid crystal (BALC) film has been proposed.⁷ It was shown for the case of bistable twisted nematic (BTN) devices, that dynamic flow is essential to the switching mechanism.

The aim of our paper is to analyze the response of the BALC film composed of asymmetric molecules, for instance, such as cyanobiphenyl, confined in the microsized volume between two horizontal and two lateral surfaces under the influence of a temperature gradient ∇T directed from the cooler to the warmer boundaries. Thus we are primarily concerned here with describing the way how the temperature gradient across the microsized BALC film can produce hydrodynamic flow. This problem will be treated in the framework of the Ericksen–Leslie theory,^8,9 accounting for the thermoconductivity equation for the temperature field T,¹⁰ whereas the Rayleigh–Benard mechanism does not produce any effect because of the small film thickness.¹¹ It should be pointed out that the thermally driven convection in a millimeter-sized horizontal layer of a nematic heated below or above and in a magnetic field has been studied for approximately 40 years.^12–14

The present paper is organized as follows: the relevant equations describing director motion, fluid flow and temperature distribution in the above system are given in Sec.II; numerical results for possible hydrodynamic regimes are given in Sec.III; conclusions are summarized in Sec.IV.

II. Formulation of the relevant equations for nematic fluids

To fix ideas and notation, we shall be considering a BALC film delimited by two horizontal and two lateral surfaces at mutual distances 2d and 2L on a scale on the order of micrometers, with the warmer upper boundary

T_{−L<x<L,z = d} = T_w (1)

where T denotes temperature, whereas for the other boundaries the temperature

T_{−L≤x≤L,z = −d} = T_{x = −L,−d<z<d} = T_{x = L,−d<z<d} = T_c (2)

is kept lower (T_w > T_c). The coordinate system defined for our task assumes that the director is in the xz plane, where is the unit vector directed parallel to the upper and lower restricted surfaces, which, in turn, coincides with the planar director orientation on the upper restricted surface (), whereas the unit vector is directed parallel to the lateral restricted surfaces, which coincides with the planar director orientation on these surfaces (), and . The bidirectionally aligned nematic state involves a gradient of θ from aligned orientation on the lower surface to planar orientation on the upper and both lateral surfaces, i.e.,

Moreover, we will assume the no-slip boundary conditions for the velocity field on these bounding surfaces, i.e.,

v_{−L<x<L,z = ± d} = v_{x = ± L,−d<z<d} = 0 (4)

where denotes velocity.

It should be noted here that these BALC layers were used in the BTN devices.⁷ In our calculations the value of the ratio d/L is chosen equal to 0.1. Such choice of the width/length ratio allows us avoid the effect of the bounding lateral walls on the director reorientation inside the BALC cell.

Note that a thin horizontal layer of quiescent LC fluid heated from below becomes unstable to convection via the Rayleigh–Benard mechanism, and this system has been used extensively for the study of a great variety of pattern-formation phenomena.^11,12 Taking into account that the size of the LC cell d ∼ 1–5 μm, in our case R ≪ R_c ∼ 1708, and the driving force is weak enough to set up of convection via the Rayleigh–Benard mechanism.

Taking into account the width of the BALC cell, one can assume the mass density ρ to be constant across the BALC cell, and thus deal with an incompressible fluid. The incompressibility condition ∇·v = 0 gives

u_,x + w_,z = 0 (5)

where u and w are the components of the vector , and .

The hydrodynamic equations describing the reorientation of the LC phase in the 2D case, when there exists a heat flux q across the BALC film, can be derived from the torque balance equation T_el + T_vis + T_tm = 0 where is the elastic torque,² is the viscous torque.² and is the thermomechanical torque,⁵ respectively (for details, see the Appendix), the Navier–Stokes equation

where σ = σ^el + σ^vis + σ^tm − P[scr I, script letter I] is the full stress tensor (ST), and , , and are the ST components corresponding to the elastic, viscous, and thermomechanical forces, respectively (see the Appendix). Here [scr R, script letter R] = [scr R, script letter R]^vis + [scr R, script letter R]^tm + [scr R, script letter R]^th is the full Rayleigh dissipation function, denotes the elastic energy density, K₁ and K₃ are splay and bend elastic coefficients, P is the hydrostatic pressure, and [scr I, script letter I] is the unit tensor. When a small temperature gradient ∇T(∼1.0 K μm⁻¹) is set up across the BALC cell, we expect the temperature field T(t, z) satisfies the heat conduction eqn (10).where is the heat flux in the BALC system, and C_p is the heat capacity of the LC system.

To be able to determine the evolution of the angle values θ(t, x, z) to the equilibrium orientation 0_eq(x, z), and the evolution of the velocity field v(t, x, z) caused both by the temperature gradient and the director reorientation to the equilibrium orientation, we consider the dimensionless analog of these equations. The dimensionless torque balance has the form

where is the scaled streamline function Ψ for the velocity field (see the Appendix), , and . The dimensionless analog of the Navier–Stokes equation takes the formwhere the set of functions σ^vis_ij(i, j = x, z), σ^el_ij (i, j = x, z) and σ^tm_ij(i, j = x, z) are given in the Appendix, whereas the dimensionless entropy balance equation takes the formwhere λ = λ_||/λ_⊥, χ ≡ χ(τ, z) = T(τ, z)/T_NI is the dimensionless temperature, is the dimensionless time, is the dimensionless distance away from the middle point of the LC cell, corresponding to z-axis, is the dimensionless space variable corresponding to x-axis, and , , and are dimensionless parameters of the LC system. Notice that the overbars in the space variables x and z have been eliminated in the previous and following equations.

Now, the reorientation of the director in the BALC film can be obtained by solving the system of nonlinear partial differential eqn (8)–(11), with the appropriate dimensionless boundary conditions for the angle

velocity

v_{−1<x<1,z = ±1} = v_{x = ±1, −1<z<1} = 0 (13)

and temperature

χ_{−1<x<1,z = 1} = χ_w, χ_{−1 ≤x≤1, z = −1} = χ_{x = −1, −1<z<1} = χ_{x = 1, −1<z<1} = χ_c (14)

Here χ_w = T_w/T_NI and χ_c = T_c/T_NI are the dimensionless temperatures corresponding to the highest and lowest values, respectively. Thus when the director n̂ is strongly bidirectionally anchored to the lower boundary and planar to the upper and two lateral restricted surfaces, the angle θ has to satisfy the boundary conditions (12) and its initial orientation is chosen equal to θ(τ = 0, x, z) = θ_el(x, z), where θ_el(x, z) is obtained from eqn (8) with Ψ_,x = Ψ_,z = χ_,x = χ_,z = 0, and the boundary and initial conditions in the form of eqn (12), and then, under the action of the viscous, elastic, and thermomechanical forces, allowed to relax to its equilibrium value θ = θ_eq(x, z).

For the case of 4-n-pentyl-4′-cyanobiphenyl (5CB), at a temperature of 300 K and density of 10³ kg m⁻³, the experimental data for elastic constants are K₁ = 10.5 pN and K₃ = 13.8 pN,¹⁵ whereas the measured data both for the rotational and six Leslie coefficients are (in [Pa s]¹⁶) γ₁ ∼ 0.072, γ₂ ∼ −0.079, α₁ ∼ −0.00066, α₂ ∼ −0.075, α₃ ∼ −0.0035, α₄ ∼ −0.072, α₅ ∼ −0.048, and α₆ ∼ −0.03, respectively. The value of the heat conductivity coefficients parallel (λ_||) and perpendicular (λ_⊥) to the director are (in [W mK⁻¹]¹⁷) 0.24 and 0.13, respectively. In the following we use the measured value of the specific heat¹⁸Cp ∼ 10³ [J (kg K)⁻¹]. In our calculations the thickness of the LC cell is equal to 5 μm, T_w = 303 K, T_c = 298 K, and T_NI = 305 K. The experimental value of the thermomechanical constant ξ was estimated as ∼10⁻¹² J Km⁻¹, based on the measurements of the liquid crystal flow in the horizontal direction.¹⁹ The set of parameters that is involved in eqn (8)–(11) has the following values: δ₁ ∼ 29, δ₂ ∼ 2 × 10⁻⁵, δ₃ ∼ 2 × 10⁻⁶, and δ₄ ∼ 1.1 × 10⁻³. Using the fact that δ_i ≪ 1 (i = 2, 3), the Navier–Stokes equations [eqn (9)and (10)] can be considerably simplified and take the form

c₁Ψ_,zzzz + c₂Ψ_,xzzz + c₃Ψ_,xxzz + c₄Ψ_,xxxz + c₅Ψ_,xxxx + c₆Ψ_,zzz + c₇Ψ_,xzz + c₈Ψ_,xxz + c₉Ψ_,xxx + c₁₀Ψ_,zz + c₁₁Ψ_,xz + c₁₂Ψ_,zz + F = 0 (15)

where c_i (i = 1,…,12) and F are the functions defined in the Appendix.

Eqn (11) also can be simplified because the parameter δ₄ ≪ 1, and the left-hand side of eqn (11), as well as the last term, can be neglected, so that eqn (11) become

(λsin²θ + cos²θ)χ_,zz + (λ − 1)sin2θθ_,zχ_,z = 0 (16)

Thus the response of the BALC film in the above setting is described by eqn (8), (15)and (16), together with the boundary conditions eqn (12)–(14), and the initial condition θ(τ = 0, x, z) = θ_el(x, z), where θ_el(x, z) is the equilibrium distribution of the angle θ across the BALC film under the influence only the elastic force.

III. Orientational relaxation of the director, temperature and velocity fields in the nematic cell

The relaxation processes of the director field , described by the angle θ(τ, x, z), the velocity u(τ, x, z), and the temperature χ(τ, x, z) are governed by eqn (8), (15)and (16), together with the boundary conditions eqn (12)–(14), and the initial condition θ(τ = 0, x, z) = θ_el (x, z). Calculations were carried out for two cases, first, when the LC sample is heated from above, with the boundary condition for the temperature field χ in the formand second, when the LC sample is heated from below, with the boundary condition for the temperature field in the form

The other boundary conditions are the same (see eqn (12), (13)) for both these cases. Fig. 1 shows the equilibrium distribution of the director field in the dimensionless BALC cell for case I, when the dimensionless temperature difference is equal to Δχ = χ_w − χ_c = 0.0162 (∼5 K). This has been calculated by solving the abovementioned system of the nonlinear partial differential eqn (8), (15)and (16), by means of Galerkin's method.²⁰ In the calculations, the streamline function is approximated by a finite sum of the orthogonal functions , where L_k denotes the Legendre polynomial of rank k, N is the order of the approximation, and the elementary functions φ satisfy the boundary conditions φ (±1) = φ′(±1) = 0. The biharmonic eqn (16) is transformed to the matrix equation A·Q = f, where the matrix A has the elements a_ijmn = (Φ_ij, DΦ_mn), with Φ_mn = φ_m(x) φ_n(z)(i, j, m, n = 0,…,N). Here DΦ_mn = c₁Φ_mn, zzzz + c₂Φ_mn, xzzz + c₃Φ_mn, xxzz + c₄Φ_mn, xxxz + c₅Φ_mn, xxxx + c₆Φ_mn, zzz + c₇Φ_mn, xzz + c₈Φ_mn, xxz + c₉Φ_mn, xxx + c₁₀Φ_mn, zz + c₁₁Φ_mn, xz + c₁₂Φ_mn, xx is the biharmonic differential operator, Q is a matrix of unknown coefficients Q_mn (m, n = 0,…,N), and the matrix f, with the elements f_ij = −(Φ_ij, F), is the scalar product (for details, see Appendix). The equilibrium distribution of the angle θ_eq (x, z) along the width of the dimensionless BALC cell (−1 ≤ x ≤ 1), for case I, and for a number of distances away from the cooler lower (χ_{−1 ≤ x ≤ 1, z = −1} = χ_c = 0.97) boundary is shown in Fig. 2. According to our calculations in case I, in which the angle θ is varied between values on the left-hand side of the lower boundary and on the right-hand side of the lower boundary, the highest value of |∇θ(x, z)| is reached in the vicinity of the lower (−1 ≤ x ≤ 1, z = −1) boundary. The distribution of the velocity field v in the BALC cell in case I for two times, τ = 0.001 (∼180 μs) after set up of case I, and, for the equilibrium distribution of v(τ, x, z), which occur after a time term τ_R = 2.05 (∼0.36 s), is characterized by maintaining of two vortices as shown in Fig. 3 and 4. Here 1 [mm] of the arrow length is equal to 0.24 [mm s⁻¹], and τ_R denotes the relaxation time. In the calculations, the relaxation criterion ε = |(θ_(m + 1) − θ_(m))/θ_(m)| was chosen to be 10⁻⁴, and the numerical procedure was then carried out until a prescribed accuracy was achieved. The direction and magnitude of the hydrodynamic flow v(τ, x, z) is influenced by both the direction of the heat flux q and the character of the preferred anchoring of the average molecular direction n to the bounding surfaces. According to our calculations of the angle θ(τ, x, z) across the BALC cell, the highest value of |∇θ| is reached in the vicinity of the middle part of the LC cell, and as a result, the biggest thermally excited velocities occur in the vicinity of the lower cooler surface. In case I, the bigger self-sustaining vortical flow in the left-hand side of the BALC cell is thermally excited in the positive sense (clockwise), whereas the smaller vortical flow in the right-hand side of the BALC cell is thermally excited in the negative sense (anti-clockwise) around their centers.

Fig. 1 The equilibrium distribution of the director field n̂_eq(x, z) in the BALC cell for case I, when the dimensionless temperature difference is equal to Δχ = 0.0162. Here is .

Fig. 2 The equilibrium distribution of the angle θ_eq(x, z) along the width of the BALC film (−1 ≤ x ≤ 1), for case I, and for a number of distances away from the lower boundary: z = −0.9 (curve 1), z = −0.7 (curve 2), z = 0.0 (curve 3), z = 0.7 (curve 4), and z = 0.9 (curve 5).

Fig. 3 The distribution of the velocity field v in the BALC cell, for case I, at the time τ = 0.001 Here 1 [mm] of the arrow length is equal to 0.24 [μm s⁻¹].

Fig. 4 The same as Fig. 3, but at the time τ = τ_R = 2.05.

The distribution of the velocity field v in the BALC cell in case II for two times, first, which occur τ = 0.001 (∼180 μs) after set up of case II, and, second, for the equilibrium distribution of v(τ, x, z) within the BALC cell, which occur after a time term τ_R = 2.05 (∼0.36 s), is characterized, as in case I, by two vortices, as shown in Fig. 5 and 6. In case II, the bigger self-sustaining vortical flow in the middle part of the BALC cell is thermally excited in the negative sense (anti-clockwise), whereas the smaller vortical flow in the vicinity of the cooler upper restricted surface of the BALC cell is thermally excited in the positive sense (clockwise) around their centers. Notice that the stationary thermally driven bi-vortical flow in case I, with the LC sample heated from above, is characterized by slightly smaller values of v(x, z) than in case II. Indeed, in case I, the highest value of |v| is equal to ∼1.2 (mm s⁻¹), whereas in case II, it is equal to ∼1.3 (mm s⁻¹).

Fig. 5 The distribution of the velocity field v in the BALC cell, for case II, at the time τ = 0.001.

Fig. 6 The same as Fig. 5, but at the time τ = τ_R = 2.05.

In order to elucidate the role of the bidirectionally aligned lower surface in maintaining of the thermally excited bi-vortical flows in the BALC cell, we have performed a numerical study of the thermally excited fluid flow v(τ, x, z) in the case of a “right-hand” tilted LC cell (case R), with

and in the case of a “left-hand” tilted LC cell (case L), withrespectively. Fig. 7 shows the equilibrium distribution of the director field n̂_eq(x, z) in the dimensionless unidirectionally aligned LC (UALC) cell, both for cases I and R, when the temperature difference is equal to Δχ = χ_w − χ_c = 0.0164 (∼5 K). Here χ_w = 0.9934 and χ_c = 0.977. In our case the temperature gradient ∇χ (∼1 K μm⁻¹) is restricted by the small temperature interval [χ_w, χ_c] belonging to the nematic phase. Our calculations of the thermally excited fluid flow in cases I and R show (see Fig. 8 and 9) that the initially established [up to 0.002 (∼360 μs)] bi-vortical flow in the UALC cell, where the first vortical flow, with the motion in the negative sense (anti-clockwise), practically fills three fourths of the left-hand part of the LC cell, whereas the second (clockwise) vortex fills the remainder of the volume, after a time τ = 0.002 is converted to a single vortical flow with the motion in the negative sense (anti-clockwise) (see Fig. 8), which is characterized by a slower speed than the vortical flow in case I.

Fig. 7 The equilibrium distribution of the director field n̂_eq(x, z) in the UALC cell, for cases I and R, when the temperature difference is equal to Δχ = 0.0162.

Fig. 8 The distribution of the velocity field v in the UALC cell, for cases I and R, at the time τ = 0.001 Here 1 [mm] of the arrow length is equal to 0.24 [μm s⁻¹].

Fig. 9 The same as Fig. 8, but at the time τ = τ_R = 2.05.

Fig. 10 shows another equilibrium distribution of the director field n̂_eq(x, z) in the UALC cell, in cases I and L. Here the temperature difference is equal to Δχ = χ_w − χ_c = 0.0162 (∼5 K). Our calculations of the thermally excited fluid flow in cases I and L shows (see Fig. 11 and 12) that the initially [up to 0.002 (∼360 μs)] maintained bi-vortical flow in the UALC cell is converted to a single vortical flow with the motion in the positive sense (clockwise) (see Fig. 12). The difference between the equilibrium vortical flows in cases (I and R) and (I and L) lies only in the directions of these thermally driven flows. In the first case it is directed in the negative sense (anti-clockwise), whereas in the second case it is directed in the positive sense (clockwise).

Fig. 10 The equilibrium distribution of the director field n̂_eq(x, z) in the unidirectionally aligned dimensionless LC cell, both for cases I and L, when the dimensionless temperature difference is equal to Δχ = 0.0162.

Fig. 11 The distribution of the velocity field v in the UALC cell, for cases I and L, at the time τ = 0.001 Here 1 [mm] of the arrow length is equal to 0.24 [μm s⁻¹].

Fig. 12 The same as Fig. 10, but at the time τ = τ_R = 2.05.

Notice that in cases (R) and (L) with UALC cells, the biggest variation of the director field is reached only in the vicinity of the lower restricted surface and close to the vertical walls (see Fig. 2), whereas the temperature gradient is kept constant across the UALC cell. Calculations show that the horizontal UALC layer, being initially in the rest, if heated both from below or above, due to interaction between the director and temperature gradients, starts moving, and, initially bi-vortical flow is converted to the single vortical flow.

In the cases of the BALC cells, the biggest variation of the director field is reached in the vicinity of the middle part of the lower restricted surface and close to the vertical walls (see Fig. 2), whereas the temperature gradient is the same as in the cases (R) and (L). As a result, an extra one strong interaction between the director and temperature gradients in the vicinity of the middle part of the lower restricted surface leads to maintaining of the self-sustaining thermally excited bi-vortical flow.

So, based on our calculations, one can conclude that the character of the anchoring to the bounding surfaces plays a crucial role in maintaining the thermally excited vortical flows in 2D LC cell.

IV. Conclusion

In summary, we have investigated the reorientational dynamics in a thin BALC cell, where the nematic sample is confined by two horizontal and two lateral surfaces, and under the influence of a temperature gradient ∇T, when the LC sample is heated both from above and from below. Our calculations, based on the classical Ericksen–Leslie theory, show that due to interaction between ∇T and the gradient of the director field ∇n̂ in the BALC cell a thermally excited bi-vortical fluid flow is maintained. The direction and magnitude of hydrodynamic flow is influenced by both the direction of the heat flux and the character of the preferred anchoring of the director to the bounding surfaces. Notice that the above-mentioned vortical flow in the BALC film can also be caused by the heating flow, for instance, across the lower surface. That heating flow can be produced by an infrared laser beam.²¹ It should be pointed out that we worked out the conditions for producing a bi-vortical fluid flow for the case of a bidirectionally aligned LC display, where the dynamic backflow is essential to the switching mechanism.

We believe that the present investigation can shed some light on the problem of control of the dynamic response of the bidirectionally aligned LC display under the influence of a temperature gradient.

Appendix: Torques and stress tensor components

The torque balance equation T_el + T_vis + T_tm = 0, can be derived from the elastic , viscous , and thermomechanical contributions, where is the elastic energy density, K₁ and K₃ are the splay and bend elastic coefficients, is the material derivative of , whereas is the viscous contribution to the total Rayleigh dissipation function [scr R, script letter R] = [scr R, script letter R]^vis + [scr R, script letter R]^tm + [scr R, script letter R]^th. Here and are the thermomechanical and thermal contributions to [scr R, script letter R]. Here α₁,…,α₆ are the Leslie viscosity coefficients, γ₁ and γ₂ are the rotational viscosity coefficients, ξ is the thermomechanical constant, and λ_||, λ_⊥ are the heat conductivity coefficients parallel and perpendicular to the director n̂, and are the symmetric and asymmetric contributions to the rate of strain tensor, , and [scr M, script letter M]₀ = ∇·n̂ is the scalar invariant of the tensor M.

The interpretation of the streamline function Ψ for the velocity field is used here for the velocity field v = v(t, x, z) of the incompressible fluid (∇·v = 0). The balance equation for the linear momentum is , where ρ is the density of the system, and σ is the the stress tensor, whereas the entropy balance is , where is the heat flux in the system, and C_P is the heat capacity of the system.

We consider here the set of dimensionless variables: both the dimensionless horizontal and vertical coordinates, dimensionless time , dimensionless temperature , where T_NI is the temperature of nematic–isotropic phase transformation, the dimensionless streamline function , dimensionless elastic energy _el, dimensionless viscous _vis, thermomechanical _tm, and thermal _th contributions to the full dimensionless dissipation function = _vis + δ_1tm + δ_5th, where and are two parameters of the system. Here the elastic and dissipation functions are:

, where , , , , , and .

Notice that the overbars in these variables and functions have been (and will be) eliminated in the following equations.

The dimensionless torque balance has the form

where . The dimensionless stress tensor as the sum of elastic, viscous, thermomechanical parts and pressure σ = σ^el + σ^vis + σ^tm − P[scr I, script letter I] can be obtained directly from the elastic contribution to the energy and Rayleigh dissipation function as, , , and , for the elastic, viscous, and thermomechanical contributions, respectively.

Straightforward calculations for the geometry n̂ = (cosθ, 0, sinθ) with the angle θ between the director n̂ and the unit vector give the following expressions for the elastic σ^el_ij, viscous σ^vis_ij and thermomechanic and σ^tm_ij components of the ST, i, j = x, z:

where the coefficients f^{k, vis}_ij are calculated as, f^{2, vis}_xx = f^{1, vis}_xx,

, f^{2, vis}_zz = −f^{1, vis}_xx

, f^{4, vis}_zz = −f^{4, vis}_xx

, ,
,
,, f^{3, vis}_zx = −f^{3, vis}_xz,

The thermomechanical stress components are

The biharmonic equation in the ST terms has the final form

where
,

c₂ = f^{3, vis}_zx + f^{2, vis}_xx − f^{2, vis}_zz

,

, and is the parameter of the system.

The dimensionless entropy balance equation can be rewritten in the form

where is an extra parameter of the system.

Acknowledgements

We acknowledge financial support by the Russian Funds for Fundamental Research (Grant No.09-02-00010-a).

References

1. D. K. Yang and S. T. Wu, Fundamentals of Liquid Crystal Devices, Wiley, New York, 2006.
2. P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd ed., Oxford University Press, Oxford, 1995.
3. R. S. Akopyan and B. Y. Zeldovich, “Thermomechanical effects in deformed nematics”, Sov. Phys. JETP, 1984, 60, 953.
4. H. R. Brand and H. Pleiner, “New theoretical results for the Lehmann effect in cholesteric liquid crystals”, Phys. Rev. A: At., Mol., Opt. Phys., 1988, 37, 2736.
5. A. V. Zakharov and A. A. Vakulenko, “Influence of the flow on the orientational dynamics induced by temperature gradient”, J. Chem. Phys., 2007, 127, 084907.
6. A. V. Zakharov and A. A. Vakulenko, “Liquid-crystal pumping in a cylindrical capillary with radial temperature gradient”, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2009, 80, 031708.
7. Y. W. Li, C. Y. Lee and H. S. Kwok, “Liquid crystal dynamic flow control by bidirectional alignment surface”, Appl. Phys. Lett., 2009, 94, 061111.
8. J. L. Ericksen, “Anisotropic fluids”, Arch. Ration. Mech. Anal., 1960, 4, 231.
9. F. M. Leslie, “Some constitutive equations for liquid crystals”, Arch. Ration. Mech. Anal., 1968, 28, 265.
10. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press, Oxford, 1987.
11. M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium”, Rev. Mod. Phys., 1993, 65, 851.
12. G. Ahlers, in Pattern Formation in Liquid Crystals, ed. by A. Buka and L. Kramer, Springer, New York, 1996, pp. 165–221.
13. E. Dubois-Violette, “Instabilites hydrodynamiques dun nematique soumis a un gradient thermique”, C. R. Acad. Sci., 1971, 21, 923.
14. P. Pieranski, E. Dubois-Violette and E. Guyon, “Heat convection in liquid crystals heated from above”, Phys. Rev. Lett., 1973, 30, 736.
15. N. V. Madhusudana and R. B. Ratibha, “Elasticity and orientational order in some cyanobiphenyls: Part IV. Reanalysis of the data”, Mol. Cryst. Liq. Cryst., 1982, 89, 249.
16. A. G. Chmielewski, “Viscosity coefficients of some nematic liquid crystals”, Mol. Cryst. Liq. Cryst., 1986, 132, 339.
17. M. Marinelli, A. K. Ghosh and F. Mercury, “Small quartz silica spheres induced disorder in octylcyanobiphenyl (8CB) liquid crystal: A thermal study”, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 2001, 63, 061713.
18. P. Jamee, G. Pitsi and J. Thoen, “Systematic calorimetric investigation of the effect of silica aerosils on the nematic to isotropic transition in heptylcyanobiphenyl”, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 2002, 66, 021707.
19. R. S. Akopyan, R. B. Alaverdian, E. A. Santrosian and Y. S. Chilingarian, “Thermomechanical effects in the nematic liquid crystals”, J. Appl. Phys., 2001, 90, 3371.
20. J. Shen, “Efficient Spectral-Galerkin method I. Direct solvers of second- and fourth-order equations using Legendre polynomials”, SIAM J. Sci. Comput., 1994, 15, 1489.
21. A. V. Zakharov and A. A. Vakulenko, “Director reorientation in a hybrid-oriented liquid-crystal film induced by thermomechanical effect”, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2009, 80, 031711.

---