Dynamics of the modulated distortions in confined nematic liquid crystals A. V. Zakharov and A. A. Vakulenko Citation: The Journal of Chemical Physics 139, 244904 (2013); doi: 10.1063/1.4851197 View online: http://dx.doi.org/10.1063/1.4851197 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/139/24?ver=pdfcov Published by the AIP Publishing
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THE JOURNAL OF CHEMICAL PHYSICS 139, 244904 (2013)
Dynamics of the modulated distortions in confined nematic liquid crystals A. V. Zakharova) and A. A. Vakulenkob) Saint Petersburg Institute for Machine Sciences, The Russian Academy of Sciences, Saint Petersburg 199178, Russia
(Received 30 October 2013; accepted 5 December 2013; published online 26 December 2013) The peculiarities in the dynamics of the director reorientation in confined nematic liquid crystals (LCs) under the influence of a strong electric field E have been investigated theoretically based on the hydrodynamic theory including the director motion with appropriate boundary and initial conditions. Analysis of the numerical results for the turn-on process provides an evidence for the appearance of the spatially periodic patterns in confined LC film, only in response to the suddenly applied strong E. It has been shown that there is a threshold value of the amplitude of the thermal fluctuations of the director over the LC sample which provides the nonuniform rotation mode rather than the uniform one, whereas the lower values of the amplitude dominate the uniform mode. During the turn-off process, the reorientation of the director to the direction preferred by the surfaces is characterized by the complex destruction of the initially periodic structure to a monodomain state. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4851197] I. INTRODUCTION
Uniform textures of nematic liquid crystals (NLCs) are produced by orienting a drop of bulk material in between two conveniently treated plates, which define usually a fixed orientation for the boundary molecules. Applying the electric field E perpendicular to a uniformly (homogeneously) oriented NLC can distort the molecular orientation aˆ with reˆ at a critical threshold field Eth given by1 spect to director n, π Eth = d
K1 , 0 a
(1)
where d is the film thickness, K1 is the splay elastic constant, 0 is the absolute dielectric permittivity of free space, and a is the dielectric anisotropy of the NLC. This form for the critical field is based upon assumption that the director remains strongly anchored (in our case, homogeneously) at the two horizontal surfaces and that the physical properties of the liquid crystal (LC) are uniform over the entire sample for E < Eth . When the electric field is switched on with a magniˆ in the “splay” geometude E greater than Eth , the director n, try, reorients as a simple monodomain.1 In the case when the electric field E Eth is abruptly applied orthogonal to an initially uniformly aligned (homogeneously) nematic LC film, ˆ reorients in order to minimize the free energy, the director, n, and the LC system is suddenly placed far from equilibrium. It responds by creating a distortion which maximizes the rate at which the LC lowers its total free energy. In this case, the final form of deformation depends on viscous, elastic, and electric torques, as well as the boundary and initial conditions and the application of the strong orthogonal electric field gives rise a) Author to whom correspondence should be addressed. Electronic mail:
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b) Email: [email protected] URL: http://www.ipme.ru
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sometimes to the appearance of a nonuniform rotation mode rather than the uniform one. The first theoretical study of the periodic structures in the LC systems imposed by the strong external field has been commonly associated with the wavelength corresponding to the mode of fastest growth predicted by a linear analysis of the nematodynamic equations.2–5 This approach, based on an eigenvalue analysis, although useful in understanding the main physical quantities in the origin of the observed periodic structures, is far less valid if we are interested in the dynamics of the modulated structures in nematic LCs. For this reason we have recently investigated theoretically the field-induced director dynamics based on the nonlinear hydrodynamic theory including the director motion with appropriate boundary and initial conditions.6 Analysis of the numerical results for the turn-on process provides an evidence for the appearance of the spatially periodic patterns in confined 4-n-pentyl-4 -cyanobiphenyl (5CB) LC film, only in response to the suddenly applied strong electric field orthogonal to the magnetic field. It has been shown that for a certain balance among the electric, elastic, and viscous torques there is the threshold value of the amplitude of the thermal fluctuations of the director over the LC sample which provides the nonuniform rotation mode rather than the uniform one, whereas the lower value of the amplitude dominates the uniform mode. During the turn-off process, the reorientation of the director to the direction preferred by the surfaces is characterized by the complex destruction of the initially periodic structure to a monodomain state. In this work we focus on the geometry where the electric field is orthogonal (or approximately orthogonal) to the horizontal boundaries (see Fig. 1). In this configuration the state of the system, immediately becomes unstable after applying the strong orthogonal electric field. When the misalignment of the director with respect to the direction preferred by the surfaces is due to the thermal fluctuations with small amplitudes, the reorientation following the sudden application of a
139, 244904-1
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J. Chem. Phys. 139, 244904 (2013)
orientation nˆ eq can be written as (for details, see the Ref. 6) θ,τ = δ1 (sin2 θ + K31 cos2 θ )θ,xx +(cos2 θ + K31 sin2 θ )θ,zz + δ1 (K31 − 1) sin 2θ θ,xz 1 + (1 − K31 ) sin 2θ θ,x2 − θ,z2 2 + δ1 (K31 − 1) cos 2θ θ,x θ,z FIG. 1. The geometry used for the calculations. The z-axis is normal to the horizontal bounding surfaces, whereas the unit vector ˆi is directed parallel ˆ to the horizontal surfaces. The electric field E, unit vector ˆi, and director, n, are in the xz-plane. The director makes an angle θ with the x-axis, and the electric field makes an angle α with the unit vector ˆi.
sufficiently strong and orthogonal electric field manifests itself by the growing of one particular Fourier mode. After switching-off the field, the long-range elastic interactions ensure that the molecules reorient themselves in the direction preferred by the surfaces. In this work, switching will be driven by a series of a voltage pulses: V > 0, for 0 < t < t0 ; V = 0, for t0 ≤ t ≤ t1 ; and V < 0, for t1 < t < t2 . The first voltage pulse corresponds to the field in the positive z-direction (see Fig. 1). Then for t0 ≤ t ≤ t1 , the field is equal to zero, and for t1 < t < t2 , the field is negative. So, the aim of this paper is to show how in the LC system of this geometry switching can be driven by the electric field and to demonstrate the kinetic pathway along which the switching proceeds. II. FORMULATION OF THE RELEVANT EQUATIONS FOR DYNAMICAL REORIENTATION OF THE DIRECTOR FIELD
The coordinate system defined by our experiment assumes that the strong electric field E is abruptly applied normal (or close to the normal) to the horizontal bounding surfaces (see Fig. 1). We consider a homogeneously aligned nematic system such as cyanobiphenyls, which is delimited by two horizontal and two lateral surfaces at mutual distances 2d and 2L on a scale in the order of tens micrometers. According to this geometry the LC system may be seen as the two-dimensional, since the director is maintained within the xz-plane (or in the yz-plane) defined by the electric field and the unit vector ˆi directed parallel to the horizontal surfaces, kˆ is a unit vector directed normal to the horizontal surfaces, and ˆj = kˆ × ˆi. We can suppose that the components of the director, nˆ = nxˆi + nz kˆ = cos θ (x, z, t)ˆi + sin θ (x, z, t)kˆ (see Fig. 1) depend only on x, z-components and time t. Here θ denotes the angle between the director and the unit vector ˆi. Our recent investigation of the field-induced reorientation of the director field under the influence of a strong electric field suggests that in order to describe the dynamical reorientation of the director correctly, we do not need to include a proper treatment of backflow.7 This means that the role of the viscous force becomes negligible in comparison to the electric, elastic, and flexoelectric contributions. In the case of the quasitwo-dimensional LC system the dimensionless torque balance equation describing the reorientation of nˆ to its equilibrium
1 2 + E (θ ) sin 2(α − θ ) − δ2 E ,z sin α sin 2θ, 2
(2)
where x = x/d and z = z/d are the dimensionless space V 2 variables, τ = γ0 1a 2d t is the dimensionless time, e1 +e3 3 1 δ1 = 04K , δ = , and K31 = K are three param2 0 a V K1 a V 2 eters of the system. Here γ 1 is the rotational viscosity coefficient, K3 is the bend elastic constant, whereas e1 and e3 are the flexoelectric constants. Here the electric field E = Exˆi + Ez kˆ = E (z) cos αˆi + E (z) sin α kˆ makes the angle α with the horizontal surfaces, and the values of which are varied in the vicinity of π2 . Notice that the overbars in the space variables x and z have been (and will be) eliminated in the last as well as in the following equations. The application of the voltage across the nematic film results in a variation of E(z) through the film which is obtained from Ref. 6
⊥ ∂ + sin2 θ E(z) sin α + δ2 θ,z sin 2θ = 0, ∂z a 1=
1 −1
E(z)dz,
(3)
, and V is the voltage applied across the where E(z) = 2dE(z) V cell. In order to elucidate the role of the thermal fluctuations in maintaining of the spatially periodic patterns in the LC sample under the influence of the strong orthogonal electric field, we have performed a numerical study of the Eqs. (2) and (3) with the strong anchoring condition for the angle θ , which read in the dimensionless form as (for details, see Ref. 6) θ (−10 < x < 10, z = ±1) = 0, (4) θ (x = ±10, −1 < z < 1) = 0. In order to observe the formation of the spatially periodic patterns developing spontaneously from homogeneous state, and excited by the strong orthogonal electric field, we consider the initial condition in the form θ (x, z, 0) = θ0 cos(qx x) cos(qz z),
(5)
which defines the thermal fluctuations of the director over the (2k LC sample with amplitude θ 0 and wavelengths qx = πd 2L + 1) and qz = π2 (2k + 1) of an individual Fourier components. Here k = 0, 1, 2, .... Having obtained the function θ (x, z, τ ), one can determine the angular velocity vector ω of the director field nˆ as ω = nˆ × n˙ˆ = −θ˙ (x, z, τ )ˆj = −ω(x, z, τ ˆj,
(6)
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J. Chem. Phys. 139, 244904 (2013)
both during the turn-on and turn-off processes, as well as the dimensionless stress tensor (ST) σ ij components 1 2 E − A sin 2θ, 4
⊥ 1 2 σzz = σxx + B sin 2θ + + sin θ E 2 , 2 a
σxx =
σxz σzx
1 = (−E 2 + A) sin 2θ, 4 1 = (E 2 + A) sin 2θ, 4
(7)
where A = γγ21 cos 2θ E 2 and B = γγ21 sin 2θ E 2 are the dimensionless functions. In our case, the dimensionless ST components take the form σij = Pδij + σijel + σijvis + δ1 σijelast , where P is the hydrostatic pressure in the LC system, σijel , σijvis , and σijelast are the dimensionless ST components corresponding to the electric, viscous, and elastic forces, respectively. Using the fact that δ 1 1, the elastic contribution to the total ST can be neglected, whereas the viscous vis = − 14 B sin 2θ , contribution to σ ij can be written as8 σxx vis vis = 14 (−E 2 + A) sin 2θ , σzx = 14 (E 2 + A) sin 2θ , and σxz vis vis σzz = −σxx . In our case, when the electric field E is applied across to the LC sample, the electric contribution to the total ST tensor σijel = 12 (Ei Dj + Di Ej ) has only one σzzel = ⊥a + sin2 θ E 2 component. Here D is the vector of electric displacement. On the other hand, the hydrodynamic pressure takes the form P = 14 (B − A + E 2 ) sin 2θ , because it has to satisfy the equation σxx,x + σxz,x + P,x = 0. A. Turn-on process in the positive sense
When a strong electric field E is abruptly applied in the positive sense at the angle α close to the right angle to the unit vector ˆi, the director moves from being parallel to the direction preferred by the surfaces to being parallel to the electric field (the turn-on process), because dielectric anisotropy is positive for all cyanobiphenyls. Now the reorientation of the
director in the nematic film under the influence of the external forces can be obtained by solving the nonlinear differential equations (2) and (3) with appropriate boundary (Eq. (4)) and initial (Eq. (5)) conditions. For the case of 4-cyano-4 pentylbiphenyl, at a temperature 300K corresponding to nematic phase, the mass density is equal to ∼103 kg/m3 and the set of δ-parameters, which is involved in Eqs. (2) and (3) takes the values6 δ 1 = 8.6 × 10−6 and δ 2 = −0.0009. These equations (2) and (3) together with the boundary (Eq. (4)) and the initial (Eq. (5)) conditions have been solved by the numerical relaxation method.9 The relaxation criterion = |(θ (n+1) (τ ) − θ (n) (τ ))/θ (n) (τ )| for calculating procedure was chosen equal to be 5 × 10−4 , and the numerical procedure was then carried out until a prescribed accuracy was achieved. It is shown that for the certain balance among the electric, elastic, and viscous torques there is the threshold value of the amplitude θ 0 of the thermal fluctuations of the director over the LC sample which provides the nonuniform rotation mode rather than the uniform one, whereas for the lower value of θ 0 dominates the uniform mode. For instance, for the case of 5CB and the angle α = 1.57(∼89.96◦ ), the periodic response appears only for the values of the amplitude θ 0 more than 0.01 (∼0.57◦ ), whereas for the lower values of θ 0 the certain balance of the torques provides only the uniform rotation mode. The evolution of both the angle θ (x, z = 0, τ ) (see Fig. 2(a), dotted curves) and the angular velocity ω(x, z = 0, τ ) (see Fig. 2(a), solid curves) profiles along the x-axis (−10 ≤ x ≤ 10), for a number of times τ = 2 (∼12 ms), 4 (∼24 ms), 6 (∼36 ms), 8 (∼48 ms), and 12 (∼72 ms), is shown in Fig. 2(a). The values of the dimensionless time (τ V 2 t) are accounted for after switching-on the elec= γ0 1a 2d tric field. In that case, for the values of θ 0 = 0.01 (∼0.57◦ ) and α = 1.57, the time propagation of the θ (x, z = 0, τ ) profile along the x-axis (−10 ≤ x ≤ 10) is characterized by the well-developed periodic structure with the lattice points at x = ±1.96 and ±5.80 and with the relaxation time τ R = 12(∼72 ms). Physically, this means that for the values of the angle α = 1.57 and the amplitude θ 0 = 0.01 (∼0.57◦ ), the balance among the electric, elastic, and viscous torques
FIG. 2. (a) The evolution both of the angle θ (x, z = 0, τ ) (dotted curves) and the angular velocity ω(x, z = 0, τ ) (solid curves) during the turn-on process (E > 0 and α = 1.57(∼89.96◦ )) along the length of the dimensionless LC film (−10 ≤ x ≤ 10), and for a number of dimensionless times τ = 2 (∼12 ms), 4 (∼24 ms), 6 (∼36 ms), 8 (∼48 ms), and 12 (∼72 ms), respectively. (b) Same as (a), but the evolutions are given along the z-axis (−1 ≤ z ≤ 1). In all these cases the amplitude of the thermal fluctuation θ 0 is equal to 0.01 (∼0.57◦ ).
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A. V. Zakharov and A. A. Vakulenko
J. Chem. Phys. 139, 244904 (2013)
FIG. 3. Plot showing the relaxation of the scaled ST components σ zx (x, z = 0, τ ) (a), σ xz (x, z = 0, τ ) (b), σ zz (x, z = 0, τ ) (c), and σ xx (x, z = 0, τ ) (d) to their equilibrium values, for a 5CB film between two electrodes, in the vicinity of the lattice point 1.96, at six scaled distances x of 0.5 (curves (1)), 1.6 (curves (2)), and 1.95 (curves (3)), 1.97 (curves (4)), 2.32 (curves (5)), and 3.42 (curves (6)), respectively, for the turn-on (E > 0) process.
enables only the non-perfect periodic patterns to be maintained with the lattice points at x = ±1.96 and ±5.80, and with the relaxation time τ R = 12 (∼72 ms). So, only at the values of the angle α = 1.57 and the amplitude θ 0 = 0.01 (∼0.57◦ ) do the optimal dimensionless wavelengths qx and qz provide the minimal values of the total energy6 W = Welast + We . Here 2 Welast δ1
ON 2 2 ON ON 2 ON sin θeq +K31 cos2 θeq = dxdz θeq,x + θeq,z +
ON ON ON dxdz (K31 − 1) sin 2θeq θeq,x θeq,z ,
(8)
is the elastic contribution to the total energy, whereas
2 ON ON cos2 θeq 2We = − dxdzE θeq (x, z) − α
ON ON ON sin α sin 2θeq θeq,z , − δ2 dxdzE θeq
(9)
is the electric contribution to the total energy. In our case, ON (x, z) value of the angle θ (x, z, τ = 12) the equilibrium θeq corresponding to the turn-on process is achieved after the dimensionless time term τ = 12. It is also shown that only for the values of qx = 0.785, qz = 64.336, α = 1.57, and θ 0 = 0.01 (∼0.57◦ ) and higher does the solution show that the periodic structure may appear spontaneously from homogeneous nematic phase under the above mentioned conditions. These values of qx and qz provide the minimal values of the total energy W. The evolution both of the angle θ (x = 0, z, τ ) (see Fig. 2(b), dotted curves) and velocity ω(x = 0, z, τ ) (see Fig. 2(b), solid curves) profiles along the z-axis (−1 ≤ z ≤ 1), for the same as in Fig. 2(a) time sequence is shown in Fig. 2(b). According to our calculations of the angle θ (x, z, τ ), the highest value of |∇θ | is reached in the vicinity of the lattice points x = ±1.96, and ±5.80. The evolution of the angular velocity ω(x, z = 0, τ ) profile is characterized by oscillating behavior of |ω(x)| along the x-axis (−10 ≤ x ≤ 10) only within the first 4 time terms, and then the range of these oscillations decreases to zero after the time term 10. Notice that in our case the value of t0 is equal to 120 ms, or 20 dimensionless time units and the highest value of ω, excited by the electric torque, is equal to ∼40 s−1 . Having obtained
the angle θ (x, z, τ ) we can calculate the components of the 2 σ ij (x, z, τ ) (i, j scaled ST components σij (x, z, τ ) = 04d a U 2 = x, z). The relaxation of the ST components σ xx (x, z = 0, τ ), σ xz (x, z = 0, τ ), σ zx (x, z = 0, τ ), and σ zz (x, z = 0, τ ), at six scaled distances x of 0.5 (curves (1)), 1.6 (curves (2)), 1.95 (curves (3)), 1.97 (curves (4)), 2.32 (curves (5)), and 3.42 (curves (6)) in the vicinity of the lattice point 1.96, during the scaled time τ up to 20 (∼0.12 s), is shown in Fig. 3. Figures 3(a), 3(b), and 3(d) show that the scaled components σ zx (τ ), σ xz (τ ), and σ xx (τ ) are characterized by an increase of |σ ij | (i = x, j = x, z) by up to 0.3 (∼6.6 Pa) only within the initial stage of the relaxation process ( τ ∼ 6 (∼36 ms)), and a fast decrease in |σ ij | down to zero, within the last stage of the relaxation process. That situation holds for all above mentioned points. The scaled normal component σ zz (τ ) (Fig. 3(c)) increases monotonically after scaled time τ ∼ 6 (∼36 ms) and saturates at the value of ∼1.5 (∼33 Pa). These calculations also show that at high values of the voltage applied across the LC sample (∼200 V), a highest value of σ zz approximately five times bigger than a highest value of the rest components σ ij (i = x, j = x, z). Calculations of the shear ST components σ zx and σ xz show that these quantities change signs with transition from the left [0.5, 1.96) to the right (1.96, 3.42] wings of the interval [0.5, 3.42], which includes the lattice point x = 1.96. Indeed, the sign of σ zx , at the scaled distances x of 0.5, 1.6, and 1.95, is positive and negative, at x = 1.97, 2.32, and 3.42, whereas the sign of σ zx , at the same scaled distances of x is negative and positive, respectively. So, these figures show how at high values of the voltage applied across the LC sample may appear the lattice point x = 1.96 in an initially uniformly aligned LC sample. Probably, we can conclude that there are the threshold values of the shear ST components σ zx and σ xz , excited by the strong electric field, which provide the appearance of the nonuniform rotation mode rather than the uniform one, whereas the lower values of these shear ST components dominate the uniform mode. That result strongly suggests that the reorientation of the director field following the sudden application of a sufficiently strong and orthogonal electric field manifests itself by appearance of the pattern formation in an initially homogeneously aligned LC sample. The lattice points xk (k = 1, . . . , n) of that periodic structure can be obtained from the equation θ (xk , z = 0, τ ) = 0.
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A. V. Zakharov and A. A. Vakulenko
J. Chem. Phys. 139, 244904 (2013)
FIG. 4. Same as in Fig. 2, but for the turn-off (E = 0) process and for a number of dimensionless times 22, 24, 26, 30, 35, and 40, respectively.
B. Turn-off process
When the electric field is removed, the director relaxes back to the direction preferred by the surfaces (the turn-off process). Now the reorientation of the director in the nematic film under the influence of the long-range elastic interactions can be obtained by solving the nonlinear differential equation (2), for the case of E = 0, and with the boundary condition equation (4) for the angle θ . The initial condition is taken in the form ON (x, z), θ (x, z, 0) = θeq
(10)
ON where θeq is defined as an equilibrium distribution of the director over the LC film obtained during the turn-on process and at the value of the angle α equal to 1.57. Figure 4 shows the evolution both of the angle θ (x, z, τ ) (dotted curves) and velocity ω(x, z, τ ) (solid curves) during the turn-off process along the length (−10 ≤ x ≤ 10) (Fig. 4(a)) and width (−1 ≤ z ≤ 1) (Fig. 4(b)) of the dimensionless LC film, for a number of times τ = 22 (∼0.132 s), 24 (∼0.144 s), 26 (∼0.156 s), 30 (∼0.180 s), 35 (∼0.210 s), and 40 (∼0.240 s). It is shown that after time τ = 40−20 = 20 (∼0.12 s), (t1 − t0 ∼ 0.12 s) when the electric field is removed, the director relaxes back to the direction preferred by the surfaces and that process is characterized by the complex destruction of the initially periodic structure (see Figs. 4(a) and 4(b), dotted curves), especially in the vicinity of the lattice points. In turn, the calculations of the angular velocity (see Figs. 4(a) and 4(b), solid curves) show that the highest value of ω is reached also in the vicinity of the lattice points, and the magnitude of |ω| gradually decreases to zero with increasing of time, up to 40. Notice that the highest value of the angular velocity excited by the elastic torque is equal to 10−3 s−1 .
C. Turn-on process in the negative sense
When the strong electric field E is abruptly applied again but in the negative sense at the angle α ∼ − π2 , the director moves from being parallel to the nˆ OFF to being parallel to the electric field. Here nˆ OFF is the final orientation of the director after the time term τ OFF , when the electric field was removed.
Now the reorientation of the director in the nematic film under the influence of the external forces can be obtained by solving the nonlinear differential equations (2) and (3) with appropriate boundary equation (4) and initial conditions. In that case the initial condition is taken in the form θ (x, z, 0) = θ OFF (x, z),
(11)
where θ OFF (x, z) is defined as the final distribution of the director over the LC film obtained during the turn-off process, when the electric field was removed. Two different scenarios of evolution of the director distribution across the LC film under the influence of the strong electric field directed in the negative sense at the angle α = −1.57 to the horizontal bounding surfaces are shown in Fig. 5. So, in these two cases the electric field was removed during τ OFF = 228 (20 ≤ τ ≤ 248) (hereafter referred to as case I) and 230 (20 ≤ τ ≤ 250) (case II) dimensionless time units, respectively. The main result of these calculations is that the final maintaining of the spatially periodic patterns, at α = −1.57, is possible only when the electric field was removed during 228 dimensionless time units or 20 ≤ τ OFF ≤ 248(0.12s ≤ tOFF ≤ 1.488s) (case I) (see Fig. 5(a)), whereas in the case II, with the longer delaying of the switching-on the electric field, for instance, during 230 dimensionless time units, the certain balance among the electric, elastic, and viscous torques provides only the uniform mode (see Fig. 5(b)). Physically, this means that the further removing of the strong electric field leads both to the further decreasing the value of the amplitude θ 0 and to the destruction of the periodic structure, and, as a result, the director is reoriented as a monodomain nematic sample when the strong electric field is again abruptly applied in the negative sense. This result confirms our previous suggestion that there is threshold value of the amplitude θ 0 which provides the nonuniform rotation mode rather than the uniform one, whereas the lower value of θ 0 dominates the uniform mode.6 Our calculation for two cases I and II also shows that during the first 2 (∼12 ms) time term (see Figs. 5(a) and 5(b)) the evolution of the angular velocity field ω is characterized by maintaining of a simple rotation of the director field nˆ only in the positive sense (anti-clockwise), whereas after time term 4 (∼24 ms) that process is characterized by the
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244904-6
A. V. Zakharov and A. A. Vakulenko
J. Chem. Phys. 139, 244904 (2013)
FIG. 5. Two different scenarios of evolution both of the angle θ (x, z = 0, τ ) (dotted curves) and the angular velocity ω(x, z = 0, τ ) (solid curves) during the turn-on process (E < 0 and α = −1.57(∼89.96◦ )) along the length of the dimensionless LC film (−10 ≤ x ≤ 10), and for a number of dimensionless times τ = 0, 2 (∼12 ms), 4 (∼24 ms), 6 (∼36 ms), and 8 (∼48 ms). Parts (a) and (b) show the sequence of times started from τ = 0 (248) and 0 (250), respectively. In our notation the first value means the dimensionless time after switching-on the electric field in the negative sense, whereas the second time means the total time after starting of the process.
complex destruction of the simple rotation of the vector ˆ In the case I, the reorientation of nˆ is characterized by n. maintaining of the vector field ω rotating in the positive (anti-clockwise) and in the negative (clockwise) directions, as well as with the rest zone, whereas in the case II, the field ω is characterized by the simple rotation of nˆ only in the positive sense (anti-clockwise). In the first case (case I), on the left (−10 ≤ x < −4.72), right (4.72 < x ≤ 10), and in the middle (−3.26 ≤ x ≤ 3.26) parts of the dimensionless interval −10 ≤ x ≤ 10, the director rotates only in the positive sense (anti-clockwise), whereas in the vicinity of the lattice points (−4.72 ≤ x < −4.18; −3.83 < x ≤ −3.26; 3.26 ≤ x < 3.83; 4.18 < x ≤ 4.72) the director rotates in the negative sense (clockwise). Moreover, there are two intervals −4.18 ≤ x ≤ −3.83 and 3.83 ≤ x ≤ 4.18, where the director does not rotates. Notice that in both cases I and II, the angular velocity decreases to zero, practically, after 6 (∼36 ms) time term. III. CONCLUSION
In summary, we have numerically investigated the peculiarities in the director reorientation during both the turnon and turn-off aligning processes in confined nematic phase. Analysis of the numerical results for the turn-on process provides the evidence for the appearance of the spatially
periodic patterns in confined 5CB LC film only in response to the suddenly applied strong electric field directed orthogonal (or approximately orthogonal) to the horizontal bounding surfaces. The main result of this calculation is that the periodic response appears only for a certain balance among the electric, elastic, and viscous torques exerted per unit LC volume, and there is a threshold value of the amplitude of the director fluctuations over the LC sample which provides the nonuniform rotation mode rather than the uniform one, whereas for the lower values of the amplitude the uniform mode appears. It would be expected that the present investigation has shed some light on the problems of the reorientation processes in nematic films confined between two plates under the presence of a strong electric field. 1 P.
G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd ed. (Oxford University Press, Oxford, 1995). 2 E. Guyon, R. Meyer, and J. Salan, Mol. Cryst. Liq. Cryst. 54, 261 (1979). 3 F. Lonberg, S. Fraden, A. J. Hurd, and R. B. Meyer, Phys. Rev. Lett. 52, 1903 (1984). 4 G. Srajer, S. Fraden, and R. B. Meyer, Phys. Rev. A 39, 4828 (1989). 5 A. Sugimura and A. V. Zakharov, Phys. Rev. E 84, 021703 (2011). 6 A. A. Vakulenko and A. V. Zakharov, Phys. Rev. E 88, 022505 (2013). 7 A. Sugimura, A. A. Vakulenko, and A. V. Zakharov, Phys. Procedia 14, 102 (2011). 8 F. M. Leslie, Arch. Ration. Mech. Anal. 28, 265 (1968). 9 I. S. Berezin and N. P. Zhidkov, Computing Methods, 4th ed. (Clarendon, Oxford, 1965).
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THE JOURNAL OF CHEMICAL PHYSICS 139, 244904 (2013)
Dynamics of the modulated distortions in confined nematic liquid crystals A. V. Zakharova) and A. A. Vakulenkob) Saint Petersburg Institute for Machine Sciences, The Russian Academy of Sciences, Saint Petersburg 199178, Russia
(Received 30 October 2013; accepted 5 December 2013; published online 26 December 2013) The peculiarities in the dynamics of the director reorientation in confined nematic liquid crystals (LCs) under the influence of a strong electric field E have been investigated theoretically based on the hydrodynamic theory including the director motion with appropriate boundary and initial conditions. Analysis of the numerical results for the turn-on process provides an evidence for the appearance of the spatially periodic patterns in confined LC film, only in response to the suddenly applied strong E. It has been shown that there is a threshold value of the amplitude of the thermal fluctuations of the director over the LC sample which provides the nonuniform rotation mode rather than the uniform one, whereas the lower values of the amplitude dominate the uniform mode. During the turn-off process, the reorientation of the director to the direction preferred by the surfaces is characterized by the complex destruction of the initially periodic structure to a monodomain state. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4851197] I. INTRODUCTION
Uniform textures of nematic liquid crystals (NLCs) are produced by orienting a drop of bulk material in between two conveniently treated plates, which define usually a fixed orientation for the boundary molecules. Applying the electric field E perpendicular to a uniformly (homogeneously) oriented NLC can distort the molecular orientation aˆ with reˆ at a critical threshold field Eth given by1 spect to director n, π Eth = d
K1 , 0 a
(1)
where d is the film thickness, K1 is the splay elastic constant, 0 is the absolute dielectric permittivity of free space, and a is the dielectric anisotropy of the NLC. This form for the critical field is based upon assumption that the director remains strongly anchored (in our case, homogeneously) at the two horizontal surfaces and that the physical properties of the liquid crystal (LC) are uniform over the entire sample for E < Eth . When the electric field is switched on with a magniˆ in the “splay” geometude E greater than Eth , the director n, try, reorients as a simple monodomain.1 In the case when the electric field E Eth is abruptly applied orthogonal to an initially uniformly aligned (homogeneously) nematic LC film, ˆ reorients in order to minimize the free energy, the director, n, and the LC system is suddenly placed far from equilibrium. It responds by creating a distortion which maximizes the rate at which the LC lowers its total free energy. In this case, the final form of deformation depends on viscous, elastic, and electric torques, as well as the boundary and initial conditions and the application of the strong orthogonal electric field gives rise a) Author to whom correspondence should be addressed. Electronic mail:
[email protected] URL: http://www.ipme.ru
b) Email: [email protected] URL: http://www.ipme.ru
0021-9606/2013/139(24)/244904/6/$30.00
sometimes to the appearance of a nonuniform rotation mode rather than the uniform one. The first theoretical study of the periodic structures in the LC systems imposed by the strong external field has been commonly associated with the wavelength corresponding to the mode of fastest growth predicted by a linear analysis of the nematodynamic equations.2–5 This approach, based on an eigenvalue analysis, although useful in understanding the main physical quantities in the origin of the observed periodic structures, is far less valid if we are interested in the dynamics of the modulated structures in nematic LCs. For this reason we have recently investigated theoretically the field-induced director dynamics based on the nonlinear hydrodynamic theory including the director motion with appropriate boundary and initial conditions.6 Analysis of the numerical results for the turn-on process provides an evidence for the appearance of the spatially periodic patterns in confined 4-n-pentyl-4 -cyanobiphenyl (5CB) LC film, only in response to the suddenly applied strong electric field orthogonal to the magnetic field. It has been shown that for a certain balance among the electric, elastic, and viscous torques there is the threshold value of the amplitude of the thermal fluctuations of the director over the LC sample which provides the nonuniform rotation mode rather than the uniform one, whereas the lower value of the amplitude dominates the uniform mode. During the turn-off process, the reorientation of the director to the direction preferred by the surfaces is characterized by the complex destruction of the initially periodic structure to a monodomain state. In this work we focus on the geometry where the electric field is orthogonal (or approximately orthogonal) to the horizontal boundaries (see Fig. 1). In this configuration the state of the system, immediately becomes unstable after applying the strong orthogonal electric field. When the misalignment of the director with respect to the direction preferred by the surfaces is due to the thermal fluctuations with small amplitudes, the reorientation following the sudden application of a
139, 244904-1
© 2013 AIP Publishing LLC
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244904-2
A. V. Zakharov and A. A. Vakulenko
J. Chem. Phys. 139, 244904 (2013)
orientation nˆ eq can be written as (for details, see the Ref. 6) θ,τ = δ1 (sin2 θ + K31 cos2 θ )θ,xx +(cos2 θ + K31 sin2 θ )θ,zz + δ1 (K31 − 1) sin 2θ θ,xz 1 + (1 − K31 ) sin 2θ θ,x2 − θ,z2 2 + δ1 (K31 − 1) cos 2θ θ,x θ,z FIG. 1. The geometry used for the calculations. The z-axis is normal to the horizontal bounding surfaces, whereas the unit vector ˆi is directed parallel ˆ to the horizontal surfaces. The electric field E, unit vector ˆi, and director, n, are in the xz-plane. The director makes an angle θ with the x-axis, and the electric field makes an angle α with the unit vector ˆi.
sufficiently strong and orthogonal electric field manifests itself by the growing of one particular Fourier mode. After switching-off the field, the long-range elastic interactions ensure that the molecules reorient themselves in the direction preferred by the surfaces. In this work, switching will be driven by a series of a voltage pulses: V > 0, for 0 < t < t0 ; V = 0, for t0 ≤ t ≤ t1 ; and V < 0, for t1 < t < t2 . The first voltage pulse corresponds to the field in the positive z-direction (see Fig. 1). Then for t0 ≤ t ≤ t1 , the field is equal to zero, and for t1 < t < t2 , the field is negative. So, the aim of this paper is to show how in the LC system of this geometry switching can be driven by the electric field and to demonstrate the kinetic pathway along which the switching proceeds. II. FORMULATION OF THE RELEVANT EQUATIONS FOR DYNAMICAL REORIENTATION OF THE DIRECTOR FIELD
The coordinate system defined by our experiment assumes that the strong electric field E is abruptly applied normal (or close to the normal) to the horizontal bounding surfaces (see Fig. 1). We consider a homogeneously aligned nematic system such as cyanobiphenyls, which is delimited by two horizontal and two lateral surfaces at mutual distances 2d and 2L on a scale in the order of tens micrometers. According to this geometry the LC system may be seen as the two-dimensional, since the director is maintained within the xz-plane (or in the yz-plane) defined by the electric field and the unit vector ˆi directed parallel to the horizontal surfaces, kˆ is a unit vector directed normal to the horizontal surfaces, and ˆj = kˆ × ˆi. We can suppose that the components of the director, nˆ = nxˆi + nz kˆ = cos θ (x, z, t)ˆi + sin θ (x, z, t)kˆ (see Fig. 1) depend only on x, z-components and time t. Here θ denotes the angle between the director and the unit vector ˆi. Our recent investigation of the field-induced reorientation of the director field under the influence of a strong electric field suggests that in order to describe the dynamical reorientation of the director correctly, we do not need to include a proper treatment of backflow.7 This means that the role of the viscous force becomes negligible in comparison to the electric, elastic, and flexoelectric contributions. In the case of the quasitwo-dimensional LC system the dimensionless torque balance equation describing the reorientation of nˆ to its equilibrium
1 2 + E (θ ) sin 2(α − θ ) − δ2 E ,z sin α sin 2θ, 2
(2)
where x = x/d and z = z/d are the dimensionless space V 2 variables, τ = γ0 1a 2d t is the dimensionless time, e1 +e3 3 1 δ1 = 04K , δ = , and K31 = K are three param2 0 a V K1 a V 2 eters of the system. Here γ 1 is the rotational viscosity coefficient, K3 is the bend elastic constant, whereas e1 and e3 are the flexoelectric constants. Here the electric field E = Exˆi + Ez kˆ = E (z) cos αˆi + E (z) sin α kˆ makes the angle α with the horizontal surfaces, and the values of which are varied in the vicinity of π2 . Notice that the overbars in the space variables x and z have been (and will be) eliminated in the last as well as in the following equations. The application of the voltage across the nematic film results in a variation of E(z) through the film which is obtained from Ref. 6
⊥ ∂ + sin2 θ E(z) sin α + δ2 θ,z sin 2θ = 0, ∂z a 1=
1 −1
E(z)dz,
(3)
, and V is the voltage applied across the where E(z) = 2dE(z) V cell. In order to elucidate the role of the thermal fluctuations in maintaining of the spatially periodic patterns in the LC sample under the influence of the strong orthogonal electric field, we have performed a numerical study of the Eqs. (2) and (3) with the strong anchoring condition for the angle θ , which read in the dimensionless form as (for details, see Ref. 6) θ (−10 < x < 10, z = ±1) = 0, (4) θ (x = ±10, −1 < z < 1) = 0. In order to observe the formation of the spatially periodic patterns developing spontaneously from homogeneous state, and excited by the strong orthogonal electric field, we consider the initial condition in the form θ (x, z, 0) = θ0 cos(qx x) cos(qz z),
(5)
which defines the thermal fluctuations of the director over the (2k LC sample with amplitude θ 0 and wavelengths qx = πd 2L + 1) and qz = π2 (2k + 1) of an individual Fourier components. Here k = 0, 1, 2, .... Having obtained the function θ (x, z, τ ), one can determine the angular velocity vector ω of the director field nˆ as ω = nˆ × n˙ˆ = −θ˙ (x, z, τ )ˆj = −ω(x, z, τ ˆj,
(6)
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244904-3
A. V. Zakharov and A. A. Vakulenko
J. Chem. Phys. 139, 244904 (2013)
both during the turn-on and turn-off processes, as well as the dimensionless stress tensor (ST) σ ij components 1 2 E − A sin 2θ, 4
⊥ 1 2 σzz = σxx + B sin 2θ + + sin θ E 2 , 2 a
σxx =
σxz σzx
1 = (−E 2 + A) sin 2θ, 4 1 = (E 2 + A) sin 2θ, 4
(7)
where A = γγ21 cos 2θ E 2 and B = γγ21 sin 2θ E 2 are the dimensionless functions. In our case, the dimensionless ST components take the form σij = Pδij + σijel + σijvis + δ1 σijelast , where P is the hydrostatic pressure in the LC system, σijel , σijvis , and σijelast are the dimensionless ST components corresponding to the electric, viscous, and elastic forces, respectively. Using the fact that δ 1 1, the elastic contribution to the total ST can be neglected, whereas the viscous vis = − 14 B sin 2θ , contribution to σ ij can be written as8 σxx vis vis = 14 (−E 2 + A) sin 2θ , σzx = 14 (E 2 + A) sin 2θ , and σxz vis vis σzz = −σxx . In our case, when the electric field E is applied across to the LC sample, the electric contribution to the total ST tensor σijel = 12 (Ei Dj + Di Ej ) has only one σzzel = ⊥a + sin2 θ E 2 component. Here D is the vector of electric displacement. On the other hand, the hydrodynamic pressure takes the form P = 14 (B − A + E 2 ) sin 2θ , because it has to satisfy the equation σxx,x + σxz,x + P,x = 0. A. Turn-on process in the positive sense
When a strong electric field E is abruptly applied in the positive sense at the angle α close to the right angle to the unit vector ˆi, the director moves from being parallel to the direction preferred by the surfaces to being parallel to the electric field (the turn-on process), because dielectric anisotropy is positive for all cyanobiphenyls. Now the reorientation of the
director in the nematic film under the influence of the external forces can be obtained by solving the nonlinear differential equations (2) and (3) with appropriate boundary (Eq. (4)) and initial (Eq. (5)) conditions. For the case of 4-cyano-4 pentylbiphenyl, at a temperature 300K corresponding to nematic phase, the mass density is equal to ∼103 kg/m3 and the set of δ-parameters, which is involved in Eqs. (2) and (3) takes the values6 δ 1 = 8.6 × 10−6 and δ 2 = −0.0009. These equations (2) and (3) together with the boundary (Eq. (4)) and the initial (Eq. (5)) conditions have been solved by the numerical relaxation method.9 The relaxation criterion = |(θ (n+1) (τ ) − θ (n) (τ ))/θ (n) (τ )| for calculating procedure was chosen equal to be 5 × 10−4 , and the numerical procedure was then carried out until a prescribed accuracy was achieved. It is shown that for the certain balance among the electric, elastic, and viscous torques there is the threshold value of the amplitude θ 0 of the thermal fluctuations of the director over the LC sample which provides the nonuniform rotation mode rather than the uniform one, whereas for the lower value of θ 0 dominates the uniform mode. For instance, for the case of 5CB and the angle α = 1.57(∼89.96◦ ), the periodic response appears only for the values of the amplitude θ 0 more than 0.01 (∼0.57◦ ), whereas for the lower values of θ 0 the certain balance of the torques provides only the uniform rotation mode. The evolution of both the angle θ (x, z = 0, τ ) (see Fig. 2(a), dotted curves) and the angular velocity ω(x, z = 0, τ ) (see Fig. 2(a), solid curves) profiles along the x-axis (−10 ≤ x ≤ 10), for a number of times τ = 2 (∼12 ms), 4 (∼24 ms), 6 (∼36 ms), 8 (∼48 ms), and 12 (∼72 ms), is shown in Fig. 2(a). The values of the dimensionless time (τ V 2 t) are accounted for after switching-on the elec= γ0 1a 2d tric field. In that case, for the values of θ 0 = 0.01 (∼0.57◦ ) and α = 1.57, the time propagation of the θ (x, z = 0, τ ) profile along the x-axis (−10 ≤ x ≤ 10) is characterized by the well-developed periodic structure with the lattice points at x = ±1.96 and ±5.80 and with the relaxation time τ R = 12(∼72 ms). Physically, this means that for the values of the angle α = 1.57 and the amplitude θ 0 = 0.01 (∼0.57◦ ), the balance among the electric, elastic, and viscous torques
FIG. 2. (a) The evolution both of the angle θ (x, z = 0, τ ) (dotted curves) and the angular velocity ω(x, z = 0, τ ) (solid curves) during the turn-on process (E > 0 and α = 1.57(∼89.96◦ )) along the length of the dimensionless LC film (−10 ≤ x ≤ 10), and for a number of dimensionless times τ = 2 (∼12 ms), 4 (∼24 ms), 6 (∼36 ms), 8 (∼48 ms), and 12 (∼72 ms), respectively. (b) Same as (a), but the evolutions are given along the z-axis (−1 ≤ z ≤ 1). In all these cases the amplitude of the thermal fluctuation θ 0 is equal to 0.01 (∼0.57◦ ).
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244904-4
A. V. Zakharov and A. A. Vakulenko
J. Chem. Phys. 139, 244904 (2013)
FIG. 3. Plot showing the relaxation of the scaled ST components σ zx (x, z = 0, τ ) (a), σ xz (x, z = 0, τ ) (b), σ zz (x, z = 0, τ ) (c), and σ xx (x, z = 0, τ ) (d) to their equilibrium values, for a 5CB film between two electrodes, in the vicinity of the lattice point 1.96, at six scaled distances x of 0.5 (curves (1)), 1.6 (curves (2)), and 1.95 (curves (3)), 1.97 (curves (4)), 2.32 (curves (5)), and 3.42 (curves (6)), respectively, for the turn-on (E > 0) process.
enables only the non-perfect periodic patterns to be maintained with the lattice points at x = ±1.96 and ±5.80, and with the relaxation time τ R = 12 (∼72 ms). So, only at the values of the angle α = 1.57 and the amplitude θ 0 = 0.01 (∼0.57◦ ) do the optimal dimensionless wavelengths qx and qz provide the minimal values of the total energy6 W = Welast + We . Here 2 Welast δ1
ON 2 2 ON ON 2 ON sin θeq +K31 cos2 θeq = dxdz θeq,x + θeq,z +
ON ON ON dxdz (K31 − 1) sin 2θeq θeq,x θeq,z ,
(8)
is the elastic contribution to the total energy, whereas
2 ON ON cos2 θeq 2We = − dxdzE θeq (x, z) − α
ON ON ON sin α sin 2θeq θeq,z , − δ2 dxdzE θeq
(9)
is the electric contribution to the total energy. In our case, ON (x, z) value of the angle θ (x, z, τ = 12) the equilibrium θeq corresponding to the turn-on process is achieved after the dimensionless time term τ = 12. It is also shown that only for the values of qx = 0.785, qz = 64.336, α = 1.57, and θ 0 = 0.01 (∼0.57◦ ) and higher does the solution show that the periodic structure may appear spontaneously from homogeneous nematic phase under the above mentioned conditions. These values of qx and qz provide the minimal values of the total energy W. The evolution both of the angle θ (x = 0, z, τ ) (see Fig. 2(b), dotted curves) and velocity ω(x = 0, z, τ ) (see Fig. 2(b), solid curves) profiles along the z-axis (−1 ≤ z ≤ 1), for the same as in Fig. 2(a) time sequence is shown in Fig. 2(b). According to our calculations of the angle θ (x, z, τ ), the highest value of |∇θ | is reached in the vicinity of the lattice points x = ±1.96, and ±5.80. The evolution of the angular velocity ω(x, z = 0, τ ) profile is characterized by oscillating behavior of |ω(x)| along the x-axis (−10 ≤ x ≤ 10) only within the first 4 time terms, and then the range of these oscillations decreases to zero after the time term 10. Notice that in our case the value of t0 is equal to 120 ms, or 20 dimensionless time units and the highest value of ω, excited by the electric torque, is equal to ∼40 s−1 . Having obtained
the angle θ (x, z, τ ) we can calculate the components of the 2 σ ij (x, z, τ ) (i, j scaled ST components σij (x, z, τ ) = 04d a U 2 = x, z). The relaxation of the ST components σ xx (x, z = 0, τ ), σ xz (x, z = 0, τ ), σ zx (x, z = 0, τ ), and σ zz (x, z = 0, τ ), at six scaled distances x of 0.5 (curves (1)), 1.6 (curves (2)), 1.95 (curves (3)), 1.97 (curves (4)), 2.32 (curves (5)), and 3.42 (curves (6)) in the vicinity of the lattice point 1.96, during the scaled time τ up to 20 (∼0.12 s), is shown in Fig. 3. Figures 3(a), 3(b), and 3(d) show that the scaled components σ zx (τ ), σ xz (τ ), and σ xx (τ ) are characterized by an increase of |σ ij | (i = x, j = x, z) by up to 0.3 (∼6.6 Pa) only within the initial stage of the relaxation process ( τ ∼ 6 (∼36 ms)), and a fast decrease in |σ ij | down to zero, within the last stage of the relaxation process. That situation holds for all above mentioned points. The scaled normal component σ zz (τ ) (Fig. 3(c)) increases monotonically after scaled time τ ∼ 6 (∼36 ms) and saturates at the value of ∼1.5 (∼33 Pa). These calculations also show that at high values of the voltage applied across the LC sample (∼200 V), a highest value of σ zz approximately five times bigger than a highest value of the rest components σ ij (i = x, j = x, z). Calculations of the shear ST components σ zx and σ xz show that these quantities change signs with transition from the left [0.5, 1.96) to the right (1.96, 3.42] wings of the interval [0.5, 3.42], which includes the lattice point x = 1.96. Indeed, the sign of σ zx , at the scaled distances x of 0.5, 1.6, and 1.95, is positive and negative, at x = 1.97, 2.32, and 3.42, whereas the sign of σ zx , at the same scaled distances of x is negative and positive, respectively. So, these figures show how at high values of the voltage applied across the LC sample may appear the lattice point x = 1.96 in an initially uniformly aligned LC sample. Probably, we can conclude that there are the threshold values of the shear ST components σ zx and σ xz , excited by the strong electric field, which provide the appearance of the nonuniform rotation mode rather than the uniform one, whereas the lower values of these shear ST components dominate the uniform mode. That result strongly suggests that the reorientation of the director field following the sudden application of a sufficiently strong and orthogonal electric field manifests itself by appearance of the pattern formation in an initially homogeneously aligned LC sample. The lattice points xk (k = 1, . . . , n) of that periodic structure can be obtained from the equation θ (xk , z = 0, τ ) = 0.
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244904-5
A. V. Zakharov and A. A. Vakulenko
J. Chem. Phys. 139, 244904 (2013)
FIG. 4. Same as in Fig. 2, but for the turn-off (E = 0) process and for a number of dimensionless times 22, 24, 26, 30, 35, and 40, respectively.
B. Turn-off process
When the electric field is removed, the director relaxes back to the direction preferred by the surfaces (the turn-off process). Now the reorientation of the director in the nematic film under the influence of the long-range elastic interactions can be obtained by solving the nonlinear differential equation (2), for the case of E = 0, and with the boundary condition equation (4) for the angle θ . The initial condition is taken in the form ON (x, z), θ (x, z, 0) = θeq
(10)
ON where θeq is defined as an equilibrium distribution of the director over the LC film obtained during the turn-on process and at the value of the angle α equal to 1.57. Figure 4 shows the evolution both of the angle θ (x, z, τ ) (dotted curves) and velocity ω(x, z, τ ) (solid curves) during the turn-off process along the length (−10 ≤ x ≤ 10) (Fig. 4(a)) and width (−1 ≤ z ≤ 1) (Fig. 4(b)) of the dimensionless LC film, for a number of times τ = 22 (∼0.132 s), 24 (∼0.144 s), 26 (∼0.156 s), 30 (∼0.180 s), 35 (∼0.210 s), and 40 (∼0.240 s). It is shown that after time τ = 40−20 = 20 (∼0.12 s), (t1 − t0 ∼ 0.12 s) when the electric field is removed, the director relaxes back to the direction preferred by the surfaces and that process is characterized by the complex destruction of the initially periodic structure (see Figs. 4(a) and 4(b), dotted curves), especially in the vicinity of the lattice points. In turn, the calculations of the angular velocity (see Figs. 4(a) and 4(b), solid curves) show that the highest value of ω is reached also in the vicinity of the lattice points, and the magnitude of |ω| gradually decreases to zero with increasing of time, up to 40. Notice that the highest value of the angular velocity excited by the elastic torque is equal to 10−3 s−1 .
C. Turn-on process in the negative sense
When the strong electric field E is abruptly applied again but in the negative sense at the angle α ∼ − π2 , the director moves from being parallel to the nˆ OFF to being parallel to the electric field. Here nˆ OFF is the final orientation of the director after the time term τ OFF , when the electric field was removed.
Now the reorientation of the director in the nematic film under the influence of the external forces can be obtained by solving the nonlinear differential equations (2) and (3) with appropriate boundary equation (4) and initial conditions. In that case the initial condition is taken in the form θ (x, z, 0) = θ OFF (x, z),
(11)
where θ OFF (x, z) is defined as the final distribution of the director over the LC film obtained during the turn-off process, when the electric field was removed. Two different scenarios of evolution of the director distribution across the LC film under the influence of the strong electric field directed in the negative sense at the angle α = −1.57 to the horizontal bounding surfaces are shown in Fig. 5. So, in these two cases the electric field was removed during τ OFF = 228 (20 ≤ τ ≤ 248) (hereafter referred to as case I) and 230 (20 ≤ τ ≤ 250) (case II) dimensionless time units, respectively. The main result of these calculations is that the final maintaining of the spatially periodic patterns, at α = −1.57, is possible only when the electric field was removed during 228 dimensionless time units or 20 ≤ τ OFF ≤ 248(0.12s ≤ tOFF ≤ 1.488s) (case I) (see Fig. 5(a)), whereas in the case II, with the longer delaying of the switching-on the electric field, for instance, during 230 dimensionless time units, the certain balance among the electric, elastic, and viscous torques provides only the uniform mode (see Fig. 5(b)). Physically, this means that the further removing of the strong electric field leads both to the further decreasing the value of the amplitude θ 0 and to the destruction of the periodic structure, and, as a result, the director is reoriented as a monodomain nematic sample when the strong electric field is again abruptly applied in the negative sense. This result confirms our previous suggestion that there is threshold value of the amplitude θ 0 which provides the nonuniform rotation mode rather than the uniform one, whereas the lower value of θ 0 dominates the uniform mode.6 Our calculation for two cases I and II also shows that during the first 2 (∼12 ms) time term (see Figs. 5(a) and 5(b)) the evolution of the angular velocity field ω is characterized by maintaining of a simple rotation of the director field nˆ only in the positive sense (anti-clockwise), whereas after time term 4 (∼24 ms) that process is characterized by the
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244904-6
A. V. Zakharov and A. A. Vakulenko
J. Chem. Phys. 139, 244904 (2013)
FIG. 5. Two different scenarios of evolution both of the angle θ (x, z = 0, τ ) (dotted curves) and the angular velocity ω(x, z = 0, τ ) (solid curves) during the turn-on process (E < 0 and α = −1.57(∼89.96◦ )) along the length of the dimensionless LC film (−10 ≤ x ≤ 10), and for a number of dimensionless times τ = 0, 2 (∼12 ms), 4 (∼24 ms), 6 (∼36 ms), and 8 (∼48 ms). Parts (a) and (b) show the sequence of times started from τ = 0 (248) and 0 (250), respectively. In our notation the first value means the dimensionless time after switching-on the electric field in the negative sense, whereas the second time means the total time after starting of the process.
complex destruction of the simple rotation of the vector ˆ In the case I, the reorientation of nˆ is characterized by n. maintaining of the vector field ω rotating in the positive (anti-clockwise) and in the negative (clockwise) directions, as well as with the rest zone, whereas in the case II, the field ω is characterized by the simple rotation of nˆ only in the positive sense (anti-clockwise). In the first case (case I), on the left (−10 ≤ x < −4.72), right (4.72 < x ≤ 10), and in the middle (−3.26 ≤ x ≤ 3.26) parts of the dimensionless interval −10 ≤ x ≤ 10, the director rotates only in the positive sense (anti-clockwise), whereas in the vicinity of the lattice points (−4.72 ≤ x < −4.18; −3.83 < x ≤ −3.26; 3.26 ≤ x < 3.83; 4.18 < x ≤ 4.72) the director rotates in the negative sense (clockwise). Moreover, there are two intervals −4.18 ≤ x ≤ −3.83 and 3.83 ≤ x ≤ 4.18, where the director does not rotates. Notice that in both cases I and II, the angular velocity decreases to zero, practically, after 6 (∼36 ms) time term. III. CONCLUSION
In summary, we have numerically investigated the peculiarities in the director reorientation during both the turnon and turn-off aligning processes in confined nematic phase. Analysis of the numerical results for the turn-on process provides the evidence for the appearance of the spatially
periodic patterns in confined 5CB LC film only in response to the suddenly applied strong electric field directed orthogonal (or approximately orthogonal) to the horizontal bounding surfaces. The main result of this calculation is that the periodic response appears only for a certain balance among the electric, elastic, and viscous torques exerted per unit LC volume, and there is a threshold value of the amplitude of the director fluctuations over the LC sample which provides the nonuniform rotation mode rather than the uniform one, whereas for the lower values of the amplitude the uniform mode appears. It would be expected that the present investigation has shed some light on the problems of the reorientation processes in nematic films confined between two plates under the presence of a strong electric field. 1 P.
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