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Field-induced stress response of nematics encapsulated in microsized volumes
Accepted Manuscript Field-induced stress response of nematics encapsulated in microsized volumes A.V. Zakharov, P.V. Maslennikov PII: DOI: Reference:
S0009-2614(17)30615-2 http://dx.doi.org/10.1016/j.cplett.2017.06.058 CPLETT 34920
To appear in:
Chemical Physics Letters
Received Date: Accepted Date:
2 May 2017 29 June 2017
Please cite this article as: A.V. Zakharov, P.V. Maslennikov, Field-induced stress response of nematics encapsulated in microsized volumes, Chemical Physics Letters (2017), doi: http://dx.doi.org/10.1016/j.cplett.2017.06.058
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Field-induced stress response of nematics encapsulated in microsized volumes A. V. Zakharov∗ Saint Petersburg Institute for Machine Sciences, The Russian Academy of Sciences, Saint Petersburg 199178, Russia. P. V. Maslennikov† Immanuel Kant Baltic Federal University, Kaliningrad 236040, Str. Universitetskaya 2, Russia. (Dated: June 27, 2017)
Abstract 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, when a strong electric field E is suddenly applied in the positive sense, provides an evidence for the appearance of the spatially periodic patterns in confined LC film. PACS numbers: 61.30.Cz, 65.40.De
∗
author to whom correspondence should be addressed. Email address:[email protected];
www.ipme.ru †
Email address:[email protected]
1
I.
INTRODUCTION
With the current sustained demand for portable interactive electronic devices with liquid crystal (LC) displays, there is considerable interest in understanding their operation from a theoretical perspective. Robust and accurate mathematical models for prototype LC displays allow simulations of hypothetical devices to be made quickly and at low cost, and this in turn can lead to new and potentially improved designs being identified. Most models used in such theoretical investigations, make several simplifications in order to arrive at a suitably tractable model for simulations. Normally the LC cell is modeled as a layer of nematic liquid crystal (NLC) sandwiched between two parallel bounding plates, across which an electric field E can be applied. Uniform textures of LC cells 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 (hoˆ with respect to director mogeneously) oriented NLC can distort the molecular orientation a ˆ , at a critical threshold field Eth given by [1] Eth = n
π d
√
K1 , ϵ0 ϵa
where d is the film thickness,
K1 is the splay elastic constant, ϵ0 is the absolute dielectric permittivity of free space, and ϵa 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 LC are uniform over the entire sample for E < Eth . When the electric field is switched on with a magnitude E greater than ˆ , in the ”splay” geometry, reorients as a simple monodomain [2]. In the Eth , the director n case when the electric field E ≫ Eth is abruptly applied orthogonal to an initially uniformly ˆ , reorients in order to minimize aligned (homogeneously) nematic LC film, the director, n the free energy, and the LC system is suddenly placed far from equilibrium. It respond 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 sometimes to the appearance of a nonuniform rotation mode rather than the uniform one [3–11]. 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 [3–5, 8]. This approach, based on an eigenvalue analysis,
2
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 [9]. 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 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. It should be noted that the nuclear magnetic resonance spectroscopy is by now also a well-established method for investigating the peculiarities in the dynamics of the director reorientation in confined LCs under crossed electric and magnetic fields [2]. In this work we focus on the geometry where the electric field (E ≫ Eth ) is orthogonal (or approximately orthogonal) to the horizontal boundaries (see Fig.1(a)). 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 sufficiently strong and orthogonal electric field manifests itself by the growing of one particular Fourier mode. After switched 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(b)). 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 shows 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.
3
FIG. 1: (a) The geometry used for the calculations. The z-axis is normal to the horizontal bounding surfaces, whereas the unit vector ˆi is directed parallel to the horizontal surfaces. The electric field ˆ , are in the xz- plane. The director makes an angle θ with the E, unit vector ˆi, and director, n x-axis, and the electric field makes an angle α with the unit vector ˆi. (b) The series of a voltage pulses. II.
FORMULATION OF THE RELEVANT EQUATIONS FOR DYNAMICAL RE-
ORIENTATION 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(a)). 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 4
ˆ is a unit vector directed normal to the horizontal surfaces, and ˆj = k ˆ × ˆi. We can surfaces, k ˆ = cos θ(x, z, t)ˆi + sin θ(x, z, t)k ˆ ˆ = nxˆi + nz k suppose that the components of the director, n (see Fig.1(a)) depend only 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 [10]. 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 quasi-two-dimensional LC system the dimensionless torque balance equation describing ˆ to its equilibrium orientation n ˆ eq can be written as (for details, see the reorientation of n the Ref.[9]) [(
)
(
)
]
sin2 θ + K31 cos2 θ θ,xx + cos2 θ + K31 sin2 θ θ,zz + [ ( )] 1 2 2 δ1 (K31 − 1) sin 2θθ,xz + (1 − K31 ) sin 2θ θ,x − θ,z + 2 1 2 δ1 (K31 − 1) cos 2θθ,x θ,z + E (θ) sin 2 (α − θ) − δ2 E ,z sin α sin 2θ, 2 θ,τ = δ1
where x = x/d and z = z/d are the dimensionless space variables, τ = dimensionless time, δ1 =
4K1 , ϵ0 ϵa V 2
δ2 =
e1 +e3 , ϵ0 ϵa V
and K31 =
K3 K1
ϵ0 ϵa γ1
(1) (
V 2d
)2
t is the
are three parameters of the
system. Here γ1 is the rotational viscosity coefficient, K3 is the bend elastic constant, ˆ= whereas e1 and e3 are the flexoelectric constants. Here the electric field E = Exˆi + Ez k ˆ makes the angle α with the horizontal surfaces, and the values of E (z) cos αˆi + E (z) sin αk 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 [9] ∂ ∂z
[(
)
]
ϵ⊥ + sin2 θ E(z) sin α + δ2 θ,z sin 2θ = 0, ϵa ∫
1= where E(z) =
2dE(z) , V
1
−1
E(z)dz,
(2)
and V is the voltage applied across the 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.(1)-(2) with the strong anchoring condition for the angle θ, 5
which read in the dimensionless form as θ (−10 < x < 10, z = ±1) = 0, θ (x = ±10, −1 < z < 1) = 0.
(3)
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),
(4)
which defines the thermal fluctuations of the director over the LC sample with amplitude θ0 and wavelengths qx =
πd (2k 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 ⃗ ˆ as of the director field n ˆ ×n ˆ˙ = −θ˙ (x, z, τ ) ˆj = −ω (x, z, τ ) ˆj, ω ⃗ =n
(5)
both during the turn-on and turn-off processes, as well as the dimensionless stress tensor (ST) σij components ) 1( 2 E − A sin 2θ, 4( ) 1 ϵ⊥ σzz = σxx + B sin 2θ + + sin2 θ E 2 , 2 ϵa ( ) 1 σxz = −E 2 + A sin 2θ, 4 ) 1( 2 σzx = E + A sin 2θ, 4
σxx =
where A =
γ2 γ1
cos 2θE 2 and B =
γ2 γ1
(6)
sin 2θE 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. 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 , the value of the voltage across the nematic film, which is 2d = 194.7 µm, was chosen to be U = 200 V , the geometry factor is L = 10d, and the set of δ-parameters, which is involved in Eqs.(1) and (2) takes the values [8] δ1 = 8.6 × 10−6 and δ2 = −0.0009. Using the fact that δ1 ≪ 1, the elastic contribution to the total ST can 6
vis be neglected, whereas the viscous contribution to σij can be written as [12] σxx = − 14 B sin 2θ, vis σxz =
1 4
vis (−E 2 + A) sin 2θ, σzx =
1 4
vis vis (E 2 + A) sin 2θ, and σzz = −σxx . In our case, when the
electric field E is applied across to the LC sample, the electric contribution to the total ST el tensor σijel = 12 (Ei Dj + Di Ej ) has only one σzz =
(
ϵ⊥ ϵa
)
+ 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 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 (E ∼ 100Eth ) 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 (1)-(2) with appropriate boundary (Eq.(3)) and initial (Eq.(4)) conditions. These equations has been solved by the numerical (
)
relaxation method [13]. The relaxation criterion ϵ = | 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 θ0th 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 θ0th dominate 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 θ0 more than 0.01 (∼ 0.57◦ ), whereas for the lower values of θ0 the certain balance of the torques provides only the uniform rotation mode. The evolution both of 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 =
ϵ0 ϵa γ1
(
V 2d
)2
t) is accounted for after switched on the electric
field. In that case, for the values of θ0 = 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 welldeveloped periodic structure with the lattice points at x = ±1.96 and ±5.80 and with the 7
1.5
1.5
0.0
0.0
12
-1.5 -10
-5
0
5
10
1.5
0
5
(x=0,z, )
-5
10
1.5 0.0
8
-1.5 -10
-5
0
5
1.0 0.0 -1.0
10
(x=0,z, ),
(x,z=0, x,z=0, )
-0.5
0.0
0.5
1.0
-1.0
-0.5
0.0
0.5
1.0
-1.0
-0.5
0.0
0.5
1.0
-1.0
-0.5
0.0
0.5
1.0
-1.0
-0.5
0.0
0.5
1.0
0.0
0.5
1.0
0.0
10
-10
6
-10
-5
0
5
10
0.5 0.0 -5
0
5
10
-1.5 1.5 0.0
E>0
8
-1.5
1.0 0.0 -1.0
6
0.5 0.0
4
-10
-0.5
10
0.1
4
0.1
0.0 -0.1
-1.0 1.5
0.0 -1.5
-0.5
12
-1.5
(a) -10
E>0 -5
0.0
2
0
5
-0.1
10
(b)
2
-1.0
-0.5
x/d
z/d
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 ∼ 100Eth 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 θ0 is equal to 0.01 (∼ 0.57◦ ).
relaxation time τR = 12 (∼ 72 ms). Physically, this means that for the values of the angle α = 1.57 and the amplitude θ0 = 0.01 (∼ 0.57◦ ), the balance among the electric, elastic, and viscous torques 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 τR =12 (∼72 ms). So, only at the values of the angle α = 1.57 and the amplitude θ0 = 0.01 (∼ 0.57◦ ) do the optimal dimensionless wavelengths qx and qz provide the minimal values of the total energy [9] W = Welast + We . Here ∫ [( )( )] 2 ON 2 ON 2 ON ON Welast = dxdz (θeq,x ) + (θeq,z ) sin2 θeq + K31 cos2 θeq δ1 ∫
ON ON ON dxdz (K31 − 1) sin 2θeq θeq,x θeq,z ,
+
(7)
is the elastic contribution to the total energy, whereas 2We = −
∫
−δ2
dxdzE ∫
2
(
)
(
ON ON θeq cos2 θeq (x, z) − α
(
)
(
)
)
ON ON ON dxdzE θeq sin α sin 2θeq θeq,z ,
(8)
ON is the electric contribution to the total energy. In our case, the equilibrium θeq (x, z) value
of the angle θ(x, z, τ = 12) 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, 8
α = 1.57, and θ0 =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 decrease 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 scaled ST components σij (x, z, τ ) =
(
4d2 ϵ0 ϵa U 2
)
σ ij (x, z, τ ) (i, j = x, z). The relaxation of the ST components
σxx (x, z = 0, τ ), σxz (x, z = 0, τ ), σzx (x, z = 0, τ ), and σzz (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) are shown in Fig.3. Figs.3(a), (b) and (d) show that the scaled components σzx (τ ), σxz (τ ), and σxx (τ ) are characterized by an increase of |σij | (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 |σij | down to zero, within the last stage of the relaxation process. That situation holds for all abovementioned points. The scaled normal component σzz (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 (E ∼ 100Eth )), a highest value of σzz approximately five times bigger than a highest value of the rest components σij (i = x, j = x, z). Calculations of the shear ST components σzx and σxz 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 σzx , 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 σzx , 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 9
1.5 (c)
(a)
0.2
1
2
1.0
3
0.5
6
4
-0.2
5
0
5
5
10
15
0.0
20
(b)
0.2
0.2
6
5
4
4
6
0
5
(d)
15
20
10
15
20
2
0.0
0.0
5
3
1
-0.2 0
FIG. 3:
10 3
1
xx
xz
2 1
zz
zx
3
0.0
5
10
15
20
4
6
-0.2
2
0
5
Plot showing the relaxation of the scaled ST components σzx (x, z = 0, τ ) (a),
σxz (x, z = 0, τ ) (b), σzz (x, z = 0, τ ) (c), and σxx (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 ∼ 100Eth ) process.
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 σzx and σxz , 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 suggest 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. 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 (1), for the case of E = 0, and with the boundary condition Eq.(3) for the angle θ. The initial condition is taken in the form ON θ(x, z, 0) = θeq (x, z),
(9)
ON where θeq is defined as an equilibrium distribution of the director over the LC film ob-
10
1.5
1.5 40
-10
-5
0
5
10
(x=0,z, )
0.0 35
-1.5 -10
-5
0
5
10
1.5 0.0
30
-10
-5
0
5
-0.5
0.0
0.5
1.0
-1.0
-0.5
0.0
0.5
1.0
-1.0
-0.5
0.0
0.5
1.0
1.5 0.0 26
-1.5 -10
-5
-10
-5
1.5
0
5
-1.0
-0.5
0.0
0.5
1.0
10
-1.0
-0.5
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.0
5
0.0
0.0
(a)
22
0
5
24
-1.5
10 1.5
-5
26
-1.5
1.5
-10
30
1.5
0.0 24
-1.5 0
35
1.5
1.5
E=0
0.0
0.0 -1.5
-1.5
10
(x=0,z, ),
(x,z=0, )
-1.0 1.5
-1.5
(x,z=0, )
40
-1.5
1.5
-1.5
E=0
0.0
0.0 -1.5
-1.5
(b) -1.0
10
22
-0.5
z/d
x/d
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.
tained 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 Fig.4(a) and (b), dotted curves), especially in the vicinity of the lattice points. In turn, the calculations of the angular velocity (see Fig.4(a) and (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 decrease 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 . Thus, the electric field is removed, a long and slow reorientation of the director field produces a negligible angular velocity field, and, as a result, the director reorientation time τoff increases by several orders of magnitude compared with τon . C. Turn − on process in the negative sense When the strong electric field E < 0 (E ∼ 100Eth ) is abruptly applied again but in the ˆ OFF to negative sense at the angle α ∼ − π2 , the director moves from being parallel to the n ˆ OFF is the final orientation of the director after the being parallel to the electric field. Here n 11
1
1
0 8
-1 -10
-5
0
5
10
-10
6
-10
-5
0
5
10
1 4
E<0
-1 -10
-5
0
5
1
10
2
0
5
10
6
0 -10
-5
0
5
10
4
0 -10
-5
0
5
E<0
1
10 2
0
0 -10
1
-5
1
1
0
(x,z=0, ),
(x,z=0, )
0
(x,z=0, ),
(x,z=0, )
1 -1
8
0
-5
0
5
-10
10
1
(a) 0
0
5
10 0
0
0 -10
-5
(b)
-5
0
5
-10
10
-5
0
5
10
x/d
x/d
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), when the electric field (E ∼ 100Eth ) is applied in the negative sense at the angle 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) (case I) and (b) (case II) 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 switched on the electric field in the negative sense, whereas the second time means the total time after starting of the process.
time term τOFF , 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 (1)-(2) with appropriate boundary Eq.(3) and initial conditions. In that case the initial condition is taken in the form θ(x, z, 0) = θOFF (x, z),
(10)
where θOFF (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 τOFF =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 ≤ τOFF ≤ 248 (0.12 s ≤ 12
tOFF ≤ 1.488 s) (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 θ0 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 confirm of our previous suggestion that there is thereshold value of the amplitude θ0th which provides the nonuniform rotation mode rather than the uniform one, whereas the lower value of θ0th dominate the uniform mode [9]. Our calculation for two cases I and II also shows that during the first 2 (∼12 ms) time term (see Fig.5(a) and (b)) the evolution of the angular velocity field ω ⃗ is characterized ˆ only in the positive sense (antiby maintaining of a simple rotation of the director field n clockwise), whereas after time term 4 (∼24 ms) that process is characterized by the complex ˆ . In the case I, the reorientation of n ˆ is destruction of the simple rotation of the vector n characterized by 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 ˆ only in the positive sense (antiII, the field ω ⃗ is characterized by the simple rotation of n 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 these cases I and II, the angular velocity decrease to zero, practically, after 6 (∼36 ms) time term. It should be noted that the electric field of the same magnitude (E ∼ 100Eth ) but directed in the positive (E > 0 see Fig.2(a)) and negative (E < 0 see Fig.5(a) (case II)) sense leads to the formation of different types of quasi periodic structures. In the first case the welldeveloped quasi-periodic structure is maintained approximately after the dimensionless time term τ ∼ 10, whereas in the second case another the non-perfect pattern is maintained practically two times faster, approximately after the dimensionless time term τ ∼ 6. 13
0.4
(2) x=3.0 (3) x=3.2
0.1
z=0
0.4
3
2
0.2
1
0.0
1
0.0
0
5
10
15
20
0
10
15
(1) x=2.0
3
3
E<0
2
-0.6
(2) x=3.0
1.0
(3) x=3.2 2
(d)
-0.8 0
FIG. 6:
20
1.5
-0.2 -0.4
5 1
(b)
1
zz
xz
xx
zx
3 2
(c)
0.6
(1) x=2.0
0.2
0.0
0.8
(a)
0.3
5
10
15
0.5
20
0
5
10
15
20
Plot showing the relaxation of the scaled ST components σzx (x, z = 0, τ ) (a),
σxz (x, z = 0, τ ) (b), σxx (x, z = 0, τ ) (c), and σzz (x, z = 0, τ ) (d) to their equilibrium values, for a 5CB film between two electrodes, at three scaled distances x of 2.0 (curves (1)), 3.0 (curves (2)), and 3.2 (curves (3)), respectively, for the turn-on (E ∼ 100Eth ) process, when the strong electric field is applied in the negative sense (E < 0 case II).
The relaxation of the ST components σzx (x, z = 0, τ ), σxz (x, z = 0, τ ), σxx (x, z = 0, τ ), and σzz (x, z = 0, τ ) to their equilibrium values, for a 5CB film between two electrodes, at three scaled distances x of 2.0 (curves (1)), 3.0 (curves (2)), and 3.2 (curves (3)), respectively, for the turn-on (E ∼ 100Eth ) process, when the strong electric field is applied in the negative sense (E < 0) (case II) are shown in Fig.6. Figs.6(a), (b) and (c) show that the scaled components σzx (τ ), σxz (τ ), and σxx (τ ) are characterized by an increase of |σij | (i = x, j = x, z) by up to 0.8 (∼17.6 Pa) (for σxz (τ ) component) only within the initial stage of the relaxation process, and a fast decrease in |σij | down to zero, within the last stage of the relaxation process. The scaled normal component σzz (τ ) (Fig.6(d)) increases monotonically after scaled time τ ∼ 3 (∼18 ms) and saturates at the value of ∼1.5 (∼33 Pa). So, we can conclude that the sudden application of a sufficiently strong and orthogonal electric field, both in the positive (E > 0) and negative (E < 0) senses, leads to normal compression of confined LC film.
III.
CONCLUSION
In summary, we have numerically investigated the peculiarities in the director reorientation and relaxation of the stress tensor components to their equilibrium values during both 14
the turn-on and turn-off processes in confined nematic phase. Analysis of the numerical results for the turn-on process, when a strong electric field E is suddenly applied in the positive sense (), provides the evidence for the appearance of the spatially periodic patterns in confined 5CB LC film. 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. Physically, this means that only the balance among the above-mentioned forces is responsible for maintaining the nonperfect periodic patterns in thick LC samples. The stress tensor (ST) relaxation following the sudden application of a sufficiently strong and orthogonal electric field allows us to conclude that there are the threshold values of the shear ST components σzx and σxz , 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 suggest 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. It should be noted that in our calculations the magnitude of the voltage applied across the LC cell was equal to ∼ 200 V . This value is large in comparison with those used in display devices (∼ 3 − 5 V ), but given the size of cells, the magnitude of the electric field in both cases is equal to ∼ 1 V /µm. 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.
15
[1] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd ed. (Oxford University Press, Oxford, 1995). [2] A. Sugimura and G. R. Luckhurst, Prog. Nucl. Mag. Res. Spectr. 94-95 (2016) 37. [3] E. Guyon, R. Meyer, and J. Salan, Mol. Cryst. Liq. Cryst. 54 (1979) 261. [4] F. Lonberg, S. Fraden, A. J. Hurd, and R. B. Meyer, Phys. Rev. Lett. 52 (1984) 1903. [5] G. Srajer, S. Fraden, and R. B. Meyer, Phys. Rev. A39 (1989) 4828. [6] D. Golovaty, L. K. Gross, S. I. Hariharan, and E. C. Gartland, Jr., J. Math. Anal. Appl. 255 (2001) 391. [7] T. Shioda, B. Wen, and C. Rosenblatt, J. Appl. Phys. 94 (2003) 7502. [8] A. Sugimura and A. V. Zakharov, Phys. Rev. E84 (2011) 021703. [9] A. A. Vakulenko and A. V. Zakharov, Phys. Rev. E88 (2013) 022505. [10] A. Sugimura, A. A. Vakulenko, and A. V. Zakharov, Physics Procedia 14 (2011) 102. [11] G. Napoli and M. Scaraggi, Phys. Rev. E93 (2016) 012701. [12] F. M. Leslie, Arch. Ration. Mech. Anal. 28 (1968) 265. [13] I. S. Berezin and N. P. Zhidkov, Computing Methods, 4th ed. (Clarendon, Oxford, 1965).
16
Highlights
Hydrodynamics of liquid crystals Liquid crystal films on a solid surface Anchoring energy
Graphical Abstract
S0009-2614(17)30615-2 http://dx.doi.org/10.1016/j.cplett.2017.06.058 CPLETT 34920
To appear in:
Chemical Physics Letters
Received Date: Accepted Date:
2 May 2017 29 June 2017
Please cite this article as: A.V. Zakharov, P.V. Maslennikov, Field-induced stress response of nematics encapsulated in microsized volumes, Chemical Physics Letters (2017), doi: http://dx.doi.org/10.1016/j.cplett.2017.06.058
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Field-induced stress response of nematics encapsulated in microsized volumes A. V. Zakharov∗ Saint Petersburg Institute for Machine Sciences, The Russian Academy of Sciences, Saint Petersburg 199178, Russia. P. V. Maslennikov† Immanuel Kant Baltic Federal University, Kaliningrad 236040, Str. Universitetskaya 2, Russia. (Dated: June 27, 2017)
Abstract 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, when a strong electric field E is suddenly applied in the positive sense, provides an evidence for the appearance of the spatially periodic patterns in confined LC film. PACS numbers: 61.30.Cz, 65.40.De
∗
author to whom correspondence should be addressed. Email address:[email protected];
www.ipme.ru †
Email address:[email protected]
1
I.
INTRODUCTION
With the current sustained demand for portable interactive electronic devices with liquid crystal (LC) displays, there is considerable interest in understanding their operation from a theoretical perspective. Robust and accurate mathematical models for prototype LC displays allow simulations of hypothetical devices to be made quickly and at low cost, and this in turn can lead to new and potentially improved designs being identified. Most models used in such theoretical investigations, make several simplifications in order to arrive at a suitably tractable model for simulations. Normally the LC cell is modeled as a layer of nematic liquid crystal (NLC) sandwiched between two parallel bounding plates, across which an electric field E can be applied. Uniform textures of LC cells 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 (hoˆ with respect to director mogeneously) oriented NLC can distort the molecular orientation a ˆ , at a critical threshold field Eth given by [1] Eth = n
π d
√
K1 , ϵ0 ϵa
where d is the film thickness,
K1 is the splay elastic constant, ϵ0 is the absolute dielectric permittivity of free space, and ϵa 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 LC are uniform over the entire sample for E < Eth . When the electric field is switched on with a magnitude E greater than ˆ , in the ”splay” geometry, reorients as a simple monodomain [2]. In the Eth , the director n case when the electric field E ≫ Eth is abruptly applied orthogonal to an initially uniformly ˆ , reorients in order to minimize aligned (homogeneously) nematic LC film, the director, n the free energy, and the LC system is suddenly placed far from equilibrium. It respond 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 sometimes to the appearance of a nonuniform rotation mode rather than the uniform one [3–11]. 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 [3–5, 8]. This approach, based on an eigenvalue analysis,
2
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 [9]. 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 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. It should be noted that the nuclear magnetic resonance spectroscopy is by now also a well-established method for investigating the peculiarities in the dynamics of the director reorientation in confined LCs under crossed electric and magnetic fields [2]. In this work we focus on the geometry where the electric field (E ≫ Eth ) is orthogonal (or approximately orthogonal) to the horizontal boundaries (see Fig.1(a)). 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 sufficiently strong and orthogonal electric field manifests itself by the growing of one particular Fourier mode. After switched 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(b)). 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 shows 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.
3
FIG. 1: (a) The geometry used for the calculations. The z-axis is normal to the horizontal bounding surfaces, whereas the unit vector ˆi is directed parallel to the horizontal surfaces. The electric field ˆ , are in the xz- plane. The director makes an angle θ with the E, unit vector ˆi, and director, n x-axis, and the electric field makes an angle α with the unit vector ˆi. (b) The series of a voltage pulses. II.
FORMULATION OF THE RELEVANT EQUATIONS FOR DYNAMICAL RE-
ORIENTATION 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(a)). 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 4
ˆ is a unit vector directed normal to the horizontal surfaces, and ˆj = k ˆ × ˆi. We can surfaces, k ˆ = cos θ(x, z, t)ˆi + sin θ(x, z, t)k ˆ ˆ = nxˆi + nz k suppose that the components of the director, n (see Fig.1(a)) depend only 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 [10]. 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 quasi-two-dimensional LC system the dimensionless torque balance equation describing ˆ to its equilibrium orientation n ˆ eq can be written as (for details, see the reorientation of n the Ref.[9]) [(
)
(
)
]
sin2 θ + K31 cos2 θ θ,xx + cos2 θ + K31 sin2 θ θ,zz + [ ( )] 1 2 2 δ1 (K31 − 1) sin 2θθ,xz + (1 − K31 ) sin 2θ θ,x − θ,z + 2 1 2 δ1 (K31 − 1) cos 2θθ,x θ,z + E (θ) sin 2 (α − θ) − δ2 E ,z sin α sin 2θ, 2 θ,τ = δ1
where x = x/d and z = z/d are the dimensionless space variables, τ = dimensionless time, δ1 =
4K1 , ϵ0 ϵa V 2
δ2 =
e1 +e3 , ϵ0 ϵa V
and K31 =
K3 K1
ϵ0 ϵa γ1
(1) (
V 2d
)2
t is the
are three parameters of the
system. Here γ1 is the rotational viscosity coefficient, K3 is the bend elastic constant, ˆ= whereas e1 and e3 are the flexoelectric constants. Here the electric field E = Exˆi + Ez k ˆ makes the angle α with the horizontal surfaces, and the values of E (z) cos αˆi + E (z) sin αk 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 [9] ∂ ∂z
[(
)
]
ϵ⊥ + sin2 θ E(z) sin α + δ2 θ,z sin 2θ = 0, ϵa ∫
1= where E(z) =
2dE(z) , V
1
−1
E(z)dz,
(2)
and V is the voltage applied across the 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.(1)-(2) with the strong anchoring condition for the angle θ, 5
which read in the dimensionless form as θ (−10 < x < 10, z = ±1) = 0, θ (x = ±10, −1 < z < 1) = 0.
(3)
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),
(4)
which defines the thermal fluctuations of the director over the LC sample with amplitude θ0 and wavelengths qx =
πd (2k 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 ⃗ ˆ as of the director field n ˆ ×n ˆ˙ = −θ˙ (x, z, τ ) ˆj = −ω (x, z, τ ) ˆj, ω ⃗ =n
(5)
both during the turn-on and turn-off processes, as well as the dimensionless stress tensor (ST) σij components ) 1( 2 E − A sin 2θ, 4( ) 1 ϵ⊥ σzz = σxx + B sin 2θ + + sin2 θ E 2 , 2 ϵa ( ) 1 σxz = −E 2 + A sin 2θ, 4 ) 1( 2 σzx = E + A sin 2θ, 4
σxx =
where A =
γ2 γ1
cos 2θE 2 and B =
γ2 γ1
(6)
sin 2θE 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. 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 , the value of the voltage across the nematic film, which is 2d = 194.7 µm, was chosen to be U = 200 V , the geometry factor is L = 10d, and the set of δ-parameters, which is involved in Eqs.(1) and (2) takes the values [8] δ1 = 8.6 × 10−6 and δ2 = −0.0009. Using the fact that δ1 ≪ 1, the elastic contribution to the total ST can 6
vis be neglected, whereas the viscous contribution to σij can be written as [12] σxx = − 14 B sin 2θ, vis σxz =
1 4
vis (−E 2 + A) sin 2θ, σzx =
1 4
vis vis (E 2 + A) sin 2θ, and σzz = −σxx . In our case, when the
electric field E is applied across to the LC sample, the electric contribution to the total ST el tensor σijel = 12 (Ei Dj + Di Ej ) has only one σzz =
(
ϵ⊥ ϵa
)
+ 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 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 (E ∼ 100Eth ) 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 (1)-(2) with appropriate boundary (Eq.(3)) and initial (Eq.(4)) conditions. These equations has been solved by the numerical (
)
relaxation method [13]. The relaxation criterion ϵ = | 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 θ0th 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 θ0th dominate 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 θ0 more than 0.01 (∼ 0.57◦ ), whereas for the lower values of θ0 the certain balance of the torques provides only the uniform rotation mode. The evolution both of 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 =
ϵ0 ϵa γ1
(
V 2d
)2
t) is accounted for after switched on the electric
field. In that case, for the values of θ0 = 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 welldeveloped periodic structure with the lattice points at x = ±1.96 and ±5.80 and with the 7
1.5
1.5
0.0
0.0
12
-1.5 -10
-5
0
5
10
1.5
0
5
(x=0,z, )
-5
10
1.5 0.0
8
-1.5 -10
-5
0
5
1.0 0.0 -1.0
10
(x=0,z, ),
(x,z=0, x,z=0, )
-0.5
0.0
0.5
1.0
-1.0
-0.5
0.0
0.5
1.0
-1.0
-0.5
0.0
0.5
1.0
-1.0
-0.5
0.0
0.5
1.0
-1.0
-0.5
0.0
0.5
1.0
0.0
0.5
1.0
0.0
10
-10
6
-10
-5
0
5
10
0.5 0.0 -5
0
5
10
-1.5 1.5 0.0
E>0
8
-1.5
1.0 0.0 -1.0
6
0.5 0.0
4
-10
-0.5
10
0.1
4
0.1
0.0 -0.1
-1.0 1.5
0.0 -1.5
-0.5
12
-1.5
(a) -10
E>0 -5
0.0
2
0
5
-0.1
10
(b)
2
-1.0
-0.5
x/d
z/d
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 ∼ 100Eth 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 θ0 is equal to 0.01 (∼ 0.57◦ ).
relaxation time τR = 12 (∼ 72 ms). Physically, this means that for the values of the angle α = 1.57 and the amplitude θ0 = 0.01 (∼ 0.57◦ ), the balance among the electric, elastic, and viscous torques 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 τR =12 (∼72 ms). So, only at the values of the angle α = 1.57 and the amplitude θ0 = 0.01 (∼ 0.57◦ ) do the optimal dimensionless wavelengths qx and qz provide the minimal values of the total energy [9] W = Welast + We . Here ∫ [( )( )] 2 ON 2 ON 2 ON ON Welast = dxdz (θeq,x ) + (θeq,z ) sin2 θeq + K31 cos2 θeq δ1 ∫
ON ON ON dxdz (K31 − 1) sin 2θeq θeq,x θeq,z ,
+
(7)
is the elastic contribution to the total energy, whereas 2We = −
∫
−δ2
dxdzE ∫
2
(
)
(
ON ON θeq cos2 θeq (x, z) − α
(
)
(
)
)
ON ON ON dxdzE θeq sin α sin 2θeq θeq,z ,
(8)
ON is the electric contribution to the total energy. In our case, the equilibrium θeq (x, z) value
of the angle θ(x, z, τ = 12) 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, 8
α = 1.57, and θ0 =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 decrease 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 scaled ST components σij (x, z, τ ) =
(
4d2 ϵ0 ϵa U 2
)
σ ij (x, z, τ ) (i, j = x, z). The relaxation of the ST components
σxx (x, z = 0, τ ), σxz (x, z = 0, τ ), σzx (x, z = 0, τ ), and σzz (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) are shown in Fig.3. Figs.3(a), (b) and (d) show that the scaled components σzx (τ ), σxz (τ ), and σxx (τ ) are characterized by an increase of |σij | (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 |σij | down to zero, within the last stage of the relaxation process. That situation holds for all abovementioned points. The scaled normal component σzz (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 (E ∼ 100Eth )), a highest value of σzz approximately five times bigger than a highest value of the rest components σij (i = x, j = x, z). Calculations of the shear ST components σzx and σxz 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 σzx , 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 σzx , 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 9
1.5 (c)
(a)
0.2
1
2
1.0
3
0.5
6
4
-0.2
5
0
5
5
10
15
0.0
20
(b)
0.2
0.2
6
5
4
4
6
0
5
(d)
15
20
10
15
20
2
0.0
0.0
5
3
1
-0.2 0
FIG. 3:
10 3
1
xx
xz
2 1
zz
zx
3
0.0
5
10
15
20
4
6
-0.2
2
0
5
Plot showing the relaxation of the scaled ST components σzx (x, z = 0, τ ) (a),
σxz (x, z = 0, τ ) (b), σzz (x, z = 0, τ ) (c), and σxx (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 ∼ 100Eth ) process.
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 σzx and σxz , 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 suggest 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. 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 (1), for the case of E = 0, and with the boundary condition Eq.(3) for the angle θ. The initial condition is taken in the form ON θ(x, z, 0) = θeq (x, z),
(9)
ON where θeq is defined as an equilibrium distribution of the director over the LC film ob-
10
1.5
1.5 40
-10
-5
0
5
10
(x=0,z, )
0.0 35
-1.5 -10
-5
0
5
10
1.5 0.0
30
-10
-5
0
5
-0.5
0.0
0.5
1.0
-1.0
-0.5
0.0
0.5
1.0
-1.0
-0.5
0.0
0.5
1.0
1.5 0.0 26
-1.5 -10
-5
-10
-5
1.5
0
5
-1.0
-0.5
0.0
0.5
1.0
10
-1.0
-0.5
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.0
5
0.0
0.0
(a)
22
0
5
24
-1.5
10 1.5
-5
26
-1.5
1.5
-10
30
1.5
0.0 24
-1.5 0
35
1.5
1.5
E=0
0.0
0.0 -1.5
-1.5
10
(x=0,z, ),
(x,z=0, )
-1.0 1.5
-1.5
(x,z=0, )
40
-1.5
1.5
-1.5
E=0
0.0
0.0 -1.5
-1.5
(b) -1.0
10
22
-0.5
z/d
x/d
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.
tained 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 Fig.4(a) and (b), dotted curves), especially in the vicinity of the lattice points. In turn, the calculations of the angular velocity (see Fig.4(a) and (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 decrease 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 . Thus, the electric field is removed, a long and slow reorientation of the director field produces a negligible angular velocity field, and, as a result, the director reorientation time τoff increases by several orders of magnitude compared with τon . C. Turn − on process in the negative sense When the strong electric field E < 0 (E ∼ 100Eth ) is abruptly applied again but in the ˆ OFF to negative sense at the angle α ∼ − π2 , the director moves from being parallel to the n ˆ OFF is the final orientation of the director after the being parallel to the electric field. Here n 11
1
1
0 8
-1 -10
-5
0
5
10
-10
6
-10
-5
0
5
10
1 4
E<0
-1 -10
-5
0
5
1
10
2
0
5
10
6
0 -10
-5
0
5
10
4
0 -10
-5
0
5
E<0
1
10 2
0
0 -10
1
-5
1
1
0
(x,z=0, ),
(x,z=0, )
0
(x,z=0, ),
(x,z=0, )
1 -1
8
0
-5
0
5
-10
10
1
(a) 0
0
5
10 0
0
0 -10
-5
(b)
-5
0
5
-10
10
-5
0
5
10
x/d
x/d
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), when the electric field (E ∼ 100Eth ) is applied in the negative sense at the angle 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) (case I) and (b) (case II) 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 switched on the electric field in the negative sense, whereas the second time means the total time after starting of the process.
time term τOFF , 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 (1)-(2) with appropriate boundary Eq.(3) and initial conditions. In that case the initial condition is taken in the form θ(x, z, 0) = θOFF (x, z),
(10)
where θOFF (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 τOFF =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 ≤ τOFF ≤ 248 (0.12 s ≤ 12
tOFF ≤ 1.488 s) (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 θ0 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 confirm of our previous suggestion that there is thereshold value of the amplitude θ0th which provides the nonuniform rotation mode rather than the uniform one, whereas the lower value of θ0th dominate the uniform mode [9]. Our calculation for two cases I and II also shows that during the first 2 (∼12 ms) time term (see Fig.5(a) and (b)) the evolution of the angular velocity field ω ⃗ is characterized ˆ only in the positive sense (antiby maintaining of a simple rotation of the director field n clockwise), whereas after time term 4 (∼24 ms) that process is characterized by the complex ˆ . In the case I, the reorientation of n ˆ is destruction of the simple rotation of the vector n characterized by 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 ˆ only in the positive sense (antiII, the field ω ⃗ is characterized by the simple rotation of n 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 these cases I and II, the angular velocity decrease to zero, practically, after 6 (∼36 ms) time term. It should be noted that the electric field of the same magnitude (E ∼ 100Eth ) but directed in the positive (E > 0 see Fig.2(a)) and negative (E < 0 see Fig.5(a) (case II)) sense leads to the formation of different types of quasi periodic structures. In the first case the welldeveloped quasi-periodic structure is maintained approximately after the dimensionless time term τ ∼ 10, whereas in the second case another the non-perfect pattern is maintained practically two times faster, approximately after the dimensionless time term τ ∼ 6. 13
0.4
(2) x=3.0 (3) x=3.2
0.1
z=0
0.4
3
2
0.2
1
0.0
1
0.0
0
5
10
15
20
0
10
15
(1) x=2.0
3
3
E<0
2
-0.6
(2) x=3.0
1.0
(3) x=3.2 2
(d)
-0.8 0
FIG. 6:
20
1.5
-0.2 -0.4
5 1
(b)
1
zz
xz
xx
zx
3 2
(c)
0.6
(1) x=2.0
0.2
0.0
0.8
(a)
0.3
5
10
15
0.5
20
0
5
10
15
20
Plot showing the relaxation of the scaled ST components σzx (x, z = 0, τ ) (a),
σxz (x, z = 0, τ ) (b), σxx (x, z = 0, τ ) (c), and σzz (x, z = 0, τ ) (d) to their equilibrium values, for a 5CB film between two electrodes, at three scaled distances x of 2.0 (curves (1)), 3.0 (curves (2)), and 3.2 (curves (3)), respectively, for the turn-on (E ∼ 100Eth ) process, when the strong electric field is applied in the negative sense (E < 0 case II).
The relaxation of the ST components σzx (x, z = 0, τ ), σxz (x, z = 0, τ ), σxx (x, z = 0, τ ), and σzz (x, z = 0, τ ) to their equilibrium values, for a 5CB film between two electrodes, at three scaled distances x of 2.0 (curves (1)), 3.0 (curves (2)), and 3.2 (curves (3)), respectively, for the turn-on (E ∼ 100Eth ) process, when the strong electric field is applied in the negative sense (E < 0) (case II) are shown in Fig.6. Figs.6(a), (b) and (c) show that the scaled components σzx (τ ), σxz (τ ), and σxx (τ ) are characterized by an increase of |σij | (i = x, j = x, z) by up to 0.8 (∼17.6 Pa) (for σxz (τ ) component) only within the initial stage of the relaxation process, and a fast decrease in |σij | down to zero, within the last stage of the relaxation process. The scaled normal component σzz (τ ) (Fig.6(d)) increases monotonically after scaled time τ ∼ 3 (∼18 ms) and saturates at the value of ∼1.5 (∼33 Pa). So, we can conclude that the sudden application of a sufficiently strong and orthogonal electric field, both in the positive (E > 0) and negative (E < 0) senses, leads to normal compression of confined LC film.
III.
CONCLUSION
In summary, we have numerically investigated the peculiarities in the director reorientation and relaxation of the stress tensor components to their equilibrium values during both 14
the turn-on and turn-off processes in confined nematic phase. Analysis of the numerical results for the turn-on process, when a strong electric field E is suddenly applied in the positive sense (), provides the evidence for the appearance of the spatially periodic patterns in confined 5CB LC film. 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. Physically, this means that only the balance among the above-mentioned forces is responsible for maintaining the nonperfect periodic patterns in thick LC samples. The stress tensor (ST) relaxation following the sudden application of a sufficiently strong and orthogonal electric field allows us to conclude that there are the threshold values of the shear ST components σzx and σxz , 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 suggest 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. It should be noted that in our calculations the magnitude of the voltage applied across the LC cell was equal to ∼ 200 V . This value is large in comparison with those used in display devices (∼ 3 − 5 V ), but given the size of cells, the magnitude of the electric field in both cases is equal to ∼ 1 V /µm. 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.
15
[1] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd ed. (Oxford University Press, Oxford, 1995). [2] A. Sugimura and G. R. Luckhurst, Prog. Nucl. Mag. Res. Spectr. 94-95 (2016) 37. [3] E. Guyon, R. Meyer, and J. Salan, Mol. Cryst. Liq. Cryst. 54 (1979) 261. [4] F. Lonberg, S. Fraden, A. J. Hurd, and R. B. Meyer, Phys. Rev. Lett. 52 (1984) 1903. [5] G. Srajer, S. Fraden, and R. B. Meyer, Phys. Rev. A39 (1989) 4828. [6] D. Golovaty, L. K. Gross, S. I. Hariharan, and E. C. Gartland, Jr., J. Math. Anal. Appl. 255 (2001) 391. [7] T. Shioda, B. Wen, and C. Rosenblatt, J. Appl. Phys. 94 (2003) 7502. [8] A. Sugimura and A. V. Zakharov, Phys. Rev. E84 (2011) 021703. [9] A. A. Vakulenko and A. V. Zakharov, Phys. Rev. E88 (2013) 022505. [10] A. Sugimura, A. A. Vakulenko, and A. V. Zakharov, Physics Procedia 14 (2011) 102. [11] G. Napoli and M. Scaraggi, Phys. Rev. E93 (2016) 012701. [12] F. M. Leslie, Arch. Ration. Mech. Anal. 28 (1968) 265. [13] I. S. Berezin and N. P. Zhidkov, Computing Methods, 4th ed. (Clarendon, Oxford, 1965).
16
Highlights
Hydrodynamics of liquid crystals Liquid crystal films on a solid surface Anchoring energy
Graphical Abstract