Investigation of Liquid Crystal Ripple Using Ericksen-Leslie Theory for Displays Subject to Tactile Force

(1)

Research Article

Investigation of Liquid Crystal Ripple Using Ericksen-Leslie

Theory for Displays Subject to Tactile Force

Y. J. Lee,

1

T. S. Liu,

1

Mao-Hsing Lin,

2

and Kun-Feng Huang

2

1_{Department of Mechanical Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan} 2_{Innolux Corporation, Tainan 74147, Taiwan}

Correspondence should be addressed to T. S. Liu; [email protected] Received 25 June 2013; Accepted 7 December 2013

Academic Editor: Farzad Khani

Copyright © 2013 Y. J. Lee et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Liquid crystal display panels subjected to tactile force will show ripple propagation on screens. Tactile forces change tilt angles of liquid crystal molecules and alter optical transmission so as to generate ripple on screens. Based on the Ericksen-Leslie theory, this study investigates ripple propagation by dealing with tilt angles of liquid crystal molecules. Tactile force effects are taken into account to derive the molecule equation of motion for liquid crystals. Analytical results show that viscosity, tactile force, the thickness of cell gap, and Leslie viscosity coefficient lead to tilt angle variation. Tilt angle variations of PAA liquid crystal molecules are sensitive to tactile force magnitudes, while those of 5CB and MBBA with larger viscosity are not. Analytical derivation is validated by numerical results.

1. Introduction

Many physical phenomena exhibited by the nematic phase liquid crystals (LC), such as unusual flow properties or the LC response to electric and magnetic fields, can be studied by treating LC as a continuous medium. Ericksen and Leslie

[1–3] formulated general conservation laws and constitutive

equations describing the dynamic behavior. Other contin-uum theories have been proposed, but it turns out that the Ericksen-Leslie theory is the one that is most widely used in discussing the nematic state. Based on the continuum theory of Ericksen-Leslie, this study constructs a theoretical model in order to investigate the tilt angle variation of LC subjected to both electric field and tactile force. The continuum theory for the nematic state flow is established in two dynamical equations—conservation laws and constitutive equations. Both equations are coupled with each other due to the properties of flow and the director of liquid crystal molecules.

Brochard et al. [4] investigated transient distortions in

a nematic film by increasing or decreasing of the magnetic

field. Lei et al. [5] used the Ericksen-Leslie equation to

describe soliton propagation in nematic liquid crystals under

shear. Leslie [6] presented a concise but clear derivation

of continuum equations commonly employed to describe

static and dynamic phenomena in nematic liquid crystals.

Gleeson et al. [7] proposed a two-dimensional theory in

which the excitations are fronts between distinct solutions of the steady-state Ericksen-Leslie equations. Lin and Liu

[8] use the Ericksen-Leslie equation to describe the flow

of liquid crystal material and prove the global existence of

weak solutions. De Andrade Lima and Rey [9] proposed

computational modeling of the steady capillary Poiseuille flow of flow-aligning discotic nematic liquid crystals; using the Ericksen-Leslie equations predicts solution multiplicity

and multistability. Nie et al. [10] used a cell gap and surface

dynamic method to derive analytical expressions of LC response time under finite anchoring energy conditions. Cruz

et al. [11] used a finite difference technique, based on a

projec-tion method, which is developed for solving the dynamic three-dimensional Ericksen-Leslie equations for liquid crys-tals subject to a strong magnetic field.

Liquid crystal display panels subjected to tactile force will show ripple propagation on screens. Tactile forces change tilt angles of liquid crystal molecules and alter optical transmission so as to generate ripple on screens. Based on the Ericksen-Leslie theory, this paper aims to develop LC dynamics subject to tactile force. Tactile force effects are taken into account to derive the molecule director equation

(2)

z F x 𝜙 𝜃 + + + + + + + + + + + ̂n z = 0

Figure 1: Homeotropic alignment LC layer sandwiched between parallel substrates, where𝑧 = 0 and 𝑧 = 𝑑 are the bottom and top substrates, respectively. Poiseuille flow Homeotropic alignment x d 2 −d 2 −ym ym f(y) 𝜃E 𝜃F 𝜃m 𝜙 y

Figure 2: Directors of LC molecules with homeotropic alignment are changed by Poiseuille flow.

of motion for liquid crystals. A theoretical model is thus proposed to analyze the tilt angle of nematic liquid crystals subject to tactile forces.

2. Derivation of Ericksen-Leslie Theory

2.1. One-Dimensional Model. Figure 1shows a homeotropic

alignment liquid crystal layer sandwiched between parallel

substrates, where𝑧 = 0 and 𝑧 = 𝑑 stand for the bottom

and top substrates, respectively [12]. The thickness between

bottom and top substrates is𝑑. The 𝑧-axis is normal to the

plane of the substrates, and the electric field𝐸 = 8.9 V is along

the𝑧-axis. 𝜃 denotes a tilt angle caused by a tactile force and 𝜙

is the tilt angle defined as the angle between the𝑧-axis and LC

directors. The dynamics of director-axis rotation is described

by an Ericksen-Leslie equation [13,14]: 𝜕 𝜕𝑧[(𝐾11cos2𝜃 + 𝐾33sin2𝜃) 𝜕𝜃 𝜕𝑧] + (𝐾₃₃− 𝐾₁₁) sin 𝜃 cos 𝜃(𝜕𝜃_𝜕𝑧)2 + (𝛼2sin2𝜃 − 𝛼3cos2𝜃)_𝜕𝑧𝜕V + 𝜀₀𝜀_𝑎𝐸2sin𝜃 cos 𝜃 = 𝛾𝜕𝜃 𝜕𝑡 + 𝑀 𝜕2_𝜃 𝜕𝑡2, (1) 0 1 2 3 4 0 0.02 0.04 0.06 0.08 0.1 T il t a n gle (deg) Cell gap (𝜇m) 1 dyne 0.5 dyne 0.1 dyne (a) 0 1 2 3 4 5 6 0 0.02 0.04 0.06 0.08 0.1 T il t a n gle (deg) Cell gap (𝜇m) 1 dyne 0.5 dyne 0.1 dyne (b)

Figure 3: Comparison of molecule tilt angles between cell gap thicknesses of (a) 3𝜇m and (b) 5 𝜇m when a tactile force is applied to LCD screen with 5CB liquid crystals.

(3)

0 1 2 3 4 0 0.01 0.02 0.03 0.04 T il t a n gle (deg) Cell gap (𝜇m) 1 dyne 0.5 dyne 0.1 dyne (a) 0 1 2 3 4 5 6 0 0.01 0.02 0.03 0.04 0.05 T il t a n gle (deg) Cell gap (𝜇m) 1 dyne 0.5 dyne 0.1 dyne (b)

Figure 4: Comparison of molecule tilt angles between cell gap thicknesses of (a) 3𝜇m and (b) 5 𝜇m when a tactile force is applied to LCD screen with MBBA liquid crystals.

where𝐾₁₁,𝐾₂₂, and𝐾₃₃are elastic constants,𝛼₂and𝛼₃are

Leslie viscosities,V is the flow velocity, 𝛾 = 𝛼₃ − 𝛼₂ is the

rotational viscosity, and𝑀 is the inertia of LC directors. If

backflows, inertial effects, and electric field energy densities

are ignored, (1) is reduced to

𝛾𝜕𝜃

𝜕𝑡 = 𝐾

𝜕2_𝜃

𝜕𝑧2 + 𝐹 × ̂𝑛, (2)

where𝐹 is the tactile force, 𝐾 is the one-constant

approx-imation with 𝐾 = 𝐾₁₁ = 𝐾₂₂ = 𝐾₃₃, and ̂𝑛 is the LC

director. Equation (2) is a torque balance equation relating

the viscosity, the elasticity, and the torque due to tactile force.

A cross product𝐹 × ̂𝑛 represents a torque expressed by

𝐹 × ̂𝑛 = |𝐹| |𝑛| sin (𝜃 − 𝜙) , (3) 0 1 2 3 4 0 0.4 0.8 1.2 T il t a n gle (deg) Cell gap (𝜇m) 1 dyne 0.5 dyne 0.1 dyne (a) 0 1 2 3 4 5 6 0 0.4 0.8 1.2 1.6 T il t a n gle (deg) Cell gap (𝜇m) 1 dyne 0.5 dyne 0.1 dyne (b)

Figure 5: Comparison of molecule tilt angles among cell gap thicknesses of (a) 3𝜇m and (b) 5 𝜇m when a tactile force is applied to LCD screen with PAA liquid crystals.

where|𝑛| = 1. Substituting (3) into (2) gives

𝛾𝜕𝜃

𝜕𝑡 = 𝐾

𝜕2_𝜃

𝜕𝑧2 + 𝐹 sin (𝜃 − 𝜙) . (4)

Liquid crystal dynamics is described in this study by consid-ering the following two cases.

Case 1. When torque due to tactile force is much larger than elastic torque,

󵄨󵄨󵄨󵄨𝐹sin(𝜃 − 𝜙)󵄨󵄨󵄨󵄨 ≫󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨󵄨}_󵄨𝐾𝜕_𝜕𝑧2𝜃₂󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨󵄨}

(4)

0 1 2 3 4 0 0.01 0.02 0.03 T il t a n gle (deg) Cell gap (𝜇m) 1 dyne 0.5 dyne 0.1 dyne (a) 0 1 2 3 4 0 0.01 0.02 T il t a n gle (deg) Cell gap (𝜇m) 1 dyne 0.5 dyne 0.1 dyne (b) 0 1 2 3 4 0 0.1 0.2 0.3 0.4 0.5 0.6 T il t a n gle (deg) Cell gap (𝜇m) 1 dyne 0.5 dyne 0.1 dyne (c)

Figure 6: Comparison of molecule tilt angles with different LC materials (a) 5CB, (b) MBBA, and (c) PAA when a tactile force is applied to LCD screen.

However, when the tilt angle is small, sin𝜃 ≈ 𝜃 and cos 𝜃 ≈ 1

and (4) becomes

𝛾𝜕𝜃_𝜕𝑡 = 𝐹 ⋅ (cos 𝜙) ⋅ 𝜃 − 𝐹 ⋅ sin 𝜙. (6)

Case 2. When torque due to tactile force is much smaller than elastic torque,

󵄨󵄨󵄨󵄨𝐹sin(𝜃 − 𝜙)󵄨󵄨󵄨󵄨 ≪󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨󵄨}_󵄨𝐾𝜕_𝜕𝑧2𝜃₂󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨󵄨}

󵄨. (7)

Equation (4) thus becomes

𝛾𝜕𝜃_𝜕𝑡 = 𝐾𝜕_𝜕𝑧2𝜃₂. (8)

Based on separation of variables, assume 𝜃(𝑡, 𝑧) =

𝜃(𝑡) sin(𝜋𝑧/𝑑), which is in turn substituted into (6) and (8).

Case 1 thus leads to an equation for𝜃 of the form

̇𝜃 = 𝑎 + 𝑏𝜃, (9)

where

𝑎 = −𝐹 sin 𝜙_𝛾 ,

𝑏 = 𝐹 cos 𝜙_𝛾 .

(10)

The solution for𝜃(𝑡, 𝑧) is written as

(5)

0 1 2 3 4 5 0 2 4 6 8 10 T il t a n gle (deg) Cell gap (𝜇m) 10 V 20 V 30 V 40 V 50 V 10 V 20 V 30 V 40 V 50 V (a) 0 1 2 3 4 5 0 2 4 6 8 10 T il t a n gle (deg) Cell gap (𝜇m) 10 V 20 V 30 V 40 V 50 V 10 V 20 V 30 V 40 V 50 V (b) 0 1 2 3 4 5 0 2 4 6 8 10 12 T il t a n gle (deg) Cell gap (𝜇m) 10 V 20 V 30 V 40 V 50 V 10 V 20 V 30 V 40 V 50 V (c)

Figure 7: Comparison of molecule tilt angles between cell gap thicknesses of 3𝜇m and 5 𝜇m when tactile force is applied to LCD screen with (a) 5CB, (b) MBBA, and (c) PAA liquid crystals under different voltages.

In Case 2, which happens when the tactile force is

negligible, the solution for𝜃(𝑡, 𝑧) in (8) is expressed by

𝜃 (𝑡, 𝑧) = 𝑐𝑒−(𝐾𝜋2/𝛾𝑑2)𝑡sin(𝜋𝑧

𝑑 ) , (12)

where 𝑐 is a constant. Since the tactile force is ignored in

Case 2, only Case 1 is further examined.

2.2. Two-Dimensional Model. The above derivation is based

on 1D LC model. Further, this study intends to use (1) to

develop an LC tilt model in two dimensions. Assume that the theoretical model satisfies the following four conditions, as

depicted inFigure 2.

Condition 1. The molecule ignores the moment of inertia;

that is,𝑀 = 0.

Condition 2. Shear rate𝑠 = 𝑠(𝑦) = 𝜕V/𝜕𝑦 = V/(𝑑/3), where V

is the flow velocity and𝑑 is the cell thickness.

Condition 3. The LC sample is homeotropic alignment, and we have the boundary conditions

𝜃 (𝑥, 𝑦, 𝑡) = 𝜃 (𝑥, ±𝑑₂, 𝑡) = 0. (13)

Condition 4. The shearing velocity belongs to Poiseuille flow

(6)

0 1 2 3 4 5 0 2 4 6 8 10 T il t a n gle (deg) Cell gap (𝜇m) 2.5 dyne 3 dyne 3.5 dyne (a) 0 1 2 3 4 5 0 2 4 6 8 10 12 T il t a n gle (deg) Cell gap (𝜇m) 2.5 dyne 3 dyne 3.5 dyne (b) 0 1 2 3 4 5 0 2 4 6 8 10 14 T il t a n gle (deg) Cell gap (𝜇m) 2.5 dyne 3 dyne 3.5 dyne 12 (c)

Figure 8: Comparison of molecule tilt angles between cell gap thicknesses of 3𝜇m and 5 𝜇m when tactile force is applied to LCD screen with (a) 5CB, (b) MBBA, and (c) PAA liquid crystals under different tactile forces.

2.5 dyne 3 dyne 3.5 dyne 0 1 2 3 0 2 4 6 8 10 T il t a n gle (deg) Cell gap (𝜇m) (a) 0 1 2 3 0 2 4 6 8 10 12 Numerical solution Analytical solution T il t a n gle (deg) Cell gap (𝜇m) (b)

Figure 9: (a) Comparison of molecule tilt angles depicted in 3𝜇m cell gap 5CB liquid crystals subject to different forces. (b) Comparison of molecule tilt angles subject to 3.5 dyne between analytical and numerical results.

According to the above assumptions, (1) becomes

𝐾 (𝜕2𝜃 𝜕𝑥2 + 𝜕2_𝜃 𝜕𝑦2) − 𝛾 𝜕𝜃 𝜕𝑡 + 𝑠 2(𝛾1− 𝛾2cos2𝜃) + 𝜀₀𝜀_𝑎𝐸2sin𝜃 cos 𝜃 + 𝐹 × ̂𝑛 = 0. (14)

Based on the Ericksen-Leslie theory, (14) represents the tilt

model of LC molecule subjected to both electric field and tactile force. In order to use separation of variables, assume 𝜃 = 𝜃(𝑥, 𝑦, 𝑡) yields

(7)

0 1 2 3 0 2 4 6 8 T il t a n gle (deg) Cell gap (𝜇m) 4 2.5 dyne 3 dyne 3.5 dyne (a) 0 1 2 3 0 2 4 6 8 10 Numerical solution Analytical solution T il t a n gle (deg) Cell gap (𝜇m) 4 (b)

where 𝑓 (𝑦) = 𝑔 (𝑦) 𝑔 (−𝑦_𝑚) ={{_{_{ { 1 or − 1, 𝑦 = −𝑦𝑚 or𝑦𝑚 𝑔 (𝑦) 𝑔 (−0.28𝑑) = 4.7503 ⋅ 𝑔 (𝑦) , 𝑦 ̸= − 𝑦𝑚 or𝑦𝑚, (16) 𝑔 (𝑦) = 2𝑦_𝑑 − sin (𝑦𝜋_𝑑 ) , (17) 𝑦_𝑚= (𝑑 𝜋) cos−1( 2 𝜋) ≈ 0.28𝑑, (18)

and𝑦_𝑚and−𝑦_𝑚are boundary conditions of function𝑔(𝑦). In

(15),𝜃_𝑚is the angle of the normal vector between the director

of LC molecules with homeotropic alignment. Molecule

0 1 2 3 4 5 0 2 4 6 T il t a n gle (deg) Cell gap (𝜇m) 2.5 dyne 3 dyne 3.5 dyne (a) 0 1 2 3 0 2 4 6 8 Numerical solution Analytical solution T il t a n gle (deg) Cell gap (𝜇m) 4 5 (b)

directions are changed by the Poiseuille flow. Values of the

angle along the𝑦-axis have extreme values at 𝑦_𝑚 and−𝑦_𝑚.

Substituting (15)–(18) into (14) gives

𝐾 [_𝜕𝑥𝜕2₂(𝑓 (𝑦) 𝜃𝑚(𝑥, 𝑡)) + 𝜕

2

𝜕𝑦2(𝑓 (𝑦) 𝜃𝑚(𝑥, 𝑡))]

− 𝛾_𝜕𝑡𝜕 (𝑓 (𝑦) 𝜃_𝑚(𝑥, 𝑡)) +_𝜕𝑦𝜕V(𝛼₃sin2𝜃 − 𝛼₂cos2𝜃)

+ 𝜀₀𝜀_𝑎𝐸2sin𝜃 cos 𝜃 + |𝐹| |𝑛| sin (𝜃 − 𝜙) = 0.

(19)

Since|𝑛| = 1, one has

(8)

0 1 2 3 0 2 4 6 8 10 T il t a n gle (deg) Cell gap (𝜇m) 2.5 dyne 3 dyne 3.5 dyne (a) 0 1 2 3 0 2 4 6 8 10 12 Numerical solution Analytical solution T il t a n gle (deg) Cell gap (𝜇m) (b)

Figure 12: (a) Comparison of molecule tilt angles depicted in 3𝜇m cell gap MBBA liquid crystals subject to different forces. (b) Comparison of molecule tilt angles subject to 3.5 dyne between analytical and numerical results.

Therefore, (19) becomes 𝐾 {[𝑓 (𝑦)𝜕2𝜃𝑚 𝜕𝑥2 + 0] + [𝜃𝑚𝜕 2_{𝑓 (𝑦)} 𝜕𝑦2 + 0]} − 𝛾𝑓 (𝑦) 𝜕𝜃_𝑚 𝜕𝑡 +𝜕V 𝜕𝑦[𝛼3sin2(𝑓 (𝑦) 𝜃𝑚) − 𝛼2cos2(𝑓 (𝑦) 𝜃𝑚)] + 𝜀₀𝜀_𝑎𝐸2sin(𝑓 (𝑦) 𝜃_𝑚) cos (𝑓 (𝑦) 𝜃_𝑚) + 𝐹 sin (𝑓 (𝑦) 𝜃_𝑚− 𝜙) = 0. (21)

As depicted inFigure 2, the director makes an angle𝜃_𝑚with

the𝑦-axis. Consequently, 𝜃_𝑚represents liquid crystal

mole-cules with homeotropic alignment caused by the Poiseuille

flow between director̂𝑛 and normal direction of LC sample.

An effective director equation of motion with the essential

physics preserved may be obtained by setting𝑦 = −𝑦_𝑚 and

𝑦 = 𝑦𝑚in (21). 0 1 2 3 4 0 2 4 6 T il t a n gle (deg) Cell gap (𝜇m) 2.5 dyne 3 dyne 3.5 dyne (a) 0 1 2 3 0 2 4 6 8 Numerical solution Analytical solution T il t a n gle (deg) Cell gap (𝜇m) 4 (b)

On boundary𝑦 = −𝑦_𝑚inFigure 2, 𝐾𝜕2𝜃𝑚 𝜕𝑥2 − 𝛾 𝜕𝜃_𝑚 𝜕𝑡 + 𝜕V 𝜕𝑦(𝛼3sin2𝜃𝑚− 𝛼2cos2𝜃𝑚)

+ 𝜀₀𝜀_𝑎𝐸2sin𝜃_𝑚cos𝜃_𝑚+ 𝐹 sin (𝜃_𝑚− 𝜙) = 0.

(22)

By using trigonometric formulas, one can rewrite (22) as

𝐾𝜕2𝜃𝑚 𝜕𝑥2 − 𝛾 𝜕𝜃_𝑚 𝜕𝑡 + 𝜕V 𝜕𝑦 × [𝛼₃(1 − cos 2𝜃𝑚 2 ) − 𝛼2( 1 + cos 2𝜃_𝑚 2 )] + 𝜀₀𝜀_𝑎𝐸2_𝜃

𝑚+ 𝐹 (sin 𝜃𝑚cos𝜙 − cos 𝜃𝑚sin𝜙) = 0.

(9)

Because sin𝜃_𝑚 ≈ 𝜃_𝑚 and cos𝜃_𝑚 ≈ 1 for small 𝜃_𝑚, (23) is

rewritten as 𝐾𝜕_𝜕𝑥2𝜃₂𝑚 − 𝛾𝜕𝜃_𝜕𝑡𝑚 +_𝜕𝑦𝜕V[𝛼₃(1 − 1₂ ) − 𝛼₂(1 + 1₂ )] + 𝜀₀𝜀_𝑎𝐸2_𝜃 𝑚+ 𝐹 (𝜃𝑚cos𝜙 − sin 𝜙) = 0. (24) Thus, 𝐾𝜕2𝜃𝑚 𝜕𝑥2 − 𝛾 𝜕𝜃_𝑚 𝜕𝑡 − 𝛼2 𝜕V 𝜕𝑦+ 𝜀0𝜀𝑎𝐸2𝜃𝑚 + 𝐹 (𝜃_𝑚cos𝜙 − sin 𝜙) = 0. (25) Hence, 𝛾𝜕𝜃_𝜕𝑡𝑚=𝐾𝜕_𝜕𝑥2𝜃𝑚₂ − 𝛼2_𝜕𝑦𝜕V+ 𝜀0𝜀𝑎𝐸2𝜃𝑚+ 𝐹 (𝜃𝑚cos𝜙 − sin 𝜙). (26) 0 1 2 3 0 2 4 6 8 10 12 T il t a n gle (deg) Cell gap (𝜇m) 2.5 dyne 3 dyne 3.5 dyne (a) 0 1 2 3 0 2 4 6 8 10 12 14 Numerical solution Analytical solution T il t a n gle (deg) Cell gap (𝜇m) (b)

Figure 15: (a) Comparison of molecule tilt angles depicted in 3𝜇m cell gap PAA liquid crystals subject to different forces. (b) Comparison of molecule tilt angles subject to 3.5 dyne between analytical and numerical results.

Substituting𝜃_𝑚 = 𝜃_𝑚(𝑡) sin(𝜋𝑥/𝑑) for separation of variables

into (26) leads to 𝛾 ̇𝜃_𝑚 = 𝐾 𝜕2 𝜕𝑥2[𝜃𝑚(𝑡) sin (𝜋𝑥_𝑑 )] − 𝛼2_𝜕𝑦𝜕V + 𝜀₀𝜀_𝑎𝐸2𝜃_𝑚+ 𝐹 (𝜃_𝑚cos𝜙 − sin 𝜙) . (27) As a result, 𝛾 ̇𝜃_𝑚= −𝐾𝜋2 𝑑2 𝜃𝑚(𝑥, 𝑡) + (𝜀0𝜀𝑎𝐸2+ 𝐹 cos 𝜙) 𝜃𝑚 + (−𝛼₂𝜕V 𝜕𝑦− 𝐹 sin 𝜙) . (28)

(10)

0 1 2 3 0 2 4 6 8 T il t a n gle (deg) Cell gap (𝜇m) 4 2.5 dyne 3 dyne 3.5 dyne (a) 0 1 2 3 0 2 4 6 8 10 Numerical solution Analytical solution T il t a n gle (deg) Cell gap (𝜇m) 4 (b)

Figure 16: (a) Comparison of molecule tilt angles depicted in 4𝜇m cell gap PAA liquid crystals subject to different forces. (b) Comparison of molecule tilt angles subject to 3.5 dyne by analytical and numerical results.

Similarly, we consider the following two cases.

Case 1󸀠 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨−𝛼2𝜕V_𝜕𝑦+ 𝜀0𝜀𝑎𝐸2𝜃𝑚+ 𝐹 (𝜃𝑚cos𝜙 − sin 𝜙)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≫󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨𝐾 𝜕2_𝜃 𝑚 𝜕𝑥2 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨, (29) 𝛾 ̇𝜃_𝑚= (𝜀₀𝜀_𝑎𝐸2_{+ 𝐹 cos 𝜙) 𝜃} 𝑚+ (−𝛼2𝜕V_𝜕𝑦− 𝐹 sin 𝜙) . (30)

Assuming ̇𝜃_𝑚 = 𝑎 + 𝑏𝜃_𝑚, one has

𝑎 = 1 𝛾(−𝛼2 𝜕V 𝜕𝑦− 𝐹 sin 𝜙) , 𝑏 = 1 𝛾(𝜀0𝜀𝑎𝐸2+ 𝐹 cos 𝜙) . (31) 𝜃_𝑚= 𝑎_𝑏(𝑒𝑏𝑡− 1) (32)

is the solution. Substituting (31) into (32) yields

𝜃_𝑚(𝑥, 𝑡) =[−𝛼_𝜀2(𝜕V/𝜕𝑦) − 𝐹 sin 𝜙 0𝜀𝑎𝐸2+ 𝐹 cos 𝜙 (𝑒 ((𝜀0𝜀𝑎𝐸2+𝐹 cos 𝜙)/𝛾)𝑡_{− 1)]sin(}𝜋𝑥 𝑑 ). (33) Consider the following.

Case 2󸀠 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨−𝛼2𝜕V_𝜕𝑦+ 𝜀0𝜀𝑎𝐸2𝜃𝑚+ 𝐹 (𝜃𝑚cos𝜙 − sin 𝜙)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≪󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨𝐾 𝜕2_𝜃 𝑚 𝜕𝑥2 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨. (34)

Since the tactile force and electric field are ignored in Case 2󸀠,

only Case 1󸀠is further examined.

On boundary𝑦 = 𝑦_𝑚inFigure 2, 𝜃_𝑚(𝑥, 𝑡) =[𝛼2(𝜕V/𝜕𝑦) + 𝐹 sin 𝜙 𝜀₀𝜀_𝑎𝐸2_{+ 𝐹 cos 𝜙} (𝑒((𝜀0𝜀𝑎𝐸 2_{+𝐹 cos 𝜙)/𝛾)𝑡} − 1)]sin(𝜋𝑥 𝑑 ) . (35)

Since𝜃_𝑚(𝑥, 𝑡) in (33) is equal to −𝜃_𝑚(𝑥, 𝑡) in (35), the tilt

angles are symmetric.

3. Discussion

Subject to both electric field of 8.9 V and tactile force,

simulation results for Case 1 are depicted in Figures 3, 4,

and 5. Comparing different LC materials including 5CB,

MBBA, and PAA subjected to 10 ms duration tactile forces

of 1 dyne, 0.5 dyne, and 0.1 dyne at 10 ms, Figures3,4, and5

show that a larger force makes the tilt angle larger regardless

of LC gap thickness. Figures6(a), 6(b), and 6(c)compare

3𝜇m thick different LC materials including 5CB, MBBA, and

PAA subjected to 5 ms duration 0.5 dyne force, respectively. The viscosity of PAA is the smallest and the tilt angle is larger than the others. Therefore, the option of liquid crystal material with smaller viscosity can reduce the ripple.

Tilt angle variations are shown under different voltages in

Figure 7and under different tactile forces inFigure 8. Using 10 V, 20 V, 30 V, 40 V, and 50 V under a tactile force of 3 dyne, it is observed that the tilt angles remain the same in Figures

7(a),7(b), and7(c). Thus, different voltages do not affect tilt

angles.

Based on (33), Figures8(a),8(b), and8(c)compare

differ-ent LC materials including 5CB, MBBA, and PAA subject to

2.5 dyne, 3 dyne, and 3.5 dyne forces in 3𝜇m and 5 𝜇m thick

cell gap LC, respectively. Accordingly, tactile forces affect tilt

angles obviously. Figures9(a), 10(a), and11(a), respectively,

(11)

Figure 17: (a) Comparison of molecule tilt angles depicted in 5𝜇m cell gap PAA liquid crystals subject to different forces. (b) Comparison of molecule tilt angles subject to 3.5 dyne between analytical and numerical results.

3.5 dyne forces in 3, 4, and 5𝜇m thick cell gap LC. Similarly,

Figures12(a),13(a), and14(a)are MBBA results and Figures

15(a),16(a), and17(a)are PAA results. In addition, Figures

9(b) to 17(b) compare analytical solutions with numerical solutions of the full coupled equations. The comparison results show that analytical solutions are consistent with numerical solutions of the full coupled equations. Therefore, the analytical derivation is validated.

4. Conclusion

The Ericksen-Leslie theory deals with the dynamics of nematic liquid crystals. Based on the Ericksen-Leslie theory, this paper contributes to LC dynamics subject to tactile force. LC molecule tilt leads to a change in the optical intensity transmitted through a liquid crystal cell. LCD panels subject to tactile force will show ripple-like propagation on screens. The present results show that the viscosity, tactile force,

factors of tilt angle variation. Tilt angle variations of PAA liquid crystal molecules are sensitive to tactile force mag-nitudes, while those of 5CB and MBBA with larger viscosity are not. Analytical derivation has been carried out in this study and analytical results have been validated by numerical results.

Acknowledgments

This work was supported by Chi Mei Optoelectronics Corp. and Innolux Display Corp. in Taiwan.

References

[1] J. Ericksen, “Conservation laws for liquid crystals,” Transactions

of the Society of Rheology, vol. 5, pp. 22–34, 1961.

[2] J. L. Ericksen, “Continuum theory of nematic liquid crystals,”

Res Mechanica, vol. 21, no. 4, pp. 381–392, 1987.

[3] F. M. Leslie, “Theory of flow phenomenum in liquid crystals,”

Advances in Liquid Crystals, vol. 4, pp. 1–81, 1979.

[4] F. Brochard, P. Pieranski, and E. Guyon, “Dynamics of the ori-entation of a nematic-liquid-crystal film in a variable magnetic field,” Physical Review Letters, vol. 28, no. 26, pp. 1681–1683, 1972. [5] L. Lei, S. Changqing, S. Juelian, P. M. Lam, and H. Yun, “Soliton propagation in liquid crystals,” Physical Review Letters, vol. 49, no. 18, pp. 1335–1338, 1982.

[6] F. M. Leslie, “Some topics in continuum theory of nematics,”

Philosophical Transactions of the Royal Society A, vol. 309, pp.

155–165, 1983.

[7] J. T. Gleeson, P. Palffy-Muhoray, and W. van Saarloos, “Prop-agation of excitations induced by shear flow in nematic liquid crystals,” Physical Review A, vol. 44, no. 4, pp. 2588–2595, 1991. [8] F. H. Lin and C. Liu, “Existence of solutions for the Ericksen-Leslie system,” Archive for Rational Mechanics and Analysis, vol. 154, no. 2, pp. 135–156, 2000.

[9] L. R. P. de Andrade Lima and A. D. Rey, “Poiseuille flow of Leslie-Ericksen discotic liquid crystal: Solution multiplicity, multistability, and non-Newtonian rheology,” Journal of

Non-Newtonian Fluid Mechanics, vol. 110, no. 2-3, pp. 103–142, 2003.

[10] X. Nie, R. Lu, H. Xianyu, T. X. Wu, and S. T. Wu, “Anchoring energy and cell gap effects on liquid crystal response time,”

Journal of Applied Physics, vol. 101, no. 10, Article ID 103110, 2007.

[11] P. A. Cruz, M. F. Tom´e, I. W. Stewart, and S. McKee, “A numerical method for solving the dynamic three-dimensional Ericksen-Leslie equations for nematic liquid crystals subject to a strong magnetic field,” Journal of Non-Newtonian Fluid

Mechanics, vol. 165, no. 3-4, pp. 143–157, 2010.

[12] X. Nie, H. Xianyu, R. Lu, T. X. Wu, and S. T. Wu, “Pretilt angle effects on liquid crystal response time,” IEEE/OSA Journal of

Display Technology, vol. 3, no. 3, pp. 280–283, 2007.

[13] I. C. Khoo and S. T. Wu, Optics and Nonlinear Optics of Liquid

Crystals, World Scientific, 1993.

[14] I. C. Khoo, Liquid Crystals: Physical Properties and Nonlinear

Optical Phenomena, John Wiley & Sons, New York, NY, USA,

(12)

Submit your manuscripts at

http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

Journal of

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied MathematicsJournal of

http://www.hindawi.com Volume 2014 Mathematical PhysicsAdvances in

Complex Analysis

Journal of Hindawi Publishing Corporation

Optimization

Journal of

Combinatorics

International Journal of

Journal of

Function Spaces

Abstract and Applied Analysis Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014 International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporation http://www.hindawi.com Volume 2014

The Scientific

World Journal

Discrete Dynamics in Nature and Society Hindawi Publishing Corporation

Discrete Mathematics

Journal of

Investigation of Liquid Crystal Ripple Using Ericksen-Leslie Theory for Displays Subject to Tactile Force

Research Article