Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 932492, 11 pages

http://dx.doi.org/10.1155/2013/932492

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

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

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.

#### Abstract

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 continuum 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 projection method, which is developed for solving the dynamic three-dimensional Ericksen-Leslie equations for liquid crystals 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 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 1 shows a homeotropic alignment liquid crystal layer sandwiched between parallel substrates, where and stand for the bottom and top substrates, respectively [12]. The thickness between bottom and top substrates is . The -axis is normal to the plane of the substrates, and the electric field V is along the -axis. denotes a tilt angle caused by a tactile force and is the tilt angle defined as the angle between the -axis and LC directors. The dynamics of director-axis rotation is described by an Ericksen-Leslie equation [13, 14]: where , , and are elastic constants, and are Leslie viscosities, is the flow velocity, is the rotational viscosity, and is the inertia of LC directors. If backflows, inertial effects, and electric field energy densities are ignored, (1) is reduced to where is the tactile force, is the one-constant approximation with , and 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 product represents a torque expressed by where . Substituting (3) into (2) gives Liquid crystal dynamics is described in this study by considering the following two cases.

*Case 1.* When torque due to tactile force is much larger than elastic torque,
However, when the tilt angle is small, and and (4) becomes

*Case 2.* When torque due to tactile force is much smaller than elastic torque,
Equation (4) thus becomes

Based on separation of variables, assume , which is in turn substituted into (6) and (8). Case 1 thus leads to an equation for of the form where The solution for is written as

In Case 2, which happens when the tactile force is negligible, the solution for in (8) is expressed by where 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 in Figure 2.

*Condition 1. *The molecule ignores the moment of inertia; that is, .

*Condition 2. *Shear rate , where is the flow velocity and is the cell thickness.

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

*Condition 4. *The shearing velocity belongs to Poiseuille flow [9].

According to the above assumptions, (1) becomes 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 yields where and and are boundary conditions of function . In (15), is the angle of the normal vector between the director of LC molecules with homeotropic alignment. Molecule directions are changed by the Poiseuille flow. Values of the angle along the -axis have extreme values at and . Substituting (15)–(18) into (14) gives Since , one has Therefore, (19) becomes As depicted in Figure 2, the director makes an angle with the -axis. Consequently, represents liquid crystal molecules with homeotropic alignment caused by the Poiseuille flow between director and normal direction of LC sample. An effective director equation of motion with the essential physics preserved may be obtained by setting and in (21).

On boundary in Figure 2, By using trigonometric formulas, one can rewrite (22) as Because and for small , (23) is rewritten as Thus, Hence, Substituting for separation of variables into (26) leads to As a result, Similarly, we consider the following two cases.

*Case 1*^{′}
Assuming , one has
Therefore,
is the solution. Substituting (31) into (32) yields

Consider the following.

*Case 2*^{′}
Since the tactile force and electric field are ignored in Case 2′, only Case 1′ is further examined.

On boundary in Figure 2, Since in (33) is equal to 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, Figures 3, 4, and 5 show that a larger force makes the tilt angle larger regardless of LC gap thickness. Figures 6(a), 6(b), and 6(c) compare 3 *μ*m 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 7 and under different tactile forces in Figure 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), and 7(c). Thus, different voltages do not affect tilt angles.

Based on (33), Figures 8(a), 8(b), and 8(c) compare different LC materials including 5CB, MBBA, and PAA subject to 2.5 dyne, 3 dyne, and 3.5 dyne forces in 3 *μ*m and 5 *μ*m thick cell gap LC, respectively. Accordingly, tactile forces affect tilt angles obviously. Figures 9(a), 10(a), and 11(a), respectively, show tilt angles of 5CB LC subject to 2.5 dyne, 3 dyne, and 3.5 dyne forces in 3, 4, and 5 *μ*m thick cell gap LC. Similarly, Figures 12(a), 13(a), and 14(a) are MBBA results and Figures 15(a), 16(a), and 17(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, thickness of cell gap, and Leslie viscosity coefficient are the factors of 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 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.

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