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

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 dealingwith tilt angles of liquid crystalmolecules. Tactile force effects are taken into account to derive themolecule 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 forcemagnitudes, while those of 5CB andMBBAwith larger viscosity are not. Analytical derivation is validated by numerical results.


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][2][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.

Derivation of Ericksen-Leslie Theory
2.1.One-Dimensional Model. Figure 1 shows a homeotropic alignment liquid crystal layer sandwiched between parallel substrates, where  = 0 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  = 8.9 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  is the tactile force,  is the one-constant approximation with  =  11 =  22 =  33 , 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 product  × n represents a torque expressed by where 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, sin  ≈  and cos  ≈ 1 and (4) becomes Equation ( 4) thus becomes Based on separation of variables, assume (, ) = () sin(/), 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 Cell gap (m) Cell gap (m) Cell gap (m) 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.

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.   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 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 n 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 Mathematical Problems in Engineering Because sin   ≈   and cos   ≈ 1 for small   , (23) is rewritten as Thus, Hence,  Similarly, we consider the following two cases.2, Since   (, ) in (33) is equal to −  (, ) in (35), the tilt angles are symmetric.

Discussion
Subject to both electric field of 8.9 V and tactile force, simulation results for Case 1 are depicted in Figures 3, 4 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

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.

Figure 1 :Figure 2 :
Figure 1: Homeotropic alignment LC layer sandwiched between parallel substrates, where  = 0 and  =  are the bottom and top substrates, respectively.

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

Figure 6 :
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.

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

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

Figure 9 :
Figure 9: (a) Comparison of molecule tilt angles depicted in 3 m 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.

Figure 10 :
Figure 10: (a) Comparison of molecule tilt angles depicted in 4 m 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.

Figure 11 :Figure 12 :
Figure 11: (a) Comparison of molecule tilt angles depicted in 5 m 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.

Figure 13 :
Figure 13: (a) Comparison of molecule tilt angles depicted in 4 m 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.

Figure 14 :
Figure 14: (a) Comparison of molecule tilt angles depicted in 5 m 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.

Figure 15 :
Figure 15: (a) Comparison of molecule tilt angles depicted in 3 m 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.

2 𝑑 2 Figure 16 :
Figure 16: (a) Comparison of molecule tilt angles depicted in 4 m 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.

, 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, and 5show that a larger force makes the tilt angle larger regardless of LC gap thickness.Figures6(a), 6(b), and 6(c) compare 3 m thick different LC materials including 5CB, MBBA, and PAA subjected to 5 ms duration 0.5 dyne force, respectively.

Figure 17 :
Figure 17: (a) Comparison of molecule tilt angles depicted in 5 m 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 5 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.
11 ,  22 , and  33 are elastic constants,  2 and  3 are Leslie viscosities, V is the flow velocity,  =  3 −  2 is the rotational viscosity, and  is the inertia of LC directors. where ) 2   +  (  cos  − sin )    2 +  cos ) .   2   +  (  cos  − sin )Since the tactile force and electric field are ignored in Case 2  , only Case 1  is further examined.On boundary  =   in Figure