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

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 𝑧=0 and 𝑧=𝑑 stand for the bottom and top substrates, respectively [12]. The thickness between bottom and top substrates is 𝑑. The 𝑧-axis is normal to the plane of the substrates, and the electric field 𝐸=8.9 V is along the 𝑧-axis. 𝜃 denotes a tilt angle caused by a tactile force and 𝜙 is the tilt angle defined as the angle between the 𝑧-axis and LC directors. The dynamics of director-axis rotation is described by an Ericksen-Leslie equation [13, 14]: 𝜕𝐾𝜕𝑧¹¹cos²𝜃+𝐾₃₃sin²𝜃𝜕𝜃+𝐾𝜕𝑧₃₃−𝐾₁₁sin𝜃cos𝜃𝜕𝜃𝜕𝑧²+𝛼₂sin²𝜃−𝛼₃cos²𝜃𝜕𝑣𝜕𝑧+𝜀₀𝜀_𝑎𝐸²sin𝜃cos𝜃=𝛾𝜕𝜃𝜕𝜕𝑡+𝑀²𝜃𝜕𝑡², (1) where 𝐾₁₁, 𝐾₂₂, and 𝐾₃₃ are elastic constants, 𝛼₂ and 𝛼₃ are Leslie viscosities, 𝑣 is the flow velocity, 𝛾=𝛼₃−𝛼₂ is the rotational viscosity, and 𝑀 is the inertia of LC directors. If backflows, inertial effects, and electric field energy densities are ignored, (1) is reduced to 𝛾𝜕𝜃𝜕𝜕𝑡=𝐾²𝜃𝜕𝑧²+𝐹×𝑛, (2) where 𝐹 is the tactile force, 𝐾 is the one-constant approximation 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 product 𝐹×𝑛 represents a torque expressed by 𝐹×𝑛=|𝐹||𝑛|sin𝜃−𝜙, (3) where |𝑛|=1. Substituting (3) into (2) gives 𝛾𝜕𝜃𝜕𝜕𝑡=𝐾²𝜃𝜕𝑧²+𝐹sin𝜃−𝜙. (4) Liquid crystal dynamics is described in this study by considering the following two cases.

Figure 1 

Homeotropic alignment LC layer sandwiched between parallel substrates, where 𝑧=0 and 𝑧=𝑑 are the bottom and top substrates, respectively.

Case  1. When torque due to tactile force is much larger than elastic torque, ≫𝐾𝜕𝐹sin𝜃−𝜙²𝜃𝜕𝑧². (5) However, when the tilt angle is small, sin𝜃≈𝜃 and cos𝜃≈1 and (4) becomes 𝛾𝜕𝜃𝜕𝑡=𝐹⋅cos𝜙⋅𝜃−𝐹⋅sin𝜙. (6)

Case  2. When torque due to tactile force is much smaller than elastic torque, ≪𝐾𝜕𝐹sin𝜃−𝜙²𝜃𝜕𝑧². (7) Equation (4) thus becomes 𝛾𝜕𝜃𝜕𝜕𝑡=𝐾²𝜃𝜕𝑧². (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 for 𝜃 of the form 𝜃̇=𝑎+𝑏𝜃, (9) where 𝑎=−𝐹sin𝜙𝛾,𝑏=𝐹cos𝜙𝛾. (10) The solution for 𝜃(𝑡,𝑧) is written as 𝜃(𝑡,𝑧)=tan𝜙𝑒^{(𝐹cos𝜙/𝛾)𝑡}sin𝜋𝑧𝑑. (11)

In Case  2, which happens when the tactile force is negligible, the solution for 𝜃(𝑡,𝑧) in (8) is expressed by 𝜃(𝑡,𝑧)=𝑐𝑒^{−(𝐾𝜋2/𝛾𝑑2)𝑡}sin𝜋𝑧𝑑, (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 in Figure 2.

Figure 2 

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

Condition 1. The molecule ignores the moment of inertia; that is, 𝑀=0.

Condition 2. Shear rate 𝑠=𝑠(𝑦)=𝜕𝑣/𝜕𝑦=𝑣/(𝑑/3), where 𝑣 is the flow velocity and 𝑑 is the cell thickness.

Condition 3. The LC sample is homeotropic alignment, and we have the boundary conditions 𝜃𝑑𝑥,𝑦,𝑡=𝜃𝑥,±2,𝑡=0. (13)

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

According to the above assumptions, (1) becomes 𝐾𝜕²𝜃𝜕𝑥²+𝜕²𝜃𝜕𝑦²−𝛾𝜕𝜃+𝑠𝜕𝑡2𝛾₁−𝛾₂cos2𝜃+𝜀₀𝜀_𝑎𝐸²sin𝜃cos𝜃+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 𝜃𝑥,𝑦,𝑡=𝑓𝑦𝜃_𝑚(𝑥,𝑡), (15) where 𝑓𝑦=𝑔𝑦𝑔−𝑦_𝑚=1or−1,𝑦=−𝑦_𝑚or𝑦_𝑚𝑔𝑦𝑔()𝑦−0.28𝑑=4.7503⋅𝑔,𝑦≠−𝑦_𝑚or𝑦_𝑚,𝑔𝑦=2𝑦𝑑−sin𝑦𝜋𝑑,𝑦_𝑚=𝑑𝜋cos⁻¹2𝜋≈0.28𝑑, (16)(17)(18) and 𝑦_𝑚 and −𝑦_𝑚 are boundary conditions of function 𝑔(𝑦). In (15), 𝜃_𝑚 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 𝑦-axis have extreme values at 𝑦_𝑚 and −𝑦_𝑚. Substituting (15)–(18) into (14) gives 𝐾𝜕²𝜕𝑥²𝑓𝑦𝜃_𝑚()+𝜕𝑥,𝑡²𝜕𝑦²𝑓𝑦𝜃_𝑚()𝜕𝑥,𝑡−𝛾𝑓𝑦𝜃𝜕𝑡^𝑚(𝑥,𝑡)+𝜕𝑣𝛼𝜕𝑦³sin²𝜃−𝛼₂cos²𝜃+𝜀₀𝜀_𝑎𝐸²sin𝜃cos𝜃+|𝐹||𝑛|sin𝜃−𝜙=0. (19) Since |𝑛|=1, one has |𝐹||𝑛|sin𝜃−𝜙=𝐹⋅sin𝜃−𝜙. (20) Therefore, (19) becomes 𝐾𝑓𝑦𝜕²𝜃_𝑚𝜕𝑥²+0+𝜃_𝑚𝜕²𝑓𝑦𝜕𝑦²+0−𝛾𝑓𝑦𝜕𝜃_𝑚+𝜕𝑡𝜕𝑣𝛼𝜕𝑦³sin²𝑓𝑦𝜃_𝑚−𝛼₂cos²𝑓𝑦𝜃_𝑚+𝜀₀𝜀_𝑎𝐸²sin𝑓𝑦𝜃_𝑚cos𝑓𝑦𝜃_𝑚+𝐹sin𝑓𝑦𝜃_𝑚−𝜙=0. (21) As depicted in Figure 2, the director makes an angle 𝜃_𝑚 with the 𝑦-axis. Consequently, 𝜃_𝑚 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 𝑦=−𝑦_𝑚 and 𝑦=𝑦_𝑚 in (21).

On boundary 𝑦=−𝑦_𝑚 in Figure 2, 𝐾𝜕²𝜃_𝑚𝜕𝑥²−𝛾𝜕𝜃_𝑚+𝜕𝑡𝜕𝑣𝛼𝜕𝑦³sin²𝜃_𝑚−𝛼₂cos²𝜃_𝑚+𝜀₀𝜀_𝑎𝐸²sin𝜃_𝑚cos𝜃_𝑚+𝐹sin𝜃_𝑚−𝜙=0. (22) By using trigonometric formulas, one can rewrite (22) as 𝐾𝜕²𝜃_𝑚𝜕𝑥²−𝛾𝜕𝜃_𝑚+𝜕𝑡𝜕𝑣×𝛼𝜕𝑦₃1−cos2𝜃_𝑚2−𝛼₂1+cos2𝜃_𝑚2+𝜀₀𝜀_𝑎𝐸²𝜃_𝑚+𝐹sin𝜃_𝑚cos𝜙−cos𝜃_𝑚sin𝜙=0. (23) Because sin𝜃_𝑚≈𝜃_𝑚 and cos𝜃_𝑚≈1 for small 𝜃_𝑚, (23) is rewritten as 𝐾𝜕²𝜃_𝑚𝜕𝑥²−𝛾𝜕𝜃_𝑚+𝜕𝑡𝜕𝑣𝛼𝜕𝑦³1−12−𝛼₂1+12+𝜀₀𝜀_𝑎𝐸²𝜃_𝑚+𝐹𝜃_𝑚cos𝜙−sin𝜙=0. (24) Thus, 𝐾𝜕²𝜃_𝑚𝜕𝑥²−𝛾𝜕𝜃_𝑚𝜕𝑡−𝛼₂𝜕𝑣𝜕𝑦+𝜀₀𝜀_𝑎𝐸²𝜃_𝑚+𝐹𝜃_𝑚cos𝜙−sin𝜙=0. (25) Hence, 𝛾𝜕𝜃_𝑚𝜕𝜕𝑡=𝐾²𝜃_𝑚𝜕𝑥²−𝛼₂𝜕𝑣𝜕𝑦+𝜀₀𝜀_𝑎𝐸²𝜃_𝑚+𝐹𝜃_𝑚cos𝜙−sin𝜙. (26) Substituting 𝜃_𝑚=𝜃_𝑚(𝑡)sin(𝜋𝑥/𝑑) for separation of variables into (26) leads to 𝛾𝜃̇_𝑚𝜕=𝐾²𝜕𝑥²𝜃_𝑚(𝑡)sin𝜋𝑥𝑑−𝛼₂𝜕𝑣𝜕𝑦+𝜀₀𝜀_𝑎𝐸²𝜃_𝑚+𝐹𝜃_𝑚cos𝜙−sin𝜙. (27) As a result, 𝛾𝜃̇_𝑚=−𝐾𝜋²𝑑²𝜃_𝑚(𝑥,𝑡)+𝜀₀𝜀_𝑎𝐸²+𝐹cos𝜙𝜃_𝑚+−𝛼₂𝜕𝑣.𝜕𝑦−𝐹sin𝜙 (28) Similarly, we consider the following two cases.

Case  1^′−𝛼₂𝜕𝑣𝜕𝑦+𝜀₀𝜀_𝑎𝐸²𝜃_𝑚+𝐹𝜃_𝑚≫𝐾𝜕cos𝜙−sin𝜙²𝜃_𝑚𝜕𝑥²,𝛾𝜃̇_𝑚=𝜀₀𝜀_𝑎𝐸²+𝐹cos𝜙𝜃_𝑚+−𝛼₂𝜕𝑣.𝜕𝑦−𝐹sin𝜙 (29)(30) Assuming 𝜃̇_𝑚=𝑎+𝑏𝜃_𝑚, one has 1𝑎=𝛾−𝛼₂𝜕𝑣,1𝜕𝑦−𝐹sin𝜙𝑏=𝛾𝜀₀𝜀_𝑎𝐸²+𝐹cos𝜙. (31) Therefore, 𝜃_𝑚=𝑎𝑏𝑒^𝑏𝑡−1 (32) is the solution. Substituting (31) into (32) yields 𝜃_𝑚()=𝑥,𝑡−𝛼₂𝜕𝑣/𝜕𝑦−𝐹sin𝜙𝜀₀𝜀_𝑎𝐸²𝑒+𝐹cos𝜙^{((𝜀0𝜀𝑎𝐸2+𝐹cos𝜙)/𝛾)𝑡}−1sin𝜋𝑥𝑑. (33)

Consider the following.

Case  2^′−𝛼₂𝜕𝑣𝜕𝑦+𝜀₀𝜀_𝑎𝐸²𝜃_𝑚+𝐹𝜃_𝑚≪𝐾𝜕cos𝜙−sin𝜙²𝜃_𝑚𝜕𝑥². (34) Since the tactile force and electric field are ignored in Case 2′, only Case 1′ is further examined.

On boundary 𝑦=𝑦_𝑚 in Figure 2, 𝜃_𝑚()=𝛼𝑥,𝑡₂𝜕𝑣/𝜕𝑦+𝐹sin𝜙𝜀₀𝜀_𝑎𝐸²𝑒+𝐹cos𝜙^{((𝜀0𝜀𝑎𝐸2+𝐹cos𝜙)/𝛾)𝑡}−1sin𝜋𝑥𝑑. (35) Since 𝜃_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, 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 μm 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.

(a)

(b)

Figure 3 

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

(a)

(b)

Figure 4 

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

(a)

(b)

Figure 5 

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

(a)

(b)

(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.

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.

(a)

(b)

(c)

Figure 7 

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

(a)

(b)

(c)

Figure 8 

Comparison of molecule tilt angles between cell gap thicknesses of 3 μm 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.

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 μm and 5 μm 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 μm 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.

(a)

(b)

Figure 9 

(a) Comparison of molecule tilt angles depicted in 3 μm 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.

(a)

(b)

Figure 10 

(a) Comparison of molecule tilt angles depicted in 4 μm 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.

(a)

(b)

Figure 11 

(a) Comparison of molecule tilt angles depicted in 5 μm 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.

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(b)

Figure 12 

(a) Comparison of molecule tilt angles depicted in 3 μm 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.

(a)

(b)

Figure 13 

(a) Comparison of molecule tilt angles depicted in 4 μm 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.

(a)

(b)

Figure 14 

(a) Comparison of molecule tilt angles depicted in 5 μm 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.

(a)

(b)

Figure 15 

(a) Comparison of molecule tilt angles depicted in 3 μm 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.

(a)

(b)

Figure 16 

(a) Comparison of molecule tilt angles depicted in 4 μm 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.

(a)

(b)

Figure 17 

(a) Comparison of molecule tilt angles depicted in 5 μm 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.

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.