Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 932492 11 pageshttpdxdoiorg1011552013932492

Research ArticleInvestigation of Liquid Crystal Ripple Using Ericksen-LeslieTheory for Displays Subject to Tactile Force

Y J Lee1 T S Liu1 Mao-Hsing Lin2 and Kun-Feng Huang2

1 Department of Mechanical Engineering National Chiao Tung University Hsinchu 30010 Taiwan2 Innolux Corporation Tainan 74147 Taiwan

Correspondence should be addressed to T S Liu tsliumailnctuedutw

Received 25 June 2013 Accepted 7 December 2013

Academic Editor Farzad Khani

Copyright copy 2013 Y J Lee et al This is an open access article distributed under the Creative Commons Attribution License whichpermits 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 ofliquid crystal molecules and alter optical transmission so as to generate ripple on screens Based on the Ericksen-Leslie theory thisstudy investigates ripple propagation by dealingwith tilt angles of liquid crystalmolecules Tactile force effects are taken into accountto derive themolecule equation of motion for liquid crystals Analytical results show that viscosity tactile force the thickness of cellgap and Leslie viscosity coefficient lead to tilt angle variation Tilt angle variations of PAA liquid crystal molecules are sensitive totactile forcemagnitudes while those of 5CB andMBBAwith larger viscosity are not Analytical derivation is validated by numericalresults

1 Introduction

Many physical phenomena exhibited by the nematic phaseliquid crystals (LC) such as unusual flow properties or theLC response to electric and magnetic fields can be studiedby treating LC as a continuous medium Ericksen and Leslie[1ndash3] formulated general conservation laws and constitutiveequations describing the dynamic behavior Other contin-uum theories have been proposed but it turns out that theEricksen-Leslie theory is the one that is most widely used indiscussing the nematic state Based on the continuum theoryof Ericksen-Leslie this study constructs a theoretical modelin order to investigate the tilt angle variation of LC subjectedto both electric field and tactile force The continuum theoryfor the nematic state flow is established in two dynamicalequationsmdashconservation laws and constitutive equationsBoth equations are coupled with each other due to theproperties of flow and the director of liquid crystalmolecules

Brochard et al [4] investigated transient distortions ina nematic film by increasing or decreasing of the magneticfield Lei et al [5] used the Ericksen-Leslie equation todescribe soliton propagation in nematic liquid crystals undershear Leslie [6] presented a concise but clear derivationof continuum equations commonly employed to describe

static and dynamic phenomena in nematic liquid crystalsGleeson et al [7] proposed a two-dimensional theory inwhich the excitations are fronts between distinct solutionsof the steady-state Ericksen-Leslie equations Lin and Liu[8] use the Ericksen-Leslie equation to describe the flowof liquid crystal material and prove the global existence ofweak solutions De Andrade Lima and Rey [9] proposedcomputational modeling of the steady capillary Poiseuilleflow of flow-aligning discotic nematic liquid crystals usingthe Ericksen-Leslie equations predicts solution multiplicityand multistability Nie et al [10] used a cell gap and surfacedynamic method to derive analytical expressions of LCresponse timeunder finite anchoring energy conditions Cruzet al [11] used a finite difference technique based on a projec-tion method which is developed for solving the dynamicthree-dimensional Ericksen-Leslie equations for liquid crys-tals subject to a strong magnetic field

Liquid crystal display panels subjected to tactile force willshow ripple propagation on screens Tactile forces changetilt angles of liquid crystal molecules and alter opticaltransmission so as to generate ripple on screens Based onthe Ericksen-Leslie theory this paper aims to develop LCdynamics subject to tactile force Tactile force effects aretaken into account to derive the molecule director equation

2 Mathematical Problems in Engineering

z

F

x

120601120579

minus minus minus minus minus minus minus minus minus minus minus

+ + + + + + + + + + +

n

z = d

z = 0

Figure 1 Homeotropic alignment LC layer sandwiched between parallel substrates where 119911 = 0 and 119911 = 119889 are the bottom and top substratesrespectively

Poiseuille flow

Homeotropic alignment

x

d

2

minusd

2

minusym

ymf(y) 120579E

120579F120579m 120601

y

Figure 2 Directors of LC molecules with homeotropic alignmentare changed by Poiseuille flow

of motion for liquid crystals A theoretical model is thusproposed to analyze the tilt angle of nematic liquid crystalssubject to tactile forces

2 Derivation of Ericksen-Leslie Theory

21 One-Dimensional Model Figure 1 shows a homeotropicalignment liquid crystal layer sandwiched between parallelsubstrates where 119911 = 0 and 119911 = 119889 stand for the bottomand top substrates respectively [12] The thickness betweenbottom and top substrates is 119889 The 119911-axis is normal to theplane of the substrates and the electric field119864 = 89V is alongthe 119911-axis 120579 denotes a tilt angle caused by a tactile force and 120601is the tilt angle defined as the angle between the 119911-axis and LCdirectorsThe dynamics of director-axis rotation is describedby an Ericksen-Leslie equation [13 14]

120597

120597119911

[(11987011cos2120579 + 119870

33sin2120579) 120597120579

120597119911

]

+ (11987033minus 11987011) sin 120579 cos 120579(120597120579

120597119911

)

2

+ (1205722sin2120579 minus 120572

3cos2120579) 120597V

120597119911

+ 12057601205761198861198642 sin 120579 cos 120579 = 120574

120597120579

120597119905

+ 119872

1205972120579

1205971199052

(1)

0 1 2 3 40

002

004

006

008

01

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(a)

0 1 2 3 4 5 60

002

004

006

008

01

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(b)

Figure 3 Comparison of molecule tilt angles between cell gapthicknesses of (a) 3 120583m and (b) 5 120583m when a tactile force is appliedto LCD screen with 5CB liquid crystals

Mathematical Problems in Engineering 3

0 1 2 3 40

001

002

003

004

005

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(a)

0 1 2 3 4 5 60

001

002

003

004

005

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(b)

Figure 4 Comparison of molecule tilt angles between cell gapthicknesses of (a) 3120583m and (b) 5 120583m when a tactile force is appliedto LCD screen with MBBA liquid crystals

where 11987011 11987022 and 119870

33are elastic constants 120572

2and 120572

3are

Leslie viscosities V is the flow velocity 120574 = 1205723minus 1205722is the

rotational viscosity and 119872 is the inertia of LC directors Ifbackflows inertial effects and electric field energy densitiesare ignored (1) is reduced to

120574

120597120579

120597119905

= 119870

1205972120579

1205971199112+ 119865 times 119899 (2)

where 119865 is the tactile force 119870 is the one-constant approx-imation with 119870 = 119870

11= 11987022

= 11987033 and 119899 is the LC

director Equation (2) is a torque balance equation relatingthe viscosity the elasticity and the torque due to tactile forceA cross product 119865 times 119899 represents a torque expressed by

119865 times 119899 = |119865| |119899| sin (120579 minus 120601) (3)

0 1 2 3 40

04

08

12

16

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(a)

0 1 2 3 4 5 60

04

08

12

16

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(b)

Figure 5 Comparison of molecule tilt angles among cell gapthicknesses of (a) 3 120583m and (b) 5 120583m when a tactile force is appliedto LCD screen with PAA liquid crystals

where |119899| = 1 Substituting (3) into (2) gives

120574

120597120579

120597119905

= 119870

1205972120579

1205971199112+ 119865 sin (120579 minus 120601) (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 thanelastic torque

1003816100381610038161003816119865 sin (120579 minus 120601)

1003816100381610038161003816≫

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579

1205971199112

100381610038161003816100381610038161003816100381610038161003816

(5)

4 Mathematical Problems in Engineering

0 1 2 3 40

001

002

003

004Ti

lt an

gle (

deg)

Cell gap (120583m)

1dyne05dyne01dyne

(a)

0 1 2 3 40

001

002

003

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(b)

0 1 2 3 40

01

02

03

04

05

06

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(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 toLCD screen

However when the tilt angle is small sin 120579 asymp 120579 and cos 120579 asymp 1

and (4) becomes

120574

120597120579

120597119905

= 119865 sdot (cos120601) sdot 120579 minus 119865 sdot sin120601 (6)

Case 2When torque due to tactile force ismuch smaller thanelastic torque

1003816100381610038161003816119865 sin (120579 minus 120601)

1003816100381610038161003816≪

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579

1205971199112

100381610038161003816100381610038161003816100381610038161003816

(7)

Equation (4) thus becomes

120574

120597120579

120597119905

= 119870

1205972120579

1205971199112 (8)

Based on separation of variables assume 120579(119905 119911) =

120579(119905) sin(120587119911119889) which is in turn substituted into (6) and (8)Case 1 thus leads to an equation for 120579 of the form

120579 = 119886 + 119887120579 (9)

where

119886 = minus

119865 sin120601120574

119887 =

119865 cos120601120574

(10)

The solution for 120579(119905 119911) is written as

120579 (119905 119911) = tan120601 (119890(119865 cos120601120574)119905) sin(120587119911119889

) (11)

Mathematical Problems in Engineering 5

0 1 2 3 4 50

2

4

6

8

10

12Ti

lt an

gle (

deg)

Cell gap (120583m)

10V20V30V40V50V

10V20V30V40V50V

(a)

0 1 2 3 4 50

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

10V20V30V40V50V

10V20V30V40V50V

(b)

0 1 2 3 4 50

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

10V20V30V40V50V

10V20V30V40V50V

(c)

Figure 7 Comparison of molecule tilt angles between cell gap thicknesses of 3 120583m and 5 120583mwhen 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 isnegligible the solution for 120579(119905 119911) in (8) is expressed by

120579 (119905 119911) = 119888119890minus(119870120587

21205741198892)119905 sin(120587119911

119889

) (12)

where 119888 is a constant Since the tactile force is ignored inCase 2 only Case 1 is further examined

22 Two-Dimensional Model The above derivation is basedon 1D LC model Further this study intends to use (1) todevelop an LC tilt model in two dimensions Assume that thetheoretical model satisfies the following four conditions asdepicted in Figure 2

Condition 1 The molecule ignores the moment of inertiathat is119872 = 0

Condition 2 Shear rate 119904 = 119904(119910) = 120597V120597119910 = V(1198893) where Vis the flow velocity and 119889 is the cell thickness

Condition 3 The LC sample is homeotropic alignment andwe have the boundary conditions

120579 (119909 119910 119905) = 120579 (119909 plusmn

119889

2

119905) = 0 (13)

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

6 Mathematical Problems in Engineering

0 1 2 3 4 50

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne35dyne

(a)

0 1 2 3 4 50

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne35dyne

(b)

0 1 2 3 4 50

2

4

6

8

10

14

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne35dyne

12

(c)

Figure 8 Comparison of molecule tilt angles between cell gapthicknesses of 3120583m and 5120583m when tactile force is applied to LCDscreen with (a) 5CB (b) MBBA and (c) PAA liquid crystals underdifferent tactile forces

25dyne3dyne

35dyne

0 1 2 30

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

(a)

0 1 2 30

2

4

6

8

10

12

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

(b)

Figure 9 (a) Comparison of molecule tilt angles depicted in3 120583m cell gap 5CB liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

According to the above assumptions (1) becomes

119870(

1205972120579

1205971199092+

1205972120579

1205971199102) minus 120574

120597120579

120597119905

+

119904

2

(1205741minus 1205742cos 2120579)

+ 12057601205761198861198642 sin 120579 cos 120579 + 119865 times 119899 = 0

(14)

Based on the Ericksen-Leslie theory (14) represents the tiltmodel of LC molecule subjected to both electric field andtactile force In order to use separation of variables assume120579 = 120579(119909 119910 119905) yields

120579 (119909 119910 119905) = 119891 (119910) 120579119898(119909 119905) (15)

Mathematical Problems in Engineering 7

0 1 2 30

2

4

6

8

10

Tilt

angl

e (de

g)

Cell gap (120583m)

4

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 10 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap 5CB liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

where

119891 (119910) =

119892 (119910)

119892 (minus119910119898)

=

1 or minus 1 119910 = minus119910119898or 119910119898

119892 (119910)

119892 (minus028119889)

= 47503 sdot 119892 (119910) 119910 = minus 119910119898or 119910119898

(16)

119892 (119910) =

2119910

119889

minus sin(119910120587

119889

) (17)

119910119898= (

119889

120587

) cosminus1 ( 2120587

) asymp 028119889 (18)

and119910119898andminus119910

119898are boundary conditions of function119892(119910) In

(15) 120579119898is the angle of the normal vector between the director

of LC molecules with homeotropic alignment Molecule

0 1 2 3 4 50

2

4

6

8

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 11 (a) Comparison of molecule tilt angles depicted in5 120583m cell gap 5CB liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

directions are changed by the Poiseuille flow Values of theangle along the 119910-axis have extreme values at 119910

119898and minus119910

119898

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

119870[

1205972

1205971199092(119891 (119910) 120579

119898(119909 119905)) +

1205972

1205971199102(119891 (119910) 120579

119898(119909 119905))]

minus 120574

120597

120597119905

(119891 (119910) 120579119898(119909 119905)) +

120597V

120597119910

(1205723sin2120579 minus 120572

2cos2120579)

+ 12057601205761198861198642 sin 120579 cos 120579 + |119865| |119899| sin (120579 minus 120601) = 0

(19)

Since |119899| = 1 one has

|119865| |119899| sin (120579 minus 120601) = 119865 sdot sin (120579 minus 120601) (20)

8 Mathematical Problems in Engineering

0 1 2 30

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

12

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

(b)

Figure 12 (a) Comparison of molecule tilt angles depicted in3120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Therefore (19) becomes

119870[119891 (119910)

1205972120579119898

1205971199092

+ 0] + [120579119898

1205972119891 (119910)

1205971199102

+ 0] minus 120574119891 (119910)

120597120579119898

120597119905

+

120597V

120597119910

[1205723sin2 (119891 (119910) 120579

119898) minus 1205722cos2 (119891 (119910) 120579

119898)]

+ 12057601205761198861198642 sin (119891 (119910) 120579

119898) cos (119891 (119910) 120579

119898)

+ 119865 sin (119891 (119910) 120579119898minus 120601) = 0

(21)

As depicted in Figure 2 the director makes an angle 120579119898with

the 119910-axis Consequently 120579119898represents liquid crystal mole-

cules with homeotropic alignment caused by the Poiseuilleflow between director 119899 and normal direction of LC sampleAn effective director equation of motion with the essentialphysics preserved may be obtained by setting 119910 = minus119910

119898and

119910 = 119910119898in (21)

0 1 2 3 40

2

4

6

8

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 13 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

On boundary 119910 = minus119910119898in Figure 2

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

(1205723sin2120579119898minus 1205722cos2120579119898)

+ 12057601205761198861198642 sin 120579

119898cos 120579119898+ 119865 sin (120579

119898minus 120601) = 0

(22)

By using trigonometric formulas one can rewrite (22) as

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

times [1205723(

1 minus cos 2120579119898

2

) minus 1205722(

1 + cos 2120579119898

2

)]

+ 12057601205761198861198642120579119898+ 119865 (sin 120579

119898cos120601 minus cos 120579

119898sin120601) = 0

(23)

Mathematical Problems in Engineering 9

0 1 2 3 4 50

2

4

6

8Ti

lt an

gle (

deg)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 14 (a) Comparison of molecule tilt angles depicted in5120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Because sin 120579119898

asymp 120579119898and cos 120579

119898asymp 1 for small 120579

119898 (23) is

rewritten as

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

[1205723(

1 minus 1

2

) minus 1205722(

1 + 1

2

)]

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601) = 0

(24)

Thus

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898

+ 119865 (120579119898cos120601 minus sin120601) = 0

(25)

Hence

120574

120597120579119898

120597119905

=119870

1205972120579119898

1205971199092

minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

(26)

0 1 2 30

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

14

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

12

14

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

(b)

Figure 15 (a) Comparison of molecule tilt angles depicted in3 120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Substituting 120579119898= 120579119898(119905) sin(120587119909119889) for separation of variables

into (26) leads to

120574

120579119898= 119870

1205972

1205971199092[120579119898(119905) sin(120587119909

119889

)] minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

(27)

As a result

120574

120579119898= minus

1198701205872

1198892120579119898(119909 119905) + (120576

01205761198861198642+ 119865 cos120601) 120579

119898

+ (minus1205722

120597V

120597119910

minus 119865 sin120601)

(28)

10 Mathematical Problems in Engineering

0 1 2 30

2

4

6

8

10

Tilt

angl

e (de

g)

Cell gap (120583m)

4

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 16 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne by analyticaland numerical results

Similarly we consider the following two cases

Case 11015840

10038161003816100381610038161003816100381610038161003816

minus1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

10038161003816100381610038161003816100381610038161003816

≫

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579119898

1205971199092

100381610038161003816100381610038161003816100381610038161003816

(29)

120574

120579119898= (12057601205761198861198642+ 119865 cos120601) 120579

119898+ (minus120572

2

120597V

120597119910

minus 119865 sin120601) (30)

Assuming 120579119898= 119886 + 119887120579

119898 one has

119886 =

1

120574

(minus1205722

120597V

120597119910

minus 119865 sin120601)

119887 =

1

120574

(12057601205761198861198642+ 119865 cos120601)

(31)

Therefore

120579119898=

119886

119887

(119890119887119905minus 1) (32)

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

120579119898(119909 119905)

=[

minus1205722(120597V120597119910) minus 119865 sin120601

12057601205761198861198642+ 119865 cos120601

(119890((12057601205761198861198642+119865 cos120601)120574)119905

minus 1)]sin(120587119909119889

)

(33)

Consider the following

Case 21015840

10038161003816100381610038161003816100381610038161003816

minus1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

10038161003816100381610038161003816100381610038161003816

≪

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579119898

1205971199092

100381610038161003816100381610038161003816100381610038161003816

(34)

Since the tactile force and electric field are ignored in Case 21015840only Case 11015840 is further examined

On boundary 119910 = 119910119898in Figure 2

120579119898(119909 119905)

=[

1205722(120597V120597119910) + 119865 sin12060112057601205761198861198642+ 119865 cos120601

(119890((12057601205761198861198642+119865 cos120601)120574)119905

minus 1)]sin(120587119909119889

)

(35)

Since 120579119898(119909 119905) in (33) is equal to minus120579

119898(119909 119905) in (35) the tilt

angles are symmetric

3 Discussion

Subject to both electric field of 89 V and tactile forcesimulation results for Case 1 are depicted in Figures 3 4and 5 Comparing different LC materials including 5CBMBBA and PAA subjected to 10ms duration tactile forcesof 1 dyne 05 dyne and 01 dyne at 10ms Figures 3 4 and 5show that a larger force makes the tilt angle larger regardlessof LC gap thickness Figures 6(a) 6(b) and 6(c) compare3 120583m thick different LCmaterials including 5CB MBBA andPAA subjected to 5ms duration 05 dyne force respectivelyThe viscosity of PAA is the smallest and the tilt angle islarger than the others Therefore the option of liquid crystalmaterial with smaller viscosity can reduce the ripple

Tilt angle variations are shown under different voltages inFigure 7 and under different tactile forces in Figure 8 Using10V 20V 30V 40V and 50V under a tactile force of 3 dyneit is observed that the tilt angles remain the same in Figures7(a) 7(b) and 7(c) Thus different voltages do not affect tiltangles

Based on (33) Figures 8(a) 8(b) and 8(c) compare differ-ent LC materials including 5CB MBBA and PAA subject to25 dyne 3 dyne and 35 dyne forces in 3120583m and 5 120583m thickcell gap LC respectively Accordingly tactile forces affect tiltangles obviously Figures 9(a) 10(a) and 11(a) respectivelyshow tilt angles of 5CB LC subject to 25 dyne 3 dyne and

Mathematical Problems in Engineering 11

0 1 2 3 4 50

2

4

6

8Ti

lt an

gle (

deg)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 17 (a) Comparison of molecule tilt angles depicted in5120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

35 dyne forces in 3 4 and 5 120583m thick cell gap LC SimilarlyFigures 12(a) 13(a) and 14(a) are MBBA results and Figures15(a) 16(a) and 17(a) are PAA results In addition Figures9(b) to 17(b) compare analytical solutions with numericalsolutions of the full coupled equations The comparisonresults show that analytical solutions are consistent withnumerical solutions of the full coupled equations Thereforethe analytical derivation is validated

4 Conclusion

The Ericksen-Leslie theory deals with the dynamics ofnematic liquid crystals Based on the Ericksen-Leslie theorythis paper contributes to LC dynamics subject to tactile forceLC molecule tilt leads to a change in the optical intensitytransmitted through a liquid crystal cell LCD panels subjectto tactile force will show ripple-like propagation on screensThe present results show that the viscosity tactile force

thickness of cell gap and Leslie viscosity coefficient are thefactors of tilt angle variation Tilt angle variations of PAAliquid crystal molecules are sensitive to tactile force mag-nitudes while those of 5CB and MBBA with larger viscosityare not Analytical derivation has been carried out in thisstudy and analytical results have been validated by numericalresults

Acknowledgments

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

References

[1] J Ericksen ldquoConservation laws for liquid crystalsrdquoTransactionsof the Society of Rheology vol 5 pp 22ndash34 1961

[2] J L Ericksen ldquoContinuum theory of nematic liquid crystalsrdquoRes Mechanica vol 21 no 4 pp 381ndash392 1987

[3] F M Leslie ldquoTheory of flow phenomenum in liquid crystalsrdquoAdvances in Liquid Crystals vol 4 pp 1ndash81 1979

[4] F Brochard P Pieranski and E Guyon ldquoDynamics of the ori-entation of a nematic-liquid-crystal film in a variable magneticfieldrdquoPhysical Review Letters vol 28 no 26 pp 1681ndash1683 1972

[5] L Lei S Changqing S Juelian P M Lam and H Yun ldquoSolitonpropagation in liquid crystalsrdquo Physical Review Letters vol 49no 18 pp 1335ndash1338 1982

[6] F M Leslie ldquoSome topics in continuum theory of nematicsrdquoPhilosophical Transactions of the Royal Society A vol 309 pp155ndash165 1983

[7] J T Gleeson P Palffy-Muhoray and W van Saarloos ldquoProp-agation of excitations induced by shear flow in nematic liquidcrystalsrdquo Physical Review A vol 44 no 4 pp 2588ndash2595 1991

[8] F H Lin and C Liu ldquoExistence of solutions for the Ericksen-Leslie systemrdquo Archive for Rational Mechanics and Analysis vol154 no 2 pp 135ndash156 2000

[9] L R P de Andrade Lima and A D Rey ldquoPoiseuille flowof Leslie-Ericksen discotic liquid crystal Solution multiplicitymultistability and non-Newtonian rheologyrdquo Journal of Non-Newtonian Fluid Mechanics vol 110 no 2-3 pp 103ndash142 2003

[10] X Nie R Lu H Xianyu T X Wu and S T Wu ldquoAnchoringenergy and cell gap effects on liquid crystal response timerdquoJournal of Applied Physics vol 101 no 10 Article ID 103110 2007

[11] P A Cruz M F Tome I W Stewart and S McKee ldquoAnumerical method for solving the dynamic three-dimensionalEricksen-Leslie equations for nematic liquid crystals subjectto a strong magnetic fieldrdquo Journal of Non-Newtonian FluidMechanics vol 165 no 3-4 pp 143ndash157 2010

[12] X Nie H Xianyu R Lu T X Wu and S T Wu ldquoPretilt angleeffects on liquid crystal response timerdquo IEEEOSA Journal ofDisplay Technology vol 3 no 3 pp 280ndash283 2007

[13] I C Khoo and S T Wu Optics and Nonlinear Optics of LiquidCrystals World Scientific 1993

[14] I C Khoo Liquid Crystals Physical Properties and NonlinearOptical Phenomena John Wiley amp Sons New York NY USA1995

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

z

F

x

120601120579

minus minus minus minus minus minus minus minus minus minus minus

+ + + + + + + + + + +

n

z = d

z = 0

Figure 1 Homeotropic alignment LC layer sandwiched between parallel substrates where 119911 = 0 and 119911 = 119889 are the bottom and top substratesrespectively

Poiseuille flow

Homeotropic alignment

x

d

2

minusd

2

minusym

ymf(y) 120579E

120579F120579m 120601

y

Figure 2 Directors of LC molecules with homeotropic alignmentare changed by Poiseuille flow

of motion for liquid crystals A theoretical model is thusproposed to analyze the tilt angle of nematic liquid crystalssubject to tactile forces

2 Derivation of Ericksen-Leslie Theory

21 One-Dimensional Model Figure 1 shows a homeotropicalignment liquid crystal layer sandwiched between parallelsubstrates where 119911 = 0 and 119911 = 119889 stand for the bottomand top substrates respectively [12] The thickness betweenbottom and top substrates is 119889 The 119911-axis is normal to theplane of the substrates and the electric field119864 = 89V is alongthe 119911-axis 120579 denotes a tilt angle caused by a tactile force and 120601is the tilt angle defined as the angle between the 119911-axis and LCdirectorsThe dynamics of director-axis rotation is describedby an Ericksen-Leslie equation [13 14]

120597

120597119911

[(11987011cos2120579 + 119870

33sin2120579) 120597120579

120597119911

]

+ (11987033minus 11987011) sin 120579 cos 120579(120597120579

120597119911

)

2

+ (1205722sin2120579 minus 120572

3cos2120579) 120597V

120597119911

+ 12057601205761198861198642 sin 120579 cos 120579 = 120574

120597120579

120597119905

+ 119872

1205972120579

1205971199052

(1)

0 1 2 3 40

002

004

006

008

01

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(a)

0 1 2 3 4 5 60

002

004

006

008

01

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(b)

Figure 3 Comparison of molecule tilt angles between cell gapthicknesses of (a) 3 120583m and (b) 5 120583m when a tactile force is appliedto LCD screen with 5CB liquid crystals

Mathematical Problems in Engineering 3

0 1 2 3 40

001

002

003

004

005

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(a)

0 1 2 3 4 5 60

001

002

003

004

005

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(b)

Figure 4 Comparison of molecule tilt angles between cell gapthicknesses of (a) 3120583m and (b) 5 120583m when a tactile force is appliedto LCD screen with MBBA liquid crystals

where 11987011 11987022 and 119870

33are elastic constants 120572

2and 120572

3are

Leslie viscosities V is the flow velocity 120574 = 1205723minus 1205722is the

rotational viscosity and 119872 is the inertia of LC directors Ifbackflows inertial effects and electric field energy densitiesare ignored (1) is reduced to

120574

120597120579

120597119905

= 119870

1205972120579

1205971199112+ 119865 times 119899 (2)

where 119865 is the tactile force 119870 is the one-constant approx-imation with 119870 = 119870

11= 11987022

= 11987033 and 119899 is the LC

director Equation (2) is a torque balance equation relatingthe viscosity the elasticity and the torque due to tactile forceA cross product 119865 times 119899 represents a torque expressed by

119865 times 119899 = |119865| |119899| sin (120579 minus 120601) (3)

0 1 2 3 40

04

08

12

16

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(a)

0 1 2 3 4 5 60

04

08

12

16

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(b)

Figure 5 Comparison of molecule tilt angles among cell gapthicknesses of (a) 3 120583m and (b) 5 120583m when a tactile force is appliedto LCD screen with PAA liquid crystals

where |119899| = 1 Substituting (3) into (2) gives

120574

120597120579

120597119905

= 119870

1205972120579

1205971199112+ 119865 sin (120579 minus 120601) (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 thanelastic torque

1003816100381610038161003816119865 sin (120579 minus 120601)

1003816100381610038161003816≫

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579

1205971199112

100381610038161003816100381610038161003816100381610038161003816

(5)

4 Mathematical Problems in Engineering

0 1 2 3 40

001

002

003

004Ti

lt an

gle (

deg)

Cell gap (120583m)

1dyne05dyne01dyne

(a)

0 1 2 3 40

001

002

003

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(b)

0 1 2 3 40

01

02

03

04

05

06

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(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 toLCD screen

However when the tilt angle is small sin 120579 asymp 120579 and cos 120579 asymp 1

and (4) becomes

120574

120597120579

120597119905

= 119865 sdot (cos120601) sdot 120579 minus 119865 sdot sin120601 (6)

Case 2When torque due to tactile force ismuch smaller thanelastic torque

1003816100381610038161003816119865 sin (120579 minus 120601)

1003816100381610038161003816≪

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579

1205971199112

100381610038161003816100381610038161003816100381610038161003816

(7)

Equation (4) thus becomes

120574

120597120579

120597119905

= 119870

1205972120579

1205971199112 (8)

Based on separation of variables assume 120579(119905 119911) =

120579(119905) sin(120587119911119889) which is in turn substituted into (6) and (8)Case 1 thus leads to an equation for 120579 of the form

120579 = 119886 + 119887120579 (9)

where

119886 = minus

119865 sin120601120574

119887 =

119865 cos120601120574

(10)

The solution for 120579(119905 119911) is written as

120579 (119905 119911) = tan120601 (119890(119865 cos120601120574)119905) sin(120587119911119889

) (11)

Mathematical Problems in Engineering 5

0 1 2 3 4 50

2

4

6

8

10

12Ti

lt an

gle (

deg)

Cell gap (120583m)

10V20V30V40V50V

10V20V30V40V50V

(a)

0 1 2 3 4 50

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

10V20V30V40V50V

10V20V30V40V50V

(b)

0 1 2 3 4 50

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

10V20V30V40V50V

10V20V30V40V50V

(c)

Figure 7 Comparison of molecule tilt angles between cell gap thicknesses of 3 120583m and 5 120583mwhen 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 isnegligible the solution for 120579(119905 119911) in (8) is expressed by

120579 (119905 119911) = 119888119890minus(119870120587

21205741198892)119905 sin(120587119911

119889

) (12)

where 119888 is a constant Since the tactile force is ignored inCase 2 only Case 1 is further examined

22 Two-Dimensional Model The above derivation is basedon 1D LC model Further this study intends to use (1) todevelop an LC tilt model in two dimensions Assume that thetheoretical model satisfies the following four conditions asdepicted in Figure 2

Condition 1 The molecule ignores the moment of inertiathat is119872 = 0

Condition 2 Shear rate 119904 = 119904(119910) = 120597V120597119910 = V(1198893) where Vis the flow velocity and 119889 is the cell thickness

Condition 3 The LC sample is homeotropic alignment andwe have the boundary conditions

120579 (119909 119910 119905) = 120579 (119909 plusmn

119889

2

119905) = 0 (13)

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

6 Mathematical Problems in Engineering

0 1 2 3 4 50

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne35dyne

(a)

0 1 2 3 4 50

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne35dyne

(b)

0 1 2 3 4 50

2

4

6

8

10

14

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne35dyne

12

(c)

Figure 8 Comparison of molecule tilt angles between cell gapthicknesses of 3120583m and 5120583m when tactile force is applied to LCDscreen with (a) 5CB (b) MBBA and (c) PAA liquid crystals underdifferent tactile forces

25dyne3dyne

35dyne

0 1 2 30

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

(a)

0 1 2 30

2

4

6

8

10

12

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

(b)

Figure 9 (a) Comparison of molecule tilt angles depicted in3 120583m cell gap 5CB liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

According to the above assumptions (1) becomes

119870(

1205972120579

1205971199092+

1205972120579

1205971199102) minus 120574

120597120579

120597119905

+

119904

2

(1205741minus 1205742cos 2120579)

+ 12057601205761198861198642 sin 120579 cos 120579 + 119865 times 119899 = 0

(14)

Based on the Ericksen-Leslie theory (14) represents the tiltmodel of LC molecule subjected to both electric field andtactile force In order to use separation of variables assume120579 = 120579(119909 119910 119905) yields

120579 (119909 119910 119905) = 119891 (119910) 120579119898(119909 119905) (15)

Mathematical Problems in Engineering 7

0 1 2 30

2

4

6

8

10

Tilt

angl

e (de

g)

Cell gap (120583m)

4

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 10 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap 5CB liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

where

119891 (119910) =

119892 (119910)

119892 (minus119910119898)

=

1 or minus 1 119910 = minus119910119898or 119910119898

119892 (119910)

119892 (minus028119889)

= 47503 sdot 119892 (119910) 119910 = minus 119910119898or 119910119898

(16)

119892 (119910) =

2119910

119889

minus sin(119910120587

119889

) (17)

119910119898= (

119889

120587

) cosminus1 ( 2120587

) asymp 028119889 (18)

and119910119898andminus119910

119898are boundary conditions of function119892(119910) In

(15) 120579119898is the angle of the normal vector between the director

of LC molecules with homeotropic alignment Molecule

0 1 2 3 4 50

2

4

6

8

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 11 (a) Comparison of molecule tilt angles depicted in5 120583m cell gap 5CB liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

directions are changed by the Poiseuille flow Values of theangle along the 119910-axis have extreme values at 119910

119898and minus119910

119898

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

119870[

1205972

1205971199092(119891 (119910) 120579

119898(119909 119905)) +

1205972

1205971199102(119891 (119910) 120579

119898(119909 119905))]

minus 120574

120597

120597119905

(119891 (119910) 120579119898(119909 119905)) +

120597V

120597119910

(1205723sin2120579 minus 120572

2cos2120579)

+ 12057601205761198861198642 sin 120579 cos 120579 + |119865| |119899| sin (120579 minus 120601) = 0

(19)

Since |119899| = 1 one has

|119865| |119899| sin (120579 minus 120601) = 119865 sdot sin (120579 minus 120601) (20)

8 Mathematical Problems in Engineering

0 1 2 30

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

12

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

(b)

Figure 12 (a) Comparison of molecule tilt angles depicted in3120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Therefore (19) becomes

119870[119891 (119910)

1205972120579119898

1205971199092

+ 0] + [120579119898

1205972119891 (119910)

1205971199102

+ 0] minus 120574119891 (119910)

120597120579119898

120597119905

+

120597V

120597119910

[1205723sin2 (119891 (119910) 120579

119898) minus 1205722cos2 (119891 (119910) 120579

119898)]

+ 12057601205761198861198642 sin (119891 (119910) 120579

119898) cos (119891 (119910) 120579

119898)

+ 119865 sin (119891 (119910) 120579119898minus 120601) = 0

(21)

As depicted in Figure 2 the director makes an angle 120579119898with

the 119910-axis Consequently 120579119898represents liquid crystal mole-

cules with homeotropic alignment caused by the Poiseuilleflow between director 119899 and normal direction of LC sampleAn effective director equation of motion with the essentialphysics preserved may be obtained by setting 119910 = minus119910

119898and

119910 = 119910119898in (21)

0 1 2 3 40

2

4

6

8

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 13 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

On boundary 119910 = minus119910119898in Figure 2

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

(1205723sin2120579119898minus 1205722cos2120579119898)

+ 12057601205761198861198642 sin 120579

119898cos 120579119898+ 119865 sin (120579

119898minus 120601) = 0

(22)

By using trigonometric formulas one can rewrite (22) as

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

times [1205723(

1 minus cos 2120579119898

2

) minus 1205722(

1 + cos 2120579119898

2

)]

+ 12057601205761198861198642120579119898+ 119865 (sin 120579

119898cos120601 minus cos 120579

119898sin120601) = 0

(23)

Mathematical Problems in Engineering 9

0 1 2 3 4 50

2

4

6

8Ti

lt an

gle (

deg)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 14 (a) Comparison of molecule tilt angles depicted in5120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Because sin 120579119898

asymp 120579119898and cos 120579

119898asymp 1 for small 120579

119898 (23) is

rewritten as

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

[1205723(

1 minus 1

2

) minus 1205722(

1 + 1

2

)]

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601) = 0

(24)

Thus

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898

+ 119865 (120579119898cos120601 minus sin120601) = 0

(25)

Hence

120574

120597120579119898

120597119905

=119870

1205972120579119898

1205971199092

minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

(26)

0 1 2 30

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

14

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

12

14

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

(b)

Figure 15 (a) Comparison of molecule tilt angles depicted in3 120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Substituting 120579119898= 120579119898(119905) sin(120587119909119889) for separation of variables

into (26) leads to

120574

120579119898= 119870

1205972

1205971199092[120579119898(119905) sin(120587119909

119889

)] minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

(27)

As a result

120574

120579119898= minus

1198701205872

1198892120579119898(119909 119905) + (120576

01205761198861198642+ 119865 cos120601) 120579

119898

+ (minus1205722

120597V

120597119910

minus 119865 sin120601)

(28)

10 Mathematical Problems in Engineering

0 1 2 30

2

4

6

8

10

Tilt

angl

e (de

g)

Cell gap (120583m)

4

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 16 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne by analyticaland numerical results

Similarly we consider the following two cases

Case 11015840

10038161003816100381610038161003816100381610038161003816

minus1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

10038161003816100381610038161003816100381610038161003816

≫

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579119898

1205971199092

100381610038161003816100381610038161003816100381610038161003816

(29)

120574

120579119898= (12057601205761198861198642+ 119865 cos120601) 120579

119898+ (minus120572

2

120597V

120597119910

minus 119865 sin120601) (30)

Assuming 120579119898= 119886 + 119887120579

119898 one has

119886 =

1

120574

(minus1205722

120597V

120597119910

minus 119865 sin120601)

119887 =

1

120574

(12057601205761198861198642+ 119865 cos120601)

(31)

Therefore

120579119898=

119886

119887

(119890119887119905minus 1) (32)

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

120579119898(119909 119905)

=[

minus1205722(120597V120597119910) minus 119865 sin120601

12057601205761198861198642+ 119865 cos120601

(119890((12057601205761198861198642+119865 cos120601)120574)119905

minus 1)]sin(120587119909119889

)

(33)

Consider the following

Case 21015840

10038161003816100381610038161003816100381610038161003816

minus1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

10038161003816100381610038161003816100381610038161003816

≪

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579119898

1205971199092

100381610038161003816100381610038161003816100381610038161003816

(34)

Since the tactile force and electric field are ignored in Case 21015840only Case 11015840 is further examined

On boundary 119910 = 119910119898in Figure 2

120579119898(119909 119905)

=[

1205722(120597V120597119910) + 119865 sin12060112057601205761198861198642+ 119865 cos120601

(119890((12057601205761198861198642+119865 cos120601)120574)119905

minus 1)]sin(120587119909119889

)

(35)

Since 120579119898(119909 119905) in (33) is equal to minus120579

119898(119909 119905) in (35) the tilt

angles are symmetric

3 Discussion

Subject to both electric field of 89 V and tactile forcesimulation results for Case 1 are depicted in Figures 3 4and 5 Comparing different LC materials including 5CBMBBA and PAA subjected to 10ms duration tactile forcesof 1 dyne 05 dyne and 01 dyne at 10ms Figures 3 4 and 5show that a larger force makes the tilt angle larger regardlessof LC gap thickness Figures 6(a) 6(b) and 6(c) compare3 120583m thick different LCmaterials including 5CB MBBA andPAA subjected to 5ms duration 05 dyne force respectivelyThe viscosity of PAA is the smallest and the tilt angle islarger than the others Therefore the option of liquid crystalmaterial with smaller viscosity can reduce the ripple

Tilt angle variations are shown under different voltages inFigure 7 and under different tactile forces in Figure 8 Using10V 20V 30V 40V and 50V under a tactile force of 3 dyneit is observed that the tilt angles remain the same in Figures7(a) 7(b) and 7(c) Thus different voltages do not affect tiltangles

Based on (33) Figures 8(a) 8(b) and 8(c) compare differ-ent LC materials including 5CB MBBA and PAA subject to25 dyne 3 dyne and 35 dyne forces in 3120583m and 5 120583m thickcell gap LC respectively Accordingly tactile forces affect tiltangles obviously Figures 9(a) 10(a) and 11(a) respectivelyshow tilt angles of 5CB LC subject to 25 dyne 3 dyne and

Mathematical Problems in Engineering 11

0 1 2 3 4 50

2

4

6

8Ti

lt an

gle (

deg)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 17 (a) Comparison of molecule tilt angles depicted in5120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

35 dyne forces in 3 4 and 5 120583m thick cell gap LC SimilarlyFigures 12(a) 13(a) and 14(a) are MBBA results and Figures15(a) 16(a) and 17(a) are PAA results In addition Figures9(b) to 17(b) compare analytical solutions with numericalsolutions of the full coupled equations The comparisonresults show that analytical solutions are consistent withnumerical solutions of the full coupled equations Thereforethe analytical derivation is validated

4 Conclusion

The Ericksen-Leslie theory deals with the dynamics ofnematic liquid crystals Based on the Ericksen-Leslie theorythis paper contributes to LC dynamics subject to tactile forceLC molecule tilt leads to a change in the optical intensitytransmitted through a liquid crystal cell LCD panels subjectto tactile force will show ripple-like propagation on screensThe present results show that the viscosity tactile force

thickness of cell gap and Leslie viscosity coefficient are thefactors of tilt angle variation Tilt angle variations of PAAliquid crystal molecules are sensitive to tactile force mag-nitudes while those of 5CB and MBBA with larger viscosityare not Analytical derivation has been carried out in thisstudy and analytical results have been validated by numericalresults

Acknowledgments

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

References

[1] J Ericksen ldquoConservation laws for liquid crystalsrdquoTransactionsof the Society of Rheology vol 5 pp 22ndash34 1961

[2] J L Ericksen ldquoContinuum theory of nematic liquid crystalsrdquoRes Mechanica vol 21 no 4 pp 381ndash392 1987

[3] F M Leslie ldquoTheory of flow phenomenum in liquid crystalsrdquoAdvances in Liquid Crystals vol 4 pp 1ndash81 1979

[4] F Brochard P Pieranski and E Guyon ldquoDynamics of the ori-entation of a nematic-liquid-crystal film in a variable magneticfieldrdquoPhysical Review Letters vol 28 no 26 pp 1681ndash1683 1972

[5] L Lei S Changqing S Juelian P M Lam and H Yun ldquoSolitonpropagation in liquid crystalsrdquo Physical Review Letters vol 49no 18 pp 1335ndash1338 1982

[6] F M Leslie ldquoSome topics in continuum theory of nematicsrdquoPhilosophical Transactions of the Royal Society A vol 309 pp155ndash165 1983

[7] J T Gleeson P Palffy-Muhoray and W van Saarloos ldquoProp-agation of excitations induced by shear flow in nematic liquidcrystalsrdquo Physical Review A vol 44 no 4 pp 2588ndash2595 1991

[8] F H Lin and C Liu ldquoExistence of solutions for the Ericksen-Leslie systemrdquo Archive for Rational Mechanics and Analysis vol154 no 2 pp 135ndash156 2000

[9] L R P de Andrade Lima and A D Rey ldquoPoiseuille flowof Leslie-Ericksen discotic liquid crystal Solution multiplicitymultistability and non-Newtonian rheologyrdquo Journal of Non-Newtonian Fluid Mechanics vol 110 no 2-3 pp 103ndash142 2003

[10] X Nie R Lu H Xianyu T X Wu and S T Wu ldquoAnchoringenergy and cell gap effects on liquid crystal response timerdquoJournal of Applied Physics vol 101 no 10 Article ID 103110 2007

[11] P A Cruz M F Tome I W Stewart and S McKee ldquoAnumerical method for solving the dynamic three-dimensionalEricksen-Leslie equations for nematic liquid crystals subjectto a strong magnetic fieldrdquo Journal of Non-Newtonian FluidMechanics vol 165 no 3-4 pp 143ndash157 2010

[12] X Nie H Xianyu R Lu T X Wu and S T Wu ldquoPretilt angleeffects on liquid crystal response timerdquo IEEEOSA Journal ofDisplay Technology vol 3 no 3 pp 280ndash283 2007

[13] I C Khoo and S T Wu Optics and Nonlinear Optics of LiquidCrystals World Scientific 1993

[14] I C Khoo Liquid Crystals Physical Properties and NonlinearOptical Phenomena John Wiley amp Sons New York NY USA1995

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

0 1 2 3 40

001

002

003

004

005

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(a)

0 1 2 3 4 5 60

001

002

003

004

005

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(b)

Figure 4 Comparison of molecule tilt angles between cell gapthicknesses of (a) 3120583m and (b) 5 120583m when a tactile force is appliedto LCD screen with MBBA liquid crystals

where 11987011 11987022 and 119870

33are elastic constants 120572

2and 120572

3are

Leslie viscosities V is the flow velocity 120574 = 1205723minus 1205722is the

rotational viscosity and 119872 is the inertia of LC directors Ifbackflows inertial effects and electric field energy densitiesare ignored (1) is reduced to

120574

120597120579

120597119905

= 119870

1205972120579

1205971199112+ 119865 times 119899 (2)

where 119865 is the tactile force 119870 is the one-constant approx-imation with 119870 = 119870

11= 11987022

= 11987033 and 119899 is the LC

director Equation (2) is a torque balance equation relatingthe viscosity the elasticity and the torque due to tactile forceA cross product 119865 times 119899 represents a torque expressed by

119865 times 119899 = |119865| |119899| sin (120579 minus 120601) (3)

0 1 2 3 40

04

08

12

16

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(a)

0 1 2 3 4 5 60

04

08

12

16

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(b)

Figure 5 Comparison of molecule tilt angles among cell gapthicknesses of (a) 3 120583m and (b) 5 120583m when a tactile force is appliedto LCD screen with PAA liquid crystals

where |119899| = 1 Substituting (3) into (2) gives

120574

120597120579

120597119905

= 119870

1205972120579

1205971199112+ 119865 sin (120579 minus 120601) (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 thanelastic torque

1003816100381610038161003816119865 sin (120579 minus 120601)

1003816100381610038161003816≫

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579

1205971199112

100381610038161003816100381610038161003816100381610038161003816

(5)

4 Mathematical Problems in Engineering

0 1 2 3 40

001

002

003

004Ti

lt an

gle (

deg)

Cell gap (120583m)

1dyne05dyne01dyne

(a)

0 1 2 3 40

001

002

003

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(b)

0 1 2 3 40

01

02

03

04

05

06

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(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 toLCD screen

However when the tilt angle is small sin 120579 asymp 120579 and cos 120579 asymp 1

and (4) becomes

120574

120597120579

120597119905

= 119865 sdot (cos120601) sdot 120579 minus 119865 sdot sin120601 (6)

Case 2When torque due to tactile force ismuch smaller thanelastic torque

1003816100381610038161003816119865 sin (120579 minus 120601)

1003816100381610038161003816≪

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579

1205971199112

100381610038161003816100381610038161003816100381610038161003816

(7)

Equation (4) thus becomes

120574

120597120579

120597119905

= 119870

1205972120579

1205971199112 (8)

Based on separation of variables assume 120579(119905 119911) =

120579(119905) sin(120587119911119889) which is in turn substituted into (6) and (8)Case 1 thus leads to an equation for 120579 of the form

120579 = 119886 + 119887120579 (9)

where

119886 = minus

119865 sin120601120574

119887 =

119865 cos120601120574

(10)

The solution for 120579(119905 119911) is written as

120579 (119905 119911) = tan120601 (119890(119865 cos120601120574)119905) sin(120587119911119889

) (11)

Mathematical Problems in Engineering 5

0 1 2 3 4 50

2

4

6

8

10

12Ti

lt an

gle (

deg)

Cell gap (120583m)

10V20V30V40V50V

10V20V30V40V50V

(a)

0 1 2 3 4 50

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

10V20V30V40V50V

10V20V30V40V50V

(b)

0 1 2 3 4 50

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

10V20V30V40V50V

10V20V30V40V50V

(c)

Figure 7 Comparison of molecule tilt angles between cell gap thicknesses of 3 120583m and 5 120583mwhen 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 isnegligible the solution for 120579(119905 119911) in (8) is expressed by

120579 (119905 119911) = 119888119890minus(119870120587

21205741198892)119905 sin(120587119911

119889

) (12)

where 119888 is a constant Since the tactile force is ignored inCase 2 only Case 1 is further examined

22 Two-Dimensional Model The above derivation is basedon 1D LC model Further this study intends to use (1) todevelop an LC tilt model in two dimensions Assume that thetheoretical model satisfies the following four conditions asdepicted in Figure 2

Condition 1 The molecule ignores the moment of inertiathat is119872 = 0

Condition 2 Shear rate 119904 = 119904(119910) = 120597V120597119910 = V(1198893) where Vis the flow velocity and 119889 is the cell thickness

Condition 3 The LC sample is homeotropic alignment andwe have the boundary conditions

120579 (119909 119910 119905) = 120579 (119909 plusmn

119889

2

119905) = 0 (13)

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

6 Mathematical Problems in Engineering

0 1 2 3 4 50

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne35dyne

(a)

0 1 2 3 4 50

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne35dyne

(b)

0 1 2 3 4 50

2

4

6

8

10

14

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne35dyne

12

(c)

Figure 8 Comparison of molecule tilt angles between cell gapthicknesses of 3120583m and 5120583m when tactile force is applied to LCDscreen with (a) 5CB (b) MBBA and (c) PAA liquid crystals underdifferent tactile forces

25dyne3dyne

35dyne

0 1 2 30

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

(a)

0 1 2 30

2

4

6

8

10

12

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

(b)

Figure 9 (a) Comparison of molecule tilt angles depicted in3 120583m cell gap 5CB liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

According to the above assumptions (1) becomes

119870(

1205972120579

1205971199092+

1205972120579

1205971199102) minus 120574

120597120579

120597119905

+

119904

2

(1205741minus 1205742cos 2120579)

+ 12057601205761198861198642 sin 120579 cos 120579 + 119865 times 119899 = 0

(14)

Based on the Ericksen-Leslie theory (14) represents the tiltmodel of LC molecule subjected to both electric field andtactile force In order to use separation of variables assume120579 = 120579(119909 119910 119905) yields

120579 (119909 119910 119905) = 119891 (119910) 120579119898(119909 119905) (15)

Mathematical Problems in Engineering 7

0 1 2 30

2

4

6

8

10

Tilt

angl

e (de

g)

Cell gap (120583m)

4

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 10 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap 5CB liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

where

119891 (119910) =

119892 (119910)

119892 (minus119910119898)

=

1 or minus 1 119910 = minus119910119898or 119910119898

119892 (119910)

119892 (minus028119889)

= 47503 sdot 119892 (119910) 119910 = minus 119910119898or 119910119898

(16)

119892 (119910) =

2119910

119889

minus sin(119910120587

119889

) (17)

119910119898= (

119889

120587

) cosminus1 ( 2120587

) asymp 028119889 (18)

and119910119898andminus119910

119898are boundary conditions of function119892(119910) In

(15) 120579119898is the angle of the normal vector between the director

of LC molecules with homeotropic alignment Molecule

0 1 2 3 4 50

2

4

6

8

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 11 (a) Comparison of molecule tilt angles depicted in5 120583m cell gap 5CB liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

directions are changed by the Poiseuille flow Values of theangle along the 119910-axis have extreme values at 119910

119898and minus119910

119898

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

119870[

1205972

1205971199092(119891 (119910) 120579

119898(119909 119905)) +

1205972

1205971199102(119891 (119910) 120579

119898(119909 119905))]

minus 120574

120597

120597119905

(119891 (119910) 120579119898(119909 119905)) +

120597V

120597119910

(1205723sin2120579 minus 120572

2cos2120579)

+ 12057601205761198861198642 sin 120579 cos 120579 + |119865| |119899| sin (120579 minus 120601) = 0

(19)

Since |119899| = 1 one has

|119865| |119899| sin (120579 minus 120601) = 119865 sdot sin (120579 minus 120601) (20)

8 Mathematical Problems in Engineering

0 1 2 30

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

12

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

(b)

Figure 12 (a) Comparison of molecule tilt angles depicted in3120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Therefore (19) becomes

119870[119891 (119910)

1205972120579119898

1205971199092

+ 0] + [120579119898

1205972119891 (119910)

1205971199102

+ 0] minus 120574119891 (119910)

120597120579119898

120597119905

+

120597V

120597119910

[1205723sin2 (119891 (119910) 120579

119898) minus 1205722cos2 (119891 (119910) 120579

119898)]

+ 12057601205761198861198642 sin (119891 (119910) 120579

119898) cos (119891 (119910) 120579

119898)

+ 119865 sin (119891 (119910) 120579119898minus 120601) = 0

(21)

As depicted in Figure 2 the director makes an angle 120579119898with

the 119910-axis Consequently 120579119898represents liquid crystal mole-

cules with homeotropic alignment caused by the Poiseuilleflow between director 119899 and normal direction of LC sampleAn effective director equation of motion with the essentialphysics preserved may be obtained by setting 119910 = minus119910

119898and

119910 = 119910119898in (21)

0 1 2 3 40

2

4

6

8

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 13 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

On boundary 119910 = minus119910119898in Figure 2

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

(1205723sin2120579119898minus 1205722cos2120579119898)

+ 12057601205761198861198642 sin 120579

119898cos 120579119898+ 119865 sin (120579

119898minus 120601) = 0

(22)

By using trigonometric formulas one can rewrite (22) as

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

times [1205723(

1 minus cos 2120579119898

2

) minus 1205722(

1 + cos 2120579119898

2

)]

+ 12057601205761198861198642120579119898+ 119865 (sin 120579

119898cos120601 minus cos 120579

119898sin120601) = 0

(23)

Mathematical Problems in Engineering 9

0 1 2 3 4 50

2

4

6

8Ti

lt an

gle (

deg)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 14 (a) Comparison of molecule tilt angles depicted in5120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Because sin 120579119898

asymp 120579119898and cos 120579

119898asymp 1 for small 120579

119898 (23) is

rewritten as

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

[1205723(

1 minus 1

2

) minus 1205722(

1 + 1

2

)]

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601) = 0

(24)

Thus

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898

+ 119865 (120579119898cos120601 minus sin120601) = 0

(25)

Hence

120574

120597120579119898

120597119905

=119870

1205972120579119898

1205971199092

minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

(26)

0 1 2 30

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

14

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

12

14

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

(b)

Figure 15 (a) Comparison of molecule tilt angles depicted in3 120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Substituting 120579119898= 120579119898(119905) sin(120587119909119889) for separation of variables

into (26) leads to

120574

120579119898= 119870

1205972

1205971199092[120579119898(119905) sin(120587119909

119889

)] minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

(27)

As a result

120574

120579119898= minus

1198701205872

1198892120579119898(119909 119905) + (120576

01205761198861198642+ 119865 cos120601) 120579

119898

+ (minus1205722

120597V

120597119910

minus 119865 sin120601)

(28)

10 Mathematical Problems in Engineering

0 1 2 30

2

4

6

8

10

Tilt

angl

e (de

g)

Cell gap (120583m)

4

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 16 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne by analyticaland numerical results

Similarly we consider the following two cases

Case 11015840

10038161003816100381610038161003816100381610038161003816

minus1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

10038161003816100381610038161003816100381610038161003816

≫

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579119898

1205971199092

100381610038161003816100381610038161003816100381610038161003816

(29)

120574

120579119898= (12057601205761198861198642+ 119865 cos120601) 120579

119898+ (minus120572

2

120597V

120597119910

minus 119865 sin120601) (30)

Assuming 120579119898= 119886 + 119887120579

119898 one has

119886 =

1

120574

(minus1205722

120597V

120597119910

minus 119865 sin120601)

119887 =

1

120574

(12057601205761198861198642+ 119865 cos120601)

(31)

Therefore

120579119898=

119886

119887

(119890119887119905minus 1) (32)

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

120579119898(119909 119905)

=[

minus1205722(120597V120597119910) minus 119865 sin120601

12057601205761198861198642+ 119865 cos120601

(119890((12057601205761198861198642+119865 cos120601)120574)119905

minus 1)]sin(120587119909119889

)

(33)

Consider the following

Case 21015840

10038161003816100381610038161003816100381610038161003816

minus1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

10038161003816100381610038161003816100381610038161003816

≪

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579119898

1205971199092

100381610038161003816100381610038161003816100381610038161003816

(34)

Since the tactile force and electric field are ignored in Case 21015840only Case 11015840 is further examined

On boundary 119910 = 119910119898in Figure 2

120579119898(119909 119905)

=[

1205722(120597V120597119910) + 119865 sin12060112057601205761198861198642+ 119865 cos120601

(119890((12057601205761198861198642+119865 cos120601)120574)119905

minus 1)]sin(120587119909119889

)

(35)

Since 120579119898(119909 119905) in (33) is equal to minus120579

119898(119909 119905) in (35) the tilt

angles are symmetric

3 Discussion

Subject to both electric field of 89 V and tactile forcesimulation results for Case 1 are depicted in Figures 3 4and 5 Comparing different LC materials including 5CBMBBA and PAA subjected to 10ms duration tactile forcesof 1 dyne 05 dyne and 01 dyne at 10ms Figures 3 4 and 5show that a larger force makes the tilt angle larger regardlessof LC gap thickness Figures 6(a) 6(b) and 6(c) compare3 120583m thick different LCmaterials including 5CB MBBA andPAA subjected to 5ms duration 05 dyne force respectivelyThe viscosity of PAA is the smallest and the tilt angle islarger than the others Therefore the option of liquid crystalmaterial with smaller viscosity can reduce the ripple

Tilt angle variations are shown under different voltages inFigure 7 and under different tactile forces in Figure 8 Using10V 20V 30V 40V and 50V under a tactile force of 3 dyneit is observed that the tilt angles remain the same in Figures7(a) 7(b) and 7(c) Thus different voltages do not affect tiltangles

Based on (33) Figures 8(a) 8(b) and 8(c) compare differ-ent LC materials including 5CB MBBA and PAA subject to25 dyne 3 dyne and 35 dyne forces in 3120583m and 5 120583m thickcell gap LC respectively Accordingly tactile forces affect tiltangles obviously Figures 9(a) 10(a) and 11(a) respectivelyshow tilt angles of 5CB LC subject to 25 dyne 3 dyne and

Mathematical Problems in Engineering 11

0 1 2 3 4 50

2

4

6

8Ti

lt an

gle (

deg)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 17 (a) Comparison of molecule tilt angles depicted in5120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

35 dyne forces in 3 4 and 5 120583m thick cell gap LC SimilarlyFigures 12(a) 13(a) and 14(a) are MBBA results and Figures15(a) 16(a) and 17(a) are PAA results In addition Figures9(b) to 17(b) compare analytical solutions with numericalsolutions of the full coupled equations The comparisonresults show that analytical solutions are consistent withnumerical solutions of the full coupled equations Thereforethe analytical derivation is validated

4 Conclusion

The Ericksen-Leslie theory deals with the dynamics ofnematic liquid crystals Based on the Ericksen-Leslie theorythis paper contributes to LC dynamics subject to tactile forceLC molecule tilt leads to a change in the optical intensitytransmitted through a liquid crystal cell LCD panels subjectto tactile force will show ripple-like propagation on screensThe present results show that the viscosity tactile force

thickness of cell gap and Leslie viscosity coefficient are thefactors of tilt angle variation Tilt angle variations of PAAliquid crystal molecules are sensitive to tactile force mag-nitudes while those of 5CB and MBBA with larger viscosityare not Analytical derivation has been carried out in thisstudy and analytical results have been validated by numericalresults

Acknowledgments

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

References

[1] J Ericksen ldquoConservation laws for liquid crystalsrdquoTransactionsof the Society of Rheology vol 5 pp 22ndash34 1961

[2] J L Ericksen ldquoContinuum theory of nematic liquid crystalsrdquoRes Mechanica vol 21 no 4 pp 381ndash392 1987

[3] F M Leslie ldquoTheory of flow phenomenum in liquid crystalsrdquoAdvances in Liquid Crystals vol 4 pp 1ndash81 1979

[4] F Brochard P Pieranski and E Guyon ldquoDynamics of the ori-entation of a nematic-liquid-crystal film in a variable magneticfieldrdquoPhysical Review Letters vol 28 no 26 pp 1681ndash1683 1972

[5] L Lei S Changqing S Juelian P M Lam and H Yun ldquoSolitonpropagation in liquid crystalsrdquo Physical Review Letters vol 49no 18 pp 1335ndash1338 1982

[6] F M Leslie ldquoSome topics in continuum theory of nematicsrdquoPhilosophical Transactions of the Royal Society A vol 309 pp155ndash165 1983

[7] J T Gleeson P Palffy-Muhoray and W van Saarloos ldquoProp-agation of excitations induced by shear flow in nematic liquidcrystalsrdquo Physical Review A vol 44 no 4 pp 2588ndash2595 1991

[8] F H Lin and C Liu ldquoExistence of solutions for the Ericksen-Leslie systemrdquo Archive for Rational Mechanics and Analysis vol154 no 2 pp 135ndash156 2000

[9] L R P de Andrade Lima and A D Rey ldquoPoiseuille flowof Leslie-Ericksen discotic liquid crystal Solution multiplicitymultistability and non-Newtonian rheologyrdquo Journal of Non-Newtonian Fluid Mechanics vol 110 no 2-3 pp 103ndash142 2003

[10] X Nie R Lu H Xianyu T X Wu and S T Wu ldquoAnchoringenergy and cell gap effects on liquid crystal response timerdquoJournal of Applied Physics vol 101 no 10 Article ID 103110 2007

[11] P A Cruz M F Tome I W Stewart and S McKee ldquoAnumerical method for solving the dynamic three-dimensionalEricksen-Leslie equations for nematic liquid crystals subjectto a strong magnetic fieldrdquo Journal of Non-Newtonian FluidMechanics vol 165 no 3-4 pp 143ndash157 2010

[12] X Nie H Xianyu R Lu T X Wu and S T Wu ldquoPretilt angleeffects on liquid crystal response timerdquo IEEEOSA Journal ofDisplay Technology vol 3 no 3 pp 280ndash283 2007

[13] I C Khoo and S T Wu Optics and Nonlinear Optics of LiquidCrystals World Scientific 1993

[14] I C Khoo Liquid Crystals Physical Properties and NonlinearOptical Phenomena John Wiley amp Sons New York NY USA1995

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Decision SciencesAdvances in

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Volume 2014

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Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

0 1 2 3 40

001

002

003

004Ti

lt an

gle (

deg)

Cell gap (120583m)

1dyne05dyne01dyne

(a)

0 1 2 3 40

001

002

003

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(b)

0 1 2 3 40

01

02

03

04

05

06

Tilt

angl

e (de

g)

Cell gap (120583m)

1dyne05dyne01dyne

(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 toLCD screen

However when the tilt angle is small sin 120579 asymp 120579 and cos 120579 asymp 1

and (4) becomes

120574

120597120579

120597119905

= 119865 sdot (cos120601) sdot 120579 minus 119865 sdot sin120601 (6)

Case 2When torque due to tactile force ismuch smaller thanelastic torque

1003816100381610038161003816119865 sin (120579 minus 120601)

1003816100381610038161003816≪

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579

1205971199112

100381610038161003816100381610038161003816100381610038161003816

(7)

Equation (4) thus becomes

120574

120597120579

120597119905

= 119870

1205972120579

1205971199112 (8)

Based on separation of variables assume 120579(119905 119911) =

120579(119905) sin(120587119911119889) which is in turn substituted into (6) and (8)Case 1 thus leads to an equation for 120579 of the form

120579 = 119886 + 119887120579 (9)

where

119886 = minus

119865 sin120601120574

119887 =

119865 cos120601120574

(10)

The solution for 120579(119905 119911) is written as

120579 (119905 119911) = tan120601 (119890(119865 cos120601120574)119905) sin(120587119911119889

) (11)

Mathematical Problems in Engineering 5

0 1 2 3 4 50

2

4

6

8

10

12Ti

lt an

gle (

deg)

Cell gap (120583m)

10V20V30V40V50V

10V20V30V40V50V

(a)

0 1 2 3 4 50

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

10V20V30V40V50V

10V20V30V40V50V

(b)

0 1 2 3 4 50

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

10V20V30V40V50V

10V20V30V40V50V

(c)

Figure 7 Comparison of molecule tilt angles between cell gap thicknesses of 3 120583m and 5 120583mwhen 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 isnegligible the solution for 120579(119905 119911) in (8) is expressed by

120579 (119905 119911) = 119888119890minus(119870120587

21205741198892)119905 sin(120587119911

119889

) (12)

where 119888 is a constant Since the tactile force is ignored inCase 2 only Case 1 is further examined

22 Two-Dimensional Model The above derivation is basedon 1D LC model Further this study intends to use (1) todevelop an LC tilt model in two dimensions Assume that thetheoretical model satisfies the following four conditions asdepicted in Figure 2

Condition 1 The molecule ignores the moment of inertiathat is119872 = 0

Condition 2 Shear rate 119904 = 119904(119910) = 120597V120597119910 = V(1198893) where Vis the flow velocity and 119889 is the cell thickness

Condition 3 The LC sample is homeotropic alignment andwe have the boundary conditions

120579 (119909 119910 119905) = 120579 (119909 plusmn

119889

2

119905) = 0 (13)

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

6 Mathematical Problems in Engineering

0 1 2 3 4 50

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne35dyne

(a)

0 1 2 3 4 50

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne35dyne

(b)

0 1 2 3 4 50

2

4

6

8

10

14

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne35dyne

12

(c)

Figure 8 Comparison of molecule tilt angles between cell gapthicknesses of 3120583m and 5120583m when tactile force is applied to LCDscreen with (a) 5CB (b) MBBA and (c) PAA liquid crystals underdifferent tactile forces

25dyne3dyne

35dyne

0 1 2 30

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

(a)

0 1 2 30

2

4

6

8

10

12

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

(b)

Figure 9 (a) Comparison of molecule tilt angles depicted in3 120583m cell gap 5CB liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

According to the above assumptions (1) becomes

119870(

1205972120579

1205971199092+

1205972120579

1205971199102) minus 120574

120597120579

120597119905

+

119904

2

(1205741minus 1205742cos 2120579)

+ 12057601205761198861198642 sin 120579 cos 120579 + 119865 times 119899 = 0

(14)

Based on the Ericksen-Leslie theory (14) represents the tiltmodel of LC molecule subjected to both electric field andtactile force In order to use separation of variables assume120579 = 120579(119909 119910 119905) yields

120579 (119909 119910 119905) = 119891 (119910) 120579119898(119909 119905) (15)

Mathematical Problems in Engineering 7

0 1 2 30

2

4

6

8

10

Tilt

angl

e (de

g)

Cell gap (120583m)

4

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 10 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap 5CB liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

where

119891 (119910) =

119892 (119910)

119892 (minus119910119898)

=

1 or minus 1 119910 = minus119910119898or 119910119898

119892 (119910)

119892 (minus028119889)

= 47503 sdot 119892 (119910) 119910 = minus 119910119898or 119910119898

(16)

119892 (119910) =

2119910

119889

minus sin(119910120587

119889

) (17)

119910119898= (

119889

120587

) cosminus1 ( 2120587

) asymp 028119889 (18)

and119910119898andminus119910

119898are boundary conditions of function119892(119910) In

(15) 120579119898is the angle of the normal vector between the director

of LC molecules with homeotropic alignment Molecule

0 1 2 3 4 50

2

4

6

8

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 11 (a) Comparison of molecule tilt angles depicted in5 120583m cell gap 5CB liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

directions are changed by the Poiseuille flow Values of theangle along the 119910-axis have extreme values at 119910

119898and minus119910

119898

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

119870[

1205972

1205971199092(119891 (119910) 120579

119898(119909 119905)) +

1205972

1205971199102(119891 (119910) 120579

119898(119909 119905))]

minus 120574

120597

120597119905

(119891 (119910) 120579119898(119909 119905)) +

120597V

120597119910

(1205723sin2120579 minus 120572

2cos2120579)

+ 12057601205761198861198642 sin 120579 cos 120579 + |119865| |119899| sin (120579 minus 120601) = 0

(19)

Since |119899| = 1 one has

|119865| |119899| sin (120579 minus 120601) = 119865 sdot sin (120579 minus 120601) (20)

8 Mathematical Problems in Engineering

0 1 2 30

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

12

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

(b)

Figure 12 (a) Comparison of molecule tilt angles depicted in3120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Therefore (19) becomes

119870[119891 (119910)

1205972120579119898

1205971199092

+ 0] + [120579119898

1205972119891 (119910)

1205971199102

+ 0] minus 120574119891 (119910)

120597120579119898

120597119905

+

120597V

120597119910

[1205723sin2 (119891 (119910) 120579

119898) minus 1205722cos2 (119891 (119910) 120579

119898)]

+ 12057601205761198861198642 sin (119891 (119910) 120579

119898) cos (119891 (119910) 120579

119898)

+ 119865 sin (119891 (119910) 120579119898minus 120601) = 0

(21)

As depicted in Figure 2 the director makes an angle 120579119898with

the 119910-axis Consequently 120579119898represents liquid crystal mole-

cules with homeotropic alignment caused by the Poiseuilleflow between director 119899 and normal direction of LC sampleAn effective director equation of motion with the essentialphysics preserved may be obtained by setting 119910 = minus119910

119898and

119910 = 119910119898in (21)

0 1 2 3 40

2

4

6

8

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 13 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

On boundary 119910 = minus119910119898in Figure 2

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

(1205723sin2120579119898minus 1205722cos2120579119898)

+ 12057601205761198861198642 sin 120579

119898cos 120579119898+ 119865 sin (120579

119898minus 120601) = 0

(22)

By using trigonometric formulas one can rewrite (22) as

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

times [1205723(

1 minus cos 2120579119898

2

) minus 1205722(

1 + cos 2120579119898

2

)]

+ 12057601205761198861198642120579119898+ 119865 (sin 120579

119898cos120601 minus cos 120579

119898sin120601) = 0

(23)

Mathematical Problems in Engineering 9

0 1 2 3 4 50

2

4

6

8Ti

lt an

gle (

deg)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 14 (a) Comparison of molecule tilt angles depicted in5120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Because sin 120579119898

asymp 120579119898and cos 120579

119898asymp 1 for small 120579

119898 (23) is

rewritten as

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

[1205723(

1 minus 1

2

) minus 1205722(

1 + 1

2

)]

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601) = 0

(24)

Thus

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898

+ 119865 (120579119898cos120601 minus sin120601) = 0

(25)

Hence

120574

120597120579119898

120597119905

=119870

1205972120579119898

1205971199092

minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

(26)

0 1 2 30

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

14

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

12

14

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

(b)

Figure 15 (a) Comparison of molecule tilt angles depicted in3 120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Substituting 120579119898= 120579119898(119905) sin(120587119909119889) for separation of variables

into (26) leads to

120574

120579119898= 119870

1205972

1205971199092[120579119898(119905) sin(120587119909

119889

)] minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

(27)

As a result

120574

120579119898= minus

1198701205872

1198892120579119898(119909 119905) + (120576

01205761198861198642+ 119865 cos120601) 120579

119898

+ (minus1205722

120597V

120597119910

minus 119865 sin120601)

(28)

10 Mathematical Problems in Engineering

0 1 2 30

2

4

6

8

10

Tilt

angl

e (de

g)

Cell gap (120583m)

4

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 16 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne by analyticaland numerical results

Similarly we consider the following two cases

Case 11015840

10038161003816100381610038161003816100381610038161003816

minus1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

10038161003816100381610038161003816100381610038161003816

≫

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579119898

1205971199092

100381610038161003816100381610038161003816100381610038161003816

(29)

120574

120579119898= (12057601205761198861198642+ 119865 cos120601) 120579

119898+ (minus120572

2

120597V

120597119910

minus 119865 sin120601) (30)

Assuming 120579119898= 119886 + 119887120579

119898 one has

119886 =

1

120574

(minus1205722

120597V

120597119910

minus 119865 sin120601)

119887 =

1

120574

(12057601205761198861198642+ 119865 cos120601)

(31)

Therefore

120579119898=

119886

119887

(119890119887119905minus 1) (32)

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

120579119898(119909 119905)

=[

minus1205722(120597V120597119910) minus 119865 sin120601

12057601205761198861198642+ 119865 cos120601

(119890((12057601205761198861198642+119865 cos120601)120574)119905

minus 1)]sin(120587119909119889

)

(33)

Consider the following

Case 21015840

10038161003816100381610038161003816100381610038161003816

minus1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

10038161003816100381610038161003816100381610038161003816

≪

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579119898

1205971199092

100381610038161003816100381610038161003816100381610038161003816

(34)

Since the tactile force and electric field are ignored in Case 21015840only Case 11015840 is further examined

On boundary 119910 = 119910119898in Figure 2

120579119898(119909 119905)

=[

1205722(120597V120597119910) + 119865 sin12060112057601205761198861198642+ 119865 cos120601

(119890((12057601205761198861198642+119865 cos120601)120574)119905

minus 1)]sin(120587119909119889

)

(35)

Since 120579119898(119909 119905) in (33) is equal to minus120579

119898(119909 119905) in (35) the tilt

angles are symmetric

3 Discussion

Subject to both electric field of 89 V and tactile forcesimulation results for Case 1 are depicted in Figures 3 4and 5 Comparing different LC materials including 5CBMBBA and PAA subjected to 10ms duration tactile forcesof 1 dyne 05 dyne and 01 dyne at 10ms Figures 3 4 and 5show that a larger force makes the tilt angle larger regardlessof LC gap thickness Figures 6(a) 6(b) and 6(c) compare3 120583m thick different LCmaterials including 5CB MBBA andPAA subjected to 5ms duration 05 dyne force respectivelyThe viscosity of PAA is the smallest and the tilt angle islarger than the others Therefore the option of liquid crystalmaterial with smaller viscosity can reduce the ripple

Tilt angle variations are shown under different voltages inFigure 7 and under different tactile forces in Figure 8 Using10V 20V 30V 40V and 50V under a tactile force of 3 dyneit is observed that the tilt angles remain the same in Figures7(a) 7(b) and 7(c) Thus different voltages do not affect tiltangles

Based on (33) Figures 8(a) 8(b) and 8(c) compare differ-ent LC materials including 5CB MBBA and PAA subject to25 dyne 3 dyne and 35 dyne forces in 3120583m and 5 120583m thickcell gap LC respectively Accordingly tactile forces affect tiltangles obviously Figures 9(a) 10(a) and 11(a) respectivelyshow tilt angles of 5CB LC subject to 25 dyne 3 dyne and

Mathematical Problems in Engineering 11

0 1 2 3 4 50

2

4

6

8Ti

lt an

gle (

deg)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 17 (a) Comparison of molecule tilt angles depicted in5120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

35 dyne forces in 3 4 and 5 120583m thick cell gap LC SimilarlyFigures 12(a) 13(a) and 14(a) are MBBA results and Figures15(a) 16(a) and 17(a) are PAA results In addition Figures9(b) to 17(b) compare analytical solutions with numericalsolutions of the full coupled equations The comparisonresults show that analytical solutions are consistent withnumerical solutions of the full coupled equations Thereforethe analytical derivation is validated

4 Conclusion

The Ericksen-Leslie theory deals with the dynamics ofnematic liquid crystals Based on the Ericksen-Leslie theorythis paper contributes to LC dynamics subject to tactile forceLC molecule tilt leads to a change in the optical intensitytransmitted through a liquid crystal cell LCD panels subjectto tactile force will show ripple-like propagation on screensThe present results show that the viscosity tactile force

thickness of cell gap and Leslie viscosity coefficient are thefactors of tilt angle variation Tilt angle variations of PAAliquid crystal molecules are sensitive to tactile force mag-nitudes while those of 5CB and MBBA with larger viscosityare not Analytical derivation has been carried out in thisstudy and analytical results have been validated by numericalresults

Acknowledgments

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

References

[1] J Ericksen ldquoConservation laws for liquid crystalsrdquoTransactionsof the Society of Rheology vol 5 pp 22ndash34 1961

[2] J L Ericksen ldquoContinuum theory of nematic liquid crystalsrdquoRes Mechanica vol 21 no 4 pp 381ndash392 1987

[3] F M Leslie ldquoTheory of flow phenomenum in liquid crystalsrdquoAdvances in Liquid Crystals vol 4 pp 1ndash81 1979

[4] F Brochard P Pieranski and E Guyon ldquoDynamics of the ori-entation of a nematic-liquid-crystal film in a variable magneticfieldrdquoPhysical Review Letters vol 28 no 26 pp 1681ndash1683 1972

[5] L Lei S Changqing S Juelian P M Lam and H Yun ldquoSolitonpropagation in liquid crystalsrdquo Physical Review Letters vol 49no 18 pp 1335ndash1338 1982

[6] F M Leslie ldquoSome topics in continuum theory of nematicsrdquoPhilosophical Transactions of the Royal Society A vol 309 pp155ndash165 1983

[7] J T Gleeson P Palffy-Muhoray and W van Saarloos ldquoProp-agation of excitations induced by shear flow in nematic liquidcrystalsrdquo Physical Review A vol 44 no 4 pp 2588ndash2595 1991

[8] F H Lin and C Liu ldquoExistence of solutions for the Ericksen-Leslie systemrdquo Archive for Rational Mechanics and Analysis vol154 no 2 pp 135ndash156 2000

[9] L R P de Andrade Lima and A D Rey ldquoPoiseuille flowof Leslie-Ericksen discotic liquid crystal Solution multiplicitymultistability and non-Newtonian rheologyrdquo Journal of Non-Newtonian Fluid Mechanics vol 110 no 2-3 pp 103ndash142 2003

[10] X Nie R Lu H Xianyu T X Wu and S T Wu ldquoAnchoringenergy and cell gap effects on liquid crystal response timerdquoJournal of Applied Physics vol 101 no 10 Article ID 103110 2007

[11] P A Cruz M F Tome I W Stewart and S McKee ldquoAnumerical method for solving the dynamic three-dimensionalEricksen-Leslie equations for nematic liquid crystals subjectto a strong magnetic fieldrdquo Journal of Non-Newtonian FluidMechanics vol 165 no 3-4 pp 143ndash157 2010

[12] X Nie H Xianyu R Lu T X Wu and S T Wu ldquoPretilt angleeffects on liquid crystal response timerdquo IEEEOSA Journal ofDisplay Technology vol 3 no 3 pp 280ndash283 2007

[13] I C Khoo and S T Wu Optics and Nonlinear Optics of LiquidCrystals World Scientific 1993

[14] I C Khoo Liquid Crystals Physical Properties and NonlinearOptical Phenomena John Wiley amp Sons New York NY USA1995

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

0 1 2 3 4 50

2

4

6

8

10

12Ti

lt an

gle (

deg)

Cell gap (120583m)

10V20V30V40V50V

10V20V30V40V50V

(a)

0 1 2 3 4 50

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

10V20V30V40V50V

10V20V30V40V50V

(b)

0 1 2 3 4 50

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

10V20V30V40V50V

10V20V30V40V50V

(c)

Figure 7 Comparison of molecule tilt angles between cell gap thicknesses of 3 120583m and 5 120583mwhen 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 isnegligible the solution for 120579(119905 119911) in (8) is expressed by

120579 (119905 119911) = 119888119890minus(119870120587

21205741198892)119905 sin(120587119911

119889

) (12)

where 119888 is a constant Since the tactile force is ignored inCase 2 only Case 1 is further examined

22 Two-Dimensional Model The above derivation is basedon 1D LC model Further this study intends to use (1) todevelop an LC tilt model in two dimensions Assume that thetheoretical model satisfies the following four conditions asdepicted in Figure 2

Condition 1 The molecule ignores the moment of inertiathat is119872 = 0

Condition 2 Shear rate 119904 = 119904(119910) = 120597V120597119910 = V(1198893) where Vis the flow velocity and 119889 is the cell thickness

Condition 3 The LC sample is homeotropic alignment andwe have the boundary conditions

120579 (119909 119910 119905) = 120579 (119909 plusmn

119889

2

119905) = 0 (13)

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

6 Mathematical Problems in Engineering

0 1 2 3 4 50

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne35dyne

(a)

0 1 2 3 4 50

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne35dyne

(b)

0 1 2 3 4 50

2

4

6

8

10

14

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne35dyne

12

(c)

Figure 8 Comparison of molecule tilt angles between cell gapthicknesses of 3120583m and 5120583m when tactile force is applied to LCDscreen with (a) 5CB (b) MBBA and (c) PAA liquid crystals underdifferent tactile forces

25dyne3dyne

35dyne

0 1 2 30

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

(a)

0 1 2 30

2

4

6

8

10

12

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

(b)

Figure 9 (a) Comparison of molecule tilt angles depicted in3 120583m cell gap 5CB liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

According to the above assumptions (1) becomes

119870(

1205972120579

1205971199092+

1205972120579

1205971199102) minus 120574

120597120579

120597119905

+

119904

2

(1205741minus 1205742cos 2120579)

+ 12057601205761198861198642 sin 120579 cos 120579 + 119865 times 119899 = 0

(14)

Based on the Ericksen-Leslie theory (14) represents the tiltmodel of LC molecule subjected to both electric field andtactile force In order to use separation of variables assume120579 = 120579(119909 119910 119905) yields

120579 (119909 119910 119905) = 119891 (119910) 120579119898(119909 119905) (15)

Mathematical Problems in Engineering 7

0 1 2 30

2

4

6

8

10

Tilt

angl

e (de

g)

Cell gap (120583m)

4

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 10 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap 5CB liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

where

119891 (119910) =

119892 (119910)

119892 (minus119910119898)

=

1 or minus 1 119910 = minus119910119898or 119910119898

119892 (119910)

119892 (minus028119889)

= 47503 sdot 119892 (119910) 119910 = minus 119910119898or 119910119898

(16)

119892 (119910) =

2119910

119889

minus sin(119910120587

119889

) (17)

119910119898= (

119889

120587

) cosminus1 ( 2120587

) asymp 028119889 (18)

and119910119898andminus119910

119898are boundary conditions of function119892(119910) In

(15) 120579119898is the angle of the normal vector between the director

of LC molecules with homeotropic alignment Molecule

0 1 2 3 4 50

2

4

6

8

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 11 (a) Comparison of molecule tilt angles depicted in5 120583m cell gap 5CB liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

directions are changed by the Poiseuille flow Values of theangle along the 119910-axis have extreme values at 119910

119898and minus119910

119898

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

119870[

1205972

1205971199092(119891 (119910) 120579

119898(119909 119905)) +

1205972

1205971199102(119891 (119910) 120579

119898(119909 119905))]

minus 120574

120597

120597119905

(119891 (119910) 120579119898(119909 119905)) +

120597V

120597119910

(1205723sin2120579 minus 120572

2cos2120579)

+ 12057601205761198861198642 sin 120579 cos 120579 + |119865| |119899| sin (120579 minus 120601) = 0

(19)

Since |119899| = 1 one has

|119865| |119899| sin (120579 minus 120601) = 119865 sdot sin (120579 minus 120601) (20)

8 Mathematical Problems in Engineering

0 1 2 30

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

12

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

(b)

Figure 12 (a) Comparison of molecule tilt angles depicted in3120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Therefore (19) becomes

119870[119891 (119910)

1205972120579119898

1205971199092

+ 0] + [120579119898

1205972119891 (119910)

1205971199102

+ 0] minus 120574119891 (119910)

120597120579119898

120597119905

+

120597V

120597119910

[1205723sin2 (119891 (119910) 120579

119898) minus 1205722cos2 (119891 (119910) 120579

119898)]

+ 12057601205761198861198642 sin (119891 (119910) 120579

119898) cos (119891 (119910) 120579

119898)

+ 119865 sin (119891 (119910) 120579119898minus 120601) = 0

(21)

As depicted in Figure 2 the director makes an angle 120579119898with

the 119910-axis Consequently 120579119898represents liquid crystal mole-

cules with homeotropic alignment caused by the Poiseuilleflow between director 119899 and normal direction of LC sampleAn effective director equation of motion with the essentialphysics preserved may be obtained by setting 119910 = minus119910

119898and

119910 = 119910119898in (21)

0 1 2 3 40

2

4

6

8

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 13 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

On boundary 119910 = minus119910119898in Figure 2

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

(1205723sin2120579119898minus 1205722cos2120579119898)

+ 12057601205761198861198642 sin 120579

119898cos 120579119898+ 119865 sin (120579

119898minus 120601) = 0

(22)

By using trigonometric formulas one can rewrite (22) as

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

times [1205723(

1 minus cos 2120579119898

2

) minus 1205722(

1 + cos 2120579119898

2

)]

+ 12057601205761198861198642120579119898+ 119865 (sin 120579

119898cos120601 minus cos 120579

119898sin120601) = 0

(23)

Mathematical Problems in Engineering 9

0 1 2 3 4 50

2

4

6

8Ti

lt an

gle (

deg)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 14 (a) Comparison of molecule tilt angles depicted in5120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Because sin 120579119898

asymp 120579119898and cos 120579

119898asymp 1 for small 120579

119898 (23) is

rewritten as

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

[1205723(

1 minus 1

2

) minus 1205722(

1 + 1

2

)]

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601) = 0

(24)

Thus

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898

+ 119865 (120579119898cos120601 minus sin120601) = 0

(25)

Hence

120574

120597120579119898

120597119905

=119870

1205972120579119898

1205971199092

minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

(26)

0 1 2 30

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

14

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

12

14

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

(b)

Figure 15 (a) Comparison of molecule tilt angles depicted in3 120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Substituting 120579119898= 120579119898(119905) sin(120587119909119889) for separation of variables

into (26) leads to

120574

120579119898= 119870

1205972

1205971199092[120579119898(119905) sin(120587119909

119889

)] minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

(27)

As a result

120574

120579119898= minus

1198701205872

1198892120579119898(119909 119905) + (120576

01205761198861198642+ 119865 cos120601) 120579

119898

+ (minus1205722

120597V

120597119910

minus 119865 sin120601)

(28)

10 Mathematical Problems in Engineering

0 1 2 30

2

4

6

8

10

Tilt

angl

e (de

g)

Cell gap (120583m)

4

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 16 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne by analyticaland numerical results

Similarly we consider the following two cases

Case 11015840

10038161003816100381610038161003816100381610038161003816

minus1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

10038161003816100381610038161003816100381610038161003816

≫

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579119898

1205971199092

100381610038161003816100381610038161003816100381610038161003816

(29)

120574

120579119898= (12057601205761198861198642+ 119865 cos120601) 120579

119898+ (minus120572

2

120597V

120597119910

minus 119865 sin120601) (30)

Assuming 120579119898= 119886 + 119887120579

119898 one has

119886 =

1

120574

(minus1205722

120597V

120597119910

minus 119865 sin120601)

119887 =

1

120574

(12057601205761198861198642+ 119865 cos120601)

(31)

Therefore

120579119898=

119886

119887

(119890119887119905minus 1) (32)

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

120579119898(119909 119905)

=[

minus1205722(120597V120597119910) minus 119865 sin120601

12057601205761198861198642+ 119865 cos120601

(119890((12057601205761198861198642+119865 cos120601)120574)119905

minus 1)]sin(120587119909119889

)

(33)

Consider the following

Case 21015840

10038161003816100381610038161003816100381610038161003816

minus1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

10038161003816100381610038161003816100381610038161003816

≪

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579119898

1205971199092

100381610038161003816100381610038161003816100381610038161003816

(34)

Since the tactile force and electric field are ignored in Case 21015840only Case 11015840 is further examined

On boundary 119910 = 119910119898in Figure 2

120579119898(119909 119905)

=[

1205722(120597V120597119910) + 119865 sin12060112057601205761198861198642+ 119865 cos120601

(119890((12057601205761198861198642+119865 cos120601)120574)119905

minus 1)]sin(120587119909119889

)

(35)

Since 120579119898(119909 119905) in (33) is equal to minus120579

119898(119909 119905) in (35) the tilt

angles are symmetric

3 Discussion

Subject to both electric field of 89 V and tactile forcesimulation results for Case 1 are depicted in Figures 3 4and 5 Comparing different LC materials including 5CBMBBA and PAA subjected to 10ms duration tactile forcesof 1 dyne 05 dyne and 01 dyne at 10ms Figures 3 4 and 5show that a larger force makes the tilt angle larger regardlessof LC gap thickness Figures 6(a) 6(b) and 6(c) compare3 120583m thick different LCmaterials including 5CB MBBA andPAA subjected to 5ms duration 05 dyne force respectivelyThe viscosity of PAA is the smallest and the tilt angle islarger than the others Therefore the option of liquid crystalmaterial with smaller viscosity can reduce the ripple

Tilt angle variations are shown under different voltages inFigure 7 and under different tactile forces in Figure 8 Using10V 20V 30V 40V and 50V under a tactile force of 3 dyneit is observed that the tilt angles remain the same in Figures7(a) 7(b) and 7(c) Thus different voltages do not affect tiltangles

Based on (33) Figures 8(a) 8(b) and 8(c) compare differ-ent LC materials including 5CB MBBA and PAA subject to25 dyne 3 dyne and 35 dyne forces in 3120583m and 5 120583m thickcell gap LC respectively Accordingly tactile forces affect tiltangles obviously Figures 9(a) 10(a) and 11(a) respectivelyshow tilt angles of 5CB LC subject to 25 dyne 3 dyne and

Mathematical Problems in Engineering 11

0 1 2 3 4 50

2

4

6

8Ti

lt an

gle (

deg)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 17 (a) Comparison of molecule tilt angles depicted in5120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

35 dyne forces in 3 4 and 5 120583m thick cell gap LC SimilarlyFigures 12(a) 13(a) and 14(a) are MBBA results and Figures15(a) 16(a) and 17(a) are PAA results In addition Figures9(b) to 17(b) compare analytical solutions with numericalsolutions of the full coupled equations The comparisonresults show that analytical solutions are consistent withnumerical solutions of the full coupled equations Thereforethe analytical derivation is validated

4 Conclusion

The Ericksen-Leslie theory deals with the dynamics ofnematic liquid crystals Based on the Ericksen-Leslie theorythis paper contributes to LC dynamics subject to tactile forceLC molecule tilt leads to a change in the optical intensitytransmitted through a liquid crystal cell LCD panels subjectto tactile force will show ripple-like propagation on screensThe present results show that the viscosity tactile force

thickness of cell gap and Leslie viscosity coefficient are thefactors of tilt angle variation Tilt angle variations of PAAliquid crystal molecules are sensitive to tactile force mag-nitudes while those of 5CB and MBBA with larger viscosityare not Analytical derivation has been carried out in thisstudy and analytical results have been validated by numericalresults

Acknowledgments

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

References

[1] J Ericksen ldquoConservation laws for liquid crystalsrdquoTransactionsof the Society of Rheology vol 5 pp 22ndash34 1961

[2] J L Ericksen ldquoContinuum theory of nematic liquid crystalsrdquoRes Mechanica vol 21 no 4 pp 381ndash392 1987

[3] F M Leslie ldquoTheory of flow phenomenum in liquid crystalsrdquoAdvances in Liquid Crystals vol 4 pp 1ndash81 1979

[4] F Brochard P Pieranski and E Guyon ldquoDynamics of the ori-entation of a nematic-liquid-crystal film in a variable magneticfieldrdquoPhysical Review Letters vol 28 no 26 pp 1681ndash1683 1972

[5] L Lei S Changqing S Juelian P M Lam and H Yun ldquoSolitonpropagation in liquid crystalsrdquo Physical Review Letters vol 49no 18 pp 1335ndash1338 1982

[6] F M Leslie ldquoSome topics in continuum theory of nematicsrdquoPhilosophical Transactions of the Royal Society A vol 309 pp155ndash165 1983

[7] J T Gleeson P Palffy-Muhoray and W van Saarloos ldquoProp-agation of excitations induced by shear flow in nematic liquidcrystalsrdquo Physical Review A vol 44 no 4 pp 2588ndash2595 1991

[8] F H Lin and C Liu ldquoExistence of solutions for the Ericksen-Leslie systemrdquo Archive for Rational Mechanics and Analysis vol154 no 2 pp 135ndash156 2000

[9] L R P de Andrade Lima and A D Rey ldquoPoiseuille flowof Leslie-Ericksen discotic liquid crystal Solution multiplicitymultistability and non-Newtonian rheologyrdquo Journal of Non-Newtonian Fluid Mechanics vol 110 no 2-3 pp 103ndash142 2003

[10] X Nie R Lu H Xianyu T X Wu and S T Wu ldquoAnchoringenergy and cell gap effects on liquid crystal response timerdquoJournal of Applied Physics vol 101 no 10 Article ID 103110 2007

[11] P A Cruz M F Tome I W Stewart and S McKee ldquoAnumerical method for solving the dynamic three-dimensionalEricksen-Leslie equations for nematic liquid crystals subjectto a strong magnetic fieldrdquo Journal of Non-Newtonian FluidMechanics vol 165 no 3-4 pp 143ndash157 2010

[12] X Nie H Xianyu R Lu T X Wu and S T Wu ldquoPretilt angleeffects on liquid crystal response timerdquo IEEEOSA Journal ofDisplay Technology vol 3 no 3 pp 280ndash283 2007

[13] I C Khoo and S T Wu Optics and Nonlinear Optics of LiquidCrystals World Scientific 1993

[14] I C Khoo Liquid Crystals Physical Properties and NonlinearOptical Phenomena John Wiley amp Sons New York NY USA1995

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014

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Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0 1 2 3 4 50

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne35dyne

(a)

0 1 2 3 4 50

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne35dyne

(b)

0 1 2 3 4 50

2

4

6

8

10

14

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne35dyne

12

(c)

Figure 8 Comparison of molecule tilt angles between cell gapthicknesses of 3120583m and 5120583m when tactile force is applied to LCDscreen with (a) 5CB (b) MBBA and (c) PAA liquid crystals underdifferent tactile forces

25dyne3dyne

35dyne

0 1 2 30

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

(a)

0 1 2 30

2

4

6

8

10

12

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

(b)

Figure 9 (a) Comparison of molecule tilt angles depicted in3 120583m cell gap 5CB liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

According to the above assumptions (1) becomes

119870(

1205972120579

1205971199092+

1205972120579

1205971199102) minus 120574

120597120579

120597119905

+

119904

2

(1205741minus 1205742cos 2120579)

+ 12057601205761198861198642 sin 120579 cos 120579 + 119865 times 119899 = 0

(14)

Based on the Ericksen-Leslie theory (14) represents the tiltmodel of LC molecule subjected to both electric field andtactile force In order to use separation of variables assume120579 = 120579(119909 119910 119905) yields

120579 (119909 119910 119905) = 119891 (119910) 120579119898(119909 119905) (15)

Mathematical Problems in Engineering 7

0 1 2 30

2

4

6

8

10

Tilt

angl

e (de

g)

Cell gap (120583m)

4

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 10 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap 5CB liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

where

119891 (119910) =

119892 (119910)

119892 (minus119910119898)

=

1 or minus 1 119910 = minus119910119898or 119910119898

119892 (119910)

119892 (minus028119889)

= 47503 sdot 119892 (119910) 119910 = minus 119910119898or 119910119898

(16)

119892 (119910) =

2119910

119889

minus sin(119910120587

119889

) (17)

119910119898= (

119889

120587

) cosminus1 ( 2120587

) asymp 028119889 (18)

and119910119898andminus119910

119898are boundary conditions of function119892(119910) In

(15) 120579119898is the angle of the normal vector between the director

of LC molecules with homeotropic alignment Molecule

0 1 2 3 4 50

2

4

6

8

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 11 (a) Comparison of molecule tilt angles depicted in5 120583m cell gap 5CB liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

directions are changed by the Poiseuille flow Values of theangle along the 119910-axis have extreme values at 119910

119898and minus119910

119898

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

119870[

1205972

1205971199092(119891 (119910) 120579

119898(119909 119905)) +

1205972

1205971199102(119891 (119910) 120579

119898(119909 119905))]

minus 120574

120597

120597119905

(119891 (119910) 120579119898(119909 119905)) +

120597V

120597119910

(1205723sin2120579 minus 120572

2cos2120579)

+ 12057601205761198861198642 sin 120579 cos 120579 + |119865| |119899| sin (120579 minus 120601) = 0

(19)

Since |119899| = 1 one has

|119865| |119899| sin (120579 minus 120601) = 119865 sdot sin (120579 minus 120601) (20)

8 Mathematical Problems in Engineering

0 1 2 30

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

12

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

(b)

Figure 12 (a) Comparison of molecule tilt angles depicted in3120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Therefore (19) becomes

119870[119891 (119910)

1205972120579119898

1205971199092

+ 0] + [120579119898

1205972119891 (119910)

1205971199102

+ 0] minus 120574119891 (119910)

120597120579119898

120597119905

+

120597V

120597119910

[1205723sin2 (119891 (119910) 120579

119898) minus 1205722cos2 (119891 (119910) 120579

119898)]

+ 12057601205761198861198642 sin (119891 (119910) 120579

119898) cos (119891 (119910) 120579

119898)

+ 119865 sin (119891 (119910) 120579119898minus 120601) = 0

(21)

As depicted in Figure 2 the director makes an angle 120579119898with

the 119910-axis Consequently 120579119898represents liquid crystal mole-

cules with homeotropic alignment caused by the Poiseuilleflow between director 119899 and normal direction of LC sampleAn effective director equation of motion with the essentialphysics preserved may be obtained by setting 119910 = minus119910

119898and

119910 = 119910119898in (21)

0 1 2 3 40

2

4

6

8

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 13 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

On boundary 119910 = minus119910119898in Figure 2

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

(1205723sin2120579119898minus 1205722cos2120579119898)

+ 12057601205761198861198642 sin 120579

119898cos 120579119898+ 119865 sin (120579

119898minus 120601) = 0

(22)

By using trigonometric formulas one can rewrite (22) as

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

times [1205723(

1 minus cos 2120579119898

2

) minus 1205722(

1 + cos 2120579119898

2

)]

+ 12057601205761198861198642120579119898+ 119865 (sin 120579

119898cos120601 minus cos 120579

119898sin120601) = 0

(23)

Mathematical Problems in Engineering 9

0 1 2 3 4 50

2

4

6

8Ti

lt an

gle (

deg)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 14 (a) Comparison of molecule tilt angles depicted in5120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Because sin 120579119898

asymp 120579119898and cos 120579

119898asymp 1 for small 120579

119898 (23) is

rewritten as

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

[1205723(

1 minus 1

2

) minus 1205722(

1 + 1

2

)]

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601) = 0

(24)

Thus

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898

+ 119865 (120579119898cos120601 minus sin120601) = 0

(25)

Hence

120574

120597120579119898

120597119905

=119870

1205972120579119898

1205971199092

minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

(26)

0 1 2 30

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

14

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

12

14

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

(b)

Figure 15 (a) Comparison of molecule tilt angles depicted in3 120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Substituting 120579119898= 120579119898(119905) sin(120587119909119889) for separation of variables

into (26) leads to

120574

120579119898= 119870

1205972

1205971199092[120579119898(119905) sin(120587119909

119889

)] minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

(27)

As a result

120574

120579119898= minus

1198701205872

1198892120579119898(119909 119905) + (120576

01205761198861198642+ 119865 cos120601) 120579

119898

+ (minus1205722

120597V

120597119910

minus 119865 sin120601)

(28)

10 Mathematical Problems in Engineering

0 1 2 30

2

4

6

8

10

Tilt

angl

e (de

g)

Cell gap (120583m)

4

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 16 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne by analyticaland numerical results

Similarly we consider the following two cases

Case 11015840

10038161003816100381610038161003816100381610038161003816

minus1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

10038161003816100381610038161003816100381610038161003816

≫

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579119898

1205971199092

100381610038161003816100381610038161003816100381610038161003816

(29)

120574

120579119898= (12057601205761198861198642+ 119865 cos120601) 120579

119898+ (minus120572

2

120597V

120597119910

minus 119865 sin120601) (30)

Assuming 120579119898= 119886 + 119887120579

119898 one has

119886 =

1

120574

(minus1205722

120597V

120597119910

minus 119865 sin120601)

119887 =

1

120574

(12057601205761198861198642+ 119865 cos120601)

(31)

Therefore

120579119898=

119886

119887

(119890119887119905minus 1) (32)

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

120579119898(119909 119905)

=[

minus1205722(120597V120597119910) minus 119865 sin120601

12057601205761198861198642+ 119865 cos120601

(119890((12057601205761198861198642+119865 cos120601)120574)119905

minus 1)]sin(120587119909119889

)

(33)

Consider the following

Case 21015840

10038161003816100381610038161003816100381610038161003816

minus1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

10038161003816100381610038161003816100381610038161003816

≪

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579119898

1205971199092

100381610038161003816100381610038161003816100381610038161003816

(34)

Since the tactile force and electric field are ignored in Case 21015840only Case 11015840 is further examined

On boundary 119910 = 119910119898in Figure 2

120579119898(119909 119905)

=[

1205722(120597V120597119910) + 119865 sin12060112057601205761198861198642+ 119865 cos120601

(119890((12057601205761198861198642+119865 cos120601)120574)119905

minus 1)]sin(120587119909119889

)

(35)

Since 120579119898(119909 119905) in (33) is equal to minus120579

119898(119909 119905) in (35) the tilt

angles are symmetric

3 Discussion

Subject to both electric field of 89 V and tactile forcesimulation results for Case 1 are depicted in Figures 3 4and 5 Comparing different LC materials including 5CBMBBA and PAA subjected to 10ms duration tactile forcesof 1 dyne 05 dyne and 01 dyne at 10ms Figures 3 4 and 5show that a larger force makes the tilt angle larger regardlessof LC gap thickness Figures 6(a) 6(b) and 6(c) compare3 120583m thick different LCmaterials including 5CB MBBA andPAA subjected to 5ms duration 05 dyne force respectivelyThe viscosity of PAA is the smallest and the tilt angle islarger than the others Therefore the option of liquid crystalmaterial with smaller viscosity can reduce the ripple

Tilt angle variations are shown under different voltages inFigure 7 and under different tactile forces in Figure 8 Using10V 20V 30V 40V and 50V under a tactile force of 3 dyneit is observed that the tilt angles remain the same in Figures7(a) 7(b) and 7(c) Thus different voltages do not affect tiltangles

Based on (33) Figures 8(a) 8(b) and 8(c) compare differ-ent LC materials including 5CB MBBA and PAA subject to25 dyne 3 dyne and 35 dyne forces in 3120583m and 5 120583m thickcell gap LC respectively Accordingly tactile forces affect tiltangles obviously Figures 9(a) 10(a) and 11(a) respectivelyshow tilt angles of 5CB LC subject to 25 dyne 3 dyne and

Mathematical Problems in Engineering 11

0 1 2 3 4 50

2

4

6

8Ti

lt an

gle (

deg)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 17 (a) Comparison of molecule tilt angles depicted in5120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

35 dyne forces in 3 4 and 5 120583m thick cell gap LC SimilarlyFigures 12(a) 13(a) and 14(a) are MBBA results and Figures15(a) 16(a) and 17(a) are PAA results In addition Figures9(b) to 17(b) compare analytical solutions with numericalsolutions of the full coupled equations The comparisonresults show that analytical solutions are consistent withnumerical solutions of the full coupled equations Thereforethe analytical derivation is validated

4 Conclusion

The Ericksen-Leslie theory deals with the dynamics ofnematic liquid crystals Based on the Ericksen-Leslie theorythis paper contributes to LC dynamics subject to tactile forceLC molecule tilt leads to a change in the optical intensitytransmitted through a liquid crystal cell LCD panels subjectto tactile force will show ripple-like propagation on screensThe present results show that the viscosity tactile force

thickness of cell gap and Leslie viscosity coefficient are thefactors of tilt angle variation Tilt angle variations of PAAliquid crystal molecules are sensitive to tactile force mag-nitudes while those of 5CB and MBBA with larger viscosityare not Analytical derivation has been carried out in thisstudy and analytical results have been validated by numericalresults

Acknowledgments

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

References

[1] J Ericksen ldquoConservation laws for liquid crystalsrdquoTransactionsof the Society of Rheology vol 5 pp 22ndash34 1961

[2] J L Ericksen ldquoContinuum theory of nematic liquid crystalsrdquoRes Mechanica vol 21 no 4 pp 381ndash392 1987

[3] F M Leslie ldquoTheory of flow phenomenum in liquid crystalsrdquoAdvances in Liquid Crystals vol 4 pp 1ndash81 1979

[4] F Brochard P Pieranski and E Guyon ldquoDynamics of the ori-entation of a nematic-liquid-crystal film in a variable magneticfieldrdquoPhysical Review Letters vol 28 no 26 pp 1681ndash1683 1972

[5] L Lei S Changqing S Juelian P M Lam and H Yun ldquoSolitonpropagation in liquid crystalsrdquo Physical Review Letters vol 49no 18 pp 1335ndash1338 1982

[6] F M Leslie ldquoSome topics in continuum theory of nematicsrdquoPhilosophical Transactions of the Royal Society A vol 309 pp155ndash165 1983

[7] J T Gleeson P Palffy-Muhoray and W van Saarloos ldquoProp-agation of excitations induced by shear flow in nematic liquidcrystalsrdquo Physical Review A vol 44 no 4 pp 2588ndash2595 1991

[8] F H Lin and C Liu ldquoExistence of solutions for the Ericksen-Leslie systemrdquo Archive for Rational Mechanics and Analysis vol154 no 2 pp 135ndash156 2000

[9] L R P de Andrade Lima and A D Rey ldquoPoiseuille flowof Leslie-Ericksen discotic liquid crystal Solution multiplicitymultistability and non-Newtonian rheologyrdquo Journal of Non-Newtonian Fluid Mechanics vol 110 no 2-3 pp 103ndash142 2003

[10] X Nie R Lu H Xianyu T X Wu and S T Wu ldquoAnchoringenergy and cell gap effects on liquid crystal response timerdquoJournal of Applied Physics vol 101 no 10 Article ID 103110 2007

[11] P A Cruz M F Tome I W Stewart and S McKee ldquoAnumerical method for solving the dynamic three-dimensionalEricksen-Leslie equations for nematic liquid crystals subjectto a strong magnetic fieldrdquo Journal of Non-Newtonian FluidMechanics vol 165 no 3-4 pp 143ndash157 2010

[12] X Nie H Xianyu R Lu T X Wu and S T Wu ldquoPretilt angleeffects on liquid crystal response timerdquo IEEEOSA Journal ofDisplay Technology vol 3 no 3 pp 280ndash283 2007

[13] I C Khoo and S T Wu Optics and Nonlinear Optics of LiquidCrystals World Scientific 1993

[14] I C Khoo Liquid Crystals Physical Properties and NonlinearOptical Phenomena John Wiley amp Sons New York NY USA1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0 1 2 30

2

4

6

8

10

Tilt

angl

e (de

g)

Cell gap (120583m)

4

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 10 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap 5CB liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

where

119891 (119910) =

119892 (119910)

119892 (minus119910119898)

=

1 or minus 1 119910 = minus119910119898or 119910119898

119892 (119910)

119892 (minus028119889)

= 47503 sdot 119892 (119910) 119910 = minus 119910119898or 119910119898

(16)

119892 (119910) =

2119910

119889

minus sin(119910120587

119889

) (17)

119910119898= (

119889

120587

) cosminus1 ( 2120587

) asymp 028119889 (18)

and119910119898andminus119910

119898are boundary conditions of function119892(119910) In

(15) 120579119898is the angle of the normal vector between the director

of LC molecules with homeotropic alignment Molecule

0 1 2 3 4 50

2

4

6

8

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 11 (a) Comparison of molecule tilt angles depicted in5 120583m cell gap 5CB liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

directions are changed by the Poiseuille flow Values of theangle along the 119910-axis have extreme values at 119910

119898and minus119910

119898

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

119870[

1205972

1205971199092(119891 (119910) 120579

119898(119909 119905)) +

1205972

1205971199102(119891 (119910) 120579

119898(119909 119905))]

minus 120574

120597

120597119905

(119891 (119910) 120579119898(119909 119905)) +

120597V

120597119910

(1205723sin2120579 minus 120572

2cos2120579)

+ 12057601205761198861198642 sin 120579 cos 120579 + |119865| |119899| sin (120579 minus 120601) = 0

(19)

Since |119899| = 1 one has

|119865| |119899| sin (120579 minus 120601) = 119865 sdot sin (120579 minus 120601) (20)

8 Mathematical Problems in Engineering

0 1 2 30

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

12

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

(b)

Figure 12 (a) Comparison of molecule tilt angles depicted in3120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Therefore (19) becomes

119870[119891 (119910)

1205972120579119898

1205971199092

+ 0] + [120579119898

1205972119891 (119910)

1205971199102

+ 0] minus 120574119891 (119910)

120597120579119898

120597119905

+

120597V

120597119910

[1205723sin2 (119891 (119910) 120579

119898) minus 1205722cos2 (119891 (119910) 120579

119898)]

+ 12057601205761198861198642 sin (119891 (119910) 120579

119898) cos (119891 (119910) 120579

119898)

+ 119865 sin (119891 (119910) 120579119898minus 120601) = 0

(21)

As depicted in Figure 2 the director makes an angle 120579119898with

the 119910-axis Consequently 120579119898represents liquid crystal mole-

cules with homeotropic alignment caused by the Poiseuilleflow between director 119899 and normal direction of LC sampleAn effective director equation of motion with the essentialphysics preserved may be obtained by setting 119910 = minus119910

119898and

119910 = 119910119898in (21)

0 1 2 3 40

2

4

6

8

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 13 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

On boundary 119910 = minus119910119898in Figure 2

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

(1205723sin2120579119898minus 1205722cos2120579119898)

+ 12057601205761198861198642 sin 120579

119898cos 120579119898+ 119865 sin (120579

119898minus 120601) = 0

(22)

By using trigonometric formulas one can rewrite (22) as

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

times [1205723(

1 minus cos 2120579119898

2

) minus 1205722(

1 + cos 2120579119898

2

)]

+ 12057601205761198861198642120579119898+ 119865 (sin 120579

119898cos120601 minus cos 120579

119898sin120601) = 0

(23)

Mathematical Problems in Engineering 9

0 1 2 3 4 50

2

4

6

8Ti

lt an

gle (

deg)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 14 (a) Comparison of molecule tilt angles depicted in5120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Because sin 120579119898

asymp 120579119898and cos 120579

119898asymp 1 for small 120579

119898 (23) is

rewritten as

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

[1205723(

1 minus 1

2

) minus 1205722(

1 + 1

2

)]

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601) = 0

(24)

Thus

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898

+ 119865 (120579119898cos120601 minus sin120601) = 0

(25)

Hence

120574

120597120579119898

120597119905

=119870

1205972120579119898

1205971199092

minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

(26)

0 1 2 30

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

14

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

12

14

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

(b)

Figure 15 (a) Comparison of molecule tilt angles depicted in3 120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Substituting 120579119898= 120579119898(119905) sin(120587119909119889) for separation of variables

into (26) leads to

120574

120579119898= 119870

1205972

1205971199092[120579119898(119905) sin(120587119909

119889

)] minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

(27)

As a result

120574

120579119898= minus

1198701205872

1198892120579119898(119909 119905) + (120576

01205761198861198642+ 119865 cos120601) 120579

119898

+ (minus1205722

120597V

120597119910

minus 119865 sin120601)

(28)

10 Mathematical Problems in Engineering

0 1 2 30

2

4

6

8

10

Tilt

angl

e (de

g)

Cell gap (120583m)

4

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 16 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne by analyticaland numerical results

Similarly we consider the following two cases

Case 11015840

10038161003816100381610038161003816100381610038161003816

minus1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

10038161003816100381610038161003816100381610038161003816

≫

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579119898

1205971199092

100381610038161003816100381610038161003816100381610038161003816

(29)

120574

120579119898= (12057601205761198861198642+ 119865 cos120601) 120579

119898+ (minus120572

2

120597V

120597119910

minus 119865 sin120601) (30)

Assuming 120579119898= 119886 + 119887120579

119898 one has

119886 =

1

120574

(minus1205722

120597V

120597119910

minus 119865 sin120601)

119887 =

1

120574

(12057601205761198861198642+ 119865 cos120601)

(31)

Therefore

120579119898=

119886

119887

(119890119887119905minus 1) (32)

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

120579119898(119909 119905)

=[

minus1205722(120597V120597119910) minus 119865 sin120601

12057601205761198861198642+ 119865 cos120601

(119890((12057601205761198861198642+119865 cos120601)120574)119905

minus 1)]sin(120587119909119889

)

(33)

Consider the following

Case 21015840

10038161003816100381610038161003816100381610038161003816

minus1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

10038161003816100381610038161003816100381610038161003816

≪

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579119898

1205971199092

100381610038161003816100381610038161003816100381610038161003816

(34)

Since the tactile force and electric field are ignored in Case 21015840only Case 11015840 is further examined

On boundary 119910 = 119910119898in Figure 2

120579119898(119909 119905)

=[

1205722(120597V120597119910) + 119865 sin12060112057601205761198861198642+ 119865 cos120601

(119890((12057601205761198861198642+119865 cos120601)120574)119905

minus 1)]sin(120587119909119889

)

(35)

Since 120579119898(119909 119905) in (33) is equal to minus120579

119898(119909 119905) in (35) the tilt

angles are symmetric

3 Discussion

Subject to both electric field of 89 V and tactile forcesimulation results for Case 1 are depicted in Figures 3 4and 5 Comparing different LC materials including 5CBMBBA and PAA subjected to 10ms duration tactile forcesof 1 dyne 05 dyne and 01 dyne at 10ms Figures 3 4 and 5show that a larger force makes the tilt angle larger regardlessof LC gap thickness Figures 6(a) 6(b) and 6(c) compare3 120583m thick different LCmaterials including 5CB MBBA andPAA subjected to 5ms duration 05 dyne force respectivelyThe viscosity of PAA is the smallest and the tilt angle islarger than the others Therefore the option of liquid crystalmaterial with smaller viscosity can reduce the ripple

Tilt angle variations are shown under different voltages inFigure 7 and under different tactile forces in Figure 8 Using10V 20V 30V 40V and 50V under a tactile force of 3 dyneit is observed that the tilt angles remain the same in Figures7(a) 7(b) and 7(c) Thus different voltages do not affect tiltangles

Based on (33) Figures 8(a) 8(b) and 8(c) compare differ-ent LC materials including 5CB MBBA and PAA subject to25 dyne 3 dyne and 35 dyne forces in 3120583m and 5 120583m thickcell gap LC respectively Accordingly tactile forces affect tiltangles obviously Figures 9(a) 10(a) and 11(a) respectivelyshow tilt angles of 5CB LC subject to 25 dyne 3 dyne and

Mathematical Problems in Engineering 11

0 1 2 3 4 50

2

4

6

8Ti

lt an

gle (

deg)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 17 (a) Comparison of molecule tilt angles depicted in5120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

35 dyne forces in 3 4 and 5 120583m thick cell gap LC SimilarlyFigures 12(a) 13(a) and 14(a) are MBBA results and Figures15(a) 16(a) and 17(a) are PAA results In addition Figures9(b) to 17(b) compare analytical solutions with numericalsolutions of the full coupled equations The comparisonresults show that analytical solutions are consistent withnumerical solutions of the full coupled equations Thereforethe analytical derivation is validated

4 Conclusion

The Ericksen-Leslie theory deals with the dynamics ofnematic liquid crystals Based on the Ericksen-Leslie theorythis paper contributes to LC dynamics subject to tactile forceLC molecule tilt leads to a change in the optical intensitytransmitted through a liquid crystal cell LCD panels subjectto tactile force will show ripple-like propagation on screensThe present results show that the viscosity tactile force

thickness of cell gap and Leslie viscosity coefficient are thefactors of tilt angle variation Tilt angle variations of PAAliquid crystal molecules are sensitive to tactile force mag-nitudes while those of 5CB and MBBA with larger viscosityare not Analytical derivation has been carried out in thisstudy and analytical results have been validated by numericalresults

Acknowledgments

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

References

[1] J Ericksen ldquoConservation laws for liquid crystalsrdquoTransactionsof the Society of Rheology vol 5 pp 22ndash34 1961

[2] J L Ericksen ldquoContinuum theory of nematic liquid crystalsrdquoRes Mechanica vol 21 no 4 pp 381ndash392 1987

[3] F M Leslie ldquoTheory of flow phenomenum in liquid crystalsrdquoAdvances in Liquid Crystals vol 4 pp 1ndash81 1979

[4] F Brochard P Pieranski and E Guyon ldquoDynamics of the ori-entation of a nematic-liquid-crystal film in a variable magneticfieldrdquoPhysical Review Letters vol 28 no 26 pp 1681ndash1683 1972

[5] L Lei S Changqing S Juelian P M Lam and H Yun ldquoSolitonpropagation in liquid crystalsrdquo Physical Review Letters vol 49no 18 pp 1335ndash1338 1982

[6] F M Leslie ldquoSome topics in continuum theory of nematicsrdquoPhilosophical Transactions of the Royal Society A vol 309 pp155ndash165 1983

[7] J T Gleeson P Palffy-Muhoray and W van Saarloos ldquoProp-agation of excitations induced by shear flow in nematic liquidcrystalsrdquo Physical Review A vol 44 no 4 pp 2588ndash2595 1991

[8] F H Lin and C Liu ldquoExistence of solutions for the Ericksen-Leslie systemrdquo Archive for Rational Mechanics and Analysis vol154 no 2 pp 135ndash156 2000

[9] L R P de Andrade Lima and A D Rey ldquoPoiseuille flowof Leslie-Ericksen discotic liquid crystal Solution multiplicitymultistability and non-Newtonian rheologyrdquo Journal of Non-Newtonian Fluid Mechanics vol 110 no 2-3 pp 103ndash142 2003

[10] X Nie R Lu H Xianyu T X Wu and S T Wu ldquoAnchoringenergy and cell gap effects on liquid crystal response timerdquoJournal of Applied Physics vol 101 no 10 Article ID 103110 2007

[11] P A Cruz M F Tome I W Stewart and S McKee ldquoAnumerical method for solving the dynamic three-dimensionalEricksen-Leslie equations for nematic liquid crystals subjectto a strong magnetic fieldrdquo Journal of Non-Newtonian FluidMechanics vol 165 no 3-4 pp 143ndash157 2010

[12] X Nie H Xianyu R Lu T X Wu and S T Wu ldquoPretilt angleeffects on liquid crystal response timerdquo IEEEOSA Journal ofDisplay Technology vol 3 no 3 pp 280ndash283 2007

[13] I C Khoo and S T Wu Optics and Nonlinear Optics of LiquidCrystals World Scientific 1993

[14] I C Khoo Liquid Crystals Physical Properties and NonlinearOptical Phenomena John Wiley amp Sons New York NY USA1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0 1 2 30

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

12

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

(b)

Figure 12 (a) Comparison of molecule tilt angles depicted in3120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Therefore (19) becomes

119870[119891 (119910)

1205972120579119898

1205971199092

+ 0] + [120579119898

1205972119891 (119910)

1205971199102

+ 0] minus 120574119891 (119910)

120597120579119898

120597119905

+

120597V

120597119910

[1205723sin2 (119891 (119910) 120579

119898) minus 1205722cos2 (119891 (119910) 120579

119898)]

+ 12057601205761198861198642 sin (119891 (119910) 120579

119898) cos (119891 (119910) 120579

119898)

+ 119865 sin (119891 (119910) 120579119898minus 120601) = 0

(21)

As depicted in Figure 2 the director makes an angle 120579119898with

the 119910-axis Consequently 120579119898represents liquid crystal mole-

cules with homeotropic alignment caused by the Poiseuilleflow between director 119899 and normal direction of LC sampleAn effective director equation of motion with the essentialphysics preserved may be obtained by setting 119910 = minus119910

119898and

119910 = 119910119898in (21)

0 1 2 3 40

2

4

6

8

Tilt

angl

e (de

g)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 13 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

On boundary 119910 = minus119910119898in Figure 2

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

(1205723sin2120579119898minus 1205722cos2120579119898)

+ 12057601205761198861198642 sin 120579

119898cos 120579119898+ 119865 sin (120579

119898minus 120601) = 0

(22)

By using trigonometric formulas one can rewrite (22) as

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

times [1205723(

1 minus cos 2120579119898

2

) minus 1205722(

1 + cos 2120579119898

2

)]

+ 12057601205761198861198642120579119898+ 119865 (sin 120579

119898cos120601 minus cos 120579

119898sin120601) = 0

(23)

Mathematical Problems in Engineering 9

0 1 2 3 4 50

2

4

6

8Ti

lt an

gle (

deg)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 14 (a) Comparison of molecule tilt angles depicted in5120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Because sin 120579119898

asymp 120579119898and cos 120579

119898asymp 1 for small 120579

119898 (23) is

rewritten as

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

[1205723(

1 minus 1

2

) minus 1205722(

1 + 1

2

)]

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601) = 0

(24)

Thus

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898

+ 119865 (120579119898cos120601 minus sin120601) = 0

(25)

Hence

120574

120597120579119898

120597119905

=119870

1205972120579119898

1205971199092

minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

(26)

0 1 2 30

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

14

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

12

14

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

(b)

Figure 15 (a) Comparison of molecule tilt angles depicted in3 120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Substituting 120579119898= 120579119898(119905) sin(120587119909119889) for separation of variables

into (26) leads to

120574

120579119898= 119870

1205972

1205971199092[120579119898(119905) sin(120587119909

119889

)] minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

(27)

As a result

120574

120579119898= minus

1198701205872

1198892120579119898(119909 119905) + (120576

01205761198861198642+ 119865 cos120601) 120579

119898

+ (minus1205722

120597V

120597119910

minus 119865 sin120601)

(28)

10 Mathematical Problems in Engineering

0 1 2 30

2

4

6

8

10

Tilt

angl

e (de

g)

Cell gap (120583m)

4

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 16 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne by analyticaland numerical results

Similarly we consider the following two cases

Case 11015840

10038161003816100381610038161003816100381610038161003816

minus1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

10038161003816100381610038161003816100381610038161003816

≫

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579119898

1205971199092

100381610038161003816100381610038161003816100381610038161003816

(29)

120574

120579119898= (12057601205761198861198642+ 119865 cos120601) 120579

119898+ (minus120572

2

120597V

120597119910

minus 119865 sin120601) (30)

Assuming 120579119898= 119886 + 119887120579

119898 one has

119886 =

1

120574

(minus1205722

120597V

120597119910

minus 119865 sin120601)

119887 =

1

120574

(12057601205761198861198642+ 119865 cos120601)

(31)

Therefore

120579119898=

119886

119887

(119890119887119905minus 1) (32)

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

120579119898(119909 119905)

=[

minus1205722(120597V120597119910) minus 119865 sin120601

12057601205761198861198642+ 119865 cos120601

(119890((12057601205761198861198642+119865 cos120601)120574)119905

minus 1)]sin(120587119909119889

)

(33)

Consider the following

Case 21015840

10038161003816100381610038161003816100381610038161003816

minus1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

10038161003816100381610038161003816100381610038161003816

≪

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579119898

1205971199092

100381610038161003816100381610038161003816100381610038161003816

(34)

Since the tactile force and electric field are ignored in Case 21015840only Case 11015840 is further examined

On boundary 119910 = 119910119898in Figure 2

120579119898(119909 119905)

=[

1205722(120597V120597119910) + 119865 sin12060112057601205761198861198642+ 119865 cos120601

(119890((12057601205761198861198642+119865 cos120601)120574)119905

minus 1)]sin(120587119909119889

)

(35)

Since 120579119898(119909 119905) in (33) is equal to minus120579

119898(119909 119905) in (35) the tilt

angles are symmetric

3 Discussion

Subject to both electric field of 89 V and tactile forcesimulation results for Case 1 are depicted in Figures 3 4and 5 Comparing different LC materials including 5CBMBBA and PAA subjected to 10ms duration tactile forcesof 1 dyne 05 dyne and 01 dyne at 10ms Figures 3 4 and 5show that a larger force makes the tilt angle larger regardlessof LC gap thickness Figures 6(a) 6(b) and 6(c) compare3 120583m thick different LCmaterials including 5CB MBBA andPAA subjected to 5ms duration 05 dyne force respectivelyThe viscosity of PAA is the smallest and the tilt angle islarger than the others Therefore the option of liquid crystalmaterial with smaller viscosity can reduce the ripple

Tilt angle variations are shown under different voltages inFigure 7 and under different tactile forces in Figure 8 Using10V 20V 30V 40V and 50V under a tactile force of 3 dyneit is observed that the tilt angles remain the same in Figures7(a) 7(b) and 7(c) Thus different voltages do not affect tiltangles

Based on (33) Figures 8(a) 8(b) and 8(c) compare differ-ent LC materials including 5CB MBBA and PAA subject to25 dyne 3 dyne and 35 dyne forces in 3120583m and 5 120583m thickcell gap LC respectively Accordingly tactile forces affect tiltangles obviously Figures 9(a) 10(a) and 11(a) respectivelyshow tilt angles of 5CB LC subject to 25 dyne 3 dyne and

Mathematical Problems in Engineering 11

0 1 2 3 4 50

2

4

6

8Ti

lt an

gle (

deg)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 17 (a) Comparison of molecule tilt angles depicted in5120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

35 dyne forces in 3 4 and 5 120583m thick cell gap LC SimilarlyFigures 12(a) 13(a) and 14(a) are MBBA results and Figures15(a) 16(a) and 17(a) are PAA results In addition Figures9(b) to 17(b) compare analytical solutions with numericalsolutions of the full coupled equations The comparisonresults show that analytical solutions are consistent withnumerical solutions of the full coupled equations Thereforethe analytical derivation is validated

4 Conclusion

The Ericksen-Leslie theory deals with the dynamics ofnematic liquid crystals Based on the Ericksen-Leslie theorythis paper contributes to LC dynamics subject to tactile forceLC molecule tilt leads to a change in the optical intensitytransmitted through a liquid crystal cell LCD panels subjectto tactile force will show ripple-like propagation on screensThe present results show that the viscosity tactile force

thickness of cell gap and Leslie viscosity coefficient are thefactors of tilt angle variation Tilt angle variations of PAAliquid crystal molecules are sensitive to tactile force mag-nitudes while those of 5CB and MBBA with larger viscosityare not Analytical derivation has been carried out in thisstudy and analytical results have been validated by numericalresults

Acknowledgments

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

References

[1] J Ericksen ldquoConservation laws for liquid crystalsrdquoTransactionsof the Society of Rheology vol 5 pp 22ndash34 1961

[2] J L Ericksen ldquoContinuum theory of nematic liquid crystalsrdquoRes Mechanica vol 21 no 4 pp 381ndash392 1987

[3] F M Leslie ldquoTheory of flow phenomenum in liquid crystalsrdquoAdvances in Liquid Crystals vol 4 pp 1ndash81 1979

[4] F Brochard P Pieranski and E Guyon ldquoDynamics of the ori-entation of a nematic-liquid-crystal film in a variable magneticfieldrdquoPhysical Review Letters vol 28 no 26 pp 1681ndash1683 1972

[5] L Lei S Changqing S Juelian P M Lam and H Yun ldquoSolitonpropagation in liquid crystalsrdquo Physical Review Letters vol 49no 18 pp 1335ndash1338 1982

[6] F M Leslie ldquoSome topics in continuum theory of nematicsrdquoPhilosophical Transactions of the Royal Society A vol 309 pp155ndash165 1983

[7] J T Gleeson P Palffy-Muhoray and W van Saarloos ldquoProp-agation of excitations induced by shear flow in nematic liquidcrystalsrdquo Physical Review A vol 44 no 4 pp 2588ndash2595 1991

[8] F H Lin and C Liu ldquoExistence of solutions for the Ericksen-Leslie systemrdquo Archive for Rational Mechanics and Analysis vol154 no 2 pp 135ndash156 2000

[9] L R P de Andrade Lima and A D Rey ldquoPoiseuille flowof Leslie-Ericksen discotic liquid crystal Solution multiplicitymultistability and non-Newtonian rheologyrdquo Journal of Non-Newtonian Fluid Mechanics vol 110 no 2-3 pp 103ndash142 2003

[10] X Nie R Lu H Xianyu T X Wu and S T Wu ldquoAnchoringenergy and cell gap effects on liquid crystal response timerdquoJournal of Applied Physics vol 101 no 10 Article ID 103110 2007

[11] P A Cruz M F Tome I W Stewart and S McKee ldquoAnumerical method for solving the dynamic three-dimensionalEricksen-Leslie equations for nematic liquid crystals subjectto a strong magnetic fieldrdquo Journal of Non-Newtonian FluidMechanics vol 165 no 3-4 pp 143ndash157 2010

[12] X Nie H Xianyu R Lu T X Wu and S T Wu ldquoPretilt angleeffects on liquid crystal response timerdquo IEEEOSA Journal ofDisplay Technology vol 3 no 3 pp 280ndash283 2007

[13] I C Khoo and S T Wu Optics and Nonlinear Optics of LiquidCrystals World Scientific 1993

[14] I C Khoo Liquid Crystals Physical Properties and NonlinearOptical Phenomena John Wiley amp Sons New York NY USA1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0 1 2 3 4 50

2

4

6

8Ti

lt an

gle (

deg)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 14 (a) Comparison of molecule tilt angles depicted in5120583m cell gap MBBA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Because sin 120579119898

asymp 120579119898and cos 120579

119898asymp 1 for small 120579

119898 (23) is

rewritten as

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

+

120597V

120597119910

[1205723(

1 minus 1

2

) minus 1205722(

1 + 1

2

)]

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601) = 0

(24)

Thus

119870

1205972120579119898

1205971199092

minus 120574

120597120579119898

120597119905

minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898

+ 119865 (120579119898cos120601 minus sin120601) = 0

(25)

Hence

120574

120597120579119898

120597119905

=119870

1205972120579119898

1205971199092

minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

(26)

0 1 2 30

2

4

6

8

10

12

Tilt

angl

e (de

g)

Cell gap (120583m)

14

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

12

14

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

(b)

Figure 15 (a) Comparison of molecule tilt angles depicted in3 120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

Substituting 120579119898= 120579119898(119905) sin(120587119909119889) for separation of variables

into (26) leads to

120574

120579119898= 119870

1205972

1205971199092[120579119898(119905) sin(120587119909

119889

)] minus 1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

(27)

As a result

120574

120579119898= minus

1198701205872

1198892120579119898(119909 119905) + (120576

01205761198861198642+ 119865 cos120601) 120579

119898

+ (minus1205722

120597V

120597119910

minus 119865 sin120601)

(28)

10 Mathematical Problems in Engineering

0 1 2 30

2

4

6

8

10

Tilt

angl

e (de

g)

Cell gap (120583m)

4

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 16 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne by analyticaland numerical results

Similarly we consider the following two cases

Case 11015840

10038161003816100381610038161003816100381610038161003816

minus1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

10038161003816100381610038161003816100381610038161003816

≫

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579119898

1205971199092

100381610038161003816100381610038161003816100381610038161003816

(29)

120574

120579119898= (12057601205761198861198642+ 119865 cos120601) 120579

119898+ (minus120572

2

120597V

120597119910

minus 119865 sin120601) (30)

Assuming 120579119898= 119886 + 119887120579

119898 one has

119886 =

1

120574

(minus1205722

120597V

120597119910

minus 119865 sin120601)

119887 =

1

120574

(12057601205761198861198642+ 119865 cos120601)

(31)

Therefore

120579119898=

119886

119887

(119890119887119905minus 1) (32)

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

120579119898(119909 119905)

=[

minus1205722(120597V120597119910) minus 119865 sin120601

12057601205761198861198642+ 119865 cos120601

(119890((12057601205761198861198642+119865 cos120601)120574)119905

minus 1)]sin(120587119909119889

)

(33)

Consider the following

Case 21015840

10038161003816100381610038161003816100381610038161003816

minus1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

10038161003816100381610038161003816100381610038161003816

≪

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579119898

1205971199092

100381610038161003816100381610038161003816100381610038161003816

(34)

Since the tactile force and electric field are ignored in Case 21015840only Case 11015840 is further examined

On boundary 119910 = 119910119898in Figure 2

120579119898(119909 119905)

=[

1205722(120597V120597119910) + 119865 sin12060112057601205761198861198642+ 119865 cos120601

(119890((12057601205761198861198642+119865 cos120601)120574)119905

minus 1)]sin(120587119909119889

)

(35)

Since 120579119898(119909 119905) in (33) is equal to minus120579

119898(119909 119905) in (35) the tilt

angles are symmetric

3 Discussion

Subject to both electric field of 89 V and tactile forcesimulation results for Case 1 are depicted in Figures 3 4and 5 Comparing different LC materials including 5CBMBBA and PAA subjected to 10ms duration tactile forcesof 1 dyne 05 dyne and 01 dyne at 10ms Figures 3 4 and 5show that a larger force makes the tilt angle larger regardlessof LC gap thickness Figures 6(a) 6(b) and 6(c) compare3 120583m thick different LCmaterials including 5CB MBBA andPAA subjected to 5ms duration 05 dyne force respectivelyThe viscosity of PAA is the smallest and the tilt angle islarger than the others Therefore the option of liquid crystalmaterial with smaller viscosity can reduce the ripple

Tilt angle variations are shown under different voltages inFigure 7 and under different tactile forces in Figure 8 Using10V 20V 30V 40V and 50V under a tactile force of 3 dyneit is observed that the tilt angles remain the same in Figures7(a) 7(b) and 7(c) Thus different voltages do not affect tiltangles

Based on (33) Figures 8(a) 8(b) and 8(c) compare differ-ent LC materials including 5CB MBBA and PAA subject to25 dyne 3 dyne and 35 dyne forces in 3120583m and 5 120583m thickcell gap LC respectively Accordingly tactile forces affect tiltangles obviously Figures 9(a) 10(a) and 11(a) respectivelyshow tilt angles of 5CB LC subject to 25 dyne 3 dyne and

Mathematical Problems in Engineering 11

0 1 2 3 4 50

2

4

6

8Ti

lt an

gle (

deg)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 17 (a) Comparison of molecule tilt angles depicted in5120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

35 dyne forces in 3 4 and 5 120583m thick cell gap LC SimilarlyFigures 12(a) 13(a) and 14(a) are MBBA results and Figures15(a) 16(a) and 17(a) are PAA results In addition Figures9(b) to 17(b) compare analytical solutions with numericalsolutions of the full coupled equations The comparisonresults show that analytical solutions are consistent withnumerical solutions of the full coupled equations Thereforethe analytical derivation is validated

4 Conclusion

The Ericksen-Leslie theory deals with the dynamics ofnematic liquid crystals Based on the Ericksen-Leslie theorythis paper contributes to LC dynamics subject to tactile forceLC molecule tilt leads to a change in the optical intensitytransmitted through a liquid crystal cell LCD panels subjectto tactile force will show ripple-like propagation on screensThe present results show that the viscosity tactile force

thickness of cell gap and Leslie viscosity coefficient are thefactors of tilt angle variation Tilt angle variations of PAAliquid crystal molecules are sensitive to tactile force mag-nitudes while those of 5CB and MBBA with larger viscosityare not Analytical derivation has been carried out in thisstudy and analytical results have been validated by numericalresults

Acknowledgments

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

References

[1] J Ericksen ldquoConservation laws for liquid crystalsrdquoTransactionsof the Society of Rheology vol 5 pp 22ndash34 1961

[2] J L Ericksen ldquoContinuum theory of nematic liquid crystalsrdquoRes Mechanica vol 21 no 4 pp 381ndash392 1987

[3] F M Leslie ldquoTheory of flow phenomenum in liquid crystalsrdquoAdvances in Liquid Crystals vol 4 pp 1ndash81 1979

[4] F Brochard P Pieranski and E Guyon ldquoDynamics of the ori-entation of a nematic-liquid-crystal film in a variable magneticfieldrdquoPhysical Review Letters vol 28 no 26 pp 1681ndash1683 1972

[5] L Lei S Changqing S Juelian P M Lam and H Yun ldquoSolitonpropagation in liquid crystalsrdquo Physical Review Letters vol 49no 18 pp 1335ndash1338 1982

[6] F M Leslie ldquoSome topics in continuum theory of nematicsrdquoPhilosophical Transactions of the Royal Society A vol 309 pp155ndash165 1983

[7] J T Gleeson P Palffy-Muhoray and W van Saarloos ldquoProp-agation of excitations induced by shear flow in nematic liquidcrystalsrdquo Physical Review A vol 44 no 4 pp 2588ndash2595 1991

[8] F H Lin and C Liu ldquoExistence of solutions for the Ericksen-Leslie systemrdquo Archive for Rational Mechanics and Analysis vol154 no 2 pp 135ndash156 2000

[9] L R P de Andrade Lima and A D Rey ldquoPoiseuille flowof Leslie-Ericksen discotic liquid crystal Solution multiplicitymultistability and non-Newtonian rheologyrdquo Journal of Non-Newtonian Fluid Mechanics vol 110 no 2-3 pp 103ndash142 2003

[10] X Nie R Lu H Xianyu T X Wu and S T Wu ldquoAnchoringenergy and cell gap effects on liquid crystal response timerdquoJournal of Applied Physics vol 101 no 10 Article ID 103110 2007

[11] P A Cruz M F Tome I W Stewart and S McKee ldquoAnumerical method for solving the dynamic three-dimensionalEricksen-Leslie equations for nematic liquid crystals subjectto a strong magnetic fieldrdquo Journal of Non-Newtonian FluidMechanics vol 165 no 3-4 pp 143ndash157 2010

[12] X Nie H Xianyu R Lu T X Wu and S T Wu ldquoPretilt angleeffects on liquid crystal response timerdquo IEEEOSA Journal ofDisplay Technology vol 3 no 3 pp 280ndash283 2007

[13] I C Khoo and S T Wu Optics and Nonlinear Optics of LiquidCrystals World Scientific 1993

[14] I C Khoo Liquid Crystals Physical Properties and NonlinearOptical Phenomena John Wiley amp Sons New York NY USA1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

0 1 2 30

2

4

6

8

10

Tilt

angl

e (de

g)

Cell gap (120583m)

4

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

10

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4

(b)

Figure 16 (a) Comparison of molecule tilt angles depicted in4 120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne by analyticaland numerical results

Similarly we consider the following two cases

Case 11015840

10038161003816100381610038161003816100381610038161003816

minus1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

10038161003816100381610038161003816100381610038161003816

≫

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579119898

1205971199092

100381610038161003816100381610038161003816100381610038161003816

(29)

120574

120579119898= (12057601205761198861198642+ 119865 cos120601) 120579

119898+ (minus120572

2

120597V

120597119910

minus 119865 sin120601) (30)

Assuming 120579119898= 119886 + 119887120579

119898 one has

119886 =

1

120574

(minus1205722

120597V

120597119910

minus 119865 sin120601)

119887 =

1

120574

(12057601205761198861198642+ 119865 cos120601)

(31)

Therefore

120579119898=

119886

119887

(119890119887119905minus 1) (32)

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

120579119898(119909 119905)

=[

minus1205722(120597V120597119910) minus 119865 sin120601

12057601205761198861198642+ 119865 cos120601

(119890((12057601205761198861198642+119865 cos120601)120574)119905

minus 1)]sin(120587119909119889

)

(33)

Consider the following

Case 21015840

10038161003816100381610038161003816100381610038161003816

minus1205722

120597V

120597119910

+ 12057601205761198861198642120579119898+ 119865 (120579

119898cos120601 minus sin120601)

10038161003816100381610038161003816100381610038161003816

≪

100381610038161003816100381610038161003816100381610038161003816

119870

1205972120579119898

1205971199092

100381610038161003816100381610038161003816100381610038161003816

(34)

Since the tactile force and electric field are ignored in Case 21015840only Case 11015840 is further examined

On boundary 119910 = 119910119898in Figure 2

120579119898(119909 119905)

=[

1205722(120597V120597119910) + 119865 sin12060112057601205761198861198642+ 119865 cos120601

(119890((12057601205761198861198642+119865 cos120601)120574)119905

minus 1)]sin(120587119909119889

)

(35)

Since 120579119898(119909 119905) in (33) is equal to minus120579

119898(119909 119905) in (35) the tilt

angles are symmetric

3 Discussion

Subject to both electric field of 89 V and tactile forcesimulation results for Case 1 are depicted in Figures 3 4and 5 Comparing different LC materials including 5CBMBBA and PAA subjected to 10ms duration tactile forcesof 1 dyne 05 dyne and 01 dyne at 10ms Figures 3 4 and 5show that a larger force makes the tilt angle larger regardlessof LC gap thickness Figures 6(a) 6(b) and 6(c) compare3 120583m thick different LCmaterials including 5CB MBBA andPAA subjected to 5ms duration 05 dyne force respectivelyThe viscosity of PAA is the smallest and the tilt angle islarger than the others Therefore the option of liquid crystalmaterial with smaller viscosity can reduce the ripple

Tilt angle variations are shown under different voltages inFigure 7 and under different tactile forces in Figure 8 Using10V 20V 30V 40V and 50V under a tactile force of 3 dyneit is observed that the tilt angles remain the same in Figures7(a) 7(b) and 7(c) Thus different voltages do not affect tiltangles

Based on (33) Figures 8(a) 8(b) and 8(c) compare differ-ent LC materials including 5CB MBBA and PAA subject to25 dyne 3 dyne and 35 dyne forces in 3120583m and 5 120583m thickcell gap LC respectively Accordingly tactile forces affect tiltangles obviously Figures 9(a) 10(a) and 11(a) respectivelyshow tilt angles of 5CB LC subject to 25 dyne 3 dyne and

Mathematical Problems in Engineering 11

0 1 2 3 4 50

2

4

6

8Ti

lt an

gle (

deg)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 17 (a) Comparison of molecule tilt angles depicted in5120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

35 dyne forces in 3 4 and 5 120583m thick cell gap LC SimilarlyFigures 12(a) 13(a) and 14(a) are MBBA results and Figures15(a) 16(a) and 17(a) are PAA results In addition Figures9(b) to 17(b) compare analytical solutions with numericalsolutions of the full coupled equations The comparisonresults show that analytical solutions are consistent withnumerical solutions of the full coupled equations Thereforethe analytical derivation is validated

4 Conclusion

The Ericksen-Leslie theory deals with the dynamics ofnematic liquid crystals Based on the Ericksen-Leslie theorythis paper contributes to LC dynamics subject to tactile forceLC molecule tilt leads to a change in the optical intensitytransmitted through a liquid crystal cell LCD panels subjectto tactile force will show ripple-like propagation on screensThe present results show that the viscosity tactile force

thickness of cell gap and Leslie viscosity coefficient are thefactors of tilt angle variation Tilt angle variations of PAAliquid crystal molecules are sensitive to tactile force mag-nitudes while those of 5CB and MBBA with larger viscosityare not Analytical derivation has been carried out in thisstudy and analytical results have been validated by numericalresults

Acknowledgments

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

References

[1] J Ericksen ldquoConservation laws for liquid crystalsrdquoTransactionsof the Society of Rheology vol 5 pp 22ndash34 1961

[2] J L Ericksen ldquoContinuum theory of nematic liquid crystalsrdquoRes Mechanica vol 21 no 4 pp 381ndash392 1987

[3] F M Leslie ldquoTheory of flow phenomenum in liquid crystalsrdquoAdvances in Liquid Crystals vol 4 pp 1ndash81 1979

[4] F Brochard P Pieranski and E Guyon ldquoDynamics of the ori-entation of a nematic-liquid-crystal film in a variable magneticfieldrdquoPhysical Review Letters vol 28 no 26 pp 1681ndash1683 1972

[5] L Lei S Changqing S Juelian P M Lam and H Yun ldquoSolitonpropagation in liquid crystalsrdquo Physical Review Letters vol 49no 18 pp 1335ndash1338 1982

[6] F M Leslie ldquoSome topics in continuum theory of nematicsrdquoPhilosophical Transactions of the Royal Society A vol 309 pp155ndash165 1983

[7] J T Gleeson P Palffy-Muhoray and W van Saarloos ldquoProp-agation of excitations induced by shear flow in nematic liquidcrystalsrdquo Physical Review A vol 44 no 4 pp 2588ndash2595 1991

[8] F H Lin and C Liu ldquoExistence of solutions for the Ericksen-Leslie systemrdquo Archive for Rational Mechanics and Analysis vol154 no 2 pp 135ndash156 2000

[9] L R P de Andrade Lima and A D Rey ldquoPoiseuille flowof Leslie-Ericksen discotic liquid crystal Solution multiplicitymultistability and non-Newtonian rheologyrdquo Journal of Non-Newtonian Fluid Mechanics vol 110 no 2-3 pp 103ndash142 2003

[10] X Nie R Lu H Xianyu T X Wu and S T Wu ldquoAnchoringenergy and cell gap effects on liquid crystal response timerdquoJournal of Applied Physics vol 101 no 10 Article ID 103110 2007

[11] P A Cruz M F Tome I W Stewart and S McKee ldquoAnumerical method for solving the dynamic three-dimensionalEricksen-Leslie equations for nematic liquid crystals subjectto a strong magnetic fieldrdquo Journal of Non-Newtonian FluidMechanics vol 165 no 3-4 pp 143ndash157 2010

[12] X Nie H Xianyu R Lu T X Wu and S T Wu ldquoPretilt angleeffects on liquid crystal response timerdquo IEEEOSA Journal ofDisplay Technology vol 3 no 3 pp 280ndash283 2007

[13] I C Khoo and S T Wu Optics and Nonlinear Optics of LiquidCrystals World Scientific 1993

[14] I C Khoo Liquid Crystals Physical Properties and NonlinearOptical Phenomena John Wiley amp Sons New York NY USA1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

0 1 2 3 4 50

2

4

6

8Ti

lt an

gle (

deg)

Cell gap (120583m)

25dyne3dyne

35dyne

(a)

0 1 2 30

2

4

6

8

Numerical solutionAnalytical solution

Tilt

angl

e (de

g)

Cell gap (120583m)

4 5

(b)

Figure 17 (a) Comparison of molecule tilt angles depicted in5120583m cell gap PAA liquid crystals subject to different forces (b)Comparison of molecule tilt angles subject to 35 dyne betweenanalytical and numerical results

35 dyne forces in 3 4 and 5 120583m thick cell gap LC SimilarlyFigures 12(a) 13(a) and 14(a) are MBBA results and Figures15(a) 16(a) and 17(a) are PAA results In addition Figures9(b) to 17(b) compare analytical solutions with numericalsolutions of the full coupled equations The comparisonresults show that analytical solutions are consistent withnumerical solutions of the full coupled equations Thereforethe analytical derivation is validated

4 Conclusion

The Ericksen-Leslie theory deals with the dynamics ofnematic liquid crystals Based on the Ericksen-Leslie theorythis paper contributes to LC dynamics subject to tactile forceLC molecule tilt leads to a change in the optical intensitytransmitted through a liquid crystal cell LCD panels subjectto tactile force will show ripple-like propagation on screensThe present results show that the viscosity tactile force

thickness of cell gap and Leslie viscosity coefficient are thefactors of tilt angle variation Tilt angle variations of PAAliquid crystal molecules are sensitive to tactile force mag-nitudes while those of 5CB and MBBA with larger viscosityare not Analytical derivation has been carried out in thisstudy and analytical results have been validated by numericalresults

Acknowledgments

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

References

[1] J Ericksen ldquoConservation laws for liquid crystalsrdquoTransactionsof the Society of Rheology vol 5 pp 22ndash34 1961

[2] J L Ericksen ldquoContinuum theory of nematic liquid crystalsrdquoRes Mechanica vol 21 no 4 pp 381ndash392 1987

[3] F M Leslie ldquoTheory of flow phenomenum in liquid crystalsrdquoAdvances in Liquid Crystals vol 4 pp 1ndash81 1979

[4] F Brochard P Pieranski and E Guyon ldquoDynamics of the ori-entation of a nematic-liquid-crystal film in a variable magneticfieldrdquoPhysical Review Letters vol 28 no 26 pp 1681ndash1683 1972

[5] L Lei S Changqing S Juelian P M Lam and H Yun ldquoSolitonpropagation in liquid crystalsrdquo Physical Review Letters vol 49no 18 pp 1335ndash1338 1982

[6] F M Leslie ldquoSome topics in continuum theory of nematicsrdquoPhilosophical Transactions of the Royal Society A vol 309 pp155ndash165 1983

[7] J T Gleeson P Palffy-Muhoray and W van Saarloos ldquoProp-agation of excitations induced by shear flow in nematic liquidcrystalsrdquo Physical Review A vol 44 no 4 pp 2588ndash2595 1991

[8] F H Lin and C Liu ldquoExistence of solutions for the Ericksen-Leslie systemrdquo Archive for Rational Mechanics and Analysis vol154 no 2 pp 135ndash156 2000

[9] L R P de Andrade Lima and A D Rey ldquoPoiseuille flowof Leslie-Ericksen discotic liquid crystal Solution multiplicitymultistability and non-Newtonian rheologyrdquo Journal of Non-Newtonian Fluid Mechanics vol 110 no 2-3 pp 103ndash142 2003

[10] X Nie R Lu H Xianyu T X Wu and S T Wu ldquoAnchoringenergy and cell gap effects on liquid crystal response timerdquoJournal of Applied Physics vol 101 no 10 Article ID 103110 2007

[11] P A Cruz M F Tome I W Stewart and S McKee ldquoAnumerical method for solving the dynamic three-dimensionalEricksen-Leslie equations for nematic liquid crystals subjectto a strong magnetic fieldrdquo Journal of Non-Newtonian FluidMechanics vol 165 no 3-4 pp 143ndash157 2010

[12] X Nie H Xianyu R Lu T X Wu and S T Wu ldquoPretilt angleeffects on liquid crystal response timerdquo IEEEOSA Journal ofDisplay Technology vol 3 no 3 pp 280ndash283 2007

[13] I C Khoo and S T Wu Optics and Nonlinear Optics of LiquidCrystals World Scientific 1993

[14] I C Khoo Liquid Crystals Physical Properties and NonlinearOptical Phenomena John Wiley amp Sons New York NY USA1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of