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Title Non-Equilibrium Steady State Response and Fluctuations of Sheared Nematic Liquid Crystals

Author(s) Jaka Fajar Fatriansyah

Issue Date 2015-09-25

DOI 10.14943/doctoral.k12016

Doc URL http://hdl.handle.net/2115/59918

Type theses (doctoral)

File Information Jaka_Fajar_Fatriansyah.pdf

Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP

https://eprints.lib.hokudai.ac.jp/dspace/about.en.jsp

Doctoral Dissertation

Non-Equilibrium Steady State Response and

Fluctuations of Sheared Nematic Liquid Crystals

Jaka Fajar Fatriansyah

Laboratory of Soft Matter Physics

Division of Applied Physics, Graduate School of Engineering

Hokkaido University,

Sapporo, Japan, August, 2015




A dissertation submitted to the Division of Applied Physics,

Graduate School of Engineering,

Hokkaido University

In partial fulfillment of the requirements for the degree of

Doctor of Engineering

Sapporo, Japan, August, 2015

Supervisor : Professor Hiroshi Orihara

2015 年度学位論文



（せん断を印加したネマチック液晶の非平衡

定常応答とゆらぎ）

北海道大学大学院工学院

応用物理学専攻ソフトマター工学研究室




Dissertation for Doctor of Engineering at Hokkaido University

Submitted on August 20, 2015

Advisory Committee

Supervisor : Prof. Hiroshi Orihara

Vice Judge : Prof. Hiroshi Akera

Vice Judge : Prof. Yuichi Murai

v

ACKNOWLEDGEMENTS

AlhamdulillahiRobbilA’lamin

The present studies were carried out in the Division of Applied Physics, Faculty of Engineering,

Hokkaido University from 2012 to 2015. The author is indebted to the many people who have been of

assistance in various forms to make this investigation possible. Foremost, the author would like to

express sincere gratitude to Professor Hiroshi Orihara, who afforded an opportunity of studying in

Hokkaido University and guided to the field of soft matter, for his continuous guidance and advice

during the course of the research. The author is also very grateful to Professors Hiroshi Akera and

Yuichi Murai for their suggestions and advice, Professor Toshifumi Hiraoki and Assistant Professor

Yuji Sasaki for the helpful comments. The author is also sincerely indebted to Soft Matter Laboratory

members for their assistance and help during experiments. The author was accepted as a JICA AUN

SEED scholarship student and would like to express gratitude to the JICA AUN SEED, JICA AUN

SEED officers and Hokkaido International Center (Sapporo ) staffs. In addition the author would like

to express gratitude to Ministry of Education, Culture, Sports, Science, and Technology of Japan

because part of this work was supported through Grant-in-Aids for Scientific Research on Innovative

Areas “Fluctuation & Structure” (No.25103006) and for Scientific Research (B) (No. 26289032).

Finally, the author wishes to express deep gratitude to my parents especially my mother, family

specially my sisters and best friends for their support and kindness.


August 2015

vi

CONTENTS

ACKNOWLEDGEMENTS v

CONTENTS vi

LIST OF ABBREVIATIONS viii

LIST OF FIGURES ix

LIST OF PHYSICAL VARIABLES xvi

CHAPTER 1: INTRODUCTION

1.1 Basic of liquid crystal 1

1.2 Order parameter and director 2

1.3 Continuum theories of NLCs 4

1.4 Director alignment under shear flow 8

1.5 Non-equilibrium steady state response of NLC under shear flow 10

1.6 Objectives and outline 15

CHAPTER 2: NON-EQUILIBRIUM STEADY STATE DYNAMICS OF NEMATIC LIQUID

CRYSTALS UNDER SHEAR FLOW AND MAGNETIC FIELDS

2.1 Background 17

2.2 Derivation of equations of motion for a monodomain NLC in shear flow and magnetic fields

18

vii

2.3 Steady and unsteady states 21

2.4 Non-equilibrium fluctuations of NLCs 35

2.5 Conclusions 44

CHAPTER 3: NON-EQUILIBRIUM STEADY-STATE RESPONSE OF A NEMATIC LIQUID

CRYSTAL UNDER SIMPLE SHEAR FLOW AND ELECTRIC FIELDS

3.1 Background 45

3.2 Experiments 45

3.3 EL theory 57

3.4 Experimental results and discussion 64

3.5 Conclusions 75

CHAPTER 4: ELECTRIC-FIELD-INDUCED FLOW-ALIGNING STATE IN A NEMATIC

LIQUID CRYSTAL

4.1 Background 77

4.2 EL theory in FAS 78

4.3 Experiments 83

4.4 Experimental results and discussion 87

4.5 Conclusions 97

Chapter 5: SUMMARY AND CONCLUSIONS 98

REFERENCES 101

LIST OF PUBLICATIONS 104

viii

LIST OF ABBREVIATIONS

LC Liquid crystals

NLC Nematic liquid crystals

EL Ericksen-Leslie

NESS Non equilibrium steady state

FDR Fluctuation dissipation theorem

ac alternating current

dc direct current

LMWLC Low molecular weight liquid crystals

FAS Flow aligning state

NFAS Non-flow aligning state

ix

LIST OF FIGURES

Figure Page

FIG. 1.1. The comparison of solid, LC and liquid phase. Solid phase

has orientational and positional order while liquid phase does not have

order at al. On the other hand LC phase has positional order.

2

FIG. 1.2. Schematic relation between and n in NLC. 3

FIG. 1.3. Dependence of order parameter S on the temperature. TNI is

the transition temperature between liquid crystal and isotropic phases.

3

FIG. 1.4. The director reorientation towards the application of the

external field.

6

FIG. 1.5. The geometry of NLC under application of shear flow. 9

FIG. 1.6. Reorientation of the director under shear flow. This figure

only depicts the flow aligning state.

10

FIG. 1.7. Simple shear flow is applied in the x direction with a velocity

gradient parallel to the z axis. A weak ac electric field is applied in the

z direction as perturbation to measure the stress response.

11

FIG. 1.8. The shear stress response 2,2 2 for shear rate 10 s-1. 14

FIG. 1.9. The shear stress response 2,2 2 for shear rate 40 s-1. 14

FIG. 1.10. The shear stress respons 2,2 2 for shear rate 80 s-1. 15

FIG. 2.1. Geometry of the system. 19

FIG. 2.2. Relation between 0n , 1e and 2e . 22

x

FIG. 2.3. Stable and unstable states for 5CB in planes parallel to the Bx

– By plane at Bz = 0, 0.2, 0.5 and 1 T with11 s . The director 0n

(bar) is given for the stable states and the rotation axis 3e (arrow) for

the tumbling states. The short bar indicates the end of the director that

lies above the plane.

25

FIG. 2.4. Relaxation rates for the stable fluctuations as a function of

magnetic field yB along 0, , 0y

B 0y

B . At the critical point Bc,

one of the relaxation rates becomes zero. This critical point also marks

the existence of non-conservative forces.

26

FIG. 2.5. Three-dimensional phase diagram for 5CB with1

1 s

. The

unstable region (the tumbling state) is colored.

27

FIG. 2.6. Stable and unstable states for 8CB in planes parallel to the Bx

– By plane at Bz = 0, 0.1, 0.5 and 1 T with11 s . The director 0n

(bar) is given for the stable states and the rotation axis 3e (arrow) for

the tumbling states. The short bar indicates the end of the director that

lies above the plane. The stable region is divided into two regions

indicated by thin red bars and thick blue ones corresponding,

respectively, to complex and real eigenvalues.

28

FIG. 2.7. Three dimension phase diagram for 8CB with11 s . The

unstable region (the tumbling state) is colored.

29

FIG. 2.8. Configuration of ie and t . 31

FIG. 2.9. Evolution of t for 5CB with 0.2, 0.2, 0.5 [T]B .

The rotation axis is given by 3 (0.0829, 0.8337, 0.5458)e .

32

xi

FIG. 2.10. Trajectory of an unstable state in 5CB with initial condition

0,0,1 n and 0.2,0.2,0.5 [T]B .

33

FIG. 2.11. Dependence on initial condition of the trajectory of an

unstable state in 8CB without a magnetic field. The initial orientations

are (a) 0,0.9, 0.4359n and (b) 0, 0.4359, 0.9n .

34

FIG. 2.12. Over-damped profiles of (a) the correlation function and (b)

the integrated response function for 8CB at 0.3, 0.1, 0.2B [T].

38

FIG. 2.13. Over-damped profiles of (a) the correlation function and (b)

the integrated response function for 8CB at 0.3, 0.1, 0.2B [T].

39

FIG. 2.14. Magnetic field dependence of (a) the correlation function

0ij

C and (b) the integrated response function ijR for 5CB in the

vorticity direction 0, , 0y

B . At 0.4634 Ty

B , the off-diagonal

components of R , 12R and 21

R have similar value.

41

FIG. 2.15. Magnetic field dependence of (a) the correlation function

0ij

C and (b) the integrated response function ijR in 5CB for

sin cos ,sin sin , cosB B with 135 , 5 .

43

FIG. 3.1. Molecular structure of 5CB. 46

FIG. 3.2. Rheometer (Physica MCR300, Anton Paar). 47

FIG. 3.3. (a) A simple shear flow is applied along the x axis with a

velocity gradient parallel to the z axis. An electric field is also applied

along the z direction. (b) We define to be the angle between the

director and the x axis.

49

xii

FIG. 3.4. Experimental setup. Electric fields are applied between the

top and bottom plates. For the electro-optical experiment we add a light

source and a microscope to observe the change of transmitted intensity.

51

FIG. 3.5. Dependence of 1,1 on E (a) and 2,2 on 2E (b) at

-110 s and -1

0 100 V mmE . Linear relations are obtained at low

ac electric fields.

52

FIG. 3.6. Molecular structure of CYTOP. 53

FIG. 3.7. Texture of 5CB under crossed polarizers at a shear rate of 10

s-1. There is no disclination.

54

FIG. 3.8. Texture of 5CB under crossed polarizers at a shear rate of 80

s-1. There appear a few disclination lines. The appearance of

disclinations may be associated with the director deviation from the

shear plane though the details are not clear.

54

FIG. 3.9. The experimental setup. 56

FIG. 3.10. Dependence of 0 on 0E calculated from the EL theory.

The angle 0 increases monotonically and tends to saturate when

1

0 200 V mmE .

60

FIG. 3.11. 1,1 as a function of angular frequency before

correction.

64

FIG. 3.12. 2,2 as a function of angular frequency before

correction.

65

FIG. 3.13. as a function of frequency . 66

FIG. 3.14. as a function of frequency . 66

xiii

FIG. 3.15. Frequency dispersion of 1,1 as a function of angular

frequency . The theoretical curves are obtained from Eq. (3.15). A

characteristic non-zero plateau, which is due to the dc electric field,

appears at high frequencies in (b), (c) and (d).

67

FIG. 3.16. Frequency dispersion of 2,2 as a function of angular

frequency . The theoretical curves are obtained from Eq. (3.23). A

plateau appears under dc electric fields as well as in the first-order

response.

68

Fig. 3.17. 0E dependence of the first-order response at very low

frequency (0.23 rad s-1 for the experiment and exactly zero for the

theory), 1,1Re ( 0) . Also shown is the dependence of plateau

height 1,1Re ( ) . For dc electric fields higher than around

-190 V mm , 1,1Re ( ) becomes larger than

1,1Re ( 0) .

70

FIG. 3.18. 0E dependence of the first-order response at a very low


theory), 2,2Re ( 0) . Also shown is the dependence of plateau

height 2,2Re ( ) .

73

FIG. 3.19. Frequency dispersion of the optical response, which

corresponds to the director response, as a function of angular

frequency . Lines are the experiment and circles are the theory. No

plateau is observed, which shows that the plateau observed in the shear

74

xiv

stress response should be ascribed to the time derivative of the director

but not to 1,1 itself.

FIG. 4.1. The dc electric field dependence of the flow alignment angle

0 in the FAS at shear rates of 10 s-1 and 40 s-1 numerically obtained

for 8CB from Eq. (4.2).

80

FIG. 4.2. (a) Experimental setup. The sample is sheared by using a

parallel-plate rheometer under dc and ac electric fields. (b) The flow is

applied in the x direction, the velocity gradient is in the z direction, and

the electric field is applied in the z direction. The setup is similar with

the experiment in the previous chapter except that we do not use the

optical apparatus.

85

FIG. 4.3. (a) Dependence of 1,1 on E at 0E =100 V mm-1 and (b)

2,2 on 2E at 0E =100 V mm-1 at a shear rate of 10 s-1 at a

frequency of -10.63 rad s . Linear relationships are obtained at low

electric fields. Both are conducted in FAS state since for NFAS the

theoretical results are not available.

86

FIG. 4.4. Frequency dispersions of the first-order harmonic response

1,1( ) at a shear rate of 10 s-1 for several dc electric fields. The

dispersions are clearly distinguishable between NFAS ((a) and (b)) and

FAS ((c)-(f)). Solid lines for the FAS are calculated on the basis of the

EL theory.

89

FIG. 4.5. 0E dependence of the peak or dip frequency of the imaginary

part, peak , where dots are experimental results, a solid line is obtained

91

xv

from calculated frequency dispersion curves using Eq. (4.11), and a

dashed line is 1/ calculated from Eq. (4.7).

FIG. 4.6. 0E dependencies of the first-order response at very low


theory), 1,1Re ( 0) , for shear rates of 10 and 40 s-1.

92

FIG. 4.7. Frequency dispersions of the second-order response

2,2 ( ) at a shear rate of 10 s-1 for several dc electric fields. The

dispersions are clearly distinguishable between the NFAS ((a) and (b))

and the FAS ((c)-(f)). Solid lines in the FAS are calculated on the basis

of the EL theory.

94

FIG. 4.8. 0E dependencies of the second-order response at a very low

frequency (0.3 rad s-1 for the experiment and exactly zero for

the thory), 2,2Re ( 0) , for shear rates of 10 and 40 s-1.

95

FIG. 4.9. 0E dependencies of 0 at shear rates of 10 and 40 s-1. It is

difficult to distinguish the FAS and NFAS based on 0 .

96

xvi

LIST OF PHYSICAL VARIABLES

n Director

S Order Parameter

df Free energy density related to the deformation of director field

1k Splay elastic constant

2k Twist elastic constant

3k Bend elastic constant

Dielectric anisotropy ( / / )

/ / Dielectric constant parallel to the director

Dielectric constant perpendicular to the director

Diamagnetic Anisotropy ( / / )

/ / Permeability constant parallel to the director

Permeability constant perpendicular to the director

ef Free energy density under an external electric field

0 Dielectric constant in vacuum

E Electric field

B Magnetic field

v Velocity

(e)

Ericksen stress

(visc)

Viscous stress

h Molecular field

1 Rotational viscosity coefficient

2 Irrotational viscosity coefficient

N Rate of change of the director with respect to the background fluid

A Symmetric part of the velocity gradient

W Antisymmetric part of the velocity gradient

xvii

P Pressure

i Leslie coefficients

f Free energy

Shear Rate

Shear Stress

0 Shear stress with no ac electric field

First order responses with respect to the ac electric field for the first

harmonics of

2,0 Second order responses with respect to the ac electric field for the zeroth

harmonics of

2,2 Second order responses with respect to the ac electric field for the second

harmonics of

Tumbling parameter

G Dynamic stability matrix

g Eigen value

ijM Mobility matrix

( )pf t Perturbation

( )rf t Random force

Bk Boltzmann constant

T Temperature

V Volume

t Time

K Regression matrix

( )tC Correlation matrix

tR Integrated response function

Angular frequency

Relaxation time

1,1

1

CHAPTER 1

INTRODUCTION

1.1. Basic of liquid crystal

Liquid crystals (LCs) are intermediate states between conventional liquids and

solid crystals [1]. They have fluidity like liquids and anisotropy like solids at the same

time. The differences between them are depicted in Fig. 1.1. In the solid phase molecules

exhibit both the positional and the orientational orders, while in the liquid phase neither

of them. As shown in Fig. 1.1, LCs generally consist of rod-like molecules, which are

necessary to generate the anisotropy. The molecules tend to point in one direction, which

is designated by a unit vector n called the director. There are various LC phases, which

are characterized by the type of ordering. Roughly speaking, there are three types of LCs.

One is the nematic shown in Fig. 1.1. The cholesteric phase is quite similar to the nematic

in a short scale, but exhibits a twisting of molecules perpendicular to the director in a long

scale. The last one is the smectic, which possesses layers that can slide over one another

like a soap film. Therefore, the smectics have a one-dimensional positional order. There

are many different types of smectic phases. In this dissertation we focused only on the

nematic liquid crystal (NLC). NLC might be characterized as the simplest LC which is

distinguished from the isotropic/liquid only by the long-range orientational order [2].

2

FIG. 1.1. The comparison of solid, LC and liquid phase. Solid phase has

orientational and positional order while liquid phase does not have order at all. On the

other hand LC phase has positional order.

1.2. Order parameter and director

The nematic phase has the uniaxial symmetry hD , which appears as a result of

spontaneous symmetry breaking of the isotropic symmetry Kh of the isotropic phase. The

direction of up cannot be distinguished from down. To quantify the degree of the

uniaxiality we can define order parameter which is non-zero in the nematic phase and

vanishes in the isotropic phase. For nematic liquid crystals consisting of rod-like

molecules, the order parameter S is expressed as follows:

213cos 1

2S , (1.1)

where is the angle between the director and the long axes of each molecule as shown

in Fig. 1.2. The bracket denotes an average over all molecules.

3

FIG. 1.2. Schematic relation between and n in NLC.

According to Eq. (1.1), in the anisotropic phase the order parameter is zero, while in

perfectly ordered phase it is unity. Figure 1.3 shows the typical value for the order

parameter of LC which has values ranging from about 0.3 to 0.9 as a function of

temperature. It should be noted that the order parameter changes discontinuously at the

transition point from the isotropic to nematic phases [3], indicating that it is of first order

phase transition.

FIG. 1.3. Dependence of order parameter S on the temperature. TNI is the

transition temperature between liquid crystal and isotropic phases.

4

We have defined the director before, but it is not rigorous. Here, we define it

rigorously by introducing the tensor order parameter, which is an extension of the scalar

order parameter given in Eq. (1.1):

1

32

ij i j ijS a a , (1.2)

where a is the unit vector parallel to a rod-like molecule. The director n is defined to be

parallel to the unique eigenvector of ijS . The ground state for nematic liquid crystals are

the uniformly aligned state. However, nematic liquid crystals are soft so that the director

can be deformed easily by application of external forces and boundary conditions. At a

constant temperature in the nematic phase the scalar order parameter S is almost constant

for usual nematic liquid crystals. Therefore, it is enough to consider only the director

change. Typical theories on the statistics and dynamics for the director will be presented

in the next section.

1.3. Continuum theories of NLCs

Many important physical phenomena of NLCs such as reorientation of director

under external fields and flow properties can be studied by regarding a NLC as a

continuous medium. Oseen and Zocher developed a static theory in the late 1920s [4].

The theoretical formulation was re-examined by Frank thirty years later with the theory

of curvature elasticity. The dynamical theories were then developed by Anzelius and

Oseen but the formulation of general conservation laws and constitutive equations which

describe mechanical properties of NLC is formulated by Ericksen and Leslie [5-7]. The

5

Ericksen-Leslie (EL) theory is the first widely accepted dynamic theory for NLC. It was

formulated using balance laws with continuum mechanics as foundation and equilibrium

theory as principle. The EL theory assumes that the system is isothermal and

incompressible in its derivation.

Frank [8], Oseen [9] and Zocher [10] derived the following excess free energy

density related to the deformation of director field:

2 2

d 1 2 3

1 1 1

2 2 2f k n n k n k n n , (1.3)

where 1k , 2k and 3k are the splay, twist and bend elastic constants, respectively. From

this equation many results are derived, which can reproduce experimentally observed

phenomena. In this dissertation, however, we treat only cases of spatially uniform director

fields, and therefore we do not further describe the details.

The couplings between the director and external fields are important for

controlling the orientation: for example the coupling with electric fields is widely utilized

for liquid crystal displays. Basically, the couplings originate from the anisotropy such as

dielectric and magnetic anisotropies. Hereafter, we will briefly explain the dielectric

anisotropy, which is closely related to our studies. The dielectric anisotropy is

defined as / / with / / and being the dielectric constant parallel and

perpendicular to the director, respectively. The free energy density under an external

electric field is given by

2

2

e 0 0

1 1

2 2f E n E , (1.4)

where 0 is the dielectric constant in vacuum and E is the applied external electric field.

6

FIG. 1.4. The director reorientation towards the application of the external fields.

Based on Eq. (1.4) we can observe that if dielectric anisotropy is positive, in

order to minimize the free energy of the system, the director n tends to be parallel to the

applied electric field, while for negative , n tends to be perpendicular to the applied

electric field. Most of NLCs have positive dielectric anisotropy except for some NLCs

such as MBBA (Methoxybenzilidene Butylanaline) and PAA (1-methoxy-4-[(4-

methoxyphenyl)-NNO-azoxy] benzene).

Here, we summarize the EL theory, which can describe the dynamics of both the

director ( )n r and the velocity ( )v r . Here, the incompressibility is assumed:

0v . (1.5)

The conservation of linear momentum and angular momentum [6] give the following

equations of motion:

7

(e) (visc)dv

dt x

, (1.6)

and

1 2n h n N n n A , (1.7)

where (e)

in Eq. (1.6) is the Ericksen stress originating from the Frank elastic energy

df and the pressure p:

(e) d

nfp

xn x

. (1.8a)

On the other hand, (visc)

is the viscous stress given by

(visc)

4 1 5 6 2 3A n n n n A n n A n n A n N n N , (1.8b)

where A is the symmetric part of the velocity gradient: 1/ 2A v v and

N is the rate of change of the director with respect to the background fluid:

dnN W n

dt

, (1.9)

with the antisymmetric part of the velocity gradient 1/ 2W v v . The

viscosity coefficients i are called the Leslie coefficients, and there is the so-called

Parodi’s relation among them derived from the Onsager reciprocal relations:

6 5 2 3 . (1.10)

In Eq. (1.7) h is called the molecular field:

f fh

x nn x

, (1.11)

8

where the free energy density f consists of the Frank elastic energy df and the electric

interaction energy density ef . In Eq. (1.7) 1 is the rotational viscosity coefficient and 2

is the irrotational viscosity coefficient, which are related to the Leslie coefficients as

1 3 2 and 2 6 5 .

1.4. Director alignment under shear flow

In this section, we examine the effect of shear flow on the director orientation on

the basis of the EL theory. Hereafter, we assume a mono-domain NLC, in which the

director is independent of the position and confined in the shear plane. Under this

assumption we can put cos ,0,sinn , where is the angle between the director

and the flow direction, as shown in Fig. 1.5. On the other hand, for the simple shear flow

the velocity field is expressed as ,0,0v z , where is the shear rate. The mono-

domain assumption means that there is no contribution from the Frank elastic energy. For

the simple shear flow, the incompressibility (Eq. (1.5)) is automatically satisfied and the

equation of motion for the velocity field (Eq. (1.6)) is also satisfied. The director obeys

Eq. (1.7), which is reduced to

1 1 1

1( cos 2 )

2t

. (1.12)

9

FIG. 1.5. The geometry of NLC under application of shear flow.

In the steady state ( / 0t ), we have

1

1 2cos / , (1.13)

which is called the flow aligning angle, if the angle exists. This can exist for 1 2/ 1 .

For 1 2/ 1 , on the other hand, the director will tumble or find another stable

orientation with the director field being spatially deformed. Thus, under a simple shear

flow, NLCs take either one of two types of states depending on the ratio of 1 to 2 : the

flow aligning state and the non-flow aligning state. Most low molecular mass NLCs

under shear flow adopt the flow aligning state except for a small number of NLCs such

as 8CB (4-n-octyl-cyanobiphenyl) [11]. The steady state of reorientation angle and the

director n is shown in Fig. 1.6.

10

FIG. 1.6. Reorientation of the director under shear flow. This figure only depicts

the flow aligning state.

1.5. Non-equilibrium steady state response of NLC under shear flow

The linear response is a powerful tool to examine thermal fluctuations. The

complete theory of linear response in equilibrium has been thoroughly developed by

Kubo [12] and is widely used. As long as the applied perturbation force is small, the

macroscopic linear response should be connected to the dynamical fluctuation of the

observed system through its correlation function multiplied by a proportionality factor.

This factor simply consists of the temperature of the equilibrium system surrounded by a

thermal bath and the Boltzmann constant. This relation is called the fluctuation-

dissipation relation (FDR) [13, 14]. This relation is a consequence of microscopic

reversibility of the system and is connected to the symmetry of the response function.

This is a monumental achievement in statistical physics. However, for a system far away

from equilibrium, the FDR is generally invalid [15-20]. However, we can have similar

11

but modified relations for some non-equilibrium steady states (NESSs). For example,

Sakaue et al. have derived a FDR for a dumbbell subjected to a simple shear flow and

discussed the symmetry of the correlation and response functions [21]. In this case, non-

conservative forces brought about by the shear flow play a crucial role, which will be

examined for our case of sheared liquid crystals in the next chapter.

The shear flow has a unique property that it easily breaks the time reversal

symmetry and brings fluids far from equilibrium [22]. In general, application of steady

shear flow can make NESSs. Orihara et al. [23] has applied a steady shear flow to a liquid

crystal as a source of non-conservative forces and measured the shear stress response to

small ac electric fields applied along the velocity gradient, as shown in Fig. 1.7. In their

experiment, they use 5CB (4-n-pentyl-4'-cyanobyphenyl) with a positive dielectric

anisotropy.

FIG. 1.7. Simple shear flow is applied in the x direction with a velocity gradient

parallel to the z axis. A weak ac electric field is applied in the z direction as perturbation

to measure the stress response.

12

For an ac electric field 0 cosE t E t with an amplitude 0E and an angular

frequency , the change of shear stress is assumed to be proportional to the square of

electric field, so the shear stress can be expressed as [7]

2

0 2,0 2,2( ) Re[ ]i tt e , (1.14)

where 0 is the shear stress with no ac electric field, 2,0 and

2,2 are the second

order responses with respect to the ac electric field for the zeroth and second harmonics

of , respectively.

Fig. 1.8 shows 2,2 as a function of 2 at a shear rate of 5 s-1. A Debye-type

relaxation is observed, which has already been explained by the EL theory [23]. On the

other hand, at high shear rates (Figs. 1.9 and 1.10) the response deviates from the Debye

type: the real part becomes negative in a certain frequency range and the imaginary part

becomes narrow. In order to examine the discrepancy, they proposed a two mode

coupling model in which there are two modes considered and they are coupled to each

other. It is obvious that if we have only one mode which obeys the usual equation of

motion the response must be of the Debye type. One mode is the orientation change in

the shear plane (in-plane mode) and the other is the out-of-shear-plane mode. The

corresponding angles are expressed by and ' , respectively. In their paper,

however, ' has not been clearly specified. This will be more clarified in the next

chapter. Under an ac electric field, the equations of motion is written as follows: [23]

13

2

1 11 12 1' Re i tda a K e

dt

, (1.15a)

2

21 22 2

'' ' Re i td

a a K edt

, (1.15b)

where 1 is the usual rotational viscosity, is the viscosity coefficient for ,

ija (i,j = 1,2) denote coefficients depending on the shear rate and iK (i = 1,2) represent

the electric force. At equilibrium, i.e., without shear flow ija components are symmetric

because they are derived from the free energy F as 2

12 21/ 'a F a . At non-

equilibrium it is possible that 12 21a a , meaning that non-conservative forces should

appear. The shear stress can be calculated by assuming that it is a linear combination of

and ' . Orihara et al. have analyzed their data in terms of the above set of equations.

Fitted curves are shown by lines in Fig. 1.9 and 1.10, which agree well with the

experimental data. As a result, they found that the eigenvalues of ija become complex,

indicating that 12 21a a . Thus, they revealed that there exist non-conservative forces in

the sheared nematic liquid crystal. A similar response has been observed in an immiscible

polymer blend where one polymer is dispersed as droplets in the other [24].

14

FIG. 1.8. The shear stress response 2,2 2 for shear rate 10 s-1.


15


In the next chapter we will derive equations of motion with non-conservative

forces from the EL theory and discuss the dynamics of response and fluctuations in a

sheared nematic liquid crystal.

1.6 Objectives and outline

This research aims to investigate the non-equilibrium steady state response of

NLCs. The NLCs are brought to non-equilibrium steady states by application of shear

flow and the linear response of shear stress to applied ac electric fields is mainly studied.

In chapter 2, the response and fluctuations of NLCs under shear flow and magnetic

fields are theoretically studied on the basis of the EL theory. We derive equations of

motion, from which the origin of the non-conservative forces observed experimentally in

Ref. [23] is clarified, and calculate the phase diagram of flow and non-flow aligning states

in the three-dimensional space of the applied magnetic field. In addition, we derive a

16

modified fluctuation dissipation relation valid for the present non-equilibrium steady state

and discuss the role of the non-conservative forces.

In chapter 3, the influence of applying a dc electric field on the shear stress

response in addition to the shear flow is investigated both experimentally and

theoretically. The non-equilibrium response under shear flow is expected to qualitatively

change because the steady orientation of director is modified by the application of dc

electric field. A special feature of this system unveiled in the experiment is discussed on

the basis of Parodi’s relation derived from the Onsager reciprocal relation.

In chapter 4, the electric field-induced transition from the non-flow aligning state

to the flow aligning state is studied by the shear stress response when the dc electric field

is increased. The NLC used in this chapter is different from that used in chapter 3: the

latter takes the flow aligning state, but the former takes the non-flow aligning state, which

can be transformed to the flow aligning state by application of dc electric fields. We

expect that the shear response is sensitive to the transition and so some phenomena

characteristic to the non-equilibrium transition are observed.

Chapter 5 is devoted to conclusions.

17

CHAPTER 2

NON-EQUILIBRIUM STEADY STATE DYNAMICS OF NEMATIC LIQUID

CRYSTALS UNDER SHEAR FLOW AND MAGNETIC FIELDS

2.1. Background

As described in the previous chapter, Orihara et al. [23] found characteristic

features of non-equilibrium steady state response under shear flow in a nematic liquid

crystal (NLC): the real part of the response takes negative values and the peak of the

imaginary part becomes narrow at high shear rates. To explain them, they developed a

two mode coupling model, where one mode is a director fluctuation in the shear plane

(in-plane mode) but the other is not identified (the out-of-shear-plane fluctuation may be

the most plausible choice). Based on this model it was shown that the eigen-values of the

dynamical matrix for the two mode become complex above a certain shear rate, indicating

the existence of non-conservative forces which are brought about by the application of

shear flow to the system. However, the two mode coupling model is entirely

phenomenological. So, in this chapter we derive the equation of motion from the EL

theory, which has been firmly established for nematic liquid crystals.

The EL theory has been used for investigating basic mechanisms and explaining

various phenomena in NLCs. In this chapter we discuss general dynamical properties of

NLCs under shear flow and magnetic fields and demonstrate the existence of non-

conservative forces [25]. In addition, we also derive the modified fluctuation dissipation

relation (FDR) for our system, which gives the relationship between response and

fluctuations at non-equilibrium under shear flow. Here, we apply magnetic fields in

18

addition to the shear flow because without a magnetic field non-conservative forces never

appear, as will be shown in subsequent sections. In the experiment by Orihara et al. it is

thought that the director deviates from the shear plane, and the deviation is necessary for

non-conservative forces to appear, which is caused by the boundary condition at the top

and bottom surfaces. It is so complicated to take into account the boundary condition that

we adopt a mono-domain sample in which the director orientation is independent of

position. Thus, we need magnetic fields to induce the deviation in the mono-domain NLC.

The boundary effects can be regarded as secondary effects when studying the non-

equilibrium fluctuations in aligned states. The EL theory has already been applied to

NLCs under shear flow and magnetic fields [6, 7]. In these theoretical investigations,

however, the magnetic field direction is restricted in the shear plane or along the vorticity

direction. We examine the stability of aligned states in any arbitrary directions of

magnetic field under shear flow.

2.2. Derivation of equations of motion for a monodomain NLC in shear flow and

magnetic fields.

We theoretically consider an incompressible mono-domain NLC subjected to

shear flow and magnetic fields on the basis of the EL theory. As shown in Fig. 1, the flow

direction and the velocity gradient are, respectively, along the x and z axes, that is, the

velocity is expressed as

19

,0,0v z , (2.1)

where is the shear rate.

FIG. 2.1. Geometry of the system.

For a monodomain NLC, the time evolution of the director n as described by the EL

theory from angular momentum balance [26] is

1 2n h n N n n A , (2.2)

with

f f

hx nn x

, (2.3)

a

dnN W n

dt

. (2.4)

20

Here, d

dt denotes the material derivative,

1

2A v v and

1

2W v v are, respectively, the symmetric and anti-symmetric parts of the

velocity gradient, 1 and 2 are viscosity coefficients. N is the rate of change of the

director with respect to the background fluid, and h is the molecular field. The molecular

field is created by the free energy of the interaction with the magnetic field:

2

1 2 1

0 0

1 1

2 2f B n B

, (2.5)

where 0 is the vacuum permeability and is the diamagnetic anisotropy defined as

/ / with / / and being the permeabilities parallel and perpendicular to the

director, respectively. Substitution of Eqs. (2.3) - (2.5) into Eq. (2.2) yields the following

equation:

1

dnn n n

dt G , (2.6a)

with

1

0 1 2G B B W A . (2.6b)

This is the starting equation in this chapter. We make numerical calculations using the

parameters of 5CB (4-cyano-4′-pentylbiphenyl) and 8CB (4′-octyl-4′-cyanobiphenyl).

5CB is stable (aligned) under shear flow without a magnetic field, while 8CB is unstable

(tumbling), because they have, respectively, positive and negative tumbling parameters,

1 2

2 1

[7].

21

2.3. Steady and unsteady states

Depending on the parameters in Eq. (2.6b), we have some stable (steady) and

unstable states under shear flow. We examine the linear stability and make phase

diagrams of steady states in the space of magnetic field. In the steady states, the properties

of fluctuations such as the magnetic field dependence of relaxation time are investigated

in detail. On the other hand, in the unstable states tumbling motion of director appears,

the trajectory of which is numerically obtained.

Steady States

As mentioned above, for steady states, if they exist, Eq. (2.6a) is reduced to the

form 0n n G , which is equivalent to the eigen-value equation

n gnG , (2.7)

where g is the eigen-value and n is the eigenvector. In a similar way, we can rewrite

Eq. (2.6a) as 1 / 0n dn dt n G and obtain

1

dnn n

dt G . (2.8)

where is a constant. The stability of the system can be examined by inserting

0n n n into Eq. (2.8). For 0n , we have 0 0n n G , implying that g in

Eq. (2.7). On the other hand, for n we have

1

d ng n

dt

G . (2.9)

22

Defining two unit vectors 1e and 2e , which form a right-handed system of 0n - 1e - 2e

together with 0n , where n can be written as 1 1 2 2n n e n e . Substitution of this

equation into Eq. (2.9) yields

FIG. 2.2. Relation between 0n , 1e and 2e .

2

1

1

iij j

j

d nM n

dt

, (2.10)

with

1

0 1 2 .

G

W A

ij ij i j

ij i j i j i j

M g e e

g e B e B e e e e

(2.11)

Now we discuss the appearance of non-conservative forces on the basis of

Eq. (2.11). First, the term 1 i je e W in Eq. (2.10) is anti-symmetric with respect to the

exchange of indices i and j as W is also anti-symmetric, though the other terms are

symmetric. The anti-symmetricity is easily proven as follows:

23

T

T T T

i j i j j i j ie e e e e e e e W W W W . (2.12)

Therefore, the matrix M can be asymmetric, for which there should be non-conservative

forces arising from the rotational flow. By introducing the vorticity vector which can be

defined as , , yz xz xyW W W , the anti-symmetric term can be rewritten as

1 1 2 1 0e e n W , indicating that the non-conservative forces disappear when 0n is

perpendicular to , which is in the y direction in our case. This result is related to the

conjecture by Orihara et al. [23], in which the observed non-conservative forces may

come from the director deviation from the shear plane. The steady state may be stable in

the case that the real parts of both eigenvalues of M are positive. As shown above, M is

not symmetric, so the eigenvalues can be complex numbers.

For both 5CB and 8CB, we calculate 0n from Eq. (2.7) for various values of B

and examine its stability. The parameters used for the calculation are as follows. For 5CB

we use 1 0.07738 Pa s , 2 0.08604 Pa s , 71.1 10 at 25 C [27], and for

8CB we use 1 0.0618 Pa s , 2 0.0556 Pa s , 71.1 10 at 37 C [28]. The

shear rate is set to be 1 s-1. It is worth pointing out that Eq. (2.6) can be expressed in terms

of scaled variables:

2

1

' ' ' ''

dnn n B B n

dt

W A , (2.13a)

with

1

0

1

' , ' , ' / , ' / .t t B B

W W A A (2.13b)

24

Thus results for other shear rates can be obtained from those for = 1 s-1.

First, we plot stable and unstable states for 5CB in planes parallel to the Bx – By

plane for Bz = 0, 0.1, 0.5 and 1 T in Fig. 2.3. For stable states, 0n is designated by a bar,

because 0n and 0n are the same due to the uniaxiality of NLC, and the short bars

attached to these bars indicate that the end of the vector lies above the plane. Unstable

states are represented by arrows which indicate the rotation axes in the tumbling state, as

described in the next subsection.

At Bz = 0, the system is stable over all the Bx – By plane including 0B . However,

there are two singular points, designated by red dots on the By axis, where the director

orientation changes discontinuously by /2. Along the By axis the eigen-values and eigen-

vectors of G are easily obtained as:

2 2

2 12 1 2 11,2 1,2

2 2

,0, for 2 2 2

n g

,

1 2

3 3 00,1,0 for yn g B . (2.14)

For each eigenvector the eigenvalues of M are also easily calculated:

2 2

2 12 2 1 2

2 1 0, 2

yB

for 1,2n ,

2 2 2 2

2 1 2 11 2 1 2

0 0, 2 2

y yB B

for 3n . (2.15)

25

FIG. 2.3. Stable and unstable states for 5CB in planes parallel to the Bx – By plane

at Bz = 0, 0.2, 0.5 and 1 T with11 s . The director 0n (bar) is given for the stable states

and the rotation axis 3e (arrow) for the tumbling states. The short bar indicates the end of

the director that lies above the plane.

Note that 2 1 , 2 1 and 2 are all negative for 5CB and so all of the eigenvalues are

real. It is readily understood that 2n is always unstable while 1n and 3n are stable,

26

respectively, for the case y cB B and

y cB B , where 1 2 2

0 2 1 / 2cB

(about 0.4634 T) and 0, , 0cB are the singular points in director orientation. This

critical point shows the transition from the state where non-conservative forces does not

exist (fory cB B ) to the state where the non-conservative force exists (for

y cB B ) .

We examine the dynamical properties around the singular points by calculating

the relaxation rates given by the eigenvalues of M divided by 1 . We show the relaxation

rates as a function of yB along 0, , 0yB 0yB in Fig. 2.4. As the critical point cB

is approached, one of the relaxation rates goes to zero from both sides, as in an equilibrium

phase transition.

FIG. 2.4. Relaxation rates for the stable fluctuations as a function of magnetic

field yB along 0, , 0y

B 0y

B . At the critical point Bc, one of the relaxation rates

becomes zero. This critical point also marks the existence of non-conservative forces.

As zB increases from zero, unstable regions appear from the singular points

(Bz = 0.1 T in Fig. 2.5), expand, merge into one region (Bz = 0.5 T), which then shrinks

27

(Bz = 1 T) and finally disappears. In the unstable region there cannot be two positive

eigenvalues as there are in the stable region. The three-dimensional shape of the unstable

region (the phase diagram) is shown in Fig. 2.5. The unstable region is divided into two

by the singular (critical) points. At high magnetic fields the system is stabilized

irrespective of the orientation of the magnetic fields. For 5CB all the eigen-values are real

in the stable region.

FIG. 2.5. Three-dimensional phase diagram for 5CB with1

1 s

. The unstable

region (the tumbling state) is colored.

For 8CB, the 0n plot and the phase diagram are shown in Figs. 2.6 and 2.7,

respectively.

28

FIG. 2.6. Stable and unstable states for 8CB in planes parallel to the Bx – By plane

at Bz = 0, 0.1, 0.5 and 1 T with11 s . The director 0n (bar) is given for the stable states

and the rotation axis 3e (arrow) for the tumbling states. The short bar indicates the end of

the director that lies above the plane. The stable region is divided into two regions

indicated by thin red bars and thick blue ones corresponding, respectively, to complex

and real eigenvalues.

29

Without a magnetic field it is unstable (strictly speaking, the stability may be marginal

because one of the eigenvalues is purely imaginary), and the application of an

infinitesimal magnetic field in the y direction can stabilize it (see Fig. 2.7). 8CB can have

complex eigenvalues of M, which arise because 2 2

2 1 0 , unlike 5CB for which

2 2

2 1 0 .

In Fig. 2.7, the two regions are distinguished by the thickness and color of the

bars; they are thin red bars for complex eigenvalues and thick blue bars for real ones. The

real and complex eigenvalues give rise to over-damped oscillation (relaxation) and under-

damped oscillation, respectively, around stable states. The over-damped region lies near

the Bx-Bz plane and it expands when the z component of the magnetic field is increased,

as shown in Fig. 2.6. These two kinds of stable states have also been observed

theoretically in the studies of liquid crystalline polymers based on Doi theory [29-31].

FIG. 2.7. Three dimension phase diagram for 8CB with11 s . The unstable

region (the tumbling state) is colored.

30

Some of the above results are in agreement with analytical ones provided by

Ericksen and Leslie [5, 32]. However, their investigations are restricted only to special

cases where the magnetic field is either in or perpendicular to the shear plane.

Unstable (tumbling) state

In the NLC, under the application of shear flow only, 5CB will undergo steady

aligning state and 8CB undergo unstable (tumbling) state although for the latter, tumbling

is only transient state in the reality. Both 5CB and 8CB can exhibit unstable states under

application of shear flow and magnetic fields. In this subsection we investigate unstable

states in detail. The matrix G in Eq. (2.7) has three eigenvalues and we assume that, for

unstable states, one is real and the other two are conjugate complex numbers. This will

be numerically confirmed for both 5CB and 8CB.

From the complex eigenvectors corresponding to the complex eigenvalues, we

can make two mutually orthogonal real vectors, 1e and 2e , which are perpendicular to the

left eigenvector of G corresponding to the real eigenvalue, 3e . If n is in the plane of 1e

and 2e , nG remains in this plane. Therefore, from Eq. (2.6a), /dn dt is also in the plane

of 1e and 2e , meaning that n is confined to this plane.

Putting 1 2cos sinn t e t e t and inserting it into Eq. (2.6a), we have

2 2

1 3 21 22 11 12

1cos sin 2 sin 0,

2

de g g g g

dt

(2.16)

31

where we have used 3 1 2e e e and defined ij i jg e e G . Therefore, we have the

equation of motion for t :

22 112 2

1 21 12

21 12 21 12 22 11

2 2

21 12 21 12 22 11 0

cos sin 2 sin2

1 1 1cos 2 sin 2

2 2 2

1cos 2 2 ,

2

g gdg g

dt

g g g g g g

g g g g g g

(2.17)

where 22 110

21 12

tan 2g g

g g

. The schematic picture is shown in Fig. 2.8.

FIG. 2.8. Configuration of ie and t .

For the system in order to become unstable it is necessary that the following inequality is

satisfied:

2 2

21 12 21 12 22 11g g g g g g , (2.18a)

32

Then the solution of Eq. (2.17) can be given by

2

1 2

0

1tan tan 1

1

AA Bt

A

, (2.19a)

with

21 12

2 2

21 12 22 11

g gA

g g g g

,

21 12

12

g gB

A

. (2.19b)

A typical evolution of t is shown in Fig. 2.9 for 5CB with 0.2, 0.2, 0.5 [T]B .

t monotonically increases with time though it is modulated, indicating that the

director should rotate or tumble around the 3e axis.

FIG. 2.9. Evolution of t for 5CB with 0.2, 0.2, 0.5 [T]B . The rotation

axis is given by 3 (0.0829, 0.8337, 0.5458)e .

33

Figure 2.10 shows a trajectory for the director, which was obtained by numerically

solving Eq. (2.6a) with an initial condition of 0 0, 1, 0n . The trajectory approaches

a circle obtained analytically in Eq. (2.19a) as time passes, indicating that it should be a

stable limit cycle. The rotation axis is calculated to be 3 (0.0829,0.8337, 0.5458)e .

FIG. 2.10. Trajectory of an unstable state in 5CB with initial condition

0,0,1 n and 0.2,0.2,0.5 [T]B .

The eigenvalues of M for 8CB without a magnetic field are purely imaginary. This means

that the trajectory is not a limit cycle and strongly depends on the initial conditions. In

this case ( 2 1/ 1 ), the solution of Eq. (2.2) is given by

2 2

2 1 2 11 1 2 10

22 1

2 1

1 1 1tan tanh tanh tan ,

1 2 1

tt

(2.20a)

and

34

1 2 1 00

2 1

1 cos 2tan tan

1 cos 2t

t

, (2.20b).

where t and t are the polar and azimuthal angles at time t, respectively, with the

polar axis in the y direction, and 0 and 0 are the initial conditions. Trajectories with

different initial conditions are shown in Fig. 2.11. Both are closed, but the shape differs

depending on the initial conditions. Note that these states are easily changed into other

states by weak perturbations such as external fields and boundary effects.

FIG. 2.11. Dependence on initial condition of the trajectory of an unstable state in

8CB without a magnetic field. The initial orientations are (a) 0,0.9, 0.4359n and (b)

0, 0.4359, 0.9n .

35

2.4. Non-equilibrium fluctuations of NLCs

In this section we discuss non-equilibrium steady states in more detail. Here, we

turn our attention to thermal fluctuations in the steady state. In our system, as has been

mentioned, there are non-conservative forces, which produce a non-vanishing probability

current violating the principle of detailed balance, leading to the breakdown of the

fluctuation-dissipation theorem [14]. At the Markovian level of description [33], we can

obtain the following Langevin equation for the director fluctuation n by adding a weak

perturbation ( )pf t and a random force ( )rf t to Eq. (2.10):

( ) ( )

1

p rd nn f t f t

dt

M . (2.21)

Note that n should be regarded as the spatial average of the director fluctuation over

the system volume V or the director fluctuation for zero wavenumber. The random force

( )rf t is assumed to be Gaussian white noise with zero mean:

( ) ( ) ( ) 11 2 1 20, r r r B

i i j ij

k Tf t f t f t t t

V

, (2.22)

where Bk is the Boltzmann constant and T is the temperature. First, let us calculate the

time correlation function ( ) ( ) (0)ij i jC t n t n . The solution of Eq. (2.21) with

( ) 0pf t is given as follows:

( )

1

1t

t pn t e f d

K

, (2.23)

where we have defined a regression matrix 1/ K M . Thus, the correlation matrix

( ) 0t n t n t t TC becomes

36

T1 2

T1 2

0 T( ) ( )

1 2 1 22

1

01

1 2 1 22

1

1

1

(0),

t t p p

t t B

t

t d d e f f e

k Td d e e

V

e

K K

K K

K

C

C

(2.24a)

with

T0

1

(0) Bk Td e e

V

K K

C . (2.24b)

By using the following relation:

T T T0 0 0

Td e e d e e d e e

K K K K K K

I K K , (2.25)

for the static correlation function (0)C , we can obtain

T

1

0 0Bk T

V I KC C K . (2.26)

From Eq. (2.26) we can calculate the static correlation function:

1 11 22 11 22 12 21

22 11 22 12 12 21 11 12 21 22

11 12 21 22 11 11 22 21 12 21

(0)

.

BC

k T

V K K K K K K

K K K K K K K K K K

K K K K K K K K K K

(2.27)

For 0t , the time-translation symmetry yields ( ) ( )t t TC C . On the other hand, since

we have no time reversal symmetry under shear flow, ( ) ( )t t TC C in general case.

37

In order to probe the system response, we apply a step force0F , which is switched

on at t = 0. Solving Eq. (2.21) after averaging, the average evolution of n t , n t ,

is given as

0 ,n t t F R (2.28a)

with

1 0

1t

tt e d

TKR . (2.28b)

The integrated response function tR can be rewritten as

1 1 1

0 0

1 1

t t

t t t t tt e e d e e d e e e

K K K K K K K

R L K L K L K L , (2.29)

where 1

1

L= I (I being the unit matrix) is the mobility matrix, which is symmetric due to

the Onsager reciprocity. It is easily confirmed that Eqs. (2.24a) and (2.29) satisfy a

modified fluctuation dissipation relation:

0 t t k V BC C R Θ/ , (2.30)

with

1 0k V BΘ L KC . (2.31)

Note that Θ becomes the scalar T in equilibrium, where non-conservative forces vanish,

and Eq. (2.30) reduces to the usual fluctuation dissipation theorem. In our case Θ is given

by the following matrix

12 21 11 221

12 21 11 22

1 /0

/ 1Θ L KC

K K K KT

K K K K

, (2.32)

38

from which the above statement can be easily confirmed, as the regression matrix

becomes symmetric without shear flow.

Typical correlation and integrated response functions for the over-damped case of

8CB at 0.3, 0.1, 0.2B [T] are shown in Fig. 2.12, where 0n is calculated to be

0.9516, 0.0882, 0.2942 and we chose the basis vectors of n as

1 0.0924, 0.996, 0e , 2 0.293, 0.0271,0.9557e .

FIG. 2.12. Over-damped profiles of (a) the correlation function and (b) the

integrated response function for 8CB at 0.3, 0.1, 0.2B [T].

39

Both functions show monotonic time dependence. Their diagonal components are not so

different, while the off-diagonal ones are quite different and asymmetric. For the under-

damped case of 8CB at 0.2, 0.2, 0B [T], similar plots are shown in Fig. 2.13, where

0n is calculated to be 0.0715, 0.9956, 0.0590 and we chose the basis vectors of n

as 1 0.9974, 0.0716, 0e , 2 0.0042, 0.0589, 0.9982e .

FIG. 2.13. Over-damped profiles of (a) the correlation function and (b) the

integrated response function for 8CB at 0.3, 0.1, 0.2B [T].

40

All the components of the correlation and response functions show under-damped

oscillations, and the corresponding components are totally different. This may be due to

the large off-diagonal components of Θ .

Finally, we examine critical behaviors such as the divergence of fluctuations. We

have observed the divergence of the relaxation time at 0, , 0cB B in 5CB,

indicating that the fluctuation intensity may diverge as well. Figure 2.14 shows the

magnetic field dependence of 0ij i jC n n divided by /Bk T V along the By axis.

Note that 0n , 1e and 2e change at the critical point: 0 cos , 0, sinf fn ,

1 0, 1, 0e and 2 sin , 0, cosf fe with flow-aligning angle

1

1 2cos / / 2f from the x axis for y cB B , and 0 0, 1, 0n ,

1 1/ 2, 0, 1/ 2e and 2 1/ 2, 0, 1/ 2e for y cB B . For

y cB B , where

there is no non-conservative force as 0n is in the shear plane, there is no cross correlation

12 210 0 0C C , and the fluctuation tilting toward the y direction, 11 0C , increases

as the critical point is approached and diverges at that point, though the fluctuation in the

shear plane, 22 0C , is constant. In contrast, for y cB B , there are non-conservative

forces and, therefore, the cross correlation always exists for any mutually perpendicular

bases 1e and 2e . All the components diverge at the critical point. The corresponding

response function R is shown in Fig. 2.14(b), which is related to 0C by

0 /k V BC R Θ (see Eq. (2.30) with t ). For y cB B , R is exactly the

41

same as 0 / BV k TC as there is no non-conservative force. While fory cB B , all the

components are different from those of 0 /V k TBC and diverge at the critical point.

FIG. 2.14. Magnetic field dependence of (a) the correlation function 0ij

C and

(b) the integrated response function ijR for 5CB in the vorticity direction 0, , 0y

B .

At 0.4634 Ty

B , the off-diagonal components of R , 12R and 21

R have

similar value.

42

It should be noted that both below and above the critical magnetic field the system

is at non-equilibrium though non-conservative forces exist only above the critical point.

The existence of non-conservative forces guarantee that the system is at non-equilibrium,

but they do not always exist for non-equilibrium steady states, as we show for the case

y cB B .

The divergence of 0C and R also occurs at the transition point from the

aligned state to the tumbling state, as shown in Fig. 2.15 for 5CB, where

sin cos ,sin sin ,cosB B with 135 , 5 . The discrepancy between

0 /V k TBC and R is not large as the direction of magnetic field is nearly in the

shear plane, that is, the director is also nearly in the shear plane, resulting in small non-

conservative forces.

43

FIG. 2.15. Magnetic field dependence of (a) the correlation function 0

ijC and

(b) the integrated response function ijR in 5CB for

sin cos ,sin sin , cosB B with 135 , 5 .

44

2.5 Conclusions

Using the EL theory, phase diagrams in the three-dimensional space of magnetic

field at a constant shear rate were obtained for 5CB and 8CB which have different flow

properties. An analytical solution was given for the tumbling state, in which the director

rotates about an axis determined as a left eigenvector of the matrix G that governs the

motion of the director. It was also shown that non-conservative forces caused by shear

flow appear when magnetic fields are applied, so that the director points out of the shear

plane. The non-conservative forces experimentally observed in the response function of

5CB [23] are thought to originate from a similar mechanism. Using the Langevin equation,

the orientational fluctuations of the director were examined in detail by calculating the

time correlation function and the response function, which are related to each other

through a modified fluctuation dissipation relation. Critical behaviors such as the

divergence of the relaxation time and fluctuations were found at the critical points where

there is a discontinuous change in director orientation and at the boundary between the

stable and unstable states.

45

CHAPTER 3

NON-EQUILIBRIUM STEADY-STATE RESPONSE OF A NEMATIC LIQUID

CRYSTAL UNDER SIMPLE SHEAR FLOW AND ELECTRIC FIELDS

3.1. Background

As discussed in the previous chapter, NLCs can be easily taken to non-equilibrium

states by applying external forces, so they have been widely used to investigate various

types of non-equilibrium phenomena [34]. Director can easily be controlled by external

fields such as electric field [35]. Experimentally, the response function in the sheared

NLC was obtained by applying a small ac electric field to measure the corresponding

shear stress response. A characteristic response, which is originated from the non-

conservative force, was clearly observed. In this chapter, we investigate both theoretically

and experimentally the influence of a dc electric field on the shear stress response in

addition to the shear flow. The application of dc electric field is expected to change the

steady orientation of the director [36-38], bringing about qualitative changes in response

[39].

3.2. Experiments

In the experiment, we used the NLC 5CB (4-n-pentyl-4'-cyanobiphenyl; Tokyo

Chemical Industry) without any further treatment. 5CB is one of the commonly used low

molecular weight liquid crystals (LMWLC) with the chemical formula C18H19N.

46

Molecular structure of 5CB is shown in Fig. 3.1 below. The material is the same with the

one that was used by Orihara et al. [23]

FIG. 3.1. Molecular structure of 5CB.

In general 5CB exhibits transitions from crystal to nematic phases at 18 C and

from nematic to isotropic phases at 35 C. Near the transition temperature physical

quantities such as the viscosity coefficients are quite sensitive to the temperature change.

Therefore, we conducted experiments at 25 C around the center of the nematic phase.

Measurements were carried out with a parallel plate rheometer (Physica MCR300, Anton

Paar) at room temperature (25 C). The rheometer is shown in Fig. 3.2. The diameter of

the rotating plate and the gap between the two parallel plates are 35 and 0.2 mm,

respectively. The rotating plate and the bottom plate were not treated with alignment

surfactant because we thought that the electric field aligned the director and the effect of

the boundaries were limited to the regions just near the boundaries.

47

FIG. 3.2. Rheometer (Physica MCR300, Anton Paar).

For the parallel plate rheometer, the shear rate is defined at the edge of the upper

plate and the shear stress at the corresponding shear rate is calculated from the mechanical

torque by assuming that the sample is a Newtonian fluid. This means that the apparent

shear stress obtained from the experiment is the average over the sample subjected to

different shear rates. Instead we can use a cone plate to avoid the position dependence of

shear rate, but in this case the electric field becomes radially position-dependent. This

situation is undesirable for our purpose.

As described above, the measured stress is averaged one. So, we average the

theoretically obtained stress to compare the experiment and theory. The shear rate at r

48

from the center for the parallel plate geometry is given as rh

, where is the angular

velocity of the rotating plate and h is the gap between the upper and lower plate. The force

exerted on the plate between r r dr is 2dF rdr , and so the corresponding torque

is given by22dM rdF r dr . By integrating this we can obtain the total torque as

R3

2 2

0 0

2 2

Rh

M r r dr d

, (3.1)

where R is the shear rate at the edge. When the sample is Newtonian, i.e., , the

torque becomes 3 34 3/ 2 / / 2 /R R RM h h . By comparing these, we get

the apparent shear stress, which correspond to the experimentally obtained one,

2

3

R 0

4 R

R d

. (3.2)

In Fig. 3.3, we show the relation among the flow direction, the velocity gradient

and the electric field, which is applied to the sample by using a synthesizer

(Model 1940, NF) and a high-voltage amplifier (Model 4005, NF). In our measurements,

dc electric fields are applied in addition to a weak ac electric field which is used with the

purpose to probe the stress response. The total field applied is thus 0 cosE E t .

However, migration of ions in the NLC sometimes becomes a problem and reduces the

electric field inside the cell. To avoid this, we use a high frequency ac electric field

modulated by the sum of the dc and ac electric fields:

02 cos cos cE t E E t t , (3.3)

49

where c is the angular frequency of the carrier signal. The induced shear stress is

proportional to the square of the applied electric field so that the square of E t can be

approximated by 2

0 cosE E t for c . In our measurements, c is chosen to

be 6280 rad s-1, which is much higher than the maximum frequency of =200 rad s-1 in

our measurements.

FIG. 3.3. (a) A simple shear flow is applied along the x axis with a velocity

gradient parallel to the z axis. An electric field is also applied along the z direction. (b)

We define to be the angle between the director and the x axis.

50

Taking into account the fact that 2

0 cosE E t =

2 2 2

0 02 cos 2 2cos 2E E E t E E t , we can put

2

0 1,1 2,0 2,2Re Rei t i tt e e , (3.4)

where 0 is the shear stress in the absence of the perturbation E , and ,i j is the stress

response of the ith order with respect to E and the jth harmonic of . Therefore,

1,1 should be proportional to E , and both 2,0 and 2,2 should be

proportional to the square of E for small values of E . The and 2 response

components of the shear stress were obtained by using a vector signal analyser

(HP89410A, Hewlett-Packard). The output of the rheometer corresponding to the torque

is a voltage proportional to the torque. The proportionality constant is obtained by using

a standard fluid (glycerin). The data from the vector signal analyzer are then acquired by

a computer and then analysed by using the IGOR software.

51

FIG. 3.4. Experimental setup. Electric fields are applied between the top and

bottom plates. For the electro-optical experiment we add a light source and a microscope

to observe the change of transmitted intensity.

The shear stress response to ac electric field under steady shear flow and electric

fields was measured by applying a small ac electric field cos( )E t . We obtained the

first-order response 1,1( ) and the second-order response 2,2 ( ) , which should be

proportional to E and 2E , respectively, for small E . The dc electric field

dependencies of the first and second-order responses are shown in Fig. 3.5, where the

measurements were done at -110 s and -1

0 100 V mmE . Linearity was confirmed to

52

hold at least up to-120 V mmE . All the measurements were performed at

-114.1 V mmE and -110 s .

FIG. 3.5. Dependence of 1,1 on E (a) and 2,2 on 2E (b) at -110 s and

-1

0 100 V mmE . Linear relations are obtained at low ac electric fields.

In addition, we make the optical response experiment to directly observe the

director reorientation to ac electric field under shear flow and dc electric fields. As will

53

be shown in the next section, the EL theory predicts that the shear stress response and the

director response behave differently at high frequencies. The optical measurement is

made for checking the theoretical prediction.

In the optical response experiment we replace the upper plate by a glass disk with

diameter 40 mm so that we can observe the transmitted light through the sample by using

a microscope. The same rheometer (Physica MCR300, Anton Paar) is used to apply shear

stress. To minimize the number of disclinations, we coat the upper plate and bottom plate

with perfluoropolymer (CYTOP) (Asahi Glass Co.Ltd.) to align the director

perpendicular to the surfaces, i.e., to obtain homeotropic alignment. The molecular

structure of CYTOP is given in Fig. 3.6. The texture under a polarizing microscope is

shown in Fig. 3.6 for the shear rate of 10 s-1 and Fig. 3.7 for the shear rate of 80 s-1. At a

low shear rate almost no disclination appears (see Fig. 3.7) while at the high shear rate a

few disclination lines appear (see Fig. 3.8).

FIG. 3.6. Molecular structure of CYTOP.

54

FIG. 3.7. Texture of 5CB under crossed polarizers at a shear rate of 10 s-1. There

is no disclination.

FIG. 3.8. Texture of 5CB under crossed polarizers at a shear rate of 80 s-1. There

appear a few disclination lines. The appearance of disclinations may be associated with

the director deviation from the shear plane though the details are not clear.

55

As a light source we used a halogen lamp (LS-LHA, Sumita Optical Glass). Polarizer and

analyzer are used to observe the transmitted light intensity, which changes due to the

birefringence change under ac electric fields. The light intensity was converted by a photo

sensor into a voltage and it was amplified (C6386, Hamamatsu Photonics). The signal is

then analyzed using a vector signal analyser (HP89410A, Hewlett-Packard).

Correction for Torque sensor of Rheometer

When we first measured the stress responses 1,1( ) and

2,2 ( ) , we noticed

a problem in the torque sensor of our rheometer. Although the EL theory predicts that

both the two responses do not go to zero and keep finite values at high frequencies of

applied ac electric fields, experimental ones become zero. We thought that this

discrepancy might be due to the reduction of sensitivity of the torque sensor at high

frequencies. To confirm this, we actually measured the sensitivity as a function of

frequency as follows.

The experimental setup is shown in Fig. 3.9. We use a parallel plate having a

circular tray above it. This tray is usually used for applying voltages to the rotating plate

by filling it with a conductive liquid. This time we fill this tray with a viscous glycerine

and immerse the tip of an L-shaped metal wire which is attached to a piezo actuator. The

diameter of the wire is 1 mm and the depth immersed is about 1 mm. We don’t mount

any sample between the rotating plate and the bottom stage. We rotate the plate at a speed

of 0.8 rpm, and apply an ac voltage to the piezo actuator so that the tip of the wire

oscillates along the flow direction with an amplitude of 28 m. The oscillation of the tip

56

in the glycerine gives rise to a drag force on the top plate, which is detected as torque. If

we assume the Stokes’ law, the torque may be proportional to the velocity of the tip.

FIG. 3.9. The experimental setup.

When the displacement of the tip is given by

0 sinx x t , (3.5)

the torque change corresponding to the above displacement can be expressed as

cosappl a t (3.6)

with a constant a. It is natural that this torque change should be transmitted to the rotating

plate. On the other hand, the torque measured with the rheometer may generally be

expressed as

cosmeas a t . (3.7)

If the torque sensor is ideal, and are constant independent of the frequency.

If not, we have to correct data by using these. We obtain function and with

57

changing , which are shown in a subsequent chapter. Both will show stronger frequency

dependence than predicted before the measurement.

3.3. EL theory

NLCs are composed of rod-like molecules with the long axes aligned statistically

parallel to each other. The average orientation of molecules is represented by a unit vector

n which is called the director. Ericksen and Leslie have formulated a continuum theory

for the velocity v and the director n of NLCs [1, 6, 7, 32, 40, 41]. Hereafter, we assume

a mono-domain (i.e., the director is independent of position) and we apply a simple shear

flow. Under these assumptions, the EL equations can be simplified and we need only the

following equations for our purpose. The angular momentum balance gives

1 2+n h n N n An , (3.8)

where h is the molecular field, N is the rate of change of the director with respect to the

background fluid, and 1

2A v v is the symmetric part of the velocity gradient.

The parameters 1 and 2 are the rotational and irrotational viscosity coefficients. The

components of the molecular field h are given by

f

hn

, (3.9)

where f is the free energy density. When subjected to an electric field, the free energy

density can be written as

58

2

2

0 0

1 1 .

2 2f E n E , (3.10)

where 0 is the dielectric constant in a vacuum, is the dielectric anisotropy defined

as / / with / / and being the dielectric constants parallel and perpendicular

to the director, respectively ( 11.5 for 5CB [42]). The rate of change of the director

is defined as

dn

N W ndt

, (3.11)

where 1

2W v v is the anti-symmetric part of the velocity gradient. The

constitutive equation for the viscous stress tensor is

(visc)

4 1 5 6 2 3A n n n n A n n A n n A n N n N , (3.12)

where i (i=1,...,6) are Leslie coefficients, in terms of which 1 and 2 are expressed as

1 3 2 and 2 6 5 . For 5CB these coefficients have been determined [27]:

1 0.00767 Pa s , 2 0.08171 Pa s , 3 0.00433 Pa s , 4 0.06642 Pa s ,

5 0.06725 Pa s , 6 0.01879 Pa s at 25oC. These values are used in numerical

calculations later. Note that, here, an NLC is treated as an incompressible fluid.

In reality, the mono-domain and simple shear flow assumptions may not be

exactly satisfied for various reasons. For example, in our experiment, no surface treatment

for aligning molecules is made so that the director and flow are spatially disturbed at least

near the surfaces. Note that our model also does not accommodate defects (e.g.,

disclinations). Furthermore, larger deformations from the flow-aligning state are reported

59

to take place [43]. For simplicity, however, we adopt the assumptions shown in Fig. 3.1(a),

where ( ) cos ,0,sinn t and ,0,0 v z with being the shear rate. The reason

is that we expect the applied dc electric field to some degree will align the director to the

same direction.

Here, we calculate the stress change due to a small ac electric field in the steady

state under constant shear flow and dc electric fields. In this case, the total electric field

in Eq. (3.10) is given as 0( ) cosE t E E t . Using Eqs. (3.9) and (3.11), Eq. (3.8)

becomes

2

1 1 1 0

1 1( cos 2 ) sin 2

2 2E t

t

, (3.13)

where the electric field is applied in the z direction. Without an ac electric field, that is,

in the non-perturbed state, the flow alignment angle 0 (as shown in Fig. 3.10) in the

steady state can be obtained by solving Eq. (3.13) by putting 0t

. Then we have 0 as

a function of 0E as follows:

2 2 22 2 2 2 2 2 2 2 2

0 0 1 2 2 0 0 0 0 1 21

0 22 2 2

0 0 2

cos2

E E E

E

.

(3.14a)

In the special case of 0 0E , 0 reduces to

1 1

0

2

1cos

2

. (3.14b)

60

The angle 0 monotonically increases with increasing electric field, as shown in Fig. 3.10,

where the values of the viscosities and the dielectric anisotropy for 5CB at 25 C,

corresponding to the experimental conditions, are used.

FIG. 3.10. Dependence of 0 on 0E calculated from the EL theory. The angle 0

increases monotonically and tends to saturate when 1

0 200 V mmE .

Expanding up to the second order with respect to E , we obtain the corresponding

change in :

2

2,0 1,1 2,2( ) Re[ ] Re[ ]i t i tt e e , (3.15)

with

0 0 01,1

1

sin 2( )

1

E E

i

, (3.16)

61

2

2,2 1,1 2 0 0 0 0 1,1 0 0 0

1

2

0 0

1( ) 1/ 2 cos 2 sin 2 cos 2

2 1

sin 2 ,

E E Ei

E

(3.17)

and

2,0 2,2 (0) , (3.18)

where the first subscript of indicates the order with respect to E and the second one

the harmonic order. The relaxation time is defined by

1

2

2 0 0 0 0sin 2 cos 2E

. (3.19)

Next, we calculate the shear stress response. The shear stress ( zx in the present

case) can be also expanded with respect to E :

2

0 2,0 1,1 2,2( ) Re[ ] Re[ ]i t i tt e e , (3.20)

where 0 is the shear stress with no ac electric field. It should be noted that the

response 1,1 appears under dc electric fields, as can be seen from Eq. (3.16); this

response vanishes for 0 0E . The stress is independent of the polarity of the electric field

as NLCs are non-polar and, therefore, the stress response depends on

2 2 2

0 0( ) 2 cos / 2 1 cos 2E t E E E t E t as shown in Eq. (3.13), clearly

62

indicating that the response 1,1 should emerge under dc electric fields in

addition to the 2 response 2,2 .

From Eqs. (3.12), (3.14a) and (3.16)–(3.19), the unperturbed shear stress 0 and

the responses 1,1 , 2,0 and 2,2 are obtained:

2 2 2 2

0 1 0 0 4 5 2 0 3 6 0 0

1sin cos sin cos sin 2

2

, (3.21)

2 2

1,1 1 0 2 3 3 0 2 0 1,1

1cos 2 cos sin

2i

, (3.22)

2 2

2,2 1 0 2 3 0 3 0 2 0 2,2

2

1 0 2 3 0 2 3 0 1,1

( ) sin 4 2 sin 2 2 cos sin ( )2

cos 4 cos 2 sin 2 ( ) ,2

i

i

(3.23)

2,0 2,2 (0) . (3.24)

Here, we have obtained one of the most important theoretical results that 1,1 and

2,2 ( ) do not goes to zero at high frequencies unlike the usual Debye-type frequency

dispersion as is seen from Eqs. (3.22) and (3.23), i.e., we have a plateau at high

frequencies. The details will be discussed in the next section. When we compare the above

theoretical result with the experimental one, the parallel-plate geometry of the rheometer

which is used in our experiment should be considered. The correction method will be

shown also in the next section “Experimental Results and Discussion”.

63

In addition, we make optical experiments with a purpose to demonstrate that the

director itself has no plateau as is readily seen from Eqs. (3.16) and (3.17), which are the

usual Debye-type function. These responses can be observed through optical experiments

as follows. When an electric field is applied, the birefringence 0( ) ( ) ( )a en n n ,

where 0n and en are the ordinary and extraordinary refractive indices, respectively, will

change and here the observation is assumed to be made perpendicular to the parallel plates

as shown in Fig. 3.4. In NLCs, 0n is independent of the director orientation . However,

en is dependent on :

/ // /2 2 2 2 1/2

/ /

( )( cos sin )

e

n nn n

n n

, (3.25)

where / /n and n are the refractive indices parallel and perpendicular to the director. On

the other hand, the transmitted light intensity of the NLC under crossed polarizers is given

by

2 2

0 sin (2 )sinan d

I I

, (3.26)

where 0I is the incoming light intensity, is the angle between the polarizer and the x

axis, d is the sample thickness and is the wavelength of the light in a vacuum. For a

small change of , , due to the application of an AC electric field, the corresponding

change of I , I , is obtained from Eqs. (3.25) and (3.26):

2 2

02 // / / 0 00 2 2 2 2 3/2

0 / / 0

( 2 cos )sinsin (2 )sin 2

2( cos sin )

an d n n n nI I

n n

, (3.27)

where 0 is the angle without any AC electric field. The intensity change can be

measured as described in the previous section.

64

3. 4 Experimental results and discussion

First, we demonstrate that corrections are necessary for the torque sensor as

described in section 3.3. We show the first order shear stress response 1,1 and the

second order response 2,2 before correction at different dc voltages and at constant

shear rate of 10 s-1 in Fig. 3.11. The theoretical curves are also shown, which are obtained

by using Eq. (3.2). The experiment and theory show good agreement at low frequencies

of ac electric field. However, at higher frequencies, we have a discrepancy between them:

the theory has a plateau but the experiment becomes zero. This is resolved by the

corrections for the torque sensor.

FIG. 3.11. 1,1 as a function of angular frequency before correction.

65

FIG. 3.12. 2,2 as a function of angular frequency before correction.

The correction curves for and obtained by the method described in the

previous section are shown in Figs. 3.13 and 3.14, respectively. Then, by using

and , we correct 1,1 as 1,1 /i

e

, which is shown in Fig. 3.15

(lines are experimental results and dots are theoretical predictions calculated using EL

theory). The corrected 1,1 has a plateau and is in good agreement with the theory.

66

The second order response 2,2 is also corrected in the similar manner and the

results are shown in Fig. 3.16.

FIG. 3.13. as a function of frequency .

FIG. 3.14. as a function of frequency .

67

FIG. 3.15. Frequency dispersion of 1,1 as a function of angular frequency .

The theoretical curves are obtained from Eq. (3.15). A characteristic non-zero plateau,

which is due to the dc electric field, appears at high frequencies in (b), (c) and (d).

68

FIG. 3.16. Frequency dispersion of 2,2 as a function of angular frequency .

The theoretical curves are obtained from Eq. (3.23). A plateau appears under dc electric

fields as well as in the first-order response.

69

Hereafter we show only corrected data. First, we discuss in detail the first-order

response 1,1( ) shown in Fig. 3.15. The responses are given at 0E =20 V mm-1,

80 V mm-1, 120 V mm-1 and 200 V mm-1. The first-order response appears only when we

apply a dc electric field 0E , which is seen from Eqs. (3.16) and (3.22). When 0E is low

( 0E = 20 V mm-1) (see Fig. 3.15(a)), the experimental stress response resembles Debye

type relaxation. However, as we increase 0E to 80 V mm-1 (see Fig. 3.15(b)), we can see

that the response remains non-zero even at high frequencies, forming the above-

mentioned plateau. The emergence of the plateau is the most remarkable characteristic of

the response to ac electric field under dc electric fields. From the EL-theory, it is easily

seen that the plateau comes from the third term with i in the bracket on the right hand

side of Eq. (3.22) because 1,1( ) in Eq. (3.22) is inversely proportional to i at high

frequencies, as can be seen from Eq. (3.16). The i term appears as a result of

differentiating the director with respect to time in Eq. (3.11). In the limit as , we

have

2 2

3 0 2 0

1,1 0 0 0

1

cos sin( ) sin 2E E

. (3.28)

It should be noted that the director response 1,1( ) vanishes at high frequencies, but

the stress response 1,1( ) remains non-zero. The experimental results for the director

response will be discussed later in the optical experiment, which convinces us that this

plateau is ascribed to the time derivative from Eq. (3.11).

70

Fig. 3.17. 0E dependence of the first-order response at very low frequency

(0.23 rad s-1 for the experiment and exactly zero for the theory), 1,1Re ( 0) . Also

shown is the dependence of plateau height 1,1Re ( ) . For dc electric fields

higher than around-190 V mm , 1,1Re ( ) becomes larger than

1,1Re ( 0) .

At higher values of 0E , the height of the plateau is larger than at low frequencies,

as shown in Figs. 3.15(b) (80 V mm-1) and (c) (160 V mm-1). The agreement between the

experimental and theoretical results for all 0E shown in Fig. 3.15 is fairly good. The

dependence of plateau height 1,1( ) on 0E is shown in Fig. 3.16 for the experiment

71

and the theoretical result in Eq. (3.22). From Fig. 3.16, we observe that when

1

0 40 V mmE the plateau is small. As we increase E0 to more than150 V mm, the

plateau height increases sharply. In the same figure, we also plot the 0E dependencies of

1,1 calculated at zero frequency and experimentally obtained from Fig. 3.17 at 0.23 rad

s-1. The agreement between them is good. The low-frequency and high-frequency

(plateau) values exchange at around E0 = 90 V mm-1. The imaginary part of the response

(Fig. 3.15) goes to zero at both low and high frequencies, and shows good agreement

between theory and experiment. The peak or dip frequency of the imaginary part is seen

to increase with increasing dc electric field, though it changes sign from positive to

negative at around E0 = 90 V mm-1 corresponding to the above-mentioned exchange of

the low and high frequency values of the response.

Next, let us discuss the second-order response 2,2 (Fig. 3.16). This response is

more complicated than the first-order response since there are contributions from the first-

order mode 1,1 as well as 2,2 , as can be seen from Eq. (3.23). It should be noted that

the second-order response appears even without a dc electric field, as shown in

Fig. 3.16(a). The stress response resembles Debye type relaxation for 0 0E . But, one

may notice that the real part becomes slightly negative at around 10 rad s-1. Orihara et al.

have previously observed this negative part and shown that it originates from non-

conservative forces due to the shear flow [16]. This is a remarkable characteristic of non-

equilibrium steady states of systems under shear flow and is observed also in an

immiscible polymer blend in which one polymer is dispersed as droplets in the other [17].

The non-conservative forces also violate the fluctuation dissipation relation (FDR). We

72

have theoretically clarified the mechanism of the appearance of the non-conservative

forces in NLCs and derived a modified FDR [18]. This was discussed in detail in chapter

2. According to the theory, the non-conservative forces can emerge only when the director

is out of the shear plane. However, we assumed that the director is confined in the shear

plane and, therefore, we cannot reproduce the negative part in the present model. The

director may tend to be in the shear plane under a dc electric field. Therefore, it is expected

that our mono domain model works better as the dc electric field is increased.

From Fig. 3.16, it is obvious that there is a plateau at every dc electric field except

for 0 0E (Fig. 3.16(a)). As we increase 0E , the plateau, which is the characteristic of the

system under DC electric fields, appears. The origin of the plateau in the second-order

response is the same as that in the first-order response. The plateau comes from the

term proportional to 2,2 in Eq. (3.23), which includes a factor 2 2

3 0 2 0cos sin .

This factor also appears in the term of the first-order response (see Eq. (3.22)).

Interestingly, this vanishes at 0 0E due to the Parodi relation 6 5 2 3 , which

is easily proved by using Eq. (3.14b). Note that the second-order response does not vanish

for 0 0E unlike the first-order response. It is obvious that the factor is in general not

zero for 0 0E . When we increase 0E to 80 V mm-1, the plateau at high frequencies rises,

as shown in Fig. 3.16(b). The plateau becomes remarkable for 0E = 120 V mm-1

(Fig. 3.16(d)). The agreement between the experimental and theoretical results is good.

The imaginary part of the response also shows good agreement between theory and

experiment.

73

Figure 3.18 shows the 0E dependence of 2,2 calculated at zero frequency and

experimentally obtained from Fig. 3.17 at 0.23 rad s-1. Also in the same figure we show

the 0E dependence of the height of the plateau. Good agreement is obtained between the

theory and experiment. The 0E dependencies are more complicated than those in the first-

order response.

FIG. 3.18. 0E dependence of the first-order response at a very low frequency

(0.23 rad s-1 for the experiment and exactly zero for the theory), 2,2Re ( 0) . Also

shown is the dependence of plateau height 2,2Re ( ) .

74

Finally, we show experimental evidence that the plateau observed in the real part

of the first-order response should be ascribed to the time derivative of the director, but

not to 1,1 itself. To do so, as was described in the previous section, we made an optical

measurement using crossed polarizers, in which the light intensity is shown to be just

proportional to the first-order response mode1,1 . Therefore, if

1,1 is of the Debye

type, there should be no plateau observed in the optical response at high frequencies,

though it is observed in the shear stress response.

FIG. 3.19. Frequency dispersion of the optical response, which corresponds to the

director response, as a function of angular frequency . Lines are the experiment and

circles are the theory. No plateau is observed, which shows that the plateau observed in

the shear stress response should be ascribed to the time derivative of the director but not

to 1,1 itself.

75

The experimental result is shown in Fig. 3.19 for 1

0 200 V mmE . Note that the

vertical axis is adjusted so that the theoretical and experimental results coincide as well

as possible, because it is difficult to determine the coefficient in Eq. (3.27). The agreement

between the experiment and theory is good, though the data are scattered to some degree.

The relaxation frequency experimentally observed is in good agreement with the one

theoretically obtained. There is no plateau in the optical response, in contrast to the stress

response, convincing us that the plateau observed in the stress response should originate

from the time derivative of the director.

3.5 Conclusions

We have investigated the NESS response of an NLC to an ac electric field under

constant dc electric field and steady shear flow. The first and second-order shear stress

responses were theoretically obtained from the EL theory assuming a mono-domain

model. It was found theoretically and experimentally that when we apply a dc electric

field both responses remain constant and non-zero even at high ac electric field

frequencies. That is, there is a plateau, which originates from the time derivative of the

director. This plateau is a remarkable feature in the shear stress response brought about

by the application of a dc electric field. Furthermore, by performing an optical

measurement, the director response is confirmed to vanish at high frequencies, strongly

supporting the above-mentioned mechanism for the appearance of the plateau. It was also

clarified that the plateau in the second-order response disappears in the absence of a dc

76

electric field, due to the Parodi relation, which is derived from the Onsager reciprocal

relation.

77

CHAPTER 4

ELECTRIC-FIELD-INDUCED FLOW-ALIGNING STATE IN A NEMATIC

LIQUID CRYSTAL

4.1. Background

In the previous chapter, the experiments were conducted with the use of 5CB (4-

n-pentyl-4-cyanobiphenyl). While in this chapter we use 8CB (4-n-octyl-cyanobiphenyl).

The difference between 5CB and 8CB is in the viscosity ratio 1 2/ where 1 3 2

(rotational viscosity) and 2 6 5 (irrotational viscosity). For 5CB 1 2/ 1 , while

for 8CB 1 2/ 1 .

According to the EL theory, when we apply shear flow to NLCs, the angle

between the director and the flow direction in a flow-aligning state is given as

1 2cos2 / . From this expression, it is clear that the solution will be only available

when 1 2/ 1 . The consequence of this equation is that for 5CB under shear flow, its

director orient in the shear plane with a flow alignment angle given by the above equation

[44], while for 8CB the director will tumble or find another stable orientation with the

director field being spatially deformed, possibly in accordance with the viscous and

elastic coefficients and the boundary condition [18, 45]. Then, under application of a

simple shear flow, NLCs will take either one of the two types of states depending on the

ratio of 1 and 2 . In this chapter, we call the state in which the flow alignment angle

exists as flow aligning state (FAS) and the state in which the flow alignment angle does

78

not exist as non-flow aligning state (NFAS). Most low molecular mass NLCs under shear

flow adopt the FAS except for small number of NLCs such as 8CB. This NFAS structure

is more common in liquid crystal polymers due to the complexity of structure [11].

NFAS can be easily transformed to FAS by applying electric fields along the

velocity gradient direction in NLCs with positive dielectric anisotropy. This may be

regarded as an electric field-induced phase transition which is of interest from the

viewpoint of non-equilibrium physics [46]. Furthermore, we expect different behaviours

of shear stress response in NFAS from those in FAS shown in chapter 3 for 5CB.

4.2. EL theory in FAS

Generally, the transition from NFAS to FAS is due to the competition between

the shear flow effect and electric field effect, where the latter compensates for the

imbalance of the torque produced by the former and finally stabilizes the director

orientation back to the shear plane in a strong enough electric field. In the FAS, the

director is assumed to stay in the shear plane [47] and to be spatially independent except

near the boundary plates. In this simple case, it is sufficient to use EL theory, in which

the order parameter is assumed to be constant, for the purpose to calculate the shear stress

responses [39] which was derived in previous chapter. Here we just rewrite some useful

equations. On the other hand, in the NFAS, it is necessary to solve EL theory in three-

dimensional space, which will make the solution more complicated. So we do not deal

with this problem here.

We have already derived the equation of motion for the FAS in the previous

chapter. So, we briefly summarize the results and note some additional considerations

79

regarding the electric-field-induced FAS in the following. Under simple shear flow with

an electric field applied in the velocity gradient direction, the angle between the

director and the flow direction obeys the following equation

2

1 1 2 0

1 1( cos 2 ) sin 2

2 2E t

t

, (4.1)

where is the shear rate, 0 the dielectric constant of vacuum and the dielectric

anisotropy. From this equation, it is readily seen that the change of , which brings about

the change of the shear stress, depends on the square of the applied electric field E t , as

described above. When 0E t E , Eq. (4.1) gives [47]

22 2 2 2 2

0 0 0 0 2 11

0

1 2

tanE E

. (4.2)

The calculated dc electric field dependence of 0 is shown in Fig. 4.1. The Leslie

coefficients for 8CB at 25 C are taken from Ref. [8] as: 1 0.0382 Pa s, 2 0.0587

Pa s, 3 0.0031 Pa s, 4 0.0520 Pa s, 5 0.0472 Pa s, and 6 0.00840 Pa s

(calculated from the Parodi relation 6 5 2 3 ) and dielectric anisotropy 6.5

is used [48]. It should be noted that all of these parameters are very sensitive to the change

of temperature. In the special case where 0 0E , 0 is reduced to the flow aligning angle

1

1 2cos / / 2f . For 8CB the viscous coefficients are given by

1 3 2 0.0618 Pa s and 2 6 5 0.0556 Pa s , and therefore we have no f .

But, Eq. (4.2) has a solution above a critical electric field given by the following equation:

80

2 2

1 22

0

cE

. (4.3)

As shown in Fig. 4.1, 0 increases with 0E , and cE increases with .

FIG. 4.1. The dc electric field dependence of the flow alignment angle 0 in the

FAS at shear rates of 10 s-1 and 40 s-1 numerically obtained for 8CB from Eq. (4.2).

The electric-field-induced transition has already been experimentally probed

using various methods such as direct observation of flow patterns and disclination density

[36, 49], rheological measurements [50], and synchrotron X-ray studies [52]. However,

it is not easy to clearly distinguish between the FAS and NFAS states by means of the

usual rheological measurements. Although the electric field dependence of shear stress in

8CB at a constant shear rate, in which an NFAS appears without electric field, was also

81

reported by Negita [47] and Patricio et al. [52], the change in the shear stress was too

small for the transition to be clearly detected. Under these circumstances, we measure the

shear stress response to an ac electric field as a probe under both shear flow and dc electric

fields [23]. The shear stress response is more sensitive to the change in director orientation

and we can demonstrate that different frequency dispersion curves in the stress response

are clearly observed and the transition from NFAS to FAS is detected by changing the dc

electric field at a constant shear rate.

Equation (4.1) with an ac electric field can be approximately solved by using the

perturbation method. Expressing ( )t as

2

0 1,1 2,0 2,2Re Rei t i tt e e , (4.4)

we have

0 0 01,1

1

sin 2( )

1

E E

i

, (4.5)

2

2,2 1,1 2 0 0 0 0

1

2

1,1 0 0 0 0 0

1 1( ) 1/ 2 cos 2 sin 2

2 2 1

cos 2 sin 2 ,

Ei

E E E

(4.6a)

and

2,0 2,2 (0) , (4.6b)

where the relaxation frequency 1 is defined by

2

2 0 0 0 0

1

1 1sin 2 cos 2E

. (4.7)

Near the critical electric field, the relaxation frequency is proportional to 0 cE E , which

is obtained from Eqs. (4.2), (4.3), and (4.7). It should be emphasized that the relaxation

82

frequency becomes zero at the transition point, but the behaviour is singular as a function

of 0E .

The shear stress can be calculated from the following equation

visc

4 1 5 6 2 3A n n n n A n n A n n A n N n N , (4.8)

where 1 2A v v and 1 2W v v are the symmetric and anti-

symmetric parts of velocity gradient respectively, /N dn dt W n is the rate of

change of the director with respect to the background fluid, and i (i=1,...,6) are the

Leslie viscosity coefficients. In the present case, cos ,0,sinn and ,0,0v z ,

where the x and z axes are taken along the flow and velocity gradient directions,

respectively.

Using t in Eq. (4.4), we can calculate the shear stress visc

zx t t

considered here as follows:

2

0 1,1 2,0 2,2Re Rei t i tt e e , (4.9)

with

2 2 2 2

0 1 0 0 4 5 2 0 3 6 0 0

1sin cos sin cos sin 2

2

, (4.10a)

2 2

1,1 1 0 2 3 3 0 2 0 1,1

1cos 2 cos sin

2i

, (4.10b)

2 2

2,2 1 0 2 3 0 3 0 2 0 2,2

2

1 0 2 3 0 2 3 0 1,1

( ) sin 4 2 sin 2 2 cos sin ( )2

cos 4 cos 2 sin 2 ( ) ,2

i

i

(4.10c)

83

2,0 2,2 (0) , (4.10d)

where 0 is the shear stress in the absence of the perturbation E , and ,i j is the stress

response of the ith order with respect to E and the jth harmonic of .

To compare the above theoretical results with the experimental ones, we need to

consider the parallel-plate geometry of the rheometer which is used in our experiment.

For this geometry, the apparent shear stress is given by

( ) 2

, ,3

0

4( ) ,

R

R

i j i j

R

d

, (4.11)

where R is the shear rate at the edge of the rotating disk. The numerically calculated

results are presented in the next section, where the superscript “(R)” on the left-hand-side

of Eq. (4.10) will be omitted for simplicity.

4.3. Experiments

The nematic liquid crystal 8CB was purchased from Wako Pure Chemical

Industries and used without any further purification. Measurements were carried out by

using a parallel plate rotational rheometer (Physica MCR300, Anton Paar). The diameter

of the rotating plate and the gap between two parallel plates were 35 and 0.2 mm,

respectively. Note that in the parallel plate geometry, the shear rate depends on position,

and shear rate is defined at the edge of the upper plate. The shear stress at the

84

corresponding shear rate is calculated from the mechanical torque by assuming that the

sample is Newtonian. In the same way as the previous chapter, we made corrections for

the measured torque. All the measurements were made at 37 C in the nematic phase.

The experimental setup and the geometry are shown in Figs. 4.2(a) and (b),

respectively. Electric fields were applied to the sample by using a synthesizer (Model

1940, NF Electric Instruments) and a high-voltage amplifier (4005, NF). In our

measurements, dc electric fields were applied to induce the FAS, and a weak ac electric

field was also applied to probe the stress response. The total field applied was thus

0 cosE E t . However, migration of ions in the NLC sometimes becomes a problem

and reduces the electric field inside the cell. To avoid this, we used a high frequency ac

electric field modulated by the sum of the dc and ac electric fields:

02 cos cos cE t E E t t , (4.12)

where c is the angular frequency of the carrier signal. As was shown above, the induced

shear stress is proportional to the square of the applied electric field so that the square of

E t can be approximated by 2

0 cosE E t for c . In our measurements, c

is chosen to be 6280 rad s-1, which is much higher than the maximum frequency of

=200 rad s-1 in our measurements. The and 2 components of the shear stress were

obtained by using a vector signal analyser (HP89410A, Hewlett-Packard).

85

FIG. 4.2. (a) Experimental setup. The sample is sheared by using a parallel-plate

rheometer under dc and ac electric fields. (b) The flow is applied in the x direction, the

velocity gradient is in the z direction, and the electric field is applied in the z direction.

The setup is similar with the experiment in the previous chapter except that we do not use

the optical apparatus.

Let us first check the linear relationships between E and1,1 , and between

2E and 2,2 . Figures 4.3(a) and (b) show E vs. 1,1 and 2E vs.

2,2 ,

respectively, at 1 1 1

010 s , 100 V mm and 0.63 rad sE . The linearities are good

and the amplitude of the ac electric field for all the measurements is determined to be

114.1 V mmE .

86

FIG. 4.3. (a) Dependence of 1,1 on E at 0E =100 V mm-1 and (b) 2,2

on 2E at 0E =100 V mm-1 at a shear rate of 10 s-1 at a frequency of-10.63 rad s .

Linear relationships are obtained at low electric fields. Both are conducted in FAS state

since for NFAS the theoretical results are not available.

87

4.4. Experimental results and discussion

The dc electric field dependence of the frequency dispersion of 1,1 at the

shear rate of 10 s-1 is shown in Fig. 4.4. The critical electric field to induce FAS from

NFAS is 168 V mmcE as calculated from Eq. (4.3). From the figure, one can observe

that the frequency dispersions below and above the critical electric field are quite different

qualitatively. In the NFAS below this field (Figs. 4.4(a) and (b)), an anomalous change is

observed at around 5 rad s-1 in both the real and imaginary parts. The origin of this

behaviour has not yet been understood since the model we use is not applicable to NFAS.

To deal with the NFAS it is necessary to introduce out-of-shear-plane configurations in

the EL theory. The results are thought to be complicated.

On the other hand, in the FAS above the critical field (Figs. 4.4(c)-(f)), the change

is smooth. The behaviour of frequency dispersion in the FAS is almost the same as that

measured in 5CB (4-n-pentyl-cyanobiphenyl), which adopts the FAS even in the absence

of a dc electric field [23]. As described in the previous chapter, in 5CB we have already

observed a characteristic behaviour under shear flow and dc electric fields: the real part

of the frequency dispersion has a plateau at high frequencies. The appearance of the

plateau is understood from Eqs. (4.5) and (4.10b) as follows. 1,1 given by Eq. (4.5)

is the Debye type which goes to zero at high frequencies, but 1,1 differs from it

because in Eq. (4.10b) there is an i term multiplied by 1,1 , which gives a non-

zero constant at the limit of as shown in the following equation:

88

2 2

3 0 2 0

1,1 0 0 0

1

cos sin( ) sin 2E E

. (4.13)

The origin of the i term is the time derivative terms in Eq. (2.4):

2 3 2 32 2

x xz zz x x z z x

dn nn dnn N n N n n

dt dt

, (4.14)

where we have put and z x for our case. From this equation it is also understood

that the plateau is not observed in the NFAS. In the NFAS the director is thought to be

out of the shear plane to some degree, indicating that xn and/or zn should be small. The

theoretical curves for the FAS calculated from Eqs. (4.5), (4.10b) and (4.11) are also

shown, and are in good agreement with the experimental results.

89

FIG. 4.4. Frequency dispersions of the first-order harmonic response 1,1( ) at

a shear rate of 10 s-1 for several dc electric fields. The dispersions are clearly

distinguishable between NFAS ((a) and (b)) and FAS ((c)-(f)). Solid lines for the FAS are

calculated on the basis of the EL theory.

90

Figure 4.5 shows the 0E dependence of the peak or dip frequency of the

imaginary part, peak , where dots represent the experimental results, the solid line is

obtained from the above calculated frequency dispersion curves using Eq. (4.11), and the

dashed line is 1/ calculated from Eq. (4.7). A slowing down of the relaxation time is

observed in the FAS, as is expected from Eqs. (4.2) and (4.7). At around a dc electric field

of 100 V mm-1 the solid line disappears because the peak changes to the dip in the

imaginary part there, and so it is difficult to obtain the peak or dip position. At the critical

point, the value of the dashed line becomes zero with the singularity of 0 cE E , but that

of the solid line is finite because it is averaged out by the integration of Eq. (4.11).

On the other hand, in the experimental results the transition point (82 V mm-1)

between NFAS and FAS seems to be located above the calculated critical point

(68 V mm-1). The reasons for this may be as follows: The FAS changes to the NFAS

before reaching the calculated critical point due to the boundary condition, and/or the

parameters used for the numerical calculations are not suitable for our sample. It should

be noted that the parameters used here (the Leslie coefficients and dielectric anisotropy)

are very sensitive to the change of sample temperature. Except for the details, good

agreement is obtained between experiment and theory in the FAS, while in the NFAS, the

relaxation frequency is relatively low and almost constant except near the critical point.

91

FIG. 4.5. 0E dependence of the peak or dip frequency of the imaginary part,

peak , where dots are experimental results, a solid line is obtained from calculated

frequency dispersion curves using Eq. (4.11), and a dashed line is 1/ calculated from

Eq. (4.7).

Next, let us examine in detail the 0E dependence of the real part of 1,1 in

the low frequency region. The experimentally obtained real part at 0.63 rad s-1 and the

theoretically calculated one at zero frequency are shown in Fig. 4.6 for the shear rates of

10 and 40 s-1. In the FAS, the response at low frequencies increases as the transition point

is approached and peaks around the transition point for both of the two shear rates. This

behaviour is also seen in the NFAS, and is attributed to the coexistence of the FAS and

92

NFAS just below the transition point. The transition electric field at the shear rate of 40

s-1 is higher than that at 10 s-1, as is expected from Eq. (4.9).

FIG. 4.6. 0E dependencies of the first-order response at very low frequency

(0.63 rad s-1 for the experiment and exactly zero for the theory), 1,1Re ( 0) , for

shear rates of 10 and 40 s-1.

Figure 4.7 shows the frequency dispersion of the second-order response

2,2 . In the NFAS (Figs. 4.7(a) and (b)) an anomalous change is seen around 3 rad

s-1 as well as in the first-order response. In the FAS (Figs. 4.7(c)-(f)), the experimental

93

behaviour is almost reproduced by the theory. In the second-order response, however, the

frequency dispersion becomes more complicated because Eq. (4.9c) contains2

1,1( ) .

There is a plateau in both the second-order response and the first-order response. In

Fig. 4.8, the 0E dependence of the real part of 2,2 in the low frequency region is

shown, where the experimental data are obtained at 0.3 rad s-1 and the theoretical curves

are calculated at zero frequency for the shear rates of 10 and 40 s-1. In the FAS, the second-

order response is negative, unlike the first-order response, for both the shear rates of 10

and 40 s-1, but the details are different: at 10 s-1 the absolute value of the response

increases near the transition point, which may be related to the slowing down, whereas at

40 s-1 it is almost constant as the measurement is taken only near the transition point.

94

FIG. 4.7. Frequency dispersions of the second-order response 2,2 ( ) at a shear

rate of 10 s-1 for several dc electric fields. The dispersions are clearly distinguishable

between the NFAS ((a) and (b)) and the FAS ((c)-(f)). Solid lines in the FAS are calculated

on the basis of the EL theory.

95

FIG. 4.8. 0E dependencies of the second-order response at a very low frequency

(0.3 rad s-1 for the experiment and exactly zero for the theory), 2,2Re ( 0) , for

shear rates of 10 and 40 s-1.

Finally, we show the shear stress without ac electric fields, 0 , in Eq. (4.9) as a

function of 0E in Fig. 4.9, from which it is difficult to distinguish between FAS and

NFAS. This clearly shows that the stress response is sensitive to the change in the director

96

field. It is worthwhile to point out that Fig. 4.9 is closely related to Figs. 4.6 and 4.8. At

the limit of 0 , Eq. (4.4) becomes

0 0 1,1 2,20 2 0E E E , (4.15)

where we have used Eq. (4.10) and note that 0 0E . Taking into account that

1,1 E and 2

2,2 E , we have 1,1 00 ' E E and

2

2,2 00 " / 4E E . From the latter equation it is easily understood that

2,2 0 should be negative because 0 0E is a convex function in the FAS, as is

seen from Fig. 4.9.

FIG. 4.9. 0E dependencies of 0 at shear rates of 10 and 40 s-1. It is difficult to

distinguish the FAS and NFAS based on 0 .

97

4.5 Conclusions

We have investigated the field-induced transition of 8CB from the NFAS to FAS

by measuring the shear stress response to an ac electric field. Totally different frequency

dispersions in the two states were observed for both the first- and second-order responses.

The slowing down of relaxation time and the increased zero-frequency response were

found near the transition point as well as in equilibrium phase transitions. The

experimental results in the FAS were in good agreement with the calculated results based

on the EL theory, including the plateau observed at high frequencies. As for NFAS, it is

necessary to elucidate the director field to calculate the stress response. Our measurement

method is expected to be useful for studying non-equilibrium phase transitions under

shear flow.

98

CHAPTER 5

SUMMARY AND CONCLUSIONS

The non-equilibrium steady state response and fluctuations of sheared LCs were

investigated both experimentally and theoretically. On the basis of the Ericksen-Leslie

(EL) theory the dynamical properties were examined. The shear stress response to an ac

electric field was measured in shear flow and dc electric fields, and the results were

analyzed in terms of the EL theory.

Using the EL theory, phase diagrams in the three-dimensional space of magnetic field

at a constant shear rate were obtained for two liquid crystals 5CB and 8CB which have

different flow properties. An analytical solution was given for the tumbling state, in which

the director rotates about an axis determined as a left eigenvector of the dynamical matrix

that governs the motion of the director. It was shown that non-conservative forces caused

by shear flow appear when magnetic fields are applied, so that the director points out of

the shear plane. The non-conservative forces experimentally observed in the response

function of 5CB were thought to originate from a similar mechanism. Using the Langevin

equation, the orientational fluctuations of the director were examined in detail by

calculating the time correlation function and the response function, which are related to

each other through a modified fluctuation dissipation relation. Critical behaviors such as

the divergence of the relaxation time and fluctuations were found at the critical points

where there is a discontinuous change in director orientation and at the boundary between

the stable and unstable states.

99

The non-equilibrium steady state response of an LC to an ac electric field under

constant dc electric field and steady shear flow was investigated. The first and second-

order shear stress responses were theoretically obtained from the EL theory assuming a

mono-domain model. It was found theoretically and experimentally that when a dc

electric field is applied both responses remain constant and non-zero even at high

frequencies of ac electric field. That is, there is a plateau, which originates from the time

derivative of the director in the viscous equation. This plateau is a remarkable feature in

the shear stress response brought about by the application of a dc electric field.

Furthermore, by performing an optical measurement, the director response was confirmed

to vanish at high frequencies, strongly supporting the above-mentioned mechanism for

the appearance of the plateau. It was also clarified that the plateau in the second-order

response disappears in the absence of a dc electric field, due to the Parodi relation, which

is derived from the Onsager reciprocal relation.

The field-induced transition from the non-flow-aligning state (NFAS) to the flow-

aligning state (FAS) was investigated by measuring the shear stress response to an ac

electric field. Totally different frequency dispersions in the two states were observed for

both the first- and second-order responses. The slowing down of relaxation time and the

increased zero-frequency response were found near the transition point as well as in

equilibrium phase transitions. The experimental results in the FAS were in good

agreement with the calculated results based on the Ericksen-Leslie theory, including the

plateau observed at high frequencies. As a result, the present measurement method can

be expected to be useful for studying non-equilibrium phase transitions under shear flow.

However, as for NFAS, it is necessary to elucidate the director field to calculate the stress

response.

100

Our current studies were only limited to the mono-domain assumption (boundary

condition is neglected). The calculation of dynamics by taking into account the boundary

condition and spatially inhomogeneous director field is expected to give abundant

features of non-equilibrium steady state response and fluctuations in sheared nematic

liquid crystals.

101

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LIST OF PUBLICATIONS

1. Jaka Fajar Fatriansyah and Hiroshi Orihara, Dynamical properties of nematic liquid

crystals subjected to shear flow and magnetic fields: Tumbling instability and

nonequilibrium fluctuations, Physical Review E 88, 012510 (2013).

2. Jaka Fajar Fatriansyah, Yuji Sasaki and Hiroshi Orihara, Nonequilibrium steady-state of

a nematic liquid crystal under simple shear flow and electric fields, Physical Review E

90, 032504 (2014).

3. Jaka Fajar Fatriansyah and Hiroshi Orihara, Electric-field-induced flow-aligning state in

a nematic liquid crystal, Physical Review E 91, 042508 (2015).

non-equilibrium steady state response and … steady state response and fluctuations of sheared...

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