INFORMATION TO USERS · 2014. 8. 6. · ACKNOWLEGEMENT xii CHAPTER 1. INTRODUCTION: PHYSICS OF LIQUID CRYSTALS 1 1.1 Introduction to liquid crystals 2 1.2 Ferroelectric liquid crystals

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A STUDY OF DIELECTRIC AND ELECTRO-OPTICAL RESPONSE OF LIQUID CRYSTAL IN CONFINED SYSTEMS

A dissertation submitted to Kent State University in partial

fulfillment of the requirements for the degree of Doctor of Philosophy

by

Hong Ding

May, 1996


UMI Number: 9706619

UMI Microform 9706619 Copyright 1996, by UMI Company. All rights reserved.

This microform edition is protected against unauthorized copying under Title 17, United States Code.

UMI300 North Zeeb Road Ann Arbor, MI 48103


Dissertation written by

Hong Ding

B.S., Sichuan University, 1985

M.S., Sichuan University, 1988

Ph.D., Kent State University, 1996

Approved by

Co-Chairs, Doctoral Dissertation Committee

Members, Doctoral Dissertation Committee

Accepted by

£ ) oaaJ UJ- Chair, Department of Physics

—.____________j Dean, College of Arts and Sciences


TABLE OF CONTENTS

ACKNOWLEGEMENT xii

CHAPTER

1. INTRODUCTION: PHYSICS OF LIQUID CRYSTALS 11.1 Introduction to liquid crystals 21.2 Ferroelectric liquid crystals 111.3 Order parameter of LC and orientation 131.4 The free energy of liquid crystal systems 21

Reference 27

2. PROPERTIES OF POLYMER DISPERSEDLIQUID CRYSTAL (PDLC) 292.1 Brief history of PDLC 292.2 Phase separation techniques 312.3 Director configuration of PDLC droplet 362.4 Optical and electric properties of PDLC 38

Reference 42

3. DIELECTRIC SPECTROSCOPY 443.1 Dielectric in an electric field 443.2 Resonance and relaxation 553 .3 Relaxation processes in liquid crystal 643 .4 Relaxation processes in ferroelectric liquid crystals 663.5 Dielectric properties of polymer 733 .6 Dielectric permittivities of heterogeneous systems 76

Reference 80

4. ELECTRO-OPTICAL RESPONSE OF POLYMERDISPERSED LIQUID CRYSTAL (PDLC) 824.1 Introduction 834.2 Theoretical model 86

iii


4.2.1 Calculation of the free energy 894.2.2 Calculation of the field free energy 964.2.3 Switching fieid 1044.2.4 Relating the model to experiment 107

4.3 Experiment 1104.4 Results and discussion 1194.5 Conclusions 125

Reference 125

DIELECTRIC RESPONSE OF LIQUIDCRYSTAL IN CONFINED SYSTEMS 1275.1 Introduction 1275.2 Dielectric properties of nematic liquid crystal in pores 129

5.2.1 Materials and sample preparation 1305.2.2 Experimental set-up 1335.2.3 Experiments 1335.2.4 Results and discussion 1355.2.5 Conclusion 155

5.3 Influence of confinement on dielectric properties offerroelectric liquid crystal (FLC) 157

5.3.1 Materials and set-up 1575.3.2 Results and discussion 1595.3.3 Conclusion 184References 186

CONCLUSION 189

iv


LIST OF FIGURES

Chapter 1

Figure 1.1

Figure 1.2

Figure 1.3

Figure 1.4

Figure 1.5

Figure 1.6

Figure 1.7

Figure 1.8

Figure 1.9

Chapter 2

Figure 2.1

Schematic representation of the isotropic phase and nematic phase.

The arrangement of molecules in the cholesteric mesophase.

Schematic representation of the smectic A and smectic C phase.

(a) Schematic cross-section of the structure illustrating how the local layer polarization turns from layer to layer, (b) Illustration of the linear coupling between tilt 9 and polarization P.

The schematic temperature dependence of the order parameter for a nematic liquid crystal from Maier-Saupe mean field theory.

a. Director fluctuation in A* phase, and its variation with temperature and (b) director fluctuation in the C* phase with constant tilt angle.

Definition of coordinates and introduction of the order parameter £ and P.

Schematic temperature dependence of the tilt angle 0 and polarization P at a second-order SmA* - SmC* transition.

Physical distortion of the director field: (a) splay, (b) twist,(c) bend.

Illustration of the SIPS process.

v


Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Chapter 3

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3 .7

Diagram illustrating the evolution of a PDLC material. through phase separation.

Illustration of the TIPS process.

Director configurations in a droplet of PDLC film a) radial b) axial c) bipolar d) toroidal.

PDLC light shutter illustrating the opaque or scattering state with randomly oriented nematic liquid crystal droplets and the transparent state with the droplets aligned by an applied electric field.

Dielectric constants e, | and e1 of nematic as a function of temperature Schematically.

Debye type relaxation for polar substances.

Cole-Cole plot for the Cole-Cole equation at a = 0.

Dispersion and loss curves for the Cole-Cole equation at a = 0.8 and 0.0 respectively.

Relaxation processes of nematic liquid crystals.

Schematic temperature dependence of dissipation peaks for the polymer.

The self consistent field approximation for effective dielectric constant of heterogeneous system.


Chapter 4

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.9

Figure 4.10

Figure 4.11

Figure 4.12

Schematic representation of nematic director configuration of elliposoidal droplet in a PDLC film: (a) before switching,(b) after switching.

The nematic unit director in cylindrical coordinates for a elliposoid droplet of PDLC at OFF state.

The nematic director in cylindrical coordinates for a elliposoid droplet of PDLC at the ON state.

Effective dielectric constant calculation. Consider a rotational ellipsoid a = b * c, a < c, a and a, are the short axis of inner and outer ellipsoid respectively.

Switching field as a function of aspect ratio X for different droplet sizes.

Contribution to switching field from the splay and bend elastic energy.

The optical and capacitance method for the thickness measurement.

Typical picture of the Scanning Electron Microscope for E7 and N65 PDLC film.

Set-up for high voltage dielectric-optical measurement.

The effective dielectric constant em of the PDLC film and dielectric constant ep of polymer binder as a function of temperature.

Dissipation of N65&E7 mixture as a function of logarithmic frequency for different concentration of E7 at room temperature.

Dissipation peaks in logarithemic frequency vs. concentration of liquid crystal E7 in the polymer.

vii


Figure 4.13

Figure 4.14

Figure 4.15

Figure 4.16

Figure 4.17

Chapter 5

Figure 5.1

Figure 5.2

Figure 5.3

Figure 5.4

Figure 5.5

Figure 5 .6

Figure 5.7

The typical curves o f capacitance and transmittance as a function of voltage for 50% N65&E7 PDLC film at temperature T—15 °C.

The dielectric constant as a function of temperature for E7 liquid crystal.

Elastic constant Kn versus temperature for E7 liquid crystal.

Switching field as a function of temperature for different aspect ratios at a fixed droplet size.

Comparison between experimental and theoretical results on the switching field as a function temperature.

Schematic picture of glass morphology, "worm holes": empty area — pores and shadow area — matrix.

Experimental Set-up for the Dielectric measurement.

The difference of dissipation for pore-size 100 A pore-size empty glass before the heating and after the heating at 450 °C for 2 hours and measurement is taken at room temperature.

Real Part of Permittivity of TL205 in Bulk sample at T=-10 °C and T=+10°C.

Imaginary part of permittivity of TL205 Bulk sample at T=-10°C and T=+10°C.

The dependence of s’ real part of permittivity o f TL205 in the pores at T=-25 °C.

Relaxation time of TL205 in porous glasses and bulk as a function of temperature.

viii


Figure 5.8

Figure 5.9

Figure 5.10

Figure 5.11

Figure 5.12

Figure 5.13

Figure 5.14

Figure 5.15

Figure 5.16

Figure 5 .17

Figure 5.18

Figure 5.19

The Frequency shift of relaxation peaks of TL205 in porous glasses with respect to free state.

Log(T) as a function of inverse of temperature for TL205 Bulk and the Activation Energy U= 0.74 (ev).

Logarithmic relaxation time as a function of reciprocal o f temperature for TL205 in different pore.

Activity energies for TL205 as a function of pore-size, for bulk sample U = 0.74 ev. The pore-size for saturation is about 1560 A.

Potential Shapes in Anisotropic Phases.

Activation Energies for TL205 LC in 500 A pores with the surface treatment and without the treatment.

Relaxation times for TL205 in 500 A pores with surface treatment and without treatment.

Dissipation frequency as a function of bias voltage for TL205 LC in 500 A pores at T=-15 °C and dissipations are all same D = 3.74.

Cole-Cole Parameter as a Function of Pore-size, for Bulk sample a is 0.09.

Frequency dependence of real and imaginary permittivities for DOBAMBC in 1000 A pore at different temperature.

The temperature dependence of soft mode dielectric strength and relaxation time for DOBAMBC in 1000A pores.

The temperature dependence of soft mode dielectric strength and relaxation time for DOBAMBC in 100A pores.

ix


Figure 5.20

Figure 5.21

Figure 5.22

Figure 5.23

Figure 5.24

Figure 5.25

Figure 5.26

Figure 5.27

Figure 5.28

Figure 5.29

The temperature dependence of soft mode relaxation frequency./j and the reciprocal of dielectric strength 1/Ae3 for DOBAMBC in 1000A pores.

The temperature dependence of soft mode relaxation frequency/3 and the reciprocal of dielectric strength 1/As3 for DOBAMBC in 100 A pores.

The temperature dependence of the reciprocal of soft mode dielectric strength and the relaxation frequency in Sm C - Sm A phase for DOBAMBC in 1000A pores.

The temperature dependence of the reciprocal of soft mode dielectric strength 1/Ae3 and relaxation frequency/3 for DOBAMBC in Sm C - Sm A phase at 100A pores.

The temperature dependence of the soft mode rotational viscosity rje and nature logarithmic rotational viscosity for DOBAMBC in 1000A pores.

The temperature dependence of soft mode rotational viscosity qe and nature logarithmic rotational viscosity for DOBAMBC in 100 A pores.

The temperature dependence of Cole-Cole parameter for DOBAMBC in 1000A and 100A pores.

The relaxation time t, of first dielectric relaxation as a function of temperature for DOBAMBC in 100 A and 1000 A pores.

Dielectric strength Ae, of first relaxation as a function temperature for DOBAMBC in 100 A and 1000 A pores.

Temperature dependence of first relaxation time t, of DOBAMBC in 100 A and 1000 A pores.

x


Figure 5.30 Relaxation times x2 of DOBAMBC in 100 A and1000 A pores as a function of temperature.

Figure 5.31 The ln (x j as a function of reciprocal of temperature for DOBAMBC in 100 A and 1000 A pores.

Figure 5.32 The dielectric strength As, as a function oftemperature for DOBAMBC in 100 A and 1000 A pores.


ACKNOWLEDGMENTS

I owe a great deal to my thesis advisors: Drs. Jack R. Kelly and J. William Doane

for their guidance and supervision through these years. Jack deserves special thanks for

always encouraging me and persistently keep me on the track with astute questions and

observations. Jack's enthusiasm and guidance have made this thesis possible. He has

taught me much about the process of doing research as well.

I wish to thank various members o f people at Physics Department and Liquid

Crystal Institute. In particular, Professor A. Saupe, J. West, O. Lavrentovich, D. Allender,

and M. Groom. Dr. Saupe has been an inspiration and a friend over several years. Thanks

also to Professor Fouad Aliev at University of Puerto Rico, who opened his laboratories

and gave of his time for many of experiments in this thesis.

Next, I would like to express my special appreciation to Tom Buer for his caring

and friendship which made the closing chapter of my life in graduate school vastly more

endurable.

Finally, to my family I extend my most heartfelt thanks for their encouragement

and unwavering support which have been a constant source o f strength in my life. This

work is dedicated to them, especially to my little boy Bill, without their love it never

would have been completed.

xii


CHAPTER ONE

INTRODUCTION: PHYSICS OF LIQUID CRYSTALS

Liquid crystals combine the properties of a solid and of an isotropic liquid. On one

hand, they can flow like ordinary liquids; on the other hand, they are orientationally

ordered, and exhibit anisotropic behavior as seen in their electrical, magnetic and optical

properties. The dielectric properties of a liquid crystal depend on the molecular

orientation. Molecular orientation by an applied field changes the dielectric properties of

the liquid crystal system, and hence changes light propagation in liquid crystal. Therefore

liquid crystals are promising materials for optical and display devices.

In this chapter, we will review some physical properties of liquid crystals, and only

focus on those aspects of liquid crystals which are essential for an understanding of the

following chapters: what are liquid crystal materials? what are the primary thermotropic

phases of liquid crystals (nematic, smectic A, smectic A*, smectic C, and smectic C*)°

What are the properties of each phase? And what physical quantities we can use to

describe each phase?


1.1 Introduction to liquid crystals

Liquid crystals are a state of matter intermediate between the solid crystalline

phase and the isotropic liquid phase, and combine the properties o f a solid and isotropic

liquid. They possess many of the mechanical properties of a liquid; for example, they can

flow like an ordinary liquid. On the other hand, they are similar to crystals in that they

exhibit anisotropy in their optical, electrical and magnetic properties. Liquid crystal

mesophases can be observed in certain organic compounds, and are usually composed of

elongated molecules. Liquid crystals can be divided into two groups: thermotropic and

lyotropic, based on whether the phase behavior is induced thermally (thermotropic) or by

the influence of solvents (lyotropic). When a substance which shows a thermotropic liquid

crystalline phase is heated, the system may pass through one or more mesophases before

it transforms from the crystal phase into the isotropic liquid. The melting point and the

clearing point define the temperature range of the mesophases.

In liquid crystalline mesophases, the molecules show some degree of orientational

order (and in some cases partial translational order as well) even though a 3 -D crystal

lattice does not exist. Therefore these phases are often called ordered fluid phases. In this

dissertation we will focus our discussion on thermotropic liquid crystals.

According to the molecular arrangement and ordering, thermotropic liquid crystals

can be further classified into the following types: isotropic, nematic, cholesteric and

smectic (however, the cholesteric is usually considered as a modified form of the nematic).

On the temperature scale the liquid crystalline phases appear as following:


Crystalline solid Liquid crystalline phase Isotropic liquid

| smetic—nematic(or Cholesteric) |

tmelting point

tclearing point

The high temperature phase of a liquid crystal is called the isotropic phase; it is

characterized by optical, electrical, and mechanical properties which are independent of

orientation. In the isotropic phase the molecules possess neither orientational nor

positional order, and the spatial average of the molecule orientational director a is zero,

where a defined as the unit vector along axis o f the molecule. However, locally the

molecules exhibit a certain degree of ordering and the two point correlation function

is non-zero for sufficiently small r which is characterized by an isotropic

correlation length E, (less than lOOA, also temperature dependent) and decays

exponentially'. As the temperature decreases and the molecules begin to correlate their

motion over longer length scales, a first order transition takes place at the

isotropic-nematic transition temperature TS1, which brings the system into an anisotropic

state — nematic phase.

Below r Nn, a certain degree of orientational long-range order exists in which the

director n points, on the average, in some particular direction. The nematic phase is the

least ordered liquid crystalline phase, being characterized by a high degree of long range


4

Isotropic Phase

n(r) points along the direction of the long axis of the molecule at r

Nematic Phase

i m m m ? ■

m m m ^

0(r) is the angular deviation of n(r) from n

Figure 1.1 Schematic representation of the isotropic phase and nematic phase.


\ \ WV

X

\../n

Figure 1.2 The arrangement of molecules in the cholesteric mesophase.


6

orientational order as shown in Figure 1.1, but no long range translational order. The

molecules of a nematic liquid crystal tend to be parallel on the average, to some common

direction. A unit vector in this preferred direction is called the nematic director n. The

parallel and perpendicular components of the macroscopic properties o f the nematic with

respect to the director exhibit different values. If the values of two components

perpendicular to each other as well as to the director are the same (complete rotational

symmetry around the director n), then the nematic is called a uniaxial phase, Otherwise, it

is called a biaxial phase2. Thermotropic nematic liquid crystals are usually uniaxial.

A distorted form of the nematic phase is the cholesteric mesophase, which is

caused by chiral molecules. The cholesteric phase is similar to the nematic phase in having

long range orientational order and no long range translational order as shown in Figure

1.2. It differs from the nematic phase in that the cholesteric director varies with a helical

form throughout the medium with a spatial period L-PI2=kI \ q01 where P is the pitch and

q0 is the wave vector. The sign of q0 distinguishes between left and right helices and its

magnitude determines the spatial periodicity. In fact, a nematic can be viewed as a

cholesteric o f infinite pitch.

If the temperature is lowered further from the nematic phase, the system can enter

another ordered phase in which a certain amount of translational order is introduced, this

is the smectic phase.

The smectic phase has not only long range orientational order but also partial long

range translational order. As many as eight smectic phases have been identified3. Here we


7

just give a few examples of relevance in later sections. In the smectic A phase the

molecules are aligned perpendicular to the layers to form a one-dimension periodic

structure with no long range translational order within the layer. This can be considered as

a two dimensional fluid with an orientational order. Thus, the layers are individually fluid

and inter-layer diffusion can occur, although with somewhat lower probability. The layer

thickness, determined from x-ray scattering data, is essentially identical to the full

molecular length in most cases. At thermal equilibrium the smectic A phase is optically

uniaxial due to the infinite-fold rotational symmetry about an axis parallel to the layer

normal. A schematic representation of smectic A order is shown in Figure 1.3a.

The major characteristics o f smectic A phase are as follows:

(a) A layered structure (with layer thickness close to the full length of the

constituent molecules for the ordinary smectic A phase). There is a quasi long

range order perpendicular to the layer.

(b) Inside each layer, the centers of mass show no long range order, each layer is a

two dimensional liquid.

(c) The system is optically uniaxial, the optical axis being the normal OZ to the

plane of the layer.

(d) The directions Z and -Z are equivalent.

The symmetry of the smectic A phase is Dm, which means the phase can not be

distinguished for chiral and racemate materials. The requirement of constant interlayer


8

Smectic A Phase

y

1 ■(a)

Smectic C Phase

IMMUllULl

(b)

Layer normalA Qrp

£n Layer

Figure 1.3 Schematic representation of the smectic A and smectic C phase.


9

spacing imposes the condition curl(ri)=0 for all macroscopic deformations of a perfect

smectic. Therefore the helix structure, which has curl(n)=qn^0 is forbidden.

Strictly speaking, a smectic A made of chiral molecules should be labeled SA*, and

considered as a phase different from the standard smectic A, with symmetry. Indeed,

the macroscopic properties of SA and SA* are not equivalent: in the SA* phase, rotatory

power and electro-clinic effect exist but not in SA.

As the temperature is lowered still further from smectic A, another phase transition

may take place in which the molecules retain their layered structure but undergo a tilt 6

with respect to the layer normal, this is called the smectic C phase. The projection of the

average molecular long axis director in the layer plane is a 2D vector called the C director.

Smectic C order is depicted in Figure 1.3b. X-ray scattering data from several smectic C

phases indicate a layer thickness significantly less than the molecular length. This has been

interpreted as evidence for a uniform tilting of the molecular axes with respect to the layer

normal. The fact that the smectic C phase is optically biaxial is further evidence in support

of a tilt angle. Tilt angles of up to 45° have been observed and in some materials the tilt

angle has been found to be temperature dependent.

The structure of a smectic C is defined as follows:

(a) Each layer is still a two dimensional liquid.

(b) The material is optically biaxial, the symmetry is specified by the point group


10

\\\\\\\\\\\\\\\\\\\\\\\\\ © \\\\\\\\\\\\\\\\\\\\\\\\\

(a> /////////////////77777T7/ / / / / / / / / / / / / / / / / / / / / / ® l l l l i i i i i n i i i i i i i i i i l i i

\ \ \ \ \ \ m i u \ v \ l\\\l\\ij\'\'\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ w \ \ \ \ \ \ \ ©

Figure 1.4 (a) Schematic cross-section of the structure illustrating how the local layer

polarization turns from layer to layer, (b) Illustration of the linear coupling

between tilt 0 and polarization P.


II

The stability of smectic A is governed by an elastic constant which keeps the long

molecular axis parallel to the smectic layer normal. However, due to the thermal energy

the director fluctuates, giving locally an instantaneous tilt angle between the director and

smectic layer normal as shown in Figure 1.6a. When the temperature approaches the A-C

transition temperature Tc , the elastic constant controlling the tilt fluctuation gets soft.

Thus the fluctuation amplitude increases drastically and its susceptibility diverges at Tc . In

the C phase, the tilt angle increases with decreasing temperature. Deep in the C phase, tilt

becomes more stable against thermal fluctuations. So the soft mode can be seen close to C

- A phase transition, but is suppressed as the temperature moves away from the transition

in either direction as shown in Figure 1.6a.

1.2 Ferroelectric Liquid Crystals

The smectic C phase produced by molecule of pure chirality is the smectic C

phase. Because of the chirality of the molecules, the mirror plane o f symmetry in the Sm

C, defined by the molecular director and the normal to smectic layer is broken. The

symmetry of the system reduces to C, allowing a net local polarization to exist as shown in

Figure 1.4.

Ferroelectricity in liquid crystal, i.e. the existence of a spontaneous electric dipole

moment in Sm C* phase, was predicted by Meyer et al in 19754. In the smectic C* phase

molecule orientational director makes an angle with the smectic layer normal, and it often


12

precesses with a finite phase angle cp from one layer to another resulting in a helical

structure with helical axis parallel to the normal to the smectic layer.

Ferroelectric liquid crystals have following general properties:

(a) The ground state o f the chiral smectic C is a helicoidal structure in which the

molecules within one layer are tilted uniformly but the direction of tilt precesses

around the normal to the smectic layers as one goes from one layer to another to

form a helix.

(b) A polarization P exists in the plane of the layers which scales with director tilt

angle 0 for small 0 (for large 0 higher order terms need to be considered).

The precession angle for the tilt is the azimuthal angle (p. The pitch of the helix is

typically a few microns4-5, so that in a thick sample, the quantity averages to

zero. So does the net value of

, since the magnitude of polarization is proportional to

0exp(icp). To study the polarization and other properties of ferroelectricity, the helix must

often be unwound by external fields ( either electric or magnetic field) to exhibit a net

polarization.

In the Fig. 1.4 for 0 > 0 we have chosen P to point into the paper. If we rotate it

around the cone to the opposite position, corresponding to a change o f sign in 0, then we

see that P has reversed its direction, now pointing out of paper. Thus, a change of sign in

0 corresponds to a sign reversal of P, and from this it follows that, at least for small 0, P

and 0 must be linearly related, or


13

P = P o k x n or P = PosinGG = PqQQ ( 1 . 1 )

where k is layer normal direction and n is molecular direction. We can only expect a linear

relation between P and 0 to hold true over a limited interval. The convention now adopted

for the sign of P can be stated extremely simply: start from the smectic A* situation with

the director n parallel to the layer normal k. Then let n tilt by 9 corresponding to the C*

phase. Then the polarization P(~ 0), is said to be positive if it follows the direction of a

right-handed screw when we rotate n out from k, i.e., in the 6 direction (direction kxn) as

show in Figure 1.4b. The sign of P has been found negative in most synthesized molecules

so far.

1.3 Order parameter of LC and orientation

In order to describe the orientational order of a nematic liquid crystal, one should

take into account two aspects: (1) the local preferred direction n, and (2) the degree of

orientational order. Therefore the orientational properties of liquid crystals are described

by a second rank tensor Q with Cartesian components 0 X] (i,j=x,y,z). For a molecule of

arbitrary shape, the element of the tensor order parameter Ot] can be written in the

generalized form6

Q,j = - < 3 a ,a ; - 5tJ > ( 1.2 )


14

Here we define a unit vector a(r) to describe the orientation of the symmetry axis of a

molecule at position site r, and ij=x, y, z refer to the space-fixed axes. 5n is the Kronecker

delta, and the brackets < > denote a volume average. The tensor Ot] is symmetric and

traceless. It vanishes in the isotropic phase and thus serves as an order parameter.

In an ordered nematic phase, Q usually has uniaxial symmetry. The symmetry axis

is defined by the eignvector of Oip i.e. n(r), corresponding to the only non-degenerate

eignvalue. By choosing a proper coordinate system, i.e., the principal axis frame, Qt] can

be expressed in a diagonal form1

Q=511 0 00 S2 2 o 0 0 S33

(1.3)

where 533 = - (5U + 5,. ). In the uniaxial nematic case, 5U = S2Z.

By choosing the z-axis parallel to the nematic director, 0 has the form

Q=- f 0 0 0 - f 00 0 5

(1.4)

where 5 is the scalar order parameter, which describes the degree of the orientational

order of the molecules. 5 is defined as7

5 = ^(3 < cos20 > - 1) (15)


S (o

rder

par

amet

er)

15

10 7

0.9 -

0.8 -4

0.7 -

0.6 -jij

0.5 -

0.4 -

0.3 -

0.2 -

Temperature (K)

Figure 1.5 The schematic temperature dependence of the order parameter for a

nematic liquid crystal from Maier-Saupe mean field theory.


16

where 9 is the angle between the direction of molecule symmetry axis and the nematic

director. Clearly, S = 1 for the fully oriented nematic phase and S = 0 for the randomly

distributed isotropic phase. The scalar order parameter S is a function of the temperature.

The schematically temperature dependence of order parameter S is shown schematically in

Figure 1.5.

In general, the nematic director n and order parameter S are spatially varying

quantities. The molecular ordering at every spatial point r is characterized by a director

n(r) pointing along the local axis of uniaxial symmetry and by a quantity S(r) giving the

local orientational order of the molecules. Thus the tensor order parameter Q(J can be

written as

Qij(x) = T ^3n ,(r)rt/r) - 5;;), i j = x ,y ,z (1.5)

The directions n(r) and -n(r) correspond to physically equivalent states due to

symmetry, but Otj and -0 tJ correspond to physically different states.

In the smectic A phase, besides the order parameter above, we can introduce

another order parameter which describes the periodic layer structure, namely the center of

mass density function along the layer normal, p(z).

In the majority of cases, the smectic C phase appears when cooling an A phase. In

such cases, the transition can be continuous and as a first approximation can be described

by a single order parameter 0 characterizing the appearance of the molecular tilt at the


17

tilt fluctuation

instantaneous tilt angle

00000000

C* phase A* phasecoCO3o353

(a)

(p large fluctuation

small fluctuation

spontaneous tilt angle

(b)

Figure 1.6 (a) Director fluctuation in A* phase, and its variation with temperature and

(b) director fluctuation in the C* phase with constant tilt angle.


18

smectic layer

Figure 1.7 Definition of coordinates and introduction of the order parameter ^ and P.


19

transition temperature Tc. On entering the C phase the system of molecules must tilt. By

symmetry, there are infinitely many tilt planes, and evidently we have the case of a

continuous degeneracy in the sense that if all molecules would tilt in the same direction,

given by the azimuthal angle cp, the chosen value of cp would not affect the free energy.

The complete order parameter thus has to have two components, reflecting both the

magnitude of the tilt 0 and its direction (p in space and can conveniently be written in

complex form®:

£ = 0e,(p (1.6)

Figures 1.6b and 1.7 illustrate the two-component order parameter ^=0e"*’

describing the smectic A-C transition. The phase variable cp is a so-called gauge variable

and is fundamentally different from 0. The latter is a "hard" variable with relatively small

fluctuations around its thermodynamically determined value (its changes are

connected to compression or dilation of the smectic layer, thus requiring a considerable

elastic energy), whereas the phase angle cp has no thermodynamically predetermined value

at all. The result is that we find large thermal fluctuations in cp around the cone for long

wavelengths compared to a molecular scale; indeed, this mode gives rise strong scattering

of visible light. The very easily excitable cone motion, sometimes called the spin mode or

the Goldstone mode, is also the motion that can most easily be induced by an applied

electric field E in the case of the ferroelectric C* phase where E couples to P. The tilt or


2 0

P(T) 0(T)

Temperature (K)

Figure 1.8 Schematic temperature dependence of the tilt angle 9 and polarization P at

a second-order SmA* - SmC* transition.


21

"soft" mode, is "hard" to excite in comparison with the cone mode, except at the A-C

transition. For T * TAC, the two motions can be considered as essentially independent of

each other; tilt mode is important for T % TAC, cone mode for T < TAC.

Although, in a typical solid ferroelectric, polarization is the natural order

parameter, in a LC the primary transition is in the tilting of the molecules, and P is only a

secondary effect of this tilting. Thus, the tilt 0 is the primary order parameter and

polarization is a secondary order parameter. Figure 1.8 shows the temperature dependence

of order parameters 0 and P at a second-order SmA* - SmC* transition.

1.4 The free energy of liquid crystal systems

In an ideal nematic single crystal, the molecules are aligned on average along one

common direction n. However, in most practical cases, because of thermal fluctuations,

the limited surface of the sample, or external fields, this ideal conformation will not exist.

For a uniaxial nematic liquid crystal, the orientational order is described by a director field

n and order parameter S. In an equilibrium state, the system has minimum free energy.

However, when the system is perturbed by some external factors such as an external field

or various surface constraints, the free energy associated with the distortion in the director

field will be added to the system. If the distance of significant variations of motion is large

compared to the molecular scale, one can describe deformations of nematic by a

continuum theory. Such a theory was first enunciated by Oseen and Zocher9, and perfected


2 2

by Frank10. In Frank theory 5 is assumed to be a constant. The contribution to the free

energy density due to the distortion of the director field can be written in vector notation

for an arbitrary deformation as9:

F= ^ { K X(V • n)2 + ^ 2(n • (V x n))2 + ^ 3(n x (V x n))2} (1.6)

where , K2 , and K2 are the Frank elastic constants which depend on S and temperature

T. The K{ , K2 , and correspond to splay, twist, and bend deformations respectively as

shown in Figure 1.9.

In 1937 Landau1112 proposed to describe second order phase transition phenomena

near the transition point, where the order parameter is small, in terms of an expression of

the free energy density in a power series in the order parameter. De Gennes1 generalized

Landau theory to include the first-order nematic-isotropic transition, where the order

parameter is not small. The de Gennes theory is qualitatively correct, but it provides a

satisfactory description of many properties of the system at the phase transition. The

expansion of the Landau-de Gennes free energy in powers of the tensor order parameter

On for a uniform uniaxial nematic is given by (in the absence of external fields)1

f= fo(T ) + ̂ A O u(r)Qj,(r) + i£ Q v(r)Oyfc(r)£ fa(r)

+ ^Q 0 ?(r)0 „ (r)]2 + 0(Q 5) (1.7)


23

Twist

Figure 1.9 Physical distortion of the director field: (a) splay, (b) twist, (c) bend.


2 4

where f 0(T) represents the free energy density of the isotropic phase; A=a0(T - T*) where

P is a temperature slightly below the nematic-isotropic transition temperature Tc and

aQ>0; B, C are coefficients weakly dependent temperature T. i, j, k =1, 2, 3 denote the

components along the three orthogonal axes of the coordinate system.

For bulk nematics, by substituting equation (1.5) for Q ^r) into equation (1.7) we

obtain13

F = f- M T ) = \ a 0( T - r ) ^ ( r ) + ± 5 S 3(r) + S \ r) (1.8)

We can extend these ideas to the free energy of the smectic C phase. If we

consider the tilt angle 0 as the order parameter for the smectic C phase, we can write the

smectic C free energy density F in terms of a Landau expansion in powers o f 0. This

expansion can not contain odd powers of 0 because in the absence of any internal

structure along the smectic layer the free energy must be independent of the sign of the tilt

(±0). Hence, we may write,

F = Fo + ±a02 + ^>e4 ++c06 + ... (1.9)

Here, for a second-order A-C transition, the general case, b is greater than zero and only

weakly depends on temperature, whereas a has to change sign in order to allow the

transition. Thus , with a=a(T-Tc) as the simplest choice,


25

F = F 0 + i a ( T - r c)02 + i be4 + ±cQ62 4 6(1.10)

A first-order /1-C transition occurs if b < 0 (Above discussion of phase transition is also

applied to nematic Landau coefficients).

As explained previously in the smectic C* phase, there are two order parameters c,

and P . £ is a two-component tilt vector order parameter £=(£, , ^ as shown in Figure

1.9. The free energy density must therefore be expanded in both order parameters, subject

to the restriction that only the different powers or power combinations which are invariant

to the symmetry operations of the C* phase be retained. We will introduce a bilinear term,

i.e., a term P£, to take into account coupling between order parameters. £=(£1 , £2)

related to our earlier order parameter as

In the following, the helix axis is taken to be along the z-direction with the smectic layers

parallel to the xy-plane. If we neglect, for simplicity, any in-plane variations in the director

n(x, y, z), thus only derivatives dldz have to be considered. The free energy density can be

written as

£ = 0e'

2 6

where e is a generalized permittivity, assumed positive and constant. The term C(PX̂2 -

Py£,) takes into account the fact that the coupling between P and 6 is chiral in character

(without chirality, finite tilt will not result in a polarization) and is a piezoelectric

coefficient. This quantity changes sign when we change from a right- to a left-handed

reference frame, which means that the optical antipode of a certain C* compound will

have a polarization of opposite sign. The term preceded by the proportionality factor A

has the same symmetry properties, and responsible for helicoidal structure. Both C and A

vanish for nonchiral or racemic materials. The Frank elastic modulus is denoted by

corresponds to a twist, and p is called the flexoelectric coefficient. We will come back to

this Classic-Landau free energy expression in chapter three to discuss some

thermodynamic properties and dynamic modes of ferroelectric liquid crystals.

In order to determine the behavior of liquid crystals in the presence of fields, one

has to consider the free energy associated with the external fields. In a static electric field,

the free energy density1 for any system is given by

where D is electric displacement and E is the applied electric field. Here "+" corresponds

to constant charge and is for constant potential.

Similarly, the free energy density in the presence of a magnetic field in SI unit is

Fe = ±jD • E = -yso£op£a-£p (113)

F m = - j B • H = PoPaptfatfp (1.14)


2 7

B is the magnetic induction, H is the applied magnetic field and (i is the magnetic

permeability and |a > 1 for paramagnetic substances; p. < 1 for diamagnetic.

R eference;

1. P.G. de Gennes, The physics of liquid crystals, Clarendon press, Oxford 1993

2. L. J. Yu and A. Saupe, Phys. Rev. Lett., 45, 1000(1980).

3. H. Sackmann and D. Demus, Fortschr, Chem. Forsch., 12,349(1969).

4. R.B. Meyer, L. Liebert, L. Strzeleki, and P. Keller, J. Phys. (Paris). Lett.,

36X69(1975)

5. P. Martinot-Lagarde, J. Phys. (Paris) Lett., 38 L I7(1977).

6. A. Saupe, Z. Naturforsch. 19a, 161(1964).

7 E.F. Gramsbergen, L. Longa, and W. H. De Jeu, Phys. Rep. 135, 195(1986).

8. J.W. Goodby and R. Blinc, etc. Ferroelectric Liquid Crystal: Principles, Properties

and Applications, Gordon and Breach Science Publishers, 1991.

9. C. Oseen, Trans. Faraday Soc., 29, 883(1933).

10. F.C. Frank, Disc. Faraday Soc., 25, 19(1958).

11. L. D. Landau, On the Theory of Phase Transition, Part I and Part II, collected

papers of L. D. Landau, edited by D. ter Haar, Gordon and Breach, Science

Pulishers, N.Y., 2nd Edition, 193(1967).


28

12. L.D. Landau and E.M. Lifshitz, "Electrodynamics o f Continous Media",

Pergamon Press, NY, (1960).

13. E.B. Priestley, P.J. Wojtowicz, and P. Sheng, Introduction to Liquid Crystal,

1979.


CHAPTER TWO

PROPERTIES OF POLYMER DISPERSED LIQUID CRYSTAL (PDLC)

Polymer dispersed liquid crystals (PDLC) form a relatively new class of a wide

variety of materials used in many types o f displays, switchable windows and other light

shutter devices1'2, and especially for fabrication of large scale flexible displays. The use of

PDLC films overcomes the two major problems normally encountered in display

technology: liquid crystal fluidity and need for light polarizers. In this chapter we give a

brief introduction to PDLCs about their history, techniques of phase separation, director

configurations inside a PDLC droplet, and electro-optical properties o f PDLC film. This

serves as a background to chapter four, where we will focus on the detailed electro-optical

properties and switching mechanism of PDLC films.

2.1 Brief history of PDLC

In a patent application published in 1976, Hilsum describes what is perhaps the

first light shutter device in which a nematic liquid is dispersed with a second medium to

induce light scattering that can be electrically controlled3. He patented a device with

29


3 0

dispersions o f glass spheres in a liquid crystal material. By electrically controlling the

birefringence of the liquid crystal he was able to match the refractive index of the glass and

liquid crystal. A similar principle was used by a group at Bell Labs where liquid crystal

was filled into microporous filters.

In 1982 Craighead et al, published a device where the second medium is a

polymer4. They made use of a microporous filter, filling the micron-size pores with a

nematic liquid crystal of positive dielectric anisotropy and sandwiching the film between

ITO coated substrates. However the contrast ratio, which is a measure of how opaque the

device is in the off state to how transparent it is in the on state, was poor for both devices

and they were never commercialized.

In 1983 Fergason succeeded in developing a film with micron size nematic liquid

crystal droplets dispersed in a polymer matrix. Initially PVA (Poly Vinyl Alcohol)\ a

water soluble polymer, is mixed with water to form a homogeneous solution. An emulsion

of liquid crystal and the homogeneous polymer solution is then made. The water is

evaporated, resulting in the forming of liquid crystal droplets in the PVA matrix film.

In 1984 phase separation methods were developed at Kent State University to

make such films. The process begins with a homogeneous solution of polymer or a

prepolymer and a low molecular weight liquid crystal. Phase separation is then induced

thermally, through polymerization or by solvent evaporation6. The results in the formation

of droplets which grow in size until the polymer solidifies. Since these phase separation


31

procedures can be applied to a broad range of polymers including thermoplastics,

thermoset polymer and UV-curable polymers, a wide variety of systems can be developed.

2.2 Phase separation techniques

The phase separation techniques used to fabricate PDLC films can be classified

into three main types: the SIPS (Solvent Induced Phase Separation), the PEPS

(Polymerization Induced Phase Separation) and TIPS (Thermally Induced Phase

Separation) processes6.

The SIPS process is illustrated in Figure 2.1, a low molecular weight liquid crystal

and a prepolymer are dissolved in a common solvent forming a homogeneous solution

represented by point I. As the solvent evaporates the solution crosses the miscibility gap

At this point labeled M, the liquid crystal becomes immiscible and phase separation occurs

as liquid crystal droplets begin to form. The droplets continue to grow until the gelation of

the polymer occurs with two phases finally reaching their equilibrium concentrations at

points A and B. If the process is not quasistatic, the equilibrium concentration (A and B)

are reached via point F. In this process the droplet size and droplet density can be

controlled by the rate of evaporation.

The PIPS process is a time dependent process. The different stages are illustrated

in Figure 2.2. The liquid crystal is dissolved in a prepolymer and curing agent. The

polymerization process is then induced thermally or photochemically (UV radiation). As


POLYMER

Miscibility GapHomogeneousSolution

LIQUID CRYSTALSOLVENT

Figure 2.1 Illustration of the SIPS process.


mix

iimmiscibility

I

gelation

ifinal set

ihomogeneous droplet droplet

solution formation purification

Time

Figure 2.2 Diagram illustrating the evolution of a PDLC material through phase

separation.


3 4

time increases the polymerization process results in the liquid crystal becoming

increasingly immiscible in the polymer. After a certain period of time phase separation

occurs and the droplets are formed. The droplet grow in size as the liquid crystal

continues to phase separate out of the polymer. The growth of the droplets ceases with

the gelation of the polymer. The polymerization process continues until the final cure. The

rate o f polymerization is controlled by the cure temperature for thermal process and light

intensity for photopolymerization. The droplet size is controlled by the rate of

polymerization and other factors such as the solubility of the liquid crystal, diffusion

coefficients and the rate of the chemical reaction.

The TIPS process is useful for thermoplastics which melt below their

decomposition temperature. In this process a binary mixture of polymer and liquid crystal

forms a homogeneous solution at elevated temperatures. A typical path for the phase

separation process is shown in Fig 2.3. Starting at point I, as the temperature decreases

the miscibility gap is crossed at point M. Phase separation occurs and droplets begin to

form. If the process is quaistatic, two phase will eventually form and have concentrations

represented by points A and B. The droplets continue to grow until the polymer hardens.

If the process is not quasistatic, the equilibrium concentrations represented by points A

and B can also be reached via point P through rapid cooling. As in the PIPS case, the

solution may also include other chemical agents to adjust the elecro-optical performance

of the display, adhesion to the substrate, etc. Cooling the homogeneous solution into the

miscibility gap causes phase separation of the liquid crystal. The droplet size is controlled


HomogeneousSolution

u .32 Miscibility

Gap

Concentration

Figure 2.3 Illustration of the TIPS process.


3 6

by the rate of cooling and depends upon a number of material parameters, which include

viscosity, chemical potential, etc. In general, larger concentrations of liquid crystals are

required for these films as compared to phase separation by polymerization.

2.3 Director configurations of PDLC droplet

Confining the liquid crystal in a droplet results in a particular director

configuration. This configuration depends on a number of factors: how the molecules are

anchored at the droplet wall, droplet size and shape, elastic constants of the liquid crystal

and the direction and magnitude of any applied electric or magnetic field. Minimization of

the free energy of the droplet determines the director configuration in a droplet.

Four director configurations have been observed7'9 as shown in Fig 2.4 Radial and

axial configurations are observed when the molecules have perpendicular (homotropic)

anchoring. Bipolar and toroidal configurations are observed when the molecules have

tangential (homogeneous) anchoring. With the application of an electric or magnetic field,

either a configuration transformation or reorientation can occur. An example of a

configuration transformation is the radial to axial transformation9. A sufficiently high field

will transform the radial configuration to axial configuration with the symmetry axis of the

axial configuration aligning along the field direction. Configuration reorientation is often

observed in bipolar droplets. The application of a field rotates the symmetry axis of the


Figure 2.4

3 7

(a) (b)

(c) (d)

Director configurations in a droplet of PDLC film a) radial b) axial c)

bipolar d) toroidal


3 8

bipolar director configuration into the field direction without changing the configuration

substantially.

Typically the configuration o f the droplet in most devices is bipolar because they

give the best scattering. For the PDLC systems studied in this work the configuration in

the droplet is bipolar.

2.4 Optical and electric properties o f PDLC

Figure 2.3 shows schematically how a PDLC film works. A PDLC display operates

on the principle of electrically controlled light scattering. The nematic liquid crystal in the

droplet is optically uniaxial with the optical axis parallel to the nematic director. It is

birefringent with an ordinary refractive index n0 for light polarized perpendicular to the

nematic director, and an extraordinary refractive index ne corresponding to the

polarization parallel to the nematic director. The polymer matrix is typically isotropic with

a single refractive index nm . In making the film the polymer refractive index nm is closely

matched to the ordinary refractive index of the liquid crystal n0 . In addition, a liquid

crystal with positive dielectric anisotropy is often used so that the nematic symmetry axis

aligns with field direction.

The refractive index that incident light probes is dependent upon the director

configuration in the droplet and angle between the optical axis (nematic director) and the

direction of the incident light. For light polarized in the plane defined by the nematic


3 9

Figure 2.5

: t * . . . I -

® 'V ' ® ■ • * # 0

► Is 4

OFF STATE ON STATE

o Incident s Scattered t transitted

PDLC light shutter illustrating the opaque or scattering state with randomly

oriented nematic liquid crystal droplets and the transparent state with the

droplets aligned by an applied electric field.


4 0

director n and the incident wavevector k, the refractive index at angle 4> (determined from

coaj) = n»k) is10

In the absence of a field the average nematic director in a droplet varies from

droplet to droplet and there is a wide distribution of n(). The incident light probes a

range of refractive indices from nD to ne and light is strongly scattered causing the film to

appear opaque.

With application of an electric field E the nematic symmetry axis reorients parallel

to the field. The incident light probes a single refractive index (the ordinary index):

and the scattering is minimized, causing the film to appear transparent.

The electro-optical properties of PDLC films are controlled by the types of

materials used, the droplet morphology and the method of the film construction. Desirable

properties include high clarity and transmission of the film on the ON and OFF states, low

driving voltage, low power consumption, fast switching times and high film resistance. All

these properties are related to each other, so it is usually not possible to change them

independently.

(2 . 1)

«() = fio * n m (2 .2 )


41

In order to quantify the optical performance of a the PDLC film, we introduce the

clarity and transmission11-12. The clarity is a measure of the sharpness o f an image viewed

through a film, and the transmission is a measure of the efficiency o f the light passage

through the film. The transmission through the film is defined as the intensity o f light

transmitted by a film divided by the incident light intensity. The clarity is defined as the

intensity of the light transmitted unscattered divided by the total light transmitted. It can

be measured with a haze meter or with an integrating sphere. The clarity of a PDLC film

in the ON state depends on the match of n0 and nm. The closer the match, the clearer the

film in the ON state. This is usually achieved by precisely adjusting the refractive index of

the matrix nm. On the other hand, liquid crystal dissolved in the binder of a PDLC varies in

its refractive index12. Also, the effective n0 of the droplet is not precisely equal to nD of the

bulk liquid crystal because the alignment is not parallel throughout the droplet.

The OFF state clarity and transmission are determined by the size and density of

the droplet and birefringence of the liquid crystal. Maximum scattering and therefore

minimum transmission and clarity are achieved when the droplet size and spacing is on the

order of the wavelength of light. Highly birefringent liquid crystals offer the largest

mismatch of the refractive indices in the OFF state. Thicker films are also more scattering;

however they also reduce the clarity in the ON state, and require higher switching fields.

The refractive index match of the liquid crystal and the polymer is also temperature

dependent. Because n0 tends to increase with temperature while nm tends to decrease, it


4 2

is usually not possible to have an exact match over the entire operating temperature range

of the film13.

One of the important parameters associated with a PDLC film is its driving

voltage. For a perfectly spherical droplet, no elastic distortion is required to align a bipolar

droplet with an electric field. In practice the droplets in PDLC materials are never

perfectly spherical and the random orientation of bipolar droplets in a PDLC film is caused

by a distribution in the shapes and orientations of slightly elongated droplets. The driving

voltage is dependent on a variety o f factors, such as dielectric properties, director

configuration, droplet shape etc. We will discuss this in more detail in Chapter 4.

References:

1. J.W. Doane, A. Golemme, J.L. West, J.B. Whitehead, Jr., and B.G. Wu, Mol. Liq.

Cryst. 165 511(1988).

2. Paul S. Drzaic, J. Appl. Phys. 60, 22142(1986).

3. Hilsum, U.K. Patent 1,442,360, July14, 1976.

4. H.G. Craighaed, J. Cheng, and S. Hackwood, Appl. Phys. Lett 40, 22(1982)

5. James L. Fergason, SID Digest of Technical Papers 16, 68(1985)

6. J.L. West, Mol. liq. Cryst. 157, 427(1988).

7. A. Golemme, S.Zumer, D.W. Allender, and J.W. Doane, Phys. Rev. Lett. 61,

2937(1988)


43

8. P.Drzaic, Mol. Cryst. Liq. Cryst 154, 289(1987)

9. J.H.Erdmann, S.Zmer, J.W.Doane, Phys. Rev. Lett. 64, 1907(1990)

10. B. Bahadur, Liquid crystals applications and uses, Vol 1, World Scientific

11. G.P. Montgomery, Jr. and N. A Vaz, Applied Optics 26, 738(1987).

12. AM . Lacker, J.D. Margerum, E. Ramos, S T. Wu and K.C. Lim, Proc. SPEE958,

73(1988)

13. N.A. Vaz and G.P. Montgomery, Jr., J. Appl. Phys. 62, 3161(1987)


CHAPTER THREE

DIELECTRIC SPECTROSCOPY

For a dielectric, one of the most important consequences of the imposition of an

external electric field is induced polarization. Dielectric spectroscopy is based on the

interaction of an electromagnetic field with the electric dipole moments of a material and

is an effective method to study molecular systems. In this chapter, we will review some

basic dielectric properties of materials; basic theory of dielectric relaxation processes of

liquid crystals; and the dielectric response of heterogeneous systems. This will be an

introduction for chapters four and five where we study the dielectric response of some

composite systems containing liquid crystal.

3.1 Dielectrics in an electric field

In non-conducting condensed materials (insulators), the constituent molecules

may have permanent dipole moments on an atomic scale. In addition to permanent dipole

moments, charges can be spatially separated over microscopic distances resulting in

induced dipoles due to the presence of an external electric field. When a material is

brought into an external electric field, for instance between the plates of a capacitor, every

44


45

portion of the material is subjected to an internal field which for the linear dielectrics is

proportional to the external electric field. In a conducting material, charge carriers such as

electrons in metals or ions in the liquid will migrate over large distances (on an atomic

scale); equilibrium is not be reached until the total field strength has become zero at all

points in the material. In the case of insulators (dielectrics), however, only very small

displacements of charges occur. When an electric field is applied, the electric forces acting

upon the charges brings about a small displacement of the electrons relative to the nuclei.

Furthermore electric field tends to orient the permanent dipoles. In both cases the electric

field gives rise to a dipole density; the electric field polarizes the dielectric.

a) Static electric fields:

Dielectrics may be broadly divided into "non-polar material" and "polar material".

In non-polar materials, when the molecules are placed in an external electric field the

positive and negative charges experience electric forces tending to move them apart in the

direction of the external field. The distance is very small (1010 -10 " m) since the

displacement is limited by restoring forces which increase with increasing displacement.

The centers of positive and negative charges no longer coincide and the molecules are said

to be polarized. The dipoles so formed are known as induced dipoles since when the field

is removed the charges resume their normal distribution and the dipoles disappear.


4 6

In polar dielectrics the molecules, which are normally composed of two or more

different atoms, have dipole moments even in the absence of an electric field. Normally

these molecular dipoles are randomly oriented throughout the material owing to thermal

agitation, so that the average dipole moment over any macroscopic volume element is

zero. In the presence of an externally applied field the molecules tend to orient themselves

in the direction o f the field.

In static case, for a linear, isotropic dielectric, the time independent polarization is

related to the electric field by1

P = XE (3.1)

where the % is the dielectric susceptibility which depends on the temperature, pressure,

chemical composition. The polarization P is related to the electric field E and electric

displacement D by

D = e 0E + P (3.2)

Using equation 3.2 may be re-written as

D =eoSrE = eE (3.3)

where s = 1 +x and is called the dielectric constant which provides the link between the

macroscopic and atomic theory of dielectrics. The s0= 8.85 x 10'12 C^/Nm2 is the


4 7

permittivity o f free space. P and E are parallel if the medium is isotropic, i.e, has the same

properties in all directions. For non-isotropic dielectrics e becomes a tensor, in its principal

axis system it can be given in the form:

If we go down further into the basic microscopic concepts of the dielectric theory,

the total polarization has several contributions: electronic, ionic, and orientational

polarizatioa We can write the total dipole moment per molecule by adding the three

polarizabilities2:

This equation is known as the Langevin-Debye formula. a e is the electronic polarizability

and a, the ionic polarizability o f the molecules. a o is the contribution due to orientation

of the molecules to the applied field. For non-interacting dipoles it is a Q=p2 I2KT. p is

average dipole moment of each molecule in the direction of the field.

Another important component of the polarization is interfacial or space charge

polarizatioa This usually arises from the presence of electrons or ions capable of

migrating over distances of macroscopic magnitude. Interfacial polarization is of particular

importance in heterogeneous or multiphase materials. Due to the differences in the

£u 0 0£//= 0 E22 0

0 0 S33(3.4)

P = (a e + a , + a 0)E (3.5)


48

electrical conductivity o f the phases present, charges move through the more conducting

phases and build up on the surfaces that separate them from the more resistive phases.

Effectively it will be apparent as an increase in the average moment of the molecules given

by P = a sE , where the a , is the interfacial or space charge polarizability. Interfacial

polarization is of importance in practical dielectric systems (our PDLC film is one

example). It is also referred to quite often as Maxwell-Wagner polarization.

In summary, the total polarization in any material is made up four components

according to the nature of the charge displaced. The average polarizability per molecule a

is the sum of the individual polarizabilities ( a e + a , + a 0 + a s) .

b) Static dielectric properties of a nematic liquid crystal

A liquid crystal molecule usually has a permanent dipole, which causes the

dielectric properties o f the liquid crystal to be strongly frequency and temperature

dependent. For uniaxial nematic liquid crystals, the dielectric tensor s can be diagonalized

with eignvalues e and e,, where e |; and e refer to the dielectric constants for

polarization parallel and perpendicular to the nematic director n, respectively.

In general, the dielectric tensor can be written asJ

e,; = s 5 , ; + T i< 2 ,; ( 3 . 6 )


4 9

where e = (e | ; + 2e. )/3, t\ = 2(e,: - e , )/S is a material constant and S is the scalar part of

the vector order parameter. In uniaxial nematics, equation (3 .6) becomes

where Ae = Si i - e is dielectric anisotropy and n0 n] are the components of the director n.

Maier and Meier4 extended Onsager’s theory of isotropic dielectrics to the nematic

phase. For a molecule with permanent dipole moment (i inclined at angle 3 with respect to

the long axis, the equations for the principal components of the dielectric permittivity

tensor in the low frequency range are

etJ = Ex5,; + Asn,rij (3.7)

eu = l+(NhF/€.0){a + ^ a aS+ F — - - [ 1 - ( 1 - 3 cos2p)^]} (3.8)

Sx = 1 + (NhF/e0){a - aS + F ~ - ^ . 1 + -̂(1 - 3 co s^ )^ }j — J

(3.9)

where N is the Avogadro number, p the density, M the molecular weight.

h = 3s/(2s+ 1) (3.10)

is the cavity factor and


Perm

ittiv

ities

5 0

iso

T em perature

Figure 3.1 Dielectric constants e and e of nematic as a function of temperature

Schematically.


51

F = 1/(1- a f) (311)

is the reaction field factor for a spherical cavity in an isotropic medium; Both o f them also

are assumed to be remain equal to their isotropic values in Maier and Meier's extension.

where s is the mean dielectric constant, a the mean polarizability, a a the polarizability

anisotropy, A/the molecular weight, S the order parameter. kB is the Boltzmann constant,

and T is the temperature. Since S is temperature dependent, the dielectric components are

temperature dependent. The temperature dependence of s for a nematic liquid crystal with

positive dielectric anisotropy is shown schematically in Figure 3.1.

The low frequency dielectric anisotropy of a molecule is determined by two

factors: 1) the polarizability anisotropy a a from electronic and ionic contributions which

for the elongated molecules o f nematogenic compounds always makes a positive

contribution (i.e, a larger contribution for the measuring field parallel to the long

molecular axis) and 2) the dipole orientational contribution. The sign o f the latter

contribution is positive if the net dipole moment of the molecule makes only a small angle

with its long axis and is negative if the angle is large ("magic angle 3 ~ 57°). We have the

expression from equations (3.8) and (3.9) like

/ = 47tATp(2e - 2)/3M (2s + 1) (3.12)

(3-13)


52

From the Equation 3.13 we can see that the dielectric anisotropy is directly proportional

to the square of dipole moment.

c) Dynamic electric fields:

The dynamic response is most easily studied with the application of a sinusoidally

varying electric field. The time dependence of the electric field strength is then given by5:

E(r) = E ° c o sq ) / ( 3 1 4 )

where E° is the amplitude and co the angular frequency of the sinusoidal variation. For

linear systems the time dependence of the dielectric displacement can also be described as

sinusoidal with frequency o, but with a constant phase difference 5 with respect to the

electric field5:

D(0 = D°cos(° (3 .16)


53

sin6(co) = s"((o)EQfDQ

then we obtain with the help of equations (3.16) and (3.17):

D(/) = e ,(o))E0cosffl/ + e ,/(co)E0sin©/ (3.18)

where s' and s" can be considered as a generalization of the dielectric constant for

sinusoidally varying fields.

When a dielectric material is subject to an alternating field the orientation of

dipoles, and hence the polarization, will tend to reverse every time the polarity of the field

changes. The component e" determines the loss of the energy in the dielectric and is called

the loss factor.

Equations (3.15) and (3.18) can be written in a more compact way by using a

complex notation. In this notation the harmonic field is represented by:

E(/) = E0e,at (3.19)

Similarly the complex dielectric displacement can be written as:

D(f) = D o e " ) (3.20)


5 4

Comparing equations (3.19) and (3.20), we see that the relation between D(t) and

E(t) can be written in the same form as the relation D=sE valid in the static case, by

introducing a complex frequency-dependent dielectric constant e’(ffl):

leading to:

D(/) = e’(oo)E(/) (3.22)

The ratio D°/£° and quantities 5 and e* depend on frequency, but due to the superposition

principle of electrodynamics, are independent o f the amplitude E° of the applied field.

Substituting Equation (3.19) into (3.22) and taking the real part of this equation,

which must be equal to the expression for D(t) given by equation (3.18), we find that the

real and imaginary part of e’ are equal to e' (©) and -e"(

55

3.2 Resonance and relaxation

Dielectric spectroscopy is the study of the dynamic interaction between an external

electromagnetic field and the dipole moments o f a material. The electric permittivity or the

susceptibility is the macroscopic manifestation of the polarization phenomena which take

place on a microscopic or atomic scale and are essentially the formation and reorientation

of dipoles within the dielectric material. Due to the fact that the dipole moments are

coupled to the molecules, the dynamics of the dipole reorientation in an alternating electric

field can provide some information about the individual or collective motion of various

parts of the molecules. For example, dielectric properties give direct information about the

orientation of liquid crystal molecules which have strong permanent and induced dipole

moments. The dipole is the probe by which the molecular level motion can be studied —

dielectric spectroscopy.

There are two different kinds of dielectric behavior in the time dependent regime:

resonance and relaxation. These processes are different from each other by their origins. In

a condensed system relaxation arises when the electric field changes too rapidly for

permanent dipoles to follow the field. When the frequency is higher, the atomic and

electronic dipoles induced by the external field will give rise to resonance.

The permanent dipole moments in molecules arise from the distribution of the

charges in the molecules in the absence of an external field. To a first approximation

molecules can be considered as composed of atoms linked together by bonds between

pairs of atoms6. In a nonmetallic solid, the atoms may be bonded with ionic, covalent, or


56

van der Waal's bonds7. The dipoles may be added vectorially for each molecule or for a

group. In organic molecules most of the atoms are arranged symmetrically, so that most of

bond dipole moments cancel each other. The net dipole of a group is the sum of the

moments which are not compensated and usually a few per group. The appropriate group

is determined by that portion of a molecule which can considered as 'rigid'.

When the permanent dipoles of a dielectric material are subjected to an alternating

electric field, the orientation of the dipoles, and hence the polarization tends to reverse

every time the polarity of the field changes. As long as the frequency remains low typically

below about 1 MHz (although specific materials may have higher frequency limits), the

polarization follows the alternation of the field without any significant lag. In this region

the permittivity of the material remains independent o f the frequency. When the frequency

is increased sufficiently, the permanent dipoles will no longer be able to rotate fast enough

and their oscillations begin to lag behind those of the field and decrease in amplitude.

When the frequency is further increased the dipoles can not follow the field and the

contribution to the dielectric constant from this molecular process becomes vanishingly

small. This usually occurs in frequency range of 10° - 10u Hz.

Relaxation phenomena are associated with the frequency dependence of the

orientation polarization. For frequencies below the infra-red the contribution of the ionic

and electronic polarizations to the total polarization P are independent of the frequency

and may be expressed as1

Pe + P, = P« = E0(£oc - 1)E (3.24)


5 7

where is the relative permittivity at frequencies which are too high for the permanent

dipoles to follow and arises solely therefore from the electronic and ionic polarizations.

Under static conditions

P = P0 + P= (3-25)

so that the orientational polarization PD is given by

P0 = P-Poo = e o ( s - £ c c ) E (3.26)

Now when a static field is applied to the dielectric, according to Debye8, P0 approaches its

final value exponentially so that the orientational polarization at any instant time t after the

application of the field id given by

P0(t) = Pc(l - e _f x) (3.27)

where t is called relaxation time of the dielectric medium; it determines the rate at which

the polarization builds up. x is independent o f the frequency but depends upon the

temperature. In the case of an alternating field E=E0e’'" we have

^ P = i[8 „ (e -8 « )E -P 0«)l (3.28)


58

>

EW

The steady-state solution o f equation 3.22 gives

P*(/) = e-° £ - - —E 1 +y

“cd "I

Z -E z - z

Figure 3.3 Cole-Cole plot for the Cole-Cole equation at a = 0.


61

10

8 - \

e' x , ,

6 - \ a=0.8

a=0.0

4

2

o

0 2 4 6 8 10

io g ( f )

Figure 3.4 Dispersion and loss curves for the Cole-Cole equation at a = 0.8 and 0.0

respectively.


6 2

These are known as the Debye8 relaxation equations. Figure 3.2 displays s' and e" as a

function of frequency for Debye relaxation.

In 1941 K.S. Cole and R.H. Cole1 suggested a graphical representation from which

it is immediately clear whether the experimental points for e'(co) and s"(co) can be described

by a single relaxation time, or if a distribution of relaxation times is necessary. This

representation, generally called the Cole-Cole plot, is obtained by plotting the

experimental values of s"(©) against those o f e'(o>). From the equations (3.33) and (3.34)

we have

(6//)2 = ( s - e /)(e/ -e=c) (3.35)

Figure 3.3 shows that imaginary part of permittivity versus the real part and is called

Cole-Cole plot. In general, for a single relaxation time the Cole-Cole plot is a semi-circle.

If there is a distribution of relaxation times then Cole-Cole plot gives a circular arc with

origin below the real axis.

The behavior of the orientational polarization of most condensed systems in a

time-dependent field can, as a good approximation, be characterized with a distribution of

relaxation times. One the most widely used empirical equation was given by K.S. Cole and

R. H. Cole in 1941'

e * (co) = £« + (e - e=c) ^ (3.36)1 + ( / cot)


6 3

a is the Cole-Cole parameter and it is readily seen that for a=0 this expression reduces to

the equation for a single relaxation time, i.e. equation 3.32. Figure 3.4 shows that the real

and imaginary part o f permittivity from equation (3.36) as a function of frequency with

a=0.8 and a = 0. We can see when a*0 there is a distribution of relaxation times and the

region of relaxation is much broader than for the single relaxation time curve.

More generally, there is a continuous distribution of relaxation time. In the

frequency range corresponding to the characteristic times for the molecular reorientation.

The complex dielectric constant can be written with5

If there is a polarization in the absence of an electric field, due to the occurrence of a field

in the past, the decrease of the orientational polarization is independent o f the history of

the dielectric, and only depends on the value of the polarization at that instant, with which

is proportional. Denoting the proportionality constant by 1/x, since it has the dimension of

a reciprocal time. The distribution function G(lnx) reduces to a delta function. The

G(ln x)d\n x (3.37)

Here G(lnx) is distribution function with:

J G(ln x)d\n x = 1 (3.38)


6 4

complex dielectric constant equation 3.37 will reduce to equation 3.32 which is single

relaxation time process — Debye situation.

3.3 Relaxation processes in liquid crystals

In the Maier-Saupe3 mean field theory, the liquid crystal molecules in the nematic

phase are considered to be long, rigid rod like molecules with strong permanent dipole

moments which make an angle (B with their l

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INFORMATION TO USERS · 2014. 8. 6. · ACKNOWLEGEMENT xii CHAPTER 1. INTRODUCTION: PHYSICS OF LIQUID CRYSTALS 1 1.1 Introduction to liquid crystals 2 1.2 Ferroelectric liquid crystals