UNIVERSITY OF CALIFORNIASanta Barbara

A Study in Pattern Formation:Electroconvection in Nematic Liquid Crystals

A Dissertation submitted in partial satisfactionof the requirements for the degree of

Doctor of Philosophy

in the subject of

Physics

by

Michael Dennin

August, 1995

Committee in charge:

Professor David S. Cannell

Professor Guenter Ahlers

Professor James S. Langer

The dissertation of Michael Denninis approved:

August, 1995

ii

ACKNOWLEDGEMENTS

As with any work that takes six years and over 200 pages to describe, there

are a large number of people who helped make it possible. First, I would like

to thank my advisors for introducing me to this fascinating subject and for

seducing me away from theory into the lab where I have spent many enjoyable

hours (as well as a few frustrating ones). I have had the pleasure of reaping

the benefits of the dedication that David Cannell and Guenter Ahlers have to

teaching and guiding their students. I am sure that the skills I have learned

under their mentorship will serve me well in my future work. I would also like

to thank Jim Langer for his time and effort as a member of my committee.

In the local Santa Barbara area, I need to thank my golf partners Toby

Falk, John Woodward, Art Bailey, and Will Nelson. This thesis may not have

been possible without them getting their acts together and graduating, so

that I could get off the golf course and into the lab. Also, my thanks to Andy

Hays, John, and Mike Warren their companionship and support through the

first year of graduate school was invaluable. As my first guide through the

world of Taylor-Vortex flow, I wish to thank Ning Li for the many hours of

interesting discussions on pattern formation. Of course, I must thank Steve

Trainoff and Art Bailey to whom I owe copious amounts of chocalate for the

patient fielding of my questions. I also wish to thank the community of St.

Mark’s parish for their wonderful support and friendship over the last seven

years.

On the international scene, I am in debt to Ingo Rehberg, Eberhard Bo-

denschatz, Stephen Morris and Barbara Frisken for the expertise they shared

on the ins and outs of nematic liquid crystals. Without their knowledge and

experience in the “Black Arts” of liquid crystal cell building these experiments

would not have been possible. Also, in particular, I wish to thank Ingo for his

patience in passing on his knowledge of electroconvection experiments. My

conversations with Lorenz Kramer, Werner Pesch, and Helmut Brand con-

tributed greatly to my understanding of the theoretical background of elec-

iii

troconvection and pattern formation, in general. Especially interesting were

my interactions with Hermann Reicke and Martin Treiber, the two theorists

to whom my experiments have so far most directly related.

Finally, I must thank my family, not only will they kill me if I fail to

mention them, but they truly have been a great support to me. I love them

all dearly and could not have done this without them. Just for the record,

that would include Mom, Dad, Elizabeth, Pete, David, Happy Cat, and, most

recently, Chris. And, of course, I thank my wife, Jeni. She is the one who had

to deal with most of my stress and worries over this piece of work. Her patience

and understanding are without equal. Not, only do I thank her for herself, but

in marrying her, I gained the wonderful support, love, and encouragement of

her family as a welcome bonus.

iv

VITA

Date of Birth: December 10, 1966

1988–A. B., Princeton University.

1988-89–Research Assistant, Department of Physics, Princeton University.

1989-91–Teaching Assistant, Department of Physics, University of California,

Santa Barbara.

1991-94–Graduate Student Researcher, Department of Physics, University of

California, Santa Barbara.

1995–Lecturer, Department of Physics, University of California,

Santa Barbara.

PUBLICATIONS

1. Origin of the Hopf bifurcation in electroconvection M. Dennin,

M. Treiber, L. Kramer, G. Ahlers, D.S. Cannell, Phy. Rev. Lett., sub-

mitted.

2. Patterns in Electroconvection in the Nematic Liquid Crystal I52 M. Den-

nin, D.S. Cannell, G. Ahlers, Mol. Cryst. Liq. Cryst. 261, 377 (1995).

3. Measurement of Material Parameters of the Nematic Liquid Crystal I52

M. Dennin, G. Ahlers, D. S. Cannell, in Spatio-Temporal Patterns, ed.

P. Palffy-Muhoray and P. Cladis (Addison-Wesley, 1994) p343.

4. Measurement of a Short-Wavelength Instability in Taylor Vortex Flow

M. Dennin, D. S. Cannell, G. Ahlers, Physical Review E, 49, 462 (1994).

v

ABSTRACT

A Study in Pattern Formation:

Electroconvection in Nematic Liquid Crystals

Michael Dennin

I have studied fundamental issues in pattern formation using electroconvec-

tion in the nematic liquid crystal I52. An electroconvection cell consists of a

nematic liquid crystal, which is doped with ionic impurities, confined between

two glass plates which have transparent electrodes on their inner surfaces.

The electrodes are used to apply an ac voltage perpendicular to the plates.

By appropriate surface treatment of the glass plates, the average molecular

alignment (the director) is made to be parallel to the plates. The nematic

liquid crystals which are used for electroconvection typically have a negative

dielectric anisotropy. As part of this thesis, I will provide a detailed prescrip-

tion for the making of electroconvection cells.

Above a critical applied voltage Vc and below a critical applied frequency

(the Lifshitz point), there is an instability to a spatially varying state with a

wavevector at a nonzero angle Θ or π−Θ to the director (oblique rolls, the two

states are degenerate). The state can either be traveling (Hopf bifurcation) or

stationary. The experimental results are divided into three sections.

The first section presents experimental measurements of the critical volt-

age, the angle between the director and the wavevector of the pattern, and

the traveling frequency of the pattern. Comparison of the experimental obser-

vations with a detailed linear stability analysis carried out by Martin Treiber

and Lorenz Kramer is made, and the agreement between the theory and ex-

periment is found to be good.

The second section presents observations of states of spatio-temporal chaos

which occur as the initial state above Vc. I observe both states which are

extended in space and localized states. These states involve the interaction of

four modes: right- and left-traveling oblique rolls.

The third section summarizes the wide variety of patterns which are ob-

vi

served well above Vc. I report on the existence of the observed patterns as

a function of conductivity of the cell, applied voltage, and applied frequency.

Also, I discuss the nature of the transitions between the different patterns.

vii

Contents

1 Introduction 1

2 Theory 12

2.1 Theoretical Tools . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Nematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Electroconvection . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Experimental Details 41

3.1 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.1 Shadowgraph . . . . . . . . . . . . . . . . . . . . . . . 42

3.1.2 Temperature Control . . . . . . . . . . . . . . . . . . . 53

3.1.3 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2 Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.3 Cell Construction . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.3.1 Glass Preparation . . . . . . . . . . . . . . . . . . . . . 73

3.3.2 Alignment . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.3.3 Sealing . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.3.4 Filling . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4 Experimental Results 98

4.1 General Methods . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.2 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

viii

4.3 Linear Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.4 Weakly Non-linear Regime . . . . . . . . . . . . . . . . . . . . 117

4.5 Nonlinear Results . . . . . . . . . . . . . . . . . . . . . . . . . 132

5 Open Issues and Future Directions 148

A Material Parameters 158

B Tables of data 171

C C-Code 197

D References 213

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

ix

Chapter 1

Introduction

One of the most challenging areas in the field of physics is the study of non-

linear, dissipative systems which are driven out of thermodynamic equilibrium

by an external stress. Particularly dramatic and fascinating is the sponta-

neous appearance of spatial and temporal structures, or patterns, at a critical

value Rc of the applied stress, or control parameter R. Pattern formation,

as this process is referred to, occurs in systems ranging in length scales from

magnetic domains in garnet crystals to the distribution of galaxies and stars,

and includes such diverse systems as the stripes and spots on animal coats,

population distributions, cloud streets, and crystal formation. Given that

patterns are ubiquitous in nature, the search for an underlying “dynamics of

patterns” which can be applied to systems with different microscopic details

is almost unavoidable. While this goal has certainly not been achieved, sig-

nificant progress has been made in the understanding of patterns through a

combination of theoretical developments and precision experiments1.

One way to appreciate the difficulties which arise in formulating a theoreti-

cal description of patterns is to briefly consider two ideas which are fundamen-

tal to the study of linear systems and thermodynamic equilibrium. We know

1A recent review of the field of pattern formation is provided in Ref. [1]. For a more

complete list of references on the phenomena discussed here, see the references in Ref. [1]

1

that any solution Ψ of a linear system can be written as a linear combination

of the eigenfunctions Fn with amplitudes an, i.e. Ψ =∑

nanFn. The Fn are

often referred to as the normal modes, or simply modes, of the system. By

definition, the eigenfunctions are independent, and their evolution is not de-

pendent on the amplitudes of the other modes (the principle of superposition).

One of the most common realizations of this idea is the use of Fourier modes,

or plane wave solutions, for many problems in electrodynamics and quantum

mechanics.

The study of systems in thermodynamic equilibrium is simplified in many

cases by the existence of a free energy. The state of the system is specified

uniquely by the minimum of its free energy. These ideas of superposition and

minimizing a free energy are essential elements of our intuitive understanding

of linear and equilibrium systems.

In contrast, spatially extended pattern forming systems are described by

nonlinear partial differential equations (PDE) where the inherent nonlinearity

destroys the principle of superposition. Even so, it is often useful to treat

solutions of these nonlinear systems as a superposition of individual modes.

For example, the eigenfunctions of a linearized version of the equations are

often used to express the solution of the full equations. Unlike in the linear

case, the evolution of each mode depends on the amplitudes of the other modes

through the nonlinear couplings. This nonlinear evolution plays an essential

role in the rich variety of pattern dynamics which are observed. For some

special pattern forming systems, which are referred to as potential- or gradient-

flow, a generalized potential which is the equivalent of a free energy exists. For

these situations, the description of the dynamics is simplified because the state

of the system is specified by the minimum of the potential. However, pattern

forming systems are inherently nonequilibrium, and it can not be assumed that

a generalized potential exists. In fact, well known cases exist for which it is

well established that no such potential exists. The absence of a potential often

results in multiple states which are equally stable, and/or a strong history

2

dependence for the state of the system. The lack of superposition and general

potentials means that much of our physical intuition is often wrong when

applied to pattern formation.

Despite the difficulties inherent in a theoretical study of patterns, a number

of tools in the field of nonlinear dynamics have been developed which enable

quantitative study of these systems[1]. First, the increase in computing power

has made available numerical studies that just weren’t possible in the past.

In addition, a number of analytic tools have also proved useful, among these

are linear stability analysis, amplitude equations[2], phase equations[3], and

simple model equations. Linear stability analysis, as the name implies, is the

analysis of a linearized version of the underlying PDEs. The goal of linear

stability analysis is the study of the onset of patterns or other instabilities as

one or more control parameters are varied. Amplitude equations are used to

study the nonlinear interactions of the amplitudes of the relevant modes of

the system. They can be derived as a perturbation of the full equations of

motion near the onset of a pattern where the amplitudes are small. Phase

equations consider the dynamics of large scale modulations of the wavevector

of the pattern. Model equations are useful for numerical simulations and for

separating the details of a system. For a review of these methods, see Ref. [1]

and the references therein. I will briefly discuss linear stability analysis and

amplitude equations in Sec. 2.1.

Essential to the development of these analytic tools has been the ability to

perform precision, quantitative experiments in systems for which the funda-

mental equations are well known. Precision experiments have both guided the

theoretical formulation, as well as, provided quantitative tests of the theory.

The solutions of the nonlinear PDEs which describe pattern forming systems

depend on the boundary conditions. So, for quantitative tests of theoretically

predicted solutions to be relevant, the experiments must maintain precise con-

trol over the boundaries. In addition, it is essentially impossible to find analytic

solutions to PDEs except in very special cases. Precision experiments provide

3

a means of “solving” these equations over a wide parameter range by simply

observing the dynamics of the patterns. Without control over the properties

of the system, it is often difficult to distinguish dynamics which are related

to the fundamental equations of the system from dynamics which are the re-

sult of “dirt” or poorly understood experimental conditions. Of course, to

even consider a comparison of the experimental “solutions” to a given theory,

knowledge of the fundamental equations for the system is needed.

Because they meet the above criteria, fluid dynamical systems have played

a large role in the advancement of our understanding of patterns[4]. The

basic equations (Navier-stokes and additional conservation laws) are well es-

tablished. The long history of experimental fluid dynamics has resulted in

systems for which the boundary conditions and fluid properties are well de-

fined and precisely controlled[5]. A number of quantitative measurement tech-

niques have been developed. Another advantage of fluid dynamical systems is

the natural existence of dimensionless control parameters. This allows experi-

mental access to a wide parameter range by the use of appropriate geometries

and/or fluids. Two classic examples of pattern forming systems for which pre-

cision experiments have been achieved are Taylor Vortex Flow (TVF)[6] and

Rayleigh Bernard convection (RBC)[7].

TVF occurs in a fluid which is confined between two concentric cylinders

with one or both of the cylinders rotating. I will consider the case where the

outer cylinder is fixed, and the inner cylinder is rotating. The fundamental

equation is the Navier-Stokes equation, and the dimensionless control param-

eter is the Taylor number T = [2r2i d

3/(ri + r2)](Ω/ν)2 where ri (ro) is the

inner (outer) cylinder radius, d = ro − ri, Ω is the rotation rate of the inner

cylinder, and ν is the kinematic viscosity. Often the applied stress is described

in terms of the Reynolds number R = (ro − ri)riΩ/ν because it is linear in

the experimentally controlled parameter Ω. Below the critical value Rc of the

Reynolds number, the flow is laminar and axially uniform. For a fixed geom-

etry and ν, Rc sets the critical rotation rate Ωc. Reasonable values of Ωc are

4

achieved by adjusting the geometry and viscosity of the fluid. Above Ωc, pairs

of counter-rotating rolls form along the axis of the cylinder. The resulting

pattern looks like donuts which are stacked along the inner cylinder. This is

an example of an effectively one dimensional system for which the pattern is

characterized by its wavenumber. It is possible to make TVF apparatus[8, 9]

with the spacing between the cylinders constant to better than 0.1% and the

cylinders concentric to 0.1%. To minimize variations of the viscosity, the tem-

perature of the fluid is kept constant to a few mK. Taking into account the

various effects, variations of Rc are on the order of 0.1%.

In RBC, the fluid is confined between two parallel plates and is heated from

below. The fundamental equations are Navier-Stokes for the velocity field,

and equations for mass conservation and energy conservation. The control

parameter is given by the Rayleigh number Ra = gα∆Td3/(νκ). Here g

is the acceleration due to gravity, α is the thermal expansion coefficient, d

the distance between the plates, κ is the thermal diffusivity, and ∆T the

temperature difference across the plates. For a given system, the geometrical

factors and fluid properties are usually held constant, and ∆T is varied. Below

the critical temperature difference ∆Tc, heat is transported by conduction,

and the system is spatially uniform. Above ∆Tc, the fluid begins to flow,

and convection rolls form which have a width roughly equal to the distance

between the plates. The patterns in RBC are intrinsically two dimensional,

and for simple periodic patterns, the pattern is described by a wavevector (or

wavevectors) which lies in a plane parallel to the plates. Depending on the

details of the system, there is a great variety of possible patterns including

straight rolls, hexagons[10], squares[11], and spirals[12].

The precision of RBC experiments is quite impressive[13]. The top and bot-

tom plate can be made parallel to 0.5 µm. The lateral temperature variations

along the top or bottom plate are typically a few mK while the temperature

difference between the plates is of the order of a few degrees and is constant

to ±0.1 mK. Combining the various effects, the convection apparatus in the

5

Santa Barbara group[13] have resolutions in the reduced control parameter

ε = (R−Rc)/Rc of 10−3 to 10−5.

A number of variations on RBC and TVF have been studied. RBC in one

dimension is studied by confining the fluid to a narrow channel. RBC[14] in one

dimension and TVF[15] have both been studied in the presence of an imposed

flow. RBC in binary fluid mixtures[16, 17, 18] adds the concentration of one

of the components of the fluid mixture as an additional variable. For binary

fluid convection, the separation ratio Ψ is a measure of the response of the con-

centration field to a temperature gradient. For negative Ψ, the concentration

behaves so as to stabilize the system against convection; whereas, for positive

Ψ, convection occurs at a lower value of Ra than for a pure fluid. Finally, the

effects of rotation about a vertical axis have been studied for RBC[19], and

about an axis perpendicular to the cylinder axis for TVF[20].

There are, of course, many other examples of experiments in pattern for-

mation, but I mention these two here to emphasize the level of precision which

can be achieved in these systems. For this thesis, the focus is on a third

fluid dynamical system: electroconvection (EC)[21] in nematic liquid crystal

(NLC),[22]. EC in NLC is a paradigm for pattern formation in an anisotropic

medium. NLC molecules have an inherent orientational order, but no posi-

tional order. The direction of the average molecular alignment is given by the

director, which is a function of space and time. A NLC cell consists of the

NLC confined between two glass plates which have been properly treated so

as to produce spatially uniform alignment of the director which is parallel to

the plates (planar alignment). Transparent conductors are evaporated onto

the glass plates, and an ac voltage is applied perpendicular to the plates. The

cells and their construction are described in detail in Sec. 3.3.

For EC to occur, the NLC must be doped with ionic impurities, and the

anisotropy in the dielectric constant εa = ε‖ − ε⊥ must be negative or only

slightly positive. Here ε‖ (ε⊥) is the dielectric constant for electric fields par-

allel (perpendicular) to the director. Above a critical value Vc of the applied

6

ac voltage, a pattern forms which consists of fluid flow in the form of con-

vection rolls and a corresponding periodic spatial variation of the director. A

great variety of spatio-temporal structures is observed, including rolls[23, 24],

traveling waves[24, 25, 26, 27, 28], defect chaos[26, 29], and chaos at onset[28].

The details of the instability mechanism are discussed in Sec. 2.3.

Another paradigm for pattern formation in an anisotropic medium is RBC

in NLC with planar alignment[30]. For both RBC in NLC with planar align-

ment and EC, the inherent anisotropy introduces a new class of patterns:

oblique rolls. Consider a pattern which consists of a set of parallel, straight

rolls with a wavenumber q. In an isotropic medium the roll axis is equally

likely to orient in any direction. The selection of a particular direction by the

roll pattern breaks the underlying symmetry of the system. For a NLC sample

with planar alignment, the director has already defined a direction. Therefore,

a given periodic roll state has a well defined wavevector with components q and

p parallel and perpendicular to the director, respectively. States with p = 0

have the roll axis perpendicular, or normal, to the director and are referred to

as normal rolls. States with p 6= 0 are referred to as oblique rolls and have a

nonzero angle between their wavevector and the director. As a control param-

eter is varied, a point where the system goes from having p = 0 to p 6= 0 is

called a Lifshitz point[31]. Because the director is not a vector, the two states

whose wavevectors have the same magnitude and form an angle θ and π − θ

with respect to the director are degenerate. These states are referred to as

the zag and zig oblique roll states. As I will show, the interaction of the two

degenerate zig and zag roll states leads to a number of interesting patterns.

The inherent anisotropy of these systems results in a greatly increased

complexity of the fundamental equations. A theoretical description of RBC

convection in NLC requires the usual equations for RBC with additional equa-

tions describing the director field. A description of EC requires equations for

the velocity field, the director field, the charge density, and the electric fields.

The anisotropic nature of the NLC translates into material parameters which

7

are tensor quantities. The thermal conductivity, electrical conductivity, dielec-

tric constant, index of refraction, and viscosity all are tensor quantities. The

director has elastic properties which introduce three elastic constants. This

list is provided here to highlight the complexity of the systems. The details of

these tensor quantities and NLC are discussed in Chapter 2.

This increased complexity serves as an ideal test of the tools developed

in the study of RBC and TVF. While some discrepancies still exist, the

quantitative agreement between the predictions for RBC in NLC and the

experimental studies is excellent[30, 32]. EC has proven to be a more dif-

ficult system, but the agreement between theory and experiment has been

quite striking. A detailed linear stability analysis[33, 34] of the model for

EC originally introduced by Helfrich[35] and to which a number of authors

have contributed (see Ref. [36] and the references therein) quantitatively pre-

dicts both Vc and the initial wavevector of the pattern. However, this model

fails to predict the traveling-roll states (Hopf bifurcation) which are observed

experimentally[24, 25, 26, 27, 28] and the experimentally observed backward

bifurcation[37], i.e. discontinuous transition.

The discrepancies between the theoretical predictions and the experimental

observations provided one of three reasons for studying EC in new NLC’s . A

number of possible reasons for the discrepancy had been suggested, including

imperfect director alignment and conductivity effects[33, 34, 38]. I surveyed

EC in a number of new NLC’s with the goal of studying these proposed effects.

As I will report (Sec. 4.2), the standard treatment of the conductivity of the

EC cells as a frequency independent parameter was found to be inadequate.

The recently introduced weak-electrolyte model (WEM)[39] extended the pre-

vious theoretical work by including the dissociation-recombination of the ionic

impurities and the effects of this on the conductivity. This model provides the

first detailed description of the Hopf bifurcation. Section 2.3 is a review of this

model, and Sec. 4.3 presents the experimental evidence for the WEM. This is

an example of the close interaction between experiment and theory which is

8

still an essential element of the study of patterns.

The second motivation for studying new NLC’s was provided by experi-

ments in EC which were the first direct observations of the effects of thermal

noise below Rc[37]. Even though fluctuations have a negative growth rate be-

low Rc, it is possible to observe fluctuating patterns for R < Rc which are

generated by a noise source which couples in to the system with the right

frequency and wavelength. Even in the absence of experimental noise sources,

thermal noise is always present in any system, and because it is white noise,

always has components which can couple to the system. Generally, the effects

of thermal noise are too small to be observed directly in RBC and TVF, but

recent work in RBC in compressed gases has also been able to observe fluctuat-

ing patterns driven by thermal noise[40]. For both of these systems, the initial

transition is backward, i.e. to a finite amplitude. One would like to study the

effects of noise in a system with a forward bifurcation where the amplitude of

the pattern grows continuously from zero at onset. Given that the detailed

calculations[33, 34] for EC predicted a forward bifurcation, it was hoped that

the right NLC could be found for which the transition was forward. Then, it

would be possible to study the effects of noise on a forward bifurcation.

Most of the drawbacks of EC are related to the difficulties involved in mak-

ing EC cells, and these difficulties provided the final motivation for studying

new materials. Precision experiments in EC are often hampered by poor align-

ment of the director and chemistry, in particular electrochemistry, in the cell

which results in unwanted impurities. These difficulties, along with suggested

methods for dealing with them, are discussed in detail in Sec. 3.3. The other

difficulty with EC as a model system is the lack of detailed knowledge of the

material parameters. As outlined in Sec. 2.3, quantitative comparison with

theory requires the knowledge of at least 14 material parameters. There is one

single-component NLC, MBBA (see Sec. 3.2 for details), for which εa < 0 and

for which the necessary parameters have been measured. Unfortunately, this

material is also highly unstable making it less than ideal for precision experi-

9

ments. Therefore, the third and final reason for studying EC in new materials:

finding a suitable replacement for MBBA for quantitative experiments.

The lack of knowledge regarding the material parameters severely limited

the usefulness of most of the new materials I was able to obtain. However,

one material, I52 (see Sec. 3.2 for details), is being marketed as a benchmark

material[41], and as such, a number of material parameters had been measured.

The work on I52 has proven useful for comparisons to the WEM, and has

revealed a wealth of interesting pattern dynamics. In addition, I extended the

knowledge of the material parameters by studying the Frederiks transition in

I52 which is discussed in Appendix A. As such, the results from EC in I52 are

the focus of this thesis.

I studied EC in I52 as a function of the applied voltage, the applied fre-

quency, and the sample conductivity. A rich variety of patterns with many

intriguing aspects (see Secs. 4.4 and 4.5) were observed. Particularly interest-

ing are the states of spatio-temporal chaos and localized states observed near

onset (Sec. 4.4). Neither of these classes of behavior occur in linear systems

which are in thermodynamic equilibrium. Nor do they occur in systems with a

generalized potential. Spatio-temporal chaos refers to variations of the pattern

that are aperiodic in time and space and results from the inherent nonlinear-

ities in the system[1]. I will report here on a system which has a continuous

oscillatory instability leading directly to spatio-temporal chaos which in princi-

ple can be compared quantitatively to coupled complex Ginzburg Landau am-

plitude equations[42]. This has not been possible in other chaotic systems[1]

because the primary bifurcation is backwards, because chaos occurs only after

secondary bifurcations, or because the system size accessible to experiments

is too small.

Localized states consist of regions of the system which exhibit a pattern

coexisting with regions of the uniform base state. This is not possible in a

system governed by a generalized potential (e.g. the free energy of an equi-

librium system), except at very special points in parameter space where there

10

are two equal minima of the potential. I will show that the localized states

in this system are stable over a wide range of parameters. Localized states in

one dimension have been studied extensively in the context of RBC in binary

fluids[43, 44, 45, 46, 47, 48, 49] but stable localized states in two dimensions

have not been observed[18]. The confirmation of the linear predictions of the

WEM [39](see Sec. 4.3) presents the unique possibility of a quantitative theo-

retical description of these states in terms of coupled amplitude equations (see

discussion in Chapter 5).

Finally, two general advantages of EC are the cell size and the intrinsic

time scales. The EC cells have thicknesses d in the range of 10 µm to 100 µm.

This makes it possible to study EC systems with aspect ratios l/d, where l is a

typical lateral dimension, on the order of 1000. By comparison, the limitations

of RBC and TVF systems are such that the aspect ratios are typically of the

order 10 to 20, and 100 is considered extremely large. The large aspect ratios

of EC are significantly closer to the theoretical idealization of a system of

infinite lateral extent[1]. In addition to achieving extremely large aspect ratios,

the ability to control the size of the electrodes allows one to make systems

of extremely small spatial extent, and to make effectively one dimensional

systems. This control over the system size is potentially useful for the study

of spatio-temporal chaos2.

The combination of thin cells and an electrical driving force results in

intrinsic time scales for EC which are typically of the order 10−3 to 10−1 s (see

Sec. 2.3). This is significantly faster than the typical RBC experiment which

has time scales of the order of minutes. Many patterns display interesting

temporal, as well as spatial, behavior. For these patterns, the relatively short

times scales of EC can simplify the measurement of the temporal dynamics of

these patterns by shortening the time required to compile reasonable statistics.

2For a nice discussion of the role of system size in studies of spatio-temporal chaos, see

Ref [1]. The possible uses for this particular system are discussed in Chapter 5.

11

Chapter 2

Theory

Section 4.3 presents a quantitative comparison between the experimental mea-

surements of the Hopf frequency and the predictions resulting from a linear

stability analysis of the weak electrolyte model (WEM)[39]. In principle, it

is possible to derive amplitude equations from the WEM which can be quan-

titatively compared with the results of Sec. 4.4. In addition, the design of

the apparatus is dependent on a number of the physical properties of nematic

liquid crystals. This chapter provides the necessary background for both an

understanding of the WEM and the experimental details.

Section 2.1 is an introduction to the techniques of linear stability analysis

and amplitude equations. Section 2.2 is a review of the physical properties of

nematic liquid crystals which are relevant to the design of the apparatus and

the models of EC. Because the WEM is an extension of the model of EC first

introduced by Helfrich[35], Sec. 2.3 has a discussion of the Carr-Helfrich model

of EC[36], and it concludes with a discussion of the changes incorporated into

the WEM[39]. None of the theoretical work described in this section is the

result of my own work. The material in Sec. 2.2 is based mainly on Ref. [1]

and the references therein. The discussion of NLC properties is based on

Ref. [22]. The description of the WEM in Sec. 2.3 is a result of extensive

discussions with Martin Treiber, and the actual theoretical work was carried

12

out by Martin Treiber and Lorenz Kramer[39].

I will denote vectors in bold-face b. When not otherwise specified, x

refers to the position vector in three dimensions with coordinates xi given

by (x, y, z), and the components bi of a general vector will refer to the same

Cartesian coordinate system. With this notation, ∂j = ∂/∂xj . For vector

expressions written in components, I will use the Einstein summation conven-

tion, the Kronecker delta function δij and the fully anti-symmetric tensor εijk

for writing cross products. Because of the central role of the dielectric tensor

εij, there is some chance of confusion. However, if it is not clear from context,

the number of indices distinguishes the dielectric tensor from εijk.

2.1 Theoretical Tools

In general, analytic solutions to nonlinear partial differential equations (PDE)

are rare. Therefore, in order to understand the behavior involved in pattern

formation, a number of methods have been developed. One option is to con-

sider a limit for which the solutions can be calculated perturbatively. Linear

stability analysis and amplitude equations are two such techniques. In linear

stability analysis, infinitesimal perturbations to a known solution of the full

nonlinear equations are considered. The full equations are expanded in terms

of the perturbations, and only terms linear in the perturbation are considered.

The stability of the known solution is determined by the growth (unstable)

or decay (stable) of the perturbations. Amplitude equations represent a per-

turbative description of the dynamics near a linear instability. This section

provides a definition of the essential elements of linear stability analysis and

amplitude equations used in the rest of the thesis and follows the development

of Ref. [1]. For a more detailed description, see Ref. [1] and the references

therein.

I will consider a general system of nonlinear (PDE) for the state U of the

13

system given by:

∂tU = G[U, ∂xU, ..., R]. (2.1)

Here R is the control parameter for the system. For EC, U includes the veloc-

ity field, the director field, and the ionic impurity concentrations, and G[U]

includes the Navier-Stokes equation, the conservation of angular momentum,

the equations of electrostatics, and charge conservation. For this case, R is

the square of the applied voltage. It is assumed that the spatially uniform

base state U0 is known and that U0 is stable for 0 ≤ R < Rc. Here Rc is the

critical value of R for which U0 first becomes unstable. For example, in EC

this state consists of v = 0 and planar alignment of the director.

To determine Rc, the state U = U0 + δU is considered where δU is an

infinitesimal perturbation. For simplicity, I will consider the case of pattern

formation in one dimension and U0 = 0. The system is assumed to be spa-

tially infinite so that the perturbation can be expanded in Fourier modes δU =∑

u(o)qi

exp(iqix+λit). If we consider an individual Fourier mode with wavenum-

ber q, the perturbed solution becomes U = U0 + δU = u(o) exp(iqx+λt) with

u(o) infinitesimal. This solution is plugged into Eq. 2.1, and the result is ex-

panded to first order in u(o),

∂t[u(o) exp(iqx+ λt)] = D[u(o) exp(iqx+ λt)], (2.2)

where D is the linear operator which results from expanding Eq. 2.1. Then,

Equation 2.2 is solved for the eigenvalues λα(q, R) of D.

The linear stability of Uo is entirely determined by the largest λα, denoted

as λ(q, R). If for all q and a given value of R, Reλ(q, R) < 0 (Re denotes

the real part), then all perturbations u(o) exp(iqx + λt) decay exponentially

in time. (Because λ(q, R) is the largest eigenvalue, if the above statement

is true for λ(q, R), then it must be true for all λα(q, R).) Because this is a

linear analysis, this situation corresponds to the state U0 being linearly stable

against all perturbation of the form δU∑

u(o)qi

exp(iqix+λit). If for some q and

R, Reλ(q, R) > 0, then the corresponding perturbation grows exponentially,

14

and U0 is said to be linearly unstable to perturbations with a wavenumber q.

These perturbations are often referred to as the unstable modes. As defined

above, Rc corresponds to the lowest value of R for which there is some value

of q ≡ qc such that Reλ(qc, Rc) = 0.

It should be emphasized that there is nothing special about the assumptions

of U0 = 0 and one dimension. In fact, the system doesn’t even have to be

a “pattern forming” system. Linear stability analysis is applicable to the

stability of any known solution of a nonlinear set of equations. The procedure

outlined above is still followed, only now U0 6= 0 is carried along as part of the

calculation. The defining characteristics are the assumption of infinitesimal

perturbations and the resulting linearization of the equations. Another option

is to do nonlinear stability analysis which considers the stability of the known

solution to finite perturbations.

The generalization to two-dimensional instabilities with a wavevector q is

also straightforward. An additional feature is that there are often two or more

degenerate wavevectors for which Reλ(qc, Rc) first becomes zero. For example,

in the case of RBC in an isotropic fluid, there is an infinite degeneracy as the

wavevectors with magnitude |qc| and arbitrary orientation are equivalent. For

EC, the two wavevectors which form an angle θ and π− θ with respect to the

director are degenerate.

Introducing the reduced control parameter

ε = (R− Rc)/Rc, (2.3)

the behavior of λ(q) for three different values of ε is plotted in Fig. 2.1. This

shows the three possibilities as a function of ε. For ε < 0, Reλ(q, ε) < 0 for all

q, and the uniform state is stable. At ε = 0, the uniform state becomes unstable

to a mode with wavenumber qc. For ε > 0, there is a band of wavenumbers

against which the pattern is unstable. Reference [1] discusses three types of

transition based on the value of qc. I will only consider the type I transitions

for which qc 6= 0. The imaginary part of λ (Im λ) determines if the transition

15

is stationary, Imλ = 0, or oscillatory, Imλ 6= 0. (An oscillatory transition is

often referred to as a Hopf bifurcation. Bifurcations and bifurcation theory

will be considered briefly in the discussion of amplitude equations.) In the

case of the oscillatory transition, the Imλ is referred to as the Hopf frequency

and denoted by ω.

q (arb. units)-2

-1

0

1

grow

th ra

te (a

rb. u

nits

) ε > 0

ε = 0

ε < 0

Figure 2.1: The figure illustrates the three possibilities for λ(q, ε). The dottedline is for ε > 0 for which there is a range of q such that λ(q, ε) > 0. For ε = 0(dashed line), a single q value, qc, exists such that λ(qc, ε = 0) = 0. For ε < 0(dashed dotted line), all q have λ(q, ε) < 0.

Another way to view the results of the linear stability analysis is the

marginal stability curve defined by Reλ(q, ε) = 0. An example of the marginal

stability curve for TVF (a one dimensional case)[50] is shown in Fig. 2.2. The

minimum of the marginal stability curve is the point (qc, Rc). For a given

value of ε, the uniform state is stable against perturbations with wavenumbers

q that are located in regions outside the curve; whereas, perturbations with

wavenumbers q located inside the curve grow exponentially. The exponential

growth of the unstable modes can not continue forever, as the nonlinearities re-

16

sult in saturation of the growth and interactions between the unstable modes.

Amplitude equations are one method used to describe the dynamics of the sys-

tem near the point (qc, Rc) where ε is small. This is referred to as the weakly

nonlinear regime, or weakly nonlinear analysis.

-0.4 -0.2 0 0.2 0.4reduced wavenumber (q - qc) / qc

0

0.1

0.2

0.3

0.4ε 1

2 2

Figure 2.2: The solid line represents the marginal stability curve calculatedfor TVF for a system with a ratio of the inner to outer cylinder radius of0.5. The uniform state is stable against perturbations outside the marginalstability curve (region 2). Perturbations inside the marginal stability curve(region 1) grow exponentially.

For a stationary transition, the simplest saturated state U(x, t) is a periodic

state with a single wavenumber qo and a uniform amplitude ao, U(x, t) =

ψ(y, z)ao exp(iqox). I am still assuming U0 = 0 and treating the case of

one dimensional patterns. Here ψ(y, z) is the eigenvector of the linearized

equations and contains the dependence of the solution on y and z. For a Hopf

bifurcation, the simplest state is a traveling wave of uniform amplitude with a

frequency ω, U(x, t) = ψ(y, z)ao exp[i(qox− ωt)]. Generally, the amplitude of

the qo mode has a slow spatial and temporal modulation due to interactions

with the band of unstable modes. For the stationary transition, one writes

17

this state of the system as

U(x, t) = [ψ(y, z)A(x, t)eiqox + cc] +O(ε), (2.4)

where cc refers to complex conjugate. The amplitude A(x, t) is assumed to be

complex with A(x, t) = |A(x, t)| exp[iφ(x, t)]. A constant φ(x, t) corresponds

to a spatial translation of the state, and ∂φ(x, t)/∂x gives the local variation

of the wavenumber away from qo.

The equation governing the dynamics of A(x, t) is referred to as an am-

plitude equation. There are a number of quantitative methods for deriv-

ing amplitude equations from the full equations of motion, but one of the

strengths of amplitude equations is that the form of the equation can be writ-

ten down based on the symmetries of the problem. For the one dimensional

case considered here, when the system is invariant under spatial translations

and A(x, t) → −A(x, t), the amplitude equation to lowest nonlinear order is

τo∂tA(x, t) = εA(x, t) + ξ2o∂

2xA(x, t) − g|A(x, t)|2A(x, t). (2.5)

The details of the full equations are contained entirely in the constants τo,

ξo, qo, and g. As mentioned, there are methods for deriving the coefficients,

but in the absence of known fundamental equations, they can be measured

experimentally.

The amplitude equations provide a natural connection to bifurcation theory

which is worth briefly discussing as the terms forward and backward bifurca-

tion are prevelant in this thesis. For this discussion, I will consider the case

where A(x, t) = A(t) is spatially uniform and real. Equation 2.5 reduces to

τo∂tA(t) = εA(t) − gA(t)3. (2.6)

In Figs. 2.3a and 2.3b, the steady state (∂tA(t) = 0) solutions for A are

plotted as a function of ε for the two cases of g > 0 and g < 0. The solid lines

represent a stable solution and the dashed lines represent an unstable solution.

In the language of bifurcation theory both of these transitions are considered

18

(a) (b)

(c) (d)

ampl

itude

(arb

. uni

ts)

0 0

0

0

ε 0 0

0 0

ampl

itude

(arb

. uni

ts)

ε

ampl

itude

(arb

. uni

ts)

ε

ampl

itude

(arb

. uni

ts)

ε

Figure 2.3: The steady state solutions of A as a function of ε for 4 standardbifurcations. The solid lines represent stable solutions, and the dashed linesrepresent unstable solutions. Notice, at ε = 0, the solution A = 0 (whichis the uniform state) always becomes unstable. (a) is a forward pitchforkbifurcation. (b) is a backward pitchfork bifurcation. (c) shows the backwardpitchfork bifurcation with a stabilizing quintic term added to Eq. 2.6. (d) isa backward transcritical bifurcation corresponding to a quadratic term beingadded to Eq. 2.6. For both (c) and (d), the dashed-dotted lines illustrate thejump in amplitude at ε = 0 and the hysteresis upon decreasing ε.

pitchfork bifurcations because of the shape of the A 6= 0 solution. This is the

result of the A→ −A symmetry which precludes a quadratic term in Eq 2.6.

As ε is varied, if the solution goes continuously from one stable branch

to another, as in Fig. 2.3a, the bifurcation is considered to be supercritical

or forward. If there is a loss of stability, as in Fig. 2.3b, the bifurcation is

subcritical or backward. In a real system for which g < 0, a stabilizing quintic

19

term is generally added to Eq. 2.6, and one has the behavior shown in Fig. 2.3c

of a finite jump in A at onset (ε = 0). The other common bifurcation in

pattern forming systems occurs when the A→ −A symmetry is broken. Then

one generically has a quadratic term in Eq. 2.6. The behavior of A for this

case is shown in Fig. 2.3d. This is referred to as a transcritical bifurcation, and

depending on the sign of the quadratic term will be subcritical or supercritical.

Figure 2.3d shows an example of a backward transcritical bifurcation. As

mentioned in the discussion of linear stability analysis, bifurcations are also

stationary or oscillatory, and oscillatory bifurcations are referred to as Hopf

bifurcations.

The initial transitions in pattern forming systems are generally character-

ized by their bifurcation class. Linear stability is sufficient for determining if

the bifurcation is a stationary or Hopf bifurcation. However, it is the nonlin-

ear coefficients of the amplitude equation which determines if the bifurcation

is forward or backward, and by definition, these can only be determined by

doing the weakly nonlinear calculation. Recall that the two unexplained fea-

tures of EC are the Hopf bifurcation, a linear property of the system, and the

backward bifurcation, a nonlinear property of the system.

Whether or not the bifurcation is forward or backward has a significant

impact on the possibility of making quantitative experimental studies of the

solutions of the amplitude equations. For the forward case, because the am-

plitude grows continuously from zero, there is a well defined range of ε for

which the the amplitude equation is a valid perturbation expansion of the full

equations. For the case of a backward bifurcation, because of the jump in

amplitude at onset, the initial amplitude might be too large for a perturbation

expansion to be valid.

At this point in a discussion of pattern formation, it is natural to make

an analogy between pattern formation and thermodynamic phase transitions.

Equation 2.5 is identical to the Ginzburg-Landau equation introduced to de-

scribe continuous phase transitions, in particular the superconducting transi-

20

tion[51]. The forward bifurcation is analogous to a second order phase tran-

sition. The amplitude of the pattern grows continuously from zero and there

is no hysteresis upon decreasing the control parameter ε. The backward bi-

furcation is analogous to a first order phase transition. The amplitude has a

finite jump at onset, and the system shows hysteresis as ε is decreased (see

Fig. 2.3). However, with respect to pattern formation Eq. 2.6 is a special case.

It is derivable from a potential of the same form as the free energy used for the

phase transition applications. As discussed in the introduction, the dynamics

of most pattern forming systems are not described by the minimization of a

generalized potential. The weakly nonlinear dynamics are still described by

a amplitude equation, often called a generalized Ginzburg-Landau equation,

but the amplitude equation is generally not derivable from a potential.

The one dimensional complex Ginzburg-Landau equations (CGL) is one of

the paradigms of generalized amplitude equations which is not derivable from a

potential. The equation has the same form as Eq. 2.5, only now the coefficients

are complex, as well as the amplitude. The CGL is the appropriate amplitude

equation when the system has a Hopf bifurcation. Of particular interest for

this thesis is the fact that the CGL in both one[52, 53, 54, 55, 56] and two

dimensions[57] has solutions which exhibit spatio-temporal chaos and localized

states. I will show in Sec. 4.4 that EC in I52 exhibits both of these interesting

behaviors. In Sec. 4.3, I show that the initial transition is a Hopf bifurcation

and described by the WEM. Therefore, in principle, the correct CGL for EC

I52 can be derived, and its solutions can be compared with the experimental

observations.

In the case of EC where the initial transition is a Hopf bifurcation, the

initial state involves an interaction of two modes. In this case, one has an

amplitude equation for each mode of interest with nonlinear couplings between

the equations. An example of such a coupling is a Hopf bifurcation in one

dimension where U(x, t) is the superposition of right and left traveling waves

U(x, t) = (ψ(y, z) [Ar(x, t)ei(qox−ωt) + Al(x, t)e

i(qox+ωt) + cc) +O(ε). (2.7)

21

For a forward bifurcation, one has two coupled complex Ginzburg-Landau

equations (CGL)

τo(∂tAr + so∂xAr) = εAr + ξ2o(1 + c1)∂

2xAr

− g(1 − ic3)|Ar|2Ar

− g1(1 − ic2)|Al|2Ar,

τo(∂tAl + so∂xAl) = εAl + ξ2o(1 + c1)∂

2xAl

− g(1 − ic3)|Al|2Al

− g1(1 − ic2)|Ar|2Al.

(2.8)

Here so is the linear group speed, the complex nature of the coefficients is writ-

ten out explicitly, and the coupling is given by the last term in each equation.

For the case of EC studied here, the initial transition is to traveling oblique

rolls, so there are actually four modes, i.e. the two degenerate oblique roll

modes each can be right- or left-traveling.

2.2 Nematics

Liquid crystals (LC) are a thermodynamic phase of matter which possesses

both fluid like properties (they flow) and crystal like properties (long range

order)[22]. For this work, the most important LC phase is the nematic liquid

crystal (NLC), or simply nematics. NLC posses orientational order in the

form of an average alignment of the molecules, but the molecules have no

positional order. The direction of the average alignment of the molecules is

referred to as the director. Upon heating a NLC, at a critical temperature

(the clearing point), there is a transition to an isotropic fluid for which the

orientational order is not present. Upon cooling, a NLC will often have a

number of transitions to LC phases which posses various degrees of positional

order (Smectic A, Smectic C, etc.), and eventually, the NLC will crystallize. I

will focus exclusively on the nematic phase, and for a discussion of the other

phases, see for instance Ref. [22] and the references therein.

22

This section focuses on the physics of NLC which are relevant to an under-

standing of EC and the experimental details: the anisotropy of the material

parameters, the elastic properties of NLC, and the nature of the viscosity

tensor. I will consider the hydrodynamic limit where the dynamics of fluid el-

ements are considered. Fluid elements represent an average over a region large

enough to contain a sufficient number of individual molecules so that a contin-

uum treatment is valid, but small enough to be treated as a point on the scale

of the entire fluid. The detailed molecular interactions will not be considered

except so far as they result in an average alignment of the molecules.

The director is represented as a unit vector n in the direction of the average

molecular alignment where states with n and −n are equivalent. The equiv-

alence of n and −n is based on the observation that the molecular alignment

does not distinguish right from left. For example, even in a NLC with a per-

manent dipole, on average, the number of molecules with the dipole aligned

in a given direction along the director is equal to the number of molecules

aligned in the opposite direction. The director field is a function of space and

time, and its dynamics must be included in a description of the NLC. Finally,

NLC have a cylindrical symmetry, so the axes perpendicular to the director

are equivalent.

The material properties of a NLC are generally anisotropic and are rep-

resented by tensors. Because of the cylindrical symmetry of the NLC, the

tensors are uniaxial with their two principle axes perpendicular and parallel

to the director. The properties considered here are the magnetic susceptibil-

ity χ, the dielectric constant ε, the index of refraction, and the electrical and

thermal conductivities σ and λ, and they can be written in the general form:

bij = b⊥δij + baninj, (2.9)

where the ni are the components of the director. For a given material property

b, b⊥ is the principle value perpendicular to the director, b‖ is the principle

value parallel to the director, and ba = b‖ − b⊥ is the “anisotropy of” the ma-

23

terial parameter. For example, εa is the anisotropy of the dielectric constant,

or the dielectric anisotropy. The anisotropy of the index of refraction is re-

sponsible for the ability to image EC patterns by the shadowgraph technique.

This is discussed in Sec. 3.1.1. The viscosity is also a tensor, but because it

involves the velocity and director field, it is not uniaxial. I will discuss the

viscosity separately.

Because of the anisotropy of χ and ε, the director can be aligned by an

external magnetic or electric field. In the absence of other forces, for χa > 0

(εa > 0), the director aligns parallel to the applied magnetic (electric) field,

and for χa < 0 (εa < 0), the director aligns perpendicular to the applied

magnetic (electric) field. (In general, χa > 0, but εa < 0 or εa > 0.) One

can see this either by directly computing the torques on the director, or by

considering the free energy density F . As an example, consider the case of a

magnetic field H at an arbitrary angle to the director n. The field produces a

magnetization M = χ⊥H + χa(H · n)n which contributes a term to F :

F = − ∫ M · dH= −1

2χ⊥H

2 − 12χa(n · H)2.

(2.10)

For χa > 0, the free energy is minimized for n and H parallel (n · H a maxi-

mum), and for χa < 0, it is minimized for n and H perpendicular (n ·H = 0).

The alignment of the director by external fields is not a statement about the

degree to which the individual molecules are aligned. The degree of molecular

ordering is determined entirely by the distance from the nematic-isotropic tran-

sition as a function of temperature, and is not affected by the field strengths

considered here. The experiments reported on here were conducted at temper-

atures well away from the clearing point where the ordering remains roughly

constant. Therefore, unless specified otherwise, “alignment” will refer to align-

ment of the director and not of the individual molecules.

Any description of EC is necessarily complicated, but two essential elements

are the elastic and viscous properties of NLC. The elastic behavior is best

24

described in terms of a free energy density F . The three basic types of elastic

deformations are splay, twist and bend. Examples of these three basic types of

deformations are given in Fig. 2.4. The elastic contribution Fd to F in terms

(a) (b)

(c)

Figure 2.4: The solid lines in the three figures represent the director orienta-tion. (a) example of a pure splay (S) deformation. (b) example of a pure bend(B) deformation. (c) example of a pure twist (T) deformation. The length ofthe lines in (c) represent the director at an angle to the plane of the paper.

of these three types of deformations is given by:

Fd =1

2[K11(∇ · n)2 +K22[n · (∇× n)]2 +K33|n × (∇× n)|2), (2.11)

where K11, K22 and K33 are referred to as the splay, twist and bend elastic con-

stants, respectively[58]. Generally, these constants are on the order of 10−11 N

with the twist elastic constant the smallest. The total free energy density F

contains additional contributions from external magnetic and electric fields,

Fm = −1

2χa(n · H)2, (2.12)

and

Fe = −1

2εoεa(n · E)2, (2.13)

respectively. Here I have not included the contributions to F which are inde-

pendent of n, and εo is the dielectric constant of the vacuum.

25

The equilibrium condition for the bulk is that the free energy F =∫

Fdx

is a minimum with respect to variations of n subject to the constrain n2 =

1. Following Ref. [22], the equilibrium condition is found using the Euler-

Lagrange equations with Lagrange multipliers. One finds

∂j

[

δF

δ(∂jni)

]

− δF

δni= −λ(x)ni, (2.14)

where δ is used to represent a functional derivative. Here λ(x) is an arbitrary

function of x (the required Lagrange multiplier). It is useful to define a new

vector function h with components

hi = ∂j

[

δF

δ(∂jni)

]

− δF

δni(2.15)

which is referred to as the molecular field. In terms of h, the equilibrium

condition is [hi + λ(x)ni] = 0, i.e. h and n must be parallel. Since the cross

product of two parallel vectors is zero, a necessary but not sufficient condition

for equilibrium, independent of λ(x), is

n × h = 0 (2.16)

I will use the notation hd, hm, he to refer to the contribution to h from the

elastic contribution to the free energy density Fd, the magnetic contribution

Fm, and the electric contribution Fe, respectively.

For EC, it is more useful to discuss distortions of the director in terms of

torques rather than free energies. The molecular field h provides a natural

way to connect the free energy density with the torque per unit volume Γ.

One finds

Γ = n × h. (2.17)

(For the rest of this section and Sec. 2.3, when I use torque Γ, I will be refering

to the torque per unit volume.) This can be derived using the variational

principle, but the easiest way to see this is to consider a specific example. For

the equilibrium configuration of a NLC subjected to a uniform magnetic field

26

H and no electric field, the torques generated by the elastic distortion must

cancel the magnetic torques. I will show that this comes out naturally from

the equilibrium condition given in Eq. 2.16, and the association of n× h with

the total torque follows.

For an arbitrary angle between n and H, the magnetization M = χ⊥H +

χa(H · n)n produces a magnetic torque

Γm = M × H

= χa(n · H)n× H.(2.18)

As discussed, the magnetic field contribution to the free energy is

Fm = −1

2χa(n · H)2 (2.19)

The equilibrium condition (Eq. 2.16) gives

n × hd + n × hm = 0

n × hd = −χa(n · H)n × H

= −Γm.

(2.20)

Because Eq. 2.20 represents the condition for equilibrium, n × hd must be

the torque generated by the elastic deformations which balances the magnetic

torque Γm. Equation 2.17 is a convenient formulation of the torque because,

as I will show below, the viscous torque is naturally written in this form as

well.

One application of the elastic torque is the alignment of NLC in the absence

of external fields[59]. For thin enough samples, if the orientation of the director

is fixed at the surface of the sample, the elastic torques generate a unique

orientation in the bulk. This is an essential feature of the construction of LC

displays and electroconvection cells. The techniques of alignment are discussed

in Sec. 3.3.2.

The hydrodynamics of NLC are governed by the Navier Stokes equation

ρ(∂tvi + (v · ∇)vi) = ∂jtji + fi, (2.21)

27

where ρ is the density, tji are the components of the stress energy tensor,

vi are the components of the velocity, and fi are body forces. The viscous

contribution to the stress tensor σij can be written[60, 61] as

σij = α1[ni(Aklnl)nj]

+α2[niNj]

+α3[njNi]

+α4[Aij]

+α5[ninkAkj]

+α6[njnkAki].

(2.22)

Here Aij = 12(∂ivj + ∂jvi), N = dn/dt − ω × n, and ω = 1

2∇ × v, with

dn/dt = ∂n/∂t + (v · ∇)n. The |αi| range from the order of 1 to 102 cP, and

are known as the Leslie coefficients[61]. There are other formulations of σij,

but this is the one which is usually used in the theory of EC. There are only

five independent αi because there is the Onsager relation α6 − α5 = α2 + α3.

There are three special cases of director orientation and fluid velocity for

which the viscous contribution to Eq. 2.21 can be reduced to the usual form for

an isotropic fluid of η∇2v. The three cases are shown schematically in Fig. 2.5.

In all three cases, the flow consists of a single nonzero velocity component vi

which has a single nonzero derivative ∂jvi where j 6= i. The η’s are linear

combinations of the α’s. For the director aligned perpendicular to both vi and

xj,

η ≡ ηa = α4/2. (2.23)

For the director aligned parallel to vi,

η ≡ ηb = (α4 + α3 + α6)/2. (2.24)

For the director aligned parallel to xj,

η ≡ ηc = (α4 + α5 − α2)/2. (2.25)

This is the case that is relevant for the simple description of the onset of EC

given in Sec. 2.3.

28

(a)

xi

xj

(b) (c)

Figure 2.5: Three special orientations of the director and flow for which a shearviscosity η is defined. The arrows represent the direction of the velocity. (a)The orientation for measuring ηa where the director is coming out of the planeof the paper (the solid dot). (b) The orientation for measuring ηb where thedirector orientation is given by the ellipsoid. (c) The orientation for measuringηc where the director is given by the ellipsoid.

The other two important viscosities are the rotational viscosities γ1 =

α3 −α2 and γ2 = α6 −α5 = α2 +α3. The γi are the linear combinations of the

α’s which appear in the expression for the viscous torque Γv on the director

field due to viscous stresses. The total torque Γ is equal to the rate of change

of the angular momentum L,

Γ = L =d

dt

∫

dx (x × ρv). (2.26)

Using Eq. 2.21 to substitute for ρ ddt

v, the viscous contribution to the torque,

Γv is just the antisymmetric part of the viscous stress tensor:

Γi = −εijkσjk

= −εijknj(γ1Nk + γ2nlAlk),(2.27)

or in vector notation,

Γv = −n × (γ1N + γ2n · A). (2.28)

Two important examples of the viscous torque are illustrated in Fig. 2.6.

For both cases, the coordinate system is as shown, with the y axis chosen to

give the usual right handed coordinate system. Also, vz(x) is the only nonzero

29

(a) α3 < 0

α3 > 0

x

z

(b) α2 > 0

α2 < 0

Figure 2.6: (a) The torque on the director (show by the ellipsoid) for the case ofthe director aligned parallel to the velocity component (vz) and perpendicularto the shear ∂xvz 6= 0. The direction of the torque depends on the sign of α3.The velocity field is show schematically by the two solid straight lines, and theinitial direction of motion of the director is given by the curved dashed linesfor the two possible signs of α3. (b) The case when the director is parallel tothe shear and perpendicular to the direction of the velocity. Here the sign ofα2 is relevant. The coordinate system is the same for both figures and the yaxis is such that it is a usual right handed coordinate system.

component of v with ∂xvz > 0. For both cases, there is only a y component of

the torque. For the situation shown in Fig. 2.6a, Eq. 2.28 gives Γy = −α3∂xvz,

and for the situation shown in Fig. 2.6b, Eq. 2.28 gives Γy = α2∂xvz. The sign

of α3 or α2 determines the direction of rotation for the director as shown in

Fig. 2.6. For rod-like NLC, α2 < 0, but α3 can be either positive or negative.

The situation in Fig. 2.6b is the case that is considered in the simple model of

EC discussed in Sec. 2.3.

Two main results of this section are used in the description of EC: the

conservation of momentum given by the Navier-Stokes equations (Eq. 2.21)

and the conservation of angular momentum given by the balance of torques,

30

Γ = 0. Taking advantage of the similar forms of Eq. 2.28 and Eq. 2.17, the

total torque Γ can be written as

Γ = n × S, (2.29)

where S is defined1 as

S = h − (γ1N + γ2n · A). (2.30)

2.3 Electroconvection

As discussed in the introduction, one of the major motivations for studying EC

in a new NLC is the attempt to resolve the discrepancies between experimental

observations and the predictions of the detailed analysis of EC presented in

Ref. [33, 34]. The model used in Ref. [33, 34] is the full three dimensional

version of the unidimensional model originally introduced by Helfrich[35]. I

will refer to the set of equations and assumptions presented in Ref. [33, 34] as

the standard model (SM). The mechanism which produces EC in the SM is

referred to as the Carr-Helfrich mechanism[36]. The essential elements of the

Carr-Helfrich mechanism are planar alignment of the director, an electric field

applied perpendicular to the director, ionic impurities in the NLC to provide

a nonzero conductivity σ, and a positive σa = σ‖ − σ⊥. In practice, an ac

electric field is used to prevent charge accumulation at the boundaries, but for

low frequencies the basic mechanism is the same in the ac case as in the dc

case. Also, εa = ε‖ − ε⊥ is generally taken to be negative, but it is possible for

εa to be slightly positive. If εa is large and positive, the Frederiks transition

dominates (see Appendix A).

The weak-electrolyte model (WEM), considered in detail in Ref. [39], is an

extension of the SM which addresses the origin of the observed Hopf bifurca-

1Equation 2.29 is the same as Eq. (2.4) in Ref. [33]. But, because the definition of h

follows Ref. [22], careful tracking of minus signs must be used when comparing the two

equations.

31

tion. The essential difference between the SM and the WEM is the treatment

of the ionic impurities and their resulting conductivity. Both models assume

equal numbers of positive and negative ions, so the fluid is electrically neu-

tral. Under the influence of the electric field, local variations of the charge

density will develop[62] which provide a body force on the fluid. The dif-

ference between the two models is the treatment of the conductivity. In the

SM, the conductivity of the sample is frequency independent, and the electric

fields and currents are related by Ohm’s Law. In the WEM, a dissociation-

recombination reaction for the positive and negative ionic species is considered,

and the resulting conductivity is an additional dynamical variable. Because

the Carr-Helfrich mechanism is central to both the SM and the WEM, I will

illustrate it using the unidimensional model of Helfrich. Then, I will discuss

the additional effects included in the WEM which produce a Hopf bifurcation.

The equations of the SM are the Navier-Stokes equation (Eq. 2.21) with the

body force fi = ρe(x)Ei(x), the conservation of angular momentum (Eq. 2.29),

incompressibility (∇ · v = 0), the equations of electrostatics,

∇ · [ε0εE(x)] = ρe(x) , ∇× E(x) = 0, (2.31)

and charge conservation

∇ · j(x) + ∂tρe(x) = 0. (2.32)

Here ρe(x) is the charge density of the ionic impurities, E(x) is the applied

electric field, ε is the dielectric tensor, ε0 is the dielectric constant of the

vacuum, and

j(x) = σE(x) + ρe(x)v (2.33)

is the electric current. I have explicitly written the x dependence of ρe to

emphasize its role as a dynamical variable; whereas, the conductivity σ is a

constant (tensor) parameter.

Before discussing the unidimensional model introduced by Helfrich, some

general comments about EC with an ac electric field are required. There are

32

two regimes of EC with an applied ac electric field (E(x, t) = E(x) cos(Ωt)):

the conduction regime and the dielectric regime. For low enough applied fre-

quencies (the conduction regime), the charge density ρe(x, t) is able to follow

the applied field (i.e. ρe(x, t) = ρe(x) cos(Ωt)), and the director and velocity

are essentially time independent. For high frequencies (the dielectric regime),

the director and velocity oscillate with the applied frequency Ω, and the hor-

izontal variation of the charge density is essentially constant in time. The

crossover between the two regimes is known as the cutoff frequency. The cut-

off frequency is set by the two time scales of the SM[33]: a charge-relaxation

time, τq = ε0ε⊥/σ⊥, and a director-relaxation time, τd = γ1d2/[(K11 +K33)π

2].

The conduction regime corresponds roughly to the conditions Ωτd 1 and

τd τq. The cutoff frequency scales with the conductivity of the sample and

ranges from a few Hz for very pure samples to kHz.

All of the work presented here, both the theory and experiments, has been

on EC in the conduction regime. Because the charge follows the applied field,

the description of the Carr-Helfrich mechanism for the dc case carries over

naturally to the ac case. Also, for the cases where there is a Hopf bifurcation,

the Hopf frequency ranges from 0.06 to 0.2 Hz. For comparison, in the dielec-

tric regime, the director oscillates with the frequency of the applied voltage,

which for my experiments ranges from 25 to 200 Hz.

The situation considered by Helfrich[35] is shown schematically in Fig. 2.7.

The director is assumed to be confined to the x-z plane, initially with planar

(n = [1, 0, 0]) alignment in the x direction. A fluctuation of the director

n = [1, 0, φ(x)] is considered, where φ(x) is assumed to be small. There

is an applied dc electric field in the z-direction Ez which is assumed to be

constant. I will follow closely the treatment of Ref. [63] which makes the

simplification of assuming εa = 0. Helfrich’s[35] original work accounts for

the more general case of εa ≤ 0. This necessitates including the dielectric

torques in addition to the elastic and viscous torques. The quantitative result

is changed, but the qualitative features of the instability are essentially the

33

x

z

α2 < 0

Ez

φ

Figure 2.7: A schematic diagram of the Carr-Helfrich mechanism. The director(shown by the ellipsoid) is assumed to have a small fluctuation (φ) away fromits initially planar alignment (horizontal dashed line). A constant electricfield Ez is applied in the positive z-direction. The current induced in the x-direction produces a separation of positive and negative ionic impurities whichthen experience a force in the z-direction. The resulting shear flow is shown bythe arrows. The situation corresponds to Fig. 2.6b with α2 < 0, and generatesa destabilizing torque. The viscous torque is balanced by the elastic torqueinduced by distorting the director.

same. The relevant equations are the ones listed above for the SM. All of the

variables are only considered to have variations in the x direction; hence, the

name unidimensional.

For the director orientated at an angle φ, Ez produces a current in the

x-direction given by Ix = σaEzφ(x) which results in a charge density ρe(x)

due to the separation of the positive and negative ions. The charge separation

produces a field in the x-direction which generates an opposing current Ix =

34

σ‖Ex. For the steady state case, charge conservation gives ∂xIx = 0, or

σ‖∂xEx = −σaEz∂xφ (2.34)

(Recall, everything is constrained to one dimension, and Ez is constant to

lowest order in φ.) Using Poisson’s equation ∇ · (εE) = ρe, the space charge

density ρe(x) generated by Ez is

ρe = ε0ε∂xEx = −ε0εσa

σ‖Ez∂xφ (2.35)

There is now a body force on the fluid of Ezρe. The velocity field which results

from the body force is equivalent to the example in Fig. 2.5c. Therefore, for

steady state flow (∂tv = 0), the Navier Stokes equation gives (assuming that

the viscous terms dominate over the term (v · ∇)v)

0 = ηc∂2xvz + Ezρe, (2.36)

or

ηc∂2xvz = −Ezρe = E2

z ε0εσa

σ‖∂xφ, (2.37)

with ηc = (α4 + α5 − α2)/2 as defined in Eq. 2.25. For both φ(x) and vz(x),

only a single Fourier mode with wavenumber q is considered, so ∂2xvz = iq∂xvz.

Therefore, Eq. 2.37 becomes

iqηc∂xvz = E2z ε0ε

σa

σ‖∂xφ. (2.38)

Recalling Eq. 2.28 and Fig. 2.6b, ∂xvz produces a torque on the director

Γv = (0, α2∂xvz, 0). For NLC used in EC, α2 < 0, so the viscous torque is

always destabilizing (see Fig. 2.6b). This torque is opposed by the elastic

torque2 Γd. In the absence of magnetic fields and for εa = 0, Γd = (n × h) =

2The result for Γd is not immediately obvious, but it is straight forward to work out

from the elastic free energy, Eq. 2.11, and use Eq. 2.15 to compute the torque. If εa 6= 0, the

additional torque due to the electric field is accounted for by including Fe when calculating

h.

35

(0,−(K33∂2xφ), 0) (see Eq.. 2.29). As Ez is increased, a critical value Ec will

be reached at which the total torque Γ = Γd + Γv = 0, which gives

K33∂2xφ = α2∂xvz. (2.39)

Combining Eq. 2.39 and Eq. 2.38, using ∂2xφ = iq∂xφ, and eliminating ∂xφ

gives

E2c =

K33σ‖ηcq2c

(−α2)ε0εσa

, (2.40)

where qc is the critical wavenumber. Experiments suggest qc scales with the cell

thickness d, and in fact, the detailed linear stability analysis[33, 34] predicts

this scaling with d. This is used to convert the expression for Ec to one for

Vc = Ecd. Taking qc ≈ π/d,

V 2c =

πK33σ‖ηc

(−α2)ε0εσa. (2.41)

Despite the fact that the value of Vc derived from this simple model is

too low, the three essential elements of the Carr-Helfrich mechanism are illus-

trated by this calculation. First, the generation of a charge density ρe(x) by

the anisotropic conductivity is essential. Second, from Eq. 2.35, the genera-

tion of the charge density requires fluctuations of the director with ∂xφ 6= 0.

The fluctuations of the director are believed to be driven by thermal noise[37].

Finally, because of the d dependence of the critical field, the actual control pa-

rameter is the voltage squared, V 2. Even when the full equations of motion are

considered, the mechanism remains essentially the same. A Hopf bifurcation

is not predicted for any combination of the parameters. The addition of the

conductivity as a dynamical variable in the WEM[39] provides a mechanism

for the Hopf bifurcation.

The WEM starts with the two species of ionic charge carriers[64] with

charges ±e and couples them through a simple dissociation-recombination

reaction[65]. The charges are considered to have number densities n+(x, t)

and n−(x, t), and constant, possibly different, mobility tensors µ± with princi-

pal values perpendicular and parallel to the director, µ±⊥ and µ±

‖ , respectively.

36

The µ±(⊥,‖) are defined in terms of the ion’s steady-state velocities in a quies-

cent fluid with an applied electric field, v±(⊥,‖) = µ±(⊥,‖)E(⊥,‖). For simplicity,

the ratio µ‖/µ⊥ is assumed to be the same for both species. The WEM ex-

presses the total space-charge density ρ(x, t), which is the same as in the

SM, as ρ(x, t) = e[n+(x, t) − n−(x, t)]. Unlike for the SM, the local conduc-

tivity tensor is an additional variable with components given by σij(x, t) =

σ(x, t)[δij + ninj(µ‖/µ⊥ − 1)]. Here σ(x, t) = e[µ+⊥n

+(x, t) + µ−⊥n

−(x, t)],

The basic equations of the WEM are two equations which couple σ(x, t)

and ρ(x, t) plus the SM equations for the director n and fluid velocity v. The

new equations for σ(x, t) and ρ(x, t) are given in Ref. [39]. The charge density

is coupled to n and v as in the SM, and σ(x, t) couples to n and v via the

conductivity tensor and the ρe(x)v part of the current. The actual boundary

conditions in the experiment are unknown, but using the frequency dependence

of the capacitance, I have experimentally determined that the thickness of the

charged boundary layers at the electrodes is small compared to the thickness

of the cell (see Sec. 4.3). In this limit, the model is insensitive to the details

of the boundary conditions[39].

As discussed, the SM has two relevant time scales: τq and τd. The consid-

eration of the individual ions introduces two new time scales: a recombination

time τrec = 1/(2krn0) for the dissociation-recombination reaction, and a mi-

gration time τmig = d2/[π2(µ+⊥+µ−

⊥)V (0)] for a charge to traverse the cell under

an applied voltage V (0) which is of order the critical voltage for low external

frequencies. Here kr is the recombination rate of the ions, n0 is the equilibrium

number density of either species of ion, and V (0) = π[(K11 + K33)/(τqσa)]1/2.

The SM is recovered in the limit of τrec/τq → 0 and τmig/τq → ∞.

Reference [39] applied the WEM to the case of normal rolls, and ref. [42]

applied the WEM to the more general case of oblique rolls. I will only consider

here the final result. The calculation can be reduced to an equation for the

amplitude of the local deviation of the conductivity from its equilibrium value

Aσ(t), and the critical SM mode An(t) which includes the deformation of

37

the director. (The velocity field and charge density are both adiabatically

eliminated.)

Aσ = λσ(R)Aσ − α2Rσ(eff)a (σ⊥τd)

−1An,

An = Rσ⊥

σ(eff)a τd

(

C1+(βΩτq)2

)2Aσ + λn(R)An,

(2.42)

Here R = (V/V (0))2 is the control parameter, and Ω is the angular frequency of

the applied voltage. The diagonal coefficients λσ(R) = −τ−1rec−τ−1

d (Rα2β)/[1+

(βΩτq)2] < 0 and λn(R) = ε/τSM

0 are the growth rates of the σ mode and the

SM amplitude, respectively. Here τSM0 is the correlation time of the SM ampli-

tude equation (O(τd)), ε = R/Rc−1 is the relative distance from threshold Rc,

and β = [(1+ εa/ε⊥)(qcd)2 +(pcd)

2 +1]/[(1+σa/σ⊥)(qcd)2 +(pcd)

2 +1] ≈ 0.85.

The dimensionless quantity C contains only SM quantities[39] and is of order

unity for our experiments[66]. The effective conductivity anisotropy σ(eff)a ≈

σa[β − εaσ⊥/(σaε⊥)]/[1 + (βΩτq)2] is proportional to the charge produced by

the Carr-Helfrich mechanism[36].

The important coupling between Aσ and An is proportional to

α2 = µ+⊥µ

−⊥γ1π

2/(σad2). (2.43)

In the limit of the SM, τmig → ∞ which gives α2 → 0, and the two modes

are decoupled. The other condition for recovering the SM, τrec → 0, gives

λσ(R) ≈ −τ−1rec → −∞ which fixes the conductivity at its equilibrium value,

Aσ = 0.

There are two main predictions of the WEM. For sufficiently high mo-

bilities and small recombination rates, there is a nonzero Hopf frequency ω

at threshold (the imaginary part of the growth rate of the 2x2 equations is

nonzero for zero real part), ω = ω√

1 − (λσ(Rc)/ω)2, where

ω = C(

πd

)3 Rc(K11+K33)1+(βΩτq)2

√

µ+⊥

µ−

⊥

γ1σa. (2.44)

The condition for a nonzero Hopf frequency can be written λσ(Rc) < ω.

By making the appropriate substitutions, this sets an upper bound on the

38

combination(

τdτq

)1/2τmig

τrec∝ (σ⊥d

2)3/2. (2.45)

The result that the condition for a Hopf frequency scales with σ⊥d2 is consistent

with experimental observations that the Hopf bifurcation tends to occur in thin

cells and at low frequencies[26, 27]. Because both τd/τq and τmig/τrec scale as

σ⊥d2, it is also expected that the regions of existence for the nonlinear patterns

might scale with σ⊥d2 (see Sec. 4.4 and Sec. 4.5).

There is an upward shift of ε relative to the SM prediction. For nonzero ω

(λσ(Rc) < ω), the shift is

∆ε = Rc−RSMc

RSMc

= −τSM0 λσ(Rc), (2.46)

and for ω = 0 (λσ(Rc) > ω), the shift is

∆ε = −ω2τSM0 /λσ(Rc). (2.47)

These conditions are equivalent when λσ(Rc) = ω: the crossover from ω = 0

to a nonzero ω. Therefore, ∆ε = −ωτSM0 represents the maximum possible

deviation between the SM and the WEM. The net result is that within the

accuracy of the measured material parameters, the WEM predicts the SM

values for Vc. Linear stability analysis also predicts the wavevector of the

pattern at onset, and in particular for EC, the angle Θ between the wavevector

of the pattern and the director is computed. The WEM also recovers the SM

prediction for Θ. Both the SM model prediction of Vc and Θ are found to agree

with experiments[33, 34]. In Sec. 4.3, I present the results of a quantitative

calculation of the WEM provided by Martin Treiber and compare them to my

experimental measurements.

Returning briefly to Eq. 2.42, the stabilizing coupling ofAσ and An provides

the mechanism of the Hopf bifurcation. From Eq. 2.42, the linear growth rate

λ for the most unstable mode is

λ = (λσ + λn)/2 ±[

(λσ − λn)2/4 −B

]1/2, (2.48)

39

where B > 0 is the product of the off diagonal coefficients in Eq. 2.42, both

of which are positive. It is the negative cross-coupling that allows for the

possibility of a nonzero Imλ. Had the coupling been destabilizing (one of the

coefficients negative), then B < 0 and λ is always real. Physically, for large

enough α2, the growth of An causes Aσ to become negative. (Recall that

λσ(R) < 0 and Aσ is the deviation from the equilibrium conductivity.) A

negative Aσ retards the growth of An, stabilizing the director mode. When

the Aσ and An have sufficiently different relaxation time scales, the feedback of

Aσ is out of phase with the growth of An and establishes an oscillation. I will

discuss this mechanism and its relation to Hopf bifurcations in other systems

again in Chapter 5.

40

Chapter 3

Experimental Details

There are three main elements to the experiment: the apparatus, the NLC’s,

and construction of the NLC cells. There are currently two EC apparatus in

use in our labs. They follow the same general design except for the temperature

regulation. The apparatus described here was designed for room temperature

operation. The second apparatus is described in Ref. [67] and was designed to

operate at temperatures up to 200C. Provided here is a detailed description

of the methods of cell construction. Included are comments on the methods

which were found not to work which will help to guide future modifications to

the cell design. In the section on the NLC’s, I discuss the relative merits of

four liquid crystals. Included is a discussion of the methods used to dope the

various NLC’s.

3.1 Apparatus

The apparatus consists of three main parts, the imaging system, tempera-

ture control stage, and electronic controls, which are shown schematically in

Fig. 3.1. The details of each part will be discussed separately. An expanded

view of the imaging system and temperature control stage is shown in Fig. 3.2.

The temperature control stage also serves as the cell holder. The electronics

41

computer

video in to pceye board ac signal out from wavetec DIO lines on labmaster A/D on daughter board

ac out to cell ac in to DAC

current to voltage conversion (Fig. 3.10)

multi-meter

analog temperature control

digital in to DAC

(Fig. 3.9)

external electronics box

Figure 3.1: Schematic of the apparatus.

are used for generating the applied ac voltage and measuring the conductivity

of the cells.

3.1.1 Shadowgraph

The shadowgraph technique[68, 69] is a well developed method of visualizing

variations in the dielectric constant ε, or index of refraction n, of a fluid.

Consider a RBC or EC cell which consists of fluid confined between two plates.

We take the z-direction to be perpendicular to the plates, and the x direction

is taken parallel to the wavevector of the pattern. There is a variation of ε

or n associated with the pattern, and for many situations, geometric optics

is sufficient to explain the shadowgraph signal produced by this variation.

Because rays are bent toward regions of high n, light propagating in the z-

42

Nikon Camera Lens

CCD Camera

lower lens

light source

cell

Translation Stage

Temperature- Control Stage

electronics in/out

shadow graph system

Figure 3.2: Schematic of the imaging system (shadowgraph), light source, andtemperature control stage.

direction through the cell is focused in a plane above the cell. This creates an

image or “shadow” of the pattern in which the bright regions correspond to

the regions of the fluid with a relatively large n and the dark regions to regions

43

with a relatively low n. (For NLC, there is an additional effect because of the

inherent anisotropy of the material which will be discussed later.)

Fermat’s principle (minimization of the optical path) provides a quantita-

tive calculation of the shadowgraph signal for the geometric optics limit[68].

This calculation clearly breaks down at the caustics, i.e. locations where the

intensity is predicted to be infinite. For any physical system, diffraction effects

prevent the intensity from actually diverging. The distance to the first caustic

above the cell is referred to as the focal length of the pattern. A more complete

calculation which includes diffraction effects has been carried out recently[69].

This physical optics approach recovers the predictions of geometric optics in

the correct limit and provides a nice intuitive picture of the shadowgraph

method in terms of diffraction. The horizontal variation of n generated by

the pattern is equivalent to a phase grating and/or an amplitude grating. An

incoming beam of light is diffracted by this grating, and the interference of the

diffracted beams with the main beam generates the intensity pattern. I high-

light here some of the relevant results of both the geometric[68] and physical

optics[69] calculations for RBC and EC. The shadowgraph apparatus used in

the EC experiments is a modification of the RBC, and the modifications are

best understood by comparing the two systems.

For the case of thermal convection in simple fluids, the variation of n is

directly proportional to variations in the fluid’s density induced by temper-

ature variations. Therefore, quantitative measurements of the temperature

field are possible using the shadowgraph technique. Because the thermal con-

vection rolls consist of regions of hot upward-traveling fluid alternating with

regions of cold-downward traveling fluid, the shadowgraph provides a quali-

tative image of the vertical average of the velocity field as well. The one to

one correspondence between the temperature variation and the variation of n

yields the nice result that the wavelength of the pattern in the shadowgraph

image corresponds to the wavelength of the physical pattern. The case of NLC

is more complicated.

44

As discussed in Sec. 2.2, NLC have a tensor ε and n with principle axes par-

allel and perpendicular to the director. Light polarized parallel to the director

is called the extraordinary beam and has an index of refraction ne = n‖ =√ε‖.

Light polarized perpendicular to the director is referred to as the ordinary

beam and has an index of refraction no = n⊥ =√ε⊥. In EC experiments,

one uses light polarized in the direction of the undistorted director. When a

pattern is present, the local director gains a z-component and makes an an-

gle φ with respect to the horizontal. In this case, the light sees an effective

index of refraction n = none(n2o cos2(φ) + n2

e sin2(φ))−1/2. It is the horizontal

variations in n which produces a diffraction grating. Because n depends on

the square of cos(φ) and sin(φ), director distortions with an angle φ and π−φ

produce equivalent n. However, they represent half a wavelength in terms

of the pattern (see Fig. 3.3). Therefore, unlike the case of RBC, the phase

grating generated by the pattern in EC has half the wavelength of the pat-

tern. Furthermore, the intensity of the shadowgraph signal due to the phase

grating is proportional to the square of the director angle. Hence, in EC, the

phase grating contribution to the shadowgraph signal is often referred to as

the nonlinear or quadratic shadowgraph effect.

As first pointed out by Rasenat, et. al.[68], there is a second contribution

to the formation of a shadowgraph image in EC which is equivalent to an am-

plitude grating. This contribution is linear in the director angle and produces

intensity variations which have the same wavelength as the pattern. This is

usually referred to as the linear effect. As with the phase grating, there is a

geometric optics explanation of this contribution. When φ is nonzero, the light

traveling through the cell has a polarization component parallel and perpen-

dicular to the director as shown in Fig 3.4. Because the velocities of these two

components are not equal, the usual Hugyen’s construction results in wavelets

which are elongated into ellipsoids instead of the usual spherical wavelets. The

envelope of the ellipsoids (shown by the dashed dotted line in Fig. 3.4) still

corresponds to a plane wave which is essentially parallel to the incident wave.

45

φ

x

mod

ulat

ion

(a) (b)

0

Figure 3.3: (a) schematically shows a director variation and the correspondingeffect on a parallel beam of light. Notice, there are two effects representedby the solid and dashed rays above the cell. The dashed rays represent apure focusing due to the phase grating effects. The solid lines represent ashift in the rays which corresponds to the amplitude grating effect. (b) showsthe director variation (solid line) and the corresponding index of refractionvariation (dashed line) which has half the wavelength for the situation shownin (a).

However, the direction of energy propagation (the ray direction) is no longer

normal to the wavefront. The ray direction is parallel to the line connecting

the origin of the wavelet and the point of tangency with the planar wavefront

(shown by the dashed line in Fig. 3.4). Because for most liquid crystals, ne is

greater than no, the component of polarization perpendicular to the director

has the greater velocity which results in a shift of the rays along the director

(Fig. 3.3). This produces a horizontal intensity variation of the light leaving

the cell, but the direction of travel of the rays is unaltered (see Fig. 3.3).

Because there is no focusing of the light, the resulting intensity modulation

∆I/I, has a constant amplitude as a function of z in the geometrical optics

limit[68]. This is not the case of course in reality. Here I use the expression

for ∆I/I given in Ref. [68],

∆I/I ≈ 4φn

1 + n(d/λ), (3.1)

where n = 1− (ne/no)2, d is the cell thickness, and λ is the wavelength of the

pattern. In contrast, the phase grating effect is a pure focusing effect within

geometrical optics and produces a modulation of the light which has a zero

46

amplitude at the surface of the cell and reaches a maximum at the caustic.

Therefore, in the geometrical optics limit, the image near the surface of the cell

is dominated by the linear effects. In practice, for extremely strong patterns,

even right near the surface of the cell the quadratic effects will often dominate

the image. However, the various effects can be separated by Fourier analysis.

k

director director

polarization direction

ve

vo

Figure 3.4: The Hugyen’s construction explanation of the shift of the opticalrays as they travel through the EC cell. The initial polarization is shown inthe lower left. The director is given by the solid line. The ellipsoid for ne > no

is shown with the corresponding ray direction as a dashed line. The dasheddotted line gives the wavefront formed by the wavelets.

The physical optics calculation predicts similar amplitude and phase grat-

ing effects. As in geometric optic, the phase grating produces an intensity

variation which is proportional to the square of the director variation and

which has twice the wavenumber of the underlying pattern. The amplitude

grating effect is linear in the director variation and has the wavelength of the

underlying pattern. However, instead of being constant as a function of z,

the amplitude of the intensity modulation due to the linear effect has a cos z

dependence, where z = 0 at the top surface of the cell. The amplitude of the

47

quadratic effect has a sin z dependence. So, the linear effect still dominates

when one focuses near the top of the cell. This behavior has been confirmed

qualitatively, but quantitative comparisons with the physical optics calculation

are still needed.

Because of the different imaging properties of RBC patterns and EC, the

principles behind the operation of the shadowgraph apparatus are quite differ-

ent, even though the designs are essentially identical. The goal in both cases

is to produce as faithful a representation of the pattern as possible. For RBC,

there is no image at the cell surface. One must image the “shadow” which

exists at some distance above the cell. For typical convection patterns, this re-

quires imaging at a distance which is meters away from the cell. An objective

lens produces an image of the intensity modulation which is either projected

on a screen, or preferably, imaged by a camera/lens system similar to the one

shown in Fig. 3.2. Notice, the objective lens creates the image of the shadow

between itself and its focal plane. This is shown schematically in Fig. 3.5.

light cell objective lens

‘‘shadow’’

focal plane ‘‘image’’

Figure 3.5: Top figure schematically shows the imaging method for RBC. Herethe objective lens is used to move the “shadow” in from a plane near “infinity”to a plane within the lens’ focal length. The bottom figure shows the methodused in EC. Here one wants to image the cell itself, not a “shadow” whichexists at a large distance away from the cell.

48

For EC, one generally wants to image near the surface of the cell where the

linear effect dominates. The intensity modulation of the light will have the

same wavelength as the pattern, and the signal is linear in the director angle.

Even if we wanted to image the “shadow” as in RBC, one still focuses close to

the top of the cell because the focal length of the pattern for the case of EC

is centimeters or less, not meters. Also, the wavelength of the patterns range

from 10 to 100 µm, so high magnification and good resolution is required.

In order to image the surface of the cell with sufficient magnification, most

EC experiments use a commercial microscope. This limits one to objective

lenses with a short focal length or very expensive microscopes. In our case,

we desired a space of at least 5 cm above the cell for the temperature control

stage. We found that the basic RBC shadowgraph design with a 5 cm focal

length objective lens and the cell located about 6.5 cm away from the lens

acts as a microscope/telescope with the required magnification (up to 30x)

and resolution (≈ 4 µm based on the Rayleigh criteria).

The shadowgraph apparatus used in the EC experiments is a modified ver-

sion of the RBC shadowgraph tower[13]. In thermal convection, the bottom

plate is generally opaque. Therefore, RBC is imaged by illuminating the sam-

ple from above and using a reflecting bottom plate. This requires that a light

source, beam splitter, etc. be included as part of the shadowgraph tower. In

EC, we do not have this limitation because the cells are transparent, and we

can illuminate the sample from below. Figure 3.2 shows that the light source is

a separate unit mounted below the sample, and the shadowgraph tower above

the cell only requires an objective lens, camera lens and camera.

The specifics of the system are as follows. The light source is a separate

unit mounted below the cell. A parallel beam is generated by locating a point

source at the focal point of a lens. In the initial design, the point source

was made by mounting a 40 µm diameter pinhole in front of a high-power red

LED. The plastic lens of the LED had been machined away so that the pinhole

was located approximately 250 µm from the active element of the LED. The

49

apparatus now uses a LED coupled into a single mode optical fiber[72] with

a diameter of 50 µm. The optical fiber is 2 m long with a 0.22 numerical

aperture and SmA connectors on each end. The LED provides 660 nm light

and is run above the rated current at roughly 100 mA [72]. The optical fiber

setup provides more uniform illumination than the pinhole/LED combination

with essentially the same amount of light. The mount for the light source can

hold either the new LED/fiber or old LED pinhole. The light is converted

into a parallel beam by a 10 mm diameter achromatic lens with a 20 mm focal

length. A dichroic sheet polarizer is placed between the light source and the

liquid crystal cell and can be rotated with respect to the cell. Another dichroic

sheet polarizer can be placed between the cell and the lens system to be used

as an analyzer which is crossed with respect to the lower polarizer.

The lens system used to image the cell from above consists of two lenses and

a CCD camera[70] which are mounted in a 1.2 m high aluminum tube. The

camera uses a 1.27 cm charge-coupled device (CCD) with 510 x 492 picture

elements. The image is digitized in the computer using a PCEYE board[71]

which produces an 8-bit gray scale. The image is divided pixel by pixel by

an image of the cell without any pattern present. This is done to remove

inhomogeneities in the lighting.

The lower lens is a 20 mm diameter achromat with a 52.7 mm focal length.

The lens is fixed in place 6.63 cm above the cell. The second lens is a Nikon

50 mm f/1.4 camera lens. The Nikon lens and CCD camera are mounted

on separate movable carriages. The design of the carriages allows the relative

position of the Nikon lens and CCD camera and the position of the Nikon lens

and camera as a unit to be adjusted independently.

The magnification is estimated using the thin lens formulas 1/f = 1/i +

1/o and M = i/o where f is the focal length of the lens, i is the image

distance, o is the object distance, and M is the magnification. Using these

formulas, the fixed achromat lens provides a magnification of approximately

4x. This also places the image of the first lens at a distance of 25 cm from

50

the lens. Because of the design of the carriages, the Nikon lens can only get

within ≈ 6 cm of the camera element for a magnification of 0.2x. This gives

a combined magnification of roughly 0.8x. With the maximum separation of

the camera and lens, the magnification of the Nikon lens is approximately

5x, for a combined magnification of 20x. For accurate determination of the

magnification, the aluminum cylinder containing the sample cell is mounted

on an x-y translation stage equipped with micrometers accurate to 1 µm. To

calibrate the magnification of the shadowgraph system, a reference object is

imaged in two measured lateral positions.

The shadowgraph apparatus functions as a microscope where the objective

lens and eyepiece (in our case, the camera lens) can be moved relative to each

other. By adjusting the location of the camera lens, one can image the focal

plane of the objective lens. The image in the focal plane of the objective lens

is the square of the modulus of the optical Fourier transform of the pattern

(the power spectrum). This follows directly from a consideration of Fraunhofer

diffraction. Two examples of images of the focal plane of the objective lens

and the corresponding image of the pattern are shown in Fig. 3.6. The image

on the left is for a relatively strong pattern, so the dominant peaks in the

power spectrum are due to the quadratic effect. Because the underlying pat-

tern is a superposition of degenerate oblique rolls, the quadratic dependence

on the director variation produces eight peaks corresponding to the sums and

differences of the fundamentals. Because of the bright central spot (the image

of the light source) and the large intensity of the second harmonics, the fun-

damental peaks are barely visible in this image. The image on the right is for

a weaker pattern. In this case, we only show the first quadrant of the focal

plane, and by enhancing the contrast of the image of the focal plane we can

pick out the fundamental spot. This highlights one of the limitations of using

the focal plane; the trick of minimizing the quadratic term by focusing on or

near the cell is not available. The other limitation is that unless one blocks off

regions of the cell, the image in the focal plane is the power spectrum of the

51

entire cell, not just the region which is convecting. There are possible benefits

of using the focal plane. For example, a number of well know image enhancing

techniques exist which involve manipulations of the spots in the focal plane.

Two examples are dark-field microscopy (blocking the central peak in the focal

plane) and the Schlieren method (block half of the focal plane).

kx

ky a b

c d

Figure 3.6: The images (a) and (b) are of the focal plane of the objective lensin the shadowgraph apparatus. Image (a) has (kx = 0, ky = 0) in the center asshown, and image (b) has (kx = 0, ky = 0) in the lower left corner of the image.The image (c) and (d) are the corresponding patterns which are observed in thecell, respectively. The patterns consist of a superposition of degenerate right-and left-traveling zig and zag rolls. Image (c) is a relatively large amplitudepattern, and the (a) highlights the dominance of the nonlinear shadowgrapheffect in the focal plane. Image (d) is for weaker convection, and (b) has beensufficiently enhanced that the spot corresponding to the fundamental is visible(the circled region in (b)).

52

3.1.2 Temperature Control

The temperature-control stage is shown schematically in Fig. 3.7 and consisted

of a cylinder of aluminum 6.78 cm high with a diameter of 9.78 cm wrapped

with 0.64 cm of insulating foam. The aluminum cylinder consists of a top and

bottom half which are screwed together. To allow for illuminating the cell

from below and viewing it from above, there is a 1.40 cm diameter hole along

the axis of the aluminum cylinder. The hole is closed at the top and bottom

with glass windows. A 0.318 cm wide and 2.54 cm high circular channel with

an inner radius of 3.56 cm is located with its midplane at the midplane of the

aluminum cylinder and surrounds the center of the cylinder. Water enters the

channel on one side of the aluminum block, flows around the cell, and exits

from the side on which it entered. (In Fig. 3.7, only the inflow connection

is visible.) The water is used to maintain the temperature of the aluminum

cylinder.

The temperature of the apparatus is regulated with an analog control sys-

tem. The cylinder’s temperature is measured with a calibrated thermistor

which is embedded in the aluminum just below the cell to the side of the

inflow channel. The thermistor serves as one arm of a standard Wheatstone

bridge. The operating temperature is set by a variable resistor which is also

part of the bridge. The error signal from the bridge is input into an integrating

amplifier, and the output of the amplifier is used to set the voltage across a

wire heater embedded in the circulating water. The integrating amplifier pro-

vides both integral and proportional control of the temperature. The integral

control eliminates long term drifts in temperature. Both the proportional gain

and the integration time constant are set by resistors and capacitors in the

amplifier. The gain and time constant can be adjusted by selecting one of 6

ranges. The wire heater has a resistance of 7 ohm. The power output from the

heater ranges from 3.5 W to 60 W, depending on the operating temperature.

The typical ohmic heating of the EC cell itself is negligible. The samples have

53

CELL

Translation Stage

Aluminum Block

Insulation

water thermistor

Electronic connections

Figure 3.7: Schematic of the temperature control stage. Main elements arethe water channel, electronics feed-through, and thermistor locations. Thealuminum block consists of two halves (divided at the horizontal white line)which are screwed together. In this view, the thermistor is going into the page,and there is another thermistor hole opposite the one shown here. The outputfor the water channel is located directly behind the input connection.There arefour BNC connectors in line with the one shown for input/output of electronics.

a resistance of roughly 8×106 ohm and are typically run at 20 V for a power of

50 µW. When operating at temperatures above 25 C, the room is an adequate

heat sink. For lower temperatures, it is necessary to pass the circulating water

through a heat exchanger in which the cold side is maintained by a Neslab

RTE-110 refrigerator.

Currently, the upper limit to the operating temperature of the experiment

is around 70C. (Above this temperature, the tubing currently used in the

circulating bath has an increased risk of leaking.) For higher temperature

work, the apparatus described in Ref. [67] can be used. The temperature sta-

bility of the aluminum is measured by a second thermistor embedded in the

aluminum just below the cell but next to the outflow of the circulating water.

Temperature stability of ±5 mK (rms) for the aluminum is easy to achieve,

and generally we operate with a temperature stability of ±1 mK (rms). Typ-

ical temperature records for two hours and for 14 hours are given in Fig. 3.8.

54

The thermistors in this apparatus were initially calibrated against a glass ther-

mometer for which the absolute temperature can be off by degrees. Recently,

the thermistors were recalibrated against a thermistor that was calibrated

against a standard platinum thermometer. The results of this calibration are

used in this thesis, and both calibrations are given in Appendix A.

0 20 40 60 80 100 120time (min)

34.912

34.914

34.916

tem

pera

ture

(o C)

0 5 10time (hours)

34.912

34.914

34.916

tem

pera

ture

(o C)

Figure 3.8: Two records of the temperature of the aluminum block. Thetop record is for a two hour time span with the temperature recorded everyminute, and the bottom record is for a 14 hour run with the temperaturemeasured every 10 minutes. In both cases, the temperature was measured tobe (34.9 ± 0.2)C with a rms stability of ±0.001 K.

55

3.1.3 Electronics

The ac voltage signal was generated by a computer controlled synthesizer

card[73]. The card was capable of generating arbitrary waveforms, but for

these experiments, only sinusoidal waveforms were used. When changing the

amplitude of the output waveform, the synthesizer card recomputed the entire

waveform. In general, this resulted in a phase jump in the output waveform.

To eliminate this problem and achieve higher resolution voltage steps, the

output of the synthesizer card was used as the input to the circuit shown in

Fig. 3.9.

17 2

14 DI/O

dat

a

12 16 15

19

Vss

20

23

24 1

18

Vss

Vdd

Vout

12-bit multiplying DAC

3 K

1 nF 100K

100K

75K

15

K

7.5K

1.

5K

1K

A

C B

D E

F

10K

10K 10

K

200

10M 10

0

100

1 µF

1 M

INA105

input stage

selection stage

output stage

Figure 3.9: Schematic of circuit used to divide voltage from quatech synthesizercard.

The circuit is divided into four sections by the dashed boxes. The resistor

and capacitor values are shown in Fig. 3.9. Unless labeled otherwise, the op

amps are all LF355.

56

The first element of the input stage is a Burr-Brown INA105 differential

amplifier which is being used as a unity gain difference amplifier. The output

of the synthesizer card is a BNC cable with the inner conductor connected

to the negative input (pin 2) of the INA105 and the shield connected to the

positive input (pin 3) of the INA105. The INA105 is connected to ground

through its pin 1. The output is at pin 6 and goes to a low pass filter with

1/(2πRC) = 53 KHz. This smoothes out the digital steps in the waveform

generated by the synthesizer. Finally, the signal is amplified by a factor of 2.

The second stage is used to select the input to the multiplying DAC. If the

points D and F are connected, the switches AB and BC must be open. For

this case, the full signal is used as the input to the multiplying DAC. If D and

E are connected, then by connecting A and B or B and C either 10% or 1%

of the full signal is used as the input to the DAC, respectively. For these two

settings, higher resolution voltage steps are achieved while the overall voltage

range is limited to ±10% or ±1% of the base signal.

The central element of the circuit is the 12-bit multiplying DAC chip, Ana-

log Devices AD7845. The input to the chip is the analog signal from the selec-

tion stage (pin 17) and a digital signal generated by the digital input/output

(DIO) lines (pins 2 - 14) on a Labmaster card[74] in the computer. The output

of the 12-bit multiplying DAC is the analog signal divided by a factor ranging

from 0 to 4096 which is set by the digital input. The 12-bit multiplying DAC

produces changes in the output without any phase jumps as the value of the

digital input is changed.

The final stage is the output stage. It contains an adder circuit for use

when only 10% or 1% of the signal was the DAC input. In those cases, the

output of the DAC is added back to the portion of the full signal which is at

D and E. In the case when the full signal is the input to the DAC, this circuit

serves as a unity gain amplifier. The last stage of the output is a high pass

filter to remove any dc component.

The signal from the circuit was amplified with a commercial power amp-

57

lifier[75]. The frequencies used for the electro-convection experiments ranged

from 25 Hz to 2000 Hz and the voltages used ranged from 0 V to 85 V (all

voltages quoted are root mean square values).

The resistance R and the capacitance C of the cell were measured using

the circuit shown in Fig. 3.10. Based on modeling the cell as a resistor and

Vin Vout

Cell

Rf

Cf C

R

Figure 3.10: Schematic of circuit used to measure capacitance and resistanceof cells.

capacitor in parallel, we define

1/R(ω) = Re[I(ω)/V (ω)] (3.2)

ωC(ω) = Im[I(ω)/V (ω)], (3.3)

where I(ω) is the current through the cell in response to an applied voltage

V (ω) = Vin cos(ωt). The circuit shown in Fig. 3.10 converts the current I to a

voltage which is digitized in the computer with an A/D converter. The phase

and amplitude of the output voltage are extracted by fitting the digitized data

to Vo(ω) = Vout cos(ωt + φ). The amplitude and phase of the output voltage

are related to the resistance and capacitance of the cell by

Vout =Rf

R

√1 + ω2R2C2

√

1 + ω2R2fC

2f

Vin (3.4)

φ = π + tan−1(ωRC − ωRfCf

1 + ω2RCRfCf

) (3.5)

58

500 1000 1500 2000

applied frequency (Hz)

0.5

1

1.5

2

gain

Figure 3.11: The circles are the measured values of the gain |Vout|/|Vin| for thecircuit shown in Fig. 3.10 with the cell replaced by a resistor Ri = 1.015 ×106 ohm. The solid line is the fit to Eq. 3.6. The result of the fit is Rf =2.226 × 106 ohm and RfCf = 5.95 x 10−4 s. Independent of the circuit, Imeasured Rf = 2.2266×106 ohm using an ohm meter, and using a capacitancebridge, Cf = 267.8 pF. This gives RfCf = 5.96 x 10−4 s.

In order to compute the resistance and capacitance of the cell we need to know

Rf and RfCf . Since we want to know their effective values in the circuit, we

calibrated Rf and RfCf by replacing the cell with a known resistor Ri =

1.015 × 106 ohm (measured using a digital multimeter). We then measured

the gain |Vout|/|Vin| and fit to the function

|Vout||Vin|

=Rf

Ri[1 + (ωRfCf )

2]−1/2 (3.6)

with Rf/Ri and RfCf as the fit parameters. A typical measurement and the

resulting fit is shown in Fig. 3.11.

The accuracy of our measurements was checked by measuring the resis-

tance and capacitance of the parallel combination of a known resistor and

59

capacitor similar in value to those of a filled cell. The measured value of the

capacitance only varied by ±0.2% over a range of 20 V. The resistance mea-

surements showed a monotonic decrease over the 20 V range of 2%. The results

for the measurements as a function of frequency are shown in Fig. 3.12 and

Fig. 3.13. One can see that this technique had difficulties at high frequencies.

We computed the values of C and R from

R =−VinRf

Vout(cosφ+ ωRfCf sinφ)(3.7)

C =Vout

VinωRf

(sinφ− ωRfCf cosφ) (3.8)

To avoid oscillations in the op amp circuit, I used RfCf = 5.95 x 10−4 s.

This meant that even at frequencies as low as 1000 Hz our technique was sensi-

tive to any errors in the measured phase shift φ. The systematic error increased

as the difference between the measured resistance and the test resistor Rf in-

creased. For the EC cells, I actually needed the conductivity σ = (d/A)(1/R)

where d is the cell thickness and A is the area of the electrode. As d could

only be determined to ≈ 10%, the error for the typical cell resistances (of the

order 86 ohm, see Fig. 3.13) were not significant.

There were cases where the cell resistance was closer to 16× 106 ohm, and

with the chosen values of Rf and Cf , the errors at high frequency could be

as large as 20%. This is shown in Fig. 3.14 where the measurement made

with this circuit is compared with a measurement made using a capacitance

bridge[76]. The 20% error is the result of an error of only ≈ 0.004 rad in the

determination of the phase of the output voltage relative to the input voltage.

This error is probably due to nonideal op amp behavior. Even this error is not

generally a problem for my measurements. The values chosen for Rf and Cf

result in the circuit being least sensitive to the phase in the range of 50 Hz to

100 Hz. As shown in Fig. 3.14, the circuit does a good job of measuring even

the 16× 106 ohm resistor in that frequency range. However, it is important to

be aware of this limitation of the circuit when measuring the frequency as a

60

function of resistance for resistances significantly greater than 10× 106 ohm.

61

0 500 1000 1500 2000applied frequency (Hz)

420

440

460

480

500

capa

cita

nce

(pF)

0 500 1000 1500 2000applied frequency (Hz)

-20

-10

0

10

capa

cita

nce

(pF)

0 500 1000 1500 2000applied frequency (Hz)

420

440

460

480

500

capa

cita

nce

(pF)

Figure 3.12: The three figures show the measured capacitance as a function offrequency for a test capacitor (top), test resistor (middle), and the resistor andcapacitor in parallel (bottom). The bottom figure also shows the capacitancefor the parallel combination of resistor and capacitor which you would expectfrom the individually measured capacitances. (solid line)

62

0 500 1000 1500 2000applied frequency (Hz)

0

200

400

resi

stan

ce (1

06 ohm

)

0 500 1000 1500 2000applied frequency (Hz)

7.5

8

8.5

resi

stan

ce (1

06 ohm

)

0 500 1000 1500 2000applied frequency (Hz)

4

6

8

10

resi

stan

ce (1

06 ohm

)

Figure 3.13: The three figures show the measured resistance as a function offrequency for a test capacitor (top), test resistor (middle), and the resistor andcapacitor in parallel (bottom). The bottom figure also shows the resistancefor the parallel combination of resistor and capacitor which you would expectfrom the individually measured resistances. (solid line)

63

0 500 1000 1500

applied frequency (Hz)

13

14

15

16

17

18

resi

stan

ce (1

06 ohm

)

Figure 3.14: Measurement of a 16.7 × 106 ohm resistor in parallel with a465 pF capacitor. The open circles were measured using a capacitance bridgeand lock-in amplifier. The closed circles were measured using the circuit ofFig. 3.10

64

3.2 Liquid Crystals

The standard NLC used for EC is 4-methoxybenzylidene-4′-butylaniline (MB-

BA), and the main NLC studied in this thesis is 4-ethyl-2-fluoro-4′-[2-(trans-

4-pentylcyclohexyl)-ethyl]biphenyl (I52)[41]. For comparison, their chemical

formulas are shown in Fig. 3.15. The other common NLC used in EC is a

mixture known as Merck Phase V. Since this is a mixture, I have not used

this particular NLC. The other two NLCs which I have studied are trans-

4n-pentyl(2/-fluor4/n-pentylphenyl)cyclohexane carboxylic acid ester (D55-F)

and trans-4n-propylcyclohexane carboxylicacid-(4-n-propylencyclohexanol) es-

ter (OS-33). These are also single component NLC’s which show promising

behavior for future study. Currently, they are difficult to obtain as the man-

ufacturer, Merck Inc.[77], is limiting the purchase of single component NLCs.

O C N

F

(a)

(b)

Figure 3.15: (a) Chemical formula for MBBA. (b) Chemical formula for I52.

65

The use of MBBA in most EC experiments is largely historical. MBBA

was one of the first single-component, room-temperature NLC with εa < 0.

Its nematic range is rather small, 20 to 40C. It was developed for use in dis-

plays which were based on the transition to the dynamic scattering mode[63].

The dynamic scattering mode occurs as a secondary bifurcation from the ini-

tial transition to EC. Being of interest for displays, a lot of information on

the alignment properties, material parameters, and doping characteristics of

MBBA were obtained. This information was useful for the original EC exper-

iments. In particular, the knowledge of the material parameters was essential

for quantitative comparison of experiment and theory.

There are a number of standard dopants used with MBBA for which the

conductivity as a function of concentration has been measured[63]. One of

the most common is a solution of 0.01% tetrabutylammonium bromide in

MBBA which provides conductivities of the order 10−7 ohm−1 m−1. Another

important parameter for EC is εa. For MBBA, εa ≈ −0.5 where the exact

value is slightly sample dependent. The absolute value of εa is large enough

that one generally observes normal rolls in EC in MBBA.

There are definite drawbacks to using MBBA. First, it is a known health

risk. One must be very careful to wear gloves and work under a hood at all

times. Second, it is highly unstable. MBBA easily decomposes in the presence

of water, and for this reason, it is difficult to obtain extremely pure MBBA. A

cell made with MBBA will initially “age” as any trace water that is present in

the cell (and often the epoxy used to seal the cell) reacts with the MBBA and

changes the properties of the cell. The cells eventually achieve an equilibrium

though the time required is often months to a year.

I52 is essentially the opposite of MBBA. It is a relatively new NLC, so most

of its material parameters are unknown[41]. It is nematic at room temperature,

but it has a wide nematic range with a melting point of 24C and a clearing

point of 103.4C. It does have a smectic B phase in the range 13 to 24C. In the

nematic phase, εa monotonically increases with increasing temperature from

66

−0.05 to 0.07, going through zero around 60C (see Appendix A). Because

εa ≈ 0, oblique rolls are the dominant pattern. Also, I52 was designed to be a

benchmark chemical and unlike MBBA, is very stable in the presence of both

light and water. I have found that some properties of I52 will change upon

extreme heating, as discussed below.

Because it is a nonpolar molecule, it is rather difficult to dope. Table 3.1

shows a list of attempted dopants1 and the success or failure. Here failure

corresponds to a conductivity < 10−9 ohm−1 m−1 for which the cutoff frequency

falls below 10 Hz and EC is not observed. As you can see from the list, I2 is

the only dopant which was found to work. Also, the concentrations required

are relatively large, O(2%); whereas, the resulting conductivities range from

10−9 to 10−8 ohm−1 m−1 and are one or two orders of magnitude smaller than

is typical for MBBA samples. It takes roughly two weeks to a month before

enough I2 dissolves and dissociates in the I52 for a solution to achieve a useful

conductivity at temperatures of roughly 30C to 60C. Because I2 is highly

volatile, the doped solutions generally remain useful for only 6 months before

the loss of I2 becomes too great. However, the relatively rapid evaporation

of I2 means that the final concentration of the solution is always unknown.

Therefore, it is not unreasonable to “recharge” a solution that is no longer

useful by adding more I2. In fact, a cell design utilizing a reservoir of NLC

allows for the placement of pellets of I2 in the reservoir which would maintain

a saturated solution. This would allow a given sample to be used for longer

periods of time.

The time it takes for the I52-I2 solution to reach a useful conductivity

can be decreased by heating the solution. However, one must be careful to

maintain the temperature at or below the clearing point. Samples of I52 heated

well above the clearing point for long periods of time underwent a permanent

change in εa. After such a heating, a value of εa > +0.2 was found for the

1I have to thank Floyd Klavetter at Uniax, Inc., Goleta, CA. for help in choosing possible

dopants for I52.

67

Table 3.1: Dopants for I52.

Chemical percent dopant success (yes/no)

tetra-butyl ammonium bromide (TBAB) 0.01 no

TBAB 0.03 no

TBAB 0.05 no

TBAB 0.1 no

sodium dodecylbenzene sulfonate 1.0 no

dodecylbenzene sulfonic acid 0.005 no

dodecylbenzene sulfonic acid 0.7 no

dioctylsulfosuccinate, sodium salt 0.3 no

Bis(ethylhexyl)hydrogen phosphate 0.1 no

cetylpyridinium, bromide 0.1 no

3-nitrophenol 0.2 no

TCNQ 0.5 no

phenylenediamine (P1) 0.6 no

tetracyanoethelyene (T1) 0.8 no

mixture of P1 and T1 0.8 no

FeCl3 0.9 no

iodine (I2) 1.0 no

I2 1.5 yes

entire nematic range. This effect was discovered while attempting to find a

method of doping I52 using various salts. When the sample was kept at high

enough temperatures to induce the change in εa, solutions with TBAB had

conductivities which approached 10−9 ohm−1 m−1. However, the change in

εa rendered this method of doping irrelevant for EC. The change in εa was

observed for pure I52 as well, so it does not seem to be an effect caused by

the dopants themselves, only the temperature. This change in εa was never

68

observed for samples kept at temperatures below the clearing point2.

The color of an I52-I2 solution serves as an indication of the conductivity.

The solutions initially are bright red. Solutions with a dark red to black

color have a conductivity of roughly 5 x 10−9 to 2 x 10−8 ohm−1 m−1. As the

sample continues to age, and the conductivity decreases, the color fades to

a pale pink. Once the sample has reached the dark red color range, a more

precise determination of the conductivity is made using a commercial cell from

DisplayTech[78] before filling a homemade cell.

For testing the conductivity of solutions, the commercials cells have the

advantages that they require less than two hours to assemble, always fill well,

and always align. Whereas, a homemade cell requires two days to make,

doesn’t always fill, and doesn’t always align. The commercial cells are also

smaller, so they require less NLC. Finally, the area of the electrode is better

defined in the commercial cells than in the homemade cell. The electrode in

the commercial cells covers a well-defined, limited region of the cell which only

contains NLC. In the homemade cells, the electrode covers the entire slide and

measurement of the geometrical factor is complicated by the sealant, gasket,

and air bubbles. This is important when measuring conductivities which are

defined as σ = (1/R)(d/A) where R is the measured resistance, d is the cell

thickness, and A is the area of the electrode (see Sec. 3.1.3).

The DisplayTech cells are 10 µm thick with an electrode that is 0.5 cm ×0.5 cm in area. The cell is shown schematically in Fig. 3.16. The bottom glass

plate of the cell extends out on one side (in Fig. 3.16, this extension is to the

right) and has two conducting strips for the attachment of wires. The wires

are connected using the same method described in Sec. 3.3.4 for the connection

of wires to the homemade cells. The cells are filled using capillary action from

one of the two holes located opposite each other. The cell is supported with

one hole at the top, and a small drop of solution is placed on the hole. After

2There have been reports by J.T. Gleeson that temperatures as low as 80C can affect

εa, but I have not observed this.

69

Figure 3.16: Schematic drawing of the top view of a commercial cell. Thegray shaded regions represent the central electrode and the two electrodesused for connecting wires to the cell. The dark circles represent the glassbeads embedded in the UV epoxy which set the cell spacing. Notice, there aretwo filling channels, one at the top and one at the bottom of the picture.

the cell has filled, the holes are wiped clean of any excess solution and sealed

with 5 minute epoxy. The cells are then placed in the apparatus where the

conductivity can be measured as a function of temperature.

In addition to measurements of σ, the commercial cells can be used to

determine if the εa of an NLC is positive or negative. If εa is positive, one will

observe the Frederiks transition instead of EC when a large enough voltage is

applied to the cell. The onset voltage for the Frederiks transition will give a

measure of the magnitude of εa when the elastic constants are known. Initial

observation of any new NLC or NLC/dopant combinations are best done in

the commercial cells. One quickly determines the alignment properties of a

rubbed polyimide and the conductivity of the solution.

Initial surveys of EC in new NLC are possible in the commercial cells.

However, when observing EC in the commercial cells, it is important to know

that the alignment layer has been assembled in the parallel orientation (see

Fig. 3.19 in Sec. 3.3.2). This is believed to result in “drifts” of the EC patterns

70

which may be confused with a Hopf bifurcation. A characteristic of the drifting

pattern is that it always travels in the same direction in the cell. Another

drawback of the commercial cells for EC is the electrode design. Because

there is a region of NLC outside of the conducting coatings (see Fig. 3.16),

EC always starts at the edge of the conductor. The apparent cause is the

fringing fields of the conductor which preferentially tilt the director away from

its planar orientation. The director field is “softer” in this region and will

tend to convect first. Finally, 10 µm is a very thin cell. The wavelength of the

pattern is close to the limit of resolution of the shadowgraph, and the validity

of many of the SM assumptions begins to break down for thicknesses at and

below 10 µm.

As mentioned, the NLC D55-F and OS-33 are candidate NLC which are

of interest. They have εa ≈ −1 which is a range of εa for which normal rolls

dominate. Both of these NLCs are easily doped with I2, but being nonpolar

molecules like I52, they are difficult to dope with any of the traditional (and

many nontraditional) salts. There was limited success in achieving conduc-

tivities in the 10−8 ohm−1 m−1 range by heating solutions of either OS-33 or

D55-F and tertrabutylammonium bromide for 1 month at temperatures well

above the clearing point. So far, no adverse effects of the high temperatures

have been observed as in I52, but a supply shortage has prevented any definite

conclusions.

The initial studies of D55-F and OS-33 have suggested the transition is

forward and stationary. One definitely observes normal rolls. One possible

drawback of these two chemicals is that their index of refraction anisotropy

∆n is rather small. This means that the power in the shadowgraph images is

lower for both D55-F and OS-33 than it is for I52 and MBBA (see Sec. 3.1.1).

Detailed studies of EC were not carried out for D55-F and OS-33 largely due

to the limits of I2 doping and availability of the NLCs.

A key element of the experiments was the ability to vary σ. In a given

sample, σ was varied by approximately a factor of 2 by changing the tem-

71

perature over the range of 25 to 65C. By varying the amount of dopant,

I achieved an additional variation in σ of up to a factor of 10 from sample

to sample. Because I varied σ for a given sample by changing temperature,

it was necessary to understand the role of the other temperature dependent

material parameters. A detailed discussion of the temperature dependence of

the material parameters is given in Appendix A.

3.3 Cell Construction

The standard EC cell consists of two glass plates separated by a spacer and

sealed by an epoxy. The glass plates are coated with a transparent conductor

and are surface treated so as to align the NLC. I will refer to these as sealed

cells. The construction of these type of NLC cells remains more of an art form

than a science because of the difficulties associated with aligning the NLC and

the potential for chemistry between the NLC, the dopant, the sealant and the

spacer. There are commercial cells available; however, they have limited uses

and were discussed in detail in Sec. 3.2. I will address both methods which

I have confirmed to work as well as ones which others have used but have

not worked for me. The second category is included as alternatives for those

people for whom my methods fail.

The cells are designed to be sturdy and easy to handle. Depending on the

NLC, they have a useful lifetime of anywhere from 6 months to years. The

main drawback to this design is that both the spacer and the epoxy often

react with either the NLC or the dopant. This results in cells which “age”.

The better cells have a relatively short time constant for the aging process

and reach an equilibrium state in a reasonable time. For many cells, the time

constant is quite long and the critical voltage in such cells drifts in a roughly

linear fashion (see Sec. 4.3). In general, the cells with a long time constant

reach a prohibitively low conductivity before achieving equilibrium. In some

situations, the aging can be solved by a better choice of epoxy and spacer. An

72

alternative is to change the cell design completely to an “open” design with

adjustable top and bottom plates.

The methods describe here are not 100% effective. For this reason, the

equipment used in cell construction has been designed to make two cells at a

time. There are four main steps to the building of a cell: preparation of the

glass, alignment treatment, sealing, and filling. The steps up to and including

sealing the cell will take about 6 - 8 hours plus an additional 6 - 10 hours

curing time for the epoxy. Filling the cells takes anywhere from 1 - 3 hours.

A list of required materials by step are giving in Table 3.2. I will discuss each

step in the process separately.

3.3.1 Glass Preparation

The first step in the construction of the cells is to acquire glass slides with

indium-tin oxide (ITO) coatings. Other transparent conductors are possible,

but ITO is the most common one and easy to obtain. Donnely[79] and Libbey-

Owen[80] are two companies which provide free samples of ITO coated glass.

The samples come in sheets of approximately 12” x 12” and are standard float

glass. Donnelly provides glass which is 1 mm thick and has a resistance of

200 ohm per square. The glass from Libbey-Owen has a thickness of 2.5 mm

and a resistance of 20 ohm per square. This thick glass is preferable because

the resulting cells are more durable, and it is easier to cut than the thinner

glass. Because of this, all glass-slide holders have been designed to hold the

thicker slides. (All but the evaporation holders can actually accommodate

slides of either thickness.) The cells are currently made from 1” x 1” pieces of

the glass.

After the glass has been cut, the slides must be cleaned. Practically every-

one has their own favorite cleaning method, and I settled on a combination of

methods that works well. The steps in the cleaning procedure are:

1. Wash your hands so that they are as clean as possible. Then, using your

73

hands, scrub the glass slides in a solution of Joy liquid detergent and

water. Rinse well with tap water to remove the soap, and then rinse

with the deionized water to remove the tap water.

2. Ultrasound the slides in a solution of 10% liquinox and 90% diluted

ammonium hydroxide. (350 ml distilled water, 50 ml NH4OH, 40 ml

liquinox) for 15 - 30 min.

3. Rinse with tap water followed by a rinse with Milli-Q[81] filtered water.

4. Ultrasound for 20 min in spectroscopic grade acetone.

5. Ultrasound for 20 min in spectroscopic grade methanol.

6. Rinse in Milli-Q[81] filtered water.

7. Dry by blowing 99.99% pure N2 gas across the slides under the Laminar

Flow hood.

A slide holder (typically Teflon) should be used for carrying the cleaned slides

and placing them in the solvents. After cleaning the glass slides, the rest of the

steps should be performed under the laminar flow hood to keep them clean.

The only exception to this is steps involving evaporations. For transporting

the cells to and from the evaporator, one should use a sealed container. There

are microscope-slide holders which are ideal for this purpose.

The next step in the preparation of the glass slides is evaporation. There

are three different types of evaporations which are used in EC cells. Depend-

ing on the type of cell which is being made, all, none or some combination

of the evaporations are used in the following order: uniform insulating layer,

“beaches”, and alignment layer. I will discuss the general techniques of evap-

oration first, and then highlight the important features of each of the three

types of evaporation.

All evaporations are performed on the side of the glass which has the

ITO coating. SiO is a convenient material to use for all three of the possible

74

evaporations. Because it is an insulator, it is ideal for both the insulating layer

and the beaches. Its aligning properties are well known, and it can be used

to obtain a number of different alignments. Also, evaporation of Si0 does not

require extremely low pressures, so it it possible to accomplish it relatively

quickly. One should be aware that even though one uses SiO as the source for

evaporating, the resulting layer is actually SiOx, i.e. some combination of SiO

and SiO2.

SiO evaporation requires the use of special boats[82]. SiO tends to sputter

when evaporating, and the boats provide baffles which prevent “large” pieces

of SiO sticking to your sample. I recommend using boats from the SM-series

of the R.D. Mathis Co. [82]. The SiO should be in pieces of roughly 1 mm in

diameter (this corresponds to a +10 mesh size) and not in powder form. When

being used for the first time, the SiO needs to be heated to the point where it

just starts to spark, and then the temperature should be lowered to just below

this point. Let the SiO sit at this temperature for roughly 20 minutes. This

allows contaminates to leave the SiO. Finally, let it cool under vacuum. After

this treatment, the SiO can be taken in and out of the evaporator as much

as necessary without repeating the procedure. Relatively high pressures are

used for evaporation, only 10−4 torr with air bleeding. (This is the required

pressure for alignment purposes. For other evaporations, the pressure is not

critical.) The boat temperature should be around 1300 - 1400C, but will vary

depending on the desired rate of evaporation. If one uses the R.D. Mathis

SM-8 boats[82], this temperature range corresponds to a current range of 255

- 300 Amps. For more information on evaporation techniques, R.D. Mathis

provides excellent technical support.

In general, the ITO slides already come with an insulating layer coating the

conducting ITO. In the context of the WEM theory of EC, this is precisely the

boundary conditions we desire: perfectly insulating boundaries, i.e. no charge

flows between the NLC and the electrodes. This implies that the samples

would have infinite dc resistance. Initial measurements of the conductivity

75

do suggest that the dc resistance is effectively infinite. Therefore, I have not

evaporated any extra insulating layers onto the glass slides.

In contrast to the insulating layer, the Si0 “beaches” were evaporated on

all of the EC cells for which results are reported in this thesis. The beaches

are used to isolate the region in which EC first occurs from the spacers used

to set the thickness of the cell. This is accomplished by reducing the voltage

across the NLC under the beaches. For studies of the Frederiks transition,

the beaches are not desired because it is more important that the voltage be

uniform everywhere than that the transition be isolated from the spacers.

The “beaches” are constructed by evaporating a 1 µm thick layer of silicon

monoxide (SiO) in 0.2 cm wide strips onto the glass slides. The strips form

a square which separates the central 0.5 cm × 0.5 cm from the outer edges of

the cell. A rate of 25 - 50 [?]/sec is used. This corresponds to 265 - 285 Amps

for the R.D. Mathis SM-8 boats, depending on the amount of SiO in the boat.

The mask is made from 0.25” thick aluminum and is shown in Fig. 3.17. The

! " # $ % & '! " # $ % & '! " # $ % & ' ! " # $ % &! " # $ % &! " # $ % &

! " # $ % &! " # $ % &! " # $ % &! " # $ % &! " # $ % &! " # $ % &

side view of cell Top view of mask

(a) (b)

Figure 3.17: (a) shows a schematic top view of the mask where the shadedregions represent the aluminum. Notice that open sections are offset fromfrom the center of the mask. This is necessary so that the beaches are alignedwhen the cell is assembled. (b) shows a side view of an assembled cell withthe beaches, represented by the hashed boxes, aligned.

mask is designed to evaporate both slides at once and to correctly position

76

the beaches so that they are aligned when the cell is constructed. There is a

small section of each beach on which no SiO is evaporated. These regions are

positioned so that they do not overlap when the cell is assembled. The asym-

metry of the two beaches can be used to determine the direction of alignment

when needed (see Sec. 3.3.2).

Due to the drop in voltage across the SiO layer, the voltage drop across

the NLC is smaller under the beaches than in the central 0.25 cm2. This

ensures that convection occurs first in the central region of the cell, and that

it does not propagate in from the spacers. Figure 3.18 shows a Dektak scan

of a typical beach edge and the resulting profile of the voltage drop across the

liquid crystal. Notice that the x- and y-axes have very different scales, so that

the beach is actually extremely gradual.

0

0.5

1

Hei

ght (

µm)

0 0.2 0.4 0.6Position, arb. origin (mm)

6

6.5

7

Vol

tage

(Vrm

s)

Figure 3.18: Shown here is the result of a Dektak scan across one edge of thebeach. The scan measures the layers relative height. Also shown is an estimateof the corresponding voltage drop across the NLC when 7 V is applied acrossthe entire cell. The estimate assumes the cell and SiO layer act as dielectricsin series.

If evaporation is being used for alignment, it is always the final evaporation

so that the entire cell will be aligned. In particular, the region of the cell

corresponding to the beaches must be aligned if they are to effectively suppress

convection. Unaligned regions of NLC generally convect first which eliminates

77

the effectiveness of the beaches. All slides should have some type of alignment,

and whenever possible, that alignment should be done by evaporation. For

more details on alignment by evaporation, as well as by other methods, see

Sec. 3.3.2.

The final stage of slide preparation is the cleaning after evaporating. It was

found that even when a cold trap was used, evaporation of material onto the

slides without cleaning afterwards seriously hindered convection. The post-

evaporation cleaning also serves as a test of the pre-evaporation cleaning. If

the slides are not well cleaned before evaporation, the SiO does not stick well

to the surface. In this case, the post-evaporation cleaning will cause the SiO

beaches to flake off, and such slides should be rejected. The cleaning consists

of:

1. Ultrasound for 3 min in trichloroethylene.

2. Ultrasound for 5 min in spectroscopic grade acetone.

3. Rinse in Milli-Q[81] filtered water.

4. Dry by blowing 99.99% pure N2 gas across the slides under the Laminar

Flow hood.

It is important to note that no alcohol is used in the post-evaporation cleaning.

If for some reason, nothing is evaporated onto the glass, one must skip the

alcohol step listed in the initial cleaning method, and this second cleaning is not

needed. If alcohol is used to clean the glass slides either after the evaporation

of material or when no evaporation is performed, EC is suppressed. This effect

has been observed by other groups and is a classic example of the “black magic”

that is sometimes needed to make a good EC cell. There is no fundamental

understanding of what the cleaning with alcohol does to ruin EC.

If SiO alignment has been used, the slides are ready to be sealed. Oth-

erwise, after the evaporation of the beaches, the slides are ready for their

alignment treatment.

78

3.3.2 Alignment

A comprehensive source on aligning techniques is the reference by Jacques

Cognard, Alignment of Nematic Liquid Crystals and their Mixtures[59]. It

contains an extensive comparison of the various types of alignment, as well as

explanations of how alignment works. I will focus here on the techniques that

I have tried. As with evaporations, all aligning layers are applied to the side

of the glass with the ITO coating.

There are two general types of alignment, homeotropic and planar. Homeo-

tropic alignment is when the director is perpendicular to the glass plates. For

our purposes, this is useful for studies of the electric Frederiks transition in

NLC with εa < 0. Also, homeotropic alignment is required to study the

behavior of EC following a Frederiks transition in NLC with εa < 0. Uniform

planar (often just called planar) alignment is when the director is parallel

to the glass plates and in a uniform direction. This is the standard type of

alignment which is used for EC. Each of these general types of alignment is

further modified by the presence of any pretilt.

In most display applications, it is required that the alignment not be per-

fectly perpendicular or parallel to the glass plates. A small nonzero angle

between the director and the glass, referred to as pretilt, is required to break

the symmetry of the state. This limits the number of defects which occur

during the transitions. For “basic” studies of EC and Frederiks transition, we

almost always desire zero pretilt. In the case of planar alignment, it is difficult

to achieve exactly zero pretilt, but pretilts less than 2 are often referred to as

zero for practical purposes.

The reasons why a given combination of surface and NLC produces either

homeotropic or planar alignment are not completely understood, but there

are some general principles which appear to hold true. Whether one has

homeotropic alignment or planar alignment depends strongly on the interac-

tion between the surface and the NLC. For example, generally it is energetically

79

favorable for a NLC to sit with one end preferentially attached to a surfac-

tant (homeotropic alignment). Whereas, with most polymers and inorganic

surfaces, it is energetically favored for the NLC to be parallel to the surface

(planar alignment). For uniform planar alignment, the molecules must also be

forced to have a particular direction as well. This usually involves some method

of establishing grooves on the surface. Because of the elastic cost when the

director is perpendicular to the grooves, one obtains uniform alignment in the

direction parallel to the grooves. Because of this added complication, uniform

planar alignment is usually harder to achieve than homeotropic alignment.

For homeotropic alignment, I have used a coating of lecithin (egg yolk). A

1% solution of lecithin in chloroform is recommended though the exact concen-

tration is not critical. It is possible to buy precisely this mixture commercially

[83].

The glass slides should be cleaned using the method described in Sec. 3.3.1.

After cleaning, use the following procedure:

1. Have a petri dish, or other shallow dish, ready with chloroform for rinsing

the slides.

2. Remove the lecithin/chloroform solution from its container using a sy-

ringe and needle. The bottle has a septum, so it is not necessary to

remove the cap, and the lecithin will remained sealed. Occasionally, dry

nitrogen should be flowed into the lecithin bottle using a syringe needle

so as to maintain the pressure of nitrogen.

3. Holding one slide, place a few drops of lecithin on the slide.

4. Take the other slide and place it on top of the first slide. Press them

together so that the lecithin completely wets the two surfaces, removing

any bubbles. (Steps 3 and 4 must be done as quickly as possible, as the

cholorform evaporates quickly)

80

5. Carefully separate the two slides and place them face up in the petri

dish.

6. Remove them from the dish and place them in a holder with a loose

cover.

7. Bake the slides for 30 min at 80C.

Visually check the slides after removal from the petri dish. If they appear ex-

tremely messy, they can be gently rinsed with chloroform from a spray bottle.

During baking, the slides should be covered in some fashion to minimize the

accumulation of dust. However, they shouldn’t be in a airtight container as

part of the heating process is the removal of the remaining chloroform. Cur-

rently, this procedure must be performed in the fume hood to avoid breathing

the chloroform. This is not ideal as the slides are exposed to dust in the air

which can lead to dirty cells. So far, this alignment technique has only been

used to look at the Frederiks transition, and there has been no observation

of problems due to dust. However, this must be kept in mind if one plans to

study Frederiks to EC transitions.

I have confirmed that this alignment method works for 5CB, 8CB, MBBA,

and I52. If a new liquid crystal is used for which lecithin does not work,

there are plenty of other methods available. The other homeotropic techniques

are essentially the same as the above method, only the specific chemical and

solvent vary.

Because most planar samples have a small pretilt, there are two types

of planar alignment: parallel and anti-parallel. Consider the slides before

assembly with the polymer side up and the direction of rubbing is either to

the right or the left. The director can either be at an angle of Θ or π − Θ

with respect to the plane of the glass slides. Now, fold the slides together for

assembly as shown in Fig. 3.19. Parallel alignment refers to a cell in which the

director has a pretilt of Θ on one plate and π−Θ on the other. This results in a

variation across the cell of the angle between the director and the plane of the

81

plates. For anti-parallel alignment, the director is at either an angle of Θ or

π−Θ on both glass slide. This results in a constant director orientation across

the cell. These two possible planar alignments are illustrated in Fig. 3.19.

(π − Θ) (π − Θ)

(b)

(a)

(π − Θ) (Θ)

Figure 3.19: (a) Schematically shows the arrangement of the glass slides andthe rubbing direction (given by arrows) for parallel alignment. Notice the di-rector will splay as a function of position across the cell to match the boundaryconditions at each plate. (b) Schematically shows the arrangement of slidesand rubbing direction for anti-parallel alignment. In this case, the director isuniform across the cell.

There are two main methods of achieving planar alignment[59]: oblique-

angle evaporation and rubbed polymers. The oblique-angle evaporation is the

preferred method. For oblique-angle evaporation, a thin layer of material is

evaporated onto the glass slide with a nonzero angle between the slide and

the direction of evaporation (see Fig. 3.20). Under the correct conditions, the

particles already deposited on the slide “shadow” the particles which arrive

later. This shadowing produces steps or ridges which run along the glass slides

and provide the uniform direction for the director field. A number of materials

can be used for the evaporation, but I have focused on SiO because of the wide

range of pretilt angles which are possible. In most cases, one wants the pretilt

to be as small as possible, and with SiO evaporation one can achieve samples

with pretilts less than 1. Pretilts up to 30 are achieved in a reproducible

82

fashion by selecting the appropriate angle of evaporation. The exact value of

the pretilt will depend on the NLC being used3. This could be used to study

the effect of a variation in the pretilt angle as a function of horizontal position

on EC patterns[84].

SiO evaporated at an angle of 30 from the vertical with approximately 12”

between the source and the substrate (see Fig. 3.20) will achieve a nominally

zero pretilt for many NLC’s. The general techniques for evaporation of SiO and

slide preparation are described in Sec. 3.3.1. When using SiO for alignment,

one wants a 200 to 500 [?] thick layer evaporated at a rate of 5 [?]/sec using

a pressure of 10−4 torr. For evaporation, there is no “rubbing” direction,

but there are still the two possible alignments: parallel and anti-parallel. In

general, the two slides for a given cell are evaporated at the same time (see

Fig. 3.20). The pretilt will be in the same relative direction for both slides. If

the cell is assembled by holding the slides in the same relative orientation as

when they were evaporated and then folding the slides together, the result is

anti-parallel alignment. If one slide is first rotated by 180 and then the slides

are folded together, the result is parallel alignment.

The evaporation method was found to work very well with 5CB, 8CB,

MBBA, D55-F, OS-33. One drawback is that it does not work for I52. I52 has

a strong tendency to align perpendicular to the evaporated layer. Some effort

was made to vary the evaporation parameters, but no values were found that

worked well. MF2 was tried as an evaporating material, but it also produced

homeotropic alignment. Further work might reveal a combination of material

and evaporation parameters that produced planar alignment for I52, but be-

cause the rubbed polymer method does work for I52, evaporation techniques

were not pursued further.

The most frustrating method of planar alignment is the rubbed-polymer

technique. The number of prescriptions for obtaining alignment seem to be as

3The values for the pretilt are from Ref. [59].

83

source

θ

subs

trate

"ridges" (higly expanded view)

source

director alignment (a) (b)

Figure 3.20: Shown here is the standard arrangement for SiO evaporation forplanar alignment. (a) shows the front view of the holder with the two glassslides (shaded regions). The alignment will be in the horizontal direction. (b)shows the side view and a magnified view of the “ridges” which result from theshadowing effect. The angle θ should be 30 to produce nominally 0 pretilt.To achieve nonzero pretilts, θ is varied.

numerous as the number of people who make EC cells. This proliferation of

techniques is probably due to the delicate balance between choice of polymer,

rubbing material, and rubbing method. For rubbed polymers, the rubbing

makes the required grooves in the surface. I will describe the method I have

found to be most successful, and then offer variations that have worked for

other people.

The solution I recommend is a polyimide solution made from 1% poly(eth-

er-imide)[85] in methylene chloride. This corresponds to 0.533 g of polymer in

40 ml of solvent. If the polymer is found to streak upon spin-coating, a drop

of mineral oil can be added to the 40 ml solution. The steps used to produce

the aligning layer are:

1. Spin coat the slide with the polymer.

84

2. Let the slide sit for 10 - 15 minutes.

3. Rub slides with a polishing cloth.

To spin coat one of the glass slides, place the slide in the spinner and coat

it with a thin layer of the polymer solution. Turn on the spinner as soon as the

slides is coated to avoid significant evaporation of the solvent. For the cleanest

possible films, the polymer solution should be applied to the glass slide using

a syringe which has a 0.45 µm syringe filter. A spinning rate of 5000 rpm

results in a film with a thickness of roughly 0.35 µm. The UCSB fluids lab

spinner is calibrated so that a setting of 30 V on the variac corresponds to

5000 rpm. The film should be essentially transparent and appear smooth to

the naked eye. If the film is excessively streaky or too thick, remove the film

with methylene chloride and repeat the spinning procedure.

Rubbing the polymer coating by hand appears to be the best method to

achieve uniform alignment. A rubbing surface is constructed by wrapping an

Extec synthetic velvet[86] polishing cloth around a 2”x4” board of length 6”.

A piece of plexiglass is clamped to the board to serve as a guide when rubbing.

The slide is placed polymer side down on the cloth and rapidly moved along the

surface 12 times. It is important to use a very light touch applying essentially

no pressure to the slide. The strokes should be as smooth and as quick as

possible with all of the strokes in the same direction. Remember to note the

direction of rubbing for each slide so that the desired parallel or anti-parallel

alignment can be achieved.

One should be aware that during the rubbing, the slide will not always

slide perfectly straight or smoothly. It may even catch on the plexiglass and

swing quite far to the side. After all, you are doing this quickly and by

hand. Based on my experience with aligning, these “mistakes” generally do

not hurt the alignment and small imperfections in the rubbing may be essential

to the aligning process. This conclusion is based in part on observations of

the quality of alignment which is achieved by rubbing machines. The best

85

machine alignment was nominally uniform but had variations perpendicular

to the rubbing direction, or streaks. The machine duplicated the rubbing by

hand in all but two regards: the motion was always regular, and the speed of

rubbing was slower. Slowing down the rubbing by hand to match the machine

speed did not introduce streaks. This suggests that it is the very regular

motion of the rubbing machine which generates the streaky alignment, and

the more irregular motion of rubbing by hand eliminates the streaks.

A wide variety of polymers have been successfully used by others. Two

highly recommended solutions[87] are

1. polyvinal alcohol (PVA) with molecular weight 85,000 - 146,000: 60%

distilled water, 40% propanol, few milligrams PVA. After spin coating,

the slides need to be baked for 30 min at 80C.

2. polyvinal formal (PVF): 0.002 g PVF in 4 g chloroform. No baking

needed.

These can be spin coated using the same method as described above. There

may be cases where another polymer in needed. For example, polyimides

are known to have problems at high temperatures. In some commercial cells

which used an unknown polyimide, the alignment of I52 changed from planar

to homeotropic when heated above 70C. The alignment of I52 using the

poly(ether-imide) recommended here was at least stable to the maximum 65C.

Polyamides are good candidates for work at significantly higher temperatures.

An alternate spin coating method is to have the substrate spinning and

drop a few drops into the center. My experience with this method of spin

coating is that it causes streaking in the film which can affect the alignment.

There are a number of techniques and rubbing materials used to rub poly-

mers to achieve uniform alignment. An alternate material is cloth (cotton)

diapers. I have found that the diapers bought in Santa Barbara did not work

that well. An alternate rubbing method is to wrap the rubbing material around

a drill chuck inserted into a drill press with the drill running at high speed and

86

press the slide firmly against the cloth[88]. One group uses a 40-7482 Rayvel

polishing cloth wrapped around a cylinder of 2 cm diameter, rotating at 1 to

10 Hz[89]. The slide moves under the cylinder at a rate of 3 mil/sec. It is pos-

sible that there was sufficient play in this rubbing machine compared to the

one built by myself to produce the irregularities required for good alignment.

The quality of the alignment can only be checked after the cell is con-

structed and filled with the NLC. The simplest way to check the alignment is to

observe the sample between crossed-polarizers. As discussed in the Sec. 3.1.1,

NLC’s are birefringent. Polarized light passing through an EC cell may have its

polarization rotated depending on the relative orientation of the polarization

direction and the director. If the light is propagating along the director axis,

as is the case for a homeotropic cell with light propagating perpendicular to

the glass plates, all polarizations of the light have the same index of refraction

and the polarization does not rotate. Therefore, independent of the relative

orientation of the polarizers and the sample, a cell with uniform homeotropic

alignment between crossed polarizer will be uniformly dark.

As discussed earlier, light which has an arbitrary angle between its polar-

ization and the director is composed of an extraordinary ray and an ordinary

ray. For light propagating perpendicular to the director, the two rays acquire

a relative phase as they travel through the cell because of their differing ve-

locities. The phase shift produces a rotation of the polarization of the light

exiting the cell relative to the initial polarization. For the two special cases of

light polarized parallel or perpendicular to the director, there is only one ray

and the polarization is not rotated. Therefore, uniform planar alignment can

be checked by placing the cell between crossed polarizers. The transmitted

intensity is zero when the director is aligned with one of the polarizers and a

maximum when the relative angle between the cell and polarizers is 45. If

the sample is not uniform, there will be light and dark regions for all relative

orientations of the cell and the polarizers.

The alignment can be measure more quantitatively using conoscopy [90].

87

A converging beam of light is passed through the cell. The ordinary and

extraordinary rays interfere with each other to produce an interference pattern

which can be quantitatively related to the director orientation. This method is

a local measurement of the director and depends on the cell thickness. Because

of our cell thicknesses, the current optical system would need to be modified

to perform conoscopy.

3.3.3 Sealing

Sealing the cells is a straightforward process but does require a judicious choice

of materials. At this stage, one should have two glass slides with some or all of

the following: an ITO coating, insulating layer, beaches, and alignment layer.

A schematic of the assembled cell which shows all of the main parts is given

in Fig. 3.21.

The first step in sealing the cell is to make a spacer. In general, you want

the spacer to form a gasket which separates the working fluid from the epoxy

as shown in Fig. 3.21. In practice, one must leave a small fill channel in the

gasket. For certain choices of material, it is possible to fill the cell even if there

is no fill channel in the gasket. This is not recommended because cells filled

by this method generally have a large number of bubbles.

Mylar is the most common choice of spacer material due to its availability,

inertness, and ease of use. It does not appear to react with most of the NLC

used in EC. Two other materials which have been tried are mica and kapton.

Kapton is as easy to work with as mylar but much harder to acquire. Mica is

fairly easy to acquire, but if one wants cells with a thickness less than 75 µm,

mica becomes difficult to work with. The main drawback of mylar is that it

absorbs I2. As discussed in Sec. 3.2, I2 is used to dope I52. The absorption

of I2 by the mylar results in a drift in the conductivity of the sample. We

tested mylar, kapton, Teflon, mica, and glass for their relative absorption I2,

and only glass did not show significant absorption. Glass spacers would be

88

a nice option, but currently the thinnest glass available for spacers is 50 µm

optical fibers. These are extremely difficult to work with and are obviously

only useful for making 50 µm thick cells. One possibility is to copy commercial

cells and to use glass beads embedded in an epoxy. But, this is not useful as

a gasket which isolates the NLC from the epoxy. Pretreating the spacer by

exposing it to I2 is also a possibility, but some experimentation is needed to

find a proper length of time for the pretreatment.

The spacer needs to be cleaned, but remember NO ALCOHOL can be used

in cleaning the spacer. This causes the same problems as cleaning the glass

slides with alcohol (see Sec. 3.3.1). The spacers should only be washed with

soap and water and dried with N2.

Figure 3.21 diagrams the cell assembly. Once the spacer is cut out, it

is placed on one of the slides. If beaches are being used, the spacer should

be centered on the beach. If there are no beaches, the spacer needs to be

placed off center on the slide. The second slide is placed on top of the spacer,

shifted relative to the bottom slide. There should be about 0.1” width of each

glass which is uncovered and on opposite sides. This is for attachment of wire

connectors. The fill channel of the spacer should be on one of the sides which

is offset. This is the side which will be used for filling. The two slides are

clamped together so that one has easy access to the fill hole. A small amount

of glass wool is placed in the location where the fill hole will be to serve as a

filter. Cells constructed with this filter are essentially free of visible dust in

the interior of the cell. The glass wool is held in place by the sealant which is

placed around the slides. The two sides used for filling are sealed first leaving

a hole on each side for filling. The side which has the filter and fill channel

should also have a small well made from the sealant to hold the NLC in the

pre-filling stage (see Fig. 3.21). Now, adjust the clamps so that the non-filling

sides are accessible and apply the sealant to the entire length of the remaining

two sides taking special care to seal the corners.

There are a number of choices for sealants. The three used most commonly

89

in this lab are Torr Seal[91], UV epoxy[92] and adhesive cement[93]. The

adhesive cement is necessary for high temperature work. The Torr Seal is the

easiest to use. It is known to react with MBBA, but a good gasket between

the sealant and the NLC will limit the effects of this reaction (as well as

the reactions of any of the other sealants with the NLC). The UV epoxy is

the one used by much of the liquid crystal industry. All three sealants have

been tested for their reactivity with MBBA and no noticeable difference was

detected. One of the difficulties is that MBBA is so naturally unstable that

it is hard to detect a difference. With I52, no difference has been observed

between different sealants because the dominant effect on the aging of the cell

is the absorption of the I2 by the spacer.

For UV epoxy the sample is cured under UV light[94] for 10 minutes with

the clamps on the cell. After this, the clamps can be removed, but the epoxy

should be cured for another 20 minutes under the light. Then, it should be

put in the oven over night at 50C. This last stage bonds the epoxy to the

glass. Torr seal is cured at 55C to 65C for at least two hours. However, the

best cells have been made with the Torr seal curing overnight. The cement is

cured in 19 - 24 hours at room temperature.

It is a good idea after the epoxy is completely cured to use a mechanical

pump and place the cell under a low pressure for roughly an hour before filing

it. This ensures that the epoxy has bonded to the glass and allows some of the

residue from the cleaning agents, epoxies, etc. to be removed. If the epoxy has

not bonded to the glass, it will clearly bulge into the cell while pumping on the

cell. This is done without any liquid crystal present. Whenever pumping on

a NLC cell, it is necessary to use a cold trap as the pump oil can contaminate

the cell.

90

3.3.4 Filling

There are a number of different filling methods4 The simplest method is to fill

the cells under atmospheric pressure using capillary action. This is the quickest

method and works fine except that it will always leave bubbles of air in the cell.

The bubbles are a real problem when using I2 doped NLCs as the air will absorb

I2. Also, depending on the final location of the bubbles, they can interfere

with the EC. However, for the commercial cells, bubbles which result from

this filling method are always far away from the electrodes. Therefore, this

is the recommended method for filling commercial cells, especially when they

are being used for conductivity measurements or preliminary measurements of

EC.

The best filling method is to use capillary action under a vacuum. In this

case, the liquid crystal is placed in the well formed by the epoxy during the

sealing stage (see Fig. 3.21). Typically, 2-3 drops from a pipet is plenty. The

sample is held at an angle so that the NLC does not touch the fill hole. The

entire sample and cell is pumped on with a mechanical pump for roughly 5

minutes. It is important to watch the solution closely during this time. I52

has an extremely large surface tension, and as bubbles form you need to make

sure that the I52 does not run over onto the top glass surface. If the sample

is not too bubbly, one can pump for longer than 5 minutes. A dessicator was

fitted with a holder that allows two cells to be filled simultaneously and was

used only for filling NLC cells. The sample was tilted by tilting the entire

dessicator.

Once the sample has been pumped on, seal off the vacuum. Keeping the

sample under vacuum, tilt the slides forward so that the NLC contacts the fill

hole. The sample will fill under capillary action. Once the cell has filled in

this fashion, there still will be bubbles, but these are at a significantly lower

4I have Ingo Rehberg and Eberhard Bodenschatz to thank for information on previously

used filling methods. All of the methods described here had been used by them.

91

pressure than air pressure. Slowly release the vacuum, and the various bubbles

will be dissolved into the NLC, and one will have a bubble free sample.

A few warnings on filling. On occasion, the cell won’t fill completely on

the first try for various reasons. An attempt can be made to fill it again by

adding some NLC to the fill hole and pumping on the sample. One should be

aware that this moves around the material which is already in the cell. In the

past, this tends to produce highly nonuniform convection in the cell. If one

is lucky, the convection in samples in which this happens eventually becomes

uniform.

The final filling method is similar to the above method but less reliable.

One uses only a single fill hole. A drop of liquid crystal is placed over the hole

and the sample is pumped on. The air inside the cell bubbles out through

the NLC and the cell reaches very low pressures. For this method, one should

pump until there are essentially no more bubbles leaving the cell. This takes

at least an hour for most cells. Then, one slowly releases the vacuum. Done

slowly enough, the NLC is forced into the cell, and if the sample has been

pumped on long enough, the resulting cell is free of bubbles. This method

does not work well with I52 because of the high surface tension. I52 makes

rather large bubbles, and when they burst, they spread the I52 all over the

top of the cell. It will work for MBBA, but care needs to be taken to watch

for extra large bubbles.

The filling method is one reason to use torr seal over the UV epoxy as a

sealant when making cells with I52. I52 wets the UV epoxy so well that it

wicks right up into the cell before you can begin pumping on it. With Torr

seal, the I52 remains in the well made from the epoxy. The other liquid crystals

discussed in Sec. 3.2 did not have this problem with the UV epoxy.

After the cells have been filled, the areas around the fill holes need to be

wiped clean. Initially, they should be wiped only with DRY cotton swabs

or tissue. NO SOLVENTS OR WATER SHOULD BE USED. Once all the

NLC has been wiped off, they should be sealed using the Hardman 5 minute

92

epoxy[95]. This has been recommended by NLC display manufacturers as

having essentially no reactions with the NLC as it cures. Once the 5 minute

epoxy has been applied, the cell can be carefully cleaned with water or acetone

to remove the remaining NLC residue.

Having sealed the cells, all that remains is to attach wires with connectors

to the electrodes. Use 28 gauge multi-strand wire. Solder the connectors

onto the wire first, and then attach the wire to the electrodes of the cell.

For connectors on the wire, use single pins from IC sockets with the plastic

removed. (These connectors can be bought in breakaway strips for single pin

use.) The connecters in the apparatus are female, so solder the wire which is

fixed to the cell to the female end of the IC pin. Heat shrink tubing is used

on the wire/connector for strain relief. The wire is attached to the electrodes

with silver conducting paint[96]. A layer of 5 minutes epoxy is applied for

extra strain relief. It is important to make sure the silver conducting paint

has dried completing before applying the epoxy. You now have a finished NLC

cell.

93

Table 3.2: Required materials for cell construction.

Step material comments/

recommendations

general needs Teflon-coated tweezers used to manipulate glass

multi-meter determining ITO side of

glass

scissors

exacto knife

laminar flow hood

powder free gloves for flow hood

protective gloves for handling NLC

evaporator

cleaning acetone (spectroscopic

grade)

methanol (spectroscopic

grade)

liquinox

ammonium hydroxide

trichloroethylene

high purity N2 gas

beakers one per solvent

Teflon slide carrier holds slides in solvents

94

Step material comments/

recommendations

evaporation SiO evaporation boat R. D. Mathis SM-series[82]

+10 Mesh SiO

sealed slide carrier (glass) biology storeroom

evaporation mask

slide holder for

evaporator

oblique and normal angles

alignment syringe/ syringe filter

spinner planar alignment

polymer solution planar alignment

rubbing material synthetic velvet

lecithin homeotropic alignment

chloroform homeotropic alignment

shallow dish homeotropic alignment

95

Step material comments/

recommendations

sealing spacer/gasket mylar

sealant torr seal[91], UV epoxy[92]

or adhesive cement[93]

UV lamp[94] if using UV epoxy

UV protection goggles

clamps

glass wool

filling mechanical pump

dessicator fit with two cell holders

pipet disposable glass

5 minute epoxy Hardman epoxy[95]

cotton swabs removal of NLC from slides

96

(*)*+*,*-*.(*)*+*,*-*.

(*)*+*,*-*.(*)*+*,*-*.

(*)*+*,*-*.(*)*+*,*-*.

(*)*+*,*-*.(*)*+*,*-*.

(*)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/(*)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/(*)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/(*)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/(*)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/(*)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/(*)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/(*)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/(*)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/(*)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/(*)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/(*)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/(*)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/(*)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/(*)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/(*)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/(*)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/

33333333

fillin

g w

ell gl

ass

woo

l

Figure 3.21: Shown here is the standard assembled cell. The hashed regionrepresents the beaches. The gray region is the spacer and the solid black isthe sealant. The two glass slides are offset so as to be able to attach theelectrodes. To fill the cells, two holes are left in the sealant along the sideswhich are offset. One of the fill holes has some glass wool placed in front of itas a filter. This is the hole that will be used for filling. In addition, there isan extra bead of epoxy (shown by a black line) extending out from the edgeof the fill hole to form a well.

97

Chapter 4

Experimental Results

4.1 General Methods

In the course of this work, well over 100 cells were made. The results presented

here for EC using I52 are from a limited number of cells which are listed in

Table 4.1. (The cells used in the Frederiks experiment are discussed separately

in Appendix A.) For the rest of the thesis, I will refer to the cells by their label.

The majority of the experiments used cells I5243 and I5263. For these two

cells, EC was studied as a function of three control parameters: applied voltage

Table 4.1: Summary of main cells.

label date filled dopant % date doped thickness

I5221 2/9/93 1.4% I2 1/26/93 10 µm

I5243 12/16/93 2.2% I2 11/21/93 28 µm

I5263 5/19/94 2.3% I2 5/6/94 28 µm

I5268 9/12/94 2.3% I2 8/16/94 52 µm

I5275 11/28/94 2.3% I2 8/16/94 54 µm

I5278 12/21/94 2.3% I2 8/16/94 30 µm

98

V , applied frequency Ω, and conductivity of the sample σ. The cells I5221,

I5268 and I5275 were used to make preliminary observations of the dependence

of EC on cell thickness d.

A separate reduced control parameter ε(σ,Ω) = (V/Vc(σ,Ω))2−1 is defined

for each value of σ and Ω. Here Vc(σ,Ω) is the critical voltage for the onset of

EC as a function of σ and Ω. For this work, all of the studies were done by

varying V at fixed values of σ and Ω. So, Vc(σ,Ω) and ε(σ,Ω) will be written

as Vc and as ε, respectively, with the σ and Ω dependence assumed. This

is consistent with definitions used in the theoretical calculations where Vc is

computed for a given value of σ and Ω, and the weakly nonlinear analysis is

done in terms of small ε = (V/Vc)2 − 1.

The patterns consisted of roll-like solutions (modes) which can be written

as An(x, t) cos(knx− ωnt), where n labels the modes. Here kn and ωn are the

wavevector and the frequency of a given mode. The An(x, t) corresponds to the

complex amplitude described by the amplitude equations discussed in Sec. 2.1:

An(x, t) = |An(x, t)| exp[φ(x, t)] where |An(x, t)| is a slowly varying function,

and φ(x, t) gives the deviations of k and ω from kn and ωn. Most commonly,

the modes of interest were the four degenerate right- and left-traveling zig and

zag oblique roll states. For these solutions, the |kn| and the |ωn| are all equal,

and the angle Θ between kn and the director is θ for the zag states and π− θ

for the zig states. Choosing the x-axis parallel to the director, Θ = tan−1 pn/qn

where qn and pn are the x and y components of kn. There were two main types

of analysis used to study these patterns.

The first method was a general use of the power spectrum of the digitized

image to determine kn, ωn and the total amplitude An of a given mode. (The

details of the image digitization are given in Sec. 3.1.1.) The images Il,m,t

consist of discrete pixels where l, m are the x and y indices and t is the time

index. I will use a continuous notation I(x, t) ≡ Il,m,t where it simplifies the

expressions without introducing ambiguities. The wavevector k is taken to

have components q and p parallel and perpendicular to the director, respec-

99

tively, and ω is the angular frequency of the pattern. For time series of images,

the discrete power spectrum is

Sp,q,w = | 1

N

∑

all l,m,t

Il,m,t exp[2πi(pl/L+ qm/M + wt/T )]|2. (4.1)

Here L, M , T are the total number of x, y, and t points, respectively, with

the total number of points N = L +M + T . For single spatial images,

Sp,q = | 1

N

∑

all l,m

Il,m exp[2πi(pl/L + qm/M)]|2. (4.2)

Here N = L+M . Again, where convenient I will use continuous notation and

write S(k, ω) ≡ Sp,q,w and S(k) ≡ Sp,q.

By Parseval’s theorem, the normalization is chosen so that∑

all k,ωS(k, ω) =

1N

∑

all x,t[I(x, t)]2, i.e. the sum of the power spectrum is equal to the mean

squared amplitude. If we consider only spatial images of the form I(x) =

An cos(knx), then, [S(kn) + S(−kn)]1/2 = An/√

2. So, for a given mode, we

define An ≡√

2([∑

kn

S(k)]+[∑

−kn

S(k)])1/2, where the summation is over a small

region around kn and −kn. The summation is performed over a small region

around the peak because the underlying pattern is generally not commensu-

rate with the finite extent of I(x), and this results in a finite width for the

peak at kn. Equation 3.1 converts the An computed in this fashion from the

shadowgraph signal to a value for the amplitude of the director variation in

radians. In addition, one determines kn from kn ≡ (∑

S(k)k)/(∑

S(k)) where

the summation is over a limited region around the peak of interest. Equivalent

definitions and methods are used to determine ωn, kn, and An from the time

series with the summation over ω in addition to k.

The other analysis technique was used in the characterization of the ob-

served spatio-temporal chaos for which the behavior of An(x, t) is of particular

interest. To extract An(x, t) of a given mode, one demodulates the images by

performing the space-time Fourier transform of a time series of images and set-

ting the Fourier transform to zero everywhere except for a small region around

100

kx

ky

zag

zig

spatial Fourier transform

inverse Fourier transform

Figure 4.1: Example of a spatial demodulation into zig and zag rolls.

the kn and ωn of interest. Taking the inverse Fourier transform of this modi-

fied function results in a complex function of space and time. The real part of

this function corresponds to An(x, t) cos[knx−ωnt+φ(x, t)]. The modulus of

the function is just |An(x, t)|. One can also demodulate single spatial images

to find the amplitude of the zig and zag rolls without distinguishing between

right- and left-traveling states. An example of spatial demodulation which

takes the real part of the inverse Fourier transform is shown in Fig. 4.1. Note,

all of the images in this chapter have the director aligned horizontally (the

101

x-axis), and the gray-scaled plots of the power spectrum have (kx = 0, ky = 0)

in the center of the image.

In analyzing the shadowgraph images, there is a difficulty which arises for

traveling rolls which is not present for stationary rolls. The effect is an artifact

of the frame grabber[71] which is used in these experiments and predominately

affects the quadratic terms in the shadowgraph signal (see Sec. 3.1.1). Fig-

ure 4.2 shows two images (left-hand side) which were taken at rather large

values of ε where the quadratic effect dominates the shadowgraph signal1.

Also shown are their corresponding power spectra (right-hand side). The top

image was taken at ε = 0.015, and the nonlinear peak corresponding to the

direction perpendicular to the director is substantially larger than the peak in

the parallel direction. The lower image is at ε = 0.070, and the two peaks have

roughly equivalent amplitudes, as expected. For ε = 0.015 the rolls are trav-

eling and for ε = 0.070 the rolls are stationary. Because the frame grabber[71]

only reads single lines not frames, each image is composed of a number of

frames. This smears out the rolls in the direction of travel, in this case, paral-

lel to the director. Therefore, the relative strength of the peak perpendicular

and parallel to the director is misleading for the case of the traveling rolls. One

can see that the relative strengths are actually equal by observing the optical

power spectrum of traveling rolls (for an example, see Fig. 3.6 in Sec. 3.1.1).

There the frame grabber plays no roll in the computation of the power spec-

trum, and the peaks perpendicular and parallel to the director clearly have

equal intensity.

1Even when focusing on the cell (see Sec. 3.1.1), diffraction effects and the finite thickness

of the cell result in the quadratic shadowgraph effect eventually dominating the image for

high enough values of ε.

102

Figure 4.2: The top image (left) is a snapshot of a superposition of right andleft traveling zig and zag rolls at ε = 0.015. The director is aligned in thehorizontal direction. In the spatial power spectrum of this image (right), thepeaks corresponding to the sum of zig and zag rolls which gives a wavevectorperpendicular to the director are clearly stronger than those parallel. (Thepeaks of interest are highlighted by circles.) The bottom image shows thesuperposition of stationary zig and zag rolls at ε = 0.070. Here the peaks inthe power spectrum have roughly equal amplitudes. Notice, for the case ofε = 0.015, the fundamental peaks are still visible.

103

4.2 Conductivity

As discussed in Chapter 2, one of the outstanding problems in EC in NLC is

the existence of traveling waves. The detailed linear stability analysis[33, 34] of

the SM predicts that the initial bifurcation is to a stationary pattern; whereas,

there are a number of experimental observations of traveling patterns[24, 25,

26, 27, 28, 37, 97]. One of the motivations for studying new liquid crystals is

to acquire information which would be useful in guiding modifications of the

SM. Two assumptions of the SM, perfect planar alignment with no pretilt and

Ohmic conductivity (see Sec. 2.3), have been considered as possible weaknesses

of the SM[33, 34, 38] which are related to its failure to predict the observed

traveling waves. It would be interesting to study both of these assumptions.

Measurements of a number of alignment techniques[59] reveal that the

pretilts are less than 3, and for many applications this is effectively a zero

pretilt. However, pretilts of less than 1 are rarely achieved, if at all. In order

to quantify the effect of pretilts on EC, a reliable measurement of this generally

small pretilt angle is necessary. In principle, by converting the shadowgraph

light source from a parallel beam to a converging beam, the well known method

of conoscopy could be used to measure the pretilt (see Sec. 3.3.2). However, for

the typical cell thicknesses used in the EC studies, conoscopic measurements

on cells of planar alignment do not have the required resolution to make ac-

curate measurements of the pretilt. Furthermore, the recent theoretical work

described in the Sec. 2.3, the WEM[39], makes a strong case for the conduc-

tivity σ as the primary source of a Hopf bifurcation. For these reasons, I have

focused my studies on the conductivity of the I52 samples.

The conductivity of the sample is due to equal numbers of positive and

negative ionic impurities[64] which generally have different mobilities, undergo

a dissociation-recombination reaction[65], and have a finite time for traversing

the width of the cell. The WEM includes these additional effects into the

SM (see Sec. 2.3 and Ref. [39]), and the resulting conductivity is frequency

104

dependent. The dopant used with I52 was ≈ 2% by weight I2. In the simple

picture, I2 forms a charge transfer complex with the benzene ring present in

I52. It is this complex which undergoes a dissociation-recombination reaction

producing positively and negatively charged species. This multi-step process

is clearly more complicated than the simple dissociation-recombination picture

used in the WEM. I measured the frequency behavior of the conductivity of

the I52 samples to check for deviations from the predictions of the WEM.

The cell is treated as a resistor and capacitor in parallel with the resistance

R and capacitance C defined by

1/R(ω) = Re[I(ω)/V (ω)] (4.3)

ωC(ω) = Im[I(ω)/V (ω)], (4.4)

where I(ω) is the current in response to an applied voltage V (ω) = Vin cos(ωt).

For the ohmic conductivity assumed by the SM, 1/R and C are independent

of frequency. In the context of the WEM, C has a high frequency limit cor-

responding to the dielectric capacitance Cd = εoεA/d. At low frequencies,

the free ions can generate boundary layers which result in an increase of C

as Ω is decreased. This behavior is demonstrated by a typical measurement

of C which is shown in Fig. 4.3. The high frequency limit is consistent with

measurements of ε made in undoped samples.

The boundary layer thickness depends on the details of the electrical bound-

ary conditions at the two electrodes, but can be deduced from the low fre-

quency rise of the capacitance. This is apparent for the case of insulating

boundaries for which analytic expressions for C and R can be derived from

the WEM[99]. Assuming a boundary layer thickness lb,

C = Cd

(

1 +(d/2lb)

1 + (d/2lb)2(Ωτq)2

)

(4.5)

R = Ro[1 + (d/2lb)2(Ωτq)

2]/(d2lb)2(Ωτq)

2. (4.6)

Here Ro is a constant and represents the asymptotic high frequency value of

R.

105

0 500 1000 1500 2000

applied frequency (Hz)

400

600

800

capa

cita

nce

(pF)

Figure 4.3: Capacitance of I5263 at T = 49C as a function of frequency.

Figure 4.4 shows a plot of the data from Fig. 4.3 and three curves com-

puted using Eq. 4.5. Even though Eq. 4.5 captures the correct low and high

frequency limits, it does not fit the detailed shape of the curve in the transition

region. Despite this, one can use these curves to estimate an upper limit of

the boundary layer thickness of lb < 0.04d ≈ 1 µm. One possible source of

the discrepancy is that the boundaries are not perfect insulators. However,

if this were true, there would be a finite dc resistance. Figure 4.5 shows a

typical measurement of R using an applied voltage of 5 Vrms as a function of

frequency. I have observed no evidence of a finite dc resistance. Figure. 4.5

clearly shows the sharp rise in R in the low frequency limit which is consis-

tent with the curve predicted by Eq. 4.5 given by the dashed line in Fig. 4.5.

However, the high frequency limit of R does not behave as expected.

One contribution to the total R of the NLC cell which is not included in the

WEM is dielectric losses in the material itself. Figure 4.6 shows the measured

R of a cell filled with undoped I52. The cell has the same dimensions and

construction as cell I5263. For comparison, the R of a mica capacitor which is

106

0 100 200 300 400

applied frequency (Hz)

400

500

600

700

800

900

capa

cita

nce

(pF)

Figure 4.4: The measured capacitance of I5263 at T = 49C compared withthree curves calculated using Eq. 4.5. The curves use a boundary layer thick-ness of 5.0 µm (solid line), 1.0 µm (dashed line), and 0.2 µm (dot-dashedline).

known to be have relatively high dielectric losses and a polystyrene capacitor

which is known to be relatively loss free are shown as well. (The data for

the polystyrene capacitor does not exist for frequencies above 200 Hz because

the resistance was too large to measure.) The loss for I52 is definitely larger

than that for the mica capacitor, but it is consistent with other measurements

of dielectric losses in NLC[63]. The rather large losses in NLC at these low

frequencies are generally attributed to long relaxation times which result from

the various intermolecular interactions which contribute to the nematic order.

In addition to losses in the NLC itself, this measurement includes any losses

inherent in the cell, for example, the mylar spacers.

One can correct the resistance predicted by Eq. 4.5 by including the re-

sistance of the undoped I52 cell as an additional resistor in parallel with the

ionic contribution given by Eq. 4.5. The solid line in Fig. 4.5 is the result of

fitting this effective resistance to the data with Ro and lb as fit parameters.

The resulting values are Ro = 9.0 × 106ohm and lb = 0.9 µm. This result for

107

0 500 1000 1500 2000

applied frequency (Hz)

0

2

4

6

8

10

12

14

resi

stan

ce (1

06 ohm

)

Figure 4.5: The measured resistance of I5263 at T = 49C. The dashed lineis computed using Eq. 4.5 with Ro = 9.0 × 106 ohm and a boundary layerthickness of 0.9 µm. The solid curve is a calculation of the resistance due tothe resistance given by the dashed line in parallel with the measured resistanceof an undoped cell (see Fig. 4.6).

lb is consistent with the capacitance measurements.

In addition to the frequency dependence, I have also measured a voltage

dependence of R. This is shown in Fig. 4.7. The capacitance was found to be

voltage independent. The voltage dependence of R can not be explained by

the WEM.

In summary, the WEM explains the high and low frequency limits of the

frequency dependence of R and C. A more detailed description of the chem-

istry of the doping process and physics of the ionic species is needed to explain

the voltage dependence of R and some of the details of the frequency depen-

dence of R and C. However, the WEM represents a clear improvement over

the SM assumption of a frequency and voltage independent σ.

108

0 500 1000 1500

applied frequency (Hz)

101

102

103

104

resi

stan

ce (1

06 ohm

)

Figure 4.6: The measured resistance of a polystyrene capacitor (open trian-gles), mica capacitor (open circles), undoped I52 cell at 44C (solid circles),undoped I52 cell at 49C (solid triangles). The solid lines are fits of the datato R = A/(f + f0), where f is the applied frequency and f0 = 70 Hz.

10 15 20

voltage (Vrms )

8.4

8.5

8.6

8.7

8.8

resi

stan

ce (1

06 ohm

)

Figure 4.7: The measured resistance the cell I5263 at 49C and and appliedfrequency of 50 Hz as a function of applied voltage.

109

4.3 Linear Behavior

As discussed in Sec. 2.3, the linear properties of EC in I52, Vc, Θ and ωτq, have

been calculated within the context of a new model, the WEM, as a function

of Ωτq, σ and d (as well as the other material parameters). I will present the

results of the measurement of these three quantities in four cells and compare

the results with the calculations. The four cells used for the linear studies are

I5221, I5243, I5263, and I5275 (see Table 4.1). Initial studies of EC were done

using the cells I5221 and I5243. These were among the earliest cells made

and quantitative data on the Hopf frequency was not acquired. The cell for

which the most extensive quantitative work was carried out was I5263 with a

thickness of 28 µm. Limited measurements were made in the thicker cell I5275

to study the d dependence of ω.

Because the WEM predicts the linear values of ω and Θ, it was exper-

imentally useful that the initial bifurcation is forward, i.e. continuous and

non-hysteretic, in the parameter range of interest. The evidence for the for-

ward bifurcation is presented in Sec. 4.4. To extract the linear values, ω and

Θ were measured as a function of ε, and the results in the limit as ε→ 0 were

compared with theory. The step size in ε was δε = 0.001 with a waiting time of

300 s (400τd). For each value of ε, a time series of images and a separate single

image were taken. The single images covered a square region 30 wavelengths

on a side. The time series consisted of 64 images taken 2 seconds (≈ 2.5τd)

apart with each image covering 6 wavelengths. From the single images, I av-

eraged Θ of the two degenerate modes (the values of Θ for the two modes

agreed to ±1 degree). I used the average ω of the four degenerate traveling

rolls computed from a weighted average of the relevant peaks in S(k, ω) (the

four frequencies differed by at most ±2%). I determined Vc using the total

power under the relevant pairs of peaks in S(k) and S(k, ω). I found that

S(k) and S(k, ω) yielded the same results for Vc.

A meaningful test of the WEM requires knowledge of the following SM

110

parameters: the three elastic constants (K11, K22, K33), six viscosities (α1,

α2, α3, α4, α5, α6), σ⊥, σ‖, ε⊥, ε‖, and d. In addition, two new parameters√

µ+⊥µ

−⊥ and λσ(Rc) are needed. The requirement that the WEM recovers

the correct SM prediction for Vc and Θ sets an upper limit on λσ(Rc) from

|λσ(Rc)|τSM0 1. The literature values for γ1, η, and ε⊥ were used, and

independent measurements of εa, K33, σ⊥ and d were made. This fixed eight of

the SM model parameters using α2 = α3−γ1 (definition of γ1), α5 = α6−α2−α3

(Onsager relation), and α4 = 2η. The remaining three viscosities (α1, α3, and

α6) and the remaining two elastic constants (K11 and K22) were fixed by

fitting Vc and Θ for a single and temperature and assuming the temperature

dependence of the unknown parameters was similar to the known ones. The

remaining SM parameter, σa/σ⊥, was determined by fitting the low frequency

behavior of Vc at each temperature. In all, six parameters were determined

from the six Vc curves and the six Θ curves. Further details and the results of

the fits are given in Appendix A. The final unknown parameter,√

µ+⊥µ

−⊥, was

fit using measured travelling frequencies ω.

Figure 4.8 is a plot of Vc for the cell I5263. For each T , I show Vc as a

function of the scaled applied frequency Ωτq. These curves were measured

with a resolution in ε of 0.001. Figure 4.8b shows the comparison of theory

and experiment for two typical curves. Figure 4.9 shows the onset curves from

I5221 and I5243. These curves were only measured with a resolution in ε of

0.005. In a number of the onset curves, there is a definite inflection point. One

of the clearest examples of this behavior is shown in Fig. 4.9b. The inflection

point[99] is always close to, but above, the Lifshitz point[31]. The Lifshitz

point is the value of Ω at which Θ goes to zero. In the measurements using

I5263, the Lifshitz point roughly coincided with the cutoff frequency, and this

work focused on the conduction regime of EC. Therefore, for most of the runs

using I5263, there is no data above the Lifshitz point, and the inflection point

was not observed.

By symmetry arguments, Θ will go to zero with a square root dependence

111

0

10

20

30ap

plie

d vo

ltage

(Vrm

s)

(a)

0 0.5 1 1.5 2 2.5

applied frequency (Ωτq)

0

10

20

30

appl

ied

volta

ge (V

rms)

(b)

Figure 4.8: (a) Onset curves for 6 temperature in I5263. Circles, squares, uptriangles, diamonds, down triangles, and open diamonds are for 29C, 34C,39C, 44C, 49C, and 59C, respectively. (b) onset curves for 29C and 44Cwith the corresponding WEM predictions (solid lines). The values of the ma-terial parameters used in the theoretical calculations are given in Appendix A.

on the relevant parameter at the Lifshitz point. For example, in thermal

convection in NLC, the Lifshitz point occurs as a function of applied magnetic

112

0

20

40

60

appl

ied

volta

ge (V

rms) (a)

0 1 2 3

Ωτq

0

10

20

30

appl

ied

volta

ge (V

rms) (b)

Figure 4.9: Onset voltages Vc. (a) The open symbols are from cell I5243 atT = 24C (squares) and T = 44C (circles). The solid symbols are from thecell I5221 at T = 44C (squares) and T = 64C (circles). (b) The data fromI5221 at T = 44C with a fit to Vc = a+b(Ωτq)

2+c(Ωτq)4. Here a = 13.2 Vrms,

b = 3.24 Vrms, and c = −0.145 Vrms.

field, and the square root behavior was found to hold for all field values[30, 32].

However, the range of validity of the square root dependence will be system

dependent. In EC, the relevant parameter is Ωτq. Two typical results for Θ as

a function of Ωτq are given in Fig. 4.10. The two curves are for very different

parameter sets. For both cases, the solid line gives the WEM prediction for Θ.

Given the large uncertainty in the material parameters for I52, the agreement

between theory and experiment is quite good.

Figure 4.11 shows the measured Hopf frequency ω and the calculation of

113

0 1 2 3

Ωτq

0

10

20

30

40

Θ (d

egre

es)

(a)

0 1 2

Ωτq

0

10

20

30

40

Θ (d

egre

es)

(b)

Figure 4.10: (a) Angle Θ at onset for cell I5263 corresponding to the onsetcurve for T = 44C shown in Fig. 4.8. Solid line represents the theoreticalcalculation. (b) Θ at onset for cell I5243 corresponding to the onset curve forT = 24C shown in Fig. 4.9. Solid line represents the theoretical calculation.The values of the material parameters used in the theoretical calculations aregiven in Appendix A.

ω as a function of temperature for each of the six values of the temperature.

Both τSM0 and 2π/ω are O(1 s), so |λσ(Rc)|τSM

0 1 implies (λσ(Rc)/ω)2 1.

Therefore, ω ≈ ω and is computed using Eq. 2.44 with only one adjustable

parameter,√

µ+⊥µ

−⊥(see Appendix A), for each of the six temperatures. Recall,

for any values of the parameters, the SM prediction ω = 0 can not be shown

on the scale used in Fig. 4.11. Also, the shape of the theoretical curves is

completely determined by the SM model parameters which are fixed from the

measurements of Vc and Θ. The precise values of√

µ+⊥µ

−⊥ simply set the overall

scale (which is definitely greater than zero) and are consistent with previous

measurements of√

µ+⊥µ

−⊥[39].

There are three predictions of the WEM which are independent of the

unknown material parameters and the fitting procedure I used. First, the

sign of the curvature of ω as a function of Ω is determined by the sign of

εa, which is measured. For εa < 0, ω increases as Ω is increased, and for

114

0 0.5 1 1.5 2 2.5

Ωτq

0.5

1

1.5

ω (s

ec-1

)

0 0.5 1 1.5 2 2.5

Ωτq

0.4

0.5

0.6

ω (

sec-1

)

Figure 4.11: Plotted here is the Hopf frequency ω as a function of Ωτq for thesix runs shown in Fig. 4.8. The six curves are shown in two separate plotsin order that they may be plotted on two different scales. The top plot isfor 29C (circles), 34C (squares) and 39C (triangles). The lower plot is for44C (diamonds), 49C (triangles), and 59C (circles). The solid lines are thecorresponding theory. It is important to note that ω = 0, the SM prediction,can not be shown on the scale used here.

εa > 0, ω decreases as Ω is decreased. Because of the drift in σ, I was able to

measure ω for two values of σ at a fixed value of T and d. The WEM predicts

that ω ∝ σ−1/2d−3. Figure 4.12 shows that the data are consistent with the

predicted σ dependence.

The cell I5275 was used to check the thickness dependence. Measurements

115

4∗10-5

6∗10-5

(a)

ω σ

1/2 (s

-1 o

hm-1

/2 m

-1/2

)

0.5 1 1.5 2 2.5

X Axis

4∗10-5

5∗10-5 (b)

0.5 1 1.5 2 2.5

Ωτq

4∗10-5

5∗10-5 (c)

Figure 4.12: (a) ωσ1/2 as a function of Ωτq for 34C in cell I5263 atσ = 0.44 × 10−8ohm−1 m−1 (open symbols) and σ = 0.37 × 10−8ohm−1 m−1

(closed symbols). (b) ωσ1/2 as a function of Ωτq for 44C in cell I5263 atσ = 0.77 × 10−8ohm−1 m−1 (open symbols) and σ = 0.65 × 10−8ohm−1 m−1

(closed symbols). (c) ωσ1/2 as a function of Ωτq for 49C in cell I5263 atσ = 1.02 × 10−8ohm−1 m−1 (open symbols) and σ = 0.85 × 10−8ohm−1 m−1

(closed symbols).

of ω at the same temperature in cell I5263 and I5275 are shown in Fig. 4.13.

Here I also scale by σ−1/2 as the conductivities were slightly different. The

high frequency limit follows the correct scaling of d−3. In the thicker cell, ω

goes to zero as Ω is decreased. Qualitatively, this behavior is predicted by the

WEM, and is related to the value of τrec. Because of the small value of εa, the

116

0.5 1 1.5 2

Ωτq

0

0.2

0.4

0.6

0.8

1

1.2

ω σ

1/2 d

3 x 1

018

Figure 4.13: Scaled ω for the two cells I5263 (triangles) and I5275 (circles) at49C.

theory is extremely sensitive to the chosen values of τrec, and because of the

drift in σ, the experimental value of Ω at which ω went to zero.

These measurements confirm that the WEM has captured the main features

of the Hopf bifurcation for EC. I have shown experimentally that for most of

the parameter regime which was considered, the linear state consists of four

degenerate modes: right- and left-traveling zig and zag modes. In the next

section, I will discuss the interactions of these modes in the weakly nonlinear

case and the ramifications of this for further tests of the WEM.

4.4 Weakly Non-linear Regime

Linear stability analysis is not sufficient to predict if a transition is forward or

backward. Furthermore, the linear stability analysis does not predict which

pattern will be selected just above onset. For example, for Ωτq less than the

Lifshitz point, there are two degenerate modes (or four in the case of a Hopf

117

bifurcation), and the state which is actually observed above ε = 0 is a result

of nonlinear interactions between these modes. In this section, I will discuss

the experimental observations of the weakly nonlinear regime. I will focus

on the range of conductivity which was discussed in Sec. 4.3; however, I will

mention some preliminary results from higher conductivities and thicker cells.

I have shown that the WEM captures the correct linear behavior of EC in

I52. In theory, it is possible to derive amplitude equations from the WEM

which should quantitatively describe the patterns discussed in this section.

As of now, this calculation has not been carried out, so comparison between

experiment and theory is not yet possible.

One difficulty in determining whether or not the transitions were forward

or backward was the drift in the conductivity of the I52 cells as a function of

time (see Sec. 3.3.3). However, the drift was slow enough that the resulting

drift in Vc was linear over relatively long time periods. To determine Vc, the

voltage was increased quasistatically with steps of ∆ε = 0.001 waiting for 15

minutes at each step. The amplitude of the pattern was measured using the

power under the relevant peaks in S(k). I determined Vc to be the value of

V halfway between the step in voltage where the measured amplitude first

becomes nonzero and the previous step. Typical results for Vc as a function

of time are shown in Fig. 4.14. The drift in the critical voltage in the various

cells was roughly 0.001 V/hr for a Vc ≈ 15 Vrms, which corresponds to a drift

in ε of roughly 1 × 10−4/hr for the typical voltages applied to the cell.

To determine if the transition is forward or backward, the voltage was both

quasistatically increased and decreased with the same step size as described

above, and the amplitude was determined using S(k). The data obtained

from increasing V were used to check the drift of Vc. Then, ε was computed

for each step in voltage for both the increasing and decreasing directions using

the computed Vc(t). If the amplitude as a function of ε was either hysteretic,

or more importantly, showed a finite jump in amplitude as a function of ε,

the transition was considered to be backward. If the amplitude grew continu-

118

11.2

11.4

11.6

11.8

Vc

(V)

0 50 100 150 200 250time (hours)

8

9

resi

stan

ce (1

06 ohm

)

Figure 4.14: Top figure shows the drift in Vc over a period of 10.5 days. Thelinear fit to Vc as a function of time (solid line) gives 1.2 × 10−3 V/hr. Thecorresponding drift in the cell’s resistance is shown in the lower figure.

ously from zero with no evidence of hysteresis, then within my resolution, the

transition was considered to be forward.

A typical measurement is shown in Fig. 4.15 for cell I5263. An example

from cell I5243 which shows a smaller range of ε is given in Fig. 4.16. The

most important features of the onset curves are that the amplitude exhibits no

large jump in value at ε = 0 and that there is no measurable hysteresis upon

decreasing ε. The value of the amplitude in radians is computed using Eq. 3.1,

119

-0.005 0 0.005 0.01 0.015 0.02

ε

0

0.1

0.2

ampl

itude

(rad

)

Figure 4.15: A typical result of an onset measurement in the cell I5263 forσ = 7.5 × 10−9 ohm−1 m−1. The open symbols (circles) are for increasing thevoltage and the closed symbols (triangles) are for decreasing the voltage.

and is only an estimate of the amplitude. For example, Eq. 3.1 does not

include any diffraction effects in the shadowgraph signal which would decrease

the measured amplitude. Figure 4.17 shows three images taken from the onset

run shown in Fig. 4.16. Also shown are the corresponding power spectra. The

images are of a limited region of the cell where convection first occurs, and

they illustrate the continuous nature of the transition. These pictures should

be compared with Fig. 4.19.

Present in both measurements is a relatively large baseline. Comparison

with the results of Ref. [37] reveal that the baseline is on the order of, but

larger than, the fluctuating amplitudes due to thermal noise. The measure-

ments shown in Figs. 4.15 and 4.16 used only a single spatial image. The result

in Ref [37] were obtained from spatio-temporal structure functions which were

computed using a large number of images. A study of the onset for this pa-

rameter range in I52 using the techniques of Ref. [37] is needed. This should

reveal a smooth transition from fluctuating patterns driven by noise to the

120

-0.004 -0.002 0 0.002 0.004

ε

0

0.01

0.02

0.03

ampl

itude

(rad

)

Figure 4.16: A typical result of an onset measurement in the cell I5243 forσ = 9.3 × 10−9 ohm−1 m−1. The open symbols (circles) are for increasing thevoltage and the closed symbols (triangles) are for decreasing the voltage.

deterministic spatio-temporal chaos. The chaos is apparent in these measure-

ments in the fluctuations of the measured amplitude as a function of ε, but

will be discussed in detail later.

There was a measurable decrease of ω with increasing ε. Figure 4.18 shows

an example of the dependence of the Hopf frequency for small values of ε. In

Sec. 4.5, I will report on a secondary transition for which ω makes a finite

jump to zero. The behavior of ω for small ε should be contained within the

CGL description (see Sec. 2.1) for which the complex coefficients provide both

a linear and a nonlinear frequency shift.

The only observations of a discontinuous transition in cells with a thickness

of 28 µm were for σ > 1.5 × 10−8 ohm−1 m−1. This transition was also to a

stationary state. However, these measurements used rather large steps in ε

of the order 0.01 (see Sec. 4.5). From the WEM (see Sec. 2.3), it is believed

that the value of the product σd2 determines when the transition becomes

stationary, and if this is so, it is likely that the value of σd2 also determines

121

b

a

c

d

e

f

Figure 4.17: Three images and their corresponding power spectra from theonset run shown in Fig. 4.16. (a) Image from ε = −1.7 × 10−3 and powerspectrum (d). (b) Image from ε = 1.0 × 10−4 and power spectrum (e). (c)Image from ε = 3 × 10−3 and power spectrum (f). The power spectra in (d)and (e) are enhanced a factor of 25 more than the one in (f). All of the imageswere scaled the same.

122

0 0.01 0.02 0.03

ε

0

0.1

0.2

0.3

0.4

ω (s

ec-1

)

Figure 4.18: Measurement of hopf freq ω as a function of ε in cell I5263 at49C, σ = 1.0 × 10−8ohm−1 m−1, and Ωτq = 1.34.

when the transition is backward. Indeed, when working with the thicker cells,

I observed a discontinuous transition to a stationary state at significantly lower

values of σ which is consistent with this assumption. Figure 4.19 shows two

images, one taken just below onset and the other taken just above onset, from

the cell I5275 for σ = 7.2 × 10−9 ohm−1 m−1. Shown here is a small region

of the cell where EC first occurs. The pattern is stationary above onset, and

the initially observed director variation has an amplitude of roughly 200 mrad.

This represents a significant jump in amplitude and strongly suggests that the

transition is backwards. The measured amplitudes of the director variation

just above onset in this case are consistent with the results of the backward

bifurcation measured in Ref. [37].

For the rest of this chapter, I will focus on the the parameter range

σ < 1.5×10−8 ohm−1 m−1. Figures 4.20a, 4.20b, 4.20c and 4.20d show four im-

ages from the cell I5278 and their corresponding power spectrum. The images

123

(b) (a)

Figure 4.19: Images from onset run in I5275 which is considered to be back-ward. (a) was taken at 12.268 V, and there is no evidence of convection. (b)was taken at 12.279 V, which represents a step in ε of 2 × 10−3. The powerspectrum gives an estimate of 200 mrad for the director amplitude in (b). Thisrepresents a large jump in amplitude at onset (see for instance, Fig. 4.16).

were taken at ε = 0.008 (4.20a) and ε = 0.005 (4.20b, 4.20c and 4.20d) with

σ = 1.5, 1.5, 1.0, and 0.6 × 10−8 ohm−1 m−1, respectively. All four images

have been displayed using the same gray scale. Figures 4.20a and 4.20b are

examples of a typical extended state which has a nonperiodic spatial and tem-

poral variation of the amplitude of the four modes. The pattern in Fig. 4.20b

is extremely weak, but the four peaks corresponding to the zig and zag modes

can clearly be seen in the power spectrum. For σ = 1.0 × 10−8 ohm−1 m−1,

I observed temporary localization of the pattern and large increases in the

magnitude of the amplitude. For the instance in time shown in Fig. 4.20c, the

pattern in the left portion of the image has localized in the direction perpen-

dicular to the director (the y-direction), but the extended state is still present

on the right. This is apparent in the power spectrum where there are still the

peaks which correspond to the zig and zag rolls, but now one sees an additional

peak, highlighted by the dashed box, which is the signature of the localized

structure. One should compare the relative amplitudes of the localized state

and the extended state in Fig. 4.20c to the images in Figs. 4.20a and 4.20b.

124

For σ = 0.6× 10−8 ohm−1 m−1, I observed no evidence of an extended pattern

in the image or the power spectrum. Only patterns localized in the y-direction

were observed, as shown in Fig. 4.20d. The patterns at all three values of σ

are examples of spatio-temporal chaos at onset where the amplitudes of the

four modes vary aperiodically in space and time.

A typical example of a spatial demodulation for the extended state found

at σ = 1.5 × 10−8 ohm−1 m−1 is shown in Fig. 4.21. Shown is the original

superposition of zig and zag rolls and the amplitudes of the corresponding zag

rolls. The spatial correlation length of the individual An(x) is computed from

the power spectrum of An(x). I find a correlation length of 25λ along the

director, and perpendicular to the director, I measure a correlation length of

20λ, where λ = 2π/|k|. Currently, this is only a rough estimate of the spatial

correlation lengths as our largest images only span a region of 60λ.

I studied the spatially-local temporal behavior of the pattern An(t), using

time series of images with a spatial extent of roughly 4λ. I demodulated

the time series to get the An(x, t). For each An(x, t), I computed An(t) =∫

An(x, t)dx. Figure 4.22 shows a 30 minute segment of An(t) for the right-

traveling zig and zag rolls. One observes periods of time where one of the

modes dominates as well as times when the modes have approximately equal

amplitudes. Computing the cross-correlation of pairs of the An(t) from a

4 hour long time series, I find that the four modes are anti-correlated with

each other. Estimating the correlation time of a single An(t) from its power

spectrum yields a value of roughly 1000τd (τd is O(1 s)) for all four modes.

Recall that the traveling frequency ω is O(1 Hz). I observed this state for

over 48 hours at ε = 0.005, and no localized states were observed. In addition,

no localization was observed as a function of ε. The next transition, which

occurred at ε = 0.1, was to an extended stationary state.

These two measurements are an extremely limited initial characterization

of the dynamics of this state. The time series measures the local amplitudes

of all four degenerate modes separately. During time periods for which the

125

a

b

d

kx

ky

c

Figure 4.20: (a), (b), (c), and (d) show typical spatial patterns (left side) whichexist for Ω/2π = 25 Hz at an ε = 0.008 for (a) and an ε = 0.005 for (b), (c)and (d) and σ = 1.5, 1.5 1.0, and 0.6 × 10−8 ohm−1 m−1, respectively. Theircorresponding power spectra are shown on the right side. All four patternsare from the cell I5278. (a) and (b) are examples of the extended state at twodifferent values of ε. (c) shows the coexistence of a localized state with theextended state of lower mean amplitude. In the power spectrum, the dashedbox is around the peak due to the localized structure and the dashed circle isaround the peak due to the zag rolls. (d) shows the localized state. All fourpatterns are time dependent, and all four images are scaled the same. Thepower spectra for images (b), (c) and (d) were scaled by the same factor, butthe power spectrum for (a) is scaled such that black represents a power 4x thepower represented by black in the power spectra (b), (c) and (d).

amplitudes of right- and left-traveling modes are equal, there are only two de-

generate modes: zig and zag standing waves. However, there are equally long

periods where the traveling modes have unequal amplitudes. During these pe-

126

Figure 4.21: The left-hand image is a snapshot from a time series at ε = 0.01,and σ = 1.5 × 10−8 ohm−1 m−1 in a cell with d = 28 µm. It shows thesuperposition of the zig and zag rolls which exists throughout the cell. Theright-hand image shows the corresponding amplitude of the zag rolls with thewhite regions corresponding to large amplitude.

0 10 20 30

time (min.)

0

0.5

1

Am

plitu

de (a

rb. u

nits

)

Figure 4.22: Plot of the temporal variation of the amplitude of the right-traveling zig (solid lines) and zag (dashed lines) rolls. The time series wastaken at ε = 0.01 and σ = 1.5 × 10−8 ohm−1 m−1.

127

riods, the resulting superposition is not a standing wave; however, the local zig

(zag) amplitude still varies periodically in time with the underlying traveling

frequency. The single snapshots measure the amplitude of this superimposed

right- and left-traveling zig (zag) state at a single instance in time.

Because the spatial demodulation represents a single instance in time, there

are a number of sources of the spatial variation of the amplitude shown in

Fig. 4.21. Obviously, one could have local zig (zag) standing waves which were

globally in phase but had a spatial variation of their amplitude. A standing

wave with a spatially uniform amplitude and a spatially varying phase would

also have an instantaneous spatial variation of the zig (zag) amplitude. Indeed,

it is possible to imagine a great variety of combinations of variations in the

relative phase and amplitudes of the four modes which would result in the

spatial variation observed in Fig. 4.21.

It is clear that for a more comprehensive understanding of this state of

spatio-temporal chaos, one requires time series of images of large spatial extent

from which the phase and amplitude of all four modes can be studied. One

important question which can only be answered in this fashion is the role of

defects in the pattern. Defects have been found to play an essential role in

other forms of spatio-temporal chaos. But, to adequately study the defects in

this system, one needs to look at the underlying four modes. Observations on

the superposition of modes (which the current spatial measurements represent)

can be misleading.

For σ = 1.0 × 10−8 ohm−1 m−1, I still observe an extended state which

is similar in amplitude to the state observed for σ = 1.5 × 10−8 ohm−1 m−1.

However, the dynamics of the An(x, t) is quantitatively different. There are

“bursts” of large amplitude convection which occur chaotically in time and in

random spatial locations. The “bursts” either result in structures which are

highly localized in the y-direction (a width of roughly λ) or result in “blobs”

of high amplitude convection. Figure 4.23 shows four instances in time from

a series taken at ε = 0.005. Figure 4.23a, 4.23b, 4.23c and 4.23d show the

128

a b

c d

Figure 4.23: Snapshots from a time series of images at ε = 0.005 and σ =1.0 × 10−8 ohm−1 m−1 in a cell with d = 28 µm. (a) is t = 324 min, (b) ist = 340 min, (c) is t = 496 min, and (d) is t = 564 min, where t = 0 mincorresponds to when the voltage was set to ε = 0.005.

extended state, a state which has localized in the central region, the extended

state again, and a “blob” state, respectively. At a fixed value of ε, this behavior

repeated in an apparently aperiodic fashion for the duration of our observations

(the longest observation period was 48 hours). An important feature of this

state was the long time scales. The state would often initially remain uniform

for up to 3 hours. This would explain why the state was not apparent in the

initial studies of cell I5263 at σ = 1.0 × 10−8 ohm−1 m−1 where the studies as

a function of ε involved waiting times of at most an hour.

129

For σ = 0.6 × 10−8 ohm−1 m−1, we observed no evidence of any extended

state. The only states which were observed (up to ε ≈ 0.1) were localized

states (worms) having a constant width in the y-direction equal to the y-

component of the wavelength of the zig and zag states observed at σ = 1.5 ×10−8 ohm−1 m−1. The worms have a distribution of lengths in the x-direction.

For low amplitudes, the worms appear to be stationary, or slowly drifting,

and blink with the underlying traveling frequency. Above a critical, as of

yet undetermined, value of the worm’s amplitude, it stabilizes into either a

left- or a right-traveling state of apparently constant amplitude. The state

is composed of a superposition of right- or left-traveling zig and zag rolls,

respectively, i.e. the worm travels in the opposite direction of the traveling

waves which compose it (see Fig 4.25 and discussion there). Just above onset,

worms grow and die throughout the cell with lifetimes on the order of 102 −103 τd. The average number and length of the worms increases with ε. At high

enough values of ε, worms are observed to extend across the cell and exist until

they travel out of the region of observation or interact with other worms. The

relatively long lifetimes of the worms near onset is consistent with the long

correlation times observed in the extended state.

The worms have a number of interesting interactions with each other. Fig-

ure 4.24 shows four images taken 1 minute apart at an ε = 0.05. The worms

are traveling in both directions and are arranged with a roughly fixed spacing

in the y-direction. As ε is increased, the spacing between the worms decreases

until the cell is filled with convection. One worm is highlighted by a white rect-

angle to emphasize its movement across the cell. During a head on collision

of worms, the overlapped regions fluctuate widely in amplitude and spread

in the direction perpendicular to the director alignment. The worms do not

generally move in the y-direction. However, if the distance in the y-direction

between two states is small enough, they are attracted to each other. Upon

“contact”, they generally repel each other which results in a rapid undulation

of the two states in the y-direction. Occasionally, the two worms at different

130

a b

c d

Figure 4.24: Snapshots from a time series at ε = 0.05 and σ = 0.6 ×10−8 ohm−1 m−1 in a cell with d = 28 µm. The images are 1 minute apart.The white rectangle highlights a worm which is traveling to the the right.

y-positions join upon contact to produce a bend in the resulting worm state.

Eventually, the combined worm moves to one of the original y-positions.

The worms appear to be a generic feature of oblique rolls for low σ and

thin cells. We have observed worms in two different cells (I5263 and I5278)

with a thickness of 28 µm. In addition, we have observed the worms in a cell

of thickness d = 52 µm (I5268). In this case, the worms were observed at a

value of σ = 0.23 × 10−8 ohm−1 m−1. We believe the worms only occur below

a critical value of σd2. This expectation still needs to be confirmed, but it

131

would explain why the worms have not been previously observed2. All of the

previous work in EC which we are aware of has been in thicker cells or at a

higher σ than we have studied here.

An image of a typical isolated worm is shown in Fig. 4.25. The worm is

traveling to the right while the rolls are traveling to the left. It appears that

the worms always travel in the direction opposite the underlying traveling wave

direction. There is a sharp increase of amplitude at the front of the worm and

a long tail at the trailing edge. The worm is a superposition of zig and zag

rolls, but nonlinear effects in the optics and the extreme localization in the

direction perpendicular to the director combine to create the optical effect of

normal rolls. The zig and zag nature of the underlying rolls is apparent in the

weak amplitude convection which extends slightly in the y-direction around

the leading edge of the state. It is also apparent when two worms interact

and the state is temporarily delocalized and has formed a blob similar to the

image in Fig. 4.23. Figure 4.25b shows an image of the same worm shown

in Fig. 4.25a, but with the light polarized in the y-direction instead of the

x-direction. The uniformity of the image shows that the underlying director

distortions remain in the x-direction, as they are for the extended state.

4.5 Nonlinear Results

The nonlinear patterns in this system show a rich variety of behavior as a func-

tion of the three control parameters V , Ωτq, and σ. As discussed in Sec. 4.4,

it is believed that the patterns also depend on d, and that the relevant control

parameters are V , Ωτq, and σd2. For the nonlinear patterns, the majority of

the detailed measurements are from the cell I5263, and I am unable to com-

ment on the d dependence except to say that the general trends are consistent

2There are previous reports of localized states for normal rolls[25], but these states did

not travel, and from the report in Ref [25], do not appear to have the same wide range of

lengths and existence

132

b

a

Figure 4.25: (a) Close up of an isolated worm in the cell I5268. (b) Image ofthe same region of the cell with the worm present, only the light is polarizedperpendicular to the director alignment. This demonstrates the lack of directorvariation perpendicular to the original alignment direction.

with this assumption. Even limiting the discussion to the three control pa-

rameters considered here, there is a rather large parameter space to study and

I will focus on two sets of data. One set was taken at a fixed Ωτq as a function

of ε and σ (Fig. 4.28) and the other one at a fixed value of σ as a function of

ε and Ωτq (Fig. 4.29).

The spatially extended patterns are labeled using three letters as given

in Table 4.2. “Worms” is used in the same sense as Sec. 4.4 and refers to

the localized states. This is the only pattern which does not follow the three

letter scheme. For a given pattern, the letters in the name are determined

right to left. I will discuss the naming process and use some examples for

illustration. The details of where in parameter space the various patterns are

found is discussed later, but Fig. 4.28 and Fig. 4.29 can be referred to if that

information is desired.

The third letter distinguishes between patterns which consist of a super-

position of zig and zag rolls and patterns which consist of only a single set of

rolls. By superposition, I mean specifically zig and zag rolls which coexist in

the same spatial location. For the rest of this chapter, I will use coexisting rolls

to refer to a pattern which consists of zig and zag rolls in separate spatial re-

gions. A pattern which consists of a single set of zig rolls which coexist with a

set of zag rolls is still considered a “single set” and is designated by an “R”. In

133

Table 4.2: Naming scheme for patterns in I52.

First Position Second Position Third Position

letter meaning letter meaning letter meaning

T traveling N “normal” S superposition of rolls

S stationary O “oblique” R single set of rolls

general, oblique roll patterns which consist of a single set of rolls always have

large regions of zig coexisting with large regions of zag. As defined here, the

distinction between superposition and coexisting is made by a visual inspection

of the pattern, and not by looking at the power spectrum, as the power spec-

trum does not reveal information about spatial locations. Figure 4.26 shows

an example of a state which would have an “S” for the third letter3 and one

which would have an “R”, and their corresponding power spectra are shown.

The second letter is decided upon differently for the superposition states

and the single roll states. For the single roll state, “oblique” and “normal” have

the definition given in Chapter 1 and refer to the angle between the wavevector

of the pattern and the orientation of the director. States with a wavevector

parallel to the director are “normal”; otherwise, the pattern is “oblique”. If

the pattern is a superposition of two rolls, “oblique” and “normal” refers to

the orientation of the resulting grid pattern. This is best determined by the

spatial power spectrum. If the two superimposed modes are degenerate, the

pattern is considered “normal” (i.e., the last two letters are “NS”). An example

image and its power spectrum is given in Fig. 4.27. If the two modes are not

degenerate, the pattern is considered to be “oblique” (i.e., the last two letters

are “OS”). One can see from the example image in Fig. 4.27 that the grid

3The superimposed patterns are visually similar to the grid patterns of Ref [98]. But,

Ref [98] does not provide power spectra, and the patterns were generated by a combined ac

and dc voltage, so the connection with the patterns observed here is not clear.

134

a

c d

b

Figure 4.26: (a) is an example of a pattern which consists of a single setof oblique rolls. In this case, there are coexisting zig and zag rolls with agrain boundary between them. (b) is the power spectrum of image (a) andit contains peaks corresponding to both sets of roll. (c) is an example of asuperposition of oblique rolls. (d) is the power spectrum of image (c) and italso contains peaks which correspond to both sets of rolls. Image (c) shows thecharacteristic grid-pattern of a state which is the superposition of two obliquemodes.

pattern of an “OS” pattern is rotated with respect to the axis defined by

the director, and the power spectrum clearly shows that the superimposed

wavevectors are not degenerate.

The first letter is determined from a time series of images and denotes

whether or not the peaks in the power spectra S(q, ω) (see Sec. 4.1) are at

ω = 0 (“S”) or at ω 6= 0 (“T”). For the case of patterns labeled by “T/S”,

there were nonzero peaks at both ω = 0 and ω 6= 0, and as is discussed later,

135

a b

c d

Figure 4.27: (a) is an example of a superposition of degenerate zig and zagrolls and would be denoted with a second letter “N”. (b) is the power spectrumof image (a) and it shows that the two wavevectors form the same angle withrespect to the director. (c) is an example of a superposition of nondegenerateoblique rolls and would be denoted with a second letter “O”. (d) is the powerspectrum of image (c) and it shows that the two wavevectors form differentangles with respect to the director.

it has not been completely determined whether or not this is the result of a

coexistence of or a superposition of traveling and stationary modes.

I will now discuss the various transitions shown in Fig. 4.28 and Fig. 4.29.

For the range of σ studied in the cell I5263 (see Fig. 4.28), the initial transi-

tion is a forward Hopf bifurcation to either the worms or TNS as discussed in

Sec 4.4. From the limited measurements in I5243 (the x’s in Fig. 4.28), the

initial transition at high values of σ appeared to be backward and to a sta-

tionary state. However, these measurements were taken with relatively large

steps in ε of 0.01, and clearly, the measurement of the initial transition must

be done with a resolution in ε which is better than 0.01. For example, in the

136

cell I5263 at 25C, I observed an initial transition to traveling rolls with a

transition to a stationary pattern at a value of ε of only 0.008. In this case,

σ = 4.8 × 10−9 ohm−1m−1 and Ωτq = 0.87. At this temperature, the value

of ε at which the transition from traveling to stationary rolls occured was de-

pendent on the applied frequency. There have also been reports of an initial

Hopf bifurcation with a transition to a stationary state at values of ε < 0.01

for EC in other NLC4. However, in the thicker cell I5275, the initial transition

was observed to be a transition to the SOR state with a resolution in ε of

0.002 as discussed in Sec. 4.4. This is consistent with the expectation that

the patterns depend on the combination σd2 and that higher values of σ and

thicker cells result in a backward bifurcation to stationary rolls as observed in

Ref. [37] and predicted by the WEM. Further quantitative work is required to

determine the value of σ (or σd2) at which the initial transition switches from

a Hopf bifurcation to a stationary bifurcation.

As noted in Fig. 4.28, the majority of this study was carried out by varying

the operating temperature. This allowed for a reproducible increase/decrease

of σ with increasing/decreasing T . As discussed in Appendix A, many other

material parameters, including εa which changes sign, are temperature depen-

dent. My observations suggest that T does not affect the qualitative nature

of the observed patterns for a given σ and d. The measurements in I5243

at T = 44C and the measurement in I5263 at T = 49C exhibit the same

nonlinear patterns as observed for the cell I5263 at T = 59C. Also, after the

cell had aged for 6 months, the worm states and the TNS states reported in

Sec. 4.4 were studied in the temperature range of 49C to 64C as opposed to

T < 54C which was used for the initial observations reported in Fig. 4.28.

However, the quantitative aspects of the pattern, i.e. ε value of the transitions,

traveling frequency and angle of the oblique rolls, does exhibit some depen-

dence on the other material parameters. Ideally, future studies will be carried

4L. Kramer, private communication

137

1 1.5 2

conductivity (10-8 ohm-1m-1)

0

0.1

0.2

ε

SORSNS

SOS

TNS

(T/S)NS

worms

T

Figure 4.28: Observed patterns as a function of ε and σ at Ωτq = 1.34. Hereσ was varied by changing the temperature T, as indicated by the arrow. Thepoints connected by lines (open circles) were taken using cell I5263 and atemperature range of 44C to 59C. The crosses are from the cell I5263 andT = 49C (The point corresponding to T = 49C from the open circle data isat σ = 1.0 × 10−8ohm−1 m−1). The x’s are from a run in the cell I5243 whereT = 44C. The solid vertical line corresponds to σ = 1.24 × 10−8ohm−1 m−1,which is the value of σ used in Fig. 4.29. The dashed vertical line is discussedin the text.

out as a function of d at a fixed temperature using a cell of variable thickness.

I will present in detail data for two values of σ from Fig. 4.28 to illus-

trate the main transitions which are observed. The first is for σ = 1.0 ×10−8 ohm−1 m−1 and is given by the vertical dashed line in Fig. 4.28. Fig-

ure 4.30 shows images of the three patterns which are observed along this line:

TNS, SNS, (T/S)NS. Also shown are the spatial power spectra of the images.

The initial pattern (TNS) consists purely of traveling rolls and undergoes a

138

0

0.05

0.1

0.15

0.2

ε

TNS

SNS

SOR

SOS(T/S)NS

(T/S)OS

0 0.5 1 1.5 2

Ωτq

0

0.05

0.1

0.15

ε

SOR

SOS

Figure 4.29: Top figure is the observed patterns as a function of ε and Ωτq atσ = 1.24×10−8ohm−1 m−1 for the cell I5263. The solid vertical line correspondsto the value of Ωτq = 1.34 which was used in Fig. 4.28. The bottom curve isthe observed patterns as a function of ε and Ωτq in the cell I5243 for the valueof σ plotted in Fig. 4.28. Here the horizontal dashed line is at ε = 0.01 whichwas the ε step size used to measure the boundaries.

secondary transition to a stationary state. What is particularly intriguing

is that the next transition is to a state with both stationary and traveling

components.

Figure 4.31 is a plot of S(k, ω) (which is a three dimensional quantity

depending on p, q, and ω) for four values of ε. As I can only plot two axes,

139

a

b

c

Figure 4.30: The left hand column are three images taken along the dashedline of Fig. 4.28. Image (a) is for ε = 0.016. (b) is ε = 0.066, and (c) isfor ε = 0.10. On the right is shown the power spectra for the correspondingleft-hand image.

140

I show a section of the full S(k, ω). One axis corresponds to the entire ω

axis. The other axis is a slice along the direction in k-space which corresponds

to the wavevector of the zag rolls. A number of features of the transition

are highlighted by this plot. First, for the length of time series used here,

S(k, ω) for ε = 0.051 shows that the right and left traveling waves had unequal

amplitudes. This was discussed in Sec. 4.4. At ε = 0.056, the transition to a

stationary state has occurred, and there is only a peak at ω = 0. At ε = 0.081,

one finds that the traveling modes reappear with a relatively small amplitude.

By ε = 0.086, there is a clear superposition of traveling and stationary modes.

The S(k, ω) for ε = 0.081 is suggestive of a continuous transition. However,

one must also consider the spatial extent of the traveling and stationary com-

ponents of the patterns. Ideally, demodulated time series of images covering

many correlation lengths should be studied; whereas, because of previously

discussed limitations, the images used to compute these S(k, ω) contained

only 8 wavelengths. Initial demodulations of the pattern using these images

of small spatial extent, and observations of movies of the pattern, strongly

suggest that the traveling rolls and stationary patterns are not superimposed

throughout space. It is clear from real time observations that localized regions

of purely stationary patterns exist. However, these regions do not remain

purely stationary modes. Regions of traveling modes move through the cell in

an apparently chaotic fashion. One can get a feel for this from image (c) in

Fig. 4.30. In this image, there are regions where the zig or zag rolls appear

to dominate (at the very least the image is fuzzier). For example, across the

top of the image, the individual oblique rolls are more apparent. However,

the right-hand edge, halfway down the image, looks more like the image (b) in

Fig. 4.30. This corresponds to a region which was observed to be stationary.

What is not so clear is the nature of the regions with traveling rolls. The

results of the demodulations of the current S(k, ω) suggest that the stationary

mode is superimposed with the traveling mode in some regions of space. But,

the images are much too small to say anything conclusive. It is a very intriguing

141

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ω (sec−1)

k

k k

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ω (sec−1) ω (sec−1)

ω (sec−1)

ε = 0.051 ε = 0.056

ε = 0.086 ε = 0.081

Figure 4.31: S(k, ω) for four values of ε taken from along the dashed line ofFig. 4.28.

thought that this pattern consists of a “stationary” background across which

localized traveling wave packets are moving. In this case, the small amplitude

in the power spectrum could be the result of localized traveling modes. Clearly,

the interaction of the stationary and traveling components of this pattern and

the overall dynamics of this state are interesting aspects of this system.

The traveling frequency corresponding to the dominant peaks in S(k, ω)

are plotted in Fig. 4.32a. The transition to a stationary pattern at ε = 0.055

occurs via a finite jump in the frequency. Similarly, at ε = 0.08, there is again

a finite jump in frequency to a value approximately equal to the value of ω

which existed below ε = 0.055. The corresponding behavior of the ampli-

142

0

0.2

0.4

0.6

ω (s

ec-1

)

-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12

ε

0

0.5

1

ampl

itude

(arb

. uni

ts)

Figure 4.32: (a) frequency ω of pattern along the dashed line of Fig. 4.28.(b) amplitude of the fundamental (circles) and the harmonic (triangles) of theshadowgraph signal measured from the power spectrum of images taken alongthe dashed line of Fig. 4.28.

tude as a function of ε for both the fundamental and the second harmonic of

the shadowgraph signal is given in Fig. 4.32b and 4.32c. There is an obvious

decrease in the amplitude of the power spectrum when the pattern becomes

stationary. Also, at the point where ω makes a finite jump, the behavior of

the fundamental peak and the second harmonic are quite different. A quanti-

tative explanation of this behavior should be possible when the physical optics

calculation has been completed for EC, but currently, there is no explanation.

The transition in the cell I5243 from SOR to SOS has none of the com-

plications of traveling modes but has some equally intriguing new features.

Figure 4.33 shows six typical images of a small section of the cell. I want to

emphasize here that there were some difficulties studying this state associated

143

a b c

d e f

Figure 4.33: Six images of oblique rolls for the run in cell I5243 shown inFig. 4.28. The images were taken at (a) ε = 0.014, (b) ε = 0.052, (c) ε = 0.067,(d) ε = 0.082, (e) ε = 0.096 and (f) ε = 0.114.

with the properties of the shadowgraph images of EC in NLC. The transition

from SOR to SOS occurred at high enough values of ε that the images were

usually dominated by the second harmonics. This is clear in the power spec-

trum of the images from Fig. 4.33 which are shown in Fig. 4.34. Here the

dominant peaks in the power spectrum are all due to the quadratic effects of

the shadowgraph, and the linear peaks are barely visible, if at all.

For the case of SOR, both the zig and zag rolls exist in the cell separated

by grain boundaries. An example of a grain boundary is given in Fig. 4.33b.

What is clearly shown by the power spectra in Fig. 4.34f is that the SOS

consist of a superposition of one set of the original zig or zag rolls and another

set of oblique rolls whose wavevector is roughly perpendicular to the original

set. For comparison, consider the power spectrum of the grain boundary which

is a superposition of degenerate zig and zag rolls (Fig. 4.34c) and the power

spectrum of the SOS state (Fig. 4.34f).

There are a number of details of the SOR/SOS transition which still need

to be measured. One issue is the angle between the original set of rolls and

the new set which grows. Typically, the angle is around 90 degrees, but there

144

c a

d f

115

Figure 4.34: Power spectra of four of the images in Fig. 4.33. The letterscorrespond to the letters in Fig 4.33. The lines in (c) and (f) highlight theangles between the two modes which are present in the images.

exists a range of angles near 90 which are allowed. I have observed angles as

large as 110 degrees as shown by the state pictured in Fig. 4.35. (Here the

fundamental peaks in the power spectrum were clearly visible.)

Another issue is the dependence of the angle Θ of the initial zig or zag roll

on ε. For small values of Θ near onset, Θ increased continuously as a function

of ε and plateaud around 35. The details of the behavior depended strongly

on σ and Ωτq. I have observed no discontinuity of Θ as ε is increased through

the SOR/SOS transition, but this needs to be studied more systematically.

By jumping from a value of ε below the SOR/SOS transition to a value

of ε which is above the transition, I observe the transition occurring first in

145

108

Figure 4.35: Image from cell I5263 at ε = 0.17 and 57.1C with a σ = 1.56 ×ohm−1 m−1, and the power spectrum of the image. The two lines in the powerspectrum are drawn through the fundamental peaks. The angles between thetwo sets of rolls is 108.

the middle of regions of zig and zag rolls and not from the grain boundaries.

It still remains to be determined whether or not the transition is forward or

backward, and what hysteresis, if any, is present. Some images (for example

Fig. 4.33d) suggest that the transition occurs due to an undulation instabil-

ity in the rolls. Calculations of such instabilities from amplitude equations

exist, so there exists the real possibility of phenomological description of this

transition using amplitude equations.

It is clear from Fig. 4.28 and Fig. 4.29 that there are a number of in-

teresting transitions which exist as a function of σ and Ωτq that were not

studied here because these studies all were done as a function of ε. For exam-

ple, the region around the point where SOS, SOR, SNS, and (T/S)NS meet

in Fig. 4.28 involves the interaction of a number of very different types of

patterns. Essentially all possible combinations of traveling, coexisting, super-

position, degenerate and nondegenerate meet at a single point. This should

lead to some extremely interesting dynamics.

Another interesting transition which is found in Fig. 4.29 is the spontaneous

break up of the (T/S)NS state into regions of differently oriented (T/S)OS.

In the SOR/SOS transition, each region had an initial orientation defined by

146

the existing zig or zag rolls. In the observed (T/S)NS to (T/S)OS transition,

the initial state is a superposition of the zig and zag rolls. Therefore, each

region had to decide, by some mechanism, whether to retain the zig or zag roll

state which formed the (T/S)NS state, and then a set of oblique rolls which

was roughly at an angle of 90 to the chosen original mode would grow. In

addition, the pattern still had both a traveling and a stationary component.

This is just one more example of the richness of patterns which occur as a

function of ε, Ωτq and σ for this system. In the future, a cell design which

incorporates the ability to easily adjust d should vastly increase the ability to

quantitatively study this rich system of interesting patterns, as well as test the

postulated dependence on σd2.

147

Chapter 5

Open Issues and Future

Directions

In the introduction, I gave three reasons for studying EC in NLC other than

MBBA and Phase V: resolution of discrepancies between theory and experi-

ment, measurements of thermal noise, and finding a replacement for MBBA

as the standard NLC used in EC. In addition, the study of the NLC I52 has

revealed a rich landscape of patterns to be explored. I will first discuss the

degree to which the search of NLC achieved each of the three goals, and then,

I will discuss possible directions to take with the fascinating system of EC in

I52.

There were two issues to be resolved between theory and experiment: the

Hopf bifurcation and the backward bifurcation. In Sec. 4.3, I showed that the

WEM[39] does an excellent job of predicting the Hopf bifurcation in the limit

of thin cells and low conductivity. Based on experiments from MBBA[68] and

my limited work using thick cells, the initial bifurcation in EC is stationary

for thick cells with a high enough conductivity. Also, there is an intermediate

range of thickness and conductivity for which the Hopf bifurcation exists for

Ω close to the cutoff frequency and goes to zero as Ω is decreased. The WEM

qualitatively predicts both of these behaviors[39], but quantitative experiments

148

are needed.

As mentioned in Sec. 2.3, one of the nice features of the WEM is that the

mechanism for producing the Hopf bifurcation is similar to the mechanisms

of other pattern forming systems which exhibit Hopf bifurcations. The charge

density (which is proportional to An for τq/τd 1) drives the instability via

the electrical volume force, but the coupling of An to Aσ provides an additional

stabilizing mechanism. This stabilizing mechanism is responsible for the Hopf

bifurcation provided that the time scale for the feedback is sufficiently slow.

This interplay between a primary instability mechanism and a slower stabi-

lizing mechanism also occurs in RBC in binary fluids and RBC in NLC with

homeotropic alignment. In RBC in a binary fluid with a negative separation

ratio, the concentration field is the slow field which provides the stabilizing

mechanism that counteracts the buoyancy-driven instability[100]. In thermal

convection in a homeotropically aligned NLC, the director field is the slow

field which stabilizes the buoyancy-driven instability[101].

The main unresolved issue with the linear predictions[42] of the WEM is the

value of τrec. The limits set by my experiments require τrec > 10 s for I2 doped

I52. Compared to typical numbers from the literature, τrec ≈ 0.05 s[102], this

is a very slow time. However, I2 doped I52 forms a charge transfer complex and

then undergoes a dissociation-recombination reaction. Since this is a multi-

step process, it is expected that the recombination times would be longer. In

addition, I52 is a nonpolar molecule which is essentially impossible to dope,

and the reported measurements have been on more polar liquid crystals which

are relatively easy to dope. For example, a solution of 0.01% TBBA in MBBA

produces a conductivity of 10−7 ohm−1 m−1. For I52 with concentrations of

up to 0.1% TBBA, the conductivity was less than 10−9 ohm−1 m−1, and it

took ≈ 2% I2 to eventually achieve conductivities of 10−8 ohm−1 m−1!! This

represents a difference of 3 to 4 orders of magnitude between I52 and MBBA

in their ability to produce dissociation for a typical organic salt. This implies

extremely small values for the rate constants of the dissociation-recombination

149

reactions in I52 and correspondingly large values for τrec. Of course, τrec

should be measured for I52 doped with I2. As discussed in Sec. 3.2, the large

uncertainty in the final concentration of I2 makes these measurements difficult.

One possibility is to determine the concentration of I2 using optical absorbance.

As predicted by the weakly nonlinear analysis[33, 34] of the SM, I observed

initial transitions which were forward for EC in I52. However, the range for

which I observed a forward bifurcation corresponds with the range for which

the WEM, not the SM, correctly describes the linear behavior[42]. Therefore,

the correct theory to compare to is a weakly nonlinear analysis of the WEM.

This remains to be done. In addition to predicting the forward bifurcation

observed in I52, it remains to be seen if the WEM can provide an explanation

for the observed backward bifurcation in MBBA[37]. These observations were

from a cell 13 µm thick, but at a higher conductivity than my studies in

I52. Also, my initial studies of EC in I52 in thick cells (see results for I5268,

Sec. 4.4) suggest that the bifurcation becomes backward for I52 as well. Thick

cells and high conductivity are the limits for which the WEM recovers the SM.

Since the SM was unable to explain the backward bifurcation, this argues that

the WEM will also be unable to explain the backward bifurcation.

One intriguing possibility is suggested by the data of Fig. 4.28. Figure 4.28

shows the secondary instability to stationary rolls approaching the initial bi-

furcation as the conductivity is increased. At this transition, there is a large

jump in the amplitude of the pattern. Both in I52 and in Merck Phase V,

this secondary transition has been observed to occur at rather small values of

ε, ε ≈ 0.01. It may be that the value of ε for which the secondary transition

occurs approaches zero, but the initial transition is always forward as pre-

dicted by the SM. At some point, the region of the forward transition might

not be resolvable experimentally, and an initially backward transition would

be observed. These ideas are clearly speculative, but definitely the behavior in

Fig. 4.28 and Ref. [103] is suggestive. For now, the evidence for the backward

bifurcation in Ref. [37] is very convincing and remains a fact that theory must

150

explain.

As a system for studying fluctuations below onset, I52 is definitely promis-

ing. The initial transition is forward as desired. The existence of the initial

transition to spatio-temporal chaos and localized states would make a study of

the role of noise below onset in this system particularly interesting. Below Vc

the pattern is linear, and the individual modes do not interact. Presumably,

it is the interaction of the modes which produces the spatio-temporal chaos

and the localization, and at the very least, the nonlinearities are essential. It

would be interesting to measure the transition from noise driven fluctuations

to the deterministic spatio-temporal chaos or a localized state.

The search of NLC revealed two other possible candidates for the study of

thermal noise. Both D55-f and OS-33 appear to have forward bifurcations to

normal rolls. For these NLC, the initial transition is also stationary. There

are two difficulties with these materials. First, they are difficult to obtain.

Second, the anisotropy in the index of refraction is relatively small for these

materials which produces a decrease in the shadowgraph signal.

The final goal was to find a replacement for MBBA. In this regard, I52

is a strong candidate. The material is reported to be highly stable, and a

number of the essential material parameters have been measured. I have mea-

sured some additional parameters using the bend electric Frederiks transition.

Measurements of the other two elastic constants are possible using different

Frederiks configurations. The large nematic range in temperature and the

variation of εa with temperature allow for the measurement of EC for a range

of material parameters using a single NLC.

From the results of Sec. 4.4 and Sec. 4.5, one is presented with a poten-

tially overwhelming number of interesting patterns to study. In the nonlinear

regime, three problems stand out. There is the transition from SNR to SOS

and the similar transition from (T/S)NS to (T/S)OS. For both of these sys-

tems, there are a number of features of the instability which require further

study. Among these, the range of angles between the two nondegenerate, su-

151

perimposed modes needs to be studied. Also, whether or not the second mode

grows continuously from zero amplitude or comes in with a finite amplitude

needs to be determined. For the (T/S)NS to (T/S)OS transition, there is the

additional issue of the dynamics by which the pattern separates into regions

of different orientation.

Another challenge is the actual nature of the (T/S)NS state. So far, I only

have the most cursory characterization of the state. As far as I know, there

is no other example of a pattern forming system for which there is a state

which is a superposition of a traveling mode and a stationary mode. The first

thing which needs to be determined is the extent to which the two modes are

superimposed, or merely coexisting. This will be relatively easy to determine

with improvements to the image taking capability.

An interesting probe of this state would be to modulate the drive frequency

at twice the traveling frequency. It has been shown that such a modulation

will stabilize a traveling state into a standing wave[103]. Given the existence

of the traveling modes, it would be interesting to study the effect of such a

stabilizing modulation on this pattern.

Finally, there is the interesting point where the four patterns SNS, (T/S)-

NS, SOS, SOR meet as a function of the control parameters. The dynamics

in this region should be quite fascinating and might be described by a phe-

nomenological amplitude equation.

All three phenomena just discussed occur at ε > 0.1 where the applicability

of amplitude equations is limited. I will close with some thoughts on the

spatio-temporal chaos and localized states which occur in the weakly nonlinear

regime.

For the spatio-temporal chaos and localized states, it is hoped that the ap-

propriate CGL will be derived from the WEM, so that quantitative comparison

between theory and experiment is possible. There are a number of examples

of spatio-temporal chaos, but there are two in particular for which compar-

ison to solutions of appropriate CGL have been made: dispersive chaos in

152

RBC in binary fluid mixtures[104], and defect turbulence in EC in NLC[26].

Even though both of these phenomena are qualitatively described by CGL

amplitude equations[57], there are difficulties with quantitative comparisons

between theory and experiment because of the relatively large amplitudes of

the patterns. For RBC in binary fluids, the chaos occurs near onset, but the

initial transition is a backward bifurcation (see Sec. 2.1). For the defect chaos,

the chaos occurs after a secondary instability for which the amplitudes are

already large. One of the great advantages of the chaotic states reported on

here is that they occur as the result of a forward bifurcation. Therefore, it is

expected that the patterns can be studied for a range of ε where the CGL are

strictly valid. Even without a quantitative derivation, many of the coefficients

in the CGL could be measured experimentally, and the coupled CGL could be

used as a phenomenological model.

There are a number of possible probes of the extended state of spatio-

temporal chaos. First, the system size can be varied. For a small enough

system, the spatial degrees of freedom can be eliminated. A number of pat-

tern forming systems exhibit dynamical chaos when the system size is small

enough[1], but one would expect that without the spatial coupling, the pattern

could become regular. By varying the system size, the effects of the spatial

coupling on the dynamics could be studied.

Reference [105] serves as a possible guide for what to expect in a small

enough system for which the spatial degrees of freedom are not relevant. Ref-

erence [105] explores the solutions of a set of coupled amplitude equations

appropriate to a system of superimposed, traveling oblique rolls with no spa-

tial variation of the amplitude. A number of regular solutions are found, as

well as at least one chaotic solution. A particularly intriguing result is the

existence of an alternating roll state which is the superposition of two stand-

ing waves that are one quarter of a cycle out of phase in time. The states of

extended chaos reported on here display local regions which temporarily con-

tain zig and zag standing waves that are out of phase with each other. A time

153

a b c d

e f g h

Figure 5.1: Time series of images from cell I5243. The applied frequency was85 Hz, σ = 1×10−8 ohm−1 m−1, and ε = 0.005. The images were taken roughly1 s apart.

series from such a region is shown in Fig. 5.1. For comparison, I show Fig. 2

from Ref. [105] in Fig. 5.2. This shows the results of a numerical simulation

of the alternating roll solution to the amplitude equations used in Ref. [105].

When comparing Fig. 5.1 to Fig. 5.2, it is important to remember that the

“alternating-roll” states observed in the experiment (Fig. 5.1) are not stable,

but exist as part of the state of spatio-temporal chaos. However, this com-

parison does suggest that equations of the form used in Ref. [105] with the

appropriate spatial gradient terms would correctly describe the dynamics of

the system. And, conversely, that a system of small enough spatial extent

would be described by the equations used in Ref.[105].

Another probe of the spatio-temporal chaos is a modulation of the applied

frequency at twice the traveling frequency. This is known to stabilize standing

wave states from traveling wave states in EC[103]. This could potentially phase

lock the system into two standing wave modes: zig and zag. The amplitude

of the modes might still exhibit spatio-temporal chaos, but the overall system

would be relatively simpler as it would only consist of two instead of four

modes.

154

Figure 5.2: Examples of the alternating roll solution from Ref. [105], Fig. 2.

One of the most striking features of the spatio-temporal chaos in this sys-

tem is the connection between the extended states and the localized states

(see Sec. 4.4). The localized states are examples of spatio-temporal chaos in

that their lengths, number density, and possibly even their velocity and am-

plitude, vary in time and space. The apparently continuous transition as a

function of conductivity from a spatially extended state, to a state which is

occasionally localized, to a purely localized state suggests that the localized

states represent a limit of the extended state of spatio-temporal chaos in which

the correlation length perpendicular to the director reaches a minimum. I use

apparently continuous because I only have three data points at this time. The

existence of this transition raises a number of interesting questions.

The intermediate range is similar in some respects to the dispersive chaos

observed in RBC in binary fluid convection[104]. Both systems exhibit bursts

of localized convection which occur randomly in space and time. Given that

the mechanism for producing the traveling waves is similar in both cases, it is

worth exploring the possible connections between these two states.

In addition to the similar “bursting” states, the possible connection be-

tween the worms and observations of localized states in RBC in binary-fluid

155

mixtures [17, 18, 43, 44, 45] should be studied. In binary-fluid convection,

the interaction of the two fields, temperature and concentration, which pro-

duces the traveling waves is found to be equally important for the production

of pulses[48, 49]. The WEM has an analogous interaction of two fields, the

slowly evolving conductivity and the faster director field [42]. A potentially im-

portant difference between the two systems is that EC in I52 is an anisotropic

system. The anisotropy clearly plays a significant role in the localization of

the pattern, as the worms are only localized perpendicular to the director.

Also, the worms are stable, and for low enough σ, they do not appear to result

from transient extended states. This situation differs from the observation of

pulses in binary-fluid convection in two dimensions[18] which is an isotropic

system. There the pulses resulted from spatially extended linear transients

and were unstable [18]. In a completely different type of system, the catalytic

oxidation of CO on platinum, localized patterns in two dimensions have been

observed[106]. This system is similar to EC in that it is an anisotropic system

because of the properties of the platinum surface.

The onset of the worms needs to be studied in detail. For example, it has

not yet been determined if the worm are the result of a forward or backward

bifurcation. Also, there are the two different regimes in which worms occur.

There are the worms which result from localization of an already existing ex-

tended pattern. This is similar to the formation of localized pulses from linear

transients in binary fluid convection[18]. But, at the lowest conductivities I

studied, the worms appear to nucleate straight from the uniform background.

Unlike other nucleation processes, once the state is formed it has a long, but

definitely finite, lifetime. As ε is increased, the lifetime of the states increases

until it is effectively infinite.

One possible cause of this interesting behavior near onset is the inherent

chaotic nature of the pattern. Recall the local time dependence of the various

modes shown in Fig. 4.22. Assume that there is a finite threshold amplitude for

the existence of a worm. The amplitudes of the four modes fluctuate around

156

until this threshold is crossed, and a worm is formed. However, the amplitude

continues to fluctuate and eventually drops below the threshold. Then, the

worm dies. At high enough ε, it becomes extremely rare that the amplitudes of

the modes fluctuate below the threshold, and the worm effectively lives forever.

If this scenario is correct, then at extremely low values of the conductivity,

because the worms appeared directly form the uniform state, the threshold to

form the worm must be extremely small. And, the growth of one worm must

rapidly suppress convection in its neighborhood.

In addition to the puzzle of their generation, there are a number of features

of the worms that will be relatively straightforward to measure. The lifetimes,

distribution of lengths, amplitudes, and velocity of the worms all need to be

measured. Also, the interactions of worms and in particular, the periodic

spacing of the worms are interesting problems to study.

In summary, the system of EC in the NLC I52 has presented us with some

unique opportunities. With the recent theoretical work of the WEM, there

exists a model which accurately describes the linear properties of the initial

transition. A relatively large number of the material parameters of I52 are

known from independent measurements, and the rest can be deduced by fit-

ting the predictions of the WEM. In principle, the WEM can be used to derive

coupled CGL equations which should describe the weakly nonlinear behavior.

Because the initial transition is forward, quantitative comparisons between

the solutions of the CGL and the experimental observations should be possi-

ble. Perhaps most exciting, the weakly nonlinear states which can be studied

in this fashion are examples of spatio-temporal chaos in an extended system

and localized states which exhibit the coexistence of the uniform state with a

pattern over a wide range of parameters. These behaviors are inherently non-

linear and nonequilibrium, and the possibility of making quantitative studies

of these phenomena in a single system is very exciting.

157

Appendix A

Material Parameters

In order to make the quantitative comparison between the WEM and the

experimental results, it was necessary to determine a number of the material

parameters of I52 as a function of the temperature T. There had been previous

measurements of a number of the parameters. In particular, εa(T ), γ1(T ), and

the bulk viscosity η(T ) had been measured[41]. By studying the bend Frederiks

transition[22, 63], I was able to confirm the measurements of εa and measure

the bend elastic constant.

For the bend Frederiks transition, the liquid crystal is aligned with the

director perpendicular to the plates (homeotropic alignment, see Sec. 3.3.2).

When an electric field is applied and εa < 0, there is a continuous transition

at V = V Fc to a state where the director is no longer normal to the plates.

The theoretical value of the onset voltage is given by

V Fc = π(

K33

ε0εa)1/2. (A.1)

As εa goes through zero, the onset voltage should diverge. Data for V Fc (T )

and for the capacitance of the liquid-crystal cell allow for determination of

εa(T ) and K33(T )/εa(T ). Obviously, the measurements can be used to obtain

K33(T ) and εa(T ) separately. As a check on our bend Frederiks measurements,

I performed the same experiments using 4-methoxybenzylidene-4′-butylaniline

158

(MBBA), a liquid crystal for which the material parameters are known,[33]

and obtained good agreement with the accepted values. The values of εa and

K33 obtained for I52 are consistent with the observed behavior in the electro-

convection experiments. The other material parameters were determined using

fits to the SM predictions of EC (see Sec. 4.3).

The Frederiks transition experiments were performed in the same appa-

ratus as described in Sec. 3.2. An additional polarizer was placed above the

cell to be used as an analyzer. Below the Frederiks transition, the director

is perpendicular to the glass plates, so light traveling through the cell sees

a uniform index of refraction. Therefore, when the analyzer is crossed with

respect to the polarizer, the intensity of light coming through the analyzer is

zero. Above the transition, the director has a component which is parallel to

the glass plates. In this case, light traveling through the cell is split into an

ordinary and extraordinary beam. These travel with different phase velocities

and upon leaving the cell the polarization is rotated. This results in regions

where the intensity is not zero.

The cells were constructed with mylar spacers and sealed with Torr seal

(see Sec. 3.3). They were filled with undoped I52 using capillary action under

a vacuum (see Sec. 3.3.4). The homeotropic alignment was achieved with

lecithin (see Sec. 3.3.2).

To determine V Fc , the intensity of the light transmitted through the NLC

cell was measured as a function of voltage. The voltage was increased in steps

of 0.05 V every 5 minutes, and then decreased at the same rate to check for

hysteresis. The cells listed in Table 4.1 were filled with I52 that came from

two separate batches. Cells for EC made before 1/94 used I52 from batch one

and cells made after 1/94 used I52 from batch two. Table A.1 lists the cells

used in the Frederiks measurements. Because the ratio d/A, where d is the cell

thickness and A is the electrode area, is used to extract εa from the capacitance

measurements, the thicker cells allowed for a more accurate measurement of

εa. Because the results from I52L5 and I52L6 differed, cell I52L10 was used to

159

Table A.1: Summary of Frederiks Cells.

label date filled batch of I52 thickness

I52L5 1/4/93 1 28 µm

I52L6 11/7/94 2 56 µm

I52L7 11/7/94 2 58 µm

I52L10 11/7/94 2 25 µm

check for thickness effects. In addition, cell I52L5 was remeasured at the time

I52L6 was used, and the earlier results in I52L5 were reproduced. This is a

testament to the stability of I52, at least at room temperature. In particular,

the lecithin coating lasted for two years, and no effects from the Torr seal

used to seal the cells were observed. Also, it shows that the calibration of the

thermistors remained constant.

Figure A.1 shows a typical transmission curve for the cell I52L5. The initial

rounding is due to a slight inhomogeneity in the cell, and V Fc is determined by

a linear extrapolation of the sharp rise in intensity to a value of zero intensity.

For this case, V Fc = 25.16 V. (The data for the other cells were similar.)

Figure A.2 shows the onset voltage as a function of temperature for cell

I52L5, and the curve is diverging as expected. In Fig. A.3, I plot (V Fc )−2

versus temperature for cell I52L5. Extrapolating this curve to zero determines

εa = 0. The results are fit quite well by a straight line and give εa = 0 at

T = 57.51C. Figure A.4 shows the plot of (V Fc )−2 versus temperature for the

cells I52L6, I52L7, and I52L10. The results are also fit by a straight line, but

now the temperature at which εa = 0 is T = 62.71C.

To obtain εa separately, I measured the capacitance of the liquid-crystal

cell both above and below the Frederiks transition and computed

εa =(C‖ − C⊥)d

εA

160

24 25 26Voltage ( Vrms )

0

0.5

1

Inte

nsity

( ar

b. u

nits

)

Figure A.1: The transmission curve for the measurement of the onset of theFrederiks transition in the cell I52L5 at T = 29.8 C. Shown here is boththe intensity of light transmitted through the cell as a function of voltage forstepping up the voltage (diamonds) and for stepping down the voltage (circles).The solid line shows the extrapolation to zero intensity used to determine V F

c .

where C‖ and C⊥ are the capacitance of the cell with the director parallel

and perpendicular to the field respectively, A is the area of the cell containing

liquid crystal, and d is the thickness of the cell. Measuring εa by this method

automatically excludes the contribution of the spacers to the capacitance of

the cell. Below the transition, one measures C‖ directly because the director

is aligned perpendicular to the glass plates. Above the transition, the director

forms an angle with respect to the field, and the high voltage limit of the

capacitance is given by[107, 108, 109]

C = C⊥ +S

V

where the slope S is a function of the material parameters and independent

of A/d.[107, 109] Extrapolating 1/V as a function of C to zero as shown in

161

30 40 50 60Temperature (°C)

0

20

40

60

80

VcF (V

rms)

Figure A.2: Results for the measurement of V Fc of the bend Frederiks experi-

ment in cell I52L5 as a function of temperature.

Fig. A.5 for T = 29.8C, one obtains C⊥.

The main sources of error in computing εa are the uncertainty in the height

and area of the cell. As such for cell I52L5, I only obtained a measurement

of εa for a single temperature. The theoretical formula for the onset voltage

is used to determine K33 from this measurement. For I52L5, at T = 29.8C, I

obtain εa = −0.051 ± 0.014 which gives K33 = (28 ± 8) x 10−12 N. For the

cells I52L6 and I52L7, I was able to obtain εa for a range of temperature as

shown in Fig. A.6. The linear behavior of εa over the range of temperatures

considered implies that K33 is temperature independent in this range.

This theoretical formula for V Fc assumes infinite anchoring strength of the

director at the boundaries. In practice, measured values of V Fc are usually

lower than the theoretical ones. Performing the experiment with MBBA gives

a rough idea of the applicability of the theoretical formula; however, there is

no guarantee that MBBA and I52 have the same anchoring strengths when

162

30 40 50 60Temperature (°C)

0

0.5

1

1.5

2

1000

(( V

cF )-2)

Figure A.3: The data from Fig. A.2 plotted as (V Fc )−2 vs. T . The solid line

is a straight line fit to the data with the point at T = 25.2C excluded. Thefit gives εa = 0 at T = 57.51C.

lecithin is used to align the director. For MBBA at T = 25.0C, I measured

V Fc = 3.95 V. The value computed using the previous measured values[33,

110, 111] εa = −0.53 and K33 = 8.4x10−12N is V Fc = 4.26 V. Using our

measured value of V Fc and our capacitance measurements for MBBA, I found

εa = −0.6 ± 0.1 and K33 = (8 ± 1) x 10−12N.

Because the second batch of I52 was used in the experiments for which

comparisons to the WEM were made, I will summarize the results for the SM

material parameters of I52 here, using the Frederiks results of batch two. The

previous measurements of the rotational viscosity γ1, the bulk viscosity η, and

ε⊥[41] were used. The data for γ1 are plotted in Fig. A.7. The data for ε⊥ are

plotted in Fig. A.8, and the data for η are plotted in Fig. A.9. The results for

εa are shown in Fig. A.6, and the result for K33 is K33 = (23.± 3.) × 10−12 N

independent of temperature.

163

20 30 40 50 60

Temperature (oC)

0

1

2

1000

((V

cF )-2)

Figure A.4: The results of (V Fc )−2 vs. T for cell I52L6 (circles), I52L7

(squares), and I52L10 (crosses). The linear fit gives εa = 0 at T = 62.71C.

By fitting1 the measured Vc and Θ curves (see Sec. 4.3), it was found

that the remaining 2 elastic constants could be expressed in terms of K33:

K11 = 0.8K33, K22 = 0.55K33 The known relationships between the viscosities

give (see Sec. 2.2): α2 = α3 − γ1 (definition of γ1) and α5 = α6 − α2 − α3

(Onsager relation). The fits to Vc and Θ curves (see Sec. 4.3) also provided

the remaining four viscosities in terms of the previously measured viscosities:

α1 = 0.1γ1, α3 = 0.1γ1, α4 = 2η, and α6 = −0.15γ1. With the temperature

dependence of the above parameters fixed by the temperature dependence of

the measured material parameters, it was found that to fit the temperature

dependence of the measured values of Vc at zero frequency required σa/σ⊥ =

0.26, 0.30, 0.34, 0.38, 0.42, and 0.45 for T = 29, 34, 39, 44, 49, and 59C, re-

spectively. At these temperatures and V = 5 Volts, we measured σ⊥ =

0.28, 0.37, 0.49, 0.65, 0.85, and 1.41 × 10−8 ohm−1 m−1. I measure σ⊥ with

1All of the fitting of the curves was done using Mathematica routines provided by Martin

Treiber for calculating the theoretical values of the parameters.

164

0 0.01 0.02 0.03

1 / V

705.5

706

706.5

707

707.5

Cap

acita

nce

(pF)

Figure A.5: 1/V versus capacitance (pF) for I52 at T = 29.8C (solid circles).The solid line is a straight line fit of the data extrapolated to 1/V = 0.

an applied frequency of 50 Hz. This frequency was chosen as it is the middle

of the range used for the EC measurements. Also, even though the conductiv-

ity is frequency dependent to fairly high frequencies (see Fig. 4.5), the ionic

contribution to the conductivity appears to have reached its high frequency

asymptotic value by 50 Hz, and the continued decrease is due to other mech-

anisms, such as the suggested dielectric loss. For comparison to the theory,

the ionic conductivity is the relevant quantity. Furthermore, the value of σ⊥

enters the theory as the scale factor for the applied frequency. The fact that

the curvature of the measured and predicted curves for Vc agree so well is

evidence that the correct σ⊥ was used.

For the two new parameters of the WEM, λσ was chosen to be small enough

that its value did not affect the calculation. This sets a lower limit on τrec of

≈ 10 s (see Chapter 5 for a discussion of this value). For each T ≥ 29C,√

µ+⊥µ

−⊥ was determined from the Hopf frequency and found to increase mono-

165

20 30 40 50 60 70

Temperature (oC)

0

0.02

0.04

0.06

ε a

Figure A.6: Shown here is εa as a function of temperature for the cells I52L6(circles) and I52L7 (triangles). The triangle represents the measurement ofεa = 0 using V F

c . The straight line is a fit to the data which gives εa =−(1.487 × 10−3(C)−1) × T + 9.257 × 10−2.

tonically with T from 0.40×10−10 m2 V−1 s−1 to 0.47×10−10 m2 V−1 s−1 which

is consistent with measured values of the mobility. Also, it is expected that

the mobility varies with temperature based on both the variation of the vis-

cosities with temperature and especially the variation of the conductivity with

temperature. The variation of the conductivity has to be the result of some

combination of changes in the mobility and the number density of ions with

temperature.

There is an important issue regarding the temperature scale used in the

thesis. The original temperature scale used in Ref. [9] and Ref. [42] was based

on calibrating the thermistors against a mercury thermometer. Since then, the

Santa Barabara group has established a standard calibrated thermistor for use

in the fluid mechanics experiments. The standard thermistor was calibrated

against a platinum resistance thermometer[112]. When the thermistors used

166

20 40 60 80 100

Temperature (oC)

0

0.1

0.2

0.3

γ 1 (P

a s)

Figure A.7: Shown here are the measured values of γ1 from Ref. [41].

to measure the temperature of the apparatus were recalibrated against the

standard thermistor, the two calibrations agreed around 80C, and differed by

at most ≈ 1.4C for 25C. The temperatures quoted in this thesis are based on

the calibration against the standard thermistor; therefore, the slight difference

in some of the temperatures quoted here from values previously reported in the

literature[42, 97]. This change does not affect the comparison of theory and ex-

periment in Ref. [42] because the shift in temperature was significantly smaller

than the uncertainties in the material parameters determined elsewhere (γ1,

ε⊥, and η), and the parameters determined by myself used the same tempera-

ture scale for both the measurement of the parameters and the measurement

of the linear properties. Figure A.10 shows both calibration curves.

167

20 40 60 80

Temperature (oC)

2.8

2.9

3

ε per

p

εperp as measured by Finkenzeller et al.

Figure A.8: Shown here are the measured values of ε⊥ from Ref. [41].

168

20 40 60 80

Temperature (oC)

0.01

0.02

0.03

η(P

a s)

Figure A.9: Shown here are the measured values of the bulk viscosity η fromRef. [41].

169

20 40 60 80

temperature (oC)

0

50

100

resi

stan

ce (K

Ω)

Figure A.10: The two curves are the calibration of the temperature scale usinga mercury thermometer (+) and the UC Santa Barbara calibrated thermistor(x). The two calibrations agree around 80C. The solid lines are the respectivefits to R(T ) = R0 exp([T −T0][A+B(T −T0) +C(T −T0)

2 +D(T −T0)3]/T ),

the standard resistance as a function of temperature for a thermistor.

170

Appendix B

Tables of data

171

Table B.1: Data for Fig. 3.8a

minutes T (C) minutes T (C) minutes T (C) minutes T (C)

0.000 34.913 30.000 34.914 60.000 34.914 90.000 34.914

1.000 34.913 31.000 34.914 61.000 34.913 91.000 34.916

2.000 34.911 32.000 34.914 62.000 34.914 92.000 34.914

3.000 34.915 33.000 34.913 63.000 34.913 93.000 34.916

4.000 34.914 34.000 34.911 64.000 34.914 94.000 34.915

5.000 34.913 35.000 34.914 65.000 34.915 95.000 34.913

6.000 34.915 36.000 34.914 66.000 34.916 96.000 34.914

7.000 34.915 37.000 34.915 67.000 34.916 97.000 34.913

8.000 34.916 38.000 34.913 68.000 34.916 98.000 34.915

9.000 34.914 39.000 34.914 69.000 34.917 99.000 34.914

10.000 34.913 40.000 34.915 70.000 34.914 100.000 34.917

11.000 34.915 41.000 34.915 71.000 34.913 101.000 34.916

12.000 34.915 42.000 34.914 72.000 34.913 102.000 34.914

13.000 34.914 43.000 34.913 73.000 34.912 103.000 34.912

14.000 34.915 44.000 34.913 74.000 34.911 104.000 34.912

15.000 34.913 45.000 34.915 75.000 34.912 105.000 34.913

16.000 34.913 46.000 34.914 76.000 34.911 106.000 34.913

17.000 34.913 47.000 34.916 77.000 34.913 107.000 34.914

18.000 34.912 48.000 34.915 78.000 34.915 108.000 34.915

19.000 34.913 49.000 34.915 79.000 34.914 109.000 34.914

20.000 34.913 50.000 34.915 80.000 34.915 110.000 34.916

21.000 34.914 51.000 34.915 81.000 34.915 111.000 34.915

22.000 34.914 52.000 34.912 82.000 34.914 112.000 34.914

23.000 34.916 53.000 34.913 83.000 34.914 113.000 34.914

24.000 34.915 54.000 34.916 84.000 34.915 114.000 34.914

25.000 34.914 55.000 34.914 85.000 34.913 115.000 34.914

26.000 34.914 56.000 34.913 86.000 34.912 116.000 34.915

27.000 34.914 57.000 34.914 87.000 34.912 117.000 34.914

28.000 34.916 58.000 34.912 88.000 34.914 118.000 34.915

29.000 34.914 59.000 34.914 89.000 34.912

172

Table B.2: Data for Fig. 3.8b

hours T (C) hours T (C) hours T (C) hours T (C)

0.000 34.914 3.500 34.914 7.000 34.915 10.500 34.914

0.167 34.914 3.667 34.915 7.167 34.913 10.667 34.912

0.333 34.914 3.833 34.914 7.333 34.913 10.833 34.914

0.500 34.913 4.000 34.914 7.500 34.915 11.000 34.915

0.667 34.915 4.167 34.913 7.667 34.913 11.167 34.914

0.833 34.913 4.333 34.915 7.833 34.913 11.333 34.914

1.000 34.914 4.500 34.913 8.000 34.914 11.500 34.914

1.167 34.913 4.667 34.914 8.167 34.913 11.667 34.914

1.333 34.915 4.833 34.913 8.333 34.913 11.833 34.913

1.500 34.913 5.000 34.913 8.500 34.913 12.000 34.912

1.667 34.915 5.167 34.914 8.667 34.914 12.167 34.916

1.833 34.913 5.333 34.914 8.833 34.915 12.333 34.914

2.000 34.913 5.500 34.914 9.000 34.914 12.500 34.916

2.167 34.915 5.667 34.914 9.167 34.915 12.667 34.914

2.333 34.913 5.833 34.914 9.333 34.914 12.833 34.913

2.500 34.913 6.000 34.914 9.500 34.914 13.000 34.916

2.667 34.913 6.167 34.913 9.667 34.914 13.167 34.914

2.833 34.913 6.333 34.915 9.833 34.914 13.333 34.913

3.000 34.912 6.500 34.914 10.000 34.915 13.500 34.915

3.167 34.915 6.667 34.913 10.167 34.916 13.667 34.914

3.333 34.914 6.833 34.914 10.333 34.914

Table B.3: Data for Fig. 3.11

freq (Hz) gain freq (Hz) gain freq (Hz) gain

100.000 2.047 800.000 0.695 1500.000 0.383

200.000 1.765 900.000 0.624 1600.000 0.360

300.000 1.465 1000.000 0.566 1700.000 0.339

400.000 1.220 1100.000 0.517 1800.000 0.320

500.000 1.036 1200.000 0.476 1900.000 0.303

600.000 0.894 1300.000 0.440 2000.000 0.289

700.000 0.784 1400.000 0.410

173

Table B.4: Data for Fig. 3.12a

freq (Hz) cap (pF) freq (Hz) cap (pF) freq (Hz) cap (pF)

25.000 467.672 300.000 466.096 1300.000 463.829

35.000 467.378 400.000 465.912 1400.000 463.853

45.000 468.123 500.000 465.425 1500.000 464.266

55.000 468.136 600.000 465.205 1600.000 463.752

65.000 468.087 700.000 464.418 1700.000 463.401

75.000 468.054 800.000 464.607 1800.000 463.861

85.000 467.929 900.000 464.923 1900.000 464.335

95.000 467.918 1000.000 464.397 2000.000 463.885

100.000 467.876 1100.000 464.054

200.000 467.194 1200.000 463.876

Table B.5: Data for Fig. 3.12b

freq (Hz) cap (pF) freq (Hz) cap (pF) freq (Hz) cap (pF)

25.000 -15.290 300.000 0.236 1300.000 0.490

35.000 -20.444 400.000 0.262 1400.000 0.453

45.000 -4.034 500.000 0.287 1500.000 0.411

55.000 7.413 600.000 0.259 1600.000 0.526

65.000 9.854 700.000 0.469 1700.000 0.555

75.000 6.750 800.000 0.352 1800.000 0.534

85.000 5.002 900.000 0.320 1900.000 0.435

95.000 3.615 1000.000 0.429 2000.000 0.496

100.000 2.926 1100.000 0.494

200.000 0.475 1200.000 0.480

174

Table B.6: Data for Fig. 3.12c

freq (Hz) cap (pF) freq (Hz) cap (pF) freq (Hz) cap (pF)

25.000 449.399 300.000 465.775 1300.000 463.760

35.000 445.643 400.000 465.407 1400.000 463.815

45.000 463.679 500.000 464.928 1500.000 464.183

55.000 475.019 600.000 464.777 1600.000 463.759

65.000 477.391 700.000 464.295 1700.000 463.439

75.000 474.389 800.000 464.551 1800.000 463.981

85.000 472.646 900.000 464.487 1900.000 464.275

95.000 471.089 1000.000 464.111 2000.000 463.965

100.000 470.213 1100.000 463.835

200.000 467.100 1200.000 463.706

Table B.7: Data for Fig. 3.13a

freq (Hz) res (MΩ) freq (Hz) res (MΩ) freq (Hz) res (MΩ)

200.000 382.179 900.000 29.775 1600.000 25.446

300.000 152.518 1000.000 36.560 1700.000 31.003

400.000 105.041 1100.000 41.829 1800.000 19.140

500.000 79.416 1200.000 33.491 1900.000 10.800

600.000 53.662 1300.000 29.801 2000.000 14.520

700.000 83.562 1400.000 23.541

800.000 43.569 1500.000 17.541

175

Table B.8: Data for Fig. 3.13b

freq (Hz) res (MΩ) freq (Hz) res (MΩ) freq (Hz) res (MΩ)

25.000 8.000 300.000 8.063 1300.000 8.114

35.000 8.005 400.000 8.083 1400.000 8.106

45.000 8.004 500.000 8.094 1500.000 8.123

55.000 8.010 600.000 8.115 1600.000 8.113

65.000 8.015 700.000 8.103 1700.000 8.112

75.000 8.015 800.000 8.118 1800.000 8.127

85.000 8.018 900.000 8.121 1900.000 8.128

95.000 8.021 1000.000 8.115 2000.000 8.127

100.000 8.023 1100.000 8.114

200.000 8.049 1200.000 8.111

Table B.9: Data for Fig. 3.13c

freq (Hz) res (MΩ) freq (Hz) res (MΩ) freq (Hz) res (MΩ)

25.000 7.880 300.000 7.660 1300.000 6.389

35.000 7.749 400.000 7.490 1400.000 6.037

45.000 7.881 500.000 7.337 1500.000 5.407

55.000 8.167 600.000 7.013 1600.000 6.167

65.000 8.361 700.000 7.328 1700.000 6.395

75.000 8.288 800.000 6.778 1800.000 5.724

85.000 8.274 900.000 6.415 1900.000 4.647

95.000 8.228 1000.000 6.653 2000.000 5.170

100.000 8.190 1100.000 6.811

200.000 7.886 1200.000 6.571

Table B.10: Data for Fig. 3.14 open circles.

freq (Hz) res (MΩ) freq (Hz) res (MΩ) freq (Hz) res (MΩ)

25.000 16.810 200.000 16.780 1600.000 16.610

50.000 16.810 400.000 16.730

100.000 16.800 800.000 16.680

176

Table B.11: Data for Fig. 3.14 closed circles.

freq (Hz) res (MΩ) freq (Hz) res (MΩ) freq (Hz) res (MΩ)

25.000 16.885 200.000 15.744 1600.000 13.292

50.000 16.720 400.000 14.996

100.000 16.345 800.000 14.161

Table B.12: Data for Fig. 4.3.

freq (Hz) cap(pF) freq (Hz) cap(pF) freq (Hz) cap(pF)

5.000 888.446 95.000 465.883 1100.000 426.618

10.000 731.673 100.000 464.043 1200.000 425.925

15.000 650.993 200.000 445.304

20.000 604.017 300.000 438.478

25.000 570.653 400.000 435.046

35.000 530.980 500.000 432.704

45.000 508.328 600.000 431.335

55.000 493.667 700.000 429.560

65.000 483.620 800.000 428.784

75.000 476.102 900.000 428.180

85.000 470.462 1000.000 427.415

177

Table B.13: Data for Fig. 4.5.

freq (Hz) res (MΩ) freq (Hz) res (MΩ) freq (Hz) res (MΩ)

5.000 12.605 95.000 8.179 1100.000 5.202

10.000 11.476 100.000 8.117 1200.000 5.050

15.000 10.759 200.000 7.409 1300.000 4.962

20.000 10.243 300.000 7.002 1400.000 4.748

25.000 9.872 400.000 6.656 1500.000 4.617

35.000 9.375 500.000 6.375 1600.000 4.555

45.000 9.042 600.000 6.103 1700.000 4.389

55.000 8.790 700.000 5.941 1800.000 4.271

65.000 8.599 800.000 5.751 1900.000 4.135

75.000 8.439 900.000 5.525 2000.000 4.110

85.000 8.290 1000.000 5.380

Table B.14: Data for Fig. 4.6 open circles.

freq (Hz) res (MΩ) freq (Hz) res (MΩ) freq (Hz) res (MΩ)

25.000 2940.000 200.000 1090.000 1600.000 316.000

50.000 2270.000 400.000 735.000

100.000 1610.000 800.000 481.000

Table B.15: Data for Fig. 4.6 open triangles.

freq (Hz) res (MΩ) freq (Hz) res (MΩ)

25.000 9100.000 100.000 50000.000

50.000 16700.000 200.000 50000.000

Table B.16: Data for Fig. 4.6 closed circles.

freq (Hz) res (MΩ) freq (Hz) res (MΩ) freq (Hz) res (MΩ)

25.000 429.200 200.000 124.400 800.000 38.730

100.000 205.800 400.000 72.730 1600.000 18.660

178

Table B.17: Data for Fig. 4.6 closed triangles.

freq (Hz) res (MΩ) freq (Hz) res (MΩ)

25.000 131.600 200.000 58.100

100.000 87.000 400.000 31.300

Table B.18: Data for Fig. 4.7.

volts res (MΩ) volts res (MΩ) volts res (MΩ)

5.652 8.825 11.314 8.652 16.972 8.483

7.065 8.789 12.728 8.611 18.392 8.432

8.480 8.744 14.142 8.568 19.806 8.401

9.894 8.697 15.557 8.526 21.220 8.357

179

Table B.19: Data for Fig. 4.8 T = 29C.

Ωτq Vc (V) Ωτq Vc (V) Ωτq Vc (V)

0.589 15.400 1.195 20.470 1.793 28.130

0.883 17.710 1.493 24.020 2.091 32.740

Table B.20: Data for Fig. 4.8 T = 34C.

Ωτq Vc (V) Ωτq Vc (V) Ωτq Vc (V)

0.446 13.230 1.120 17.800 2.464 33.750

0.675 14.450 1.570 22.240

0.891 16.040 2.016 27.590

Table B.21: Data for Fig. 4.8 T = 39C.

Ωτq Vc (V) Ωτq Vc (V) Ωτq Vc (V)

0.339 12.060 0.846 14.630 1.858 23.150

0.508 12.730 1.185 16.970 2.196 26.900

0.678 13.560 1.524 19.970 2.534 31.950

Table B.22: Data for Fig. 4.8 T = 44C.

Ωτq Vc (V) Ωτq Vc (V) Ωτq Vc (V)

0.383 11.680 1.156 16.030 1.925 23.360

0.639 12.770 1.412 17.940 2.182 25.540

0.895 14.230 1.669 20.020 2.439 29.300

Table B.23: Data for Fig. 4.8 T = 49C.

Ωτq Vc (V) Ωτq Vc (V) Ωτq Vc (V)

0.493 11.650 1.378 16.760 2.264 26.270

0.726 13.000 1.673 19.070 2.560 30.420

1.083 14.770 1.968 22.290

180

Table B.24: Data for Fig. 4.8 T = 59C.

Ωτq Vc (V) Ωτq Vc (V) Ωτq Vc (V)

0.358 10.600 1.422 16.320 2.487 28.510

0.710 11.910 1.777 20.150

1.067 13.700 2.132 24.430

Table B.25: Data for Fig. 4.9 I5243 T = 44C .

Ωτq Vc (V) Ωτq Vc (V) Ωτq Vc (V)

0.463 12.190 0.834 18.080 1.205 25.470

0.556 13.200 0.927 19.890 1.297 27.400

0.649 14.270 1.019 21.740 1.390 29.300

0.741 16.270 1.112 23.580

Table B.26: Data for Fig. 4.9 I5221 T = 44C.

Ωτq Vc (V) Ωτq Vc (V) Ωτq Vc (V)

0.270 12.530 1.351 19.280 2.432 26.870

0.540 13.790 1.621 21.290 2.702 28.470

0.811 15.450 1.891 23.340 2.972 30.130

1.081 17.270 2.161 25.120 3.242 31.890

Table B.27: Data for Fig. 4.9 I5221 T = 64C.

Ωτq Vc (V) Ωτq Vc (V) Ωτq Vc (V)

0.025 10.350 0.402 11.250 0.905 14.100

0.050 10.300 0.503 11.750 1.005 14.850

0.101 10.350 0.603 12.300 1.257 16.850

0.201 10.550 0.704 12.900 1.508 19.050

0.302 10.850 0.804 13.500 1.759 20.700

181

Table B.28: Data for Fig. 4.9 I5243 T = 24C.

Ωτq Vc (V) Ωτq Vc (V) Ωτq Vc (V)

0.463 15.880 1.483 41.100 1.854 53.030

0.834 22.300 1.576 42.800 1.946 57.030

1.205 31.100 1.668 47.070

1.390 38.280 1.761 49.960

Table B.29: Data for Fig. 4.10 I5263 T = 44C.

Ωτq Θ (deg.) Ωτq Θ (deg.) Ωτq Θ (deg.)

0.383 38.500 1.412 25.900 2.439 6.200

0.639 36.400 1.669 22.100 2.690 0.000

0.895 33.500 1.925 18.700

1.156 29.700 2.182 13.100

Table B.30: Data for Fig. 4.10 I5243 T = 24C.

Ωτq Θ (deg.) Ωτq Θ (deg.) Ωτq Θ (deg.)

0.463 37.020 1.483 15.500 1.854 0.000

0.834 33.300 1.576 8.700 1.946 0.000

1.205 24.900 1.668 0.000

1.390 18.140 1.761 0.000

Table B.31: Data for Fig. 4.11 T = 29C.

Ωτq ω (s−1) Ωτq ω (s−1) Ωτq ω (s−1)

0.589 0.659 1.195 0.884 1.793 1.285

0.883 0.726 1.493 1.126 2.091 1.481

182

Table B.32: Data for Fig. 4.11 T = 34C.

Ωτq ω (s−1) Ωτq ω (s−1) Ωτq ω (s−1)

0.446 0.507 1.120 0.667 2.464 1.101

0.675 0.536 1.570 0.826

0.891 0.609 2.016 0.957

Table B.33: Data for Fig. 4.11 T = 39C.

Ωτq ω (s−1) Ωτq ω (s−1) Ωτq ω (s−1)

0.339 0.467 0.846 0.535 1.858 0.722

0.508 0.471 1.185 0.601 2.196 0.841

0.678 0.514 1.524 0.680 2.534 0.955

Table B.34: Data for Fig. 4.11 T = 44C.

Ωτq ω (s−1) Ωτq ω (s−1) Ωτq ω (s−1)

0.639 0.501 1.412 0.526 2.182 0.607

0.895 0.494 1.669 0.572 2.439 0.630

1.156 0.512 1.925 0.574

Table B.35: Data for Fig. 4.11 T = 49C.

Ωτq ω (s−1) Ωτq ω (s−1) Ωτq ω (s−1)

0.493 0.423 1.378 0.490 2.264 0.528

0.726 0.443 1.673 0.504 2.560 0.523

1.083 0.478 1.968 0.509

183

Table B.36: Data for Fig. 4.11 T = 59C.

Ωτq ω (s−1) Ωτq ω (s−1) Ωτq ω (s−1)

0.358 0.388 1.422 0.394 2.487 0.380

0.710 0.396 1.777 0.403

1.067 0.386 2.132 0.399

Table B.37: Data for Fig. 4.12a open symbols.

Ωτq ωσ−1/2 Ωτq ωσ−1/2 Ωτq ωσ−1/2

0.912 3.972e-05 1.641 5.414e-05 2.371 7.150e-05

1.094 4.281e-05 1.824 5.509e-05 2.553 7.473e-05

1.277 4.318e-05 2.006 6.282e-05

1.459 4.693e-05 2.188 6.679e-05

Table B.38: Data for Fig. 4.12a closed symbols.

Ωτq ωσ−1/2 Ωτq ωσ−1/2 Ωτq ωσ−1/2

0.446 3.094e-05 1.120 4.067e-05 2.464 6.719e-05

0.675 3.271e-05 1.570 5.039e-05

0.891 3.713e-05 2.016 5.835e-05

Table B.39: Data for Fig. 4.12b open symbols.

Ωτq ωσ−1/2 Ωτq ωσ−1/2 Ωτq ωσ−1/2

0.636 3.525e-05 1.484 4.487e-05 2.333 4.982e-05

0.848 3.855e-05 1.696 4.321e-05 2.545 4.865e-05

1.060 3.894e-05 1.909 4.496e-05

1.272 4.224e-05 2.121 4.593e-05

184

Table B.40: Data for Fig. 4.12b closed symbols.

Ωτq ωσ−1/2 Ωτq ωσ−1/2 Ωτq ωσ−1/2

0.383 3.591e-05 1.156 4.129e-05 1.925 4.631e-05

0.639 4.047e-05 1.412 4.246e-05 2.182 4.900e-05

0.895 3.988e-05 1.669 4.620e-05 2.439 5.088e-05

Table B.41: Data for Fig. 4.12c open symbols.

Ωτq ωσ−1/2 Ωτq ωσ−1/2 Ωτq ωσ−1/2

0.483 3.851e-05 1.450 4.471e-05 2.417 4.750e-05

0.725 4.091e-05 1.692 4.129e-05 2.659 4.826e-05

0.967 4.218e-05 1.934 4.395e-05

1.209 4.218e-05 2.176 4.408e-05

Table B.42: Data for Fig. 4.12c closed symbols.

Ωτq ωσ−1/2 Ωτq ωσ−1/2 Ωτq ωσ−1/2

0.493 3.898e-05 1.378 4.512e-05 2.264 4.859e-05

0.726 4.084e-05 1.673 4.645e-05 2.560 4.819e-05

1.083 4.405e-05 1.968 4.685e-05

Table B.43: Data for Fig. 4.13 triangles.

Ωτq ωσ−1/2d−3 Ωτq ωσ−1/2d−3 Ωτq ωσ−1/2d−3

0.365 0.856 1.021 0.992 1.677 1.069

0.538 0.897 1.239 1.022 1.896 1.058

0.802 0.967 1.458 1.028

185

Table B.44: Data for Fig. 4.13 circles.

Ωτq ωσ−1/2d−3 Ωτq ωσ−1/2d−3 Ωτq ωσ−1/2d−3

0.276 0.000 0.829 0.276 1.390 0.855

0.415 0.000 0.968 0.353 1.530 0.952

0.553 0.000 1.110 0.486 1.670 0.971

0.691 0.000 1.250 0.719 1.950 1.029

Table B.45: Data for Fig. 4.3.

hours Vc (V) hours Vc (V) hours Vc (V)

0.510 11.354 120.120 11.495 163.520 11.550

47.400 11.423 138.800 11.530 257.000 11.656

Table B.46: Data for Fig. 4.3.

hours res MΩ hours res MΩ hours res MΩ

0.000 7.500 168.000 8.890 257.000 9.680

96.000 8.400 240.000 9.450

186

Table B.47: Data for Fig. 4.15 triangles.

ε amp. (rad) ε amp. (rad) ε amp. (rad)

2.041e-02 0.452 9.105e-03 0.086 -2.242e-03 0.018

1.984e-02 0.287 8.556e-03 0.099 -2.794e-03 0.017

1.927e-02 0.366 7.965e-03 0.092 -3.388e-03 0.018

1.870e-02 0.418 7.397e-03 0.064 -3.932e-03 0.018

1.812e-02 0.521 6.787e-03 0.079 -4.503e-03 0.017

1.755e-02 0.227 6.249e-03 0.075 -5.067e-03 0.017

1.697e-02 0.308 5.662e-03 0.063 -5.629e-03 0.016

1.638e-02 0.707 5.097e-03 0.055 -6.186e-03 0.017

1.583e-02 0.182 4.524e-03 0.057 -6.740e-03 0.018

1.527e-02 0.200 3.947e-03 0.059 -7.303e-03 0.017

1.465e-02 0.340 3.386e-03 0.043 -7.846e-03 0.017

1.409e-02 0.144 2.815e-03 0.045 -8.411e-03 0.018

1.356e-02 0.663 2.261e-03 0.033 -8.983e-03 0.019

1.304e-02 0.139 1.724e-03 0.027 -9.552e-03 0.018

1.244e-02 0.233 1.119e-03 0.032 -1.011e-02 0.019

1.187e-02 0.131 5.890e-04 0.028 -1.066e-02 0.018

1.134e-02 0.121 -9.000e-06 0.021 -1.122e-02 0.018

1.079e-02 0.334 -5.390e-04 0.020 -1.179e-02 0.019

1.020e-02 0.114 -1.108e-03 0.021 -1.235e-02 0.017

9.667e-03 0.127 -1.683e-03 0.021

187

Table B.48: Data for Fig. 4.15 circles.

ε amp. (rad) ε amp. (rad) ε amp. (rad)

-6.902e-03 0.016 2.505e-03 0.044 1.239e-02 0.147

-6.653e-03 0.017 3.106e-03 0.048 1.289e-02 0.158

-6.242e-03 0.017 3.618e-03 0.034 1.340e-02 0.203

-5.818e-03 0.019 4.020e-03 0.036 1.391e-02 0.204

-5.372e-03 0.019 4.429e-03 0.049 1.442e-02 0.254

-4.876e-03 0.017 4.905e-03 0.050 1.496e-02 0.350

-4.278e-03 0.019 5.422e-03 0.048 1.546e-02 0.364

-3.855e-03 0.018 5.998e-03 0.065 1.595e-02 0.301

-3.247e-03 0.018 6.465e-03 0.065 1.646e-02 0.195

-2.727e-03 0.021 6.898e-03 0.091 1.699e-02 0.253

-2.347e-03 0.020 7.397e-03 0.101 1.753e-02 0.316

-1.765e-03 0.017 7.931e-03 0.095 1.804e-02 0.584

-1.205e-03 0.018 8.403e-03 0.111 1.856e-02 0.191

-6.910e-04 0.021 8.876e-03 0.084 1.906e-02 0.800

-2.170e-04 0.024 9.344e-03 0.119 1.955e-02 0.464

1.500e-04 0.032 9.878e-03 0.123 2.006e-02 0.496

5.760e-04 0.031 1.038e-02 0.120 2.056e-02 0.410

1.017e-03 0.039 1.089e-02 0.147 2.107e-02 0.697

1.505e-03 0.036 1.139e-02 0.127 2.159e-02 0.980

2.000e-03 0.036 1.189e-02 0.168

188

Table B.49: Data for Fig. 4.16 circles.

ε amp. (rad) ε amp. (rad) ε amp. (rad)

-8.556e-03 0.006 -4.014e-03 0.005 6.030e-04 0.011

-8.062e-03 0.006 -3.521e-03 0.005 1.065e-03 0.016

-7.529e-03 0.005 -3.042e-03 0.005 1.577e-03 0.021

-7.095e-03 0.006 -2.609e-03 0.006 2.037e-03 0.020

-6.650e-03 0.006 -2.145e-03 0.006 2.512e-03 0.020

-6.205e-03 0.006 -1.697e-03 0.006 2.996e-03 0.025

-5.738e-03 0.006 -1.296e-03 0.006 3.454e-03 0.027

-5.262e-03 0.006 -8.010e-04 0.006 3.886e-03 0.031

-4.844e-03 0.006 -3.690e-04 0.006 4.348e-03 0.031

-4.422e-03 0.006 1.190e-04 0.009

Table B.50: Data for Fig. 4.16 triangles.

ε amp. (rad) ε amp. (rad) ε amp. (rad)

-4.843e-03 0.005 1.616e-03 0.022 7.254e-03 0.040

-4.187e-03 0.005 2.192e-03 0.025 7.996e-03 0.040

-3.437e-03 0.005 2.684e-03 0.026 8.672e-03 0.035

-2.815e-03 0.006 3.238e-03 0.026 9.243e-03 0.032

-2.330e-03 0.006 3.752e-03 0.029 9.978e-03 0.044

-1.860e-03 0.006 4.373e-03 0.020 1.060e-02 0.044

-1.007e-03 0.006 4.807e-03 0.024 1.123e-02 0.040

-3.190e-04 0.005 5.434e-03 0.027 1.185e-02 0.029

2.100e-04 0.013 5.917e-03 0.034 1.247e-02 0.048

6.480e-04 0.015 6.564e-03 0.037

189

Table B.51: Data for Fig. 4.18 solid triangles.

ε ω (s−1) ε ω (s−1) ε ω (s−1)

2.067e-03 0.343 2.484e-02 0.260 4.791e-02 0.010

4.125e-03 0.342 2.695e-02 0.287 5.001e-02 0.010

6.196e-03 0.340 2.903e-02 0.264 5.212e-02 0.000

8.267e-03 0.355 3.114e-02 0.264 5.425e-02 0.000

1.034e-02 0.352 3.324e-02 0.264 5.636e-02 0.000

1.239e-02 0.331 3.535e-02 0.231 5.847e-02 0.000

1.450e-02 0.313 3.742e-02 0.008 6.061e-02 0.000

1.653e-02 0.306 3.953e-02 0.000 6.272e-02 0.000

1.860e-02 0.297 4.159e-02 0.000 6.487e-02 0.000

2.069e-02 0.306 4.370e-02 0.010 6.702e-02 0.000

2.278e-02 0.295 4.579e-02 0.000 6.914e-02 0.000

Table B.52: Data for Fig. 4.18 solid circles.

ε ω (s−1) ε ω (s−1) ε ω (s−1)

6.920e-02 0.000 4.377e-02 0.000 1.876e-02 0.335

6.706e-02 0.000 4.168e-02 0.012 1.671e-02 0.317

6.495e-02 0.000 3.962e-02 0.010 1.462e-02 0.340

6.281e-02 0.000 3.751e-02 0.224 1.256e-02 0.339

6.071e-02 0.011 3.331e-02 0.281 1.049e-02 0.334

5.856e-02 0.010 3.121e-02 0.263 8.407e-03 0.338

5.648e-02 0.000 2.914e-02 0.264 6.353e-03 0.334

5.220e-02 0.010 2.709e-02 0.294 4.293e-03 0.360

5.011e-02 0.000 2.500e-02 0.310 2.342e-03 0.301

4.800e-02 0.000 2.292e-02 0.321 2.400e-04 0.375

4.588e-02 0.000 2.082e-02 0.304

190

Table B.53: Data for Fig. 4.18 open triangles.

ε ω (s−1) ε ω (s−1) ε ω (s−1)

2.067e-03 0.347 2.695e-02 0.266 5.212e-02 0.009

4.125e-03 0.337 2.903e-02 0.263 5.425e-02 0.010

6.196e-03 0.334 3.114e-02 0.244 5.636e-02 0.000

8.267e-03 0.344 3.324e-02 0.230 5.847e-02 0.000

1.034e-02 0.355 3.535e-02 0.261 6.061e-02 0.000

1.239e-02 0.329 3.742e-02 0.000 6.272e-02 0.000

1.450e-02 0.333 3.953e-02 0.000 6.487e-02 0.009

1.653e-02 0.319 4.159e-02 0.009 6.702e-02 0.000

1.860e-02 0.297 4.370e-02 0.009 6.914e-02 0.000

2.069e-02 0.305 4.579e-02 0.000 7.126e-02 0.000

2.278e-02 0.295 4.791e-02 0.010

2.484e-02 0.279 5.001e-02 0.010

Table B.54: Data for Fig. 4.18 open circles.

ε ω (s−1) ε ω (s−1) ε ω (s−1)

6.920e-02 0.000 4.588e-02 0.000 2.292e-02 0.303

6.706e-02 0.009 4.377e-02 0.000 2.082e-02 0.316

6.495e-02 0.000 4.168e-02 0.010 1.876e-02 0.313

6.281e-02 0.010 3.962e-02 0.013 1.671e-02 0.310

6.071e-02 0.010 3.751e-02 0.211 1.462e-02 0.335

5.856e-02 0.010 3.542e-02 0.246 1.256e-02 0.335

5.648e-02 0.007 3.331e-02 0.235 1.049e-02 0.339

5.433e-02 0.009 3.121e-02 0.246 8.407e-03 0.339

5.220e-02 0.000 2.914e-02 0.250 6.353e-03 0.364

5.011e-02 0.000 2.709e-02 0.252 2.342e-03 0.367

4.800e-02 0.000 2.500e-02 0.298 2.400e-04 0.393

191

Table B.55: Data for Fig. 4.28.

transition σ 10−8ohm−1m−1 ε

SOR 1.733 0.024

1.562 0.035

1.332 0.053

1.325 0.055

1.200 0.073

SOS 1.733 0.239

1.562 0.148

1.332 0.118

1.325 0.112

1.200 0.094

SNS 1.562 0.030

1.332 0.047

1.325 0.045

1.200 0.045

1.136 0.044

0.993 0.054

0.884 0.067

0.749 0.085

T/SNS 1.332 0.218

1.200 0.128

1.136 0.087

0.993 0.089

0.884 0.086

0.749 0.093

0.682 0.098

0.650 0.106

crosses 1.670 0.029

1.670 0.110

x’s 2.200 0.010

2.200 0.100

192

Table B.56: Data for Fig. 4.29a.

transition Ωτq ε

SNS 0.338 0.037

0.608 0.038

0.877 0.040

1.148 0.036

1.418 0.043

1.688 0.046

SOS 1.148 0.101

1.418 0.108

1.688 0.140

(T/S)NS 0.338 0.102

0.608 0.098

0.877 0.102

SOR 1.418 0.090

1.688 0.075

(T/S)OS 0.338 0.154

0.608 0.172

0.877 0.122

Table B.57: Data for Fig. 4.29b.

transition Ωτq ε

SOS 0.675 0.080

0.811 0.090

0.946 0.120

1.081 0.110

1.216 0.090

1.351 0.080

1.621 0.110

1.891 0.100

193

Table B.58: Data for Fig. A.3

C 1000 (1/V )2 C 1000 (1/V )2 C 1000 (1/V )2

54.640 0.159 50.000 0.437 34.120 1.325

54.020 0.199 43.980 0.756 29.780 1.576

51.440 0.343 40.170 1.005 25.150 1.904

Table B.59: Data for Fig. A.4 I52L6.

C (1/V )2 C (1/V )2 C (1/V )2

23.590 2.311e-03 39.040 1.372e-03 54.020 5.410e-04

28.880 1.993e-03 44.020 1.082e-03 59.070 2.040e-04

33.910 1.680e-03 49.100 8.030e-04

Table B.60: Data for Fig. A.4 I52L7.

C (1/V )2 C (1/V )2 C (1/V )2

28.880 1.937e-03 44.020 1.051e-03 59.070 1.580e-04

33.910 1.640e-03 49.100 7.560e-04

39.040 1.340e-03 54.020 4.610e-04

Table B.61: Data for Fig. A.4 I52L10.

C (1/V )2 C (1/V )2 C (1/V )2

28.880 1.936e-03 39.040 1.335e-03 49.100 7.430e-04

194

Table B.62: Data for Fig. A.6 I52L7.

C εaC εa

C εa

28.880 0.047 39.040 0.035 49.100 0.018

33.910 0.045 44.020 0.024 54.020 0.014

Table B.63: Data for Fig. A.6 I52L6.

C εaC εa

C εa

23.590 0.059 39.040 0.034 54.020 0.004

28.880 0.051 44.020 0.026

33.910 0.050 49.100 0.019

Table B.64: Data for Fig. A.7.

C γ1 (Pa s) C γ1 (Pa s) C γ1 (Pa s)

20.000 0.331 65.800 0.056 84.600 0.031

25.000 0.262 75.200 0.042 94.000 0.021

Table B.65: Data for Fig. A.8.

C ε⊥C ε⊥

C ε⊥

20.000 3.038 65.800 2.881 84.600 2.739

25.000 3.024 75.200 2.841

Table B.66: Data for Fig. A.9.

C η (Pa s) C η (Pa s) C η (Pa s)

20.000 0.032 65.800 0.007 84.600 0.005

25.000 0.025 75.200 0.006 94.000 0.004

195

Table B.67: Data for Fig. A.10 mercury thermometer.

C KΩ C KΩ C KΩ

21.600 128.062 46.000 41.733 71.200 15.582

26.500 100.001 50.900 34.390 75.300 13.397

31.000 79.995 56.000 28.108 80.100 11.269

36.000 64.350 60.800 23.191

41.000 51.725 65.700 19.096

Table B.68: Data for Fig. A.10 calibrated thermistor.

C KΩ C KΩ C KΩ

19.863 128.120 29.060 83.593 38.144 56.009

22.075 115.369 32.004 73.252 40.875 49.855

23.927 105.788 34.949 64.328 43.832 44.035

196

Appendix C

C-Code

This chapter contains the code for the driver used to run the Qua Tech[73]

synthesizer card. The driver functions in wave.c perform exactly like the func-

tions of the same name in the Qua Tech manual, and I refer the reader there

for a full understanding of the board’s operation. The four main pieces of in-

formation required by the board to function are: the number of points used to

define the waveform, the values of the points defining the waveform, the period

between points, and the values in the BMP port which special board settings.

The values for the BMP port are stored using a pointer unsigned *bitmap.

This is initialized in the routine winit() which must be called before using

the board. The board has two modes: single cycle and continuous. Currently,

the only difference between this driver and the one described in the manual

is that the driver given here only contains code for continuous mode. The

library and documentation is stored on the NeXT network in the directory:

/LocalLibrary/QNX4.2.

197

/∗∗∗∗∗∗∗∗∗∗∗ all code is for QNX 4.2 and is divided into four files

∗ makefile, wvlib.h, wave.c, wvconv.c, gen func.c and divide.c

∗ First: this is the code for makefile

∗∗∗∗∗∗∗∗∗∗/# makefile for wvlib.lib

#

# RCS Info follows:

#

# $Header$

#

# $Log$

#

# Additions to CFLAGS from default.mk

DEBUGON = −g

DEBUGOFF = −Q

CFLAGS = −O −ml −w3 −T1 $(DEBUGON) −I /usr/local/include

# Additional defs for creation of library

WLIB = wlib

LIB.o = $(WLIB) $(LIBRARY)

# Addtional macros for this makefile

LIBRARY=wvlib.lib

OBJS = wave.o divide.o gen func.o wvconv.o

HDRS=wave.h

DEST=/usr/local/lib

# Dependence rules for this makefile

$(LIBRARY): $(OBJS) # all files in lib

wave.o : wave.c wave.h # .o files depend on .h and .c files

divide.o: divide.c wave.h

gen func.o: gen func.c wave.h

wvconv.o: wvconv.c wave.h

wave.h:

# How to construct .o from .c

198

.c.o:

$(COMPILE.c) $<

$(LIB.o) −+$@ # add .o to lib

# Special call for removing .o files (call make clean)

clean:

rm ∗.o# Special call for installing the executable in filesystem

install:

mv /usr/local/lib/wvlib.lib /usr/local/lib/wvlib.lib.old

mv /usr/local/include/wave.h /usr/local/include/wave.h.old

cp wvlib.lib /usr/local/lib

cp wave.h /usr/local/include

/∗∗∗∗∗∗ WAVE.H

∗∗∗∗∗ ∗//∗ HEADER FOR THE WAVE.C FUNCTIONS THAT CONTROL THE QUATECH

SYNTHESIZER FREQUENCY GENERATOR ∗/#define ZERO 0

#define UZERO 0u

/∗ for setting ceratin bits to 1 ∗/

#define BIT0 0

#define BIT1 2

#define BIT2 4

#define BIT3 8

#define BIT4 0x10

#define BIT5 0x20

#define BIT6 0x40

#define BIT7 0x80

#define BIT8 256

#define BIT9 512

#define BIT10 1024

#define BIT11 2048

#define BIT12 4096

/∗ addresses for the iio ports ∗/

199

#define BASE 0x2E0 /∗ base port ∗/#define DL BASE /∗ low data byte ∗/#define DH BASE + 1 /∗ high data byte ∗/#define NL BASE + 2 /∗ npts low byte ∗/#define NH BASE + 3 /∗ npts high byte ∗/#define BMP BASE + 4 /∗ bit map port ∗/#define RPM BASE + 5 /∗ run/ program mode ∗/#define CSM BASE + 6 /∗ continous vs. single cycle ∗/#define INTRPT BASE + 7 /∗ interrupt reset ∗/#define CRT0 BASE + 8 /∗ cntr 0 data ∗/#define CRT1 BASE + 9 /∗ cntr 1 data ∗/#define DELAY BASE + 0x0a /∗ delay cntr data ∗/#define CRTM BASE + 0x0b /∗ cntr mode ∗/#define RST M BASE + 0x0c /∗ reset data cntr ∗/

/∗ control words for counters: set cntr to recieve low byte first

then the high byte ∗/

#define CRT0CW 0x34 /∗ control word for cntr 0 ∗/#define CRT1CW 0x74 /∗ control word for cntr 1 ∗/#define DELCW 0xba /∗ control word for delay cntr ∗/

/∗ define min/max values ∗/

#define MINC 2u /∗ min. cntr value ∗/#define MAXC 65536u /∗ max cntr value ∗/#define MIND 1u /∗ min delay value ∗/#define MAXD 65536u /∗ max delay value ∗/#define VDO 0u /∗ val of delay when turned off ∗/#define MINPTS 2u /∗ min npts value ∗/#define MAXPTS 2048u /∗ max npts value ∗/#define MINDAT −2047 /∗ min value of data ∗/#define MAXDAT 2047 /∗ max value of data ∗/

/∗ define common settings ∗/

#define INIT BITMAP UZERO /∗ initial bit map settings ∗/#define CYCLE ZERO /∗ sets mode to single cycle ∗/#define PROG ZERO /∗ sets mode to program ∗/

200

#define RUN BIT7 /∗ sets mode to run ∗/#define CONT BIT7 /∗ sets mode to continous ∗/#define DLYON BIT6 /∗ bitmap on settings ∗/#define CNTRS BIT4

#define EXTCLK BIT5

#define INTEN BIT7

#define INTCLK UZERO

#define DLYOFF UZERO

#define INTDE UZERO

#define DOFF BIT11 /∗ data offset ∗/

/∗ Conversion factors for board ∗/

#define CF 0.0025 /∗ conversion factor - currently set for

an open-circuit (high impedance load) ∗/#define RCF 10000000. /∗ conversion factor for freq ∗/#define RSS 0.0000001 /∗ step size in secs for cntrs ∗/

/∗ parameters needed to use max NPTS of Quatech ∗/

#define NPTS 2048

#define dNPTS 2048. /∗ used when NPTS is needed as a double ∗/#define MAXV 5.12 /∗ max voltage of output for simple functions ∗/#define MAXV2 2.56 /∗ max voltage when using sums of functions∗/#define MAXV3 4.50 /∗ max voltage used on modulation experiment ∗/

/∗ function definitions ∗/

extern void winit(unsigned ∗);extern void wstop();

extern void wstart(unsigned ,unsigned ∗);extern void set delay(unsigned ,unsigned ∗);extern void set cntr(unsigned ,unsigned, unsigned ∗);extern void def wave(unsigned ,int ∗);extern void dat conv(int ,double ∗ ,int ∗);extern double freq conv(double ,int ,unsigned ∗ ,unsigned ∗);extern double freq conv2(double ,unsigned ∗ ,unsigned ∗ ,unsigned ∗);extern double gen func(double ,double ∗ ,unsigned);

extern void divide(int);

201

/∗∗∗∗∗∗ WAVE.C

∗∗∗∗∗∗ ∗/#include <stdio.h>

#include <conio.h>

#include <math.h>

#include "wave.h"

/∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ Collection of programs that control the QUATECH Synthesizer

∗ wave form generator. Programs include:

∗∗ winit: initializes the board

∗ wstop/ whold : stops the board

∗ wstart/ wcont : starts the board

∗ set delay : sets the delay cntr

∗ set cntr : sets the period between points

∗ def wave : defines the wave form

∗∗ Many programs require as an input ”unsigned ∗bitmap”:

∗ this pointer is used to keep track of the values of the BMP port.

∗ every program using these functions should have an unsigned

∗ variable bitmap, and should call WINIT before using any of the

∗ other routines. This makes sure that the various settings

∗ are kept track of during the course of running the program.

∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/

/∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ WINIT() sets up board in initial configuration given by

∗ values in wave.h - uses min values for cntrs/ npts

∗ takes as input the pointer bitmap which keeps track of the

∗ BITMAP settings

∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/

void winit(bitmap)

unsigned ∗bitmap;

202

void wstop();

wstop(); /∗ set board to program mode ∗/

∗bitmap = INIT BITMAP; /∗ sets bitmap to initial bit map ∗/outp( BMP , ∗bitmap); /∗ sets bit map ∗/outp( CSM , CONT ); /∗ sets continous mode ∗/outp( NL, MINPTS ); /∗ sets npts ∗/outp( NH, ZERO );

outp( INTRPT, BIT7); /∗ resets interrupt ∗/set cntr( MINC, MINC, bitmap); /∗ sets cntrs ∗/set delay( MIND, bitmap); /∗ sets delay value, no - delay ∗/

/∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ WSTOP() : sets to program mode

∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/

void wstop()

outp( RPM, PROG);

/∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ WSTART(): resets the data counter and delay, then starts the

∗ waveform going

∗ input : delay value and bitmap pointer

∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/

void wstart(val, bitmap)

unsigned val, ∗bitmap;

set delay(val, bitmap);

203

outp( RPM, RUN); /∗ start running ∗/

/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ SET DELAY ( delay, bitmap): sets delay time, turning it on or off

∗ input : int - period of delay

∗ int - delay on or off

∗ unsigned pointer - bitmap keeps track of BMP settings

∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/

void set delay( val, bitmap)

unsigned val, ∗bitmap;

unsigned valh, vall;

int sw;

outp( RST M, BIT4 ); /∗ dummy write to reset cntr ∗/

if (val == UZERO) /∗ set delay on/off ∗/sw = DLYOFF;

else sw = DLYON;

if ( val < MIND) /∗ make sure val is in the correct limits ∗/val = MIND;

if ( val > MAXD)fprintf( stderr, "delay greater than max - set to max \n");

val = MAXD;

∗bitmap = (∗bitmap & ∼BIT6) | (sw & BIT6); /∗ sets delay bit in bitmap ∗/outp( BMP, ∗bitmap); /∗ sends correct bit map to board ∗/

vall = val;

valh = val 8;

outp ( CRTM, DELCW); /∗ sets delay cntr mode ∗/

204

outp ( DELAY, vall); /∗ sets low byte ∗/outp ( DELAY, valh); /∗ sets high byte ∗/

/∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ SET CNTR (RATE0, RATE1) : sets counters using rate0 and rate1

∗ unsigned ∗bitmap - keeps track of BMP settings

∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/

void set cntr(rate0, rate1, bitmap)

unsigned rate0, rate1, ∗bitmap;

unsigned r0h, r0l, r1h, r1l;

if (rate1 < MINC)∗bitmap &= ∼CNTRS; /∗ sets for 16 bit cntr ∗/rate1 = MINC;

else

∗bitmap |= CNTRS; /∗ sets for 32 bit cntr ∗/

/∗ set rates to within limits ∗/

if (rate1 > MAXC)fprintf( stderr, "rate1 greater than max in SET_CNTR - set to max \n");

rate1 = MAXC;

if (rate0 < MINC)

rate0 = MINC;

if (rate0 > MAXC) fprintf( stderr, "rate0 greater than max in SET_CNTR - set to max \n");

rate0 = MAXC;

205

r0l = rate0;

r0h = rate0 8;

r1l = rate1;

r1h = rate1 8;

outp(CRTM, CRT0CW); /∗ sets cntr 0 mode ∗/outp( CRT0, r0l); /∗ set low byte ∗/outp( CRT0, r0h); /∗ set high byte ∗/

outp(CRTM, CRT1CW); /∗ sets cntr 1 mode ∗/outp( CRT1, r1l); /∗ set low byte ∗/outp( CRT1, r1h); /∗ set high byte ∗/

outp(BMP, ∗bitmap); /∗ sets correct bitmap ∗/

/∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ DEF WAVE( NPTS, DATA)

∗ NPTS - unsigned number of points

∗ DATA - data points, signed integer from -2047 to 2047

∗ if NPTS is invalid returns a 1, if a data point is outside of range

∗ it is set to max or min

∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/

void def wave(npts, data)

unsigned npts;

int ∗data;

unsigned i;

unsigned dh, dl, n1h, n1l;

/∗ set npts cntr to npts and rst cntr with dummy write ∗/

n1l = npts − 1;

n1h = n1l 8;

outp( NL, n1l);

outp( NH, n1h);

outp( RST M, n1l);

206

/∗ adjust data, then write it to data port ∗/

for (i = 0; i < npts; i++)

/∗ if ( data[i] < MINDAT )

data[i] = MINDAT;

if (data[i] > MAXDAT)

data[i] = MAXDAT;

∗/dl = DOFF − data[i];

dh = dl 8;

outp(DL, dl);

outp(DH, dh);

/∗ set npts cntr to npts - 1 ∗/

outp( NL, n1l);

outp( NH, n1h);

/∗∗∗∗∗∗∗ WVCONV.C

∗∗∗∗∗∗ ∗/

/∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ This file contains programs which are used to convert

∗ data and frequencies to the correct values to be used

∗ by the QUA TECH WSB - 10 synthesizer generator

∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/

/∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ dat conv() : converts an array of voltages into the correct

∗ integer values for the program def wave.

∗ input: data array (doubles), npts

∗ output: data array (integers) both arrays must have a

∗ size declared in the main program

∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/

207

#include <stdio.h>

#include <math.h>

#include "wave.h"

void dat conv(npts, dat, idat)

double ∗dat;

int ∗idat, npts;

int i;

double x;

for (i = 0; i < npts; i++)

x = dat[i]/ CF;

x = x+0.5;

idat[i] = x;

/∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ freq conv(): converts a frequency into rate1 and rate0

∗ to be used by the QUATECH board

∗ input: freq npts

∗ output: rate0, rate1

∗ return value: actual frequency outputted

∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/

double freq conv( freq, npts, rate0, rate1)

double freq;

unsigned ∗rate0, ∗rate1;

int npts;

double per, perp, temp;

208

per = 1./ freq; /∗ compute period of waveform ∗/perp = per/ npts; /∗ compute period per point ∗/

temp = RCF ∗ perp;

∗rate1 = (temp / MAXC) + 1;

∗rate0 = temp/ ∗rate1;

temp = (double) (∗rate0) ∗ (∗rate1);

per = temp ∗ npts;

per = per / RCF;

freq = 1./ per;

return(freq);

/∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ freq conv2(): converts a frequency into npts, rate1 and rate0

∗ to be used by the QUATECH board

∗ input: freq npts

∗ output: rate0, rate1, npts

∗ return value: actual frequency outputted

∗ Used to get a finer range in frequency by adjusting npts used

∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/

double freq conv2( freq, npts, rate0, rate1)

double freq;

unsigned ∗rate0, ∗rate1;

unsigned ∗npts;

double per, perp, temp, temp2, temp3;

per = 1./ freq; /∗ compute period of waveform ∗/

/∗ compute if rate1 will be needed (only used for long periods ∗/

temp = RCF ∗ per/ MAXPTS;

∗rate1 = (temp / MAXC) + 1;

209

/∗ compute rate0 and npts ∗/

temp2 = floor(temp/ (double)∗rate1 + 0.5);

temp3 = floor(RCF ∗ per / ( (double) (∗rate1) ∗ temp2) + 0.5);

if (temp3 > 2048.) temp2 += 1;

temp3 = floor(RCF ∗ per / ( (double) (∗rate1) ∗ temp2) + 0.5);

∗rate0 = temp2;

∗npts = temp3;

temp = (double) (∗rate0) ∗ (∗rate1);

per = temp ∗ temp3;

per = per / RCF;

freq = 1./ per;

printf("freq = %f\n", freq);

return(freq);

/∗∗∗∗∗∗ GEN FUNC.C

∗∗∗∗∗ ∗/

/∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ FUNCTION: GEN FUNC converts ”real” frequencies and voltages

∗ to values usuable by the Quatech board, then outputs the

∗ waveform to the board and starts it.

∗ INPUT: freq : double - frequency of the waveform

∗ data : pointer to array of doubles - data array

∗ npts : unsigned - number of data points

∗ returns the actual value of the frequency

∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/

#include <stdio.h>

#include <math.h>

210

#include "wave.h"

double gen func(freq, data, npts)

unsigned npts;

double ∗data, freq;

unsigned dval, rate0, rate1, bitmap;

int ∗idat;

/∗ allocate memory and convert frequencies/ data ∗/

idat = (int ∗)malloc( npts ∗ sizeof(int));

freq = freq conv(freq, npts, &rate0, &rate1);

dat conv(npts, data, idat);

dval = 0; /∗ sets delay to zero ∗/

/∗ output to quatech board ∗/

winit(&bitmap); /∗ initialize board ∗/def wave( npts, idat); /∗ program data array into board∗/set cntr( rate0, rate1, &bitmap); /∗ set counters - i.e. freq ∗/wstart(dval, &bitmap); /∗ start waveform outputting ∗/

/∗ return freq ∗/

return(freq);

/∗∗∗∗∗∗∗∗ DIVIDE.C

∗∗∗∗∗∗ ∗/

#include <i86.h>

#include <stdio.h>

#include <math.h>

#include "labmaster.h"

#include "wave.h"

211

void divide(div)

int div;

int factor, flow, fhigh;

char far ∗base;

extern char LM SEG;

/∗ init(); ∗/

factor = div;

base = ((char far ∗) MK FP(LM SEG, 0));

∗(base + 15) = 0x80; /∗ sets all dio ports for output ∗/

flow = factor & 255;

fhigh = factor 8;

fhigh = fhigh & 15;

∗(base + 13) = ∼flow; /∗ sets low bits to port A ∗/∗(base + 14) = ∼fhigh; /∗ sets high bits to high bits of port c ∗/

212

Appendix D

References

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213

Swinney, in Hydrodynamic Instabilities and Transitions to Turbulence,

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214

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215

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216

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219

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[66] There is a printing error in Eq. (38) of Ref. [39]. The factor (1− εa

εqLnnq

2)

should be replaced by (Lnn − εa

εqq2).

[67] C. Bijiang, Master’s Thesis (1995).

[68] S. Rasenat, G. Hartung, B. L. Winkler, and I. Rehberg, Experiments

in Fluids 7, 412, (1989).

[69] S. Trainoff, D. S. Cannell, and G. Ahlers, physical optics shadowgraph,

in progress.

[70] Sony HVM-200 CCD camera.

[71] PCEYE Video Capture System manufactured by Chorus Data Systems,

Inc., P.O. Box 370, 6 Continental Blvd., Merrimack, NH 03054.

[72] The LED used is General Fiber Optics part 80-0663-2001 and the optical

fiber is 16-505-mc-2-sma/2. 1 Washington Ave., Fairfield, NJ. 07004.

(201) 239 3400. Contact: Dennis Drewing.

220

[73] WSB - 10 arbitrary waveform synthesizer manufactured by Qua Tech,

Inc., 478 East Exchange St., Akron, OH 44304.

[74] Labmaster DMA manufactured by Scientific Solutions, Inc., 6225

Cochran Rd., Solon, OH 44139-3377.

[75] Hewlett Packard 6827A Bipolar Power Supply and Amplifier.

[76] 1615-A General Radio capacitance bridge and Model HR-8 Princeton

Applied Research Lock-in

[77] To order from Merck use the New York branch: EM Industries,

Hawthorne, NY. (914) 592 4660. An alternate source for liquid crys-

tals is American Kokyo Kasei, Inc. 9211 N. Harborgate St., Portland,

OR. 97203. (503) 283 1681.

[78] Cells were obtained from Display Tech Inc., 2200 Central Ave, Boulder,

CO 80301.

[79] Donnelly Applied Films Corp. 6797 Winchester Circle, Boulder, CO.

80301. (303) 530 1411.

[80] Libbey Owens Ford, 811 Madison Ave, PO Box 794, Toledo OH., 43695-

0799. (419) 247 3931.

[81] The Milli-Q filter systems used is the product of Millipore.

[82] A good supplier for evaporation equipment is R. D. Mathis Co., 2840

Gundry Ave. PO Box 6187, Long Beach, Ca. 90806. (310) 426 7049.

[83] Sigma Chemicals, catalog no. P-2772, 1- 800 325 3010.

[84] W. Zimmermann, private communication.

221

[85] ULTEM 1000 resin from General Electric Company Plastics group. One

Plastics Ave. Pittsfield, MA 01201. 800 GEPLAST. (ULTEM is a regis-

tered trademark of General Electric.)

[86] Extec synthetic velvet polishing cloth #XL17422. Excel Technologies,

Inc. (203) 741 3435. Reorder from: Max ERB Instrument Co., 2112 W.

Burbank Blvd., Burbank, Ca. 91506. (213) 849 2374.

[87] B. Frisken, private communication.

[88] S. Morris, private communication.

[89] S. Kai, private communication.

[90] For a description of conoscopy, see for instance, Ernest E. Wahlstrom,

Optical Crystallography 4th Ed. (Wiley and Sons, Inc., New York, 1969).

[91] Torr Seal reorder number 953-0001 from Varian Vacuum Products, 800

8 VARIAN, or Physics Dept. store room.

[92] Norland 91 UV epoxy. Norland Products, Joyce Kilner Ave, New

Brunswick, NJ 08902. (908) 545 7828.

[93] Sauereisen Cements, No.1 Paste. 160 Gamma Drive, Pittsburgh, PA

15238. (412) 963 0303.

[94] Norland Products Mercury Spotlamp catalog #5500. (see [92].

[95] Hardman 5 minute epoxy, Red package #04001. Hardman Industries,

Belleville, NJ 07109. (201) 751 3000. Reorder: Litco. (714) 893 5081.

[96] GC Electronics Silver Print (22-202).

[97] M. Dennin, G. Ahlers, and D. S. Cannell, in Spatio-Temporal Patterns,

edited by P. E. Cladis and P. Muhoray (Addison-Wesley, 1994) p. 353.

222

[98] S. Kai, K. Yamaguchi, K. Hirakawa, “A new pattern in a nematic liquid

crystal.” Japan. J. Appl. Phys. 14, 1385, (1975).

[99] M. Treiber, private communication.

[100] D.T.J. Hurle and E. Jakeman, “Soret-driven thermosolutal convection.”

J. Fluid Mech. 47, 667 (1971).

[101] H. Lekkerkerker, “Oscillatory convective instabilities in nematic liquid

crystals.” J. Phys. France Lett. 38, 277 (1977).

[102] For typical recombination constants for 5CB, see for example, A. Sug-

imura, N. Matsui, Y. Takahashi, H. Sonomura, et al., “Transient currents

in nematic liquid crystals.” Phys. Rev. B 43, 8272, (1991). There is a

report of an anomalously large recombination time (27,000 sec) in ex-

tremely low conductivity MBBA, G. Briere, R. Herino, and F. Mondon,

Mol. Cryst. Liq. Cryst. 19, 157, (1972).

[103] M. de la Torre-Juarez and I. Rehberg, “Four-wave resonance in electro-

hydrodynamic convection.” Phys. Rev. A 42, 2096, (1990).

[104] J. Glazier, P. Kolodner and H. Williams, ”Dispersive chaos.” J. Stat.

Phys. 64, 945–960, (1991).

[105] M. Silber, H. Riecke, and L. Kramer, “Symmetry-breaking Hopf bifur-

cation in anisotropic systems.” Physica D 61, 260, (1992).

[106] H. H. Rotermund, S. Jakubith, A. von Oertzen, and G. Ertl, “Solitons

in Surface Reaction.” Phys. Rev. Lett. 66, 3083, (1991).

[107] S. W. Morris, P. Palffy-Muhoray, and D. A. Balzarini, “Measurements

of the bend and splay elastic constants of octylcyanobiphenyl.” Mol.

Cryst. Liq. Cryst. 139, 263, (1986).

223

[108] H. J. Deuling, “Deformation of nematic liquid crystals in an electric

field.” Mol. Cryst. Liq. Cryst. 19, 123, (1972).

[109] T. Uchida, Y. Takahashi, Mol. Cryst. Liq. Cryst. Lett. 102, 35, (1984).

[110] H. W. De Jeu, W. A. P. Claassen, and A. M. J. Spruyit, “The determi-

nation of the elastic constants of nematic liquid crystals.” Mol. Cryst.

Liq. Cryst. 37, 269, (1976).

[111] H. Kneppe, F. Schneider, and N. K. Sharma, “Rotational viscosity γ1 of

nematic liquid crystals.” J. Chem. Phys. 77, 3203, (1982).

[112] Calibration of the UCSB Fluids Group standard thermistor was done

against a platinum resistance thermometer over the range 18C to 50C

by Steve Trainoff.

224