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dissertation entitled

Dielectric Study of the Glass
Transition in Orientationally-Disordered Crystals

presented by

Matthew A. Miller

has been accepted towards fulﬁllment 1
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Date 61.19/98

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DIELECTRIC STUDY OF THE GLASS TRANSITION IN ORIENTATIONALLY-
DISORDERED CRYSTALS

By

Matthew A. Miller

A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1998

ABSTRACT

DIELECTRIC STUDY OF THE GLASS TRANSITION IN ORIENTATIONALLY-
DISORDERED CRYSTALS

By

Matthew A. Miller

The glass transition in supercooled liquids occurs when the time required for
equilibration processes in the liquid exceeds some experimental timescale. Below the
glass transition temperature T3, the material is designated a glass and displays
thermodynamic properties more appropriate for a solid than a liquid. Despite
considerable theoretical and experimental effort in the past century, a fundamental
description of the microscopic mechanisms underlying glass formation has not been
realized. Alternative approaches are welcomed which offer the possibility of studying
the glass transition in systems displaying reduced structural complexity.

This thesis focuses on studying the glass transition dynamics in a class of
materials known as Orientationally-Disordered Crystals (ODC's). The molecules of
ODC's possess translational order while retaining orientational degrees of freedom,
giving rise to an orientational glass transition which displays the same characteristic
features of the glass transition in fully-disordered liquids. Thus, ODC's are of great
interest as model glass systems.

Using the technique of dielectric spectroscopy, we have investigated the
dynamical behavior of two particular ODC's near their respective glass transitions:
Cyclo-Octanol and Ethanol. We ﬁnd that the orientational glass transition in each

material displays the same two characteristic features of the structural glass transition in

fully-disordered liquids: 1) rapidly increasing relaxation times as the temperature is
lowered toward the glass transition temperature TS, and 2) increasingly non-Debye
relaxation behavior with decreasing temperature. For ethanol in particular, we have
performed a quantitative comparison of the dynamics in the supercooled liquid (SCL) and
rotator crystal (RP) phases, and ﬁnd that rotational motion is the dominant dynamical
process governing the glass transition in both phases, and that diffusive processes
contribute to a lesser extent.

Finally, we have performed a detailed examination of the shape of the a-
relaxation in the SCL and RP crystal phases of ethanol for a frequency range covering ten
decades. We ﬁnd that the high frequency power-law behavior in the RP crystal is
consistent with the expectations of a well-known scaling form used for structural glasses,
while the limited amount of SCL data appears to show steeper power-law behavior than

expected.

To my wife, Lori.

iv

ACKNOWLEDGEMENTS

First and foremost, I would like to thank my advisor, Norman Birge. Without his
friendship, guidance, and tireless efforts to obtain ﬁnancial support for my research, I
surely would not have completed this journey.

I thank Charles Moreau for his friendship, engaging conversations, and our
competitive "death-matches". I thank David Hoadley for teaching me nearly everything I
know about computers and securing a job outside of academia.

I also would like to thank the machine shop guys - Jim Muns, Tom Hudson, and
Tom Palazollo - for their highly professional work and endless patience with us graduate
students. I am also indebted to Scott and Manfred over in the chemistry glass shop for all
their fabrication - and repair - work on the many sample cells that I provided them with.

I thank Monica J imenez-Ruiz and Javier Bermejo for their extensive contributions
to the ethanol work, and Rod Lambert and Diandra Leslie-Pelecky for guidance and
contributions to the cyclo-octanol studies.

Of course, a great deal of thanks goes to my family — my wife, Lori, in particular
— for supporting me along the way and putting up with me during the difﬁcult times. I
would like to thank my parents for constantly encouraging me to achieve goals that
sometimes seemed beyond my reach.

Last, but certainly not least, I would like to acknowledge the persistent efforts of my
mother-in-law, JoAnn, whose well-wom quote of “Matthew, just put it in HIGH!” no

doubt had a signiﬁcant impact on my success as a graduate student.

TABLE OF CONTENTS

LIST OF TABLES ................................................................................... ix
LIST OF FIGURES .................................................................................. x
Chapter 1: INTRODUCTION ...................................................................... 1
Chapter 2: GLASS PHENOMENOLOGY ....................................................... 7
I. Supercooled Liquids ................................................................ 7
II. The Glass Transition in Supercooled Liquids .................................. 8
a. Thermodynamic Aspects ................................................ 8
b. Dynamics of Supercooled Liquids near T3 ........................... 11
1. Rapidly increasing timescales for motion ........................ ll
2. Non-Debye Relaxation ............................................... 14
111. Overview of Experimental Studies of the Glass Transition... ... . . . .. ......18
IV. Theories of the Glass Transition in Supercooled Liquids .................... 21
a. Introduction .............................................................. 21
b. Theories postulating low temperature phase transitions ........... 22
c. Theories predicting critical behavior at T > T8 ...................... 28
V. Conclusions ....................................................................... 3 1
Chapter 3: ORIENTATIONALLY-DISORDERED CRYSTALS ......................... 34
I. The need for Model Glass Systems ........................................... 34
11. History ............................................................................ 35
III. Structure and Dynamics of ODC's ............................................ 36
IV. Example ODC's .................................................................. 38

vi

V. Motivation ......................................................................... 41
Chapter 4: EXPERIMENTAL TECHNIQUES ................................................ 44
I. Dielectric Spectroscopy ......................................................... 44
II. Review of Dielectric Theory ..................................................... 44
a. Polarization Mechanisms Dielectrics .................................. 44
b. Behavior in Static Fields - Static Dielectric Constant... ............46
c. Time-Varying Fields - Complex Dielectric Constant .............. 52
III. Dielectric Measurement System ............................................... 57
a. Goals of Dielectric Spectroscopy .................................... 57
b. The Dielectric Sample Cell ............................................ 58
0. Low Frequency Measurement Circuit ............................... 59
d. High Frequency Measurement Circuit ............................... 61
e. Cryostat and Temperature Control ................................... 63
IV. Sample Fabrication Methods ................................................................ .64
a. Fractional Vacuum Distillation ....................................... 64
b. Nitrogen Atmosphere Loading ........................................ 66

Chapter 5: DIELECTRIC STUDIES OF CYCLO-OCTANOL..............................68

I.

II.

III.

Motivation ........................................................................ 68
Dielectric Measurement of Cyclo-Octanol .............................................. 68
a. Experimental Notes .................................................................... 68

b. Summary of Low Frequency (lmHz - lOkHz) Measurements...70
c. High Frequency (lkHz - 10 MHz) Measurements...................74

Discussion of Results ........................................................... 82

vii

Chapter 6: DIELECTRIC MEASUREMENT OF ETHANOL ............................ 85

1. Motivation ........................................................................ 85
II. Dielectric Measurement of Ethanol ......................................................... 86
a. Experimental Notes .................................................... 86
b. Dielectric Response of SCL and RP Crystal Phases ............... 87
III. Detailed Study of Shape of the Relaxation in Ethanol ..................... 97
a. Motivation ............................................................... 97
b. High Frequency Power-Law Behavior ............................... 98
c. Dixon Scaling of Ethanol Dielectric Data ................................. 106
Chapter 7: NONLINEAR DIELECTRIC MEASUREMENT ............................ 109
1. Motivation and Background ................................................... 109
II. Nonlinear Measurement Circuit ............................................. 111
III. Present Status of Experimental Work ....................................... 112
Chapter 8: SUMMARY AND FUTURE DIRECTIONS .................................. 114
Appendix A: DATA CORRECTION METHODS ......................................... 118

viii

LIST OF TABLES

Table 6.1. Cole-Davidson ﬁtting parameters for the RP crystal phase of ethanol in
the temperature range 96 — 110 K ................................................... 92

Table 6.2. Cole-Davidson ﬁtting parameters for the SCL (96 — 100 K) and
high-temperature liquid (164 — 181 K) ............................................. 92

Table 6.3. VTF ﬁt parameters of the mean relaxation time in the RP
crystal and SCL phases of ethanol ................................................... 96

Table 6.4. Parameters obtained in the double CD ﬁtting of the RP crystal phase of
ethanol ................................................................................. 100

Table 6.5. Parameters obtained in the double CD ﬁtting of the SCL phase of
ethanol ................................................................................. 100

Table 6.6. Measured relaxation widths and comparison of scaling-predicted
and actual high-frequency power laws for the RP crystal
phase of ethanol ..................................................................... 105

Table 6.7. Measured relaxation widths and comparison of scaling-inferred

and actual high-frequency power laws for the SCL
phase of ethanol ...................................................................... 105

ix

Figure 2.1.

Figure 2.2.

Figure 2.3.

Figure 2.4.

Figure 2.6.

Figure 3.1.

Figure 3.2.

Figure 3.3.

Figure 4.1.

Figure 4.2.

Figure 4.3.

Figure 4.4.

Figure 4.5.

LIST OF FIGURES

Thermodynamic signatures of the glass transition ............................... 9

The use of slower cooling rates results in the glass transition occurring at
successively lower temperatures .................................................. 10

The curvature of the mean relaxation time on an Arrhenius plot varies
widely among materials. “Fragile” liquids show strong curvature,
while “strong” liquids display nearly arrhenius behavior ..................... 13

Imaginary part of the dielectric constant of supercooled ethanol at
T=102 K. Fits to the Debye, Cole-Davidson (oc=0.796), and KWW

([3=0.850) functions have been included for reference .......................... 16
Graphical illustration of the Kauzmann Paradox ................................ 22
The translational order and rotational disorder of molecules in a

typical rotator phase crystal ........................................................ 37
Schematic diagram of the phases of cyclohexanol .............................. 38

Experimental heat capacity of cyclohexanol in the supercooled Sl phase.
The sudden drop in C9 at ~ 150 K is due to the orientational glass

transition. (Data from: M. Mizukami et al., Solid. St. Comm. 100, 83-88
(1996)) ................................................................................. 40

Polar molecules in an external electric ﬁeld ...................................... 45

The response of induced and orientational polarization mechanisms to a
constant electric ﬁeld turned on at t = 0 .......................................... 54

Frequency response of a dielectric obeying the Debye equations. This

example assumes so = 10, em = 2.0, and t = 1.0 .................................. 56
Dielectric sample cell used for measurements ................................... 58
Circuit used for low-frequency (lmHz — lOkHz) measurements ............. 59

Figure 4.6.
Figure 4.7.
Figure 5.1.

Figure 5.2.

Figure 5.3.

Figure 5.4.

Figure 5.5.

Figure 5.6.

Figure 5.7.

Figure 5.8.

Figure 5.9.

Figure 6.1.

Schematic of F our-Terminal-Pair Measurement .................................. 61
Fractional Distillation System ...................................................... 65
Schematic diagram illustrating the various phases of cyclo-octanol .......... 69

a) e' and b) e” for the supercooled S1 phase of cyclo-octanol as
measured by Leslie-Pelecky et al. in the low-frequency regime ............... 71

Temperature dependence of parameters extracted from the KWW
ﬁtting of Figure 5.2 data. (a) "streching exponent", B, and (b) mean
relaxation time 1: ...................................................................... 72

Scaling of the dielectric data using a form given by Dixon, et a1
Cyclo-octanol begins to show slight deviations from structural glass
formers (e. g. glycerol) at high frequencies ........................................ 73

The imaginary part of dielectric response of cyclo-octanol covering a
frequency range of ten decades. The appears of the high temperature
B-relaxation is readily apparent at high frequencies ............................. 76

Some additional examples of high-temperature B relaxation processes
similar to those observed in cyclo-octanol. (Data ﬁom: G. P. Johari,
J. Chem. Phys. 58, 1776 (1973)) ................................................... 77

The dielectric loss (8") curves for cyclo-octanol as obtained by R. Brand.
The inset to the ﬁgure contains dielectric measurements of the stable

crystalline phase S3 at the four lowest temperatures shown on the main
plot. (Data from: R. P. Brand et al., Phys. Rev. B 56, 5713 (1997)) .......... 78

The dielectric loss (8") at a temperature of 160 K as obtained by R. Brand.
In contrast to our measurements, Brand identiﬁes three separate processes
labeled as a, B, and 7. (Data from: R. P. Brand et al, Phys. Rev. B 56, 5713
(1997)) ................................................................................. 79

Plot of the mean relaxation times for the a, B, and y processes shown in
Figure 5.7. Similar to the results of Leslie-Pelecky et al [3], the behavior of
the a-pI‘OCCSS was found to be non-Arrhenius. The B and y processes exhibit
thermally activated (Arrhenius) behavior. (Data from: R. P. Brand et al, Phys.
Rev. B 56, 5713 (1997)) ............................................................. 80

The various phases of ethanol ...................................................... 85

xi

Figure 6.2. (a) s’ and (b) 8" versus frequency for temperatures of 96, 98, and 100 K
in the supercooled liquid phase of ethanol. Solid lines are ﬁts to the
Cole-Davidson function ............................................................. 89

Figure 6.3. (a) e' and (b) 8" versus frequency for temperatures of 96, 98,
100, 102, 104, 106, 108, and 110 K in the RP crystal phase of

ethanol. Solid lines are ﬁts to the Cole-Davidson function ..................... 90

Figure 6.4. (a) 8' and (b) 8" versus frequency for temperatures of 164, 173, and
181 K in the high temperature liquid phase of ethanol. Solid lines are
ﬁts to the Cole-Davidson ﬁmction plus a term of the form 4nodJm in a"
due to do conductivity .............................................................. 91

Figure 6.5. Plot of the Cole-Davidson width exponent, or, versus temperature for the
SCL and RP crystal phases of ethanol ............................................. 93

Figure 6.6. Plot of so - 8,, versus temperature. The observed decrease in 80-800 at low
temperatures is not due to partial crystallization of the sample ............... 94

Figure 6.7 . Temperature dependence of the mean relaxation time for the
liquid (above TM) , supercooled liquid, and rotator crystal
phases of ethanol ..................................................................... 95

Figure 6.8. Double CD ﬁtting of 98 K (top) and 100 K (bottom) RP crystal ethanol.
Single CD ﬁt is included for reference ........................................... 101

Figure 6.9. Double CD ﬁtting of 102 K (top) and 104 K (bottom) RP crystal data.
Fits to the single CD form are included for reference .......................... 102

Figure 6.10. Double CD ﬁtting of 106 K RP (top) and 98 K SCL (bottom). Fits to
the single CD form are included for reference ................................. 103

Figure 6.11. Double CD ﬁtting of 100 K SCL data. Fits to the single CD form are
included for reference ............................................................. 104

Figure 6.12. Dixon scaling of the RP crystal and SCL ethanol dielectric data.
The RP crystal scales in accordance with structural glasses
(represented here by glycerol) while the SCL shows systematic
deviations at high frequencies .................................................... 107

Figure 7.1. Circuit for nonlinear measurement ................................................ 111

Figure A. 1. Circuit diagram for impedance measurement using I-IP4192A ............... 123

xii

Figure A.2. Transmission line correction of capacitance data for a 511 pF Air
capacitor. Correction of the data obtained at 1 meter should
match the data obtained at the instrument front panel (L=0 meters).
Clearly, the capacitance data is being over-corrected ........................ 127

Figure A.3. Transmission line correction of conductance data for a 511 pF Air
capacitor. Correction of the data obtained at 1 meter should
match the data obtained at the instrument front panel (L=O meters).
Instead, the conductance data is being under-corrected ...................... 128

xiii

Chapter 1

INTRODUCTION

Around 3000 BC, Mesopotamian potters discovered, quite by accident, that
common desert sand could be liquiﬁed and subsequently cooled to reveal a smooth, shiny
substance that we now call silicate glass. Today, in addition to these pioneering silicates,
there exists a great many glassy materials that ﬁnd use in applications ranging from the
mundane to the complex. Decorative glassware for dining, extremely pure optical
communication ﬁbers, and high strength, heat resistant window panes for the Space
Shuttle are just a few examples illustrating the wide range of applications [1]. In fact, the
manufacture and sale of glassy materials has developed over the last 200 years into a
multi-billion dollar industry. This reﬂects the fact that few other materials can claim to
have a similar impact on human cultural and technological development.

Achieving the glassy phase can be accomplished by any of several routes, but the
simplest and most technologically useful glasses result from the vitriﬁcation of liquids.
The process of glass formation begins by cooling a liquid from a temperature well above
its crystallization point. The primary requirement to be satisﬁed is that the liquid must be
cooled sufficiently fast so as to avoid the onset of crystallization at the liquid’s freezing
point. This process is called supercooling the liquid and the resultant product designated
a supercooled liquid. The amorphous glassy phase is produced by continuing to cool the
supercooled liquid until it falls out of equilibrium, and measurement of the material’s

thermodynamic properties will yield values more appropriate to a solid than a liquid. The

point at which the liquid falls out of equilibrium is called the glass transition
temperature, T8, and corresponds roughly to a liquid viscosity of 1013 poise.

Detailed study of the glass formation process began well over 100 years ago, and
has grown to be a subject of great interest to the physicist, chemist, and material scientist.
This long period of study has led to a number of technological improvements, and we
have become extremely proﬁcient at manipulating the chemical, electrical, and optical
properties of glass to ﬁt a wide range of applications. However, many people are
surprised to learn that the actual microscopic mechanisms of glass formation are only
poorly understood, and remain largely a mystery.

There are several reasons for why a fundamental understanding of the glass
transition has not yet been achieved. First, the glassy state is amorphous in nature and
lacks a periodic crystalline lattice that can be treated analytically using the powerful
mathematical techniques developed during solid state physics research in the 1950’s and
60’s. Secondly, the glass transition is not an equilibrium phenomena, making theoretical
treatment more difﬁcult. Finally, despite the existence of certain general features
characterizing real glasses, there is no universal agreement on which features are
considered essential to glassy behavior.

Numerical modeling of the glass transition is also difﬁcult, resulting in computer
simulations that occur on picosecond timescales. Experimental studies, on the other
hand, are carried out on the order of seconds to minutes, and invariably encounter an
equilibration timescale that increases extremely rapidly near the transition. The net effect
is that present-day theory and experiment are separated by a wide gap. The lack of a

consistent theory that encompasses the experimentally observed features of the transition

to the glassy state continues to provide motivation for scientiﬁc study. Perhaps the best
way to summarize the current state of glass transition research and expectations for the
future is to quote Nobel Laureate W. P Anderson, who, in a recent issue of Science [2]

stated the following:

“...perhaps the deepest unsolved problem facing condensed matter physics is the

nature of glass and the glass transition. This could be the next major breakthrough...”

In this thesis, the glass transition problem is approached through experimental
examination of the dynamical behavior of a class of materials known as Orientationally-
Disordered Crystals (ODC’s). In addition to possessing a true structural glass transition
in the liquid state, these materials also display an orientational glass transition in a state
where the molecular centers of mass are arranged on a periodic lattice. Thus, with their
restricted degrees of freedom, ODC’s are of great interest as model glass-forming
systems. Speciﬁcally, the technique of dielectric spectroscopy is used is to study the
glassy behavior of two ODC’s in detail: Cyclo-Octanol (C3H150H) and Ethanol
(C2H50H).

The ﬁrst issue addressed in this thesis is the feasibility of ODC’s as model glass
systems. This thesis will show that model glass systems, with their restricted degrees of
freedom, are a useful tool for both experimental and theoretical investigations of glass
transition phenomena.

The second major goal of this work is to provide a quantitative comparison of the
dynamical behavior in two structurally different phases near their respective glass

transition points in a single material. Our recently published work on ethanol [3] allows

one to identify which degrees of ﬁeedom — orientational or translational — are most
relevant to the glass transition process. Until now, such a comparison has never been
carried out due to lack of an appropriate material having sufﬁcient polymorphism and
phase stability to allow thorough experimental investigation.

Finally, the last portion of this thesis addresses the question of whether or not a
true phase transition underlies the glass transition, which is a question that presently has
no deﬁnitive answer. To date, there is no experimental evidence to suggest that the glass
transition results from anything more than kinetic slowing of the liquid. However, based
on thermodynamic arguments, some theories predict the existence of a second order
phase transition at or near the glass transition point, and thus the question remains open.
To address this question, we are attempting to measure the nonlinear dielectric constant
of the orientationally-disordered phase of ethanol. The nonlinear dielectric constant is
sensitive to the grth of a correlation length, which is a prime indicator of the presence
of a second order phase transition.

The remainder of this document takes the following organizational form: Chapter
2 provides a detailed introduction to the phenomenology of the glass transition in
supercooled liquids, and reviews the most relevant theoretical and experimental progress
to date. Theories of the glass transition involving free volume, excess entropy, and the
more recent “mode coupling” approach will be examined. Results from dielectric,
speciﬁc heat, light scattering, x-ray diffraction, and neutron diffraction studies of the
glass transition are also summarized and discussed. Chapter 3 covers the history and
structural properties of Orientationally-Disordered Crystals. Experimental aspects are

treated in Chapter 4, including dielectric spectroscopy, measurement instrumentation and

sample preparation techniques. Chapter 5 begins with a review of dielectric studies of
Cyclo-Octanol, and then presents high-frequency measurements as an extension of
previous data. A comparison of the results is made with a similar study carried out
simultaneously by another research group. Chapter 6 focuses exclusively on the
dielectric spectroscopy of ethanol, and results are presented and discussed. The
background and strategy of the nonlinear measurement of ethanol is treated in Chapter 7.
Finally, Chapter 8 summarizes the results of these experiments and discusses their

relevance to current and future glass transition research.

REFERENCES

 

[1] Education Department, The Coming Museum of Glass, One Museum Way
Corning, NY 14830. [Online] Available.

http://www.corningglasscenter.com, March 27, 1997.

[2] P.W. Anderson, Science 267, 1616 (1995).

[3] M. A. Miller, M. J. Ruiz, F. J. Bermejo, and N. O. Birge, Phys. Rev. B 57, June 1
(1998)

Chapter 2

GLASS PHENOMENOLOGY

I. Supercooled Liquids.

The process of glass formation from the liquid state depends critically on the
ability to bypass ﬁrst-order crystallization that normally intervenes and results in an
ordered, low energy crystalline phase. The process of avoiding crystallization by using
high rates of cooling is called “supercooling” the liquid. Thus, the general term
“supercooled liquid” refers to any liquid that exists below its thermodynamic freezing
temperature Tm. The thermodynamic properties of the supercooled liquid state are
exactly those that would be expected upon extrapolation of the equilibrium liquid
properties to temperatures below Tm. It should be emphasized that the supercooled liquid
phase is metastable in nature, and is susceptible to crystallization in the presence of a
proper catalytic event.

The degree to which a given liquid can be effectively supercooled depends
strongly on the rate at which crystal nucleation and growth occurs in the liquid. The
liquid will fully crystallize when small regions of crystalline material (called crystallites)
form in the liquid and grow beyond a certain critical size. The nucleation of crystallites
can occur either heterogeneously or homogeneously, however in practice the dominant
process is heterogeneous nucleation, in which crystallites develop in the presence of a
catalyst (e.g. suspended impurities or defects in the walls of the container housing the
liquid). Typical liquids contain anywhere from 105 to 106 impurities per cm3 [1] and

nucleation occurs via the most effective “seed” impurity present. Homogeneous

nucleation, which occurs even in pure liquids free from impurities, results when a
ﬂuctuation in the liquid brings together a critical number of molecules into crystalline
conﬁguration. This nucleation mechanism is almost never responsible for crystallization
as observed in the laboratory, as all real world samples and containers exhibit both
impurities and defects to some extent.

Turnme [2] has identiﬁed several factors inﬂuencing the crystallization process,
some of which are directly controllable in the laboratory. The most critical factors are the
prevailing cooling rate —dT/dt, the viscosity of the liquid at the crystallization
temperature Tm, the purity of the liquid, and volume of the liquid sample. These factors
work in combination to determine the material’s overall susceptibility to crystallization.
Rapid cooling and high viscosity are effective at suppressing crystallization because the
molecules lack the sufﬁcient time and mobility necessary for rearrangement onto a
crystalline lattice. Pure liquids provide few sites for nucleation, and small volumes allow
for efﬁcient removal of thermal energy from the sample.

11. The Glass Transition in Supercooled Liquids.

II. a. Thermodynamic Aspects.

Continuing to cool the supercooled liquid at a rate prohibiting crystallization will
eventually result in the occurrence, at a temperature denoted Tg, of a phenomenon known
as the glass transition. Figure 2.1 shows the typical behavior of a supercooled liquid near

Tg. As the material is cooled through T3, derivative thermodynamic quantities such as the

speciﬁc heat Cp = T[§§-] and thermal expansion coefﬁcient or = _1_[6_V] will
61‘ p V 6T p

decreases more or less suddenly from values characteristic of the liquid to values more

appropriate for a solid. Quantities such as the entropy S and speciﬁc volume V show a

characteristic “leveling off” at Tg.

C,0t

 
 
      

 

 

 

S,V

 

 

 

T T

9

Figure 2.1. Thermodynamic signatures of the glass transition.

Kauzmann [3] pointed out that for a given liquid, the temperature T3 at which the
glass transition occurs is not a rigidly deﬁned quantity and instead locates the center of a
temperature interval over which the observed changes occur. Figure 2.2 shows how the
location of Tg and the width of the temperature interval characterizing the transition
depend on the rate of cooling used in the experiment. The use of high cooling rates will
result in the glass transition occurring at a slightly higher temperature (Tg') and
encompassing a broader temperature range relative to what would be observed using

slower cooling rates (ng, Tg3).

C,oc

          

 

i

 

 

T

(ca-'0,
«3'1,
(0—1,

Figure 2.2. The use of slower cooling rates results in the glass transition
occurring at successively lower temperatures.

At ﬁrst glance, the characteristic signatures of the glass transition might resemble

the appearance of a second order thermodynamic phase transition. However, as observed

10

in the laboratory, the glass transition is most certainly not a thermodynamic phase
transition of any kind. Rather, it is an entirely kinetic phenomenon reﬂecting the fact that
at T8 the time required for structural relaxation processes in the liquid becomes greater
than some experimental timescale. In other words, if measurement of some material
property reﬂects the contribution of some structural degree of freedom, then a meaningful
contribution will occur only if that degree of freedom has been given the necessary time
to fully “relax” and attain equilibrium during the duration of the measurement.

II. b. Dynamics of Supercooled Liquids near T3.

Experimental data suggest that glass forming liquids display two nearly universal
dynamical features near the glass transition: 1) a rapidly increasing timescale for
reorientational and diffusional motion, and 2) relaxation behavior which is non-
exponential (or equivalently, non-Debye) in nature. These features will be described in
the following sections.

11. b. l. Rapidly Increasing Timescales for Molecular Motion near T3.

The response of a material to an externally applied perturbation such as an
electric ﬁeld, heat pulse, or mechanical shear can provide insight into the dynamical
processes occurring within the supercooled liquid. The response of the material is a
superposition of several relaxation processes that occur on varying timescales. The
primary “or-relaxation” of the material is a structural relaxation, and roughly corresponds
to overall molecular rotation. Secondary relaxation processes called B-relaxations [4,5]
occur above and below Ts, and are representative of intramolecular conformation changes

as well as relaxation of interstitial regions between clusters of molecules. These B-

processes usually occur at temperatures below T3 where a-relaxation is frozen out, and

11

provide contributions which are much smaller in magnitude than the or-process. B-
relaxation processes of intramolecular origin are divided into two categories depending
on the timescale on which they occur: slow processes (BS) due to internal molecular
conformations and nearly temperature-independent fast processes (Bf) associated with the
vibrational properties of the material.

The most striking feature of the glass transition in supercooled liquids is related to
the “freezing out” of the a-relaxation at T8. This is manifested by an enormous increase
(~ 13 orders of magnitude) in the viscosity 7} and structural relaxation time t which
occurs between the melting temperature Tm and the glass transition temperature T3. The
two quantities n and r are usually assumed to be proportional to one another as described

by the Maxwell theory of viscosity, where the proportionality constant is the high

frequency shear modulus:

_l
(2.1) t— G

m

Figure 2.3 shows the temperature dependence of the structural relaxation time
for the (it-relaxation process for two typical glass forming liquids. The temperature
dependence of 1: is often non-Arrhenius, and is often described by the Vogel-Tamman-
Fulcher (V TF) [6] phenomenological ﬁtting form given by
(22) 1(T) = To eXp(A/(T-To))

Except in the high and low temperature extremes, the VTF form has successfully
characterized the temperature dependence of the relaxation time and viscosity for a wide
variety of glass forming liquids, and provides a quantitative measure of a material’s

departure from Arrhenius-like behavior (To = 0). It is interesting to note that the VTF

12

form predicts that a divergence should occur at T0 < Tg , which has resulted in theoretical
models suggesting the possible existence of an “ideal glass transition” having
thermodynamic origin at or near To. Theories of the glass transition in supercooled

liquids will be reviewed in section IV of this chapter.

103 r
102 r
101 r — Fragile (e.g. Toluene)
100 L Strong (e.g. SiOz)

 

 

 

 

10-1
1o-2
104
104

'“"l

1: (sec)

10-5
10*
10-7
10-5
10-9

 

 

WWW ' ”'1 'TY'T‘F """1 r ""T ""“1 ""7 ""1

104° ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Tng

Figure 2.3. The curvature of the mean relaxation time on an Arrhenius plot varies
widely among materials. “Fragile” liquids show strong curvature,
while “strong” liquids display nearly Arrhenius behavior.

Angel] [7] has suggested classifying glass formers based on the degree of
curvature exhibited on an Arrhenius plot of the relaxation times. As shown in Figure 2.3,

the degree of curvature varies widely among liquids, with strongly bonded systems

13

typically displaying Arrhenius-like behavior and weakly interacting systems showing
strongly non-Arrhenius responses. In Angell’s classiﬁcation scheme, glass-forming
liquids are indexed by their ﬁagility m, deﬁned by

(2.3) m a [ d loge) / d (Ts/r) 1m.

For example, one of the best known structural glass formers is silicon dioxide (SiOz),
which has a fragility of m = 20 due to nearly Arrhenius-like relaxation behavior arising
from a three-dimensional network structure of strong covalent bonds. On the opposite
end of the spectrum are liquids composed of weakly interacting molecules such as
toluene (m = 107), which show much steeper temperature dependence in their relaxation
times and correspondingly higher fragility values.

11. b. 2. Non-Debye Relaxation in Supercooled Liquids.

The second nearly universal feature of supercooled liquids near the glass
transition is the appearance of increasing non-Debyeness in the a-relaxation as T3 is
approached upon cooling from the melt.

The a-relaxation in supercooled liquids has been studied in the linear response
regime using a wide variety of approaches [8] which are able to resolve the dynamical
properties of the material near T8 in either the frequency domain or the time domain. The
relaxation behavior of a material is characterized by a response function f(t) which
describes the time domain response of the system to an external perturbing agent. The
frequency domain response can be derived through Fourier transformation of f(t):

(2.4) f(m) ~ F { - df/dt}
The simplest type of response function assumes that the perturbed system will regain its

equilibrium conﬁguration in an exponential manner characterized by a single relaxation

l4

time 1:. This is the so-called Debye relaxation ﬁrnction, which is expressed in the time

domain as
(25) fDebyc(t) ~ eXPI-(t/TH

The frequency domain representation is obtained using eq. 2.4:

(2.6) fnebyrw) ~ [ em" [—%(fncbyc(t))]dt

-®

1

(2-7) fDebyc((D) ~ .
1 — 1cot

 

However, the response of a glass forming liquid rarely obeys eq. 2.5, and instead
displays a characteristic departure from Debye behavior in the form of a relaxation which
is typically broadened relative to eq. 2.5. Figure 2.4 shows the actual imaginary part of
the relaxation response of Ethanol (CszOH, Tg = 97 K) as obtained from measurements
of the complex dielectric constant e(m,T) = 8' + is" in a frequency-domain dielectric
spectroscopy experiment (dielectric spectroscopy is outlined in detail in Chapter 4). The

Debye form of the dielectric constant is given by the Debye equations:

2.8 e'=e +
( ) 1+002‘t2

m

 

(01(8 —8 )
2.9 "=-————° °°
( ) 8 l+03212

To illustrate the departure from Debye-like behavior, the Debye response of eq. 2.9 is
also plotted on Figure 2.4, along with Cole-Davidson (0t=0.796) and KWW (B=O.850)

ﬁts to the data. The width of the ethanol relaxation is clearly broadened.

15

 

- Ethanol Data

1; --------- Debye Response
in} --. ----- Cole-Davidson

——— wa

 

 

 

.\
7"
,2

101 y

u
'x
4’

I
O
U
C
O
C
'
t a
O
O
C
O
0
O
. O
O b . ‘, C
‘
1 o L . .‘ .
b “ .
C
. . C
C
‘
b § '
. '
‘ C
C
.
I
.. ‘
* a
O

101.r °

..1 r ...1+L.I r 1111 I r Errr I I . -..41..I . .rrrrrrl a 1....111

10'3 10'2 10'1 10° 101 102 103 104
Frequency (Hz)

 

Figure 2.4. Imaginary part of the dielectric constant of supercooled ethanol at
T=102 K. Fits to the Debye, Cole-Davidson (0t=0.796), and KWW
(B=O.850) functions have been included for reference.

To account for the broadened shape, phenomenological ﬁtting functions such as
the Kohlraush-Williams—Watts (KWW) [9] and Cole-Davidson (CD) [10] forms have
been applied with good success in the peak region of the relaxation. In the frequency
domain, the KWW function is the Fourier-transform of a stretched exponential time

domain function:

(2.10) fmm ~ expl wt)" 1

16

The stretching exponent B, which can vary between 0 and 1, provides a measure of the
degree of departure from a Debye relaxation (B = 1). Furthermore, it can be shown [11]
that B is related to width of the asymmetrically shaped imaginary part of the relaxation
shown in Figure 2.4. More speciﬁcally, B can be related to the normalized width w,
which is the full width of the relaxation at half maximum (W) in decades divided by the
width of a Debye relaxation (WD = 1.14 decades):

(2.11) w = W / WD

(2.12) (1 - B) = 1.047 (1 - w“)

The Cole-Davidson (CD) function also gives an estimate of non-Debyeness, and

is used extensively in frequency-domain dielectric spectroscopy studies:

(2.13) 8(0)) = a... + (co — 2.9/(1 - im/prD)“

Here, the exponent or ranges between 0 and 1, providing a measure of the non-
exponentiality of the relaxation. The CD form reduces to the Debye form for the case of
or = 1.

Bohmer et al. [12] have studied the degree to which the fragility is correlated with
non-exponentiality of the relaxation process for a given material. By plotting the fragility
of the material versus the stretching exponent B obtained at T8, he suggests that a general
correlation exists between the two which arises from the nature of the bonding
mechanism present. “Fragile” liquids (high m values) typically show non-Arrhenius
relaxation behavior and highly non-exponential relaxations, while “Strong” liquids (low
m values) exhibit Arrhenius relaxation behavior and relaxation which is closer to

exponential.

l7

Richert [13] has suggested two possible explanations for the non-exponential
nature of the relaxation process in supercooled liquids. The ﬁrst approach assumes that
the supercooled liquid consists of many individual heterogeneous regions each of which
relax in a nearly exponential manner but with differing relaxation times. The non-
exponential relaxation behavior of the material is then the result of averaging the
distribution of exponential relaxations governing behavior in the individual sub-regions.
The second approach assumes that the sub-regions of the liquid are all homogeneous and
that the relaxation of each region is intrinsically non-exponential.

III. Overview of Experimental Studies of the Glass Transition.

Experimental study of the dynamical aspects of the glass transition in supercooled
liquids has been carried out using a variety of methods over a very long period of time.
The common goal of dynamical studies is to determine the response of a material in the
presence of an external perturbing agent. Each technique probes the dynamical response
on a characteristic timescale, and is sensitive to certain degrees of ﬁ'eedom such as
molecular reorientation or vibration. Some techniques provide greater ﬂexibility than
others, and in many cases the results of independent methods can be combined to form
more or less complete picture of the material behavior at various timescales.

There are several ways to categorize the methods that are used in studies of the
glass transition. The approach that will be taken in this thesis will be to categorize
experimental methods based upon the timescale which they are most successful in
examining. To this end, we can divide the range of supercooled liquid behavior into

three dynamical regimes: (1) slow processes such as those associated with molecular

18

reorientation and diffusion, (2) intermediate processes associated with intramolecular
conformation, and (3) fast dynamics associated with phonon processes.

Slow dynamics (~10'3 seconds and longer) which are typically associated with
the a-relaxation have been examined with a variety of methods including dielectric
spectroscopy [14-17], speciﬁc heat spectroscopy [18,19], frequency-dependent shear
modulus [20], and ultrasonics [21]. Dielectric spectroscopy (see Chapter 4) uses
alternating electric ﬁelds to probe polarization processes in a material, and is capable of
studying dynamical behavior over an extremely wide range of timescales (10'll — 106
seconds). Speciﬁc heat spectroscopy is a relatively new technique that investigates the
frequency-dependence of the speciﬁc heat in response to small temperature oscillations
of the system about a point of equilibrium. Ultrasonic experiments typically measure
density relaxation in response to acoustical stimuli.

The intermediate dynamical regime is more difﬁcult to quantitatively deﬁne, but
roughly corresponds to timescales of approximately 10'3 - 10'8 seconds. Probes that are
effective in this range include dielectric spectroscopy and ultrasonics. This timescale is

characteristic of B-relaxation processes that are typically easiest to observe at

temperatures below T8. Below T3, the large response of the a-relaxation is suppressed
(frozen out), leaving faster processes such as molecular side-group rotations in a more
favorable position for observation.

The fast process regime occurs on a timescale that is shorter than approximately
10'9 seconds. Among the tools available to the experimenter are inelastic (Brillouin) light
scattering (10'9 — 10'12 sec), inelastic neutron scattering (10’10 - 10'12 sec), and dielectric

spectroscopy. The Mode-Coupling Theory (MCT) of viscous liquids (see section IV.c.)

l9

has recently made predictions regarding the nature of fast relaxation processes above T ,
and many of these experimental techniques have focused on verifying these predictions.
Examples of this include neutron scattering studies [22] of the so-called “Boson Peak”,
which is a collection of vibrational states giving rise to a distinct peak in the relaxation
spectra of supercooled liquids in the 0.1 - 1 THz frequency range. The crossover from
structural relaxation (or-process) to fast vibrational excitations have been studied by
dielectric [23] and light scattering [24] experiments, and incoherent neutron scattering
[25] has been used to study the crossover from Debye to non-Debye behavior in the or-
relaxation.

One principal focus of recent studies is determination of the mean relaxation time
t(T) for various relaxation processes. Wu et al. [26] have recently compared the mean 0t-
relaxation time as determined by various techniques including speciﬁc heat spectroscopy
and dielectric spectroscopy carried out on the same material. It is a remarkable fact that
while each technique probes a different property of the material, the measurements show
the same qualitative behavior and in some cases the measured relaxation times are
virtually identical.

The similarity of relaxation phenomena as measured by various techniques has
prompted to some to search for “universalities” in the behavior of supercooled liquids
near the glass transition. Dixon et al. [27] have proposed an empirical scaling form
which appears to successfully collapse the dielectric spectroscopy data for a variety of
simple liquids onto a single “master curve”. Whether this scaling form suggests a deeper

universality underlying the behavior of supercooled liquids remains an open question.

20

The scaling form and its inherent advantages and disadvantages will be discussed in more
detail in Chapter 5.
IV. Theories of the Glass Transition in Supercooled Liquids.

IV. a. Introduction.

Theoretical study of the slowing dynamics of supercooled liquids approaching the
glass transition has been a topic of intense study for many years. Early studies postulated
the existence of a true phase transition at a temperature T < Tg based on reasonable
extrapolations of various thermodynamic quantities to temperatures below T8. More
recently, attempts have been made using ﬁrst-principle hydrodynamical arguments to
characterize the behavior of liquids at temperatures well above T3 in hope of gaining a
basic understanding of the processes involved in the initial slowing of the liquid. It is
useful then, to attempt to classify theories of viscous slowdown and the glass transition
into separate categories based upon the temperature range in which they are deemed most
appropriate. Following an approach by Ediger et al. [28], theories of supercooled liquid
behavior can be classiﬁed as one of two types: (1) Theories which postulate a low-
temperature phase transition as being responsible for the apparent divergence, at a
temperature To < T3, of the viscosity and mean relaxation times as predicted by the VTF
form (eq. 2.2), and (2) Theories which attempt to characterize the initial slowing of the
high temperature liquid, resulting in the prediction of divergent behavior near a high

temperature critical point Tc.

21

IV. b. Theories predicting low temperature phase transitions.

In 1948, Walter Kauzrnann [3] ﬁrst proposed that a thermodynamic phase
transition might be responsible for the glass transition in supercooled liquids. The
argument, generally referred to as the “Kauzrnann Paradox”, involves examination of the
supercooled liquid entropy S as shown schematically in Fig. 2.5 for three cooling rates

R1,R2, and R3.

supercooled
liquid

R _....——-—-—- crystal

 

 

 

 

TkTE’Té’T‘é’ T

Figure 2.6. Graphical illustration of the Kauzrnann Paradox.

The argument goes as follows: For a given rate of cooling R1, the laboratory glass

transition will be seen to occur at a temperature T3“), at which point the speciﬁc heat

22

drops suddenly and the liquid entropy changes slope and levels off as indicated.
However, if a slower rate R2 of cooling is used, the liquid has more time to equilibrate
and the glass transition will occur at a temperature T30) where T80) < T3“). Following
this line of reasoning, one can imagine cooling slower and slower until eventually the
entropy of the supercooled liquid becomes equal to the entropy of the low energy
crystalline phase. The temperature TX at which this occurs is called the Kauzrnann
Temperature and further extrapolation of the liquid entropy for an inﬁnitely slow rate of
cooling leads to the undesirable situation of negative liquid entropy at temperatures well
above 0 K. Such a situation is a clear violation of the 3“' law of thermodynamics, and has
been termed the “Entropy Crisis” in the glass transition literature. Kauzrnann proposed
that a mechanism, such as a thermodynamic phase transition, must exist in the vicinity of
TR which would prevent the liquid from entering a regime where its entropy is less than
that of the crystal.

Support was given to the phase transition idea from the predicted divergence of
the relaxation times (or equivalently, the viscosity) by the VTF form (eq 2.2) at a
temperature To, which in many cases was striking similar to TK. The Kauzrnann Paradox
and its implication of an impending entropy crisis in supercooled liquids fueled the
efforts and imagination of theorists in the 1950’s and 1960’s. This time period was
largely dominated by two prevalent theories of the glass transition: Free-Volume Theory,
and Adam-Gibbs Theory.

Free-Volume Theory was ﬁrst introduced by Cohen and Tumbull [29,30] in the
late 1950’s and later received a modiﬁed interpretation in terms of percolation theory

[31,32]. In its simplest form, the theory imagines that an individual molecule moves

23

through a ﬂuid comprised of other identical molecules. At high temperatures, the
molecule is relatively free to move about, and interacts with other molecules through
binary collision events. As the liquid temperature decreases and the density increases,
each molecule becomes enclosed in a transient “cage” formed by neighboring molecules.
In this scenario, the molecule undergoes many collision events with the molecules
comprising the cage before ﬁnally escaping via diffusion. The free volume available to
the molecule, designated by Vf, is deﬁned to be the volume v of the transient cage minus
the minimum physical volume vc of the molecule:

(2.14) Vf= v - vc

Diffusive displacement, or the ability of a molecule to escape the cage, depends on the
formation of voids through statistical ﬂuctuation of the free volume. It can be shown [30]
that if the temperature dependence of the free volume is assumed proportional to T — To,
then the viscosity of the liquid will assume a VTF form (eq. 2.2). As the temperature
approaches To, the glass transition will occur as the cage effect strengthens and the
number of available escape routes from the cage becomes vanishingly small.

The free-volume approach is intuitively appealing, however many theoreticians
viewed it with skepticism due to the rather imprecise deﬁnition of what exactly
constituted the “free volume” for a molecule. In the early 1960’s, Adam and Gibbs [33]
sought to explain the glass transition phenomena using fundamental thermodynamic
arguments, with a belief that the entropy of a system was the principal variable governing
glass formation. They proposed a molecular theory of the glass transition based on the
notion that the liquid could be divided into small subsystems called cooperatively

rearranging regions (CRR’s). The basic premise of the theory was that molecular

24

packing in viscous liquids was too restrictive for individual molecules to reorient without
the cooperation of their neighbors. The relaxation response of a material to a
perturbative stress would then consist of a cooperative movement, or rearrangement, of
groups of molecules. An individual CRR was deﬁned to be a subsystem of the sample
containing 2 molecules, which could rearrange itself into another independent
conﬁguration if given a sufﬁcient ﬂuctuation in energy. The macroscopic system was
considered to be an ensemble of N equivalent, non-interacting CRR’s, with an

isothermal-isobaric partition function given by

(2.15) z = Z w(z, E, V) exp(—E/kT) exp(—PV / kT)
EN

where w is the degeneracy of energy-level E and volume V of the subsystem. If the sum
in eq. 2.15 is taken only over values of E and V that permit a transition, then the result
will be 2', an expression for the partition function of rearrangeable subsystems. The
Gibbs free energy can be then be deﬁned in each case as

(2.16) G = zu = —lenZ

(2.17) G' = 211': -—kT 1n 2'

where p. and u’ are chemical potentials. If it is assumed that n of the N subsystems are in

states allowing a transition, then the cooperative transition probability Wz(T) is given by

I

(2.18) wz(r) oc ii = Zi— = exp[-—(G' — G)/kT]

Substitution of 2.16 and 2.17 into 2.18 yields the transition probability of a cooperative

region as a function of its size 2:

(2.19) w, (T) = A exp (—zAu/kT)

25

Here Au = u’ - n represents the potential energy hindering the cooperative rearrangement
and A is a frequency factor whose dependence on T and z is assumed negligible in

comparison to that encountered in the exponential. The average transition probability

W(T) is determined by summing over all possible non-vanishing values of Wz(T)
corresponding to different values of 2. Before the sum can be carried out however, it
should be noted that there is a critical lower limit 2‘ to the size of cooperative regions that
can contribute non-zero transition probabilities. For a given temperature, 2’ is
determined by the smallest cooperative region that contains two conﬁgurations: one in
which the region resides before transition, and another into which it may move. The sum
is then expressed as

(2.20) Wm = 2A exp[—zAp/kT]

The summation of this geometrical series is

— z'Au]

(2'21) Wm = 1-exp(-Att/kT) exp[ kT

 

 

and for Au / kT >>1 the denominator is nearly 1 and essentially independent of

temperature. Absorbing it into a new temperature independent prefactor A gives

 

(2.22) W0“) = K exp(— if“)

Based on conﬁgurational entropy arguments, it can be shown that the critical size 2‘ can

expressed as

(2.23) z =

 

26

where NA is Avogadro’s number, s; is the critical conﬁgurational entropy and Sc is the
conﬁgurational entropy of the macroscopic system. NA, Au, k, and so. can be absorbed

into a new constant C and the average cooperative transition probability becomes

 

(2.24) Wm = X exp[ F: j

The ﬁnal step involves supplying a suitable expression for the conﬁgurational entropy Sc,
which is the difference between the liquid and crystal entropy and depends on
temperature. The temperature dependence of Sc can be determined by integration of the

excess speciﬁc heat ACp = Cpliquid — Cpg'm:

AC
p dT
T

 

(2.25) Sc(T) = j

k

Providing a functional form for ACp is difficult, but it has been shown [34] that the
particularly simple form ACp = 0L / T , where 0. is a material dependent constant,
describes the behavior of many molecular liquids. Using this form, the integration in eq.
2.25 is straightforward and yields

“(T-TR)

(2.26) s,(T)= T T

This result can be substituted into eq. 2.24, to yield the ﬁnal forrrr of the transition

probability:

(2.27) Wm = ‘A‘ apt—('1‘???)
K

27

The mean relaxation time for the system is deﬁned by t a 1/W (T) , and inversion of eq.

2.27 leads to the VTF expression (eq. 2.2) with the identiﬁcation that D = C/ A , TK = T0 ,

and 1:0 =1/A:

 

(2.28) r(T) = To exp( 131%) j

The emergence of the VTF expression represents the major success of the Adam-
Gibbs theory. Based on the idea of cooperatively rearranging subsystems, the prediction
of non-Arrhenius behavior of the relaxation time agrees well with the experimentally
established fact that many liquids display deviations from thermally-activated behavior.
However, the theory in its original form provides little explanation for the observed non-
exponential shape of the relaxation as discussed previously in section II.b.2. Recently,
claims have been made [3 5] that non-exponential behavior will result if the the Adam-
Gibbs approach is modiﬁed to include interactions between the CRR’s.

IV. c. Theories predicting critical behavior at T > Tg.

Early work on the liquid-glass transition focused almost exclusively on the
temperature region near Tg and placed little, if any, attention on liquid behavior at higher
temperatures. However, within the last decade much theoretical (and experimental) study
of the liquid-glass transition has shifted the focus toward the high-temperature regime Ts
< T < Tm and away from the traditional low-temperature regime near T3. The reason for
this shift in focus is primarily due to the advent of Mode-Coupling Theory (MCT), which
is an outgrowth of hydrodynamical liquid theory developed by Leutheusser [36] and

Bengtzelius et al. [37,38] in the mid-1980’s.

28

The rearrangement of molecules in a supercooled liquid is directly related to
ﬂuctuations in the local density. MCT asserts that the detailed nature of these structural

rearrangements is contained in the time-dependent density autocorrelation function,
F(q, t). This quantity describes the time-dependence of the relaxation of density

ﬂuctuations having wavevectorq. F(q,t) is also known as the intermediate scattering

function, and is expressed as follows:

(229) F(c‘1,t)=%(Sp°(?1,t)59(?1,0))

where the microscopic density in the liquid p(q, t) given by

(2.30) 9(51, 0 = ZCXP(i¢1'?,-(t))

1:1
where N is the number of particles and 'r}(t) denotes the position of particle j at time t.
The use of the intermediate scattering function F(q,t)as the starting point for
determining the dynamical behavior of the material is also of great practical importance

to experimentalists, as this quantity can be measured in neutron and light scattering

experiments.

Assuming the liquid is isotropic in nature, the equations of motion for F(('i, t) can

be written in a form resembling a damped harmonic oscillator:

(2.31) I3‘(q,t) + Q’(q)F(q,t) + j [M°(q,t — t') + 02(q)m(q,t — t')] F(q,t')dt' = 0

0
Here, 02 (q) is a microscopic frequency, M° (q,t — t') is a ﬁrnction that describes the short

time dynamics of the system, and m(q,t — t') represents the dynamics on longer

timescales. In the idealized [38] version of MCT, M°(q,t —t') is assumed to be sharply

29

peaked around t=0, and m(q,t — t') is assumed to be quadratic in the correlation functions

F(q, t). The compicated mode-coupling equations, whose solution gives the full time-
dependence of the intermediate scattering function, emerge upon substituting these forms
for M0 (q, t —t') and m(q,t — t') into eq. 2.31.

Solution of the idealized mode-coupling equations leads to predictions for the
dynamical behavior of supercooled liquids. One of the main predictions is that there
exists a critical temperature T6 at which the diffusion constant D (or equivalently, the
inverse of the a-relaxation time 1:) vanishes as a power-law:

(2.32) D at 1:" cc (T—Tc)’ , y > 1.5

The critical temperature Tc is predicted to lie in the region between T3 and Tm, and such
behavior has been veriﬁed recently in the computer simulation of a binary Lennard-Jones
system [39]. Experimental investigations of real systems [40] reveal the existence of
power law behavior at high temperatures (Tc > Tg ) but fail to follow the power law close
to Tc.

The fact that the experimentally determined a-relaxation times fail to show the
predicted divergence at T6 has prompted re-examination of the MCT and its assmnptions.
In the extended [41] version of MCT, an additional term is included in the deﬁnition of
M0 (q,t—t') so as to allow for activated "h0pping" processes which are ignored in the
idealized version of the theory. It is claimed that the addition of hopping processes
restores ergodicity in the liquid and results in no divergence of 1 near Tc.

The second notable feature of MCT is its predicition of distinct relaxation
processes occuring on varying timescales within the liquid. As discussed earlier in this

chapter, it is well known that distinct 0t- and B-relaxation processes exist in real liquids.

30

MCT has attracted a signiﬁcant amount of attention in the glass literature due to its
prediction of crossover behavior between the various relaxation regimes. Stimulated by
this prediction, many current experimental efforts have focused on studying such
crossover behavior [25].

V. Conclusions.

As described in this chapter, the phenomenology of supercooled liquids is now
well developed. The three major theoretical approaches (Free-Volume, Adam-Gibbs, and
MCT) have provided useful starting points for understanding the dynamics of
supercooled liquids. However, no single theory has yet been able to capture all of the
salient features of the glass transition.

At present, MCT appears to demand the most attention in the literature due to its
success in qualitatively describing the high temperature behavior of liquids. However,
because it is presently unable to account for the observed behavior of real liquids in the
lower temperature regime (T < Tc) there is considerable disagreement regarding the
ability of MCT to provide a fundamental theory of the glass transition. For the number of
experimental studies conﬁrming the predictions of MCT, there is an equal number
producing results which appear inconsistent with the theory [28,42]. It is possible that
future versions of MCT (e. g. revisions of the more complex extended version) will
resolve these issues and provide a more quantitative description of the dynamics at low

temperatures.

31

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[4] G. P. Johari, Ann. New York Acad. Sci. 279, 117 (1976).

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[6] H. Vogel, Phys. Z. 22, 645 (1961); GS. Fulcher, J. Am. Ceram. Soc. 8, 339 (1925);
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[7] C. A. Angell, J. Non-Cryst. Solids 131-133, 13 (1991).

[8] L. Wu, P. K. Dixon, and S. R. Nagel, J. Non-Cryst. Solids 131-133, 32 (1991).

[9] G. Williams and D. C. Watts, Trans. Faraday Soc. 66, 80 (1970).

[10] D.W. Davidson and RH. Cole, J. Chem. Phys. 18, 1417 (1951).

[11] P. K. Dixon, Phys. Rev. B 42, 8179 (1990).

[12] R. Bohmer, K. L. Ngai, C. A. Angell, D. J. Plazek, J. Chem. Phys. 99, 4201 (1993).
[13] R. J. Richert, J. Non-Cryst. Solids 172-174, 209 (1994).

[14] M. Shablakh, R. M. Hill, and L. A. Dissado, Chem. Soc. Faraday Trans. 2 78, 625
(1982)

[15] L. Wu, Phys. Rev. B 43, 9906 (1991).

[16] M. A. Miller, M. J. Ruiz, F. J. Bermejo, and N. O. Birge, (accepted for publication in
Phys. Rev. B 57, June 1, 1998).

[17] D. L. Leslie-Pelecky and N. O. Birge, Phys. Rev. B 50, 13250 (1994).
[18] N. O. Birge, Phys. Rev. B 34, 1631 (1986).

[19] P. K. Dixon, Phys. Rev. B 42, 8179 (1990).

[20] Y. H. Jeong, Phys. Rev. A 36, 766 (1986).

[21] Y. H. Jeong, S. R. Nagel, and S. Bhattacharya, Phys. Rev. A 34, 602 (1986).

32

[22] F. J. Bermejo, A. Criado, and J. L. Martinez, Phys. Lett. A 195, 236 (1994).

[23] P. Lunkenheirner, A. Pimenov, M. Dressel, Yu. G. Goncharov, R. Bohmer, and A.
Loidl, Phys. Rev. Lett. 77, 318 (1996).

[24] H. Z. Cummins, G. Li, Y. H. Hwang, G. Q. Shen, W. M. Du, J. Hernandez, and N.
J. Tao, Z. Phys. B 103, 501 (1997).

[25] J. Colmenero, A. Arbe, and A. Alegria, Phys. Rev. Lett. 71, 2603 (1993).

[26] L. Wu, P. K. Dixon, S. R. Nagel, B. D. Williams, and J. P. Carini, J. Non-Cryst.
Solids 131-133, 32 (1991).

[27] P. K. Dixon, L. Wu, S. R. Nagel, B. D. Williams, and J. P. Carini, Phys. Rev. Lett.
65, 1108 (1990).

[28] M. D. Ediger, C. A. Angell, and S. R. Nagel, J. Phys. Chem. 100, 13200 (1996).
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33

Chapter 3

ORIENTATIONALLY-DISORDERED CRYSTALS

I. The Need for Model Glass Systems in Studies of the Glass Transition.

The difﬁculties involved in studying the glass transition in supercooled liquids
were brieﬂy remarked upon in Chapter 1, and can essentially be linked to the inherent
complexity of the liquid state itself. In general, six degrees of freedom must be
associated with each constituent molecule of the liquid: three translational degrees to
locate the molecule with respect to some origin and three more to describe its orientation.
The situation is further complicated when attempts are made to account for the oﬁen
transient nature of interactions between the molecules. Thus, the need to treat both
translational and rotational degrees of freedom constitutes a signiﬁcant barrier for
theoretical and computer-simulation studies of the glass transition. The result has been
the formation of a large gap between theory/simulation and experiment. Computer
simulations of the glass transition in systems containing perhaps several hundred particles
over timescales of 10°10 to 10'12 seconds is clearly far from what is examined in real
experiments. To emphasize this point, Angell [1] has recently classiﬁed various features
of the glass transition according to the level of difﬁculty involved in their simulation by
computer. Despite the rapid advances in calculation speed of today’s modern computers,
he has concluded that much of the interesting phenomena occurring in supercooled
liquids will continue to remain beyond reach for the foreseeable future.

Recently, attempts have been made to simplify the situation by studying the

properties of model glass systems, which are materials that retain the essential features of

34

the structural glass transition in fully-disordered liquids, yet are somehow structurally
less complex. The primary requirement of a model glass is the existence of a glass
transition (or glasslike) transition in an environment which is less complex than the
liquid. One particular class of materials called Orientationally-Disordered Crystals
(ODC’s) possess unique structural properties that make them excellent candidates for
model glasses.

II. History of ODC's.

In the early 1930’s, Pauling [2] ﬁrst proposed that some molecular crystals could
display a metastable phase characterized by molecules having translational order and
rotational disorder. Experimental veriﬁcation of Pauling’s theory occurred in 193 8, when
Timmerrnans [3] observed that the behavior of several molecular solids could be
explained if it were assumed that the nearly-spherical molecules comprising the solid
were executing essentially free rotations. Shortly thereafter, dielectric measurements by
Smyth [4] showed that this unique phase of matter displayed static dielectric constant
values slightly higher than those observed for comparable temperatures in the high-
temperature liquid state. The large values of the static dielectric constant, which is a
quantity sensitive to the rotational motion of molecules in the sample, suggested that the
molecules in the rotationally-disordered phase were executing rotations in a nearly free
manner — even more free than the rotation of molecules in the liquid state.

The selection of an appropriate name to characterize the material and its
rotationally-disorderd phase has proven diﬁicult and resulted in much confusion in the
literature. Originally, the high degree of mechanical plasticity possessed by the phase

resulted in the material being referred to as a “plastic crysta ” [3]. This title has also been

35

used to refer to the phase itself, along with other descriptive names such as
“orientationally-disordered crystal” and “rotator crystal”. The currently accepted labeling
scheme refers to the actual rotationally-disordered phase of the material as a “rotor phase
crystal” and the material itself is classiﬁed as an “orientationally-disordered crystal”.

The existence of glassy behavior in the supercooled rotator phases of many
ODC’s was discovered by Suga and Seki in the late 1960’s. By performing a number of
thermodynamic experiments [5-9] studying the heat capacity and enthalpy relaxation of
orientationally-disordered crystals, the Suga group discovered that the rotator phase of
many substances exhibited an orientational glass transition possessing the same
thermodynamic signatures as the glass transition in fully-disordered liquids (e.g.
anomalous behavior in the speciﬁc heat). The term “glassy crystal” was proposed to
represent the supercooled rotator phase below Tg , however this name has given way to
the present terminology of “orientational glass”. The prototype materials studied by the
Suga group will be discussed in greater detail in section IV of this chapter.

111. Structure and Dynamics of ODC’s.

Upon supercooling from the high-temperature liquid phase, the occurrence of
ﬁrst-order crystallization processes may result in the formation of a phase that is
metastable relative to the lowest energy crystalline state. This behavior is due to
differences in the growth rate of crystallites in the metastable and lowest energy
crystalline states, and is commonly known as Otswald’s “Law of Stages” [10]. Put more
generally, Otswald’s Law states that the supercooled liquid will tend not to transform
directly into the lowest energy crystalline phase, but rather into a phase that is next most

stable to the liquid. The rotator phase of ODC’s is precisely such a metastable state, and

36

 

is characterized by molecules that are tr'anslationally ordered and rotationally disordered.
The crystal structure of the rotator phase is usually fee. or b.c.c. due to packing
constraints and energy arguments.

Most materials possessing rotator phases typically have molecular units which are
globular or nearly spherical in shape. Figure 3.1 provides a two-dimensional
representation of the actual three-dimensional character of a typical rotator phase crystal.
In the absence of any applied ﬁelds, each molecule rigidly attaches itself to a lattice site,
yet retains its rotational ﬁeedom. Similar to the liquid phase, the rotator phase can be
supercooled to reveal a glass-like transition in the orientational degrees of freedom which

is similar to the structural glass transition in supercooled liquids.

 

 

Figure 3.1. The translational order and rotational disorder of molecules in
a typical rotator phase crystal.

37

IV. Example ODC’s.

Suga’s pioneering work on ODC’s began with studies of cyclic alcohols [5], all of
which are characterized by a main hydrocarbon ring supplemented by an OH radical.
Rotator phases have been observed [6] in cycloheptanol (C7H13OH), cyclohexanol
(C6H110H), cyclopentanol (C5H90H), and cyclooctanol (C3H150H) to name a few.
However, Suga ﬁrst discovered glassy behavior in cyclohexanol, and thus cyclohexanol

emerged as the prototypical model glass system before any of the other cyclic alcohols.

Structural Supercooled Liquid Liquid

glasy—
Orientational S“ ercooled SI 51
glass S2

S3
T.= 150 K 245 K 265 K 298 K

 

 

Figure 3.2. Schematic diagram of the phases of cyclohexanol.

Following the labeling scheme of Suga, Figure 3.2 provides a useﬁrl schematic
diagram illustrating the behavior of cyclohexanol as its temperature is varied. Upon very
slow cooling, the liquid undergoes a ﬁrst-order transition at 298 K to S3, the stable
crystalline phase. Moderate cooling rates (~ 10 K/min) will not result in transition to S3,

but rather a transformation to 81 will occur. The 81 phase is a rotator phase (f.c.c.

38

 

structure, lattice constant a = 8.6 angstroms) and is characterized by translational order
and rotational disorder. S] can then be supercooled to form an orientational glass at T3 =
150 K. Due to its metastability, the supercooled SI phase is susceptible to crystallization
resulting in either 82 or S3, depending on the cooling rate of the experiment. Of course,
the transition at 298 K can be avoided altogether through the use of high cooling rates
(~20-30 K/min) resulting in the fully-disordered supercooled liquid. However, the
supercooled liquid phase of cyclohexanol is rarely examined in experiments. In addition
to the high cooling rates required for formation, the supercooled liquid demonstrates poor
stability against crystallization [5].

The orientational glass transition in cyclohexanol occurs when the rotational
ability of the cyclohexanol molecules vanishes. The result is a glass transition associated
with the orientational degrees of freeedom that displays the same thermodynamic
signatures as the structural glass transition in fully-disordered liquids. The similarity in
behavior can be seen in a plot of the speciﬁc heat versus temperature in the supercooled
S1 phase of cyclohexanol as shown in Figure 3.3. Here, the orientational glass transition
at ~ 150 K involves a drop in the speciﬁc heat of approximately 20 J mol'1 K’l
corresponding to the freezing of rotational degrees of freedom. Below T3, only
vibrational mechanisms contribute to the speciﬁc heat, and the speciﬁc heat resembles
that of a crystalline solid.

Despite its lengthy history, cyclohexanol is still of interest to experimentalists
studying the glass transition. For example, Gangasharan and Murphy [11] have recently
investigated the supercooled S1 relaxation phenomena using dielectric spectroscopy;

NMR studies of the molecular motion have been carried out by Kuhns et al.[12], and

39

Mizukami et al.[13] have reported calorimetric evidence suggesting that supercooled S1

exhibits more than one type of or- and B-relaxation.

 

zoo - ! . - . '9 .

loop

cpm/(J tK"-m01 "1

 

 

 

o ‘ ' ‘ L130 ii.izoo7
_ ’—rvrx

 

Figure 3.3. Experimental heat capacity of cyclohexanol in the supercooled S1
phase. The sudden drop in CF at ~ 150 K is due to the orientational
glass transition. (Data from: M. Mizukami et al., Solid. St. Comm.
100, 83-88 (1996)).

Currently, the search for model glass systems has shifted away from cyclohexanol
toward other materials. This is due in part to the fact that cyclohexanol is quite difﬁcult
to maintain in the supercooled S] phase. Cyclo-octanol, on the other hand, has an
additional two hydrocarbons which provide greater resistance to crystallization. Thus,

because of its structural similarity to cyclohexanol and the additional stability provided

by its rotator phase, cyclo-octanol has received considerable attention in the past 5 years.

40

Cyclo-octanol and its usefulness as a model glass system will be discussed in detail in
Chapter 5.

Even more recent has been the conﬁrmation (or "re-discovery") [14] of an
experimentally accessible rotator phase in ethanol (C2H50H). As will be greatly
expanded upon in Chapter 6, ethanol can be readily prepared in both the supercooled
liquid and rotator phases - a very desirable feature which invites comparison of the
dynamical aspects of the glass transition in each phase.

V. Motivation.

The rotator phase of ODC’s provides a unique environment in which to isolate
and study the dynamical aspects of the glass transition without the additional complexity
of translational disorder. It is the purpose of this thesis to experimentally investigate the
properties of two ODC’s in particular: cyclo-octanol and ethanol.

First, we wish to determine if certain universalities discovered in dynamical
response of supercooled liquids also exist in the supercooled rotator phase of ODC’s.
This may help to determine which features of the glass transition are truly universal, and
which are linked to the particular structural characteristics of a material. Previous low-
frequency dielectric studies performed by our group on the rotator phase of cyclo-octanol
[15] have indicated that the material possesses the same general features seen in
structural glass forming liquids. Chapter 5 of this work extends the previous range of
our dielectric measurements on cyclo-octanol to much higher frequencies in an attempt to
provide a detailed description of the a-relaxation. Recent studies [16] of supercooled
liquids have suggested that universal behavior exists in the high-frequency response of a

material, and we wish to investigate if this holds for ODC’s as well.

41

The second goal of this work is to provide a quantitative comparison of the
dynamical aspects of the glass transition in two structurally dissimilar phases. A
comparison of this type may provide critical insight as to what degrees of freedom are
most relevant to the glass transition, and has never been possible before due to lack of an
'appropriate material. As discussed in end of section IV of this chapter, the intrinsic phase
polymorphism of ethanol provides a perfect opportunity for such a comparison. Chapter
6 presents the comparison of the or-relaxation dynamics in the supercooled liquid and

rotator phases of ethanol.

42

 

REFERENCES

 

[1] C. A. Angell, Comp. Mat. Sci. 4, 285-291 (1995).

[2] L. Pauling, Phys. Rev. 36, 430 (1930).

[3] J. Timmermans, J. Chim. Phys. 35, 331 (1938).

[4] C. P. Smyth, Trans. Faraday Soc. 42, 175 (1946).

[5] K. Adachi, H. Suga and S. Seki, Bull. Chem. Soc. Japan 41, 1073-1087 (1968).

[6] K. Adachi, H. Suga and S. Seki, Bull. Chem. Soc. Japan (Short Communications) 43,
1916 (1970).

[7] H. Suga and S. Seki, J. Non. Cryst. Sol. 16, 171-194 (1974).
[8] O. Haida, H. Suga, and S. Seki, J. Chem. Thermodynamics 9, 1133-1148 (1977).
[9] H. Suga, Journal de Chimie physique 82, 275-282 (1985).

[10] J. Sherwood, The Plastically Crystalline State, Chapter 1, John Wiley & Sons, New
York (1979).

[11] M. Gangasharan and G. Murphy, J. Chem. Phys. 99, 271 (1993).
[12] P. L. Kuhns and M. S. Conradi, J. Chem. Phys. 80, 5851-5857 (1984).
[13] M. Mizukami, H. Fujimori, and M. Oguni, Solid. St. Comm. 100, 83-88 (1996).

[14] A. Srinivasan, F .J . Bermejo, A Andres, J. Dawidowski, J. Zuniga and A. Criado,
Phys. Rev. B 53, 8172-8175 (1996).

[15] D.L. Leslie-Pelecky and NO. Birge, Phys. Rev. Lett. 72, 1232 (1994); Phys. Rev.
B 50, 13250 (1994).

[16] N. Menon and SR. Nagel, Phys. Rev. Lett. 74, 1230 (1995).

43

Chapter 4

EXPERIMENTAL TECHNIQUES

1. Dielectric Spectroscopy.

One of the most useful and productive techniques available to the experimentalist
interested in characterizing the orientational and translational motion of molecules in a
polar material involves measurement of the material's dielectric properties. This
technique is referred to as dielectric spectroscopy, and consists of measuring the
frequency and temperature dependent response of an ensemble of dipolar molecules to an
externally applied electric ﬁeld of frequency f. The method provides an estimate of the
time required for reorientation of the molecule, or the so-called relaxation time.

II. Review of Dielectric Theory.

II. a. Polarization Mechanisms in Dielectrics.

When a dielectric material is placed into an external electric ﬁeld, such as that
produced in the region separating parallel plates of a capacitor, a separation of charge
occurs and the material becomes polarized. The resulting polarization is measured in
terms of a dipole moment per unit volume, and depends strongly on the chemical makeup
of the material. The macroscopic polarization in the material is actually a superposition
of the effects of several microscopic polarization mechanisms occurring simultaneously
on the molecular level. These microscopic polarization mechanisms can be divided into
two general classes: induced polarization and orientational polarization.

Orientational polarization occurs when molecules with intrinsic electric dipole

moments attempt to reorient themselves in the direction of an externally applied electric

44

ﬁeld as shown in Fig. 4.1. Molecules having permanent dipole moments are termed
polar molecules, and the material as a whole is designated a polar dielectric. In practice,
this polarization mechanism is limited by thermal agitation of the molecules, which

prohibits uniform alignment with the ﬁeld.

Q QQ GEE
Qa0ﬁ 363%

Figure. 4.1. Polar molecules in an external electric ﬁeld.

Induced polarization occurs even in non-polar materials, and is a result of
“stretching” the charge distribution of a given atom or molecule in such a manner so as to
cause a physical separation of the centers of positive and negative charge. For example,
in a typical atom with no external ﬁelds present, the location of the positively charged
nucleus and center of the surrounding negatively charged electron cloud coincide, and
thus no dipole moment exists. However, in the presence of an electric ﬁeld, the
positively charged nucleus is displaced relative to the surrounding electron cloud,
resulting in an induced dipole moment. The size of the induced moment depends on the
magnitude of the local electric ﬁeld present as well as the atom or molecule’s
polarizability, or. The magnitude of the dipole moment induced in an individual atom or

molecule by the local electric ﬁeld can then be expressed as

45

(4.1) u = or F

where F is the local electric ﬁeld present at the location of the molecule.

For N non-polar atoms in a volume V, the total polarization due to induced dipoles is
obtained by summing the contributions of all dipoles over the volume of the sample.

The two classes of polarization mechanisms, orientational and induced, combine
to give the ﬁnal macroscopic polarization observed in the material. However, upon
application of the external electric ﬁeld, the contribution of each mechanism to the total
polarization occurs at differing rates. The electronic and molecular polarization occurs
rapidly (~1O'12 see) because these processes only involve a shift of the equilibrium charge
distribution. The orientational polarization occurs much more slowly, as interactions
with other molecules hinder the rotation of individual dipoles as they attempt to align
with the ﬁeld. As a result, the total polarization in the material approaches a maximum
only at very long times.

11. b. Behavior in Static Fields - Static Dielectric Constant.

In attempting to understand the general behavior of dielectrics when subjected to
electric ﬁelds, a natural starting point is to consider their behavior in the presence of
ﬁelds that do not vary with time. Such an approach introduces fundamental concepts and
provides a natural stepping stone for the more advanced treatment involving time-varying
ﬁelds.

Let us begin by considering a typical parallel plate capacitor assembly having
opposite charges of magnitude 6 on each plate, which give rise to a uniform electric ﬁeld

E in the region between the plates. When the region between the plates is devoid of

46

material and a vacuum exists, the magnitude of the electric ﬁeld can be expressed in

terms of the charge density on the plates. In the cgs system of units, this is given by

(4.2) E =4m

vacuum

If a material with dielectric constant e is subsequently inserted between the plates, the
electric ﬁeld decreases by a factor of the dielectric constants, and is given by

(4.3) E = 1’19
8

The drop in ﬁeld strength is due to the polarization mechanisms discussed in section II.a,
which create a distribution of bound charge producing an electric ﬁeld pointing in the
opposite direction of the ﬁeld produced by the free charge on the capacitor plates. To
account for polarization in the dielectric, we introduce the electric displacement D, which
is deﬁned in terms of the free charge density as

(4.4) D = 4m:

and is related to electric ﬁeld E and polarization P by the relation

(4.5) D = SE = E +41tP

Rearrangement of terms leads to an expression for the dielectric constant in terms of the

electric ﬁeld and polarization present in the dielectric:

-553
E

(4.6) s —1

To determine a for a particular material, an accurate expression must be provided
for the polarization P. To obtain this, we must ﬁrst have knowledge of the local electric
ﬁeld F that each molecule experiences, which in general is not the same as the

macroscopic ﬁeld E. The difference is due to dipole-dipole interactions that modify the

local environment and cannot be neglected. One way of deriving an expression for the

47

local ﬁeld is to consider a microscopic spherical region surrounding the molecule whose
overall size is large compared to the molecular dimensions but small compared to the size
of the sample. In this picture, the interactions of dipoles with one another can be
calculated, and Lorentz showed [1] that for a cubic lattice of polarizable atoms the
dipoles inside the spherical region produce zero electric ﬁeld. The local ﬁeld inside the
sphere is then due to contributions from all sources external to the spherical region.
Using techniques from electrostatics (see refs. 2,3), the result for the local ﬁeld F

experienced by an individual dipole can be expressed as

_(e+2)E
"T—

(4.7) F
As discussed earlier, the total dipole moment of a molecule consists of both intrinsic and
induced portions. The induced portion of the moment can be described in terms of the
molecular polarizability 0t and local ﬁeld F. The resulting contribution, P1, to the

macroscopic polarization P for N molecules in a volume V is then given by

NaF
4.3 p __ __

For non-polar substances P = P1 , and this result can be directly substituted into Eq. 4.6 to

give the Clausius-Mossoti relation for the static dielectric constant:

(e—1)_ 41rN0t
(e+2) ’ 3v

 

(4.9)

The Clausius-Mossoti equation is a satisfactory description of the behavior of the
dielectric constant in non-polar gases and very dilute liquids, but cannot be applied to the
case of polar liquids. In many liquids and polycrystalline solids, the effect of

orientational polarization far exceeds polarization induced by distortion of the charge

48

distribution, and must be accounted for in any theory attempting to provide an accurate
description of the behavior.

In the absence of any applied ﬁelds, individual dipoles are randomly oriented and
no net polarization exists in the material. Application of a ﬁeld causes the dipoles to
rotate in an attempt to align with the ﬁeld. The examination of a system of rigid dipoles
under the inﬂuence of an external electric ﬁeld was ﬁrst carried out by Debye [4], who

began with an expression for the polarization due to permanent dipoles in the material:
(4.10) P2 :1; u<cos0>

Here, (cos 0) is the mean value of the cosine of the angle of inclination of a dipole with
respect to the external ﬁeld. Debye then assumed that the individual moments would
distribute themselves about the ﬁeld direction based on Boltzmann statistics, and

integrating over all possible directions gives

_ E __“_T__ 11F
(4.11) (cosG>—coth[kT] uF—L( AT)

where L(uF / kT) is the Langevin function originally used to calculate the mean magnetic
moment of a gas consisting of particles with permanent magnetic dipoles. For uF / kT
<< 1, corresponding to local ﬁelds that are not too strong, the value of the function

approximates to uF / 3kT, and substitution of this result into eq. 4-10 gives:

NuzF
3kTV

 

(4.12) P2 4

To obtain the Debye expression for the dielectric constant in a material where both

induced and orientational polarization is accounted for, we combine the effects of

49

induced and orientational mechanisms to obtain the total polarization and substitute the

result into Eq. 4.6. The result for the dielectric constant is given by:

_ 2
(4.13) 8 1—4“N[a+ “ J
8

 

 

+2’ 3v 3kT

This expression is in good agreement with the measured dielectric constant of many
gases and dilute liquids, but fails to provide reasonable results in the case of dense
liquids. In fact, a major shortcoming of the equation is the prediction of ferroelectric
behavior in dense liquids when the temperature falls below a certain critical value.

The failure of the Debye equation is due to the fact that the Lorentz local ﬁeld
does not accurately describe the interaction of dipoles when the density is high, and thus
the expression for the local ﬁeld at each molecule is incorrect. Onsager [5] produced a
dramatically improved theory of dielectric polarization by considering a different
approach to calculating the dipolar interactions and arrived at a more accurate expression

for the local ﬁeld F. His treatment represented the molecule as a point dipole of moment

1/3
m in a spherical cavity of radius a =(431N) . The local ﬁeld F within the cavity now
1:

consisted of two parts G and R, the ﬁrst of which is the ﬁeld produced in the empty

cavity by the external applied ﬁeld,

38
(4.14) G—[Zs )E

 

+1
and the second part being a reaction ﬁeld generated in the cavity from the polarization

induced by the dipole:

_ 2(8-1)
(4.15) 11.0“”a3 m

50

The dipole moment m consists of both an induced part and a permanent part, given by
(4.16) m = u+0tF

The local ﬁeld is now obtained by F = G + R, and an approach similar to that used by
Debye gives the ﬁnal expression for the Onsager formula. A detailed derivation of the
Onsager equation can be found in a number of references, including F rolich [3] and
Smyth [4]. The ﬁnal result is as follows, where a... is the contribution of fast electronic

polarization processes, u is the molecular dipole moment, and N0 = N/V:

(a — so, )(28 + 8,0) _ 41rN0u2

4.17
( ) e(e,, +2Y 91a

 

The Onsager expression is superior to the Debye form, and provides sufﬁcient
agreement with the experimentally determined static dielectric constants of many non-
hydrogren bonded dipolar liquids. Unfortunately, even eq. 4.17 fails for hydrogen bonded
liquids such as water and ethanol because the formation of a hydrogen bond between
molecules changes the local environment so as to be inconsistent with Onsager's
expression for the local ﬁeld. The inadequacy eq. 4.17 can be illustrated numerically for
the case of ethanol (CszOH). Because solution of eq. 4.17 is simplest for the dipole
moment u, we can substitute the known values for a, 8.0, and No at room temperature (298
K) and compare the result with the actual measured dipole moment. In Debye units (1 D
= 10'18 esu-cm), the Onsager expression applied to ethanol at 298 K (e = 24.3, a... = 1.85,
and N0 = 1.0572e+22 m'3) leads to a dipole moment of u = 2.96 D. This is considerably

larger than the experimentally measured moment of 1.67 D [6].

51

Thus, the failure of the Onsager expression has led to more sophisticated developments
such as that by Kirkwood [7] which attempt to describe the intermolecular interactions in
greater detail.

II. c. Time-Varying Electric Fields — The Complex Dielectric Constant

The application of time varying electric ﬁelds gives rise to dielectric dispersion
and loss that is not present in the case of static ﬁelds. Dispersion and loss can be
qualitatively understood by considering the reaction of an individual molecule to an
external electric ﬁeld that varies with time. When the frequency of the ﬁeld is very low,
even bulky molecules interacting strongly with their neighbors are able to stay in phase
with the ﬁeld, and the measured dielectric constant will yield values close to that of the
static dielectric constant. On the other hand, if the frequency of the ﬁeld is very high,
then the period of the ﬁeld is much shorter than the time required for a molecule to
complete rotation and the molecules cannot reorient themselves quickly enough to align
with ﬁeld. Orientational polarization does not contribute to the total polarization in the

dielectric at high frequencies and only induced polarization mechanisms contribute. The

resulting value of the dielectric constant, 8 is often referred to as the “optical”

dielectric constant. Finally, there exists a band of intermediate frequencies where the
dipolar response is measurable but lags behind the external electric ﬁeld. This
characteristic phase lag gives rise to energy dissipation in the dielectric, and to account
for it a complex dielectric constant, s' = s' + is ", is introduced. The real part 8'

represents the dielectric constant discussed in the previous section, and s" is called the

loss. The real and imaginary parts are related through mutual transformation relations

called the Krarners-Kronig relations [8].

52

To provide a quantitative description of the behavior in time-varying ﬁelds, an
accurate description of the time-dependent polarization P(t) in the material must be
speciﬁed. The simplest description that quantitatively displays the phenomena of
dispersion and loss was ﬁrst examined by Debye [4]. In his theory, he assumed that the
total polarization (P) in the material consisted of a sum of induced (PI) and orientational
(P2) effects:

(4.18) P = P1+ P2

Furthermore, he assumed that the orientational contribution P2 would approach its
equilibrium value in an exponential fashion given by

(4.19) dP2/dt = (P — P1 - P2) / r

(4.20) P2 = (P — P1) [1 — exp(-t / 1)]

where r is designated to be the macroscopic relaxation time in the dielectric describing
the decay of the polarization upon removal of the external ﬁeld A pictorial
representation of the response of the orientational and electronic polarization to an

electric ﬁeld turned on at t = O can be seen in Figure 4.2.

53

 

      

P2

P .....

 

 

Figure 4.2. The response of induced and orientational polarization mechanisms to a
constant electric ﬁeld turned on at t = 0.

Recall from the previous section that the equilibrium polarization P can be expressed in
terms of the static dielectric constant (which we will explicitly refer to as as) and the
induced polarization P1 can be expressed in terms of the optical dielectric constant 8...:
(4.21) (s, — 1) = 41tP / E

(4.22) (so, - 1) = 4nP1 /E

Subtracting 4.22 from 4.21 gives

(4.23) P — P1 = (ss — s..)E / 47:

The time-varying electric ﬁeld E can be written as a complex quantity, where 0) is the

angular frequency of the oscillation:

54

(4.24) E(t) = Eo exp(icot)

The above equations for P - P1 and E can then be substituted into (4.19) to give the
differential equation for the orientational polarization contribution:

(4.25) dP2 / dt = (8s - so.) Eo exp(icot)/47t - P2 / r

The solution of (4.25) shows that the ratio P2 / E is a complex quantity, indicating that the
orientational polarization is out of phase with the driving ﬁeld:

(4.26) P2(t) = (ss - sw)E(t) / [41: (1 -— imr)]

The total polarization in the dielectric is then given as

(4.27) P(t) = P; + P2(t) = 4n[ (sco — l) + (ss — s..)/(1+i03r) ] E(t)

The dielectric susceptibility, x6, can be obtained using the general relation P = x, E, and it
can be seen that 1,, is complex:

(4.28) xc = 4n[ (3.. - 1) + (85 — sm)/(1+imt) ]

The frnal expression for the dielectric constant in alternating ﬁelds can be obtained using
the relationship between 8 and xe:

(4.29) s = 1 + 47me

(4.30) s = .90° + (ss - so.) / (1 + icor)

Separation of (4.30) into real and imaginary parts and introducing s = s' + is" gives the
Debye equations for the dielectric constant and loss:

(4.31) s' = e..+ (e. - e...) / (1 + (021:2)

(4.32) s" = (e, - e...)m / (1 + (1)2152)

The frequency response of a dielectric obeying the Debye equations (4.31) and (4.32) is

shown in Fig. 4.3 for the case so = 10, s... = 2.0, and r = 1.0.

55

 

10- sunscreen-000...

 

 

10'2 10'1 10° 101 102
(0 (rad/s)

Figure 4.3. Frequency response of a dielectric obeying the Debye equations. This
example assumes so = 10, 8.0 = 2.0, and r = 1.0.

An expression for the macroscopic relaxation time t can be determined by maximizing
eq. 4.32 with respect to co, the result being given as
(4.33) 1: = “(speak
where (speak is the angular frequency locating the maximum of s". Thus, measurement of
a material’s dielectric response under the inﬂuence of a time-varying electric ﬁeld
provides an estimate of the mean relaxation time.

The Debye equations provide a useful way of understanding the behavior of

dielectrics in time varying ﬁelds and assume that the orientational polarization process is

56

governed by a single relaxation time t. However, as discussed earlier in Chapter 2, the
measured dielectric response of most supercooled liquids is typically non-Debye in
nature, showing a relaxation spectrum that is broadened relative to what is expected from
the Debye theory.

III. Dielectric Measurement System.

III a. Goals of Dielectric Spectroscopy.

The primary goal of dielectric spectroscopy is to accurately measure the complex
dielectric constant, e = s’ + is", as a function of temperature and applied electric ﬁeld
frequency. The most practical method of generating an external electric ﬁeld in a
material involves driving a capacitor containing the material with a voltage that varies
harmonically in time. Using this method an electric ﬁeld is generated in the region
between the capacitor plates and a current ﬂows through the sample, in this case a polar
liquid. For a given temperature and frequency, the impedance of the sample can be
determined by measuring the voltage across, and current through the sample. As will be
discussed in the next two sections, knowledge of the sample impedance at a particular
temperature and frequency allows determination of s = s' + is".

The frequency range of the measurement dictates the type of measurement
technique used. Fortunately, study of the alpha relaxation in supercooled liquids does not
require the use of extremely high frequency techniques (> 10 MHz). For this study, lock-
in ampliﬁer techniques were utilized for frequencies in the range 1 mHz to 10 kHz, and
an impedance analyzer was employed for higher frequencies in the range lkHz to

lOMHz. These techniques will be discussed in the following sections.

57

III. b. The Dielectric Sample Cell

The sample cell used for the dielectric measurements was a glass-sealed coaxial
capacitor similar in design to that used by Adachi, et al. [9]. Shown in Figure 4.4, the
capacitor consists of stainless steel inner and outer conductors separated by two ceramic

Macor spacers.

electrodes

 

 

 

 

 

 

inner
conductor
outer
conductor
”ceramic
spacer

Figure 4.4. Dielectric sample cell used for measurements.

The gap between inner and outer conductors was 2.54e-4 meters, resulting in a nominal
empty cell capacitance of Co = 49.7pF. The four electrodes at the top of the capacitor
were fashioned from 0.030-inch diameter tungsten rods. The material choice of tungsten

was necessary for glass-to-metal bonding, and the largest possible diameter was chosen

58

 

so as to minimize lead inductance as well as provide increased surface area during glass
bonding process.

The entire capacitor assembly was then encased in a form-ﬁtting envelope of
Pyrex glass using facilities and personnel at the Scientiﬁc Glassblowing Laboratory
located in the Chemistry building at MSU. Pyrex was chosen over other materials due to
its resiliency to thermal cycling stresses over temperatures ranging from liquid Helium
(4K) to room (300K), as well as other factors such as availability and cost.

111. c. Low Frequency Measurement Circuit

In the range lmHz to 10kHz, the real (s’) and imaginary (8") parts of the
dielectric constant were measured using a Stanford SR850 Dual Phase Lock-in Ampliﬁer

in conjunction with a Stanford SR570 Current-to-Voltage Ampliﬁer as shown in Figure

I \ l Lockin

Sample_|_ _L_ / * Ampliﬁer
1

Current
Ampliﬁer

4.5.

 

 

 

 

 

 

 

Figure 4.5. Circuit used for low-frequency (lmHz — lOkHz) measurements.

There are several reasons why this measurement circuit was chosen over a more

traditional ratio-transformer bridge, which was available for use in the form of a General

Radio 1616 Precision Capacitance Bridge. First, the lockin-ampliﬁer setup allows for
measurement at much lower frequencies (~ lmHz) than the capacitance bridge (~10 Hz).
This allowed for a more complete determination of the peak region of the a-relaxation.
The experiment is also a high impedance measurement, since the capacitance of the full
cell is typically in the range 10'10-10'12 pF and measurement frequencies extend to the
mHz realm. The virtual ground conﬁguration of the current ampliﬁer eliminates
charging of the parasitic capacitance between the sample and the current ampliﬁer,
allowing a 2-terminal measurement instead of the typical 4-terminal conﬁguration. In
this scheme, the internal oscillator of the SR850 sinusoidally drives the capacitor
containing the sample liquid. Because real dielectrics exhibit loss, the current in the
sample has two components: 1) a component which is in-phase with the driving voltage
and related to energy dissipation in the dielectric, and 2) a component which is 900 out-
of-phase with the driving voltage related to the molecules ability to remain in phase with
the external electric ﬁeld. This behavior can be accounted for by introducing a complex
dielectric constant s' = s' + is" into the standard current-voltage relationship for a
capacitor containing a dielectric:

(4.34) I = iVoors'C0

Here, Vo is the magnitude of the applied voltage, 0) = 21rf, and Co is the capacitance of the
cell when empty. The SR570 Current-to-Voltage ampliﬁer transforms the current into a
complex voltage that can be measured by the lock-in ampliﬁer, where B is the gain of the

current ampliﬁer:

(4.35) v0“, = vx + ivy = (31 = BiVotns’Co

60

Substituting the expression for the complex dielectric constant and solving for s' and s"

 

 

gives
V
(4.36) s'= ’
BOJVOCO
(4.37) s": V“
Bwvoco

III. (I. High-Frequency Measurement Circuit

For frequencies ranging from 1 kHz to 10 MHz, a Hewlett-Packard 4192A
Impedance Analyzer was used in a standard four-terminal pair conﬁguration to measure
the capacitance C and conductance G of the sample as shown in Figure 4.6. The four
terminal pair arrangement avoids measurement of the sample voltage and current using
the same electrical leads, and is essential for minimization of the parasitic capacitance

and inductance which occurs in the measurement circuitry at frequencies above 1 MHz.

 

 

¥

 

 

.. (L0 a
I. (L0 ' 2

V. U

o >[

Figure 4.6. Schematic of Four-Tenninal-Pair Measurement.

 

\/

 

 

 

 

 

 

\"2—\_./

 

61

Among its many features, the HP4192A provides the experimenter with the
ability to auto-correct for phase shifts (in the measurement of impedance) that occur in
the four coaxial cables connecting the sample to the instrument. The correction assumes
an individual cable length of exactly 1m, and custom cables were fashioned to allow
implementation of this feature.

The HP4192A operates by ﬁrst determining the sample impedance from
measurements of the voltage and current using an autobalancing bridge technique [10].
Next, the instrument determines C and G by modeling the sample as either a parallel or
series combination of capacitance and resistance. The model to be used is selected by the
experimenter using a selector switch on the instrument’s front panel. In this study the
parallel model was chosen to be the most accurate representation of the sample. It should
be noted, however, that either choice is acceptable since measurements were not seen to
vary substantially between the two types of models.

The ﬁnal step involves determination of s' and 8" based on C and G. The
relationship between the various quantities is given as follows, where Co is the

capacitance of the cell when empty and (o = 27rf:

 

C
4.38 '=—
( ) a c.
G
4.39 '=
( ) a (”C0

The process of determining s'and s" using 4.38 and 4.39 is actually more complex
that it appears. The measurements of C and G usually need to be corrected for errors
which are introduced by lead inductance and non-standard coaxial cable lengths. These

corrections are nontrivial, and are crucial in obtaining accurate measurements. The

62

interested reader is referred to Appendix A for a summary of empirical and theoretical
attempts to understand data corrections for the dielectric experiments.

III. e. Cryostat and Temperature Control

In order to provide a controlled environment for measurement, the capacitor
containing the sample was suspended inside a nested cryocan arrangement. The use of a
high thermal conductivity copper can nested inside a brass can provided the thermal
isolation necessary to achieve sample cell temperature stability to within 0.05 K of the
desired target temperature.

The cryocan arrangement was then attached to an insert consisting of two hollow
stainless steel tubes. The tubes provided a conduit for electrical connections and allowed
introduction of helium exchange gas between the cryocans for temperature control. The
entire assembly was then immersed in a dewar capable of holding either liquid nitrogen
or helium as dictated by the requirements of the experiment.

Temperature control was achieved through the use of helically wrapped heater
wires on both the sample cell and copper cyrocan. The cell heater consisted of
approximately six feet of enameled high-resistivity wire (Thermo-Alumel, 23 Q/ft)
giving a total resistance of 140 Q. The copper can heater was considerably more robust,

and consisted of 10 feet of enameled Nichrome wire (50 Q/ft) for a total resistance of 500

Q. The cell and copper can heaters were controlled using programmable power supplies

0(epco model ISO-ATE for copper can heater, MSU E-Shop 0-60V for sample cell) in

conjunction with a proportional-integral-derivative (PID) algorithm of our own design.
Measurement of temperature was carried out using platinum (Pt) resistance

thermometers (RTD’s). Cell temperature was determined by measuring the resistance of

63

an Omega model F3105 attached to the glass envelope at the cell’s midpoint. Similarly,
the copper can temperature was determined using an Omega model F100 embedded in
the upper lid of the can. A homemade (MSU E-shop) conductance bridge provided
resistance measurements for the sample cell RTD, and a Keithley 196 Digital Multimeter
was used to read the copper can RTD.

IV. Sample Fabrication Methods.

The materials examined in this study consisted of hydrogen-bonded liquids, and
due to their tendency to absorb water considerable attention was given to obtaining
samples of high purity. Two techniques were used for sample preparation: 1) Fractional
Vacuum Distillation, and 2) Nitrogen Atmosphere Loading. Both methods involved
ﬁlling the capacitor cell in a controlled environment free from impurities.

IV. a. Fractional Vacuum Distillation.

The technique of Fractional Vacuum Distillation (FVD) yields samples of very
high purity and is illustrated in Figure 4.7. The system consists of ﬁve major components
including solvent bottle, cold trap, sample cell, roughing pump, and diffusion pump. The
liquid to be puriﬁed is ﬁrst independently vapor loaded into a solvent bottle containing a
small amount of desiccant material (CaH2) which aids the process of water removal. The
solvent bottle and sample cell are then attached using 9mm Fischer-Porter valves to a
cold trap submerged in a liquid nitrogen bath. The function of the cold trap is to freeze
out impurities such as pump oil that may be introduced from the pumping station. The
contents of the solvent bottle are then frozen using liquid nitrogen and a vacuum of
approximately 10'6 Torr is produced in all regions using the difﬁrsion pump backed by

the roughing pump. Because CaH2 reacts with hydrogen bonded liquids to produce H2

gas, it is often necessary to subsequently thaw the solvent bottle and repeat the entire
process. This is referred to as a “F reeze-Pump-Thaw” cycle, and is repeated several

times until no measurable quantity of gas is drawn from the solvent bottle.

Diffusion I I Roughing]

 

 

 

 

 

 

Solvent
Bottle
Sample
® __, Fischer-Porter ,. C611

valve

Figure 4.7. Fractional Distillation System.

When a sufﬁcient vacuum exists, the solvent bottle/cold trap/sample cell is sealed
off from the pumping apparatus by means of a PP valve on the cold trap. The liquid in
the solvent bottle is then vaporized by gently heating a bath of warm water surrounding
the bottle. Simultaneous cooling of the sample cell with liquid nitrogen forces the vapor
to travel through the connecting tubing and condense in the cell. This process continues
until the sample cell is ﬁlled to a level deemed appropriate by the experimenter, at which

point the vapor transfer is stopped by closing the FP valve located above the sample cell.

65

The ﬁnal step is to freeze the contents of the sample cell, pump off any excess gas
that may be present, and transport the cell to the MSU Chemistry Glassblowing
Laboratory for subsequent sealing using a natural gas torch.

IV. b. Nitrogen Atmosphere Loading.

When conditions do not dictate the controlled precision of FVD, a less complex
and much less time consuming method may be used to load the sample cell called
Nitrogen Atmosphere Loading. Using this technique the unopened bottle of sample
liquid, an independent transfer container, and sample cell are initially placed inside a
standard polyethylene glove bag (Aldrich AtmosBagm). The bag is then sealed with 2”
duct tape and purged several times with nitrogen gas so as to remove any water vapor that
may be present prior to initiating the ﬁlling process. The bag is then inﬂated with
nitrogen so as to obtain a comfortable working space, and the sample cell ﬁlled with
liquid by hand. The valve at the top of the sample cell is then closed, and the cell is
removed from the bag and attached to the portable pumping station used for F VD. The
contents of the sample cell are then frozen, and the remaining nitrogen gas pumped off.

Finally, the cell is transported to MSU Chemistry Glassblowing Laboratory for sealing.

66

REFERENCES

 

[1] J .D. Jackson, Classical Electrodynamics, John Wiley & Sons, New York,
1975, pp152-155.

[2] P. Debye, Polar Molecules, Chemical Catalog, New York, 1929.
[3] H. Frohlich, T heory of Dielectrics, Oxford University Press, 1949.

[4] GP. Smyth, Dielectric Behavior and Structure, McGraw-Hill Book
Company, Inc., New York, Chapter 1, 1955.

[5] L. Onsager, J. Am. Chem. Soc. 58, 1486 (1936).

[6] CRC Handbook of Chemistry and Physics, CRC Press Inc., Boca Raton, Florida,
1988, p. E-59.

[7] OP. Smyth, Dielectric Behavior and Structure, McGraw-Hill Book
Company, Inc., New York, 1955, pp. 28-34.

[8] J .D. Jackson, Classical Electrodynamics, John Wiley & Sons, New York,
1975,p.311.

[9] K. Adachi, H. Suga, and S. Seki, Bull. Chem. Soc. Jpn. 41, 1073 (1968).

[10] M. Honda, The Impedance Measurement Handbook, Section 2-3, Yokogawa-
Hewlett-Packard LTD, 1989.

67

Chapter 5

DIELECTRIC STUDIES OF CYCLO-OCTANOL

1. Motivation.

The dielectric studies of cyclo-octanol represent the ﬁrst effort of the Birge group
to construct a phenomenological approach toward understanding the glass transition.
By studying the glass transition in a material which is structurally less complex than a
fully-disordered supercooled liquid, we hoped to identify which aspects of the glass
transition are most "universal" and which depend on the detailed structural nature of a
material. The intrinsic translational periodicity and rotational disorder characteristic of
the rotator phase in orientationally-disordered crystals provides an ideal environment for
such studies.

The sections that follow present dielectric spectroscopy measurements performed
on cyclo-octanol covering ten decades in frequency. Due to the necessity for two
independent experimental techniques, the dielectric measurements naturally divide into
two overlapping studies: a low-frequency set which covers the range 1 mHz to 10 kHz,
and a high-frequency set which extends the measurements to 10 MHz.

11. Dielectric Measurement of Cyclo-Octanol.

II. a. Experimental Notes.

The sample material for the low and high frequency studies consisted of cyclo-
octanol (C3H150H) purchased from Aldrich Chemical Company, Milwaukee, WI. The
original purity of the stock cyclo-octanol was given as 99.5%, but additional efforts were

made to ensure a clean sample through vacuum distillation (see Chapter 4, section Na).

68

The distilled sample was then introduced to the coaxial capacitor cell and sealed under
vacuum using facilities at the MSU Chemistry Scientiﬁc Glassblowing laboratory. The

sealed sample cell was then attached to the cryostat and measurement electronics.

Liquid

    
 
    

S1

S3 - C . stallino

 

 

TQ=1 50K 220K 265K 288K

Figure 5.1. Schematic diagram illustrating the various phases of cyclo-octanol.

The phase diagram for cyclo-octanol has been previously studied [1,2] using
using dielectric measurements, and a schematic diagram is shown in Figure 5.1. On
cooling from the high temperature liquid, the sample will undergo a transition to the
orientationally-disordered S1 phase at 288 K. The S1 phase has f.c.c. structure with a
lattice constant of 9.56 Angstroms. Slow cooling in the 81 phase will result in transition
at 265 K to the 82 phase, which is also orientationally-disordered but has a dielectric
response which is negligible compared to that of S1. The transition at 265 K can be
bypassed through the use of moderate cooling rates of 5-10 K/min, resulting in the

supercooled S1 phase and eventually the orientational glass transition at T1; = 147 K.

69

Virtually all of the dielectric measurements presented in this chapter were obtained from
the supercooled S1 phase. It should also be noted that transition to the stable crystalline
phase S3 occurs below 220 K. Time-dependent transitions to lower energy
conﬁgurations can also occur in the region 220 — 265 K if cooling rates are not
sufﬁciently high.

11. b. Summary of Low-Frequency (lmHz - lOkHz) Measurements.

The initial dielectric experiments on cyclo-octanol were carried out by Leslie-
Pelecky and Birge [3] in 1991-1994 here at Michigan State University. The work
investigated the or-relaxation dynamics of cyclo-octanol in the low-frequency regime
over a temperature range of 166 - 205 K. This section will provide a short review of the
low-frequency results, as they are important in establishing the motivation for extension
of the dielectric studies to higher ﬁequencies.

Using an established cooling protocol discussed elsewhere [3], Leslie-Pelecky and
Birge measured the complex dielectric constant 8* = s' + is" of the supercooled S1 phase
over a frequency range 1 mHz - 10 kHz for temperatures 166 - 205 K. The measured
dielectric response is shown in Figure 5.2, where the solid lines passing through the data
are ﬁts to the KWW function described previously in Chapter 2. Other empirical ﬁtting
forms such as Cole-Davidson [4] were also tried, but neither provided better ﬁts to the
data. For frequencies approximately 2-3 decades and higher above the main peak in s",
empirical ﬁtting of the relaxation data with the KWW and others fail to provide an
accurate representation of the data [5]. However, the KWW can be used in the main peak

region to extract parameters such as the mean relaxation time (I ~ l/fpcak) and degree of

non-Debyeness (via the KWW stretching exponent B and eq. 2.10). Figures 5.3 (a) and

70

(b) show the temperature dependence of the width of the relaxation relative to a Debye
process (B=1) and the mean relaxation time I. These two graphs indicate that the
dynamical response of cyclo-octanol displays the same two nearly mriversal features
found in structural glass forming liquids: 1) Rapidly increasing relaxation time for
molecular motion as the temperature is decreased, and 2) increasingly non-Debye

relaxation behavior as the glass transition is approached.

 

 

 

It

  
 
  

 

 

, A A.

*'—.a—1—- - r—1———'
10" 10 10 10 ‘ 10° 101 10' 10 10‘ 10‘

V .

 

Frequency (Hz)

Figure 5.2. a) s' and b) s" for the supercooled S1 phase of cyclo-octanol as
measured by Leslie-Pelecky et al. in the low-frequency regime.

71

0.78 -
0.76 ~ ' ° .
0.74 -
0.72 . '
0.70 ~

068’

 

 

0.66 r I I A L r I 1 I
47 49 51 53 55 57 59

1 000/T

.3 .5
O O
M (a)
Y 'I I 'I
O

A
o
a.
v T

' (b)

1 (seconds)
8

A
Q
.5
1
O

J . I A 1 A I r l L A 1

4B 50 52 SA 55 5B 60
1000/T

 

 

Figure 5.3. Temperature dependence of parameters extracted from the KWW
ﬁtting of Figure 5.2 data. (a) "streching exponent", B, and (b) mean
relaxation time t .

72

The search for turiversalities in the relaxation behavior of cyclo-octanol continued
with scaling of the dielectric data [3] using an empirical form proposed by Dixon et al.
[5]. Despite the rather complex look of the scaling form, it has been successﬁrl in
collapsing the dielectric data for a wide variety of glass forming materials onto a single
"master" curve. The results of the scaling for cyclo-octanol can be seen in Figure 5.4,
and it appears that the data scale in accordance with that of structural glasses for all

regions except in the high frequency wing where small deviations appear.

cub

 

l
o i— OII .I-Om‘
A l \\
2 “1 x
\ '2 i .
1e -3 E \a,‘\
*0.
vo .4 I: ‘3
g ,5 - Cyclo-Octanol ’N‘
‘; J5 i, Glycerol "r...”
7i "-
i- l L l 1 l 4
_8 - . . . r .
-2 o 2 4 6 a

-1 -1
w (1+w )log1o(flfp)

Figure 5.4. Scaling of the dielectric data using a form given by Dixon et a1.
Cyclo-octanol begins to show slight deviations from structural glass
formers (e.g. glycerol) at high ﬁ'equencies.

73

It was originally thought that this discrepancy might be related to fact that the
measuring apparatus was optimized for low frequencies, and that data at the high end of
the measurement range might be less reliable.

The decision to extend the dielectric measurements was based on the need to
obtain a more detailed picture of the shape of the relaxation at higher frequencies.
Additional measurements in the higher frequency regime would conclusively determine if
the discrepancy observed on the scaling plot was a real feature of the data. This was
important to know because some people have assumed that the scaling form and its
resulting “master” curve is a universal feature of glass forming liquids, despite recent
arguments to the contrary [7]. Based on the validity of the scaling form, predictions have
been made of divergent behavior in the static dielectric constant at low temperature [8].
Thus, accurate knowledge of the high frequency behavior of a material is of great
importance.

II. c. High-Frequency (lkHz — 10 MHz) Measurements.

For a long time, dielectric studies of glass forming liquids focused their attention
on studying the low frequency (kHz region) dynamical response of the alpha relaxation.
The availability and ease of low frequency measurement techniques in conjunction with
the fact that one could learn a great deal from such studies meant that few experiments
targeted the high frequency response of a material. However, recent interest in the high-
frequency regime coupled with advances in dielectric spectroscopy have produced studies
[9] probing the dynamics of glass formers at frequencies well into the high GHz regime.
These experimental techniques have provided a wealth of new data regarding the high

frequency response of supercooled liquids near the glass transition. While our

74

 

experimental setup cannot achieve such extremely high frequencies, its 10 MHz range
still provides enough ﬂexibility to allow examination of the dynamics several decades
above the peak of the alpha relaxation.

The high frequency tail of the a-relaxation in supercooled liquids is known to
obey a shallow power-law that extends for many decades in frequency above the main
peak [5]. In an attempt to determine the form of the expected power-law behavior of
cyclo-octanol at high frequencies, the dielectric measurements were extended to 10 MHz
using the high frequency measurement circuitry described in Chapter 2, section III.d.

Figure 5.5 shows the results of the high frequency measurements combined with
the low frequency measurements of Leslie-Pelecky and Birge, and the data sets agree
well in the overlap frequency range. It can be clearly seen that the high frequency
response of cyclo-octanol is most deﬁnitely not like a power law, but rather the data
display a characteristic “shoulder” at all temperatures investigated. This feature is due to
an additional relaxation process, or B-relaxation, which overlaps the primary or-
relaxation.

Secondary relaxation processes such as the B-relaxation in Figure 5.5 typically
occur in the temperature region below T8 where the (it-relaxation is frozen out. The
origin of B-processes have been studied extensively by Johari [9] and Johari and
Goldstein [10], and quite often can be associated with a molecule’s ability to overcome
internal energy barriers and convert itself into different structural conﬁgurations. The
appearance of B-relaxations at temperatures above T8 is not as common, but several cases
have been documented in the literature. Examples of liquids displaying high temperature

B-relaxations include chlorobenzene-decalin and toluene-pyridine [9] (shown in Figure

75

5.6) and benzyl chloride-toluene [1 1]. With regards to the latter study, the secondary

relaxation process was explicitly identiﬁed as a rotation of the CH2C1 subgroup.

 

 

 

 

 

101 T 0 V A
. .0 . II I v I: :0 o
I O O A 9
. " 0 ' : v . O 0 170 K
V O I
. . V : : V' A . ' 175 K
O
- p . . 2 v 180 K
I A ‘ .
10° .- ' ' . v , - 185 K
' I o ' ° 0 190 K
' e
V . g . V ‘5 9.
:w V o . I - v ‘ 0..
o I V A
I O
V o. I ”V," [0,.
‘10-1 f . o I-III'WV;“A::°OO.
: O. . "" “A‘...
» V .... :‘I. "' A. .Q
o v
'o I. 'v
.0...-
0..
10'2 r
r ..-I r r .1. I r Irrrrrl r I .I I

 

 

104 10r2 10'1 10° 101 102 103 10‘ 105 10‘3 107
Frequency(Hz)

Figure 5.5. The imaginary part of dielectric response of cyclo-octanol covering a
frequency range of ten decades. The appears of the high temperature
B-relaxation is readily apparent at high ﬁequencies.

76

 

IO C y r I r I
E 1 - 43.9 mole % tolune - pyridine
5.0 2 - 17.2 mole % cholorbenzene - decalin,

A 111.14

I

  

I 1T1 YfI'

  
  
   

A l ILAAAI

 

 

 

 

£-
a 1

O.l _.
Mi 1
T= 133.7 K J
L Tg= 131 K .
r 4
0.02- 2 _

0.01 l l L L r r
-i o I z 3 4 s 6

Log f(Hz)

Figure 5.6. Some additional examples of high-temperature B relaxation processes
similar to those observed in cyclo-octanol. (Data from: G. P. Johari,
J. Chem. Phys. 58, 1776 (1973)).

Interest in identifying the source of the high temperature B—relaxation in cyclo-
octanol is not limited solely to the Birge group at MSU. Concurrent with our high
frequency measurements, Brand [12] produced an independent study of the fast dynamics
in cyclo-octanol covering an extremely wide frequency range (10'6 - 1012 Hz). As can be

seen in Figures 5.7 to 5.9, Brand found evidence for two secondary processes (labeled B

and 7) both exhibiting thermally activated dynamics (i.e. Arrhenius behavior in the

77

7"—

 

    

 

 

 

 

 

 

 

l I I
1.-2K _
0 — _

II
8
'1 _ -l
2 ”' ”0..
I I I -2.0 o l
I I I I I I I I
-6 -4 -2 0 2 4 6 8 10 12

'0910 [V(HZ)l

Figure 5.7. The dielectric loss (8") curves for cyclo-octanol as obtained by R. Brand.
The inset to the ﬁgure contains dielectric measurements of the stable
crystalline phase S3 at the four lowest temperatures shown on the main plot.
(Data from: R. P. Brand et al., Phys. Rev. B 56, 5713 (1997)).

78

 

 

 

 

 

log10 [v(Hz)]

Figure 5.8. The dielectric loss (8") at a temperature of 160 K as obtained by R. Brand.
In contrast to our measurements, Brand identiﬁes three separate processes

labeled as or, B, and 7. (Data from: R. P. Brand et al., Phys. Rev. B 56, 5713
(1997)).

79

 

I0910 [<r>(s)l

 

 

 

 

1 000 / T(K)

Figure 5.9. Plot of the mean relaxation times for the or, B, and y processes shown in
Figure 5.7. Similar to the results of Leslie-Pelecky et al. [3], the behavior of

the a-process was found to be non-Arrhenius. The B and y processes exhibit
thermally activated (Arrhenius) behavior. (Data from: R. P. Brand et al. ,

Phys. Rev. B 56, 5713 (1997)).

80

temperature-dependence of their relaxation times). Comparison of Figure 5.5 with Figure
5.7 shows that our labeled B-process appears to correspond well with Brand’s y-
relaxation. For example, comparison of our 175 K data to Brand's 176 K set shows an
inﬂection point corresponding to the ﬁrst noticeable contribution of the secondary
process at approximately 103 Hz. The B-process in Brand's study becomes increasingly
obscured by the large or-response at higher temperatures and is simply not visible in our
data.

Given our data for the supercooled S1 phase, it is very difﬁcult to quantitatively
determine the temperature dependence of the secondary relaxation processes. The
optimal situation would be to subtract the ct-response away leaving only the response of
secondary processes. The difﬁculty with this approach lies in the fact that none of the
empirical ﬁtting functions (Cole-Davidson, KWW, etc.) accurately describe the or-
relaxation at frequencies higher than a few decades above the main peak. Because of the
power-law nature of the (rt-relaxation at higher frequencies, the subtraction process using
empirical ﬁtting forms will not effectively remove the or-response at high frequencies.
The remaining response would not only contain the secondary processes, but also a
contribution from the a-relaxation as well.

Brand's determination of the temperature-dependence of the secondary processes
avoided the aforementioned problems by noting that both of the secondary relaxation
processes were also present in the stable crystalline (83) phase. The (it-relaxation is not
present in the ordered S3 phase, thus making it possible to accurately extract the

temperature dependence of the secondary relaxations using a Cole-Cole [13] ﬁtting

81

function. The solid line passing through the data of Figure 5.8 shows the result of adding
together the Cole-Davidson ﬁt of the main peak plus the Cole-Cole ﬁts of the secondary
processes B and y. The resulting line agrees remarkably well with the actual data except
in the region ~ 0.1 - 1 Hz, where as expected, the Cole-Davidson has underestimated the
a-response.

Speciﬁc identiﬁcation of the source of the B-relaxation (as labeled in our study) is
not clear-cut for cyclo-octanol. However, because the response is also observed in the
stable crystal, it is most likely associated with the intrinsic ﬂexibility of the 8-member
carbon ring. Based on structural considerations, it has been suggested [14] that the main
ring undergoes a continuous deformation resulting in multiple molecular conformations
contributing to the appearance of the B-relaxation. A detailed NMR study, such as that of
Kuhns et al. [15] for cyclohexanol, would no doubt prove useful in further study of the
molecular motion in cyclo-octanol.

111. Discussion of Results.

The primary success of the dielectric studies of cyclo-octanol was establishing the
fact that the orientational glass transition in ODC’s displayed the same characteristic
signatures of the structural glass transition in fully-disordered liquids: 1) an increasing
width of the relaxation as the temperature is lowered, and 2) faster-than-Arrhenius
behavior in the relaxation time as the temperature is lowered. However, extension of the
dielectric data to higher frequencies revealed the presence of a beta relaxation that
overlapped the tail of the primary alpha relaxation, preventing a more detailed study of its
shape. The origin of the beta relaxation is somewhat unclear, but is attributed to the

conformational ability of the 8-member carbon ring. Thus, it is clear that while cyclo-

82

octanol is a good model glass, the additional structural complexity of the molecule

suggests that it is not an ideal model glass.

83

REFERENCES

 

[1] M. Shablakh, L. Dissado, R. Hill, J. Chem. Soc. Faraday Trans. 79, 369 (1983).
[2] H. Forsman and O. Andresson, J. Non-Cryst. Solids 131-133, 1145 (1991).

[3] D.L. Leslie-Pelecky and NO. Birge, Phys. Rev. Lett. 72, 1232 (1994);
Phys. Rev. B 50, 13250 (1994).

[4] D.W. Davidson and RH. Cole, J. Chem. Phys. 18, 1417 (1951).

[5] P. K. Dixon, L. Wu., S. R. Nagel, B. D. Williams, and J. P. Carini, Phys. Rev. Lett.
65, 1108 (1990).

[7] It has been suggested that the log-log form of the scaling tends to suppress detail in
the dielectric data - thus small features of the data become "washed ou " after scaling.

[8] N. Menon and S. R. Nagel, Phys. Rev. Lett. 74, 1230 (1995).

[9] G. P. Johari, J. Chem. Phys. 58, 1766 (1973).

[10] GP. Johari and M. Goldstein, J. Chem. Phys. 53, 2372 (1971).

[11] L. Wu, Phys. Rev. B 43, 9906 (1991).

[12] R. Brand, P. Lunkenheimer, and A. Loidl, Phys. Rev. B 56, 5713-5716 (1997).
[13] K. S. Cole and R. H. Cole, J. Chem. Phys. 9, 341 (1941).

[14] Private communication with M.F. Thorpe, Michigan State University.

[15] P. L. Kuhns and M. S. Conradi, J. Chem. Phys. 80, 5851 (1984).

84

Chapter 6

DIELECTRIC STUDIES OF ETHANOL

1. Motivation.

The inherent complexity of cyclo-octanol and its overlapping relaxation processes
indicated that the search for an ideal model glass system should focus on structurally
simple materials. It has been known for two decades [1] that ethanol (C2H50H) exhibits
interesting phase polymorphism, and appears to be quite unique in that it can be made to
assume a variety of phases simply by varying the temperature in a controlled manner.

Figure 6.1 shows the various phases of ethanol that can be produced upon cooling from

the high temperature liquid.
structural . .
su ercooled li uid qulIld

rotator phase (RP)

- C i ’ ‘
orientational
glass (OG)

 

stable crystal

 

Tm = 159 K
T = 97 K
g (melting point)

Figure 6.1. The various phases of ethanol.

85

The fully-disordered liquid phase transforms into a structural glass (SG) upon
rapid cooling at a rate not less than 6 K/min. The orientationally-disordered crystalline
phase transforms from a rotator phase (RP) to an orientational glass (OG) upon cooling at
a moderate rate. Finally, the stable, orientationally-ordered crystal (monoclinic, lattice
constant 5.377 angstroms [2]), is produced either by very slow cooling of the liquid or by
annealing the RP or supercooled liquid. Recent X-ray, neutron diffraction and Raman
spectroscopic [2,3] studies have conﬁrmed the existence of an orientational glasslike
transition in the RP crystal at nearly the same temperature (T8 = 97 K) observed for the
supercooled liquid-to-structural glass transition. The close similarity of high-frequency
dynamical behavior in the orientational glass and structural glass phases has been
established by neutron time-of-ﬂight spectroscopy and low-temperature speciﬁc heat
measurements [4]. These properties, combined with the fact that the RP crystal can be
readily prepared from the structural glass suggest that ethanol provides a unique
opportunity to investigate and compare the relaxation dynamics of both phases near their
respective glass transitions. Such a comparison may help to clarify the role of
translational and rotational degrees of freedom in the dynamical response of a material
approaching the glass transition.

11. Dielectric Measurement of Ethanol.

II. a. Experimental Notes.

Sample material for the dielectric study consisted of 200 proof, dehydrated ethyl
alcohol (ethanol) as obtained from the Quantum Chemical Company of Tuscola, Illinois.
Transfer of ethanol from the sealed factory bottle to the capacitor cell was carried out

using the nitrogen atmosphere loading method described in Chapter 4, section IV.b.

86

Because the goal of the experiment was to measure the complex dielectric
constant (8 = s' + is") of the supercooled liquid and rotator crystal phases near their
respective glass transitions, a rigid experimental cooling protocol was established which
yielded reproducible results between runs. The protocol allowed production and
measurement of all phases seen in Fig. 6.1 within a single experimental cooldown. This
protocol can be summarized as follows. Rapid cooling of the liquid at a rate greater than
6 K/min bypasses crystallization at TM = 159 K [5], and allows entry into the
supercooled-liquid regime. Continuing to cool at the prescribed rate leads to the structural
glass transition at Tg = 97 K. The RP crystal (body-centered cubic, lattice constant a
=5.37 angstroms [2]) is then formed by warming and annealing in the temperature range
102 K to 110 K. The exact temperature and annealing time required for conversion to the
RP crystal depends on the warming rate, as well as the initial cooling rate into the glassy
state [1].

The dielectric response of the RP crystal and SCL phases were investigated over
frequency range of lmHz to 10 MHz, utilizing both the low-ﬁequency and high-
frequency measurement techniques described in Chapter 4, sections III.c and III.d.

II. b. Dielectric Response of the SCL and RP Crystal Phases.

Figures 6.2 (a) and 6.2 (b) show the real (8’) and imaginary (8”) parts of the
dielectric constant for the supercooled liquid as a function of frequency for temperatures
of 96, 98, and 100 K. Figures 6.3 (a) and 6.3 (b) show the corresponding data taken in
the RP crystal at temperatures of 96, 98, 100, 102, 104, 106, 108, and 110 K. Although
the glass transition temperature is usually quoted as Tg=97 K, it should be emphasized

that all the data were taken in the metastable equilibrium SCL and RP, respectively. The

87

data at 96 K, in particular, were observed not to change over more than an hour,
indicating that no structural relaxation was taking place. Obtaining data over a broad
temperature range in the SCL phase was hindered by the fact that time dependent
transitions to the RP crystal occur above 102 K. Thus, in order to broaden the range of
liquid phase data, additional measurements were made in the temperature range 164 K to
181 K. Figure 6.4 (a) and 6.4 (b) show the high-temperature liquid phase data, where the
contribution of dc conductivity to the response at low frequencies is readily apparent.

The solid lines passing through the data in Figures 6.2-6.4 are ﬁts to the Cole-
Davidson ﬁtting form [6] given by
(6.1) 8(0)) = e... + (e0 — e...)/(1 - im/6,CD)°‘
where a) = 21rf. It is well known that ﬁtting relaxation data with empirical forms such as
the Cole-Davidson or Kohlraush-Williams-Watts (KWW) function [7] does not
accurately describe the relaxation at all frequencies. In particular, at frequencies higher
than two decades or so above the peak, the data typically obey a shallow power-law
dependence extending to very high ﬁ'equency [8,9]. (The high frequency behavior and
detailed examination of the shape of the dielectric relaxation will be examined in the next
section). Nevertheless, the Cole-Davidson form can be used in the main peak region to
provide estimates of parameters characterizing the dynamical response such as the mean
relaxation time (t z l/prD), and 01, the degree of departure from a Debye-govemed
relaxation process (01 = 1). The Cole-Davidson ﬁt parameters as obtained for the RP

crystal and SCL are summarized in numerical form in tables 6.1 and 6.2, respectively.

88

 

11o . ......., . W, . -.....r . ...-.., . --. A.
,

'00 ' '

.

 

80‘. A 98K _‘

I 1 ‘
70? 00K j

60 7 1
50 f (a) j
40 f ' a
so ,~ 1
207 i
10 f -

 

 

 

40 ” ‘
20 ' ‘

10” ‘

 

 

l A L L A A A l

104 10-3

10'2 10'1 100 101
Frequency (Hz)

 

Figure 6.2. (a) s' and (b) 8" versus frequency for temperatures of 96, 98,
and 100 K in the supercooled liquid phase of ethanol. Solid
lines are ﬁts to the Cole-Davidson function.

89

 

100 1 ‘ 96 K

" ‘ - " ‘W 98 K
100 K ‘
102 K ‘
104 K ‘
106 K ‘
108 K -
110 K

 

ODDODCI.

 

 

 

o . _ ,.__—‘- _,

111 1 ”111111 1 Hirinl 1 1....111 . .11....1 1 Hum] 1 1L...”
103 10'2 10'1 10° 101 102 103
Frequency(Hz)

 

 

 

Figure 6.3. (a) s' and (b) 8” versus frequency for temperatures of 96, 98,
100,102,104,106,108, and 110 K in the RP crystal phase of
ethanol. Solid lines are ﬁts to the Cole-Davidson function.

90

 

 

 

 

 

 

 

60 IIII] I I IIIIIII I I IIIIIII I I ITTIII] I I IIIIIII I
(a)
\\
\\
\\
10 ’— ~‘
t.
1111] l llllllll 14L IIILLII I
I Tr] 7 I IIIIII' I I IfIHTII I
u
20 ‘\
- _.
-
00 15 0
10 " s‘
_ ‘\
5 _ “
\‘ s
_ . ‘2 :2“: ‘ “‘2‘
iUJl—I 11111111 I I lllllll I llIIILIl I lIlIIIII L

Frequency(Hz)

Figure 6.4. (a) s' and (b) 8" versus ﬁ'equency for temperatures of 164, 173, and
181 K in the high temperature liquid phase of ethanol. Solid lines are
ﬁts to the Cole-Davidson function plus a term of the form 4nooJ0) in
s” due to do conductivity.

91

 

 

 

 

 

 

 

 

T(K) 1 (80¢) a s... 80 - 800
96 424.1 0.710 3.60 78.1
98 58.6 0.744 3.77 82.0
100 12.1 0.760 3.86 85.2
102 3.0 0.796 3.85 85.1
104 0.95 0.811 4.21 92.9
106 0.35 0.821 4.34 91.1
108 0.13 0.822 4.44 85.5
110 0.05 0.815 4.53 72.2

 

 

 

 

 

 

 

Table 6.1. Cole-Davidson ﬁtting parameters for the RP crystal phase of ethanol in the
temperature range 96 — 110 K.

 

 

 

 

 

 

 

 

 

 

T(K) 1 (sec) 01 s... 80 - as a (sec")
96 347.2 0.776 3.22 80.2
98 21.76 0.790 3.60 91.0
100 2.27 0.819 3.88 94.0
164 6.05e—8 0.987 6.16 48.9 676
169 4.18e-8 0.986 6.07 46.9 931
173 3.04e-8 0.998 6.28 44.9 1210
177 2.29e-8 1.000 6.47 43.2 1520
181 1.90e-8 0.925 5.37 42.7 1886

 

 

 

 

 

 

Table 6.2. Cole-Davidson ﬁtting parameters for the SCL (96 — 100 K) and
high-temperature liquid (164 — 181 K).

92

 

 

Figure 6.5 shows the temperature dependence of the width parameter, or, for both
the SCL and RP crystal. Both phases display a characteristic feature common to many
glass-forming materials, namely a decrease of 01 with temperature indicating deviation

from idealized Debye behavior [10].

 

 

 

 

 

 

 

 

0.82 . A ° ’ °
0
0.80 - r . 1
0.78 -
d 0.76 ~ '
s 0 RP Crystal
0.74 ~ ° 4 SCL
0.72 . i
O
0.70 ‘ r ' ‘rr ‘ * ' r ‘ ‘ ‘
96 98 100 102 104 106 108 110
T(K)

Figure 6.5. Plot of the Cole-Davidson width exponent, 01, versus temperature for the SCL
and RP crystal phases of ethanol.

Figures 6.2 and 6.3 show that the rotational contribution to the static dielectric
constant, so — 8..., is similar in the two phases, indicating that the molecules undergo
complete rotation in the SCL as they do in the RP crystal. Figure 6.6 plots the static
dielectric constant versus temperature, and reveals a surprising feature of the data. In
both the SCL and RP, there is a marked decrease in so — 800 at low temperature which is

not typically seen in the response of polar liquids. This type of behavior can usually

explained by partial crystallization of the sample during cooling, which results in a

93

reduction of the total number of molecules capable of rotations. However, the observed
decrease in Figure 6.6 can not be due to partial crystallization of the sample during the
measurement because the data in Figure 6.2 were taken on warming from the SG, and the

data in Figure 6.3 were veriﬁed to be reproducible on warming from the 0G.

 

 

 

 

 

 

 

 

98 r
‘ O
O
90 ‘ ‘ .1
o 0 °
8
03
I 82 - O ..
O
O.)
O
0 RP Crystal
74 ~ A SCL 4
O
66 r I I x_ I r I r I I I r I g
94 96 98 100 102 104 106 108 110 112
T(K)

Figure 6.6. Plot of so — 8.0 versus temperature. The observed decrease in so-s0° at low
temperatures is not due to partial crystallization of the sample.

In particular, the only known crystalline phases of ethanol are the rotator phase
we are studying and the stable crystal, which has a very small dielectric response. Thus,

progressive transformation to the stable crystal is not consistent with the data.

94

Figure 6.7 shows the temperature dependence of the mean relaxation time in both
the RP crystal and SCL phases. The mean relaxation time can be determined by I ~
l/prD, where 00,,CD locates the peak of s". The solid lines are ﬁts to the Vogel-Tamman-
Fulcher equation [11] given by

(6.2) I = to exp(A/(T-To))

Here, to is an attempt time and To is the Vogel temperature at which the relaxation time

diverges. Results of the VTF ﬁtting for each phase are listed in table 6.3.

 

 

 

 

 

 

 

 

 

7-
1.
7.
‘1
'1.
10.7:- At '1
k
i .4‘
10‘8 r' 7
10a . I r J . I r I x I r
5.0 6.0 7.0 8.0 9.0 10.0 11.0

1000rr (K '1)

Figure 6.7. Temperature dependence of the mean relaxation time for the
liquid (above Tm) , supercooled liquid, and rotator crystal phases
of ethanol.

95

 

Parameter A To To

 

SCL 596 75.6 6.8e-11

 

 

RP crystal 529 73.5 2.6e-8

 

 

 

 

Table 6.3. VTF ﬁt parameters of the mean relaxation time in the RP crystal and SCL
phases of ethanol.

Examination of Figure 6.7 shows that the mean relaxation times in the two phases
are nearly equal at T=96 K, and display a temperature-dependent shift toward a longer
average relaxation time in the RP crystal phase relative to the SCL with increasing
temperature. This manifests itself experimentally as a pronounced shift of the RP crystal
relaxation peak toward lower frequency relative to the relaxation peak in the SCL at the
same temperature. From a practical point of view, this behavior provides a useful method
of determining if the sample has completely converted itself from the SCL to the RP
crystal: for a given temperature simply perform successive measurements of the dielectric
constant until the peak in s" ceases to move to lower frequency.

The behavior of the relaxation times can be explained by considering the degrees
of freedom available to the material in passing through the glass transition in each phase.
The 0G -) RP transition involves the liberation of purely rotational degrees of freedom,
while the SG '9 SCL transition involves liberation of both rotational and translational
degrees of freedom. As the temperature increases above 96 K, the additional
translational freedom present in the SCL provides the constituent molecules with a less
restrictive local environment, leading to increasingly shorter relaxation times in the SCL

relative to the RP crystal for a given temperature. Recent speciﬁc heat measurements

96

 

[1,12] performed on both the SCL and RP crystal near their respective glass transitions
support this argument: the 0G -) RP crystal transition involves an increase of about 22
Jmof'K' and the 8G9 SCL transition involves an increase in the speciﬁc heat of
approximately 31 Jmol'lK". Presumably the extra 9 Jmol"K’1 in the SCL speciﬁc heat
not present in the RP crystal correspond to translational degrees of freedom. These
results suggest that the rotational degrees of freedom are the dominant contributor to
structural relaxation processes near the glass transition in both the RP crystal and SCL,
and that ﬂow processes such as mass diffusion associated with translational degrees of
freedom in the SCL contribute to a lesser extent.

TH. Detailed Study of the Shape of the Alpha Relaxation in Ethanol

III. a. Motivation.

The dielectric measurements of ethanol indicate that the material does not exhibit
overlapping relaxation processes such as those observed in the behavior of cyclo-octanol.
The (it-relaxation appears to be "clean" out to the highest measurement frequencies of our
experiment, and suggests that a detailed examination of the entire shape of the or-
relaxation is possible. As was alluded to brieﬂy near the end of Chapter 5, section II.b,
the high-frequency behavior of supercooled liquids is known to obey a shallow power
law. Based upon extrapolation and analysis of the "master" curve obtained from the
Dixon scaling form, Menon et al. [13] have proposed that the high frequency power law

behavior of a material will be described by
(6.3) s"(f) oc f‘° , 0' = 0.72 (1+ w-‘)—1
where w is the full width at half max (W) of the relaxation in decades normalized by the

width of a Debye relaxation process (WD = 1.14 decades):

97

31’. )1".
wD 1.1

(6.4) w =

The Dixon scaling form and Eq. 6.3 suggest that speciﬁc power law behavior will
result for a given relaxation width, and we would like to determine if this holds true for
the case of ethanol.

III. b. Determination of High-Frequency Power Law Behavior.

To extract the power law characterizing material behavior at high frequency, the
data must be ﬁt in an appropriate manner. The Cole-Davidson form, which was used to
successfully in section II.b to characterize the low-frequency behavior in the peak region
of the a-relaxation, cannot adequately represent the data at high frequencies. However,
after publication [14] of our low-frequency results, we constructed an improved ﬁtting
ﬁrnction which can ﬁt our full range of experimental data and appears to be quite superior
to ﬁtting with the single Cole-Davidson. The ﬁtting form utilizes two Cole-Davidson
functions having the same mean relaxation time but differing in their amplitudes and
widths. The Cole-Davidson with the larger amplitude and smaller width ﬁts the peak
region, while the broader, lower amplitude CD is used to extract information about the

high-frequency power law a. The mathematical form of the ﬁtting function is then taken

to be:

+ As1 + As2
°° (l—icor)" (l—iarr)° ’

 

(6.5) 8((0) = s 0) = 271f

The assignment of uncertainties to the data is difﬁcult for a dielectric experiment
because most of the uncertainty is due to systematic rather than statistical error. For the
ﬁts, we have assumed that the uncertainty for each data point consists of a constant and

proportional term. The constant term (typically 0.005) reﬂects the base accuracy and

98

noise level of the instruments. The proportional term (~.02) accounts for the possibility of
relative errors on different gain settings as well as the fact that no empirical model can
accurately represent the data over the entire frequency range. With regards to the latter,
ﬁtting without the proportional term will over-weight the measurements in the peak
region of s" (where the dielectric response is the largest) at the expense of the data in the

tail region where the response is much smaller.

The results of the double CD ﬁtting for the RP crystal and SCL can be seen
numerically (tables 6.4 and 6.5) and graphically (ﬁgures 6.8 through 6.11) on the pages
that follow. In each of the ﬁgures, s" has been plotted on a logarithmic scale to
emphasize the ﬁt to the data in the high frequency power law regime where 3" << s"peak.

Fits using the single CD form have also been included to illustrate the improvement

provided by the double CD form.

99

 

T(K)

A81

A82

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

01 s... t o
98 74.3 .835 3.42 50.70 6.3 0.336
100 78.5 .853 3.48 10.27 6.7 0.346
102 78.3 .880 3.43 2.63 7.2 0.364
104 85.8 .889 3.60 0.85 7.6 0.359
106 82.7 .910 3.63 0.32 9.1 0.386
Table 6.4. Parameters obtained in the double CD ﬁtting of the RP crystal phase of
ethanol.
T(K) A81 01 8.0 t 482 o
98 81.1 0.879 3.31 20.34 9.3 0.419
100 79.9 0.932 3.43 2.21 15.5 0.472

 

 

 

 

 

 

 

Table 6.5. Parameters obtained in the double CD ﬁtting of the SCL phase of ethanol.

100

 

 

CD and 200 Fitting of 98K RP Crysnl Dielectric Data

 

 

 

 

 

 

 

 

 

 

 

 

 

 

100 j i 102
90 - O epsreal i
1'L --------- epsreaIZCd J
80 .
K‘s I epsrm 1 10‘
70 _ ‘1. 5‘" ------- epsim2cd ‘
60 _ 9 ‘I‘ ——— epsumcd
‘2‘ .......... epsrealcd _, 100
50 ~ q I?- . -
’ ’- \'\.‘I' .0)
h ‘- \
40 ’ ‘-\ \\.-"n.' . 104
_ \ 3
30 . ‘. \\ m..-
20 t .' \\ 5". l
\.‘. \\ a...‘ 4
10 ~ .., x ~~~~ 1 10"
“unnumuuwwn... ........ --.....'.:.i
O \\
\
-10 . -.. ..I . - .I . -- ..I A.-.-...I A ... -.I . -..I . ......s.l - -----.I . ---.-.. 10.3
10-3 10'2 10" 10° 10‘ 1o2 103 10‘ 105 10'5
Frequency (l-Iz)
CD and ZCD Fitting 0f100 K RP Crystal Dielectric Data
100 . 1 102
90 PITA - epsreal ‘
80 1. I '2‘ ......... epsreancd q 1
E" ‘4, I epsrm 10
70 v__ ------- epsim2cd l
60 . E. —-- epsimcd 1
. ’7‘ .-..._.... epsrealcd 1 100
50 r 8 \\'\l 1 =0)
, .
40 t r. \\\"u.
30 L t,- \ ‘5... 1 10-1
‘ \\\ lu‘l. l
20 ' ‘ \\ '1.“
. \ ‘ .2
10 * \ \\ l 10
» Mauuuuuumuévuuoowwmwwi
0 ’ \\ 1
\
-10 . . I - ...1 I ._.- -. I . .I .--...I - - ...I A --...I - - --- 104
104’ 10'2 10'1 10° 101 102 103 10" 105 106
Frequency (Hz)

Single CD ﬁt is included for reference.

101

Figure 6.8. Double CD ﬁtting of 98 K (top) and 100 K (bottom) RP crystal ethanol.

Figure 6.9.

CD and 200 Fitting of102 K RP Crystal Dielectric Data

 

 

 

 

 

 

 

 

 

 

 

 

 

 

100 r ‘1 102
1
go LI-"Wﬂ . epsreal 1
80 _ .3 --------- epsrealch . 1
' I] i I epsrm 3 10
70 i / 9. ------- epsim2cd
l f .
60 r 1 ‘ -—— epsimcd ‘
1 2 Fix. ---------- epsrealcd 1 10°
50 1' ii. \\I l =
I, \IL‘I. 00
40 i I \\ .II'
i '9. \ “I. ‘3 10'1
30 F 2‘ \\ 5'... ‘
‘0 \\ .
2° " "e \ "' .
2.“. \\ ._ 10.2
10 ' \ ;
Muueeeeeeeeeeeeuoaqeooeeo-vm- aaaaa A
0 - \ ‘
. \
-10 . -r4.m..1 ._F. ..I - . l ......I . A......I ....L.I . r....l . .... .I . ....r 10.3
10'3 10'2 10'1 10° 10‘ 102 103 104 105 10‘5
Frequency (Hz)
CD and 2CD Fitting of104 K Rotator Crystal Dielectric Data
100 L-.-......- i 102
. 06.“ 1
90 I ' - epsreal ;
80 r i, --------- epsreal2cd 1 10‘
' l E ' i
70 - I '2 K ' epslm I
r / g ------- epsum2cd j
60 - {I ‘i R ——— epsimcd - 0°
'4‘ 5. \'-. ---------- epsrealcd 1
50 r g \H,‘ ‘ =
’ i. \ 00
40 r '9 \\ '5,
l \ '51., ‘ 10-1
30 ” ‘9 \ I‘. f
3‘ \\ I“
20 7 '1 \
‘0 \\ - -2
10 - ‘4... \ 3 1°
* ”“u””””>~m’ﬁwﬁ‘“"ﬂ
0 - \ ‘
'10 . . r A .4 - --.-.r . ...-..-r . .....r . .-. ..r . ..r . . .....r \ ... 10.3
103 10'2 10'1 10° 101 1o2 103 104 105 106
Frequency (l-lz)
Double CD ﬁtting of 102 K (top) and 104 K (bottom) RP crystal data. Fits to

the single CD form are included for reference.

102

CD and 200 Fitting of106 K RP Crystal Dialectic Data

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

100 r 1
,_-._...-.A..“... 102
9° ‘ “5.x 0 epsreal
80 P 9 ......... epsrea'ch
‘ - 1 101
» I epsrm ;
70 r ------- epsim2cd ‘
60 - I : ——— epsrmcd
/ E \\ ‘ .......... epsrea|cd ‘1 10°
_ 50 " f, b \ 'l ‘ =
w I ‘7. \\“.- l w
40 1’ l, \ ‘l‘ ‘
' 1 \ ‘l. -1
' \ I i 10
30 h I \\ “I l
T \ ‘1
20 r ’0. \\ ‘
. a.“ \\ 1 104,
10 P ’9. \ 3
Weemnwr 1': n w n‘
0 P \ ‘
1 \
.1 . .r..-..1 ..- “I . .......l . -......I . -......l . --r....l . .-.....l . .\....

-10 ‘ ‘ “ 10-3
10-3 10-2 10'1 10° 101 102 103 10‘ 105 10"
Frequency (Hz)

CD and 20D Fitting of 98 K SCL Dielectric Data
100 - 7 102

l
90
i 0 epsreal 4
80 ‘ ......... epsrealzcd 1 101
70 I epstm ‘
» ------- epsrm2cd ,
5° ——— epsimcd 1 10°
_ 50 --~-~~~ epsrealcd 1 =
00 00
4o
1 1 '1
30 , 0
'I. J
20 “'4“
10 - K \ 1 10°2
0.. \
' “ummuneh‘eee ------------- -::Wf'"‘ﬂl
0 r \ ‘
\\ ‘
-10 . ... I ... I . .r....l . .......I . ......l . .4.....I 4......x . .r-...l K-A..-. 104
10'3 10'2 10'1 10° 101 102 10:1 10‘ 105 106

Frequency (Hz)

Figure 6.10. Double CD ﬁtting of 106 K RP (top) and 98 K SCL (bottom). Fits to the
single CD form are included for reference.

103

 

CD and 20D Fitting of100 K SCL Dielectric Data

 

 

 

 

 

 

100 FL'.'.:'.'-.:;.‘. - 102
. .t 1
90 L ', 0 epsreal 1
_ ‘ l
80 b , x ......... epsrealzcd 10‘
> 2 \_ I epsnm l
70 f / 1‘ \ ------- epsim2cd l
60 7/ 3 ﬂh ——— epslmcd . o
g {5" -------~- epsrealcd : 10
.. 5° " i \\ . I =
w ‘o I w
40 ‘ '3 \\ I‘I‘
E; \\ "ln. ‘ 10'1
30 ’- 3‘ \\ .‘ 4
20 _ ‘l \\ ~.
'3. \\
l 1 -2
10 _ ﬁg \\ N. l 10
'n‘Ooomooooooooooébqwm-nwnwwmm73a
o - \ .
\\
-10 A+h4l “1 .4 A r rul A r. ““1 r i.....l i i.... ‘1 .M1 J_A.AU.. ...r r 103
10-3 1o-2 10'1 1o° 101 102 1o3 10‘ 1o5 106
Frequency (Hz)

Figure 6.11. Double CD ﬁtting of 100 K SCL data. Fits to the single CD form are
included for reference.

The measured values of the high frequency power law were then be compared
with the expectations of eq. 6.3 for both the RP crystal and SCL data. To obtain the
expected values of a from scaling, the normalized width (w) of the relaxation at each
temperature had to be determined. To do this, the full width at half maximum (W) of the
relaxation for a given temperature was determined by hand using the double CD ﬁts to
the a" data, and then normalized by the width of a Debye relaxation (1.14 decades) to
give the relative width w. The measured relative widths w, along with the expected and
measured values of the high frequency power law 0' are summarized in tables 6.6 and 6.7

for the RP crystal and SCL, respectively. We observe that the the measured power laws

104

for the RP crystal are in fairly good agreement with the expectations from scaling,

however the power law for the SCL appears to be slightly steeper than expected.

 

 

 

 

 

 

 

T(K) W(decades) W 0' measured Cscaling
98 1.262 1.107 0.336 0.371
100 1.248 1.095 0.346 0.377
102 1.233 1.082 0.364 0.3 86
104 1.227 1.076 0.359 0.389
106 1.219 1.069 0.3 86 0.393

 

 

 

 

 

Table 6.6. Measured relaxation widths and comparison of scaling-predicted and
actual high-frequency power laws for the RP crystal phase of ethanol.

 

 

 

 

T(K) W(decades) W O'measured oscaling
98 1.238 1.086 0.419 0.383
100 1.223 1.073 0.472 0.391

 

 

 

 

 

Table 6.7. Measured relaxation widths and comparison of scaling-inferred and

105

actual high-frequency power laws for the SCL phase of ethanol.

 

 

111. c. Dixon Scaling of Ethanol Dielectric Data.

The a-relaxation data for both the RP crystal and SCL phases of ethanol were
scaled using Dixon's formalism, which was introduced earlier in Chapter 5, section II.b.
The results of the scaling for the RP crystal and SCL phases are shown in Figure 6.12.
As can be seen from the ﬁgure, the RP crystal data appear to scale in accordance with
that of true structural glasses (in order to avoid clutter on the scaling plot we have
represented the so-called "master curve" using data from glycerol) while the SCL data
show deviations at the higher frequencies. This behavior is consistent with the results of
the power law comparsion carried out in the previous section, where it was shown that
the measured high-frequency power law for the RP crystal was in agreement with the
prediction of eq. 6.3. The steeper slope of the SCL power law manifests itself as the
observed deviation on the scaling plot.

It is not clear why the RP crystal scales in accordance with structural glasses
while the SCL appears to show deviations. In fact, this result is perhaps opposite to what
might have been expected. However, because of the limited amount of SCL data
available (due to the temperature range over which the phase is stable for measurement),
we caution against a hasty dismissal of the scaling form's implied universality in the

dielectric response of glass forming liquids.

106

   

 

 

 

 

4— RPCwam
Glycerol
a: SCL
21 -3r
Zn 1
3°,
2 '5’
O)
o
:' ,7. "v-
3 x4“
15",
-9- ."'.

 

l.l.141il.1114111.l.1.lilrlllal.l
-11

4-3-2-1012345678910111213
w '1 (1+w ") Log10 (f/fp)

 

Figure 6.12. Dixon scaling of the RP crystal and SCL ethanol dielectric data.
The RP crystal scales in accordance with structural glasses
(represented here by glycerol) while the SCL shows systematic
deviations at high ﬁ'equencies.

107

REFERENCES

 

[1] O. Haida, H. Suga, and S. Seki, J. Chem. Therrnodyn. 9, 1133 (1977).

[2] A. Srinivasan, F.J. Bermejo, A. de Andres, J. Dawidowski, J. Zuniga, and A. Criado,
Phys. Rev. B 53, 8172 (1996).

[3] R. Fayos, F .J. Bermejo, J. Dawidowski, H.E. Fischer, and M.A. Gonzalez, Phys. Rev.
Lett. 77, 3823 (1996).

[4] M.A. Ramos, S. Vieira, F.J. Bermejo, J. Dawidowski, H.E. Fischer, H. Schober,
M.A. Gonzalez, C.K. Loong, and D.L. Price, Phys. Rev. Lett. 78, 82 (1997).

[5] A typographical error in ref. 1 giving Tm as 169 K has been propagated in some
papers, e.g. refs. 6 and 7.

[6] D.W. Davidson and RH. Cole, J. Chem. Phys. 18, 1417 (1951).
[7] G. Williams and D. C. Watts, Trans. Faraday Soc. 66, 80 (1970).

[8] PK. Dixon, L. Wu, S.R. Nagel, B.D. Williams, and J .P. Carini, Phys. Rev. Lett. 65,
1108 (1990).

[9] R. Brand, P. Lunkenheimer, and A. Loidl, Phys. Rev. B 56, 5713-5716 (1997).

[10] MD. Ediger, C.A. Angell, and SR. Nagel, J. Phys. Chem. 100, 13200-13212
(1996)

[1 l] H. Vogel, Phys. Z. 22, 645 (1961); GS. Fulcher, J. Am. Ceram. Soc. 8, 339 (1925);
G. Tamman and W. Hesse, Z. Anorg. Chem. 156, 245 (1926).

[12] F.J. Bermejo, A. Criado, R. Fayos, R. Femandez-Perea, H. E. Fischer, E. Suard, A.
Guelylah and J. Zuniga, Phys. Rev. B 56, 11536 (1997); F.J. Bermejo, H.E. Fischer,
M.A. Ramos, A. de Andres, J. Davidowski, and R. Fayos in Complex Behaviour of
Glassy Systems, M. Rubi et al.. (Eds) Springer Lecture Notes in Physics, Springer
Verlag, Berlin 1997, p. 44.

[13] N. Menon and S. R. Nagel, Phys. Rev. Lett. 74, 1230 (1995).

[14] M. A. Miller, M. J. Ruiz, F. J. Bermejo, and N. O. Birge, (accepted for publication in
Phys. Rev. B 57, June 1, (1998)).

108

Chapter 7

NONLINEAR DIELECTRIC MEASUREMENT

1. Motivation and Background.

As described in Chapter 2, the relaxation behavior of many supercooled liquids is
described quite accurately by the VTF form. The apparent divergence of the VTF
equation (eq. 2.2) at a temperature To has prompted a lengthy debate as to whether or not
the glass transition is due to a thermodynamic phase transition. To date, no simulation
[1] or experimental [2,3] study has observed evidence for such a transition, suggesting
that the glass transition is simply the result of kinetic slowing of the liquid. This in fact
may be true, however the possibility exists that the reason experimental studies have
failed to observe a phase transition is because they are simply unable to get close enough
to To.

The idea to measure the nonlinear dielectric susceptibility in glass forming
materials has its root in the ﬁeld of spin glass systems [4], which are random spin
systems with competing ferromagnetic and antiferromagnetic interactions. Prominent
examples of spin glasses include AgMn and CuMn. In these systems, the random
placement of the ions provides disorder, and the presence of the RKKY [5] interaction
gives rise to both ferromagnetic and antiferromagnetic interactions. Because of the slow
dynamics near the spin glass transition, research on spin glasses was plagued by many of
the same problems faced by studies of the liquid-glass transition. However, a major

breakthrough occurred when it was shown that the nonlinear magnetic susceptibility of

109

AgMn increased rapidly near the transition [6,7]. The equation of state for the system

can be expanded in terms of the magnetic ﬁeld:
(7.1) %=xo—me’+O<H‘)
Here, M is the magnetization, H is the magnetic ﬁeld, x0 is the linear magnetic

susceptibility and you is the nonlinear magnetic susceptibility. The nonlinear

susceptibility, which is related to the correlation length in a random system, is observed

to diverge at a critical temperature Tf:

_ T-T, "
(7-2) xm-[ T, ]

 

Because of its sensitivity to a growing correlation length in the material, a diverging
nonlinear susceptibility is a characteristic signature of a second-order phase transition.
We are interested in carrying out a similar experiment for glass forming materials
near the glass transition. The idea is to measure the nonlinear dielectric constant for a
properly chosen material to determine if a similar divergence can be observed. For

dielectrics, the nonlinear dielectric constant is deﬁned by
(7.2) % = e' —eNLE2 + 0(E‘)
where D is the electric displacement, E is the electric ﬁeld, a. is the linear dielectric

constant and eNL is the nonlinear dielectric constant. If a diverging nonlinear dielectric

constant is observed, it may provide a conclusive answer to the elusive phase transition
question. The absence of such a divergence in a carefully chosen material would also be
meaningful, as it would raise fundamental questions regarding the ability of experiments

to ever observe behavior consistent with a phase transition near To.

110

II. Nonlinear Measurement Circuit.

Attempting a measurement of this sort is difﬁcult because the nonlinear dielectric
constant is much smaller in magnitude than the linear dielectric constant. By applying a
large sinusoidal electric ﬁeld of frequency f to a capacitor containing the sample, one can

measure the response of the system at f as well as higher harmonics 3f, 5f, etc. The
lowest order nonlinear signal occurs at 3f, and corresponds to em. However, because the

nonlinear signal is much smaller in magnitude than the linear signal, a measurement
scheme must be devised which can subtract the large linear response from the signal.
The circuit we plan to use to measure the nonlinear dielectric constant is shown in Figure

7.1.

 

 

 

H ADC

 

 

Reference

 

Figure 7 . 1. Circuit for nonlinear measurement.

lll

Subtraction of the linear dielectric response is accomplished by a driving a
reference capacitor nearly 180 degrees out of phase with respect to the capacitor
containing the sample. Both the sample and reference capacitor are driven by a 2-
channel IOTech digital-analog converter (DAC) which can perform the required phase
shiﬁing digitally. The output signal from the current-to-voltage ampliﬁer is then read by
a IOTech analog-to-digital converter (ADC) that is synchronized to the DAC.
Information about the response of the system at the driving frequency and higher
harmonics is obtained by fast-Fourier transformation (F FT) of the data using a personal
computer.

111. Present Status of Experimental Work.

To minimize the noise level present in the measurement, synchronization of the
DAC output and ADC sampling is crucial. To accomplish this, we trigger the ADC
sampling using an appropriate division of the DAC's internal clock signal. To set the
ADC sampling rate, we divide the DAC's internal 100 kHz clock signal using a
programmable divide-by-N counter (Harris model CD4059AE, 24 pin DIP) and supply
the resulting signal to the external trigger of the ACD. To verify that this method
produces a low-background environment, we have performed extensive stimulus-
response testing of the DAC and ADC. In general the synchronization between the
instruments appears good, but we have encountered additional sidebands in the frequency
response which are of undetermined origin. The sidebands appear unresponsive to
ﬁltering, and must be eradicated before the measurement can be proceed further. Once
the desired low-noise background is achieved, we are conﬁdent that the measurement can

be performed.

112

REFERENCES

 

[1] R. M. Ernst, S. R. Nagel, and G. S. Grest, Phys. Rev. B 43, 8070 (1991).
[2] L. Wu, Phys. Rev. B 43, 9906 (1991).

[3] P. K. Dixon, S. R. Nagel, and D. Weitz, J. Chem. Phys. 94, 6924 (1991).
[4] K. Binder and A. P. Young, Rev. Mod. Phys. 58, 801 (1986).
[5] M. A. Ruderman and C. Kittel, Phys. Rev. 96, 99 (1954).

[6] P. Monod and H. Bouchiat, J. Phys. (Paris) Lett. 43, L45 (1986).

[7] L. P. Levy, Phys. Rev. B 33, 4963 (1988).

113

Chapter 8

SUMMARY AND FUTURE DIRECTIONS

In this work we have used dielectric spectroscopy to study the glass transition in
the orientationally-disordered crystals cyclo-octanol and ethanol. In general, we ﬁnd that
the orientational glass transition in each material displays the same two characteristic
features of the structural glass transition in fully-disordered liquids: 1) rapidly increasing
relaxation times as the temperature is lowered toward the glass transition temperature Tg ,
and 2) increasingly non-Debye relaxation behavior with decreasing temperature. The
appearance of these features in a system lacking translational freedom suggests that these
materials may be of great interest as model glass systems for theoretical and numerical
simulation studies of the glass transition.

However, in addition to displaying the general characteristics of glassy behavior,
an ideal model glass system must display structural simplicity. The behavior of a
material is critically linked to its structural properties, and a complete dynamical picture
can only be achieved by examining the dielectric properties over as wide a frequency
range as possible. More speciﬁcally, the extension of the cyclo-octanol dielectric
measurements to higher frequency revealed the appearance of a complicating B-
relaxation process that overlapped the primary or-relaxation at temperatures above T3.
Because the B-relaxation was also observed in the stable crystal as well, it is believed to

be associated with the intrinsic ﬂexibility of the cyclo-octanol molecule. Thus, while

114

cyclo-octanol satisﬁes the basic requirements for a model glass, its structural complexity
and overlapping relaxation processes suggest that it is not an ideal model glass.

The subsequent study of ethanol provided important insights into the nature of the
glass transition which could not be gained from cyclo-octanol. First, the intrinsic
polymorphic character of ethanol allowed for a quantitative comparison of the glass
transition dynamics in the supercooled liquid and rotator crystal phases. The results of
the comparison indicate that rotational motion is the dominant dynamical process
governing the glass transition in the SCL phase, and that diffusive processes contribute to
a lesser extent. Such a result may imply that theories of the glass transition based solely
on the diffusive properties of liquids are too general in form, and tend to exclude
important contributions from rotational processes.

The structural simplicity of ethanol also provided an opportunity to examine the
shape of the (rt-relaxation without the additional complexity of superimposed relaxation
processes (as was the case with cyclo-octanol). Dixon scaling of the RP crystal data
showed good agreement with the "master curve" characterizing a wide variety of
materials. Equivalently, the power law nature of the relaxation at high frequencies was
shown to agree well with the scaling-based predictions of Menon et al. The SCL phase
appeared to show deviations from the structural glass scaling in the high frequencies, and
also displayed steeper power law behavior than the RP crystal.

Based on our results, we conclude that the RP crystal phase of ethanol appears to
be the best example of a physical model glass system discovered so far. Thus, the
construction of a minimal theoretical model of the glass transition utilizing restricted

degrees of freedom appears relevant to the behavior of more complex structural glasses.

115

A model such as that explored by C. Renner et al. [1] should be simple enough to allow a
detailed numerical and analytical treatment.

Finally, we have constructed the framework required for the difficult nonlinear
measurement. After the resolution of certain experimental obstacles, performing this
measurement on an extremely fragile liquid (e.g. propylene carbonate, etc.) may resolve
the question of whether or not a thermodynamic phase transition underlies the glass

transition in supercooled liquids.

116

REFERENCES

 

[1] C. Renner, H. Lowen, and J.L. Barrat, Phys. Rev. E 52, 5091(1995).

117

Appendix A

DATA CORRECTION METHODS

I. Introduction.

Introductory physics textbooks oﬁen leave one with the impression that obtaining
accurate measurements of a material's dielectric constant is a simple matter of dividing
the measured capacitance with the dielectric in place by that of some appropriate empty
cell capacitance. Unfortunately, real-world experiments are complicated by a number of
factors including the parasitic effects of lead capacitance and inductance, as well as
phase-shifts due to extended cabling between the sample and measuring instrumentation.

These effects become particularly important at high frequencies ( > 1 MHz), but
also affect the integrity of measurements made at low frequencies as well. Thus,
corrections must be applied in order to obtain accurate results in any dielectric
spectroscopy experiment. Considerable effort has been put forth in determining a
practical scheme for data correction as well as attempting to understand the nature of data
corrections from a theoretical perspective. Section II of this appendix will describe the
actual empirical correction scheme used for all dielectric data presented in this thesis.
Section III of this appendix will outline in detail our theoretical attempt to derive high-
frequency data corrections for the Hewlett-Packard HP4192A Impedance Analyzer.
These calculations are based on transmission line theory and reasonable assumptions

regarding the analyzer's measurement circuitry. Unfortunately, despite a seemingly

118

satisfactory analysis, the resulting theoretical correction scheme was unable to account
for the observed behavior in the experiment.
11. Empirical Dielectric Data Correction.

II. a. Correction of Low-frequency Data.

It was initially thought that empty cell measurements alone would sufﬁce for the
correction of parasitic elements in the analysis of the actual dielectric data. In general,
the empty cell dielectric measurement can be described by a frequency-dependent

magnitude A(f) and phase ¢(f), where these quantities are deﬁned as:

y 8

(AA) W) = emu—J = uni-8;]

W + vf
(A.2) A(f) = ——
(”Col-“lo

As deﬁned in Chapter 4, [3 is the gain of the current ampliﬁer, Co is the capacitance of the
empty cell, and V0 is the voltage of the driving oscillator.

For an ideal capacitor with no dielectric in place, measurement at any frequency should
yield zero for Vx and a constant for V,. Thus, ¢(f) = 0 and A(f) = 1 (see eqs. 4.36 and
4.37). However, for the capacitor used in our experiments, the observed empty cell
measurements showed a tendency toward negative phase values and increasing
amplitudes at the high end of the measurement range. Theoretically, with knowledge of
the empty cell phase and magnitude in hand, correction of data taken with the dielectric
in place can then be accomplished by ﬁrst performing a rotation through angle (I) which

corrects for phase errors, and subsequently correcting magnitude using A(f):

119

(A3 V; _ c084) —sin¢ V,
') V; _ sin¢ cos¢ Vy

 

._ v;
(A.4) v, -—A(f)
(A 5) V"- V;

' ”Am

After the data has been corrected for phase and magnitude, 8' and s" can be determined

from eq. 4.36 and 4.37:

 

V.
(A6) 8' = ’
B‘DVoCo

(A7) 8" = B—mgli—C;

The problem we encountered with correction of the low-frequency data was the
following: Because the high-frequency experiment was performed ﬁrst, only high-
frequency empty cell data was taken before the sample cell was ﬁlled and permanently
sealed. When the experiment was converted to the low-frequency measurement scheme,
we had no opportunity to perform low-frequency measurements on the empty cell. Thus,

we had to search for an alternative method of constructing an appropriate "empty cell"
data set. To do this, we needed to ﬁnd a structural phase of the material where e' was
essentially frequency independent and s" was zero. Fortunately, at low temperature the
stable crystalline phase provided these characteristics. In the end, the measurements used
to correct the rotator phase and supercooled liquid data at a given temperature were
actually a normalized version of the crystalline dielectric data at a temperature where the

sample was known to have no loss. For example, our low-frequency ethanol

120

measurements were corrected using normalized 90 K crystal data. The only modiﬁcation
to the data correction equations shown above is the inclusion of an appropriate
normalization factor in A(t). This method proved to be adequate for data correction, and
as will be described in the next section, was used for high-frequency data correction as
well.

11. b. Correction of High-Frequency Data.

The correction of high-frequency data obtained from the HP4192A Impedance
Analyzer followed the same procedure given in section II.a. for the low-frequency data,
however the correction equations are expressed in different variables. Thus, for
completeness and future reference the equations will be outlined in this section.

For the empty sample cell, the impedance analyzer measures the capacitance C
and conductance G of the sample. The admittance Y of the empty cell is then:

(A.8) Y = G —i(oC
The magnitude and phase are then expressed as

G2 + (02C2
(10C0

 

(A.9) A(t) =|Y| =
(A.10) ¢(f) = tan"[£)
coC
Similar to the section II.a, the data correction process involves both phase correction and
magnitude correction using an appropriate set of empty cell data. Unfortunately, the
actual empty cell data available was unable to correct the measurements. We believe that
the reason for the failure is not due to the correction method described above, but rather
because the necessary correction depends upon the impedance of the sample, which of

course is much different for a ﬁlled capacitor and varies with applied ﬁeld frequency and

121

temperature. Thus, we again turned to crystalline phase in order to construct the "empty
cell" set. For example, correction of the high frequency ethanol data used crystal
measurements taken at 87 K. The correction scheme is then identical to that outlined in

the previous section:

All G' _ cos¢ -sin¢ G
( ' ) mC' _ sind) cos¢ (0C

._ 0'

(A.12) G _—A(f)
(A.13) circa—“19;
A“)

Here, the A(f) in eq. A.12 and A.13 must be appropriately normalized. Aﬁer correction

of the data for phase and magnitude, s'and s" can be extracted using eq. 4.38 and 4.39:

 

 

(A.14) e’=C
0

(A.15) 8": G
coC0

III. Theoretical Attempts to Correct High-Frequency Data.

The high-frequency measurements (1kHz - 10 MHz) were made exclusively with
the Hewlett Packard HP4192A Impedance Analyzer. To determine the behavior of the
HP4192A over its listed ﬁequency range, we designed a homemade 512 pF parallel-plate
air capacitor of high stability and measured its frequency response. The behavior of the
impedance analyzer can be described as follows: At frequencies less than approximately
1 MHz, the measured capacitance C and conductance G of the air capacitor was
negligibly different from 512 pF and 0 mho (lmho = 1 (2"), respectively, as expected.

However, for frequencies above 1 MHz, the measured capacitance would fall steadily

122

while the conductance measurements took on negative values. The reason for this
behavior is due to inductance effects in the sample leads and phase shifts in the coaxial
cables. For a standard length (1 meter) of coaxial cabling between the impedance
analyzer and DUT (Device Under Test), the HP4192A provides an internal correction
procedure which automatically compensates for phase and magnitude errors occurring at
high frequency. Thus, we wanted to determine the form of the correction used by the
analyzer. If it were possible to do so, we would be able to construct our own correction

scheme that would work for cabling of arbitrary length.

 

 

 

 

 

 

 

 

”L
j:

 

   

 

 

 

 

 

 

 

 

Figure A. 1. Circuit diagram for impedance measurement using HP4192A.

A schematic diagram of the HP4192A's impedance measurement circuit can be

seen in Figure A.1. To measure the impedance of the sample, the analyzer measures the

current through and voltage across the DUT using a four-terminal lead arrangement

123

consisting of four coaxial cables (1-4 in the diagram). To carry out the analysis of this

circuit, we make the following assumptions:

(1) HP4192A current meter has 50 Q input impedance, thus, no reﬂected wave will
occur at the input to the meter.

(2) HP4192A voltmeter has inﬁnite impedance and draws no current.

(3) All quantities vary in time as em.

(4) All cables have same length.

(5) For each cable, x = 0 is located at the impedance analyzer, x = L is at the DUT.
(6) In Figure A.l, Current leads are labeled 1 and 4, voltage leads are 2 and 3.

(7) Measured current and voltage are 10 and V0, respectively.

(8) The impedance of the coaxial cable is 20 = 50 Q/meter.

(9) k = 21V)», where it is the wavelength of the electromagnetic wave in the coaxial cable.

From assumption 1, we can write the voltage and current for cable #4 as:

(A.16) v,(x) = v, e-‘kx

(A.17) I4(x) -_- —%—e'“°‘

0

At the impedance analyzer (x = 0), the current in cable #4 will be

(A.18) 14(0) = —10 :> V4 = IOZ0

Thus, the current and voltage expressions can now be written as

(A.19) V4(x) =10Zoe'ik"

(A.20) 14(x) = —Ioe'“"‘

Using assumption #2 we can write expressions for V2(x) and V3(x), which have similar

form and will satisfy the boundary conditions at the volmeter:

124

(A.21) V2.3(x) = v,.,(eikx + e'ik")

V23 ikx -ikx
(A22) I.,.(x)=-—Z~—<e —e >

o
The volmeter reading will then be the voltage difference between cables 2 and 3 at x = 0:
(A23) V,(0) - V3(0) = V0 2 2V2 — 2V3 = V0

The actual voltage drop at the sample location (x = L) is:

(A.24) v, = v2 (L) — v, (L) => v8 = (v2 — v,)(eikL + e""‘L)

Substituting the result from eq. A.23 into A.24 gives:

(A.25) V = 1,29 (en‘L + 6“): V0 cos(kL)

S

As it turns out, we do not need to know anything about the wave in cable #1, so
we will not introduce expressions for V1(x) and boundary conditions on the current and
voltage at the junction of cables 1 and 2. However, we do need to apply boundary
conditions at the junction of cables 3 and 4 in order to ﬁnd the current through the sample

1,:
(A.26) V3 (L) : V4 (L) 2 V3 (eikL + e-ikL)= IoZoe'M‘

(A.27) I,(L) + I,(L) + 18 = o :3 %(e“‘L —e'“"-)— roe-i“L +1. = o

0
Substituting for V3 in eq. A.27 using A26, and solving for 1, yields:

_1 e-ikL
(A.28) Is = 'To—ka

(cm. _ e~ikL)+ Ice-ix].
e + e

erkL -r|d..

ild. _ -ikL
(A.29) 15:10e'”[1—i—e—.—]
+6

125

 

(A 30) I — ZIOe-Zikl. _ Ice-ZikL
. s eikL +e’ikL cos(kL)

The actual impedance of the sample, 2,, can be found by dividing eq. A.25 by A.29:

 

a.» z gunman»
. . I,

“ -2ikL
Ioe

Deﬁning the impedance as measured by the HP4192A as Zrn E V% , we can write the

0

actual sample impedance in terms of 2",:

cos2 (kL)

(A.32) Z, = Zm( a“, J = Zm cosz(kL)e2ikL
e

Now that we have the actual sample impedance expressed as a function of the measured
impedance, we must supply expressions for Z8 and Zm. Since the analyzer models the

sample in terms of a capacitance C in parallel with a conductance G, we can substitute

the general form Z'1 = G — iorC into eq. A32:
(A33) Gm 4ow = (G, —i(oCs)cosz(kL)e2ikL
(A34) Gm — imCm == (Gs — imc, )cos2 (kL)[cos(2kL) + isin(2kL)]

Solving A.34 for G8 and Cs provides the expressions for the correction formula:

 

 

1 .
(A35) G -mmm cos(2kL)-a)Cmsrn(2kL)]

___1___I:Cm cos(2kL) + 94”- sin(2kL)]
c ) C0

A.36 C =
( ) s 032(kL

 

 

Eq. A35 and A36 can then be applied to any set of measurements (Cm, Gm) taken at

frequency to = 21tf to obtain the corrected values.

126

Despite the apparent consistency of the analysis given above, eq. A35 and A.36
were unable to satisfactorily correct our measurements. As shown in Figures A2 and
A3 for measurements made on a 511-pF Air capacitor using 1 meter cables, the
capacitance measurements tended to become over-corrected and the conductance

measurements under-corrected. Thus, we must assume that our analysis is somehow

 

 

 

 

 

 

 

ﬂawed.
30° ” —0— Transm'ssion line correction for L=1 m
........ Ehta taken for L=O m(no cables)
.-.¢--- Data taken w ith L=1 mcabies
700
[C
O.
v
0)
O 600
c
(U
a:
r
(U 500
O
400
300 ‘ ‘ ‘ 1 A l A 1
3.09005 3.0e+oos 5.7e+ooe 8.3e+006 1.1e+007

Frequency (Hz)

Figure A.2. Transmission line correction of capacitance data for a 511 pF Air
capacitor. Correction of the data obtained at 1 meter should
match the data obtained at the instrument front panel (L=O meters).
Clearly, the capacitance data is being over-corrected.

127

 

—0— Data taken using L=1 m cables plus transm'ssion line correction

 

 

 

 

 

 

........ Datatakenusingnocables(L=0)
.-..-.- Data taken using L=1 mcables
’2‘ 0
CD
(I)
8
' -100
E
v
a)
‘c’ 150
.9 .
0
3:
s -200
0
-250 "
3_Oe+oos 3.0e+oos 5.7e+oos 8.3e+006 1.1e+007

Frequency (Hz)

Figure A.3. Transmission line correction of conductance data for a 511 pF Air
capacitor. Correction of the data obtained at 1 meter should
match the data obtained at the instrument front panel (L=0 meters).
Instead, the conductance data is being under-corrected.

128

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