-_
7 cf)
Impulsive Stimulated Thermal Scattering
of Glass-Forming Liquids
by
Dora Marie Paolucci
B.S. (Chemistry and Mathematics) University of Richmond (1993)
Submitted to the Department of Chemistry
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 1998
@ Massachusetts Institute of Technology 1998. All rights reserved.
Auth
or ..
....
,
Certiified by .............
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Department of Chemistry
21, 1998
I rMay
....................................
Keith A. Nelson
Professor of Chemistry
Thesis Supervisor
Accepted by .....
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INSTiJ'I
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Dietmar Seyferth
Chairman, Departmental Committee on Graduate Students
This doctoral thesis has been examined by a committee of the Department of Chemistry
as follows:
Professor Irwin Oppenheim
1
Chairman
Professor Keith A. Nelson
Thesis Supervisor
Professor Robert Silbey
7?.
Impulsive Stimulated Thermal Scattering
of Supercooled Liquids
by
Dora Marie Paolucci
Submitted to the Department of Chemistry
on May 21, 1998, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
Impulsive Stimulated Thermal Scattering is a time domain laser spectroscopy that
is applied to studying the relaxation dynamics of liquids in the supercooled temperature
regime. Characterizing the relaxation dynamics of supercooled liquids is essential to
developing an understanding of the physics of the liquid-glass transition. Mode Coupling
Theory is a popular theory that has had some success in explaining experimental
observations of the dynamics of certain supercooled liquids.
Glass forming liquids are categorized according to fragility. To date, many of the
experimental and theoretical studies of glass forming liquids have focused on liquids with
fragile characteristics. To extend the field of research, ISTS studies are conducted on
glycerol, a glass-forming liquid of intermediate strength. Other experimental studies of
glycerol have given conflicting results, and the applicability of Mode Coupling Theory is
questioned. ISTS is added to the other experimental techniques to measure the DebyeWaller factor and relaxation times in the regime from a few nanoseconds to hundreds of
microseconds. There is no MCT anomaly in the temperature range measured. Also, the
shape of the relaxation spectrum is constant, without the enhanced stretching commonly
observed at the crossover temperature Tc.
ISTS is also applied to characterization of the dynamics of a glass-forming
polymer, Polypropylene Glycol. Like many polymers, Polypropylene Glycol shows two
features in the slow relaxation portion of the dielectric spectrum. In ISTS, only the
feature which is strongly coupled to the density is observed, at least quantitatively. A
comparison of the results for two molecular weights shows that the relaxation observed in
ISTS is independent of molecular weight, and therefore related to the main a peak in the
dielectric spectrum, which is also independent of molecular weight. Some temperature
dependence is observed in the Debye-Waller factor and the stretching of the relaxation
spectrum. There is no Debye-Waller factor anomaly as predicted by MCT in the
temperature range measured.
The conclusion reached from ISTS experiments on non-fragile glass forming
liquids is that, despite the success MCT has enjoyed in predicting the behavior of
relatively simple systems, there is not sufficient detail incorporated in the theory to
address the complexity of networked materials (glycerol) and polymers.
Thesis Supervisor: Keith A. Nelson
Title: Professor of Chemistry
Acknowledgments
I would like to thank my advisor, Keith Nelson, for his constant enthusiasm for
doing science. He is always a source of valuable advice, and always believes a problem
can be solved. I am very thankful for the respect he has shown me. I would also like to
thank my undergraduate advisor, Sam Abrash, for teaching me physical chemistry and
starting me on the research path. The enthusiasm of Mr. Elmer and Mr. Moran, my high
school chemistry and biology teachers, is largely responsible for my interest in science.
Over the years, I have been privileged to work with a wide variety of people. I
owe Laura Muller many thanks for introducing me to the world of lasers and liquids. She
is a remarkably patient mentor, and always understands that sometimes you just need to
"walk like an Egyptian". This research on supercooled liquids has greatly benefited from
the work of earlier students in the Nelson group.
In particular, I would like to
acknowledge Yongwu Yang for his work on the hydrodynamic treatment of ISTS, and
the experiments on fragile liquids. Many of the figures in Chapter 4 are the results of his
work. Rebecca Slayton has provided essential help and encouragement in this last year,
and will be continuing the experimental efforts in this field. Tanya, Alex, Ariya, and
Randy have also been helpful picosecond lab collaborators. Ciaran Brennan has given
invaluable advice about an astonishing number of lab-related things, and has been a good
friend as well. I would also like to thank Lisa, John, John, Marc, Dutch, Greg, Richard,
and Tim for all their help. A great many people have passed through the Nelson Group
and the other groups in the basement of building 2, making it an exciting place to work,
and I thank them all. When I started graduate studies at MIT, I was very fortunate to
have a wonderful group of classmates. Many times, we donned our helmets and tackled
those nasty integrals together, with a spirit of teamwork. Dave Rovnyak in particular has
been a good friend and fellow Richmond alum. I wish everyone the best.
I am very appreciative of the good and faithful friends who have supported and
encouraged me, particularly in this last year. Anand Mehta has always been there for me,
through all the ups and the downs. I don't know how I could have done this without him.
Amy, Jen, (and Tessa) have been very understanding (and warm and fuzzy) housemates,
particularly in these last crazy thesis-writing weeks. Along with the other residents, they
have helped make the Berskshire St. house a warm and safe place. I would also like to
thank everyone involved in eaps-vball, the Women's Volleyball Club, and the various
Chemistry hockey teams for all the good times we have enjoyed.
Most of all, I would like to thank my parents, who have given me so much. From
the day I ran down "the path" to kindergarten, my parents have always encouraged me to
tackle any challenge that came my way. I would like to extend very special thanks to
them and the rest of my family for all of their love over the years.
Table of Contents
List of Abbreviations
9
List of Figures
10
List of Tables
14
Chapterl Introduction
15
Chapter 2 Mode Coupling Theory
20
2.1 Mode Coupling Theory Background
20
2.2 Experimental Methods of Testing Mode Coupling Theory
33
Chapter 3 Impulsive Stimulated Thermal Scattering
36
3.1 Overview and Theory
36
3.2 Experimental
38
3.2.1 Nd:Yag Laser
38
3.2.2 Generation of the Excitation Grating
41
3.2.3 Probing the Grating
46
3.2.4 Detecting the Signal
48
3.2.5 Temperature Control
49
3.2.6 Automation
50
3.3 Heterodyne Signal
51
3.4 Data Analysis Methodology
57
3.4.1 Accounting for Detector Response
57
3.4.2 Time Domain Fits to the Data
66
3.4.3 Alternate Analysis of the Relaxation Mode
66
Chapter 4 Review of Impulsive Stimulated Scattering on Fragile Liquids
68
Chapter 5 Impulsive Stimulated Thermal Scattering on Glycerol
76
5.1 Experimental background
76
5.2 Experimental
81
5.3 Results
82
5.3.1 Acoustics
85
5.3.2 Thermal Diffusivity
91
5.3.3 Relaxation and Debye-Waller factor measurements
94
5.4 Discussion
102
5.5 Conclusions
108
Chapter 6 Impulsive Stimulated Thermal Scattering on Polypropylene Glycol
111
6.1 Overview
111
6.2 Experimental
115
6.3 Results
115
6.3.1 Acoustics
118
6.3.2 Relaxation measurements
125
6.4 Discussion
139
6.5 Conclusions
146
Chapter 7 Summary and Future Directions
148
Appendix A Selected Matlab codes
153
Appendix B Selected References
157
List of Abbreviations
ISS
ISTS
ISBS
MCT
BS
PCS
KWW
VTF
PPG
PPG425
PPG4000
salol
Impulsive Stimulated Scattering
Impulsive Stimulated Thermal Scattering
Impulsive Stimulated Brillouin Scattering
Mode Coupling Theory
Brillouin Scattering
Photon Correlation Spectroscopy
Kohlrausch-Williams-Watts
Vogel-Tamman-Fulcher
Polypropylene Glycol
Polypropylene Glycol, Molecular Weight 425
Polypropylene Glycol, Molecular Weight 4000
phenyl-salicylate
CKN
Ca 4 K 6(NO 3) 14
YAG
cw
RTD
T,
Tc
To, TTF
Yttrium Aluminum Garnet
Continuous Wave
Resistance Thermal Detector
Glass Transition Temperature
Mode Coupling Theory Crossover Temperature
Vogel Temperature
List of Figures
Figure 1-1: Angell plot illustrating the concept of fragility
16
Figure 2-1: Illustration of the relaxation function and relaxation spectrum as
predicted by Mode Coupling Theory.
23
Figure 3-1: Illustration of the optical setup for generation and probing of the grating.
43
Figure 3-2: Phase mask design showing patterns obtained for ISTS, ISRS, and other
applications.
44
Figure 3-3: Illustration of a simple imaging system for generation and probing of a
grating using a phase mask.
47
Figure 3-4: Comparison of simulated data for ISTS signal with different amplitudes of
homodyne and heterodyne contribution.
54
Figure 3-5: Simulated data with a heterodyne component (3.3.2) and fit to homodyne
ISTS signal (3.1.3). Data are presented on a linear scale.
55
Figure 3-6: Simulated data with a heterodyne component (3.3.2) and fit to homodyne
ISTS signal (3.1.3). Data are presented on a log scale.
56
Figure 3-7: Example of raw ISTS data taken with two different detectors.
58
Figure 3-8: Measured Impulse Response of the Antel detector.
62
Figure 3-9: ISTS data taken with the Antel detector.
63
Figure 3-10: Response of the Hamamatsu Detector to a quasi-cw probe.
64
Figure 3-11: ISTS data taken with the Hamamatsu detector.
65
Figure 4-1: Temperature Dependence of the Relaxation Time in Salol. Includes a
comparison to Dynamic Light Scattering Data.
69
Figure 4-2: Temperature Dependence of the Debye-Waller factor in Salol, in the q--0
70
limit.
Figure 4-3: Temperature Dependence 3 in Salol. Data obtained from Dynamic Light
71
Scattering is included for comparison.
Figure 4-4: Temperature Dependence of the Debye-Waller factor in CKN, in the q->O
72
limit.
Figure 4-5: Temperature Dependence of P in CKN. Data obtained from other
experimental techniques is included for comparison.
73
Figure 4-6: Temperature Dependence of the Debye-Waller factor in butylbenzene
74
Figure 5-1: ISTS data and fits from glycerol at q = 0.305 tm-'.
83
Figure 5-2: ISTS data and fits from glycerol at q = 1.19 tm-'.
84
Figure 5-3: Speed of sound in glycerol at several wavevectors.
86
Figure 5-4: Acoustic damping in glycerol at several wavevectors.
87
Figure 5-5: Acoustic frequency vs. wavevector at several temperatures.
89
Figure 5-6: Acoustic damping vs. q2 at several temperatures.
90
Figure 5-7: Thermal diffusivity vs. temperature in glycerol.
92
Figure 5-8: Relaxation times (log scale) vs. temperature in glycerol at all
wavevectors.
94
Figure 5-9: Relaxation times at all wavevectors, with a best fit to Arrhenius
temperature dependence.
95
Figure 5-10: Vogel-Tamman-Fulcher fit to relaxation times (In scale) in glycerol.
96
Figure 5-11: Temperature Dependence of P in glycerol at all wavevectors.
97
Figure 5-12: The relaxation function in glycerol at several temperatures and
wavevectors.
99
Figure 5-13: Debye-Waller factor in glycerol for all wavevectors.
100
Figure 5-14: Debye-Waller factor vs. temperature at the two smallest wavevectors,
to highlight the data at the lowest temperatures.
101
Figure 5-15: Relaxation times in glycerol fit to a power law with no restrictions.
106
Figure 5-16: MCT power law fit with T, restricted to 225 K.
107
Figure 6-1: ISTS data from PPG4000. The data shows the growth of the structural
relaxation feature as the sample cools.
116
Figure 6-2: ISTS data from PPG425, showing the structural relaxation appearing at
intermediate temperatures.
117
Figure 6-3: Speed of sound in PPG425 at several wavevectors.
119
Figure 6-4: Speed of sound in PPG4000 at several wavevectors.
120
Figure 6-5: Comparison of the speed of sound in PPG425 and PPG4000.
121
Figure 6-6: Acoustic damping in PPG425.
122
Figure 6-7: Acoustic damping in PPG425 at one wavevector, highlighting the low
temperature shoulder.
123
Figure 6-8: Acoustic damping in PPG4000 at one wavevector.
124
Figure 6-9: Log plot of relaxation times in PPG425 obtained by ISTS and other
experimental methods.
126
Figure 6-10: Comparison of Relaxation Times for PPG425 and PPG4000.
127
Figure 6-11: Vogel-Tamman-Fulcher fit to relaxation times in PPG425.
128
Figure 6-12: VTF fit to relaxation times in PPG4000.
129
Figure 6-13: Average relaxation time in PPG425 fit to a MCT power law for the
temperature dependence.
130
Figure 6-14: Average relaxation times, with data at temperatures above 230 K fit to a
MCT power law temperature dependence.
131
Figure 6-15: Stretching Parameter 3 from PPG425 as a function of temperature at
several wavevectors.
132
Figure 6-16: Stretching Parameter 1 from PPG425 as a function of temperature at
several wavevectors, including measurements from VV and VH Photon
Correlation Spectroscopy.
133
Figure 6-17: Stretching parameter P in PPG4000 at several wavevectors.
134
Figure 6-18: The Debye-Waller Factor in PPG425 as a function of temperature.
136
Figure 6-19: The Debye-Waller Factor in PPG425 at different wavevectors.
137
Figure 6-20: Debye-Waller factor measured for PPG4000.
138
List of Tables
Table 3-1: Simulated and Fit parameters for ISTS signal with a mixture of
homodyne and heterodyne signal.
52
Table 5-1: Experimentally determined exponents for the susceptibility spectrum of
glycerol.
79
Chapter 1
Introduction
The physics of supercooled liquids and the glass transition has been the subject of
a stimulating dialog between theory and experiment in recent years. 1-3 A supercooled
liquid is a substance that has been cooled below its freezing point without crystallization.
As the material is cooled further, the viscosity dramatically increases, accompanied by a
corresponding increase in the characteristic time for relaxation processes. At a certain
temperature, the glass transition occurs, where changes in the thermodynamic properties
are observed.
Despite the change in thermodynamic properties, however, the glass
transition is not a traditional first or second order phase transition, but an experimentally
defined kinetic transition. So far, no conventional order parameter has been defined to
describe behavior near the glass transition. The characteristic time scale for structural
relaxation at the glass transition is on the order of many seconds, corresponding to a
viscosity of 10"'poise.
Supercooled liquids and glasses can be formed from a wide variety of chemical
substances. Commonly studied glass forming liquids included molecular liquids such as
salol and glycerol, ionic melts such as Ca 4K6(NO3) 4 (CKN), and polymers. Other glasses
have been less studied from the theoretical perspective, but have important practical
applications, including the common "window" glass, and metallic glasses.
Glass forming liquids are often categorized by the temperature dependence of the
viscosity through the supercooled regime. The two extremes are termed "strong" and
"fragile". A strong liquid displays an Arrhenius temperature dependence of the viscosity.
The viscosity of the fragile liquid shows extreme deviation from Arrhenius behavior.
Although there are liquids showing continuous behavior between the two extremes, the
liquids are generally grouped into the three categories of strong, intermediate, and fragile.
Figure 1-1 shows an example of the temperature dependence of the viscosity for a strong,
an intermediate, and a fragile liquid.
122
10
K2
0
8E
-
6
2
glycerol
ui"
-2
0.5
0.6
I
0.7
-6
"-8
-10
0.8
0.9
1.0
TV T
Figure 1-1: Angell plot showing glass-forming liquids of
varying fragility. SiO 2 is a strong liquid, glycerol is an
intermediate liquid, and ortho-terphenyl (OTP) is a fragile
liquids. The y axes show the viscosity or relaxation time,
while the x axis is in terms of the reduced temperature T/T,
so all three liquids show the same viscosity at the glass
transition. This figure is reproduced from Ref. 4
Fragile liquids have been the most studied to date, particularly salol, CKN, and
ortho-terphenyl. Although in general, fragile liquids are more challenging experimentally
and prone to crystallization, it is easier to simulate their behavior. Strong liquids are
seldom studied experimentally, as the glass transition temperature is often extremely
high, and the liquids must be further heated if the goal is to study fast dynamics.
Intermediate liquids are gaining more and more attention as the natural extension of the
work that has been completed for the fragile materials. Polymers are also gaining more
attention from a fundamental perspective, as they form glasses easily, and have viscosity
behaviors ranging from fragile to intermediate.
Much empirical work had previously
been done due to the practical uses of polymers.
Despite the dramatic differences in chemical composition, there are several
properties or behaviors of supercooled liquids and glasses that are universally observed.
The theoretical and experimental efforts to explain and characterizes these dynamics
amount to a search for the physical mechanism underlying the dramatic increase in the
relaxation time through the supercooled regime. Mode Coupling Theory, a kinetic theory
of the glass transition, postulates that non-linear feedback between high wavevector
modes creates a bottleneck situation which describes the physics underlying the glass
transition.
Impulsive Stimulated Thermal Scattering (ISTS) is one of many experimental
tools that is employed in the study of the dynamics of supercooled liquids. The current
dynamic range of ISTS measurements is about 5 decades, from nanoseconds to
milliseconds. The location of this range is intermediate between several light scattering
techniques, and includes some of the range covered by dielectric techniques.
Most
importantly, in several well studied glass forming liquids, ISTS covers the frequency
range where central experimental predictions are made. ISTS can test these predictions
without extrapolation of frequency or temperature.
The main difficulty
in studying the glass transition
experimentally
is
characterization of the dynamics of a glass forming liquid over the entire temperature
range, spanning the extremely wide range of time scales. As a result, to cover the entire
spectrum, the data from different experiments are often combined. Whenever there is a
change in the experiment, the nature of the measurement must be reevaluated. In many
cases, it is far from clear that two experiments probe exactly the same dynamics in the
material. Each experiment has some observable or property that it measures, and will
give information on modes that strongly affect the value of that specific property. If a
mode is not strongly coupled to the specific observable of the experiment, it will be
attenuated, or not observed at all.
Often, achieving a complete understanding of each
experiment is itself a challenge.
One of the recurring difficulties in testing Mode Coupling Theory is that the
predictions of the theory focus on the time and temperature dependence of the density of
the material.
ISTS and neutron scattering are the only techniques that are well
understood to directly measure density variations in time or frequency. These techniques
do not cover the entire relevant frequency range, so other experiments must be
incorporated to completely span the dynamics. As a result, there is always the open
question of how well the measurement truly relates to the prediction for the density
dynamics. Comparison of ISTS results with results from other experiments may help
elucidate the similarities and differences between several techniques.
The research included in this thesis has several goals. The first is to measure the
density dynamics of a glass forming liquid with intermediate strength, glycerol, and to
determine how well Mode Coupling Theory explains the results. This is an extension of
previous ISTS studies on fragile liquids. The second goal is to begin work on applying
ISTS as a technique for studying polymer dynamics. As appropriate, Mode Coupling
Theory predictions are also evaluated for the polymer results. In the course of studying
these materials, the opportunity arises to compare and contrast the measurements made
with ISTS and other experimental techniques.
Chapters 2, 3, and 4 provide background information. Chapter 2 outlines Mode
Coupling Theory, experimentally testable predictions, and describes several experimental
techniques commonly used to test Mode Coupling Theory. Chapter 3 gives details on the
Impulsive Stimulated Thermal Scattering experiment and data analysis. Chapter 4 briefly
reviews some data obtained from ISTS from fragile glass forming liquids.
The results of ISTS studies on non-fragile glass forming liquids, glycerol and
Polypropylene Glycol, are presented in Chapters 5 and 6.
Chapter 5 presents the
experimental results for glycerol and a Mode Coupling Theory analysis of the results.
Two molecular weights of Polypropylene Glycol are studied in Chapter 6. The results are
compared and analyzed relative to other data on polypropylene glycol. Where applicable,
the data are compared to Mode Coupling Theory predictions. A summary of the results
and suggestions for future efforts are contained in Chapter 7.
References
1
Structure and Dynamics of Glasses and Glass Formers,Vol. 455, edited by C. A.
Angell, K. L. Ngai, J. Kieffer, T. Egami, and G. U. Nienhaus (Materials Research
Society, Pittsburgh, PA, 1997).
2
DisorderedMaterials and Interfaces, Vol. 407, edited by H. Z. Cummins, D. J.
Durian, D. L. Johnson, and H. E. Stanley (Materials Research Society, Pittsburgh,
PA, 1996).
3
Supercooled Liquids: Advances and Novel Applications, Vol. 676, edited by J. T.
Fourkas, D. Kivelson, U. Mohanty, and K. A. Nelson (American Chemical
Society, Washington, DC, 1997).
4
M. D. Ediger, C. A. Angell, and S. R. Nagel, J. Phys. Chem. 100, 13200 (1996).
Chapter 2
Mode Coupling Theory
2.1
Mode Coupling Theory Background
One of the major theories of the liquid-glass transition, Mode Coupling Theory
(MCT) has received much theoretical and experimental attention in recent years. 1-3
Several major reviews of Mode Coupling Theory, its mathematical formulation, and the
experimental evidence to support or contradict the theory are available in the literature.
The review by Cummins et. al.4 summarizes the mathematical results and presents a wide
variety of experimental evidence.
The presentation of Gotze et. al.5,6 begins with a
discussion of typical features in the relaxation of supercooled liquids. Subsequently, the
theory is presented in a detailed mathematical treatment in response to these observations.
Mode Coupling Theory and its predictions are summarized in this section for
completeness and future reference in the discussion of results and comparison to
literature.
Mode Coupling Theory attempts to explain the glass transition as a kinetic
phenomenon controlled by non-linear coupling between modes. Each mode, labeled by
wavevector q, is described by an equation of motion for the density autocorrelation
function, (I(t).
(t) =
(2.1.1)
q0)
(P
S
(2.1.2)
(t) + QO,(t) + Mq(t -t')
(t')dt' = 0
where Q is a characteristic microscopic oscillation frequency and S, is the static structure
factor.
The substance of MCT is contained in the memory function
(2.1.3)
Mq(t - t') = C [yq(t - t') +mq(t - t')]
The memory function contains two terms, a regular damping term, and the relaxing term,
m,. In the simplest version of the theory, modes are quadraticly coupled by a coupling
constant in the relaxing term.
(2.1.4)
mq(t)=
V(2)(q,ql,q2)
q ,q2
qt (t)
q2 (t )
The theory becomes richer, albeit less mathematically tractable, with increased
complexity in the coupling constant. Without the relaxing term, or in the weak coupling
limit where the coupling constants approach zero, (2.1.2) reverts to the standard harmonic
oscillator equation of motion with frequency Qiq and damping rate y,, which describes the
behavior of a simple, non-viscous liquid.
(2.1.5)
q,(t) + q qbq (t')+
q,(t) = 0
The MCT equations (2.1.2) now form a closed set of equations that can in
principle be solved numerically. The coupling constant V(2 ) is a function of the static
structure factor, and contains the temperature dependence in the MCT formulation. The
temperature dependence of the coupling results in a slowing down of the dynamics
underlying the glass transition. As the system is cooled, a critical value V(2 )c is reached at
a critical temperature T,. At this point, the feedback between the modes is strong enough
to cause structural arrest.
A single schematic equation can be obtained by restricting the modes to modes at
q = qo near the peak of the static structure factor, where the coupling is strongest. Each
relaxation function q,has identical time dependence due to symmetry, so the single MCT
equation is
(2.1.6)
(t) + n 7(t) + C20(t)+ IV(2)b 2 (t-
(t)
(t')dt' = 0
As with the case of the full set of equations, at the critical value V(2)c, reached at T., the
structural relaxation slows down, resulting in an ergodic to non-ergodic transition at To.
Experimentally testable predictions are made in the vicinity of this transition, often
formulated in terms of a separation parameter
(2.1.7)
a = (T - T)/T,.
Figure 2-1 displays 0(t) and 0((o) for both the idealized and the extended theory,
discussed below. In a qualitative, physical picture of relaxation, MCT predicts a two step
relaxation function in time, or two peaks in the frequency spectrum.
There is a fast
relaxation, called the P relaxation. The second feature is a slow relaxation, which is
referred to as structural relaxation, or a relaxation.
The a relation is linked to the
coupling term and thus shows a dramatic temperature dependence.
The a relaxation
extends to infinite time, or zero frequency at Te, according to the discussion above.
_
I
I~
010S
06.
I'I
nr.
J0
020
Os
o"
•
04
01
0'
0'
u'
0
0 106 4 I.
10
I
Figure 2-1: Illustration of the relaxation function in time,
and the relaxation spectrum in the frequency domain. The
upper illustrations represent idealized Mode Coupling
Theory, while the lower figures are derived from the
extended theory. The figure is reproduced from Ref. 4
Approaching T , the relaxation time and the viscosity diverge to infinity,
consistent with structural arrest. MCT predicts that this divergence follows a power law
temperature dependence
(2.1.8)c
Ocr
17 IT- Tc-r
Viscosity data at high temperatures have supported this power law behavior, but with a
measured TC value significantly above the glass transition temperature T,. 4,7
This
immediately challenges the theory, as the viscosity does not diverge to infinity, but has
measurable values down to the glass transition temperature, where the viscosity reaches
1013 poise. In addition, the a relaxation, which is arrested at Tc in the idealized theory, is
clearly observed in experiments at temperatures approaching the glass transition, below
the To value determined by the viscosity prediction.
To address this issue, Mode Coupling Theory has been extended to include
thermally activated process which restore ergodicity below Tc.
The viscosity and a
relaxation continue to vary smoothly through Te, which is now referred to as a crossover
temperature.
This version of the theory is referred to as extended Mode Coupling
Theory, as opposed to idealized Mode Coupling Theory. Many of the results of the
idealized theory are not greatly modified by the extended theory, and most experimental
tests have only considered the idealized version of the theory.
The strength of MCT is that there are several experimentally testable predictions
derived from the solution of the MCT equation. One is the power law temperature
dependence of the viscosity, mentioned above (2.1.8).
As the relaxation time is
proportional to the viscosity, it is also expected to exhibit power-law temperature
dependence.
Many of the predictions of MCT relate to the time dependence of the relaxation
function, or equivalently, the frequency dependence obtained by Fourier transformation.
The relevant quantities are defined by
(0o) = 0'(W) + iq"()
(2.1.9a-c)
'(mc)
=j(t)
cos(ot)dt
0
(t) sin(ot)dt
0"() = 0
The imaginary part of the susceptibility, X, is related to the imaginary part of the
relaxation function,
t, by
a frequency factor
(2.1.10)
X"(Co) oc coo"(Co)
The prediction made by MCT concerning T, which is experimentally observable
at, above, and below TC describes an anomaly in the variation of the Debye-Waller factor
with temperature. The Debye-Waller factor is defined as the strength of the ax relaxation.
In the frequency domain, this corresponds to the integrated area of the quasielastic line,
the relaxation spectrum.
(2.1.11)
fq= I
f "(o))d
I|m< ,.
=c
, d(lnw)
In o<ln
q
where oc is a cutoff frequency, set to a value high enough to completely include the ac
relaxation in the integration. In the time domain,
(2.1.12)
fq = q(t Y )
Again, the turn-on time is chosen before the beginning of the slow relaxation. As long as
any elastic features are well separated from the relaxation in time or frequency, the exact
value of the cutoff time or frequency should not affect the measure value of the DebyeWaller factor.
It may seem contradictory to speak of the "strength" of the a relaxation below To,
as in the idealized theory the a relaxation is arrested at Tc. In this context, the concept of
ergodicity gives a more physically intuitive picture of the Debye-Waller factor. Given
the initial condition 4(t=0) = 1, the Debye-Waller factor, or the non-ergodicity parameter,
is a measure of how far the long time limit of (t) is from zero. The amplitude of this
non-zero long time limit is a measure of the non-ergodicity of the system. The increased
strength of the ca relaxation implies a corresponding decrease in the strength of the 3
relaxation. These concepts can be visualized with the aid of Figure 2-1.
In the low wavevector limit, the Debye-Waller factor can also be determined from
acoustic measurements. 8
(2.1.13)
fo =-1
=- ,
M0 and M are the elastic modulus in the limit of zero frequency and infinite frequency
respectively. co and c, are the zero and infinite speed of sound, respectively. Similar to
the cutoff frequency, in practice, zero frequency means frequency much lower than the c
relaxation frequency range, and infinite frequency is of the order of co, above in (2.1.11)
and (2.1.12), i.e., above the a relaxation spectrum.
Since the Debye-Waller factor is a wavevector-dependent quantity, f, and fo will
differ in absolute magnitude, but the temperature dependence, regardless of the method of
measurement, is predicted to be the same.
The prediction derived from MCT for the
temperature dependence of the Debye-Waller factor (2.1.13) is written in terms of the
separation parameter o, and the critical amplitude hq.
h
(2.1.14)=f
(,OTT)
(o < 0, T > T )
fq(T)= fq
At low temperatures near the glass transition, there is a smooth transition from the glassy
behavior, which is similar to the temperature dependence of the Debye-Waller factor for a
classical harmonic solid, Inf, oc -T.
The strength of the relaxation decreases as yo1/2 as
the material warms, and at the crossover Tc, the relaxation strength reaches a steady level.
This prediction cannot be exactly tested in the formulation stated above. Making
a measurement of zero for the Debye-Waller factor would require an experiment with
infinite frequency or time resolution, in order to select the appropriate cutoff frequency.
Taking into account practical experimental considerations, the prediction is restated in
terms of the effective Debye-Waller factor.
Sfq(T) = f + h
f,,() =
+ ()
+ (a) (a > 0,T < T)
(o < o, T> T
The linear temperature dependent term is weak and is rarely included in the experimental
analysis.
Other predictions made by MCT concern the susceptibility spectrum (2.1.10).
Although such experiments are not included in this thesis, these are also theoretically
testable through ISTS, by measuring the acoustic modulus spectrum, given a large
enough wavevector range. These predictions are often tested by frequency domain light
scattering. 4
As predicted by MCT, the susceptibility spectrum should show two features, a
high frequency feature referred to as P relaxation, and a lower frequency peak
corresponding to a relaxation.
If the frequencies are well separated, a minimum is
observed between the features.
The frequency window where this minimum can be
observed is a function of temperature.
In practice, accurate characterization of the
minimum requires measurements over a wide frequency range including both the ca and P
relaxation features.
Due to the temperature dependence of the a relaxation, at high
temperatures the features overlap and cannot be distinguished. As the a relaxation slows
down with decreasing temperature, the features are well-separated, but the slow
relaxation eventually moves beyond the low-frequency limit of the experiment, and the
minimum can no longer be analyzed.
The shape of the minimum in the susceptibility spectrum is determined in MCT
by the exponent parameter k, which is specific to the material, and independent of
temperature and wavevector.
The magnitude of k is indicative of the degree of
cooperativity, as seen in the original formulation of the equation of motion.9
(2.1.16)
(t) + yI(t) +
~j YI\/ J CIt
\L' jI.T Y/\L
J
+ 4n(t) F0 (t -t')(t')dt'
- t') 0(t') dt = 0
The minimum is described by
(2.1.17)
"()=
[b(wm +±a
(
n
n
(a + b)
The critical exponents a and b are related to the exponent parameter X and are restricted to
the values 0 < a <1/2 and 0 < b
(2.1.18)
1. The relation between the exponent parameters is
A= F_2-a
=F2(l+b)
F(-2a) F(1+2b)
where F is the gamma function.
Both the frequency and amplitude of the susceptibility minimum vary as power
laws relative to Tc.
(2.1.19a-b)
z",, oc(T- T)
Co"c (T-T Te
There are equivalent power law relationships for the decay of the autocorrelation
function in the time domain. The time domain signature of the high and low frequency
features in the frequency domain is a two step decay of the autocorrelation function. The
crossover between the two decays occurs at a time t, which is a function of temperature.
(2.1.20)
t, = t o / 10
1
"
to is a microscopic time constant, the inverse of the microscopic frequency Q0 , and is
independent of temperature and wavevector.
For times longer than to but shorter than the timescale of the cX relaxation time ta,
the decay function follows the power law decay
(2.1.21)
~,(t)
=
fqC
- Bh,(t/ta)
The time constant t, varies strongly with temperature. (2.1.21) describes a power law
asymptotic region of the a relaxation. In general, however, the ca relaxation is often well
described by a Kohlrausch-Williams-Watts stretched exponential function.
q(t) oc e
(2.1.22)
A similar relationship is predicted for the [3 relaxation, at times longer than the
microscopic time to but shorter than t,.
(2.1.23)
qq(t) = f,
+ hq (t/to)
The above power law decay describes the decay from the initial value )(0) to the plateau
valuefqc, which is the same as the Debye-Waller factor (2.1.14)
In the above expressions, the only role temperature plays in the shape of the
relaxation function is in the scaling time t,, which varies as a power law relative to Tc
(2.1.20).
The wavevector dependence is contained in the amplitudes fq and hq. The
exponents a and b are independent of time, temperature, and wavevector.
These
properties imply that the susceptibility spectra, or the relaxation functions, obtained at
different temperatures can be superimposed simply through scaling by the relaxation
time. The shape of the spectrum or relaxation function, determined by the power law
forms in (2.1.21) and (2.1.23), is the same regardless of the temperature, above T c. This
also implies that the shape of the a relaxation function, determined
stretched exponential parameter 3 (2.1.22) is also the same.
by the KWW
3 should not change with
temperature above Tc. This behavior is referred to as time-temperature superposition.
The exponents in these equations are the same as those in the frequency domain,
and the crossover time t,
is analogous to the frequency at the minimum of the
susceptibility spectrum. It should be reiterated that these predictions in the idealized
theory are for temperatures above Tc, consistent with the idealized theory prediction of
structural arrest of the ocrelaxation at T c.
Another point that merits reiteration is the fact that Mode Coupling Theory is
formulated in terms of the autocorrelation function for density fluctuations (2.1.1). From
this, all of the above predictions for time decay of the autocorrelation function, or the
properties of the susceptibility spectrum, are meant to apply to density fluctuations. A
valid experimental test of MCT must involve a quantity that is strongly coupled to
density fluctuations.
Before discussing experimental tests of mode coupling theory, some quantitative
comments on time scales and frequency ranges will be made. A starting point on the fast
time scale and high frequency extreme are molecular vibrations.
A typical molecular
vibration occurs in the tens of terahertz range, with a vibrational period of tens of
femtoseconds. These motions are beyond the range typical of experiments designed to
test MCT, and are not relevant to MCT, which is a monatomic liquid theory.
Intermolecular motions occur on a longer time scale of hundreds of femtoseconds, with
frequencies of a few terahertz.
These modes have been detected in time domain
experiments, and appear as the boson peak and other high frequency features in frequency
domain experiments. As these motions are also not included explicitly in MCT, no direct
discussion of their properties is necessary, although this contribution to the spectrum may
be considered in data analysis procedures. MCT predictions begin after the microscopic
time to = Q-..
The 3 and ca relaxations in MCT occur at longer times and lower frequencies in
the supercooled liquid regime. Several characteristic spectra are reproduced in the review
by Cummins et. al.4 The a and P relaxation features first make their appearance at
frequencies of a few terahertz at temperatures well above the glass transition. As the
liquid is cooled, the P relaxation remains in the terahertz or hundreds of gigahertz range,
with the 3 contribution to the decay of the autocorrelation function occurring within a few
picoseconds.
The a relaxation is dramatically temperature dependent.
It has been
measured from the high frequency limit where it becomes separated from the 3 feature, to
very low frequencies less than one hertz near the glass transition. These low frequency
measurements correspond to relaxation times on the order of many seconds.
The wide range in frequency or time required to characterize the c and P
relaxations throughout the entire supercooled temperature range is experimentally
challenging.
The only technique to date with this wide dynamic range is dielectric
spectroscopy. Various neutron and light scattering spectroscopies characterize the high
frequency, high temperature regions, with frequencies from THz down to MHz. ISTS in
its current incarnation covers the time range from a few nanoseconds to milliseconds.
Fortunately, this time range corresponds to the time scale of the ca relaxation in the
temperature range where T, is expected. This range, however, is not nearly fast enough to
observe the
P relaxation,
or the crossover between a and
1relaxation.
2.2
Experimental Methods of Testing Mode Coupling Theory
Neutron scattering is one of the most common techniques used to study the liquid-
glass transition. 10 Several methods are combined for a frequency range from 100 MHz to
10 THz. The accessible wavevectors are of the order of inverse angstroms, corresponding
to interatomic length scales. In this wavevector and frequency regime, neutron scattering
directly probes atomic and molecular motions.
The measured quantity in neutron
scattering is the structure factor, S(q,o), or its Fourier transform, S(q,t). 1I This quantity
is the same as the correlation function discussed in MCT. Coherent neutron scattering
measures the pair correlation function, while incoherent scattering measures the self
correlation function.
Neutron scattering is often used to test the MCT prediction for the anomaly in the
Debye-Waller factor.
The measured frequency range often does not extend to low
enough frequencies to permit integration of the spectrum. Instead, the Debye-Waller
factor is obtained by measuring the plateau height in the decay of the correlation function
in time. 10 As the full relaxation is not observed, this can be a source of uncertainty in the
measurement.
In addition to measuring the Debye-Waller factor, the MCT predictions regarding
the behavior of the correlation function are investigated by neutron scattering.1 2 The o
relaxation is most commonly fit to a Kohlrausch-Williams-Watts (2.1.22) function, and
the invariance of the Kohlrausch 3 exponent with temperature is verified. However, due
to the limited time duration of the signal, this test can only be conducted for temperatures
well above Te, and the shape of the relaxation at and below Tc cannot be investigated.
Light scattering studies overlap with neutron scattering measurements in the THz
regime, and extend into the MHz frequency range. The exact mechanism for depolarized
light scattering at high frequencies is not completely understood,4 but qualitative
agreement with neutron scattering at high frequencies is normal. It is often assumed that
the light scattering active modes are strongly coupled to density fluctuations, but the light
scattering spectrum may include contributions from molecular rotational motions.
The most common measurements made with light scattering test the MCT
predictions concerning the shape of the susceptibility minimum (2.1.17). The exponent
parameters X, a, and b are determined, and the power law temperature dependences of the
susceptibility minimum and the minimum frequency are tested.
Dielectric spectroscopy has an impressive frequency range from less than 1 Hz up
to hundreds of GHz. 13 The drawback of testing MCT using dielectric spectroscopy is
that the modes that are active in dielectric spectroscopy are coupled to the dipole
moment, not necessarily to density fluctuations. In fact, deviations between the dielectric
and light scattering spectroscopies can be observed, particularly at high frequencies.
However, given the broad frequency range, dielectric spectroscopy is particularly
valuable for studying the a relaxation throughout the supercooled temperature range.
The glass transition has been studied using other tools, such as viscosity
measurements, NMR relaxation times, specific heat spectroscopy, and other techniques.
These experiments will be discussed as pertinent results are cited.
References
1
Structure and Dynamics of Glasses and Glass Formers, Vol. 455, edited by C. A.
Angell, K. L. Ngai, J. Kieffer, T. Egami, and G. U. Nienhaus (Materials Research
Society, Pittsburgh, PA, 1997).
2
DisorderedMaterials and Interfaces, Vol. 407, edited by H. Z. Cummins, D. J.
Durian, D. L. Johnson, and H. E. Stanley (Materials Research Society, Pittsburgh,
PA, 1996).
3
Supercooled Liquids: Advances and Novel Applications, Vol. 676, edited by J. T.
Fourkas, D. Kivelson, U. Mohanty, and K. A. Nelson (American Chemical
Society, Washington, DC, 1997).
4
H. Z. Cummins, G. Li, W. M. Du, and J. Hernandez, Physica A 204, 169 (1994).
5
W. Gotze and L. Sjogren, Reports on Progress in Physics 55, 241 (1992).
6
Liquids, Freezing, and the Glass Transition, Vol. 1, edited by J. P. Hansen, D.
Levesque, and J. Zinn-Justin (Elsevier Science Publishing Company, Amsterdam,
1991).
7
P. Taborek, R. N. Kleiman, and D. J. Bishop, Phys. Rev. B. 34, 1835 (1986).
8
M. Fuchs, W. Gotze, and A. Latz, Chem. Phys. 149, 185 (1990).
9
E. Leutheusser, Phys. Rev. A 29, 2765 (1984).
10
W. Petry and J. Wuttke, Transport Theory in Statistical Physics 24, 1075 (1995).
11
N. H. March and M. P. Tosi, Atomic Dynamics in Liquids (Dover Publications,
Inc., New York, 1976).
12
J. Wuttke, W. Petry, and S. Pouget, J. Chem. Phys. 105, 5177 (1996).
13
A. Pimenov, P. Lunkenheimer, and A. Loidl, Ferroelectrics176, 33 (1996).
Chapter 3
Impulsive Stimulated Thermal Scattering
3.1
Overview and Theory
Impulsive stimulated scattering is a time domain laser spectroscopy that can
excite and probe a wide variety of material modes. 1 Impulsive Stimulated Thermal
Scattering, as implied by the nomenclature, results from heating of the material.
The
mode or modes that are probed consist of the response of the material to the heating.
Using ISTS, relaxation dynamics of supercooled liquids are observed in the nanosecond
to millisecond time range.
Two picosecond infrared laser pulses at wavelength ke cross at an angle
e
in the
sample, impulsively heating the material in a grating pattern created at wavevector
A
2 sin(2)
Heating results from absorption into vibrational overtones. A probe beam diffracted at
Bragg angle monitors the density evolution of the grating, as the index of refraction
varies with the density. The signal shows the density response dynamics.
The theory of ISS, and its relationship to frequency domain light scattering, has
been developed in detail. 1 In the limit of ideal time and wavevector resolution, the ISS
signal is directly proportional to the square of the Green's function (impulse response
function) describing the material. In the case of ISTS on supercooled liquids, the material
under consideration is a complex fluid in the hydrodynamic (q-0O) limit.
The
generalized equations of hydrodynamics are solved for the Greens function response of
density to sudden heating, GpT(q,t) 2 . In the Debye approximation, the result is
(3.1.2) G,,(q,t) = A[e -r ' -e-r It. cos(2xrco,t)] + B e -rH' -e
0R
where FH is the thermal diffusion rate, FA is the acoustic attenuation rate, cA
=
wA/q
describes the speed of sound and acoustic frequency, co is the zero frequency speed of
sound, and tR is the relaxation time. The relaxation dynamics of supercooled liquids
universally show a non-Debye relaxation, and are more commonly described by a
Kohlrausch-Williams-Watts (KWW) stretched exponential function. Then the ISTS
signal is
(3.1.3) I(q,t) oc G,, (q,t)2 = {A[e-'
- e" - cos(2rco ,t)] + B[e-
'' _
r
')]}2
In principle, this expression can be generalized to include other relaxation
contributions, but this form has been found to give good fits to the data at low frequencies
without any additional terms. For the KWW form, the average relaxation time is given
by
(3.1.4)
(r) = FR
The ISTS signal contains three modes, acoustic, structural, and thermal, with
respective amplitudes A, B, and A+B.
These modes are analogous to the Brillouin,
Mountain, and Rayleigh modes observed in frequency domain light scattering.
The
relaxation strength is called the Debye-Waller factor or the non-ergodicity parameter, and
is given by the amplitude ratio
2
fCoo = I--
(3.1.5)
c.
B
A+B
This expression is valid when the acoustic, structural, and thermal modes satisfy
the condition,
A>>FR>>H.
This measurement is equivalent to the integration of the
Mountain mode in light scattering or the integration of the quasielastic peak in neutron
scattering. This measurement is used to test the predictions of Mode Coupling Theory.
3.2
Experimental
3.2.1
Nd:YAG laser
The excitation source for all ISTS measurements in this thesis is a Nd:YAG laser,
originally manufactured by Spectron, and modified to produce the desired output. The
detailed cavity design is described elsewhere. 3 The original continuous wave (cw) laser
is modified by incorporating a mode-locker, q-switch, and Pockel's cell for cavity
dumping.
When operating optimally, the output consists of 120 ps pulses with 700
microjoules per pulse, at a variable repetition rate from tens of Hz to 1 kHz. Frequently,
when the experiment doesn't demand short pulses, the cavity length is detuned for a pulse
duration in the vicinity of 200 ps, achieving more stable operation.
The mode-locking frequency is generated by a radio frequency source, which is
amplified for the mode-locker. The driver for the Pockel's Cell, manufactured by Medox,
includes a frequency divider. The mode-locking frequency is variably divided by the
Medox controller to determine the repetition rate of the laser and produce a trigger for the
Stanford digital delay generator. One output from the delay box triggers the q-switch.
This same output is again variably delayed by the Medox controller to fire the Pockel's
cell. The other outputs are available to control the probe laser timing, or serve as triggers
for detection systems.
Optimal operation of the laser requires careful attention to several diagnostics.
Since the laser cavity contains multiple elements with several adjustments each, no one
diagnostic is sufficient to optimize the laser for energy, stability, and pulse duration.
Among the properties that must be monitored are q-switch profile, pulse duration, energy,
and cw-steady-state level. Once the laser is running well, it will run for several days
without serious adjustment.
The temporal shape and stability of the q-switch envelope are the best diagnostics
of overall laser energy and stability. The cavity alignment should be optimized so that
the envelope rises as quickly and as high as possible. Care should be taken to iteratively
optimize the cavity alignment and the alignment of the signal onto the photodiode. A
large area photodiode should be used to eliminate this complication. The cw-steady-state
level maintained between q-switch bursts should be set to achieve the desired balance
between energy and stability. The higher this level, the more stable the output, but at the
expense of output energy.
Once a stable q-switch envelope is achieved, cavity-dumping should be optimized
by Pockel's cell alignment and timing. The timing should be set to sharply cut off after
the highest pulse in the q-switch envelope.
The alignment of the crystal should be
adjusted to minimize any trailing or leading pulse, as seen on a photodiode monitoring
the laser output. Typically, the output easily saturates a photodiode, however, saturation
by the peak pulse may be necessary to see the trailing or leading pulses. An extinction
ratio of 500:1 can be achieved. Mode quality is also an indication of good Pockel's cell
alignment.
The lower the time constant controlling the timing, the better the laser
alignment, as a low time constant is indicative of a quickly rising q-switch envelope.
The pulse duration depends on the relationship between the mode-locking
frequency and the cavity length, and is roughly determined from a scanning
autocorrelator.
pulsing".
Autocorrelation is necessary to ensure that the laser is not "double-
The Medox Pockel's cell currently in use was not originally designed for
internal cavity operation. The crystal is parallel cut, and can create etalon effects in the
cavity. This results in a "double" pulse in time, two 100 picosecond pulses separated by
100 ps. This behavior has been observed with both a streak camera and an autocorrelator.
With a wedge cut crystal in the cavity, this behavior could not be duplicated.
One
"quick-and-dirty" method of estimating the pulse duration is to watch the stability of the
cw level. Unfortunately, this is least stable when the pulse is shortest, or the laser is
double pulsing. If a short pulse is not necessary, the cavity length can be optimized for
stability. The laser has not been demonstrated to operate with pulses longer than 300 ps.
Pulse duration has been found to be largely insensitive to minor adjustments in the cavity
alignment.
Major alignments, such as TFP angle, Pockel's cell position, Brewster angle for
the mode locker and q-switch, and Bragg angle for the mode locker should only be
adjusted in extreme situations. Under typical operating conditions, the only adjustments
that should be made are to the end mirrors, Pockel's cell alignment, cavity length, and
timing.
3.2.2
Generation of the excitation grating
The "traditional" method of generating the diffraction grating in ISTS experiments
is to manually align the excitation beams at the desired angle. The probe is also manually
aligned to be incident on the grating at Bragg angle.
Figure 3-1 illustrates the optical
configuration. The excitation beam is split into two arms with a 50% beamsplitter. The
two arms are aligned onto the sample, and focused in the direction perpendicular to the
wavevector. Timing can be controller with the translation stage, but is not a sensitive
adjustment using 100 ps excitation pulses.
The probe is focused with an additional
cylindrical lens parallel to the wavevector to produce a spot as close to round as possible.
To generate small wavevectors, with angles less than 2 degrees, a combination of a
dichroic beamsplitter and an additional 50% beamsplitter for the excitation may be used.
Alignment is optimized by visual or electronic observation of the signal. The signal
follows a long path from the sample to the detector, to facilitate separation of the signal
from scattered light from the transmitted probe beam. Transmitted excitation light can be
separated using a color filter.
All data in this thesis were obtained using the method described above.
An
alternative to the "traditional" method of manually aligning the two arms of the excitation
to cross at the desired angle in the sample is to use a phase mask. A phase mask is a
piece of fused silica with a pattern etched to a specified depth. Figure 3-2 shows the
design for a set of masks designed for several applications, including ISTS. The drawing
contains the geometry of the design for the 4 inch darkfield chrome on quartz mask. The
patterns consist of an array of vertically oriented bars and spaces with the widths of the
bars equal to the spacing between the bars. The value in microns for the width and
spacing of each pattern is given by the number in the box. An example for 3 microns is
shown immediately at the bottom of the picture.
When an optical beam is passed through the phase mask, multiple higher orders of
diffraction are created. The diffraction angle is determined by the feature size of the
mask and the wavelength of the light. The relative amplitude of the diffraction compared
to the zero order is also determined by the wavelength of the light and the etch depth of
the mask.
For an etch depth equal to the light wavelength, virtually no light is
transmitted in the zero order.
_~~~
~
_~ _~_
___
Excitation Beam
Probe Beam
Cylindridal Lens
V
Translation Stage
ST=50 %
R=50%
Translation Stage
Translation Stage
Cylindrical Lens
Sample
Diffracted
Signal
Figure 3-1: Illustration of the optical setup for generation
and probing of the grating. Not drawn to scale.
__
~_
_ C_ ___ _
___
5 mm
4mmI[mm
H
1mm
12
41231F
] F 61
m
4 or 10 mm
as indicated
H
2 mm
1 253011
1 1125]0
JpJ
12.5 13.5 14.51 5.5 16.51 8
5 mm
10
4mmI MM
21..5
H
12
14.5 5.51 6.51
37 35]
10
14
20
LII6122 11261
25 1
35
111 131 161 20
I
1 mm
12 jl14
S10
121 15
18 20
5mm
2
H
~~~~~~~~3h1~~~13
[H
F9
3
43
6
o
84o
1130
1 mm
1.
10 mmI
H1
5
2
50
100
H
1 mm
I8F1E196
EWW
SmmIE
H
1mm
3.00
ll'''' H
3,00
Figure 3-2: Phase mask design showing patterns obtained
for ISTS, ISRS, and other applications.
40
22 24
25
30
The +1 and -1 orders of diffraction are recombined with an imaging system to
create a grating with fringe spacing determined by the phase mask feature size and the
properties of the imaging systems. All wavelength components within the bandwidth of
the optical pulse create a grating with the same wavevector. This has the advantage of
eliminating the wavevector spread associated with the bandwidth of an ultrafast pulse. If
excitation and probe beams pass through phase masks with the same feature size, the
probe is automatically incident at Bragg angle regardless of the wavelength.
This
eliminates one major source of error in the alignment by the traditional method. Another
advantage of the phase mask method of generating gratings is that since the probe is split
in two, one arm can act as a "finder" beam for locating the diffracted signal in space.
This is particularly useful if the signal is weak, or the probe wavelength is not visible by
eye. Alternatively, the "other" probe beam can be attenuated and used for heterodyne
detection.
Figure 3-3 shows a simple imaging system using a phase mask. The mask is
placed at a distance F from the first lens, the lenses are separated by the sum of their
focal lengths, F + F2, and the sample is a distance F2 from the second lens. More
complex imaging systems can be designed, but have not yet been applied to ISTS
experiments. If F = F2, the imaging is 1:1, and the grating wavelength is equal to 2
times the spacing on the mask. Other imaging ratios can be selected to change the
wavevector of the grating.
3.2.3
Probing the grating
The probe source for the experiments in this thesis is a continuous-wave Argon
laser (Lexel 3500) operated with an etalon for a single line (514.5 nm), single mode
beam. The samples are transparent at this wavelength, reducing the possibility of signal
created by probing effects. The output is electro-optically gated to produce a quasi-cw
square pulse of arbitrary duration, to reduce heating of the sample. Careful alignment of
the gate device will achieve a pulse of constant amplitude after some initial ringing. Very
long pulses on the order of several milliseconds did not remain flat due to temperature
changes in the gate device, so to measure very long time signal gating was not used. In
this case, the sample was exposed to the beam for as limited a time as practical,
particularly at low temperatures.
reduce heating effects.
The laser was run at the lowest possible power to
Phase mask
Lens with f=F1
Excitation beam
Excitation beam
Probe
beam
"Lens
I
l
with f=F2
i
VSample
i Diffracted Signal
Figure 3-3: Illustration of a simple imaging system for
generation and probing of a grating using a phase mask.
Recently, a diode laser is implemented as a probe. The diode used is an SDL
543 l-G1, at 830 nm with a maximum output of 200 mW. The beam is reasonably well
collimated to a spotsize of 1 by 4 mm, using a collimation package from Thorlabs. A
quasi-cw pulse is produced by triggering the diode laser power supply. The diode laser
has the advantages of simple, reliable operation. Photodiode materials are often more
sensitive at 830 nm than for visible wavelengths. Since the wavelength is similar to the
1064 nm excitation wavelength, one phase mask etched at either 800 or 1064 nm
provides sufficient diffraction efficiency for both wavelengths. The major disadvantage
of using the diode laser is that the signal is no longer visible by eye, even with an IR
viewer and IR sensitive card. This difficulty is largely overcome by using the "finder"
beam generated by the phase mask.
3.2.4
Detecting the Signal
The signal is detected electronically with a fast photodiode and a digitizing
oscilloscope.
A Tektronix Digital Signal Analyzer (DSA) 602 with 11A72 1 GHz
bandwidth plug-in is used for all data acquisition function. One of two fast amplified
photodiodes was used to detect the signal. A photodiode manufactured by Antel (ARXSA) has a bandwidth from hundreds of MHz to dc, with a gain of 35000 Volts/Watt.
Alternatively, a photodiode from Hamamatsu, with bandwidth from 2 GHz to 3 MHz and
gain of 250,000 Volts/Watt is used. The high gain of the Hamamatsu diode makes it very
easy to detect the signal, but the low frequency roll-off prevents measuring any long time
dynamics. The use of these two diodes, and the procedures used to merge the data, will
be discussed further in the section on Data Analysis.
3.2.5
Temperature Control
All of the samples for the experiments in this thesis were contained in teflon-
coated aluminum sample cells. The windows of the cell are held in place with a flange
and o-ring, to permit some contraction of the sample volume as the material cools. The
pathlength of the cell is approximately one inch, so the depth of the optical grating is
always determined by the angle and the excitation spot size.
A resistance thermal
detector is sealed in the cell, immersed directly in the liquid. Using this design, the
sample maintains good optical quality throughout the supercooled temperature range, but
usually develops cracks around the glass transition temperature.
The sample is mounted on a cold finger in a cryostat.
Either a closed cycle
helium refrigerator, or a liquid nitrogen cold finger were used for these experiments. The
sample is directly attached to a copper block at the base of the cold finger, using indium
or copper mixed in vacuum grease to achieve good thermal contact. The copper block
contains resistive heaters for temperature control.
A platinum resistor monitors the
temperature of the copper block.
A Lakeshore temperature controller is used to monitor the temperatures of the
sample and the copper block. The temperature data are fed to a computer through a GPIB
interface.
3.2.6
Automation
The ISTS experiment on supercooled liquids is particularly well suited for
automation. Once the signal is acquired and optimized, little manual adjustment of the
laser or optics is required. A typical temperature range at a single wavevector for a
supercooled liquid will span 150 to 200 degrees, with temperature increments of 5 K or 1
K depending on the conditions. It takes at least 10 minutes per temperature for the
sample to reach thermal equilibrium, resulting in an experiment that takes 8 or more
hours, the majority of which is spent equilibrating the temperature.
A Labview® code has been written to automate the process of changing the
temperature, determining the equilibrium temperature, and acquiring data with the
oscilloscope. Labview is a graphical programming language designed to interface with
laboratory equipment. The temperature controller and oscilloscope both have a GPIB
interface. Briefly, the user creates an input file with the temperatures for data acquisition.
The user also initializes and saves the acquisition parameters for the oscilloscope. Then
the program takes over the experiment.
The temperature is set, and the oscilloscope
begins averaging the data once thermal equilibrium is reached. Temperature readings are
also taken during the 20 to 30 seconds it takes for the oscilloscope to complete averaging
to ensure that the sample temperature is not drifting or cycling. Although data analysis
has not yet been incorporated into the routine, it is also possible to create an interface
between Matlab® and Labview, making it possible to automate a rough fit of the data
while the sample is equilibrating at the next temperature.
Phase masks create even more possibilities for automation. The mask could be
translated automatically to change from one wavevector to another.
Given careful
imaging of the signal onto the detector, no manual adjustments would be required. This
would be particularly valuable if an experiment required measurements at multiple
wavevectors at exactly the same temperature.
3.3
Heterodyne Signal
When a heterodyne signal is deliberately desired and controlled, it can be used as
a tool for amplifying a weak signal, or favorably changing the time dependence of the
signal. However, when there is scattered light, there can be uncontrolled heterodyning
between the scattered light and the signal. Any contribution from a heterodyne signal
will have an effect on the parameters determined from the fit to the ISTS signal. The
fitting routine assumes that the signal is purely
(3.3.1)
S(t) oc GP,(t)l
However, if there is significant scattered light, the signal with a cw probe may
become
(3.3.2)
S(t) oc c lGp,,(t
)
+ c2 Gp,,(t)l2
with experimentally uncontrolled amplitudes. A third component, unlisted in (3.3.2) is a
dc signal which cannot be distinguished from scattered light. This contribution does not
affect the analysis, and only affects the experiment by possibly saturating the detector.
A signal with c, = c2 = 0.5 has been simulated using values of the parameters
typical of glycerol at low temperature and wavevector. The modes are clearly separated in
time, permitting measurement of the structural relaxation time and the Debye-Waller
factor. As c2 is of the order of magnitude of the scattered light, and c, is very small due to
the small diffraction efficiency, the amplitudes selected above are not unreasonable.
Theoretically, c, should average to zero because there is no fixed phase relationship
between the signal and the scattered light. It is assumed here that there is at least some
phase preference permitting observation of the heterodyne signal. The lack of control
over the phase of the light prohibits conducting the experiment to observe the heterodyne
term overwhelmingly. The original values, and the fit results are listed in Table 3-1.
Table 3-1: Simulated and Fit parameters for ISTS signal
with a mixture of homodyne and heterodyne signal. Errors
in the fit results are +/-5% or less
A
Original Value
0.4
Fit Result
0.56
Deviation
-
FA
16
27
+70%
_) A
FH
B
FR
P
t
DWF
50
50
0%
0.0001
0.77
0.02
0.60
0.03
0.66
0.000067
0.61
0.027
0.59
0.04
0.52
-33%
+35%
-2%
+33%
-21%
Due to geometric considerations, particularly the proximity of the signal to the
transmitted probe, heterodyning is more likely at low wavevectors. Heterodyne signal is
also more likely to have a significant contribution at lower temperatures, as the decreased
compressibility approaching the glassy state is accompanied by a decrease in diffraction
efficiency. The scattered light level also increases at low temperatures.
One signature of a heterodyne component to the signal would be an unusually
small value of the thermal diffusivity.
However, this measurement depends on the
accuracy in the determination of the wavevector. If the extent of heterodyning varies
within a wavevector, for example with translation of the sample, unusual variations in the
rate of thermal diffusion would be observed. The acoustic frequency is unaffected, and
the perceived damping is high. Once again, a large value of FA/q 2 could be an indicator,
but is subject to error in the wavevector and in the damping rate.
Determination of
damping rates at low temperatures and low wavevectors is also prone to walk-off errors.
The relaxation time is lengthened, but the value of 3 is relatively unaffected. As the
relaxation time should be independent of wavevector, these measurements can be
examined for evidence of heterodyne contributions. The large discrepancy in the DebyeWaller factor is of serious concern.
amplitude factors.
This change is the natural result of additional
1.0
0.8
cl=0; c2=1
............. cl=0.5; c2=0. 5
0.66-60.4
0.2
S 0.0
.*
I
I
0
1000
'
I
I
'
2000
'
3000
I
4000
'
I
5000
0.8
1.0
0.6
0.4
....
0.2
0.0
1E-3
0.01
0.1
1
10
100
1000
10000
Time (pts)
Figure 3-4: Comparison of simulated data for ISTS signal
with different amplitudes of homodyne and heterodyne
contribution. The data are simulated according to Equation
(3.3.2).
100 000
Simulated Data and Fit
cl = 0.5; c2 = 0.5
1.0-
0.8-
0.6-
0.4-
0.2-
0.0 -
I
_
_
1000
1000
I
2000
3000
4000
I000
5000
log Time (ps)
Figure 3-5: Simulated data with a heterodyne component
(3.3.2) and fit to homodyne ISTS signal (3.1.3). Data are
presented on a linear scale, with the inset showing the
acoustic signal. The simulation and fit parameters are
presented in Table 3-1.
1
Simulated Data and Fit
cl = 0.5; c2 = 0.5
1.0
0.8
0.6
0.4
0.2
0.0
*1...................................I_______
________~~
1E-3
0.01
0.1
1
10
100
1000
_____
10000 100000
log Time (ps)
Figure 3-6: Simulated data with a heterodyne component
(3.3.2) and fit to homodyne ISTS signal (3.1.3). Data are
presented on a log scale, with the inset showing the
acoustic signal. The simulation and fit parameters are
presented in Table 3-1.
3.4
Data Analysis Methodology
3.4.1
Accounting for detector response
The derived function for the ISTS signal assumes ideal time resolution.
In
practice, this is not the case, and the observed signal is affected by the excitation pulse
duration, the probe pulse profile, and the response function of the experimental detection
system. Specifically, the observed signal is given by 4
(3.4.1)S,,h,,(t) oc fR(t - t') Acw
O(t - t')G,(t' -t") A, (t")dt"
dFt'
For all of the experiments in this thesis, the limiting factor in the time resolution is
the finite response time of the electronic detection system, which is much longer than the
pump pulse duration. When using a quasi-cw probe pulse profile and a detector with dc
response, the convolution over the pump has no effect on the signal.
The response function of the detection system is determined by measuring the
response of the system to a pulse much shorter than the rise and fall times of the system.
The detection system includes the digitizing oscilloscope, amplified photodetector, and
any connectors and cables. This measurement was made for both a sub-picosecond pulse
and the excitation pulse for the ISTS experiments. The response to the picosecond pulse
was slightly longer in duration with less ringing, and this response was used in the data
analysis.
The Antel amplified photodiode has a frequency response from dc to several
hundred MHz. The fall-off of the response starting above 200 MHz strongly affects the
acoustic ISTS data. The impulse response must be deconvolved from the data to obtain a
more accurate representation of the material response. The Hamamatsu detector has a
level high frequency response, but a low frequency roll-off at 3 MHz. In both cases, the
oscilloscope digitizes the signal with a bandwidth of 1 GHz.
differences in the appearance of
Figure 3-7 shows the
identical data from glycerol, taken with the two
detectors.
Comparsion of ISTS Data from Antel and Hamamatsu Detectors
Antel
0
100
200
30 0
)
400
50C
Hamamatsu
-100
0
100
200
300
400
Time (ns)
Figure 3-7: Example of raw ISTS data taken with two
different detectors. The Antel data show attenuation of the
high frequency component of the signal. The Hamamatsu
data show a low frequency roll-off in the long time signal.
500
Accounting for detector response may not be necessary if the object of the
experiment is simply to measure the parameters such as acoustic frequency and thermal
decay rate. However, for ISTS experiments, one of the values determined is the ratio of
high frequency to low frequency response. In this case, even though a high frequency
oscillation may easily be observed and measured, the amplitude of the high frequency
response relative to dc can be significantly affected by the bandwidth characteristics of
the detection system.
Deconvolution of the detection response is also important in making very accurate
measurements of acoustic frequency and especially the damping rate near the upper
frequency limit of the detection system. In the frequency domain, the Brillouin peak,
represented by a Loretzian lineshape, is multiplied by a decreasing frequency response.
When fitting the data, this can result in a systematic underestimation of the frequency.
The damping will also be affected since the lineshape is no longer strictly Lorentzian.
The convolution described in Equation (3.4.1) can be rewritten as a product in the
frequency domain.
(3.4.2)
S0I,,()
c [R(co) x G,.(wc)]
Z
One option for deconvolving the impulse response is to transform into the frequency
domain, and do the division in Equation (3.4.2) to obtain GT(co). The frequency domain
response R(co) of the system is given by the Fourier Transform of the temporal impulse
response. However, practical considerations limit the accurate measurement of the
relative amplitudes R(o) at high and low frequencies determined solely from the fast
temporal response. At the low frequency end, the data must continue indefinitely in time
to determine the dc amplitude. To measure the high frequency spectrum, the data points
must be closely spaced. The duration of the impulse response is on the order of a few
nanoseconds. Quantitatively, the sampling rate must be on the order of 1 point per 50
picoseconds to accurately measure the impulse response. For the Fourier transform to
include frequencies as low as a few kilohertz, the total duration of the signal must be on
the order of 1 millisecond. A 1 millisecond duration signal, sampled at a rate of 100
samples per second, has 107 points. The oscilloscope cannot acquire an arbitrarily large
number of points with an arbitrarily long time duration for the data. Acquiring a shorter
waveform and padding with zeros can give the appropriate properties of the data, but the
number of points is computationally unwieldy.
To take the frequency domain approach, the data must also be Fourier
transformed. The ISTS data at short time commonly consist of 1024 or 2048 points at a
sampling rate of 2 Gs/sec. In order to perform the discrete division with the response
function, this data must have the same time axis as the response function. The data must
be interpolated to the sampling rate of the impulse response, and extended to the same
long time limit, on the order of 1 millisecond. Another potential pitfall is that if any
ringing or aliasing occurs in the Fourier transform, the high frequency components will
be amplified by the division of the response function which approaches zero at very high
frequencies.
The data after the division must subsequently be passed through a low
bandpass filter designed to not recreate the original problem of high frequency
attenuation. Finally, the data should be reduced to the original sampling rate so they can
be easily fit and manipulated.
The approach selected is to correct the data by deconvolution in the time domain.
Matlab was used for its flexibility in data manipulation, its graphical capabilities, and its
built-in convolution functions. Several key Matlab codes are included in the Appendix of
this thesis.
The procedure involves resampling the short time, acoustic part of the ISTS data
up to the sampling rate of the real data for the impulse response. The measured impulse
response is deconvolved from the resampled data. The data are subsequently resampled
back down to the original rate. Without resampling, the ISTS data only have a few points
in the duration of the impulse response. To maintain the low sampling rate, the impulse
response would only be represented by 2 or 3 points. This method is prone to error in the
measurement of the impulse response. In addition, the impulse response (Figure 3-8)
cannot be fit to a functional form with a high degree of accuracy. Figure 3-9 shows the
results of deconvolution of ISTS data from the Antel detector, at an acoustic frequency
near 300 MHz. Note that the deconvolution doesn't seem to completely recover simple
liquid response, with complete modulation of the first acoustic period. The reason for
this is not completely understood. Perhaps the measured impulse response does not give
sufficient quantitative information on the relative amplitudes of dc and high frequency
response. As the data from the Hamamatsu detector show complete modulation, it is
assumed that this is still an electronic, not a physical, effect.
Antel Response to Picosecond Pulse
0.180.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
-1.6
-0.8
0.0
0.8
1.6
2.4
3.2
4.0
Time (ns)
Figure 3-8: Measured Impulse Response of the Antel
detector. This is the response used in the deconvolution of
the ISTS data.
4.8
Comparsion of Original and Deconvolved Data
Antel Detector
Original
"
-100
0
100
200
300
400
_--
-100
0
100
200
300
500
Deconvolved
400
500
Time (ns)
Figure 3-9: ISTS data taken with the Antel detector. The
deconvolved data show an increased depth of modulation in
the acoustic signal.
The above discussion focuses on attenuation of the acoustic response in ISTS due
to finite bandwidth of the detection system. In addition, the opposite problem on the low
frequency end is encountered when using the Hamamatsu detector with a low-frequency
roll-off in the amplifier. Figure 3-10 shows the response of the Hamamatsu detector to a
quasi-cw pulse. The detector sees the rise and fall of the pulse, but doesn't give the dc
signal.
Response of Hamamatsu Detector
to 200 [ts quasi-cw pulse
I0
0
-50
0
/
/I
50
200
250
300
Time (jis)
Figure 3-10: Response of the Hamamatsu Detector to a
quasi-cw probe, 200 [ts in duration.
At present, no rigorous correction is made for this problem. Instead, the data are
treated as if the probe pulse has an intensity that decreases at long times. The slope of the
decay is estimated from the response of the Hamamatsu detector to the flat quasi-cw
probe (Figure 3-10). The data are divided by this decaying function. Data from the
Hamamatsu detector are only used until the end of the acoustic signal, after which the
data from the Antel detector are used.
The correction to the high frequency data
improves accuracy in the combination of the data from the two detectors, but does not
necessarily correct for different amplitudes of the acoustic and slow components in the
Hamamatsu data.
Comparsion of Original and Corrected Data
Hamamatsu Detector
Original
_-
0
0
100
300
200
400
-
500
CCorrected
1
-100
0
100
200
300
400
Time (ns)
Figure 3-11:
ISTS data taken with the Hamamatsu
detector. The corrected data show a long time signal more
compatible with the data from the Antel detector in Figure
3-9.
500
3.4.2
Time Domain fits to the data
The time domain data are fit to the signal derived from hydrodynamics theory. A
Levenberg-Marquat algorithm 5 is used to conduct a non-linear least squares fit to the
data.
Whenever possible, the data are fit with equal weights to all points.
When
experimental conditions such as a poor baseline warrant, the higher quality portions of the
data are weighted more heavily.
3.4.3
Alternate analysis of relaxation mode
An alternative method to study the relaxation dynamics and measure the Debye-
Waller factor would be to analyze the data using a method similar to one employed for
neutron scattering measurements of the autocorrelation function. 6 Take the square root of
the data to obtain
(3.4.3)
S' oc (A + B)e - "' - Ae-r'"cos(2roAt) - Be - (r
The data at long time can be fit to a single exponential decay to obtain the
quantities (A+B) and FH.
The data are subtracted from the fit of the thermal decay,
resulting in
(3.4.4)
- (r
S" oc Ae-r" cos(2coAt)+ Be
')
'
At times longer than a few times the acoustic damping FA, the signal is simply the decay
function
(3.4.5)
S"'= b(t)= Bexp(-(
1
Pt)
)
The relaxation function, in the form
(3.4.6)
- In = Fft -ln B
can be fit to a power law equation to determine the parameters B, FR, and p. This method
of analyzing the relaxation function should be far less dependent on the response of the
detector at high frequencies, since the acoustic mode is not included in the analysis. The
relaxation function (3.4.5) can be tested for the prediction of time-temperature
superposition. The relaxation can also be tested for deviation from KWW behavior.
References
1
Y.-x. Yan and K. A. Nelson, J. Chem. Phys. 87, 6240 (1987).
2
Y. Yang and K. Nelson, J. Chem. Phys. 103, 7722 (1995).
3
S. Silence, Ph.D. Thesis, Massachusetts Institute of Technology, 1991.
4
A. Duggal, Ph.D. Thesis, Massachusetts Institute of Technology, 1992.
5
W. H. Press, W. T. Vetterling, S. A. Teukolsky, and B. P. Flannery, Numerical
Recipes in C (Cambridge University Press, Cambridge, 1992).
6
J. Wuttke, W. Petry, and S. Pouget, J. Chem. Phys. 105, 5177 (1996).
Chapter 4
ISTS of fragile glass-formers
Previous ISTS experiments have focused on fragile glass-forming liquids,
specifically phenol salicylate (salol), 1 Ca 4K6(NO3) 14 (CKN), 2 and n-butylbenzene. 3 Salol
and CKN have been several of the most studied supercooled liquids. In general, Mode
Coupling Theory has had good success in describing the dynamics of these systems.
Several figures are reproduced in this chapter for illustration of experimental tests of
Mode Coupling Theory, and for future reference.
As discussed in the previous Chapters, ISTS measures the a relaxation in the time
domain, in the temperature range surrounding common values of Tc found in fragile glass
forming liquids. The a relaxation has been observed in salol from 280 to 220 K (Tg =
215). Notice that since salol is a fragile liquid with relatively short relaxation times until
the temperature is quite near the glass transition, the relaxation time can be measured to
fairly close to the glass transition (Figure 4-1).
In salol, the change in slope of the Debye-Waller factor is clearly observed from
ISTS measurements (Figure 4-2). The value of To = 266 K is within experimental error
of values obtained from neutron scattering. It is also interesting to note that a change in
the behavior of P occurs in the temperature vicinity of Tc (Figure 4-3).
This is not
predicted by Mode Coupling Theory, as the a relaxation is theoretically arrested at Tc.
However, this observation is in agreement with the physical picture of a change in the a
relaxation dynamics at Tc.
These data also serve as evidence that time-temperature
superposition does not hold below Tc.
Similar results are obtained from measurements on CKN (Figures 4-4 and 4-5).
In this case, the "cusp" is more rounded than in salol.
This effect is attributed to
thermally activated processes, which are included in the extended version of Mode
Coupling Theory.
temperatures near T.
As with salol, a change in the behavior of P is observed at
A crossover temperature is also observed in butylbenzene, as
shown by the Debye-Waller factor data in Figure 4-6. The 1 value also shows a decrease
in the vicinity of Tc.4
Average Relaxation Time in Salol
10°
100
10 -
10-2
10-3
10-4
10 -5 -
1010 -6
-7
10
10 -
10o
200
220
240
260
280
300
320
340
360
380
400
T (K)
Figure 4-1: Temperature Dependence of the Relaxation Time in
Salol. Includes a comparison to Dynamic Light Scattering Data
Debye-Waller Factor of Salol
0.3 L
230
240
250
260
270
280
T (K)
Figure 4-2: Temperature Dependence of the Debye-Waller
factor in Salol, in the q-*0 limit. A value of Tc = 266 K is
obtained from a fit to the Mode Coupling Theory prediction.
Stretching Parameter P vs Temperature
1.0
0.9
S0
0
Og
0.8
0.7
o q=B m-'
0.6
IT
11-
0.5
a
q=B m-I
V
q=B m-1
*
q=1.260 ptm
*
DLS
-
M PCS
0.4 1
200
[
I
220
I
I
240
I
I
I
260
I
280
I
300
350
T (K)
Figure 4-3: Temperature Dependence 3 in Salol. Data obtained
from Dynamic Light Scattering is included for comparison. A
change in behavior is observed near Tc = 266 K obtained from
Debye-Waller Factor measurement.
400
Debye Waller Factor for CKN
0.66
0.64
0.62
0.60
A
0.0
0.58
A
0.56
I
360
I
I
I
I
400
380
T (K)
Figure 4-4: Temperature Dependence of the Debye-Waller
factor in CKN, in the q->0 limit. A value of T c = 378 K is
obtained from a fit to the Mode Coupling Theory prediction.
A
I
420
Stretching Parameter P of CKN
0.8
0.7 -
0.6
X++
0.5
M
v
~
WI-----
O q=0.232 im
A q=0.235 Vm
A-M
0.4
-
q=0.227 Vm
-
AC q=0.336 lm
O q=0 423 im'
T
_
0.3
X q=0.623 jlm
-
+ q=0.896lm
--- NS
A PCSI
0.2
* PCS2
I
340
I
360
I
380
,
I
400
420
*
440
DLS
,
I
,
460
T (K)
Figure 4-5: Temperature Dependence P in CKN. Data obtained
from other experimental techniques is included for comparison.
A change in behavior is observed near Tc = 377 K obtained
from Debye-Waller Factor measurement.
Debye-Waller Factor in Butylbenzene
0.54 0.520.500.480.460-"
0.44U
0.420.400.38- 140
U
U
142
144
146
148
150
152
154
156
Temperature
Figure 4-6: Temperature Dependence of the Debye-Waller
factor in butylbenzene. The figure is reproduced from Ref. 3.
158
References
1
Y. Yang and K. Nelson, J. Chem. Phys. 103, 7732 (1995).
2
Y. Yang and K. A. Nelson, J Chem. Phys. 104, 5429 (1996).
3
L. Muller, Ph.D. Thesis, University of Texas, 1995.
4
DisorderedMaterials and Interfaces, Vol. 407, edited by H. Z. Cummins, D. J.
Durian, D. L. Johnson, and H. E. Stanley (Materials Research Society, Pittsburgh,
PA, 1996).
Chapter 5
Impulsive Stimulated Thermal Scattering
Study of Glycerol
5.1
Experimental background
Glycerol has been extensively studied experimentally, in an attempt to extend
Mode Coupling theory analysis to a glass-forming liquid of intermediate strength.
Experimental techniques used to measure the properties of glycerol include neutron
scattering, 1- 8 light scattering, 2,9, 10 dielectric
spectroscopy, 11-14
non-resonant
burning,15 specific heat spectroscopy,1 6 NMR8,17 and other techniques.18
hole
The
experiments most pertinent to testing MCT are summarized here, along with their results.
R~ssler et. al.9 conducted frequency domain light scattering experiments to
measure the low frequency Raman spectrum, in the frequency range from several hundred
GHz to 10 THz. In this frequency range, a minimum in the susceptibility spectrum is
observed for the temperature range from 298 K to 403 K.
To analyze the data in the framework of MCT, the susceptibility spectrum, and
the data at different temperatures are scaled according to
z"min.
The resulting data are fit
to the MCT prediction for the shape of the minimum, with parameters b = 0.71 and a =
0.342. If the MCT restriction for the exponents is lifted, a better fit, particularly at higher
temperatures, is obtained with b = 0.37 and a = 1.16. This violates the MCT restrictions
for the magnitude of a and the relationship between a and b. Agreement of the data with
the scaling relations for
Z"mn
and o,,i,, can be found for a = 1.16 from the unrestricted fit.
A value of Tc - 300 K is obtained from the power law fits. If the parameters from the
restricted fit are used for the power law fit to viscosity measurements, Tc = 320 K is
obtained. The best fit to the viscosity data results in y = 2.5 and T, = 310 K.
From the large value of a = 1.16 obtained in the unrestricted fits to the
susceptibility, it becomes evident that the strong vibrational contribution to the spectra in
the THz regime complicates the MCT analysis. To obtain a more realistic analysis, the
vibrational component of the spectrum, obtained from low temperature data, is subtracted
from the spectrum. The resulting master fit with the MCT restriction results in the same
parameters a = 0.342 and b = 0.71. The best unrestricted fit is much closer to the MCT
restriction, with b = 0.49 and a = 0.69. The resulting power law temperature dependence
is insensitive to the change in exponents.
The TC value of 310 K is significantly farther above the glass transition than
values obtained for more fragile glass-forming liquids. For glycerol, Tg = 185 K and Tc =
310 K yields Tc
1.7 Tg, where typical values for fragile glass-formers are T, - 1.2 T,.
The MCT predictions for the high frequency side of the minimum are tested
directly and independently of the rest of the minimum, using the data minus the ca
relaxation spectrum. The resulting data show a crossover from a power law with a - 1 to
a = 0.34 closer to the frequency minimum. The conclusion is that MCT is only valid up
to the frequency where the boson peak or the vibrational contribution complicates the
high frequency spectrum.
Simultaneous studies of neutron and light scattering are conducted to compare the
two techniques and test the predictions of MCT. 2 The data from both techniques coincide
at high frequencies greater than 1.4 THz, where the spectra are constant with temperature
both above and below the glass transition.
A free fit to the susceptibility minimum, discounting the MCT restriction, gives
an exponent of a = 1, corresponding to the boson peak rather than the P relaxation. Data
at lower temperatures show the power law low frequency wing of the 3 feature, and fits to
this data for a more limited temperature and frequency range result in the parameters a =
0.54 and b = 0.43 without the MCT restriction and a = 0.32 and b = 0.61 with the MCT
restriction.
The latter fit poorly represents the minimum frequency.
relaxation only results in b = 0.54 and k = 0.76.
A fit to the c
A power law fit to the minimum frequency using the a value obtained from the
restricted MCT fit yields T, ; 225 K. However, this temperature cannot be used to fit the
X"min
data. A power law fit to the viscosity indicates a Tc value greater than 300 K.
The light scattering data were more extensively analyzed in the framework of
MCT. 19 An additional term for linear coupling is incorporated in the memory function
(5.1.1)
m(t) = v(t)
+v 2
(t) 2
The coupling constants are allowed to vary with temperature, and the resulting calculated
spectra are similar to the experimental spectra from glycerol. The exponent parameters,
however, are not in agreement with the original MCT prediction. The value of To used in
the calculations is between 223 and 233 K. No more specific prediction of Tc is made.
The exponent parameters obtained are a = 0.314, b = 0.591, and X = 0.730. This fit
includes the crossover from P relaxation dynamics to vibrational dynamics, as well as the
a relaxation to p relaxation crossover.
Glycerol has also been studied by dielectric spectroscopy.12,13, 20
There is a
pronounced difference between the dielectric loss E"and the light or neutron scattering
susceptibility at high frequencies in the THz regime. 13 The explanation offered is that the
vibrational motions that contribute to the high frequency feature in the light scattering
spectra are not coupled strongly to dipolar reorientations, which are probed in dielectric
spectroscopy.
The spectra below the minimum are in good agreement.
From this
agreement, it is concluded that "In the a-relaxation regime the reorientation of the
glycerol molecules involves the tear and repair of hydrogen bonds which leads to the
observed strong coupling of structural and dipolar relaxation".13,16
The susceptibility minimum is fit to the MCT prediction with exponenets a =
0.325 and b = 0.63, although the frequency range does not extend very far above the
minimum. The power law relations are tested for the frequency of the minimum (Vmin),
the magnitude of the minimum (E"mi), and the maximum of the a relaxation (vmx).
Consistent fits are obtained with TC = 262 K.
Another dielectric study yielded Tc =
248.8 K from a power law fit to the peak frequency. 21
Table 5-1: Experimentally determined exponents for the
susceptibility spectrum of glycerol.
Reference
Method
Exponents
Tc (K)
Wuttke et al. 2
Wuttke et al.
Rissler et. a19
Rissler et. al
Franosch et al. 19
Light
Light
Light
Light
Light
225
Schonhals, et.
al. 2 1
Lunkenheimer et
al. 13
Dielectric
Spectroscopy
Dielectric
Spectroscopy
a = 0.32; b = 0.61
a= 1
a = 0.342; b = 0.71
a = 1.16
a = 0.314; b = 0.591;
k = 0.730
y= 3.65
Scattering
Scattering
Scattering
Scattering
Scattering
a = 0.325; b = 0.63;
k = 0.705
320
300
223-233
MCT
restricted
yes
no
yes
no
yes
248.8
yes
262
yes
Glycerol has also been studied by the more unusual method of specific heat
spectroscopy. 16 Specific heat spectroscopy measures the real and imaginary parts of the
product of the constant pressure specific heat and the compressibility, Cp.
IKdoes not
vary with either temperature or the frequency in the ranges studied in this experiment, so
the temperature and frequency dependences observed are attributed to the properties of
cp. The real part shows a change in behavior from liquid-like to glass-like, with a lower
cp in the glass. The transformation occurs at lower temperatures when the measurement
is made at lower frequencies. This behavior is analogous to the change in the speed of
sound with temperature observed in ISTS.
The imaginary part of the heat capacity is treated as a susceptibility spectrum.
Fits are generated to the Fourier transform of a KWW function, with P = 0.65±0.03 for all
temperatures in the range from 200 to 219 K. The peak frequency from the spectrum is
fit to the power law equation
(5.1.2)
v= vo[(T-)/
0
]a
with parameters To = 169 K and a = 15.0. This equation is mathematically equivalent to
the prediction of Mode Coupling Theory for the temperature dependence of the peak
frequency of the a relaxation.
However, the To value obtained is below the glass
transition. In fact, the data are below the TC = 1.2 Tg seen for many fragile glass forming
liquids. This result should not be discounted however, as MCT makes no statement as to
the location of T. relative to the glass transition temperature.
The acoustic properties of glycerol have been studied by an ultrasonic
experiment. 22
The sound velocity and attenuation are measured as a function of
temperature at 2 MHz, 10 MHz, and 27 MHz, very close to the frequencies associated
with lowest wavevector ISTS measurements. The observed behavior shows the expected
frequency dependent change from liquid to glass behavior in the speed of sound. There is
also a temperature and frequency dependent maximum in the acoustic attenuation rate
corresponding to temporal overlap of the acoustic period with the relaxation time.
The longitudinal compliance is calculated from the acoustic parameters.
The
resulting susceptibility spectrum is fit to a KWW function with P = 0.60±0.05.
The
maximum frequencies from these spectra are combined with the maximum frequencies
from the specific heat data, and again fit to Equation (5.1.2). The fit parameters are To =
175.5 K and a = 12.5. The same data are fit to a VTF equation with B = 2310 and TTF
=
129.
From the variety in the results described above, it is clear that there is a need for
additional testing of MCT predictions for glycerol. ISTS is used to study the density
dynamics specifically. So far, evidence of To obtained from the behavior of the DebyeWaller factor has not been reported, 23 and this study seeks that evidence.
5.2
Experimental
Glycerol
(99.5+%, <0.1% water, under nitrogen, Aldrich) was used without
further drying or purification. The sample was transferred under nitrogen into a specially
designed Teflon-coated cell that incorporates moveable windows to reduce sample stress
and cracking. The sample was mounted onto a cold finger outfitted with resistive heaters.
A resistance thermal detector immersed in the sample measures the temperature.
Data
were taken at temperature intervals of 5 K in the high and low temperature regions, and at
1 K intervals in the region where structural relaxation is well separated from the acoustic
and thermal modes.
For all temperatures, the sample equilibrated until the standard
deviation of 10 consecutive resistance measurements was less than 0.005K. The sample
maintained good optical quality over the entire temperature range from the high
temperature liquid to the glass transition. Cracking of the sample prevents measurement
at temperatures below the glass transition.
The ISTS experimental design is described in detail in Chapter 3. The excitation
pulses are 200 ps, 500 VLJ at X=1.064 Lm, with the pulse duration deliberately increased at
small wavevectors for increased stability. Cylindrical focusing maximizes the excitation
spot size along the wavevector direction, with the spot size selected to generate 100
fringes or more at each wavevector. The large number of fringes in the direction of the
wavevector is necessary to avoid walkoff of the propagating acoustic waves. The height
of the grating, in the direction perpendicular to the wavevector, is approximately 100 jtm.
The experimental wavevector range is 0.08 Lm-' to 1.2 Im-', corresponding to fringe
spacings from 74 jm to 5 jLm. All wavevectors are determined by measuring the acoustic
frequency in an ethylene glycol sample, for which the speed of sound and its temperature
dependence are well known.2 4
At larger wavevectors, the angle is also measured
geometrically with a mirror mounted on a rotation stage.
The two measurements agree
to within 0.001 jtm).
A continuous wave argon laser is the probe.
Electro-optical gating reduces
sample heating. The diffracted signal is detected with a fast photodiode and a digitizing
oscilloscope, with 1024 shots averaged for each temperature. For frequencies below 200
MHz, the Antel diode with uniform frequency response to dc is used. At frequencies 300
MHz and higher the Hamamatsu diode acquires the high frequency acoustic component
of the data. Data analysis procedures have been described in Chapter 3.
5.3
Results
Figure 5-1 shows ISTS data at a small wavevector, in the temperature range where the
acoustic, structural, and thermal modes are well separated. Figure 5-2 shows additional
data at a higher wavevector. Note the change in the x axis (log scale), and the higher
temperature range where structural relaxation can be observed. Over the accessible range
of wavevectors in this study, the structural relaxation in glycerol is measured from tens of
nanoseconds to hundreds of microseconds.
The data are analyzed according to ISTS theory to extract acoustic, structural and
thermal diffusion parameters.
A Levenberg-Marquardt least squares fit to the time
domain signal expression determines the acoustic, thermal, and structural relaxation
parameters along with mode amplitudes.
ISTS Signal from Glycerol
q=0.305 im-'
T=350 K
cH
1E-3
0.01
0.1
1
10
100
1000
1E-3
0.01
0.1
1
10
100
1000
log Time (ps)
Figure 5-1: ISTS data and fits from glycerol at q = 0.305
ptm'. The data show the progression from simple liquid to
complex liquid with structural relaxation to simple glass
behavior.
ISTS Signal from Glycerol
q= 1.19
Tm-K
T=375 K
1E-3
0.01
0.1
1
10
100
1E-3
0.01
0.1
1
10
100
1E-3
0.01
0.1
1
10
100
H
r/
r/c
log Time (ps)
Figure 5-2: ISTS data and fits from glycerol at q = 1.19
tm 1-. The time axis is considerably shorter than for the
smaller wavevector in Figure 5-1.
5.3.1
Acoustics
The acoustic frequency and attenuation rate can be determined from a Lorentzian
fit to the power spectrum at all temperatures except for the region of maximum damping,
where structural relaxation interferes with the spectrum. The acoustic values can also be
obtained from a time domain fit to the data.
The deviation between the acoustic
frequency obtained in the time or frequency domain is less than 0.1%, so there is no
advantage to one method over the other. There is a larger difference in the values of the
acoustic attenuation, due to imperfections in the frequency response of the detection
apparatus or heterodyne effects. Acoustic attenuation cannot be measured accurately at
the smallest wavevector due to finite spot size effects.
At small wavevectors, the
attenuation can also be affected by uncontrolled heterodyning of the signal with scattered
light. It is impossible to determine the relaxation time corresponding to the maximum
acoustic attenuation with good accuracy. There is considerable uncertainty in the acoustic
attenuation measurements near the maximum attenuation.
In glycerol, the acoustic
oscillations are so strongly damped that only a few oscillations are visible. The data
presented in Figures 5-1 and 5-2 show this behavior. In addition, structural relaxation is
overlapping in time with the acoustic frequency, and must be included in the fit even
though the acoustic and structural relaxation timescales are not well separated. There is
also an uncertainty of a few degrees in the temperature where the maximum damping
occurs.
Speed of Sound in Glycerol
3500
3000-
A
q=0.305 gm-'
A
q=0.457 ±m-'
*
q=0.552 ptm-
O q=0.785 gm-'
#
2500-
q=1.036 gm-'
O q=1.186gm-'
2000 -
1500-
I
200
220
I
240
260
I
300
I
180
200
220
240
260
280
300
320
I
I
360
380
340
360
380
Temperature
Figure 5-3: Speed of sound in glycerol at several
wavevectors. All wavevectors show a change from high
temperature liquid to low temperature glass behavior. The
transformation temperature range is a function of
wavevector.
Acoustic Damping in Glycerol
1200
L
1000
A
q=0.305 gm-'
A
q=0.457 gm"
0
q=0.552 tm-'
A
0
q=0.785 gm'
A
'
O
q=1.036 gm-1
q=1.186 Rlm'
800
A
600
cj)
0
400
200
0
180
220
240
340
360
Temperature
Figure 5-4: Acoustic damping in glycerol at several
wavevectors. The maximum damping occurs at temporal
overlap with structural relaxation, and is a function of
wavevector.
The speed of sound shows the predicted temperature dependence, approaching
linear temperature dependence at the high and low temperatures (Figure 5-3).
At
frequency dependent dispersion occurs, showing
the
intermediate
temperatures,
increasing stiffness of the material on the time scale determined by the acoustic
frequency. Figure 5-4 shows the acoustic attenuation as a function of temperature. There
is a baseline of acoustic attenuation at high and low temperatures due to viscosity and
thermal diffusion.
The attenuation reaches a maximum at the wavevector-dependent
temperature where structural relaxation is occurring on the timescale of the acoustic
frequency. There is no evidence of increased acoustic damping at very low temperatures,
which would be attributed to the slowing down of the P relaxation into the MHz regime
near the glass transition.
The acoustic parameters are analyzed as a function of wavevector in Figure 5-5 to
confirm the linear dispersion relation for the speed of sound at high and low
temperatures. At 340 K in the liquid regime, and at 210 K near the glass transition, the
frequency varies linearly with wavevector. At 275 K, where structural relaxation occurs
on a time scale close to the acoustic frequencies in ISTS, the frequency does not vary
linearly with wavevector.
From the linear fit to the data, a slight deviation can be
observed. The deviation is very slight due to the limited wavevector range.
The acoustic attenuation varies as the wavevector squared at high and low
temperatures, where damping is due to thermal dissipation and viscosity (Figure 5-6).
The values obtained from the lowest wavevectors are not included in the linear fits at
high and low temperatures, as there is substantial error due to spot size or heterodyne
effects.
At intermediate temperatures where the structural relaxation increases the
damping rate, this relation does not hold.
Acoustic frequency vs. wavevector
800
700
600
500
400
300
200
-
100 0.
0.0
0.2
0.4
I
0.8
0.2
0.4
0.6
0.8
1.0
1.0
v
T=210K
*
T=275 K
.
T = 340 K
1.2
1.2
Wavevector (tm 1 )
Figure 5-5: Acoustic frequency vs. wavevector at several
At all temperatures, there is a linear
temperatures.
dispersion relation for the sound wave.
1.4
1.4
Acoustic damping vs. q
300
250
200
*
340 K
O
320 K
A
275 K
v
230 K
*
210K
A
0
150
A
O
A
100
A
50
SVV
V
R
0 n
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
q2 (Pm)-2
Figure 5-6:
Acoustic damping vs. q2 at several
temperatures. At high and low temperatures there is a
linear relationship. The data at intermediate temperatures
are affected by structural relaxation.
1.6
1.8
5.3.2
Thermal Diffusivity
The thermal decay rate is determined from the long time, single exponential decay
of the data. The thermal diffusivity is given by IH/q 2, and should be independent of
wavevector.
The thermal diffusivity varies weakly with temperature throughout the
majority of the supercooled regime (Figure 5-7). As the temperature approaches the glass
transition, the thermal diffusivity begins to increase to a value characteristic of the glass
state. The temperature where this occurs depends on wavevector, as the material must
behave like a glass on the time scale determined by the thermal decay rate. This behavior
is similar to the change in behavior of the speed of sound, except it occurs at a
temperature much closer to the glass transition as the time scale is much slower.
The
scatter and the apparent dip in the data at this same temperature range are due to difficulty
in fitting the data when the structural relaxation overlaps in time with the thermal
diffusion feature.
Thermal Diffusivity in Glycerol
0.30 -
N q=0.088 mO q=0O.117 gm-
0.25 -
A q=0.305 gm-1
>
0.20i
a
q=0.457 gm-
*
q=0.552 gm-
*
"
q=1.036 gm
<>o q=1.186 gm-
0.15-
Ht
S
0.10-
0.05
-
180
200
220
240
260
280
300
320
340
360
Temperature
Thermal diffusivity vs. temperature in
Figure 5-7:
glycerol. The increase at low temperatures signal the onset
The apparent wavevector
of the glass transition.
dependence is discussed in the text.
380
There is some deviation from the invariance of the thermal diffusivity with
wavevector at the smallest wavevectors.
One source of this discrepancy could be
inaccuracy in the determination of the wavevector. Another potential source of error at
very long times is a sloping baseline due to a intensity variation in the probe pulse. For
data where the baseline did not remain flat at zero, the data for the fit were cut off at an
earlier time. This could lead to a systematic error in the measurement of the thermal
decay rate. Occasionally, at long times and low temperatures, a very long time rise is
observed in the signal. In general, this effect decreases in intensity if the probe power is
decreased, and is attributed to heating of the sample by the probe. Every effort is made to
eliminate this problem, but it is not unusual to still have a non-zero baseline at long times
and low wavevectors. Again, the data for the fit are cut off at an earlier time. One of the
last two effects is considered the most probable source of the discrepancy in the
measurement of the thermal diffusivity at low wavevectors.
As discussed in Chapter 3, serious error can result from a contribution to the
signal from heterodyning with scattering light. However, it is expected that this would
result in low measurement for the thermal diffusion rate, and thus a low value of the
thermal diffusivity. As the error shown above is in the opposite direction, it is concluded
that this is not the source of the error. In addition, the values obtained for the thermal
diffusivity indicate that there is most likely not a large heterodyne contribution to the
signal.
5.3.3
Relaxation and Debye-Waller factor measurements
The structural relaxation feature appears at intermediate times and temperatures in
the data, and is fit to a Kohlrausch-Williams-Watts (KWW) stretched exponential
function.
The average relaxation times in Figure 5-8 are determined from the time
constant FR and the stretching parameter, P. There is a departure from Arrhenius
temperature dependence, as shown by the log plot of relaxation times in Figure 5-9. The
temperature dependence of the average relaxation time fits a Volger-Tamman-Fulcher
(VTF) form (Figure 5-10).
The parameters B = 2200 K and TVTF = 133 K agree very
well with results from dielectric spectroscopy. 14
Relaxation Times in Glycerol
100 - U
U
10
A
q=0.305 gm
A
q=0.457 pm
O
q=0.119 pm
*
q=1.036 pm
*
q=0.086 pm
*
q=0.552 pm
O
q=1.186 pm
0.1
220
225
230
235
240
245
250
255
260
265
Temperature
Figure 5-8: Relaxation times (log scale) vs. temperature in
glycerol at all wavevectors.
270
Arrhenius Fit to Relaxation Times
100
U,-
10
0.1
o.i
S------- Arrhenius Fit
0.01
~1
3.6
3.8
4.0
4.2
4.4
1000/T
Figure 5-9: Relaxation times at all wavevectors, with a best
fit to Arrhenius temperature dependence. The data deviate
from Arrhenius behavior.
4.6
Volger-Tamman-Fulcher Fit to
Relaxation Times in Glycerol
SIn <,>
_-
VTF Fit
To = 133 K
B = 2210 K
15
To=2x10- s
A
V
-2 -
-4_
220
230
240
250
260
270
Temperature
Figure 5-10: Vogel-Tamman-Fulcher fit to relaxation times
(In scale) in glycerol. Fit parameters are To = 133 K,
B=2210 K.
Stretching Parameter
P in Glycerol
0.70
Sq=0.552 pm
q=0.086 gim
q=0.119 pm
*
q=1.036 lm
q=0.305 ptm O q=1.186 pm
q=0.457 pm ----- p = 0.60
0.65
0.60-
0.55 -
0.50
I
210
'
I
220
'
I
230
*
I
240
*
I
*
250
I
260
I
*'
270
Temeprature (K)
Figure 5-11: Temperature Dependence of 3 in glycerol at
all wavevectors. Error bars are representative for +/- 5 %.
The dotted line indicates the average over all wavevectors
and temperatures.
280
Within error bars, the values of 3 are independent of temperature throughout the
range studied (Figure 5-11).
The average for all temperatures and wavevectors is
indicated by the dashed line through the data. Dielectric data show a slight increase of 0
from 0.55 at Tg to 0.8 near 300 K. 14 This is not inconsistent with ISTS results, but the
experimental error prevents a definitive statement supporting a temperature dependence
of p.
There is significant scatter in the measurements at different wavevectors. This
may be indirectly attributable to the variation of detector response with frequency, as
discussed in more detail below in reference to Debye-Waller factor measurements. The
quality of the fit to the acoustic damping can affect the fit to the "rise" in the data, which
determines the value of p. Figure 5-12 shows only the relaxation function, obtained by
subtracting the thermal decay from the square root of the data, as described in Chapter 3.
Data at different wavevectors and temperatures can be superimposed by scaling according
to the relaxation time, verifying the applicability of time-temperature superposition.
Structural Relaxation in Glycerol
0.80.70.60.5 O
0.4•v,=
d
0.3
-
0.2I
T=230; q=0.119 tm-'
0.1-
+tO
T=240; q=0.457 m-'
0.0-0.1 I
T=260; q=1.036 im-'
aI
"a
II
IE-3
II
I
0.01
''''"I1
1 1
'
0.1
'"aI I
1
I I I III"II
I
I
10
I
I
100
Log Time (ps)
Figure 5-12: The relaxation function in glycerol at several
temperatures and wavevectors. The data are obtained by
subtracting the thermal diffusion mode from the square root
of the ISTS signal.
The Debye-Waller factor is measured by the relative amplitudes B/(A+B). There
is no quantifiable temperature dependence within the error of the data (Figure 5-13) for
the entire temperature range of the measurements. There is no quantifiable temperature
dependence in any of the individual wavevectors.
The line indicating the average
magnitude of the Debye-Waller factor at all temperatures and wavevectors is included for
reference. In the temperature range studied, we do not observe a square root cusp, or any
other anomalous or discontinuous behavior in the temperature dependence.
The two
smallest wavevectors, where the acoustic frequencies are lowest and the detection
response should be the most predictable, are presented separately for clarity at the lowest
temperatures (Figure 5-14). The average for these two wavevectors is the same as for all
wavevectors within the error bars.
Debye Waller Factor in Glycerol
0.75 -
0.70 -
0.65 -
qsr+
0.60 -
*
q=0.086 m -'
*
q=0.552 tm-
O
q=0.119 tm -
*
q=1.036
A
q=0.305
A
q=0.457 pm
0.55
220
230
240
250
260
pm -
Sq=1.186 m-l
--.-DWF=0.66
270I
270
Temperature (K)
Figure 5-13: Debye-Waller factor in glycerol for all
wavevectors. There is no evidence of a MCT cusp in the
temperature range presented. The dashed line indicates the
average of all values.
100
m -'
280
Debye Waller Factor in Glycerol
0.700.68
T
-r
i
0.66
0OO
.. .. 0
......
----------It.-...........
IF --- -
..
In
0
0.64-
.... ......
---------
O
0.62
0.60
.
q=0.086 ptm
o
q=0.119 tm-'
DWF = 0.65
0.58 -....
215
220
225
230
235
240
245
Temperature
Figure 5-14: Debye-Waller factor vs. temperature at the
two smallest wavevectors, to highlight the data at the
lowest temperatures. The average value is the same as the
average for all points.
There appears to be some variation in the data with wavevector.
Since the
wavevectors in ISTS are several orders of magnitude smaller than the wavevectors in
neutron scattering where the Debye-Waller factor does vary with wavevector, the
possibility that the variation is due to the changing wavevector is rejected. In addition,
the acoustic measurements have confirmed that ISTS is in the hydrodynamic regime.
Since the acoustic frequency does vary with wavevector, the scatter in the DebyeWaller factor is attributed to a systematic attenuation of the acoustic mode signal by the
imperfect frequency response of the detector. In general, if there is attenuation of the
acoustic mode signal, the value deduced for A would be artificially small, resulting in an
apparent increased value of the Debye-Waller factor. In addition, the frequency response
of the detector does not necessarily vary smoothly on the scale of a few MHz, and this
can account for some of the error from point to point as well. Use of the two detectors
creates additional variation in the data. However, the data at each wavevector can be
viewed individually, at least qualitatively.
5.4
Discussion
ISTS measurements indicate an effective Debye-Waller factor that is independent
of temperature.
The magnitude of the Debye-Waller factor is compared to the
measurements for salol and CKN (Chapter 4).25
The average value of f0 = 0.66 is
significantly larger than the value fo = 0.36 above T, observed for salol. In fact, the
amplitude for glycerol is larger than the amplitude for salol even 20 degrees below Tc of
the latter. The amplitude is also larger than f = 0.575 in CKN.
Recall that the Debye-Waller measures the height of the plateau of the
autocorrelation function. The plateau occurs at the crossover between the fast and slow
relaxation features. A large Debye-Waller factor indicates that more relaxation occurs
through the slow relaxation process. When the Debye-Waller factor is discussed in terms
of integrated area of the relaxation spectrum, the Debye-Waller factor is the area of the
slow relaxation normalized by the entire integrated area of the both features.
The comparison of the Debye-Waller factor amplitudes in glycerol and in the
fragile liquids indicates that the amplitude of structural relaxation is larger in glycerol
than in these fragile systems. Conversely, it can be stated that fast relaxation is stronger
in fragile liquids than in glycerol.
It is interesting to observe however, the unusual
strength of the boson peak in glycerol, and the difficulty that has presented in the MCT
analysis of the susceptibility minimum. A low amplitude of the P relaxation relative to
102
the entire spectrum, which follows from a large Debye-Waller factor measurement, can
contribute to the difficulties in the MCT analysis of the high frequency susceptibility
spectra.
Our results for the Debye-Waller factor indicate that there is no Mode Coupling
Theory crossover behavior in the temperature region 225 K to 265 K. This temperature
region includes several previously cited Tc values. Tc = 262 K and T, = 248.8 K have
been obtained from a power law fits to of the dielectric data. 13,2 1 TC = 225 K has been
suggested, although again not convincingly, from light scattering measurements. A range
of T c = 223 to 233 K was used in fitting the light scattering spectrum with an expanded
MCT coupling term. Several high temperature values, greater than 300 K, also have been
reported.
To make the most conservative statement, these ISTS measurements only rule out
the possibility of a marked anomaly in the temperature region from 228 K to 268 K. It is
also unlikely that a cusp similar to those observed in salol and CKN occurs in the
temperature range from 270 to 275 K, as we do not observe the square root temperature
dependence, or any other significant variation with temperature in this temperature range.
The temperature range for these measurements barely extends down to 1.2 Tg = 222 K,
where T c is observed for fragile liquids. As MCT predicts a flat Debye-Waller factor
above Tc, within the quality of the data, we do not rule out the possibility of Tc at 225 or
lower. Efforts are being made to extend the ISTS measurements in glycerol to lower
wavevectors and temperatures to address this possibility.
If the crossover behavior occurs at a high temperature above 300 K, depending on
the amplitude of the temperature dependent term (hq), the data may appear flat within
experimental error at temperatures 40 to 70 degrees below the crossover. In fact, the
glycerol measurements are not inconsistent with the shape and amplitudes of the fit to
salol, with a high Tc. This possibility is also consistent with the large magnitude of the
Debye-Waller factor.
It is noted, however, that a crossover at high temperatures means
that the time scale for relaxation at To is considerably shorter in fragile liquids. From
dielectric datal4 at T c = 360 K for CKN, the peak frequency is on the order of 10 kHz,
103
while at 300 K in glycerol the peak frequency is greater than 100 MHz. The frequency
ranges covered by the Debye-Waller factor measurements from glycerol and the fragile
liquids (Chapter 4) are very similar, so if the process underlying the Debye-Waller factor
anomaly occurs as a function of a characteristic relaxation time, it would certainly occur
in the frequency range covered by the ISTS glycerol data. From a physical standpoint,
any relationship between the T, anomaly in fragile liquids and a high temperature T. in
glycerol is considered extremely unlikely.
A third possibility is that the Debye-Waller factor anomaly is not dramatic enough
to be observed within the error of the data. It has been suggested that with the inclusion
of activated processes in extended Mode Coupling Theory, the square-root behavior
survives, but the cusp is "softened". It has also been suggested that processes involving
the breaking and formation of hydrogen bonds obscure the MCT dynamics in glycerol. 26
A more complicated coupling scheme is necessary to describe the light scattering data. 19
At this point, the possible impact of the coupling scheme used in the light scattering
analysis on the Debye-Waller factor prediction is not known. It might be assumed that
the coupling of more modes and the inclusion of more temperature dependent parameters
in the theory might also have the effect of softening the cusp.
The possibility of crossover behavior without an observable anomaly in the
Debye-Waller factor can be addressed by considering the temperature dependence of the
stretching parameter, which is here considered a measure of the shape of the
susceptibility spectrum.
The temperature dependence of 0 can be used to test the
prediction of time-temperature superposition. Idealized Mode Coupling Theory predicts
that the slow relaxation scales with temperature above T, and is arrested below To.
Extended MCT restores ergodicity below T, by including "hopping" or activated
processes.
This might imply, however, that since the mechanism for relaxation is
different, there would be a change in the shape of the susceptibility spectrum below T.
For both salol and CKN, there is observation of a change in the value of P near or
below TC. 25 In both cases, 3 decreases, indicative of enhanced stretching, or the inclusion
of additional modes in the relaxation. There is no similar observation in the data for
104
glycerol. The lowest wavevectors may appear to show some slight decrease in 3, but this
decrease is well within the error bars. Again, extension of ISTS measurements to lower
wavevectors and lower times can address this question further.
Addressing the possibility of a crossover in the temperature range 223 to 233 K,
we revisit the data obtained for the average relaxation time. Mode Coupling Theory
predicts a power law temperature dependence of the relaxation time above Tc. The best
fit of relaxation times from ISTS to power law temperature behavior results in Tc = 188
K, with an exponent y = 10 (Figure 5-15). The value of y is much higher than common
values for fragile liquids, but does not violate the MCT restrictions for the exponents a
and b. The scaling temperature obtained is actually very close to Tg = 185 K. These
results are actually quite similar to results from ultrasound and specific heat
measurements, which cover a wide frequency range, and are closely related to ISTS. 22
Without a wider dynamic range, a free fit to the MCT prediction for the power
law temperature dependence of the relaxation time, allowing data below Tc to be
excluded, is not attempted. The relaxation time data above 235 K are fit with a fixed Tc
of 225 K for comparison with light scattering analysis (Figure 5-16).2 The resulting
exponent is y = 4.3, which is closer to the values used in other scaling analyses (Table 51).
As expected, the measured times become faster than the power law prediction
approaching Tc. However, these results are far from conclusive given the limited range of
relaxation times used in this analysis. As the MCT prediction holds for times above Tc,
we are unable conduct a similar analysis to address the question of a high temperature To.
105
Relaxation Times in Glycerol
MCT Power Law fit
MCT power law t=t0 *(T-TC)Y
y=10
T = 188 K
-2 -
-4-
2I
220
230
240
250
260
Temperature
Figure 5-15: Relaxation times in glycerol fit to a power
law with no restrictions. The TC value is unusually low and
the exponent is unusually high.
106
2
270
Relaxation Times in Glycerol
-
MCT power law fit with y = 4.3
00
00
S
6
A
V
0
2.0
2.5
3.0
3.5
ln(T-225K)
Figure 5-16: MCT power law fit with T; restricted to 225
K. The exponent y = 4.1 is somewhat larger than the values
obtained in light scattering measurements.
4.0
5.5
Conclusions
The dynamics of glycerol have been characterized over the time range from
nanoseconds to milliseconds. The structural relaxation dynamics are well fit by a KWW
stretched exponential at all temperatures where the structural relaxation timescale is well
separated from the acoustic frequency and the thermal decay rate. The relaxation times
follow a VTF
temperature dependence.
The relaxation measurements permit
determination of the Debye-Waller factor over a 40 degree temperature range from 1.2 Tg
to 1.45 Tg. No evidence of a MCT crossover temperature Tc is observed in that range.
The shape of the relaxation spectrum, determined by the stretching exponent 3, does not
change with temperature. A lower temperature for T, cannot be ruled out from the
available ISTS data, but no anomalies are observed in the specific heat spectroscopy
study, which cover the temperature range from the lowest ISTS measurement down to
Tg.16
It should not be surprising that Mode Coupling Theory analyses of glycerol have
yielded conflicting results. There is no consistency in the values of Tc obtained from the
different power laws. There is also no reported evidence of a square root cusp in the
Debye-Waller factor measured by neutron scattering. 23 In a review of Mode Coupling
Theory, 26 it is clearly stated that Mode Coupling Theory is not expected to apply to
intermediate and strong glass forming liquid. "The known fragile systems have a simple
microscopic structure. It is perhaps a reasonable guess, that all glass formers exhibit a
cross over temperature Tc. However, in the complicated systems the strongly temperature
dependent mechanisms of bond breaking and of hydrogen bridge formation presumably
mask the more subtle effects responsible for Tc."
The conclusion reached from these results is that MCT in its current form is
significantly limited in its ability to accurately describe the dynamics of a network
forming liquid of intermediate strength. MCT predictions alone have been inadequate to
describe the minimum in the susceptibility spectrum without complicated additional
analysis. 2,9
Incorporation of additional coupling terms seems to improve the ability of
108
the theory to describe the susceptibility spectrum.19 The strong boson peak can be taken
as an indication that high frequency dynamics, likely related to the breakage and
formation of bonds, are more prevalent in glycerol than in fragile liquids. ISTS cannot
directly elucidate the microscopic mechanism underlying structural relaxation. However,
all results point to a relaxation mechanism that is less strongly dependent on temperature
than in fragile liquids.
It is very likely that this process involves the hydrogen bonding
that is prevalent in glycerol, and as a result, MCT, which at present does not include this
process, fails to describe the behavior of glycerol. The present results along with earlier
reports may motivate attempts to generalize the theory sufficiently to include important
microscopic degrees of freedom.
References
1
F. J. Bermejo, A. Criado, A. de Andres, E. Enciso, and H. Schober, Phys. Rev. B
53, 5259 (1996).
2
J. Wuttke, J. Hernandez, G. Li, G. Coddens, H. Z. Cummins, F. Fujara, W. Petry,
and H. Sillescu, Phys. Rev. Lett. 72, 3052 (1994).
3
J. Wuttke, W. Petry, and S. Pouget, J. Chem. Phys. 105, 5177 (1996).
4
M. Soltwisch and D. Quitmann, Journalde Physique 40, 666 (1979).
5
J. Dawidowski, F. J. Bermejo, R. Fayos, R. Fernandez Perea, S. M. Bennington,
and A. Criado, Phys. Rev. E 53, 5079 (1996).
6
D. C. Champeney and R. N. Joarder, Molecular Physics 58, 337 (1986).
7
H. J. M. Hanley, G. C. Straty, C. J. Glinka, and J. B. Hayter, MolecularPhysics
62, 1165 (1987).
8
F. Fujara, W. Petry, R. M. Diehl, W. Schnauss, and H. Sillescu, Europhysics
Letters 14, 563 (1991).
9
E. Rossler, A. P. Sokolov, A. Kisliuk, and D. Quitmann, Phys. Rev. B 49, 14967
(1994).
109
10
D. A. Pinnow, S. J. Candau, J. T. LaMacchia, and T. A. Litovitz, Journalof the
Acoustical Society ofAmerica 43, 131 (1967).
11
D. W. Davidson and R. H. Cole, J. Chem. Phys. 19, 1484 (1951).
12
P. Lunkenheimer, A. Pimenov, B. Schiener, R. Bohmer, and A. Loidl,
Europhysics Letters 33, 611 (1996).
13
P. Lunkenheimer, A. Pimenov, M. Dressel, Y. G. Goncharov, R. Bohmer, and A.
Loidl, Phys. Rev. Lett. 77, 318 (1996).
14
A. Pimenov, P. Lunkenheimer, and A. Loidl, Ferroelectrics176, 33 (1996).
15
B. Schiener, R. Bohmer, A. Loidl, and R. V. Chamberlin, Science 274, 752
(1996).
16
N. O. Birge and S. R. Nagel, Phys. Rev. Lett. 54, 2674 (1985).
17
J. P. Kintzinger and M. D. Zeidler, Berichte der Bunsen-Gesellshaft , 98 (1972).
18
T. Christensen and N. B. Olsen, Phys. Rev. B 49, 15396 (1994).
19
T. Franosch, W. Gotze, M. R. Mayr, and A. P. Singh, Phys. Rev. E 55, 3183
(1997).
20
P. Lunkenheimer and A. Loidl, J. Chem. Phys. 104, 4324 (1996).
21
A. Schonhals, F. Kremer, A. Hofmann, E. Fischer, and E. Schlosser, Phys. Rev.
Lett. 70, 3459 (1993).
22
Y. H. Jeong, S. R. Nagel, and S. Bhattacharya, Phys. Rev. A 34, 602 (1986).
23
W. Petry and J. Wuttke, Transport Theory in Statistical Physics 24, 1075 (1995).
24
A. Duggal, Ph.D. Thesis, Massachusetts Institute of Technology, 1992.
25
Y. Yang, Ph.D. Thesis, Massachusetts Institute of Technology, 1996.
26
Liquids, Freezing, and the Glass Transition,Vol. 1, edited by J. P. Hansen, D.
Levesque, and J. Zinn-Justin (Elsevier Science Pub. Co., Amsterdam, 1991).
110
Chapter 6
ISTS Study of Relaxation in
Polypropylene Glycol
6.1
Overview
Many polymers easily form supercooled liquids To some extent, the behavior of
supercooled polymers is similar to other supercooled liquids such as molecular or ionic
systems. For example, the viscosity will change dramatically with temperature through
the supercooled regime. In some ways, however, the behavior is different.
One common difference
between supercooled polymers and supercooled
molecular liquids is the "double a peak" feature in the dielectric loss spectrum. As
discussed for glycerol, and as predicted by Mode Coupling Theory, the typical relaxation
spectrum for a supercooled molecular or ionic liquid consists of a high frequency P
relaxation, and a lower frequency, strongly temperature dependent a relaxation. The
relaxation spectrum of a polymer, which is most frequently measured by dielectric
spectroscopy, shows a similar 0 relaxation feature, but in the a relaxation regime, there
are often two peaks, commonly referred to as a and a' peaks.
In Polypropylene Glycol (PPG) the a relaxation occurs at the same frequency
regardless of molecular weight. The frequency of the a' relaxation depends strongly on
molecular weight, and is closer to the a relaxation in a lower molecular weight.1 In PPG,
the a relaxation is faster than the c' relaxation.
Presumably, the main a relaxation
represents the motion underlying the glass transition, as the glass transition temperature
changes very little with molecular weight. 2 Although there is a difference in absolute
frequency between the features at moderate temperatures, the temperature dependence of
the two features seems to be the same, at least for some temperature range.
At low
temperatures, there is a change in behavior, and the a' relaxation slows down more
quickly as the polymer cools. It is unclear whether or not the two features will merge in
the high temperature, as high frequency dielectric results are not available.
The increased complexity of the relaxation spectrum provides an additional
challenge to MCT analysis of polymers as supercooled liquids approaching the glass
transition. It must be clearly stated that the oa nomenclature used to describe the slow
feature in the dielectric spectra of polymers has its origins in the field of dielectric
measurements. It is not completely obvious that there is a direct connection between
features called oa and P in dielectric spectroscopy and the slow and fast processes in Mode
Coupling Theory. In general, caution should be taken in the comparison of experimental
results, as the language used to describe the features in the relaxation spectrum is not
entirely consistent in the literature, particularly between the fields of condensed matter
physics and polymer science and engineering.
MCT does not specifically predict the entire shape of the slow relaxation regime.
The KWW function has commonly been found to fit the data, but is not directly derived
from MCT.
A double ca peak, or a two stage slow relaxation, is not immediately
inconsistent with all predictions of the theory. A minimum may still observed between
fast and slow relaxations. The lower frequency side of the minimum, corresponding to
the high frequency wing of the a relaxation, may still follow the predicted power law
dependence.
Equivalently, the initial part of the decay of the autocorrelation function
from the plateau value may follow the predicted power law behavior. Predictions about
the fast relaxation feature may still apply.
The Debye-Waller factor prediction may still have the same meaning with a more
complicated a relaxation spectrum. Recall that in the idealized theory, the Debye-Waller
factor is a measure of how far the relaxation decays from the normalized value 4(O) = 1,
to the plateau value which persists indefinitely below Tc. If the Debye-Waller factor
prediction is viewed as describing decreasing strength of the fast relaxation processes as
the temperature is reduced, it becomes apparent that the particular characteristics of the a
relaxation are not pertinent, particularly as the oa relaxation theoretically disappears
entirely below Tc in the idealized version of MCT.
112
Although theoretically the double a relaxation may have little relevance to the
MCT predictions, there may be experimental implications in testing the predictions. To
measure the Debye-Waller factor by integrating the spectrum of the a relaxation, the
experiment must have sufficient dynamic range to completely include both a relaxation
features throughout the required temperature range.
More serious difficulties are
encountered in some of the scaling law predictions. A single characteristic frequency or
relaxation time of the a relaxation is not now well defined, particularly in the time
domain. The two a relaxation features both depend strongly on temperature, but not
necessarily in the same way. Clearly, there is now some difficulty in selecting the data
for fits to the power law prediction for the temperature dependence of the relaxation time.
Possible interpretations are to use either of the two peak frequencies, or an effective peak
frequency for the entire a regime. Testing of predictions regarding the minimum of the
susceptibility spectrum is unaffected, as the minimum only includes the high frequency
wing of the slow relaxation.
The principle of time-temperature superposition should be carefully considered.
If time-temperature superposition holds, the overall shape of the slow relaxation spectrum
scales with temperature and time independently. This implies that all aspects of the
double peak spectrum must scale with time and temperature, including both peak
frequencies, the shapes of both features, and their relative amplitudes.
A Mode Coupling Theory analysis of the dynamics in Polypropylene glycol, MW
4000, has been conducted using a combination of long time and high frequency light
scattering techniques. 3
The long time data from Photon Correlation Spectroscopy
measures the decay of the correlation function (t). Overall, the data are fit to a KWW
stretched exponential. However, there are some deviations both at short and long times.
There is additional signal intensity at short times, before the data level off for a
period of time prior to the KWW decay. This observation is made at low temperatures
near or below the glass transition, and is assigned to P relaxation. The time dependence
is fit to a power law decay with exponent a = 0.23. If these data represent the MCT fast
relaxation, only the extremely long time tail of the feature is observed on the microsecond
113
time scale.
A direct comparison to the depolarized light scattering susceptibility,
presented in the same study, show a susceptibility minimum at GHz and higher
frequencies at all temperatures, with the fast feature at frequencies above the minimum.
Deviations from a single KWW decay are also observed at long times.
The
original P value obtained from the fits is P = 0.39. When the long time data are fit
separately from the main relaxation, the 3 values are P = 0.43 for the main relaxation and
P = 0.7 for the long time tail. The long time 3 = 0.7 decay is assigned to the same motion
underlying the dielectric a' peak, although due to the scatter in the data it is unclear that
the KWW function provides an adequate fit. The time scales of tens of seconds are in
good agreement with the dielectric data.'
A Mode Coupling Theory analysis of the susceptibility minimum is also
attempted from a combination of Brillouin and Raman scattering data at frequencies
above 1 GHz.
The minimum is only completely characterized at high temperatures, as
the slow relaxation moves to frequencies much lower than 1 GHz. The values of the
exponent a, obtained from a MCT fit to the minimum, are higher than allowed by the
MCT restrictions.
The values obtained for Tc from power law fits to the minimum
susceptibility and frequency are 265 K and 180 K, using the exponents obtained from the
high frequency and low frequency measurements, respectively.
Clearly, the exponent
needs to be determined less ambiguously before T, can be determined well. Neither of
these values agree with the value of Tc = 236 K reported from a combination of PCS and
viscosity measurements. 4
Polypropylene Glycol is chosen for ISTS analysis for several reasons.
motivation is to characterize the relaxation dynamics of a polymeric system.
One
MCT
predictions are tested using the relaxation data. Recall that MCT is not formulated for a
complex liquid, let alone a polymer, so any success of the theory in predicting the
behavior of the dynamics might be somewhat surprising. ISTS is used as a quantitative
experimental method for measuring the relaxation dynamics of a supercooled polymer,
and to test the principle of time-temperature superposition.
114
While time-temperature
superposition is incorporated into MCT, it originated empirically as part of the discussion
and treatment of relaxation in polymers.
6.2
Experimental
Polypropylene glycol (PPG) was purchased from PolySciences (MW 4000) and
Aldrich (MW 425). Polypropylene glycol MW4000 (PPG4000) was dried at 100 degrees
C under a light vacuum for 24 hours. The warm polymer was forced through a 0.2 tm
filter appropriate for high temperature liquids. The liquid was transferred into a tefloncoated aluminum cell with moveable windows, and an RTD immersed in the liquid. To
avoid thermal degradation, the lower molecular weight sample was dried at 50 degrees C
in a vacuum oven for several days. It was similarly filtered and transferred to a sample
cell. The same samples were used to collect all data.
The other experimental details are the same as described in Chapter 3 and in the
experimental section of Chapter 5. For PPG4000, only the Antel diode was used to
collect the data.
The Hamamatsu diode was used for the highest wavevectors for
PPG425.
6.3
Results
Figures 6-1 and 6-2 present ISTS data from PPG425 and PPG4000.
At all
temperatures, the data show a fast acoustic response and a long time thermal decay. At
intermediate times and temperatures, a slow-rising feature appears in the data, which
represents structural relaxation. The data presented are collected at a single wavevector,
but are representative of data at all wavevectors, with the acoustic and thermal timescales
varying as q and q2, respectively.
115
ISTS data from PPG4000
E-3001
0.1
1
10
100
1T=340K00
1E-3
0.01
0.1
1
10
100
1000
1E-3
0.01
0.1
1
10
100
1000
T=230K
1E-3
0.01
0.1
1
10
100
1000
1E-3
0.01
0.1
1
10
100
1000
Log time (pts)
Figure 6-1: ISTS data from PPG4000. The data show the
growth of the structural relaxation feature as the sample
cools. The feature disappears from the data as the
relaxation slows beyond the thermal diffusion time.
116
ISTS data from PPG425
T=320 K
1E-3
100
... 0.01
T=270 K
i
I. . ... ..
I
.
..
I. I .I
. .. I..II
.
. . . . ... I
.
. .. I
.
. I
IE-3
0.01
0.1
1
10
100
1E-3
0.01
0.1
1
10
100
Log Time (ps)
ISTS data from PPG425, showing the
Figure 6-2:
structural relaxation feature appearing at intermediate
temperatures.
117
. . . . .
6.3.1
Acoustics
The measured speed of sound in PPG ( Figures 6-3 and 6-4) compares very well
with the velocity of sound obtained from Brillouin Scattering (BS) measurements, taking
the difference of wavevector into consideration.5, 6 ISTS and BS data both show a
temperature region with dispersion, with a lower temperature range for ISTS due to its
lower frequency and wavevector ranges. From the data at high temperatures, ISTS can
measure the slope of the zero frequency speed of sound, dVo/dT.
The ISTS result
averaged for all wavevectors is -3.4 (m/s)/K, and at the lowest wavevector only is 3.3(m/s)/K. This result is higher than the BS measurement of -2.1 (m/s)/K, but due to the
high BS frequencies, Vo in the BS experiment consists of only a few points, at higher
temperatures than the ISTS values. 6 Also, the cited data were later corrected for a
calibration error, which might also have some impact on the measured slope. 5 The
analogous infinite frequency, low temperature slope measured with BS is -4.6 (m/s)/K.
There are not sufficient data at high enough frequencies from ISTS for comparison.
118
Speed of Sound in PPG425
2800 m q=0.261 Jm
26002400 -
22002000
O
q=0.585 tm'
A
q=0.817 Pm-
A
q=1.36
m-'
-
18001600
1400
1200180
200
220
240
260
280
300
320
340
Temperature (K)
Figure 6-3:
wavevectors.
Speed of sound in PPG425 at several
119
360
Speed of Sound in PPG4000
28002600 2400 2200 2000 1800
-
*
q=0.169 tm
O
q=0.485 tm
*
q=1.21 CVm-
1600 1400 -
12001000
lI '
180
I
200
220
240
260
280
300
320
340
360
Temperature (K)
Figure 6-4: Speed of
wavevectors.
sound in PPG4000 at several
120
380
Comparison of Speed of Sound
in PPG425 and PPG4000
2800 2600-
Oo0
0
2400-
22002000 -
1800-
*
PPG425 q=1.36 tm-'
O
PPG4000 q=1.21 tm
-
1600140012001000
I
'
'
I
'
I
'
I
'
I
'
I
'
I
'
I
'
I
'
I .
180 200 220 240 260 280 300 320 340 360 380
Temperature
Figure 6-5: Comparison of the speed of sound in PPG425
and PPG4000. The wavevectors are similar, but not
identical.
The speeds of sound in PPG4000 show virtually identical behavior to PPG425.
Figure 6-5 presents data from both molecular weights as a function of temperature. The
data cannot be compared quantitatively, as the wavevectors are different for the two
samples, and there may be small difference in the material density. Brillouin Scattering
measurements also report that the speed of sound at fixed q does not vary with the
molecular weight in PPG, considering polymers of molecular weight 425, 1025, 2025,
and 4000. 5 It is interesting to note that the speed of sound is similar despite a very large
different in the viscosities of the different molecular weight polymers at room
temperature.
Acoustic Damping in PPG425
700
600 -
r:ri
r
mm
500 -
o
#
400 -
o q=0.585 gm1-
4-.
Ct
0E
300 S
,
Ct
q=0.261 tm -
*
O
U*
0
0
o o A0
200
AAA
A
q=0.817 pm
A
q=1.36
tm-i
AOM
100A
4
0
OAA
0
'
I
220
'
240O
I
I
I
I
I
260
200
'
260
2 lO
'
IO
280T
'
as 4
I
300
'
I
'
320
Temperature
Figure 6-6: Acoustic damping in PPG425. There is a main
peak at all wavevectors, and some indication of a shoulder
on the low temperature side of the maximum.
I
340
-
Acoustic Damping in PPG425
150
aa
120
c
A q=1.36 tm -
rCl
. ,
C
30 -
A
220
A A
aA
240
260
280
300
320
340
Temperature
Figure 6-7:
Acoustic damping in PPG425 at one
wavevector, highlighting the low temperature shoulder.
The acoustic damping rate in PPG425 passes through a maximum at the
temperature where the relaxation timescale coincides with the acoustic frequency.
In
PPG425, at temperatures lower than those corresponding to the maximum acoustic
damping, there is some indication of a second peak, or a shoulder in the damping rate.
Figure 6-7 shows the wavevector with the clearest indication of this behavior. A similar
feature is observed in measurements of the derivative of the real part of the longitudinal
modulus at constant wavevector measured by Brillouin scattering. 7 The Brillouin data
are analyzed in terms of Debye relaxation, and no insight is offered on the physical origin
of the low temperature deviation from the Debye fit. Other Brillouin scattering data do
not show a low temperature feature. 5
Acoustic Damping in PPG4000
200AA
rj
150-
S
A
AAA
A
A
A
a
100-
a
S
S q=0.967 im
AA
50-
AZA
aAa
ap
zzzzz~z
C3
nAaA
180
200
20
240
260
28'
I 0
0
3
40
360
380I
I
180 200 220 240 260 280 300 320 340 360 380
Temperature
Figure 6-8:
wavevector.
Acoustic damping in PPG4000 at one
-
The acoustic damping in PPG4000 is presented in Figure 6-8. Similar to PPG425,
there is a possibility of a low temperature feature in the damping, but in this case there are
not sufficient data to draw any conclusions. The quality of the data at other wavevectors
is poorer. Previous results from PPG4000 also give an extremely subtle indication of a
second feature in the acoustic damping.8 As with PPG425, this behavior is not observed
in Brillouin Scattering measurements of acoustic attenuation. 5
6.3.2
Relaxation Measurements
The average relaxation time is obtained from the fit to the structural relaxation
feature in the ISTS data. Over all wavevectors, relaxation times can be measured from
tens of nanoseconds to tens of microseconds.
For both PPG molecular weights, this
corresponds to observation of structural relaxation for temperatures 25 to 50 degrees
above the glass transition.
Figure 6-9 presents the average relaxation times in PPG425 on a log plot, with
PCS results included for comparison. The times from ISTS do not appear to extrapolate
at low temperature to coincide with measurements from PCS. As it is unclear whether
the two techniques probe the same dynamics, no conclusion can be drawn from the
discrepancy. Also, the relaxation time is changing very rapidly with temperature in this
regime, and error in temperature measurements may cause deviations in the relaxation
time comparison. There also may be differences in the sample, such as water content.
125
Average Relaxation Time in PPG425
10000
-
q=0.261 tm-1
1000
q=0.585 tm -'
q=0.817 gm-
100
q=1.36 tm-1
ISTS acoustics
10-
PCS (VV)
Vt-
V
PCS (VH)
10.1
0.01
210 215 220 225 230 235 240 245 250 255 260 265 270 275
Temperature
Figure 6-9: Log plot of relaxation times in PPG425
obtained by ISTS and other experimental methods. The
high temperature ISTS data are determined from the
maximum in the acoustic damping. ISTS data appear to
extrapolate to faster relaxation times at low temperatures,
compared to the PCS results.
126
Comparison of Relaxation Times
in PPG425 and PPG4000
4
2
PPG425
o
0
PPG4000
V
-2
rnrn
-4
4.0
4.1
4.2
4.3
4.4
4.5
1000/T (1/K)
Figure 6-10: Comparison of Relaxation Times for PPG425
and PPG4000. The data are presented on logarithmic axes.
The left and right axes are offset by one order of magnitude
for clarity. The scale is the same for both data sets. The
dashed lines are Arrhenius fits to the relaxation time.
The relaxation time data for PPG425 and PPG4000 are simultaneously presented
for comparison in Figure 6-10. The temperature dependence of the relaxation times is
non-Arrhenius in this temperature range, as shown in Figure 6-10. The deviation from
Arrhenius behavior is more apparent, at the lower molecular weight. This illustrates the
need for a wide dynamic range in experiments that address the temperature dependence of
the relaxation time. Particularly for a strong or intermediate liquid, relaxation times for
only a limited range can appear to have an Arrhenius temperature dependence.
127
The data fit a Vogel-Tamman-Fulcher form for the temperature dependence. The
results for PPG425 and PPG4000 are displayed in Figure 6-11 and Figure 6-12. The VTF
results for B and To are very similar to results from VTF fits to viscosity data. 9 For the
temperature range from 243 K to 233 K, the reported parameters are B = 955 K, To = 174
K for PPG425 and B = 898 K, To = 176 K for PPG4000.
VTF Fit to Relaxation Times in PPG425
32-
--VTF fit
..
0A
V
VTF FIT
-1
IL
M'
-2 -'g
-3
To=
170 ±5 K
B=
1020 +20 K
T
5e-14 s
--
-4
220
225
230
235
240
245
Temperature
Figure 6-11: Vogel-Tamman-Fulcher fit to relaxation times
in PPG425. The results are To = 170 K and B = 1020 K.
128
250
VTF Fit to Relaxation Times in PPG4000
2
1
0-
I.
-1
"'Ii.
-2
-3
-4
-1
-4
-3
-2
220
"
225
230
'
I
'
235
230
"
240
i2
245
250
Temperature
Figure 6-12: VTF fit to relaxation times in PPG4000. Fit
results are included on the Figure.
The relaxation times are also fit to a power law temperature dependence for
comparison with Mode Coupling Theory predictions (Figure 6-13).
When relaxation
times from all temperatures are included in the fit, the power law fit results in Tc = 205 K.
When only higher temperature data are included in Figure 6-14, above 235 K, the Tc
measured is 217 K. As expected, the fit restricted to higher temperatures seems to
129
provide a better representation of the highest temperature data. It is expected that further
restrictions in the temperature range of the data would result in progressively higher
values of To. Since there is no obvious point at which to begin restricting the data
included in the fit, it is impossible to extract a meaningful measure of Tc from the
relaxation time data. It is noted that the higher temperature T, is accompanied by a value
of the exponent that is more typical of MCT fits.
MCT Power Law Fit to PPG425
43-
2
1
m
•
<R>
-- MCT Power Law Fit
o
"mm
s
S
MCT Power Law fit
-
y= 7.3
= 205 K
-1
VT
- -2 -
-3
-4
220
225
230
235
240
245
Temperature
Figure 6-13: Average relaxation time in PPG425 fit to a
MCT power law for the temperature dependence. All data
are included in the fit.
130
250
MCT Power Law fit to PPG425
Selected Temperature Range
4-
*~
3U
2-
*~~
m
,
1-
<T R>
i
=L
V
-s
-----
0-
MCT Power Law Fit
•MCT
Power Law Fit
y = 4.3
-1
Mmk
v
T= 217K
T > 230 K
'I
-2
Ui a m'm.
.
-3
-4
I~
I
220
~
*
225
230
I
235
240
245
250
Temperature
Figure 6-14: Average relaxation times, with data at
temperatures above 230 K fit to a MCT power law
temperature dependence.
The intermediate time, intermediate temperature data representing structural
relaxation are fit to a Kohlrausch-Williams-Watts stretched exponential function. The
value of 0 reflects the degree of stretching, or the extent of the deviation from Debye
behavior, where P is 1. The measurements of P for PPG2425 show some systematic
variation with temperature, with p decreasing with decreasing temperature (Figure 6-15).
There is some scatter in the data at different wavevectors, but each wavevector shows a
similar temperature dependence. The values appear to decrease to coincide with the P
values obtained from PCS in the VH geometry (Figure 6-16). 9 It is also noted that the
magnitude of 3, near 0.5 for the temperature range from 230 to 240 K, is similar for the
two molecular weights. This is taken as further evidence that the structural relaxation
observed in ISTS is similar for both molecular weights.
Stretching Parameter 1 in PPG425
0.75 0.70
T
-
0.65
A
A"
-
aa
0.60-
A
0.55 *,E
0.50 0.45 0
0.40-
0.35
*
q=0.261 ttm-
o
q=0.365 rtm'
9
q=0.496 im'
O
q=0.585 tm-
A
q=.817 [m'
A
q=1.36 pm -'
I
220
225
235
230
240
245
Temperature
Figure 6-15: Stretching Parameter P from PPG425 as a
function of temperature at several wavevectors.
132
250
Stretching Parameter 3 in PPG425
0.75 -
0.70 0.65 -
A
AX
0.600.55 "
q=0.261 gm '
0.50 -
q=0.365 gm'
q=0.496 .m'
0.45
q=0.585 pm
-
"
q=0.817 gm')
q=1.36 gm"
0.40-
PCS (VV)
+
PCS (VH)
0.35
210
210
215
I
215
20
220
'
230
25
225
Temperature
240I
235
240
I
245
Figure 6-16: Stretching Parameter P from PPG425 as a
function of temperature at several wavevectors, including
measurements from VV and VH Photon Correlation
Spectroscopy.
133
250
Stretching Parameter 3 in PPG4000
0.55 -
0.50-
a
*
4-j
A!
M
d)
S
0.45
-
oJ2
Q:
0.40-
0.35
2I
220
*
q=0.169 tm
A
q=0.485 jm -1
*
q=1.21 tm-'
'
I
I
'
225
230
I
235
'
2
240
2
245
Temperature
Figure 6-17: Stretching parameter 3 in PPG4000 at several
wavevectors.
There is a decrease in the value of 3 with
decreasing temperature.
The Debye-Waller factor in PPG425 shows some slight variation with
temperature, with higher values at lower temperatures. In Figure 6-18, the temperature
dependence is obscured by the scatter among different wavevectors. In Figure 6-19, the
data obtained at the two highest wavevectors, where data from two detectors are
combined, are rescaled for better agreement with the measurements at lower wavevectors
in the temperature range where data from multiple wavevectors could be analyzed. It is
not improbable that in the merging of data from two detection systems, a systematic error
is incurred in the relative amplitudes of the acoustic and structural relaxation modes.
However, it is stressed that there is no rigorous basis for this adjustment.
The Debye-Waller factor in PPG4000 is presented in Figure 6-20. As only one
detector was used to obtain the ISTS data for PPG4000, no rescaling of the Debye-Waller
factor is attempted. The overall magnitude of the temperature variation of the DebyeWaller factor is similar for the two molecular weights.
Debye Waller Factor in PPG425
0.80 0.75 -
q=0.261 pm
q=0.365 lpm
0.70 -
q=0.496 lpm"
q=0.585 ipm-
0.65 -
1
q=0.817 ipm
q=1.36 vpm
0.60 0.55 -
U
AI
0.50 0.45 0.40 -
220
225
225
235
230
240
245
250
Temperature
Figure 6-18: The Debye-Waller Factor in PPG425 as a
function of temperature. There is no distinct evidence of
the anomaly predicted by Mode Coupling Theory.
136
Debye Waller Factor in PPG425
0.70
(rescaled high wavevectors)
q=0.261 Ltmz
0.65 -
q=0.365 lam '
q=0.496 lm'
q=0.585 lam'
0.60
E
0
q=0.817 m-'m
0o
q=1.36 lpm-'
*)
-1
L -..-
0.55
A
0.50
0.45
220
225
230
235
240
245
Temperature
Figure 6-19: The Debye-Waller Factor in PPG425 at
different wavevectors. The measurements from the highest
wavevectors are arbitrarily rescaled to 90% to better agree
with the measurements at lower wavevectors.
137
250
Debye Waller Factor in PPG4000
0.70 -
0.65
-
*
q=0.169 tm'
O
q=0.485 pm -
A
q=1.21 pm'
0
0.60-
0
iga
_L
0.55
0.50
0
-
12
225
-
230
235
240
Temperature
Figure 6-20: Debye-Waller factor measured for PPG4000.
245
6.4
Discussion
In ISTS, there is no compelling evidence of the second, a' feature as a structural
relaxation mode. Although the available dielectric data cover a wide frequency range
from a few hertz into the megahertz range, higher frequency data are not available.
Brillouin scattering is conducted at higher frequencies, but the range is somewhat limited
and does not map out all of the features in the spectrum. In particular, the behavior of the
a and a' features at high temperatures is not available. We can only assume that the
features converge at high temperatures, and separate at some point, or that one feature is
not observable in light scattering or ISTS. As the liquid cools, the a' relaxation slows
down more rapidly than the a relaxation. The a' relaxation depends on molecular weight.
It follows from this scenario that in a higher molecular weight polymer, where the a and
a' peaks are more separated at lower temperatures, the a' relaxation splits off from the
main a peak at higher temperatures. Note that this scenario implies deviation from time
temperature superposition at high frequencies.
In ISTS measurements, the timescale where structural relaxation can be observed
is that where the relaxation time is slower than the acoustic frequency but faster than the
thermal decay.
Acoustic damping provides evidence of structural relaxation at times
moderately faster than the acoustic frequency. The acoustic frequency increases by about
a factor of 2 between the liquid and the glass, and the thermal diffusion rate varies only
weakly with temperature until very close to the glass transition. Thus, the time where
structural relaxation is visible is nearly the same for all temperatures at a give
wavevector. Any structural relaxation feature that contributes to the density dynamics in
the experimentally defined time window can appear at any temperature.
To completely analyze ISTS results for the presence of two relaxation features, a
semi-quantitative comparison is made to dielectric results. Exact agreement between
dielectric peak frequencies and ISTS relaxation times is not expected, but the order of
magnitude can be used for comparison to the ISTS results. Dielectric peak frequencies
are reported for PPG4000 and PPG880.1 It is assumed that frequencies for PPG425 will
139
follow the molecular weight trend, and be the same for the ca relaxation and slightly faster
for the a' relaxation. At T = 235, the a peak frequency is 100 kHz for both weights, and
the a' peak occurs at 10 kHz for PPG880 and 100 Hz for PPG4000. At T = 222 K, the
features have slowed down to 3 kHz for the acpeak and 200 Hz and 3 Hz for the PPG880
and PPG4000 a' peaks. The relaxation times from ISTS are in agreement with the a
relaxation frequency within an order of magnitude.
For ISTS, there are temperatures and wavevectors where both relaxations in
PPG425 should be visible if both motions couple to the density dynamics. Presumably,
the effect of two features on the ISTS data would be a poor KWW fit to the structural
relaxation feature. However, it is noted that the sum of two Kohlrausch functions can
still be well represented by a Kohlrausch function if the characteristic relaxation times are
closely spaced. The widest separation of the features is at lower temperatures. At the
lowest wavevector, where the time window for structural relaxation is widest, the data
have been closely examined for deviations from the KWW fit.
No extraordinary
discrepancies are observed, including when the thermal decay is subtracted from the data
and the relaxation function is observed directly. At temperatures approaching the glass
transition, there is occasionally a very long time rise to the data mixed with the end of the
thermal decay. On an experimental basis however, this can be attributed to a variation of
the probe intensity with time.
If the a' feature were observed in the ISTS experiment, it could conceivably be
observed at higher temperatures, particularly in the PPG4000 data.
The a and a'
frequencies are well separated at the higher molecular weight at low temperatures.
Presumably, at some temperature the two features converge, implying that at an
intermediate temperature, the separation is one or two orders of magnitude. If this
condition occurs when the main a relaxation is faster than the acoustic frequency, only
the a' relaxation would be observed as structural relaxation.
This behavior is not
observed at any temperature or wavevector.
There is some evidence of a second relaxation feature in the acoustic damping rate
as a function of temperature. The initial interpretation of this result in ISTS is that a
140
shoulder in the damping rate is indicative of a second maximum in the relaxation
spectrum. This second maximum would actually be a higher frequency feature in the
relaxation spectrum measured at constant temperature. When considering the acoustic
damping rate measured at constant wavevector, "lower" frequencies features appear at
higher temperatures, and a "higher" frequency feature appears at lower temperatures,
where it has slowed down to the acoustic frequency timescale.
The ISTS damping rate data are not of sufficient quality or quantity to make
numerical measurements of relaxation times from acoustic damping. However, estimates
are that the main damping feature occurs near 400 MHz at a temperature near 270 K. The
second feature also has a frequency near 400 MHz at a temperature near 255 K.
Numerical values for dielectric peak frequencies are not available in this regime for
comparison, however, both the a and a' frequencies are below 1 MHz at 255 K.1 From
extrapolation of the dielectric results, the main peak in the acoustic damping rate is
assigned to the a relaxation. A peak due to the a' relaxation would occur at higher
temperatures, and is not observed.
Further
information
cannot
be
gained
from
the
Brillouin
Scattering
measurements, as the frequency range is over an order of magnitude higher, and the
acoustic damping displays a broad maximum at temperatures greater than 300 K. If there
is a second feature contributing to the damping at higher frequencies and temperatures, it
is unlikely that the two features could be distinguished, as they likely merge at high
temperatures.
The breadth of the acoustic absorption spectrum, however, does not
contradict the possibility of multiple, or complex relaxation processes.
The recurrence in the damping occurs at a temperature below the peak, so it is
due to a process that occurs at higher frequencies. Thus, this behavior is not assigned to
the a' relaxation as observed in dielectric spectroscopy.
If anything, the additional
damping should be assigned to the slowing down of a "p-like" relaxation into the ISTS
frequency regime. Also, the frequency for this feature does not change nearly as rapidly
with temperature as the temperature dependence of the a relaxation.
The frequency
remains in the hundreds of MHz for about 30 degrees in PPG425. There is no evidence
of a feature in the hundreds of KHz to 1 MHz regime in dielectric data at temperatures
down to 209 K. l 0 No similar shoulder is evident in the acoustic damping data for
glycerol (Chapter 5), salol, or CKN."1
Although this seems somewhat slow for P
relaxation as reported for non-polymeric materials, it is consistent with PCS results
reporting the tail of the 3 relaxation on a microsecond time scale. 4 This would also imply
that the 3 relaxation in PPG has a different character than in the non-polymer glassforming liquids. The description of this high frequency feature should not be considered
synonymous with the MCT fast relaxation. In fact, the susceptibility minimum, which is
analyzed in terms of MCT predictions, occurs at frequencies greater than 1 GHz for all
temperatures above the glass transition. 3
The most definitive statement regarding the a and a' relaxation that can be made
from ISTS data is that if the a' relaxation is strongly coupled to density fluctuations, it is
indistinguishable from the a relaxation on the timescales from nanoseconds to
microseconds in both molecular weights.
There is also the strong possibility that
although the motion associated with the a' feature is strongly associated with a dipole
moment in the polymer, it does not make a strong contribution to the density dynamics.
If the a' relaxation is not coupled to the density dynamics, the MCT analysis of
the relaxation dynamics in a polymer is simplified. To date, we are not aware of the
observation of two slow features in an experiment that directly probes the density
dynamics. There is some evidence in the PCS measurements of a very slow relaxation
near the glass transition,3 but the relationship between PCS data and density dynamics is
not completely clear. With one slow contribution to the density dynamics, the relaxation
in the polymer follows the MCT scenario of a fast and a slow process. The fast process is
represented by the motions contributing to the ISTS acoustic response, and is above the
frequency range of the dielectric experiments on the polymer in the liquid state. The
Debye-Waller factor will still be measured by the relative amplitudes of slow, structural
relaxation.
Time-temperature superposition will still be indicated by a temperature-
invariant value of the stretching parameter 0, which represents the width of the relaxation
spectrum.
142
The relaxation time measurements in both molecular weights of polypropylene
glycol do not adhere well to the MCT predictions. Due to limited dynamic range, the
power law fits of the relaxation time to the power law prediction are inconclusive. A
power law adequately describes the temperature dependence of the relaxation time, but a
value of TC cannot be reliably extracted from the data.
A few comments need to be made regarding the viscosities and the relative
fragilities of the different molecular weights. At room temperature, PPG4000 is several
orders of magnitude more viscous than PPG425.
Traditionally, the glass transition
temperature occurs when the viscosity reaches a value of 1013 poise. Considering these
two endpoints, PPG425 would be more fragile than PPG4000, as the viscosity must
undergo a more dramatic change between room temperature and the glass transition. The
T, values reported, however, rely on calorimetric measurements 2 , and not on a measured
viscosity. In the range accessible through ISTS, the relaxation times show a very similar
temperature dependence. The VTF parameters are also very similar. The Arrhenius fit to
the relaxation times for PPG4000 is slightly better. However, this kind of judgment is
difficult to make without an extremely wide dynamic range. To our knowledge, there are
no measurements of the viscosity, or the relaxation time over the entire range from room
temperature to the glass transition. From the evidence available, we assume that PPG425
is more fragile than PPG4000, but the difference is not dramatic. The difference in the
fragility is not as large as between a fragile liquid such as salol, and an intermediate
liquid such as glycerol. The fragility of both molecular weights is somewhere between
these materials.
There is no observation of an anomaly in the Debye-Waller factor in the
temperature region accessible through ISTS, however, in both molecular weights, there is
some temperature dependence. The slight temperature dependence could be an indication
of several possibilities. The decreasing Debye-Waller factor as the material warms could
simply be an extension of the glassy behavior into the supercooled regime, as observed in
neutron scattering. 12
The ISTS measurements cannot rule out the possibility of a
crossover temperature above the temperature range where the Debye-Waller factor
measurements are made. A steadily decreasing value of P is observed in both molecular
weights, which would be somewhat similar to observations in fragile glass forming
liquids.
The temperature range in this study does include 1.2Tg,
240 K for both
polymers, and there is no clear anomaly at that temperature. Power law MCT fits have
resulted in TC = 236, 4 which also is not supported by ISTS measurements. Of course, the
possibility that the anomaly is softened to the point of not being observable always exists.
There is no evidence of a distinct anomaly of any sort in the Debye-Waller factor
in either molecular weight. The possibility remains that there is a MCT square-root cusp
in the Debye-Waller factor at temperatures above the temperature range presented. The
scatter in the data prevents quantitative analysis of the temperature dependence, such as a
square root dependence. The data for PPG425 are only suggestive of this behavior with
the rescaled wavevectors. The strongest conclusion that can be reached is that there is
evidence of some increase of the Debye-Waller factor as the material cools.
The absolute magnitudes of the Debye-Waller factor are similar for the two
molecular weights. The evidence from the relaxation times and stretching parameter 0
already indicate that a similar motion underlies the structural relaxation observed in ISTS
for both weights. Similar Debye-Waller factors indicate that the relaxation has similar
strength in both molecular weights.
The temperature dependence in the Debye-Waller factor mirrors the temperature
dependence of P. A smaller value of P indicates more stretching in the relaxation
spectrum. This may naturally imply a larger integrated area of the relaxation spectrum,
i.e. a larger Debye-Waller factor. Thus, an increase in the Debye-Waller factor might be
consistent with a decrease in 3, provided the integration limits completely include the
relaxation feature.
The values of P for PPG4000, particularly at q = 0.169 tpm'
show a clearer
decrease with T. In the case of PPG4000, this is most likely due to an increased width in
the a relaxation at low temperatures. The a' relaxation at these temperatures is slower
than the thermal diffusion, even at this small wavevector.
Only the tail of the a'
relaxation would fall in the ISTS time window, and would only have an impact on the
144
shape of the structural relaxation if the motions strongly contribute to the density
variation. These measurements do not agree with PCS results, which report a KWW
behavior with P = 0.20 at 242 K. At decreasing temperatures, there appear to be two
features. By T = 208 K, the data revert to a single KWW decay with P = 0.39. This
value is in better agreement with the ISTS results. As the experiment only shows the
long time tail of the decay at high temperatures, it is possible that the low value is
reflective of two features that are closely spaced or overlapping in time.
There is a striking similarity between both the Debye-Waller factor measurements
and the 0 values obtained for the two molecular weights. The relaxation times also nearly
coincide, given the slight difference in the glass transition temperature. For these results,
we can conclude that the origin of the structural relaxation mode in ISTS is closely
related to the main a peak in dielectric loss, which has also been found to be independent
of molecular weight. By comparison of the dielectric frequency with the ISTS relaxation
time, it is also clear that the main a relaxation is the mode that is active in ISTS.
For several of the measurements made in PPG, there appears to be some
systematic variation of the measurement with wavevector, despite expectations that the
value should be independent of wavevector. In the zero wavevector regime of ISTS, the
Debye-Waller factor, the relaxation time, and the stretching parameter P should be
independent of wavevector. There are several sources of error that can be introduced in
an explanation of the apparent wavevector dependence.
The frequency response of the detector is one candidate to explain variation of the
Debye-Waller factor, relaxation time, or P with wavevector. As the wavevector changes,
the signal moves to a different area of the frequency response curve of the detection
system. Also, at the highest wavevectors in PPG425, two different detectors may be used
to completely span the frequencies in the signal.
Despite every effort to accurately
account for bandwidth characteristics, error introduced in these processes cannot be
totally eliminated.
In the case of a polymer that is known to exhibit a complex and stretched
relaxation spectrum, the variation with wavevector may be a result of the experimental
145
limitations of ISTS. As the wavevector is changed in the ISTS experiment, the frequency
"window" for observation of the structural relaxation changes as well.
At the high
frequency end, the acoustic frequency increases linearly with wavevector. On the low
frequency end, the thermal diffusion rate varies as wavevector squared. The combination
of these two conditions leads to a wider time or frequency window at low wavevectors
than at high wavevectors. When the Debye-Waller factor is considered as an integration
measurement, a larger frequency window for the integration would result in a larger
integrated area, and thus a higher Debye-Waller factor.
6.5
Conclusions
ISTS measurements have characterized the structural relaxation dynamics of
PPG425 and PPG4000. The relaxation times, relaxation shape, and Debye-Waller factors
are quantitatively similar for both molecular weights.
Thus, we conclude that the
structural relaxation mode that is active in ISTS, i.e. coupled to the density dynamics, is
most closely related to the main a feature in dielectric spectroscopy. MCT predictions do
not adequately describe the relaxation dynamics. There is no evidence of a crossover
temperature in the range studied, either by an anomaly in the Debye-Waller factor
measurement or by a divergence of relaxation times. There is some enhanced stretching
at low temperatures, but it is impossible to determine if this is a result of some
contribution from an additional feature, or an indication of crossover behavior. As with
glycerol, we conclude that Mode Coupling Theory in its current form is not adequate to
describe the dynamics of a polymer liquid.
146
References
1
A. Schonhals and E. Schlosser, Physica Scripta T49, 236 (1993).
2
G. P. Johari, A. Hallbrucker, and E. Mayer, Journalof Polymer Science, Polymer
Physics Edition 26, 1923 (1988).
3
R. Bergman, L. Borjesson, L. M. Torell, and A. Fontana, Phys. Rev. B 56, 11619
(1997).
4
D. L. Sidebottom, R. Bergman, L. Borjesson, and L. M. Torell, Phys. Rev. Lett.
68, 3587 (1992).
5
C. H. Wang and Y. Y. Huang, J. Chem. Phys. 64, 4847 (1976).
6
Y. Y. Huang and C. H. Wang, J. Chem. Phys. 62, 120 (1975).
7
H. H. Krbecek, W. Kupisch, and M. Pietralla, Polymer 37, 3483 (1996).
8
A. R. Duggal and K. A. Nelson, J. Chem. Phys. 94, 7677 (1991).
9
C. H. Wang, G. Fytas, D. Lilge, and T. Dorfmuller, Macromolecules 14, 1363
(1981).
10
K. Pathmanathan, G. P. Johari, and R. K. Chan, Polymer 27, 1907 (1986).
11
Y. Yang, Ph.D. Thesis, Massachusetts Institute of Technology, 1996.
12
W. Petry and J. Wuttke, Transport Theory in StatisticalPhysics 24, 1075 (1995).
147
Chapter 7
Summary and Future Directions
ISTS studies have been conducted on glycerol and Polypropylene Glycol of two
molecular weights. The relaxation dynamics have been characterized as a function of
temperature for times from a few nanoseconds to hundreds of microseconds.
On a
qualitative level, these liquids behave similarly to fragile liquids. Three hydrodynamic
modes are observed, a fast acoustic mode, slow thermal diffusion, and temperature
dependent structural relaxation.
In glycerol, the results are analyzed according to Mode Coupling Theory
predictions. The Debye-Waller factor is measured for a temperature range from 1.2 Tg to
1.45 Tg, and no evidence of a MCT square root cusp is observed in this range. The
stretching parameter, P, is constant with temperature in this same range, indicating that
time-temperature superposition holds.
temperature dependence.
The relaxation times are fit to a power law
A fit to all the data results in a To value below the glass
transition temperature. When the data are restricted to high temperature values, the data
are not inconsistent with TC = 225 K, but this should not be considered a definitive
measurement of T. Without data at longer times and lower temperatures, we are unable
to rule out the possibility of a crossover temperature below 225 K. However, due to the
conflicting evidence from ISTS and other experiments, we conclude that Mode Coupling
Theory is very limited at present in its ability to explain the dynamics of a material with
complex interactions, such as the hydrogen bonding that is prevalent in glycerol.
ISTS is also used to characterize the relaxation dynamics in two molecular
weights of Polypropylene glycol. The dielectric spectrum shows two features in the a
relaxation regime.
ISTS results only indicate one relaxation process, which is
quantitatively similar for both molecular weights.
The measured relaxation times,
stretching parameters, and Debye-Waller factors are similar for both molecular weights.
148
We conclude that the dynamics observed in ISTS are related to the main a relaxation in
dielectric spectroscopy. There are weak observations of a more complicated relaxation
scenario, such as decreased values of P at low temperatures, but no quantitative
conclusions are drawn. The second relaxation does not appear to be strongly coupled to
the density dynamics probed by ISTS.
As with glycerol, there is no evidence of a Mode Coupling Theory crossover
temperature in either molecular weight. Although the Debye-Waller factor is weakly
temperature dependent, there is no clear evidence of an anomaly in the temperature range
studied.
There is a decrease of P with temperature, but this observation cannot be
definitively interpreted as an indication of crossover behavior. Power law fits to the
relaxation times cannot clearly determine a To value in the supercooled regime. Mode
Coupling Theory applied to Polypropylene Glycol appears to suffer from the same
limitations as in the application to glycerol.
The dynamics in a material with very
complex interactions are not adequately described by the theory. Additional theoretical
considerations may help to extend MCT capabilities into this regime.
Continuing efforts to apply Mode Coupling Theory to the dynamics of glass
forming liquids would benefit from expanding the frequency range of ISTS. Expanding
to include higher wavevector measurements can permit analysis of the minimum in the
acoustic modulus spectrum. The acoustic modulus can be computed from the acoustic
parameters obtained from ISTS and ISBS measurements. This has been done to some
extent for salol and PPG, using a picosecond pulsed probe version of the ISTS
experiment. 1,2
The difficulty in manually generating large angles can now be bypassed by using
phase masks, and the data could be improved with measurements at more wavevectors.
Instead of taking data at a range of temperatures at a single wavevector, the sample can be
kept at a fixed temperature while the wavevector is changed by switching patterns on the
phase mask. This greatly reduces the uncertainty due to temperature variation in the
measurement of the modulus. In order to explore the minimum in the susceptibility
spectrum, the frequency range of the acoustic measurements made with ISTS or
149
Impulsive Stimulate Brillouin Scattering (ISBS) must be extended well into the GHz
regime, to a frequency of 30 GHz or higher. Generating an acoustic frequency of 30
GHz, assuming a speed of sound near 3000 m/s, would require a wavelength of about 100
nm.
In order to generate a grating with the appropriate fringe spacing, ultraviolet
excitation wavelengths must be used. At present, the technology is not available to
generate the necessary excitation pulses.
The practical limit for time resolution with current femtosecond technology is
sub-100 femtosecond. Frequencies into the THz regime can be observed, if the acoustic
wave can be generated. The highest practically generated wavevector would involve the
third harmonic of Ti:Sapphire pulses, at 266 nm.
This excitation could be used to
generate fringe spacings on the order of hundreds of nanometers, corresponding to
acoustic frequencies near 10 GHz. Femtosecond excitation and probing would provide
sufficient time resolution for these acoustic frequencies.
The sample should be
transparent to avoid photochemical effects. Assuming sufficient energy in the excitation
and probe, ISBS signal could be obtained from transparent samples.
A simpler, but limited way of extending the frequency range of ISTS
measurements would continue with a cw probe and use a streak camera for detection.
Steak camera detection, with femtosecond excitation pulses, can provide time resolution
into the tens of picoseconds, extending the frequency range to a few GHz.
This
improvement would be sufficient for some rudimentary evaluation of the susceptibility
minimum at certain temperatures.
Moving away from Mode Coupling Theory, ISTS can be used to continue to
explore polymer dynamics. Many polymers easily form glasses without crystallization,
and thus are an attractive area to extend ISTS measurements, both for practical purposes
and theoretical advancement.
The contribution of rotational motions is an important question in the
interpretation of light scattering data. Orientational relaxation times can be made in the
time domain by conducting Optical Kerr effect (OKE) experiments. OKE experiments
have been conducted for salol, and questions arise in the comparison of the OKE
150
relaxation times with ISTS relaxation times. 3 Additional insight may be gained from
OKE and ISTS or ISBS experiments on the same sample. Sulfur Monochloride, S2C12 is
a glass-forming liquid which is also strongly polarizable. As such, it is an attractive
candidate for measuring the temperature dependence of orientational relaxation times
extending down to the glass transition. Recent experiments have characterized the Kerr
signal in the femtosecond and picosecond regime. 4
A cw probe experiment with
electronic detection may be useful for measuring longer relaxation times near and
possibly below the glass transition. If ISTS or ISBS signal can also be obtained from
S2C1 2, some questions about the relationship between ISTS relaxation times and
orientational relaxation times may be answered.
The use of phase masks and the presence of a controlled "carrier" beam for
heterodyne detection creates additional possibilities for experiments related to ISTS.
Experiments that previously did not generate sufficient signal levels for observation with
a cw probe may be feasible using heterodyne detection. Samples that absorb very weakly
may be studied, either through increasing the ISTS signal by heterodyning, or by ISBS.
A variation of ISBS with crossed polarizations of the excitation light can measure shear
acoustics. 5
Heterodyne detection may simply be used to reduce the amount of excitation and
probe energy. Recently, the applicability of ISTS measurements of the properties of thin
films for practical application has been demonstrated. 6 In the traditional sense, however,
ISTS measurements in liquids still requires excitation energy only available in laboratory
lasers. Incorporating phase masks and heterodyne detection, it may be possible to detect
signal generated from weaker powered, miniturized diode pump lasers.
A possible
extension of this work would be a commercially viable ISTS apparatus for real-time
monitoring of polymer manufacturing processes.
In general, expansion of the characterization of relaxation dynamics, whether by
measuring different properties, or by increasing the range of the experiment, can have
significant positive impact on the understanding of the behavior of complex systems.
Hopefully, the dialog between theory and experiment will continue, improving the
physical understanding of relaxation dynamics in complex materials.
References
1
Y. Yang, Ph.D. Thesis, Massachusetts Institute of Technology, 1996.
2
A. Duggal, Ph.D. Thesis, Massachusetts Institute of Technology, 1992.
3
R. Torre, P. Bartolini, and R. M. Pick, Phys. Rev. E 57, 1912 (1998).
4
J. Fourkas, (1998).
5
S. Silence, Ph.D. Thesis, Massachusetts Institute of Technology, 1991.
6
J. Rogers, Ph.D. Thesis, Massachusetts Institute of Technology, 1995.
152
Appendix A
Selected Matlab Codes
Deconvolution of Antel Impulse Response
function data=deconvolveimp(fname,simp,tstep)
%function to deconvolve impulse responsefrom data
data=readdata(fname);
N=length (data (:,1))
data=data (:N-1, :);
renorm=max (data (:,2));
trep=data (:, 1);
data=zerotime (data) ; % set zero of time
time=data (:, 1);
signal=data (:,2);
signal=zerobeg (signal) ; % zero the baseline
signal=signal/max(signal);
n=sum(time<0);
N=length (time);
s=signal (n :N) ; % take only positive time data
tdata=time (2) -time (1) ; % calculate sampling interval
% resample ratio is the ratio of data sampling interval to impulse
% sampling interval
R=tdata/tstep
% resample the signal to the same sampling intervalas the impulse response
s=resample (s, round (R) , 1) ;
% This is the time consuming step
sconv=[ s ' s imp' ' ; % combining the impulse response and data
[x,y]=deconv(sconv,simp);
x=x(1:length(x)-l);
s=resample (x, 1, round (R) ); % resample data back down to the originalrate
s=s*renorm/max (s) ; % multiply data back to originalamplitude
newsignal=[0.1*signal(l:n-) ' s']';
% put back the originalt<O signal
data=[trep(l:round( 0 .95*length(trep)))
newsignal(l:round(0.95*length(newsignal)))];
% cut off the end of the data, because often some noise/ringingis picked up
% in the deconvolution process
Correction for Hamamatsu roll-off
function data=hamcorrect (data)
% correct datafrom Hamamatsu detector for low frequency rolloff
% decay constant is determinedfrom single exponentialfit to early time
% response to quasi-cwpulse
% do not trust this datafor times longer than 1 microsecond
1);
data=zerotime (data);
T=data (:,
time=data ( :, 1) ;
N=length(time);
n=sum(time<=0);
signal=data (:,
2);
decay=ones(size(signal));
decay(n:N)=exp(-time(n:N)/2.2e-06);
signal=signal./decay;
data=[T signal];
Combination of two data files with different time spacing
function [data]=combine(datal, data2)
% take signals of different time spacingand combine them
% data should already be turned "right-side-up"
% input order is short, long
timel=datal (:,1);
signall=datal(:,2);
signall=zerobeg(signall);
time2=data2 (:,1);
signal2=data2(:,2);
signal2=zerobeg(signal2);
N1=length (timel) ;
N2=length (time2) ;
%findamplitude of the end of the short signal
levell=mean(signall(N1-50:N1));
tl=timel (N1-50);
t2=timel (N1) ;
indexl=sum(time2<tl);
index2=sum(time2<t2);
%findamplitude of overlappingportion of long signal
level2=mean(signal2 (indexl :index2));
signall=signall*level2/levell; %scaletomatch
signal=[signall' signal2 (index2+1:N2) ']; %combinesignal
time=[timel' time2 (index2+1:N2) ']; %combine time axes
data= [time; signal] ';
ISTS semi-automated fitting routine. Least-squares fitting code is called as an
executable.
% scriptto fit ISTS data, semi-automatedwith presetfile names
% optionfor range of temperaturesor single temperature
% yfit.exe must be in the present directory
T=input('Enter the temperature filename (in single quotes)
or temperature integer>');
directory=input('Enter the directory containing the combined
data>', 's');
directory=strcat(directory,'\');
directory2=input('Enter the destination directory>', 's');
directory2=strcat(directory2,'\');
% only save parameterfiles for multiple temperaturefits
if isstr(T)
% read in the temperatures as integers
temps=readdata(T);
% initialize matricesforfit parameters
% set up Nx2 matrices with temps in thefirst column
A=initialize(temps);
ga=A;wa=A;tO=A;gh=A;B=A;gr=A;beta=A;chi=A;
continue=l;
I=1;
while and (I<=length(temps), continue==l)
fr=input('Do you want to edit fitin.rise?
(no=O;yes=l)');
if fr==l
! notepad.exe fitin.rise
end
T=int2str(temps(I,1))
fname=strcat(directory,T,'all.dat')
outfile=strcat(directory2,T,'fit.dat')
fidl=fopen(fname, 'r')
fid2=fopen(outfile,'w')
fclose('all');
if fidl+fid2==7
eval(['! yfit.exe ' fname ' -s ' outfile ' -f
fitin.rise'])
[fitdata,A,ga,wa,tO, gh,B,gr,beta,chi]=readparams...
(outfile,I,A,ga,wa,tO,gh,B,gr,beta,chi);
data=readdata (fname);
fitdata=fitdata';
figure (1)
zoom on
subplot(2,1,1), plot(data(:,l), data(:,2),
fitdata(:, ) ,fitdata(:,2))
N=length (data ( :, 1));
n=sum(data (:,1)<=0)
nl=sum(data(:,l)<data(N,1)/100);
subplot(2,1,2), plot(data(l:nl,l),data(l:nl,2),...
fitdata(l:nl,l),fitdata(l:nl,2))
figure (2)
zoom on
semilogx(data(n:N,1), data(n:N,2), fitdata(n:N,1),
fitdata(n:N,2))
continue=input('Keep going? (n=0,y=l)');
end
I=I+1;
end
w=input('Save parameters? (no=0;yes=l) ');
if w==0
w=input('Save parameters? (no=0;yes=l) ');
end
if w==1;
n=sum (A( :,3) >0);
A=A(1:n,:)
ga=ga(1l:n,:);
wa=wa(1:n, :);
t0=t0(1:n,:);
gh=gh(l:n,:);
B=B(1:n,:);
gr=gr(l:n,:);
beta=beta(l:n,:);
chi=chi(1:n, :);
writeparams(directory2,A,ga,wa,t0,gh,B,gr,beta,chi)
DWF=[A(:, 1)';A(:,2)';(B(:,3)./(A(:,3)+B(:,3)))']';
dwffile=strcat(directory2,'dwf.dat');
fid=fopen(dwffile, 'w');
fprintf(fid,
'%d %d\n',DWF);
fclose(fid);
end
else
if input('Do you want to edit fitin.rise?
(no=0;yes=l)')==1
I notepad.exe fitin.rise
end
T=int2str(T);
fname=strcat(directory,T,'all.dat');
data=readdata (fname);
outfile=strcat(directory2,T,'fit.dat')
eval(['!
yfit.exe
'
fname '
-s
' outfile
'
fitin.rise'])
% savefit andparameters,getfitdata
fitdata=writeonefit(outfile,directory2,T);
figure(1); zoom on
plot(data(:,1),data(:,2),fitdata(:,1),fitdata(:,2))
end
156
-f
Appendix B
Selected References
Appendix B contains selected references, organized by subject matter. The
references to particular citations are found at the end of each chapter. This Appendix
contains a more comprehensive list of applicable sources, including some that are not
cited in individual chapters.
Impulsive Stimulated Scattering Experiment
1.
Yan, Y.-x. and K.A. Nelson, Impulsive stimulated light scattering. I. General
Theory of Impulsive Stimulated light scattering. II. Comparisonto frequencydomain light-scatteringspectroscopy. Journal of Chemical Physics, 1987. 87(11):
p. 6240.
2.
Yang, Y. and K. Nelson, Impulsive Stimulated Light Scatteringfrom Glassforming Liquids: I. Generalizedhydrodynamics approach.Journal of Chemical
Physics, 1995. 103(18): p. 7722.
Mode Coupling Theory and Hydrodynamics and General Liquid-Glass Transition
3.
Angell, C.A., Formationof Glassesfrom Liquids and Biopolymers. Science,
1995. 267: p. 1924.
4.
Angell, C.A., et al., eds. Structure and Dynamics of Glasses and Glass Formers.
Materials Research Society Symposium Proceedings, ed. M.R. Society. Vol. 455.
1997, Materials Research Society: Pittsburgh, PA.
5.
Angell, C.A., Perspective on the Glass Transition.Journal of Physical Chemistry
of Solids, 1988. 49(8): p. 863.
6.
Bengtzelius, U., W. Gotze, and A. Sjolander, Dynamics of Supercooled Liquids
and the Glass Transition.Journal of Physics C: Solid State Physics, 1984. 17: p.
5915.
7.
Cummins, H.Z., et al., Relaxationaldynamics in supercooled liquids:
experimental tests of the mode coupling theory. Physica A, 1994. 204: p. 169.
8.
Ediger, M.D., C.A. Angell, and S.R. Nagel, SupercooledLiquids and Glasses.
Journal of Physical Chemistry, 1996. 100(31): p. 13200.
9.
Fourkas, J.T., et al., eds. SupercooledLiquids: Advances and Novel Applications.
ACS Symposium Series. Vol. 676. 1997, American Chemical Society:
Washington, DC.
10.
Gotze, W. and L. Sjogren, Relaxation Processesin Supercooled Liquids. Reports
on Progress in Physics, 1992. 55: p. 241.
11.
Hansen, J.P., D. Levesque, and J. Zinn-Justin, eds. Liquids, Freezing,and the
Glass Transition.. Vol. 1. 1991, Elsevier Science Publishing Company:
Amsterdam. 503.
12.
Kim, B., Stretching, Mode Coupling, and the Glass Transition.Physical Review
A, 1992. 46(4): p. 1992.
13.
Kim, B. and G.F. Mazenko, Mode Coupling, Universality,and the Glass
Transition. Physical Review A, 1992. 45(4): p. 2393.
14.
Petry, W. and J. Wuttke, QuasielasticNeutron Scattering in Glass Forming
Viscous Liquids. Transport Theory in Statistical Physics, 1995. 24: p. 1075.
Experimental Studies of Fragile Glass-Forming Liquids
15.
Yang, Y. and K. Nelson, Impulsive stimulatedlight scatteringfrom glass-forming
liquids: II. Salol relaxationdynamics, nonergodicityparameter,and testing of
mode coupling theory. Journal of Chemical Physics, 1995. 103(18): p. 7732.
16.
Yang, Y. and K.A. Nelson, Impulsive stimulated thermal scatteringstudy of alpha
relaxation dynamics and the Debye-Wallerfactor anomaly in CKN. Journal of
Chemical Physics, 1996. 104(14): p. 5429.
17.
Cummins, H.Z., et al., Light scatteringspectroscopy of the liquid-glass transtion:
comparison with the idealized and extended mode coupling theory. Physica A,
1993: p. 207.
18.
Cummins, H.Z., et al., Light-scatteringspectroscopy of the liquid-glass transition
in CaKN03 and in the molecular glass Salol: Extended-mode-coupling-theory
analysis. Physical Review E., 1993. 47(6): p. 4223.
19.
Muller, L., Nonlinear Spectroscopic Studies of Liquids, Ph. D. Thesis in
Chemistry. 1995, University of Texas: Austin, TX.
158
20.
Schonhals, A., et al., Anomalies in the scaling of the alpha relaxationstudied by
dielectric spectroscopy. Physica A, 1993. 201: p. 263.
21.
Torre, R., P. Bartolini, and R.M. Pick, Time-resolved optical Kerr effect in a
fragile glass-forming liquid, salol. Physical Review E, 1998. 57(2): p. 1912.
Experimental Studies of Glycerol
22.
Wuttke, J., W. Petry, and S. Pouget, Structuralrelaxationin viscous glycerol:
Coherent neutron scattering.Journal of Chemical Physics, 1996. 105(12): p.
5177.
23.
Wuttke, J., et al., Neutron and Light ScatteringStudy of Supercooled Glycerol.
Physical Review Letters, 1994. 72(19): p. 3052.
24.
Birge, N.O. and S.R. Nagel, Specific Heat Spectroscopy of the Glass Transition.
Physical Review Letters, 1985. 54(25): p. 2674.
25.
Franosch, T., et al., Evolution of structuralrelaxationspectra of glycerol within
the gigahertz band. Physical Review E, 1997. 55(3): p. 3183.
26.
Jeong, Y.H., S.R. Nagel, and S. Bhattacharya, Ultrasonicinvestigation of the
glass transition in glycerol. Physical Review A, 1986. 34(1): p. 602.
27.
Lunkenheimer, P., et al., High-frequency dielectric spectroscopy on glycerol.
Europhysics Letters, 1996. 33(8): p. 611.
28.
Lunkenheimer, P., et al., Fast Dynamics of Glass-FormingGlycerol Studied by
Dielectric Spectroscopy. Physical Review Letters, 1996. 77(2): p. 318.
29.
Pimenov, A., P. Lunkenheimer, and A. Loidl, BroadbandDielectricSpectroscopy
of Glycerol and CKN. Ferroelectrics, 1996. 176: p. 33.
30.
Schonhals, A. and E. Schlosser, Relationshipbetween segmental and chain
dynamics in polymer melts as studied by dielectric spectroscopy. Physica Scripta,
1993. t49: p. 236.
Experimental Studies of Polypropylene Glycol
31.
Duggal, A.R. and K.A. Nelson, Picosecond-microsecondstructuralrelaxation
dynamics in poly(propylene glycol): Impulsive stimulated light-scattering
experiments. Journal of Chemical Physics, 1991. 94: p. 7677.
159
32.
Duggal, A.R. and K.A. Nelson, Resolution of conflicting descriptions of
poly(propylene glycol) relaxationdynamics through Impulsive Stimulated
Scattering experiments. Polymer Communications, 1991. 32(12): p. 356.
33.
Johari, G.P., Dielectric relaxation in the liquid and glassy states of
poly(propylene oxide) 4000. Polymer, 1986. 27: p. 866.
34.
Jones, D.R. and C.H. Wang, DepolarizedRayleigh scatteringand backbone
motion ofpolypropylene glycol. Journal of Chemical Physics, 1976. 65(5): p.
1835.
35.
Pathmanathan, K., G.P. Johari, and R.K. Chan, Effect of water on relaxations in
the glassy and liquid states ofpoly(propylene oxide) of molecular weight 4000.
Polymer, 1986. 27: p. 1907.
36.
Pathmanathan, K. and G.P. Johari, The effect of water on dielectricrelaxations in
the glassy states ofpoly(propylene oxide) andpropylene glycol. Polymer, 1988.
29: p. 303.
37.
Sidebottom, D.L., et al., Observation of Scaling Behavior in the Liquid-Glass
TransitionRange from Dynamic Light Scattering in Poly(propylene glycol).
Physical Review Letters, 1992. 68(24): p. 3587.
38.
Baur, M.E. and W.H. Stockmayer, DielectricRelaxation in Liquid Polypropylene
Oxides. Journal of Chemical Physics, 1965. 43(12): p. 4319.
39.
Bergman, R., et al., Dynamics aroundthe liquid-glasstransition in
poly(propylene glycol) investigatedby wide-frequency-range light-scattering
techniques. Physical Review B, 1997. 56(18): p. 11619.
40.
Borjesson, L., J.R. Stevens, and L.M. Torell, Brillouin scatteringstudies of
structuralrelaxationin polypropylene glycol. Polymer, 1987. 28: p. 1803.
41.
Huang, Y.Y. and C.H. Wang, Brillouin,Rayleigh, and depolarizedRayleigh
scatteringstudies ofpoly(propylene glycol). I Journal of Chemical Physics, 1975.
62(1): p. 120.
42.
Wang, C.H. and Y.Y. Huang, Brillouin-Rayleighscatteringstudies of
poly(propylene glycol). III. Journal of Chemical Physics, 1976. 64(12): p. 4847.
43.
Wang, C.H., et al., Laser light beatingspectroscopicstudies of dynamics in bulk
polymers: poly(propylene glycol). Macromolecules, 1981. 14: p. 1363.
160