Toward ultra-broadband photoacoustic spectroscopy of
supercooled liquids
_
______
MASSACHUSETTS INSTITUTE
OF TECHNOLOLGY
by
Kara Jean Manke
JUN 23 2015
I
B.A., Macalester College (2007)
LIBRARIES
Submitted to the Department of Chemistry
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Chemistry
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
@ Massachusetts
Institute of Technology 2015. All rights reserved.
Signature redacted
A uth or .......
......................................
Department of Chemistry
May 14, 2015
Signature redacted
C ertified by ................................................................
Keith A. Nelson
Professor of Chemistry
Thesis Supervisor
Signature redacted
Accepted by ......................................................
Robert W. Field
Haslam and Dewey Professor of Chemistry
This doctoral thesis has been examined by a Committee of the Department of
Chemistry as follows:
Professor Robert G. Griffin..
Signature redacted
Chairm n, Thesis Committee
Professor of Chemistry
Professor Keith A. Nelson...
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Thesis Supervisor
Professor of Chemistry
Professor Jianshu Cao ...............
Signature redacted
U
Member, Thesis Committee
Professor of Chemistry
Toward ultra-broadband photoacoustic spectroscopy of supercooled
liquids
by
Kara Jean Manke
Submitted to the Department of Chemistry
on May 14, 2015, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Chemistry
Abstract
Theoretical understanding of the dramatic slow-down of structural relaxation dynamics in
supercooled liquids leading to the glass transition is limited in part by the lack of data in these
systems. Between the melting point and the glass transition, the viscosity of supercooled
liquids can change by 16 orders of magnitude or more and few experimental techniques can
access this broad a range of frequencies and time scales.
Photoacoustic spectroscopy is an ideal technique for probing the dynamics of supercooled
liquids because it provides direct access to mechanical relaxation. Currently, photoacoustic
spectroscopy of supercooled liquids has been carried out in the range of 10 MHz to 1 GHz
with impulsive stimulated light scattering (ISS) and 8 GHz to 200 GHz with picosecond
ultrasonics. In this work, I present advances in photoacoustic techniques with the goal of
extending the range of frequencies accessible by these methods.
To achieve higher-frequency acoustic generation, experiments on semiconductor superlattices (SLs) composed of alternating layers of gallium arsenide (GaAs) and aluminum
arsenide (AlAs) and multiple quantum well structures (MQW) of indium gallium nitride
(InGaN) and gallium nitride (GaN) as photoacoustic transducers are presented. The results
demonstrate that InGaN/GaN SLs can be employed as broadband photoacoustic transducers, generating acoustic frequencies up to 2.5 THz. A simple transient reflectivity technique
for the detection and reconstruction of short (- 2 ps) acoustic strain pulses is also presented.
Using a time-domain analogue of Brillouin light scattering, data on the acoustic velocity
and attenuation rate of supercooled liquid DC704 at frequencies of
-
6 GHz and
-
12 GHz,
a region which can be hard to access with both ISS and PU approaches, is shown. Finally,
the slow rise or "Mountain mode" component of ISS signal from DC704, which arises from
slow components of the density response at timescales from 10- 4 s to 10- 7 s is examined.
Comparison with the broadband compliance spectrum of DC704 demonstrates that the slow
rise signal does not directly reflect the relaxing elastic compliance, but contains contributions
from other slow degrees of freedom that couple to the density.
Thesis Supervisor: Keith A. Nelson
Title: Professor of Chemistry
5
6
Acknowledgments
I have to admit, one of the reasons I wanted to become a scientist is because it seemed
like the closest thing in the real world to being a wizard. After seven years as a graduate
student at MIT, I can't say I've learned how to change a rabbit into a teapot or pull money
out of someone's ear (if only!), but looking back, I believe my personal and professional
transformation has been nearly as magical. My years here have been full of more stress,
frustration, self-doubt, and hard work than I ever could have imagined. But all of the effort
has paid off in wisdom, self-confidence, and self-understanding, and I could not have grown
so much were it not for the support of the amazing group of people I have found at MIT.
For someone with magical aspirations, Keith Nelson is the perfect advisor and role model.
To Keith, nothing is impossible. He dreams up an infinite number of questions and for each
question imagines an infinite number of possible answers. Every conversation I've had with
him has been a lesson in scientific creativity and problem solving. And, like any good
storybook wizard, what is most admirable about Keith is how genuinely he cares about the
success and well being of those around him, and strives to only use his powers for good.
I have also had the benefit of interacting with my thesis committee members Jianshu Cao
and Robert Griffin. I learned everything I know about statistical mechanics from being a
student in, and later a grader for, Jianshu's statistical mechanics class, and our discussions
about my research have been equally as informative. My meetings with Bob have been
encouraging and helpful with navigating graduate school and my future career.
My scientific development has been guided by amazing mentors over the years, including
Christoph Klieber, Alexei Maznev, Kung-Hsuan Lin, Vasily Temnov, Jeremy Johnson, and
Darius Torchinsky. Christoph taught me everything I know about picosecond ultrasonics
and how to run an experiment, and the rate at which he can generate ideas rivals that of even
Keith. Alex, our photoacoustics guru, has provided me with invaluable scientific guidance
over the years. If ever I am unsure of what the next step is, I just ask myself "what
would Alex do?" and that usually points me in the right direction. Vasily provided me with
close mentorship while I crafted my first first-author paper, and Kung-Hsuan was a patient
instructor who led me through the construction of my first optical set-up. Finally, Jeremy
and Darius have been my ever-cheerful and friendly guides to the science of glass-forming
liquids and the impulsive stimulated scattering measurements.
More recently, I have had the pleasure of working closely with Doug Hyun Shin and
Jeffrey Eliason. Doug's work ethic and wonder for science have made him a delightful mentee,
and I am confident that he will continue to make excellent progress in the development of
picosecond ultrasonics techniques. Jeff has been an engaging co-conspirator on acoustics
measurements, and I am proud to be graduating alongside him this year. The entire Nelson
group, past and present, has made the lab an incredibly fun and pleasant place to work.
I also have to thank the many other scientific teachers, advisors, and collaborators I have
had over the years, without whom the work presented in this thesis would not have been
possible: Tina Hecksher and Jeppe Dyre of Roskilde University; Chi-Kuang Sun of National
Taiwan University and Jen-Inn Chyi of National Central University; Victor Shalagatskyi
and Denys Makarov the Universite du Maine; Seung-Hyub Baek and Cheng-Beom Eom of
the University of Wisconsin-Madison; and my colleagues in the Solid-State Thermal Energy
Conversion (S3TEC) EFRC at MIT. Finally, I have to thank Dave Terry, Libby Shaw,
7
Timothy McClure, technicians at MIT's MTL and CSME self-user facilities.
Outside the lab, my professional development has been shaped by affiliation with many
inspiring people and student groups, including the ChemREFS; Nathan Sanders, Shannon
Morey, and the many participants and organizers of the Communicating Science Conference;
and the AAAS Mass Media Fellowship, my amazing group of friends among the 2014 class
of Fellows, and the support and encouragement of Scott Hensley, Nancy Shute, Joe Palca,
and Deborah Franklin during my summer at NPR.
Perhaps the greatest personal challenge I faced during my time at MIT was my diagnosis
with, and ongoing recovery from, chronic illness. When my health was at its worst, I often
wondered whether I had the strength and fortitude to finish my degree, and it is thanks to
all the wonderful people here that I had the courage to continue. I was overwhelmed by the
emotional and material support offered by those around me when I needed it most: Keith
for giving me the time and space to heal; Susan Brighton, Li Miao and Blanche Staton for
helping me navigate the administrative process of medical leave; Vikki Brown for her endless
wisdom and encouragement; and Jenn Scherer and Steph Teo for letting me store my stuff
for free in their apartment. When I returned from medical leave, Patrick Wen, Nicole Davis,
and Kevin Jones gave me a place to stay for over a month while I searched for a new place,
and Christina Hanson and Malinky the cat became my new cheerful, upbeat roommates.
But perhaps what buoyed me the most were the amazing group of comrades who accompanied me along the journey, giving me a reason to go on even when things were at their
darkest. Patrick Wen and Dylan Arias, you have been my companions from day one, endless
fonts of calm and humor. If only Characters was still open I would take you back for beers
and a round of pool. Lemon, you are my rock. Is that cheesy? Maybe, but you'll just have
to deal with it, cause it's true. Casi Newell, I feel like we have grown up together. I am
so proud of how far we both have come, and that our friendship has not only survived the
journey, but also grown deeper and stronger. One day, we will move to a commune and raise
chickens and cats. Ben Ofori-Okai, do you realize how many people owe their happiness and
success to your patient and unwavering support? No, you don't, and perhaps that makes
you even greater. Steph Teo and Jenn Scherer, without your friendship I may have forgotten
what it means to have two X chromosomes. Now, let's go dress shopping and get manicures.
Harold Hwang, underneath all the antics (which you know I secretly love) you have been an
immensely kind and generous friend. Nate Brandt, I would have been much more homesick
without a fellow bicycle-loving Minnesotan to talk to, and Colby Steiner, I am always up
for a coffee and some gossip. Jian Cui, we've had our ups and downs, but you've seen me
through some of the toughest moments of my life, and for that I will always be grateful.
Last, but certainly not least, my hero The Great Sharly Fleischer (or would you prefer I call
you Amanda?). You helped me through my fear of aligning the terrifying RegA and my fear
that my career would be forever stymied by health challenges. Your depth of caring for all
creatures, animal and human alike, is truly awe-inspiring.
Finally, I have to thank my wonderful friends and family back home, without whom
none of this would have been possible: my parents, Brian and Jean Manke, who are always
available on the other end of the phone whenever I need a sympathetic ear, who support me
through sickness and health and have made numerous trips to visit me in Boston even though
neither particularly enjoy travel; my siblings, Neil and Terren Manke, who continuously
remind me of what is truly important in life; the rest of my amazing family, who have been
8
my tireless cheerleaders since the day I was born; Andrew Hookom, my best friend for so
long we might as well be family; Linse Lahti, Andrea Nelson and Micaela Mooers, sisters
in spirit; and my grandparents, Lyle Kragh, Esther Kragh, Herman Manke, and Shirley
Manke, whose boundless love continues to light my way.
Kara Jean Manke
May 14th, 2015
9
10
For Grandma
12
Contents
1
Overview
17
2
Dynamics of supercooled liquids and the glass transition
21
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2
Phenomenology of supercooled liquids and the glass transition . . . . . . . . . 22
2.3
Thermodynamics of the glass transition
. . . . . . . . . . . . . . . . . 22
2.2.2
Kinetics of the glass transition
2.2.3
Relaxation dynamics in supercooled liquids
. . . . . . . . . . . . . . . . . . . . . . 24
2.3.1
Mode coupling theory equations . . . . . . . . . . . . . . . . . . . . . . 34
2.3.2
Mode coupling theory predictions . . . . . . . . . . . . . . . . . . . . . 36
2.3.3
Tests of mode coupling theory . . . . . . . . . . . . . . . . . . . . . . . 43
Introduction to photoacoustic spectroscopy
3.1.1
. . . . .
47
Impulsive stimulated scattering . . . . . . . . . .
49
. . . . . . . . . . . .
.
Introduction to stress and strain
.
.
45
3.2.1
The ISS experiment
.
3.3
Introduction . . . . . . . . . . . . . . . . . . . . .
50
3.2.2
Qualitative characteristics of ISS signal
.
53
3.2.3
Experimental set-up for ISS measurements
56
. . . . . . . . . . . . . . .
57
Picosecond ultrasonics
.
3.2
45
.
3.1
. . . . . . . . . . . . . . . 28
Mode coupling theory of supercooled liquids . . . . . . . . . . . . . . . . . . . 33
.
3
2.2.1
13
3.5
4
57
Acoustic detection
. . . . . . . . . . . . . .
61
3.3.3
Liquid sample design . . . . . . . . . . . . .
67
3.3.4
Experimental set-up for picosecond ultrasonics measurements
69
.
3.3.2
.
. . . . . . . . . . . . .
.
Acoustic generation
The longitudinal compliance spectrum of DC704
71
3.4.1
The compliance minimum . . . . . . . . . .
73
3.4.2
Fast dynamics and the Boson peak . . . . .
74
Summary and conclusion . . . . . . . . . . . . . . .
75
.
.
.
.
.
3.4
3.3.1
Coherent narrow-band acoustic generation and detection in semiconductor
superlattices
. .. .. . .. . .
.
77
4.2
Phonon lifetimes in GaAs/AlAs superlattices
. ... . . .. . .
80
4.2.1
Experimental methods . . . . . . . .
. .. .. . .. . .
82
4.2.2
Results and analysis . . . . . . . . .
. .. .. . .. . .
83
.
.
.
.
Broadband acoustic generation in InGaN/GaN multiple quantum wells
87
. . . . . . . . . .
88
4.3.2
Results and analysis . . . . . . . . .
. . . . . . . . . .
89
Conclusions and outlook . . . . . . . . . . .
. . . . . . . . . .
92
.
.
.
.
Experimental methods . . . . . . . .
.
4.3.1
.
4.4
.
Introduction . . . . . . . . . . . . . . . . . .
.
4.1
4.3
Measurement of shorter-than-skin-depth acoustic pulses in a metal film
via transient reflectivity
95
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 95
5.2
Theory of transient reflectivity detection . . . . . . . . . .
. . . . . . . . . . 97
5.3
Experimental Methods . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 101
5.4
Results and Analysis . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 104
5.4.1
Response Fitting . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 106
5.4.2
Acoustic Broadening . . . . . . . . . . . . . . . . .
. . . . . . . . . . 107
5.4.3
Pulse Reconstruction . . . . . . . . . . . . . . . . .
. . . . . . . . . . 107
.
.
.
.
.
.
5.1
.
5
77
14
5.5
6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Time-domain Brillouin scattering measurements on DC704 near the compliance loss minimum.
Introduction . . . . . . . . . . .
. 111
6.2
Experimental Methods .......
. 113
6.3
Data Fitting and Analysis
. 116
.
.
6.1
Acoustic velocity . . . .
. 116
6.3.2
Acoustic attenuation . .
.119
6.3.3
Modulus and Compliance
. 123
.
.
6.3.1
Results and Discussion . . . . .
. 126
6.5
O utlook . . . . . . . . . . . . .
. 131
.
.
6.4
Contributions of mechanical and specific heat relaxation to the ISTS slow
rise signal in liquid DC704
133
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.2
A hydrodynamics model of ISS Signal
. . . . . . . . . . . . . . . . . . . . . . 135
. . . . . . . . . . . . . . . . . . . 141
7.2.1
Acoustic Mode and Elastic Modulus
7.2.2
Slow Rise Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.3
Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.4
Data Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.5
7.4.1
Errors in the Acoustic Oscillations
. . . . . . . . . . . . . . . . . . . . 145
7.4.2
Heat Accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.4.3
Fitting Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.5.1
Structural Relaxation Times . . . . . . . . . . . . . . . . . . . . . . . . 154
7.5.2
Debye-Waller Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.5.3
Comparison of Structural and Mechanical Relaxation Behavior
157
7.5.4
Additional slow dynamics in the slow-rise signal . . . . . . . .
160
.
7
I 11
15
7.6
C onclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
16
Chapter 1
Overview
Experimental studies of the of the propagation of sound waves through materials can reveal
important information about their mechanical and thermal properties. Two approaches for
the generation and detection of acoustic waveforms with specified frequencies and shapes
are electroacoustic transduction or photoacoustic transduction.
Electroacoustic transduc-
tion arises from the conversion of alternating electrical currents into mechanical motion,
and is responsible for the sound waves generated by common devices such as speakers and
ultrasound machines. While this approach is excellent for the production of acoustic waves
at frequencies from 10-3 Hz to 10 9 , which includes infrasound (1 mHz to 20 Hz), the range
of human hearing (20 Hz to 20 kHz), and ultrasound (20 kHz to 1 GHz), they are not capable of generating sound waves at the highest frequencies that can propagate in condensed
materials, which can extend to 10-20 THz in crystals. This upper frequency is limited by
the speed of electronic transitors.
With ultrafast lasers capable of generating optical pulses that are picoseconds or femtoseconds in length, photoacoustic transduction is ideal for studying the behavior of sound
waves at higher frequencies. Photoacoustic techniques are currently able to generate and detect acoustic waves at frequencies ranging from 10 MHz to ~ 1 THz (107 Hz to 1012 Hz), and
have been applied to the study of thermal diffusion, phonon propagation, and mechanical
response in a wide array of systems. The work in this thesis is focused on the advancement
17
of photoacoustic techniques to access a broader range of acoustic frequencies, with the goal
of measuring the mechanical responses of glass-forming liquids across the entire Brillouin
zone.
Chapter 2 describes the phenomenogical behavior of glass-forming liquids and their scientific interest. If a liquid is cooled below its melting point and crystallization is avoided,
it becomes supercooled.
Any liquid can be supercooled under the right conditions, but
glass-formers usually refer to those that are readily supercooled. Supercooled liquids exhibit
strongly temperature-dependent relaxation dynamics, with structural relaxation times varying by orders of magnitude over relatively small temperature ranges. The rapid change in
relaxation dynamics is manifest in strongly frequency- and temperature- dependent behavior of relaxing properties such as the elastic moduli, specific heat, and dielectric constant.
At the glass transition temperature T9 , structural relaxation times in supercooled liquids
become so long that they exceed the usual experimental timescale (conventionally defined
as relaxation times - exceeding 100 s or a viscosity exceeding 1013 Poise), and the liquid
is considered an amorphous solid or a glass.
This kinetically defined glass transition is
also accompanied by changes in experimentally observed thermodynamic properties of the
material, such as the thermal expansion coefficient and the isobaric heat capacity [47].
While the first-order phase transition from a liquid to a crystal is well understood, there
is no agreed-upon model to describe the glass transition. In addition to solving a physical
problem of fundamental scientific interest, such a predictive theory would be of immense
value in the development of novel polymer materials, whose structural properties are central
to their function. Part of the difficulty in constructing such a model arises from the challenge
of acquiring experimental data on the dynamics of supercooled liquids and glasses over such
a broad range of time scales, and broadband photoacoustic spectroscopy would facilitate
these endeavours.
An introduction to mechanical spectroscopy in general and photoacoustic spectroscopy
in particular are provided in chapter 3. Two broad categories of photoacoustic measurements
are described: impulsive stimulated light scattering (ISS), which can access acoustic frequen18
cies in liquids ranging from 10 MHz to 1 GHz, and picosecond ultrasonics (PU), which can
access acoustic frequencies in liquids from 10 GHz to 200 GHz. In the ISS technique, the
crossing of two laser pulses creates an optical interference pattern which can excite density
and thermal modes within a system. The thermal and density variations induce changes in
the optical index of refraction of the material which can be detected with the time-dependent
diffraction of a probe beam. In PU measurements, laser heating of a thin metal transducer
film leads to thermal expansion and strain pulse generation.
My colleagues have utilized these techniques to map the elastic compliance spectrum of
glass-forming liquid DC704. Combined with data from piezoelectric gauge measurements,
the results span 13 decades in frequency, and were utilized to test the predictions modecoupling theory (MCT), the only description of the glass transition that is based on a firstprinciples rather than phenomenological approach. However, gaps remain in key portions of
the spectrum. The Boson peak, a high-frequency (- 1 - 2 THz) feature in the light-scattering
spectra of glasses and amorphous solids, cannot yet be reached by acoustic measurements.
An understanding of the nature of this phenomenon could shed light on the microscopic
mechanisms leading to the glass transition. In addition, data are sparse in portions of the
frequency spectrum that are key to testing MCT predictions.
Chapters 4 and 5 describe efforts to extend the PU technique to achieve the generation and detection of acoustic frequencies approaching 1 THz and beyond.
Chapter 4
explores acoustic generation in semiconductor superlattices (SLs). In SLs, alternating layers of different semiconductor materials impose a folding of the Brillouin zone, allowing
access to higher-frequency acoustic modes.
Frequencies up to 2.5 THz are demonstrated
in an InGaN/GaN superlattice, and preliminary attempts at applying these techniques to
amorphous solids are presented.
As acoustic detection is as key as generation, chapter 5
demonstrates how a relatively straightforward reflectivity measurement can be utilized to
detect acoustic strain pulses at frequencies of hundreds of GHz.
Currently, acoustic frequencies between
-
1 and
-
15 GHz in liquids are difficult to
access using either the ISS or the PU techniques. However, data in this region are critical
19
to testing one of the central predictions of MCT. Chapter 6 presents work with the goal of
extending the range of PU measurements to lower frequencies so that this gap can be closed.
Finally, chapter 7 provides an analysis of a slow structural relaxation component of ISS
signal. This component reflects the relaxation of the elastic modulus and specfic heat at
timescales between 10-
4
and 10-7 s, corresponding to frequencies between 10 kHz and 10
MHz. If information on the frequency-dependent elastic modulus could be reliably extracted
from this feature, it would allow access to five decades in frequency with a single photoacoustic technique. However, comparison with available acoustic data indicates that the relaxing
specific heat has a significant impact on the dynamics observed in this signal.
20
Chapter 2
Dynamics of supercooled liquids and
the glass transition
Introduction
2.1
Glasses and amorphous solids are ubiquitous in nature. Many plastics, optical fibers, and
semiconductors are composed of glassy or amorphous materials [47]. Metallic glasses, an active subject of materials science research, have been found to be stronger and more corrosionresistant than crystalline metal [68, 95j. Many foods and biomolecules are preserved in the
glassy state 1201, and some organisms even survive complete dehydration or sub-freezing
conditions in the glassy state, which effectively halts all bio-reactions in a cell [42].
From a practical point of view, the creation of glass is relatively straightforward. Any
liquid has the potential to form a glass when cooled, as long as crystallization can be
avoided, and the term glass-forming liquid refers to liquids that readily form glasses. As the
temperature falls below the melting point TM, molecular motions in the liquid slow down;
the resulting supercooled liquid has a higher viscosity and longer response time than the
normal liquid. Eventually, the molecular motions become so slow that they are no longer
observable on an experimental time scale. At this point, the material exhibits the mechanical
properties of a solid, while retaining the molecular disorder of a liquid.
21
Though humans have known how to make glass for hundreds of years, the fundamental
physics underlying the rapid slow-down in molecular motion between the melting point and
the glass transition remain poorly understood. This chapter begins with a phenomenological
description of the behavior of supercooled liquids and the glass transition. Mode-coupling
theory (MCT), the only theory of the glass transition that relies on a first principles molecular description of the liquid, is then introduced. MCT predicts a physically non-intuitive
connection between the fast, local molecular motions of the liquid state and the slow, correlated motions that characterize the supercooled and glassy states. Finally, the results of
recent experimental tests of the mode coupling theory and underline the need for further
study in the density response of glass forming systems are summarized.
2.2
Phenomenology of supercooled liquids and the glass transition
2.2.1
Thermodynamics of the glass transition
Supercooled liquids and glasses occupy a unique territory between the liquid and the solid
phases.
Liquids and solids are both condensed phases, but liquids are characterized by
their ability to change shape and to flow, while solids have a rigid shape and structure.
Many solids also exhibit long-range order on the molecular level, while liquids are inherently
disordered.
Figure 2-1 illustrates the temperature dependence of a liquid's volume and enthalpy as it
is cooled below its melting point. First-order phase transitions are characterized by a change
in symmetry as molecules rearrange from a disordered liquid state to an ordered crystalline
state, as well as a discontinuity in many of the physical properties including the volume and
the enthalpy.
A supercooled liquid maintains the ability to change shape and flow and remains disordered on the molecular level. However, the decrease in kinetic energy at lower temperatures
relative to the potential energy barriers that restrain rearrangement of local intermolecular
22
Liquid
-C
+4
Supercooled Liquid
Freezing
Crysta I
Temperature
TM
Figure 2-1: A schematic illustration of the temperature dependence of a liquid's volume and
enthalpy as it undergoes crystallization or the glass transition. Freezing, a first-order phase
transition, is characterized by a discontinuity in volume and enthalpy as well as a change
in material symmetry, while the transition from a supercooled to a glassy state features a
rapid but continuous change in the thermal expansion coefficient ap = (a In V/JT)p and
the isobaric heat capacity cp = (&H/0T),. The glass transition is weakly dependent on
temperature, with faster cooling rates resulting in a higher glass transition temperature (Tgi)
and a different molecular glass structure than slower cooling rates (Tg2). Figure adapted
from [471.
23
geometries is evident as a drastic increase in the viscosity. As the temperature approaches the
glass transition temperature, T, the viscosity of a supercooled liquid eventually becomes so
high that all large-scale rearrangements of intermolecular organization, called a-relaxation,
exceed practical measurement timescales. At this point, the liquid undergoes a transition
into a disordered solid or glassy state.
The transition from supercooled liquid to glass, however, does not involve a collective
rearrangement or change in the symmetry of the material, and also is not accompanied by
a discontinuity in the volume or enthalpy. Instead, the glass transition is characterized by a
sudden but continuous shift in the rate of change of the volume with respect to temperature,
also known as the thermal expansion coefficient ap = ( ln V/&T)p, and in the rate of
change of the enthalpy with respect to temperature, also known as the isobaric heat capacity
cp = (OH/OT)p. At the glass transition temperature, these parameters shift to values that
are similar to those of the corresponding crystalline state. If this change did not occur, and
the properties of the supercooled liquid were extended to lower temperatures, then at some
reduced temperature the entropy of the liquid state would appear to drop below that of the
crystalline state, constituting an entropy crisis at which the disordered glass would appear
to have a lower entropy than the ordered crystalline solid. Finally, unlike first-order phase
transitions, the cooling rate can affect both the temperature of the glass transition and the
structure of the resulting glass.
Attempts to categorize the glass transition in terms of other types of thermodynamic
phase transitions have also been unsuccessful [135, 145]. As a result, the glass transition is
often decribed by its kinetic features.
2.2.2
Kinetics of the glass transition
Supercooled liquids undergo a dramatic increase in viscosity as they approach Tg, and the
glass transition can be defined in a practical way as the temperature at which the liquid
viscosity exceeds 1013 Poise. At this point, the liquid is no longer able to sample all structural
configurations and drops out of equilibrium on practical experimental timescales. In this
24
14
Tg 1
12
CD
C
.)
C,,
C)
10
8
6
4
0)
-j
2
0
~f50
Tm
i
200
250
300
Saloi
350
400
450
T(K)
Figure 2-2: The viscosities of glycerol [1571 and salol [981 as a function of temperature.
Between the melting point TM and the glass transition Tg, the viscosities increase over 10
orders of magnitude.
25
141210
10/
8-
I:y
0-
IN.
d0
oOI
0.2
.21
Al1
0.4
0.6
0.8
-2
1.0
TI T
Figure 2-3: An Angell plot depicting the viscosity of common glass-formers as a function
of scaled inverse temperature. "Strong" glass-formers show a nearly Arrhenius temperature
dependence (indicated by the straight line), while "fragile" glass-formers show markedly
super-Arrhenius behavior. Figure adapted from [5].
way, glass formation can be thought of as a transition of the liquid from an ergodic to a
non-ergodic state. Figure 2-2 demonstrates how the viscosities of two common glass-formers,
salol and glyercol, change in excess of 10 orders of magnitude between TM and Tg
Understanding the nature of this rapid increase in viscosity remains one of the great
questions in the physics of the glass transition. Figure 2-3 shows a plot of the log of the
viscosity of common glass-formers as a function of the inverse temperature, where the inverse
temperature has been scaled by Tg. This plot, named an "Angell" plot after it's creator C.
A. Angell [3], illustrates how the viscosity of nearly all glass-forming liquids deviates from
the expected Arrhenius temperature dependence, which is indicated with a solid black line
26
in the plot. This Arrhenius functional form is given by
7(T) = A exp(
kBT
)
(2.1)
where kB is the Boltzmann constant and A and E are temperature-independent constants.
Many functional forms have been used to describe this super-Arrhenius behavior of the
viscosity [6], with the Vogel-Tammann-Fulcher (VFT) being among the most versatile. The
VFT formula is given by
rq(T) = A exp
DT ]
I(T - To)_
(2.2)
where A, B, and To are constants. While empirical data on temperature-dependent viscosities can be fit well by the VFT formula, there is no good physical interpretation for the
meaning of To.
Glass-forming systems are often classified according to their fragility, which is a measure
of the deviation from Arrhenius behavior [4, 21, 5]. The fragility of a liquid can be quanitfied
by the fragility index m, which is given by the slope of the viscosity curve as it approaches
T9 [120].
M log 7(T)
a(Tg/T) T-_
(2.3)
"Strong" glass-formers have a low fragility index and a nearly Arrhenius relationship between
the temperature and the viscosity. These systems are characterized by strong directional
bonds which make them resistant to structural change as they pass through the glass transition. Silica glass (SiO 2 ) and germanium dioxide (GeO 2 ) are among the archetypal examples.
"Fragile" glass-formers have a higher fragility index and their viscosity exhibits a stronger,
super-Arrhenius temperature dependence near Tg. Fragile glass-formers are often organic
liquids held together by weak, non-directional dispersion forces, such as o-terphenyl, and
are highly prone to structural reorganization above Tg.
27
2.2.3
Relaxation dynamics in supercooled liquids
Unlike simple liquids, supercooled liquids exhibit complex, non-exponential relaxation. To
understand this behavior, let us consider how a supercooled liquid responds to a perturbation
from its equilibrium state.
In general, the linear response of an observable A to an applied external force f(t) can
be expressed as [96, 181]
AA(t) = j
K(t,)f(T)dr
(2.4)
where K(t) is the response function, which supplies the weighted response of the system to
f (t) at each point in time. The response of A can also be expressed in terms of the two-point
time correlation function [32, 129]
C(t) = (A(t)A(O)).
(2.5)
The response function and the time correlation function are related by
1
K(t) = - kBT U(t)0(t)
(2.6)
where kB is Boltzmann's constant and 9(t) is the Heaviside function.
In the study of supercooled liquids, two of the most commonly discussed response functions describe the density and orientational relaxation.
The density response of a liquid
to local temperature and pressure fluctuations or driven pressure waves can be accessed
through light-scattering spectroscopy and acoustic spectroscopy, respectively, while dielectric spectroscopy can be used to measure the orientational relaxation of dipoles in response
to an applied electric field.
An example of the complex, non-exponential relaxation dynamics in supercooled liquids
is provided in figure 2-4 [30]. This plot shows simulations of the intermediate self-scattering
function (ISSF) of supercooled selenium; the ISSF gives the time correlation function of
28
I
I
I
I
1105K
'
0.5
550K
01S
10~2
10~
10
10
10
10
10
t (ps)
Figure 2-4: The intermediate self-scattering function of selenium above and below T ~
300 K, as computed from molecular dynamics simulations [30]. As the temperature drops,
two timescales emerge in the density relaxation: a fast, temperature-independent relaxation, called the #-relaxation, and a slow, temperature-dependent relaxation, called the
a-relaxation. The simulations correspond to temperatures of, from top to botoom, 105 K,
200 K, 255 K, 290 K, 330 K, 355 K, 400 K, 445 K, 495 K, and 550K. Figure reproduced
from [30].
29
P-relaxation, ps timescales
temperature-independent
a-relaxation, ps-ks timescales
temperature-dependent
Figure 2-5: An illustration of the fast and slow structural relaxation processes of a supercooled liquid. At short times, collective rearrangement is prohibited; individual molecules
(pictured in blue) are trapped within "cages" formed by neighboring molecules (pictured
in purple), and can only sample local configurations (0-relaxation). At longer times, these
cages break apart, and the molecules can undergo collective rearrangement (a-relaxation).
Figure adapted from [76].
density fluctuations
4(q,t) = (6 pq(0)Spq(t))
(2.7)
From the plot, two relaxation regimes emerge: a fast relaxation at picosecond timescales,
often referred to as the /-relaxation, followed by a slow relaxation at longer timescales, often
referred to as the a-relaxation.
The
#-relaxation
is relatively constant with temperature,
while the a-relaxation changes over many orders of magnitude between TM and Tg. As the
temperature drops, a plateau appears between the two regimes, which becomes more and
more pronounced as Tg is approached.
To understand the physical nature of these two relaxation regimes, imagine a liquid that
has undergone a rapid deformation (figure 2-5). Structural relaxation to an equilibrium state
requires long-range collective rearrangement of molecules. At fast timescales, this collective
rearrangement is prohibited; individual molecules are trapped within "cages" formed by
neighboring molecules. While they are free to explore the lowest energy configurations on
30
a local scale, they cannot relax on a large scale. This local relaxation consititutes the fast
-relaxation.
At longer times, these cages break apart, and the liquid can again explore
all available configurations.
This overall structural relaxation constitutes the slower a-
relaxation.
The relaxation dynamics of supercooled liquids can also be described in the frequency
domain. The frequency-dependent response function, which gives the susceptibility of an
observable to an applied force, is related to the time-domain response function through the
Fourier transform
x(w) =
eiwtK(t)dt.
(2.8)
The complex susceptibility spectrum includes real and imaginary components
x(w) = x'(w) + ix"(w)
(2.9)
which are related to one another through the Kramers-Kronig transformation.
The dielectric spectrum of glycerol is shown in figure 2-6 [137]. Dielectric spectroscopy
probes the orientational relaxation of dipoles perturbed by an applied electric field, and
so the real and imaginary parts of the susceptibility spectrum provide the reactive and
dissipative parts of the complex permittivity c. As in the time domain, two distinct relaxation regimes appear.
The dissipative component of the spectrum shows a temperature-
independent /-relaxation peak centered around 1012 Hz and a broad a-relaxation peak that
extends to lower frequencies as the temperature drops.
The relaxation peaks observed in the dielectric spectrum are generally too broad and
assymetric to be fit to a simple exponential curve, and currently no analytical form for the
relaxation can be justified by first principles.
However, experimental measurements and
numerical simulations of the a-relaxation can be fit well with the stretched exponential or
Kohlrausch-Williams-Watts (KWW) function [165]
[-
(2.10)
[_ KWW
31
"
<D(t) = exp
a)
204K
184K
213K
22K5-
195K
234
501
253K 23
295K
lycerol
0O
323K
0
T (K) 400
413K
10
200 T (K) 400
-
- 0.8
000
51
0.4
1 016
100
103
106
109
101
109
1012
v (Hz)
b)
184K
195K
204K 213K 223K 234K 253K
A
7
10 1
10
2
1000 / [T(K)]%%
4
6
10
-
102
-
10~
. .
10~3
Glyr-
Glycerol-
-5-5
100
10 3
10 6
v (Hz)
Figure 2-6: The real (a) and imaginary (b) components of the dielectric spectrum of glycerol,
which provides the reactive and disspative parts of the complex permittivity. The spectrum
shows one temperature-independent peak centered around 1012 Hz and another broad peak
that extends to lower frequencies as the temperature is lowered. Figure reproduced from
[137].
32
where TKWW is the characteristic relaxation time and the stretching parameter /3 can vary
between 0 and 1. While there is no direct frequency-domain analogue to the KWW function,
the Havriliak-Negami form provides a good fit to susceptibility spectra[74],
X(U)) = X00 +
where
THN
(1
Cc(2.11)
0
+ (iwTHN)C)(
is the characteristic relaxation time, xo is the high-frequency limit, Xo is the
low-frequency limit, and the stretching parameters a and / can vary between 0 and 1.
2.3
Mode coupling theory of supercooled liquids
Numerous phenomenological models have been proposed to explain the behavior of supercooled liquids and the glass transition. For example, many have observed a connection the
between the non-Arrhenius viscous slow-down in glass-forming systems and the values of
the high-frequency shear and bulk moduli [53, 120, 52, 1551. While these models satisfy the
need for a qualitative physical description of the behavior leading to the glass transition,
they lack the power of a quantitative predictive theory.
To date, the mode coupling theory (MCT) provides the only truly first-principles description of the glass transition. In this approach, equations of motion for the density correlator
are derived using the Mori-Zwanzig projection operator method [180, 1151, and mode coupling approximations are employed to solve for its time-evolution.
The solutions predict
a number of key properties of glass-forming systems, including an ergodic to non-ergodic
transition and a quantitative link between the fast and slow dynamics.
The following sections present a brief description of the mode coupling equations and the
assumptions underlying their derivations, followed by an overview of their key predictions.
This discussion largely follows that in reference [16], and a more thorough consideration
can be found therein. Other helpful summaries of the mode coupling theory of the glass
transition can be found in [154, 86, 1301.
33
2.3.1
Mode coupling theory equations
To understand the structural relaxation in glass-forming systems, we would like to find an
expression for the time-evolution of the intermediate scattering function,
F(q, t) = -J(bp(q,t)p*(q, t))
N
(2.12)
which gives the time correlation function of density fluctuations Sp(q, t)
N
6p(q, t) =
exp[iq - rj (t)].
(2.13)
j=1
In these expressions, q is the wavevector, q is the wavevector magnitude, N is the number of
particles, rj is the position of the jth particle, and 6p is the deviation from the equilibrium
density.
The Mori-Zwanzig projection operator formalism provides a method for deriving the
equations of motion for slowly-varying correlators. It does so by projecting the time evolution of a system onto selected slow variables, eliminating the contributions from fast variables
whose time scales are too rapid to be relevant [73, 15]. In glass-forming systems, we can
assume that the particle positions rj (t), and likewise the density fluctuations 6p(q, t), vary
only slowly in time. This assumption is justified by two key observations. First, the microscopic structure of glass-forming liquids is relatively insensitive to temperature, even as the
viscosity changes over many orders of magnitude. Second, at low temperatures approaching
Tg, the a-relaxation is significantly slower than the microscopic relaxation.
Application of the projection operator method, taking the density fluctuations as slow
variables, yields the following expression for the time-evolution of the density correlator
<>(t)
+ Q2(<(t) +
j
dt' [Mre9(q, t - t') + M(q, t - t')] <I(q, t) = 0
(2.14)
where <D(q, t) is the normalized intermediate scattering function, F(q, t)/F(q, 0).
Equation 2.14 takes the form of a generalized Langevin equation [961, whe squared fre-
34
quency is given by
Q22 = q2kBT
mS(q)
(2.15)
S(q) = F(q, 0) is the static structure factor, m is the mass of the particles and kB is
Boltzmann's constant.
Here, the memory function has been divided into two parts. The
first part of the memory function, Mre 9 (q, t), provides the evolution of <b(q, t) at short
times, and can be approximated to first order as a Gaussian function. It is referred to as the
"regular" part because it is also present in the case of normal liquids. The second part of the
memory function, M(q, t), is a slowly varying function that provides the long time relaxation
dynamics of the system. In the mode coupling approximation, it is assumed that this slow
contribution to the memory function arises from the coupling between density fluctuation
modes, which is consistent with the interpretation of structural relaxation as requiring longrange cooperative rearrangement of particles. The resulting four-point correlator is simplified
by assuming that it can be factored into the product of two two-point correlators. Taken
together, these approximations yield the following expression for M(q, t):
M(q, t)
-
2(2))2
dkVE (q, k, Iq - kj)<D(k, t)<D(lq - ki, t).
(2.16)
The vertex V(2 ) is related the the static structure factor through
V(2) (q, k, Iq - ki) = n S(q)S(k)S(lq - kj)
where n = V/N is the particle density and c(k)
q [kc(k) + (q - k)c(lq - kI)]
=
(2.17)
n(1 - 1/S(q)).
Equations 2.14 to 2.17 make up the "ideal mode-coupling equations", which can be
solved for the behavior of the density correlator. The only input to these equations is the
static structure factor S(q) which varies as a function of temperature.
Examination of the structure of these equations yields a qualitative explanation for the
dramatic viscous slow-down leading to the glass transition. We can see that the decay of
the correlator <b(q, t) is determined by the amplitude of the memory function M(q, t), which
35
in turn depends on both the magnitude of the correlator 4(q, t) and the magnitude of the
static structure factor S(q).
As the temperature is lowered, S(q) becomes more sharply
peaked, increasing the value of M(q, t) and resulting in a slower decay of D(q, t). The slower
decay of 1(q, t) further increases M(q, t), thus creating a non-linear feedback mechanism in
which small changes in temperature can result in large changes in the viscosity [63, 60]. This
feedback mechanism can be triggered by any outside forces which sharpen S(q); therefore,
small changes in the pressure or ordering of a system can also have a large impact on the
structural relaxation time [100].
2.3.2
Mode coupling theory predictions
The mode coupling equations are quite complex, and it is difficult to obtain an analytical solution for the density correlator given a general form for the static structure factor.
Therefore, study of the mode-coupling theory predictions usually proceeds by solving the
equations for a specific form of S(q), the simplest being one in which S(q) is modeled as a
delta-function at the main peak S(q) = 6(q - qo) [14, 141], or through numerical solution
of the equations using a hard-sphere model for the liquid [101].
The solutions for these
"schematic" models are naturally limited in scope to only a subset of glass-forming systems; nevertheless, these studies have revealed a number of general properties of the MCT
solutions.
Figure 2-7 (a) and 2-7 (b) show the predictions of the ideal MCT equations for behavior
of the density correlation function 1(t) and associated suscepibility spectrum X(w) for a
series of temperatures above and below T, [66]. These curves were obtained from numerical
solutions of the MCT equations for a schematic model of a liquid.
We can see that the
MCT predictions for the behavior of the correlators show two distinct relaxation regimes,
separated by a plateau in the time correlation function and a minimum in the susceptibiliy
spectrum, and have the same general form as the simulations and data in figures 2-4 and
2-6.
One notable prediction of the ideal mode coupling theory is that there exists a critical
36
temperature T, below which the density correlator does not decay to zero. This transition
from an ergodic liquid phase to a non-ergodic glassy phase has been dubbed the "ideal glass
transition". However, it differs from the experimentally observed glass transition in a few
key aspects. The ideal glass transition exhibits a sudden shift from ergodic to non-ergodic
behavior at a well-defined temperature Tc, whereas the actual glass transition involves continuous retardation of the dynamics until they exceed the laboratory timescale around 1013
Poise. Further, the experimental glass transition temperature Tg is not well-defined but can
vary slightly with cooling rate, and experiments show that it occurs at significantly below
the predicted T, [64].
Extended versions of MCT include couplings between density and current fluctuations in
the memory function. These couplings introduce "hopping terms" which restore ergodicity
to the system below T, [46, 65]. Solutions to these extended MCT equations are shown in
2-7 (c) and 2-7 (d) [66]. Despite the lack of a distinct ergodic to nonergodic transition, the
extended MCT equations still predict a dramatic slowing of the relaxation times as T, is
approached.
The following sections lay out some of the most notable quantitative MCT predictions
for the behavior of the correlator. In general, these predictions only hold near Tc; therefore,
we can define a reduced temperature
(Te - T)
(218
Te
and note that the following MCT predictions only hold for small values of a.
Debye-Waller Factor
In non-ergodic states, the long-time limit of the density correlator <b(t
+
00)
= fq is referred
to as the non-ergodicity parameter or the Debye-Waller factor. In ergodic states, the DebyeWaller factor can also be defined as the height of the plateau region between the o- and
#-relaxation
regimes at T. MCT predicts that the value
37
fq exhibits a square-root "cusp"
.
-
( a )
(a)
..(c)
.
Ideal MCT
0.8
InM
Ex t
Extended MCT
0.8
0.6
0.64-a
4-a
0.4
-
0.4
D,
0.2
0.2
E
A
1
106
104
102
F
1
108
102
(d)
Ideal MCT
(b)
106
104
108
f (units of Q1)
f (units of Q-)
S I
I
I
I
I
I|
Extended MC T
0.20
-
0.20
C'
A
D'
0. 15
-
0.15
A
HED C 8
0.10
0. 10
-
SF
0
E
0.05
E
1
0.05
.~~~~1
10-8
10-6
10-4
10-2
10~1
1
w (units of Q)
10 2
10~4
10-6
w (units of Q)
I 1I
Figure 2-7: The behavior of the density correlator <D(t) and the susceptibility x(w) for
different values of T (A-G, B', C', D', E', F'), obtained by iteratively solving the ideal and
extended MCT equations of motion for a schematic model of a liquid [66]. The curves
labeled A-G correspond to temperatures above the critical temperature Tc, while the curves
marked with a prime correspond to temperatures below Tc. Figure adapted from [66].
38
as the liquid passes below T,
f~c
T > Tc,
fq(T) =
f j + hqU.1/2
where
fj is the Debye-Waller factor at Tc and
(2.19)
T < Tc
hq is a critical amplitude. This square-root
cusp has been observed in experimental studies of the fragile liquids [177, 178], but was not
evident in some less fragile liquids [1221.
Power-law dynamics
MCT also predicts that the correlator exhibits asymptotic power-law behavior in the vicinity
of the plateau region. In other words, the tail end of the
-relaxation should exhibit a power-
law time-dependence which can be represented by
<b(t)
where
To
= fqc + hq(t/TO)-a
(2.20)
is a microscopic relaxation time and the power law exponent a is restricted to
0 < a < 0.5.
Similarly, the initial decay of the a-relaxation shows a power law time-
dependence given by
<D(t) =
where B is a constant and
T,
fqc +
Bhq(t/Ta)b
(2.21)
is the a-relaxation time. Note the sometimes confusing fact
-relaxation while the power-law exponent b
that the power-law exponent a governs the
governs the /-relaxation.
MCT also predicts that the a-relaxation time should show a power-law dependence as
the critical temperature Tc is approached:
r, = ro(T - T).
39
(2.22)
The exponent -y should be related to the power-law exponents a and b through the relation,
1 + 1
2a
2b'
(2.23)
Strikingly, MCT predicts a quantitative relationship between the power-law exponents for
the fast and slow relaxation. This relationship can be expressed as
A
-
F(1 - a) 2
(1 - 2a)
-
F(1 + b) 2
(1 + 2b)
(2.24)
where IF is the F-function. Thus, a, b, and -y can be determined by knowing the values of
any two of the three parameters.
The a- and
-relaxation regimes can be separated by many orders of magnitude in time,
and they seem to arise from different underlying physical mechanisms; therefore, this prediction of a quantitative connection between the fast and slow dynamics is unexpected and
unique to MCT. As a result, tests of MCT often hinge on probing this proposed connection
between the two relaxation regimes, and require data across a large range of timescales and
frequencies.
The power-law behavior can also be represented in the frequency domain. MCT predicts
the susceptibility spectrum to exhibit power-law behavior in the region of the susceptibility
minimum. These power-law dependencies can be expressed as
X")
acw
(2.25)
for the low frequency wing of the /-relaxation peak, and
x"(w) cX Wb
(2.26)
on the high frequency side of the a-relaxation peak. Interpolation of these two expressions
40
yields an explicit formula for the behavior of the susceptibilty spectrum near the minimum,
X"(W) = aXmin
+b
a( Wmin)
w
+b( w)a
Wmin)
(2.27)
In this expression, x"(w) is the value of the susceptibilty minimum between the a- and
/3-relaxation peaks and Wmin is the frequency where the susceptibilty minimum occurs. The
value and position of the susceptibility minimum are given by:
X"in(Wmin) oc IT - TC11/2
(2.28)
and
Wmin
c IT - Tcll/ 2a.
(2.29)
Scaling laws and time-temperature superposition
The temperature dependence of the correlator near the plateau region is entirely contained
within the characteristic relaxation times, TO(T) and r(T), while the shape of the decay,
<(t)
-
f
ta and <D(t) -
f
c t-b, does not change.
Therefore, we expect the density
correlator to scale with temperature according to TO(T), r(T), and fq(T). In the frequency
domain, the susceptibility near the minimum also scales according to X in(w) and Wmin.
This asymtotic scaling behavior is perhaps most evident in the imaginary part of the
susceptibility spectrum. If the data from figure 2-7(b) are scaled by the susceptibility minimum x"'in(w) and the frequency at the susceptibility minimum Wmin, the curves at different
temperatures all collapse onto one master plot (figure 2-8). We can see that the overlap of
the data is strongest near the susceptibility minimum.
A similar scaling behavior is predicted in the a-relaxation regime. MCT predicts that
the a-relaxation scales with
T-(T),
and that the shape of the relaxation can be described
by a temperature-independent master function F.
<D(t, T) = F(t/r-a(T))
41
(2.30)
10
10
I
II
I
I
D
E
10-1/
A
F'
101
A
1
10-2
w,
B
102
(units of Q)
Figure 2-8: The susceptibility curves obtained by iteratively solving the ideal MCT equations
of motion for a schematic model of a liquid at various temperatures above T, (A - G)
and below T, (F'), scaled by the susceptibility minimum x"in(w) and the frequency at the
susceptibility minimum Wmin. The scaled curves show good overlap near the susceptibility
minimum. The dotted lines indicate the power-law behavior of the low-frequency wing of
the /-relaxation and the high-frequency wing of the a-relaxation. Figure reproduced from
[66].
42
This scaling behavior in the a-relaxation regime is often referred to as time-temperature
superposition (TTS) [121], and can be seen in the imaginary part of the dielectric spectrum
of glycerol (figure 2-6), where the a-relaxation peak appears to keep the same form but
shift to lower frequencies as the temperature is dropped. While no analytical form for F(t)
has been found, numerical simulations and experimental measurements indicate that it is fit
well by the KWW function in the time domain (equation 2.10) and the Havriliak-Negami
function in the frequency domain (equation 2.11).
2.3.3
Tests of mode coupling theory
While the discussion of the mode coupling theory thus far has focused on the time-evolution
of the density correlator <D(t), it is important to note that many of these predictions can
be extended to the behavior of other correlators that couple to the density fluctuations.
Through the so-called a-scale coupling, the characteristic a-relaxation times of two variables
coupled to the density fluctuations, A and B, should be related to a universal relaxation
time r(T) through the relations TA(T) = CAT(T) and TB(T) = CBr(T), where CA and CB
are slowly varying parameters that can be considered constants in the asymtotic region [64].
These couplings may break down below Tc. As a result, tests of the mode couple theory
are not limited to techniques that directly probe the density fluctuations, such as light or
neutron scattering spectroscopy [39, 40, 71] or mechanical spectroscopy [75, 92], but are
extended to techniques that probe other correlators, such as the orientational or transverse
current relaxation [84, 59].
Comprehensive reviews of the experimental tests of mode-coupling theory can be found
in [64, 45] and the details are covered here. In general, the MCT predictions have been found
to hold for fragile liquids such as Cao. 4Ko.6 (NO 3 ) 1.4 (CKN) and o-terphenyl, but appear to
break down for stronger liquids such as glycerol.
These discrepancies may also indicate
that the "schematic" models used to obtain MCT solutions are a good approximation for
dispersive and van der Waals liquids, but cannot take into account the directional forces
present in hydrogen-bonded or network glass-formers. In addition, discrepancies between
43
measurements of coupled quantities and theoretical predictions are often attributed at least
speculatively to the the complexity of the coupling between those quantities and the density,
which may vary with temperature in unexpected ways.
Conclusive statements about the validity of the MCT predictions are stayed by the
limited time and frequency ranges accessible by most techniques. Currently, broadband dielectric spectroscopy is the only method capable of mapping the relaxation in supercooled
liquids across its full 17-decade timescale [137]. However, recent advances in photoacoustic
spectroscopy will soon make broadband mechanical spectra accessible as well. In the following chapter, I discuss the need for acoustic techniques that can probe the mechanical
response of materials across a wide range of frequencies, and describe two of the most useful
photoacoustic methods: impulsive stimulated scattering and picosecond ultrasonics.
44
Chapter 3
Introduction to photoacoustic
spectroscopy
3.1
Introduction
Mechanical spectroscopy measures the deformation or strain of a material in response to an
applied mechanical stress. The applied mechanical energy may either be stored, as in an
elastic solid, or it may be absorbed and dissipated, as in a viscous material. For viscoelastic
materials, which exhibit both viscous and elastic behavior, the stress response is strongly
dependent on its frequency or time scale [81].
For example, if a viscoelastic liquid with
a characteristic relaxation time TR is subjected to a high-frequency stress c-(w), such that
wTR
> 1, the structure of the liquid will not have time to respond through viscous flow,
and it will store energy as in an elastic solid. If the stress is much lower in frequency, such
that wrR
< 1, the material will flow like a viscous liquid. The energy storage ability of the
material is given by the compliance J, which gives the ratio between applied mechanical
stress and the resulting strain, or by the modulus M, which is the inverse of the compliance,
M = J-1.
Mechanical spectroscopy of viscoelastic materials has many applications, ranging from
the industrial development of novel polymer materials to the geophysical understanding of
45
volcanic glasses and melts [56, 160]. As discussed in chapter 2, experimental study of the
density response of glass-forming liquids also has the potential to elucidate the microscopic
mechanisms leading to the glass transition.
Mechanical spectroscopy is most commonly carried out with piezoelectric ceramics or
crystals, which convert AC driving fields into physical deformations. Using this technique,
the mechanical responses of materials have been measured at frequencies from 10-3 Hz to
10 9 Hz. The upper limit on this range is constrained by the speed of electronic transistors,
which currently do not exceed a few tens of GHz.
The development of ultrafast lasers, which are capable of producing light pulses with
durations ranging from nanoseconds to attoseconds, has enabled access to higher frequencies through photoacoustic transduction.
In addition to opening up a greater frequency
range, photoacoustic techniques have the advantage of being non-contact and usually nondestructive, lending themselves to the study of a wider array of samples.
Optical radiation can excite electronic or vibrational modes in a material, leading to mechanical stress through heating and thermal expansion or through the inverse piezoelectric
effect. Optical fields can also distort electron clouds in highly polarizable molecular liquids,
creating compressive forces through the electrostrictive effect. In optical pump-probe techniques, a short laser "pump" pulse imposes a sudden stress on a material.
The resulting
strain response induces a change in the dielectric constant, which can be detected with a
second optical "probe" pulse.
The highest frequency phonons that can propagate through a material are given by
v/2a, where a is the dimension of the unit cell and v is the speed of sound in the direction
of propagation [90]. For a typical crystal with lattice constant a ~ 5 and speed of sound
v ~ 10, 000 m/s, the highest frequency of sound propagation is 10 THz corresponding to an
acoustic period of 100 fs. Therefore, photoacoustic techniques have the potential to probe
acoustic modes throughout the Brillouin zone, and the frequencies accessible by photoacoustic techniques are limited by the transducer response rather than the laser pulse duration.
Currently acoustic frequencies reaching ~1 THz can be achieved with photoacoustic tech-
46
niques.
This chapter describes how mechanical spectroscopy in general, and photoacoustic spectroscopy in particular, can be applied to study the dynamics of supercooled liquids spanning
a wide range of frequencies. Broadband mechanical spectroscopy of glass-forming liquids has
been a goal of the Nelson group for many years, and recently my colleagues produced the
first mechanical spectrum of a supercooled liquid (Dow Corning 704 silcone diffusion pump
oil or DC704) spanning over 13 decades in frequency.
The resulting data were utilized to
perform a direct test of the MCT predictions for supercooled liquids [76]. The mechanical
spectrum of DC704 was constructed by a team of collaborators including Jeppe Dyre and
Tina Hecksher of Roskilde Universitat in Denmark, who collected data in the mHz to kHz
range using piezoelectric gauge measurements [75, 86], Jeremy Johnson and Darius Torchinsky, former members of the Nelson group who collected data in the MHz to low-GHz range
using a photoacoustic technique known as impulsive stimulated light scattering (ISS) [154],
and Christoph Klieber, another former member of the Nelson group who collected data in
the GHz range using a photoacoustic technique called picosecond ultrasonics (PU) [91, 92]. I
also contributed to the collection of GHz-frequency data under the mentorship of Christoph
Klieber.
The work presented in chapters 4 through 7 of this thesis aims to extend the range of
photoacoustic measurements on liquids beyond what is currently accessible by the ISS and
PU techniques. As this work is built heavily upon the current methodologies, sections 3.2
and 3.3 of this chapter introduce the ISS and PU measurements in more detail. Section 3.4
then summarizes the results from the broadband mechanical study of DC704 and outlines
the motivation for further advancement of our current photoacoustic methods.
3.1.1
Introduction to stress and strain
A full description of a three-dimensional stress can be obtained by writing the second order
tensor where each component o'j describes the force per unit area parallel to the
j
direction
and acting on a plane perpendicular to i on a cubical element. Similarly, the components
47
y
I
-
-x
--
z
Bulk Compression
Bulk Longitudinal
Simple Shear
Figure 3-1: Illustration of bulk compressional deformation, bulk longitudinal deformation,
and simple shear deformation of a cubic element. Black lines indicate the shape of the
unperturbed element, green lines indicate the shape of the element after strain deformation,
and red arrows indicate the direction of the applied stress. (a) Bulk compression: equal
force is applied to all faces of a cubical element. The bulk modulus is denoted with K.
(b) Bulk longitudinal: applied stress and strain is limited to one dimension. The bulk
longitudinal modulus is denoted with M. (c) Simple shear: a force parallel to Z is applied
to face perpendicular to y of a cubical element. The shear modulus is denoted with G.
of the three dimensional strain tensor y
describe the deformation of a cubical element in
each direction. However, there are three specific types of uniform stress and deformation for
which the stress and strain tensors simplify considerably, as illustrated in figure 3-1. The
associated moduli are referred to as the bulk modulus K, the bulk longitudinal modulus M,
and the shear modulus G, and are related through,
4
3
The photoacoustic techniques described here will focus on the measurement of the longitudinal modulus and compliance, in which the applied stress and deformation are restricted to
one direction and so only a single component of the stress and strain tensors, o33 and /33,
needs to be considered.
The compliance J gives the ratio between an applied mechanical stress and the resulting
strain. For a viscoelastic material, in which the compliance can vary with the timescale or
frequency of the applied stress u(t), the strain 7(t) can be written [81]
y(t) =
77M=
-0
J(t - t')
48
dt
o-(t')dt'
(3.2)
In the photoacoustic experiments described here, we generally assume a step-function stress
applied at time zero. In this case, the relationship between the stress and strain simplifies
to
q(t)
=
uJ(t).
(3.3)
Likewise, the stress response of a material to a sudden strain is given by the modulus, which
is simply the inverse of the compliance
0(t) = qM(t).
(3.4)
For periodic stress and strain, the longitudinal compliance and modulus can be represented
in the frequency domain, J(w) = J'(w) + iJ"(w) and M(w) = M'(w) + iM"(w). The real
parts of the compliance and modulus, J'(w) and M'(w), are proportional to the energy stored
during each deformation cycle, and the imaginary parts J"(w) and M"(w) are proportional
to the energy dissipated for each cycle.
This discussion of the ISS and PU techniques will focus on measurement of the longitudinal compliance J(w) and bulk longitudinal modulus, M(w) in the frequency domain. The
speed and attenuation rate of longitudinal waves are often referred to as simply the "speed
of sound" and the "sound attenuation rate", and this convention will be used throughout
this thesis.
While this discussion is focused on the generation of longitudinal acoustic waves through
photoacoustic techniques, it is important to note that over the past ten years, significant advances have also been made in the generation of shear acoustic waves through photoacoustic
techniques, and likewise measurement of the shear modulus G [126, 155].
3.2
Impulsive stimulated scattering
Impulsive stimulated light scattering (ISS) is an optical pump-probe technique in which two
pulsed laser beams are crossed inside or at the surface of a sample. Interference between the
49
two beams creates a transient grating pattern with period A and wavevector q given by
A
--
q
2sin(0/2)
(3.5)
where A is the wavelength of the laser and 0 is the beam crossing angle. Within the sample,
the optical grating imposes a sudden stress on the material through either laser absorption
and heating or through the electrostrictive effect. The density response is then probed
through the diffraction of a second laser beam [55, 173, 174].
The ISS technique was first developed in the late seventies [54, 159], and since then has
been utilized to study a wide range of time-dependent phenomena, including the lifetimes
of excited electrons [133], thermal transport in semiconductors [87, 88], and mechanical
relaxation in supercooled liquids [1551.
This discussion will focus on the application of
the ISS technique to the study of structural relaxation phenomena in liquids. Readers are
encouraged to consult Jeremy Johnson's doctoral thesis [861 for a more general description
of the ISS technique and its many applications. The following discussion summarizes the
relevant details of the ISS technique from [154, 86].
3.2.1
The ISS experiment
A schematic illustration of a typical ISS experiment on a liquid is pictured in figure 3-2. A
spatially periodic phase mask (PM) splits a pulsed pump beam and continuous-wave (CW)
probe beam into their first-order diffraction components, and the beams are recombined in
the sample by a pair of lenses [112]. The relative polarizations of the beams can have a
significant impact on ISS excitation, and so the polarization of each beam is controlled by a
waveplate. This discussion will focus on experiments in which all four beams are vertically
polarized (VVVV signal), generating longitudinal acoustic waves.
More details on how
changing the pump and probe polarizations can result in the generation of both longitudinal
and transverse waves, as well as allow for the detection of molecular orientational alignment
in liquids, can be found in [61, 127, 1551.
50
Detector
PM
Sample
N-2
Pbe Mm)
Pump (1035nm)
A=
A
2sin(0/2)
X
Figure 3-2: A schematic illustration of a typical ISS experiment on a liquid sample. A phase
mask (PM) splits a pulsed pump beam and continuous-wave (CW) probe beam into their
first-order diffraction components, and a pair of lenses recombines the beams in a sample.
The optical interference pattern of the pump beams excites a density grating within the
sample through laser heating and thermal expansion (impulsive stimulated thermal scattering or ISTS excitation) and/or through electrostriction (impulsive stimulated Brillouin
scattering or ISBS excitation). The sudden change in density launches a pair of counterpropagating acoustic waves, which generate a standing wave pattern. The standing wave
pattern and subsequent relaxation of the density grating are probed through diffraction of
the probe beam, which is heterodyned with a reference beam to increase the signal-to-noise
ratio and give access to both amplitude and phase grating information. The polarization
of each beam is controlled by a waveplate, and the reference beam phase and intensity are
controlled by a neutral density (ND) filter. Figure adapted from [76].
51
In an isotropic liquid sample, ISS excitation by parallel polarized beams can proceed via
two mechanisms. If the wavelength of the pump beam is on resonance with an optical absorption within the sample, some of the laser light will be absorbed. The subsequent heating
and thermal expansion imposes a sudden periodic stress UT, which launches counterpropagating acoustic waves with wavevector q given by equation 3.5. The change in temperature
is related to the absorbed energy, which is proportional to the laser intensity I according to
(lx)
6T(x) oc I(x) = I, cos2
= 1, (1 + cos(qx))
(3.6)
and the amplitude and spacial profile of the periodic stress is given by,
O-T(X) = cr,6T(x) oc ci cos(qx)
(3.7)
where c is the bulk elastic modulus and r, is the thermal expansion coefficient. Propagation
of acoustic waves and the thermal diffision of heat from grating peaks to nulls creates strainand temperature-induced changes in the refractive index, and these changes are monitored
through the time-dependent diffraction of a probe beam.
Even in the absence of laser absorption,
ISS can excite acoustic waves through elec-
trostriction. The electric field imposed by the laser beams distorts molecular electron clouds
within the sample, pushing nuclei into new positions. The electrostrictive force imposed by
two vertically polarized pump beams with electric field magnitudes E, can be expressed as
Fx oc E,,q7 1
22
sin(qx)(t)
(3.8)
where 71122 is a material property that is related to the photoelastic constants of the sample
[82]. It is important to note that, in contrast to thermal excitation, the electrostrictive force
is proportional to the wavevector q of ISS excitation. In addition, while the laser heating
imposes a step-function stress to the sample, which does not dissipate until the thermal
grating has diffused, the electrostrictive stress is only applied for the duration of the pump
52
laser beams, and so can be represented as a delta-function. In experiments, the difference
between the step-function stress and the delta-function stress manifests as a 90 degree phase
lag between acoustic waves generated through thermal expansion and electrostriction [86].
In general, I will refer to the electrostrictive generation of a density grating as impulsive
stimulated Brillouin scattering (ISBS) excitation, and the thermal generation of a density
grating as impulsive stimulated thermal scattering (ISTS) excitation; however, most ISS
signals in liquids arise from a combination of both excitation mechanisms.
Use of heterodyne detection in ISS measurements increases the signal-to-noise ratio and
allows for the separation of amplitude and phase grating components of the signal [118, 1121.
Heterodyning is achieved by combining the diffracted probe beam with a reference beam
whose amplitude is controlled by a neutral density (ND) filter and whose phase is adjusted
by changing the orientation of a glass plate in the beam path, whose rotation varies the
optical thickness. The heterodyne signal intensity Is is given by
Is = ID+ IR + 2
where
ID
IRID cos 0
(3.9)
is the diffracted beam intensity, IR is the reference beam intensity, and 0 is the
optical phase difference between the diffracted and reference beams. When ID < IR, the
first term is negligible and the second term is a constant. The intensity of the modulated
heterodyne signal is then proportional to
V/IR,
and can be increased by raising the intensity
of the reference beam. The signal-to-noise of the measurement can be increased further by
recording signal at both positive and negative phase and subtracting the two, eliminating
any common sources of noise.
3.2.2
Qualitative characteristics of ISS signal
Impulsive stimulated scattering in liquids can be thought of as a time-domain analogue to
traditional frequency-domain light scattering measurements; however, rather than probing
random density fluctuations in the sample, it probes impulsively driven, coherent density
53
waves.
The frequency-domain light scattering spectra of a simple liquids exhibit a Rayleigh
peak, which arises from diffusion of thermal fluctuations, and a Brillouin peak, which arises
from propagating acoustic waves. In viscous liquids, a third peak, called the mountain mode,
is observed when collective structural relaxation occurs at frequencies that fall between the
acoustic frequency and thermal diffusion [116, 31].
The same three features can be observed in time-domain ISS experiments. Characteristic
ISS signal from a supercooled liquid at a series of different temperatures is pictured in figure
3-3.
At all temperatures, one observes fast oscillatory behavior arising from propagating
acoustic waves on top of a slowly decaying thermal background. However, at intermediate
temperatures, one can also observe a rise in the signal arising from slow thermal expansion.
The ISS signal is strongly dependent on both the sample temperature and the grating
spacing.
At relatively high temperatures (i.e. above TM) or wide grating spacings, the
structural relaxation time is much shorter than the acoustic period, or
wATR <
1. In this
regime, the liquid rapidly responds to applied acoustic stress, and the acoustic waves are only
weakly damped. As the temperature drops and the relaxation time approaches the acoustic
period
WATR
~ 1, the acoustic energy is converted into structural relaxation, and the acoustic
oscillations become heavily damped. At still lower temperatures, the structural relaxation
time falls between the acoustic period and the thermal diffusion time. In this regime, the
liquid responds to the acoustic forces as an elastic solid.
However, structural relaxation
from the applied thermal stress can be observed as a slow rise in the signal intensity as
thermal expansion converts the thermal grating into a density grating.
lower temperatures or smaller grating spacings,
WATR
Finally, at much
> 1, the structural relaxation time is
longer than both the acoustic period and the thermal diffusion time. The liquid responds to
the applied stress like an elastic solid, and the acoustic oscillations are again weakly damped.
The frequency
WA
and damping rate
FA
54
of the acoustic oscillations can be used to
(a')
300K
(b')
.(b)
255K
CO
I-)
-O238K
CDo
()
('
(d)
(d')
21 OK
IE9168I-7
1E4
IE-5
Time (sec)
1E-4 1W
1&2 101I~
102
104
10'
ReqencyQ-")
Figure 3-3: ISS signal in DC704 at the indicated temperatures and at a grating spacing
A = 40 km. The right side of the figure shows a schematic representation of the acoustic
loss spectrum at each temperature. At T = 300 K, the relaxation time is much shorter than
the acoustic period, or WATR < 1. The liquid rapidly responds to applied acoustic stress, and
the acoustic waves are only weakly damped. At T = 255 K, the relaxation time approaches
the acoustic period WATR - 1, and acoustic oscillations are heavily damped as the acoustic
energy is converted into structural reorganization. At T = 238 K, the structural relaxation
time falls between the acoustic period and the thermal diffusion time, and the relaxation
from the applied thermal stress can be observed as a slow rise in the signal intensity. Finally,
at T = 210 K, WATR >1, the liquid responds to the applied stress like an elastic solid, and
the acoustic oscillations are again weakly damped. Figure reproduced from 1154].
55
calculate the bulk longitudinal modulus through the relations
M'(WA) = p
W2
- i2
2 A
(3.10)
(3.11)
M"(WA) = p 2w 2 I
q2
where p is the density of the liquid and q is the excitation wavevector. Currently, grating
spacings from A = 100
tm to A = 1
Lm can be generated, corresponding to acoustic
wavevectors from q = 0.06 rad/im- 1 to q
with a speed of sound on the order of v
=
=
6 rad/ Lm- 1 . Therefore, for a viscous liquid
2000 m/s, the ISS technique gives access to
the longitudinal modulus and compliance at acoustic frequencies ranging from 20 MHz to 2
GHz.
The thermal expansion or slow rise portion of ISS data also contains information on the
structural relaxation of the liquid at longer timescales or lower frequencies than the acoustic
data.
However, because structural relaxation in liquids involves the relaxation of many
parameters, including the compliance and specific heat, it is not immediately clear how this
signal is related to the compliance. Chapter 7 presents a detailed examination of the slow
rise component of ISS data and its relationship to a relaxing compliance and specific heat.
3.2.3
Experimental set-up for ISS measurements
The data presented in chapter 7 was collected using the ISS experiment. In our experimental
set-up, the pulsed pump beam is provided by a High
Q FemtoRegen regeneratively
amplified
laser with a central wavelength of A ~~1035 nm, a repetition rate of 1 kHz, and an output
energy of 400
J per pulse. While this laser is capable of generating laser pulse durations
less than 300 fs through the use of an internal compressor, highly compressed pulses are not
required for the ISS acoustic generation and can lead to unwanted nonlinear optical effects
including sample damage. Therefore, the compressor was bypassed within the laser and the
output pulse duration was ~ 60 ps. The 830 nm CW probe beam was provided by a Sanyo
DL-8032-001 diode laser. The pump and probe beams were focused by Thorlabs near-IR
56
achromatic lenses. The heterodyne light intensity was detected with a Cummings Electronics
Labs Model 3031-0003 amplified photodiode with a bandwidth of 10 kHz to 3 GHz for the
short-time responses, and a New Focus Model 1801-FS detector with a bandwidth from
DC to 125 MHz for the long-time response. All data was and processed with a Tektronix
TDS-7404 digitizing oscilloscope with a 4 GHz bandwidth. A diagram of the laser table and
the experimental components can be found in [86].
3.3
Picosecond ultrasonics
In the picosecond ultrasonics technique, ultrashort laser pulses are converted into GHzfrequency acoustic waves through the action of thin metal or semiconductor transducer
films. Photoacoustic transduction can occur through either laser absorption and heating
or through the inverse piezoelectric effect. The discussion in this chapter will focus on the
thermal generation mechanism, while the piezoelectric mechanism will be covered in more
detail in chapter 4.
3.3.1
Acoustic generation
To understand acoustic generation in the picosecond ultrasonics technique, let us first consider an acoustic wave launched at the free surface of a metal film (figure 3-4). An ultrashort
laser pulse (typically having a full width half max (FWHM) of < 1 ps) incident on a metal
film will penetrate into the film at a length scale given by the optical absorption length (,
which will be referred to as the optical skin depth. The optical energy excites free electrons
which rapidly thermalize among themselves (on a femtosecond time-scale) through electronelectron collisions [131].
Energy is then transferred to the lattice (on a picosecond time
scale) through electron-phonon collisions, raising the temperature of the excited regions.
Assuming that the two-dimensional area illuminated by the pulse A is significantly larger
in lateral dimensions than the optical skin depth (,
57
the energy deposited per unit volume
(a)
(b
Ballistic
(n
Electrons
C
Energy
(AbsorbedElcrn
Deposited
.- Energy
C
I-
C,,
Limited by Light
Penetration
-10
Broadened by
Thermal Diffusion
0
10
Transducer Film
time in ps
Substrate
Figure 3-4: Picosecond ultrasonic strain generation. A laser pulse incident on a the surface
of a metal film penetrates a distance given by the optical skin depth (. Free electrons absorb
the laser light and transfer their energy to the lattice, and the resulting thermal expansion
launches two counterpropagating strain pulses, one of which reflects from the free surface
with a change in sign. (a) Assuming electron and thermal diffusion are negligible, the total
strain will have a bipolar waveform with exponential tails, where the lengths of the tails
are determined by the optical skin depth ( (black line). Thermal diffusion broadens the
peaks (red line). (b) Hot electron diffusion carries the absorbed energy farther into the film.
The resulting strain pulses are significantly broader than what would be predicted from the
optical skin depth alone. Figure reproduced from [911.
58
by the laser at a depth z in the film can be written [153]
W(z) = (1 - R)
Q
(3.12)
-Z/(
where R optical reflectivity of the surface, A is the illuminated area, and
Q is
the energy of
the laser pulse. The increase in temperature is then given by AT = W(z)/C, or
AT = (1 - R)Q e-z/(
A(C
(3.13)
where C is the specific heat per unit volume. The sudden change in temperature generates
an isotropic thermal stress given by
-3KAT.
(3.14)
where K is the thermal expansion coefficient and K is bulk modulus.
Because we have
assumed that the thermal stress only varies in the z-direction, we only expect strain in
the z-direction as well; in other words, the only non-zero components of the stress and
strain tensors are U33 and 733, respectively. The strain can then be found by evaluating the
equations of elasticity
0-33
6pU 3
Pt2
=
3
=
=
Kq 3 3
U33
-
3KKAT(z)
(3.15)
(3.16)
z
-
6
T733
1+v
U33
(3.17)
6z
where v is Poisson's ratio, u 3 is the displacement in the z direction, and p is the density.
Under the initial conditions that the strain is zero everywhere and the stress is zero at the
59
free surface, the strain wave equation is given by
M 33(z, t)
=
(1 - R)
At long times
1733
Qr, 1+v
e-z/C (i
t
_
2
-vt/C- e-lz-vtl/C sign(z - Vt)] .
(3.18)
(318
can be divided into two parts: a static thermal expansion at the surface of
the metal film, and two counterpropagating acoustic strain pulses with longitudinal sound
velocity v given by
V
311
vK
+v p
(3.19)
The overall shape of generated acoustic waveform depends on the geometry of the sample and
the transducer. When an acoustic wave travels through an interface between two materials,
,
the acoustic reflectivity and transmission is governed by the acoustic impedances Zi and Z 2
which can be calculated from the density and the speed of sound of each material according
to Z,
=
piv1 and Z2 = P2v2. The acoustic reflectivity is given by
r12 =
Z1 + Z2
(3.20)
and the acoustic transmission is given by
t12 =
1
-
r12 =
2Z2
2
(3.21)
Z1 + Z2
In the case of acoustic generation at the surface of a metal, one strain pulse travels into the
material while the second reflects from the free surface with a change in sign. Because the
acoustic impedance of a metal is around four orders of magnitude larger than the impedance
of air, we can assume that the acoustic wave is entirely reflected from the metal/air interface,
leading to a bipolar acoustic waveform with exponential tails, as shown in figure 3-4. In
practice, fast electron diffusion and heat conduction carry laser-deposited energy further
into the material than would be predicted by the optical skin depth.
60
When these effects
are taken into account, the acoustic waveform has the same overall shape but the peaks are
smoothed out and elongated [167].
In picosecond ultrasonics experiments on liquids, acoustic waves are rarely generated
at an air/metal interface. Instead, the transducer film is layered between an optical substrate and a liquid sample. The acoustic impedance of metals is on the same order as the
optical substrate materials (i.e. fused silica or sapphire), limiting the reflection from the
metal/substrate interface. In addition, the metal transducer layer is thin, between 20 nm
and 200 nm. At these distances, nonequilibrium electron diffusion carries energy throughout the film. Therefore, to a first approximation the acoustic waveform traveling into the
liquid sample takes the form of a unipolar rectangular pulse whose length in time can be
approximated by r = d/v, where d is the thickness of the metal film and v is the speed of
sound in the metal. Likewise, the peak frequency of the pulse can be approximated from
f
= 1/r. For an acoustic pulse generated in a 20 nm thick aluminum film (vAl = 6420 m/s),
the peak frequency lies at
3.3.2
f
~ 300 GHz.
Acoustic detection
After generation in a metal transducer film, the frequency component rj(w) of the acoustic
strain pulse travels through a liquid sample with a velocity v and attenuation length a.
The complex modulus at frequency w and temperature T can be calculated from v and a
according to
M (T, w) = p (T) b (T, w) 2
(3.22)
where p is the density, f is the complex acoustic velocity defined by
and
= 41
(3.23)
- + ia.
V
(3.24)
4 is the complex wavevector
61
Optical detection of the propagating strain pulse can be achieved through one of two mechanisms. The strain pulse induces a change in the dielectric constants of a material which can
be detected by monitoring the reflectivity of an optical probe. The strain pulse can also be
detected interferometrically by monitoring the displacement of the surface of a metal film.
These two techniques, which here will be referred to as time-domain Brillouin scattering
(TDBS) and picosecond ultrasonic interferometry (PUI) are discussed below.
Time-domain Brillouin scattering
A propagating strain pulse r(x, t) induces a change in the local index of refraction n of
magnitude [134]
An(x, t) =
2
(3.25)
Pn 3 q(X, t)
where P is the photoelastic constant of the material. In an optically transparent material,
an optical probe indicent on the strain pulse will be partially reflected as it travels between
the unperturbed material with index of refraction n and the perturbed material with index
of refraction n
+ An.
In the TDBS technique, the partial reflection of an optical probe from a propagating
strain pulse is combined with a reference reflection from a static surface.
As the strain
pulse moves through the sample, the partial reflection moves in and out of phase with the
reference, and the intensity of the combined reflection is modulated by constructive and
destructive interference. The frequency
f of the
intensity modulation is given by
cos 0
f =
(3.26)
where n is the index of refraction of the material, v is the acoustic velocity, A is the wavelength of the optical probe, and 0 is the angle of incidence of the optical probe.
As the
acoustic pulse propagates through the material, the magnitude of the strain decreases with
an attenuation rate rac = av, which in turn decreases the value An. As the perturbation in
the refractive index decreases, so too does the magnitude of the intensity modulation.
62
To Detector
(a)
Reference Reflection
Partial Reflection
I
PUMP
0
Ti(t)
Substrate
PROBE
LiquidSample
Transducer
(b)
1
0.8
0.6
+D
Ca)
0.4
0.2
'-
0
N -0.2
E
0
C
-0.4
-0.6
-0.8
-1I
0
1000
500
1500
time (ps)
Figure 3-5: Time-domain Brillouin scattering detection of an acoustic strain pulse. (a)
Experimental geometry. A propagating strain pulse induces a local change in the index of
refraction of the liquid (equation 3.25) which partially reflects an incident probe beam. The
partial reflection is combined with a reference reflection from the metal transducer film.
As the strain pulse propagates, the partial reflection moves in and out of phase with the
reference reflection, modulating the intensity of the total reflectivity at a frequency given by
equation 3.26. The acoustic wave attenuation is given by the decay in the signal modulation
intensity. (b) TDBS signal in liquid DC704 at 242 K.
63
Characteristic TDBS signal in liquid DC704 is pictured in figure 3-5(b).
The signal
exhibits a decaying oscillation which can be fit to the functional form
F = Ae-l-/Ta sin(wt -
/) + Be-/th + C
(3.27)
where the oscillation decay time rac gives the acoustic attenuation rate I'A = 1/Tac, and
the frequency w = 27rf can be used to calculated the acoustic velocity.
In many metal
transducers, laser absorption and heating induces a temperature jump in the metal film,
which results in an instantaneous jump in the reflectivity of the reference beam followed by
a gradual decay as the heat diffuses away from the excitation spot. This thermal component
is accounted for by adding a second exponential decay with magnitude B and decay rate
Tth
to the fitting function.
Like ISBS, TDBS is a time-domain analogue to frequency-domain Brillouin light scattering experiments. Like these methods, it only provides access to the speed and attenuation
rate of the acoustic wave at a single frequency determined by n, v, A, and 0. While repeated
measurements at different probe wavelengths A and incidence angles 0 can provide access to
a range of frequencies, the index of refraction n and acoustic velocity v in liquids limits this
range to ~ 5 - 25 GHz. TDBS experiments on liquid DC704 are presented in chapter 6.
TDBS detection can be modified to access higher acoustic frequency ranges by detecting
the strain pulse in a second optical substrate such as a lens. The acoustic detection frequency
f can be adjusted by choosing substrates with different acoustic velocities or refractive
indices. In this experiment, the substrate is placed on the detection side of the liquid; after
propagating through the liquid layer, the strain pulse passes into the optical substrate, and
can be detected by TDBS. The speed of sound v can be determined from the acoustic timeof-flight through different thickness liquid layers, and the attentuation o can be determined
from the magnitudes of the acoustic oscillations.
Detection in a substrate has provided
access to acoustic frequencies from 25 GHz to 100 GHz.
mechanism can be found in reference [911
64
More details on this detection
Detection
Film
Generation
Transducer
Strain
Probe
Pump
Generation Side
Substrate
Detection Side Substrate
DC704
Variable
Thickness
Figure 3-6: Experimental geometry for acoustic detection via picosecond ultrasonic interferometry. Generation and detection transducers are placed on either side of a liquid layer
of varying thickness. An acoustic strain pulse propagates through the liquid layer and displaces the detection film; this displacement is detected with a Sagnac interferometer. Figure
reproduced from [761.
Picosecond ultrasonic interferometry
The acoustic strain pulse can also be detected directly via interferometry.
The sample
geometry for this detection mechanism is pictured in figure 3-6(a). The liquid layer is placed
between two metal transducers, one for acoustic generation and a second for detection. After
propagating through the liquid layer, the strain pulse displaces the detection film, and this
displacement can be measured via Sagnac interferometry or other types of interferometric
techniques [79, 148, 911. In the plane-wave approximation, the strain r(x, t) is proportional
to the time derivative of the displacement u(x, t),
ld
d
1 du(xt)
-u(x, t) = v
r(x, t) = dx
dt
(3.28)
where v is the acoustic velocity.
Different frequency components of the strain i/(w) move through the liquid with different
complex velocities f)(w) (equation 3.23); as a result, the shape of the strain pulse changes as
the wave propagates through different liquid thicknesses. This effect can be seen in figure 3-
65
1 .2.
0
0 .6a-
2 0 .4U)
0..2-
I
0
1
Onm(1 0
-
0.81-
T14
-
0.6
C
11
113 nm(
--
275 nm (12 3)
---
Z97
125
0.2F-
4
-
-
710
0.4-
45nm(i
T133
5)
203 nm (T 3)5)
358 nm (11
434 nm (T1
0)
T14
0
0
50
100
150
200
250
Time (ps)
Figure 3-7: PUI signal in DC704 at 200 K. The acoustic strain pulse was generated in a 30
nm Al transducer film on a sapphire substrate and detected in a 100 nm Al film on a sapphire
substrate. The displacement of the Al film after strain propagation through liquid layers
of different thickness is shown in the top diagram. The acoustic strain r(t) was calculated
from the displacement signal u(t) using equation 3.28; solid lines in the displacement curves
indicate portions of the signal that were used in calculating the strain shapes. The strain
pulse attenuates and broadens after passing through thicker liquid layers. Figure reproduced
from [76].
66
7, which depicts the shape of the displacement and the strain of a broadband acoustic pulse
after propagating throught different thicknesses of liquid DC704. Strain pulses broaden as
higher frequency components experience higher damping rates.
After propagating through an additional distance Ad = d2 - d1 , the "output" strain
12(W)
is related to the "input" strain through [91]
2(W) = e ik(w)Ad
(w)
(3.29)
,
(3.30)
If we define the transmitted strain as
krans(W) =
~
i/i(w)'
then the complex velocity through different thicknesses of liquid layers is given by
cv
iwAd
0(W) - - ijd(3.31)
In~htrans(w)
k
Unlike TDBS measurements, which only provide information at a single acoustic frequency,
PUI is capable of measuring b(w) over a broad range of frequencies in a single measurement. However, the sensitivity at individual frequencies is often lower. Currently, the PUI
technique has been utilized to measure mechanical properties in supercooled liquids up to
frequencies of 200 GHz [76].
3.3.3
Liquid sample design
Picosecond ultrasonics experiments on liquids pose many challenges in sample design and
implementation.
The acoustic impedance mismatch between liquids and solids can limit
the magnitude of acoustic pulses that are transmitted into liquid samples, and the high
acoustic damping rates in liquids further limit the length scales over which acoustic waves can
propagate. Therefore, TDBS and PUI experiments, in which the acoustic waves propagate
through a liquid layer and into another material, require liquid layers at thicknesses on
67
Cryostat
Heat Sink
Temperature
Sensor A
Sample Holder Top
Part with Beveled
Opening for Probe
Laser Beam
Six Screws for
Adjustment of
Substrate Tilt
Liquid Film
'Detection'
Side
Substrate
'Generation'
Sample Holder
Bottom Part
Side
Substrate
Temperature
Sensor B
Figure 3-8: Illustration of the sample design used in picosecond ultrasonics experiments
on liquids. The sample is held by copper sample holder which is designed to attach to a
Janis ST-100 cryostat cold finger. The liquid sample is squeezed between a flat "generation"
substrate (usually coated with a metal transducer film) and a curved or canted detection
substrate which is either optically transparent or coated with a second metal transducer layer
for PUI detection. The tilt of the detection substrate can be controlled by adjusting the
tightness of six screws. With this sample design, liquid thicknesses from tens of nanometers
to several microns can be achieved. Figure reproduced from [91].
68
the order of nanometers to microns.
In addition, the elucidation of supercooled liquid
dynamics requires measurements over large temperature ranges, and so liquid samples must
be compatible with cryostat mounts.
Klieber et al. [91] developed a robust sample design for picosecond ultrasonics experiments on liquids (see figure 3-8). In this design, a thin transducer film is deposited on an
optically transparent substrate. The liquid sample is then squeezed between the transducer
substrate and a second substrate which is either optically transparent (for detection via
TDBS in a substrate) or coated with a thin metal film (for detection via PUI). The second substrate is composed of a lens or canted crystal, creating a liquid layer with thickness
varying from tens of nanometers to tens of microns. The tilt of the second substrate can be
controlled by adjusting the tightness of six screws.
3.3.4
Experimental set-up for picosecond ultrasonics measurements
The experiments presented in chapters 4, 5, and 6 utilize variations on the picosecond
Figure 3-9 shows a diagram of the experimental set-up used for
ultrasonics technique.
these measurements. Ultrafast laser pulses were generated with a Coherent RegA amplified
Ti:Sapphire laser system which operated at a central wavelength of A
-
790 nm, a pulse
duration of 300 fs, and a repetition rate of 250 kHz. The RegA amplifier was pumped with a
Verdi V18 (central wavelength A = 532, output power 16 W) and seeded by a Mira oscillator
(central wavelength A
-
790 nm, repetition rate 80 MHz). The output beam was split into
excitation and variably delayed probe pulses. The excitation beam was modulated by an
acousto-optic modulator at
-
90 kHz to facilitate lock-in detection. In some experiments,
a Beta Barium Borate (BBO) crystal was used to frequency-double the pump beam, the
probe beam, or both, to a wavelength of
-
390 nm. After reflection or transmission from
the sample of interest, the probe beam was directed to a photodiode.
The output of the
photodiode was combined with that of a second reference photodiode for balanced detection
and fed into a lock-in amplifier.
69
Photodiodes
Lock-in
Reference
Signal
{Verdi
4-
BBO
Mira
2 meter
delay stage
Sa mple
Trigger
RegA
AOM
Figure 3-9: : Experimental set-up for picosecond ultrasonics measurements. The output of
a Coherent RegA amplified Ti:Sapphire laser system (A ~ 790 nm, pulse duration ~ 300
fs, and repetition rate of 250 kHz) was split into pump and variably delayed probe pulses.
The pump pulses were modulated at a frequency of 90 kHz by an acousto-optic modulator
and focused onto the sample. Depending on the sample of interest, the pump, probe, or
both were freqeuncy-doubled by a Beta Barium Borate (BBO) crystal to a wavelength of
A - 390 nm. The probe was focused on the sample and the reflection or transmission was
directed to a photodiode. The output of the photodiode was combined with a reference
photodiode for balanced detection and fed into a lock-in amplifier.
70
Si
Si
0
Si
0
Figure 3-10: The molecular structure of tetramethyl tetraphenyl trisiloxane (trade name
Dow Corning 704 silicone diffusion pump oil, or DC704), a Van der Waals liquid and fragile
glass former.
3.4
The longitudinal compliance spectrum of DC704
Mechanical spectroscopy offers the most direct test of MCT predictions because longitudinal acoustic waves couple directly to the density current fluctuations in liquid. To this
end, my colleagues have applied the ISS and PU techniques to construct the mechanical
spectrum of glass-forming liquid DC704 across the full range of frequencies accessible by
photoacoustic methods. These data have been combined with mechanical data collected at
lower frequencies to map the longitudinal modulus and compliance spectrum of DC704 at
frequencies from 10-2 Hz to 1011 Hz. The construction this spectrum was carried out by
a team of collaborators including Tina Hecksher, Jeremy Johnson, Darius Torchinsky, and
Christoph Klieber and myself (see section 3.1, These combined results will be published
in reference [761, and further details on these experiments and analysis can be found in
[154, 91, 86, 75, 92].
These data support the fundamental predictions of MCT. The a-relaxation peak in
the loss spectrum was fit to a single functional form at all temperatures, supporting the
prediction of time-temperature superposition chapter 2. In addition, above the predicted
MCT critical temperature T, ~ 240 K, the high-frequency wing of the a-relaxation and the
low-frequency wing of the
-relaxation exhibit the expected power-law behavior, where the
power-law exponents a = 0.3 and b = 0.5 confirm the predicted relationships between the
71
PSG+PBG
0.55
PBGres. NAI
0.5-
S
S
(0
0
+
"
DC704
V
0.4
a- 0.35
*
"V
"
0.45
TDBS PU
ISTS
*
*
*
*
V
0
c
0Go
-
v
O
*
o
*
0.3
0
0.25
V
*
*
8.a....
0.
0.2
*
*
v 246K
o 244K
0
Now*-
0.15
-v
10~1
o
.0
3
0
,-,10-2
o go
U
S*
*,~,q
0
0
V
%
V
o 242K
240K
o 236K
234K
232K
+ 230K
* 228K
v 226K
* 225K
+ 224K
*222K
+
09+
(0j0-
S
315K
310K
305K
300K
290K
285K
280K
275K
268K
264K
256K
248K
220K
*218K
*216K
215K
O.
V
10-4 V
.
. . ...- j
.
. . .... j
10-310-210-1100 10
.
. ....- j
. . . ...- j
.
. . _j
.
.
..- i
.
. . _j
.
. ._j
. ..
...- j
102 103 104 105 106 107 108 log 101010111012
frequency [Hz]
Figure 3-11: The broadband longitudinal compliance spectrum of glass-forming liquid
DC704. Data from 10-2 Hz to 10 4 Hz were collected using a piezoelectric bulk modulus
gauge (PBG) and shear modulus gauge (PSG), which were combined to find the longitudinal modulus according to equation 3.1 [751. Resonance overtones of the PBG measurements
provided limited data in the 10 5 Hz to 106 Hz range. Nanosecond acoustic interferometry
(NAI) was used to access frequencies between 106 Hz to 10 7 [86]. The ISS experiment described in section 3.2 provided data from 10 7 Hz to 109 Hz [154, 861. Picosecond ultrasonics
generation (section 3.3) with TDBS detection provided data around 1010 Hz, and PUI detection provided data at the highest frequencies [91, 92]. Figure reproduced from reference
[76]. A thorough discussion of the measurement techniques can be found therein.
72
fast and slow relaxation,
F(1 - a) 2
(1 - 2a)
_
1(1 + b)2
(1 + 2b)
(3.32)
While the broadband mechanical spectrum of DC704 has enabled unprecedented tests of
MCT and demonstrates the impressive range of current acoustic techniques, it also illustrates
the frequency regimes that are currently inaccessible by photoacoustic methods.
These
include frequencies of 10 4 Hz to 107 Hz (below the low-frequency limit of the ISS technique),
109 Hz to 1010 Hz, (between the high high-frequency limit of the ISS technique and the lowfrequency limit of the PU technique), and above 1011 Hz (above the high-frequency limit of
the PUI technique). As discussed below, access to these gaps in the acoustic spectrum will
enable more accurate tests of MCT in glass-forming liquids and a better understanding of
the anomalous high-frequency dynamics in amorphous solids.
3.4.1
The compliance minimum
The idealized MCT predictions for the relationship between the fast and slow relaxation are
only expected to hold above the critical temperature T,
-
240 K. Below this temperature,
the idealized MCT model breaks down because of the onset of thermally assisted hopping
(see section 2.3.2). In these experiments on DC704, the values of a and b were determined
from fitting the minimum in the loss compliance spectrum to the interpolation formula
J"(w) = Jmun a (W)
a+ b
Wmin
+ b (w
Wmin
.
(3.33)
While the interpolation formula provided good fits, examination of the data shows that the
compliance minimum above T, - 240 K lies in the frequency region between the ISS and
PU techniques, where data are sparse. Ideally, additional data should be collected in this
frequency region to confirm the values of the power-law exponents a = 0.3 and b = 0.5 above
T, ~ 240 K. In chapter 6, I will present work on extending the PU technique to access the
mechanical response of DC704 in this frequency regime.
73
3.4.2
Fast dynamics and the Boson peak
The fast dynamics in supercooled liquids are believed to involve motions among multiple
potential energy minima within existing intermolecular or interionic geometries, in contrast
to slower relaxation which involves exchange of molecular or ionic positions as occurs in
diffusion. Our current high-frequency data are consistent with the mode-coupling description of the fast relaxation, which applies for temperatures well above the glass-transition
temperature Tg [921. However, experiments at higher frequencies are needed to fully map
the behavior of the fast dynamics; indeed, at some temperatures the data at 200 GHz barely
extend beyond the susceptibility minimum between the fast and slow relaxation.
In addition, a microscopic description of the fast dynamics requires further experimentation at temperatures near and below Tg. Dielectric, light-scattering, and optical Kerr effect
data have shown evidence of an additional, temperature-dependent constant loss component
in the high-frequency region of the susceptibility spectrum that is believed to arise from the
anharmonicity of the local molecular vibrations [27, 1251. The presence of this excess loss
remains to be confirmed by mechanical spectra.
As acoustic wavelengths drop below
-
10 interatomic spacings, the disordered structure
of glasses and amorphous solids causes the acoustic plane wave model to break down, and
the behavior of acoustic vibrations becomes unclear [114, 41]. In this same frequency range,
which corresponds to ~1 THz in silica glass, the vibrational density of states in disordered
solids exhibits an excess of modes over the Debye-model prediction, which appears as a broad
feature in inelastic X-ray and neutron scattering spectra and is referred to as the Boson
peak [132, 10, 138]. Though located at high frequencies, the Boson peak has been linked to
supercooled liquid dynamics, and an understanding of its origins could have relevance to the
glass transition [143]. Many theories have been postulated for the origin of the Boson peak,
including that it is analogous to the Van Hove singularity of transverse acoustic phonons
in crystals [381, and that it is related to the cross-over between propagating acoustic waves
and localized vibrational modes [1131. Chapters 4 and 5 present work on extending the PU
technique to access higher frequency acoustic responses.
74
3.5
Summary and conclusion
Broadband mechanical spectroscopy of supercooled liquids and glasses may help elucidate
the microscopic mechanisms leading to the glass transition as well as clarify the nature of the
Boson peak in amorphous solids. Photoacoustic techniques are ideal for these measurements
because they provide a non-contact, non-destructive means for probing the mechanical properties of materials at frequencies ranging from 10 MHz to 1 THz. With ultrafast laser pulses
now below 100 fs in length, photoacoustic transduction has the potential to probe acoustic
propagation in materials to the end of the Brillouin zone.
In the following chapters, I will present experimental advances in photoacoustic techniques with the aim of extending both the upper and lower frequency limits of these methods.
The focus will be on the application of these techniques to the study of viscoelastic liquids,
though these developments also have applications in the study of mechanical properties and
thermal and electron transport in a wide variety of materials.
In chapter 4 and chapter
5, I present advances in the picosecond ultrasonics technique that improve the generation
chapter 4 and detection chapter 5 of acoustic waves at frequencies exceeding 100 GHz. In
chapter 6, I present work on extending the ISS and picosecond ultrasonics techniques to
close the gap of approximately one decade in frequency that lies between these two methods. Finally, chapter 7 includes a detailed examination of the "slow rise" mode in ISS data
and its relationship to the elastic compliance.
75
76
Chapter 4
Coherent narrow-band acoustic
generation and detection in
semiconductor superlattices
Content in section
4.3 adaptedfrom:
A. A. Maznev, K. J. Manke, K.-H. Lin, K. A. Nelson, C.-K. Sun, and J.-I. Chyi. "Broadband terahertz ultrasonic transducer based on a laser-driven piezoelectric semiconductor
superlattice." Ultrasonics 52, p. 1-4 (2012).
4.1
Introduction
Experimental access to the behavior of phonons at frequencies approaching 1 THz and
beyond could shed light on a number of interesting physical phenomena. As discussed in
section 3.4.2, the Boson peak and fast relaxation in supercooled liquids and glasses occur
at frequencies between a few 10s of GHz and 2 THz.
While the Boson peak has been
studied at frequencies above 1 THz with inelastic neutron and X-ray scattering spectroscopy
[132, 10, 139, 138], mechanical measurements on these systems currently do not exceed 740
GHz [1041, and data above 100 GHz are sparse [179, 49, 93, 8]. Further exploration of the
sub-THz and THz mechanical frequency response of these systems is crucial for constructing
77
an accurate microscopic description of these phenomena.
The behavior of phonons in crystalline materials also displays interesting properties in
this frequency regime. Theoretical models of phonon attenuation assume that the lifetimes
of high-frequency phonons are dominated by three-phonon scattering processes, while lowfrequency phonons decay through scattering with the thermal phonon bath [162, 33, 1051.
The crossover between these two relaxation processes is expected to lie in the sub-THz range
[44], but this transition has yet to be observed experimentally [110]. Further, phonons at
frequencies above 100 GHz appear to exhibit a lower thermal boundary resistance than
predicted by the acoustic mismatch model [147, 150, 29], and additional experimental data
are required to understand this anomaly.
The highest acoustic frequency generated from a traditional picosecond ultrasonics experiment on a metal film is 440 GHz [179], with few measurements exceeding ~ 200 GHz. At
these frequencies the acoustic wavelengths are typically in the single-digit nanomater range,
and diffuse scattering from nanometer-scale interface roughness can significantly attenuate
acoustic waves [164]. Metal films with high-quality surfaces and thickness less than ~ 10
nm are difficult to fabricate using traditional deposition methods.
With the development of sophisticated nanoscale fabrication over the past 20 years, many
have worked to improve the picosecond ultrasonics technique by replacing metal transducer
films with metal or semiconductor nanoparticles [109, 124] or with single- or multi-layer
semiconductor structures [169, 12]. The key advantage of using semiconductor materials is
the excellent quality of the single-crystal layers and interfaces.
While single semiconductor layers have been used to produce broadband acoustic pulses
[163], superlattice (SL) structures can produce higher signal levels at well defined frequencies.
The periodic structure of SLs imposes a zone folding of the acoustic branches into the into
the mini-Brillouin-zone (mini-BZ), allowing access to higher frequency Raman-active modes
in light scattering experiments [89]. In time-domain measurements, coherent oscillations are
excited with an optical pump beam and detected with a time-delayed probe beam. Acoustic
generation in semiconductor SLs can occur through a deformation potential mechanism
78
or through the inverse piezoelectric effect [9]. Acoustic oscillations modulate the effective
optical constants of the SL structure, resulting in changes in the transmission or reflection
of a probe beam.
The nature of the excitation and mechanisms depend on the material
properties of the SL, and will be discussed in more detail in the context of specific SL
structures in sections 4.2 and 4.3.
The frequency of acoustic waves generated in a SL is determined by the periodicity of the
structure, with the fundamental frequency approximately equal to v/d, where v is the speed
of sound and d is the SL period. Therefore, one way to achieve high acoustic frequencies is to
reduce the SL period. Acoustic waves with frequencies of 1 to 1.2 THz have been generated
in SLs comprising alternating layers of Indium Gallium Nitride (InGaN) and Gallium Nitride
(GaN) 11461 and of Gallium Arsenide (GaAs) and Aluminum Arsenide (AlAs) [144, 158, 80]
with periods of
-
5 nm. In many cases, SL structures also yield higher order acoustic modes
and frequencies near the harmonics of the fundamental frequency, and the amplitudes of
higher harmonics have been found to increase with a more asymmetric ratio of the layer
thicknesses [34]. A SL designed to yield a large number of harmonics could enable broadband
spectroscopy with a single transducer structure. In acoustic spectroscopy of viscous liquids
and phonon lifetimes, one rarely encounters sharp resonances; broad relaxation-type features
are by far more common.
Thus even a sparse coverage over a wide range of harmonic
frequencies would provide valuable information.
This chapter presents photoacoustic measurements on GaAs/AlAs and InGaN/GaN SL
structures demonstrating their ability to generate acoustic waves at sub-THz and THz frequencies'. In section 4.2, which presents this work carried out in collaboration with Alexei
Maznev, asymmetric GaAs/AlAs SLs with 16-nm periods are shown to generate acoustic
frequencies at ~ 320 GHz and ~ 640 GHz. The data are then used to produce preliminary
estimates of the phonon lifetimes in GaAs/AlAs SLs at these frequencies. Section 4.3 then
explores acoustic generation in InGaN/GaN multiple quantum wells (MQWs) in work carried
out under the mentorship of Kung-Hsuan Lin. Preliminary attempts at measuring acoustic
'Work in this chapter performed in collaboration with Alexei Maznev, Christoph Klieber, and KungHsuan Lin.
79
attentuation in amorphous silica at high frequencies using a highly asymmetric 10-period
InGaN/GaN MQW were unsuccessful; however, the results demonstrated the generation of
four acoustic harmonics at frequencies up to 1.4 THz. Further measurements on a second
sample with the same MQW structure but no amorphous silica layer by Alexei Maznev
revealed the generation of seven harmonics reaching frequencies up to ~ 2.5 THz.
4.2
Phonon lifetimes in GaAs/AlAs superlattices
Superlattice structures comprising alternating layers of gallium arsenide (GaAs) and aluminum arsenide (AlAs) can be manufactured with high interface quality and low defect density due to their well-matched lattice constants (aGaAs
=
0.5653 nm and aA1As
=
0.5660 nm
[140]), and as a result have become model systems for studying the behavior of phonons in
nanolayered materials [62, 26]. In GaAs/AlAs SLs, light absorption and carrier excitation
in the GaAs layers (the AlAs layers are transparent at optical wavelengths) creates a deformation potential and mechanical stress; the rapid change in stress yields a driving force
that initiates coherent phonon oscillations.
The acoustic generation is strongest at zero wavevector q
=
0, or the zone center of the
mini-BZ, while detection is strongest where the acoustic wavevector is twice the wavevector
of the optical probe, q
=
2
kprobe.
In theory, there would be zero overlap between the
generated and detected frequencies a perfect, infinite SL. however, in real SLs finite size
effects and light absorption ensure that there there is some non-zero overlap between the
generated and detected frequencies, and acoustic modes at both the zone center and the
detection wavevector have been observed in a number of pump-probe experiments [11, 1231.
Semiconductor SLs can be used to study phonon lifetimes in one of two ways, as illustrated in figure 4-1. In the first approach, the SL structure is used strictly as a transducer
and receiver material.
As in the studies on liquids (see section 3.3.3), the sample of in-
terest is sandwiched between two SL structures. Acoustic waves are generated in one SL,
transmitted through the sample, and then detected in the second SL. While this method
80
)
(
Transparent
Sample
Substrate
(varying thickness)
Detectior
superlatti :e
Generation
superlattice
(b)
PROBE
Substrate
PUMP
Thick superlattice
Figure 4-1: Sample configurations for measuring phonon lifetimes with semiconductor SLs.
(a) The sample of interest is sandwiched between a detection SL and a generation SL.
Acoustic waves generated in the first SL propagate through the sample and are detected in
the second SL. (b) The phonon lifetime in a thick SL structure is measured by monitoring
the decay of acoustic oscillations.
has been demonstrated using GaAs/AlAs SLs grown on either side of a ~ 1 pIm GaAs layer
1123], it has not yet been applied to the measurement of phonon lifetimes; the challenges
lie in fabricating a sample of varying thickness (which is necessary for separating phonon
attenuation in the material of interest from that arising from sample-specific effects, such as
scattering from interface roughness) and in the weak signal-to-noise ratio often observed in
SL experiments.
A simpler approach is to monitor the decay of phonon vibrations in an SL structure
with a large number of periods. If the SL is thick enough, one can assume that the acoustic
waves cannot propagate outside of the SL structure within the time frame of the experiment,
and the decay rate of acoustic oscillations in experimental signal is proportional to the
attenuation rate of the phonon modes. One major drawback of this method is that it can
only be used to find phonon lifetimes in SL structures themselves, and not a general sample.
However, given the dearth of experimental data on phonon lifetimes at sub-THz frequencies,
measuring phonon attenuation in SL structures is still useful. In this section, I will use the
81
(T
Detector
PROBE
GaAs Substrate
GaAs/AlAs superlattice
-219 periods, 3.5 rn total
Figure 4-2: Sample illustration and experimental configuration. A 790 nm pump laser beam
(optical penetration depth (GaAs = 700 nm) excites the q = 0 phonon modes of an 8 nm
GaAs/ 8 nm AlAs superlattice structure. The phonons are detected my monitoring the
reflectivity change of a 790 nm probe laser beam.
second configuration to estimate phonon lifetimes in GaAs/AlAs SLs at 320 GHz and 640
GHz.
4.2.1
Experimental methods
GaAs/AlAs samples were provided by our colleagues in Eugene Fitzgerald's group at MIT.
Alternating layers of GaAs and AlAs were grown on (100) GaAs substrates by metal-organic
chemical vapor deposition (MOCVD) at a temperature of 750 C. A 500 nm GaAs layer was
deposited onto the substrate, followed by the SL structure. Four SL samples comprising 219
periods of 8 nm GaAs/8 nm AlAs, 219 periods of 14 nm GaAs/2 nm AlAs, 219 periods of
2 nm GaAs/14 nm AlAs, and 876 periods of 2 nm GaAs/2 nm AlAs were fabricated. Each
SL structure had a nominal thickness of
-
3.5 ptm.
Acoustic waves were generated and detected using the output of an amplified Ti:Sapphire
system with a central wavelength of 790 nm, a pulse duration of 300 fs, and a repetition
rate of 250 kHz.
The beam was split into excitation and variably delayed probe pulses.
The excitation beam was modulated by an acousto-optic modulator at 93 kHz frequency
to facilitate lock-in detection and focused to a 100 jim diameter spot at the sample. The
probe beam was focused to a 25 jim diameter spot at the center of the excitation spot. The
82
(a) 3 x0
-'7
5
-(b)
Spectrum
2'
4 -
330 GHz
Raw Data
25
.'.-
L.
1.5
a,
-11
Derivative
20
0
20 40 60
time (ps)
80
0
100
1.0
0.5
frequency (THz)
1.5
Figure 4-3: Measured signal from the 8 nm GaAs/8 nm AlAs SL sample. (a) Data showing
the reflectivity change and its derivative as a function of time. (b) The FFT of the reflectivity
data shows a clear frequency component at 330 GHz, corresponding to the first q = 0 phonon
mode of the superlattice.
probe beam reflection was directed to a photodiode whose output was fed into a lock-in
amplifier. The excitation beam was was directed toward the sample at a 20 degree angle
of incidence, which had no measurable effect on the measurement: the tilt of the excitation
pulse corresponded to about 25 fs time difference across the probe spot diameter, which was
insignificant compared to the laser pulse duration.
4.2.2
Results and analysis
No acoustic signal was observed from the SLs comprised of 2 nm GaAs/2 nm AlAs and 2
nm GaAs/14 nm AlAs. We surmise that this is likely due to the thin layers of GaAs, which
is the primary material contributing to acoustic generation.
Figure 4-3 shows reflectivity signal from the 8 nm GaAs/8 nm AlAs SL sample. The
raw data are dominated by a step-like response due to electronic excitation within the SL
structure. The derivative of the signal eliminates the slowly varying background and reveals
sub-THz acoustic oscillations at short times. Only acoustic oscillations at the fundamental
zone center frequency,
f
= 330 GHz (VGaAs = 4719 m/s and
83
VA1As
= 5718 m/s [691), were
observed, and the oscillations are short-lived.
As expected, the asymmetric 14 nm GaAs/2 nm AlAs SL yielded harmonics of the
fundamental acoustic frequency. Reflectivity signal from this sample is shown in figure 44. The spectrum shows primary peaks at 320 GHz and 640 GHz, corresponding to the
fundamental and second-harmonic acoustic modes at zero wavevector.
The fundamental
mode is slightly lower in frequency than in the 8 nm GaAs/8 nm AlAs because of the
lower acoustic velocity in the thicker GaAs layers.
The 320 GHz and 640 GHz modes
are both branched by two satellite peaks corresponding to acoustic modes at the detection
wavevector, q =
2
kprobe = 47r/790 nm. These modes are illustrated in a schematic diagram
of the dispersion curve in figure 4-4(d). The peak at
-
920 GHz occurs at too low a frequency
to be the third-harmonic acoustic mode, and is likely an artifact.
The phonon lifetimes of the 320 GHz and 640 GHz acoustic modes were estimated by
calculating the amplitude of the Fast Fourier Transform (FFT) of the signal for shifting
time windows. The Fourier amplitudes were then fit to a simple exponential decay (figure
4-5). The fits yielded decay times of 1280 ps for the 320 GHz phonon mode and 500 ps
for 640 GHz phonon mode. Phonon lifetimes are commonly defined as one half of the 1/e
amplitude decay time [33]. Therefore, the experimental phonon lifetimes are T320GHz = 640
ps and
T640GHz=
250 ps.
As discussed in the introduction, the phonon attenuation in bulk semiconductors is dominated by phonon-phonon scattering procresses. However, the layered structure in SLs can
add additional attenuation due to interfacial scattering (scattering from imperfections at
the layer boundaries) or inhomogeneous broadening from slow changes in the periodicity of
the lattice. The latter two sources of attenuation are not inherent to the SL materials themselves but rather depend on the quality of individual samples. Therefore, attenuation due
to phonon-phonon scattering can be deemed intrinsic scattering, while interfacial scattering
and inhomogeneous broadening can be deemed extrinsic scattering.
In determining the inherent phonon lifetimes in SLs, it is clearly necessary to separate
the contributions from intrinsic and extrinsic scattering.
84
Intrinsic phonon lifetimes are
(a)
jo1-,
4-
(b)
e
3 .5 3-
2 5
Raw Data
-I
x.
.,
2
3
2 .5
15
2
2CO
50[
0 5-
0,5
1000
500
0
1
time (pm)
0
0-
-0.5-
-0
0
300
200
100
400
500
600
800
700
900
0
1000
20
(d)
x10
(c)
40
60
100
80
time (ps)
time (ps)
1
0.91.4-
320 GHz
12-
0.6
1
C1C)
E 0.8
t
C0.5UO.4-
640GHz
06
)
Q
0.7
0.4
0.2
042
02
.1
02
0
MM
0
01
0.2
0.3
0.A
0.5
0.6
0.8
0.7
frequency (THz)
0.9
1
t/d
2k
wavevector
Figure 4-4: Generation of the second harmonic acoustic modes in an14 nm GaAs/2 nm AlAs
SL structure. The sample and experimental configuration are identical to that pictured in
figure 4-3(a), but with different individual GaAs and AlAs layer thicknesses. (a) Data
showing the reflectivity change and its derivative as a function of time. (b) A close-up view
of the signal derivative showing acoustic oscillations with a peaked shape, indicating the
presence of higher harmonics (c) The FFT of the reflectivity data shows primary frequency
components at 320 GHz and 640 GHz, corresponding to first- and second-harmonic acoustic
oscillations at the zone center. Each primary peak also exhibits two satellites at the primary
optical detection frequency q = 2 kprobe. (d) An illustration of the GaAs/AlAs band diagram
in the mini-BZ. Blue circles indicate zone center acoustic modes and red squares indicate
acoustic modes at the coherence scattering wavevector.
85
Phonon Lifetimes (Approximate)
3.5
0
-
320 GHz
320 GHz fit
640 GHz
640 GHz fit
1000
1200
3 .
-
cd
2 5
E
< 1.5
0 1
LL
0.50-
200
400
600
800
time (ps)
Figure 4-5: An approximate determination of 320 GHz and 640 GHz phonon lifetimes in
GaAs/AlAs superlattices. The time-dependent Fourier amplitudes were determined from
shifting 200 ps windowed FFTs of the signal and fit to exponential decays. The fits yeilded
decay times of 1280 ps for the 320 GHz phonon mode and 500 ps for the 640 GHz phonon
mode, corresponding to phonon lifetimes of T320GHz = 640 ps and 7640GHz = 250 ps.
86
strongly temperature dependent, while extrinsic phonon lifetimes are largely temperature
independent. Therefore, one method for determining the relative contributions from each is
to examine the temperature dependence of the phonon lifetimes.
The data presented here were used as preliminary results in a more comprehensive study
of phonon lifetimes in GaAs/AlAs SLs carried out by my colleagues Alexei Maznev and
Felix Hofmann. They conducted temperature-dependent measurements on phonon lifetimes
in order to separate contributions from intrinsic and extrinsic scattering mechanisms. Comparison of the phonon lifetimes at RT and at 79 K yielded an intrinsic lifetime of re = 950 ps
for an 8 nm GaAs/8 nm AlAs SL and [110] and ri, = 890 ps for a 14 nm GaAs/2 nm AlAs
SL [77].
4.3
Broadband acoustic generation in InGaN/GaN multiple
quantum wells
A number of experiments have demonstrated that high frequency acoustic generation via
the laser-induced inverse piezoelectric effect [72] is more efficient than acoustic generation
via the deformation potential mechanism. Therefore, SL structures composed of strained
piezoelectric crystals are expected to be more efficient photoacoustic transducers than other
semiconductor SLs.
GaN and InN have a wurzite crystal structure with large piezoelectric constants along
the [0001] orientation [94] (see figure 4-6).
Coherent layers of InGaN grown on a pure
GaN substrate experience compressional strain (aGaN = 0.3189 nm,
aInN = 0.3544 nm,
CInN
CGaN =
0.5185 nm [97],
= 0.5703 nm [1]), creating a piezoelectric field [36]. Excitation by an
optical pulse above the band-gap of the InGaN quantum wells excites carriers in the material
which very quickly move toward the positively charged sides of the layers, neutralizing the
electric field and partially releasing the stress. The stress release stimulates an expansion in
the layers and launches a set of counterpropagating acoustic waves.
This section examines acoustic generation in InGaN/GaN multiple quantum well (MQW)
87
(b)
(a)
Ga or In atom
0 N atom
C
GaN
(C)
E
_E
PUMP
GaN
GaN
InGaN
InGaN
E
GaN
Figure 4-6: Mechanism of acoustic generation in GaN/InGaN quantum well structures. (a)
Group III nitrides (including GaN and InN) have a wurzite crystal structure with large
piezoelectric constants along the [00011 orientation. (b) Coherent layers of InGaN grown
on a pure GaN substrate experience compressional strain due to lattice mismatch (aGaN =
0.3189 nm, CGaN = 0.5185 nm [97], aInN = 0.3544 nm, CInN = 0.5703 nm [1]), creating a
piezoelectric field. (b) The introduction of an optical pump beam with energy above the
band-gap of the InGaN/GaN MQWs excites carriers which shield the intrinsic electric field,
leading to rapid stress and expansion through the inverse piezoelectric effect.
structures. While preliminary attempts to use MQW structures to measure acoustic attenuation in amorphous silica were unsuccessful, InGaN/GaN MQWs with a strongly asymmetric
well-barrier width ratio were capable of generating up to seven harmonics covering a broad
spectral range.
4.3.1
Experimental methods
InGaN/GaN MQWs were provided by our colleagues in Chi-Kuang Sun's group at National
Taiwan University. The sample comprised 10 periods of 3 nm Ino.1Gao. 9 N/19 nm GaN quantum wells/barriers grown on a 2- im-thick wurtzite GaN buffer layer on a (0001) sapphire
substrate via metal organic chemical vapor deposition (MOCVD). The SL structure was
capped with 150 nm of GaN deposited by the same method and a 100 nm layer of amorphous silica deposited by sputtering.
A second sample featuring a 140 nm GaN capping
layer and no amorphous silica layer was also grown.
The experimental set-up was similar to that described in section 4.2.1. However, both
pump and probe beams were frequency-doubled to 395 nm by a beta barium borate (BBO)
88
crystal to account for the higher bandgap in InGaN/GaN MQWs. In addition, because of
the low reflectivity of GaN, transmission signal rather than detection signal was detected.
4.3.2
Results and analysis
Figure 4-7(b) shows the transmission signal from an InGaN/GaN MQW sample. The signal
is dominated by the step-like response due to electronic excitation in the InGaN quantum
wells. The time-derivative of the signal eliminates the slowly-varying background and reveals
acoustic oscillations at a fundamental frequency of fi
the acoustic velocity along the c-axis of GaN, v
d
=
=
=
360 GHz, which is consistent with
8020 m/s [48], and the MQW period of
22 nm. The oscillations decay as coherent phonon wavepackets leave the SL structure.
The Fourier spectrum of the signal (figure 4-7(d)) contains four peaks spanning frequencies
from 360 GHz to 1.4 THz, corresponding to the phonon modes at zero reduced wave number
q = 0 in the mini-BZ (figure 4-7(c)).
The data also reveal a recurrence of acoustic oscillations at 75 ps. The timing of the
recurrence indicates that it is due to the phonon wavepacket traversing the GaN capping
layer, reflecting off the GaN/SiO 2 interface, and returning to the SL structure to modulate
the transmission of the probe beam. While the hope was to also observe signal arising from
the reflection of the acoustic wavepacket from the SiO2 /air interface, there is no evidence
of acoustic oscillations at 100 ps, the expected arrival time of these reflections. In addition,
the spectrum of the reflected wavepacket shows only peaks at the first two frequencies, fi
and f2. The loss of the higher frequency components is likely due to poor quality of the
GaN/SiO 2 interface. Depending on the nature of the interface roughness, losses due to diffuse
scattering may increase as fast as the fourth power of frequency. The acoustic impedance
mismatch between GaN and silica may also contribute to the lack of reflected signal from the
SiO 2 /air interface. Silica glass has an acoustic impedance of ZGaN
(PSiO 2 = 2.2 g/cm- 3,
x
=
=
13 X 106 kg -1
6000 m/s) while GaN has an acoustic impedance of
106 kg s-1 m- 2 along the c-axis (PGaN
=
6.1 g/cm- 3, V
=
8000 m/s).
m-
ZGaN =
2
49
According to
equation 3.21, this will result in a 57% reflective loss each time the acoustic wave travels
89
(a)
Sample Structure
~V~iU1IW~ IPROBE
GaNHIfI
sapir
substre
Ibuffer~jIII
(2
o
GaN
cap
)
(us1e
50 nm)
4
1(100 nm)
(
PUMP
3 nm ln0,Ga.,N/19 nm GaN
superlattice
-10 periods, 220 nm total
14 f4
raw data
1.2N-
f3
8
f2
100
50
echo
0
generation
0.
fi
time derivative
75
50
25
0
100
wavevector
125
time (ps)
(d)
(e)
echo
generation
spectrum
spectrum
f2
fl
flspectrum
f3
2
E
E
M
f4
0.0
0.5
1.0
1.5
2.0
2.5
0.0
frequency (THz)
0.5
1.0
1.5
2.0
2.5
frequency (THz)
Figure 4-7: Broadband acoustic generation in InGaN/GaN MQWs and preliminary attempt
at measuring acoustic attenuation in amorphous SiO 2. (a) Sample diagram and experimental
configuration. The sample comprised 10 periods of 3 nm Ino.jGao. 9 N/19 nm GaN quantum
wells/barriers grown on a 2-km-thick wurtzite GaN buffer layer on a (0001) sapphire substrate via metal organic chemical vapor deposition (MOCVD). The MQW structure was
capped with 150 nm of GaN deposited by the same method and 100 nm of amorphous silica
deposited by sputtering. Acoustic waves were excited by a 395 nm pump beam and detected
through the transmission of a time-delayed, 395 nm probe beam. (b) The derivative of the
experimental signal (raw signal shown in the inset) shows the initial large amplitude oscillations from acoustic wave generation in the SL, followed by a recurrence of oscillations at
70 ps from acoustic propagation and reflection through the GaN layer. (c) A simulation of
the GaN/InGaN band diagram, estimated from the acoustic velocity in GaN v = 8020 m/s
and the MQW period d = 22 nm. (d) The Fourier spectrum of the acoustic generation
shows four frequency components up to -1.4
THz. (d) The Fourier spectrum of the acous-
tic reflection shows only the first two frequency components.
90
Sample Structure
(a)
PROBE
cap
buffe
(2 itm)
sapphire
subste
(150 nm)
PUMP
3 nm InGa . 9N/19 nm GaN
superlattice
(b)
~-10
periods, 220 nm total
(c)
-
10
x3
Cou
0
a10
E
rdue
reucedwavenumber
ULL
0
5
10
15
20
0
25
1
2
3
4
5
frequency (THz)
time (ps)
Figure 4-8: Broadband acoustic generation in InGaN/GaN MQWs. (a) Sample diagram and
experimental configuration. The sample and experimental configuration were identical to
that described in figure 4-8, but without the a 100 nm layer of amorphous silica. (b) The
experimental signal and acoustic oscillation trace; the latter was obtained by multiplying
the signal by a factor of 30 and subtracting the slowly varying background. (c) The Fourier
spectrum of the acoustic oscillations shows frequency components up to ~2.5 THz. The
inset schematically shows dispersion curves of the periodic structure within the first miniBrillouin zone; the first several laser-excited modes are shown by filled circles. Figures (b)
and (c) reproduced from 1111].
between the GaN and the SiO 2 layers.
This experiment was repeated by Alexei Maznev on a similar sample which lacked the
SiO 2 layer, and the results showed even higher harmonic acoustic generation in these structures. Figure 4-8(b) shows the transmission signal from this experiment [1111. Subtracting
the slowly varying background reveals acoustic oscillations, again at a fundamental frequency
of
f
= 360 GHz. The shape of the acoustic signal, with sharp maxima and smooth minima,
agrees with theoretical calculations for a similar asymmetric InGaN/GaN SL [351, and the
presence of higher harmonics is indicated by the non-sinusoidal shape of the signal. If the
91
acoustic impedance mismatch between quantum wells and barriers is small, the number of
oscillations at the fundamental frequency is expected to be equal to the number of quantum
wells in the structure, in agreement with the data. The Fourier spectrum of the signal (figure
4-8) contains seven peaks spanning frequencies from 360 GHz to 2.5 THz, corresponding to
the phonon modes at zero reduced wave number q = 0 in the mini-BZ.
Kung-Hsuan Lin continued these measurements on similar samples featuring thinner
layers of GaN and SiO 2 , and was able to observe acoustic propagation through a silica layer
[1041. He reports attenuation coefficients of 3.17 x 10 7 m-
1
at 370 GHz and 5.16 x 10 7 M
1
at 740 GHz.
4.4
Conclusions and outlook
As these experiments show, semiconductor SLs provide an efficient mechanism for the generation of THz-frequency acoustic waves. However, our difficulty in detecting acoustic propagation through an SiO 2 layer underscores a serious challenge for THz acoustic spectroscopy:
it is not enough to generate THz acoustic waves; they also need to be transmitted into and
through a material of interest. These challenges are heightened when THz acoustic measurements are applied to liquids. Generation of THz acoustic waves requires crystalline SL
structures with atomically flat interfaces. However, these semiconductor crystalline materials have acoustic impedances that are on the order of 5 to 10 times higher than liquids,
resulting in significant acoustic losses during transmission. While losses due to the acoustic
impedance mismatch may be mitigated with the introduction of impedance matching layers (a common practice employed in ultrasonic acoustic measurements on the human body
[161]), these layers would have to be exceptionally thin and smooth to augment acoustic
transmission. Further, the brittle nature of crystalline semiconductor SLs and substrates
creates difficulties in integrating them into the liquid sample structures discussed in section
3.3.3.
Preliminary attempts at incorporating superlattice structures into liquid samples,
such as by gluing them to round glass substrates, have so far been unsuccessful.
92
One promising new direction in THz acoustics is the recent observation of coupling between THz-frequency optical and acoustic waves in piezoelectric materials [7, 25]. Reed et
al. first predicted that THz radiation should be emitted when shock waves or solitons pass
between materials with different piezoelectric constants [128]; they later demonstrated this
effect experimentally by observing THz radiation coherently generated by acoustic waves
traveling through a GaN/AlN interface [7]. A similar coupling has been observed between
THz radiation and THz-frequency phonons in BaTiO 3 /SrTiO
3
superlattices 125].
Inter-
conversion between THz-frequency radiation and phonons may provide a more sensitive
mechanism for the detection of THz phonon and thus enable THz acoustic spectroscopy on
a wider variety of samples.
93
94
Chapter 5
Measurement of
shorter-than-skin-depth acoustic
pulses in a metal film via transient
reflectivity
Content adapted from:
K. J. Manke, A. A. Maznev, C. Klieber, V. Shalagatskyi, V. V. Temnov, D. Makarov, S.H. Baek, C.-B. Eom, and K. A. Nelson. "Measurement of shorter-than-skin-depth acoustic
pulses in a metal film via transient reflectivity." Appl. Phys. Lett 103, 173104 (2013).
K. J. Manke, A. A. Maznev, C. Klieber, V. V. Temnov, D. Makarov, S.-H. Baek, C.-B.
Eom, and K. A. Nelson. "Detection of shorter-than-skin-depth acoustic pulses in a metal
film via transient reflectivity." AIP Conf. Proc. 1506, 22 (2012).
5.1
Introduction
As discussed in the previous chapter, semiconductor superlattice structures can be used
to extend the range photoacoustic measurements to THz frequencies.
[67, 169, 146, 111].
However, while the capabilities of these structures are impressive on their own, ultilizing
95
them in experiments can prove challenging. Superlattice structures with nanometer-thick
layers are difficult to manufacture reliably, and the need for properly matched substrates
makes it difficult to integrate them into the complex sample designs required for liquid and
glass experiments. Therefore, we continue to seek more robust methods for the generation
and detection of high-frequency acoustic pulses.
One of the simplest methods for the optical detection of ultrashort acoustic signals is
measurement of the transient reflectivity of a thin metal film 1153]. As discussed in section
3.3, an acoustic strain pulse propagating through a metal film will reflect from the metal-air
interface with a change in sign, and the arrival of the pulse at the surface can be detected
through the strain-induced change in the complex optical index of refraction. If the change in
reflectivity were due only to the acoustic strain at the free metal surface, the time-dependent
reflectivity measured would precisely reproduce the acoustic strain profile. However, because
the pulse penetrates into the metal, the reflectivity begins to change as the acoustic pulse
approaches the free surface. Therefore, the reflectivity is described by a convolution over
the optical penetration depth and the strain effect on the complex refractive index as a
function of depth [153, 501.
Since the acoustic pulse moves toward and then away from
the free surface as a function of time, the measured time-dependent reflectivity change is
given by the same convolution. Here, the conversion between the distance of the acoustic
pulse from the surface z and the measurement time t is mediated by the acoustic velocity
v, t = z/v. As a result, it is generally accepted that the ability to resolve a short acoustic
pulse is limited by the optical skin depth of the metal film [1681, and aluminum, which has
a short skin depth of
-
7 nm in the visible range, is considered the material of choice for
transducer films in picosecond ultrasonics [179, 28].
Recently Mante et al. [109] demonstrated the time-dependent reflectivity detection of
acoustic pulses that showed features significantly shorter in duration than the optical absorption depth divided by the speed of sound in semiconductor InP. Similarly, time-resolved
plasmonic interferometry has demonstrated that it is possible to measure acoustic pulses
shorter than the skin depth of a surface plasmon in gold [151, 152]. The question remains
96
whether this transient reflectivity approach can resolve shorter-than-skin-depth pulses in
commonly used metal films and whether the acoustic strain profile can be extracted from
the intricate waveforms observed by Mante et al.
This chapter investigates the detection of ultrashort acoustic pulses in metal films by
transient reflectivity1 . The reflectivity response of any strongly absorbing material to a
shorter-than-skin-depth acoustic pulse can be decomposed into two terms: the first term
contains a sharp step, caused by the change in sign during the acoustic reflection at the free
interface, and gives access to the acoustic pulse shape with sub-skin-depth spatial resolution;
the second term is slowly varying, and is not sensitive to the fine features of the acoustic
pulse shape.
The relative magnitudes and signs of these two terms are dictated by the
photoelastic and optical constants of the material.
For some select materials, particularly gold, the magnitude of the first, discontinuous
term is large enough to allow the acoustic pulse shape to be extracted from transient reflectivity signal [1081. This effect is demonstrated by detecting ultrashort acoustic pulses that
are generated in very thin (4-12 nm) layers of SrRuO 3 (SRO) and that propagate from those
layers to the free surfaces of gold films. The results indicate that the detection bandwidth
of the transient reflectivity method is limited not by the optical skin depth but by the microstructure and the roughness of the metal films. Furthermore, the acoustic pulse profile
from these transient reflectivity measurements can be extracted using a pulse reconstruction
algorithm developed by Vasily Temnov and Victor Shalagatskyi.
5.2
Theory of transient reflectivity detection
Consider a photoacoustic pump-probe experiment like that depicted schematically in figure
5-2. A longitudinal acoustic pulse with a time-dependent strain profile of r/33(z, t) is incident
on the free surface of a sample. The arrival of the acoustic pulse at the sample-air interface
induces a change in the complex reflection coefficient of the probe light, which is given by
'Work in this chapter performed in collaboration with Alexei Maznev, Christoph Klieber, Vasily Temnov,
and Victor Shalagatskyi
97
A=2,B=0
A= 0, B=-1
| --OK----
4
I
,
C
.,
__J
-20
0
time (ps)
-2
20
-
-2 L
2
A=2,B=1
Q)
0
-2L-
-20
20
0
time (ps)
20
A=2,B=-1
0
0
time (ps)
-21
-J
20
-20
0
time (ps)
20
Figure 5-1: The response function G(t) (see equation 5.4) for different combinations of
the coefficients A and B. The coefficient A governs the magnitude of the slowly-varying
response, while the coefficient B governs the magnitude of the step function. When A and
B have opposite signs, a sharp step appears in the response function, which allows short
acoustic pulses to be detected.
98
[153, 133]:
[0 0
.47r
=1
2
di
1 2d
.47r
exp
-n
where h = n + ir, is the complex refractive index and dA/d1
(
R(t)
3 3 (z, t)dz,
=
(5.1)
dn/dq + idK/dr is the
complex photoelastic coefficient. The reflection of the acoustic pulse q(t) from the sample-
air interface can be taken into account according to 7133 (z, t) =
(t + z/v) - r(t - z/V),
where we define z = 0 as the metal surface and t = 0 as the time at which the center of the
acoustic pulse reaches the surface. Using this expression, we can rewrite equation 5.1 in the
time domain,
3i (t) =i
j
47rv
[1 - (A + iB)e 4rflv I'-tI /A] sign(t' - t)r(t')dt'. (5.2)
where we have defined a the complex dimensionless parameter A+ iB
=
2(dii/d) / (1 -2)
which depends on both the photo-elastic coefficient di/d1 and the index of refraction h.
The real part of this expression gives the reflectivity change S(t) measured in our pumpprobe experiment,
_____
S(t) = 6R(t)
R
_
-
2Re
fi(t)i
(
Irf
-
8irv
A
f
_0
G(t' - t)r7(t')dt',
(5.3)
where we have defined a material response function,
G(t)
=
[A sin(wBrt) + B cos(wBrt) sign(t)]estI/Tkin.
Here we have introduced the Brillouin frequency,
Tskin.
WBr
(5.4)
[103], and the acoustic time-of-flight,
In our detection material, gold, the complex index of refraction is i! = 1.67
+ 1.94i
[107] at probe wavelength A = 393 nm. Thus, we can calculate the Brillouin frequency,
WBr =
47rnv/A
=
27r x 29 GHz, and the acoustic time-of-flight Tskin =
through the skin depth
6
skin
6
skin/V =
4.8 ps
= A/(47rrn) = 16.5 nm.
The response function G(t), which describes the reflectivity variation induced by an
infinitely short acoustic pulse, represents the sum of two contributions. The first sine-term
99
in equation (5.4) is a smooth function whose duration rskin is determined by the skin depth,
whereas the second cosine-term contains a sharp step when the center of the acoustic pulse
reaches the surface (t
=
0). Fig. 5-1 shows both terms of the response function, as well as
examples of a total response corresponding to B/A
1/2 and B/A = -1/2.
=
One can see
that the shape of the transient reflectivity response is controlled by the ratio B/A:
B =
x
A
0X+1'
n2*21.
-1 =d
(5.5)
d
+' 2
It is also instructive to analyze the time derivative dS(t)/dt of the transient reflectivity
signal S(t):
dt
A (2B i(t)+
-
F(t' - t)r(t')dt)
,
(5.6)
with
F(t)
[(AWBr
-
skin)
cOs(WBrt) - (BWBr + Tsin) sin(WBrjti) e-ski".
For a short acoustic pulse with duration
Tac << Tskin,
(5.7)
the slowly varying integral term in
equation (5.6) is proportional to Tac. Consequently, in the short-pulse limit the first term
is dominant and the time derivative of the reflectivity response yields the acoustic strain
profile. Indeed, in the signal derivative shown in the inset in figure 5-2, one can see a unipolar
pulse of 2 ps in duration. However, the pulse is not short enough for the slow component of
the signal derivative to become negligible.
We see that in order to detect a short acoustic pulse, one needs a material and probe
wavelength combination that yields a large value of B. One can see that B vanishes for a
weakly absorbing medium with vanishing K, hence a strongly absorbing material is desired.
On the other hand, B also vanishes in the limit , >> n.
is not necessarily preferred.
Thus a smaller skin depth
Because detailed data on the wavelength dependence of the
photoelastic constant dA/d71 are typically lacking, one has to resort to trial-and-error in
search for the best material
/
wavelength combination. As is shown below, gold works well
100
at 400 nm, yielding B/A = -0.59
5.3
0.03.
Experimental Methods
We used transient reflectivity to detect ultrashort acoustic pulses at a gold/air interface.
The experimental geometry for the measurements is pictured in figure 5-2(a). The samples
consisted of a very thin layers of SrRuO 3 (SRO) sandwiched between a gold layer and an
SrTiO 3 (STO) substrate. The single crystal SRO layers [57], with thicknesses of 4 nm and
12 nm, were grown using 90 degree off-axis RF magnetron sputtering [58] at a rate of 1
nm/min2 . Polycrystalline gold films with thicknesses of
200 nm and a primarily (111)
orientation, as determined by electron backscattering diffraction (figure 5.3), were then
deposited on top of the SRO layer 3 . The sputtering conditions were varied to produce two
sets of gold films with quantitatively different material properties. High-resolution scanning
electron microscope (SEM) images of the two gold films (figure 5.3) indicate that the first
set of gold films has an average grain diameter of
films has an average grain diameter of
-
-
400 nm, while the second set of gold
40 nm.
A Ti-sapphire oscillator with a regenerative amplifier was employed to generate laser
pulses with a central wavelength of 786 nm and duration of 300 fs. The laser output was
split into separate pump and probe beams and the probe beam was frequency-doubled to
a wavelength of 393 nm. Illumination of the SRO transducer layer with the pump pulse
through the 1 mm thick STO substrate launched an acoustic strain pulse through the gold
layer, which was detected at the gold/air interface through the transient reflectivity of the
variably delayed probe pulse.
101
SRO
(a)
Au
200 nm
probe
STO
Substrate
Ipump
(b) 10
o- 7
x
11
(t+z/v)
z
-
to detector
x10-,
X
Cn
z
~
+
2ps
-2
30
40
50
time (ps)
60
0
0
20
40
time (ps)
60
80
Figure 5-2: (a) Schematic diagram of the sample structure and the experimental geometry;
(b) the transient reflectivity signal S(t) detected at the free surface of the gold layer shows
the arrival of the acoustic pulse at a delay time of 46 ps. The inset shows the time derivative
dS(t)/dt of the reflectivity signal.
102
a) {1001
b) {1iao
vn
X0
Figure 5-3: Pole figures showing the crystal orientation distribution of (a) gold film 1 and
(b) gold film 2, as determined by electron backscattering diffraction.
103
a)
b)
Figure 5-4: SEM images collected at the free surface of (a) gold film 1, showing an average
grain diameter of ~ 400 nm and (b) gold film 2, showing an average grain diameter of
~ 40 nm.
5.4
Results and Analysis
The reflectivity signal S(t) from gold film 1 is depicted in Fig. 5-2(b).
The sharp spike
at zero pump-probe delay time arises from the optical generation of hot electrons at the
gold/SRO interface followed by their ultrafast diffusion through the 200 nm thin gold layer
on a sub-picosecond time scale [24]. The barely visible slow rise in the background signal is
due to the subsequent thermal diffusion [149, 152, 99]. The small peaks at delay times of
18 ps and 35 ps are caused by multiple roundtrips of the residual pump reflections within the
1 mm thick (STO) substrate (nsTo = 2.34 [22] at A = 800 nm). The optical signal at 46 ps
marks the arrival of the acoustic strain pulse at the gold/air interface after propagating
through the layer of (111) gold at the speed of sound v = 3.45 nm/ps [2]. It contains a
fast 2-ps transient on top of a slowly varying component.
The duration of the transient
corresponds to the expected acoustic pulse duration generated in the SRO film given the
2
The SRO samples were prepared by Seung-Hyub Baek, a member of Chang-Beom Eom's research group
at the University of Wisconson-Madison.
3
Gold films were deposited by Denis Makarov at the Institute for Integrative Nanosciences, IFW Dresen.
104
a)
10-7
S
1.5,x10
b)
1.51 X
CI)
1
C
0.5
C,
0.5[
0
Mi
0
Cc
1
-0.5F
-0.5
40
time (ps)
C
1.2
U)
0
U-
60
-i440
80
X 1016
d)
---I 4nn m SRO_
-- 12 n m SRO
.
1
C
10-
4
UI)
C
U)
C
0.80.6-
V
60
~-8
time (Ds)
80
10 0
-- 4 nm SRO
-12 nm SRO
3
2
0.40
U-
0.2
1
I-
C
200
400
600
Frequency (GHz)
Ito
200
400
Frequency (GHz)
6( 0
Figure 5-5: (a) and (b) Transient reflectivity response of gold film 1 and gold film 2, respectively, to an acoustic strain pulse generated in a 12 nm SRO transducer; (c) and (d) the
Fourier transforms of the acoustic pulses generated in 4 nm and 12 nm thick SRO layers for
each gold film
105
speed of sound in SRO 6.31 m/s [170]. In gold, this duration corresponds to a distance of
7 nm, significantly shorter that the skin depth of 16.5 nm [107].
A comparison of the transient reflectivity responses of the two gold films is shown in
figure 5.4(a and b).
The overall signal strength is similar from the two films, but the
magnitude of the short central feature is significantly smaller in gold film 2. The Fourier
transform spectra of the time-derivatives indicate that the detection bandwidth of gold film
1 extends to ~ 350 GHz, and the detection bandwidth of gold film 2 extends to ~ 300 GHz.
Applying a shorter strain pulse by exciting a 4 nm thick transducer layer only extends the
bandwidth of gold film 1 to ~ 450 GHz. Under ideal conditions, we would expect 12 nm
and 4 nm SRO films to launch acoustic strain pulses with central frequencies of ~ 500 GHz
and
-
1.5 THz, respectively. Therefore we can conclude that the bandwidth is most likely
limited by the detection mechanism or by acoustic broadening within the film.
5.4.1
Response Fitting
To obtain an estimate of B/A in gold, we fit equation 5.6 to the time derivative of the
reflectivity using a simulated strain pulse. Because the low acoustic impedance mismatch
between STO and SRO suppresses the reflected wave (ZSRO = PSROvSRO = 41X106 kg
m-
2
s--I and ZSTO = PsToVsTo = 40x10 6 kg m- 2 s- 1 ) 113], we expect the initial strain
pulse to have a rectangular shape with FWHM ~ dSRO/VSRO = 1.90 ps. However, acoustic
dispersion and the difference in the arrival time at the gold-air interface due to nanometer
surface roughness cause the initial rectangular pulse shape to spread [152]. Therefore, we
assumed a Gaussian pulse shape and allowed the width to vary to account for the overall
acoustic pulse broadening. The fit parameters were the ratio B/A and the FWHM of the
Gaussian profile. The data pictured in Fig. 5-2 show an average over 25 scans; to test the
repeatability of our fits, we divided the raw data files into five sets of five scans each and fit
each set independently.
The results of fitting the full data sets for gold film 1 and gold film 2 are shown in Fig. 56(a and b). For gold film 1, the best results were obtained for B/A = -0.59
106
and FWHM
= 2.54 ps, and for gold film 2, the best results were obtained for B/A
=
-0.60 and FWHM
= 4.25 ps. The statistical fitting error for B/A was 0.01; however, based on measurements
of other samples we would suggest a more conservative error estimate of 0.03. Using our
fit value for B/A, we obtained a ratio of x=1. 9 at a wavelength of 393 nm. We expect a
significant variation in the photoelastic constants of gold at visible wavelengths due to an
interband transition in this region [50]. The known photoelastic coefficients of other metals
such as nickel and chromium have a ratio on the same order of magnitude as what we have
observed here,
5.4.2
lxi
~ 1 [133].
Acoustic Broadening
Acoustic pulse broadening in polycrystalline thin films may be caused by acoustic damping in
the material, surface roughness, variations in grain orientation, or grain-boundary scattering
[142]. For broadening that arises exclusively from surface roughness, the Gaussian width of
the acoustic pulse can be related to the root mean square (orms) surface roughness of the
film Tpulse = 0rms/VAu. AFM analysis indicates that gold film 1 (Orms = 1.28 nm) actually
has a slightly higher surface roughness than gold film 2 (Orms = 1.14 nm); this amount
of roughness would result in broadening of
0.3 ps, which is smaller than our estimated
broadening. Therefore, surface roughness variability between the two films is unlikely to be
the cause of additional acoustic broadening in gold film 2. In addition, orientation analysis
by electron backscatter microscopy shows that there is a greater variation in grain orientation
in gold film 1 than in gold film 2. However, the difference in sample morphology, evidenced
by Fig. (3), is still likely the source of the additional acoustic broadening witnessed in gold
film 2.
5.4.3
Pulse Reconstruction
Quantitative acoustic measurements require a reliable algorithm for the reconstruction of
an acoustic pulse profile r(t).
It was demonstrated recently that for ultrafast acousto-
plasmonic measurements, the strain profile can be recovered from the plasmonic analog of
107
4
Sample #2
Sample #1
X 0-7
4
-
-
2
Cz
experiment
simulation
0
-10
2
CO
0
0
time (ps)
_A I
-NO
20
10
simulation
-10
0
10
20
time (ps)
X 103
10
1X
-
experiment
-
simulation
-
2
-
.E
-:
experiment
-2
-2 [
4
--
S0
C-
U6
b)
x 10-
2
-
a)
experiment
simulation
'-0
-2
C-2
C)
C,
Cz
-10
-$0
0
time (ps)
10
-10
20
---simnulation
--reconstruction
-
4
-10
0
time (ps)
10
20
^x 10
x 10-3
-
.
)
"a
C,
2
C
P6
-
4
simulation
reconstruction
2
P,
0
0
0
-10
0
time (ps)
10
20
-10
0
time (ps)
10
20
Figure 5-6: (a) A comparison of the simulated reflectivity profiles and the experimental data.
(b) The time-derivative of the reflectivity data and the fit that was obtained for Eq. 5.6. (c)
A comparison of the simulated strain pulse and the reconstructed strain pulse.
108
Eq. (5.6) using an iterative numerical method [152].
However, in the case of reflectivity
measurements, the coefficients of the sine and cosine terms in Eq. (5.7) are too large and
the iterative solution of Eq. (5.6) diverges.
A more general method for pulse reconstruction was developed by Victor Shalagatskyi
and Vasily Temnov, and here is applied to simulate the strain pulse from the transient
reflectivity signal. After taking the Fourier transform of Eq. 5.3 and applying the convolution
theorem, the reconstructed strain in the Fourier domain is given by 7(w)
=
(-j)S(w)/G(w)
with
2
[0(
G(w)
=
G(t)e-iwdt = -2iw (
J~oc(W2
2
WiBr/Tskin)A + (w - Wr 2 1/T 2)
22
r
kin)B
r
+
- W2 + j/ 2. )2 + (2WBr/Tskin )2
(5.8)
Applying the inverse Fourier transform delivers the desired acoustic strain profile
)= 87r,
(5.9)
__COO G
e
)
j
iwtdw
r2v (w)
shown in Fig. 3(c). The reconstructed waveform matches the Gaussian pulse obtained by
curve fitting fairly well. However, the reconstruction algorithm did not use any assumptions
about the acoustic pulse shape and can be employed to recover arbitrary acoustic strain
profiles.
The reconstructed waveform contains barely visible slow oscillations of the baseline which
become more prominent when noisier data are analyzed. The spectral function G(w) possesses three zeros, w
=
0 and w
=
twO=
27r x 56 GHz, resulting in singularities in
,q(w) = (-)S(w)/G(w) in the presence of noise with non-zero spectral components at these
frequencies.
The singularity at w
=
0 is prevented by subtracting the slow background
from the experimental signal S(t) to guarantee f S(t)dt
=
0 (S(w
=
0)
=
0).
other roots w = twO result in a background modulation with a period of 27r/wo
The two
=
18 ps.
This modulation does not present a problem for the analysis of shorter-than-skin-depth
acoustic pulses but may become a nuisance for longer waveforms containing significant frequency components at wo. The existence of non-zero roots of G(w) requires
109
Wsr
-1/7s2
(A/B)(2WBr/Tskin) and can be avoided if different probe wavelengths and/or a different
metal are used.
5.5
Conclusion
The results demonstrate that transient reflectivity can be used to detect shorter-than-skin
depth acoustic pulses in common metals such as gold. The ability to resolve short acoustic
pulses is not limited by the the skin depth of the material; rather, it is controlled by the
parameter B/A that depends on the photoelastic constants and the the complex index
of refraction at the detection wavelength.
We have demonstrated a pulse reconstruction
algorithm capable of recovering the acoustic pulse profile from transient reflectivity data
which will advance quantitative ultrafast acoustic spectroscopy.
110
Chapter 6
Time-domain Brillouin scattering
measurements on DC704 near the
compliance loss minimum.
6.1
Introduction
The acoustic response of liquids at frequencies from 1 GHz to 10 GHz is difficult to access
with both the ISS and PU techniques. In ISS, acoustic frequencies above 1 GHz generally
require the generation of optical grating spacings below
-
1 -2
nml. To achieve interference
fringe spacings of this magnitude, the optical beams must cross at steep incidence angles
that cannot be attained with the common-path optic design pictured in figure 3-22.
In
TDBS detection, which uses a similar detection mechanism as ISS detection but in a rotated
geometry, frequencies below 5 to 10 GHz (depending on the liquid sample, temperature, and
probe wavelength) are similarly limited by the need to detect at steep incidence angles.
This region is of special interest in the broadband mechanical studies of DC704 and other
'For example, in liquid DC704, the acoustic velocity is v ~ 2000 m/s at the MCT T. of 240 K. A grating
spacing of A = 1 m will launch an acoustic wave with frequency v/A = 2 GHz.
2
From equation 3.5, for an optical pump wavelength of A = 1035 nm, a grating spacing of 1 pim can be
achieved with an angle of 600.
111
supercooled liquids because the susceptibility minimum between the fast and slow relaxation
often falls within this frequency range near the predicted MCT critical temperature T. The
position of the susceptibility minimum wmin is used to predict the MCT T, through the
relation
Wmin ~ [T - Tc11/( 2 a)
(6.1)
where a is the power-law exponent describing the low-frequency behavior of the fast relaxation component. In addition, the predicted MCT relationship between the power-law
exponents of the fast and slow relaxation are only expected to hold above the MCT critical
temperature Tc. Below this temperature, MCT predictions break down with the onset of
thermally assisted hopping.
As a result, the power-law exponents a and b are often de-
termined by fitting the minimum in the susceptibility spectrum between the fast and slow
relaxation components to the MCT interpolation formula:
J"(w) =
min
a- b
a
Wmin Imin
(6.2)
+ b (w)].
In the broadband spectrum of DC704 (see section 3.4), data in the in the 1 GHz to 15 GHz
region are sparse. Based on the available data points, an MCT T,
-
240 K was predicted,
and the interpolation formula fit the data well with power law exponents a = 0.3 and
b = 0.5 at temperatures between 240 K and 250 K. However, additional data in the region
would confirm these fits. Further, Dielectric and light-scattering spectra suggest that the
susceptibility minima of other fragile glass-formers such as o-terphenyl [43], CKN [102, 106]
and propylene carbonate [106] lie in this frequency range at temperatures near their expected
values of T,. In addition, as the temperature increases and the fast and slow relaxation peaks
in the susceptibility loss spectrum overlap to a greater extent, the susceptibility minimum
between them becomes increasingly shallow, and the power-law exponents become more
difficult to determine accurately.
This chapter presents time-domain Brillouin scattering (TDBS) measurements on DC704
at acoustic frequencies of ~ 6 GHz and
-
12 GHz. In TDBS experiments, the frequency in
112
a liquid
f
is given by:
(6.3)
f = 2nv
A C
where n is the refractive index of the sample, v is the speed of sound in the liquid, A
is the optical detection wavelength, and 9 is the detection angle. The refractive index n
and acoustic velocity v are fixed by the choice of sample; therefore, changing the detection
frequency
f
requires that either the optical detection wavelength A or the detection angle 9
be altered.
In these experiments, the incidence angle 9 is increased to achieve detection of lower
frequency acoustic waves than previously obtained. While these measurements are similar
in nature to TDBS measurements at normal incidence angles, a number of complications
arise when the technique is extended to lower frequencies. At low temperatures and low
frequencies, the acoustic propagation lengths in supercooled liquids grow to the order of
tens of microns and optical decoherence effects must be considered when calculating the
acoustic attenuation from TDBS interference signal.
The long propagation lengths and
low signal-to-noise ratio in these measurements also contribute to extended data collection
times. These issues are analyzed and solutions proposed. Finally, the limits of the MCT
interpolation formula (equation 6.2) in fitting the asymtotic power-law behavior of the fast
and slow relaxation at higher temperatures are explored.
6.2
Experimental Methods
The schematic diagram of the sample structure in TDBS experiments on DC704 is pictured
in figure 6-1.
A thin layer of DC704 was held between two round substrates in a copper
sample holder like that illustrated in figure 3-8.
The acoustic transducer consisted of a
200 nm single-crystal SrRuO 3 (SRO) layer grown on a round 27' offcut (010) SrTiO 3 (STO)
113
DC704
Layer
PROBE
PUMP
sapphire
substrate
SrM 0
substrate
SrRuO3
(200 nm)
Figure 6-1: Sample structure and experimental configuration for TDBS measurements on
DC704. The sample comprised a thin layer of DC704 held between a thin layer of SrRuO 3
deposited on an SrTiO 3 substrate and a sapphire substrate. An optical pump beam excited
a broadband acoustic strain pulse in the 200 nm SrRuO 3 layer, which propagated through
the DC704 liquid layer and was detected by an optical probe beam at a 450 angle of incidence
to the sample.
114
substrate 3 using 90 degree off-axis RF magnetron sputtering 4 [581. A thin (- 1 mm) sapphire
substrate was placed behind the STO substrate to provide support and prevent the crystal
from cracking. A few drops of DC704 were placed on the SRO transducer and held in place
with a sapphire lens.
The liquid DC704 sample was acquired from Sigma Aldrich. Volatile impurities were removed by heating the sample under vacuum, and larger impurities were removed by straining
the liquid through a 0.22 Vim millipore filter.
Sample cooling was achieved with a Janis ST-100 model cold finger crysostat controlled
by a Lakeshore model 332 temperature controller. The temperature was measured with a
factory-calibrated silicon diode which was mounted to the sample holder. A layer of indium
foil was placed between the sample holder and the cold finger to ensure good thermal contact
with the cold finger.
The measurements were conducted using the experimental set-up described in section
3.3.4. Acoustic waves were generated and detected using the output of an amplified Ti:Sapphire
system with a central wavelength of 777.5 nm, a pulse duration of 300 fs, and a repetition
rate of 250 kHz. The beam was split into excitation and variably delayed probe pulses. The
excitation beam was modulated by an acousto-optic modulator (AOM) at a frequency of 93
kHz to facilitate lock-in detection and focused to a 100 tm diameter spot at the sample.
The probe beam was focused to a 25 pim diameter spot at the center of the excitation spot.
The sample was mounted at a nominal 450 angle of incidence to the incoming probe beam.
The probe beam reflection was directed to a photodiode, whose output was fed into a lock-in
amplifier.
To minimize the effect of cumulative laser heating on the sample temperature 5 , the pump
intensity was reduced to the minimum power that still yielded detectable signal. The pump
3
An offcut substrate was chosen to facilitate a separate set of experiments on shear acoustic wave generation (see [126] for details on this technique). However, the crystalline orientation of the substrate is not
relevant for these measurements.
4
SRO samples were provided by our collaborators Dr. Kwang-Hwan Cho and Prof. Chang-Beom Eom
in the Materials Science and Engineering department at the University of Wisconson-Madison.
5
For a detailed discussion of this issue of cumulative heating in TDBS measurements on liquids, see
chapter 4 of Christoph Klieber's doctoral thesis [911.
115
power was adjusted by changing the intensity of the electronic signal sent to the AOM,
which modifies the intensity of the first order diffraction beam used in the measurements.
The probe power was adjusted by changing the rotation of a A/2 waveplate before a polarizer
in the beam path. Measurements were taken with 15 mW of pump power and 0.2 mW of
probe power corresponding to laser fluences of 0.78 mJ cm- 2 and 0.37 mJ cm
Data were collected with the probe wavelength at A
to A
=
=
2
, respectively.
777.5 nm and frequency-doubled
387.5 nm. Temperature-dependent data were taken at 5 K intervals between 210 K
and 240 K and between 260 K and 325 K and at 2 K intervals between 240 K and 260 K.
6.3
Data Fitting and Analysis
Characteristic TDBS signals at temperatures between T
=
240 K and T
=
260 K are shown
in figure 6-2. The signal waveform exhibits damped acoustic oscillations at the Brillouin
scattering frequency
f
and with a decay rate
FA.
The background signal also shows a slow
exponential decay as heat diffuses away from the excitation spot in the SRO film. At each
temperature and detection wavevector, the waveform was fit to the expression
F - Ae-'At sin(wt - (b) 4-
where A, B, and C are constants, w
=
27rf,
#
Re-rHt
is a phase offset and FH is the thermal decay
rate. Because fast electron diffusion and heating can affect the reflectivity of the transducer
layer at short times, the fitting window was adjusted to exclude the initial excitation. Data
fitting was carried out using the Trust-Region nonlinear least squares algorithm.
6.3.1
Acoustic velocity
Rearranging equation 6.3 provides an expression for calculating the acoustic velocity in the
liquid as a function of the Brillouin scattering frequency,
V
fA
2n cos 6
116
(6.5)
i
1
T =242 K
-.--
experiment
fit
0~
Z)
--0
experiment
fit
1500
1000
500
0
--
=246 K
1 -T
C--
-1
1500
1000
500
-10
T =254 K
--
experiment
fit
C-Z)
-1-0
0
.2
1500
1000
500
1
0
T = 258 K
-
-1-
0
500
1000
experiment
- fit
1500
time(ps)
Figure 6-2: TBDS data in DC704 at temperatures between 240 K and 260 K. Data were
collected at a probe wavelength of A = 387.5 nm and nominal incidence angle of 25' in the
liquid (450 in air). Red lines indicate fits to the data using equation 6.4.
117
2600
1
---12
2400--
1
16 GHz (TDBS 0.121 gm)
GHz (TDBS 0.132 m)
8 GHz (TDBS 0.242 gm)
-a-~6 GHz (TDBS 0.295 gm)
~--~1.4 GHz (ISS 1.15 pm)
2200C')
E
E 2000-
C
0
cn1800-
c
0..
(1600-
0L
C')
1400-
1200200
250
300
350
400
temperature (K)
Figure 6-3: The acoustic velocity in DC704 as a function of temperature and acoustic
wavelength. TDBS data at wavelengths of 0.132 urm (- 12 GHz) and 0.295 jxm (~ 6 GHz)
are from the present study. TDBS wavelengths of 0.121 tm (- 16 GHz) 0.242 tm ( 8 GHz)
and ISS data at wavelength 1.15 Rm (~ 1.4 GHz) are taken from [92]. Dotted lines indicate
the infinite frequency c,, and zero frequency co speeds of sound determined in [92].
118
Therefore, an accurate determination of the acoustic velocity requires the wavelength of
the probe light, index of refraction in the liquid, and angle inside the liquid. The spectrum of
the probe light was measured with an Ocean Optics spectrometer and the central wavelength
was found to be A = 777.5 nm for the red probe light and A = 387.5 nm for the frequencydoubled blue probe light. The temperature-dependent index of refraction in DC704 can be
well approximated by [92]
n(Tg < T ~ 400K) = 1.750 - 4.9
between the glass transition temperature Tg
x
10-
4
K- 1
x
T
(6.6)
210 K and Tg ~ 400 K.
-
The probe beam had a nominal angle of incidence in air of 45'. The angle of incidence was
determined accurately by measuring the frequency of acoustic oscillations in the sapphire
substrate; the angle of incidence in sapphire was then calculated from equation 6.3 and
known properties of sapphire 6 . The Brillouin scattering frequency in sapphire was 46.42
GHz, yielding an incidence angle of 9 sapp = 23.70. The angle of incidence inside the liquid
was then calculated with Snell's law using the temperature-dependent refractive index in
DC704:
sin ODC704
=
sapp
slnlsapp.
(6-7)
nDC7O4
The speed of sound as a function of temperature and probe wavelength is shown in figure
6-3, along with previous speed of sound data from [92]. The data at acoustic wavelengths of
0.295 tm appear to confirm the small degree of acoustic dispersion observed at wavelengths
of 0.242
6.3.2
rm and 0.121 km.
Acoustic attenuation
The scaled acoustic damping rate
(PA
x A, where FA is the acoustic attenuation rate and A
is the wavelength) as a function of temperature are shown in figure 6-4 along with previous
TDBS and ISS data from [92]. The data from the current study show good agreement with
6
A = 777.5 nm,
Vsapp
=11.2 km/s [166], nsapp = 1.76 (117]
119
10
U)
E
0U
E
O
-e-~16 GHz (TDBS 0.121 gm)
-e-~12GHz (TDBSO.132Lm)
-*--8 GHz (TDBS 0.242 gm)
-B-~6 GHz (TDBS 0.295 gm)
--1.4 GHz (ISS 1.15 gm)
102
200
250
300
temperature (K)
350
400
Figure 6-4: The scaled acoustic damping rate in DC704 as a function of temperature and
acoustic wavelength. TDBS data at wavelengths of 0.132 tm (- 12 GHz) and 0.295 Lm
(- 6 GHz) are from the present study. TDBS wavelengths of 0.121 .tm (~ 16 GHz) 0.242
km (- 8 GHz) and ISS data at wavelength 1.15 im (- 1.4 GHz) are taken from [92].
120
(a ) 1 . ...
.
.
,
--
0.8
0.8
0.6
0.6
E
0.4
-- Interference Intensity
0.2
-
00
5
10
15
20
distance (gm)
0.4
0.2
Attenuation at 210K
-Attenuation
Interference Intensity
at 210K
-Attenuation
at 240K
-Attenuation
Sapphire Attenuation
-
.(b).
., ..
1.
......
......
...
at 240K
25
0
30
5
10
15
20
distance (jim)
25
30
Figure 6-5: Simulations of the intereference fringe intensity of TDBS measurements on
DC704 as a function of acoustic propagation distance. The simulated interference intensity
was calculated from equation 6.8 assuming zero damping in the acoustic pulse. The magnitude of the acoustic wave as a function of propagation distance was modeled as a simple
exponential with a decay rate of a, where a =FA/V. (a) Interference fringe intensity and
acoustic propagation length in DC704 at A = 0.132 jim. (a) Interference fringe intensity
and acoustic propagation length in DC704 at A = 0.295 jim. The green line indicates the
TDBS interference decay in a sapphire substrate.
previous results where they overlap.
Optical coherence effects
In TDBS measurements, the acoustic attenuation rate FA at the detection wavelength is
determined by a decay in the magnitude of the oscillations in the signal intensity. Acoustic
waves at high temperatures and high frequencies have high damping rates and so only
propagate a distance of a few microns or less. However, at lower temperatures or lower
frequencies, this distance can grow to several microns in length.
Oscillations in the TDBS signal arise from interference between light reflecting from
a propagating acoustic wave and light reflecting from the metal transducer. If these two
sources grow too far apart, decoherence of the probe beam may affect the magnitude of
the interference oscillations.
The intensity of interference fringes I as a function of the
121
separation between sources L is given by [134]
.2
I oc L2
AkL
(2L)
(6.8)
(kL)2
and the width of the distribution of optical wavevectors is given by
Ak =
2wrn AA
A A
(6.9)
where n is the index of refraction in the material, A is the optical wavelength and AA is
optical spectral width.
Equation 6.8 was employed to estimate the relative intensities of optical interference
fringes in liquid DC704 as a function of acoustic propagation distance L, assuming no acoustic damping. This result was compared to the experimental acoustic propagation lengths
measured in DC704 at different detection wavevectors and temperatures. In simulations of
the interference intensity, the index of refraction of DC704 was estimated using equation
6.6.
The central wavelengths A and widths AA of the fundamental and second harmonic
probe beams were measured with a spectrometer, and determined to be A = 387.5 nm and
AA = 1.5 nm (TDBS wavelength 0.132 pm) and A = 777.5 nm and AA = 9 nm (TDBS
wavelength 0.295 pm). The damping coefficient was calculated from a = IPA/v), where
PA
and v are the fit values for the acoustic damping rate and velocity (from equation 6.4).
The amplitude of the acoustic wave as a function of propagation distance was modeled as a
simple exponential with a decay rate of a.
The results of the simulations are pictured in figure 6-5. For detection with the blue
probe, the decay in the interference intensity appears to be at least an order of magnitude
slower than the acoustic decay, and so we assume that decoherence in the probe does not
play a significant role in the decay of the TDBS intensity oscillations.
However, at the
lower acoustic frequencies detected by the red probe, the acoustic decay rate is lower, and
the coherence length is also shorter due to the greater spectral width of the probe. As a
result, decoherence of the probe may affect the low-temperature values of FA at a TDBS
122
wavelength of 0.295 km.
To confirm the simulations of the interference length, the acoustic propagation in the
sapphire substrate was measured at a probe wavelength of A = 777.5 nm. Acoustic damping
in sapphire is very low compared to liquids [281, and so decay in the TDBS oscillations
in sapphire is assumed to arise solely from probe decoherence.
The decay in the TDBS
oscillations in sapphire is shown in green in figure 6-5(b), and is similar in magnitude to
the simulated interference intensity. For this reason, 0.295 pIm TDBS data at temperatures
below 240 K were deemed unreliable.
The issue of probe decoherence could be solved by narrowing the spectral width of the
probe beam.
A narrower spectral width could be achieved by placing a bandpass filter
with a FWHM < 1.5 nm in the probe path.
Similarly, a narrower portion of the probe
spectrum could be selected by inserting a diffraction grating to separate out the spectral
components and filtering out a narrow range with a slit, in a manner similar to that utilized
in a spectrometer. One additional option would be to replace the variably-delayed pulsed
probe with a CW probe and fast detector.
This possibility will be discussed further in
section 6.5.
6.3.3
Modulus and Compliance
The complex elastic modulus at temperature T and frequency v can be written in terms of
the density p(T) and the complex acoustic velocity 'b(T, v),
M (T, v) = p (T) b (T, v)
2
(6.10)
where the complex acoustic velocity is defined by,
Ib 27rv
2ir=
(6.11)
and q is the complex wavevector,
2rv
V
123
+ ia.
(6.12)
0.5
I
I
I
o
215 K
* 220 K
* 225 K
v 230 K
o 240 K
o 240 K
o 242 K
m 242 K
0 244K
V
0.45
V
03
V
0
000
00
0.4H
0
*
'OC'
VVVV
S
00
V
VV
7
V
0
77V
U
-
0.35F
0o
0
V7
0
0
V
0
V
0
V7
V
vvy
v
o
V
0
0.3F
V vV
0o
00
o
VV7
0 0
CD
90
VV~
vv
0
0
V
V
00
00
00
V
0
0.25F
+
*7
*
*
a0.
*
0
7@00
0
q%vv 000 %4
v
00
%
00000
00
0000VV
V0
0
v
0 7
v vIcP
0
4*
V70
0
130
V
0
M
0
o0o
CIem
vv7Vywav
*
* * 0 0000
V
VVVV
9V
0
0.2H
0000
V
00 V00000000
V vvvvvvvvvvvy
0
0 C010 00330000
0.1
000000*
0
a
V
0
0
0 0OOcOM
0
0 00 0000aco
V V VVVVW7
a 0 [3D[30aC
0
0 0 0 0 0 00o0000000
252 K
254 K
256 K
* 256 K
* 258 K
+ 260 K
v 264 K
* 265 K
* 270 K
* 275 K
* 275 K
v 280 K
v 280 K
o 285K
* 285 K
O 290 K
* 290 K
* 295 K
300K
v300 K
* 305 K
* 310 K
+ 315 K
v 320 K
o 325 K
-
0.15
0
244 K
v 246 K
v 246 K
o 248 K
* 248 K
* 250 K
10
10 8
109
10
10 11
1012
frequency (Hz)
Figure 6-6: The real component of the complex elastic compliance for liquid DC704 as a
function of temperature. Previous data are represented with open symbols and new data
are represented with filled symbols. Previous data taken from [76].
124
-
10
o
V
0
0
0
000000
0
V
0
0
VV
V13
10-2
0
V
0
)00
00
00GocP
V
0
VVVVVV
1
00
00
C)
Cb I
t
L
0
V
1
so
V
od
1 %V
a0
.7
0
I
gI
40
I 0
00
00
a
0
0
V
4
Cz
0
0
*
242 K
* 242 K
0 244 K
* 244 K
v 246 K
v
246 K
o
*
*
248 K
248 K
250 K
*
v
252 K
254K
o
256 K
*
256K
258 K
*
*
v
VV
V
V V
V
V
VVV
4
*
*
4
+
Evv0
v
v
-
10
0 0
4 4
o
0
*
*
*
*
00
0
0
0
0
0
0
0
v
v
0
*
*
*
0
v
*
10 7
108
109
1010
1011
260K
264 K
265 K
270 K
275 K
275 K
280K
280K
285 K
285K
290K
290K
295K
300 K
300K
305 K
310 K
315 K
320 K
325 K
-
0*
%
0 00
0 oN
0
VO
.
- 00000 0
%
VV0
-V
VV
VVV
-
V
0
6W
V
VV
K
K
K
K
K
~
-
0
a
0
220
0 225
v 230
o 240
* 240
-
0 1
%
01
VO
No
Wvy
00000
215 K
0
.
I
-
0
1012
frequency (Hz)
Figure 6-7: The imaginary component of the complex elastic compliance for liquid DC704
as a function of temperature. Previous data is represented with open symbols and new data
is represented with filled symbols. Previous data taken from 1761
125
Here, v is the acoustic velocity and a is the attenuation coefficient. The elastic compliance
is defined as the inverse of the modulus:
1
M
-(6.13)
Finally, the temperature-dependent density in DC704 can be described by [92]
p (T > Tg ) =
1070 kg m-3
100k
3(6.14)
1+ 7.2 x 10-4 K~ 1 x (T - 298 K)
Equations 6.10 through 6.14 were used to calculate the complex elastic compliance from
the acoustic velocity and attenuation data.
The real and imaginary components of the
compliance are pictured in figures 6-6 and 6-7, along with compliance data taken from [76]
(previous data are represented with open symbols and new data are represented with filled
symbols). The new data agrees well with the past results.
6.4
Results and Discussion
In glasses and supercooled liquids at temperatures near T, the a- and ,3-relaxation peaks
are separated by many orders of magnitude allowing an unambiguous determination of the
power-law exponents governing of behavior of the relaxation wings. At higher temperatures,
the value of
T,
drops while -T8 remains constant, and the relaxation peaks in the suscepti-
bility loss spectrum begin to overlap. In this region, the power-law exponents a and b can
be obtained by fitting the MCT interpolation formula (equation 6.2) to the susceptibility
minimum.
However, as the overlap between the a- and #- peaks increases, the susceptibility minimum becomes increasingly shallow.
This effect is evident in the imaginary compliance
spectrum of DC704 (figure 6-7). At temperatures around Tc ~ 240 K, the spectrum exhibits
a clear minimum between the fast and slow relaxation; however, as the temperature increases
to 260 K and above, the presence and position of the susceptibility minimum becomes dif126
ficult to ascertain. Therefore, it is reasonable to expect that at a certain temperature (or
likewise, spacing between r and r), the interpolation fit will no longer provide accurate
values for a and b.
The efficacy of the interpolation formula in fitting the compliance minimum in DC704
is confounded by the limited amount of data in this range. Therefore, to explore the limits
of the interpolation formula, simulated plots mimicking the DC704 compliance loss curves
were simulated and fit the minimum to the interpolation formula.
The a-peak was simulated by a Cole-Davidson function:
a(w)
a
=
(1 + iwr)
(6.15)
3
The stretching exponent 0 was set to a value of 0.5 in concordance with the findings of
time-temperature superposition. At each temperature, the value of TQ was determined by
the VFT fit to the a-relaxation times in the compliance spectrum,
Ta = To exp
with log ro
=
-15.0,
[TDTo _
D = 6, and To = 183 K. The
(6.16)
-peak was simulated by a power law
with exponent a = 0.6. The maximum frequency of the simulated data was set to 100 GHz,
which corresponds approximately to the maximum frequency in the experimental compliance
spectrum. The relative magnitudes of the a- and 3- features were adjusted to approximate
the DC704 compliance data.
The simulated data were then fit to the MCT interpolation formula,
J"(w) =
a
""
+b
)
[a(
Wmin)
+b(
)j
(6.17)
Win)
The simulated data and interpolation fits for temperatures between 235 K and 260 K are
depicted in figure 6-8, along with the fit values for a and b. Figure 6-9 depicts the difference
between the simulated and fit values for a and b as a function of the difference between T,
127
T = 235 K
T =250 K
10
10
a-fit =0.61
a-fit =0.60
b fit =0.50
E
b-fit =0.47
10--2
102
E
----
---
Alpha (Cole-Davidson)
Beta (power-law)
Alpha + Beta
-
- - - Interpolation Fit
--- Interpolation Fit
10
0
A'pha (Cole-Davidson)
Bata (power-law)
Alpha + Beta
---
108
10-3
10
10 6
Frequency (Hz)
T = 255 K
T = 240 K
10-1
10~
''
''a_-fit
b-fit =0.48
E
10
10 10
10
Frequency (Hz)
a-fit =0.63
=0.61
b-fit =0.44
-2
10-2
-Alpha
-
(Cole-Davidson)
--
Beta
-Alpha
+Beta
--- Interpolation Fit
-- -Interpolation Fit
106
Alpha (Cole-Davidson)
(power-law)
-
Beta (power-law)
Alpha +Beta
---
108
Frequency (Hz)
10
1010
108
10
Frequency (Hz)
1010
T = 260 K
T = 245 K
10~
10
a-fit =0.65.
a-fit = 0.61
b-fit =0.41
b-fit =0.47
10
-2
E
-Alpha
-- - -
10 0
10-3
-
(ColeDavidson)
Alpha (Cole-Davidson)
Beta (power-law)
-Alpha
+ Beta
- - - Interpolation Fit
Beta (power-law)
-
Alpha + Beta
Interpolation Fit
10 8
Frequency (Hz)
10 1
10
108
Frequency (Hz)
1010
Figure 6-8: Simulations of the compliance loss spectrum in DC704 and interpolation formula
(equation 6.2) fits to the compliance minimum. The a-peak was simulated by a ColeDavidson function with a stretching parameter # = 0.5 and the 3-peak was simulated by a
power law with exponent a = 0.6. The relative magnitudes of the a- and 0- features were
adjusted to approximate the DC704 compliance data. The interpolation formula is able to
accurately predict the simulated behavior up to a temperature of T-250 K.
128
1
0.8-
-
0.6
0.6----------------
----
0.4-
02
'
3
'
5
4
log(t ) - log( P)
'
0.2-
6
7
Figure 6-9: A comparison of the simulated values (dotted lines) and fit values (filled circles)
of the power-law exponents a and b as a function of the difference between the a- and #relaxation times.
129
o 240 K
a 244 K
0 248 K
-Interp.
Fits
E
10~2
0
11100
0 0
10
8
109
10
101
frequency (Hz)
Figure 6-10: The imaginary compliance spectrum of DC704 near the compliance minimum
at 240 K, 244 K, and 248 K. Open symbols indicate data from previous studies, and closed
symbols indicate the current measurements. Red lines show interpolation formula fits to the
data with b = 0.5 and a = 0.28.
and -r3.
The interpolation fits are able to predict accurate values for a and b up to a temperature
of - T = 250K; above this temperature, the fit values start to diverge from the simulated
values. These results indicate that while the interpolation formula likely provides reasonable
fits at temperatures between 240 K and 250 K, caution should be employed when applying
the formula at higher temperatures.
Figure 6-10 shows a plot of the compliance loss minimum of DC704 at 240 K, 244 K,
and 248 K with interpolation formula fits. In the interpolation fits, b was set to 0.5, in
130
accordance with the fits to the a-relaxation in the compliance loss spectrum, and a was set
to 0.28 in concordance with the predicted MCT relationship
A
F(l - a) 2
(I - 2a)
F(1 + b) 2
(1+ 2b)
(6.18)
The interpolation formula fits show good agreement to the data, in support of MCT.
6.5
Outlook
Time-domain Brillouin scattering experiments at wide scattering angles provide a means
to access the mechanical response of supercooled liquids in the region between 1 and 15
GHz. However, the low signal-to-noise levels of the reflectivity modulation coupled with
the relatively long acoustic decays contribute to extended data collection times which can
make experiments at multiple acoustic frequencies unfeasible. The data pictured in figure
6-2 consist of an average of 150 separate scans of the probe delay stage, and each data set
took
-
70 minutes to collect. The crystat also must be allowed to stabilize for 30 to 60
minutes between each scan in order to reach the set temperature. As a result, a full set of
temperature-dependent scans at a single detection wavevector can take a week or longer to
collect.
Photodetectors and oscilliscopes are now capable of detecting optical signals at frequencies up to 30 GHz, well above the signal modulation frequency observed in TDBS measurements. Therefore, lock-in detection with a variably-delayed pulsed probe beam could be
replaced with real-time detection of the modulation of a CW probe beam with a fast photodetector and oscilloscope, in a set-up similar to that used in ISS detection. This change
would radically reduce the data collection times. As CW laser diodes are relatively inexpensive and available in a wide array of wavelengths, they would also provide access to additional
acoustic frequencies, and as they generally have a narrower bandwidth than pulsed sources
(the spectral bandwidth of the 830 nm diode utilized in ISS experiments is AA ~ 1.5nm at
FWHM), optical decoherence would pose less of challenge at lower frequencies.
131
132
Chapter 7
Contributions of mechanical and
specific heat relaxation to the ISTS
slow rise signal in liquid DC704
7.1
Introduction
In ISTS experiments, an impulsive thermal grating launches counterpropagating acoustic
waves whose frequency and attenuation rate provide access to the elastic moduli of liquids
at frequencies between 10 MHz and 1 GHz. In addition to this transient acoustic response,
ISTS signal from liquids also exhibits a "slow rise" component that reflects slower changes
in the density of the liquid in response to the applied stress and/or heating. This slow rise,
which is analogous to the "Mountain Mode" in light scattering techniques [116, 31], may
provide valuable insight into the relaxation dynamics of liquids at frequencies between 10
kHz and 10 MHz (corresponding to timescales between 10-4 and 10-7 s) allowing access to
over six decades in frequency with a single technique.
It is tempting to assume that the temperature of the excited regions increases and reaches
a steady state on a much faster timescale than the density response.
Then the density
would be responding to a new constant temperature, and slow changes in density could be
133
attributed to a relaxing elastic compliance. However, a number of experiments [19, 18, 17,
85, 70] show that in supercooled liquids, the increase in temperature in response to a heat
input is dependent on the time- and frequency-scale of the heat input; in addition, this
frequency dependence is similar to that of the modulus and compliance. Therefore, changes
in temperature and density may occur simultaneously, and the density response would then
reflect a complex interplay between a relaxing specific heat and a relaxing compliance.
Though the slow rise component of the ISTS signal has been used to study relaxation
dynamics in a number of liquids [172, 51, 175, 178, 122, 61], a detailed examination of the
relative roles of the relaxing compliance and relaxing specific heat on the slow response has
yet to be carried out; this is in large part due to the complexity of the problem, and to the
lack of data on the behavior of the specific heat or compliance in this region.
With the broadband mechanical spectrum of DC704 in hand [76], it is now possible
to compare the density response of the slow rise component with the relaxation of the
elastic compliance. The results may elucidate the relative roles of the specific heat and the
compliance in the slow rise data.
Before doing so, it is necessary to obtain a more detailed understanding of the contributions of the density and the temperature to ISTS signal. In this chapter, I will review
the phenomenogical theory that has been developed to model ISTS signal in liquids. ISTS
signal is generally fit by an expression derived using a hydrodynamics approach in which the
viscosity term has been generalized to account for time-dependent relaxation behavior. In
this model, the slow rise in the ISTS signal is fit to a KWW stretched exponential function
whose relaxation time is equal to the viscosity relaxation time TR scaled by the ratio of the
infinite and zero frequency moduli,
T, =
(c2 /c2)r.
However, in deriving this expression,
the specific heat is assumed to be a constant.
Using this model, I will provide a detailed analysis of the slow rise in ISTS signal from
DC704 at temperatures near the MCT critical temperature of T, = 240 K. A comparison of
the KWW viscosity relaxation time of the slow rise TR with that of the elastic compliance
77
3
Tj.
determined from [76] reveals that rR is consistently larger than Tj, by a ratio of
134
TR -
Earlier comparisons of the shear modulus, bulk modulus, and specific heat relaxation times
of DC704 at lower temperatures and frequencies [83] indicate that the relaxation times of
these parameters show a similar temperature dependence, and that the ratios among them
are within one order of magnitude. Therefore, a ratio of three between the slow rise and
compliance relaxation times is evidence that the contribution of a time-dependent specific
heat cannot be neglected when modeling the slow rise signal.
7.2
A hydrodynamics model of ISS Signal
The signal intensity of an ISS experiment can be represented as [1721
S(q, t) = -FoGpp(q, t) + QoGpT(q, t)
(7.1)
The first term, Gpp(q, t), is the density-density response function and gives the response
of the sample to laser-induced electrostrictive stress Fo (ISBS signal).
The second term,
GpT(q, t), gives the density response of the sample to laser-deposited heating proportional
to Qo (ISTS signal). In order to extract quantitative information from ISS signal, we need
to consider functional forms for Gpp(q, t) and GpT(q, t).
Local changes in the density 6p and temperature 6T of a liquid sample can be obtained
by evaluating the linearized hydrodynamic equations for conservation of mass, momentum,
and energy [172]. However, this approach is only valid for describing the long-time or lowfrequency behavior of the system, while the ISS technique is capable of probing density
changes at timescales that are short relative to the relaxation times of supercooled liquids.
Therefore, more accurate expressions for 6p and temperature 6T in the ISS experiment
can be obtained by generalizing the hydrodynamics equations to account for a relaxing or
time-dependent viscosity [231.
Yang et al. [176] have developed a model for ISS signals in liquids based on a generalized
hydrodynamics approach [23], and the relevant details of this model are reproduced here.
The continuity equation, which describes the conservation of mass in the system, is given
135
by
cp(q, t) - poiqvll (q, t) = 0
(7.2)
where v is the particle velocity and q is the wavevector. The conservation of momentum, or
Navier-Stokes equation, is expressed as
po
a 6vj (q, t) -
kBTO
iq S
p(q, t) -iq
kBTO
porJT(q, t)
+ poq 2
j
dt#L(q, t - t')v 1 (q, t)
=
iqF(q, t)
(7.3)
where S(q) is the static structure factor, To is the local temperature, K is the thermal
expansion coefficient, and F is the electrostrictive force imposed by the laser.
viscosity term,
viscosity.
#L(q,
Here, the
t), has been generalized to account for a time-dependent or relaxing
Since we are primarily interested in the density response, the particle velocity
term vii can be eliminated by combining Equations 7.2 and 7.3,
1
a2
q2
at2
kBT
kBTO
S
Sp(q,t)+6p(q, t) +
S( q)
S(q)
U pot6T(q, t)
a2
ft
dt$L(qt - t')-
-
6pg p(qt) =-F(qt) (7.4)
The expression for the conservation of energy can be written
poc a6T(q, t) - c,
at
K-Y at
p(q, t) + (q 2 3T(q, t)
=
Q(q, t)
(7.5)
where c, is the constant volume specific heat, -y is the specific heat ratio cp/c, ( is the
thermal conductivity, and
Q
is the thermal energy deposited by the laser.
Equations 7.4 and 7.5 form a coupled set of partial differential equations describing the
temporal responses of the density and temperature.
136
Taking the Laplace transform and
writing the result in matrix form G- 1 X = X 0 , gives
+ SL(q,
s) +
q2
kBTO
S(q)
-s(7y
-
1)cv
Ep(q,
kBTO
s)
POKS(q)
pocvs
+ Aq 2
ST(q, s)
+ L (q, s)
-F(q, s) + 6p(q)
(7.6)
Q(q, s) - 6p(q)
c + 3T(q)pocJ
where 3p(q) and 6T(q) are the initial values of 6p(q, t) and ST(q, t), respectively. Then the
solutions for 6p(q, s) and 6T(q, s) (represented as column vector X), can be found by taking
the inverse of G- 1 , through X = GX0 , yielding the following expressions for the Green's
functions Gpp(q, s) and GpT(q, s), GTp(q, s), GTT(q, s) [176, 154] :
Gpp(q, s)
{
(q}
=
GpT(q, s) =
GTp(q, s)
GTT(q, s) = --
SPocV
sq2 + (q}
Poc,
(7.7)
-
(7.8)
)s
=
2
qL(q, s)s +
+
(7.9)
P-O
(7.10)
po cv S (q)f
pocV
where A is given by the determinant of G- 1 :
=
detG-
1
=
q2
pocV
)
q2
+ kBT)
('+ SOL (q, s) +(pocv
Sgq)
(
2
) + SPOCv(7
s + Aq2)+soo(
kB- O
)
A
(7.11)
S(q)
By identifying the roots of A, we can transform the functions for Gpp(q, s) and GpT(q, s) back
into the time variables. To do this, we must first define a functional form for the generalized
viscosity, qL (q, t). Simple relaxation behavior can be described by a single-exponential or
137
Debye relaxation function, which is characterized by a single relaxation time -rR. As we
have seen, the viscosity relaxation near the glass transition is more accurately modeled by
a stretched exponential functions such as the Cole-Cole relation, the Havriliak-Negami relation, or Kohlrausch-Williams-Watts (KWW) relation [471. However, these functional forms
do not allow us to arrive at analytical solutions for G,,(q, t) and GpT(q, t). Therefore, we
will begin by assuming that the time-dependence of the viscosity takes a single exponential
form, and later generalize the solution to account for more complex relaxation dynamics.
In Laplace space, the Debye relaxation of the viscosity can be described as
2
liM O'L (q, t) = vL
q-+O
1
2
x-
+ STR
(7-12)
TR
where VL is the background viscous damping, vL = (IB + 1ris)/p, and c2
the infinite- and zero-frequency speeds of sound, respectively.
We can define the thermal
diffusivity, y = (/(poc) = ITH q 2 . Then, under the assumptions that VL
and
IHTR
and c2 are
< q 2, FH < coq,
< 1, the expression for A can be factorized:
A = (s +
H)(s+iwA + FA)(s -iwA+PA)
IP
[+
21
CATRJ
(7.13)
These four roots describe the nature of the ISS excitations in the system. The first and last
root, which correspond to diffusive modes, describe the density changes in the system due to
slow thermal diffusion and structural relaxation, respectively. The other two roots describe
acoustic waves which propagate at frequency
acoustic oscillation frequency
WA
WA = CAq
WA
and with a damping rate of FA , where the
is given by
=
c0q
D2
ID+
+ (coqTR2- 1/2
(7.14)
where
D =
2
c2c2
00
- (coqTR)-2
138
(7.15)
and the acoustic damping rate IA is given by
1
2
IPA = -q2
cd++Xl1( c2 -c2
-
2v)])+ +
CA
2
q2
)o02
rR1+ (WATR)
Note that the standard hydrodynamics approach only predicts three modes:
diffusion mode and two propagating acoustic modes.
(7.16)
a thermal
The inclusion of a time-dependent
viscosity terms adds the fourth structural relaxation mode. In the hydrodynamics limit, or
when the experimental timescale is long compared to
TR,
the generalized solutions should
simplify to the standard hydrodynamics model. Indeed, when
TR
-+ 0, the fourth structural
relaxation root
With the roots of A in hand, we can transform the expressions for GpT(q, s) and Gpp(q, s)
back into the time domain. We find that the density-temperature response function is given
by
GpT(q, t) = A [e
cos (WAt)] + B
-
-eH
[e-
A-tCrR
(7.17)
where
KC2 C2q2 2
A =
2 -2
(7.18)
TR + COCA
B = -
2q2(c
Bp cq2 T(C
c-)
(7.19)
CO)
and the density-density response function can be expressed as
Gyp(q,t) =
C2q2 2 cici
2 -2q
ciq2T$+
-r
4 q
q22
_TCOC
-e
4 cA CA
+(1
FAt sin
(WAt)
- c c 2)q 2 [
cAq
2
-e~FAt cos (wAt) +e-0
+ cOcA
At/TR
(7.20)
We see that the functional form for the ISTS signal shows the expected terms for the
acoustic oscillation, slow rise, and thermal decay. The ISBS signal has a similar functional
form without the thermal decay terms because the electrostrictive process does not involve
laser heating.
139
In order to reach an analytical solution for GpT(q, t) and G,,(q, t), we have made the
assumption that the viscosity relaxation can be represented as a simple exponential decay.
To model relaxation in glass-forming systems more accurately, we can replace the singleexponential relaxation in equations 7.17 and 7.20 with the KWW stretched-exponential
function:
+ B [e-PHt - -(tTs)"] (7.21)
G pp(q, t) =
GpT(q, t) = A [e-rH
-
c2q22+c-2 q2-2
A TR + COCA 4
e FAt sin (wAt)
cq2T+c4c-
CA
c0 A)q 2TR [-FAt
(2
+
cAq 71
cos (wAt) + C_
(7.22)
COCA
where 3 is the stretching parameter and the structural relaxation time is given by
TRC/c2.
FH
Further, it is important to note that these expressions only hold when vL
< c0q, and
IHrR
<
T=
<
q2
,
_--At cos (wAt)]
1, or when the acoustic, slow rise, and thermal decay modes are
well-separated in time. Experimentally, separation of the three modes can be accomplished
by adjusting the excitation wavevector.
The second part of the ISBS signal is generally extremely small compared to the first part,
so in an ISBS measurement only the acoustic response is typically observed. This is because
in ISBS the steady-state density is the same as the initial density, so the slow relaxation
term represents a relaxation of an extremely small deviation back toward equilibrium while
in ISTS the slow relaxation term describes the gradual change toward a new steady-state
density that corresponds to the sample at an increased temperature.
In the numerator of the first term, the first product ciq2wi
= WiTi = (2rrR/A)2
< 1
for any wavevector at which the temporal separation conditions noted above are satisfied,
where rA is the acoustic oscillation period. The second product, c c2
140
< 1, can be nelgected.
The ratio of the first term to the second is approximately,
T 7 R <1
2/r TR
2I
(.3
TA)
2R/2
2q
21
2
2
=
T27
R )22q
2
CA q TR
__
A
2
2
TA
q WATR
TA
TR
/
2
7rTR
TA
showing that the second term in G,, is negligible. Thus while the acoustic part of the ISS
signal may contain significant contributions from both Gprho and GpT, the slow in signal
can be attributed to GpT only. We therefore only consider GpT in further treatment of the
slow rise.
7.2.1
Acoustic Mode and Elastic Modulus
From the dispersion relation,
pos 2 + M(s)q 2 = 0
(7.24)
we can write expressions for the real and imaginary parts of the elastic modulus in terms of
the acoustic frequency WA and damping rate FA,
2
-r2
2
M'(WA) = P
(7.25)
,
(7.26)
2Aq2 A
M"(W A) = p
q
Using these relations, the relaxation spectrum of the elastic modulus, M*(w) = M'(w)
+
=(7-R T2A
iM"(w) can be constructed from the acoustic response in the signal. We can further use
the generalized hydrodynamics model to relate the elastic moduli to the viscosity relaxation
[175]. According to equation 7.13, the acoustic dispersion relation can be written,
+
s[#L(q, s)q 2
+ PH(
-
1)]
2 = 0.
(7.27)
cO
(7.28)
Comparison to equation 7.24 yields,
M(s)
p
=
s[#L(q, s)
+ X(-y - 1)] +
141
with the real and imaginary parts,
M'(WA)
M"(WA)
= -WAIm[#L(q, WA)] + cO,
= WARe[OL(q, WA)]
+ U[lL + X(
P
(7.29)
-
1)]
(7.30)
We can see that when the viscosity relaxation is modeled with Debye relaxation behavior, the
imaginary part of the elastic modulus has a Lorentzian shape with a maximum at WATR
1.
As with the slow rise component of the data, the exponential relaxation function can be
replaced with a KWW function to more accurately replicate the experimental relaxation
behavior.
7.2.2
Slow Rise Mode
The Green's functions GPP and GpT describe the density response of the liquid to applied
thermal or elastic stress. Therefore, it would appear that the slow rise components of G,
and GpT should correspond to the relaxing compliance J(t). However, in ISTS signal, the
likelihood that changes in temperature occur on the same timescale as changes in the density
through a time-dependent specific heat may confound the density response. Indeed, we can
see from equation 7.21 that the amplitude of the slow rise B in ISTS signal is in part dictated
by the magnitude of the specific heat cp,
B
= -
,q 2 T(cA - 4c0)
-4.
2
CP C2q2
C c7q2 T1 + cOc-A
(7.31)
Measurements of the longtitudinal specific heat cl [37] of DC704 at 214 K indicate that the
long-time, low frequency value (co = 1.65 x 106 J (K M 3)) is higher than the short-time,
high-frequency value (c'
= 1.35 x 106 J (K M3 )) by ~ 20% [70]. While we might expect the
exact ratio to differ at temperatures near T,
-
240 K, a difference in cp of this magnitude
would likely have an impact on the slow rise signal.
142
Experimental Methods
7.3
To examine the impact of the specific heat on the slow rise signal in more detail, ISS signal of
DC704 was collected in the temperature and frequency region in which the density relaxation
time r
and
lies between the acoustic attenuation and thermal decay times (i.e.
TRFH
rLRA
> 1
< 1), yielding a slow rise in the signal. The data presented here were originally
collected by Darius Torchinsky, and a more thorough discussion of the experimental methods
can be found in his doctoral thesis [154]. Relevant details for this analysis are presented
here.
An ISS set-up was used as described in section 3.2. ISS excitation was achieved with
a High
Q
FemtoRegen regeneratively amplified laser operating at a central wavelength of
A ~ 1035 nm, a repetition rate of 1 kHz, and an output energy of 400
J0 per pulse. The CW
probe beam was generated with Sanyo DL-8032-001 diode lasing at 830 nm with a maximum
average power of 150 mW. For the short-time acoustic response, the scattered probe light
was detected with a Cummings Electronics Labs Model 3031-0003 amplified photodiode
with a bandwidth of 10 kHz to 3 GHz. The longer-time structural relaxation and thermal
response was recorded with New Focus Model 1801-FS detector with a bandwidth from DC
to 125 MHz. All data were and processed with a Tektronix TDS-7404 digitizing oscilloscope
with a 4 GHz bandwidth.
Prior to taking measurements, the optical phase of the heterodyne beam was adjusted to
maximize the acoustic signal. Data were collected at both positive and negative heterodyne
phase, and the negative phase signal was subtracted from positive phase signal to remove
common-mode electronic noise and improve the signal-to-noise ratio [112]. The pump and
probe polarizations were also adjusted to minimize signal contributions from orientational
flow [61].
DC704 was purchased from Sigma Aldrich and strained through a 0.22 Rm millipore
filter to remove large impurities. The liquid sample was held within a 5 mm fused quartz
cuvette manufactured by Starna Cells and and mounted inside a sample holder provided by
143
Janis to ensure good thermal contact with the cold finger. Sample cooling was achieved with
a Janis ST-100 model cold finger crysostat controlled by a Lakeshore model 331 temperature
controller. The temperature was measured with a factory calibrated Lakeshore model PT102 platinum resistor which was suspended within the liquid sample.
The ISS signal was recorded at grating spacings of 11.5 rim, 24.5 rim, 50
min, and 100
gm and at 2 K temperature intervals between 220 K and 250 K. At each grating spacing,
the acoustic wavevector was calibrated through measurements on ethylene glycol, for which
the acoustic speed of sound is well known. Between each 2 K change in temperature, the
sample was allowed to equilibrate for 30 minutes before measurements were taken. Reported
data consist of averages over 2,000 to 4,000 recorded traces, depending on the signal-to-noise
levels at each temperature and wavevector.
7.4
Data Fitting
In the analysis of the data, we begin by assuming the generalized hydrodynamics model of
ISS signal developed by Yang et al [176] and detailed in section 7.2. the signal intensity in an
ISS experiment can be written as the sum of contributions from electrostrictive excitation
(ISBS signal) and thermal excitation (ISTS signal):
S(q, t) = -FoG pp(q, t) + QoGpT(q, t)
(7.32)
In measurements on DC704, the ISS signal is likely dominated by laser absorption and
subsequent sample heating (ISTS excitation). The third overtone of the benzene's "beating"
vibrational mode lies near the ISS excitation wavelength of A ~ 1035 nm 1119]. In addition,
ISBS signal scales with the excitation wavevector, and these experiments were conducted at
relatively small wavevectors. Therefore, we start with a fit function of
A [e- Ht
-
e-'Al cos
(wAt)]
+ B [e-rHt
144
_
-(t
+ Ce-FAt sin (WAt)
(7-33)
which is identical to equation 7.21 but with an additional term to account for contributions
to the acoustic mode from ISBS excitation. Here, PH is the thermal decay rate,
acoustic decay rate,
WA
PA
is the
is the acoustic oscillation frequency, r is the relaxation time of the
slow rise mode, and 3 is the stretching parameter.
While equation 7.33 reproduces the qualitative aspects of the DC704 ISS data well, a
couple of adjustments were required to achieve quantitative fits. These deviations from the
model can be attributed to less-than-ideal experimental conditions, rather than a fault in the
underlying theory. The alterations to equation 7.33 are described in the next two sections,
and a comparison of the fits to this original function and the adjusted function are shown
in figure 7-1.
In addition, it is important to note that equation 7.33 only applies under the conditions
vLq2 « c0q, PH « c0q, and FHTs < 1, or when the acoustic, slow rise, and thermal responses
are well separated in time. When the timescales overlap, the responses may interact in a
way that is not captured by this model. Therefore, after carrying out initial fits on all data
sets, the acoustic period, slow rise time, and thermal decay time were compared. Data sets
at wavevectors and temperatures where these times occured within one order of magnitude
of each other were discarded.
7.4.1
Errors in the Acoustic Oscillations
The relative amplitudes of the thermal (A + B) and structural components (B) of in the
ISS spectrum can be used to calculate the Debye-Waller Factor, which, according to MCT
predictions, should exhibit a square-root cusp at the critical temperature TC. The most
accurate estimates for A and B can be obtained by fitting the full data set, including the
acoustic oscillations. However, a number of experimental factors may influence the relative
amplitude of the acoustic oscillations, leading to an inaccurate fit of A under this approach.
For the 11.5 [m grating spacing, no single detector had the bandwidth to measure both
the high frequency acoustic oscillations and the slow structural and thermal responses. In
these cases, the full data sets had to be pieced together based on signal collected from two
145
separate detectors. Even taking this step, detector response may drop off at low and high
frequencies.
Paolucci et al.
cite variation in detector response at high frequencies as a
possible source of scatter in measurements of the Debye-Waller factor in glycerol [122].
In addition, the acoustic portion of the signal can have a contribution from ISBS excitation. Acoustic waves generated by ISBS have the same characteristic frequency as those
generated by ISTS but occur out of phase, so it is difficult to tease out the relative contributions of the two. Further, the initial phase of the acoustic oscillations not only depends
on the relative contributions from ISTS and ISBS excitation, but also on the strength of
coupling to the structural relaxation, which varies as a function of the ratio of the real and
imaginary acoustic moduli.
For these reasons, the magnitude of the acoustic oscillations may not provide an accurate
value of A. However, an estimate of the Debye-Waller factor can still be obtained by fitting
the acoustic portion of the signal independently, and then determining the ratio of the
structural relaxation portion of the signal (B) to the thermal decay portion of the signal
(A
+ B). In this approach, the fitting function can be written
(A + B)e--Ht
where D and
7.4.2
#
-
Be(tl/s' 3 + De-rAt cos (wAt +
/)
(7.34)
are the magnitude and phase of the acoustic oscillations, respectively.
Heat Accumulation
The long-time behavior the data collected at grating spacings of 50 gm and 100 pm are not
well fit by the model function. Rather than exhibiting a rise followed by a exponential decay
to zero, the signal appears to decay in a linear fashion and does not fully reach baseline.
This effect is more pronounced at 100 Rm than at 50 ltm.
From the 1-D heat equation for an impulsive, spatially periodic heat source, we find
that the thermal decay rate is proportional to the square of the wavevector, PH = aq2
where a is the thermal diffusivity [861. Therefore, it is possible that at these larger grating
146
spacings, the thermal decay rate has become so long that the thermal grating produced by
one excitation pulse does not have time to fully decay before the next laser pulse arrives,
leading to cumulative heating.
The cumulative heating effect can be eliminated by reducing the repetition rate of the
pump laser, or can be accounted for by adjusting the fitting function. In his doctoral thesis,
Aaron Schmidt lays out a detailed methodology for analyzing the cumulative heating in a
sample from a periodic heat source [136], and here this approach is applied to model the
effect of cumulative heating in ISTS signal.
The temperature response of a sample to a time-varying heat input q(t) can be described
by the convolution:
e(r, t)
q(t) * h(r, t)
=
(7.35)
q(t')h(r, t - t')dt'
=
(7.36)
where h(r, t) is the unit impulse response of the system.
For the ISTS experiment, each input pulse has approximately the same magnitude, and
is separated by time T, which is determined by the rep rate of the laser. Therefore, the
time-varying heat input can be described as:
00
q(t) =
Q6(t - mT - To)
E
(7.37)
M=_00
where T is the time between pulses, To is the time offset between t
first pulse, and
Q
=
0 and the arrival of the
is the energy per pulse. The temperature response of the system,
e(r, t)
is then given by:
e(r, t)
=
Q
/
6(t' - mT - To)h(r, t - t')dt'
147
(7.38)
Applying the convolution property
J
f(x)J(x - xo)dx = f(xo),
o00
(7.39)
the temperature response of the system can be simplified to
1(r, t)
=
Q
h(r, t - mT - To).
(7.40)
m=-oo
If we assume that the thermal decay is described by a single exponential function, h(t)
=
e-at, then the steady-state heating solution will simply be a sum over a single exponential
function at different times, or, equivalently, different amplitudes.
Setting To = 0 and setting h(t) = 0 for t < 0, we can write
00
O(r, t)
Q
=
e-a(t+mT)
(7.41)
m=O
Qe-at
=
Y
e-amT
(7.42)
m=O
The sum can be evaluated using the geometric series
N-1
r
rN
=
(7.43)
1- r
n=O
which gives the solution
N-1
1
e
-a(N+1)T
-
(7.44)
V_________
-amT
11- eaT
m=O
Finally, taking the limit as N -* oc gives
-
1
ea(N+1)T
1-
e-aT
)
1
148
1
-
e-aT
(7.45)
Therefore, the temperature of the system as a function of time is given by:
E)(t) =Q
_e-aT
(7.46)
e -t
We see that for a signal described by a single exponential thermal decay, the cumulative
heating effect increases the overall magnitude of the signal, but does not change its functional
form.
However, the cumulative heating effect can still pose a problem for the analysis of ISTS
data.
If the thermal grating signal from one excitation pulse is non-zero when the next
excitation pulse arrives, the detector will throw away the remaining signal, assuming it is
a DC offset. In this case, the signal appears to reach zero within the detection period, but
only the early part of the thermal decay is represented. In this case, the decay would appear
to be more linear than exponential, which is what we observe in the 50 tm and 100 tm
data.
This issue can be addressed by adding a constant offset to the fitting function, and
adjusting the amplitude of the thermal portion of the signal so that the relative amplitudes
of (A + B) and B remain unchanged. The cumulative heating fit function then becomes
(A + B)
_
HT
Ht
- Be-(tl')' + De--At cos
(WAt
+
4)
+ to
(7.47)
where
to = (A + B) (le
7.4.3
(7.48)
T
Fitting Procedure
Owing to the numerous fitting parameters and the large time range spanned by the data,
fitting was carried out in three steps. First, the slowly-varying components of the data,
comprising the structural relaxation and the thermal decay, were selected and fit for A,
B, FH, and
Ts.
Then, with these parameters fixed at their fit values, the rapidly-varying
components of the data, comprising the acoustic oscillations, were fit for D,
149
rA, WA,
and 0.
10
24.5 gm
10
24.5 tm
E
E
5
5
- -Data
- - Fit
0
1E -3
1E-2
0.1
1
10
100
-- Data
-- - Fit
0
1000
1E-3 1E-2
0.1
1
10
100
1,000
Time (ps)
Time (gs)
20
50pgm
E
0-1
15
15
10
E 10
5
5
0
1E-3
50g
-Data
Fit
-~-
1E-2
0.1
1
10
100
-- Data
0
1000
-Fit
1E-3 1E-2
0.1
Time (gs)
10
1
10
100
1000
Time ( is)
10
100 pm
E
E
5
5
-- Data
-Fit
0
1E-3 1E-2
0.1
1
10
100
-Data
-Fit
0
1000
Time (jis)
1E-3
1E-2
10
0.1
100
10 00
Time (ps)
Figure 7-1: ISS signal in DC704 at 238 K and grating spacings of 24.5 [tm, 50 Im, and 100
rim. Plots on the left show the raw data with the fits to equation 7.33. At 50 Rm, and 100
jim, the slow rise and thermal decay fits begin to break down due to heat accumulation.
Plots on the right show the raw data with fits to equation 7.34, which are consistent with
the data at long times.
150
Finally, the full data set was selected, and all parameters fit simultaneously. At each grating
spacing, data at the lowest temperature were fit first, and the results of these fits were used
as starting parameters for the fits at the next highest temperature.
In all fits, the period between laser pulses was set to T = 1000s (1 kHz laser rep
rate). The value of the stretching parameter was set to 3 = 0.5 for consistency with the the
broadband mechanical study of DC704, which showed that its a-relaxation dynamics were
well fit by a KWW function and displayed time-temperature superposition with a stretching
parameter of / = 0.5 [76].
Figure 7-1 shows the ISS signal in DC704 at grating spacings of 24.5 Im, 50 tm, and 100
tm and at a temperature of 238 K. As expected, increasing the grating spacing decreases
the acoustic frequency and thermal decay rate while leaving the slow rise time unchanged.
The plots on the left show the fits to the data using the original model fitting function,
equation 7.33, and the plots on the right show the fits using the fitting function adjusted for
cumulative heating, equation 7.47. One can see that the original function does a poor job of
replicating the thermal decay signal at 50 jxm and 100 Rm, but the adjusted function is able
to replicate the signal nicely. In addition, the adjusted fitting function is more successful at
capturing the amplitudes of the acoustic oscillations.
As mentioned in the previous section, the thermal decay rate from a spatially periodic
heat source is proportional to the square of the wavevector, FH = aq2 , where a is the
thermal diffusivity. This relation can be used to check the accuracy of the fits using the
cumulative heating model. Figure 7-2 shows a plot of inverse thermal decay rate 1/PH versus
the inverse wavevector squared 1/q 2 for temperatures from 232 K to 244 K. The thermal
decay times all show a linear relationship with 1/q 2 , though there is a large degree of scatter
at 100 Rm. A linear fit to the data at 240 K provides a thermal diffusivity of a = 7.85 x 10-2
mm 2 /s, which is comparable to the thermal diffusivity of engine oil at room temperature,
a = 7.38 x 10-2 mm 2 /s [78].
Aside from the 100 Rm data, the thermal decay times do
not change with temperature, indicating that the thermal diffusivity has little temperature
dependence in this region.
151
350
-CO
0
N
+ 236 K
LI
2500-
* 238 K
240 K
242 K
2000 -
0 244 K
+
3000-
X 232 K
A
- Linear Fit
x
1500-
1000500-
0
50
100
150
200
250
300
1/q2 (gm-2
Figure 7-2: The inverse thermal decay rate 1/FH versus the inverse wavevector squared 1/q 2
for temperatures from 232 K to 244 K in DC704. The 240 K data can be fit by a linear
function, 1/PH = 12.7 x 1/q 2 , predicting a thermal diffusivity of a = 7.85 x 10-2 mm 2 /s.
152
103
0 11.5 gm
0
-O 10
50 gm
A 100 gm
A
A
E
-
24.5 gm
5A
A
_0
A
x
W
10
A
-~~
z
-7
10
,
C/)
10-8
225
230
235
240
Temperature (K)
245
Figure 7-3: The fit structural relaxation times -s as a function of temperature. The values
of r at 24.5 m and 50 -r show close agreement, while those at 50 jim and 100 jim are
consistently lower.
The fit thermal decay time is 800
s at 50 tim and around 3200
the period between laser pulses is only 1000
s at 100 jim, while
s, indicating that these data sets are clearly
affected by cumulative heating. Due to the very limited thermal decay window provided
by the 100 jim data, these data we re-fit with the thermal decay time fixed at the value
predicted by the measured thermal diffusivity,
153
Tth =
1/FH = 3200 jis.
Analysis and Results
7.5
7.5.1
Structural Relaxation Times
Figure 7-3 shows the fit structural relaxation times T, for all measured grating spacings. The
values ofTr at 24.5 pm and 50 jim um show close agreement, while those at 11.5 ptm and 100
tm are consistently lower. The discrepancy in the 11.5 Rm data may be attributed to the
fact that the full data set was pieced together from two separate photodetectors (see section
7.4.1), and so the signal response may have changed over the slow rise region. Likewise, the
100 jim slow rise response may have been affected by the cumulative heating.
To visualize the slow rise relaxation function, the acoustic and thermal relaxation components were subtracted from the overall signal, and the remaining component was scaled
by a factor of -1/B.
These normalized slow rise signals, along with their KWW fits, are
pictured in figure 7-4. The KWW fits show close agreement with the experimental signals.
7.5.2
Debye-Waller Factor
The Debye-Waller factor, or the strength of the alpha relaxation, is given by the integral
under the imaginary part of the relaxation curve or the amplitude of the real part of the
relaxation function. MCT predicts that the value of the Debye-Waller factor should show a
square-root "cusp" at the critical temperature T.
In the generalized hydrodynamics description of ISS signal, the Debye-Waller factor can
be calculated from the relative amplitudes of the structural relaxation and thermal decay
modes
fq(T) =
B
B
=
(A +B)
In the low-frequency limit, WATr
q2
2(C 2-_ c2)
cAq 2
IA
+
k
.
(7.49)
COCA
> 1, the acoustic sound speed equals the infinite frequency
sound speed, CA = cc,, and this expression simplies to
2
fq-+o(T) = 1 -
-.
coo
154
(7.50)
-230
-232
1
%.
.+-
CO
%
C
%
CD)
K
K
K
-234
K
-236
-238
-240
K
K
K
-242
-244
-246
K
K
K
-
1.2
N S0.2-
0
z -o0.2-
sponding
~ ~ ~ ~~~~ ~
10-2
10-1
~
~ ~~~~~~- KWW fits.Teswrsecmoetwssoadbyretgasiutdcre
10 0
10 1
10 2
103
time (ps)
Figure 7-4: The normalized slow rise component of the ISS signals from DC704 and corresponding KWW fits. The slow rise component was isolated by creating a simulated curve
featuring only the acoustic and thermal components of the signal and subtracting this curve
from the full experimental data set. The data at 230 K, 232 K, 234 K, 236 K were taken at
a grating spacing of 50 tm and the data at 238 K, 240 K, 242 K, 244 K, and 246 K were
taken at a grating spacing of 24.5 ptm.
155
1
-
0.9
0.8 -
0
A
0
A1A
A
A
A
U
U- 0.6 --
00.30.20.1224
0 11.5 gm
0 24.5 gm
-50gm
A 100gm
calc
228
232
Figure 7-5: The Debye-Waller factor, or
the signal B/(A + B), as a function of
the Debye-Waller factor calculated from
measured values are significantly higher
236
240
Temperature (K)
244
248
ratio of the structural and thermal components of
temperature. The dotted line shows the value of
the acoustic speeds of sound (equation 7.50). The
than those predicted from the acoustic data, and
there is no evidence of a square-root cusp at T,
156
-
240 K.
Figure 7-5 compares the values of the Debye-Waller factor calculated from the amplitude
ratio B/(A+B) with those calculated from the acoustic sound speeds. In evaluating equation
7.50, fit values for c,,(T) and co(T) between 220 K and 350 K from Klieber et al. [92] were
utilized:
c'(T) = (3840
10)[m/s] - (6.9
0.04) [m/sK] x T[K]
(7.51)
co(T) = (2470
10) [m/s] - (3.52
0.02) [m/sK] x T[K]
(7.52)
The Debye-Waller factor shows no evidence of a square root cusp as predicted by MCT.
In addition, the values are all consistently larger than those predicted from the zero- and
infinite- frequency speeds of sound. In past work, the square root cusp has been observed in
measurements on salol [175] and CKN [178], but was not evident in experiments on glycerol
[122].
7.5.3
Comparison of Structural and Mechanical Relaxation Behavior
A comparison of a-relaxation times from mechanical and slow rise data is shown in figure
7-6. For the slow rise data, R was calculated from
Ts
according to
2
TR =
For the compliance data,
Tj
(7.53)
cTs
was taken from Hecksher et al.[76]. A VFT fit to the mechanical
values for TJ, reported by [76], is also included in the plot. While the scaled relaxation time
data look qualitatively similar, the ratio of rR to rj varies between 2 and 5, with an average
value around 3 (figure 7-7).
Jakobsen et al. [831 present comparisons of characteristic relaxation times of the shear
and adiabatic bulk moduli, the dielectric constant, and the longitudinal specific heat of
DC704 at temperatures from 210 K to 230 K and at frequencies from
-
10-2 Hz to ~ 10 4 Hz.
They found that the relaxation times of these parameters all show a similar temperature
dependence, and that the ratios between the parameters are all within an order of magnitude.
157
10 4
Compliance
---VFT fit
10
0
Slow Rise 11.5 pm-
1 Slow Rise 24.5 jim
0 Slow Rise 50 pm
A
n100
E
-
10 -4
0)
19
e/I
10
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6
4.8
1 000/T (K)
Figure 7-6: Comparison of the elastic compliance relaxation time with the slow rise relaxation time. The values for the compliance relaxation times and the VFT fit are taken from
[761.
158
0.5 0 --
- - - - - - - - - - - - - - - - - - - - - - -
0
-
S
A
4 Slow Rise 24.5 gm
- Slow Rise 50 lm
* Compliance
VFT Fit
~ 5
-1 ---230
240
235
Temperature (K)
245
Figure 7-7: The ratio of the slow rise relaxation times to the compliance relaxation times.
Based on these data, which are for frequency and temperature ranges just below the ones
examined here, it appears likely that the slow rise or Mountain mode signal reflects relaxation
in at least some degrees of freedom that are different from those that contribute to the
mechanical relaxation dynamics. In principle, all the thermally accessible degrees of freedom
that do not change the density, may not contribute significantly to structural relaxation.
.
should contribute to the specific heat while some, such as molecular orientational motions
Early ISS experiments on glycerol and polypropylene glycol by Yan [172, 171] and Duggal
[51], respectively, also indicate a discrepancy between the relaxation time measured by the
slow rise and that determined through acoustic measurements. However, it is important to
note that their analysis was carried out before the generalized hydrodynamic description of
the ISS signal was formulated. In addition, their comparisons were based on extrapolations
from the narrow window of acoustic data accessible by the ISS technique, and, in the case
of Yan, extrapolations of the slow rise that was only partially measured in data and did not
extend nearly to its long-time limiting signal level.
159
7.5.4
Additional slow dynamics in the slow-rise signal
To obtain a rough estimate of the contribution of non-mechanical relaxing degrees of freedom
to the slow rise dynamics, the full experimental data sets were fit to a modified version of
equation 7.47 in which the specific heat term cp in the amplitude of the slow rise mode B was
replaced with a KWW relaxation function with stretching parameter 3, and the value of
r
was set to the value of rJ obtained from a VFT fit to a-relaxation times in the compliance
spectrum,
(A + B)
_
(e-Ht
1-
e-rHT
t/Jr)"
--(c.c2
- C
et/rd#,
)
+
De-FAt cos
(WAt - #) + to.
(7.54)
with
ro exp
T-1
where log To = -15.8,
To
IT - To.
(7.55)
D = 6, and To = 183 K. The choice of the KWW functional form is
based on the limited empirical data provided by Birge et al. [19, 18] in which the specific
heat relaxation spectra of glass-formers glycerol and propylene glycol are best fit with a
KWW function and exhibit time-temperature superposition at frequencies between 10-1
and 104 Hz. A comparison of the shear modulus, bulk modulus, and specific heat relaxation
spectra of DC704 in [83] indicates that they all exhibit the same functional form; therefore,
the stretching exponent 3 was set to a value of 0.5 in concordance with the compliance
spectra. Thus, the only fit parameter for the slow-rise data was the add, TO.
Figure 7-8 shows the fit values of
TC
in comparison to the experimental values of TJ. The
values of TC are consistently larger than rj by approximately an order of magnitude. This
is consistent with the comparisons between the elastic moduli and specific heat relaxation
times reported by [83].
The ratio between rc and rj also appears to increase with temperature, suggesting a
possible decoupling between the degrees of freedom contributing to mechanical relaxation
and others that do not. As discussed in section 2.3.3, MCT predicts couplings between
160
(a)
10-
3
VFT Fit
.+
" Experimental
CC 24.5 lm
10-4
S105
0
-6
E
C50 pm
-
10
C
(1)
10 81
228
230
232
234
236
238
240
244
242
246
248
T (K)
(b) 102
101
C)
100
--------------- --------------------------- ------------
0
VFT Fit
-.
Experimental
rm
C 24.5
+
1021
228
230
1
232
234
236
238
240
242
50 pm
244
246
248
T (K)
Figure 7-8: The fit value for rc (equation 7.54) in comparison to rj
161
the characteristic a-relaxation times of different variables, which may break down below
T. While data comparing the relaxation times of different degrees of freedom near T, are
limited, a decoupling of the relaxation times of the shear and longitudinal moduli of fragile
glass-former triphenylphosphite has been reported at its MCT critical temperature T, = 231
K [156]. However, the data are too preliminary and the fitting model is too crude to make
any conclusive statements about the presence of decoupling.
7.6
Conclusion
The structural relaxation dynamics observed in ISTS slow rise signal from DC704 were
compared with the mechanical relaxation dynamics reported in the broadband compliance
spectrum (section 3.4) [76]. The slow rise dynamics are slower than would be expected from
the relaxing elastic compliance, indicating that additional slowly relaxing degrees of freedom
are coupled to the slow rise response.
Without a rigorous theoretical treatment, we cannot equate the additional relaxation
time to a specific relaxing variable.
However, preliminary estimates of the timescales of
the additional relaxation processes contributing the slow rise signal indicates that they
are comparable in magnitude to experimental measurements of th
in DC704 at lower temperatures and frequencies.
relaxing speciflc hat
Measurements on other glass-forming
systems with different intermolecular bonding degrees of freedom (such as hydrogen-bonded
glycerol) may help elucidate the nature of these contributions.
162
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