Studies of Kinetic Glass Transition in a Triblock
Copolymer Micellar System
by
Wei-Ren Chen
Submitted to the Department of Nuclear Engineerng
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Radiological Sciences
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
July 2004
© Massachusetts Institute of Technology 2004. All rights reserved.
Author..............................................................
Department of Nuclear Engineerng
July 28, 2004
Certified by .......................
Sow-Hsin Chen
Professor
Thesis Supervisor
Read by .........................................
Sidney Yip
Professor
Thesis Reader
Accepted by ...................................................
Jeffrey A. Coderre
Chairman, Department Committee on Graduate Students
2
Studies of Kinetic Glass Transition in a Triblock Copolymer
Micellar System
by
Wei-Ren Chen
Submitted to the Department of Nuclear Engineerng
on July 28, 2004, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Radiological Sciences
Abstract
If a liquid is cooled sufficiently below the melting point, it becomes metastable with
respect to the crystalline state. However, if nucleation is suppressed, one can supercools the liquid without resulting in crystallization. The characteristic time for
structural relaxation increases rapidly and at a certain point, it becomes comparable
to the duration of experiment time scale. At this point, the liquid is being arrested
structurally. We define this structurally arrested matter a glass.
Dynamically, a glass transition can be viewed as a transition from ergodic to
nonergodic state of matters. In a hard-sphere system, mode coupling theory (MCT)
predicts occurrence of a glass transition when the volume fraction exceeds a certain
value due to the excluded volume effect.
However, Recent MCT calculations for a hard-sphere system with a short-range
attraction show that one may observe a new type of structurally arrested state originating from clustering effect, called the "attractive glass", as a result of the attractive
interaction. This is in addition to the well-known glass-forming mechanism due to the
cage effect in the hard sphere system, called the repulsive glass. The calculations also
indicate that, if the range of attraction is sufficiently short compared to the diameter
of the hard sphere, within a certain interval of the volume fraction and the effective
temperature, the two glass-forming mechanisms can compete with each other. For
example, by varying the effective temperature at appropriate volume fractions, one
may observe respectively, the glass-to-liquid-to-glass re-entrance or the glass-to-glass
transitions. Here we present experimental evidence for both transitions, obtained
from small-angle neutron scattering (SANS) and photon correlation spectroscopy
measurements taken from dense L64 copolymer micellar solutions in heavy water.
We show, a sharp transition between the two types of glass can be triggered by varying the temperature in the predicted volume fraction range. In particular, according
to MCT, there is an end point (called A3 singularity) of this glass-to-glass transition
line, beyond which the long-time dynamics of the two glasses become identical. Our
findings confirm this theoretical prediction. Surprisingly, although the Debye-Waller
factors, the long-time limit of the coherent intermediate scattering functions, of these
3
two glasses obtained from PCS measurements indeed become identical at the predicted volume fraction, they exhibit distinctly different intermediate time relaxation.
Furthermore, our SANS results on the local structure obtained from volume fractions
beyond the end point are characterized by the the same features as the repulsive glass
obtained before the end point. A complete phase diagram giving the boundaries of
the structural arrest transitions for L64 micellar system is given.
Furthermore, SANS experiment shows that, in the region of disordered micellar
liquid phase, as the hydrostatic pressure increases, the effective micellar interaction
potential changes in response to the pressure perturbation, resulting in a significant
increase in the low-k part of the SANS intensity distribution, characterizing the formation of fractal clusters. In addition to the formation of fractal clusters, starting
from a structurally arrested state, the applied pressure induces the matrix to evolve
from an initial attractive glass state through an intermediate state and ends up in a
final liquid state. Inspired by the random phase approximation, an additional structure factor term derived from fractal clusters is added to the adhesive hard sphere
structure factor to approximate the variation of the effective interaction potential.
Based on this idea, a model for the scattering intensity distribution is developed
which successfully explains the measured SANS intensity distributions.
One of the great challenges in soft matter sciences is to understanding the structurally arrested state of matter. Although it still remains as an open question, it
is well-acknowledged that the understanding of the interaction potential is of fundamental importance. Due to the fact that colloidal particles can be produced with
well-defined chemical and physical properties such as shape, size and most importantly, the tunable interaction potential. Combining with theoretical predictions,
they serves as convenient model systems to provide key information about the relationship between the rich phase behaviors and various interaction potential, such as
van der Waals forces, short-range attraction et al. The experimental studies of kinetic
glass transition in a micellar system presented in this thesis may provide useful information to further experimental investigations and theoretical calculations on this
rapidly expanding field of research.
Thesis Supervisor: Sow-Hsin Chen
Title: Professor
4
Acknowledgments
Although all the experimental work reported in this thesis is entirely my own responsibility, I would like to thank many people who acted as a source of help and inspiration during this endeavour. In particular, I gratefully acknowledge the constant
and invaluable academic and personal support received from my three supervisors
throughout this research. First of all, my thesis supervisor: Prof. Sow-Hsin Chen:
for his constant support. Without his dedicated supervision, this work would have
been far more difficult to be completed on time. Secondly, I would also like to thank
the members of my thesis committee: Prof. Piero Baglioni, Prof. Alice P. Gast,
Prof. Francesco Mallamace and Prof. Sidney Yip: who greatly enriched my knowledge with their exceptional insights into complex fluids, scattering theory and liquid
theory. Thank you all for making constructive comments, criticisms and suggestions
for improving my research in MIT. Above all, thank you for educating me how to
become a researcher.
I also humbly acknowledge the support received from Dr. Charles J. Glinka and
Mr. Bryan Greenwald of Center for Neutron Research at National Institute of Standard and Technology and Dr. Pappannan Thiyagarajan of Intense Pulsed Neutron
Source at Argonne national Laboratory: for their generosity of allowing me to use
their beam lines. Especially, I must thank Mr. Denis G. Wozniak, the Engineering
Associate of SAND at IPNS, for his substantial and tireless assistance in the numerous
trips I made to IPNS for the past five years.
I am also deeply indebted to the many others who provided help, support and
encouragement in various ways. Here I include my best friends at MIT: Yun Liu,
Xiao Yong and Poe-Jou Chen: who have been a substantial source of inspiration and
profound insight during this research. Dr. Emiliano Fratini and Dr. Antonio Faraone:
who purified the L64 sample for experiment and gave outstanding advice throughout
the years. Those long nights at NIST for experiment and the driving during the
storm of the century will not be forgotten. I am also indebted to the fellow graduate
students: Ciya Liao, Li Liu, Dazhi Liu and Jianlan Wu. I also wish to thank the
5
Institute of Nuclear Energy Research (INER) of Taiwan for the financial support of
my research in MIT. Further gratitude goes to Dr. Chun-Keung Loong of IPNS for
introducing me to Dr. Furusaka of KEK, Japan, who offered me an opportunity to
participate in the J-PARC project. My apologies to those that I have inadvertently
overlooked or forgot to mention in my acknowledgements but who equally deserve.
Lastly, I would like to thank my family for their support.
Above all, I cannot
express my full gratitude to my parents: Chau-Kun Chen and Tsai-Wei Liu for being
so patient, understanding and supportive. I am also greatly indebted to my wife,
Mei-Fang Lin, and my children, Kevin Chen and Katherine Sze-Wen Chen for the
love they have been giving me and enduring the ordeal with me during our live in
Ithaca and Boston.
I dedicate this thesis to my grandfather, An-Ding Liu.
6
Contents
1 Introduction
21
1.1 L64/D 2 0 Triblock Copolymer Micellar System .............
24
1.2
24
Effective Micelle-Micelle interaction
Potential
.............
1.3 Adhesive Hard Sphere System: Experimental Evidence and the Prediction of Mode-Coupling Theory ...................
.
2 The Mode-Coupling Theory of the Liquid-Glass Transition
2.1
Introduction
2.2
Mode-Coupling Theory of the Liquid-Glass Transition
2.3
Validity of Mode-Coupling Theory on Colloidal Systems
. . . . . . . . . . . . . . . .
.
26
29
. . . . . . . ......
29
........
30
.......
33
3 Theory of Small Angle Neutron Scattering and Photon Correlation
Spectroscopy
41
3.1
Basic Cross Section Formula of SANS ..................
41
3.1.1 Calculation of the Intra-Particle Structure Factor P(k) ....
46
3.1.2
48
3.2
Calculation of the Inter-Particle Structure Factor S(k)
Scaling plot of SANS intensity distribution
3.3
SANS Experiment
............................
3.4
PCS Measurement
...............
.
....
.............
55
61
..........
61
4 Temperature Effect on Kinetic Glass Transition in L64/D 20 Micellar
Solutions
63
7
4.1
Intermediate Scattering Function (ISF) Measured by Photon Correlation Spectroscopy (PCS) .........................
64
4.2
Fitting Results of SANS Data ......................
68
4.3
Scaling Plots of SANS Data .......................
73
5 Pressure Effect on Microstructure of Micellar Systems
5.1
Introduction
. . . . . . . . . . . . . . . .
5.2
Pressure Effect on the Microstructure of L64/D 2 0 Micellar System
6 Conclusions
.
85
. . . . . . . ......
85
.
87
97
Bibliography
101
8
List of Figures
1-1 The phase diagram (in temperature-weight fraction plane) of Pluronic
L64 in D2 0 solution. The phase diagram contains the CMC-CMT line,
cloud point line, critical point, percolation line, the equilibrium liquidto-hexagonal liquid crystal phase boundary (dot line), the equilibrium
crystal-to-crystal phase boundary (dash line), and the reentrant kinetic
glass transition KGT lines, separating the liquid phase and two different glasses. H, D and V represent equilibrium hexagonal, lamellar and
cubic phases respectively ..........................
22
2-1 The hard sphere potential (panel A) and the prediction of MCT for a
system of spherical particles interacting via the hard sphere potential
(panel B). ..................................
35
2-2 The adhesive hard sphere (AHS) potential
................
36
2-3 The schematic phase diagram of the AHS system predicted by MCT on
temperature and volume fraction phase plane. The glass lines separate
the liquid regions, where particles are able to diffuse, from the glass
regions. The vertical black dashed line represents the hard-sphere glass
line. In this case, the re-entrant shape of the glass line creates a pocket
of liquid states that are stabilized by the short-range attraction. The
location of the theoretical glass-glass transition line and its terminus
(indicated as A 3 singularity) are also shown ...............
3-1
The general geometry of SANS.
.....................
3-2 The schematic sample structure: divided sample into Np cells. ....
9
38
..
42
44
3-3 The upper panel shows the typical SANS intensity distribution in an
absolute scale and its model fit taking into account the effect of resolution function. Symbols represent the experimental data and the dash
line the theory convoluted with the resolution function. The inset gives
the same data but plotted a in log-log scale. It can be seen that the
Porod's law is satisfied at large k, as is expected in a two-phase system
with a sharp interface. The lower panel gives the normalized intra-
particle structure factor P(k) (circles) and the inter-particle structure
factor S(k) (squares) used to fit the data in the upper panel The S(k) is
calculated by solving the OZ equation with a square-well inter-micellar
potential and the P(k) is calculated using the modified cap-and-gown
model as the polymer segmental distribution function in a micelle. The
observed SANS data is the product of these two functions and therefore it is clear that the interaction peak in the SANS data is primarily
due to the first diffraction peak in the inter-micellar structure factor.
56
3-4 The resolution functions of NG 7 NIST ( = 5 A with A\A/ = 10%,
the distance between sample holder and detector is 3 m.) and SAND
IPNS respectively.
The functions is plotted for k = 0.01, 0.05, 0.1, 0.15
and 0.2 A-l respectively ..........................
57
3-5 The scaling plot of SANS intensity distributions of the micellar solution at
= 0.460 and at different temperatures.
It is seen from the
figure that the system is indeed characterized by a unique temperaturedependent length scale. The fact that the scaling peak is considerably
broader than the resolution function indicates that the system is in
the liquid state. The inset shows the plot of the scaling height of all
the scaled SANS intensity distributions as a function of temperature,
indicating no sign of phase transition
10
...................
60
4-1 Theoretical phase diagram predicted by MCT calculations using the
short-range attractive square well potential with
£
= 0.03. The calcula-
tions predict the attractive glass-to-liquid-to-repulsive glass re-entrant
transition, the attractive glass-to-repulsiveglass transition and the end
point of the glass-glass transition line. Symbols (circles and squares)
represent the effective temperature T*, determined by fitting the experimental SANS data taken in the liquid state with the method explained
in the text.
................................
65
4-2 The ISFs of L64 micellar solutions at different concentrations and temperatures measured by PCS .......................
67
4-3 The theoretical fits to SANS data for Pluronic L64/D 2 0 micellar solutions at various concentrations and temperatures. Symbols are experimental data and lines are the fits. The scattering intensities increase
at the higher temperature liquid phase as compared to lower temperature liquid phase and the positions of the peak shifts toward smaller
k due to the enhanced self-association as the consequence of increased
hydrophobicity of the polymer segments at higher temperatures.
. .
69
4-4 Variation of the aggregation number N as a function of temperature at
different concentrations. It can be seen that the degree of self-assembly
increases as temperature increases at a given concentration. It is consistent with the fact that the PPO core becomes less hydrophilic at
higher temperatures.
We also note that at a given temperature, N
decreases as concentration increases.
11
..................
70
4-5 The ratio of the volume fraction to weight fraction as a function of
polymer weight fraction, obtained from analyses of SANS data. This
is an important calibration curve which enables us to compare the experimentally determined phase diagram with the theoretical phase diagram predicted by mode coupling theory. We note that the hydration
level of the micelle decreases as the polymer concentration increases. It
predicts that at weight fraction 0.54 the micelle will be completely dry,
since density of L64 is close to unity. Denoting the concentration (in
weight fraction) by c and the volume fraction by A, then the empirical
relation is 0/c= 1.96- 1.78c.......................
71
4-6 The effective temperature T* of the liquid state as a function of temperature, obtain by fitting SANS intensity distributions from different concentrations. For the indicated concentrations, T* generally increases as
temperature increases slightly except in the glass region which appears
as a gap in the figure..........................
12
72
4-7 SANS intensity distributions and their scaling plots at q = 0.522,
where the liquid-to-glass-to-liquid transition is observed, at a temperatures range spanning from 288 K to 343 K. The top panel gives the
SANS intensity distributions as a function of k. The broader peaks
(288 K to 303 K, 325 K to 343 K) represent the liquid states and
the narrower ones (306 K to 321 K) represent the glassy state. The
scattering intensities increase at the higher temperature liquid phase
(the left hand side peak, 325 K to 343 K) as compared to lower temperature liquid phase (the right hand side peak, 288 K to 303 K)
and the positions of the peak shifts toward smaller k due to the enhanced self-association (larger aggregation number) as the consequence
of increased hydrophobicity of the polymer segments at higher temperatures. The scaling plots of SANS intensity distributions are given in
the bottom two panels. It is seen clearly that, there are two distinct
degrees of disorder (judging from the width of the peak), which depend
on temperature. While the narrower peak, which is resolution limited,
represents the glassy state, the broader peak, which is much broader
than the resolution, represents the liquid state with a broader distribution of the inter-particle distances. The inset gives the peak height
of the scaling plots as a function of temperature. The variation of the
peak heights indicates the re-entrant liquid-to-attractive glass-to-liquid
transition.
.................................
13
74
4-8 SANS intensity distributions and the scaling plots at 5 = 0.532, at
a temperature range spanning from 290 K to 340 K. The top panel
gives the SANS intensity
distributions
as a function of k.
It shows
the similar features as the top panel of Fig. 4-7: As temperature
increases, a liquid-to-glass-to-liquid transition is observed. However, as
the temperature is increased to 340 K, another disorderedstate peaked
at k = 0.082 A-l is observed. The other two panels show the scaling
plots of the SANS intensity distributions. A temperature dependent
degree of disorder characterizes the system. While the narrowest peak
(340 K) is resolution limited, the slightly broader peak (298 K to 322
K) is also nearly resolution limited, but lower in intensity. From the
inset given in the bottom panel, the liquid-to-glass-to-liquid-to-glass
76
re-entrant transition can be seen clearly. ................
4-9 SANS intensity distributions and the scaling plots at
= 0.536, at
a temperature range spanning from 285 K to 343 K. It exhibits all
the features found in Fig. 4-8. However, the high temperature liquid
state is only observed at 331 K, when the temperature increases to 333
K, the system transforms to the repulsive glass state. Figure 4-8 and
Figure 4-9 together provide convincing evidence of the predicted reentrant glass-to-liquid-to-glass transition. The only difference between
them is that the transition temperature between different amorphous
states are different. The inset shows the re-entrant transition more
clearly.
77
..................................
14
4-10 SANS intensity distributions and the scaling plots at
= 0.538, at a
temperature range spanning from 295 K to 343 K. From the SANS
intensity distributions given in the top panel, it can be seen that the
much broader peaks (liquid state) disappear. Variation of temperature
triggers the transition between the two amorphous solid states with
different degrees of disorder. The variation of the peak heights of the
scaling plots as a function of temperature given in the inset shows a reentrant repulsive glass-to-attractive glass-to-repulsive glass transition
for the first time. .............................
79
4-11 SANS intensity distributions and the scaling plots of SANS intensities
at
X
= 0.544, which is predicted to be the volume fraction where the
A3 singularity point is located, at a temperature range spanning from
285 K to 343 K. From the top panel, it can be seen that all the peaks
of the SANS intensity distributions are located at the same k value
(0.0836 A- 1 ). Furthermore, from the scaling plots given in the bottom
two panels, it can be seen that all the scaled intensities collapse into
one single master curve. It suggests that the two different types of
glass become identical in their local structure at this volume fraction.
The inset given in the bottom panel shows that all the scaling peaks
have an identical height (about 140), indicating the two glasses indeed
have the same degree of local order.
...................
80
4-12 SANS intensity distributions and the scaling plots of SANS intensities
at
= 0.546, a volume fraction beyond the A 3 point, at a temperature
range spanning from 283 K to 343 K. Like Fig. 10, all the scaled intensities are again characterized by a unique length scale and collapse into
one single master curve, independent of temperature, showing identical
local structure of these two glasses. ....................
15
81
4-13 The experimental phase diagram of L64/D 2 0 micellar system. The
solid line represents the equilibrium phase boundary of disorder micellar liquid states and the hexagonal liquid crystalline states. The dash
line gives the kinetic glass transition boundary which is determined by
SANS and PCS. The Symbols represent the phase points where parts
of the experimental data were taken. The triangles represents liquid
state (L), the circles the attractive glass (AG), the squares the repulsive
glass (RG). This figure gives several important pieces of information
about this system: Within the region where the true ground state is
the hexagonal liquid crystalline phase, only the metastable attractive
glass is observed. The repulsive glass is found in the region where the
volume fraction is larger than 0.536. It is interesting to see that there
is a pocket of the attractive glass imbedded in between two separate repulsive glass regions spanning the volume fraction range between 0.536
and 0.544. Furthermore, it is important to note that the two different glasses become identical in their local structures and the long-time
dynamics when the volume fraction exceed (A 3 ) = 0.544. ......
83
4-14 The upper panel gives the scaling plots of SANS intensities at c = 42.5
wt% for the aging experiment (see text for the details), the height of
the scaling peak as function of aging time is given in the bottom panel.
84
5-1 Illustrations of the effect of the applied pressure on the microstructures of a micellar liquid: (a) A typical particle configuration of a
micellar liquid at 0 = 0.4 without applying pressure and its associated
static structure factor. (b) A particle configuration of the mixture of
the liquid phase and the pressure-induced fractal aggregates and its
associated static structure factor. ....................
16
88
5-2 The SANS intensity distribution in an absolute scale (open symbols)
obtained at L64 44.0 wt%, 326 K and 70 bar, and its model fitting
taking into account the resolution correction (solid line). The red dash
line gives the interparticle structure factor SAHs(k) obtained by solving the OZ equation with a square-well inter-micellar potential. The
green dash line represents the structure factor SGEL(k) of the fractal geometric structure.
The blue dash line indicates the normalized
intra-particle structure factor P(k) calculated using the modified capand-gown model as the polymer segmental distribution function in a
micelle. The observed SANS data is the product of these functions (see
text).
91
..................................
5-3 Top Panel: The SANS intensity distribution I(k) in an absolute scale
obtained at L64 44.0 wt%, 326 K are shown as data points for different applied pressures. The lines show the model fittings to the data
that gives the fitting parameters in Table I. The inset gives the same
data but plotted a in semi-log scale. it indicates clearly that, as pressure increases, the low-k part of I(k) increases rapidly, signaling the
formation of fractal clusters. Bottom Panel: The scaling plot of first
interaction peak of the SANS intensity distributions given in the top
panel. It shows that the matrix is characterized by a unique length
scale at various applied pressures. The scaling peak is considerably
broader than the resolution function indicates that the system is in
the liquid state. The inset shows the scaling height of all the scaled
SANS intensity distributions remains unchanged as pressure increases.
17
93
5-4 Top panel gives the SANS intensity distribution I(k) in an absolute
scale obtained at L64 44.0 wt%, 310 K for different applied pressures
and the inset gives the same data but plotted a in semi-log scale. Similar to the Figure 5-3, the formation of the fractal clusters is observed
as applied pressure increases.
However, as well as the formation of
the fractal aggregation, the associated scaling plots (Bottom panel)
indicates the matrix of the colloidal system transits from an attractive
glass state (with a scaling height of 120) [25, 26] through an intermediate state (with a scaling height about 80) then finally to a liquid state
(with a scaling height of 60).
......................
95
5-5 The scaling plots of SANS intensity distribution of 44.0 wt% L64/D 2 0
micellar solution taken at ambient pressure, 70 bar and 280 bar and at
different temperature.
..........................
18
96
List of Tables
5.1
Results for the fitting parameters of the L64 44.0 wt% at 326 K for
three different applied hydrostatic pressures
19
..............
92
20
Chapter
1
Introduction
The material world we are living in is abundant with colloidal systems. Our daily
life is surrounded by stuffs made of colloids, either occurred naturally or prepared
artificially [1]: Examples include inorganic materials such as inks and paints; many
foodstuffs, like milk or yogurt, which provide the proteins required for everyday life;
Even the very stuff we are made of, like cells, blood and bones, also contains colloidal
particles.
Despite the diversity of the colloidal systems, in general they share a unique
feature: unlike atomic or molecular systems, the interaction potential existing among
the colloidal particles is in general short-ranged namely, small compare to the particle
size. Colloids exhibit rich variety of phase behaviors: for example, the existence of
liquid-liquid phase separation, e.g. binodal line with the associated critical point and
the percolation transition line. These phenomena can be traced back to some of the
manifestations of the short-range nature of the interaction potential. Figure 1-1 gives
an example of a phase diagram of such complex liquid: the L64/D 2 0 micellar solution
which is the main theme of this thesis.
Among all the different regions of the phase diagram, probably the existence of
the structural arrest states (glassy states) is the most interesting one. They are
observed in many colloidal systems and their vital importance to our daily life is
reflected from the following list: glue, shampoo, medical oral gel are all examples of
the glassy states. However, despite their ubiquity, their scientific investigation is a
21
ou
I
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..... Equilibrium Liquid-Crystal Boundary
Equilibrium Crystal-Crystal Boundary
Kinetic Glass Transition Boundary
--_o-
70
CriOical Point
60
.
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0 oo0 o ooooo°°
0
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Cloud Point Line
o
50
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RGI
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PercolationoLine
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20
CMC-CMT
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5
10
15
20
25
30
I
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35
40
45
50
55
60
65
L64 Weight Fraction (%)
Figure 1-1: The phase diagram (in temperature-weight fraction plane) of Pluronic
L64 in D 2 0 solution. The phase diagram contains the CMC-CMT line, cloud point
line, critical point, percolation line, the equilibrium liquid-to-hexagonal liquid crystal
phase boundary (dot line), the equilibrium crystal-to-crystal phase boundary (dash
line), and the reentrant kinetic glass transition KGT lines, separating the liquid phase
and two different glasses. H, D and V represent equilibrium hexagonal, lamellar and
cubic phases respectively.
22
recent development. Although the structural arrest states are amorphous states of
soft matter, they are notoriously ill-characterized structurally and dynamically: They
can not be categorized as gaseous, liquid nor crystalline states.
During the last decade, a number of theoretical explanations based on a thermodynamic perspective have been proposed to interpret their physical nature. Examples
include the percolation transition and liquid-to-solid transition. However, compared
with experimental evidence, none of them is completely satisfactory. On the other
hand, from a kinetic viewpoint, if a liquid has been supercooled below the melting
point without resulting in crystallization, the dynamics of its constituent particles
will slow down dramatically and eventually its structure will be frozen. Instead of
transition into a crystalline solid, the true equilibrium ground state, it undergoes a
so-called structural arrest transition or kinetic glass transition (KGT) and turns into
an amorphous glassy state, in which the local particle configuration is deviated from
the thermodynamical equilibrium state and its motion is hindered by presence of the
long-lived cage formed by its neighbors. If one acknowledges its nonergodic nature,
the transition which causes the colloidal system to transform into an amorphous state
can be regarded as a consequence of a kinetic glass transition (KGT). By studying the
liquid-glass transition in a colloidal system interacting via a short-ranged attractive
interaction, recent mode coupling theory (MCT) calculations [2] provide a promising
theoretical tool to draw a fundamental coherent picture of this familiar yet ill-defined
phenomena.
The KGT in supercooled liquids has been studied extensively in the past two
decades [3, 4, 5]. For a class of systems which the interaction potential among the
particles can be modelled as a hard sphere interaction, many valuable insights into the
physics of KGT have resulted from the ideal MCT calculations. The major theoretical
predictions are summarized as follows: At low volume fractions, the behavior of the
hard sphere system is fluid-like. As the volume fraction increases, the test-particle
time correlation function (self - ISF) and the density-density correlation function
(ISF) of the particles exhibit a two-stage relaxation process. The initial decay of the
self - ISF corresponds to rattling of a typical particle confined within a transient
23
cage formed by its neighbors, followed by a slow decay, resulting from relaxation of the
cage itself and the escape of the trapped particle by re-arranging its nearest neighbor
configuration. The latter process leads to a possibility of particle diffusion through
coupling to the structural relaxation process. The system will be dynamically much
slower when its volume fraction is near a certain value, and beyond which the system
will be 'arrested' in an amorphous state, rather than condensing into a crystal.
1.1
L64/D2 0 Triblock Copolymer Micellar System
The main goal in this research is to study the observed kinetic glass transition in
L64/D 2 0 solution, a triblock micellar copolymer system, one member of Pluronic
family, used extensively in industrial applications [6]. After necessary purification
procedure [7]to remove hydrophobic impurities, the polymer is dissolved in deuterated
water (D 2 0).
Pluronic is made of polyethylene oxide (PEO) and polypropylene
oxide (PPO ). The chemical formula of L64 is (PEO)
13 (PPO) 3 0 (PEO)1 3 ,
having
a molecular weight of 2990 Dalton. At low temperatures, both PEO and PPO are
hydrophilic, so that L64 chains readily dissolve in water, and the polymers exist as
unimers. However, as the temperature increases there is a decrease in the probability
of hydrogen-bond formation between water and polymer molecules, PPO tends to
become less hydrophilic faster than PEO. This creates an unbalance of hydrophilicity
between the end-block and the middle-block of the polymer molecule and consequently
the copolymer molecules acquire surfactant properties in the aqueous environment
and self-assembleto form micelles. Thus the micellar formation is initiated at a
well-definedCritical Micellar Temperature - Concentration (CMT - CMC) line.
1.2
Effective Micelle-Micelle interaction Potential
For the L64/D 2 0 micellar system, the excluded volume effect, which characterizes
the strength of the interaction, dominates the effective interaction among micelles. To
a first approximation, micelles can be modeled as hard spheres. Both of the polymer
24
chains and the hydrated water molecules of a micelle contribute to the excluded
volume. The intermicellar distance is comparable with micellar diameter at high
volume fractions.
However, as the temperature further increases, water becomes progressively a poor
solvent to both PPO and PEO chains, and the effective micelle-micelleinteraction
becomes attractive in addition to the excluded volume repulsion. Above the CMC CMT line, the decreasing solubility of D2 0 solvent molecules in PEO corona as
temperature increases leads to the attraction between the micelles. When two micelles
touch each other, some D 2 0 molecules in the corona region are squeezed out and in
the penetrated region, the overlapping between the PEO coronas of two micelles
give rise to the characteristic of short-range intermicelle attraction at the micellar
surface [8]. More specifically, there are two different hydrogen bonds existing in the
system: the hydrogen bonds among D2 0 molecules and the hydrogen bonds between
D 2 0 molecules and L64 polymers. It is clear that the responses of these two hydrogen
bonds to the thermal fluctuation is different and the increasing short-range attraction
can be viewed as the consequence of the interplay between these two hydrogen bond
networks as a function of temperature.
The evidence for the increased short-range
micellar attraction as a function of temperature comes from the existence of a lower
consolute critical point at C = 5.0 wt% and T = 330.9 K [9] and a percolation line
detected by a jump of zero-shear viscosity of more than two orders of magnitude
[10]. All the observed phase behavior shown in Figure 1-1 can be attributed to the
competition between these two hydrogen bonds.
With the presence of the micellar inter-penetration, the effective potential is not
merely hard sphere-like. In order to describe the additional short-range attractive
interaction, we model a single micellar particle as a hard sphere with an adhesive
surface. Based this idea, a adhesive hard sphere potential is used to described the
effective interaction potential of the L64/D 2 0 copolymer micellar system.
25
1.3
Adhesive Hard Sphere System: Experimental
Evidence and the Prediction of Mode-Coupling
Theory
Previously, we reported an observation of a temperature and concentration dependent
structural arrest transition, or a kinetic glass transition (KGT), line, distinct from
a percolation line [11], in L64/D 2 0 copolymer micellar system showing many characteristics of a system having a short-range attractive interaction [10]. We showed
that at a fixed polymer concentration, as we approach KGT from the low temperature side, the time evolution of the photon correlation function (PCS) measured in
a dynamic light scattering experiment, exhibits a fast-slow, relaxation scenario. The
intermediate scattering function (ISF) begins with a short-time Gaussian-like relaxation, soon turning into a logarithmic decay, followed by a plateau region and then
a power-law decay, before evolving into a final, slow, alpha relaxation. The fact that
there is a logarithmic time decay region preceding the plateau, suggests that the system state is in the vicinity of a cusp-like bifurcation singularity of a A 3 type, predicted
by mode coupling theory (MCT) for a system with a hard-core plus a short-range
(square well) attractive potential [2]. By studying the KGT observed at high polymer concentrations by SANS and PCS, we reported the first experimental evidence
of the existence of liquid-to-glass-to-liquid re-entrant and glass-to-glass transitions
(transition between two types of structurally arrested (glassy) states) in L64/D 2 0
copolymer micellar system. Within a certain range of micellar volume fractions, a
sharp transition between these two types of glass is observed by varying the temperature. Furthermore, we found an end point of this transition line beyond which the
two glasses become identical in their local structure and their long-time dynamics.
These experimental results are presented in chapter 4.
Moreover, it is well known that temperature and pressure are two important thermodynamic variables. By changing temperature, the interplay between these two
distinct glass-forming mechanisms gives rise to the possibility of the system under26
going a re-entrant and glass-to-glass transitions.
However, although many valuable
insights into these fascinating phase behaviors due to the variation of temperature
have resulted from our previous experiments, to our best knowledge, the effect of
applied pressure to the KGT in a colloidal system still remains unexplored. Aiming
on solving this puzzle, a series of SANS investigation have been launched recently.
Analysis of the SANS results indicates the interaction potential is altered by the
perturbation of applied pressure. The short range attraction is enhanced and hence
triggers formation of fractal clusters. The detail results are shown in chapter 5.
27
28
Chapter 2
The Mode-Coupling Theory of the
Liquid-Glass Transition
2.1
Introduction
Although the liquid-glass transition resembles a second-order phase transition, with
the liquid transforming continuously into an amorphous solid with no latent heat,
it exhibits no diverging correlation length, symmetry change, or obvious order parameter. It is therefore generally not considered as a conventional thermodynamic
phase transition and is better understood as a dynamical phenomenon, an ergodicnon-ergodic transformation related to a singularity in the underlying dynamics. The
challenge has been to find a theoretical framework capable of predicting such a transformation and of simultaneously providing a detailed description of the relaxation dynamics of liquids and its evolution with decreasing temperature. In 1984, Leutheusser
[12] and Bengtzelius, G6tze and Sjilander [13] showed that a particular version of a
mode-coupling theory of liquids could lead to a dynamical singularity with characteristics resembling those of the liquid-glass transition. Subsequent analysis of this
theory (now usually called MCT) by G6tze and Sj6gren and co-workers [4]led to several detailed predictions for the dynamics of supercooled liquids which have stimulated
much of the recent research in the glass transition field. One notable characteristic of
the new approach is an extension of interest from the very slow structural relaxation
29
close to the glass transition temperature TG, to include higher frequencies and higher
temperatures, bringing into play a number of new experimental approaches.
In this chapter the MCT of the liquid-glass transition will be presented, and then
how MCT leads to some of its specific predictions will be described as well.
2.2
Mode-Coupling Theory of the Liquid-Glass Tran-
sition
First of all, define the normalized density correlation function (also called Density
Correlator)
as:
q(t)
(
Yq exp(- i-.
*())
exp(-i
Al (t)))
(2.1)
where () represents the thermal average, Sq = 1 + 4rp f r 2g(r) sin(kr) dr the static
structure factor, g(r) the pair correlation function. MCT begins with the equation
of motion for the normalized density correlation function bq(t) which can be derived
from the Zwanzig-Mori formalism:
Oq(t)+ Qnq (t) +
f
Mq(t - t')q(t')dt'
=0
(2.2)
The memory kernel Mq(t), namely the correlation function of the fluctuating force
Fq(t), is then separated into a 'fast' (regular) component ?q6 (t), a contribution from
the collisions among individual particles, and a time-dependent part Q mq (t).(The
frequency Q2 = v2q 2 /Sq is the normalized second moment of dynamic structure factor
S(q, w), v is the thermal velocity, and Sq denotes the static structure factor.) Equation
(2.2) then becomes
t')q(t')dt' = 0
Oq(t)+ Qq2q(t)+ q() (tt30
(2.3)
The third term of the equation (2.3) is called the Enskog approximation, with yq being the collision frequency of the uncorrelated binary collisions. The time-dependent
decay is treated by coupling different hydrodynamic mode such as density, current and temperature fluctuations. Since the fluctuating force occurs primarily between pairs of particles, the dominant contribution to Fq(t), the Fourier transform
of Fq(r12)6P(r1, t)Sp(r 2, t), can be approximated as a sum of density-fluctuation pairs
Pql(t)pq2 (t) with q +
q2
= q. With a factorization approximation applied to the re-
sulting four-point correlators, one obtains for the leading-order contribution to mq(t):
mq(t)
mq(t) = 22- J
dqld3q2
(2)6 (q V q , , q2)ql(t)0q2(t)5(q
+ q +2)
(2.4)
(2.4)
From the kinetic theory of dense atomic fluids, the two-mode vertices (or coupling
constants V(q, q,
q42))
are completely determined by the static structure factors Sq
via
V(q,q1, q2) = q4 Sqqlq2 (
1Cql + V.t2Cq2
where Cq is the direct correlation function, related to Sq by Sq = l
(2.5)
, p is the
number density. It is important to note that equation (2.5) for the coupling constants
V contains only the static structure factors Sq and not the intermolecular potentials.
This is a crucial simplification since for some potentials (e.g. hard spheres) the
potential is singular, but the structure factors are nevertheless well behaved. The
origin of this simplification lies in treating the fluctuating force not as the gradient
of the intermolecular potential (which may be undefined), but instead, via Newton's
equation, as the time derivative of the current.
Equations (2.3), (2.4), and (2.5) constitute the basic (or idealized) version of MCT,
given as a closed set of equations. If the intermolecular potential is known, the structure factor Sq can be computed with well-established approximation methods (e.g.
via the Ornstein-Zernike Equation with the Percus-Yevick approximation) and used
to evaluate the vertices V(q, ql, q2). The equations are then solved self-consistently for
a discrete set of wavevectors. One essential aspect of these equations is, in equation
(2.2), rather than substituting an empirical function with adjustable parameters for
31
mq(t), a formal procedure is given by equations (2.4) and (2.5) for its computation
with no free parameters.
It was shown by Leutheusser [12], and Bengtzelius, G6tze and Sj6lander [13]
respectively that the memory function contribution mq(t) leads to an ergodic-tononergodic transition when the control parameters, such as density and temperature,
reach at a critical values. At which points the diffusivity vanishes and the shear
viscosity diverges.
The t -- oo limit of
Oq(t)
is called the non-ergodicity parameter fq. It can be
shown that the equation for the density correlator reduces to a static equation, i.e.
the bifurcation equation,
fq
1
1 -1 f
2-
dk
(2r) V(q, ql, q2)fqlfq2
(2.6)
At high temperatures, where the coupling constants V(q, q, q2 ) in equation (2.4) are
small, decays to zero rapidly following the initial microscopic transient. The system
is ergodic, and fq = 0. With decreasing temperature, the coupling constants increase
smoothly, until the glass transition singularity is reached where Oq(t) no longer decays
to zero, even for infinite time. The density fluctuations are then partially frozen in,
since Oq(t) only decays from 1 to fq with 0 < fq
<
1. The non-ergodicity parameter
fq is called the Debye-Waller factor.
Smooth variation of the coupling constants with decreasing temperature (or increasing density) leads to a critical temperature T, at which fq changes discontinuously from zero to a non-zero critical value fqC.Solving the MCT equations for t -- oc
and T - T,, one finds that, for T < Tc (or equivalently 0 > c),
fq(t) = fq + hvwhere
oc (T, - T) /Tc is called the separation parameter.
(2.7)
This square-root cusp
in fq(t) is the first general prediction of MCT. Since, at T = T, Oq(t
oo) jumps
discontinuously from 0 to fq, the dynamical singularity at T constitutes a bifurcation.
Because there are many control parameters in the theory (all of the Vq), many types
32
of bifurcation are possible.
Another important prediction of mode-coupling theory concerns the relaxation
dynamics of supercooled liquids at the temperature range T > T,. As the temperature increases (or volume fraction increases), the q(t) exhibit a two-stage relaxation
process, which reflects the growing strength of the 'cage effect', i.e. temporary localization of a particle in the transient cage formed by its neighbors. The initial decay,
corresponding to rattling of a typical particle confined within a transient cage formed
by its neighbors, relaxes towards a "plateau", which is followed by a slow decay, resulting from relaxation of the cage itself and the escape of the trapped particle by
re-arranging its nearest neighbor configuration. The latter process leads to a possibility of particle diffusion through coupling to the structural relaxation process. The
system will be dynamically much slower when its volume fraction is near a certain
value, and beyond which the system will be 'arrested' in an amorphous state, rather
than condensing into a crystal.
2.3
Validity of Mode-Coupling Theory on Colloidal
Systems
The experimental results show that the application of MCT to colloidal systems
is highly successful. In 1986, Pusey and van Megen showed that a suspension of
spherical particles has an equilibrium freezing-melting transition which is consistent
with that expected for a system interacting via the ideal hard-sphere potential, as
given in Figure 2-1A. For a spherical colloidal system, at volume fractions of less
than 0.49, only the fluid is found. Between 0.49 and 0.55 the equilibrium state is
two-phase coexistence of a fluid and an FCC crystal.
However, if one is able to
manipulate a colloidal system to avoid crystallization at the volume fraction of 0.49,
for example by artificially creating a polydispersity in sizes of a few percent, at a
critical volume fraction 0, which was predicted to be 0.516 (but determined to be
0.58 experimentally)
(see Figure 2-1B) [13, 14, 15, 16], the ergodic-to-non-ergodic
33
transition (or kinetic glass transition, KGT) can be observed [17]. At the KGT, both
the particle diffusion and the long-time density fluctuations freeze and the system
undergoes an ergodic-to-nonergodic transition. Furthermore, the predicted two-step
relaxation of dynamics is also observed in different colloidal systems. Subsequent
studies of colloidal glass have yielded quantitative agreement between the experiments
and the MCT predictions. However, in spite of these characteristic features confirmed
by laboratory experiments [17, 18] involving several model hard-sphere systems, there
have been some anomalous dynamical observations, which cannot be interpreted in
terms of the theory based on the hard sphere potential alone [19].
It is natural to ask, can a more complete picture of the KGT be obtained by
modeling the interparticle potential more accurately? The answer stems from a slight
modification of the hard sphere interaction potential and is attracting a great deal
of attention
lately.
Recent MCT calculations
[2, 20, 21] show that if a system is
characterized by a hard core plus an additional short-range attractive interaction,
for example, by an adhesive hard sphere system (AHS) (see Figure 2-2), a different
structural arrest scenario emerges. Theoretically, the phase behavior of the AHS
is characterized by an effective temperature T* = kBT/u, the volume fraction of
the particles
and the fractional attractive well width
= A/R, where -u is the
depth of the attractive square well, A the width of the well and R the diameter of the
particle. In this case, for a given a, aside from the volume fraction q, a second external
control parameter, the effective temperature T*, is introduced into the description of
the phase behavior of the system and the loss of ergodicity can take place either by
increasing the volume fraction or by changing the effective temperature.
The predicted phase diagram based on the AHS is given schematically in Figure
2-3. As one can see, it is possible for the system to undergo a re-entrant (glassto-liquid-to-glass) transition by varying the effective temperature.
At high effective
temperatures and at sufficiently high volume fractions, the system evolves into the
well-known structurally arrested (glassy) state, called a "repulsive glass", as a result
of the cage effect, a manifestation of the excluded volume effect due to the existence of
the hard core. However, at relatively low effective temperatures, an "attractive glass"
34
A K,
A
Hard Sphere Potential
V(r)
b
.
r
R
A
B
Liquid
00
0
0 00
0
0
000
00
Repulsive Glass
0o 00
/0.516
h
r
f-
Volume Fraction +
Figure 2-1: The hard sphere potential (panel A) and the prediction of MCT for a
system of spherical particles interacting via the hard sphere potential (panel B).
35
Square-Well Short Range
Attractive Potential
V(r)
r
R+A
Figure 2-2: The adhesive hard sphere (AHS) potential.
36
can form in which motion of the typical particle is constrained in stead by the cluster
formation with neighboring particles.
With this insight, we may divide spherical
colloidal systems into two categories: the one-length-scale hard- sphere system, in
which the glass formation is dictated by the cage effect; and the two-length-scale
AHS, in which the two glass-forming mechanisms may coexist and compete with each
other. Furthermore, MCT shows that in an AHS with sufficiently small ratios of
the range of the attractive interaction to the hard-core diameter, variation of the
control parameters allows the transition between these two distinct forms of glass.
Moreover, in an AHS, aside from the hard-core diameter, the range of the attractive
well, should come into play (parameter
). The MCT calculations show that, with
sufficiently small , variation of the two control parameters, T* and
, allows the
transition between these two distinct forms of glass. Thus there is a branch of the
KGT line across which transitions between the attractive glass and the repulsive glass
are predicted. Of particular interest is the occurrence of an A 3 singularity at which
point the glass-to-glass transition line terminates. MCT suggests that the long-time
dynamics of the two distinct structurally arrested states become identical at and
beyond this point.
Several ongoing experimental investigations have confirmed some of the theoretical predictions, such as the re-entrant glass-to-liquid-to-glass transition phenomenon
[22, 23, 24] and the logarithmic relaxation of the glassy dynamics in the liquid states
in the vicinity of the A 3 singularity [11], which considered to be the signatures of
the glassy dynamics in the two-length-scale system. More recently, a series of experimental investigations focusing on a dense micellar system first report the glassto-glass transition line and its associated end point and the detailed analysis leads
to the physical insight into the slow dynamics in the two glassy states [25, 26, 27].
The quantitative agreement between experiment results and MCT calculations for an
AHS justifies that the L64/D 2 0 micellar systems can be treated with a framework
of glass transition satisfactorily.
In the following chapters, we first present detail SANS intensity distribution analysis and the SANS scaling plots, experimental methodology of PCS measurements,
37
T*=kBT/u
Volume Fraction
+
Figure 2-3: The schematic phase diagram of the AHS system predicted by MCT on
temperature and volume fraction phase plane. The glass lines separate the liquid
regions, where particles are able to diffuse, from the glass regions. The vertical black
dashed line represents the hard-sphere glass line. In this case, the re-entrant shape of
the glass line creates a pocket of liquid states that are stabilized by the short-range
attraction. The location of the theoretical glass-glass transition line and its terminus
(indicated as A 3 singularity) are also shown.
38
then discuss experimental results of the KGT of the micellar system, as a function of
temperature and pressure, at different volume fractions, and report the first observation of the glass-glass transition and its end point in a colloidal system.
39
40
Chapter 3
Theory of Small Angle Neutron
Scattering and Photon Correlation
Spectroscopy
In general, a small angle neutron scattering (SANS) experiment (given schematically
in Figure 3-1) a neutron beam with intensity Io (number of neutrons per unit area
per second) incidents on a sample cell of volume V containing N particles in solution,
is scattered in a small cone around the forward direction. The scattering intensity Is
is measured at distance r, an angle 0 with a detector of area r2 dQ. What measured
in the experiment is the ratio
lr 2 dQ
du(k)
(3.1)
which with a dimension of area and therefore is called differential cross section, a
essential quantity in a SANS experiment. The following section gives a brief introduction of its importance.
3.1
Basic Cross Section Formula of SANS
Define the differential scattering cross section per unit volume as
41
Detector
Element
Area
r2dQ
Sample
Neutron Beam X,
Figure 3-1: The general geometry of SANS.
dE
d k)
1 du
V dQ(k)
IK b exp(ik r)
(3.2)
where () represents the average of all the accessible equilibrium particle configurations, bl is the bound scattering length of the Ith nucleus in the sample, rl the
position of the Ith nucleus.
Equation (3.2) can be viewed as a consequence of the interference of different
scatterers contained in the sample and can be separated into coherent and incoherent
parts, namely
dZ
d(k)
_dECoh
_dEin
dQ (k)+ dQ3.3)
(33)
where
d
(k) = I(k) = V
42
b
° h exp(ik
(34)
dEinc
1 N (inc)
(bi
dQ
- V
in c
9nc
n 4 - nH4w
(3.5)
bcoh= b7
(3.6)
and
inc _=(
b-2)1/2
(37)
The average is taken over different spin and isotopic states for each species. In coherent scattering there is a strong interference between scattered waves from different
scatterer, therefore, it tells us something about the correlations between the positions
of different nuclei, i.e about the structure of a material and how it evolves with time.
On the other hand, if there is no change in neutron energy at scattering, incoherent scattering is completely isotropic, namely uniform in all directions, and contains
no information regarding the interference. Therefore, the information of structure
only contained in the coherent scattering and therefore the incoherent scattering is
generally treated as background and removed before any further analysis.
In a two phases system such as the L64/D 2 0 micellar solution, the constitute
particles (micelles) can be distinguished from the continuous solvent (D 2 0) clearly.
Thus, they can be considered as the scattering centers. If the sample can be divided
into Np cells which center at different micelle particles (see Figure 3-2). The position
of the jth nucleus in the ith cell can be represented as ri = Ri +
. Therefore,
equation (3.4) can be rewritten as
) V
exp(Z
j) E' bexpoi
j):cell
(3.8)
define a form factor of each cell by
F/(k) =
5 b}exp(icell i
43
·j)
(3.9)
Figure 3-2: The schematic sample structure: divided sample into Np cells.
44
In terms of Fi(k), I(k) can be expressed as
I(k) = V
EZ
i=l i'=1
(iei-X)
IF
(k) F(k)) exp [i T~
(3.10)
by introducing the local scattering length density (sld) of ith particle defined as
Pi(T) =
Z
b(
(3.11)
-
i
we can rewrite equation (3.9) as
Fi(k) =
Fd3r
exp(it -T)pi(-)
=
particle d 3r exp(iT
) [pi(t)
particle
- Ps]+
ell id 3 r exp(i'
T)p,( 3 .12 )
where p, is the scattering length density of the solvent.
For a monodisperse spherical system, the form factor is identical for each particle
and therefore equation (3.10) can be further expressed as
1(k)
=
Np IF(k)l2 I
V ()
=
N
p
E exp [i
= npP(k)S(k)
wherenp=
N
VP
(3.13)
is the number density of the particles, P(k) = F(k) 12 the intra-particle
structure factor and S(k) = N (Pl
N1
exp [i
(
-
]) the inter-particle
structure factor. Equation (3.13) indicates that the differential cross section per unit
volume is proportional to the number density of the particles and a product of the
intra-particle structure factor P(k) and the inter-particle structure factor S(k).
It can be shown that, for a system of mono-dispersed micelles in D2 0 solvent, the
absolute intensity I(k) (in unit of cm -1 ) can be express by the following formula:
I(k) = cN [
bIp..J2
bi- pvp] P(k)S(k)
(3.14)
where c is the concentration of polymer (number of polymers/cm 3 ), N the aggregation
45
number of polymers in a micelle, Yi bi sum of coherent scattering lengths of atoms
comprising a polymer molecule, Pw the scattering length density of D 2 0,
VP
the
molecular volume of the polymer, P(k) the normalized intra-particle structure factor
and S(k) the inter-micellar structure factor.
3.1.1
Calculation of the Intra-Particle Structure Factor P(k)
We assume the micelle has a compact spherical hydrophobic core of a radius a, consisting of all the PPO segments with a polymer volume fraction in the core qp = 1
(namely a dry core), and a diffuse corona region consisting of PEO segments and the
solvent molecules. The original cap-and gown model assumes that the radial distribution profile in the corona region is a Gaussian form. However, our analysis of SANS
data at high concentrations shows that we need a more diffuse corona region than
previously assumed. In this paper, we therefore take the radial profile of the polymer
in the corona region to be a power-law decay with the power n to be determined by
geometrical constraints. The overall polymer segmental distribution in a micelle can
thus be expressed as
qp(r)=
1
for O<r<a
(a)n
for r >a
(3.15)
The core radius a and the power n are related by two geometrical constraints.
The total volume of polymer segments in the core is given by the product of the
aggregation number N and the PPO segmental volume vppo
$r7a3= NvPO
where v
O
(3.16)
= 95.4 x 30A3 .
Similarly, the total volume of polymer segments outside the core is given by the
product of the aggregation number N and the volume integral of the PEO segmental
distribution
47r ( (a)nr2dr = NvPEO
46
(3.17)
where vEO = 72.4 x 26 A3 .
Combining these two constraints, we can write
vppo
VppE0
3 fa (a)nr2dr
3
a
VPEO
r
3
- ___
n-3
=
0.658
(3.18)
where 0.658 is the ratio of the known molecular volumes of PPO and PEO segments. From this equation, we obtained a unique value of n = 7.56.
The particle form factor is then calculated by the following formula
F(k)
=J
T)
d3r(pp - p) exp(i'
4(PPw
)
l(a)
+
(3.19)
sin(kax)x
- 6 .56
dx]
where jl(k) is the spherical Bessel function of order one.
It is convenient to define a normalized form factor by dividing F(k) by its value
at k=0
F(0) =
(pp _ p)(VCore + Vcorona)
47a3
=
(Pp - Pw)
3
(1
(3.20)
VPEO )
Vppo
Therefore, the normalized form factor is
F(k) _ F(k)
(3.21)
F(0)
3PPO
=PPO +VPEO
[jil(ka)+ - sin(kax)x-6.56d
ka
11
ka
which depends on only one parameter, the core radius a. Note that a is tied to the
aggregation number N uniquely through 3.16. The normalized intra-particle structure
factor is then calculated as P(k) = F(k) 2.
47
3.1.2
Calculation of the Inter-Particle Structure Factor S(k)
The intra-particle structure factor S(k) of an fluid system is uniquely determined by
the inter-particle interaction potential.
The model for the inter-micellar structure
factor S(k): A square well potential with a repulsive core of diameter R' and the real
particle diameter R is used to model a hard sphere with an adhesive surface layer
which is introduced to describe the inter-micellar attractive interaction. The pairwise
potential
is defined as [28]
+oo
=ln(12Tr) for R' <r < R
kBT
We define here E = Rthe effective temperature.
for O < r < R'
0
(3.22)
for r>R
as the fractional well width parameter and T* = kBT/u as
Then the inter-micellar structure factor is a function of
four parameters: the real particle diameter R, the volume fraction 0, the fractional
well width parameter E and the effective temperature T*. It should be noted that real
particle diameter R is tied uniquely to the aggregation number N and the volume
fraction X through a relation 0 =
·
The Ornstein-Zernike (OZ) equation can be expressed in terms of Fourier transforms as
h(k) = c(k) + pc(k)h(k)
(3.23)
where h(k) and h(k) are the 3D Fourier transforms of the total correlation function
h(r) = g(r) - 1 and the direct correlation function c(r) respectively, i.e.
h(k) =
dT exp(iT
c(k) = dT exp(it ·
T)h(r)
(3.24)
)c(r)
(3.25)
dr cos(kr)H(r)
(3.26)
It can be shown by integrating by parts that
h(k) = 47
/0
48
where
H(r) = j
dtth(t)
(3.27)
and similarly,
c(k) = 4 j
dr cos(kr)C(r)
(3.28)
where
C(r) = j
dttc(t)
(3.29)
The direct correlation functions depend on the form of the interaction potential,
for a given p, T and 0q(r),the OZ equation can be solved under certain approximations
for h(r), g(r) and c(r). One of them is the Percus-Yevick (PY) approximation: it
assumes that
c(,r)=
-exp
[kBT
g(r)
(3.30)
For the adhesive hard sphere potential given in equation (3.22), which vanishes beyond
R, PY approximation predicts
c(r) = 0,
r>R
(3.31)
Then from equations (3.28) and (3.29)
1 - pc(k) = 1 - 47p
dr cos(kr)C(r)
(3.32)
Since there is no long-range order in a fluid system, equations (3.24) must converge
for all real value of k, so that h(k) is finite. Furthermore,
from equations
(3.23) one
concludes
S(k) = 1 + ph(k) = [1- pc(k)]-1
(3.33)
Therefore, the right-hand side of equation (3.32) cannot be zero for any real k. In
this case, Baxter [29] shows that equation (3.32) can be factorized into the form
1 - pE(k) = 1 - 47rp
cos(k)C()
(-k)
drcos(kr)C(r) = Q(k)Q(-k)
49
(3.34)
(3.34)
where
= 1
R
2o
Q(k) -I - 2p J
P(ik)Q()
dr exp(ikr)Q(r)
(3.35)
Q(r) is a real function. Substituting equation (3.35) into equation (3.34) gives
C(r)= Q(r)- 2pJ
R
drQ(t)Q(t- r)
(3.36)
for 0 < r < R. Also from equations (3.32) and (3.33), one gets
Q(k) [ + ph(k)] = [Q(-k)] -
1
(3.37)
Substituting the forms of equations (3.35) and (3.26) of Q(k) and h(k) into the lefthand side of equation (3.37) and performing the integration, it follows that for r > 0
-Q(r)+H(r)- 2p
dtQ(t)H( Ir- t) = 0
(3.38)
By differentiating equations (3.36) and (3.38), they can be expressed in terms of c(r)
and h(r), it is found that
rc(r) = -Q'(r) + 27p
R
dtQ'(t)Q(t - r)
(3.39)
dt(r - t)h(Ir - tl)Q(t)
(3.40)
for 0 < r < R, and
rh(r) = -Q'(r) +2p j
for r > 0.
For the adhesive hard sphere potential given in equation (3.22), the OZ equation
is solved under the following conditions:
h(r) = -1
= -1+
for 0 < r < R'
X
(
r
6(r
- R) for R' < r < R
12
50
(3.41)
and
c(r) = 0 for r > R
(3.42)
where is A a dimensionless parameter to be determined. Substituting equation (3.41)
into equation (3.40) gives
Q'(r)= ar + R for 0 < r < R, where
a = 1 - 27rp>
AR 2
12
6(r- R)
(3.43)
Rdtt)
0o
(3.44)
dtQ(t)
and
(3.45)
27rpRttt)
/13= Rifo dttQ(t)
Integrating equation (3.43) with the boundary condition of Q(R) = 0 gives
Q(r) =
a
2 2 - R22) +R(r
(r -R ) +R(
2
R) +AR 2
- R)
12
12
(3.46)
Combining with equations (3.44) and (3.45), a and l}therefore can be solved respectively as
a =(1-2 - )
(1 -
)2
(3.47)
and
= AX(1
- )
(3.48)
Furthermore, the Mayer f-function is defined as
f(r) = exp [-u(r)] - 1
(3.49)
In terms of the Mayer f-function, the PY approximation can be rewritten as
1 + h(r) = [1+ f(r)] p(r)
51
(3.50)
where
p(r) = + h(r) - c(r)
(3.51)
From equations (3.41) and (3.49), one found that R' -* R, p(r) is continuous at
r = R.
f(r) = -1+ 12 6(r-R),
O<r<R
(3.52)
r>R
-,
from the OZ equation, it can be seen that for R' < r < R, the PY approximation is
p(R) = AT
(3.53)
From (3.51), p(R) can be obtained by adding r to both sides of equation (3.40)
followed by subtracting equation (3.39) and setting r = R, i.e.
Rp(R) = R + 2rp
dt(R - t)h(R - t)Q(t)
(3.54)
For R' < r < R, from equations (3.41), (3.46) and (3.47), and noting that Q(r) is
continuous at r = 0, it follows that
p(R)=a+
+p+R2 Q()
(3.55)
Therefore, the parameter A can be determined by solving the following equation
p(R) = AT
1+/2 1 --1A2
(1- ) 2
1- 0
12
(3.56)
and it turns out that the physical root is the smaller one.
Thus, for a given volume fraction 0 = c,3
obtained:
(1 + 2¢ -
-1)2
(1- )4
52
the following parameters can be
3(2 + )2 - 2(1 7 + 2 ) +[2 (2+ )
/
2(1 -
/U =
Aq(1-
)
6(A -
A-2 F;)
)4
q
A =
+(I -
r
1
q
12e
)
exp(-
kBT
)+
(1 -
)
(1 + /2)
3(1 - )2
Define a dimensionless quantity Q = kR, the structure factor S(Q) can be obtained directly from c(Q) by
S(Q)
=
1 + 47pc(Q)
(3.57)
where c(Q) is obtained by performing the Fourier transform of the direct correla-
tion function c(r).
The structure factor is given explicitly as follows:
1
S(Q)
- 1 = 24$ [af2 (Q) +/3f 3 (Q) + I2¢af5(Q)] +
f3(EQ)]+±
4q2A2ze2 [f2(sQ) 4IA
- 2f(Q)]
2q 2A2 [fl(Q)- _2fl(EQ)]
-
2q£ [fl(Q) (1- )2 f((1 -E)Q)] 24X [f2(Q) - (1 -
E) 3 f2((1 -E)Q)]
(3.58)
where
fi(x)
f2 (x)
f3 (x)
1 - cos x
(3.59)
x2
sin x - x cos x
(3.60)
x3
2xsinx-
(x 2 - 2)cosx- 2
53
(3.61)
f(x ) :(4x
3-
24x)sinx- (x4 - 12x2 + 24)cosx+ 24
(3.62)
X6
It should be noted that the real volume of a micelle is significantly larger than the
volume of N dry polymer chains. It consists of the volume of the N dry polymer
chains plus the volumes of the associated hydration water per polymer chain in the
corona region. The real volume of the micelle, rather than the polymer volume, determines the thermodynamic quantities such as the osmotic compressibility and the
scattering intensities. It further determines the dynamical properties, such as diffusivity and viscosity, of the polymeric micellar solution [30]. Obtaining the real volume
of a micelle in a self consistent way is not a trivial matter. It requires the knowledge
of the detailed microstructure such as hydration number per polymer chain. This
information on the associated solvent molecules is not available from the theoretical
predictions and can be extracted accurately from the analysis of the scattering in-
tensity distributions through the volume fraction 0 and the aggregation number N.
It is important to note that an absolute SANS intensity distribution can be fitted
uniquely with four parameters: the aggregation number N, the volume fraction
,
the fractional well width parameter E and the effective temperature T*. The normalized intra-particle structure factor P(k) is the function of N only. The inter-micellar
structure factor S(k) is the function of all the four parameters.
An example of SANS intensity distribution, shown an absolute scale together with
its model fitting, for a 48.5 wt.% micellar solution at 326 K is given in the upper
panel Figure 3-3 to illustrate how to to analyze SANS intensity distributions by using
the model described above. Since we were dealing with polymeric micellar solutions
having very sharp scattering peaks in this experiment, it is essential that we apply
the resolution broadening to the theoretical cross-section when fitting the intensity
data. This was accomplished by convolving the theoretical scattering intensity obtained from the model described previously with a Gaussian-approximation resolution
function:
Imease(k) =
I(t)R(k, t)dt
54
(3.63)
where
R(k, t)= -V/
is the Gaussian-approximation
exp
[ (k-t)
2
(3.64)
resolution function determined by
R, the width of
the resolution function which is determined by the neutron wavelength spread, collimation effect and detector resolution [31]. Figure 3-3 gives the resolution functions of
NG7 NIST (solid lines,wave length A = 5 A with AA/A = 10%, the distance between
sample holder and detector is 3 m.) and the SAND IPNS. The lower panel gives the
normalized intra-particle structure factor P(k) (dotted line) and the inter-particle
structure factor S(k) (solid line) for this case. The observed SANS data is proportional to the product of these two functions. The fact that the first diffraction peak of
S(k) occurs at relatively smooth tail part of P(k) implies that the interaction peak
in the SANS intensity distribution is primarily due to the first diffraction peak in the
inter-micellarstructure factor S(k).
3.2
Scaling plot of SANS intensity distribution
At high enough polymer concentration, SANS intensity distribution from the L64/D 2 0
micellar system generally consists of a single, sharp interaction peak. Since there is
a single peak in the observed SANS intensity distribution, we can assume that the
system is characterized by a single length scale A =
1
kmax
where
kma,,
is the peak
position of the intensity distribution. It is well known that the absolute intensity in
a two-phase system (the micelles and the solvent) is given by a 3D Fourier transform
of the Debye correlation function F(r) and the Debye correlation function in this case
must be of the form r (A) (scaling). Therefore
I(k) =<
where <
12 >
2> X
2j(kr)r (A)
dr47rr
is the so-called invariant.
By making a transformation of variables
55
(3.65)
I
/U I
I
I
I
I
I
C = 48.5 wt%
60- _
T=333
Fitting
e~
--
K
I
=0.532 11
T = 1.48
-
+
50
50
E
I
-Y 40
I
I
.3V
U
i
I
;JU
I
0.01
I
~
I
20
10
0.1
(A)
/
,
I,
10
0.00 0.02
0.04 0.06
0.08 0.10 0.12
0.14 0.16 0.18 0.20
J.0
3.0
°
0
P(k)
S(k)
2.5
o0
2.0
1.5
0
0
1.0
0
0.5
0
0
0
0
0
nn
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
k (A-')
Figure 3-3: The upper panel shows the typical SANS intensity distribution in an
absolute scale and its model fit taking into account the effect of resolution function.
Symbols represent the experimental data and the dash line the theory convoluted with
the resolution function. The inset gives the same data but plotted a in log-log scale.
It can be seen that the Porod's law is satisfied at large k, as is expected in a two-phase
system with a sharp interface. The lower panel gives the normalized intra-particle
structure factor P(k) (circles) and the inter-particle structure factor S(k) (squares)
used to fit the data in the upper panel The S(k) is calculated by solving the OZ
equation with a square-well inter-micellar potential and the P(k) is calculated using
the modified cap-and-gown model as the polymer segmental distribution function in a
micelle. The observed SANS data is the product of these two functions and therefore
it is clear that the interaction peak in the SANS data is primarily due to the first
diffraction peak in the inter-micellar structure factor.
56
350
·
I
1
300
-
~~~~~~~~~~~~~~~~
NG7 NIST
SAND IPNS
250
200
150
100
50
n
II
.X
!
0.00
0.05
0.10
0.15
k(A)
0.20
0.25
1
Figure 3-4: The resolution functions of NG 7 NIST ( = 5 A with AA/A = 10%, the
distance between sample holder and detector is 3 m.) and SAND IPNS respectively.
The functions is plotted for k = 0.01, 0.05, 0.1, 0.15 and 0.2 A-
57
1
respectively.
r
x =
A = kmaxr
Y
k
(3.66)
dx4-xx 2j o (xy)F(x)
(3.67)
k
kmax
it is straight forward to show that
<A 2 >
A
Thus it can be seen that the scaled intensity
magnitude of the scattering vector, y =
k
k3
axI(k) is
a unique function of the scaled
. Therefore, if we plot the scaled intensity
distributions at different temperatures as a function of y, they should collapse into
one single master curve in the single phase amorphous state.
More specifically, take Figure 3-4 as an example in our case, the physical meaning
of the scale intensity
,kl(k)can be interpreted as follows:
The invariant < 772> is mathematically defined as
< 2 >=
k2I(k)dk
1
(3.68)
From Figure 3-4 one can see that the minimum wave vector signaling the onset of the
Porod's law is at k
0.09 A-1 . According to the inset, Porod's law I(k) = Ak-4 is
valid, when k = 0.09 A- 1 . I(k 1 ) = 1 cm -1 = 10-8 A-1, thus A = 6.5 x 10 -13 A-5.
Therefore, the integral in (T2 ) can be divided into two parts and the scale intensity
xI(k) can be rewritten as
k3k(>k)
k3 axI (k)
kmaxI(k)
< 2 >
o
k 2 I(k)dk
k3 ax (k)
212
22
['fo k 2 I(k)dk
+ fk
k2I(k)dk]
k axI (k)
[k k 2I(k)dk +
k2 6.5 O~d
l0-13 dk]
58
kgx (k)
271 fk
k2
(3.69)
(3.69)
max
i(k)dk + 3.3 x 10-13
Furthermore, < r72 > can be evaluated numerically and result is 4.7 x 10- 12 A- 4 .
Since the contribution from the Porod's part is only 3.3 x 10-13 A- 4 , which is about
7.0 % of < 12 >,
kl(k)
can thus be approximated as
k3 xI(k)
< 72 >
1
kax I(k)
l k2 i(k)dk
As we see, P(k) varies slowly within the k range of the first diffraction peak of
S(k), and is negligibly small beyond k, the scale intensity
kxI(k)
can thus be well
approximated by substituting equation (3.14) into equation (3.70) and canceling the
common factor P(k) from the numerator and the denominator of the equation to
obtain the result:
k3 I(k)
< 72>
k 3 S(k)
fakl 2S(k)dk
for k < k
(3.71)
Therefore, without losing generality, the scale intensity is proportional to the interparticle structure factor S(k) in the region of its first diffraction peak.
The mean particle separation distance which characterizes the unique length scale
in the liquid state can best be visualized by the scaling plot. As an example, Figure
3-5 shows a scaling plot of a micellar solution at polymer weight fraction of 35.0
wt.% at a series temperatures ranging from 304 K to 322 K. All the scaled intensity
distributions are temperature independent and indeed collapse into one single master
curve. According to our measurement, this temperature independent scaled plot is
observed for every polymer solution which the polymer weight fraction is between
25.0 wt.% and 35.25 wt.%. Figure 3-5 also indicates that the system is in the liquid
state by the fact that all the scaled peaks are considerably broader than the resolution
function of the instrument.
Examples and discussion of more scaling plots are given in the next chapter.
59
160
35.0 wt% (4 = 0.460)
160
. . . . .304K
140
A
-
120
A
140
120
100
o
X
100
~V
l c
E
.~Y
yE
80
.
312K
60
314K
20
318K
40E
316K
320K
322K
324K
326K
20 -v
0
300305310315320325330335
60 -
310K
'
E80
r
306K
308K
Temperature (K)
332K
20
20
0.2
' '-
0.4
0.6
......... RF
0.8
1.0
1.2
1.4
1.6
1.8
k/kmax
Figure 3-5: The scaling plot of SANS intensity distributions of the micellar solution
at 0 = 0.460 and at different temperatures. It is seen from the figure that the system
is indeed characterized by a unique temperature-dependent length scale. The fact
that the scaling peak is considerably broader than the resolution function indicates
that the system is in the liquid state. The inset shows the plot of the scaling height
of all the scaled SANS intensity distributions as a function of temperature, indicating
no sign of phase transition.
60
3.3
SANS Experiment
SANS measurements were performed at NG 7, 40 m SANS spectrometer in the NIST
Center for Neutron Research, and at SAND station at the Intense Pulsed Neutron
Source (IPNS) in Argonne National Laboratory. At NG 7, incident monochromatic
neutrons of wave length A = 5 A with AA/A = 10% was used. Sample to detector
distance was fixed at 6 m, covering the magnitude of wave vector transfer (k) range
0.008 A-1 to 0.3 A- 1 . At IPNS, a pulse of white neutrons was selected with an effective
wave length range from 1.5 A to 14 A. In SAND all these neutrons are utilized by
encoding their individual time-of-flight and their scattering angles determined by their
detected position at a 2D area detector. The 2D area detector has an active area of
40 x 40 cm2 and the sample to detector distance is 2 m. This configuration allows a
reliable k-range covering from 0.004 A -
1
to 0.6 A-1. Sample liquid was contained in
a flat quartz cell with 1 mm path length.
3.4
PCS Measurement
The Photon Correlation Spectroscopy (PCS) measurements were made at a scattering
angle 0 = 90° , using a continuous wave solid state laser (Verdi-Coherent) operating
at 50 mW (A = 5120 A) and an optical scattering cell of a diameter 1 cm in a
refractive index matching bath. The intensity data were also corrected for turbidity
and multiple scattering effects. PCS data have been taken using a digital correlator
with a logarithmic sampling time scale which allows an accurate description of the
intermediate Scattering Function (ISF), from 1
sec up to 100 seconds. Following
the method used for colloidal hard spheres [18] we measure the correlation function
9g(2)(k,T) = ((I (k, 0) I (k,
where I(k,
T)
T)))
/ ((I (k, 0)))2
(3.72)
is the intensity of light scattered at wave vector k and at a delay time
T.
The first bracket denotes the time average, and the second bracket denotes the positional average over different parts of the sample. Since the system shows a structural
61
arrest transition, and therefore a nonergodic behavior, particular care has been taken
in averaging over many different positions in the sample for the measurement of the
long time part of the time correlation function. For each sample and each temperature we have performed more than 200 positional average measurements, observing a
large scattering area corresponding to three or more independent Fourier components
and changing the position of the sample in order to observe different scattering volumes. The ISF g(1)(k,
T)
can be obtained from equation (3.72) in a straightforward
way using the Siegert relation.
62
Chapter 4
Temperature Effect on Kinetic
Glass Transition in L64/D2 0
Micellar Solutions
Figure 4-1 gives part of the predicted phase diagram based on the AHS with E = 0.03.
In this case, it is possible for the system to undergo a re-entrant (glass-to-liquid-toglass) transition (for example at
= 0.532) by varying the effective temperature.
At high effective temperatures and at sufficiently high volume fractions, the system
evolves into the well-known structurally arrested (glassy) state, called a "repulsive
glass", as a result of the cage effect, a manifestation of the excluded volume effect
due to the existence of the hard core. However, at relatively low effective temperatures, an "attractive glass" can form in which motion of the typical particle is hindered
in stead by the cluster formation with neighboring particles due to the short-range
attractive interaction. Moreover, in an AHS,aside from the hard-core diameter, an
additional length scale, the range of the attractive well, should come into play (parameter
). With this insight, we may divide spherical colloidal systems into two
categories: the one-length-scale hard sphere system, in which the glass formation is
dictated by the cage effect alone; and the two-length-scale AHS, in which the two
glass-forming mechanisms may coexist and compete with each other, depending on
the values of the controlled parameters. The MCT calculations show that, with suffi63
ciently small E, variation of the two control parameters, T* and , allows the transition
between these two distinct forms of glass. Thus there is a branch of the KGT line
across which transitions between the attractive glass and the repulsive glass are predicted. Of particular interest is the occurrence of an A 3 singularity at which point the
glass-to-glass transition line terminates. MCT suggests that the long-time dynamics
of the two distinct structurally arrested states become identical at and beyond this
point.
4.1
Intermediate Scattering Function (ISF) Measured by Photon Correlation Spectroscopy (PCS)
ISFs were measured for a set of volume fraction within the interval 0.525 <
E
< 0.546,
where the re-entrant phenomenon of the glass-to-liquid-to-glass transition, glass-toglass transition and the A 3 singular point were predicted for the case of E = 0.03,
as shown in Figure 4-1. Figure 4-2 A & 4-2 B show ISFs obtained by PCS for two
volume fractions,
= 0.525 and
X
= 0.535 respectively. On increasing temperature,
the system, starting from a liquid state at a lower temperature, where the ISF decays
to zero at long time, approaches a KGT, characterized by a diverging a relaxation
time. At KGT the ISF tends to a finite plateau at long time (F(k, t -, oc) = fk
>
0).
Just before such an ergodic-to-nonergodic transition taking place, the ISF measured
in an ergodic state exhibits a logarithmic relaxation (indicated by a straight line fit) at
an intermediate time followed by a power-law decay before the final a relaxation sets
in. This is in agreement with the MCT prediction. The non-ergodic state (T = 300
K, for
= 0.535), as indicated by the ISF having a finite plateau at long time,
represents the attractive glass for which the Debye-Waller factor (DWF, the height
of the plateau)
fk
is about 0.5. As discussed above, an important prediction of the
MCT calculation for an AHS is a suggestion that there exists an attractive branch
of KGT line near the cusp singularity separating two different glass phases [32, 2].
Motivated by this prediction, we extended the study of the micellar system to higher
64
L IL
2.50
m
i=
2.25
II
H-
2.00
Z)
1.75
a)
0.
E
1.50
a)
a)
0
a)
1.25
1.00
0.75
rn =n
0.510 0.515 0.520 0.525 0.530 0.535 0.540 0.545 0.550
Volume Fraction
Figure 4-1: Theoretical phase diagram predicted by MCT calculations using the
short-range attractive square well potential with E = 0.03. The calculations predict
the attractive glass-to-liquid-to-repulsiveglass re-entrant transition, the attractive
glass-to-repulsive glass transition and the end point of the glass-glass transition line.
Symbols (circles and squares) represent the effective temperature T*, determined
by fitting the experimental SANS data taken in the liquid state with the method
explained in the text.
65
temperatures. Starting from a glass state, upon increasing the temperature further,
the measured ISF reveals a unexpected re-entry from the glass to the liquid state, as
seen by the ISF again decaying to zero at higher temperatures.
The ISFs measure at
X
= 0.538 and
D. Aside from the volume fraction
= 0.540 are given in Figure 4-2 C & 4-2
, the glass transition can be triggered by varying
the effective temperature T* as well. Due to the fact that the depth of the potential
well -u is temperature dependent and increases on heating to a certain extent, the
effective temperature T* actually decreases as temperature rises for certain interval
of temperatures.
Variation of temperature thus makes the transition between two
distinct types of glass possible. Because the long-time limit of the ISF (DWF) reflects
the degree of localization of the density fluctuation, these two different types of glass,
if they have different degree of local disorder, can be identified by their different values
of DWF. From Figure 4-2 C & 4-2 D it is obvious that all the ISFs can be grouped
into two distinct sets of curves having two different values of DWF - one at
fk
= 0.5
(attractive glass) and the other at fk = 0.4 (repulsive glass). These two panels, 4-2
C & 4-2 D, confirm the predicted glass-to-glass transition.
Figure 4-2 E shows the ISFs measured at
= 0.544, which, according to MCT, is
the end point of the predicted glass-to-glass transition line. Our measured ISFs verify
this prediction, showing two nearly identical values of DWF, for the two glassy states,
with an average value of fk
'
0.45 . Surprisingly, although the DWFs of these two
types of glasses are identical, there is a significant difference between the dynamics
of their intermediate time relaxations (the 3 relaxation region).
The ISFs measured at 0 = 0.546, which is a volume fraction beyond the A 3 point,
is shown in Figure 4-2 F. This figure shows a similar features as in Figure 4-2 E, with a
merged DWF of fk - 0.4. It indicates that beyond the volume fraction
X
= 0.544, the
system exists in a repulsive glass state, suggesting a critical point-like characteristics
of the A 3 point.
66
r
I OO
I .L
_OIOO
OO''
6O6, 6 6OO 6666'66666
i'OO-N
q
I
I
I
I
I
0O
T-o
0
0
a)
EO
00
00
IrO
6
S
io
_
O
_
Xo
0
Figure 4-2: The ISFs of L64 micellar solutions at different concentrations and temperatures measured by PCS
67
4.2
Fitting Results of SANS Data
SANS experiments were performed on Pluronic L64/D 2 0 micellar solutions at different concentrations and temperatures.
The experiments were carried out using NG7
SANS spectrometer at NIST Center for Neutron Research and the SAND station at
IPNS, Argonne National Laboratory. Part of the experimental results as well as their
theoretical fits are given in Figure 4-3. Symbols are experimental data and lines are
the fits. The fits are in an absolute intensity scale taking into account the effects of
the instrument resolution and the incoherent background. It can be seen that the
peak intensity is generally higher for the sample at the higher temperature liquid
phase. Furthermore, the position of the peak shifts toward smaller k as temperature
increases. This is due to the enhanced self-association of the micelle (larger aggregation number) as the consequence of the increased hydrophobicity of the polymer
segments at higher temperatures.
It is important to note here that in spite of the
micelle growth as temperature increases, SANS data analyses show that the volume
fraction of the micelles remains constant, for a given weight fraction of L64, at all
temperatures studied.
The variation of the aggregation number N as a function of temperature for different concentrations is given in Figure 4-4. It can be seen that due to the fact that
the PPO core in the micelle becomes less hydrophilic when heated, the micelle grows
as temperature increases for a given concentration. It is to be noted that at a fixed
temperature, the aggregation number decreases as a function of concentration.
Figure 4-5 gives the calibration curve between the ratio of the volume fraction
to
the weight fraction c as a function of c, obtained from fitting approximately 150 SANS
intensity distributions. It should be noted here that the fitting results show that the
volume fraction q is a unique function of concentration, independent of temperature.
The figure shows that at high concentration, the ratio of 0 to c approaches unity,
indicating that the hydration level of the micelle goes down as the concentration goes
up. The mathematical relation between them is /c
= 1.96 - 1.78c, predicting that
the micelle will be dry at a weight fraction of 0.54 since the density of L64 is close to
68
90
35.0 wt%
=0.480
0
80
30 K
-
=0.03
70 .
i
§Er 0-0
_so
O.
70
N=8
324 K
~
....
-r
.
N
100
38.0 Wt%
= 0.4780
: 0.03
80
,-,
--
~
80
~~~
[1 ~:
T =0.93
50
304K
-
ri0.85
N =94
332 K
-
80
so
E
=127
wt%
=0.03
70
325 K
E
=0.91
183
37.0
90
315K
r
=0.82
N=10
r = 0.92
N =12
40
30
20
20
0,00oo 0.03
0.06
o.00
0.12
0.15
0.00
.16
k (A')
0.03
0.06
k( 1)
940
0.09
0.12
0.15
0.10
k (Ak)
100
40.0 wt%
=0.54
0
70
E
100.00
40.00.03
wt% 0.00
80
0.52
.0.03
.
0
70
o
so
:
N=71
32 K
= 0.94
N=138
-
:
...
50
.
0.12
-r
0.09
40
E
20
10
10
:
70
3K
.
s0
_
40
30
20
10
0.00
0.03
0.06
0.09
0.12
0.15
o L-;~---~'~:
0.00
0.03
0.18
. >ri..... ...
0.08
o.o
k (A )
0.12
T
0.1s5
,,. I.
.18
0o
800
.- :r..
0.00
0.03
0.00
k (A)
=
0.095
N=59
324 K
70
E
-
4.0
wt%
4 0%
0.530
= 0.03
90
297 K
T=1.19
80
0.18
K
29
r-.11
N=47
324 K
-
70 -
N=99
0.15
0,
48.0
t#
= 0.528
=0.03
o
302K
'
-
.o09 0.12
k(A7')
100
44.0 wt%
= 0.518
305 K
r
0. s
= 87
324 K
T-1.08
N 113
.
30
20
42.5wl%
=0.512
=0.03
s
0
0.1S
0.1K
N06.
304K
-o 2
N=115
so
40
90
304 K
N=97
-so.
-:s
40
30:
10
-
20
0
1.20
0o
f T;i~T-t_
00L
o.00oo 0.03
0.0
o.o
012
0.12
0.18
0.00
0.03
0.08
0
. 32
0.591
= 0.03
90
21K
70
0.12
0.15
0.18
0.03
0.08
:
T= 1.10
N=35
329 K
T'
1.40
0S
50.0 Wt%
= 0.535
0,03
-
70
1
..
S
2
_;
0.09
0.12
-N.=9.......
0.15
0.18
1)
k (A'
294K
T=1.20
N= 42
324 K
T= 1.22
E
2
0.00
k(A1)
k (A 1)
80
0.09
0
51 5 wt%
=0 0.=0.03
-
70
i
285
K
2=0.038
r=1.04
N=37
331K
S
.E
21 40
50
40
150
30
30
1
S
120
20
O
(
.00
0.03
0.0
k (A1)
0.09
k(
0.12
1)
0.15
0.18
0.00
0.03
0.0
0.09
0.12
0.15
.18
k (A')
Figure 4-3: The theoretical fits to SANS data for Pluronic L64/D 2 0 micellar solutions
at various concentrations and temperatures. Symbols are experimental data and lines
are the fits. The scattering intensities increase at the higher temperature liquid phase
as compared to lower temperature liquid phase and the positions of the peak shifts
toward smaller k due to the enhanced self-association as the consequence of increased
hydrophobicity of the polymer segments at higher temperatures.
69
180
160
35.0wt% (=0.460)
e
36.0wt% (=0.479)
0
A
A
37.0 wt%
37.5 wt%
40.0wt%
42.5 wt%
o
44.0 wt% (4=0.518)
o
140
z
0
o
(4=0.483)
((=0.493)
(=0.504)
(4=0.512)
o
0
0
0 46.0 wt% (4=0.528)
O 48.0 rwt/ (=0.531)
0 50.0 wt% ((=0.535)
120
E
z
o
0 100
o
0o EC A A 0
o OC
e
aA
A
o
o
-
o)
n0
C
0 oa
80
60
0o oo0
*°S
4.CO
40
,,
-JB
10
10
20
30
40
50
60
70
Temperature(°C)
Figure 4-4: Variation of the aggregation number N as a function of temperature at
different concentrations. It can be seen that the degree of self-assembly increases as
temperature increases at a given concentration. It is consistent with the fact that the
PPO core becomes less hydrophilic at higher temperatures. We also note that at a
given temperature, N decreases as concentration increases.
70
unity.
r
1A
.4U
1.35
1.30
1.25
NC 1 .2 0
1.15
1.10
1.05
I nn
0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54
L64 Weight Fraction
Figure 4-5: The ratio of the volume fraction to weight fraction as a function of
polymer weight fraction, obtained from analyses of SANS data. This is an important
calibration curve which enables us to compare the experimentally determined phase
diagram with the theoretical phase diagram predicted by mode coupling theory. We
note that the hydration level of the micelle decreases as the polymer concentration
increases. It predicts that at weight fraction 0.54 the micelle will be completely dry,
since density of L64 is close to unity. Denoting the concentration (in weight fraction)
by c and the volume fraction by A, then the empirical relation is 0/c = 1.96 - 1.78c.
Figure 4-6 summarizes the variation of the fitted effective temperature parameter
T*, as a function of temperature T, for several volume fractions studied. It can be
seen clearly that as the T increases, the T* increases also. There are no data shown
in the glass regions, which appears as gaps in the plots, due to lack of an appropriate
theory for the structure factor of a system in a nonergodic state.
71
4
1
r
1.3
1.2
1.1
1.1
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.6
0.5
0.7
0.6
0.5
1.4
1.4
1.3
1.3
1.2
1.1
1.2
1.1
T*
=
1.4
1.4
1.3
1.2
1.0
1.0
0.9
0.8
0.7
0.6
0.9
0.8
0.7
0.5
0.5
1.4
1.3
1.2
1.4
1.3
0.6
1.2
1.1
1.1
1.0
0.9
0.8
0.7
0.6
0.5
1.4
1.3
1.0
0.9
0.8
0.7
0.6
0.5
1.4
1.3
1.2
1.2
1.1
1.0
1.1
1.0
0.9
0.8
0.7
0.6
0.9
0.8
0.7
0.6
0.5
0.5
280 290 300 310 320 330 340 350 290 300 310 320 330 340 350 290 300 310 320 330 340 350
Temperature
(K)
Figure 4-6: The effective temperature T* of the liquid state as a function of temperature, obtain by fitting SANS intensity distributions from different concentrations. For
the indicated concentrations, T* generally increases as temperature increases slightly
except in the glass region which appears as a gap in the figure.
72
4.3
Scaling Plots of SANS Data
As we go above the concentration 35.25 wt.%, the situation changes dramatically.
As an example, Figure 4-7 shows the scaling plots for the 45 wt% concentration
sample at different temperatures.
The volume fraction of micelles for this polymer
concentration is determined by SANS data analysis to be at 0 = 0.522, independent
of temperature.
In view of the fact that the 2D SANS patterns shows an isotropic
ring, we conclude that the sample stays amorphous in the entire temperature range
studied at this concentration. As mentioned before, the scaling peak characterizes the
position and the height of the first diffraction peak of S(k). It is well known that the
position of the first diffraction peak of S(k) reflects the mean inter-particle separation
of the system and the height of the peak reflects the degree of local order surrounding
a typical particle. Thus the height and the width of the scaling peak can be used
to visualize the degree of local order surrounding a given particle in an amorphous
state. Therefore, a sudden sharpening at given temperature signals the onset of the
liquid-to-amorphous solid transition. As one can see, there are two distinct degrees of
disorder, that depends on temperature. While the narrower peak, which is resolution
limited, represents the glassy state, the broader peak, which is much broader than the
resolution, represents the liquid state with a broader distribution of the inter-particle
distances. This figure indicates that the system shows a re-entrant liquid-to-attractive
glass-to-liquid transition as temperature increases. The sharpness of the scaling peaks
which are resolution limited indicates that the nearest neighbor distance in the glassy
state (ranging from 306 K to 321 K) is more uniform than that in the liquid state
(288 K to 303 K, 325 K to 343 K). The inset gives the peak height of the scaling
plots as a function of temperature.
The variation of the peak heights indicates the
re-entrant liquid-to-attractive glass-to-liquidtransition.
SANS intensity distributions and their scaling plots for a sample at 48.5 wt%,
or
= 0.532, at a series of temperatures ranging from 291 K to 340 K are given in
Figure 4-8. The scenario is similar to the previous example: A temperature dependent
degree of disorder again characterizes the system. As the temperature rises, the
73
45.0 wt% (4 = 0.522)
100
288 K
o 294 K
80
303 K
306K
315 K
321K
·
·
60
·c
40
-0
:8
v~,325K
.
_8
ffi.ff.
o 334 K
c,'
.
o
343K
20
0
).00
i1fn
A
V
0.02
0.04
0.06
0.08
0.10
0.12
I
I
....
I
140
120
100
I
0
a
F'
*
.A
v
o
o
-·
60
28 K
294 K
303K
306K
315 K
321 K
325 K
334K
343 K
,sW
sM
40
-
0.116
k(A)
I
80
v
0.14
20
I~~~axrmdt~~~IR~mr' L
··
nl ~~~rorrmmm~~~'
co
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
k(A-)
k( )
160
1.00----'
140
A
EU
S~
288 K
40
120
V
100
80
60 40
'I
0.11
o
O
AG
lO
120
o
.
303K
-06K
3
0
315K
o
L
40
294 K
7
t321K
ao~e
L
20
0 290 300 310320330
340......
A,,
325K
o
334 K
343K
-
RF
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
8
1.E
k/kmS.
Figure 4-7: SANS intensity distributions and their scaling plots at X = 0.522, where
the liquid-to-glass-to-liquid transition is observed, at a temperatures range spanning
from 288 K to 343 K. The top panel gives the SANS intensity distributions as a
function of k. The broader peaks (288 K to 303 K, 325 K to 343 K) represent the
liquid states and the narrower ones (306 K to 321 K) represent the glassy state. The
scattering intensities increase at the higher temperature liquid phase (the left hand
side peak, 325 K to 343 K) as compared to lower temperature liquid phase (the right
hand side peak, 288 K to 303 K) and the positions of the peak shifts toward smaller k
due to the enhanced self-association (larger aggregation number) as the consequence
of increased hydrophobicity of the polymer segments at higher temperatures. The
scaling plots of SANS intensity distributions are given in the bottom two panels.
It is seen clearly that, there are two distinct degrees of disorder (judging from the
width of the peak), which depend on temperature. While the narrower peak, which
is resolution limited, represents the glassy state, the broader peak, which is much
broader than the resolution, represents the liquid state with a broader distribution of
the inter-particle distances. The inset gives the peak height of the scaling plots as a
function of temperature. The variation of the peak heights indicates the re-entrant
liquid-to-attractive glass-to-liquidtransition.
74
system experiences a liquid-to-glass-to-liquid transition. However, in addition to all
the similarities, when the temperature increases to 340 K. the system is driven into
another glassy state. The differences between these two disordered glassy states from
the scaling height: While the narrowest peak (340 K) is resolution limited, the slightly
broader peak (298 K to 322 K) is also nearly resolution limited, but lower in the
scaled intensity. Since the difference in local structure of different states is reflected
in their scaling plots, we conclude that the degree of disorder are different for these
two amorphous states. It can be interpreted that, by varying the temperature, the
system shows a liquid-to-glass-to-liquid-to-glass transition.
Figure 4-9 gives the same plots taken at concentration of 51.5 wt%, or
for different temperatures.
= 0.536,
Generally it delivers the same message as Figure 4-8: it
shows the re-entrant transition. However, it can be seen that the transition temperature between different states for these two different volume fractions are different.
Especially at the high temperature region. For example, for the case of 0 = 0.536,
the high temperature liquid state is only observed at 331 K. When the temperature
increases to 333 K, the system transforms to the repulsive glass state.
The peak
heights of the scaling plots as a function of temperature are given in the insets of
Figure 4-8 and Figure 4-9. The transition temperatures between different amorphous
states can be visualized clearly from them. These two figures give a firm evidence of
the re-entrant glass-to-liquid-to-glass transition which is in good agreement with the
prediction of MCT for an AHS system. Furthermore, the KGT boundaries determined
by SANS agree with the results obtained from the latest specific heat measurements
[33].
According to the MCT calculations for an AHS with sufficiently short-range attraction, a glass-to-glass transition is predicted [2]. Although the transitions between
different amorphous glassy states are not uncommon in pure substances [34],yet there
is no detail investigation of the glass-to-glass transition in a micellar system so far
except for some recent reports [25, 26, 27]. Combining with the PCS (see Figure
4-2 C & 4-2 D), a concrete evidence of such a transition
in a micellar system was
revealed by SANS. From SANS intensity distributions, taken at
75
= 0.538, given in
60so~~
48.5 wt% (4 = 0.532)
80
70 -o
-v
50
291 K
295 K
K
312K
322 K
'5 ·
~~~~ ~~~~~298
__h~~~~~~~~~~~
a.329
v
40
v
Y
30-
viA~
·
>^^;D
K
V
333 K
'*
340
K
20 0
0.00
0.06
0.04
0.02
0.08
0.10
0.12
0.14
0.16
k(A)
140
A.
N
120
291 K
295 K
298 K
312 K
322K
o
·
·
Y
·
~"
V
-
a 329K
100
80'
333 K
340 K
v
·
·^
60
40 20
,
0.02
0.00
0.04
0.08
0.06
0.10
0.12
I
I
0.14
0.16
k(A-)
160
I
14U= .
6o I
1
120 -
cN
100
V
-
X
80
60
Y
40
g
.
I
I
I
...
I
I
.
I
291 K
AC
120
E1 1
.
-
J
L
20
280290300310:
20
0.2
Tempera
0.4
0.6
1.0
0.8
1.2
1.4
1.6
1.8
klkmax
Figure 4-8: SANS intensity distributions and the scaling plots at 9 = 0.532, at a
temperature range spanning from 290 K to 340 K. The top panel gives the SANS
intensity distributions as a function of k. It shows the similar features as the top
panel of Fig. 4-7: As temperature increases, a liquid-to-glass-to-liquid transition is
observed. However, as the temperature is increased to 340 K, another disordered
state peaked at k = 0.082 A-' is observed. The other two panels show the scaling
plots of the SANS intensity distributions. A temperature dependent degree of disorder
characterizes the system. While the narrowest peak (340 K) is resolution limited, the
slightly broader peak (298 K to 322 K) is also nearly resolution limited, but lower in
intensity. From the inset given in the bottom panel, the liquid-to-glass-to-liquid-toglass re-entrant transition can be seen clearly.
76
51.5 wt% (
50
--
= 0.536)
I
I
40
I
I~~~~~~~~~~~~~~~~~~~
I
285 K
290K
;~~ 300 K
*
310K
f,~.
· cP<~.^
30 I-
323 K
K
331 K
328
20 io
o tJ .fD.
10 _
CMrr·
YLI·
n
, -·ILZB~YY~-
U_
0.00
0.02
0.04
0.06
oKo
333K
33 K
340K
0
A
0.08
0.10
--
L
.
----
0.12
0.14
0.11
k(Ak)
I OU
A
140
cm
120
V
100
·
140~~~
_
·
·
A
·
*
0
80
E
60
0
,,
0.00
1 n
140
A
.E
0.02
x~
-
0.06
0.08
0.10
0.12
333K
338K
°
340K
343K
0.14
-
0.11
I
I
I
285 K
140
'
,AG
-
1
, '.
80
^> . wso
^.9
4L
60
20
33043.
"
Temperature (K)
20
~~
_
0.4
a
O
280 2.00031.320
0.6
.
l....._
~~_
0.8
1.0
290
K
3300K
310 K
323 K
*
328 K
o
331K
.~~~,-"·~~
80
0
0. 2
0.04
I
100
40
_
k(A-)
I
120
V
,-
20
20 0
_
300K
310 K
323 K
328 K
331 K
AG
*
40 -
285 K
290K
-
1.2
1.4
0
333 K
33 K
A
340K
v
343 K
RF.
1.6
-
-
1.8
k/k max
Figure 4-9: SANS intensity distributions and the scaling plots at
= 0.536, at a
temperature range spanning from 285 K to 343 K. It exhibits all the features found
in Fig. 4-8. However, the high temperature liquid state is only observed at 331 K,
when the temperature increasesto 333 K, the system transforms to the repulsiveglass
state. Figure 4-8 and Figure 4-9 together provide convincing evidence of the predicted
re-entrant glass-to-liquid-to-glass transition. The only difference between them is that
the transition temperature between different amorphous states are different. The inset
shows the re-entrant transition more clearly.
77
Figure 4-10, the much broader peaks (liquid state) disappear and variation of temperature triggers the transition between the two amorphous solid states with different
degrees of disorder. By increasing temperature, the variation of the peak heights of
the scaling plots given in the inset shows a re-entrant repulsive glass-to-attractive
glass-to-repulsive glass transition.
Perhaps the most important prediction of the AHS system is the existence of the
end point of the glass-to-glass transition line, the so-called A 3 singularity. Comparing
with the critical point of the equilibrium states, we find a certain degree of similarity
between them. Therefore, it is intriguing to speculate the extent to which one can
draw the analogy between the A 3 singularity and the ordinary equilibrium critical
point. In the case of
be
= 0.03, the volume fraction of the A 3 point is predicted to
(A 3 ) = 0.544. SANS intensity distribution and its associated scaling plot are
shown in Figure 4-11. As one can see, all the scaling intensity curves collapse into
one single master curve, indicating that first diffraction peaks of the structure factor
for all states are identical. It suggests that the local structures of the two glasses are
identical. The inset given in the bottom panel shows that all the scaling peaks have
the identical height (about 140), indicating the two glasses indeed have same degree
of local order.
Increasing
further to 0.546 (Figure 4-12), all the scaled intensities are again
characterized by a unique temperature-independent
length scale and collapse into one
single master curve, independent of temperature, showing an identical local structure
of the two glasses. This is proof that the MCT predictions are accurate.
Figure 4-13 summarizes the essential results of the extensive SANS and PCS data
analysis. It contains the known equilibrium, liquid-to-hexagonal crystalline phase
boundary (solid line) [35], the experimentally determined KGT lines (dash lines) and
the phase points where parts of the experimental data are taken (symbols). This figure gives several important information about this system: First, only the metastable
attractive glass is observed within the region where the true lowest free energy state
is the hexagonal liquid crystalline phase. Next, the repulsive glass only exists in the
region where the volume fraction is larger than 0.536. It is interesting to see that
78
52.5 wt% (4 = 0.538)
50
,
295 K
300 K
313K
315K
320 K
325K
330 K
333 K
336 K
343 K
·
o
40 -
A
*
*
a
·
v
0
o
30 20
,
10 0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
k(A-)
160
140
1120
V
0
a
o
100
·
*
80o-
Y
60 -8
-b
0
o
40 20
0
0.00
16
14.0
0.02
V
x
0.06
0.08
0.10
0.14
0.12
0
Ao
O
'
·
295 K
° 300K
aRG313K
,
20R-G"oF
.
12o
.=-"
*·
-
40
20.
·
0
410
0.116
k(A)
20
10C0_
6
0.04
10
12
0A
295 K
300K
313 K
315 K
320 K
325 K
330K
333K
336K
343 K
'
-
0
2002903003103203303400
Tempratur
20
(K)
.
320K
325 K
330 K
333 K
336 K
343 K
RF
n~~~~~~~~~~~~~~~~~~~
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
k/kmax
Figure 4-10: SANS intensity distributions and the scaling plots at 5 = 0.538, at
a temperature range spanning from 295 K to 343 K. From the SANS intensity
distributions given in the top panel, it can be seen that the much broader peaks
(liquid state) disappear. Variation of temperature triggers the transition between
the two amorphous solid states with different degrees of disorder. The variation of
the peak heights of the scaling plots as a function of temperature given in the inset
shows a re-entrant repulsive glass-to-attractive glass-to-repulsive glass transition for
the first time.
79
54.4 wt% ( = 0.544)
50
It
S
285K
290 K
294 K
'
o
40
30
°
_·
20
~
*
z
~,v~*
310K
*
322K
328 K
·,~
~334K
*
343K
A
10
0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.11
k(A-)
160
140
cA
120
V
100
X
80
'
60
40
-,
oQ~
-
A
v
v~~
*
285K
290K
294K
308K
310 K
322K
328 K
334K
343 K
~
20
0
0..00
0.02
0.04
0.06
°
~16600
° °
.
140
120
V
100
'
X
80
yE
60
40
0.10
0.12
0.14
0.11
k(A)
160
CA
0.08
140 DDO
- F120
..
.
A
.
-RG
10l30
.
~~~~
290K
.
*
-~~
·
285K
A
294 K
v30
3
K
-
310 K
l 040
7
·
.......
;
20
-- - ----.
,
334K
K
343K
RF
-
-
o
20200300310320330340300
?
Tempeature (K)
j _~
.
20
0
0).2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
k/kmax
Figure 4-11: SANS intensity distributions and the scaling plots of SANS intensities
at = 0.544, which is predicted to be the volume fraction where the A 3 singularity
point is located, at a temperature range spanning from 285 K to 343 K. From the
top panel, it can be seen that all the peaks of the SANS intensity distributions are
located at the same k value (0.0836 A- 1 ). Furthermore, from the scaling plots given
in the bottom two panels, it can be seen that all the scaled intensities collapse into one
single master curve. It suggests that the two different types of glass become identical
in their local structure at this volume fraction. The inset given in the bottom panel
shows that all the scaling peaks have an identical height (about 140), indicating the
two glasses indeed have the same degree of local order.
80
54.6 wt% (4 = 0.546)
50
283 K
'
285 K
290 K
40
30 -
_~~~30~~~~~~
-
20
295 K
K
K3
308K
316K
~306
3~
.
0
330 K
343K
5
10
0
0.02
0.00
0.04
0.06
0.08
0.10
0.12
0.14
0.16
k(A)
I bU
140
o
120
N
V
J
$=-
v
100
0
80
v·
I
0.02
0.04
0.06
0.08
F
100
RG
120
-
80
60
Y
0.10
0.12
0.14
0.16
k(Ak)
12-
C,
308K
316 K
330K
343K
i
9.
IOU
V
306K
,
40
20
0
0.00
N
285K
290K
295 K
F~~~~~~~~~~~
640
""
285 K
o
^
I
eo
60
20
oO
2
z28
40
00
v
-1
#
it
0
I
a
Y·
i316
290 K
295K
306 K
308K
-
K
300310 320330340350
330K
K
343K
RF
Temeratur (K)
20 -
0.2
g
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
k/kMx
Figure 4-12: SANS intensity distributions and the scaling plots of SANS intensities at
q = 0.546, a volume fraction beyond the A3 point, at a temperature range spanning
from 283 K to 343 K. Like Fig. 10, all the scaled intensities are again characterized
by a unique length scale and collapse into one single master curve, independent of
temperature, showing identical local structure of these two glasses.
81
there is a pocket of the attractive glass imbedded in-between two separate repulsive
glass regions spanning the volume fraction range between 0.536 and 0.544, where
the re-entrant glass-to-glass transition is observed. From Figure 4-2, Figure 4-11 &
Figure 4-12, the two different glasses become identical in their local structures and
the long-time dynamics when the volume fraction exceeds (A 3 ) = 0.544. Furthermore, judging from the DWF and the peak height of the scaled intensity, the merged
identical glassy state is the repulsive glass.
Provided with the structure factor derived from a given potential, MCT calculations have successfully predicted the discrete transition in the liquid state, namely
the structural arrest transition. Due to the nonergodic nature of the glassy state,
the analytical structure factor of the arrested state is unavailable. Therefore, a major assumption in MCT calculations is, when the KGT is triggered by varying the
external control parameters, there is no significant difference in the local structure.
However, from our SANS experiments, when crossing the KGT boundaries at various
temperatures for all the volume fractions studied, instead of smooth change in the
structure factor, a significant difference is observed. As an example, an evolution
of the structure factor as a function of time is given in Figure 4-14. Initially, the
sample is held in the liquid state (303 K, one degree away from the KGT line), then
transferred to another thermal bath (306 K), which drives the sample from the liquid
state to attractive glass state. The top panel gives the associated scaling plots of
the SANS intensity distributions and the height of the scaling peaks is given in the
bottom panel. As predicted by the MCT, the structure factor in the glassy state is
identical as the one in the liquid state. However, as time evolves, the scaling height
increases gradually from 60 to 120 (within 5 minutes), corresponding to a structure
factor which can not be derived from the equilibrium liquid theory. Therefore, the
deviation between the MCT and our SANS experiments can be explained as that
our SANS data is reflecting the aging effect of the sample in the time scale of our
measurements.
82
350
I
,
I
,
I
,I
r
/RRG
340
330
L
...~"....__...
7ro--- ' ° --
Pe 320
.10
t~i
1.
-,~
~ ~
310
~
d
-.
m"
=_/ - ~ -, a?'R(
D
Ad
}
eAG
"~.
·z~~~~~~~~~~~~~~~~~~~~~~~
Disordere
Miea
Pas(DP
Q.
E 300
I290
9Rn
-v
_
II
m
I
0.447
0.48
0.49
0.50
l .........
m
0.51
0.52
I
I
0.53
0.54
0.55
Volume Fraction (p)
Figure 4-13: The experimental phase diagram of L64/D 2 0 micellar system. The solid
line represents the equilibrium phase boundary of disorder micellar liquid states and
the hexagonal liquid crystalline states. The dash line gives the kinetic glass transition
boundary which is determined by SANS and PCS. The Symbols represent the phase
points where parts of the experimental data were taken. The triangles represents
liquid state (L), the circles the attractive glass (AG), the squares the repulsive glass
(RG). This figure gives several important pieces of information about this system:
Within the region where the true ground state is the hexagonal liquid crystalline
phase, only the metastable attractive glass is observed. The repulsive glass is found
in the region where the volume fraction is larger than 0.536. It is interesting to
see that there is a pocket of the attractive glass imbedded in between two separate
repulsive glass regions spanning the volume fraction range between 0.536 and 0.544.
Furthermore, it is important to note that the two different glasses become identical
in their local structures and the long-time dynamics when the volume fraction exceed
O(A3)
= 0.544.
83
42.5 wt% AT = 3.0K (303K to 306K)
J.A
q]f
I U
120
*
v~~~v
{"
100
v
7 min
60
5min
6 min
A 8min
*
min
10min
12 min
14min
o
16 min
v
o
c°o3
x
_E
o
°
"Vt
80
V
A
1 min
2 min
3min
4 min
o
O
0
·v
d3
P8~~~
v
40
0
Ela
20
9
18 min
20min
22min
24 min
26min
28min
a
lil
30min
n
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
klkmax
140
120
100
A
V
V
x
-A
O A V O O °
O
OoAV
O
80
E
x
60
E
40
20
0
5
10
15
20
25
30
Time (min)
Figure 4-14: The upper panel gives the scaling plots of SANS intensities at c = 42.5
wt% for the aging experiment (see text for the details), the height of the scaling peak
as function of aging time is given in the bottom panel.
84
Chapter 5
Pressure Effect on Microstructure
of Micellar Systems
5.1
Introduction
The study of complex, structured liquids has been expanding at a very rapid pace
over the past few decades [36]. In addition to purely scientific interest, they are
intimately related to numerous industrial applications such as biotechnology, food
science, cosmetic manufacturing and pharmaceutical industry and therefore a fully
qualitative understanding of their structural configurations and dynamic properties is
essential. Among all the structured liquids, numerous researches have been devoted
to investigate the colloidal systems. The major reason is, alongside with its extensive
industrial applications, it is relatively modeled theoretically and can be prepared with
well-defined geometry experimentally. It is well acknowledged that various interactions among colloidal particles such as Coulomb repulsion, excluded volume effect,
van der Waal forces and depletion attraction give rise to the rich variety of phase behaviors observed in a colloidal system and hence the study of the transition between
different phases in colloidal systems has been attracting a great deal of attention for
a long time. Recently, with the advance of MCT calculations [37], considerable attention has been paid to different non-ergodic states of matter in different colloidal
systems
[25, 26, 27, 22, 23, 38]. Unambiguously,
85
the equilibrium
structure
of the
solvent plays a key role in the kinetic glass transition.
Take L64/D 2 0 as an example to illustrate this point [25, 26, 271: as temperature
rises, the decreasing solubility of the micellar corona PEO segments in D2 0 solvent
leads to an effective attraction between the micelles. When two micelles touch each
other, some D2 0 molecules are squeezed out of the corona region and the overlapping
between the PEO segments in coronas of two micelles give rise to the characteristic
short-range intermicelle attraction [8]. Microscopically, there are two different hydrogen bonds existing in the copolymer-solvent system: the hydrogen bonds among
D2 0 molecules and the hydrogen bonds between D 20 molecules and oxygen atoms in
L64 copolymers. Our experiments suggest that the responses of these two hydrogen
bonds to the temperature variation are quantitatively different and the increasing
short-range attraction can be viewed as the consequence of the interplay between the
probability of formation these two types of hydrogen bonds as a function of temperature.
Temperature and pressure are thermodynamic variables used to manipulate the
phase behavior in structured liquids. In the L64/D 2 0 micellar system, several fascinating phase behaviors are observed by varying the temperature.
Although the
pressure effect on the microstructure of structured fluids has been investigated pre-
viously [39], to our best knowledge,the response of the structural arrest transition
in a micellar system to the applied pressure still remains unexplored. To provide
the answer, a series of SANS experiments have been carried out to investigate the
microstructure altered by the applied pressure. SANS measurements were performed
at SAND station at the Intense Pulsed Neutron Source (IPNS) of Argonne National
Laboratory. A pulse of white neutrons was selected with an effective wave length
range from 1.5 A to 14 A. All these neutrons are utilized by encoding their individual
time-of-flight and their scattering angles determined by their detected position at a
2D area detector. This configuration allows a reliable k-range covering from 0.004
A-1 to 0.6 A-1 . Sample was contained in a flat quartz cell with 1 mm path length. A
SANS hydraulic pressure cell was used to pressurize the L64/D 2 0 micellar solution.
86
5.2
Pressure Effect on the Microstructure of L64/D20
Micellar System
The effect of applied pressure on the microstructure of the micellar system is represented schematically in Figure 5-1. Panel (a) gives the 'snapshot' of a typical
configuration of spherical micellar liquid ( = 0.4) at ambient pressure, in which the
effective intermicellar interaction is modeled as a pairwise hard core plus a short-range
attractive square well potential with depth u = 1 kBT, and fractional width E = 0.03
[30]. The inset provides its associated structure factor S(k) describing the local order.
At the presence of high applied pressure, the hydrogen bonds between D 2 0 molecules
and micellar corona regions are disturbed. The decreased hydrophilicity in the corona
region makes the contrast of the hydrophobicity between the PPO core and PEO
corona less significant. The effective intermicellar interaction potential changes accordingly: as a consequence, the PEO chains tend to entangle with each other and
short-range attraction depth increases, namely the micelles become more 'sticky' in
response to the pressure disturbance: as soon as two micelles make contact, they may
stick together. Furthermore, in addition to this enhanced short-range attraction, the
movement of the single polymer coils from one micelle to its neighbors may result in
a long-range interaction.
At this stage, due to enhanced short-range attraction and additional long-range
interaction, a coagulation transition may occur and cause part of the micelles aggregate to form a loose open cluster called the colloidal floc. An interesting feature of
the colloidal floc is, as the structure of a colloidal floc is magnified, the appearance
does not change. Due to this unique nature, it has been proven that the colloidal floc
can be best described by using a theory of fractals, a mathematical concept dealing
with the structure of same appearance at any length scale [40]. It is shown by Chen
et al that [41], with a static structure factor
1
DF(D - 1) sin [(D - 1) tan-l (ks)]
(kR) D
1 +-(k0)2] [(- / ) ]
(kR)o
[1
87
(5.1)
(a)
A
w
9
ae
W
*
*
e
ee
.
*
0
*
I,1W.
·
a.
i-
I
9a;
a
a
a
a
e**
i a la
a
0
*
as
+iI I e ·
#*
4
*
k
r
*
*
*I
a*
*
a!
ia· i i
(b)
*
*
*
*9
i*
0
a.
i.
CY
6n
*
. i
F
k
A&, kB~cI
ll
a.. ia
! ,S
e
S
a
*
*
i
a
*
a, (aa!2
a~la
a
Figure 5-1: Illustrations of the effect of the applied pressure on the microstructures of
a micellar liquid: (a) A typical particle configuration of a micellar liquid at 4= 0.4
without applying pressure and its associated static structure factor. (b) A particle
configuration of the mixture of the liquid phase and the pressure-induced fractal
aggregates and its associated static structure factor.
88
derived from fractal pair correlation function g(r), SANS measurements are able to
provides essential parameters of a growing fractal cluster in a colloidal system such
as fractal dimension of the packing of the micellar clusters D, the correlation length
of the fractal cluster ( and its aggregation number NFRA [41]. With the presence of
the high pressure, the corresponding particle configuration is illustrated in Panel (b):
Several flocculation transitions arise from the alteration of the equilibrium structure
of the D 2 0 in response to the applied pressure and form fractal clusters. However,
most of the system still remains as micellar liquid state.
If we consider the long-
range interaction as an additional weak perturbation to the original AHS potential.
Random phase approximation (RPA) suggests that the resulting structure is of the
following form [42]:
S(k) =
where i(k)
+
SAHs(k)
PS(k)(k)
SAHS(k) [1 - 3pSAHS(
k(k)]
(5.2)
is the Fourier transform of the long-range perturbation and §AHS(k) is
the structure factor of an AHS given in Equation (3.58). Due to the complexity of
the long-range interaction, a complete knowledge of i(k) may not be available at the
present time. However, with the presence of the fractal clusters, the perturbation
term of Equation (5.2) can be approximated by Equation (5.1). Bearing with this
picture, a model for the SANS cross section is proposed as follow to fit the SANS
absolute intensity distribution
I(k) =
cNA 2
P(k) [LSAHS(k)
+ qFARSFRA(k)]
(5.3)
where c is the concentration of polymer (number of polymers/cm 3 ), N the aggregation
number of polymers in a micelle, A the contrast of coherent scattering lengths between
atoms comprising a polymer molecule and D 2 0 solvent. P(k) is the normalized
intraparticle structure factor calculated by the modified cap-and-gown model [25].
The structure factor of an ANS SAHS(k) is obtained by solving the OZ equation in
PY approximation to the first order in a series of small E expansion [30]. SFRA(k)
is structure factor of fractal aggregates composed of particles of radius R [41]. For a
89
sample of certain weight fraction, the corresponding micellar volume fraction
determined uniquely by SANS technique [25].
FRA/L
X
can
gives the volume ratio of
the fractal cluster to the micellar liquid matrix. The corresponding static structure
factor is given in the inset. A significant increase of the low angle scattering of SANS
intensity distribution is observed due to the formation of fractal clusters.
To illustrate the model fitting, a SANS intensity distribution in an absolute scale
is shown in Figure 5-2 as an example. Open symbols give the experimental data and
the solid line gives the model fitting taking into account the resolution correction.
Phenomenologically, the interaction peak in the SANS intensity distribution is primarily reflecting the width of the first diffraction peak of SAHs(k) and the significant
increase in the low angle scattering can be traced back to the contribution of SFRA.
Furthermore, in our previous reports [25, 27], we show that the dimensionless scaled
intensity k3xI(k)
where kmaxis the peak position of the SANS intensity distribution
and < 12 > is the invariant defined as < 7 2 >
fA
k 2 1(k)dk, is proportional to
the structure factor S(k) within the region of the first diffraction peak. It is well
known that the position of the first diffraction peak of S(k) reflects the mean interparticle separation of the system and the height of the peak reflects the degree of
local order surrounding a typical particle. The scaled intensity is used to visualize
evolution of the degree of local order on the molecular level as a function of applied
pressure.
Figure 5-3 gives SANS intensity distributions and their model fitting (top panel)
obtained at 44.0 wt% and 326 K and their associated scaling plots (bottom panel), as
a function of applied pressure. As the applied pressure increases, the SANS intensity
keeps decreasing and the position of the peak shift toward larger k. However, the
scaling plots suggest the system remains in the liquid phase in the whole pressure
range. The extracted fitting results are given in the Table 5-1: As pressure increases,
the aggregation number N becomes smaller, namely the size of the micelle reduces.
It can be explained as follow: The pressure perturbation smears out the contrast of
the hydrophobicity between the PPO core and PEO corona, induces removing D2 0
molecules out of the corona region (hydration number H decreases) and hence makes
90
102
101
E
t. 100
2
10-1
2
i
-
0.00
0.03
0.06
0.09
0.12
0.15
0.18
0.21
0.24
1'
k(A )
· r2
lU-
100
10-1
SFRA~k
S,,~w(k)
-2
10
.......... P k
10-3
.. ......
1-4
0.00
0.03
0.06
0.09
0.12
0.15
0.18
0.21
0.24
1'
k(A )
Figure 5-2: The SANS intensity distribution in an absolute scale (open symbols)
obtained at L64 44.0 wt%, 326 K and 70 bar, and its model fitting taking into
account the resolution correction (solid line). The red dash line gives the interparticle
structure factor SAHS(k) obtained by solving the OZ equation with a square-well
inter-micellar potential. The green dash line represents the structure factor SGEL(k)
of the fractal geometric structure. The blue dash line indicates the normalized intraparticle structure factor P(k) calculated using the modified cap-and-gown model as
the polymer segmental distribution function in a micelle. The observed SANS data
is the product of these functions (see text).
91
Table 5.1: Results for the fitting parameters of the L64 44.0 wt% at 326 K for three
different applied hydrostatic pressures.
H
N
E
u/kBT
OL
Papplied(bar)
0
70
700
850
0.518
0.516
0.496
0.490
0.03
0.03
0.021
0.018
0.88
0.93
1.34
1.61
102.88
93.42
83.24
77.59
Papplied(bar)
R (A)
D
0
70
700
850
17.79
16.42
9.37
8.11
NFRA-(=
J (A)
qFRA/0L
49.93 N/A
N/A
N/A
N/A
48.87
46.50
45.45
201
245
386
0.021
0.044
0.057
20.62
36.83
366.60
2.14
2.17
2.76
the corona region less penetrable ( decreases). The short-range attraction between
micellar corona regions thus enhances accordingly (u increases). In the fractal clusters,
as pressure increase, more micelles form fractal clusters through flocculation transition
and the fractal clusters are growing ( = 201 OFRA/OL
= 0.021 -
386, NFRA = 20.62 --+ 366.60 and
0.057) and becoming more compact (D = 2.14 --+ 2.76). From
the bottom panel, it can be seen clearly that all the scaled intensity distributions are
pressure-independent and collapse into one single master curve. Although part of the
micelles aggregate to form fractal clusters, the matrix of the sample indeed remains
in the liquid state in the whole pressure range by the fact that all the scaled peaks
are considerably broader than the resolution function of the instrument.
For the same sample, as the temperature goes down to 310 K, the scenario is
different. As an example, Figure 5-4 gives the SANS intensity distributions (top
panel) obtained at 44.0 wt% and 310 K and their associated scaling plots (bottom
panel), as a function of applied pressure.
Similar to Figure 5-3, as the pressure
increases, formation and growth of the fractal clusters (when Papplied= 70 bar
280 kpsi, R = 42.7 A NFRA = 6.40
-
41.6 Ai, D = 2.12 -
2.63, 5 = 201 A -- 342 A and
45.64) are observed by fitting the SANS intensity distributions for
k < 0.02 A-1. However, from the scaling plot as illustrated in the bottom panel,
the matrix experiences a phase transition:
92
as reported previously [25, 26, 27], an
80
70
60
'-
50
E
0 40
-
30
20
10
0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
1'
k(A- )
4 An
14
12
A
10
cJ
V
Cy)
E
8
6
4
2
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
k/kma x
Figure 5-3: Top Panel: The SANS intensity distribution I(k) in an absolute scale
obtained at L64 44.0 wt%, 326 K are shown as data points for different applied pressures. The lines show the model fittings to the data that gives the fitting parameters
in Table I. The inset gives the same data but plotted a in semi-log scale. it indicates
clearly that, as pressure increases, the low-k part of I(k) increases rapidly, signaling
the formation of fractal clusters. Bottom Panel: The scaling plot of first interaction
peak of the SANS intensity distributions given in the top panel. It shows that the
matrix is characterized by a unique length scale at various applied pressures. The
scaling peak is considerably broader than the resolution function indicates that the
system is in the liquid state. The inset shows the scaling height of all the scaled
SANS intensity distributions remains unchanged as pressure increases.
93
attractive glass state is observed at the ambient pressure. As pressure is applied,
the effective intermicellar interaction potential is altered by the variation hydrogen
bonds network of D 2 0 solvent, thus the micelles respond by a redistribution of their
spatial configuration and relax from the original attractive glass state through an
intermediate state with less degree of local order and finally to a liquid state. These
effect of pressure can be observed clearly from the scaling plots given in the bottom
panel: The degree of local order decreases as pressure increases.
Figure 5-5 gives the scaling plots of SANS intensity distribution of 44.0 wt% L64/D 2 0
micellar solution taken at ambient pressure, 70 bar and 280 bar respectively and at
different temperature. It can be seen that, with the presence of applied pressure, the
KGT boundaries become less well-defined: intermediate states are observed between
the attractive glass state and the liquid state.
94
100
o
El
v
A
80
-
l
l
·
l
60
Obar
1
70bar
140bar
280bar
280bar fitti
o
50
60
E
0
20
oo
C)~
40
20
130
V
o0
o
20
.(00
0
0.005 0.010 0.015 0.021
k (A-')
X XvX~
n 1
0.00
B~r
uuuu-
-
0.02
A
.
0.06
0.04
_
=AOM"M
0.08
41ER
n*MOffl l
L '
0.10
0.12
0.14
0.16
0.18
0.20
1'
k(A )
140
II
I
__
~·c~
I~
J =
I~l
0
Obar
70bar
v
14obar
120
A
100
120
o
i
A 280 bar
......... RF
00
cJ
V
E
-
91
80
60
80
L64 44.0 wt% 310 K
40
i'
40
20
V ;
o
60
I
40
I
I
I
Pressure (bar)
20
r~r~3~e~
13
0.2
I
50 10015020025030
~
nnz~/i
, "
-~~~E
0.4
v
A
'
0.6
1.0
0.8
1.2
1.4
1.6
1.8
k/kmax
Figure 5-4: Top panel gives the SANS intensity distribution I(k) in an absolute scale
obtained at L64 44.0 wt%, 310 K for different applied pressures and the inset gives
the same data but plotted a in semi-log scale. Similar to the Figure 5-3, the formation
of the fractal clusters is observed as applied pressure increases. However, as well as
the formation of the fractal aggregation, the associated scaling plots (Bottom panel)
indicates the matrix of the colloidal system transits from an attractive glass state
(with a scaling height of 120) [25, 26] through an intermediate state (with a scaling
height about 80) then finally to a liquid state (with a scaling height of 60).
95
An
4
14U
A
C;.
120
100
V
80
60
E
40
v~
I~
20
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
I
0
L64 44.0 wt%
bar
70
1.6
1.8
k/kmax
140
A
C\J
I
I
I
I
120
c
100
V
,i
I
40
60
E
_
40
_X
*0 0'<~
.C
oa
20
0
0 .2
CO
8.
00
X
_
0.6
*
321K
323K
.329K
280 290 300 310 320 330 340
0.4
293 K
298 K
303K
306K
o0
**8
Temperature
(K)
.
a
a
v
. ' 310K
'.
80
I
*
RF
X
P-.I
0.8
1.0
..
-
1.2
1.4
1.6
1.8
1.2
1.4
1.6
1.8
k/kmax
"
A
14U
120
A
cv
100
80
C-
E
60
40
20
0
0.2
0.4
0.6
0.8
1.0
k/kmax
Figure 5-5: The scaling plots of SANS intensity distribution of 44.0 wt% L64/D 2 0
micellar solution taken at ambient pressure, 70 bar and 280 bar and at different
temperature.
96
Chapter 6
Conclusions
In conclusion, we use PCS to show that L64/D 2 0 micellar system follows the overall
structural arrest transition behavior predicted by MCT, calculated using a square well
potential with a short-range attraction relative to the micellar size. In particular, we
show experimentally, the existence of a glass-to-glass transition line which starts at
point C*, where the two glass phases and the liquid phase coexist, and ends at point
A3 , where the two glass phases merge. With this model potential, we are able to
extract the corresponding volume fraction for a certain polymer weight fraction by
SANS model fitting and therefore pinpoint the exact volume fractions where the C*
point and the A 3 singularity are located and map out the whole KGT boundaries in
the L64/D 2 0 micellar system. The SANS experiment further shows that while the
local structures of the attractive and the repulsive glasses are in general different, they
become identical at the predicted volume fraction of the A 3 singularity, independent
of temperature.
However, our PCS results indicates that the relaxation of the two
glasses are different in the intermediate time region even at the A 3 point.
Moreover, SANS reveals that when the system transforms from liquid state into
glassy state, initially the structure factor in the glassy state is similar to the one of the
liquid state, which is consistent with the assumption of MCT. Therefore, we conclude
the distinct difference in local structure observed in an extensive set of SANS intensity
distributions is due to the aging effect occurring during the measurement time.
We propose a model for SANS intensity distribution which allows us to extract
97
information on the effect of pressure on the microstructure of the L64/D 2 0 micellar
system. Under an increased pressure, the distribution of the hydrogen bonds among
D 2 0 molecules and between the D 2 0 molecules and the polymer segments are altered.
This in turn changes the range and the strength of the attraction between micelles.
By fitting the first interaction peak of the SANS intensity distribution, our model
suggests that the micelles becomes much stickier upon increasing the pressure. In
the liquid state, a significant increase in the low angle scattering of SANS intensity
distribution is observed when the pressure is applied, signaling the formation of the
fractal clusters. Furthermore, as pressure increases, the fractal clusters become larger
and denser. In the attractive glass state, in addition to the formation of fractal
clusters, a transition of the matrix from the initial attractive glass state to a new
intermediate state then finally to a liquid state is observed when pressure increases
from ambient pressure to 280 bar. We are in the process of investigating it at the
present time.
The following is a list of publications related to my work toward to the degree,
which includes publications included in this thesis and publications not included in
this thesis.
Wei-Ren Chen, Yun Liu, Francesco Mallamace, Pappannan Thiyagarajan and
Sow-Hsin Chen, "Studies of Structural Arrest Transition in L64/D 2 0 Micellar Solutions." Journal of Physics: Condensed Matter, 2004. accepted, in press.
Wei-Ren Chen, Francesco Mallamace, Charles J. Glinka, Emiliano Fratini and
Sow-Hsin Chen, "Neutron- and Light-scattering Studies of the Liquid-to-Glass and
Glass-to-Glass Transitions in Dense Copolymer Micellar Solutions." Physical Review
E., 68:041402, 2003.
Sow-Hsin Chen, Wei-Ren Chen and Francesco Mallamace, "The Glass-to-Glass
Transition and Its End Point in a Copolymer Micellar System." Science, 300:619,
2003.
Francesco Mallamace, Wei-Ren Chen, Piero Tartaglia, Antonio Faraone and SowHsin Chen, "Viscoelastic processes in non-ergodic states (percolation and glass transitions) of attractive micellar systems." Physica A., 330:206 2003.
98
Wei-Ren Chen, Sow-Hsin Chen and Francesco Mallamace, "Observation of Liquidto-Glass and Glass-to-Glass Transitions in L64/D20 Triblock Copolymer Micellar
System." Molecular Simulation, 29:611, 2003.
Wei-Ren Chen, Francesco Mallamace and Sow-Hsin Chen, "Kinetic Glass Transition in a Copolymer Micellar System with Temperature-Dependent
Short-Range
Attractive Interaction.", in Self-Assembly, Edited by B. H. Robinson, ISO Press,
2003
Sow-Hsin Chen and Wei-Ren Chen, "Neutron and Light Scattering Studies of the
Liquid-to-Glass and Glass-to-Glass Transitions in a Copolymer Micellar System", in
Slow Dynamics In Complex Systems, Edited by Michio Tokuyama and Irwin Oppenheim, Springer Verlag, 2004.
Francesco Mallamace, Wei-Ren Chen and Sow-Hsin Chen, "Observation of a Reentrant Kinetic Glass Transition in a Micellar System with Temperature-Dependent
Attractive Interaction." European Physical Journal E., 9:283, 2002.
Wei-Ren Chen, Francesco Mallamace and Sow-Hsin Chen, "Small-Angle Neutron
Scattering Study of the Temperature-Dependent Attractive Interaction in Dense L64
Copolymer Micellar Solutions and Its Relation to Kinetic Glass Transition." Physical
Review E., 66:021403, 2002.
Wei-Ren Chen, Francesco Mallamace and Sow-Hsin Chen, "Observation of LiquidGlass-Liquid and Glass-Glass Transitions in the L64/D 2 0 Copolymer Micellar Sys-
tem." Annual Report of NIST Center of Neutron Research, 32, 2002.
99
100
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