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The kinetics of surface-mediated phase separation in the quasi-binary mixture of guaiacol-glycerol-water

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The kinetics of surface-mediated phase separation in the quasi-binary mixture of guaiacol-glycerol-water
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Shi, Qingbiao, 1963-
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viii, 120 leaves : ill. ; 29 cm.

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Binary fluids ( jstor )
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Kinetics ( jstor )
Light scattering ( jstor )
Metastable atoms ( jstor )
Molecules ( jstor )
Phase diagrams ( jstor )
Silanes ( jstor )
Water temperature ( jstor )
Wetting ( jstor )
Dissertations, Academic -- Physics -- UF
Physics thesis Ph.D
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Thesis:
Thesis (Ph. D.)--University of Florida, 1994.
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Includes bibliographical references (leaves 116-119).
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Typescript.
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Vita.
Statement of Responsibility:
by Qingbiao Shi.

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THE KINETICS OF SURFACE-MEDIATED PHASE SEPARATION
IN THE QUASI-BINARY MIXTURE OF GUAIACOL-GLYCEROL-WATER









By


QINGBIAO SHI


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


1994




THE KINETICS OF SURFACE-MEDIATED PHASE SEPARATION
IN THE QUASI-BINARY MIXTURE OF GUAIACOL-GLYCEROL-WATER
By
QINGBIAO SHI
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1994


Dedicated to my grandmother
FENG-HWA FU


ACKNOWLEDGMENTS
First of all, I would like to thank my supervisor Andy Cumming for his
efforts and patience throughout the whole period that I worked on the project. I
was very much in a limbo when he took me on board. Starting from the ground
up, I learnt from him not only the basic light scattering techniques and the
setups, but also how to be an experimentalist. Furthermore, he was a constant
source of ideas and encouragement which were essential to push this project
through.
Also, I want to express my deep thanks to Wade Robinson, our resident
engineer in the lab, who had helped a great deal in putting the lab together and
in keeping it running smoothly. He had to endure the almost constant pestering
from me and other students to get this or that done, sometimes even just to finish
other's dirty work. Without his patience and help, our lab would not have been
running. Thanks are also in order to my fellow students in the lab, especially to
James Ellis Teer, who has done all the silane treatments of the windows, and to
Bill Rippard, who did most of the microscopy studies for the project.
There are some other people in the University of Florida I would like to
thank: first to the other members on my supervisory committee, Jim Dufty, John
Klauder, Neil Sullivan and Randy Duran, for patiently sitting through both my
oral exam and then thesis defense, and providing insightful suggestions and
comments; Chuck Hooper, who was always ready to help if needed; and the
fellow graduate students in the physics department, especially Mike Jones and
Laddawan Rumsuwan, for their help throughout the years.
On a more personal side, I want to express of deep gratitude to Richard
Trogdon and Suzy Spencer, for their friendship and help, especially when I first
came to this country and everything was foreign and you were an alien. Their
m


patience and warmheartedness were indispensable for me to get over the initial
"culture shock" and the language barrier.
Heartfelt thanks go to Rob and Cynthia, for their friendship over the
years, even after they had moved away. Their enthusiasm and encouragement
are deeply appreciated, especially when I felt down and out.
Finally, to a special friend, C. J., I want to say thanks.
IV


TABLE OF CONTENTS
gage
ACKNOWLEDGEMENTS iii
ABSTRACT vii
CHAPTERS
1 INTRODUCTION 1
1.1 Phase Diagram 4
1.2 Mean Field Theory of Metastability and Unstability 9
1.3 Cahn's Linear Theory of Spinodal Decomposition 14
1.4 Later Stage Coarsening 20
1.4.1. Droplet Coalescence 21
1.4.2. Lifshitz-Slyozov Theory 24
1.4.3. Hydrodynamic Growth Mode in Concentrated
Mixture 26
1.4.4. Scaling Hypothesis for Structure Function at
the Later Stage 29
1.5 Experimental Results 33
1.6 Wetting Phenomena and Phase Separation 36
2 LIGHT SCATTERING METHOD: CONCEPTS 42
3 APPARATUS 48
3.1 Light Scattering Apparatus 49
3.2 Temperature Regulation and Quench System 57
4 EXPERIMENTAL PROCEDURE 65
4.1 Sample Preparation 66
4.2 Phase Diagram of Guaiacol-Glycerol-Water Mixture 67
4.3 Sample Cell and Carrier 70
4.4 Data Acquisition and Processing 73
4.5 Treatment of Glass Surface with Trichlorosilanes 75
5 RESULTS AND DISCUSSIONS 81
5.1 Data Analysis 81
5.2 Phase Separation Kinetics 94
5.3 Gravitational Effects 100
5.4 Surface Treatments and Their Effects 103
v


5.5 Discussions 109
6 CONCLUSIONS 112
APPENDIX
SAMPLE CELL CARRIER DESIGN 114
REFERENCES 116
BIOGRAPHICAL SKETCH 120
vi


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
THE KINETICS OF SURFACE-MEDIATED PHASE SEPARATION
IN THE QUASI-BINARY MIXTURE OF GUAIACOL-GLYCEROL-WATER
By
Qingbiao Shi
April, 1994
Chairman: Dr. Andrew W. Cumming
Major Department: Physics
The kinetics of phase separation in a quasi-binary mixture of guaiacol-
glycerol-water has been investigated using a time-resolved light scattering
method following a sudden quench into the thermodynamically unstable state of
the mixture. The mixture was confined between two optically transparent
windows with thickness less than 1 mm. In addition to the common growth
mode in the bulk, where the average domain size, L, grows as t1/3 (where t is
time), a novel growth mode was observed that the domain size L grows as ft,
where b increases from 1.1 to about 1.5, depending on the quench depth. As the
same novel mode had been observed earlier in a polymer mixture, it is thus
established that it is generic to all classes of binary fluids.
Experimental results from both the light scattering and video-microscopy
suggested that the novel growth mode was closely associated with wetting
phenomena of the two separating fluid phases near the glass surfaces. It was
also dependent on the properties of the glass surface, which could be altered
with a self-assembled monolayer of silane molecules. This experimental
Vll


evidence confirmed that the novel mode of phase separation was surface
mediated.
vin


CHAPTER 1
INTRODUCTION
The phenomena of phase-separation have been observed in a wide range
of systems with two or more components. It has been an increasingly important
process in material manufacturing and processing. Therefore comprehensive
understanding is needed in both physical and chemical aspects of the
phenomena. Over the last 30 years, experimental investigations have been
conducted in alloys [Gaulin et al., 1987] and solid solutions [Hono and Hirano,
1987], simple binary solutions [Goldburg, 1983] and glass forming mixtures
[Tomozava et al., 1970], and more recently, in liquid-polymer solutions [Sasaki
and Hashimoto, 1984] and polymer-polymer blends [Hashimoto, 1988; Bates and
Wiltzius, 1989], etc. On the theoretical side, efforts have been concentrated on
elucidating the growth mechanisms that govern the kinetics of phase separation,
especially those growth kinetics that are universal, depending only the general
properties of the phase transition, e.g., the symmetries of the system and the
order parameter, but not on specifics of the system.
Significant progress has been achieved so far in our understanding of the
kinetics of phase separation, both in experiment and theory, especially in the last
two decades. In classical theories, phase separation proceeds either through
spinodal decomposition or through nucleation and growth [see, e.g., Gunton et
al., 1983]. The former leads to the decay of an thermodynamically unstable state,
through long wavelength fluctuations of infinitesimal amplitude. The latter is
1


2
due to the instability against localized (droplet-like) fluctuations with finite
amplitude, and that leads to the decay of a metastable state. This distinction is
clear in the mean field theory, as the unstable and the metastable states are
separated by a so-called spinodal line in the phase diagram. Although later
development has shown that the spinodal line is not well defined and the
transition is not clear cut, it is still very helpful to distinguish these two from a
thermodynamic point of view.
This chapter is a brief summary of the theoretical models, and a general
comparison of their results with the experiments. As a number of excellent and
extensive reviews have been written in the field [e.g. Gunton et al., 1983], here I
will only discuss the key elements, and all the discussions will be concentrated
on binary fluids from now on. First the phase diagram is presented along with
the equilibrium properties, its general features can be readily understood in
terms of some elementary thermodynamic arguments. And then I will discuss
the thermodynamic unstability and metastability in terms of mean-field theory.
Next Cahn's linear theory of spinodal decomposition is described, from which
we can clearly see the difference of the early stage of decay from the unstable or
metastable state. Some nonlinear modification is also mentioned, and it will be
helpful later on to understand how the domain size is measured in a scattering
experiment. Section 1.4 is a description on three key growth modes at later
stages that have been clearly identified so far: droplet coalescence, evaporation
and condensation [Lifshitz and Slyozov, 1961] and the hydrodynamic mode due
to E. Siggia [Siggia, 1979]. Also in the same section, the dynamic scaling theory is
discussed [Lebowitz et al., 1982], And then a general comparison between
experiment and theory is presented in section 1.5. Finally, we review the recent
experimental observations and a general discussion on the wetting phenomena,
which provided the motivations for the present work.


Temperature
3
1-PHASE REGION
(Miscible)
FIG. 1.1 The coexistence (solid) line and the spinodal line (dashed) in
classical theory (mean field theory) for binary fluid. Two types of quenches
are differentiated: (1) into the metastable state which is bounded by the
coexistence and spinodal lines; (2) into unstable state bounded by the spino
dal curve. In the first case the system is quenched into the metastable state
by the depth ST, which corresponds to an initial supersaturation <5C= C1 CA.
The phase separation eventually results in two phases, a major phase with
concentration CA, and a minor phase of CB. The volume ratio of the two
phases is given by the ratio 5C/(AC-5C), where AC = CB- CA.


4
1.1 Phase Diagram
A generic phase diagram of binary fluids is shown in Fig. 1.1. The
coexistence curve, also known as the binodal, divides it into two regions: the
miscible region (one phase region or disordered phase) and the immiscible
region (two-phase region or ordered phase). For binary fluids, the phase
transition between the miscible phase and the immiscible phase is of second
order, with the critical point at temperature Tc and critical concentration Q- It is
well known that the equilibrium properties of the fluids around the critical point
belong to the universality class of the 3-d Ising model, where the properties
around the critical point can be determined through:
the correlation length £ = <^o| t \ ~v, (1.1a)
where the critical exponent v = 0.63, and t= (T Tc)/Tc, the reduced temperature,
and
the isothermal compressibility Kj ~ 111 ~y- (1-lb)
with the corresponding exponent y= 1.3. All the other critical exponents can be
determined once two of them are known.
The qualitative features of the phase diagram can be readily understood in
terms of the free energy F, the internal energy U and the entropy S:
F=U-TS
where T is the temperature in Kelvin. The equilibrium state has minimal free
energy for given external conditions. For most fluids, the interactions between


5
the molecules are the van der Waals force, which is an attractive force due to the
fluctuating induced dipole moments of the molecules. In most cases, the
attractive force between the same species of molecules is much stronger than that
of different species. Then the internal energy U will be minimized when the
same species of molecules stay together, thereby favoring the phase separation.
On the other hand, the entropy term has the temperature as a prefactor, so it is
more important at high temperature. The entropy term will dominate over the
internal energy above a certain temperature, the system will try to maximize its
entropy and thus lower the free energy, then the fluids are miscible. At the other
end, when the temperature is low, then the entropy term has minimal effects on
total free energy F, and the internal energy term is more important, which is
typically minimized when the like molecules stay together, so the fluids will try
to keep away from each other. This explains the immiscibility at low
temperature.
Most binary fluid systems have a similar phase diagram as Fig. 1.1, which
is known as the upper critical solution temperature (UCST) type. There are other
classes of mixtures whose phase diagrams are quite different; Fig. 1.2 lists several
other common types. The opposite type to UCST is called the lower critical
solution temperature (LCST) and behaves as shown in Fig. 1.2b, which has the
immiscible phase at high temperature. This kind of behavior is due to some
specific interactions between the different species of molecules in the mixture,
e.g., hydrogen bonding. Some binary fluids can even show both UCST and LCST
behavior, as described in Fig. 1.2c and 1.2d. Here I just want to mention
particularly the closed-loop type of phase diagram as Fig. 1.2c, which shows a
miscibility gap between the two critical temperature and Tc/, because the
system involved in this study Guaiacol-Glycerol-Water (GGW) shows this
type of phase behavior.


Temperature Temperature
6
a
B
i-i
d)
CL,
g
a>
H
Concentration
Concentration
Vh
4-*
a>
a
6
ai
H
Concentration
Concentration
FIG. 1.2. Four types of phase diagrams of binary fluids are shown here
schematically. The shaded area and beyond are the two-phase regions
bounded by the coexistence curve, (a) is the most common, espcially if
the fluids are nonpolar; All the other types usually involve some types
of specific interaction, e.g., hydrogen bonding between the molecules.
The phase boundary is determined by the delicate balance of energy
and entropy due to these interactions among the fluid molecules.


7
The closed-loop phase diagram, also known as a reentrant phase diagram,
is due to the hydrogen bonding among the unlike species of molecules. One of
the peculiarities of the hydrogen bond is that it is highly oriented; the bond will
break if it sways beyond 10 degrees away from its optimal orientation axis. Due
to this feature, hydrogen bonding effectively freezes the orientational degrees of
freedom of the molecules, and hence decreases the compositional entropy of the
mixture. At low temperature, the entropic contribution to the free energy will be
minimal, so the hydrogen bonding between the unlikes strongly favors mixing.
The close loop phase diagram can be understood as follows: at high temperature
the fluids are miscible and the entropy is high. In this phase, the molecules mix
together and are oriented randomly with respect to each other, and the entropy is
maximized. As the temperature is lowered, internal energy due to van der Waals
attraction becomes more important. At some point, it will dominate over the
entropy contribution to the free energy and the fluids become immiscible,
because the like molecules stay together which lowers internal energy. Now if
we lower the temperature even further, the hydrogen bonding comes into play.
And the mixture becomes miscible again by forming hydrogen bonds between
the unlike molecules if the temperature is low enough, so the fluids reenter the
miscible phase.
The close loop type of phase diagram can be elegantly fitted with the
Walker-Vause (WV) model [Walker and Vause, 1980; Walker and Vause, 1983],
which is an Ising-like lattice model, but each spin site can assume q different
states. The Hamiltonian H of the model is
H= 1 [K1(l-8ss)5Ci0. + K2(l-5Ssi)(l-50i0)]
11 1 1
(1.2)


8
where the summation is over the nearest neighbor lattice sites. There are
two sets of variables: sj and aj, the spin variable sf can be +1 (spin up) or -1 (spin
down), which can also be designated as molecule species A or B respectively. The
other site variable, cr/, can take values 1, 2, 3, q, which designates different
orientational states of the molecules. If the two neighboring A and B molecules
are in the same orientation state, i.e. = <7g, then the energy is Kj/ if not, the
energy will be K.2- If we let Kj < fy, the Hamiltonian will simulate approximately
the case of hydrogen bonding between the neighboring sites when they are in the
same CT-state, and the number q is a measure of directionality of the hydrogen
bond.
The phase diagrams of many binary fluid systems that exhibit the close
loop have been fitted successfully by the WV model [Walker and Vause, 1983;
Johnston, 1983], including the GGW mixture used in this study. From the
molecular structures of guaiacol (2-Methoxyphenol) and glycerol (1,2,3-
Propanetriol) and Water (Fig. 1.3), we can see the presence of multiple hydroxyl
groups -OH, and the hydrogen bonding is a dominating factor in the mixing of
the components. Strictly speaking, the system is a quasi-binary system, with a
small amount of H2O used in the mixture (less than 6% of the glycerol mass).
Without the presence of this small amount of water, the guaiacol and glycerol
will be miscible in any respective amounts. Only when a small amount of water
is added to the mixture, a miscibility gap opens up in the phase diagram. The
size of the immiscibility loop depends on the amount of water added. As it is
well known, glycerol is highly hygroscopic; it will readily absorb the moisture
from the surrounding air if the container is left open. On the other hand, the
solubility of guaiacol in water is rather low; 70 ~ 80 cm3 of water can only
dissolve 1 g of guaiacol at room temperature. Therefore, most of the water


9
present will dissolve in glycerol, and glycerol-water effectively forms one
component in the quasi-binary mixture, with guaiacol as the other .
OCH3
Guaiacol (2-Methoxyphenol)
CH2 OH
CH OH
CH2 OH
Glycerol (1,2,3-Propanetriol)
FIG.1.3 The molecular structures of guaiacol and glycerol. Due to the presence
of the hydroxyl group OH, hydrogen bonds can form between these groups of
neighboring molecules. Water is also a strong hydrogen-bonding molecule.
1. 2. Mean Field Theory of Metastabilitv and Unstabilitv
Let us go back to Fig. 1.1 and explain how a phase separation experiment is
done in general. Assume a binary mixture is made up of two fluids, A and B,
and the concentration of A is Cj.
Initially, the fluid mixture is set at temperature T which is close to the
coexistence curve but in the one-phase region. The fluid is miscible and is
optically homogeneous. The temperature is suddenly and quickly changed to Tf,


10
inside the two-phase region. The sudden change in temperature is called a
quench. Once inside the two-phase region, the fluid is either unstable or
metastable. It will demix into the final equilibrium (stable) phases, which will
have concentrations Ca and Cg that are at the coexistence curve at the
temperature Tf. The final state of the fluid mixture is phase separated, and it
consists of 2 phases of fluids: one is A-rich (concentration Ca), the other A-poor
or B-rich (concentration Cg). To study the kinetics of phase-separating fluids,
various experimental techniques can be used to monitor the growth of the
domains after a quench.
The phase separation process starts as soon as the fluid is inside the two-
phase region of the phase diagram. It is a nonequilibrium process because it
concerns the transition from an unstable or metastable state back to equilibrium.
Phase-separation kinetics is the study of this process of how the initially
homogeneous fluid mixture demixes into heterogeneous phases. Some technical
jargon should be made clear here. The temperature difference AT = I Tf- T, I is
called the quench step, but a more relevant quantity is <5T = Tco Tf, the quench
depth (see Fig. 1.1); Tco is the temperature at which the fluid first crosses the
coexistence curve. If the fluid is of critical concentration Cc, then Tc0 = Tc, the
critical temperature, as shown in Fig. 1.1 by quench (2).
In the mean-field-approximation picture, the two-phase region is further
divided into two regions: unstable and metastable. The boundary line between
the two is shown as the dashed line in Fig. 1.1. This line is the so-called spinodal
line. The fundamental assumption in the mean-field approximation is the
existence of the free energy F as an analytic function of thermodynamic variables
temperature T and concentration C, even inside the two-phase region. This may
not to be true in general, because these are not equilibrium states inside the two-
phase region. But as a starting point, the mean field picture is still very useful in


11
understanding the phase transition, and even some general features in the phase
separation.
The mean field theory starts with the Ginzburg-Landau-like free energy
functional F (C, T), where C is the concentration in the case of binary fluids, and T
is the temperature. In general, an order parameter y/ corresponding to the phase
transition will take the place of the concentration C, e.g., the magnetization M in
the case of ferromagnetic-paramagnetic transition.
The Ginzburg-Landau free energy functional F(C,T)
(1.3)
where f (C,T) = hi (C C0)2 + Ui(C C0)4
2 4
(1.4)
and a = A(T-TC), A > 0, k, u are positive constants. The actual thermodynamic
equilibrium state is determined through the minimization of F:
= 0, as the
equilibrium means the lowest energy state possible, therefore most stable. Notice
that a changes from positive to negative as T is varied from the upper side of Tc.
Then the shape of f(C,T) as a function of C changes as T crosses Tc: At T > Tc, F
has only one minimum at C = Co; (see Fig. 1.4 for details). At T < Tc, Co is no
longer an absolute minimum, instead two minima appear at
(1.5)
The coexistence curve (binodal line) is determined by the equation:


12
P =
and we can introduce a susceptibility
(1.6)
(1.7)
and the line defined by x = 0 is the spinodal line. Inside the two-phase region of
the phase diagram (see Fig. 1.1), the area between the binodal and spinodal line
corresponding to the metastable state, since with x >0 ,a small fluctuation of the
concentration CO) will increase the energy, an energy barrier which needs to
overcome before the system can decompose and phase separation can occur. In
contrast, the area inside the spinodal line corresponds to the unstable state,
because with x < 0 any spontaneous small fluctuation will drive the system to
lower free energy, and the system can phase separate without having to
overcome an energy barrier.
The phase separation process is thus the decay of the unstable or
metastable state into equilibrium states. Due to the difference in metastability
and unstability, the phase separation proceeds differently in general. If the fluid
is quenched into the unstable region of its phase diagram, the phase separation is
said to proceed via spinodal decomposition. On the other hand, it proceeds via
nucleation and growth after being quenched into the metastable state. From the
phase diagram, we can see that the unstable region usually has the initial
concentration fairly close to the critical concentration Cc, while the metastable
part corresponds to concentration further away from Q, for off-critical quenches.


Free Energy Density f(C,T)
13
Concentration
FIG. 1.4 The Ginzburg-Landau form of the free energy density f(C, T)
(eq. (4)) at tempertures around critical temperature Tc. At T > Tc,f has
only one minimum at Cg. But at T < Tc, double minima are present,
indicating two phases. The coexistence curve (binodal line) can be
derived from f(C,T) by eq.(1.6) and the spinodal line by the susceptibility
X = 0 (see eq.(1.7)).


14
It is important to determine the validity of the mean-field theory.
Remember the basic assumption in the theory is that the fluctuations in the
system is small compared to the mean value (statistical average). In many cases,
it is not valid around the critical point of a phase transition, where the
fluctuations are significant. Furthermore, the existence of a free-energy functional
in a nonequilibrium state, as inside the two-phase region, is somewhat in doubt.
Therefore the validity of the mean field treatment is uncertain, especially near the
spinodal line. In fact, it has been shown using the renormalization group method
that the spinodal line shifts as a function of the renormalization size L [Langer,
1974; Binder et al., 1978; Kaski et al., 1983]. It is now generally accepted that the
spinodal line is not sharp, and the transition from metastable and unstable states
is gradual. However, despite all these problems, the mean field theory still
provides a simple and elegant picture about the phase transition and its
dynamics.
1. 3 Cahn's Linear Theory of Spinodal Decomposition
Cahn's treatment of spinodal decomposition is one of the pioneering steps
toward the understanding of the general phenomena of the dynamics of phase
transition. In a series of papers [Cahn, 1961; 1962; 1966; 1968], Cahn outlined a
linearized theory of spinodal decomposition, and concluded an exponential
growth of the order parameter occurs at the early stage following the quench into
the unstable region of the phase diagram. In the case of phase separation in
binary fluids, the order parameter is the local concentration C(r), which is
conserved (the molecules of each species do not simply disappear.), in contrast to
some other systems where the order parameter is not conserved. A simple case


15
with nonconserved order parameter is the Ising-like spin system, where spins are
allowed to flip. This system can be made to simulate the binary fluid system
when the spin flipping is prohibited (so-called spin-exchange models), then the
number of up-spins or down-spins is conserved.
The linearized theory can be generally regarded to describe the phase
separation as a diffusion process. The diffusion constant D is proportional to
which is the susceptibility, as defined in eq. (1.7). Inside the unstable region,
where ^ < 0, D is negative, the diffusion is along the concentration gradient,
therefore achieving phase separation. Conservation of the order parameter is
expressed in the continuity equation:
(1.8)
where j is the interdiffusion current
(1.9)
j = M V/i (r),
and M is the mobility, and /i(r) is the local chemical potential, which can be
related to the free energy F :
[i (r) =
8C(r)
(1.10)
So if we use the Ginzburg-Landau free energy functional F shown in (1.3), we get
the following equation
^^- = MV2{-iC V2C + ^ }
(1.11)


16
Cahn linearized this nonlinear equation about the average concentration Co, and
obtained
where
du( r)
dt
= -M
K
V2 +
u( r)
(1.12)
u(r) = C(r) C0 (1.13)
2
Note in the long wavelength limit, the term K V can be ignored. Then we have
a diffusion equation, with a diffusion constant
D = M¡^-\ (1.14)
\dC2 Co
Fourier transform eq.(l-12) respect to space leads to the following
= CO (k) u (k)
dt
here u (k) is the Fourier component of u(r) and
oik) = -MKk2(k2 + £-1 1 )
\5C2 jco
(1.15)
(1.16)
and the shape of co as a function of k is shown in Fig. 1.5. Inside the spinodal
region where x is negative, co is positive for k < kc where
(1.17)


17
So the long wavelength fluctuations will grow exponentially in the spinodal
region. Notice co reaches its maximum at km = kc/i2 .
The experimentally more relevant quantity is the structure function S (k, t)
= \u (k)fj, which is the Fourier transform of the two-point correlation function.
It can be measured by the scattering methods commonly used in physics (X-ray,
light, etc.). Cahn's theory thus predicted an exponential growth
S (k,t) = S (k,0) exp( 2 oXk) t) (1.18)
Therefore at the initial stage, the spinodal decomposition should undergo an
exponential growth in time in the scattering intensity for k < kc, and the intensity
profile has a peak at km which does not change during this stage of growth.
Due to the fast growth, the peak wave number km will soon dominate over the
other length scale. As a result, km characterizes the typical size of the domain
seen in spinodal decomposition experiments.
The linearization can only hold at the very early stage of spinodal
decomposition when the fluctuation amplitude is small, and beyond that
nonlinear effects will become important. This exponential-growth stage passes
too quickly to be observed experimentally in most of the physical systems such
as binary fluids, with the exception of the polymer blends, where due to its high
viscosity, the dynamics is slowed down tremendously.
Here it is convenient to point out the difference in dynamics if the order
parameter y/ is not conserved. The corresponding equation to eq.(1.12) can be
2
obtained by replacing M V in eq. (12) by M,


co(k)
18
FIG. 1.5 The growth rate as predicted by the linear theory for (a) a
unstable quench, (b) a metastable quench. In the linear theory, the peak
which has maximum rate is wave number km, which doesn't change with
time. But if nonlinear effect are incorporated, km will shift toward smaller k
as time proceeds.
dt
= -M
K V2+(-^-)
Co
vKr)
(1.19)
and the corresponding rate of growth


19
oXk) = M
(1.20)
Here the maximal growth occurs at k = 0 instead of at some wave number km-
An important extension to Cahn's linear theory was made by Cook [Cook,
1970]. He observed it was necessary to add a noise term to eq. (1.12) to have a
correct statistical description of the dynamics in alloys. The noise arises from the
random thermal motion of the atoms or molecules in the system. Within the
context of linear theory, the noise term does not affect the major results we have
discussed so far.
Attempts have been made to include the nonlinear effects. Immediately
we encounter an equation that includes higher order correlation functions, and it
leads to the typical hierarchy of coupled equations. So some kind of truncation
approximation has to be employed. One of the simplest schemes is due to
Langer, Bar-on and Miller (LBM) [Langer et al., 1975], which assumes the odd
order-n two-point functions are zero, and if we only keep the first higher order
correlation function S^(k), which is related to the normal, or second order, two-
point correlation function S(k) (= S2(k)) as follows
S4a) = 3 (m 2) s(k)
(1.21)
with
(1.22)
then the correction to the linear theory is the replacement of
Co
by


20
(1.23)
Consequently kc will decrease as (u 2(f)) increases with time, (see eq. (1.17)), and
the peak km in the scattering profile will shift toward to lower k, indicating
growth of the domain structure. This coarsening behavior is qualitatively
accurate for the early stage of spinodal decomposition.
Another effect that is specific to fluids is hydrodynamic motion, which has
not been included into consideration so far. Efforts have been made to take it
into account [Kawasaki and Ohta, 1978], and it brought significant modification
to the LBM results. But its agreement with experimental data is still only
qualitative. Hydrodynamics will have a significant effect at the later stage of
coarsening, and we will discuss it in the next section.
1.4 Later Stage Coarsening
It is difficult to develop a theory for the later stage, that is the coarsening
process of phase separation. The main difficulties stem from the nature of
nonequilibrium and nonlinearity in the phase separation process itself. So far
our understanding at the later stage of phase separation remains qualitative.
Most of the theories attempt to find universal growth laws and mechanisms in
some simple scenarios. But in real systems there is no way to make these
distinctions, many of these mechanisms are at work at the same time. The
following discussion in section 1.4.3 is largely based on the paper by Siggia
[Siggia, 1979], who discussed the key coarsening mechanisms responsible at the
later stage of phase separation, especially the hydrodynamic effects.


21
The division of the phase separation process into early and later stages is
somewhat vague. Qualitatively the later stage starts after domains of two
distinct phases have formed, i.e., the concentrations of the domains have reached
Ca and Cg, respectively, so the driving force is mainly derived from the
minimization of the domain wall, namely the reduction of the interface area and
thickness.
1.4.1 Droplet Coalescence
Droplet coalescence, also known as coagulation, is the recombination
process of small droplets (usually of liquids) upon encounter. As a result, larger
droplets form as more and more small ones bump into each other and recombine
into larger ones. It is readily observed in colloids and aerosols, in which
aggregates form. In binary fluids, we can imagine the minor phase may form a
suspension of small droplets in the medium of the majority phase at later stage,
(in the case of Fig. 1.1, the majority phase is the A-rich phase with concentration
Ca, and the minor phase is B-rich with concentration Cg). Small droplets have
larger surface-to-volume ratio compared to larger ones, so the energy can be
reduced by recombining into large droplets.
Consider a droplet with radius Ri, fixed at the origin and surrounded in a
mist of droplets with radius R2, with number density n0. The motion of the
droplets are Brownian, i. e., diffusive. And the number of droplets that collide
into the drop in the origin and thus recombine can be calculated through the
diffusion equation:
= nli ir? \
dt r23r| dr I
(1.24)


22
with the boundary conditions
n =n0
n = 0
n =n0
The number of collisions per unit time
as
for r>Ri+R.2 at t = 0
for r = Ri + R2 t > 0, (1.25)
for r >
can be calculated (see, e.g., Levich, 1962)
I = 4 n (Ri + R2 ) D2n0, (1.26)
where D2 is the diffusion constant of droplets with radius R2, which are related
through the Einstein relation
D2 = ^ (1.27)
6 n pvR2
where p is the density, v is the kinetic viscosity of the medium, and ks is the
Boltzmann constant.
In reality the droplet at the origin is moving as well; then the number of
collision per unit time per volume is
11,2 = 4 n (Rj + R2 ) (D2 + Di)nin0,
(1.28)


23
Assuming that droplets are approximately monodisperse at a given
moment, then the number density will decrease due to the recombination as
= -16 nDRn2
dt
(1.29)
where we have set Dj = D2 = D, and Ri = R2 =R.
From eq. (1.27), (D R) is independent of the time, we can find
R3= 12 (DR) vt (1.30)
where v is the volume fraction of the droplets. We get from this simple model of
droplet coalescence that the droplet size grows as tJl3 approximately.
In the above simplistic droplet coalescence model, the droplets are treated
as free particles and the medium is static, and all the interactions between the
droplets are neglected. The reality is that both medium and the droplets are
fluids, and they are undergoing hydrodynamic motion. At the level of droplets,
they have to squeeze aside the liquid along the way upon approach. We can
imagine that will slow down the recombination rate of droplets. In his paper
[Siggia, 1979], Siggia took into account both the effects of the hydrodynamics and
the van der Waals forces between the droplets. He found the correction to eq.
(1.30) as following
3 (DR) v t
log(R)
(1.31)
So at the later time R still grows as t1!3 asymptotically.


24
In the above argument we have explicitly assumed the spherical droplets,
which means that the above coalescence model can only apply when the volume
fraction of the minor phase v is small, so the minor phase forms the isolated
droplets. In this picture, the process is distinctively nucleation and growth.
However, it has been argued [Voorhees and Glicksman, 1984; Huse, 1986] that
the R ~ fll3 law still applies, when the morphology is far from spherical, as in the
case of spinodal decomposition.
1.4.2. Lifshitz Slvozov Theory
As mentioned before, the Lifshitz-Slyozov (LS) theory [Lifshitz and
Slyozov, 1961] is one of the few well established results in the field of phase
transition kinetics. The change from metastable to stable phases occurs as a
result of fluctuations, which form a new phase or nuclei out of the original
homogeneous medium. There are two main factors contributing to the energy of
these nuclei, and thus determine their stability. They are the free energy of the
nuclei and the surface energy due to the creation of the interface. The former is
negative in the metastable state, because the nuclei usually have energy of the
equilibrium state, and it is proportional to the volume of each nucleus. On the
order hand, the creation of an interface always costs energy. It is positive and
proportional to the surface area of the individual nucleus. The competition
between the two terms results that those nuclei smaller than a critical radius Rcr
unstable, due to their lower volume-to-surface area ratio, and thus tend to
evaporate, while those larger than RCr are stable and thus grow larger. The
larger nuclei grow at the expense of the smaller ones, thus Lifshitz-Slyozov
theory is also known as the evaporation-condensation mechanism.


25
The coarsening process is still governed by diffusion. In the limit of small
supersaturation, the diffusion gradient will be small. And we can use the steady
state approximation to the diffusion equation
V2 C = 0 (1.32)
with appropriate boundary conditions. From the solution, one can obtain the
growth rate equation (see, e.g., Lifshitz and Pitaevskii, 1981)
dJL D iC(i-RcA (1.33)
dt R c R 1
where 8C(t) = C(t) is the supersaturation, and AC= Cg C^. If the radius of
the droplet R > Rcr, then dR/dr > 0, it grows. And if R < Rcr, then dR/dt <0, it
shrinks and eventually evaporates. The critical radius Rcr is related to the
supersaturation through
Rcr(t)= cc/SC(t) (1.34)
where a is a constant proportional to the surface tension a
The distribution function f(R,t) satisfies the continuity equation
(1-35)
where v(R) is the radial velocity, given by the eq. (1.34), and f(R)dR is the number
density of nuclei with radius between R and R + dR. And finally, we have the
conservation of the solute


26
^0.'..y.(O=4jrR3N (136)
AC 5C(t) 3
where <5Co is the initial supersaturation, N is the number of the nuclei droplet
per unit volume and ^-R 3 N is the average volume of the nuclei..
Lifshitz and Slyozov were able to obtain the asymptotic solution to the
above coupled equations. Besides the distribution function f(R,t), they found the
growth law
R \t) = Da t
9
(1.37)
This LS theory is only strictly true at small supersaturation, when the
volume ratio of the minor phase to the major phase is small. With increased
volume ratio, significant deviation from LS theory has been found. Hence,
unlike the previous model of droplet coalescence, LS mode of growth is specific
to the decay of metastable state, i. e., to the nucleation and growth.
1.4.3. Hydrodynamic Growth Mode in Concentrated Mixture [E. Siggia, 19791
The observation of phase separation of binary fluids reveals two types of
distinct morphologies at later stage. One type is usually seen in systems with
small volume ratio, so there is clearly a minor phase and a major phase. In this
case, the minor phase grows as isolated spherical droplets in the medium of the
major phase. However, in the systems where the volume ratio of the two phases
is comparable, as in spinodal decomposition of a near critical concentration, a


27
different type of morphology emerges: the structure of a jumbled network of
interpenetrating tubes. An idealization of the structure is two interconnecting 3-
d networks of A and B phases, like those that have been clearly identified in the
amphiphilic systems. This change of morphology is gradual as the volume ratio
is increased.
At the later stage of growth, we expect that two distinct phases have
formed that and the interface between the two phases is well defined and has a
definite surface tension, cr. In the concentrated mixture, the two phases exhibit
the morphology of an interpenetrating tube-like network. Siggia has pointed out
that the tube-like structure is unstable against pinching. This can be readily
understood from the Laplace formula
(1.38)
where p is the pressure difference between two sides of the tube, and rj and are
the principal radii of curvature at a given point on the surface of the tube. As
shown in Fig. 1.6, when a tube is pinched at a given point, the pressure p at the
point will increase due to the decrease of r's, that will push the fluid away from
the point of pinching, and decrease the radius even further.
Siggia proposed a novel mechanism of growth due to the instability of the
tube-like structure in binary fluids. Image a long wavelength disturbance along
the axis of a tube of radius R, where the wavelength / R. According to eq.
(1.38), it will lead a pressure gradient ~ o/R l along the tube axis, that will tend to
transfer the liquid from the neck to the bulges. The corresponding average
velocity due to the Poiseuille flow is


28
v ~ 0.1 o R/l r¡
(1.39)
Take this as the rate of growth dR/dt, then
R ~ (0.1 a/r¡) t
(1.40)
The prefactor 0.1 is only based on rough estimations, and r] is the shear
viscosity of the fluid. As a results of the hydrodynamic flow, the coarsening
(a) (b)
(c)
FIG. 1.6 The tube-like structure is unstable against the pinching, which
will lead to the eventual breakdown of the tube, (a) shows a section of
tube which forms the interface between the two phases; (b) As the tube
is pinched, the pressure inside the tube at the pinching site increases
according to eq. (38), which consequently induce a pressure gradient that
push the fluid away from the pinching site (as indicated by the arrows),
and that further shrinks the tube, and eventually the tube breaks down, as
shown in (c).


29
induces the growth to be linear with time. It is not essential to assume tube-like
structure in the above argument. At the concentrated mixture, any kind of
nonflat interface may induce this type of coarsening.
The above mechanism is observed in the spinodal decomposition of
binary fluid. Up to now, the general consensus on the process is that early on the
diffusive mode dominates, i.e., the domain grows as fl/3, when the interface may
not yet be well defined and the hydrodynamic growth mechanism by surface
tension is not possible. Gradually there is a crossover from the diffusive growth
to the faster hydrodynamic growth regime, and domain size grows as t. In the
nucleation and growth process, the surface tension does not come into play, and
consequently the growth should be diffusive for all times.
1.4.4. Scaling Hypothesis for Structure Function at the Later Stage
The renormalization group has brought a revolutionary leap to our
understanding of critical phenomena and phase transitions. At the basic level, it
is recognized that the correlation length | is divergent at the critical point. As a
result around the critical point, the only relevant length scale is £, and at length
scales L the physical system is scale invariant. Or simply put, the system
looks self-similar at any scale L exactly at critical point, where £ - >. Then the
thermodynamic functions scale around the critical point with respect to j. And
these functions only depend on some very general properties of the Hamiltonian
of the system, like the symmetry of the system and the space dimensions of the
system and order parameter, but not on the details of the interactions in the
system. This enables us to classify phase transitions into different broad
universality classes, in which all the systems in the same class have the same
properties around the critical point.


30
So far the quantities involved are the equilibrium quantities such as the
heat capacity, susceptibility, etc. It is tempting to extend the same type of
argument into the dynamic aspect of the phase transitions. But so far it has
proven to be much more difficult. The dynamic properties of system seem to be
more system specific then the static quantities around the critical point.
However, it is still desirable to apply similar concepts to the dynamic aspects of
the criticality, which is the so-called dynamic scaling hypothesis.
The most important function in the kinetics of phase-separating is the 2-
point structure function S(k,t). The basic idea of the dynamic scaling hypothesis
in this case is as follows: After an initial period of transient time following the
quench, a characteristic length scale l(t) is well established which represents the
average size of the domain structure. The l(t) is the only relevant length in the
problem, and it plays the similar role as the correlation length £ in the critical
phenomena. Then the structure function S(k,t) is scale invariant if all its lengths
are scaled relative to l(t).
The definition of the characteristic length scale l(t) may vary. For example,
in experiment where S(k,t) is measured, it is convenient to choose the inverse of
the peak position, k^ft). In the case of computer simulation studies, kj'1 is
usually used, where kj is the first moment of the structure function S(k,t). In this
study, we will stick to the peak position km-J(t). Let us first introduce a
normalized structure function S(k,t):
S(k,t)
S(k,t)
5>2s(Jt,o
k
so that ^ k 2S(k,t) is independent of time.
k


31
Then the scaling hypothesis assumes the renormalized structure function
has the following form
S(k,t)= k^F(k/km) (1.41)
where d is the dimensionality of the system. The functional form of F(x) is
universal to some degree. In the phenomenological theories of Furukawa
[Furukawa, 1977; 1978; 1979; 1985.], a family of functions F(x) is proposed
(1 + y/2) x2
y! 2 +x2+r
(1.42)
where the constant yis related to the dimension d: (1) Critical quench ,y= 2d; (2)
Off-critical quench, y= d+1.
As a result of the scaling hypothesis (1.42), the time t only enters the
structure function through km(t). Otherwise the structure function S(k,t) doesn't
depend on time explicitly. In addition, if we let k= km in equation (1.42), we will
have
S(km) ~kj (1.43)
If plotting on log-log axes, it is a straight line with slope -d.
Another straightforward result from eq. (1.41), if it is true, is that the ratio
of the second moment of the structure function, kzft), to the square of the first
moment kj(t),
r(t) =
kht)
(1.44)


32
should be independent of time t.
The scaling behavior was noticed first in the computer simulations of
phase separation process in Ising-like lattice models, where the structure
function satisfies (1.42) at later stage. In the simple binary fluids, it has been
shown experimentally that it holds approximately. But the function form F(x)
can vary with temperature, quench depth, and concentrations. We should only
take (1.42) as a first order approximation, as various type of corrections may be
present in real systems.
I have summarized the key mechanisms related to the phase separation in
the binary fluids. Rather than attempting to give an exhaustive list of the results,
I chose to concentrate on several key elements and well established models
instead. In terms of results, I have ignored most of the contributions from the
computer simulation studies, except just to say that most of the results are
consistent with the experiments.
Our understanding of the phase separation process is still very limited,
e.g., we are unable to calculate some of the most important and basic quantities
like the correlation functions and thus the structure factor, which can be directly
measured experimentally. The intrinsic problems about the process are its
nonlinearity, which we still do not know how to deal with analytically, and the
fact that it is a nonequilibrium process. Most of our current knowledge in this
field is qualitative, restricted to the certain modes of growth laws. It is not
surprising to see that new phenomena still being discovered over the years. The
phenomena of spinodal decomposition and nucleation and growth are far richer
and complex than the current models have suggested.


33
1.5 Experimental Results
As I have mentioned, the phase separation phenomena have been studied
in a variety of systems. Among these, the most detailed studies have been
conducted in alloys, binary fluids and polymer blends. Here we will concentrate
on the results from binary fluids, and furthermore limit our discussion mostly to
the spinodal decomposition process in critically quenched systems.
The binary fluid systems used most in the phase separation experiments
are 2, 6 lutidine water (L-W), and isobutyric acid water (I-W), as pioneered by
Goldburg and coworkers [Chou and Goldburg, 1979; 1981], and Knobler and
coworkers [Wong and Knobler, 1978; 1979; 1981; Knobler and Wong, 1981]. Later
the French group headed by Beysens [Guenoun et al., 1987] used a quasi-binary
system, deuterated cyclohexane (D-cyclohexane) cyclohexane methanol, which
has the advantage of density matching between methanol and the D-cyclohexane
-cyclohexane mixture of the right proportion, thereby minimizing the effect of
gravity.
All three groups used time-resolved light scattering to probe the evolution
of the phase separation. I will discuss the details of the techniques later.
Following a temperature or pressure quench of the system into the unstable state,
the mixture will turn milky and thus starts to scatter light strongly. Because
fluids are isotropic, the scattered light forms a ring, known as the spinodal ring,
which brightens and decreases in its diameter as time proceeds. Experimentally
the scattered light intensity profile I(cj,t) is measured, where c¡, the momentum
transfer, is a function of the scattering angle 6. We can determine the ring
diameter c\m(t) and the peak intensity l(cjmrt). When the multiple scattering effect
is small and can be ignored, is proportional to the structure function S(q,t),


34
then c¡m-1 represents the average length scale of the structures growing in the
phase separating mixture, and therefore we can directly compare with theories.
If the quench depthes are well within the critical region, it is natural to
use the scaled variables k and t,
k = q £ and r = Df/£2 (1.45)
where £ and D are the correlation length and diffusion coefficient, respectively.
In these dimensionless variables, the evolution of km vs. r falls remarkably onto a
single curve for quenches of different depths. Experimental results from
different binary systems showed excellent agreement [Goldburg, 1983]. In the
two opposite limits km~ 1 and Jtml the growth can be approximated as a
power law:
km =Az~b
and the kinetic exponents are
b = 0.30.1, for 0.6 > k,> 0.3,
b = 1.1 0.1 for 0.1 >ltm> 0.08.
There is a gentle crossover between the two limits. This is just what the
theories predicted for the coalescence process. In the beginning the phase
separation is diffusive, and the length scale L grows as L ~ t1/3. At later stage
when interfaces are well developed, the interfacial tension will induce the
hydrodynamic flow, and that will speed up the coarsening process, and the


35
growth law approaches L ~ t, just as pointed out by Siggia. However, it should
be pointed out that the agreements are only qualitative, there are large
discrepancies in the estimation of the constant A between the theories and
experiments.
In the simple binary fluid systems, the linear regime in early stage as
predicted by Cahn has not been observed experimentally. Only in the polymer
blends do we have convincing evidence of the existence of this linear regime,
where the structure function grows exponentially with time [Bates and Wiltzius,
1989; Hashimoto, 1988;]. Evidently the high viscosity of the polymeric systems
slows down the dynamics of phase separation significantly, which makes it
feasible to study the early stage of phase separation.
A more difficult comparison between experiments and theories is in peak
intensity behavior and the structure function S(krt). Apart from linear Cahn
theory, there are no satisfactory calculations up to now which can be used to fit
the experimental results, especially at the later stage. However, most of the
experimental results were tested to see if they satisfy the dynamic scaling
hypothesis (eq. (1.42)). (see § 1.4.4).
Define a normalized structure function S(k,t) in terms of the scattering
intensity as
S(k,t) =
Kk,t)
The range of integration covers most of the ring area, where the peak in I(k,t) is
centered. The S(k,t) defined in such way was shown to follow roughly the scaling
hypothesis (eq. (1.42)), especially at the later stage when r> 10, with the scaling


36
function F(q/qm(t)) exhibiting no time dependence. But generally the function
form F(x) is believed not to be universal, it shows some dependence on the
concentrations of the samples, and even on the quench depth. Later experiments
[Hashimoto et al., 1986a; 1986b; Izumitani et al., 1990; Takenaka et al., 1990] on
polymer blends exhibit similar behaviors.
From the scaling hypothesis (eq. (1.42)), the peak intensity
I(km,t)~ S(km,t) = kj F(l)
where d=3 for bulk samples. At later stage, km ~ r'1 for critical mixtures, and
therefore I(km, x) ~ r3. This power law was confirmed in experimentally.
The exponential growth predicted in Cahn's theory was not observed, the
LBM theory and extensions due to Kawasaki and Ohta, who included
hydrodynamic interaction, have some qualitative features similar to the
experimental observation, but did not fit the data properly. In general, it can be
said that those results are consistent with the theoretical results although there
might still be intricacies that need to be clarified.
1.6 Wetting Phenomena and Phase Separation
Wetting is a phenomenon associated with the interfaces between different
phases of matter, of which the most familiar is the contact angle, e.g., a sessile
liquid drop on a solid substrate (see Fig 1.7). At equilibrium the contact angle 6
is determined by the surface tensions a of different phases through the Young
equation:


37
cos e =vs-OLS'
glv
(1.46)
where the subscript V, L, S stand for the three phases, vapor, liquid and solid
(substrate) respectively.
Fig. 1.7. Partial and complete wetting: (a) Partial wetting, the contact angle
6 is finite, the liquid phase form isolated droplets on solid substrate;
(b) Complete wetting, angle 6=0. the liquid phase form a continuous
layer of film on the substrate, separating the vapor and the substrate.
The similar phenomena can be observed in binary fluids when they are
fully phase separated. Now instead of the vapor and the liquid phase, we have
two liquid phases a and /}, in contact with a solid substrate, which is usually the
container wall. The Young equation still applies, and the contact angle 6 can be
measured at the edge of the meniscus as shown in Fig. 1.8.
From the Young equation, we can see that
|as" tfps| ^ ap '
(1.47)


for any finite contact angle 6, where the subscript s stands for the substrate or
solid phase. This case is usually called the partial wetting. If the equality holds,
i. e.,
|as aps| = aap, (1-48)
then the contact angle must be zero. In this case, it is more energetically
favorable to have the two phases with the highest surface tension physically
separated by a layer of the third phase, as shown in Fig. 1.8b. This case is known
as complete wetting. To be more concrete, we assume cas > aps, then the ¡5 phase
will be sandwiched between the a phase and the substrate, and we will say that
the /? phase wets the substrate.
Fig. 1.8 Wetting in binary fluids with the container wall; (a) Partial wetting in
fully phase separated binary fluids, the finite contact angle 6 can be measure at
the edge of the meniscus; (b) Complete wetting: a layer of phase (5 completely
covers the substrate, insulating phase a from it.


39
The interests in wetting phenomena increased significantly after Cahn
[Cahn, 1977] pointed out that there is always a phase transition from partial to
complete wetting upon the approach to a critical point. His argument was
simple and intuitive, based on the thermodynamics of the phase transition.
Upon approach to the critical point inside the two-phase region, the interfacial
tension a,ap vanishes as the inverse square of the bulk correlation length £ i.e.f
Oaf) ~ ~f2v 0-49)
with v = 0.63 is a universal exponent, and the t is the reduced temperature. On
the other hand the difference in surface tension Ous Ops should vanish as the
order parameter close to the substrate (in the case of vapor and liquid, the
density difference between the two phases), i. e., it goes to zero as t^, with =
0.8. Then according to Young's equation (1.46), the contact angle
cos 6 ~ t ^'2v ~ t'05, (1.50)
thus it will reach unity as the critical point is approached. That point marks the
transition from partial to complete wetting, and it is known as the wetting
temperature Tw. In the case of binary fluids, in the two phase regions of the
phase diagram, there is a temperature Tw that marks the boundary.
In the last decade we have seen remarkable advances in our
understanding about the wetting phenomena, especially on the wetting
transition. Most of the effects have been concentrated on the equilibrium
properties. But gradually the dynamic aspect of the wetting effect started to
catch certain attentions among physicists, especially in the topics of liquid or
polymer spreading [Chen and Wada, 1989; Heslot et al., 1989]or dewetting


40
[Redon et al., 1991] on solid substrates The study of the effect of wetting on the
dynamics of phase separation started relatively recently. Because usually Tw is
at least several degrees Celsius away from the critical point, the final state in the
quench experiments will be within the complete wetting regime. As a result,
there is a complete wetting layer of the preferentially adsorbed phase next to the
solid substrate (usually the container wall). It is of great interest to understand
the growth of this wetting layer. Early on it was pointed out theoretically
[Lipowsky and Huse 1986] that the thickness l(t) of the wetting layer will grow as
a power law at the late stage of the phase separation if it is diffusion limited, i.e.,
l(t) ~fb, (1.51)
where b is equal to 1/8 and 1/10 respectively for nonretarded and retarded van
der Waals force. It is much slower than the coarsening process, e.g., Lifshitz -
Slyozov theory, largely because after the initial wetting layer formed on the
substrate, it leaves a depletion zone right next to the wetting layer. Any further
growth of the wetting layer will have to diffuse through this zone, which act like
an energy barrier in slowing down the process. The type of slow growth law was
confirmed in some computer simulations, even though it hasn't been observed
experimentally in binary fluid systems.
More recently Guenoun, Beysens and Robert [Guenoun et al., 1990]
undertook optical microscopy study on the morphology and domain growth in a
binary fluid mixture undergoing spinodal decomposition near the container wall.
Besides observing the morphology changes due to the wetting effect, they found
the wetting layer thickness / grows with time roughly like t1/2. But that result
should be taken with certain caution, as it is well known that the kinetic
exponent extracted from microscopic observation is unreliable, (see, e. g., Chou,


41
1979). More recently, light scattering was used by Wiltzius and Cumming
[Wiltzius and Cumming, 1991] on a quenched binary polymer mixture, there
they discovered a new "fast" mode where the length scale L grows as f3/2 with
time. This fast mode was attributed to the wetting effect of the container wall on
the fluids. But up to the moment we have still no theoretical understanding that
can explain the nature of this fast growth process.
Our work had been motivated by that of Wiltzius and Cumming. First of
all, we sought to establish the generality of the fast mode by studying a simple
low-molecular-weight binary system, guaiacol-glycerol-water (GGW), and we
gave a detailed characterization of the fast mode and its relations to the other
physical quantities of the system. From there, we hoped to gain a deeper
understanding of the phenomena which might shed light on certain aspects of
phase separation processes, especially on the role of wetting. Also through these
experiments, we hoped to provide interested theorists enough experimental
input and incentive to develop a comprehensive theoretical model.


CHAPTER 2
LIGHT SCATTERING METHOD: CONCEPTS
Light scattering has been widely used nowadays in almost every
discipline of natural science, particularly in physics and chemistry, where it has
become one of the basic tools in studying the structures and the dynamics of
many systems. Over the past century, light scattering method had enjoyed a
steady development starting with the classical work of Lord Rayleigh and others.
The advent of lasers as a primary light source brought a revolution to this field,
during which the dynamic light scattering method was invented and widely
applied to a whole range of systems. Nowadays laser light scattering, along
with X-ray and neutron scattering, are the standard tools for studying the
structures of various materials in science and engineering. As light scattering is
the most convenient and inexpensive among the scattering methods, it is more
widely available and used. Over the years, there have been excellent reviews
and books published on the subject of light scattering [Berne & Pcora 1976; Chu,
1974; Chu, 1991; Kerker, 1969; van der Hulst, 1957], where its principles have
been well elucidated and its applications to a varieties of physical and chemical
systems outlined. Therefore, here in this chapter I will limit myself to a brief
review the basic concepts and some of the common notations involving light
scattering. Furthermore, in relevance to our experiments we will confine our
discussion here largely to elastic light scattering.
42


43
Fig. 2.1. Scattering geometry. lg, Is, and It are incident, scattered and
transmitted light intensity respectively, 6 is the scattering angle, ko
and ks are respectively the incident and scattered wave vectors,
whose magnitudes are 2;r/As, where As = Ao/n, with Ao and n being
the wavelength in vacuo and the refractive index of the scattering
medium.
Fig. 2.1 shows the typical light-scattering geometry, where the incident
light beam impinges on the scattering medium (shaded area). Most of the
light will transmit through the medium without any scattering, which is the
transmitted light with intensity It, and a small portion of light is scattered that
has the intensity Is at angle 6. In this geometry we have ignored variations
with respect to the azimuthal angle, which is appropriate for systems which
are isotropic, like most liquids. Experimentally we can measure not only the
intensities and the direction (the scattering angle), but also the frequency
(energy) changes of the scattered photons, as there can be energy exchange
between the scattered photon and the scattering medium. Whereas the
scattered intensity can be related to the structures of particles or


44
inhomogeneities in the medium, the optical spectra reveal the dynamical
motions. In the experiments we are concerned with here, we are interested
mostly in the structures of domains and their growth, therefore
experimentally we only measure the scattered light intensity Is and its profile,
i. e., its variation over the scattering angle 6. This is known as elastic light
scattering, where we are not concerned with the change of energy of the
scattered photons, only their numbers (intensity). Equivalently we assume
that the photon energy (frequency) isn't changed from the scattering, or the
photon scatters elastically. In contrast, those light scattering experiments
which also probe the energy (frequency) change of the photons are generally
called dynamic light scattering. I will limit the discussion here only to elastic
scattering.
We want to measure the variation of the scattered light Is(6, t).
Conventionally the more relevant quantity is the momentum transfer q
which is defined as
(2.1)
q = ks ko,
where and ks are ko are the wave vectors of the scattered and the incident
photon, respectively, as shown in Fig. 2.1. Because the magnitudes of ks and
ko are the same, and equal to 2/r/As, the magnitude of q is simply
q = ^ sin(&) = 4ft n sin(@~)
(2.2).
Therefore, q is a function of the scattering angle 6, but it has the dimensional
of inverse length. In isotropic medium, the scattered intensity Is only
depends on the magnitudes of q, i. e., q.


45
The structures and dynamics of a system are characterized by its
structure factor. The dynamic structure factor S(q,iu) is defined as the Fourier
transform of the time-dependent correlation function G(r, t), which is
(2.3)
where (pit, t) is the local order parameter, and for binary fluids, (p is the local
concentration C of one of the molecular species. The angular brackets denotes
an equilibrium ensemble average, and
<5 (2.4)
is the local deviation or fluctuation of the order parameter from the
equilibrium value. Then S(q,co) is defined through G(r, t) as
(2.5)
In concert with S(q,), the static structure factor is defined as
(2.6)
What makes the scattering methods so important in determining the
structures and dynamics experimentally is the following relation: the
frequency-resolved scattering intensity Isiq,() is proportional to S(q,io) if
multiple scattering can be ignored, and in the absence of frequency-resolution,
the scattering intensity /s(q) is proportional to the static structure factor So(q).


46
This relation is the basis for understanding and interpreting any experimental
data. Here I will not bother with its derivation from first principles, as it has
been dealt with in most books on scattering, (Stanley [Stanley, 1971] gives a
simple and neat derivation), but rather try to emphasize certain important
points associated with it.
The above relation is strictly true in the single scattering limit, namely
when multiple scattering is negligible. An incident photon scatters only once
at most inside the scattering medium before exiting [ see, e.g., van der Hulst,
1957]. Multiple scattering poses a problem too complicated to deal with
mathematically, although theories have existed to treat double scattering
analytically, but even that isn't straightforward. Therefore light scattering
experiments are usually conducted in the single scattering limit, where
multiple scattering probability is small. The contribution from multiple
scattering can be monitored qualitatively by measuring the beam attenuation
or the turbidity r of the scattering medium, which is defined through
h
h
e
- r d
/
(2.7)
where d is the path length through the scattering medium. In the single
scattering limit, the turbidity is small.
In studying the kinetics of phase separation in binary fluids, keeping at
low turbidity limits the quench depth AT, which indicates how deep the
system can be quenched into its 2 phase regime (see Fig. 1.1). Deep quench
will turn the fluid too turbid to be in the single scattering limit. Turbidity
could also be a problem at late stage of phase separation process.


47
To study the kinetics of the phase separation process, we measure the
scattered light profile Is (c¡) at different times following a quench into the 2-
phase region of the phase diagram, and thus observe its evolution with time.
From each scattering profile at specific time, we can extract a length scale L
that is indicative of the average size of the domains growing in the fluids, and
thence the growth law of L.


CHAPTER 3
APPARATUS
We have been using time-resolved light scattering to study the kinetics of
phase separation in GGW mixtures. The experimental setup of our light
scattering apparatus resembled that described in Cumming et al. [Cumming et
al., 1992]. We started by choosing a CCD (charge-coupled device) camera as the
detector for measuring the intensity of the scattered light. Compared to other
light detectors, e.g., photomultiplier, etc., CCD cameras offer distinctive
advantages for our purpose. The main component of the camera, the imaging
area, is a silicon chip with a 2-d array of imaging elememts called pixels. An off-
axis paraboloidal mirror and optical components were used to focus and adjust
the light beam onto the CCD chip, which has an imaging area about 2 cm 2 cm.
The apparatus was noted for its simplicity in design and relatively small number
of components. Here I will first present the setup of our light scattering
apparatus, and then discuss its major components in detail, where its design will
be emphasized. Afterwards I will describe the temperature control system, that
enabled us to change the temperature quickly, whereas still have a good
temperature stability. These two systems constituted the essential elements for
our experiments.
48


49
3.1 Light Scattering Apparatus
The initial setup of our light scattering apparatus was very similar to
that used by Cumming et al. [Cumming et al., 1992]. A schematic plan view
diagram of the apparatus is shown in Fig. 3.1, where all the components
shown are on a horizontal plane. Of these components listed, the two plano
convex lenses (condenser#l: Product No. 01LPX311, focal length /= 330 mm,
diameter 145 mm; condenser#2: Product No. 01LPX127, focal length / = 60
mm, diameter 42 mm) and the paraboloidal mirror (Product No. 02-POA-019,
diameter 95 mm, off-axis distance 83.6 mm) were purchased from Melles
Griot, and the other optical components, including the 5-axis lens positioner
(model 740) on which the paraboloidal mirror is mounted, were purchased
from Newport. Our light source was a 5mW power HeNe laser from Jodon
Inc. (model # HN-5HFP). Non-essential components (not shown in Fig. 3.1)
included two XYZ translation stages (model # 460-XYZ) on which the two
lenses were mounted, and a dual-axis translation stage (model # 405) on
which the CCD camera head rested. These components enabled us to finely
tune the positions of the lenses and the camera within the precision of 1
micron in order to achieve the full resolution of our apparatus. Other non-
essential components used in the apparatus were several types of rod/clamp
assemblies (model # 305, 340,345, 370) that anchored optical elements onto the
optical table, and a beam steering instrument (model # 670) which adjusted
the laser beam direction. All of the above listed non-essential elements were
purchased from Newport.
As I have mentioned above, we chose a CCD camera as the light
detector that measures the scattered light from a phase separating fluids. The
main component of the camera is a CCD chip, which is one of the silicon


50
array imaging devices developed since the late 60s. A CCD chip consists of a
2-dimensional array of imaging elements called pixels. A pixel on a CCD chip
beam stop
Condenser #2
Paraboloidal Mirror
CCD Camera
16 bits Dynamic Range Condenser #1 ||F
5 mW HeNe Laser
5 Axis Lens
Positioner
Fig. 3.1 Top view of the elastic light scattering apparatus. All major
components are listed. The light from a 5 mW laser impinges on the
sample via an attenuator. The sample is positioned at the focus of an
off-axis paraboloidal mirror capable of collecting light scattered up to
50 away from incidence. The dashed line indicates the axis of the
paraboloidal mirror. Light ray (1) indicates the transmitted beam,
while 2 and 3 the scattered rays. The scattered light ray column is
reduced to appropriate size by a pair of confocal condenser lenses before
sending onto a CCD camera.


51
is essentially a quantum well. As photons strike the silicon chip and they
create electron-hole pairs, the holes leak away and the electrons are trapped in
the well. Therefore the number of electrons accumulated in the well is
proportional to the number of photons that had struck the pixel, and thus
measuring the intensity of light is a matter of counting the number of
electrons in the well, that is done by a set of registers inside the camera. Our
camera was purchased commercially as a package from Photometries. The
whole package consisted of a camera head (model No. CH250), where the CCD
chip was mounted, an electronic control unit (model No. CE 200A) and liquid
cooling unit (model No. LC200). It came also with a interface board so that
the camera could be operated from a Macintosh II fx computer. The CCD chip
in the camera was a Thomson CSF chip of size 1024 x 1024 pixels. Each pixel
had dimension of 19 pm by 19 pm.
For our purpose, CCD chips offer great advantages over other detectors.
First of all, because it consists of a 2-dimensional lattice of pixels, it can
measure the whole scattered light profile in a single exposure, thus
eliminating the complications associated with having to take the
measurements sequentially. This feature was particularly suitable for
studying the early stages of the phase separation. With a CCD camera as the
detector, the measurement of the scattered light profile only takes as little as a
fraction of a second. Another advantage of CCD chip is its high sensitivity,
which is determined by two factors: the quantum efficiency (Q.E.) and the
noise level. Q. E. is the probability of a incoming photon being detected by a
device or detector, for a CCD chip it can reach up to 80%, which is an order of
magnitude higher than the sensitivity of photographic films. For the CCD
chip used in the present experiments, its Q. E. was at about 40% at the
wavelength of the NeHe laser, which is 632.8 nm. With regard to noise, the


52
CCD camera head used in the present experiments was Peltier cooled to about
-40 C, at that temperature the noise level was significantly reduced compared
to that at room temperature. Other benefits of a high quality CCD camera
include the wide dynamic range and extremely good linearity. Dynamic
range is defined as the ratio of the device saturation charge to the system
noise level, expressed in electrons. Our camera had 16 bits of dynamic range.
Linearity is the strict proportionality of the number of electrons accumulated
to the intensity of light, and in our camera, the deviation from the linearity
within the saturation limit was below 0.1%.
The setup of the rest of the optical components was designed to collect
the scattered light and reduce it to appropriate size before sending onto the
CCD camera. For the former, we used an off-axis paraboloidal mirror whose
reflective surface was an inside section of a spherical paraboloid, the curved
surface swept by a parabola undergoing a full revolution around its axis. We
positioned the mirror in relation to the sample cell so that the spot where the
laser beam impinged on the sample cell was also the focal point of the
paraboloidal mirror. In the scattering plane as shown in Fig. 3.1, the
transmitted laser beam was the lactus rectum of the parabola (shown in Fig.
3.1 as light ray #1), and the revolution axis of the parabola intersected the
transmitted beam perpendicularly right at the spot where the beam impinged
onto the sample cell. Here the scattering volume was the volume of the
GGW mixture in the sample cell that directly illuminated by the incident
laser beam, whose beam spot was about 2 to 3 mm in diameter, and our
sample cell thickness was 0.8 mm, therefore the volume was of about 4 to 7
mm3. In this configuration, the scattered light rays emanated from the
scattering volume in the sample cell and then reflected from the mirror, they
became a parallel ray bundle (see Fig. 3.1). In this way, the mirror served to


53
columnate the scattered light rays, whereupon they could be passed through a
pair of convex lenses which share a focus, and then onto the CCD camera.
The pair of confocal lenses are used to reduce the cross section of the parallel
rays to the size of the CCD chip, so all of the light rays collected by the mirror
would land on the CCD camera.
The paraboloidal mirror collects the light scattered up to 50, which
corresponds to photon momentum transfer magnitude up to 86,000 cm-1 or 8.
600 pm-1. To achieve the full resolution of the CCD chip, we required that all
light rays emanating from the scattering volume in a given direction or a
given momentum transfer q be directed to a single pixel on the chip, i.e., all
the parallel rays from the scattering volume must map onto a single pixel.
This was the real technical challenge of this apparatus for achieving the best
resolution. For adjustment and fine tuning, we first aligned the mirror and
the condensers on an optical axis by a laser beam. Each of the lenses was
mounted on an XYZ stage (from Newport), and it enabled us to fine tune the
positions of these optical components. Naturally, the CCD camera had to be
adjusted along the optical axis to yield the best resolution. The most
important adjustment was the 5-axis lens positioner on which the
paraboloidal mirror was mounted. Its fine tuning was done in combination
with the calibration of the CCD chip.
The calibration of the apparatus consisted of finding the mapping from
momentum transfer c¡ to pixel positions on the CCD chip. For that purpose
we put a grating in the position of the sample cell. The interference pattern
by a laser beam is well known. Because of the high coherence of laser light,
we were able to trace every order of interference maxima from the image
taken by the CCD camera. In order to fully calibrate every pixel on the CCD
chip, we built a rotor with a low speed motor (60 revolutions per minute) to


54
rotate the grating in the plane perpendicular to the incident laser beam. For
every revolution, the interference maxima swept through 360 and traced a
set of coaxial light cones, which can be imaged on the CCD by making an
exposure of appropriate duration, in our case that is half of the period of the
rotation, about 0.5 second. From the image taken by the CCD camera, we
could find the exact pixel location for each order of the interference maxima,
which also satisfy the positive interference condition (see, e. g., Hecht, 1987)
d sin 9m = m A (3.1)
with m = 0, 1, 2, 3,.... where d is the grating spacing, and angle 6 is the
scattering angle, and A is the wavelength of the incoming light, i. e., 632.8 nm
in our case. Thus we could calculate the momentum transfer cj for each
maximum because c¡ is a function of scattering angle 6. In this way, we
calibrated all the pixels that the interference maxima had swept through on
the CCD chip. From these maxima, we found a fitting equation from their
pixel positions to the corresponding c¡'s. The c¡ of these pixels in between two
neighboring maxima could be calculated from this fitting equation. The CCD
image of the rotating grating also provided a good assessment about the
alignment of the paraboloidal mirror, and allowed further fine tuning to
reach the best resolution possible. We have used several gratings of different
spacing ranging from 100 lines/mm to 20 lines/mm for the calibration, the
resultant calibration maps differed by less than 0.02 pm'1 in the c\ values,
which was less than a pixel on the CCD. Fig. 3.2 shows a calibration curve
from the grating images.
The setup of the light scattering apparatus was modified slightly from
the initial design (Fig. 3.1) after a period of preliminary experimentations. A


55
schematic diagram of the final setup is shown in Fig. 3.3. The main reason
for the modification was to change the orientation of the sample cell from
vertical as in Fig 3.1 to horizontal as shown in Fig. 3.3, in order to minimizing
the gravitational effect on the process of phase separation and annealing,
which turned out to be of particular importance on the fast mode.
200 400 600 800 1000
Pixel Position on CCD chip
Fig. 3.2 Calibration curve for the CCD chip. It maps each pixel in a row
on the CCD chip onto its corresponding q value. It was made form a 20
line per mm grating.


56
Our sample cells shape like a disk, and the fluids are confined between two
disk windows about 1 inch in diameter and usually about 0.8 mm thick (see
the next chapter for details of the sample cell geometry.). The thickness could
be changed if necessary. The GGW mixture used in this study has a density
mismatch about 10% in their pure forms, which is relatively large among the
fluid mixtures extensively used for the similar studies (L-W, I-W, etc.). In the
initial orientation, the large density mismatch could induce sedimentation
and convection of the heavier phase sinks to the bottom and lighter phase
rises to the top, which causes macroscopic phase separation. That eventually
Paraboloidal Mirror
CCD Camera
5mW Laser
Sample
Fig. 3.3 Side view of the final setup of the elastic light scattering
apparatus. The major change was the orientation of the sample cell,
and correspondingly other optical components had to be readjusted for
the purpose. But the same components were used as the initial set up
(Fig. 3.1), and the design principle and concepts were kept the same.


57
creates concentration gradient in the sample cell along the vertical direction.
A change of the orientation eliminated most of these problems. The details
of the gravitational effects will be discussed in chapter 5.
The final setup (Fig. 3.3) used most of the original components, the
major optical components were the same, so was the design principle. The
alignments and calibration procedure are almost exactly the same as described
above.
The whole light scattering apparatus was mounted on a Newport
optical table (model No. RS-48-12) equipped with pneumatic vibration
isolation legs (model No. XL4A-23.5).
3.2 Temperature Regulation and Quench System
To study the dynamics of the phase transition, we needed a
temperature regulation system which could quench the temperature from
one point to another and then stabilize at that temperature quickly. This
requirement is especially stringent for binary fluids, whose interdiffusion
constant D is normally several orders of magnitude larger than that of other
systems frequently used in the studies, like alloys and glasses. Because of this,
it was initially believed that it was not experimentally feasible to study
spinodal decomposition in these liquid mixtures, as the time scales are
typically very small due to the large diffusion constant. However, this
difficulty could be circumvented by making measurements very close the
critical point, where D vanishes as
n = Tc
(3.2)


58
where r¡* is the high frequency kinematic viscosity, and £ is the correlation
length, which diverges at critical point Tc according to eq. (1.1a).
Further restrictions on the quench depth AT = I Tf- Tc I (see Fig. 1.1)
come from the light scattering experiments. Only in the single scattering
limit, the scattered light intensity profile I(q, t) is strictly proportional to the
structure function S(q, t), whose physical meaning is well understood.
Beyond this limit, the multiple scattering sets in and can render the
experimental data hard to interpret. Therefore, we had to require the quench
depth to be shallow enough that most of experimental data are within the
single scattering limit.
The restrictions posed above limits the quench depth AT to be fairly
small. For usual binary systems, e. g., 2, 6 lutidine water (L-W), AT is limited
to be below 10 mK. For our system of GGW, which has a much higher
viscosity comparing to that of L-W, the single scattering limit will be met if
the quench depth is below 100 mK or 0.1 K.
In relation to this quench depth, we needed to have a temperature
control system that had a temperature stability at least one order of
magnitude lower, that is, 10 mK. We set out to design a simple control
system that met this requirement, and also could quench relatively quickly.
These two goals were often in conflict, as stability meant resistance to changes
or fluctuations, and we had to strike a balance between the two factors.
We built two separate circulation circuits with independent circulation
pumps and thermal regulation circuitry set up at two temperatures. One was
at anneal temperature T, (initial temperature), which was about 0.1 C away
from the critical temperature Tc but on the one phase side, and the other was
set at quench temperature Tf (final temperature). For the GGW mixture close


59
to the LCST, it happens that T, < Tc and Tf> Tc. Water was used as the
temperature regulation medium. A schematic diagram of the circuit is
shown in Fig. 3.4. Each circuit had a heat exchanger made of copper tubing
that submerged in a Neslab RTE 210 refrigerated bath/circulator, which had a
stability better than 0.1 C. Within the circuit loop, the temperature was
further regulated by a Lakeshore DRC 93C temperature controller through a
resistive heater, which was made from a resistive wire wrapped around a
segment of copper tubing of diameter 7/8 inch and about 10 inches in length,
and then sealed with a thermally conducting epoxy. Each circulation circuit
also included a Millipore filter of mesh size 0.22 pm to rid air bubbles and dirt
from the circulation water, and the water circulation was sustained by a Little
Giant pump (model#: 3-MD) with 9 gallon per minute pumping power. This
two stage temperature control system achieved temperature stability of 2.5
mK or better over the duration of most of our experiments, which could last
from several minutes to hours. Fig. 3.5 shows a typical temperature readings
during a period about one hour and a histogram of the data, where the
distribution of the temperature fluctuation fits well with a gaussian, from the
fit we had got the average to be 29829.4 Ohms with standard deviation of 0.9
Ohm, with corresponds to 0.7 mK in temperature. The temperature was
monitored through a YSI thermistor, which had a nominal resistance of
30,000 Ohms at 25 C.
A pair of four-way ball valves (Whitey 4-way crossover valves,
material: stainless steel, Cat. # 45YF8) were used to switch the water flow
tothe sample cell from one circuit to another, thus making the sudden
quench of temperature from T, to Tf, They were interconnected in such a way
that one circuit flow pass through the sample cell, while the other circuit flow
was shunted. The roles of the two water flow circuits could be reversed by a


60
Lake Shore temperature
Controller
Regulated Bath
Fig. 3.4 One half of the temperature regulation system It is a closed
water circulation system, and the temperature was controlled through
the heat exchange with the regulated bath, and a wire heater that was
regulated by a Lakeshore temperature controller. A water filter with
small pore size ridded the dirt and air bubble of the circulating water.
Similar circuits were used to regulate and pump water at each of T, and
Tj, but for clarity, only one side is shown.


THERMISTOR RESISTANCE (ohms)
61
29.840x1 Q3
29.835
29.830
29.825
29.820
TIME (second)
25.127 C
25.131 C
25.135 C
25.139 C
25.143 C
Fig. 3.5 Temperature stability over a period of a hour, (a): Temperature
readings over that period; (b): historgram of the data in (a), with a gaussian fit.


bZ
(A)
Warm Out Warm In
(B)
Fig. 3.6 Two configurations of a pair of 4-way valves which rerouted
circulation waters to the sample cell. The solid arrow ^ indicates
flow of hot water while the dashed arrow the flow of cooling
water. (A) Annealing configuration, cool water flows through the
sample while hot water is shunted. (B) Quench configuration, hot
water flowing through the sample cell.


Temperature (Ohm)
63
Fig. 3.7 Temperature profile of a quench with time. The insert
window is the blowup of the profile first 50 seconds, when an
equilibrium was reached. That part of temperature readings can be
fitted with an exponential function with time constant about 5 seconds.
simple switching of the valves. A schematic presentation of the flow
directions in two configurations is shown in Fig. 3.6, where the temperature
of the cool water is T,, and hot water Tf and the two circles present the two 4-


64
way valves. The scheme allows for a minimal exchange volume between the
two flow circuits while keeping the majority of the plumbing with constant
temperature flow. After the switching the valves, the temperature increased
exponentially, with the time constant about 3 ~ 5 sec., upon approach to final
temperature Tf. A typical quench profile is shown in Fig. 3.7, the time
constant for the quench was 5 seconds. It effectively reached the final
temperature within 40 seconds after the quench.


CHAPTER 4
EXPERIMENTAL PROCEDURES
The binary fluid system we have chosen for the kinetics study of was
guaiacol-glycerol-water (GGW), as mentioned earlier in Chapter 1 where we
discussed its phase diagram. Strictly speaking, it is a ternary system, but
because of the relatively small quantity of water (H2O) presented and the
physical and chemical properties of the three components, we argued (see §
1.1) that the mixture was quasi-binary and almost all of its physical properties
(e.g., critical exponents, etc.) were binary like. This assumption was confirmed
by the experimental studies on critical properties of the system [Johnston,
1983; Johnston et al. 1985].
We chose GGW mainly because of its high viscosity of glycerol, which
in its pure form is 1000 times more viscous than the common molecular fluid
like water. Wiltzius and Cumming [Wiltzius and Cumming, 1991] observed
the fast mode in a polymer mixture, which was highly viscous that its critical
dynamics was slowed down drastically in comparison to the binary fluids
used extensively for previous phase separation studies, e.g. L-W, I-W, etc..
Even in the polymer mixture, the fast mode was observed only within a time
window of 1000 seconds following the quench. If we rescale this time with
viscosity to the simple binary fluid systems like L-W, it will be reduced to the
order of seconds, which roughly equals the quench time. Therefore, to
65


66
observe the fast mode in a simple binary fluids, we had to search for a system
with relatively high viscosity, and GGW fulfilled the requirement. In
addition, GGW also provides the convenience of tunable Tc with water
content in the mixture, with 5 to 6% of water, Tc happens to be between 20 to
30 C, which is in the neighborhood of room temperature. There are other
binary mixtures involving glycerol, but their critical temperatures are either
close to 0 C or above 80 C. Therefore, GGW was most suitable
experimentally besides being most well studied, and the chemicals were
readily available and inexpensive.
4.1 Sample Preparation
The glycerol and guaiacol used were purchased from Aldrich. Glycerol
was of spectrophotometric grade, and 99.5+% in purity. Guaiacol was of 98%
of purity. Distilled water was purchased from VWR Scientific, and it was a
product of EM Industries, Inc.. All three were used without further
purification. The glycerol is highly hygroscopic, it will readily absorb moist
from the air if left open, and guaiacol is light and air sensitive. Therefore
their containers were only opened inside an argon environment (glove box)
during the preparation of the mixtures. We used 15 ml vials as the container
for the mixture, with Teflon lined caps and parafilm each vial was carefully
sealed and then stored in a refrigerator, which provided an environment of
relatively stable ambient temperature below the phase separation threshold of
the mixture and with minimal exposure to light. If it happened, air and light
contamination could be readily detected as the color of the mixture turned
increasingly dark over a period of a week or two.


67
We prepared the GGW mixture in an argon glove box, which had a
filter system that recycled the argon gas inside the box constantly and rid the
moisture and oxygen from the environment. First we mixed the water with
glycerol, the water was never more than 6% of the water-glycerol mixture by
mass. Throughout this thesis, we define the water content x of a mixture as
water content x = Mass of water (4.1)
Mass of water + Mass of glycerol
in its glycerol-water mixture. Then the mixture was shaken and agitated, and
finally stirred with a magnetic stirrer for at least 3 hours before mixing with
guaiacol. After the finally adding guaiacol to the mixture, it was tumbled and
stirred at a temperature well below its LCST for at least a day before use.
4.2 Phase Diagram of Guaiacol-Glvcerol-Water Mixture
The general features of the phase diagram of GGW have already been
discussed in section 1.1, and its general miscibility behavior was explained
qualitatively in terms of thermodynamics and peculiar interactions among
the molecules. A schematic phase diagram of GGW mixture is shown in Fig.
4.1. The critical behavior the mixture has been investigated by Johnston and
coworkers [Johnston et al., 1985], and they found when the mixture had over
5% of water content x (defined by eq.(4.1)), the two critical points, LCST and
UCST, were then essentially independent, and each had the same critical
exponents as 3-D Ising model (see § 1.1), which is the universal class to which
the binary fluids belong. For our experiments, we only concerned with the


68
0.0 0.2 0.4 0.6 0.8 1.0

Fig. 4.1 A schematic phase diagram of GGW mixtures with different
water contents x. Without water (x=0), glycerol and guaiacol are fully
miscible in any proportion. A miscibility gap opens up when x > 1.4%,
and the coexistence curve is a closed loop centered around temperature
To= 63.1C. Inside the loop, the mixture phase separates into two
phases. The size of the loop increases with the water content x, as
shown schematically with two loops for x is about 2% and 5%
respectively. Each loop has two critical temperatures Tcu (UCST) and
Tci (LCST), when x > 5%, the loop is large enough that the two
temperatures are well separated and independent from each other.
Our experiments were restricted in the neighborhood of Tc¡ (LCST)
on the phase diagram, where the mixture was initially equilibrated at
Tj, then quenched to Tf inside the coexistence loop and started phase-
separating, which we studied by means of light scattering.


69
Volume Percentage of Guaiacol (%)
Fig. 4.2 Coexistence curves in the neighborhood of LCST for several
GGW mixtures with different water concentrations. The LCST Tc/ is
very sensitive to the water content in the mixture, as it decreases
significantly with the increase of the water content.
LCST. The water content of the mixtures we have used was about 5.9%,
where the difference Tcu Tc\ is over 60 C.
We mapped out the phase boundary (coexistence curve) around the
LCST for several mixtures with different water contents, and the results are
shown in Fig 4.2. For each water content, we made a whole series of mixtures
with different mass ratios of glycerol-water and guaiacol, ranging from
80%:20% to 20%:80%, contained in 15 ml vials. Then the vials were
submerged in a refrigerated Neslab water bath. At any temperature within


70
the range of the miscibility gap (Tcj < T < Tcu), some vials would phase
separate while other still be homogeneous. At each temperature, we
measured the volume ratios of the two phases in those vials that had been
fully phase-separated. From the volume ratio and the original concentration
of the mixture, we determined the phase boundary. Before taking
measurements at the next temperature, these phase separated samples had to
be remixed again, preferably at a temperature where they were miscible. The
results from three series of mixtures are plotted in Fig. 4.2. As we were only
interested in the part of phase diagram in the neighborhood of the LCST Tc¡,
we limited our measurements to the relevant area.
As we can see from Fig. 4.2 that the phase diagram around the LCST is
remarkably flat. This feature was found in earlier experimental studies
[Johnston, 1983] and the WV model. It made difficult to find the critical
concentration Cc Fortunately the phase diagram is approximately symmetric,
therefore it is reasonable to assume the midpoint of the flat part as the critical
concentration, which is about 50% of guaiacol by volume and 48% by mass.
4.3 Sample Cell and Carrier
Our sample cells consisted of a sandwich of a fused quartz window and
a sapphire window separated by a Teflon gasket, with the GGW mixture in
between, and clamped in a stainless steel carrier with four screws (see Fig. 4.3).
Both of windows had diameters of 1 inch, but the sapphire window was 0.5
mm in thickness while the quartz was about 1.6 mm (1/16 inch). We chose
Teflon gaskets because of its chemical stability, after we had numerous
experiences with the corrosiveness of guaiacol, which dissolved several kinds


71
of epoxies that we could find. The Teflon gaskets were made from virgin
skived Teflon sheet purchased from PTFE Industries Inc., these sheets have a
superior surface smoothness compared to those commonly available. We
purchased these Teflon sheets of various thicknesses in order to have
samples of different thickness, which ranged from 0.2 mm to 1.0 mm.
Scattered Light
Sample Cell
Quartz Window
Regulated Water
Thermal!)
Teflon Gasket
HeNe Laser Beam
Fig. 4.3 A cutaway view of sample cell and heat exchange chamber
configuration as in the light scattering apparatus. The stainless steel
parts are slant shaded, the GGW mixture (dark shaded) is enclosed
between a sapphire window and a fused quartz window by a Teflon
gasket, the sapphire window is in contact with the thermally regulated
waterflight shaded). A laser beam incidents upwards from below,
as shown in Fig. 3.3.


72
The sample cell was then mounted onto a heat exchange chamber,
with the sapphire-window side in contact with the circulation water of
constant temperature, and the fused quartz window exposed to air (see Fig.
4.3). Sapphire is known to have a much higher heat conductivity and
hardness than fused quartz or any other types of glass because of its
crystallinity. The hardness of sapphire enabled us to use thin window that
still could stand the force of clamping without breaking. Also because of the
low thermal conductivity and larger thickness, the heat exchange across the
quartz window should be minimal compared to the sapphire side.
Consequently the sample was able to reach thermal equilibrium with the
circulating water quickly and was not much affected by the ambient
temperature on the quartz window side. In addition to the quartz window,
we chose the water content of the GGW mixture so that its LCST, Tci was
close to the room temperature of the our laboratory, the heat exchange
between air and the sample through the fused quartz window should be
negligible.
The GGW mixture was loaded into the sample cell inside a refrigerated
purge box, which was modified from a small commercial refrigerator
purchased from Sears, equipped with gloves and a see-through window.
Before and throughout the loading process, the box was constantly purged
with a dry nitrogen flow. Thus it provided a dry environment at a
temperature several degrees lower than the room temperature, which
prevented phase separation in the mixture from handling or accidental
temperature fluctuation. A sample cell of 0.8 mm thick contained
approximately 10 pi of the GGW mixture, and most of our studies were
performed on samples of that thickness. It usually had an air bubble about 3 ~
5 mm in diameter inside, which facilitated the mixing and homogenizing the


73
mixture once inside the cell if necessary. Due to the relatively small volume
of the sample cells, the critical temperature Tc/ of each sample cell was
different even though GGW mixture was drawn from the same container,
and the variation Tc¡ of could be as large as 0.5 C. This was understandable
as we knew that Tci was very sensitive to the water content of the mixture
(see § 4.1). There might be some change of water content during the transfer
of the mixture from the container to the sample cell, when the mixture was
exposed to the ambient environment for a however brief moment. After the
sample cell was sealed and clamped, Tci was still noticeably drifting
downwards over time on the average of about 0.005 C per week, presumably
due to the absorption of moisture by glycerol in the mixture through the
gasket. But overall, the Teflon gaskets gave a reasonably good seal.
4.4 Data Acquisition and Processing
Before every quench, the sample was first allowed to equilibrate at
initial temperature T, which was about 0.1 to 0.2 C below the critical
temperature Tc for at least two hours. Normally this time was between 6 to 12
hours if not more. Then it was quenched to a final temperature Tf inside the
miscibility gap by 0.01 to 0.15 C. The quench was effected by simultaneously
turning a pair of 4-way valves, which resulted the higher temperature water
being routed to the heat exchange chamber and the sample cell instead of the
lower temperature water (see §3.2). Following the quench, we measured the
scattered light intensity with the CCD camera, with equal interval of time
between measurements for every sequence that consisted of 20
measurements, the interval subsequently increased exponentially for later


74
sequences as the phase separation proceeded, in deference to the power law
behavior observed in the measurable quantities over time. The exact time of
the interval depended also on the quench depth AT, which determines the
dynamic time scale To through the relation
To = P/D,
(4.2)
where £ is the correlation length, which relates to AT as Eq. (1.1a) (see. § 1.1),
and D is the interdiffusion constant.
The temperature of the sample was read with a Yellow Springs
Instruments (YSI) thermistor with a nominal resistance of 30 kQ at 25 C. It
was located inside the heat exchange chamber next to the sapphire window of
the sample cell. The abruptness of the temperature quench and the
temperature stability following the quench could be monitored in this way,
and a typical temperature profile has been shown in Fig. 3.7.
The structure functions were calculated by averaging together 20 or 40
adjacent rows of pixels on the CCD image; the row corresponding to the
scattering plane, and same number rows on each side of the plane. This
averaging helped to reduce the laser speckle noise in the structure function
profile, and increase the sensitivity to low light levels at large momentum
transfer by improving the photon counting statistics. Before every quench,
the background noise was measured, and later subtracted from all subsequent
data set to eliminate the effects of scattering from imperfections on the glass
surfaces, dust and dirt, and other static sources, such as the dark current from
the CCD chip. Each measured structure function was also corrected for the
fact that the scattered light from scattering volume had to go through two
interfaces, the first between the liquid mixture to quartz glass and the second


75
between quartz glass and air, before exiting the sample cell; at each interface,
the scattered light has certain probability of being reflected, and that
probability increases with the scattering angle 6. Thus the scattered light
intensity directly measured by the CCD camera was reduced, especially at
larger angle 6, by as much as 14%. The correction could be simply calculated
from the Fresnel formulas [See, e.g. Hecht, 1987], and thus we called the
correction the Fresnel factor. In summary, the final structure factor S(q) is
calculated from measured light intensity 1 as
s( )=f(gWg(g) (4.3)
F(q)
where /g is the background, and F is the Fresnel factor just mentioned.
4.5 Treatment of Glass Surface with Trichlorosilanes
With the observation of the fast mode in the GGW mixtures [Shi et al.,
1993], it has been established that it is generic to a broad class of fluid mixtures,
as now the same phenomena had been seen both in a polymer system and a
simple low-molecular-weight binary fluids. Furthermore, we had evidence
that this new mode of phase separation is mediated by solid substrate surface
(container wall), i.e., driven by the wetting dynamics of the three phases
involved: two phase-separating liquid phases and the solid substrate phase.
We will discuss our results and supporting evidence for this picture in detail
in the next chapter, here I only want to briefly state our motive for chemically
altering the surface properties of the substrate and the method that we used.


76
Based on our assumption that the fast mode is a surface-mediated
phenomenon, we would expect the modifications the surface properties of
the substrate will change the interactions among the three phases. It was well
known that the glass surface chemically treated with a trichlorosilane
compound became hydrophobic, and in phase separated binary fluids, it was
experimentally observed that it effectively inverted the preferentially
absorbed phase on the glass surface [Dixon et al., 1985], and in addition, the
modification could result in shifting the wetting temperature Tw up to 10 C
[Abeysuriya et al., 1987; Durian and Franck, 1987]. The same phenomenon
was seen in our phase separated GGW: when a glass surface was just cleaned
with sulfuric acid, the glycerol-rich phase (or lower phase, because glycerol is
denser than guaiacol) will preferentially wets the surface; after it was treated
with phenethyltrichlorosilane (PET), the guaiacol-rich phase (upper phase)
wets the glass surface. Using two types of capillary tubes, one was acid cleaned
and the other treated with PET, situated across the phase separation meniscus,
we experimentally demonstrated that the first case had capillary rise, while in
the PET treated tubes had capillary sink, as shown schematically in Fig. 4.4.
Therefore, the silane treatment of the glass surface effectively changed the
surface tensions between the solid substrate and the two liquid phases. Based
on our assumption that the fast mode is surface-mediated, we expected it
should affect the dynamics of wetting and thus the fast mode.
We treated the window surfaces, both sapphire and quartz, with a self-
assembled (SA) monolayer of silane compounds, and we chose the method
mainly because of the simplicity of the procedure. The SA monolayer
method was developed rapidly over the last ten years chiefly by chemists for


77
Silane Treated Untreated
Guaiacol-rich phase
Fig. 4.4 Schematic representation of the capillary rise experiments. In
fully phase-separated GGW mixture, glycerol-rich phase has a higher
density, thus at the bottom while the guaiacol-rich phase at the top. In
untreated capillary tube (on the right), glycerol-rich phase has a higher
affinity to the tube wall, thus resulted in a rise. After the tube was
treated with silane, the guaiacol-rich phase has a higher affinity
instead, and that resulted in a capillary sink.
studying thin organic films [Ulman, 1991], which have the potential
applications in a whole range of fields, like molecular electronics and
nonlinear optics, etc.. The substrates used for the monolayer treatment had
included metals, metal oxides, and most often glasses. The silane compounds
were specially apt to form monolayers on the glass substrates, mainly because
these silanes usually were surfactants like soaps, they had a polar head group
centered on the silicon (see Fig. 4.5A), and a nonpolar alkyl (hydrocarbon) tail


78
of various length. They spontaneously assemble into monolayer films on an
appropriate substrate immersed in the solution, with the head groups binding
to the substrate and the alkyl tails stretching out. The substrate surface is
usually polar. The polar head group is attracted to the substrate surface and
then chemically binds to the surface where a molecular monolayer thus
forms eventually, e.g., for glass, a fully hydroxylated surface yields better
monolayer film. A simplified scheme of the process is shown on Fig. 4.5.
This monolayer effectively insulates the original substrate from anything that
comes in contact, and thus drastically changed its surface properties.
The self assembled monolayer has a higher stability compared to the
organic films deposited in other ways, e.g., Langmuir-Blodgett films, because
the molecules are chemically bonded to the surface. For our experiments, it is
important to have the film stable enough so it could withstand to the attack
by the GGW mixture, at least for a period of several weeks.
We have used the four trichlorosilanes for the treatments of the glass
windows: ethyltrichlorosilane (C2H5 SCI3XETS), decyltrichlorosilane (C10H21
- SCI3 )(DTS), octadecyltrichlorosilane (C18H37 SCI3XOTS) and
phenethyItrichIorosilanefCHs-C2H4 SCI3XPET), all these chemicals were
purchased from Huls America Inc.. The first three compounds belong to the
same homologous series and thus shared very similar molecular structures,
and the only differences are in the lengths of the alkyl chains, which increase
from 2 carbons in ETS to 18 carbons OTS. PET has a phenyl ring at the end of
the tail, and therefore should have higher affinity for guaiacol molecule than
glycerol. This had been confirmed in our capillary rise experiments.
The windows (both sapphire and quartz) were first cleaned in the so
called "Piranha" solution made from sulfuric acid and hydrogen
peroxide(30%) at 3:1 volume ratio. Caution must be taken in handling this


79
solution, as it reacts violently with any organic matter, which it literally
"eats" away, and it should only be handled inside a fumehood. The windows
were submerged in the "Piranha" solution for about an hour before being
taken out. Afterwards they were rinsed thoroughly with deionized water,
ethanol and acetone in sequence. Then they were dried in a nitrogen stream
at room temperature. After the cleaning with the "Piranha" solution, the
window becomes more hydrophilic compared to the original surface, as the
glass surface is saturated with OH groups [Dixon et al., 1985], and
hydroxylated. These windows could be used to make sample cells or further
treated with the trichlorosilanes.
To treat the windows with the trichlorosilanes, we made a solution of
each compound. As solvent we used 70% hexadecane with 30% chloroform
by volume, and then we added 1% of trichlorosilane by volume to make the
solution. Because all trichlorosilanes react with water to form polymers, all
the processes involving them must be done inside a dry glove box. After
windows had dried, they were submerged in the trichlorosilane solution for
about 10 minutes before being taken out again. Then as an indication of a
successful treatment, the windows should be dry and clean, and no droplets of
solvent were left on the surface as they were withdrawn from the solution.
Thus the surface modification enabled us to change the surface tension
between the fluid phases and the solid substrate, as established
experimentally. The light scattering experiments were then carried out on
these treated windows, the results of which will be discussed in the next
chapter.


80
Fig. 4.5 Trichlorosilane monolayer formation on glass surface. Here
we use OTS as an example, it has a 18 carbon tail and a strongly polar
head centered at a silicon, (a) Before the treatment with silane, the glass
surface is fully hydroxylated, i.e., saturated with OH groups, which
attracted the polar heads of the silane molecules, (b) After the
treatment, a monolayer of OTS formed, each OTS molecule is
chemically bonded to the glass surface in addition to the nearest
neighbors.


CHAPTER 5
RESULTS AND DISCUSSIONS
This chapter is primarily concerned with the methods of data analysis and
the results of our experiments. Our main efforts will be on the fast mode, whose
properties I will try to outline. However, I find it necessary and useful to give an
adequate description on the slow mode, and present the evidence of it being the
phase separation mode in the bulk, which had been thoroughly investigated by
the previous researchers in the field (see § 1.5). We will discuss the fast mode in
full detail and its properties, including some results from the video microscopy
studies. These fast mode results are compared with those from polymer mixture
by Cumming and coworkers [Cumming et al., 1992]. Finally, we will discuss the
surface treatments and their effects on the fast mode.
5.1 Data Analysis
So far we had limited our investigation to the spinodal decomposition
process in a GGW mixture with critical concentration Cq, which is 47.9% of
guaiacol by mass, 50.3% by volume. For critical quenches, we observed the well
known spinodal rings associated with the bincontinuous interpenetrating
network morphology seen in phase separation via spinodal decomposition. The
81


Scattered Light Intensity (CCD Count)
82
0.1 1
Momentum Transfer q (pm'1)
Fig. 5.1 Structure functions at various times after a quench of 0.111C
above Tc The 4 functions of later times have been scaled up by a factor
of 2 consectively to avoid overlapping of the data.


83
(A)
(B)
(C) (D)
Fig. 5.2 A sequence of video micrographes of GGW mixture under
going a phase separation. At time t =0, the mixture was quenched
into the miscibility gap by a depth of 0.145 C. Before the quench the
mixture was homogeneous and structureless. Structures similar to
those in spinodal decomposition gradually started to emerge after
the quench These eight micrographes were taken 25,60,190,210,
382,415,595, and 740 seconds in sequence, respectively, following
the quench. The longer white bar shown on the upper lefter corner
in A, B, C, and H is 60 pm in length, while the shorter bar in D, E,
F, G is 50 pm in length. This difference of was simply due to the
switching from a 32X to a 1 OX objective in microscope. Micrographes
A to F show the growth of the fast mode, (Continued to the next)


84
(G) (H)
(Continued from the previous page), while the microscope was focused at
about 20 pm above the surface of the quartz window. G and H show the
slow mode growth, when the microcroscope was focused on a plane about
300 pm above the surface. At later stage, the fast mode morphology was
quite anisotropic, as seen in E and F, it looks like one of the fluid was
spreading across the plane while replacing the other fluid. E was taken
using the schlieren technique, the fuzziness of the graph was partly due
to the slow mode, which was outside the focal plane of the microscope.
The fast mode was almost gone by the time of the graph G was taken, only
a remnant of it could be seen in the center lower region of G.


85
major difference of our experiments are the observation of two spinodal
rings, instead of one as in most of other experiments. Fig. 5.1 shows a time
sequence of the scattered light profiles, or structure functions, it clearly shows
the emergence and development of two peaks and their evolution. These
data had been averaged over every 10 neighboring pixels, in order to reduce
number of data points in each structure function to about 100, so that each
data point could be clearly represented in the Figure. Also as a result of the
averaging, the noise in the data had been reduced. The two peaks correspond
to the two length scales presented in the system, as can be seen in Fig. 5.2,
which is a time-series of video-micrographes of a sample phase separating.
The first six micrographes show the morphology and its growth of the fast
mode, and the microscope was focused on a plane about 20 pm above the
inner quartz window surface, the last two show that of the slow mode after
the fast mode had finished, at a location about the middle of the sample cell.
These micrographes clearly show the two distinct length scales presented in
the sample, although at different locations in the sample.
As we were most interested in finding the time evolution of the peaks
in the present research, we only wanted to extract two quantities from each
scattering profile I(q) in a time sequence: the peak position and peak intensity.
The peak position is directly related to the average domain size, and from its
evolution we get the growth law. The peak intensity is a quantity crucial in
testing the scaling hypothesis and other properties. So we chose two simple
functions for the model line shape: for the fast mode, we used a gaussian, i.
e.,
S(q) = If exp! --^1 +/o,
I 2 cr I
(5.1)


86
where If, c¡f, a and I0 are the fitting parameters, q¡, anda are the peak position
and peak width, respectively, and the sum of If, and Io is the peak intensity;
for the slow mode, we chose one of the Furukawa functions (eq. (1.43)), and
let y=4, then
(5.2)
where Is and qs are slow mode peak intensity and peak position, respectively.
We chose the model function largely because at large q qs, S(q) ~ q'4 which
is known as the Porod law [Porod, 1951; Porod, 1982]. This characteristic tail is
due to scattering from sharp interfaces in three dimensional systems. In our
experiments, we would expect to see the q'4 tail at the late stage of phase
separation, when the slow mode peak position qs is fairly small, mainly due
to the limited range of q window of our apparatus. At this stage, well defined
interfaces should have developed and thinned. Fig. 5.3 illustrates such an
asymptotic approach to the q-4 behavior, where the tails (q qs) of the
structure functions become increasingly parallel to the q~4 line, which is a
straight line on a log-log plot, at later stages of phase separation.
Our primary interest was in the peak positions as functions of time,
qf(t) and qs(t), and the peak intensities as functions of time, Ij(t) and 7s(f). The
model functions (eq. (5.1) and eq. (5.2)) served reasonably well individually as
peak finders when fitted to the experimental data with a least square fitting
routine. For the fast mode, we usually limited the fitting in range q-space to
the interval between 1000 cm-1 and q¡+ a, which corresponds approximately


Intensity
o/
Fig. 5.3 Late time structure functions of a quench of depth 0.09C.
As the phase separation proceeded, the tail of the structure function
approached asymptotically cf4, satisfying the Porod law. The peak
in the structure function corresponds to the slow mode.


88
to the point where the structure function falls to half of its peak value. The
lower limit of 1000 cm-1 was due the beam stopper (see Fig. 3.1), which served
the purpose of reducing the transmitting beam intensity It by several order of
magnitude before reaching the CCD camera, and it also blocked the scattered
light within a small angle which corresponded to 1000 cm'1 in q. The slow
mode peak was fitted with the Furukawa function eq. (5.2), usually within a
range of in /-space where the corresponding intensities were above one half
of peak value. Fig. 5.4 illustrates the fitting of both model functions to fast
and slow mode, respectively, in the structure functions following a typical
quench. Both fitting functions worked reasonably well, within their limited
range of /-space, they adequately fitted the data, so that the fitting parameters
qf, qs, If, and Is could be determined as functions of time. Among the fitting
parameters, the peak positions qj and qs were the most robust, especially qf,
simply due to the gaussian function.
A word of precaution is needed here. In our fitting scheme, the two
peaks were fitted separately, although with two different model functions.
That is, we fitted the same structure function twice, once for the fast mode
peak, and the other for the slow mode. Each time we got the peak position
and intensity as fitting parameters for one peak. Ideally, if we have one
model function that could fit the structure function in the entire /-space, i.e.,
both of the peaks at once, we should have obtained the fitting parameters in a
single fit. In comparison, our two-step fitting process would introduce certain
errors in the resulting fitting parameters. For example, the peak position and
peak intensity for the slow mode, qs and Is, could be strongly influenced by the
tail of the fast mode peak, when the peaks were not well separated, as the fast
mode intensity was usually far greater than that of the slow mode. Even


Intensity (CCD Count)
Fig. 5.4 The structure functions at various times as indicated after a quench
of 0.082 C. In the first three functions, at 48, 82, and 116 seconds, the slow
mode peak hadn't fully developed. At 328 seconds, the fast mode has gone
forward into the beam stop. The two types of fits, gaussian annd Furukawa
functions, are illustrated for these data set. Each successive structure
function has had 2000 CCD count added to give the figure greater clarity.


90
when the fast mode peak had already gone into the beam stop, its tail at the
location of the slow mode peak could still be substantial compared to the slow
mode peak itself. Therefore, the fitting parameter qs could be reduced and Is
increased artificially simply due to our two-step fitting process, especially
when the two peaks were just emerging and thus not well separated. In that
case, the error introduced could be too large to be ignored. Thus among the
fitting parameters, the fast mode parameters qf and If were more accurate
than their counterparts of the slow mode, because in the same structure
function, the fast mode intensity was usually far greater.
Sometimes the sum of the two model functions (eq. 5.1 and eq. 5.2)
yielded good results in fitting the structure function in the entire g-space
available in our experiment. One of the examples is shown in Fig. 5.5, the
model function fitted the data very well, and the two peaks were fitted in a
single step. To give greater clarity of the data and the fit function, we have
added an off-set to the data, the earlier four structure functions have been off
set by 200 CCD count successively, and the four others at later times by 1000.
At later times, the fit function showed progressively larger deviation from
the data at the valley between the peaks and at the tail, part of it could be
possibly attributed to the multiple scattering effect, as the quench depth was
large enough, we would expect the contribution of the multiple scattering to
be progressively larger as the phase separation proceeded. Also clearly seen in
Fig. 5.5 is that the fast mode peak, which was fitted with a gaussian model
function, evolved at a much faster rate toward q = 0 (beam stopper) than the
slow mode, which was fitted with the Furukawa function (eq. (5.2)). Thus the
former was conveniently named fast mode, while the later slow mode. We
will give a detailed discussion on the kinetics of the two modes next.


Scattered Light Intensity (CCD Count)
91
1 03 1 o4 1 o5
Momentum Transfer q (cm1)
Fig. 5.5 Structure functions of after a quench of 0.126 C into the
miscibility gap and the fitting with the sum of gaussian and Furukawa
function in the entire range of q-space available in our experiments.


Full Text
LD
H80
. 6565a.
UNIVERSITY OF FLORIDA
3 1262 08556 9894



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