UNIVERSITY OF MINNESOTA This is to certify that I have examined ...

UNIVERSITY OF MINNESOTA
This is to certify that I have examined this copy of a doctoral thesis by
Kwanho Chang
and have found that it is complete and satisfactory in all respects,
and that any and all revisions required by the final
examining committee have been made.
David C. Morse, Christopher W. Macosko
Name of Faculty Advisors
Signature of Faculty Advisors
Date
GRADUATE SCHOOL

Block Copolymer Surfactants in Immiscible
Homopolymer Blends
A THESIS
SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL
OF THE UNIVERSITY OF MINNESOTA
BY
Kwanho Chang
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
David C. Morse and Christopher W. Macosko, Advisors
December 2005

c○ Kwanho Chang 2005

To my parents, Sonya, and Kwanjoo
i

Acknowledgments
This thesis would have not been completed without collaboration and support of many
people. First and foremost of them, I would like to thank my advisors, Professors David
Morse and Chris Macosko for their incessant guidance and encouragement throughout the
entire period of my Ph.D. education. I have been incredibly lucky to have an opportunity
to explore both theoretical and experimental research areas. Their advice and valuable
discussion from outstanding knowledge and insight in polymer science has been critical to
expand my experience.
With respect to polymer synthesis, I have greatly appreciated the help by Professor
Timothy Lodge and Professor Frank Bates. All polymers used in this thesis have been
synthesized in their labs. I also would like to specially thank Professor Daniel Joseph in
the Aerospace Engineering and Mechanics Department. He generously allowed me to use
his spinning drop tensiometer over two years.
I have enjoyed being a part of both Morse and Macosko research groups and would like
to thank all group members for helping my research in overall directions. Discussion with
Hyun Jeon was very helpful to initialize my research in polymer blends. I also thank her
for introducing my future wife, Sonya, to me. For all the generous assistance and support
of my research, I would like to thank Jongwhi Lee, Jonathan Schulze, Pieter Spietal,
Jennifer Dean, Chris Tyler, Drew Davidock, Kasiraman Krishnan, Jianbin Zhang and
YongIl Kwon. Thanks go out to Dave Hultman and David Giles for modifying a spinning
drop tensiometer, and my officemates Lisa Lim, Lifeng Wu, Yoichiro Mori, Mickael Castro,
ii

David Ackerman, Hyunwoo Kim for the most precious time in my life.
Finally, I never thank enough my mom for her endless love and encouragement. Her
letters sent me for six years are just two hundred. I do not find any proper words for
appreciating my love, Sonya. Also, I thank my best friends, Byung Wook and Seungho,
for their support.
I would like to acknowledge IPRIME and the RTP company for financial support and
the Minnesota Supercomputing Institute (MSI) for computation time.
iii

(261words)
Abstract
Block copolymers have been often used as emulsifying agents in binary homopolymer
blends. We have used both theory and experiment to study the competition between
interfacial adsorption and micellization of copolymer in mixtures of copolymer and two
immiscible homopolymers.
Self-consistent field theory (SCFT) has been used to characterize the elastic properties
of a block copolymer monolayer dividing A- and B-rich homopolymer domains in the
context of the Canham-Helfrich theory, and to study the thermodynamics of two phase
systems in which one of the coexisting phases contains micelles.
Interfacial tension has been measured between polyisoprene and polydimethylsiloxane
in the presence of poly(isoprene-b-dimethylsiloxane) copolymer using a spinning drop tensiometer.
In the case of a symmetric block copolymer, we observed a decrease of interfacial
tension by three orders of magnitude. Experimental measurements of interfacial tension
were compared with the SCFT predictions for a system in which swollen spherical micelles
are present in one phase. The predicted dependence of interfacial tension on the block
composition f A agreed well with the experiment results for sufficiently asymmetric copolymers,
with f A > 0.65. However, for nearly symmetric copolymers with 0.5 < f A < 0.6,
interfacial tension was much lower than these predictions. For those nearly symmetric
copolymers, the formation of a bicontinuous microemulsion phase has been observed.
Interfacial tension and the cmc have also been measured in a system of polybutadiene
and polystyrene with poly(styrene-b-butadiene) block copolymer. The cmc has been deiv

termined by both X-ray scattering and transmission electron microscopy. In this system,
measured interfacial tension was strongly affected by the slow diffusion of block copolymer.
v

Contents
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii
1 Introduction and Background 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Overview of The Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Phase Behaviors in Oil/Water/Nonionic Surfactant Systems . . . . . 7
1.3.2 Interfacial Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.3 Wetting/Nonwetting Transition of Interface . . . . . . . . . . . . . . 12
1.3.4 Helfrich Bending Elasticity . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.5 Stability of Surfactant Monolayer . . . . . . . . . . . . . . . . . . . . 17
1.3.6 Thermal Fluctuation . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Self Consistent Field Theory 28
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Formulation of SCFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.1 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.2 Homogeneous Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.3 Interfacial Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.4 Nondimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . . 32
vi

2.3 Numerical Implementation of SCFT . . . . . . . . . . . . . . . . . . . . . . 33
2.3.1 Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.2 Numerical Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4 Iteration Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.1 Relaxation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4.2 Newton-Raphson Algorithm . . . . . . . . . . . . . . . . . . . . . . . 41
3 Interfacial Bending Elasticity 44
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 Interfacial Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.1 Mechanical Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2.2 Thermodynamic Variables . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2.3 Helfrich Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.4 Lifshitz Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3 SCFT Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 Symmetric Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.5 Asymmetric Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4 Swollen Micelles and Interfacial Tension 70
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 Micellization - Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.1 Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.2 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2.3 Critical Micelle Concentration . . . . . . . . . . . . . . . . . . . . . 79
4.2.4 Interfacial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2.5 Helfrich Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
vii

4.3 SCFT Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.4 Results - Micelle Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.5 Results - Helfrich Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.6 Other Possible Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5 Transport of Block Copolymer Surfactant Between Two Homopolymer
Phases 101
5.1 No Micelles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.2 Micelles in Matrix, c I c < c II
c K . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2.1 Micellar Diffusion (No Exclusion Zone) . . . . . . . . . . . . . . . . 106
5.2.2 Molecular Diffusion (Exclusion Zone) . . . . . . . . . . . . . . . . . . 107
5.3 Micelles in Wrong Matrix, c II
c < c I c/K . . . . . . . . . . . . . . . . . . . . . 113
6 Interfacial Tension of PI/PDMS with PI-b-PDMS Copolymer 118
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.2.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.2.2 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.2.3 Interfacial Tension Measurement . . . . . . . . . . . . . . . . . . . . 125
6.2.4 Small Angle X-Ray Scattering . . . . . . . . . . . . . . . . . . . . . 126
6.2.5 Dynamic Mechanical Spectroscopy . . . . . . . . . . . . . . . . . . . 127
6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.3.1 Bare Interfacial Tension . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.3.2 Dependence on Copolymer Concentration . . . . . . . . . . . . . . . 128
6.3.3 Dependence on Copolymer Composition . . . . . . . . . . . . . . . . 130
6.3.4 Dependence on Homopolymer Molecular Weight . . . . . . . . . . . 134
viii

6.3.5 A Larger Copolymer . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.3.6 Bicontinuous Microemulsion Phase . . . . . . . . . . . . . . . . . . . 136
6.4 Analysis of Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.4.1 Estimate of Diffusion Time . . . . . . . . . . . . . . . . . . . . . . . 144
6.4.2 Can Micelles Reach the Interface? . . . . . . . . . . . . . . . . . . . 145
6.4.3 Experimental Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.5 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.5.1 Equilibrium vs. Nonequilibrium . . . . . . . . . . . . . . . . . . . . . 152
7 Micellization and Interfacial Tension in Viscous Ternary PS/PB/P(S-b-
B) Blends 158
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.2.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.2.2 Blending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.2.3 Preparation of SDT Samples . . . . . . . . . . . . . . . . . . . . . . 163
7.2.4 Interfacial Tension Measurement . . . . . . . . . . . . . . . . . . . . 164
7.2.5 Transmission Electron Microscopy (TEM) . . . . . . . . . . . . . . . 166
7.2.6 Small Angle X-ray Scattering (SAXS) . . . . . . . . . . . . . . . . . 166
7.2.7 Dynamic Mechanical Spectroscopy . . . . . . . . . . . . . . . . . . . 166
7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.3.1 CMC in PS4K/SB Binary Mixtures . . . . . . . . . . . . . . . . . . 167
7.3.2 Bare Interfacial Tension . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.3.3 Interfacial Tension Reduction . . . . . . . . . . . . . . . . . . . . . . 174
7.3.4 Blend Morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
ix

7.4.1 Kinetics Below the CMC . . . . . . . . . . . . . . . . . . . . . . . . 182
7.4.2 Kinetics Above the CMC . . . . . . . . . . . . . . . . . . . . . . . . 182
7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
8 Summary and Outlook 189
8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
8.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
A Flory-Huggins Interaction Parameter between PS and PB 196
B Estimate of Diffusion Coefficient of P(S-b-B) Block Copolymer in the
Homopolymer Phase. 202
C The Effect of the Uniaxial Extensional Flow on Interfacial Tension 206
C.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
C.1.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
C.1.2 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
C.1.3 Extensional Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . 209
C.1.4 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
C.1.5 Shear Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
C.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
C.2.1 Extensional Viscosity of the pure Homopolymers . . . . . . . . . . . 212
C.2.2 Shrinkage of Multilayers . . . . . . . . . . . . . . . . . . . . . . . . . 214
C.2.3 Extensional Viscosity of Multilayers . . . . . . . . . . . . . . . . . . 215
C.2.4 Interfacial Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
x

List of Figures
1.1 Schematic phase prism of a water-oil-nonionic surfactant (C i E j ) system in
a vertical axis of temperature. A water-rich two phase, a bicontinuous microemulsion
middle phase, and the oil-rich two phase are denoted by 2¯, 3,
and ¯2, respectively. The three test tubes on the right-hand side illustrate
how the phases are observed in experiment. An upper critical solution temperature
T β is not necessarily located at higher temperature than T u .[73].
Redrawn from Strey.[76] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2 A schematic phase diagram along an isopleth of the phase prism. T o = T m
and φ ∗ s is the minimum surfactant concentration to reach a uniform middle
phase. The lower is φ ∗ s , the more efficient is the surfactant. Reproduced
from Wennerström et al. .[86] . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Isothermal diagram of water/n-octane/C 10 E 5 mixtures at the mean temperature
T = 44.6 ◦ C . Note the 1-phase region near the middle phase enclosed
by the lamellar phase. Reproduced from Strey.[77] . . . . . . . . . . . . . . 11
1.4 Schematic diagram of the individual interfacial tension curves of oil/water(γ ab ),
water/emulsion(γ ac ), and oil/emulsion(γ bc ) on a log scale as a function of
temperature. Redrawn from Sottmann and Strey.[80] . . . . . . . . . . . . 12
1.5 Reduced oil-water interfacial tension γ ∗ as a function of reduced temperature
τ ∗ . The full line is calculated from Eqn. 1.2. The definition of γ ∗ and τ ∗
are given in ref. [80]. Redrawn from Sottmann and Strey.[80] . . . . . . . . 13
xi

1.6 Lens of C 8 E 3 -rich middle phase floating on the interface between the lower
water- and upper decane-rich phase at T = 21.8 ◦ C . Reproduced from
Kahlweit et al. .[75] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.7 Phase stability of spherical, cylindrical, and lamellar micelles. The crosshatched
region is a two-phase coexistence of spheres and cylinders. Sphere
is the most stable structure along the emulsification failure line. In this plot,
r = ρ ˜C o and x = −¯κ/(2κ + ¯κ). Reproduced from Safran.[62] . . . . . . . . 15
1.8 The comparison of the topology among different structures. Only (C) has
g = 1 where as others have g = 0, where g is the number of handles.
Reproduced from Taddei.[65] . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1 Interfacial tension γ vs. the number of spatial nodes in a simulation box
with the fixed size of 800 Å. The adaptive grids dramatically increase the
accuracy of the calculation compared to the case with the same number of
grid points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2 The interfacial tension difference ∆γ as a function of h 2 for a uniform grid. 37
2.3 The chemical potential difference ∆µ A and ∆µ C as a function of (∆t) 2 . . . 38
2.4 Interfacial tension γ with various time steps ∆t for a uniform grid. There
is little effect of time step on γ. . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5 The chemical potential difference ∆µ A and ∆µ C of the homopolymer A and
block copolymer C determined at both boundaries of the simulation cells
vs. the spatial step size h. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
xii

3.1 Schematic view of the state of a two-phase system at fixed pressure as a
function of T and of a copolymer activity Z C ∝ e µ C/kT , which is defined for
this purpose so that Z C is proportional to the concentration of copolymer
dissolved in one of the two coexisting phases. The Lifshitz point is the
intersection of a line of critical points, for which the copolymer concentration
tends to increase with decreasing temperature, with a saturation line e µ∗ C (T) ,
for which copolymer concentration decreases with decreasing temperature
as the copolymer becomes less soluble. . . . . . . . . . . . . . . . . . . . . 52
3.2 SCFT results for interfacial tension γ vs. half curvature C/2 = 1/R for
spherical interfaces with R ≥ 100 Å, for system with the copolymer of a
fixed size of B block f B χN C = 10 and symmetric homopolymers of fixed
lengths α A = α B = f B , with b A /b B = 1, for several values of f A . Solid
lines are quadratic fits of the curvature dependence at small curvatures to
Eqn. 3.8, from which κ + and τ can be extracted. . . . . . . . . . . . . . . 56
3.3 SCFT results for interfacial tension γ vs. curvature C = 1/R for cylindrical
interfaces with R ≥ 100 Å, for the same mixtures as those studied in the
previous figure. κ can be obtained. . . . . . . . . . . . . . . . . . . . . . . 56
3.4 The non-dimensional bending rigidity [κ + ] of a sphere vs. χN in symmetric
mixtures with various values of α = α A = α B at fixed β = 1. . . . . . . . . 59
3.5 The normalized bending rigidity [κ] vs. χN for symmetric systems with
several values of α. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.6 The normalized Gaussian rigidity [¯κ] vs. χN for symmetric mixtures with
several values of α. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.7 The ratio of the Gaussian rigidity to bending rigidity −[¯κ]/[κ] vs. χN for
symmetric mixtures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
xiii

3.8 Bending parameter τ of block copolymer monolayer vs. a block ratio f A of
the A block for systems with f B χN = α A χN = α B χN = 10. The system in
this plot is identical to that in Fig. 3.2 and 3.3. f bal
A
is always approximated
as a limiting value linearly extrapolated to C = 0 as shown here. . . . . . . 63
3.9 Bending rigidity κ + of sphere vs. a block ratio f A of the A block for systems
with f B χN = α A χN = α B χN = 10. In this article, κ + is always
approximated as a limiting value linearly extrapolated to C = 0 as shown
here. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.10 The variation of the balance point f bal
A
with fixed α = 1. The solid line
indicates the balance point f L at an isotropic Lifshitz point. . . . . . . . . 64
3.11 The variation of the balance point f bal
A
for asymmetric mixtures with a fixed
minority block size f B χN C = 10. The solid line indicates the balance point
f L at an isotropic Lifshitz point. . . . . . . . . . . . . . . . . . . . . . . . . 64
3.12 The variation of the balance point f bal
A
with asymmetric statistical segment
length b. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.13 Rigidities κ + , κ, and ¯κ vs. β = α B /α A for systems with a copolymer
minority block of size f B χN = 10, a B homopolymer of length α B χN = 5,
and b A = b B , for varying size A homopolymers. Values for κ, κ + , and ¯κ are
given in units with kT = 1 on the left scale, and corresponding values [κ],
[κ + ], and [¯κ] on the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
xiv

4.1 Schematic phase diagram of a ternary system comprised of homopolymers A
and B and diblock copolymer C, in which an A rich phase of swollen micelles
(or, equivalently, microemulsion droplets) coexists with a phase of nearly
pure B. The dotted line b-c is a line of the critical micelle concentration
in a single A-rich phase. Micelles are drawn above this line. A point b
corresponds to the cmc in two phase ternary system. . . . . . . . . . . . . 73
4.2 Dependence of concentrations of free copolymer and of copolymer within
micelles upon the total concentration of copolymer within the micellar phase,
along the coexistence line. . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3 Dependence of copolymer chemical potential upon total concentration of
copolymer within the micellar phase, along the coexistence line. The dotted
line shows the continuation of ideal solution behavior beyond the cmc . . . 75
4.4 Schematic of the ratio γ/γ 0 of macroscopic interfacial tension γ to its value
γ 0 in the absence of copolymer, vs. the total volume fraction of copolymer
within the A-rich micellar phase. As in Fig. 4.3, the solid line shows the
behavior of a micellar solution, and the dotted line shows the continuation
of ideal solution behavior, in the absence of micelles. . . . . . . . . . . . . 76
4.5 Copolymer volume fraction φ cmc
C
at the critical micelle concentration of a
two-phase ternary system (solid lines), and of a binary mixture of copolymer
C in A (short dashed lines), vs. volume fraction f A of the corona
block within the copolymer, for systems with homopolymer sizes α A χN =
α B χN = 10, for three values of χN CB . Also shown is the copolymer volume
fraction φ ∗ C
(dot dashed line) at which the macroscopic interfacial tension
of the two phase system extrapolates to zero. Note that φ ∗ C
always exceeds
the cmc φ cmc
C
of ternary two-phase system, but φ ∗ C
drops below the cmc of
the corresponding binary system for nearly symmetric copolymers. . . . . 87
xv

4.6 Micelle core radius at the cmc for the ternary two-phase system and binary
system considered in Fig. 4.5, vs. volume fraction f A of the corona block.
The hollow and solid symbols represent the radii of unswollen and swollen
micelles, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.7 Macroscopic interfacial tension γ cmc at the cmc, normalized by the value γ 0
for a bare interface, vs. volume fraction f A of the copolymer corona block,
for the same parameters as those used in Fig. 4.5. Leibler’s prediction based
on the unswollen micelle is shown with a dotted line for comparison. . . . 89
4.8 Macroscopic interfacial tension γ cmc at the cmc, normalized by the bare tension
γ 0 , vs. copolymer corona volume fraction f C for system with f B χN =
10 and various values of the ratio β of copolymer molecular weights. . . . . 90
4.9 Dependence of critical micelle concentration φ cmc
C
on χN BC = f B χN for
systems with f A = 0.6 and α A χN = α B χN = 10. Symbols show predictions
for the ternary two phase system by SCFT (filled triangles) and predictions
for a binary system by SCFT (filled circles) and by the approximate strong
LOW theory (open triangles). The dashed line is the function Ce −χN BC,
with C = 376. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.10 Comparison of SCFT predictions for copolymer cmc φ cmc
C
of a ternary twophase
system (filled triangles) and a binary system (open triangles) to the
strong stretching predictions for the binary system by Leibler (dot dashed
line) and LOW (solid lines) for the systems considered in Fig. 4.5. . . . . . 91
4.11 The prediction of γ/γ o from direct micelle simulation (hollow circle) and
from Helfrich bending elasticity (solid circle) as a function of f A . They are
consistent for f A < 0.55 but start to deviate for larger f A . Surprisingly, the
quadratic function (solid line) of γ/γ o (Eqn. 4.23) with τ ′ and κ + evaluated
at C = 0 accurately fits the entire results from the direct micelle simulation. 92
xvi

4.12 The comparison of micelle radii obtained from direct micelle simulation (hollow
circle) and from Helfrich bending elasticity (solid circle) as a function of
f A . They are consistent for f A < 0.55 but the latter starts to underestimate
the radii for larger f A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.13 The normalized slope [τ ′ ] as a function of χN in a range of 0.25 ≤ α ≤ 2 . 94
4.14 The prefactor of γ flat ,
τ ′2
8κ + γ o
, normalized by a bare interfacial tension γ o as
a function of χN CB in a range of 0.25 ≤ α ≤ 2 . . . . . . . . . . . . . . . . 94
5.1 A schematic diagram of concentration profile of surfactant perpendicular to
an interface z = 0 below the cmc. . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2 A schematic diagram of concentration profile of surfactant and micelle perpendicular
to an interface z = 0 without exclusion zone (Q < 1) above the
cmc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.3 A schematic diagram of concentration profile of surfactant perpendicular to
an interface z = 0 for Q > 1 above the cmc. A solid line is for the case of
S ≫ 1 and a dotted line for the case of S ≪ 1. . . . . . . . . . . . . . . . . 108
5.4 A schematic diagram of concentration profile of surfactant when surfactant
is initially added to the wrong phase (I) (a) when Eqn. 5.21 and Eqn. 5.22
are satisfied (b) not satisfied. . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.1 The transient diameter of the PI drop in the PDMS matrix as a function
of time. (a) PI1 and PI6 in the absence of block copolymer (b) PI4 in
PDMS1/IDMS6 and in PDMS2/IDMS12. The concentrations of IDMS6
and IDMS12 in the corresponding PDMS matrices are 0.1 wt % and 0.2 wt
%, respectively. The angular velocity is 6300 rpm. . . . . . . . . . . . . . . 128
xvii

6.2 Bare interfacial tension between PI and PDMS1 in the absence of block
copolymer as a function of molecular weights of PI. The solid line is the
SCFT prediction using χ = 0.175. . . . . . . . . . . . . . . . . . . . . . . . 129
6.3 The interfacial tension reduction of PI4/PDMS1 and PI4/PDMS2 as a function
of the concentration of IDMS1 and IDMS12, respectively. The dashed
lines are guides to the eye. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.4 The interfacial tension reduction between PI4 and PDMS1 as a function of
IDMS1. The solid lines are the SCFT prediction with four different χ’s. . . 131
6.5 Interfacial tension in PI4/PDMS1/IDMS system as a function of the block
ratio f A of IDMS. The quadratic dependence of interfacial tension on f A
agrees well with the SCFT prediction and the swollen micelle theory (hollow
dots) for f A > 0.65. Leibler’s theory[23] (a dotted line) predicts that a wider
range of f A will cause vanishing interfacial tension. . . . . . . . . . . . . . 133
6.6 Data and calculations of Fig. 6.5 plotted on a logarithmic scale. Near a
balance point, measured values are much lower than those predicted by
SCFT. The concentrations of block copolymer are all greater than the cmc. 133
6.7 The time-dependent diameter of a PI4 drop in a PDMS1 matrix at an
angular velocity of 6300 rpm with two different concentrations of IDMS7
(f A = 0.6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.8 Interfacial tension in the system of PDMS1/PI/IDMS1 as a function of
molecular weight of PI. The concentration of IDMS1 is 0.1 wt %. . . . . . 137
6.9 Interfacial tension in the system of PDMS2/PI/IDMS13 as a function of
molecular weight of PI. The concentration of IDMS13 is 0.2 wt %. The
minimum of the interfacial tension is not ultra-low, but much greater than
that in Fig. 6.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
xviii

6.10 The variation of the diameter of the PI5 drop in the PDMS2 matrix at a
concentration of 0.2 wt % of IDMS13. The rotational speed is 6300 rpm. . 138
6.11 The ternary isopleth blends of the system PDMS1/PI4/IDMS3 for the various
concentrations of block copolymer: 5, 25, 32, 45, 57, and 70 wt %
from the leftmost vial. The samples are turbid below 30 wt % due to the
macrophase separation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.12 The ternary isopleth blends of the system PDMS2/PI5/IDMS13 for the
various concentrations of block copolymer: 10, 20, 30, 40, 50, and 70 wt %
from the leftmost vial. The sample with 20 wt % of IDMS13 turns bluish. 139
6.13 The SAXS profiles for the ternary isopleth blends of PDMS1/PI4/IDMS3
as a function of the concentration of IDMS3. All curves are presented on
arbitrary linear scales for clarity. The primary peak continuously moves
to a small q region as the mixtures are diluted with the homopolymers.
Two peaks are distinct between 50 % and 55 % indicating a bicontinuous
microemulsion phase coexist with a lamellar phase. . . . . . . . . . . . . . 140
6.14 The SAXS profiles for the ternary isopleth blends of PDMS2/PI5/IDMS13
as a function of the concentration of IDMS13. The primary peaks of IDMS13
appear at lower q than those of IDMS3 in Fig. 6.13 at the same concentration
due to the twice longer block copolymer. . . . . . . . . . . . . . . . . . . . 140
6.15 The variation of the domain spacing of a lamellar phase (ξ L ) and a bicontinuous
microemulsion phase (ξ µE ) along the isopleth in the system of
PDMS1/PI4/IDMS3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.16 The ratio of the pure lamellar spacing ((ξ Lo ) to the spacing at q = q ∗
multiplied by a weight factor α as a function of the concentration of IDMS3.
ξ µE /ξ L = 1.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
xix

6.17 The proposed phase diagram of the system PDMS1/PI4/IDMS3 in Fig. 6.13.
Large three phase region exists below 30 % of block copolymer. . . . . . . 143
6.18 Time t to reach a center of spinning drop tensiometer drop by molecular
diffusion as a function of interfacial tension γ. . . . . . . . . . . . . . . . . 146
6.19 The time-independent interfacial concentration and the partition coefficient
as a function of the block ratio f A in the block copolymer. The effect of the
micelle diffusion on c int
f
is not considered. . . . . . . . . . . . . . . . . . . . 148
6.20 The radii R of unswollen/swollen micelles and the dimensionless ratio Q
with respect to f A . R is taken from the micelle simulation with f B χN c = 12
in Fig. 4.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.1 GPC curves of poly(styrene-b-butadiene) block copolymer before and after
fractionation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7.2 The sample preparation for a spinning drop tensiometer in a the heating
oven. The heating tape is removed for clear demonstration. . . . . . . . . . 165
7.3 The experimental setup to inject a PB drop into a PS matrix. The region
near a needle of a syringe is blown up at the right corner. . . . . . . . . . . 165
7.4 TEM images of the binary blends of PS4K/SB1 at (a) 0.125 wt% (b) 0.25
wt% (c) 0.5 wt% (d) 1 wt% (e) 2 wt% (f) 3 wt% (g) 4 wt% (h) 6 wt%. The
scale bars are 50 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.5 The critical micelle concentration in the binary system of PS4K/SB1 determined
by TEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.6 TEM images of the binary blends of PS4K/SB2 at (a) 0.125 wt % (b) 0.25
wt % (c) 0.5 wt % (d) 1 wt %. The scale bars are 100 nm. . . . . . . . . . 171
7.7 TEM images of micelles in the PS4K matrix (a) 0.5 wt % SB3 and (b) 1 wt
% SB4. The scale bars are 50 nm and 100 nm, respectively. . . . . . . . . 172
xx

7.8 The normalized SAXS intensity in the binary system of PS4K/SB1. . . . . 173
7.9 The critical micelle concentration in the binary system of PS4K/SB1 determined
by SAXS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7.10 Bare interfacial tension between PS4K and PB as a function of the molecular
weights of PB. The experimental results are bound by the upper and lower
limits in the SCFT prediction. The dotted line is given with χ = 0.054 and
the solid line is with χ = 0.080, respectively. . . . . . . . . . . . . . . . . . 175
7.11 Bare interfacial tension between PS12K and PB. . . . . . . . . . . . . . . . 175
7.12 The interfacial tension reduction of PS4K and PB13K vs. the concentration
of SB1 that is premixed with PS4K (circle) or PB13K (triangle). The solid
line is the SCFT prediction with χ = 0.064 and the dashed line is with
χ = 0.080. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
7.13 The change in a diameter of the PB drop as a function of time (min) in
spin-up experiments. The data are plotted on a linear scale. . . . . . . . . . 178
7.14 The change in a diameter of the PB drop as a function of time (min) in
spin-up experiments. The data are plotted on a logarithmic scale. . . . . . . 178
7.15 TEM images of the ternary blends of PS4K/PB13K/SB1 at various concentrations
of SB1. (a) 1 wt% (b) 2 wt% (c) 4 wt% (d) 6 wt%. The scale bar
corresponds to 400 nm in (a) and (b), and 200 nm in (c) and (d), respectively. 180
7.16 The variation of Q as a function of the block ratio f A . The molecular weights
of the homopolymers are fixed and c o m /c c = 5. f A of SB1 is indicated as an
arrow. essentially identical to the CMC. . . . . . . . . . . . . . . . . . . . . 183
7.17 The dependence of Q on the micelle concentration. . . . . . . . . . . . . . 183
xxi

A.1 The Flory-Hugggins interaction parameter χ between PS and PB with various
% 1,2 of PB in the literature. All values are based on a fixed v = 81.5
cm 3 /mol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
C.1 The extensional viscosity of the PS homopolymer at different temperatures
of 170, 180, 185 ◦ C . The extensional rate is 0.01 s −1 . Curves are averages
of five different runs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
C.2 The extensional viscosity of the PP homopolymer at 170, 180, and 185 ◦ C
. The extensional rate is 0.01 s −1 . . . . . . . . . . . . . . . . . . . . . . . . 211
C.3 The extensional viscosity η ǫ of the pure PS homopolymer and the 2 wt %
binary mixtures of PS(SEE) at extensional rates ˙ǫ of 0.01, 0.05, and 0.1 s −1
at 180 ◦ C . Note that η ǫ of PS(SEE) is greatly reduced from that of PS due
to SEE block copolymer. The extensional viscosity of PS is consistent with
3η + (t), which is given by a solid line. η ǫ of PS(SEE) is a weak function of
˙ǫ approaching that of the pure PS. . . . . . . . . . . . . . . . . . . . . . . . 213
C.4 The extensional viscosity of the PP homopolymer at extensional rates of
0.01, 0.05, and 0.1 s −1 . Experimental temperature is 180 ◦ C . The solid line
represents 3η + (t) of PP in the LVE regime. . . . . . . . . . . . . . . . . . . 213
C.5 Shrinkage in width of multilayer samples during annealing in the RME on
a semi-logarithmic time scale. . . . . . . . . . . . . . . . . . . . . . . . . . 214
C.6 The extensional viscosity of the PS/PP 32-multilayer at extensional rates of
0.01, 0.05, and 0.1 s −1 . Experimental temperature is 180 ◦ C . . . . . . . . 216
C.7 Summary of the extensional viscosities of all samples used in this paper at
˙ǫ = 0.01s −1 . Block copolymer reduces the extensional viscosity by a factor
of two. Three times zero shear viscosities in the LVE region are given for
the validity of the measurements. . . . . . . . . . . . . . . . . . . . . . . . 216
xxii

C.8 Interfacial tension extracted from the multilayer samples as a function of
time with ˙ǫ = 0.01, 0.05, and 0.1 s −1 . The solid line corresponds to equilibrium
interfacial tension γ eq measured with a pendent drop method.[15] . . 218
C.9 Maximum interfacial tension vs. the extensional rate. The linear dependence
of γ on ˙ǫ can be easily observed. . . . . . . . . . . . . . . . . . . . . 218
C.10 Interfacial viscosity η ex = (γ − γ eq )/˙ǫ as a function of time for ˙ǫ = 0.01,
0.05, and 0.1 s −1 . At t → 0, γ → γ eq . . . . . . . . . . . . . . . . . . . . . . 219
C.11 Interfacial tension in the presence of block copolymer. It is interesting that
block copolymer increases interfacial tension. Only one experiment was
conducted with ˙ǫ = 0.05s −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
xxiii

List of Tables
6.1 Material Characteristics of Homopolymers . . . . . . . . . . . . . . . . . . . 123
6.2 Material Characteristics of Block Copolymers . . . . . . . . . . . . . . . . . 124
7.1 Material Characteristics of Homopolymers . . . . . . . . . . . . . . . . . . 160
7.2 Material Characteristics of Block Copolymers . . . . . . . . . . . . . . . . 160
7.3 The Reduction of Interfacial Tension . . . . . . . . . . . . . . . . . . . . . 179
A.1 Summary of the Interaction Parameters Between PS and PB . . . . . . . . 198
C.1 Extensional Viscosity at t = 10 s. . . . . . . . . . . . . . . . . . . . . . . . . 212
xxiv

Chapter 1
Introduction and Background
1.1 Introduction
Polymer blending has been of great interest in the polymer industry during the past three
decades because it provides a convenient way to develop new materials. A major advantage
is that it is a time and cost effective route to commercialization of products compared with
the synthesis of an entirely new polymer. Polymer blends have achieved a sizable growth in
automotive and electric/electronic device markets, which together account for over 90 % of
the total volume for polymer blends. Packaging, medical, household, and sports/recreation
goods make up the remainder.[1, 2]
With the exception of a few miscible polymer systems that have been studied or
commercialized,[3] most polymers are inherently immiscible. Long polymer chains have
a low entropy of mixing, so that even a small positive enthalpic contribution to the total
free energy can cause the polymer blends to macroscopically phase separate. Accordingly,
a challenging task among researchers has been to attain a fine and uniform dispersion of
components in the melt state which usually leads to enhancement of the interfacial strength
in the solid state.
The external flow field generated by mechanical mixing is a very powerful tool in
accomplishing the above goals through the domain break-up process. For a Newtonian
drop under simple shear or extensional flow in the Newtonian matrix, the droplet size is a
1

function of the viscosity ratio η r = η d /η m , where η d and η m is the viscosity of the droplet
and the matrix, respectively, and the capillary number Ca = dη m ˙ǫ/2γ where d is a drop
diameter, ˙ǫ is a deformation rate, and γ is interfacial tension.[4]
The minimum stable size of an isolated drop achieved by the balance between the shear
force and interfacial tension force, known as the Taylor limit,[5, 6] is
d = γ
˙ǫη m
16η r + 16
19η r + 16
(1.1)
for η r < 2.5. There have been many experiments to apply this idea to polymer blends
containing a small amount of the droplet phase by controlling η r and ˙ǫ. The diameter d
in Eqn. 1.1 has often provided an asymptotic limit of the drop size obtainable in dilute
homopolymer blends. However, the morphology of highly viscous polymer blends depends
upon the complex flow history during mixing.[7, 8, 9]
An additional issue in achieving fine dispersion is the problem of the stability of polymer
blend morphology. Most immiscible homopolymer blends with concentrated droplets
are not stable against thermal annealing and suffer from drastic coalescence of droplets.
One simple method to reduce interfacial tension and improve the thermal stability as well
as decreasing the drop size (Eqn. 1.1) is to add a small amount of AB block copolymers
to A and B homopolymer blends. [10, 11, 12, 13] Sometimes AB copolymer is replaced
by AC or CD copolymer when specific interaction between monomers exists.[14, 15] The
role of block copolymer is analogous to that of a small molecule surfactant in emulsifying
the mixtures of oil and water. Since each block has a favorable interaction with the corresponding
homopolymer phase, block copolymers preferentially segregate at the interface
and reduce the excess free energy of the interface. The traditional strategies for compatibilization
(emulsification) of polymer blends with block copolymers. Copolymers can be
either premade or created in situ from the reaction during mixing.[2, 7, 16] Although the
latter is more often used in industry due to the kinetic advantage of directly creating block
2

copolymer at the interface, the underlying principles that govern the thermodynamics of
polymer blends should be identical in both cases.
Block copolymers can modify interfacial properties in three ways:[2, 7, 9] 1) reduce
interfacial tension, thus promoting drop break-up, 2) suppress particle coalescence and thus
stabilize morphology, and 3) increase the interfacial adhesion in the solid state by bridging
the distinct domains. In contrast to the linear dependence of the size of a Newtonian droplet
on the interfacial tension (Eqn. 1.1), recent theoretical [17] and experimental [8, 10, 11, 18]
studies point out that the relatively small decrease in interfacial tension is not enough
to account for the huge reduction of the drop size. Rather the suppression of particle
coalescence has a dominant effect on the morphology development during melt blending.
Lyu et al. showed that the minimum interfacial coverage of block copolymer required to
prevent drop coalescence is about 20 - 50 % of that required to form a lamellar phase
of pure block copolymer [10] and nearly independent of shear rate. They attributed this
result to the steric repulsion between block copolymer brushes.[8, 11, 19]
When the copolymer concentration exceeds the critical micelle concentration in either
homopolymer phase, block copolymer added into the homopolymer blends starts to form
micelles rather than segregating at an interface.[11, 20, 21, 39] Since micellization causes
the chemical potential of block copolymer to become independent of copolymer concentration,
the interfacial tension reduction will level off without further accumulation of block
copolymer at an interface. Thus, from an economic point of view, excess block copolymers
beyond the CMC are wasted as micelles.
Since early 1980s, numerous theoretical studies have investigated the reduction of interfacial
tension by addition of copolymer to binary homopolymer blends.[22, 23, 24, 28, 29, 30,
39, 40, 41] or the formation of a spherical micelle in a selective solvent[39, 45, 46, 49, 50, 51]
or a pure homopolymer melt.[39, 47, 48, 52, 53] Self consistent field theory (SCFT) has
been used to predict the equilibrium density profiles of each component in addition to
3

earlier work based on analytical approximations.[39, 44, 45, 46, 48] Both numerical SCFT
and analytical approximations have predicted that sufficiently high coverages of copolymer
surfactants should be able to drive interfacial tension to zero.[22, 23, 24, 28, 29, 30, 40, 41]
These results have inspired several researchers to directly measure interfacial tension using
several techniques, including the spinning drop,[31] the pendant drop,[20, 33, 34, 35, 36]
and the breaking thread method.[9, 37] These methods have been applied to various polymer
pairs mixed with diblock,[20, 32, 33] triblock,[34] graft,[34] random,[35] three-arm[36],
and gradient[40] copolymers. However, none of these experiments achieved the vanishing
interfacial tension theoretically predicted for symmetric copolymers. Most measurements
showed ∼ 50 % reduction of the original bare tensions [8, 20, 37, 38] up to a maximum
reported reduction of ∼ 90 %.[34, 56]
Most theoretical studies have only focused on either interfacial adsorption[1, 24], or
the properties of micelles in binary solutions [4, 5, 9, 48, 49, 52] although both interfacial
adsorption and micellization are always competitive in the ternary polymer blends.
Leibler attempted to link both phenomena by equating the chemical potential of block
copolymer at the CMC with that in a copolymer monolayer of copolymer adsorbed to a
flat interface. He concluded that vanishing interfacial tension can precede micellization as
long as block copolymer has the block ratio f A ranged from 0.31 to 0.69.[39] Unfortunately,
his theory neglected an important feature of micelles in ternary polymer systems, that is,
a micelle can swell by emulsifying the minority homopolymer phase in its core. In Chapter
4, we show that this effect quantitatively changes the results.
More recent studies have begun to recognize that hindered block copolymer diffusion
could have a significant effect on interfacial tension measurement. [36, 55, 56] In experiments
in which copolymers were premixed with either the drop or the matrix, they all
observed that measured interfacial tensions were greatly dependent on the homopolymer
phase to which copolymers were initially added. Some reported that the time to reach
4

steady state was retarded by several orders of magnitude in highly viscous system compared
with the case in the absence of copolymer.[54, 57]
Interpretation of the experimental data without an adequate understanding of thermodynamics
and kinetic factors often leads to very perplexing or apparently inconsistent
conclusions. An example is the effect of the block copolymer length on reducing the domain
size in immiscible blends. Cigana et al. reported that small molecular weight poly(styreneb-ethylene-butylene-b-styrene)
(SEBS) triblock copolymers were more efficient in producing
PS/EPR blends with smaller domain size and higher impact strengths.[58] However,
other researchers found that the drop morphology after mechanical mixing was nearly insensitive
to the molecular weight of copolymer, but more efficiently stabilized with large
copolymer.[9, 11, 25] Macosko and coworkers have suggested the existence of an optimum
molecular weight of block copolymer to achieve the maximum stability upon annealing.
[8, 10, 12, 13]
1.2 Overview of The Thesis
This thesis pursues the ultimate goal of finding an optimum thermodynamic condition
to emulsify immiscible homopolymer blends by adding block copolymer surfactant, and
largely consists of two parts: the prediction of self consistent field theory (SCFT) in Chapter
2, 3, and 4, and the experimental verification of the theoretical results in Chapter 5, 6
and, 7.
The rest of this thesis is organized as follows. For better understanding of the thermodynamics
of ternary polymer blends, we turn our attention to analogous systems of
oil/water/nonionic surfactant in the subsequent sections of this chapter. Based on wellknown
phenomena in the small molecule surfactant system, we try to find a fundamental
physics to govern the polymer system. Chapter 2 describes the underlying assumption and
numerical implementation of SCFT in a grand canonical ensemble for ternary polymer
5

systems containing A, B homopolymers and AB block copolymer.
In Chapter 3, SCFT is used to calculate elastic parameters of the Canham-Helfrich
theory. The simulations have been conducted for spherical and cylindrical interfaces of
varying curvature in order to extract the bending rigidities and spontaneous curvature.
The resulting quantities are used to search for a balance point where the spontaneous
curvature of a monolayer vanishes.
In Chapter 4, we study the thermodynamic competition between interfacial adsorption
and micellization of block copolymers. Independent simulations are carried out at a
flat interface and swollen micelles at equal chemical potentials in order to calculate the
interfacial tension obtained at concentrations above the CMC.
Chapter 5 is an analysis of the diffusion of block copolymer which is initially mixed with
one of the homopolymer phases and the effect of transport limitations upon the measured
interfacial tension.
In Chapter 6, interfacial tension is measured between 1,4 polyisoprene (PI) and poly
(dimethyl siloxane) (PDMS) in the presence of poly( 1,4 isoprene-b-dimethyl siloxane) (IDMS)
block copolymer. The spontaneous curvature of a copolymer monolayer is controlled by
changing the block ratio of IDMS block copolymer or the molecular weight of the PI
homopolymers.
In Chapter 7, we measure interfacial tension of polystyrene and 1,2 polybutadiene (PB)
by adding poly(styrene-b- 1,2 butadiene) block copolymer in the PS matrix in Chapter 7.
The critical micelle concentration is independently identified using transmission electron
microscopy and small angle X-ray scattering. The highly viscous PS matrix has a significant
effect on the interfacial tension reduction.
Finally, a summary of a discussion of thesis and the future directions are given in
Chapter 8. In the Appendices, we present 1) the methods to determine the Flory-Huggins
interaction parameter focusing on the system of polystyrene and polybutadiene, 2) the way
6

to estimate the diffusion coefficient of block copolymer in the pure homopolymer phase, and
3) the preliminary experiments of measuring interfacial tension in the uniaxial extensional
flow with multilayer samples.
1.3 Background
This subsection is devoted to the description of the phase diagram of oil/water/surfactant
emulsions and a relevant theory based on the bending elasticity of an interface in a surfactant
micelle.
1.3.1 Phase Behaviors in Oil/Water/Nonionic Surfactant Systems
The most carefully studied model oil/water/surfactant mixtures are mixtures of alkanes
(C k H 2k+2 ), water, nonionic (C i E j ) alkyl polyether surfactants, which have been systematically
studied by Strey and coworkers. [74, 76, 78] Here, C i and E j denote (CH 2 ) i H
and [(CH 2 ) 2 O] j OH, respectively. We now briefly discuss the ternary phase behavior of
these mixtures, focusing on the region in which the presence of a small concentration of
surfactant causes the formation of a microemulsion phase. Specifically we mostly focus on
the macroscopic two phase region between oil and water containing a sufficient amount of
amphiphiles. A representative composition-temperature prism is reproduced in Fig. 1.1.
The solubility pattern of surfactant (C) in water (A) or in oil (B) along the binary edge of
A-C or B-C is characterized by an upper and lower critical solution temperature denoted
T α and T β , respectively, within this phase prism. Consequently, the relative distribution of
nonionic surfactant in the oil-rich and water-rich phases in ternary mixtures is also sensitive
to the changes in temperature. At low temperatures, close the bottom of the prism, most
surfactants are dissolved in the (lower) water-rich phase as swollen micelles and constitute
a micellar aqueous phase coexisting with the (upper) excess phase of nearly pure oil. This
macroscopic two phase system is denoted by or Winsor I.[71] As temperature increases,
2¯
7

surfactants become less soluble in the water-rich phase, and finally phase separate at a
lower critical end temperature T l which is an end point of a critical line emanating from
T β passing through the plait points P.
Figure 1.1: Schematic phase prism of a water-oil-nonionic surfactant (C i E j ) system in a
vertical axis of temperature. A water-rich two phase, a bicontinuous microemulsion middle
phase, and the oil-rich two phase are denoted by 2¯, 3, and ¯2, respectively. The three test
tubes on the right-hand side illustrate how the phases are observed in experiment. An
upper critical solution temperature T β is not necessarily located at higher temperature
than T u .[73]. Redrawn from Strey.[76]
With further rising temperature, the three phase coexistence region (3 or Winsor III)
opens as the water-rich phase moves toward the pure A corner and the surfactant-rich phase
moves toward the pure B corner along the round trajectory. Such a middle phase remains
stable until the temperature reaches an upper critical end temperature T u originating from
8

T α passing through the plait points Q. The middle phase becomes completely soluble in
the oil-rich phase in which surfactants form inverted swollen micelles. Above T = T u ,
there is only the micellar oil-rich phase coexisting with the excess water-rich phase (¯2 or
Winsor II). The schematic pictures of the resulting samples are shown on the right hand
side of the phase prism at the corresponding temperature. The volume of the middle
phase depends on the amount of surfactants in the mixture. Near the mean temperature
T m = (T u + T l )/2 between two end critical temperatures, the three phase body has a
maximum size so that surfactant can have a maximum efficiency in emulsifying oil and
water.[68, 70] As long as there exist two distinct (disconnected) critical end points, which
is true of C i E j surfactants with j ≥ 1, the formation of the middle phase is generally
observed in any surfactant system regardless of the strength of surfactant.
Although temperature has been controlled to induce a 2¯-3-¯2 phase transition as an
example in a nonionic surfactant system, a similar phase sequence can be also obtained at
fixed temperature by changing the structure of surfactants,[74, 78] or by adding a fourth
component such as a salt,[82] cosurfactant e.g. alcohols,[83, 85] or another polar solvent[77].
The 2¯-3-¯2 phase sequence and the three phase body can also be illustrated in a phase
diagram along an isopleth with an equal volume of oil and water. The schematic diagram
given in Fig. 1.2 is often referred to as a ‘fish cut’ and is roughly symmetric with respect
to the mean temperature T m . It consists of a ‘head’, which corresponds to a three phase
coexistence region, and a ‘tail’, which corresponds to a concentrated homogeneous middle
phase. By increasing the surfactant concentration beyond φ ∗ s
in the ‘tail’ portion, one can
lead to a uniform middle phase (µE) followed by a lamellar phase (L α ). The ‘head’ portion
of a fish is fully described by two parameters: temperature interval, T u −T l , and the amount
of surfactant, φ ∗ s , needed to achieve homogeneous mixture. The latter basically determines
the height of a three phase triangle body (dark region) in Fig. 1.1. The location and size
of a ‘head’ portion strongly depends on the type of a surfactant. In general, a head shrinks
9

Figure 1.2: A schematic phase diagram along an isopleth of the phase prism. T o = T m and
φ ∗ s is the minimum surfactant concentration to reach a uniform middle phase. The lower
is φ ∗ s, the more efficient is the surfactant. Reproduced from Wennerström et al. .[86]
and moves to a lower concentration of surfactant as a surfactant becomes strong. A fish
diagram is a characteristic feature in a small molecule surfactant system that is caused by
complex interaction between water and surfactant.
On the other hand, Fig. 1.3 shows an isothermal Gibbs phase triangle, which is a
horizontal slice of the phase prism at a particular temperature (T = T m ).[77] One can
easily notice that the homogeneous bicontinuous middle phase, denoted by 1, in a fish
‘tail’ has a narrow stable region around the isopleth whereas the lamellar phase is stable
over a much wider water-to-oil ratio.[72]
1.3.2 Interfacial Tension
Since the mid 1970s people have noticed that the 2¯-3-¯2 phase transition is closely related
to the variation of interfacial tension between T l and T u . [68, 69, 70, 74, 78, 80] As shown
in a schematic diagram in Fig. 1.4, at T < T l the water and surfactant phases become
identical and form a single micellar phase so that interfacial tension γ ac between the water
10

Figure 1.3: Isothermal diagram of water/n-octane/C 10 E 5 mixtures at the mean temperature
T = 44.6 ◦ C . Note the 1-phase region near the middle phase enclosed by the lamellar
phase. Reproduced from Strey.[77]
(a) and surfactant (c) phase vanishes in a region. As temperature rises from T < T l , γ ac
2¯
continuously increases from zero at T = T l and a surfactant-rich phase separates out. Then
γ ac coincides with the interfacial tension γ ab between the oil (b) and water phase at T =
T u , when the oil and surfactant phases become identical. Conversely, interfacial tension
γ bc between the oil and surfactant phase increases from zero at T = T u as temperature
decreases, and becomes equal to γ ab at T = T l . When two tension curves, γ ac and γ bc ,
intersect at T = T m , γ ab becomes a minimum. Empirical observation by Shinoda,[70] and
Kahlweit and Strey[74] indicate that if two critical end points are close enough, T l ≃ T u ,
such a minimum can be ultra-low because γ ac and γ bc meet at a very low value. Thus the
minimum of γ ab can be observed in any ternary surfactant system in which the middle
phase forms near T m regardless of the strength of surfactant.
Inferring from the experimental observation that the phase behaviors of nonionic surfactant
systems are associated with some characteristic length scale,[79] Strey et al. showed
11

Figure 1.4: Schematic diagram of the individual interfacial tension curves of oil/water(γ ab ),
water/emulsion(γ ac ), and oil/emulsion(γ bc ) on a log scale as a function of temperature.
Redrawn from Sottmann and Strey.[80]
that interfacial tensions in various systems can collapse into a single master curve for a
wide range of temperature when they are normalized by relevant length scales of internal
domains.[80] All measured tensions were successfully fit to a semi-empirical expression
derived by Leitao et al. [60]
γ ∗ = τ ∗2 + 1 (1.2)
as shown in Fig. 1.5.
1.3.3 Wetting/Nonwetting Transition of Interface
When the two phase micellar system or ¯2) traverses the three phase coexistence region
(2¯
(3) through the end critical temperature T l or T u , a surfactant monolayer separating the
oil-rich and water-rich phases undergoes one of two different types of bicontinuous middle
phase transition. First, weak surfactants such as small alcohols (C i E 0 ) and C 4 E 1 form a
12

Figure 1.5: Reduced oil-water interfacial tension γ ∗ as a function of reduced temperature
τ ∗ . The full line is calculated from Eqn. 1.2. The definition of γ ∗ and τ ∗ are given in ref.
[80]. Redrawn from Sottmann and Strey.[80]
middle phase that always wets an interface in any temperature range between T u and T l .
When the water/oil interface is wet by a macroscopically thick layer of the middle phase,
the interfacial tension γ ab is simply given by Antonoff’ rule[84]
γ ab = γ ac + γ bc (1.3)
as shown with a dashed line in Fig. 1.4. In this case, interfacial tension is usually not so
low (∼ 1 mN/m).
On the other hand, strong surfactants C i E j with j ≥ 2 do not wet the interface except
for a very small region of temperature near the critical end point. Instead, the middle
phase is often observed to form small thin blisters along the oil/water interfaces as shown
in Fig. 1.6. The presence of blisters is critical for the accurate measurement of ultra-low
interfacial tension because they act as the surfactant reservoirs.
In this case, the measured interfacial tension γ corresponds to the actual interfacial
13

Figure 1.6: Lens of C 8 E 3 -rich middle phase floating on the interface between the lower
water- and upper decane-rich phase at T = 21.8 ◦ C . Reproduced from Kahlweit et al.
.[75]
tension γ ab between the water- and oil-rich phases, and γ ab becomes lower than the sum
of γ ac and γ bc ,[80, 81]
γ ab < γ ac + γ bc (1.4)
as shown with a solid line in Fig. 1.4 at a surfactant chemical potential that is characteristic
of the three phase coexistence region.
1.3.4 Helfrich Bending Elasticity
When a small amount of surfactant is added to a mixture of oil and water, surfactant
adsorbs to the oil/water interface, resulting in decrease of interfacial tension. The reduction
of interfacial tension is limited at some critical concentration of surfactant by the formation
of micelles in one of the bulk phases. As first noted by Schulman,[59] surfactants in a selfassembled
monolayer essentially stay in the saturated state, in which interfacial tension is
zero because interfacial area per molecule should be adjusted to minimize the overall free
energy. When interfacial tension becomes sufficiently low, the free energy of a surfactant
monolayer (membrane) is governed primarily by the bending energy associated with the
14

Figure 1.7: Phase stability of spherical, cylindrical, and lamellar micelles. The crosshatched
region is a two-phase coexistence of spheres and cylinders. Sphere is the most
stable structure along the emulsification failure line. In this plot, r = ρ ˜C o and x =
−¯κ/(2κ + ¯κ). Reproduced from Safran.[62]
curvature of the interface.[2, 61] For a weakly curved interface with the principal radii of
curvature C 1 and C 2 , Helfrich bending elasticity is given by
∫
H =
]
1
dA[
2 κC2 + ¯κK − κC o C
(1.5)
in which C = C 1 + C 2 , K = C 1 C 2 , and C o are the mean, Gaussian, and spontaneous
curvatures, respectively, and in which κ and ¯κ are the mean and Gaussian rigidities. Since
the free energy of a surfactant monolayer varies depending upon the monolayer curvature,
Eqn. 1.5 is very useful in predicting a phase transition between competing structures
formed from monolayers. Safran et al. examined the phase transition between spherical,
cylindrical, and flat surfaces in ternary oil/water/surfactant mixtures by comparing the
15

corresponding bending energy per area (interfacial tension) γ = H/A, given by[62, 63]
γ s =
2κ + ¯κ
R 2
− 2κC o
R
for a sphere with C = 2/R and K = 1/R 2 , and
γ c = κ
2R 2 − κC o
R
for a cylinder with C = 1/R and K = 0, and
γ l = 0
for a lamellar phase with C = 0 and K = 0. From Eqn. 1.5, they found that the monolayer
structure that minimizes the Helfrich bending energy along the emulsification failure line
is a sphere with the curvature ˜C o = κC o /(κ + ¯κ/2). When a micellar phase coexists with
an excess phase, the volume of material emulsified in the core of each micelle can be freely
adjusted by transfer of material to or from the excess phase. In this case, the optimal mean
curvature can be obtained with either a spherical or cylindrical surface, but the Gaussian
rigidity favors a spherical surface over a cylindrical surface of equal mean curvature when
¯κ < 0. If the system enters a 1-phase micellar region from the emulsification failure line,
however, the volume(V )-to-surface(A) ratio ρ of a micelle is determined by a stoichiometry
such that 3(V/A) s = R for a sphere and 2(V/A) c = 2R/3 for a cylinder where R is a radius
of a sphere. By setting ρ = 3V/A, Fig. 1.7 shows that a dimensionless ratio r = ˜C o ρ can
drive a phase transition between different structures in the absence of thermal fluctuation.
If r > 1, excess bulk phase is simply rejected (emulsification failure) so as to maintain
the saturation state in a micelle. As one decreases r from the unity by increasing the
amount of surfactant or by decreasing total volume of the internal bulk phase, spherical
micelles may be transformed into either cylinders or lamellar phase, depending on the ratio
x = −¯κ/(2κ + ¯κ). In this thesis, we are primarily interested in the effect of copolymer
16

upon macroscopic interfacial tension along an interface between a micellar phase and an
excess homopolymer phase, and consider only spherical micelles.
In fact, during the 2¯-3-¯2 transition, the spontaneous curvature C o of a surfactant monolayer
dividing the oil-rich and water-rich phases continuously varies with temperature and
changes sign at the mean temperature.[78] A surfactant monolayer with high spontaneous
curvature forms a spherical (inverted) swollen micelle in the two phase (2¯, ¯2) region. If
the curvature becomes sufficiently small, however, a phase transition into a bicontinuous
microemulsion phase is induced with the diverging length scale ζ ∼ 1/C o between T l and
T u .
1.3.5 Stability of Surfactant Monolayer
Eqn. 1.5 can be rewritten in a more useful functional form to account for the stability of
a membrane surface as
H = 1 2
∫
dA
[κ +
(
C + − κC o
κ +
) 2
+ κ − C − 2 − κ +
( κCo
κ +
) 2
]
(1.6)
in which C ± ≡ C 1 ± C 2 , κ + ≡ κ + (1/2)¯κ, and κ − = −(1/2)¯κ. For a special case of a flat
surface with C o = 0, Eqn. 1.6 is reduced to
H = 1 ∫
dA [ κ + C 2 2 + + κ − C ] − . (1.7)
2
The above equations indicate that both κ + and κ − must be positive for a weakly curved
surface to be stable. Otherwise, a monolayer will break into infinitesimally small spheres
(C 1 = C 2 ) if κ + < 0, or will transform into a multiply connected structure of an infinite
minimal surface (C 1 = −C 2 ) via concentrated ’holes’ if κ − < 0.
The density of ’holes’ is also related to the integral ∫ dAK of the Gaussian curvature
over closed surface, which is a topological invariant given by the Gauss-Bonnet theorem
∫
dAK = 4π(1 − g), where g is the number of handles on a surface.[64] For example, all
shapes in Fig. 1.8 have the same topology of g = 0 except a torus (C) of g = 1. Therefore,
17

if ¯κ − < 0 or ¯κ > 0, the Gaussian contribution to the free energy difference between a
monolayer and a minimal surface increases linearly with g.
Figure 1.8: The comparison of the topology among different structures. Only (C) has
g = 1 where as others have g = 0, where g is the number of handles. Reproduced from
Taddei.[65]
Eqn. 1.6 also yields an important result for the interfacial tension γ flat of a macroscopic
flat interface coexisting with a phase containing swollen spherical micelle of curvature
˜C o = (κC o )/κ + . Interfacial tension is the free energy per unit area for creating more
macroscopic interface. Assuming the area per molecule is similar in the flat macroscopic
interface and in the curved interface of a micelle, the creation of more macroscopic interface
thus requires the destruction of an equal amount of micellar interfacial area. Therefore,
interfacial tension can be the free energy penalty to transfer surfactant molecules from
micelles to a flat interface, and obtained from Eqn. 1.6 [78]
γ flat = 1 2 κ + ˜C o
2
. (1.8)
Eqn. 1.8 stresses that γ flat has a quadratic dependence on the micelle curvature ˜C o and
has a vanishing minimum at a balance point, where the spontaneous curvature vanishes.
The balance point corresponds to the mean temperature in nonionic surfactant system.[80]
The Helfrich theory has been extended to ternary polymer blends in which block
copolymer coexists with two immiscible solvents[89] or homopolymers.[39, 43, 90, 91, 92]
Cantor[89] and Leibler[39] studied the elastic properties including C o and K + of block
18

copolymer adsorbed at the spherically curved interface by assuming the constant density
profile of copolymer brushes. Cantor was concerned with the scaling behaviors of K + and
Leibler was particularly interested in the stability of the droplet with nonzero interfacial
tension and prediction of the emulsification failure. Wang and Safran[90, 91] improved the
results of Cantor and Leibler in homopolymer blends by considering the chain end distribution
of copolymer at saturated state in the strong segregation limit. They analytically
derived κ, ¯κ, and C o as a function of the block ratio in both states of the melt and swollen
brush, and predicted the phase transition of copolymer in a plot of the surface-to-volume
ratio vs. the block ratio, which is similar to Fig. 1.7.[90] Matsen used numerical SCFT to
calculate the elasticity of symmetric (mono or polydisperse) block copolymer in homopolymer
blends with equal lengths and concluded that block copolymer should be reasonably
short so as to keep −¯κ ∼ kT, which is favorable to form a bicontinuous microemulsion
phase.[43, 92]
1.3.6 Thermal Fluctuation
Beyond the mean field approximation of a rigid surface, deGennes and Taupin[93] considered
the entropy effect on a flexible surface by introducing a concept of a ’persistence
length’ ξ κ , which is given by
( ) 4πκ
ξ κ = a exp
αkT
(1.9)
where a is a cut-off length of a molecular size, and α = 3.[64] Rearranging Eqn. 1.9 gives
the renormalized bending rigidity κ(ξ κ ) as
κ(ξ κ ) = κ o − αkT ( )
4π ln ξκ
a
(1.10)
, which implies that the rigidity of a monolayer vanishes beyond the persistence length.
They argued that the membrane is effectively flat at length scales smaller than ξ κ , but
becomes strongly wrinkled above ξ κ . Therefore, a lamellar phase should have a mean
19

igidity of κ ∼ kT in order to melt into a randomly fluctuating surface because ξ κ exponentially
depends on κ. Also, they regarded a membrane surface as consecutive uncorrelated
’patches’ with an area ξ 2 κ . In this view, in order to obtain a one-phase mixture of
oil/water/surfactant with a domain spacing ξ κ , the free energy cost γ ξ 2 κ to couple adjacent
random patches should be comparable to kT or less, that is,
γ ξ κ 2 ∼ kT (1.11)
, which has been verified in many experiments. [78, 80] If γ ξ 2 κ kT, the system phase
separates into macroscopic domains.
However, careful neutron scattering experiments by Porte et al. showed a lamellar phase
could melt into the sponge phase (L 3 ) in the AOT/brine system at a much smaller length
scale than ξ κ .[67] It convinced researchers that the Gaussian rigidity ¯κ primarily governs the
topological transition of a lamellar phase to a L 3 phase. Partly for this reason, Morse[94, 95]
and Golubovic[96] argued that it was the renormalized Gaussian rigidity that controlled the
competition between a lamellar and bicontinuous structure. The renormalized Gaussian
rigidity is given by
( )
¯κ(ξ¯κ ) = ¯κ o + α′ kT
4π ln ξ¯κ
a
(1.12)
where α ′ = 10/3.[64] By comparing both length scales of ξ κ and ξ¯κ , they both observed
that a phase transition of a lamellar phase into a L 3 phase occurs when ¯κ+ 10 9
κ > 0, which
is the criterion that Matsen used for the formation of a bicontinuous microemulsion in
ternary polymer systems.[92] The subject of the fluctuation effect on a membrane surface
is actually beyond the scope of this thesis. A more detailed review is given in ref [97].
20

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[76] Kahlweit, M.; Strey, R.; Busse, G. Phys. Rev. E 1993, 47, 4197
[77] Strey, R. Ber. Bunsenges. Phys. Chem. 1993, 97, 742
[78] Strey, R. Coll. Ploym. Sci. 1994, 272, 1005
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[79] Sottmann, T.; Strey, R. J. Phys.: Condens. Matter 1996, 8, A39
[80] Sottmann, T.; Strey, R. J. Chem. Phys. 1997, 106, 8606
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1979, 83, 1984
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1987, 91, 1137
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27

Chapter 2
Self Consistent Field Theory
2.1 Introduction
Self Consistent Field Theory (SCFT) is a mean field theory of spatially inhomogeneous
polymer melts. In SCFT, the statistical mechanics of interacting many polymer chains is
reduced to a single chain problem. Instead, the interaction between nearest neighboring
chains is simplified as an unknown chemical potential field that bias the random walk of
a Gaussian chain. The chemical potential field for each type of monomer at each point in
space depends upon the local composition and the Flory-Huggins interaction parameter χ,
and must be determined self-consistently.
In this chapter we review SCFT for the multicomponent polymer blends and discuss
conventions and numerical methods used in Chapter 3 and 4. For generality, we consider a
mixture of polymer species labeled by indices i and j that are constructed from monomers
that are labeled by indices α or β. Polymers of type i contain the total number of N i
monomers, whose position within the chain is parameterized by a variable 0 < s i < N i .
2.2 Formulation of SCFT
2.2.1 Field Equations
SCFT requires the calculation of a statistical weight q i (r,s) for each species that is proportional
to the constrained partition function of the subchain containing monomers 0 to
28

s of species i, when monomer s is constrained to point r, in a self-consistently determined
chemical potential field. This function satisfies a modified diffusion equation
[
]
∂q i (r,s)
= − − b2 α(s)
∂s 6 ∇2 + ω α(s) (r) q i (r,s) (2.1)
with an initial condition q i (r,0) = 1 for all r. Here, b α is a statistical segment length and
ω α (r) is a chemical potential field for α monomers, and α(s) is the type index for monomer
s, which has different values within different blocks. An analogous function q † i
(r,s) for
the subchain containing monomers s to N i satisfies a corresponding equation with the
l.h.s. multiplied by −1, with a final condition q † i
(r,N) = 1. In case of an immiscible
homopolymer blend we assume no flux at a unit cell boundary such as ∂q i (r,s)/∂r = 0.
The chemical potential field ω α (r) is given
ω α (r) = ∑ β
χ αβ φ β (r) + ξ(r) , (2.2)
where χ αβ is a binary Flory-Huggins interaction parameter, for which χ αβ = χ βα and
χ αα = χ ββ = 0, φ α (r) is a local volume fraction of α monomers, and ξ(r) is a Lagrange
multiplier field, which is chosen so as to satisfy the incompressibility constraint,
∑
φ α (r) = 1 . (2.3)
The local volume fraction φ β (r) is given by
α
φ β (r) = ∑ i
¯φ i
Q i
∫ Ni
0
ds
N i
q i (r,s)q † i (r,s)δ β,α(s) , (2.4)
where ¯φ i = M i N i v/V is the overall volume fraction of i molecules, M i is the total number
of i molecules, v is a monomer reference volume, and
Q i ≡ 1 ∫
dr q i (r,N i ) . (2.5)
V
The s integral in Eqn. 2.4 is constrained to blocks of monomer type β by the the Kronecker
δ function δ β,α(s) .
29

SCFT may be implemented either in the canonical ensemble, in which one specifies a
value of M i or ¯φ i for each species [1], or in grand-canonical ensemble, in which one specifies
a chemical potential µ i for each species [2, 3]. The chemical potential and molecular volume
fraction of a species i are related by the identity
¯φ i = Q i e µ i/kT
. (2.6)
Eqn. 2.6 implicitly introduces a convention for the chemical potential in which µ i vanishes
in a hypothetical standard state consisting of pure species i, with ¯φ i = 1, in a vanishing potential
field ω α (r) = 0 for all α, for which q i (r,N i ) = 1 and Q i = 1. In the grand-canonical
implementation of SCFT, local monomer densities are thus calculated by replacing the
factor of ¯φ i /Q i in Eqn. 2.4 by the activity e µ i/kT . The total grand canonical free energy is
given by
Φ
kT
= −V ∑ i
+ 1 ∑
2
αβ
¯φ i
N i v − ∑ α
∫ dr
v χ αβ φ α (r)φ β (r) ,
∫ dr
v ω α(r)φ α (r) (2.7)
where ¯φ i is replaced by Eqn. 2.6.
In an incompressible liquid, SCFT predictions for the volume fraction fields φ i,α (r) and
Helmholtz free energy F = Φ+ ∑ i µ iM i are invariant under a spatially homogeneous shift
ξ(r) → ξ(r) + v δP (2.8)
of the Lagrange multiplier field, which corresponds to a shift δP in hydrostatic pressure.
Such a shift causes corresponding trivial shifts
ω α (r) →
ω α (r) + v δP
µ i → µ i + N i v δP (2.9)
in chemical potentials. The solution of the SCFT equations for a two-phase system is thus
30

unique only to within such homogeneous shifts in ξ(r), unless a value is specified for either
ξ or for the corresponding macroscopic pressure P in one of the two bulk phases.
2.2.2 Homogeneous Mixtures
Calculation of an interfacial excess free energy γ requires knowledge of the pressures in
the surrounding homogeneous phases as functions of chemical potential. In homogeneous
mixtures, SCFT reduces to a form of Flory-Huggins theory. The chemical potential in a
homogeneous mixture, in the convention used here, is given by
∑
µ i = kT ln(¯φ i ) + N i f i,α ω α , (2.10)
where f i,α is the fraction of monomers of type α on a chain of species i, and ¯φ i and ω α are
homogeneous values of the fields. The macroscopic pressure P = −Φ/V for a homogeneous
phase may be obtained from Eqn. 2.7, which yields
Pv
kT = ξ + ∑ i
¯φ i
N i
+ 1 2
α
∑
χ αβ ¯φα ¯φβ . (2.11)
By using Eqn. 2.11 to express µ i as a function of P, rather than ξ, noting that M i =
¯φ i V/N i v and ¯φ α = ∑ i ¯φ i f i,α , and evaluating the sum G = ∑ i M iµ i , we may confirm that
the corresponding Gibbs free energy per monomer is given by an expression
Gv
V kT = ∑ i
which is consistent with Flory-Huggins theory.
αβ
( )
¯φ i ¯φi
ln + 1 ∑
χ αβ ¯φα ¯φβ + Pv (2.12)
N i e 2
Using the Flory-Huggins theory, one can derive the equilibrium distribution of block
copolymer molecularly dissolved in two coexisting A- and B-rich homopolymer phases.
αβ
Setting the equal chemical potential of copolymer µ A C = µB C
(Eqn. 2.10) in both phases at
fixed bulk pressure P (Eqn. 2.11) gives the partition coefficient K ≡ φ A c /φ B c
( 1
ln K ≃ χN c (f A − f B ) + N C − 1 )
N A N B
(2.13)
in the infinitely dilute copolymer solution, in which φ A A ≃ φB B ≃ 1.
31

2.2.3 Interfacial Tension
Interfacial tension γ is defined as the excess grand canonical free energy per unit area. For
a system of volume V with a curved Gibbs dividing interface with a radius of curvature R,
which separates an A-rich domain (phase I) of volume V I for r > R from a B-rich domain
(phase II) of volume V II for r < R, interfacial tension is given by γA = Φ+P I V I +P II V II ,
which yields
γ
kT
= ∫ dr
Av
+
∫ dr
Av
[
e µ i/kT
− ∑ q i (r,N i ) − ∑ ω α (r)φ α (r)
N
i i
α
⎡
⎤
(2.14)
⎣ 1 ∑
χ αβ φ α (r)φ β (r) + v
2
kT {P A Θ(r − R) + P B Θ(R − r)} ⎦ ,
αβ
where A is an area, and Θ(x) = 1 for x > 0 and Θ(x) = 0 for x < 0. For a cylindrically
or spherically curved interface, the one dimensional integral over system volume per unit
]
interfacial area is given by
∫ dr
A = ∫
(
dz 1 +
R) z d−1
(2.15)
where r = R + z, and the dimension d of the system is d = 1 for a flat surface, d = 2 for a
cylinder, and d = 3 for a sphere, respectively.
2.2.4 Nondimensionalization
The modified diffusion equation can be non-dimensionalized by introducing reduced variables
t ≡ s/N and ˜r = r/R o , with R 2 o = Nb 2 , where N is a reference degree of polymerization,
for which we take N = N C , and b is a reference statistical segment length, for which
we take b = b A in this work. This yields the non-dimensionalized diffusion equation
with ˜∇ 2 ≡ R 2 o∇ 2 and
∂q i (˜r,t)
∂t
[
= − − 1 ]
6 (b α
b )2 ˜∇2 + ˜ω α (˜r) q i (˜r,t) , (2.16)
˜ω α (˜r) ≡ Nω α (˜r) = ∑ β
˜χ αβ φ β (˜r) + ˜ξ(˜r) (2.17)
32

where ˜χ αβ ≡ Nχ αβ and ˜ξ(˜r) ≡ Nξ(˜r). By non-dimensionalizing the self-consistency
conditions in terms of the same variables, it may be shown that the self-consistent solution
for q i near a curved interface is a function of t, ˜r, the chemical potentials µ A , µ B and µ C ,
and the dimensionless parameters α A , α B , f A , b B /b A , ˜χ αβ , and C √ Nb alone.
The grand-canonical free energy per interfacial area may be expressed in non-dimensional
form as an integral
∫
Φ
AkT = b
N 1/2 v
∫
b
+
N 1/2 v
{
dẑ − ∑ i
e µ i/kT
α i
q i (˜r,α i ) − ∑ }
˜ω α (˜r)φ α (˜r)
α
dẑ ∑ ˜χ αβ φ α (˜r)φ β (˜r) , (2.18)
αβ
where the integrand is a function of dimensionless distance ẑ from the interface given by
dẑ =
(
1 + z ) d−1 dz
R N 1/2 b
. (2.19)
A and R is the area and radius of the Gibbs dividing surface, and α i = N i /N, so that
α C = 1 with our convention N = N C . To obtain the excess free energy density γ, we must
subtract from bulk pressure in the neighboring phases from the integrand, subtracting
the bulk pressure of a given phase throughout the domain on the corresponding side of
the Gibbs dividing surface, and thus cancel contributions to Φ from points far from the
interface. This yields an interfacial tension of the form
γ
kT =
b
N 1/2 v [γ](α A,α B , b A
b B
,f A ,χN,[C],µ C ) (2.20)
in which [γ] is a dimensionless interfacial tension, and [C] ≡ CN 1/2 b is a dimensionless
curvature.
2.3 Numerical Implementation of SCFT
In this work, the modified diffusion equation for q i (r,s) is numerically implemented using
a finite difference method in one dimensional real space. Since the chemical potential
33

field ω α (r) is initially unknown, it is determined in a self-consistent manner using proper
iteration schemes.
2.3.1 Finite Difference Method
The modified diffusion equation (Eqn. 2.1) is spatially discretized in a unit cell with a flat,
cylindrical, or spherical geometry and time integrated with the Crank-Nicholson technique
as previous studies of the ternary polymer blends.[1, 4, 5] The Crank-Nicholson method is
unconditionally stable and has a truncation error of order O((∆t) 2 +h 2 ) in time and space
where ∆t and h are the time and spatial step, respectively.[6]
The time and spatial grid points are labeled as (s j ,r k ) where s j = j(∆t) for j =
0,... , N i
∆t of a molecule of type i, and where r k = kh for k = 0,... ,n for uniform grid
points. In order to increase the numerical accuracy and save the computational time,
the adaptive grid points are also used by adjusting the distribution of n + 1 node with
a stretching function S(k), which is a mixture of a hypertangent and a linear function,
satisfying S(0) = 0 and S( n 2 ) = 1,
⎧
⎨ tanh[A( 2k −1)]+B( 2k −1)
n n
tanh(A)+B
+ 1 : r k ≥ R, k = 0,... , n 2
S(k) =
⎩ tanh(A 2k n )+B 2k n
tanh(A)+B
: r k < R, k = 0,... , n 2
where R is a radial position of a Gibbs dividing surface, and A and B are the parameters
to control the node concentration near R. Such adaptive grids were usually adopted for
the simulation of a weakly curved membrane in Chapter 3 by fixing R at the center of a
simulation cell with A = 3 and B = 0 whereas the fixed grids were used for the micelle
simulation in Chapter 4 because a micelle radius is spontaneously determined given the
chemical potential of block copolymer in a grand canonical ensemble.
Approximating the first and second derivative of q i (r,s) with respect to r as
∂q i
∂r
= 1 (q i,k+1 − q i,k−1 ) (2.21)
L 1
( ) ]
h1
+ 1 q i,k + q i,k−1 (2.22)
h 2
∂ 2 q i
∂r 2 = 1 L 2
[
h1
h 2
q i,k+1 −
34

in the spatial steps h 1 and h 2 with arbitrary sizes defined by
h 1 = r k − r k−1 h 2 = r k+1 − r k (2.23)
L 1 = h 1 + h 2 L 2 = 1 2 h 1(h 1 + h 2 ) (2.24)
transforms Eqn. 2.1 to a tridiagonal band matrix of
C j+1
k−1 qj+1 i,k−1 + Cj+1 k
q j+1
i,k
+ Cj+1 k+1 qj+1 i,k+1
(2.25)
= C j k−1 qj i,k−1 + Cj+1 k
q j i,k + Cj+1 k+1 qj i,k+1
where each coefficient C j k
is given by
(
C j+1
k−1
= b 2 α h 2 − (d − 1)L )
2
(2.26)
r k L 1
(
C j+1
k
= −
[b 2 αL 1 + 6L 2 ω α,k + 2 )]
(2.27)
∆t
(
C j+1
k+1
= b 2 α h 1 − (d − 1)L )
2
(2.28)
r k L 1
(
C j k−1
= −b 2 α h 2 − (d − 1)L )
2
(2.29)
r k L 1
(
C j k
=
[b 2 α L 1 + 6L 2 ω α,k − 2 )]
(2.30)
∆t
(
C j k+1
= −b 2 α h 1 − (d − 1)L )
2
. (2.31)
r k L 1
This finite difference equation is solved under no flux boundary condition at the unit cell
boundary. If a system is defined in terms of the mean curvature C = (d − 1)/R instead
of r, the inverse radial distance r −1 = (R + z) −1 can be replaced by r −1 =
−∆ ≤ z ≤ ∆.
2.3.2 Numerical Accuracy
C
d−1+Cz where
The numerical accuracy of a SCFT code with a finite difference method greatly depends on
the way to discretize the nodes in the simulation box and the way to integrate the system
in time. In this subsection, therefore, we examine the sensitivity of interfacial tension γ
35

7
∆ t = 0.5
6.5
γ o
6
10 6 γ /kT (Å −2 )
5.5
5
4.5
adaptive grids
fixed grids
4
0 500 1000 1500 2000 2500 3000 3500
the number of grids (#)
Figure 2.1: Interfacial tension γ vs. the number of spatial nodes in a simulation box
with the fixed size of 800 Å. The adaptive grids dramatically increase the accuracy of the
calculation compared to the case with the same number of grid points.
and the chemical potentials µ i to the variation of the spatial and time steps in the case
of χ = 0.1, N A = N B = f A N C = 100, and b α = 6Å at an arbitrary chemical potential of
block copolymer.
Fig. 2.1 compares interfacial tension γ calculated on fixed grids with that on adaptive
grids. As total number of nodes increase in the system, or decreasing fixed grid step, γ
increases by about 50 % and approaches a plateau. On the other hand, the adaptive grids
dramatically improve the numerical accuracy giving the better estimate of γ with fewer
nodes.
Fig. 2.2 and Fig. 2.3 show the dependence of the interfacial tension difference ∆γ
relative to the plateau value γ o marked in Fig. 2.1 and the chemical potential difference
of phase I and II, ∆µ A (= µ I A − µII A ) and ∆µ C(= µ I C − µII C
), at both boundaries of the
simulation box on the uniform spatial grid step h for a uniform grid and the time step ∆t,
respectively. As expected for a Crank-Nicholson method, these quantities have the second
36

3
2.5
∆ t = 0.5
10 6 ∆ γ / kT ( Å −2 )
2
1.5
1
0.5
0
0 1 2 3 4 5
h 2 [ Å 2 ]
Figure 2.2: The interfacial tension difference ∆γ as a function of h 2 for a uniform grid.
order accuracy in space and time.
On the contrary, interfacial tension and the chemical potential difference are nearly
independent of the time and spatial steps, respectively, as shown in Fig. 2.4 and Fig. 2.5.
For consistency, we applied the same time step ∆t = 0.25 for all SCFT calculations in this
thesis, but adaptive grids were used for the membrane simulation, whereas h = 1Å for the
micelle simulation.
2.4 Iteration Scheme
In this thesis, two types of the one dimensional SCFT calculations are carried out, in both
grand canonical ensemble. One is the calculation of an entire (spherical) micelle, in which
the size of a whole micelle is allowed to adjust so as to minimize the grand canonical free
energy. This is presented in Chapter 4. The other is the calculation of an interface with a
flat, cylinder, or spherical geometry, in which we impose an extra constraint on the position
of the interface within a simulation cell. Two iteration schemes have been employed to
obtain the solutions, depending on the type of calculation: relaxation method is used only
37

10 −1 10 −2 10 −1 10 0 10 1
10 −2
∆ µ A / kT, ∆ µ C / kT
10 −3
10 −4
10 −5
10 −6
∆ µ A
∆ µ C
(∆ t) 2
Figure 2.3: The chemical potential difference ∆µ A and ∆µ C as a function of (∆t) 2 .
7
6.5
10 6 γ /kT ( Å −2 )
6
5.5
5
4.5
h = 0.5 Å
h = 1 Å
4
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
∆ t
Figure 2.4: Interfacial tension γ with various time steps ∆t for a uniform grid. There is
little effect of time step on γ.
38

∆ t = 0.5
10 −2
∆ µ A / kT , ∆ µ C / kT
10 −3
10 −4
10 −5
∆ µ A
∆ µ C
10 −1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
10 −6
h [ Å ]
Figure 2.5: The chemical potential difference ∆µ A and ∆µ C of the homopolymer A and
block copolymer C determined at both boundaries of the simulation cells vs. the spatial
step size h.
for a swollen micelle simulation and a Newton-Raphson method has been used both for
unswollen micelle simulation and for all interface simulations.
2.4.1 Relaxation Method
Recently Drolet and Fredrickson developed a powerful Picard-type relaxation algorithm,
which was originally used to screen multiblock copolymer morphology in real space.[7, 8]
In this thesis this method is extended to ternary polymer blends in a grand canonical
ensemble.
Initial guess of ω α (r) and ξ(r) are obtained from the analytical solutions for the binary
homopolymer blends with infinite molecular weights given by Helfand and Tagami.[9] Let
ω j α(r) and ξ j (r) be the approximate fields at the beginning of the jth iteration.
1. Given ω j α(r), new density profiles φ j+1
α (r) are calculated using Eqn. 2.1 and Eqn. 2.4.
39

2. The potential fields ωα
j+1 (r) are locally updated with a proper linearized mixing rule
ω j+1
α (r) = ω j α(r) + λ δω j α(r) (2.32)
where the field difference δω j α is given by
δω j α (r) = ∑ β
χ αβ φ j+1
α (r) + ξj (r) − ωα j (r) . (2.33)
The weight factor λ, which takes the value between 0.1 and 0.2, decides the speed to
approach a local minimum of the total free energy along the local gradient direction.
3. Finally a local pressure field ξ j+1 (r) is updated using Eqn. 2.2 and Eqn. 2.3. For
instance, in the ternary system of A/B/AB
ξ j+1 (r) = 1 2 (ωj+1 (r) + ωj+1(r) − χ) (2.34)
A
B
where χ = χ AB . At this step, the chemical potential fields ω α (r) and a Lagrange multiplier
ξ(r) are not forced to have vanishing spatial average as suggested by Rasmussen and
Kalosakas.[10]
4. The whole procedure is repeated until max(|δωα|) j drops below a set tolerance (10 −8 ).
Through this thesis, most SCFT results were obtained using a Newton-Raphson method
which is described in detail in the next subsection because the relaxation method usually
showed the limited stability, especially in the flat geometry, only when χN of the system
was small enough. However, we had to rely on only this iteration scheme for the purpose
of simulating highly swollen micelles of nearly symmetric block copolymer emulsifying the
minor homopolymer phase in the core (Chapter 4) since in this case the Newton-Raphson
method did not converge due to the extremely shallow minima of the free energy. In this
case, the number of iteration steps usually exceed several millions as the block ratio f A
approaches 1/2.
40

2.4.2 Newton-Raphson Algorithm
A Newton-Raphson iteration scheme is based on the first order Taylor expansion of the
residual equations by assuming the variation of the equations is locally linear near the
solutions. Different variants of Newton-Raphson iteration were used for micelle simulations,
in which we solve only the basic SCF equations, and for simulations of interfaces, in which
we solve SCF equations subject to a constraint on the interface position.
We first describe the algorithm used in micelle simulations. In SCFT, the constraints of
the chemical potential fields ω α (r) in Eqn. 2.2 and the incompressibility in Eqn. 2.3 can be
used to set up the residual vector R (α,k) with a dimension of C(n+1) where α = A,B,... , C
and k = 0,... ,n in case of the canonical system. For example, in the ternary system of
A/B/AB with C = 2, R (α,k) can be defined by
R (A,k) = χ(φ A,k − φ B,k ) − (ω B,k − ω A,k ) (2.35)
R (B,k) = φ A,k + φ B,k − 1 . (2.36)
from which the Jacobian matrix J (α,k)(β, l) can be established as
J (α,k)(β, l) = ∂R (α,k)
∂ω β,l
(2.37)
with a dimension of 2(n + 1) × 2(n + 1) and further reduced to
J (A,k)(β, l)
J (B,k)(β, l)
(
∂φA,k
= χ − ∂φ )
B,k
− δ k, l (δ β,B − δ β,A ) (2.38)
∂ω β,l ∂ω β,l
(
∂φA,k
= δ k, l + ∂φ )
B,k
(2.39)
∂ω β,l ∂ω β,l
where β = A,B,... , C and l = 0,... ,n. The Jacobian matrix was numerically obtained in
this thesis by perturbing ω β,l by an amount of δω β . Finally, the chemical potential fields
are updated as
ω j+1
α,k = ωj α,k − J −1
(α,k)(β, l) R (β,l) (2.40)
41

for the next iteration step j + 1 until the solutions converge.
We next discuss the iteration scheme for simulations of interfaces. For a flat interface,
the position of an interface is arbitrary because the pressure is identical through an interface.
Thus we fix the location of an interface at the center of a simulation box imposing
an extra constraint of equal amount of A and B homopolymers in the system,
R (B,n+1) = ¯φ A − ¯φ B (2.41)
by slightly adjusting the chemical potential of the B homopolymer, µ B .
For a weakly curved phase, the radius of curvature is a very sensitive function of the
pressure (or chemical potential) difference. The position of a curved interface can also
be determined in a similar manner by matching the prescribed curvature with the Gibbs
dividing surface of no excess A monomers such as
I
R (B, n+1) = V I ¯φ A + V II
II ¯φ A − V ¯φ A (2.42)
where V I and V II are the volume of the I and II phases with the constraint of V = V I +V II ,
and ¯φ I
A
and ¯φ
II
A
are the bulk volume fractions of A monomer in the corresponding phases.
To satisfy the extra constraint, µ B is updated using the expanded (2n + 3) × (2n + 3)
Jacobian matrix as
µ j+1
B
= µj B − J −1
(B, n+1)(β,l) R (β,l) (2.43)
where l = 0,... ,n + 1.
42

Bibliography
[1] Nalani, J.; Hong, K. M. Macromolecules 1982, 15, 482;
[2] Matsen, M. W. Phys. Rev. Lett. 1995, 74, 4225; Matsen, M. W. Macromolecules 1995,
28, 5765
[3] Matsen, M. W. J. Chem. Phys. 1999, 110, 4659
[4] Noolandi, J.; Hong, K. M. Macromolecules 1984, 17, 1531
[5] Shull, K. R.; Kramer, E. J. Macromolecules 1990, 23, 4769
[6] Burden, R. L.; Faires, J. D. Numerical Analysis (Brooks/Cole Publishing Company)
1997
[7] Drolet, F.; Fredrickson, G. H. Phys. Rev. Lett. 1999, 83, 4317
[8] Drolet, F.; Fredrickson, G. H. Macromolecules 2001, 34, 5317
[9] Helfand, E.; Tagami, Y. J. Chem. Phys. 1972, 56, 3592
[10] Rasmussen, K, ∅; Kalosakas, G. J. Polym. Sci.: B 2002, 40, 1777
43

Chapter 3
Interfacial Bending Elasticity
Abstract
Self-consistent field theory (SCFT) has been used to calculate the Helfrich bending elastic
constants for monolayers of A/B diblock copolymers at interfaces between immiscible A
and B homopolymer liquids. We focus here on the properties of saturated monolayers,
with vanishing interfacial tension. We also determine values of the volume fraction f A
of the A-block within the copolymer for which the monolayer has vanishing spontaneous
curvature in systems with asymmetric homopolymers. The procedure used here to define
and calculate the curvature-dependent interfacial tension is somewhat different than that
used previously by Matsen.
3.1 Introduction
Mixtures of two immiscible homopolymers and a compatibilizing diblock copolymer surfactant
exhibit phase behavior that is often closely analogous to that of mixtures of oil, water,
and a small molecule surfactant. Some such polymer mixtures form equilibrium structures
consisting of domains of nearly pure A or B homopolymer separated by copolymer monolayers,
including both disordered microemulsion phases and swollen ordered phases. When
the characteristic interfacial radii of curvature are sufficiently large (i.e., much larger than
the monolayer thickness), the thermodynamic competition between phases of similar com-
44

position that differ in the geometrical arrangement of interfaces can be described by the
Canham-Helfrich theory of interfacial bending elasticity.[1, 2] The chemical potential of
surfactant in such structures tends to remain very close to the value at which the microscopic
interfacial tension vanishes [3, 4, 5], creating so-called saturated monolayers, for
reasons that were first articulated by Schulman.[6]
Of particular interest for some purposes is the identification of surfactants that form
saturated monolayers with vanishing spontaneous curvature, which are referred to here as
balanced monolayers. Both theory and experiments with small molecule surfactants [7, 8]
suggest that minimum possible macroscopic interfacial tension between oil- and waterrich
(or A- and B-rich) phases (which can be several orders of magnitude below that
obtained with no surfactant) is obtained when balanced monolayers are formed. The creation
of balanced monolayers is also believed to be a necessary condition for the existence
of three-phase coexistence of excess A- and B-rich phases with a balanced bicontinuous microemulsion,
within which surfactant monolayers divide space into A- and B-rich domains
with nearly equal volume fractions. In mixtures of oil, water, and some small nonionic
C i E j surfactants, for which the conformation of the water-soluble block of the surfactant
is very sensitive to changes in temperature, balanced monolayers can be created by adjusting
the temperature, leading to the formation of a balanced bicontinuous microemulsion at
a specific temperature. Balanced bicontinuous microemulsion phases have been observed
for ternary polymer systems of A and B homopolymers mixed with both symmetric AB
copolymers [9, 10, 11] and carefully chosen AC [12, 13] block copolymers. The design
of balanced bicontinuous microemulsions for systems with asymmetric homopolymers is
of interest as a strategy for the design of polymer alloys with interpenetrating network
structures.
The bending elasticity of a copolymer monolayer has been considered in several previous
theoretical studies [3, 4, 14]. The Helfrich elastic constants were calculated by Wang
45

and Safran [3, 4] using strong stretching theory, and by Matsen [14] using numerical SCFT,
as also done here. Matsen proposed a general method for calculating the Helfrich elastic
constants, but gave results only for symmetric systems containing homopolymers of equal
molecular weights and a symmetric block copolymer, for which the monolayer has a vanishing
spontaneous curvature by symmetry. Matsen devoted as least as much attention to
systems with nonzero interfacial tensions as to saturated monolayers. Here, we give results
for asymmetric as well as symmetric mixtures, and focus on more fully characterizing the
case of saturated monolayers that is relevant to self-assembly. In mixtures with asymmetric
homopolymers, we emphasize the identification of values of the copolymer composition
required to form a balanced monolayer. The SCFT calculations presented here are also
based upon a definition of the excess free energy of a curved monolayer somewhat different
from that used by Matsen, as discussed in more detail below.
3.2 Interfacial Thermodynamics
We consider an interface in an incompressible ternary mixture of two immiscible homopolymers,
A and B, and an AB diblock copolymer, under conditions for which system phase
separates into two phases rich in A and B, respectively. Monomers of types A and B
occupy an equal volume v, and have statistical segment lengths b A and b B . Let N A , N B ,
and N C be the degrees of polymerization of the two homopolymers and the copolymer
(C), respectively. Let f A be the volume fraction of the A block within the copolymer, and
f B = 1 − f A . Let α A ≡ N A /N C , α B ≡ N A /N C , and β ≡ N A /N B .
It is convenient to formulate both the classical thermodynamics and the SCFT calculations
for a monolayer in the grand canonical statistical ensemble. The grand canonical
free energy Φ of any system is
Φ = F − ∑ i
µ i M i (3.1)
where F is the Helmholtz free energy, and µ i is the chemical potential and M i the total
46

number of molecules of species i in the system. For any macroscopic phase of volume V
and pressure P, Φ = −PV .
Interfacial tension γ of a flat macroscopic interface is the interfacial excess grandcanonical
free energy per unit area of interface, so that Φ = −PV + Aγ for a system of
volume V with an interface of area A. Here, we consider a curved interface, and define γ
in the same manner, by defining
Φ ≡ −P I V I − P II V II + Aγ (3.2)
for some choice of Gibbs dividing surface that separates an A-rich domain (phase I) from
a B-rich domain (phase II), where V I and V II are the volumes of the domains, P I and P II
are the pressures of bulk phases I and II at the temperature and chemical potentials of
interest, and A is the area of the dividing surface.
In the case of a flat interface, the value of γ is independent of our choice of a convention
for position of the dividing surface. Mathematically, this is a result of the fact that in this
case, for which P I = P II = P, both the sum P I V I + P II V II = PV and the area A are
independent of the position of the interface. For a curved interface, however, P I and P II
are generally unequal (for reasons discussed below), and A and γ both depend upon the
choice of the dividing surface. The relations among interfacial excess quantities that are
obtained below are, however, valid for any choice of dividing surface.
3.2.1 Mechanical Equilibrium
For a curved surface to be in mechanical equilibrium, the total free energy Φ must be an
extrema with respect to normal displacements of the interface at fixed chemical potentials.
For a flat surface, this mechanical equilibrium condition reduces to the statement that
pressures must be equal on either side of the interface.
Consider the mechanical equilibrium condition for a curved interface. For simplicity, we
restrict ourselves to situations in which the inner B-rich domain (II), is either a spherical
47

or cylindrical domain of radius R. Differentiating Eqn. 3.2 with respect to R while holding
temperature and all chemical potentials fixed yields the condition
0 = ∂Φ
∂R = (P I − P II ) ∂V II
∂R + ∂(γA)
∂R
, (3.3)
The required derivative of volume V II with respect to radius is area: ∂V II /∂R = −∂V I /∂R =
A. The derivative of the area is given by ∂A/∂R = AC, where
C = d − 1
R
(3.4)
is the mean curvature, in which d = 3 for a sphere of radius R and d = 2 for a cylinder.
Substitution in Eqn. 3.3 then yields a pressure difference
P II − P I = Cγ + ∂γ
∂R
(3.5)
If we parameterize γ at fixed temperature and chemical potential as a function of C, rather
than R, this becomes
P II − P I = Cγ −
C2 ∂γ
d − 1 ∂C . (3.6)
This mechanical equilibrium condition reduces to the usual Young-Laplace condition for
the pressure drop across a curved interface in the limit γ ≫ |C ∂γ
∂C
| in which the curvature
dependence of γ is negligible. It further reduces to the condition P I = P II in the limit
C = 0, thus recovering the usual condition for macroscopic phase coexistence only in the
case of a flat interface.
3.2.2 Thermodynamic Variables
A straightforward extension of the Gibbs phase rule may be used to show that interfacial
tension γ of a mechanically stable spherical or cylindrical interface in an incompressible
ternary mixture is a function of three parameters, which we may take to be temperature
T, block copolymer chemical potential µ C , and interfacial curvature C. To show this, we
first specify the state of such a system in terms of five degrees of freedom, which may
48

e taken to be temperature, three chemical potentials (which must be the same in both
phases), and interfacial curvature. The requirement that the interface be in mechanical
equilibrium removes one degree of freedom, leaving four degrees of freedom, one of which
is the curvature. The usual Gibbs phase rule, which yields three degrees of freedom for a
ternary two-phase system, corresponds to the case C = 0 of a flat interface.
The restriction to incompressible fluids reduces the number of physically relevant degrees
of freedom by one more: Changes in the pressure of an incompressible mixture have
no effect upon the state of the system other than to cause trivial shifts in the values of the
chemical potentials, without changing γ or any other interfacial excess properties. This
trivial dependence on hydrostatic pressure is discussed in the context of SCFT in Chapter
2.2.1. The state of a mechanically stable curved interface in an incompressible ternary
mixture may thus be specified as a function of three variables, e.g., T, µ C , and C.
In order to unambiguously define a functional relationship γ(T,µ C ,C) for an incompressible
system, however, one must introduce a standard state for chemical potential (as
always) and a convention to fix a value of the pressure in one of the two bulk phases. In numerical
work, for flat interfaces, we evaluate γ(T,µ C ) in the macroscopic state in which the
bulk pressure vanishes in both phases. For curved interfaces, we have chosen to calculate
γ(T,µ C ,C) in a state in which µ A is kept fixed at the value obtained in the macroscopic
two-phase with P I = P II = 0 at the prescribed value of µ C , in which µ B must be adjusted
so as to create a stable interface with the chosen curvature. The results obtained with
this convention are expected to be very similar to those that would be obtained by instead
requiring that P = 0 in the A-phase for all values of C: Changing µ B at fixed µ A changes
the pressure in the A-rich phase only as a result of changes in the osmotic pressure arising
from changes in the concentration of dissolved B homopolymer, which is generally very
small in the situations of interest.
An interface in which γ = 0 will hereafter be referred to as a saturated interface.
49

Let µ ∗ C (T) denote the value of µ C at which γ = 0 for a flat interface. In phases that
contain an extensive area of surfactant monolayers between A- and B-rich domains, such
as a microemulsion or swollen lamellar phase, the interfaces are generally very nearly
saturated. [6, 5]
3.2.3 Helfrich Expansion
We consider a Helfrich expansion of γ about the flat saturated state, at fixed temperature
T, as a function of the principal curvatures of the interface and of the deviation
δµ ≡ µ C − µ ∗ C . (3.7)
Expanding to quadratic order in curvature and linear order in δµ yields an expansion of
the form
γ = −τC + 1 2 κC2 + ¯κK + Γ ∗ δµ + ΛδµC (3.8)
where C = R −1
1 +R −2
2 is the mean curvature and K = 1/(R 1 R 2 ) is the Gaussian curvature
of a surface with principal radii of curvature R 1 and R 2 . For a sphere, C = 2/R and
K = 1/R 2 , while for a cylinder C = 1/R and K = 0. The coefficients κ and ¯κ are bending
and Gaussian rigidities, respectively, while
τ ≡ ∂γ
∣
∂C
∣
µC =µ ∗ C
(3.9)
is a parameter that controls the asymmetry of the monolayer. The coefficient of the term
linear in δµ is given by the Gibbs adsorption equation
Γ ∗ = − ∂γ
∂µ C
∣
∣∣∣C=0
(3.10)
in which Γ ∗ ≡ Γ ∗ C
is the excess number density (copolymers per area) of copolymers in a
saturated flat interface. The coefficient Λ is the second derivative Λ ≡
∂2 γ
∂µ C ∂C .
50

The interfacial tension of a cylindrical or spherical surface at the constant saturation
chemical potential µ C = µ ∗ C
may be expressed as a harmonic function
γ = −τC + 1 2 κ′ C 2 (3.11)
where κ ′ = κ for a cylindrical surface or κ ′ = κ + ≡ κ + ¯κ/2 for a spherical surface. The
minimum of interfacial tension with respect to C at µ C = µ ∗ C
is given for a cylindrical
surfaces by the ratio τ/κ, which is often referred to as the spontaneous curvature (though
“spontaneous cylindrical curvature” would arguably be more appropriate). The corresponding
spontaneous curvature for a spherical surface is given by τ/κ + . The spontaneous
curvatures τ/κ and τ/κ + correspond to the thermodynamically preferred radii of swollen
cylindrical or spherical micelles in an A matrix that coexists with an excess B-rich phase.
In the absence of a coexisting excess phase, micelles generally adopt smaller radii as the
result of a stoichiometric constraint on the amount of available B homopolymer.
A saturated monolayer for which τ = 0 will be referred to here as a balanced monolayer.
Symmetric mixtures, with f A = 1/2, N A = N B , and b A = b B , must form balanced
monolayers, by symmetry. In systems with asymmetric homopolymers, with N A ≠ N B
and/or b A ≠ b B , let f bal
A
denote the value of f A necessary to create a balanced monolayer.
3.2.4 Lifshitz Point
Consider an incompressible ternary system containing two coexisting phases rich in A
and B. For a specified set of three molecules, the behavior of such a system depends
on only surfactant chemical potential µ C and temperature. For simplicity, we assume in
the following discussion that χ monotonically decreases with increasing T, and ignore the
possibility of three-phase coexistence.
In the absence of copolymer, A and B are completely miscible above a binary critical
temperature T c . As T is decreased below T c , a two-phase region of Gibbs phase triangle
emerges from the binary critical point (a), as shown in Fig. 3.1. At temperatures slightly
51

Figure 3.1: Schematic view of the state of a two-phase system at fixed pressure as a function
of T and of a copolymer activity Z C ∝ e µ C/kT , which is defined for this purpose so that
Z C is proportional to the concentration of copolymer dissolved in one of the two coexisting
phases. The Lifshitz point is the intersection of a line of critical points, for which the
copolymer concentration tends to increase with decreasing temperature, with a saturation
line e µ∗ C (T) , for which copolymer concentration decreases with decreasing temperature as
the copolymer becomes less soluble.
52

elow T c , the two phase region generally terminates at a critical point (b) that moves to
higher surfactant concentration with decreasing T. Within the two phase region, each
tie-line may be characterized by a value of µ C and a corresponding value of the interfacial
tension γ(T,µ C ,C = 0) of a flat macroscopic interface. This interfacial tension generally
decreases with increasing µ C and, at temperatures only slightly below T c , vanishes only at
the critical point, where the the compositions of the coexisting phase merge. At temperatures
below a Lifshitz temperature T L , however, the macroscopic interfacial tension may
be driven to zero at chemical potential µ ∗ C
below the value at the critical point (c), producing
a saturated monolayer between distinct A- and B-phases. At temperatures T < T L
for which such a saturated monolayer can exist, increasing temperature causes the critical
point c to move to lower copolymer concentrations and the saturation tie line to move
toward higher copolymer concentrations, until they merge at the Lifshitz point, denoted
by L in Fig. 3.1. By examining instabilities of the disordered phase, the Lifshitz point
can also be identified as the point along the line of critical points (a − L) at which the
divergence of the structure factor S(q) at q = 0, which signals an instability to macroscopic
phase separation, is first preempted by divergence of S(q) at q ≠ 0, signalling an instability
towards formation of a mesophase. As the Lifshitz point is approached along the
saturation line µ ∗ C
(T), the compositions of the two coexisting phases must merge, and so
the Helfrich elastic constants of a saturated monolayer must approach zero at the Lifshitz
temperature.
Broseta and Fredrickson [15] have considered thermodynamics of symmetric mixtures,
with f = 1/2, α A = α B = α, and b A = b B . For such systems, the line of critical points
must stay within an isopleth φ A = φ B . In this case, a stable Lifshitz point occurs at
(χN C ) L = 2(1 + 2α2 )
α
(φ C ) L = 2α2
1 + 2α 2 (3.12)
for all α < 1. [15, 16]. For α > 1, they found that the appearance of the Lifshitz point
53

is preempted by the appearance of a region of coexistence between three homogeneous
phases.
A Lifshitz point will generally also exist in systems with slightly asymmetric homopolymers
and/or an asymmetric copolymer. For each choice of a set of values N A , N B , and N C ,
however, there exists a unique value of f A , given by the limit lim T →TL fA bal (T), for which
a balanced saturated monolayer will be formed at temperatures infinitesimally below T L .
Fredrickson and Bates [16] have proposed the identification of such balanced Lifshitz points
as a strategy for the formulation of balanced bicontinuous microemulsions in mixtures with
asymmetric polymers. They also proposed an approximate criterion for identifying such
points: They required that the critical fluctuation mode at the Lifshitz point not involve
any fluctuation in surfactant concentration. This is equivalent to requiring that the slope
of the tie-lines infinitesimally close to the critical point at T = T L should be parallel to
the binary A-B homopolymer edge of the Gibbs phase triangle. This criterion was found
to be satisfied for any critical point with
f A = √ β/(1 + √ β). (3.13)
This criterion does not appear to us to be equivalent to the requirement τ = 0 near the
Lifshitz point, but the two criteria are shown to yield similar results in section 3.4.
3.3 SCFT Methodology
SCFT calculations of γ for spherically and cylindrically curved interfaces with a radius
R have been carried out using the grand-canonical formulation, which is discussed in
Chapter 2.2.2. Calculation of γ for a weakly curved interface requires the construction of
a spherically or cylindrically symmetric solution to the SCF equations within an annular
domain R −
< r < R + , where r is a radial coordinate, in which R − and R + are far
enough from the dividing surface so that the solution is insensitive to changes in these cell
54

oundaries. To calculate the elastic parameters for an expansion about a saturated flat
interface, for a specified set of molecules at a specified value of χ, µ ∗ C
is first determined
by calculating interfacial tension γ(µ C ,C = 0) of a flat interface for several values of µ C .
In these simulations, the chemical potentials µ A and µ B are chosen for each value of µ C so
as to satisfy the condition P I = P II = 0. The required values of chemical potential can be
calculated from Flory-Huggins theory. Once µ ∗ C
is known for a specific system, calculations
of γ(µ ∗ C
,C) are conducted for both cylindrical and spherically curved interfaces at several
values of R or C. For each such calculation, the values of µ A and µ C are kept fixed at the
values obtained for a saturated flat monolayer at zero bulk pressure, and µ B is adjusted
in each calculation as to obtain a prescribed radius of curvature R for the Gibbs dividing
surface. We take the Gibbs dividing surface to be the equimolar surface for either type of
monomer, defined so that there is a vanishing interfacial excess of either A or B monomers,
when local A or B monomer concentrations are taken to include contributions from both
one homopolymer and one block of the copolymer. Values of κ and κ + are obtained for
each system of interest by fitting SCFT results of γ(µ ∗ C
,C) for cylindrical and spherical
surfaces, separately, to Eqn. 3.8.
For both flat and curved interfaces, a Newton-Raphson iteration scheme is used to
solve the grand canonical SCF equations constrained simultaneously with a constraint
requiring that the dividing surface have a radial position R = (R + + R − )/2 in the middle
of the unit cell. The macroscopic chemical potential µ B is adjusted so as to satisfy this
additional constraint, and each iteration thus involves a simultaneous adjustment of the
SCFT chemical potential fields ω α at all spatial grid points and the macroscopic chemical
potential µ B . In grand-canonical ensemble, it is necessary to force the dividing surface to
take a specified position within the unit cell in order to obtain a unique, stable solution
even for a flat interface, because the position of a flat interface within the unit cell is
otherwise indeterminate.
55

0.1
0.05
f A =0.5098
0
f A =0.5455
−0.05
f A =0.5833
γ / γ ο
−0.1
−0.15
−0.2
0.003
γ / γ ο
f A =0.5098
f A =0.6667
−0.25
−0.001
0 0.0025
R −1 [Å −1 ]
−0.3
0 0.002 0.004 0.006 0.008 0.01
R −1 [Å −1 ]
Figure 3.2: SCFT results for interfacial tension γ vs. half curvature C/2 = 1/R for
spherical interfaces with R ≥ 100 Å, for system with the copolymer of a fixed size of
B block f B χN C = 10 and symmetric homopolymers of fixed lengths α A = α B = f B ,
with b A /b B = 1, for several values of f A . Solid lines are quadratic fits of the curvature
dependence at small curvatures to Eqn. 3.8, from which κ + and τ can be extracted.
0.04
0.02
0
f A =0.5098
f A =0.5455
γ / γ ο
−0.02
−0.04
f A =0.5833
−0.06
0.003
f A =0.5098
−0.08
γ / γ ο
f A =0.6667
−0.1
−0.001
0 0.0025
R −1 [Å −1 ]
−0.12
0 0.002 0.004 0.006 0.008 0.01
R −1 [Å −1 ]
Figure 3.3: SCFT results for interfacial tension γ vs. curvature C = 1/R for cylindrical
interfaces with R ≥ 100 Å, for the same mixtures as those studied in the previous figure.
κ can be obtained.
56

Fig. 3.2 and Fig. 3.3 show examples of results for γ vs. 1/R at µ C = µ ∗ C
, for spherical
and cylindrical geometries respectively, for a series of systems in which N A = N B = f B N C
and f B χN C = 10, and in which the size of the A block of the copolymer is varied to change
f A . For both spherical and cylindrical deformations, the Helfrich expansion is found to be
accurate for this choice of parameters for surfaces with radii R 150 Å, below which γ
starts to deviate significantly from Eqn. 3.8. The radius at which this deviation becomes
significant is comparable to that of a ”dry” binary micelle of the copolymer used here, in
which no B homopolymer is contained in the core. The minimum in γ vs. 1/R for nearly
symmetric copolymers corresponds to the thermodynamically preferred (i.e., spontaneous)
curvature for a swollen cylindrical micelle or spherical micelle. Surprisingly, as well as
1/R, f A strongly affects this preferred curvature so that it exceeds the range of values for
which the Helfrich theory remains valid when f A ≥ 0.58. The Helfrich theory is thus found
to be valid over a rather wide range of curvatures in the case of symmetric copolymer,
as noted previously by Matsen, [14] but is useful for asymmetric monolayers only over a
rather limited range of f A , because of the rapid variation of spontaneous curvature with
f A .
We have shown in Chapter 2.2.4 that, as noted previously by Matsen [14], the SCF
expression for interfacial tension of a spherical or cylindrical surface of radius R may be
expressed in the dimensionless form
γ =
b
N 1/2 [γ] (3.14)
v
where we have chosen N = N C as a reference degree of polymerization and b = b B as
reference statistical segment length, and where [γ] is dimensionless interfacial tension that
can depend only upon the dimensionless arguments χN, f A , α A , α B , b B /b A , µ C /kT, and
the dimensionless radius R/(N 1/2 b). Following Matsen, we express the Helfrich expansion
57

for a membrane with µ C = µ ∗ C
in nondimensional form as
[γ] = 1 2 [κ][C]2 + [¯κ][K] 2 − [τ][C] (3.15)
where [C] ≡ C N 1/2 b and [K] ≡ KNb 2 are dimensionless mean and Gaussian curvatures,
and
κ =
¯N 1/2 [κ]
¯κ = ¯N 1/2 [¯κ] (3.16)
τ =
¯N
1/2
N 1/2 b [τ] ,
where ¯N ≡ Nb 6 /v 2 , and where [κ], [¯κ], and [τ] are dimensionless elastic parameters that
depend only on χN, f A , α A , α B , and b B /b A ,
As noted in Introduction, the definition of the excess free energy density γ for a curved
interface used here is slightly different from that used by Matsen.[14] Solutions to the SCF
equations are always extrema of a corresponding SCF free energy functional, and can thus
only describe mechanical equilibrium states. In order to simulate curved interfaces with
a range of curvatures, Matsen anchored the chemical potentials to the values at the coexistence
condition for a flat interface in all simulations, but instead stabilized each curved
interface with desired curvature by adding to the SCF free energy functional an additional
term whose sole purpose is to exert a stabilizing force on the interface. The prefactor of this
added term is treated as a Lagrange multiplier, which can be adjusted so as to provide any
required force and thereby allow the interfacial curvature to be controlled independently of
the pressure difference across the interface. Matsen required that the macroscopic chemical
potentials µ A , µ B , and µ C always be chosen so as to satisfy the condition P I = P II for
macroscopic phase coexistence. As a result, the magnitude of the Lagrange multiplier that
he introduced to restore mechanical stability must vanish in the limit of a flat interface,
but must become nonzero for any other curvature. The form chosen by Matsen for the
58

0.35
0.3
α = 1
0.5
0.25
0.25
0.2
[κ + ]
0.15
0.1
0.05
0
0 5 10 15 20 25 30 35 40 45 50
χ N C
Figure 3.4: The non-dimensional bending rigidity [κ + ] of a sphere vs. χN in symmetric
mixtures with various values of α = α A = α B at fixed β = 1.
term added to the free energy introduced δ-function spikes in the SCF chemical potential
fields ω A (r) and ω B (r) along the dividing surface along which φ A (r) = φ B (r) in his definition.
The addition of a such a δ-function contribution to the SCF chemical potentials must
perturb the structure calculated for any curved interface. Specifically, it must produce a
discontinuity in the gradients of the monomer volume fraction fields at the dividing surface.
This construction seems conceptually awkward to us, since no such ”magic finger” exists
to stabilize the interface in any physical situation. The procedure used here, in which a
curved interface is stabilized by a pressure difference between the neighboring phases, is
instead more natural for the physical situation in any self-assembled system.
3.4 Symmetric Mixtures
In this section, we discuss the bending elasticity of saturated monolayers in symmetric
mixture with f A = 1/2 and symmetric homopolymers, with α A = α B = α and b A = b B = b.
Fig. 3.4-3.6 show variation of the dimensionless elastic constants [κ], [κ + ], and [¯κ] as
59

0.35
0.3
0.25
α = 1
0.5
0.25
0.2
[κ]
0.15
0.1
0.05
0
0 5 10 15 20 25 30 35 40 45 50
χ N C
Figure 3.5: The normalized bending rigidity [κ] vs. χN for symmetric systems with several
values of α.
0
−0.05
−0.1
[ − κ ]
−0.15
−0.2
α = 1
0.5
0.25
−0.25
0 5 10 15 20 25 30 35 40 45 50
χ N C
Figure 3.6: The normalized Gaussian rigidity [¯κ] vs. χN for symmetric mixtures with
several values of α.
60

− [ − κ ] / [κ]
1
α = 1
0.5
0.8
0.25
0.6
0.4
0.2
0
0 5 10 15 20 25 30 35 40 45 50
χ N C
Figure 3.7: The ratio of the Gaussian rigidity to bending rigidity −[¯κ]/[κ] vs. χN for
symmetric mixtures.
functions of χN, for several values of α. Here, we show data only for values of α ≤ 1,
which corresponds to the range in which Broseta and Fredrickson find a stable Lifshitz
point. Within this range, [κ], [κ + ] and [¯κ], all extrapolate to zero at the value of (χN) L
given in Eqn. 3.12, and appear to vary linearly with χN near the Lifshitz point. Note that
the Gaussian rigidity ¯κ is negative for all χN > (χN) L , as required for a flat membrane,
or a lamellar phase, to be stable against the formation of a bicontinuous minimal surface
structure.
Fig. 3.7 shows the ratio −[¯κ]/[κ] for saturated monolayers with α ≤ 1. In the strong
stretching limit, Wang and Safran predict a value of −[¯κ]/[κ] = 4
15
= 0.266 for the bending
rigidity of a monolayer in the dry brush (melt state) limit [3, 4], and −[¯κ]/[κ] = 17
30 = 0.5667
in the limit of a strongly stretched but fully solvated brush with a parabolic concentration
profile. Our numerical results for moderately segregated monolayers appear to be qualitatively
consistent with these strong stretching predictions: For χN C > 20, we obtain values
in the range 0.2 < −[¯κ]/[κ] < 0.6, with higher values of −[¯κ]/[κ] in systems with lower
61

values of α, for which the monolayer is more swollen. For all α < 1 and χN C > 20, the
ratio −[¯κ]/[κ] also satisfies the condition −[¯κ]/[κ] < 10/9, which has been proposed [17, 18]
as a criterion for systems in which, upon swelling, a lamellar phase of balanced membranes
will ultimately melt into a bicontinuous microemulsion rather than a dispersion of spheres.
3.5 Asymmetric Mixtures
In this subsection we switch our attention to asymmetric blends, in which α A ≠ α B ,
f A ≠ 1/2, and/or a A ≠ b B . Note that we redefine α B ≡ N B /(2f B N C ) because we fix only
the B block length of the copolymer rather than total copolymer length.
Figs. 3.8 and 3.9 show how the elastic parameters τ and κ vary with f A by changing
only the A block length of the copolymer in a series of systems with N A = N B and
f B N C = N B , as in Figs. 3.2 and 3.3. A linear dependence between τ and f A holds over
approximately the same range of f A 0.58 as that for which the spontaneous curvature
was found to lie within the range of validity of the Helfrich expansion.
Figs. 3.10 and 3.11 show the the balance point f bal
A
at which τ = 0 a function of the
ratio β = α A /α B for several different sets of systems with b A = b B . Fig. 3.10 shows the
results for a set of systems for which the size of the B homopolymer and the B block of
the copolymer are equal, with several values of χN B = f B N C χ, while we have varied N A
to vary β and the size of the A block of the copolymer to vary f A . Fig. 3.11 presents
f bal
A
vs. β for several values of the ratio f BN C /N B for systems in which the B block of
the copolymer has a fixed size f B χN C = 10. In both cases, increasing β by increasing N A
leads to an increase in f bal
A . This is because increasing N A decreases the tendency of the A
homopolymer to swell the A brush, which must be compensated by an increase in the size
of the A block. The sensitivity of f bal
A
to changes in β decreases as the brush is made drier
by increasing either χN C or α B , In the limit of a completely dry brush, we would expect
fA bal to approach 1 2 , independent of β. 62

0.1
0.09
0.08
0.07
0.06
τ
0.05
0.04
0.03
0.02
0.01
0
0.5 0.6 0.7 0.8
f A
Figure 3.8: Bending parameter τ of block copolymer monolayer vs. a block ratio f A of
the A block for systems with f B χN = α A χN = α B χN = 10. The system in this plot
is identical to that in Fig. 3.2 and 3.3. fA
bal is always approximated as a limiting value
linearly extrapolated to C = 0 as shown here.
2.4
2.2
2
κ +
1.8
1.6
1.4
0.5 0.6 0.7 0.8
f A
Figure 3.9: Bending rigidity κ + of sphere vs. a block ratio f A of the A block for systems
with f B χN = α A χN = α B χN = 10. In this article, κ + is always approximated as a
limiting value linearly extrapolated to C = 0 as shown here.
63

0.6
α B = 0.5
0.55
f A
bal
0.5
0.45
0.4
f L
f B χ N C = 5
10
20
0.35
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
β
Figure 3.10: The variation of the balance point fA
bal with fixed α = 1. The solid line
indicates the balance point f L at an isotropic Lifshitz point.
0.6
f B χ N C = 10
0.55
0.5
f A
bal
0.45
0.4
f L
α B = 0.25
0.5
1
0.35
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
β
Figure 3.11: The variation of the balance point fA
bal for asymmetric mixtures with a fixed
minority block size f B χN C = 10. The solid line indicates the balance point f L at an
isotropic Lifshitz point.
64

0.6
0.55
0.5
f A
bal
0.45
0.4
f L
b A /b B = 1
2
0.35
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Figure 3.12: The variation of the balance point fA
bal
length b.
β
with asymmetric statistical segment
Eqn. 3.13 for the value of f A at which the balanced Lifshitz point is obtained is shown
as a solid line in Figs. 3.10 and 3.11. It appears to provide a reasonably accurate approximation
to the balance point that is obtained here by setting τ = 0, which remains useful
over the the entire range of values of χN C and α B explored here.
Fig. 3.12 explores the effect of changes in statistical segment length upon the balance
point f bal
A for system containing a B block of fixed size f BN C χ = 10 and α B = f B .
Increasing b A /b B by a factor of 2 is found to cause a rather slight increase in f bal
A . The
sign of the change may be understood by noting that an increase in b A decreases the free
energy cost of stretching the A block of the copolymer, by decreasing the spring constant
kT/b 2 A
in the Gaussian stretching energy. Thus it causes a tendency for the monolayer to
curve around the A block that may be compensated by increasing f A .
At last, results for κ, ¯κ, and κ + are shown in Fig. 3.13 as a function of β for a series of
asymmetric systems with a fixed size of the B block f B χN = 10, α B = 0.5, and b A = b B .
Note that all three rigidities seem to extrapolate to zero at a value of β slightly below the
65

3
2
κ, κ + , − κ
1
0
κ
κ +
− κ
−1
−2
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
β
Figure 3.13: Rigidities κ + , κ, and ¯κ vs. β = α B /α A for systems with a copolymer minority
block of size f B χN = 10, a B homopolymer of length α B χN = 5, and b A = b B , for varying
size A homopolymers. Values for κ, κ + , and ¯κ are given in units with kT = 1 on the left
scale, and corresponding values [κ], [κ + ], and [¯κ] on the right.
lower limit of the range shown in this plot (beyond which the simulations become difficult
to converge), indicating the existence of the ’balanced’ Lifshitz point at this value of χN
and its dependence on β.
3.6 Conclusions
We have used numerical SCFT to investigate the bending elasticity of block copolymer
monolayers in melts of immiscible homopolymers. The method used to calculate the free
energy of an interface as a function of an imposed curvature is somewhat different than
that employed previously by Matsen. In both methods, a force must be exerted upon an
interface in order to impose an arbitrary curvature. Here, the required force is provided
by a pressure difference.
The Helfrich expression for the free energy of a curved interface is a Taylor expansion
about a flat reference state. This expansion is useful for describing equilibrium structures
66

only if it remains accurate at curvature comparable to the spontaneous curvature. The
Helfrich expansion for a symmetric monolayer is found to be accurate over a surprisingly
wide range of imposed curvatures, extending almost to the curvature of a ”dry” micelle,
with no homopolymer in its core. However, this expansion appears to be useful only over a
rather limited range of values of the copolymer block ratio f A because of a rapid variation
of spontaneous curvature with f A . For the moderately segregated systems considered here,
with χN C ≃ 20, the spontaneous radius of curvature of a spherical micelle is significantly
greater than the core radius of a dry micelle only for 0.5 f A 0.6 (for micelles in an A
matrix), which limits the potential usefulness of the expansion to roughly the same range
of values.
The three elastic constants κ, ¯κ, and τ were shown to all continuously approach zero at
the analytically predicted Lifshitz point for symmetric systems with α ≤ 1. The Gaussian
rigidity for moderately segregated symmetric systems was found to always lie with the
range 0 < −¯κ < (10/9)κ (with a κ > 0) in which the lamellar phase is mechanically
stable but is expected to be susceptible to melting into a bicontinuous microemulsion upon
swelling, as the result of effects of interfacial fluctuation that are not captured by SCFT.
The value f bal
A
of the copolymer composition at which the spontaneous curvature vanishes
has been determined for a variety of systems with asymmetric homopolymers. Knowledge
of this ’balance point’ is potentially useful for the design of optimal polymeric surfactants.
The sensitivity of f bal
A
to changes in the ratio β decreases as the brush becomes drier
either because of an increase in χN or an increase in α B . The analytical approximation
for f bal
A
proposed by Fredrickson and Bates, which was obtained from an analysis of the
critical mode at the Lifshitz point, is found to provide a reasonably accurate estimate even
far from the Lifshitz point.
67

Bibliography
[1] Canham, P. B. J. Theoret. Biol. 1970, 26, 61
[2] Helfrich, W. Z. Naturforsch. C 1973, 28, 693
[3] Wang, Z.-G.; Safran, S. A. J. Phys. France. 1990, 51, 185
[4] Wang, Z.-G.; Safran, S. A. J. Chem. Phys. 1991, 94, 679
[5] De Gennes, P. G.; Taupin, C. J. Phys. Chem. 1982 86, 2294
[6] Schulman, J. H.; Montagne, J. B. Ann. N.Y. Acad. Sci. 1961, 92, 366
[7] Strey, R. Coll. Ploym. Sci. 1994, 272, 1005
[8] Sottmann, T.; Strey, R. J. Chem. Phys. 1997, 106, 8606
[9] Bates. F. S.; Maurer, W.; Lipic, P. M.; Hillmyer, M. A.; Almdal, K.; Mortensen, K.;
Fredrickson, G. H.; Lodge, T. P. Phys. Rev. Lett 1997, 79, 849
[10] Washburn, N. R.; Lodge, T. P.; Bates. F. S. J. Phys. Chem. B 2000, 104, 6987
[11] Hillmyer, M. A.; Maurer, W.; Lodge, T. P.; Bates. F. S.; Almdal, K. J. Phys. Chem.
B 1999, 103, 4814
[12] Jeon, H. S.; Lee, J. H.; Newstein, M. C. Macromolecules 1998, 31, 3340; 34, 6557
[13] Lee, J. H.; Ruegg, M. L.; Balsara, N. P.; Zhu. Y.; Gido, S. P.; Krishnamoorti, R.;
Kim, M.-H. Macromolecules 2003, 36, 6537
68

[14] Matsen, M. W. J. Chem. Phys. 1999, 110, 4659
[15] Broseta, D.; Fredrickson, G. H. J. Chem. Phys. 1990, 93, 2927
[16] Fredrickson, G. H.; Bates, F. S. J. Polym. Sci.: Part B: Polym. Phys. 1997, 35, 2775
[17] Morse, D. C. Phys. Rev. E 1994, 50, R2423
[18] Golubovic, L. Phys. Rev. E 1994, 50, R2419
69

Chapter 4
Swollen Micelles and Interfacial
Tension
Abstract
We combined self-consistent field theory (SCFT) and the Helfrich theory of interfacial
bending elasticity to analyze the thermodynamics of a ternary two-phase systems of two
immiscible homopolymers A and B and a small amount of diblock copolymer AB. We
focus on the effect of micellization upon the interfacial tension in a two-phase state in
which copolymer micelles form in the A rich phase. SCFT has been utilized to calculate the
critical micelle concentrations (cmc’s) and macroscopic interfacial tensions for the interface
between A- and B-rich phases. The behavior of systems of nearly balanced copolymers,
which tend to form highly swollen micelles, is also discussed within the context of the
Helfrich theory, using elastic constants obtained from SCFT simulations of weakly curved
monolayers.
4.1 Introduction
There have been numerous efforts to emulsify or compatibilize immiscible homopolymer
blends by adding small amounts of block copolymer. Added block copolymers adsorb at
interface between homopolymer phases, and stabilize the dispersion of minor phases by
70

educing interfacial tension, which promotes droplet break-up, and by suppressing droplet
coalescence.[1, 2] Once the concentration of copolymer dissolved in either phase exceeds
the critical micelle concentration (cmc), additional block copolymer instead form micelles,
preventing further increase in the interfacial coverage of copolymer at the interfaces and
any further decrease in the interfacial tension.
Although interfacial adsorption and micellization are always competitive in the ternary
polymer blends, most theoretical studies of these phenomena have focused either on interfacial
adsorption, while ignoring the possibility of micellization [3, 4], or on predictions
of the cmc in binary mixtures of homopolymer and copolymer.[5, 6, 7, 8, 9, 10] Leibler
has studied the saturation of interfacial tension γ above the cmc in a two-phase ternary
mixture [11], by equating the chemical potential at the cmc with that of the copolymer adsorbed
to a macroscopic interface, while using a strong stretching theory to describe both
the micelle and the adsorbed monolayer. Leibler concluded that a vanishing macroscopic
interfacial tension, γ = 0, should be obtained below the cmc for systems with a rather
wide range of values of block copolymer compositions f A . However, Leibler calculated
the free energy of formation of a micelle using a model appropriate to a binary system
of copolymer in a homopolymer matrix, and thus neglected the fact that a micelle in an
A-rich phase that coexists with an excess phase of the nearly pure B homopolymer may
swell by emulsifying the B homopolymer within its core. Allowing for the formation of
swollen micelles (which are sometimes referred to as microemulsion droplets [12, 13]) will
generally reduce the calculated free energy of formation of a micelle, thereby reducing the
predicted value of the copolymer chemical potential at the cmc, and raising the predicted
saturation value of γ above the cmc.
In this Chapter we numerically solve SCFT to examine the effect of the formation of
spherical micelles on macroscopic interfacial tension within a ternary two-phase mixture,
while allowing the formation of swollen micelles. In contrast to Leibler, we find that micelles
71

always form at the concentration of a copolymer below that needed to drive the interfacial
tension to zero, yielding a nonzero interfacial tension above the cmc, except in the limit of
a perfectly balanced surfactant monolayer with vanishing spontaneous curvature.
4.2 Micellization - Theory
We consider a ternary system of two immiscible homopolymers, A and B, and an AB
diblock copolymer (denoted C), under condition for which system phase separates into
two phases rich in A and B, respectively, with a small concentration of copolymer C.
Let N A , N B , and N C denote the degrees of polymerization of the two homopolymers
and copolymer, respectively, and f A denote the volume fraction of A monomers within the
copolymer. Let α A ≡ N A /N C , α B ≡ N B /N C , and β ≡ N A /N B . For specificity, we consider
the case in which, above the critical micelle concentration, spherical micelles preferentially
form in the A-rich phase. In the case of symmetric homopolymers with β = 1 and equal
statistical segment lengths, this corresponds to the case f A > 1/2.
4.2.1 Phenomena
The dilute region of the ternary phase diagram for such a system is shown schematically
in Fig. 4.1. The copolymer is assumed to form micelles in the A-rich phase above cmc line,
shown by a dotted line c-b, which extends from the cmc of the binary A-C system (point c)
to the cmc of an A-rich phase that coexists with a B-rich phase (point b). The dash-dotted
tie line that intersects the phase boundary at point b divides the two-phase region into a
lower region of very low copolymer concentration in which neither of the two coexisting
phases contains micelles, and an upper region of higher copolymer concentration in which
a micellar A-rich phase coexists with a micelle-free B-rich phase. Below this cmc tie-line,
the boundaries of the A and B-rich phases are reasonably well described by Flory-Huggins
theory. Above the cmc, the limit of solubility of B within the micellar phase becomes
72

Figure 4.1: Schematic phase diagram of a ternary system comprised of homopolymers
A and B and diblock copolymer C, in which an A rich phase of swollen micelles (or,
equivalently, microemulsion droplets) coexists with a phase of nearly pure B. The dotted
line b-c is a line of the critical micelle concentration in a single A-rich phase. Micelles are
drawn above this line. A point b corresponds to the cmc in two phase ternary system.
73

higher than that predicted by Flory-Huggins theory as a result of the solubilization of the
B homopolymer within the cores of swollen micelles. Also, the upper long-dashed two
phase boundary (the emulsification failure line) is almost straight because a micelle radius,
or the ratio of a core volume to an interfacial area, is nearly constant due to the fixed
chemical potentials until the emulsification failure to exceed the limit of solubility of the
B homopolymer in the core occurs.
The chemical potential µ C of copolymer within the A-rich phase is related by an ideal
solution law,
µ C = µ 0 C + kT ln(ρf C /ρ0 C ) , (4.1)
to the number concentration ρ f C
of ”free” copolymers dissolved as single molecules (outside
of any micelles) within the A-rich phase. Here, ρ 0 C
is a standard state concentration and
µ 0 C
is a corresponding standard state chemical potential for copolymer dissolved in A. A
similar relation exists between the chemical potential µ B of the B homopolymer and the
concentration of B dissolved within the A matrix. Above the cmc line, the concentrations
and the corresponding chemical potentials of free B and C molecules within the matrix of
a two-phase system always remain very close to those obtained at the cmc.
The resulting saturation of ρ f C (or the volume fraction φf C ) and µ C to nearly constant
values at concentrations above the cmc are shown schematically in Fig. 4.2 and Fig. 4.3.
In a two-phase system, micelles generally form in only one phase (here, the A-rich phase)
because the saturation of µ C caused by micellization in the A-rich phase prevents µ C from
ever reaching the higher value required for micellization in the B-rich phase, unless the
values of µ cmc
C
in the two phases are extremely similar.
Interfacial tension γ of a macroscopic interface between coexisting A- and B-rich phases
monotonically decreases with increasing µ C while the interfacial coverage of adsorbed
copolymer increases. As shown in Fig. 4.4, interfacial tension extrapolates to zero at a
74

φ c
m
φ c
(f,m)
φ c
f
φ c
Figure 4.2: Dependence of concentrations of free copolymer and of copolymer within micelles
upon the total concentration of copolymer within the micellar phase, along the
coexistence line.
µ c - µ c
o
φ c
Figure 4.3: Dependence of copolymer chemical potential upon total concentration of
copolymer within the micellar phase, along the coexistence line. The dotted line shows the
continuation of ideal solution behavior beyond the cmc
75

1
γ / γ o
0
φ c
cmc
φ
*
c
φ c
Figure 4.4: Schematic of the ratio γ/γ 0 of macroscopic interfacial tension γ to its value γ 0
in the absence of copolymer, vs. the total volume fraction of copolymer within the A-rich
micellar phase. As in Fig. 4.3, the solid line shows the behavior of a micellar solution,
and the dotted line shows the continuation of ideal solution behavior, in the absence of
micelles.
saturation chemical potential µ ∗ C or a corresponding concentration ρ∗ C
of free dissolved
copolymer. If µ cmc
C
is less than the saturation chemical potential µ ∗ C
, then γ will decrease
only until the cmc is reached, and thereafter remain very close to the nonzero value
γ cmc obtained at the cmc. If µ cmc
C
were somehow to exceed µ ∗ C
, however, then it would
be possible to drive the interfacial tension to zero at a copolymer concentration below the
cmc.
Leibler estimated µ ∗ C
and µcmc
C
from the free energies of flat interfaces and spherical
micelles using a simple strong-stretching calculation. The most important limitation of
his calculation, as noted in the introduction, is that he did not allow for the possibility
of swelling of the micelle cores. Leibler found µ cmc
C
< µ ∗ C
for sufficiently asymmetric
copolymers, with |f A − 0.5| > 0.19, but µ ∗ C < µcmc C
for more symmetric copolymers, with
0.31 < f A < 0.69. These calculation suggested that it might be possible to produce
76

macroscopic interfaces with γ = 0 using block copolymers with a rather wide range of
values of f A . The nearly symmetric copolymers for which Leibler found µ ∗ C < µcmc C
also,
however, prefer to form monolayers with large radii of curvature, and thus form highly
swollen micelles. Any theory that prohibits swelling of such micelles will overestimate the
free energy of formation of a micelle in a ternary system, and thereby overestimate µ cmc
C .
We show in what follows that, if the micelles of nearly symmetric copolymers are allowed
to swell to their preferred size by emulsifying the B homopolymer, µ cmc
C
always remains
less than or equal to µ ∗ C
, causing the macroscopic interfacial tension to always saturate to
a non-negative value.
4.2.2 Thermodynamics
It is convenient to formulate both the classical thermodynamics of this system and the
corresponding SCFT calculations in grand canonical ensemble. The grand canonical free
energy Φ of any system is given by the difference
Φ = F − ∑ i
µ i M i (4.2)
where F is the Helmholtz canonical free energy, and µ i is the chemical potential and M i
the total number of molecules of species i in the system of interest, for i = A,B,C. For
any macroscopic system of volume V and pressure P, Φ = −PV .
The grand-canonical free energy of a dilute solution of micelles at a specified set of
chemical potentials and a specified number density ρ m of nearly monodisperse micelles
may be expressed as a function
Φ
V = −P h (T,µ) + ρ m
[
Φ m (T,µ) + kT ln(ρ m /ρ 0 m e)] (4.3)
where P h is the pressure of a hypothetical homogeneous (i.e., micelle-free) phase at the
specified set of chemical potentials and Φ m (T,µ) is the extrapolated excess grand-canonical
free energy per micelle of a hypothetical micellar solution with a standard state micelle
77

concentration ρ 0 m . Here and hereafter, we use µ, with no subscripts, to refer to a set of
chemical potentials µ ≡ {µ A ,µ B ,µ C }. The true equilibrium grand-canonical free energy
at a specified T and µ is obtained by minimizing Eqn. 4.3 with respect to variations in the
number density ρ m of micelles yielding
ρ m = ρ 0 m exp [−Φ m (µ,T)/kT] . (4.4)
Substituting this back into Eqn. 4.3 produces a macroscopic pressure P ≡ −Φ/V = P h +
ρ m kT in which the micelles simply contribute an ideal-solution osmotic pressure of ρ m kT.
In Eqn. 4.3, we have not attempted to account separately for different ”species” of
micelles containing different numbers of molecules of C copolymer or solubilized B homopolymer.
The free energy Φ m (T,µ) should thus be understood to be an excess grandcanonical
free energy per a micelle of fluctuating size and composition. The numbers of
copolymer and homopolymer molecules within each micelle can fluctuate by exchanging
molecules between the micelle and the surrounding matrix. Correspondingly, ρ m is the
number density of micelles of any size or composition. This formulation assumes only that
it would be possible to accurately count the total number of micelles in any microscopic
state of a micellar solution, but does not require us to exactly define the distribution of
molecules in a system by deciding how many molecules of each type reside ”within” each
micelle, and how many are part of the surrounding matrix.
Let the excess aggregation number of species i within a micelle, denoted M ex
i , be
defined to be difference between the average number of molecules of species i in a system
(or simulation) that is constrained to contain one micelle and that in a system of equal
volume and equal chemical potentials with no micelles. These excess aggregation numbers
are related to the corresponding excess free energy Φ m by the thermodynamic identity
M i = − ∂Φm (µ,T)
∂µ i
(4.5)
by dropping the superscript ex out for brevity. In an incompressible fluid, the excess
78

aggregation numbers for the three species are related by the fact that the total number
of monomers are conserved regardless of the introduction of a micelle so that they must
satisfy 0 = ∑ i N iM i for i = A,B,C. This definition thus yields positive values for the
copolymer excess M C , and (in an A-rich phase) for the excess M B of solubilized B, but
negative values for the ”excess” M A of matrix homopolymer.
Fluctuations in the excess number of molecules of each type within a polydisperse
ensemble of micelles can be related to the susceptibility of the average aggregation numbers
to changes in the chemical potential by the statistical mechanical identity
〈δM i δM j 〉 = kT ∂M i
∂µ j
= −kT ∂2 Φ m (µ)
∂µ i ∂µ j
(4.6)
in which δM i is the deviation of the ”actual” excess number of molecules of type i in a
micelle from its average value M i . This is a special case of the equilibrium fluctuationresponse
theorem.
The interfacial tension γ of a flat macroscopic interface between A- and B-rich phases is
also most conveniently expressed in grand-canonical ensemble, as an excess grand canonical
free energy per unit area
γ = f int − ∑ i
µ i Γ int
i (4.7)
where f int is excess interfacial Helmholtz free energy per unit area, and Γ int
i
is the interfacial
excess number density of molecules of type i. Interfacial excess quantities, such as f int
and Γ i must be defined using some choice of a Gibbs dividing surface. For a flat interface,
however, the interfacial tension γ is independent of one’s choice of convention for the
position of this surface, because the bulk grand-canonical free energy per unit volume −P
must be the same for two bulk phases separated by a flat interface.
4.2.3 Critical Micelle Concentration
A useful approximation for the critical micelle concentration may be obtained by simply
neglecting the translational entropy of the micelles, and thus removing the logarithmic term
79

in Eqn. 4.3. In this simplified expression, the formation of micelles is favorable whenever
Φ m (T,µ) ≤ 0. The cmc line (b − c) in Fig. 4.1 thus corresponds in this approximation to
the set of chemical potentials for which
0 = Φ m (T,µ) . (4.8)
In this approximation, the thermodynamic analysis of micellization becomes mathematically
identical to the analysis of the limit of solubility of a second phase (e.g., a precipitate)
in a dilute solution.
When the translational entropy of the micelles is taken into account, the critical micelle
concentration ceases to be a sharp phase transition. Instead, in the limit of sufficiently
large micelles, it marks the position of a rapid but continuous crossover, which approaches
the behavior of a true phase transition. For finite micelles, the definition of the cmc
is necessarily somewhat arbitrary. We choose to define the cmc of copolymer to be the
concentration of free copolymers at the chemical potential µ cmc
C
of unimers is equal to that of copolymers within micelles, i.e., for which
for which the concentration
ρ cmc
C = ρ f C = ρ mM C . (4.9)
By substituting this definition into Eqn. 4.4, we find that
0 = Φ m + kT ln(ρ cmc
C /ρ0 m M C) (4.10)
at the cmc, where Φ m (T,µ) is evaluated at the chemical potential for which ρ f C = ρcmc C .
By using Eqn. 4.5 for the derivative of Φ m with respect to µ C , one may easily show
that the value of µ C at which Eqn. 4.10 is satisfied differs from that at which the simplified
condition of Eqn. 4.8 is satisfied by an amount δµ C ≃ kT ln(ρ cmc
C
/ρ0 m M C)/M C . The
corresponding shift in ρ cmc
C
, which arises from our neglect of the translational entropy of a
micelle in Eqn. 4.3, is small in the limit M C ≫ 1 of a large micelle, and corresponds to a
very small fractional change in ρ cmc
C
under all circumstances of interest here.
80

4.2.4 Interfacial Model
We have emphasized above that micelles in a ternary system generally swell by emulsifying
the B homopolymer within the core of the micelle. In a two-phase system coexisting with
a reservoir of bulk B, nearly symmetric copolymers can form highly swollen micelles (or,
equivalently, microemulsion droplets). A core of each micelle is rich in the B homopolymer
and has a composition similar to that of the coexisting B-rich phase surrounded by a
weakly-curved monolayer with a structure also similar to that of a macroscopic saturated
interface. This suggests a view of the free energy of highly swollen micelles as a sum of
bulk and interfacial contribution, in which the latter is a function of curvature.
Now let the phase I and II be the region of an A-rich matrix and a B-rich core,
respectively, separated by a Gibbs dividing surface of area A. The excess free energy
Φ m ≡ Φ+P I V of a spherical or cylindrical micelle with a core region of volume V II = V −V I
may be expressed as a sum
Φ m = (P I − P II )V II + Aγ (4.11)
of the excess interfacial free energy Aγ plus an excess bulk free energy of the core, which
arises from the fact that the grand-canonical free energy density −P is generally different in
the core and in the matrix. The simplified condition for the cmc, which we use throughout
this section, requires that Φ m = 0 at the cmc.
We focus in this chapter on the case of a micellar phase that coexists with a second
excess phase. The condition for mechanical equilibrium between the two macroscopic
phases separated by a curved monolayer with a radius of curvature R is given by
P II − P I = Cγ −
C2 ∂γ
d − 1 ∂C . (4.12)
where the mean curvature C = (d−1)/R with d = 3 for a sphere or d = 2 for a cylinder. If
an interface is flat, the pressure in the micellar A-rich phase be equal to that in the B rich
phase. Note that the local pressure within the matrix of the micellar phase slightly differs
81

from the macroscopic pressure of the whole system as a result of an osmotic pressure ρ m kT
arising from the translational entropy of the micelles, which we ignore here. Because the
chemical potentials in B-rich core of a highly swollen micelle must be the same as those in a
coexisting B-rich macroscopic phase in equilibrium, the local pressure within the core must
equal to that in the B-rich phase which also equals to that in matrix, implying P I = P II .
Combining this requirement with Φ m = 0 at the cmc yields a requirement that
γ = 0 (4.13)
at the cmc of a two-phase system implying interfacial tension within a micelle is essentially
zero. Combining the conditions P I = P II and γ = 0 with the mechanical equilibrium
condition yields the additional requirement that
∂γ
∂C ∣ = 0 , (4.14)
µ
where C is the mean curvature and the derivative is evaluated with all chemical potentials
held constant. Eqn. 4.14 is a statement of the fact that, when the micellar phase coexists
with an excess B-rich phase, the preferred micelle radius is that which minimizes the excess
free energy density of the interfacial monolayer.
Though we focus in this thesis primarily upon two-phase systems, it is interesting to
briefly consider the case of a ternary micellar solution that does not coexist with a second
phase (1φ region in Fig. 4.1). Combining Eqn. 4.11 for Φ m with the mechanical equilibrium
condition, which requires that the micelle radius minimize Φ m , yields
Φ m = A d
[
γ + C ∂γ ]
∂C
(4.15)
where d = 3 for a sphere or d = 2 for a cylinder. Combining this with the simplified cmc
condition, Φ m = 0, yields the requirement that
γ = −C ∂γ
∂C
82
(4.16)

at the cmc. Substituting this back into the mechanical equilibrium condition (Eqn. 4.12)
yields a pressure difference
P II − P I = −
d
d − 1
C2
∂γ
∂C =
d Cγ (4.17)
d − 1
everywhere along the cmc line. In a single-phase solution, the preferred micelle radius is
restricted by a stoichiometric constraint on the amount of available the B homopolymer,
which causes the micelle interface to adopt a smaller radius, or higher curvature, than the
spontaneous curvature adopted along the phase-coexistence line. This leads to a situation
in which ∂γ/∂C > 0, and in which a negative relative pressure P II < P I develops in the
core of the micelle, so as to prevent the micelle from expanding to the radius corresponding
to the spontaneous curvature. Along the cmc line, the micelle radius is thus expected to
steadily decrease, and magnitude of the pressure difference P I −P II to steadily increase, as
one moves away from the emulsification failure line and towards the point c of the binary
A-C system as shown in Fig. 4.1.
4.2.5 Helfrich Theory
Combining the CMC and mechanical equilibrium conditions with the Canham-Helfrich
theory,[15] which is expanded about zero curvature and δµ = µ C − µ ∗ C
while assuming a
spherical interface, yields the conditions
0 = γ = −τC + 1 2 κ +C 2 − Γ ∗ δµ − Λδµ C C (4.18)
0 = ∂γ
∂C = −τ + κ +C − Λδµ (4.19)
1
in which κ + ≡ κ + 2¯κ is the bending rigidity for a spherically curved interface. Eqn. 4.19
yields an equilibrium curvature
C ≃ τ
κ +
(4.20)
83

with corrections of O(τ 2 ). In the same limit, Eqn. 4.18 and Eqn. 4.19 yield a chemical
potential shift
−Γ ∗ δµ = 1 2 κ +C 2 (4.21)
that vanishes in the limit τ → 0, or C → 0.
The interfacial tension γ flat of a macroscopic flat interface between a micellar A-rich
solution and an excess B-rich phase can be obtained by evaluating Eqn. 4.18 with C = 0
and replacing δµ by Eqn. 4.21. This yields a predicted interfacial tension
γ flat = −Γ ∗ δµ = 1 2 κ +C 2 (4.22)
that varies quadratically with the spontaneous spherical curvature C ≃ τ/κ + , and that
vanishes in the limit of a balanced monolayer.
It is reasonable to assume that τ linearly depends on any variable to affect the spontaneous
curvature near at a balance point. For instance, the nonionic surfactant system has
utilized temperature to control the spontaneous curvature of a micelle which linearly varies
close a mean temperature where c o vanishes.[14] Analogously, for the polymeric system the
polymer chain length can be used as a control parameter instead of temperature since a
Flory-Huggins interaction parameter χ is usually a monotonic function of temperature unlike
the complex interaction between water and surfactant. Assuming a constant bending
rigidity κ + ≃ κ + | C=0 and τ ∝ f A − fA bal , Eqn. 4.22 is rewritten as
where a balance point f bal
A
γ flat = (τ ′ ) 2
2κ +
(f A − f bal
A )2 (4.23)
is a volume fraction of A block within block copolymer giving
the zero spontaneous curvature and τ ′ = dτ/df A | C=0 . Eqn. 4.23 completely characterizes
γ flat with two parameters: the ”sensitivity” (τ ′ ) 2
κ +
of a parabolic curve to a curvature and
a position of a zero minimum which is a balance point f bal
A .
84

4.3 SCFT Methodology
We carry out two separate SCFT simulations in grand-canonical ensemble in order to
calculate the excess free energy Φ m of an isolated micelle within an A-rich matrix, and the
corresponding interfacial tension γ of a flat interface at common chemical potentials for
sequentially increasing µ C . To obtain the coexisting conditions appropriate to a two-phase
system the values of µ A and µ B are taken, for each value of µ C , to follow a line a − b of a
phase boundary in Fig. 4.1 adjusting a pressure P = 0 in both phases. The required values
of µ A and µ B are calculated using Flory-Huggins theory, which is the homogeneous limit
of SCFT.
SCFT simulations of a spherical micelle and a macroscopic flat interface are carried
out in a one dimensional unit cell of the corresponding geometry with a finite-difference
discretization. In the micelle simulations, use of grand-canonical ensemble is convenient
because it allows the micelle to automatically adjust its size, composition, and structure
so as to minimize the relevant free energy: Solutions of the SCF equations for the grandcanonical
ensemble may be shown to be extrema of the excess grand free energy Φ m (µ,T)
at fixed chemical potential and temperature.
In the simulation of a flat interface, a mechanically stable interface may be formed
only at a coexistence condition which requires equal pressures in both phases. In grandcanonical
ensemble, however, the position of the interface within the simulation cell is
indeterminate so that small pressure difference can easily cause the instability. To obtain
a unique, stable solution, we impose on the SCFT equations a requirement to specify Gibbs
dividing surface for the interface within the unit cell. The combined set of equations are
solved by Newton-Raphson iteration (as discussed in Chapter 2).
The number density ρ m of micelles at each value of µ C may be calculated from Eqn. 4.4,
in which we approximate Φ m by the excess grand-canonical free energy calculated for
85

an isolated stationary micelle by SCFT. Values for the critical micelle concentration are
obtained from Eqn. 4.9. The standard-state micelle concentration ρ 0 m is taken to be the
hypothetical state in which the excess volume fraction of copolymer contained within
the whole micelles extrapolates to unity, i.e., a dense unswollen micelle such as ρ 0 m =
1/(V C M C ), where V C is the volume occupied by a copolymer chain. Fig. 4.2-Fig. 4.4,
previously illustrated as schematic plots, are actually the results of SCFT calculations for
a system with f A = 0.635, f B χN C = 10, and N A = N B = 100, for which φ cmc
C
= 0.0051.
The crossover at the cmc in the derivatives of µ C and φ f C
with respect to total copolymer
volume fraction at the cmc is continuous, but appears quite sharp, which gives a welldefined
apparent cmc at a concentration that agrees well with that obtained from Eqn. 4.9.
4.4 Results - Micelle Simulations
Fig. 4.5 shows the effect of micelle swelling upon the cmc, by comparing SCFT predictions
for the copolymer volume fraction φ cmc
C
at the cmc of a ternary two-phase system (point
b in Fig. 4.1) and at the cmc of a corresponding binary system (point c in Fig. 4.1) in
which the micelles are necessarily unswollen. Data is shown for three different values of
χN CB = f B χN C , where N CB is the size of the core block, in systems with homopolymers
of N A = N B = 100. Each line in this figure thus corresponds to the systems with a series
of copolymers with the same core block χN CB but the varying corona block length. In
either binary or ternary system, the cmc exhibits a roughly exponential dependence on
χN CB , and a weaker dependence on the corona block length.
It is important to note that the cmc of the ternary two-phase system is always lower
than that of the corresponding binary system as a result of the freedom of the ternary
system to minimize the micelle free energy by emulsifying B so as to relax the curvature of
the copolymer layer. The difference between the cmc of the swollen micelles in a ternary
mixture and that of unswollen micelles in a binary mixture is greatest for symmetric
86

φ c
cmc
10 -1 χ N
10 -2
CB = 10
χ N CB = 12
10 -3
cmc
φ c (binary)
cmc
φ c (ternary)
*
φ c
10 -4
0.5 0.6 0.7 0.8 0.9
f A
χ N CB = 8
Figure 4.5: Copolymer volume fraction φ cmc
C
at the critical micelle concentration of a twophase
ternary system (solid lines), and of a binary mixture of copolymer C in A (short
dashed lines), vs. volume fraction f A of the corona block within the copolymer, for systems
with homopolymer sizes α A χN = α B χN = 10, for three values of χN CB . Also shown is
the copolymer volume fraction φ ∗ C
(dot dashed line) at which the macroscopic interfacial
tension of the two phase system extrapolates to zero. Note that φ ∗ C always exceeds the
cmc φ cmc
C
of ternary two-phase system, but φ ∗ C
drops below the cmc of the corresponding
binary system for nearly symmetric copolymers.
copolymers due to the highest degree of swelling, but is always modest by less than a
factor of two in the present cases.
This modest difference between the cmc’s of swollen and unswollen micelles have a
profound effect on the predictions of interfacial tensions. The dotted-dashed lines in Fig. 4.5
show the saturation concentration φ ∗ C
of free copolymer within the A-rich phase at which
macroscopic interfacial tension extrapolates to zero as shown in Fig. 4.4. The cmc in
the ternary two-phase system thus always remains less than or equal to the saturation
concentration φ ∗ C
, implying that the interfacial tension will saturate to a nonzero (or at least
non-negative) value above the cmc. The predicted cmc of a binary system does, however,
drop below the saturation concentration as f A approaches 0.5, in a manner consistent with
87

R (Å)
500
450
χ N CB = 8
400
10
350
12
300
250
200
150
100
50
0.5 0.6 0.7 0.8
f A
Figure 4.6: Micelle core radius at the cmc for the ternary two-phase system and binary
system considered in Fig. 4.5, vs. volume fraction f A of the corona block. The hollow and
solid symbols represent the radii of unswollen and swollen micelles, respectively.
what Leibler found in a more approximate calculation. The values of f A for which the
saturation concentration and the binary cmc become equal are surprisingly close to the
value of 0.69 obtained by Leibler. However, Leibler’s prediction of a wide range of values of
f A for which φ cmc
C
> φ ∗ C
is clearly an artifact of his two-component model for the micelles.
The convergence of φ cmc
C
and φ ∗ C at f A = 1/2 implies interfacial tension for a flat
interface at and above cmc extrapolates to zero in the limit of the symmetric system.
Fig. 4.6 shows the core radii of micelles in the same set of ternary two-phase systems
and binary systems as those considered in Fig. 4.5. These radii are calculated using a
statistical segment length of 6Å for both monomers. In the ternary two-phase system, the
micelle radius swells dramatically and diverges as f A approaches 1/2.
In Fig. 4.7 we shows the SCFT predictions of macroscopic interfacial tension saturated
at the cmc for the same set of systems as those considered in two previous figures. As
suggested in Eqn. 4.23, the interfacial tension converges to zero at a balance point f bal
A =
88

0.25
γ cmc / γ o
0.2
0.15
0.1
χN CB = 8
10
12
Leibler
0.05
0
0.5 0.55 0.6 0.65 0.7 0.75 0.8
f A
Figure 4.7: Macroscopic interfacial tension γ cmc at the cmc, normalized by the value γ 0
for a bare interface, vs. volume fraction f A of the copolymer corona block, for the same
parameters as those used in Fig. 4.5. Leibler’s prediction based on the unswollen micelle
is shown with a dotted line for comparison.
1/2 in the symmetric limit, in which the micelle radius diverges, and exhibits a parabolic
dependence on f A . For comparison Leibler’s prediction is given with a dotted line showing
a much wider region of f A for which interfacial tension is saturated earlier than the critical
micelle concentration.
The position of a balance point is also affected by the asymmetry of the system. Fig. 4.8
shows the same plot as the previous one, but for the systems with asymmetric homopolymers
(β ≠ 1). In each case, there is a one value of the corona volume fraction f A for which
the interfacial tension extrapolates to zero.
Fig. 4.9 and Fig. 4.10 compare the cmc’s in both binary and ternary systems calculated
by SCFT with those in the binary mixtures by simplified analytic models. The theory of
Leibler, Orland, and Wheeler (LOW) [5] is based on a simple strong stretching calculation
of the free energy of a copolymer micelles in a single A homopolymer matrix. The
89

0.6
γ sat /γ o , γ flat /γ o
0.5
0.4
0.3
0.2
β = 0.5
1
2
0.1
0
0.3 0.4 0.5 0.6 0.7 0.8
f A
Figure 4.8: Macroscopic interfacial tension γ cmc at the cmc, normalized by the bare tension
γ 0 , vs. copolymer corona volume fraction f C for system with f B χN = 10 and various values
of the ratio β of copolymer molecular weights.
10 0 8 9 10 11 12 13 14 15 16
10 -1
f A = 0.6
SCFT (b)
SCFT (t)
LOW
10 -2
φ c
cmc
10 -3
10 -4
10 -5
χ N CB
Figure 4.9: Dependence of critical micelle concentration φ cmc
C
on χN BC = f B χN for systems
with f A = 0.6 and α A χN = α B χN = 10. Symbols show predictions for the ternary
two phase system by SCFT (filled triangles) and predictions for a binary system by SCFT
(filled circles) and by the approximate strong LOW theory (open triangles). The dashed
line is the function Ce −χN BC, with C = 376.
90

0.014
0.012
0.01
SCFT (b)
SCFT (t)
Leibler
LOW
0.008
φ cmc
0.006
0.004
0.002
0
0.5 0.6 0.7 0.8
f A
Figure 4.10: Comparison of SCFT predictions for copolymer cmc φ cmc
C
of a ternary twophase
system (filled triangles) and a binary system (open triangles) to the strong stretching
predictions for the binary system by Leibler (dot dashed line) and LOW (solid lines) for
the systems considered in Fig. 4.5.
density profiles are assumed uniform in the core and corona regions, but some penetration
of homopolymer into the core and corona is allowed. The theory denoted ’Leibler’ is a
more simplified version of the LOW theory that ignores the effect of the homopolymer
penetration and the translational entropy of a micelle. [11] These analytic theories capture
the dominant exponential dependence of the cmc on χN CB at the value of f A = 0.6 in
Fig. 4.9, but exhibit much too weak a dependence on f A in Fig. 4.10.
4.5 Results - Helfrich Theory
The interfacial tension reduction is thermodynamically inhibited by the onset of micellization.
Therefore, interfacial tension γ cmc saturated at the CMC (Fig. 4.4) should be
identical to γ flat predicted by the Helfrich membrane theory of the bending elasticity of
a micelle coexisting with two bulk phases. Fig. 4.11 tests the validity of Eqn. 4.23 in two
ways. First, a parabolic curve of Eqn. 4.23 constructed with two parameters τ ′ and κ +
91

0.7
γ / γ o
0.6
0.5
0.4
0.3
γ flat
γ cmc
τ 2 /2κ +
0.2
0.1
0
0.5 0.55 0.6 0.65 0.7 0.75 0.8
f A
Figure 4.11: The prediction of γ/γ o from direct micelle simulation (hollow circle) and
from Helfrich bending elasticity (solid circle) as a function of f A . They are consistent for
f A < 0.55 but start to deviate for larger f A . Surprisingly, the quadratic function (solid
line) of γ/γ o (Eqn. 4.23) with τ ′ and κ + evaluated at C = 0 accurately fits the entire
results from the direct micelle simulation.
evaluated at a balance point f bal
A = 1/2 is compared with τ2 /2κ + in combined Eqn. 4.20
and Eqn. 4.22 directly obtained in each membrane simulation. The two approaches are
reasonably consistent for f A < 0.55 but start to deviate thereafter. Second, interestingly
Eqn. 4.23 agrees fairly well with γ cmc from the direct micelle simulation, that is, the
quadratic relationship between γ and f A can be extended to the system with highly asymmetric
block copolymer. The Helfrich membrane theory, however, fails even for moderately
asymmetric systems. Both Fig. 4.7 with the symmetric homopolymers and Fig. 4.8 with
the asymmetric homopolymers clearly show that γ cmc remarkably follows the quadratic
curvature dependence of the Helfrich theory.
Fig. 4.12 compares the radii of swollen micelles calculated by both micelle and Helfrich
theories as a function of f A . The radius of a micelle is defined as R = ( 3
4π (M BV B + M C V C )) 1 3
in a micelle simulation, or as the spontaneous spherical radius of curvature of R = 2κ + /τ in
92

R (Å)
1000
900
800
700
600
500
400
300
200
100
unswollen micelle
swollen micelle
2 κ + / τ
0.5 0.6 0.7 0.8
f A
Figure 4.12: The comparison of micelle radii obtained from direct micelle simulation (hollow
circle) and from Helfrich bending elasticity (solid circle) as a function of f A . They are
consistent for f A < 0.55 but the latter starts to underestimate the radii for larger f A .
the Helfrich membrane theory, respectively. The systems are identical to those in Fig. 4.11.
One can easily notice that the radii determined by the Helfrich membrane theory are bigger
than those of swollen micelles by the micelle simulation for f A < 0.55 because the simplified
cmc condition Φ m = 0 used in the Helfrich theory causes micellization in the ternary
systems to occur at a few kT higher energy states than those at the cmc in the micelle
simulation at which the translational entropy of micelles are considered. However, the radii
become smaller than even those of unswollen micelles for highly asymmetric copolymers.
This is consistent with the observation that, in Fig. 4.11 and in Chapter 2, the Helfrich
membrane theory is accurate only for f A < 0.6.
We can observe more interesting features of a block copolymer monolayer in the plots
of [τ ′ ] in Fig. 4.13 and τ ′2
8κ + γ o
in Fig. 4.14 as a function of χN CB . Here γ o is bare interfacial
tension of homopolymers in the absence of block copolymer and [τ ′ ] is evaluated
by changing only the corona A block while fixing the core block so as to keep the cmc
93

2.5
2
1.5
α = 2
1
0.7
0.5
0.25
[τ’]
1
0.5
0
0 5 10 15 20 25 30
χ N CB
Figure 4.13: The normalized slope [τ ′ ] as a function of χN in a range of 0.25 ≤ α ≤ 2
5
4.5
(τ’) 2 /(8κ + γ o )
4
3.5
3
2.5
2
1.5
1
0.5
α = 2
1
0.7
0.5
0.25
0
0 5 10 15 20 25 30
χ (N CB )
Figure 4.14: The prefactor of γ flat ,
τ ′2
8κ + γ o
, normalized by a bare interfacial tension γ o as a
function of χN CB in a range of 0.25 ≤ α ≤ 2
94

nearly constant. Both quantities directly indicate how sensitively γ flat and c o respond to a
small curvature induced by f A ≠ f bal
A . If one assumes γflat is an exact harmonic function
in a range of 0 ≤ f A ≤ 1 which satisfies γ flat = γ o at f A = 0 and f A = 1,
τ ′2
8κ + γ o
= 1
becomes a criterion to decide the relative ’flatness’ of a parabolic γ flat curve. When the
length of the homopolymer is equal to or less than that of the copolymer (α ≤ 1), both
τ ′2
8κ + γ o
and [τ ′ ] always continuously decrease to zero as the system approaches the Lifshitz
point (χN CB ) L = (1 + 2α 2 )/α. However, for α > 1 these terms sharply diverge at low
χN CB implying the interface becomes unstable because small deviation from a balance
point can lead to a dramatic increase of the interfacial tension, which makes it harder to
compatibilize the large homopolymer with short copolymer. With small block copolymers,
say χN CB < 7, the parabolic γ flat curve is greatly affected by the relative length of homopolymer
to block copolymer but it becomes insensitive to α with larger copolymers due
to the increasing rigidities of interfaces as shown in Chapter 2.
4.6 Other Possible Phenomena
In this chapter, we focus primarily upon the formation of swollen spherical micelles and its
relation to interfacial tension in ternary two-phase systems. There are, however, several
other possible phenomena that should be considered.
Cylindrical Micelles It is possible for dilute solutions of block copolymers to form
cylindrical, rather than spherical, micelles. In binary systems of A homopolymer and
AB copolymer, analytic strong stretching calculations by Mayes and de la Cruz [16] and
numerical SCFT calculations by Matsen [17], and Janert and Schick [18] have predicted the
formation of either cylindrical micelles [16] or highly swollen periodic hexagonal mesophases
[17, 18] in some systems of symmetric and slightly asymmetric copolymers.
In an analysis of the behavior of highly swollen micelles in ternary systems, which was
based upon the Helfrich theory of weakly curved saturated interfaces, Safran and Turkevich
95

[19] found that spherical micelles are always preferred over cylinders along the emulsification
failure line. In this theory, a transition from spherical to cylindrical micelles can,
however, be induced in a homogeneous micellar phase by decreasing the volume fraction
of B. Within the context of the Helfrich theory the preference for spherical micelles is a
result of a contribution of a negative Gaussian rigidity ¯κ, which favors the formation of
spherical surface over a cylindrical curvature with the same mean curvature.
The Helfrich theory is valid only for systems containing monolayers with low spontaneous
curvatures, which tend to form highly swollen micelles near the emulsification
failure line. In the opposite limit of highly asymmetric copolymers, however, we also expect
spherical micelles, even in a binary system, as a result of packing constraints. These
considerations suggest that spherical micelles will always be preferred over cylindrical micelles
in a two-phase system.
The above considerations also suggest that addition of B homopolymer to a binary
system that forms cylindrical micelles must generally cause a transition from cylindrical to
spherical micelles somewhere along the line of critical micelle concentrations that extends
from the binary system to the emulsification failure line. We have not systematically
pursued this here. As an example, however, we have compared the cmc’s for cylindrical and
spherical micelles in a single binary system with f A = 1/2, χN C = 20 and α A = α B = 1/2,
for which we find that spherical micelles are preferred over cylinders even in absence of
B homopolymer. Cylindrical micelles would presumably be more favorable for symmetric
copolymers with larger values of χN CB or α, for which the corona block would be less
strongly swollen by homopolymer. [16]
Lamellar Phases: The model of a non-interacting spherical micelle discussed above predicts
an infinite micelle radius in a system with symmetric copolymer and homopolymers.
This prediction is a result of our implicit assumption that excluded volume interactions
between swollen micelles remain irrelevant even as their radii diverge. Systems with bal-
96

anced surfactants are expected to instead form either a lamellar phase or a bicontinuous
microemulsion.
Monolayers of symmetric block copolymers with molecular weights less than that of either
homopolymer are known to exhibit a weak entropic attraction when separated by
distances for which opposing copolymer brushes slightly overlap. This attraction can
prevent swelling of the lamellar spacing, and instead possibly lead to a broad region of
three phase coexistence of a collapsed lamellar phase with two phases of essentially pure
homopolymers.[20] Thompson and Matsen [21] have used numerical SCFT to study the
effective interaction between symmetric monolayers within a ternary lamellar phase, and
identified a line of values of α as a function of χN at which the interaction between monolayers
changes from effectively repulsive (at lower α) to attractive (at higher α). This
critical value of α decreases very slowly with increasing χN, varying from α ≃ 1.3 near
the critical point to α = 0.8 at χN > 100. Symmetric systems with α < 0.8 are thus
not expected to form a collapsed lamellar phase. The attraction between brushes that
can cause the collapse of a lamellar phase may analogously cause ordered dense crystals
of spherical or cylindrical micelles under similar conditions, but the interactions between
spheres and cylinders have not been so carefully studied.
Bicontinuous Microemulsion: When the interactions between monolayers in a lamellar
phase are repulsive, the lamellar spacing may increase with decreasing the surfactant
concentration, eventually leading, within the context of SCFT, to an unbinding transition
at which the lamellar spacing diverges at a nonzero surfactant volume fraction. Systems
for which SCFT predicts such an unbinding transition are are, however, likely to form a
bicontinuous microemulsion phase over a narrow range of surfactant concentrations near
the predicted unbinding concentration. The bicontinuous microemulsion phase is believed
to be stabilized by interfacial fluctuations that cannot be correctly described by SCFT.
[22] This, however, lies far beyond the scope of the present thesis.
97

4.7 Conclusion
In this chapter, we formulated micellization of block copolymer in a grand canonical ensemble
and connect it to Helfrich bending elasticity of a block copolymer monolayer in a
micelle for the two-phase ternary polymer system. While a micelle radius is simply determined
by a stoichiometry of available B and C molecules in a single phase system, a micelle
coexisting with the excess B phase should swell until a spontaneous spherical curvature
is accomplished at equal pressure across an interface along the emulsification failure line.
Also, due to the slightly reduced excess free energy of a micelle, the interfacial saturation
is always preempted by micellization producing non-negative interfacial tension.
In the limit of highly swollen micelles, a weakly curved interface within a micelle
is successfully described by Helfrich bending elasticity of a block copolymer monolayer.
Combining with the simplified cmc condition in which the translational entropy of a micelle
is ignored, the macroscopic interfacial tension saturated at the cmc was found to have a
quadratic dependence on the spontaneous spherical curvature and vanish at a balance
point, which was verified with both micelle simulation and the Helfrich membrane theory.
These parabolic curves of saturated interfacial tension were significantly affected by the
relative length of homopolymer to block copolymer if block copolymers are small but start
to become nearly constant with increasing copolymer lengths in the context of [τ ′ ] and
τ ′2
8κ + γ o
. As long as α < 1, both terms continuously decrease to zero at the Lifshitz point
but sharply diverge for α > 1.
Although here we only focus on the formation of spherical micelle, this approach may
be extended to other structures of micelles and to phase transitions between them.
98

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[1] Linc, H.; Favis, B. D.; Yu, Y. S.; Eisenberg, A. Macromolecules 1999, 32, 1637 .
[2] Macosko, C. W.; Guegan, P.; Khandpur, A. K.; Nakayama, A.; Marechal, P.; Inoue, T.
Macromolecules 1996, 29, 5590
[3] Noolandi, J.; Hong, K. M. Macromolecules 1982, 15, 482; Noolandi, J.; Hong, K. M.
Macromolecules 1984, 17, 1531
[4] Shull, K. R.; Kramer, E. J. Macromolecules 1990, 23, 4769
[5] Leibler, L.; Orland, H.; Wheeler J. C. J. Chem. Phys. 1983, 79, 3550
[6] Whitmore, M. D.; Noolandi, J. Macromolecules 1985, 18, 657
[7] Mathur. D.; Hariharan, R.; Nauman, E. B. Polymer 1999, 40, 6077
[8] Lent, B. V.; Scheutjens, J. M. H. M. Macromolecules 1989, 22, 1931; Evers, O. A.;
Scheutjens, J. M. H. M.; Fleer, G. J. Macromolecules 1990, 23, 5221;
[9] Linse, P.; Malmsten, M. Macromolecules 1992, 25, 5434
[10] Shull, K. R. Macromolecules 1993, 26, 2346
[11] Leibler, L. Makromol. Chem., Macromol. Symp. 1988, 16, 1
[12] Palmer, K. M.; Morse, D. C. J. Chem. Phys. 1996, 105, 11147
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[13] Shull, K. R.; Kellock, A. J.; Deline, V. R.; MacDonald, S. A. J. Chem. Phys 1992,
97, 2095
[14] Strey, R. Coll. Ploym. Sci. 1994, 272, 1005
[15] Helfrich, W. Z. Naturforsch. C 1973, 28, 693
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[22] Fredrickson, G. H.; Bates, F. S. J. Polym. Sci.: Part B: Polym. Phys. 1997, 35, 2775
100

Chapter 5
Transport of Block Copolymer
Surfactant Between Two
Homopolymer Phases
The measurements of interfacial tension between viscous homopolymers are often seriously
affected by slow kinetics in the system when block copolymer is selectively added to either
homopolymer phase. Although equilibrium interfacial tension is uniquely determined by
thermodynamics, measured interfacial tension can depend on the actual interfacial concentration
of block copolymer at steady state during experiments. Thus the purpose of this
chapter is to provide the comprehensive understanding about the effect of limited transport
of block copolymer on interfacial tension depending on the various initial conditions.
It should be stated that the primary authorship of this chapter belongs to my advisor,
Prof. Morse, but I have included it here because the analysis of kinetics associated with
transport of block copolymer in Chapter 6 and 7 has used the results given in this chapter.
Here we consider the transport of AB block copolymer surfactant that is initially
homogeneously mixed throughout the matrix of the nearly pure homopolymer A (phase
I) into the drop of the initially pure homopolymer B (phase II). This configuration is
closely related to the initial state of the interfacial tension measurement in a spinning drop
tensiometer, which is described in the following two chapters.
101

1
0.8
c f (z) / c o
0.6
0.4
II
c f
I,int
I
0.2
c f
II,int
0
z
0
Figure 5.1: A schematic diagram of concentration profile of surfactant perpendicular to an
interface z = 0 below the cmc.
The problem may involve dynamics over several time regimes. Initially, some relatively
short time τ int is required to build up a block copolymer monolayer at the interface. Once a
monolayer is established, a steady state is achieved at all times τ int ≪ t ≪ τ eq if we assume
that the rate of accumulation of copolymer at the interface is negligible compared to the
rate of flux to the interface from phase I or from the interface into phase II. At much
longer time t > τ eq copolymer finally diffuses to the center of the drop and an equilibrium
concentration of copolymer is obtained throughout the drop. We focus primarily in what
follows upon the copolymer diffusion in the intermediate time regime τ int ≪ t ≪ τ eq with
a negligible rate of accumulation at the interface, and negligible resistance to transport
through the interface. In this period, the problem becomes equivalent to that of diffusion of
a substance from one semi-infinite medium from one into another across an essentially flat
boundary. [1] To discuss this time regime, we thus consider the one-dimensional problem
of copolymer transport across a flat interface at z = 0 and consider the evolution of the
copolymer concentration c(z) along a direction perpendicular to the interface. Phase I is
102

taken to occupy the half space z > 0 with an initial copolymer concentration c o , while
phase II occupies the half space z < 0 with a vanishing initial concentration, as shown in
Fig. 5.1.
To discuss transport under conditions in which micelles may be present, we must distinguish
at each point a local concentration c f (z) of molecularly dissolved ”free” copolymer
from a local concentration c m (z) of copolymer in micelles, such that c(z) = c f (z) + c m (z).
We adopt a slightly simplified view of micellization in which the concentration of free
molecules is equal to the critical micelle concentration (cmc), c f (z) = c c , where c c is the
cmc in the phase of interest, at any point with c(z) > c c . This description assumes both
that the equilibrium between free copolymer and micelles can be adequately described as
a simple limit of solubility (which can be shown to be equivalent to neglecting the comparatively
small translational entropy of the micelles), and that the formation and dissolution
of micelles occur essentially instantaneously, so that local equilibrium is always maintained.
The assumed constancy of c f (z) in regions in which micelles are present has an important
implication for transport: There can be no diffusive flux of free molecules in regions with
micelles since there is no gradient in the concentration of free copolymer. All transport
of copolymer must thus occur by micellar diffusion alone in regions with c(z) > c c and by
diffusion of free molecules alone in regions with c(z) < c c .
Let D I f
and DII
f
be the diffusion coefficients of free copolymer in phases I and II,
respectively, and D I m
and DII m
be the diffusivities of micelles in the corresponding phases.
Let c I c
and cII c
be the critical micelle concentrations in phases I and II, and c I f,int
and cII
f,int
be the concentrations of free copolymer infinitesimally close to the interface in phases I
and II, respectively. Let K be the partition coefficient for free copolymers in either phase,
defined by c I f,int = KcII f,int
, so that the copolymer is more soluble in phase I if K > 1.
In what follows we consider three cases: First, we review as background the relatively
simple case in which there are no micelles in either phase. Next, we discuss the case in
103

which c o > c I c , so that micelles are initially present in phase I, and in which the value µI c
of the copolymer chemical potential required to form micelles in phase I (i.e., the value
of µ at the cmc in phase I) is lower than the corresponding value µ II
c
in phase II. The
condition µ I c < µII c
is equivalent to the condition c I c
/K < cII c . In this case, micelles do not
appear in phase II, and the mechanism by which copolymer is transported from phase
I into phase II is shown to depend to depend upon the magnitude of the diffusivity of
micelles in phase I relative to the diffusivity of free copolymer in phase II. Finally we
consider the case in which c o > c I c
but µII c
< µ I c , or cII c
< c I c /K, in which micelles are
initially present only in phase I, but in which micelles would appears only in phase II in
thermodynamic equilibrium.
5.1 No Micelles
When c o ≪ c c in both phases, so that no micelles are present in either phase, transport
occurs by Fickian diffusion of dissolved copolymer. In this case we thus consider the
evolution of the copolymer concentration c(z) for a one-dimensional diffusion across an
interface, in which the concentrations on either side of the interface satisfy a boundary
condition c I f,int = KcII f,int , where cI f,int = c f(z = 0 + ) and c II
f,int = c f(z = 0 − ).
It is known[1] that there exists an analytical solution to this problem in which the
concentration c(z) changes with time everywhere except at the interface, where the c I f,int
and c II
f,int
are actually independent of time. To confirm this, we assume a time-independent
concentration on either side of the interface, and consider the well known error-function
solution for free diffusion from or into a semi-infinite slab from an interface held at fixed
concentration. For the problem at hand, in which the copolymer concentration approaches
c o deep in phase I and vanishes deep in phase II, this proposed solution yields a total flux
104

J I (t) to the interface and J II (t) from the interface that both decrease as 1/ √ t, given by
√
Df
I J I (t) =
πt (co − c I f,int )
√
Df
II
J II (t) =
πt cII f,int . (5.1)
Requiring that J I (t) = J II (t) yields a constant interfacial concentration of copolymer in
phase I
c I f,int =
co
1 + P
(5.2)
where
√
P =
√ DII f
Df
I
1
K . (5.3)
For P ≫ 1, the interfacial concentration thus stays at the much lower concentration than
that far from an interface, c I f,int ≪ co . If the diffusion occurs into a finite drop (rather than
a semi-infinite slab) the concentration remains at this value until the entire drop reaches its
equilibrium concentration at t ≃ τ eq . Conversely, for P ≪ 1, the interfacial concentration
c I f,int reaches a value very close to co , even at times much less than τ eq . Therefore, a high
interfacial concentration may be quickly achieved for systems in which the solubility is
much higher in the matrix than in the drop, so that K ≫ 1, or in which the diffusivity in
the matrix is much higher than that in the drop, so that D I f ≫ DII f .
5.2 Micelles in Matrix, c I c < c II
c K
We next consider the case c o > c I c , in which micelles are initially present in phase I, but
in which µ I c < µII c , or cI c /K < cII c , so that the formation of micelles in phase I is preferred
over the formation of micelles in phase II. In this case, micelles never exist in phase
II. Two qualitatively different situations can arise in phase I: One possibility is that
c(z) is greater than c I c throughout phase I, so that c f(z) = c I c
for all z > 0 at all times
of interest. In this case, transport in phase I is governed by micellar diffusion alone, as
105

3
c m
o
c f (z) / c c
I , cm (z) / c c
I
2
1
II
I
c c
I
0
z
0
Figure 5.2: A schematic diagram of concentration profile of surfactant and micelle perpendicular
to an interface z = 0 without exclusion zone (Q < 1) above the cmc.
a result of the absence of a gradient in the concentration of free molecules. The other
possibility is that there exists a region near the interface in which c(z) < c I c, from which
micelles are completely excluded. This will be referred to in what follows as an ”exclusion
zone”. Whether an exclusion zone is created depends upon a combination of kinetic and
thermodynamic properties of both phases, as discussed below.
5.2.1 Micellar Diffusion (No Exclusion Zone)
We first consider the case in which there is no exclusion zone, so that c(z) > c I c for all z > 0.
In this case, surfactant in phase I is carried all the way to the interface by micellar diffusion
alone. Since the interfacial concentration c I f,int is equal to cI c , the interfacial concentration
in phase II must be c II
f,int = cI c/K as shown in Fig. 5.2. The flux into phase II is thus
given again by the solution for the previous case c(z) < c I c, which gives
√
Df
II c I c
J II (t) =
πt K
(5.4)
106

for a system with an interfacial concentration c II
f,int /cI c /K. Postulating an analogous concentration
profile for micelles in the micellar phase, controlled by a diffusivity D I m of a
micelle with a time-independent interfacial concentration c m (0 + ) in phase I yields the flux
J m (t) =
√
D I m
πt [co m − c m(0 + )] (5.5)
that also decreases as 1/ √ t. By equating the prefactors of 1/ √ t in Eqns. (5.4) and (5.5),
we find
c m (0 + )
c o m
= 1 − Q (5.6)
where
Q ≡
√
D II
f
D I m
c c
c o m
1
K . (5.7)
A non-negative interfacial concentration, c m (0 + ) > 0, is obtained only for Q < 1. This
requirement Q < 1 is equivalent to the requirement that the flux that would be obtained
by micelle diffusion to an absorbing boundary, with c m (0 + ) ∼ 0, must exceed the rate
of leakage of copolymer into phase II, as given by Eq. (5.4). Note that the condition
involves both the micellar diffusivity D I m
in phase I and the molecular diffusivity DII
f
in
phase II, but not molecular diffusivity D I f
in phase I, since no molecular diffusion occurs
in phase I. If the condition Q < 1 is satisfied, the concentration of free copolymer near
the interface can thus reach the cmc over times much faster than the equilibrium time τ eq
needed to fill the drop, thus rapidly establishing the highest interfacial coverage and the
lowest interfacial tension that thermodynamics can allow.
5.2.2 Molecular Diffusion (Exclusion Zone)
If Q > 1, micelle diffusivity alone cannot provide the flux of copolymer into domain II
needed to maintain a concentration c I f,int = cI c
at the interface. As a result, the concentration
c f (z) of copolymer near the interface must drop below the cmc, and cause the
formation of a region near the interface from which micelles are completely excluded. We
107

3
h (t)
d m (t)
c m
o
c f (z) / c c
I , cm (z) / c c
I
2
1
II
I
c c
I
0
z
0
Figure 5.3: A schematic diagram of concentration profile of surfactant perpendicular to an
interface z = 0 for Q > 1 above the cmc. A solid line is for the case of S ≫ 1 and a dotted
line for the case of S ≪ 1.
may thus divide phase I at any time t into an ”exclusion zone” 0 < z < h(t) in which
c f (z) < c c and c m (z) = 0, and a micellar zone z > h(t) in which c f (z) = c c and c m (z) > 0.
A schematic of this behavior is shown Fig. 5.3. As shown there, the micelle concentration
c m (z) must vanish along the moving edge z = h(t) of the exclusion zone. Within the
micellar zone z > h(t), there will generally be a ”depletion region” of some width d m that
precedes the edge of the exclusion zone, in which the concentration c m (z) is significantly
less than c o m, within which micellar diffusion is significant. For copolymer that starts in
a micelle to reach the interface between phases I and II, it must first be transported by
micellar diffusion to the moving edge of the diffusion zone, where the micelle dissolves, and
then undergo molecular diffusion across the exclusion zone.
When an exclusion zone exists, we are no longer able to find an exact analytic solution
to the problem. Significant insight can be gained, however, by analyzing two possible
limiting cases:
108

One limit is the situation in which the width d m of the depletion region is very narrow
compared to the width h(t) of the exclusion zone. This corresponds to the case of negligible
micelle diffusivity. In this limit, the presence of the narrow depletion region is irrelevant,
and the magnitude of the flux of copolymer to the interface at z = 0 is controlled by
the rate at which essentially immobile micelles that dissolve at the edge z = h(t) of the
exclusion zone can diffuse across the exclusion zone.
The opposite limit occurs when the width h(t) is very narrow compared to the width of
the depletion region. The edge of the exclusion zone thus acts in this case as an essentially
stationary absorbing boundary, along which c m (z) = 0, along a surface that remains very
near z = 0. In this case, the micelle concentration c m (z) is thus well approximated by the
standard solution for diffusion of an initially homogeneous concentration of micelles to an
absorbing boundary, in which the width of the depletion region increases as √ Dm I t. In
this case, the overall flux of copolymer to the interface is controlled by micelle diffusivity,
and the presence of a very narrow exclusion zone is essentially irrelevant. We will show
in what follows that this limit is obtained in the limit of a vanishing cmc, in which the
diffusion of free copolymer is limited by the limited solubility of free copolymer.
Limit i) Negligible Micelle Diffusivity
We first consider the case of negligible micelle diffusivity D I m
→ 0, in which the width of
the depletion zone goes to zero. In this limit the flux of copolymer to the interface, which
occurs by diffusion of free molecules across the exclusion zone, is provided by dissolution of
micelles at the boundary z = h(t) between the depletion and micellar zones. The micellar
zone is thereby eaten away, in a process analogous to that which occurs (in the absence of
convection) when a soluble solid is immersed in solvent.
In this limit, the flux of free copolymers at z = h(t) must equal the rate
J I (t) = c o dh(t)
m
dt
109
(5.8)

at which micelles are dissolved along the moving boundary h(t). If we make a quasi-steady
state approximation in which we neglect accumulation of copolymer within the depletion
zone, and thus approximate c f (z) within the depletion zone by a linear function of z, the
diffusive flux throughout this region is
J I (t) = Df
I c I c − c I f,int (t)
h(t)
. (5.9)
By considering a candidate solution in which c I f,int
(t) is postulated to be independent of
time, and equating the above two expressions for the flux, we obtain an ordinary differential
equation for h(t), which has a solution
h(t) =
√
2D I f t (cI c − cI f,int )/co m (5.10)
Combining this with Eq. (5.8) yields a flux
√
Df
I J I (t) =
2t (cI c − c I f,int )co m . (5.11)
Note that in this case the flux again decreases with time as 1/ √ t, but that the prefactor
is a nonlinear function of c c , c I f,int , and co m.
Matching the prefactor of 1/ √ t in the r.h.s. of Eq. (5.11) to the corresponding prefactor
on the r.h.s. of Eq. (5.1) for the flux into domain II again yields a solution with a time
independent value for c I f,int
. In this case, we find that the ratio
x ≡ c I f,int /cI c , (5.12)
satisfies a quadratic equation
0 = Rx 2 + x − 1 (5.13)
in which
R ≡ 2 π
D II
f
D I f
c I c
c o m
1
K 2 . (5.14)
110

The solution x = (−1 + √ 1 + 4R)/2R has the limits
x =
{ 1 − R for R ≪ 1
R −1/2 for R ≫ 1
. (5.15)
At times t ≪ τ eq , the systems with essentially immobile micelles and R ≫ 1 thus exhibit an
interfacial concentration c I f,int ≪ cI c much less than the cmc as the result of the relatively
high diffusivity in the domain II. In the extreme limit of D II
f
→ ∞, c I f,int
= 0 at an
absorbing boundary. On the other hand, the systems with R ≪ 1 exhibit c I f,int ≃ cI c .
Both cases lead to apparently time-independent values for the interfacial concentration,
and thus a time-independent interfacial tension, at times t ≪ τ eq .
When micelle diffusivity is small but nonzero, the width d m (t) of the depletion region
may be estimated as follows: The gradient dc m /dz of the micelle concentration at
z infinitesimally greater than h(t) may be calculated by matching the flux Dmdc I m /dz of
micelles to edge of the exclusion zone to the rate at which micelles dissolve along this edge,
which is equal to the total flux J I (t). By approximating dc m /dz ≃ c o m/d m (t) and matching
fluxes we obtain a width
d m (t) ∼ Sh(t) ≃ √ √
S Dmt I (5.16)
where S is a dimensionless ratio
S ≡
D I mc o m
D I f (cI c − c I f,int ) . (5.17)
The approximation considered above, in which we neglected micelle diffusivity, is value
when S ≪ 1. In this case, d m (t) remains much less than both the width h(t) of the
exclusion zone and the width √ Dm I t of the region that would be depleted in the limit of
an infinitesimal exclusion zone. In the opposite case S ≫ 1, we obtain an exclusion zone
that is much narrower than the depleted region, which is discussed below.
111

Limit ii) Negligible Solubility
We now consider the situation h(t) ≪ d m (t). This is found to occur when S ≪ 1. Since
the micelle diffusivity D I m is always much less than the free molecule diffusion D I f , the
limit S ≫ 1 may be obtained only in systems in which the cmc c I c, and thus the difference
c I c − cI f,int
, is much less than the total concentration. This limit is thus best understood in
practice as a limit of very low molecular solubility, in which the exclusion zone in which
c f (z) < c I c must become very narrow in order to produce a concentration gradient large
enough to support the mass flux provided by micellar diffusion.
In this case, the edge of the exclusion zone acts as a nearly stationary absorbing
boundary along a surface h(t) that stays very near z = 0. In the limit h(t) → 0 of interest,
c m (z) may thus be approximated by the error-function solution for diffusion of an initially
homogeneous distribution to an absorbing boundary at z = 0. This yields a depletion
region of width
d m (t) ∼
√
D I m t (5.18)
and a total flux
J I (t) =
√
D I m
πt co m . (5.19)
The concentration c I f,int
of free molecules at the interface may be obtained by matching
Eq. (5.19) to Eq. (5.4) for the flux into phase II. This yields a value
c I f,int
c I c
= 1 Q , (5.20)
for the dimensionless ratio x = c I f,int /cI c . A reasonable approximation for x for Q > 1 and
any value of S might be obtained by using the larger of the values for x obtained from
Eq. (5.20) and from the solution of Eq. (5.13), or by an appropriate interpolation between
these values.
112

5.3 Micelles in Wrong Matrix, c II
c < c I c /K
Finally, we consider the situation in which we put micelles in what is, from a thermodynamic
point of view, the wrong phase: We consider the situation in which copolymer
is initially mixed into phase I with a concentration c o > c I c, but in which µ II
c
< µ I c, or
c II
c
< c I c /K so that the global equilibrium state is one in which appear only in phase II.
This might occur, for example, if one mixed a diblock copolymer with f A < 1/2 into a
phase of nearly pure A, forcing the formation of inverted micelles in which the B core block
is longer than the A corona, and tracked the diffusion of the copolymer into phase B, in
which the longer B block becomes the corona.
In this case, the concentration c II
f,int
in phase II near the interface can never exceed
the critical micelle concentration c II
c . As a result, the concentration of free molecules near
the interface in phase I can never exceed c II
c K. Since, by assumption, c II
c K < c I c, the
interfacial concentration of free molecules in phase I must remain below c I c , forcing the
formation of an exclusion zone. In this case there must thus always an exclusion zone near
the interface in phase I.
Two possibilities remain: One possible outcome is an interfacial concentration c II
f,int <
c II
c . In this case no micelles appear in phase II. (More precisely, in this case no micelles
can appear in phase II until times t > τ eq required to for copolymer to diffuse over the
entire volume of phase II). The other possibility is one in which c II
f,int = cII c , and in which
micelles exist in phase II near the interface.
When no micelles are present in phase II, the mathematical model needed to describe
transport is identical to that developed above to describe the situation in which there is an
exclusion zone in phase I (Q > 1) and no micelles in phase II, but in which c II
c
> c I c/K:
The only distinction between the two situations is the value of c II
c , which is irrelevant until
micelles actually appear in phase II.
113

4
h (t)
d m (t)
c m
o
c f (z) / c c
I , cm (z) / c c
I
3
2
1
c c
II
II
I
c c
I
c c
II K
0
z
0
(a)
4
h (t)
d m (t)
c m
o
c f (z) / c c
I , cm (z) / c c
I
3
2
1
c c
II
II
I
c c
I
c c
II K
0
z
0
(b)
Figure 5.4: A schematic diagram of concentration profile of surfactant when surfactant is
initially added to the wrong phase (I) (a) when Eqn. 5.21 and Eqn. 5.22 are satisfied (b)
not satisfied.
114

Micelles appear in phase II near the interface if the model that neglects this possibility
predicts an interfacial concentration c II
f,int ≥ cII c , or c I f,int ≥ cII c K. Since we do not have
an exact expression for c II
f,int
for arbitrary values of S, we can discuss only the limiting
behaviors. In the limit S ≪ 1 of vanishing micelle diffusivity, we find by setting c II
f,int ≥ cII c
in Eqn. 5.13 that micelles appear when
2 Df
II
π Df
I
c II
c
c o m K + 1 <
cI c
Kc II c
(5.21)
Note that c I c /KcII c
> 1 by assumption, so the rhs always exceeds one. In the opposite limit
S ≫ 1, we find from Eqn. 5.20 that micelles form when Q < c I c/(c II
c K), or
√
D II
f
D I m
c I c
c o mK <
cI c
c II c K
(5.22)
For both large and small S, the formation of micelles within phase II near the interface is
favored by a low diffusivity in phase II, since a high diffusivity in phase II tends to suppress
the concentration at the interface by sweeping copolymer rapidly into the copolymerpoor
interior of phase II. When micelles do appear in phase II, they are formed by a
creation reaction that occurs at the interface. When this happens, the chemical potential
of copolymer at the interface reaches the maximum value allowed by thermodynamics,
which in this case corresponds to a concentration c II
f,int = cII c , during a time regime in
which most of phase II contain no copolymer.
If micelles appear near the interface in phase II, the structure of micellar zone and
exclusion zone that exists in phase I must be repeated in phase II. There must a micellar
zone near the interface in phase II, within which c m (z) > 0 and c f (z) = c II
c , and an
semi-infinite micelle-free zone further from the interface, within which c m (z) = 0 and
c f (z) < c II
c . In this case, transport of copolymer between the two phases must thus
involve: micellar diffusion to the edge z = h(t) of the exclusion zone in phase I, dissolution
of the micelles at this edge, molecular diffusion across the exclusion zone, reformation of
115

micelles on the phase II side of the interface, micellar diffusion across the micellar zone
in phase II, dissolution of the micelles at the edge of the micellar region in phase II, and
molecular diffusion into the micelle-free interior of phase II.
The evolution of such a system over very long times can follow different scenarios,
depending upon the relative volumes of phases I and II, and the initial concentration
of copolymer in phase I. The simplest scenario is for the micellar region of phase II to
expand until it fills all of phase II, the micelle-free region region to expand until it fills
phase I, and transport of copolymer from phase I to phase I to then continue until the
concentration of free molecules in phase I reaches c f = c II
c K throughout. This leaves
the system in a final state in which there are micelles throughout phase II and none in
phase I. A complicated scenario must occur when the excess of the initial copolymer
concentration above the final concentration c I c
K in phase I is insufficient to fill phase II
at a homogeneous concentration c I c (i.e., when difference co − c II
c K times the volume of
phase I is less than c II
c
times the volume of phase II). In this case, the final state will
contain no micelles in either phase, and so any micelles that form in phase II near the
interface during the exchange of copolymer between phases must ultimately re-dissolve.
116

Bibliography
[1] Crank, J. The Mathematics of Diffusion (Oxford:Clarendon Press) 1975
117

Chapter 6
Interfacial Tension of PI/PDMS
with PI-b-PDMS Copolymer
Abstract
Equilibrium interfacial tension γ has been measured with a spinning drop tensiometer
for 1,4 polyisoprene (PI) and polydimethylsiloxane (PDMS) homopolymers mixed with
poly(isoprene-b-dimethylsiloxane) (IDMS) block copolymers. Measurements have been
conducted for 12 copolymers in which the volume fraction f A of the PDMS block of the
copolymer is varied from 0.49 to 0.73. The interfacial tension is found to be independent of
copolymer concentration above a critical concentration of order 0.1 wt % copolymer, which
we associate with the formation of micelles or some other structure of aggregated copolymer.
In the case of a symmetric copolymer, the measured interfacial tension is three orders
of magnitude below that obtained in the absence of copolymer. Experimental results have
been compared to the predictions of numerical self-consistent field theory for an interface
between a PDMS phase containing swollen spherical micelles and a phase of nearly pure
PI. Agreement between this theory and experiments is excellent for sufficiently asymmetric
copolymers, with f A > 0.65. Measured values of γ are much lower than predictions for
0.5 < f A ≤ 0.6, however, and exhibit an apparently discontinuous change with increasing
f A over a narrow range 0.60 < f A < 0.65. The existence of a bicontinuous microemulsion
118

phase has been identified by results of small angle X-ray scattering experiments on ternary
blends containing symmetric copolymer.
6.1 Introduction
When small amounts of surfactant are added to a mixture of oil and water, the oil-water
interfacial tension initially decreases, and the amount of adsorbed copolymer increases,
with increasing concentration of dissolved surfactant. Both the interfacial tension and the
interfacial coverage generally saturate, however, when the surfactant concentration exceeds
a critical value, above which additional surfactant begins to form aggregates in one of the
bulk phases. [1, 2]
This saturation of macroscopic interfacial tension is observed independent of the form
taken by surfactant aggregates: It occurs whether surfactant aggregates into swollen spherical
or cylindrical micelles, or into monolayers within a bicontinuous microemulsion. The
value of the saturated macroscopic interfacial tension depends, however, upon the structure
and free energy per molecule of the aggregates. Extremely low interfacial tensions can
be obtained in systems for which the thermodynamically preferred form of aggregation is
a surfactant monolayer with a vanishing spontaneous curvature (i.e., a vanishing thermodynamically
preferred value of interfacial mean curvature). Systems of such ”balanced”
surfactants generally form either a bicontinuous microemulsion or a lamellar phase, either
of which may exhibit regions of two and three phase coexistence with phases of excess
oil and/or water. In some mixtures of oil, water, and nonionic alkyl-polyethylene oxide
surfactants [3, 4, 5], in which the monolayer spontaneous curvature is a sensitive function
of temperature, the macroscopic interfacial tension between oil and water phases reaches
a minimum at a ”mean” temperature T m at which the monolayer spontaneous curvature
passes through zero, with a minimum value that can be a few orders of magnitude below
the bare oil-water interfacial tension.
119

In mixtures of immiscible A and B homopolymers with small amounts of AB diblock
copolymer, the copolymer plays a role analogous to that of small molecule surfactants in
oil-water mixtures.[6] Some symmetric ternary mixtures have been shown to form balanced
bicontinuous microemulsions, [7, 8, 9]. Direct measurements of interfacial tension have not
previously been conducted for these systems. The spontaneous curvature of monolayers
of AB block copolymer in mixtures of A and B homopolymers cannot, however, be easily
controlled by varying the temperature. Here, we instead vary spontaneous curvature by
conducting experiments at fixed temperature with a sequence of mixtures in which we vary
both the molecular weight of one block of the copolymer and the molecular weight of one
of the two homopolymers.
One potential advantage of the study of block copolymers as model surfactants is
the ability of numerical self-consistent field theory (SCFT) to provide rather accurate
predictions of free energies of polymeric monolayers and micelles. We recently used to
SCFT to study the interfacial tension between a phase of nearly pure B homopolymer and
a coexisting A-rich micellar phase containing spherical micelles of AB diblock copolymer,
in which the micelles may be swollen with emulsified B. For mixtures of homopolymers
with equal degrees of polymerization, and monomers with equal statistical segment lengths,
this model predicts that the interfacial tension that vanishes for a symmetric copolymer,
with f A = 1/2, and varies quadratically around with the volume fraction f A of the A block.
One goal of the present study is to quantitatively test the predictions of this theory.
From an experimental point of view, the most important difference between smallmolecule
and macromolecular surfactants is the difference in equilibration time scales,
rather than any difference in equilibrium behavior. The low diffusivity of any macromolecular
surfactant in a polymeric matrices make it difficult to establish a state of true phase
equilibrium in a macroscopic apparatus for measuring interfacial tension. In both the pendant
drop method [10] and the spinning drop method used here [11, 12], the tension of
120

an interface between a macroscopic (e.g., millimeter) drop and a surrounding matrix is
inferred from the deformation of the droplet caused by gravitational or centrifugal forces.
To guarantee that such a system reaches a state of global thermodynamic equilibrium,
it would be necessary to allow time for copolymer to diffuse across the entire drop and
matrix, and thereby establish homogeneous copolymer chemical potential throughout the
sample.
The ideal samples for interfacial tension measurement are thus macroscopically phase
separated mixtures that have been allowed time to fully equilibrate. Such samples have
been used in the small molecule surfactant system.[4] For polymeric surfactants, however,
it can be difficult or impossible to obtain a state of global equilibrium by diffusion of the
copolymer. In samples in which the copolymer is initially mixed into one of two immiscible
homopolymers, it can be difficult to assure that copolymer diffusion will have sufficient
time to establish an equilibrium concentration of copolymer at the interface, or to confirm
that this has occurred. Thus, it has been challenging to measure equilibrium interfacial
tension in the presence of block copolymer. Over the last decade, several researchers
have noted that the slow diffusion of block copolymer can affect the results of interfacial
tension measurements.[10, 13, 14] The results of pendant drop tensiometer experiments
differ significantly depending upon whether the copolymer is initially premixed with the
drop or the matrix material.
The goals of this study are to measure equilibrium interfacial tensions in a system
whose characteristics have been chosen to minimize the effect of slow copolymer diffusion
upon the measured interfacial tension, and to compare the results to the predictions of the
SCFT presented in Chapter 4.
121

6.2 Experiment
All of the experiments reported in this chapter have been carried out on spinning drop tensiometer
samples containing a drop surrounding a drop of poly(dimethylsiloxane) (PDMS)
surrounded by a matrix of 1,4 polyisoprene (PI) mixed with poly( 1,4 isoprene-b-dimethylsiloxane)
(IDMS). The copolymer was always initially mixed with the PDMS matrix.
6.2.1 Materials
All polymers used in this study were synthesized by anionic polymerization.[15] Isoprene
(Aldrich) monomers were stirred with calcium hydride (CaH 2 ) overnight and purified with
n-butyl lithium twice under vacuum for 2 hours each. Polymerization of polyisoprene (PI)
homopolymer/block was initiated with sec-butyl lithium in 10 vol % cyclohexane solution
and carried out at 40 ◦ C for 6 hours. The solution of PI homopolymer was terminated
in degassed methanol, washed with deionized water several times and finally dried in a
vacuum oven until the weight remained constant.
For the synthesis of poly(dimethylsiloxane) (PDMS) homopolymer, hexamethylcyclotrisiloxane
(D 3 , Aldrich) was stirred over calcium hydride at 80 - 90 ◦ C for 4 hours
after which it was degassed 3 times and further purified with dibutylmagnesium under
vacuum for 1.5 hours. After initiation with sec-butyl lithium, at least 18 hours was required
to assure the complete activation of D 3 in 10 vol % cyclohexane at 25 ◦ C and
afterwards actual polymerization started upon adding an equal volume of tetrahydrofuran
(THF). After 2 - 3 hours, the reaction was terminated with 10-fold molar excess of
chlorotrimethylsilane (Aldrich). The conversion was approximately 40 to 45 %. The final
solution was redissolved in cyclohexane after the majority of THF was removed. It was
then immediately washed with 5 to 10 wt % sodium bicarbonate (NaHCO 3 ) aqueous solution,
and then washed several times in deionized water. The products filtered with 0.2
µm pore filter disks were further dried in a vacuum oven for two to three weeks in order to
122

Table 6.1: Material Characteristics of Homopolymers
sample code M n (g/mol) a M w /M n N b η o (Pa·s) c γ o (mN/m) d
PI1 2900 1.08 42 1.3 2.6
PI2 4600 1.07 68 3.0 3.1
PI3 6000 1.09 88 4.1
PI4 6600 1.06 97 7.2 3.4
PI5 7700 1.08 113 8.9 3.4
PI6 9600 1.06 141 11.7 3.5
PI7 15800 1.06 231 47.0 3.6
PDMS1 6200 1.08 84 0.1
PDMS2 8000 1.10 107
a The number average molecular weights determined by GPC and converted using
polystyrene standards. b Based on a referece volume v = 126.1 Å. c Zero shear viscosity
measured with parallel plate rheometry at 298K. d Bare interfacial tension with PDMS1.
remove the residual materials such as solvent and unreacted D 3 monomer. The molecular
characteristics of homopolymers are listed in Table 6.1.
Poly(isoprene-b-dimethylsiloxane) (IDMS) block copolymer was polymerized by the
sequential addition of isoprene and D 3 monomers in the same way described above for
homopolymer, except that the IDMS solution was precipitated in a 3:1 vol mixture of
methanol and 2-propanol. Before adding D 3 monomers for the second block, an aliquot
of PI block was removed and quenched in methanol for analysis. The molecular characteristics
of that block are summarized in Table 6.2. In order to synthesize pairs of block
copolymers with the same length of PI, but PDMS blocks of different length, a small
amount of the IDMS solution was cannulated out during the synthesis of the DMS block
and terminated in a separate pre-flamed flask. The IDMS pairs made in this way are
IDMS2/IDMS9, IDMS4/IDMS5, IDMS6/IDMS7, and IDMS11/IDMS12. A small portion
of PDMS homopolymer can be produced during the polymerization of IDMS, [16] but we
did not identify a distinct peak associated with PDMS homopolymer in gel permeation
chromatography (GPC) curves. The sensitivity of this GPC measurement to homopolymer
is limited, however, by the fact that a small homopolymer peak can be buried in the tail
123

Table 6.2: Material Characteristics of Block Copolymers
sample code M PI (g/mol) a M w /M n fA b γ (0.1 %) c γ (0.2 %) γ (0.3 %) γ (0.4 %)
IDMS1 4800 1.06 0.49 0.0011 0.0014 0.0013
IDMS2 4400 1.15 0.50 0.0031
IDMS3 4000 1.08 0.50 0.0007
IDMS4 4800 1.09 0.53 0.0060
IDMS5 4800 1.11 0.57 0.0094
IDMS6 4900 1.07 0.58 0.0084
IDMS7 4900 1.09 0.60 0.050 0.016 0.020
IDMS8 5200 1.09 0.61 0.12 0.064
IDMS9 4400 1.13 0.63 0.17 0.064
IDMS10 4400 1.13 0.66 0.26 0.25 0.22
IDMS11 4700 1.05 0.71 0.435 0.43
IDMS12 4700 1.04 0.73 0.50 0.49
IDMS13 9300 1.07 0.50
a The number average molecular weights of PI blocks determined by GPC and converted
using polystyrene standards. b The volume fraction of PDMS within IDMS. c Interfacial
tension (mN/m) in the system of PI4/PDMS1/IDMS.
of the IDMS peak due to the similar hydrodynamic volumes. No further fractionation was
performed.
The number (M n ) and weight (M w ) averaged molecular weights were determined with
GPC in THF for PI based on the PS standards. Those of PDMS and IDMS were determined
with GPC in toluene based on the refractive index increment dn/dc by way of both a
refractive index detector and a light scattering detector. NMR analyses were used to
determine a block ratio f A = N CA /N C of IDMS, where N CA is a block length of PDMS
block and N C is a total length of IDMS. Also, NMR was used to calculate 1,4 fractions of
PI homopolymer and blocks which were at least 94 % and double check molecular weights
of PDMS homopolymer. Hereafter a symbol A consistently stands for PDMS and B for
PI.
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6.2.2 Sample Preparation
IDMS block copolymers were premixed with pure PDMS homopolymer at room temperature
(298K) at a concentration of 0.1 - 0.4 wt %, which is above the cmc (Fig. 6.3). A
total volume of a prepared sample was 1 mL. The samples were kept in a vacuum oven
until block copolymer disappeared visually. This took 4 to 5 hours at room temperature
without mechanical stirring.
6.2.3 Interfacial Tension Measurement
The interfacial tension was measured with a spinning drop tensiometer (SDT). An electric
motor rotates a horizontal glass tube containing a liquid drop in a denser matrix at a
constant angular velocity. The PDMS matrix (density ρ A = 0.970 g/mol)[17] was poured
into a glass tube, a 6 mm inner diameter, 15 cm long. One end of the glass tube is sealed
by a plug equipped with a set of stepped calibration posts. We used either of two different
plugs, which contained terraced cylindrical posts that range from 0.3 mm to 0.5 mm and
0.5 mm and 2 mm, respectively, depending on the expected final diameter of the drop.
Vacuum was pulled through the glass tube for a few hours until no air bubbles came out
of the gap between the O-rings around a plug and a glass wall. Then a pure PI drop (ρ B
= 0.90 g/mol)[17] with an appropriate diameter was injected in the middle of the PDMS
matrix using a glass pipette, and another plug with a threaded hole at the center was
quickly pushed into the glass tube until the first drop of the PDMS matrix was pushed out
through the hole in the plug. The hole was then tightly sealed with a bolt and an O-ring
before the PI drop could rise significantly and the glass tube was loaded into the SDT.
In an attempt to reduce the effects of the slow diffusion of block copolymer into the
drop, interfacial tension measurements were carried out on samples that were stored for
one to four weeks after injection of the pure PI drop into a PDMS/copolymer matrix.
Assuming a cylindrical drop of length l and diameter d with l/d > 4 capped with
125

hemispherical ends, the interfacial tension γ can be calculated as [18]
γ = 1
32 ∆ρω2 d 3 (6.1)
where ω is an angular velocity. The density difference is ∆ρ = ρ A − ρ B = 0.07 g/mol.
The initial drop size and angular velocity was chosen for each sample so as to give l/d ∼ 5
at a speed of 3000 to 6000 rpm. The rotation of the sample was synchronized with the
frequency of a stroboscope in order to obtain still images of the rotating drop with a video
camera. Since γ is proportional to d 3 , it was critical to get images of sharp drop edges.
The stroboscope minimized the effect of off-centered movements of both the drop and
calibration posts.
During the measurements, both spin-up and spin-down experiments were carried out
to confirm that the system had reached hydrodynamic equilibrium.[11, 12] Also, a few
samples were run at two different angular velocities to confirm that the same value of γ
was obtained at different rotation rates. Individual measurements were reproducible within
ca. ± 5%. This error is primarily due to the uncertainty in diameter measurement.
The observation of an apparently time-independent drop diameter is not necessarily
sufficient to prove that a tensiometer sample has reached thermodynamic equilibrium. The
analysis presented in Chapter 5 shows that, under some conditions, it is possible for the
concentration of copolymer at an interface, and the corresponding interfacial tension, to
remain at apparently time-independent values at times much less than that required to
establish a global thermodynamic equilibrium. An analysis of the kinetics of copolymer
transport in this experiment is given in Sec. 6.4.
6.2.4 Small Angle X-Ray Scattering
Small Angle X-ray scattering (SAXS) experiments were conducted using a 6m beamline
at University of Minnesota.[19] Ternary blend samples of PDMS1/PI4/IDMS3 were prepared
by adding various concentrations of block copolymers to 50:50 vol mixture of the
126

homopolymers. Dilute block copolymer samples were mechanically stirred with a magnetic
bar overnight. The concentrated samples were solution blended with benzene at a concentration
of 20 wt % of polymers and vacuum dried for a few days at room temperature. All
isotropic 2D scattering patterns were azimuthally integrated to produce 1D plots of the
intensity I vs. the magnitude of the scattering wavenumber q = 4π/λ sin(θ/2) where θ is
the scattering angle and λ is the wavelength (1.54 Å) of the incident radiation.
6.2.5 Dynamic Mechanical Spectroscopy
The zero shear viscosity η o of each homopolymer was measured with a TA instruments
ARES rheometer using a steady rate sweep mode at room temperature with 25 mm parallel
plates, a 1 mm gap, and 10 % strain.
6.3 Results
6.3.1 Bare Interfacial Tension
Bare interfacial tension γ o between PI and PDMS1 in the absence of block copolymer was
measured as a function of molecular weight of the PI homopolymer, denoted M PI . In the
absence of copolymer, the drop diameter typically became independent of time within 20
to 30 seconds in both spin-up and spin-down experiments as shown in Fig. 6.1 (a). Fig. 6.2
shows γ o as a function of PI molecular weight. The interfacial tension reaches an apparent
plateau value of 3.6 mN/m for the highest PI molecular weights. This plateau value for γ o
is in a reasonable agreement with the literature value (3.2 mN/m),[20] which was measured
using commercial grade homopolymers with a density difference ∆ρ = 0.064 g/cm 3 , rather
than 0.07 g/cm 3 obtained for our sample.
The solid line in Fig. 6.2 is a SCFT prediction for this series of samples. The SCFT
calculations were conducted using the reported molecular weights and a Flory-Huggins
parameter χ = 0.175 based on a reference volume v = 126.1 Å3 . This value of χ was
127

3
1
2.5
0.8
f A = 0.73 (IDMS12)
D (mm)
2
1.5
1
PI6 (9.6K)
PI1 (2.9K)
D (mm)
0.6
0.4
f A = 0.58 (IDMS6)
0.5
0.2
0
0 1 2 3 4 5
t (min)
(a)
0
0 100 200 300 400 500 600 700
t (min)
(b)
Figure 6.1: The transient diameter of the PI drop in the PDMS matrix as a function of
time. (a) PI1 and PI6 in the absence of block copolymer (b) PI4 in PDMS1/IDMS6 and in
PDMS2/IDMS12. The concentrations of IDMS6 and IDMS12 in the corresponding PDMS
matrices are 0.1 wt % and 0.2 wt %, respectively. The angular velocity is 6300 rpm.
chosen to fit measurements of the initial decrease in interfacial tension with small amounts
of added copolymer and the cmc observed in experiments that are reported in the following
subsection. Our chosen value of χ = 0.175 is is slightly larger than the value χ = 0.146
at T = 298K reported by Cochran et. al [19], which was obtained by fitting T ODT of two
different symmetric PI-b-PDMS block copolymers with the equation χ = A/T + B and
assuming (χN C ) ODT = 10.495.[21] SCFT predictions of the bare interfacial tension using
χ = 0.175 underestimate the interfacial tension at high PI molecular weights by about 20
%. Use of the lower literature value would make the discrepancy slightly worse.
6.3.2 Dependence on Copolymer Concentration
Fig. 6.3 shows the reduction of interfacial tension γ with different block copolymers on a
logarithmic scale. The dashed lines are just guides for eyes. When the highly asymmetric
block copolymer IDMS12, with f A = 0.73, was added to PI4/PDMS2, γ became saturated
128

5
4
γ (mN/m)
3
2
1
0
2 4 6 8 10 12 14 16
M PI (kg / mol)
Figure 6.2: Bare interfacial tension between PI and PDMS1 in the absence of block copolymer
as a function of molecular weights of PI. The solid line is the SCFT prediction using
χ = 0.175.
at 0.44 mN/m. This is an 87 % reduction from the bare interfacial tension, which is
comparable to the maximum reduction found in previous studies. [10, 22]
However, when the symmetric block copolymer IDMS1 was added to PI4/PDMS1,
we obtained a strikingly low interfacial tension of approximately 0.0015 mN/m, at all
concentrations above 0.05 wt % of block copolymer. In this case, the interfacial tension is
reduced by more than three orders of magnitude. This dramatic decrease has commonly
been observed for balanced small molecule surfactants, and predicted theoretically for
balanced surfactants of either large or small molecule, but had not been observed previously
in a polymeric system.
The value of copolymer concentration beyond which γ becomes independent of concentration
is close to 0.5% for these two systems. This apparent cmc is associated with the
aggregation of copolymer into self-assembled structures, though this saturation of interfacial
tension is expected whether the aggregates are spherical micelles, as appears likely for
129

f A = 0.73, or a bicontinuous microemulsions, as appears to be the case for the symmetric
copolymer.
The predictions of SCFT for the cmc and the slope dγ/dφ c of interfacial tension vs
concentration below the cmc are extremely sensitive to the choice of χ parameter. The cmc
is known to vary as e −χN Cf B
. Fig. 6.4 compares our results for IDMS1 to SCFT predictions
for several slightly different χ values which are all defined using a reference volume v =
126.1 Å3 . All of the other SCFT results presented in this chapter use the value χ = 0.175
chosen to fit this data.
In the presence of block copolymer, the transient behavior in the spinning drop tensiometer
is much slower than in the absence of copolymer. Fig. 6.1 (b) shows representative
transient behaviors of the diameter of the PI4 drop during each spin-up and spin-down
experiment with 0.1 wt % of IDMS6 in PDMS1 and 0.2 wt % of IDMS12 in PDMS2,
respectively. The time required for the drop to reach its final diameter is several hours
at concentrations above the cmc, while less than a minute is required in the absence of
copolymer. Allowing the apparatus to spin for 24 hours was found to be sufficient to obtain
a time-independent value in all but a few samples that contain block copolymer. (The
exceptions are are discussed below).
6.3.3 Dependence on Copolymer Composition
We have measured the interfacial tension at concentrations above the apparent cmc for each
of the 12 PI-b-PDMS copolymers IDMS1-IDMS12 listed in Table 6.2. These copolymers
all have PI blocks of similar size, corresponding to χN CB ≃ 12, but varying PDMS corona
block. The volume fraction of PDMS in this series varies from 0.49 to 0.73. All of these
experiments were carried out with homopolymers PI4 and PDMS1. Measurements were
taken with copolymer concentrations varying from 0.1 wt % to 0.4 wt % in order to confirm
that, within this range, the interfacial tension is generally independent of concentration.
130

10 1 0 0.05 0.1 0.15 0.2 0.25 0.3
10 0
f A = 0.73 (IDMS12)
γ (mN/m)
10 -1
10 -2
10 -3
f A = 0.49 (IDMS1)
10 -4
φ c (%)
Figure 6.3: The interfacial tension reduction of PI4/PDMS1 and PI4/PDMS2 as a function
of the concentration of IDMS1 and IDMS12, respectively. The dashed lines are guides to
the eye.
10 1 0 0.05 0.1 0.15 0.2 0.25 0.3
γ (mN/m)
10 0
10 -1
10 -2
χ=0.15
0.17
0.175
0.18
10 -3
10 -4
φ c (%)
Figure 6.4: The interfacial tension reduction between PI4 and PDMS1 as a function of
IDMS1. The solid lines are the SCFT prediction with four different χ’s.
131

Fig. 6.5 show the saturated interfacial tension as a function of f A . Fig. 6.6 shows the
same data using a logarithmic axis for interfacial tension. For most copolymers, values
are shown for more than one concentration. It should be apparent that γ is not a smooth
function of f A . Starting from an ultra-low interfacial tension of order 10 −3 mN/m near
the symmetric copolymer f A = 0.49 (IDMS1), γ increases relatively slowly to about 10 −2
mN/m near f A = 0.6, and then changes by roughly an order of magnitude over a narrow
range 0.6 < f A < 0.65. For f A > 0.65, γ again rises smoothly with f A . This seemingly
discontinuous dependence on f A is more obvious in Fig. 6.6.
The open circles in Fig. 6.5 are the results of SCFT predictions of γ for two systems.
The required calculations were similar to the micelle simulations presented in chapter 4,
in which we have determined the cmc from simulations of a swollen spherical micelle and
used a separate SCFT simulation of a flat interface to determine the interfacial tension at
the cmc. The solid line is the result of an expansion of the Helfrich theory for γ around a
balance point f bal
A , of the form γ = τ ′2
2κ +
(f A − f bal
A )2 (6.2)
in which we have used SCFT predictions for the balance point f bal
A
= 0.48, the bending
rigidity κ + = 1.92kT, and for τ ′
= ∂τ/∂f A = 0.270 kT Å−1 . This simple parabolic
expansion agrees extremely well with the results of the two micelle calculations shown by
open circles. Leibler’s prediction[23] of interfacial tension, in which the micelle core was
not allowed to swell, is also shown as a dotted line in the same plot for comparison.
For sufficiently asymmetric copolymers, with f A > 0.65, the SCFT predictions are
remarkably consistent with the measured interfacial tensions. However, SCFT predictions
of γ for f A < 0.6 are significantly higher than the measurements, particularly near the
apparent discontinuity at 0.6 < f A < 0.65.
In the transition region between 0.6 < f A < 0.65, it was very difficult to obtain
132

γ (mN/m)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.1 %
0.2 %
0.3 %
0.4 %
swollen micelle
Leibler
0
0.3 0.4 0.5 0.6 0.7 0.8
f A
Figure 6.5: Interfacial tension in PI4/PDMS1/IDMS system as a function of the block
ratio f A of IDMS. The quadratic dependence of interfacial tension on f A agrees well with
the SCFT prediction and the swollen micelle theory (hollow dots) for f A > 0.65. Leibler’s
theory[23] (a dotted line) predicts that a wider range of f A will cause vanishing interfacial
tension.
10 0 0.3 0.4 0.5 0.6 0.7 0.8
10 -1
γ (mN/m)
10 -2
10 -3
10 -4
0.1 %
0.2 %
0.3 %
0.4 %
swollen micelle
Leibler
f A
Figure 6.6: Data and calculations of Fig. 6.5 plotted on a logarithmic scale. Near a balance
point, measured values are much lower than those predicted by SCFT. The concentrations
of block copolymer are all greater than the cmc.
133

eproducible results, because the relaxation of the drop diameter became dramatically
slower, and because the results depend on concentration. Fig. 6.7 shows the relaxation of
the drop diameter for IDMS7 (f A = 0.6). In this case, the drop diameter continued to
decrease throughout 3 days of observation, and the results depended significantly upon the
concentration of copolymer. This transient behavior is very different from that observed
for either f A > 0.65 or f A < 0.6, as shown in Fig. 6.1 (b).
0.6
0.55
0.5
D (mm)
0.45
0.4
0.1 wt %
0.35
0.3
0.2 wt %
0.25
0 1000 2000 3000 4000
t (min)
Figure 6.7: The time-dependent diameter of a PI4 drop in a PDMS1 matrix at an angular
velocity of 6300 rpm with two different concentrations of IDMS7 (f A = 0.6).
6.3.4 Dependence on Homopolymer Molecular Weight
An alternative way to control the spontaneous curvature of a monolayer is to change
the relative lengths of the homopolymers. In this subsection we examine the effect of
varying molecular weight M PI of polyisoprene on interfacial tension. We have measured
the interfacial tension between PDMS1 matrix that initially contains 0.1 wt % of the
symmetric block copolymer IDMS1 and a series of PI homopolymers.
The results are shown in Fig. 6.8. The results show an essentially quadratic dependence
134

on the M PI , with a very low minimum in γ at a balance point between 6 K and 6.6 K. The
solid lines are an approximate SCFT prediction, of the form γ = τ ′2
2κ +
(β − β bal ) 2 analogous
to Eqn. 6.2, in which β = N PI /N PDMS and τ ′ ≡ ∂τ/∂β. The displayed prediction uses
parameters M PI,bal = 6.2 K, κ + = 2.00 kT and τ ′ = ∂τ/∂β = 0.0237 kT Å−1 that were
obtained from SCFT calculations similar to those presented in Chapter 3.
6.3.5 A Larger Copolymer
Another series of measurements was carried out with copolymer IDMS13, PDMS2 matrix,
and PI homopolymers of varying molecular weight. IDMS13 is a symmetric copolymer
with a molecular weight nearly twice that of IDMS1, and a PI block size roughly twice
that of any of the other copolymers used in our experiments. IDMS13 was initially added
the PDMS2 matrix at a concentration of 0.2 wt %.
Results for the interfacial tension inferred from the final observed drop radii in experiments
that were each run for several days are shown in Fig. 6.9 for systems with several
different PI homopolymers. The minimum value of γ is much higher than observed with
IDMS1, and overall agreement with SCFT predictions for the equilibrium interfacial tension
is disappointing. These results are not, however, reliable measurements of equilibrium
interfacial tension, because drop diameter may not have reached its true steady state.
In spinning drop experiments with IDMS13, equilibration of the drop size was dramatically
slower than in experiments with the smaller symmetric copolymer. Fig. 6.10
shows the time-dependence of the drop diameter in a system with PDMS2, IDMS13, and
PI5. In this example, the drop diameter appears to still be noticeably decreasing after the
tensiometer has been left running for one week. The relaxation time for the drop radius
in this experiment appears to be at least 50 times greater than the relaxation time of a
few hours observed in IDMS1. We find it difficult to understand this dramatic difference
in time scales on the basis of the difference in diffusivities alone for two molecules whose
135

molecular weights differ by only a factor of two. We do not understand the reason for this
dramatic difference in equilibration times.
6.3.6 Bicontinuous Microemulsion Phase
Some systems containing symmetric diblock copolymers, like some systems of balanced
small molecule surfactants, have been found to form a bicontinuous microemulsion phase.
In order to determine whether this also occurs in some of the systems studied here, we
have studied the phase behavior of mixtures of the symmetric homopolymer IDMS1 with
equal volume fractions of PI4 and IDMS3, as well as mixtures of the larger symmetric
copolymer, IDMS13 with equal volume of PI5 and PDMS2. These systems correspond to
the minima of the parabolic tension curves in Fig. 6.8 and Fig. 6.9.
Photographs of the samples are shown in Fig. 6.11 and Fig. 6.12, respectively. Mixtures
of PDMS1/PI4/IDMS3 were turbid at block copolymer concentrations up to 25 wt %,
indicating phase separation into two or three coexisting phases. The sample with 32
wt % IDMS1 was slightly bluish, which is a characteristic signal of the formation of a
uniform bicontinuous microemulsion phase µE with internal structures large enough to
induce Rayleigh scattering. At higher concentrations, samples of this mixture become
totally transparent. A similar sequence is observed for PDMS2/PI5/IDMS13. In this case,
however, the 10 wt % is turbid, and 20 wt % sample is bluish. These observations are
consistent with the existence of two or three phase coexistence of a microemulsion with
PDMS and/and PI rich phases at low copolymer concentrations ( below 25 % for IDMS3
and 10 % for IDMS13), the formation of a microemulsion ( near 32 wt % for IDMS3), and
the formation of an optically clear lamellar phase at higher concentrations. This is the
normal sequence for a system that forms a balanced bicontinuous microemulsion.
We have also carried out small angle X-ray experiments on these samples. One dimensional
SAXS intensities are shown as a function of scattering wavenumber q in Fig. 6.13
136

10 1
10 0
2 4 6 8 10
γ (mN/m)
10 -1
10 -2
10 -3
10 -4
M PI (kg/mol)
Figure 6.8: Interfacial tension in the system of PDMS1/PI/IDMS1 as a function of molecular
weight of PI. The concentration of IDMS1 is 0.1 wt %.
10 1
10 0
2 4 6 8 10
γ (mN/m)
10 -1
10 -2
10 -3
10 -4
M PI (kg/mol)
Figure 6.9: Interfacial tension in the system of PDMS2/PI/IDMS13 as a function of molecular
weight of PI. The concentration of IDMS13 is 0.2 wt %. The minimum of the interfacial
tension is not ultra-low, but much greater than that in Fig. 6.8.
137

1
0.8
D (mm)
0.6
0.4
0.2
0
0 2000 4000 6000 8000 10000
t (min)
Figure 6.10: The variation of the diameter of the PI5 drop in the PDMS2 matrix at a
concentration of 0.2 wt % of IDMS13. The rotational speed is 6300 rpm.
and Fig. 6.14 for both sets of mixtures. In both systems, at high concentrations, we see
the scattering peak characteristic of a lamellar phase. In Fig. 6.13, as the concentration of
IDMS3 decreases, the peak wavenumber q ∗ decreases until φ c = 0.59. The corresponding
variation of the domain spacings ξ = 2π/q ∗ with copolymer concentration is shown given
in Fig. 6.15. Then another peak suddenly appears in front of a primary lamellar peak,
consistent with the formation of a bicontinuous microemulsion phase coexisting with a
lamellar phase. Within a range of between 0.49 < φ c < 0.58, it appears that neither peak
position changes with concentration, which we interpret as two-phase coexistence between
a lamellar and microemulsion phase. The ratio of the domain spacing of a bicontinuous
microemulsion phase to that of a lamellar phase (ξ µE /ξ L ) is 1.1 as compared in Fig. 6.16.
Ideally this ratio should be 1.5 in case of the structure with the perfect minimal surface.[24]
In fact, these double peaks also appear with an ideal ratio of ξ µE /ξ L = 1.5 in Fig. 6.14
at the lower concentration of IDMS13 (φ c ∼ 0.45). Since the primary peaks for φ c <
138

Figure 6.11: The ternary isopleth blends of the system PDMS1/PI4/IDMS3 for the various
concentrations of block copolymer: 5, 25, 32, 45, 57, and 70 wt % from the leftmost vial.
The samples are turbid below 30 wt % due to the macrophase separation.
Figure 6.12: The ternary isopleth blends of the system PDMS2/PI5/IDMS13 for the various
concentrations of block copolymer: 10, 20, 30, 40, 50, and 70 wt % from the leftmost
vial. The sample with 20 wt % of IDMS13 turns bluish.
139

10 −4 I (a.u.)
18
16
14
12
10
8
6
4
2
0
φ c
0%
5%
10%
20%
30%
32%
35%
40%
48%
50%
55%
57%
60%
70%
80%
90%
100%
0 0.01 0.02 0.03 0.04 0.05 0.06
q (Å −1 )
Figure 6.13: The SAXS profiles for the ternary isopleth blends of PDMS1/PI4/IDMS3 as a
function of the concentration of IDMS3. All curves are presented on arbitrary linear scales
for clarity. The primary peak continuously moves to a small q region as the mixtures are
diluted with the homopolymers. Two peaks are distinct between 50 % and 55 % indicating
a bicontinuous microemulsion phase coexist with a lamellar phase.
14
φ c
10 −4 I (a.u.)
12
10
8
6
4
2
0
10%
20%
30%
35%
40%
45%
48%
50%
60%
70%
80%
90%
100%
0 0.01 0.02 0.03 0.04 0.05 0.06
q (Å −1 )
Figure 6.14: The SAXS profiles for the ternary isopleth blends of PDMS2/PI5/IDMS13
as a function of the concentration of IDMS13. The primary peaks of IDMS13 appear at
lower q than those of IDMS3 in Fig. 6.13 at the same concentration due to the twice longer
block copolymer.
140

0.4 are hidden inside the low q intensity, further analysis of the morphology transition
is not possible. However, a bicontinuous microemulsion phase unambiguously forms with
IDMS13 even though interfacial tension is not ultra-low. Returning to Fig. 6.13, if the
concentration of the homopolymers increases in the blends, a uniform bicontinuous microemulsion
phase is encountered between 0.31 < φ c < 0.49, and finally a wide region of
three phase coexistence, denoted 3 µE , opens for φ c < 0.3. A 3 µE phase can be identified
by both the turbid blends and a constant peak position at q ∗ = 0.17 Å −1 as clearly shown
in Fig. 6.11 and Fig. 6.12.
A macroscopic two phase region could not be verified here, but it should be stable
below the cmc. Surprisingly, the proposed diagram of the system PDMS1/PI4/IDMS3
in Fig. 6.17 agrees very well with the theoretical diagram for a small molecule surfactant
system with κ = kT by Daicic et al. [25] In addition, the semiquantitative relationship,
γ ξ 2 ≃ kT, between the domain spacing and interfacial tension in the small molecule
surfactant system,[3, 4] is also valid in this system using ξ µE = 52.5 nm in a 3 µE region
and γ ≃ 0.0007 mN/m.
In contrast to the previous literature of the ternary polymer blends [7, 8, 9, 11, 26]
in which a uniform bicontinuous microemulsion phase was observed at low concentration
of φ c ≃ 0.1 in the limit of unbinding transition of a lamellar phase,[7] the present system
has a denser microemulsion phase at a much higher concentration of φ c = 0.30 without
unbinding transition. Such a uniform bicontinuous microemulsion phase is often found at
the similar concentration in the nonionic surfactant system depending on the strength of the
surfactants.[27] We conjecture the different location of the Lifshitz point given by (φ c ) L =
2α 2 /(1+2α 2 ) and (χN c ) L = 2(1+2α 2 )/α is implicitly related to those results. Within the
mean field approximation[28] the Lifshitz point are expected to be (φ c ) L = 0.08 for α = 0.2
in the most literature, but (φ c ) L = 0.47 for α = 0.66 in our system. Although our system is
far from the Lifshitz point (χN c ) L = 5.7, these predictions are reasonably consistent with
141

60
50
40
ζ (nm)
30
20
10
ζ µE
ζ L
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
φ c
Figure 6.15: The variation of the domain spacing of a lamellar phase (ξ L ) and a bicontinuous
microemulsion phase (ξ µE ) along the isopleth in the system of PDMS1/PI4/IDMS3.
1
0.8
ζ L
ζ Lo
/ ζ ∗
0.6
0.4
ζ µE
0.2
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
φ c
Figure 6.16: The ratio of the pure lamellar spacing ((ξ Lo ) to the spacing at q = q ∗ multiplied
by a weight factor α as a function of the concentration of IDMS3. ξ µE /ξ L = 1.1.
142

2 φ 3 µE µE µE
+
L
L
0 10 20 30 40 50 60 70 80 90 100
φ c (%)
Figure 6.17: The proposed phase diagram of the system PDMS1/PI4/IDMS3 in Fig. 6.13.
Large three phase region exists below 30 % of block copolymer.
the experimental data. Moreover, in case of the system PDMS2/PI5/IDMS13, (φ c ) L = 0.26
for α = 0.42 so that a bicontinuous microemulsion phase could be located at more dilute
regime.
6.4 Analysis of Kinetics
In this section, we analyze the kinetics of copolymer diffusion in the spinning drop tensiometer,
in order to estimate the time necessary for the copolymer chemical potential to
reach its equilibrium value at the PI/PDMS interface. The analysis is based on that of
Chapter 5. That analysis suggests that the copolymer chemical potential at the interface
of the drop should reach its equilibrium value of µ I c , corresponding to the cmc in PDMS,
either if the copolymer has time to diffuse to the center of the drop and fill the drop to
its equilibrium concentration, or if there is no exclusion zone in the PDMS phase near
the interface even at early times, in which case the chemical potential at the interface is
expected to reach µ cmc in a time comparable to that required to establish an interfacial
143

monolayer.
6.4.1 Estimate of Diffusion Time
We first estimate the time for free copolymer to diffuse to the center of the PI drop. This
requires an estimate of the diffusion coefficient D f of block copolymer in the PI matrix.
At the low molecular weights used in our experiments, we may use the Rouse model to
predict the dependence of diffusivity on molecular weight. We thus require estimates of
the monomer friction coefficient for the PI and PDMS blocks of IDMS copolymer in the
PI matrix. We are not aware of any direct measurement of friction coefficient for a PDMS
tracer in PI. The reasoning used to estimate this friction coefficient, which is based on
a review of the literature on friction coefficient in mixtures, is discussed in Appendix B.
Briefly, we assume that the segmental dynamics of IDMS block copolymer is governed by
the solvent (homopolymer) viscosity rather than the friction coefficient ζ i of each block,
because this assumption has been found to provide a reasonable estimate in systems that
are chemically similar to PI and PDMS. The monomer friction coefficient PI homopolymer
at 298 K has been reported to be [29] ζ PI = 1.5×10 −6 dyn·s/cm for a reference volume of 99
cm 3 /mol. Because the friction coefficient for pure PI (i.e., a PI tracer in the PI matrix) is of
order 10 2 times larger than of pure PDMS, we assume that the friction coefficient for PDMS
in PI is less than or equal to that of PI in PI. With this assumption, the tracer diffusion
coefficient of block copolymer in the PI homopolymer must lie between that calculated by
ignoring the friction of the PDMS block, giving D o = D PI = kT/Nζ PI ≃ 5 × 10 −10 cm 2 /s,
where N corresponds to the length of PI block (ca. 4800 g/mol) and that obtained taking
the monomer friction coefficient of PDMS to equal that of PI, which yields a diffusivity
roughly half as large.
The diffusion coefficient of PDMS can be directly obtained in ref. [30] or [31] as
D PDMS = 1 × 10 −7 cm 2 /s for M PDMS = 5200 g/mol.
144

The time for copolymer to diffuse to the center of an essentially cylindrical droplet of
diameter d is given by
t = d 2 /4kD PI
f (6.3)
A numerical factor k = 5.8 is obtained by identifying t with the relaxation time of the
slowest decaying mode in an eigenmode expansion of the concentration field in a cylindrical
geometry. For this diffusion problem the concentration at the edge of the drop is held
constant and then the eigenmodes are zeroth order Bessel functions. The final radius
of a drop in a spinning drop tensiometer depends upon its interfacial tension, and is
substantially different for samples with widely differing interfacial tensions. Assuming the
diffusion coefficient D PI
f
of block copolymer in the PI phase as D PI
f
= D o (1 − f A ) with
increasing PDMS blocks, Fig. 6.18 shows the characteristic time t as a function of γ. This
curve is obtained by combining Eqn. 6.3 with Eqn. 6.1 for the relationship between γ and
drop diameter d = 2R for the current system. Fig. 6.18 demonstrates that only nearly
symmetric block copolymers, which produce γ < 0.01 mN/m, corresponding to f A < 0.6,
can diffuse to the center of the drop in less than about 10 hours.
6.4.2 Can Micelles Reach the Interface?
We now consider the question of whether the concentration of free copolymer in PDMS near
the PI/PDMS interface reaches the cmc during the early stages of diffusion, at times much
less than time required to fill the drop with copolymer, or whether it remains depressed
below the cmc at early times, leading to the formation of an exclusion zone with no micelles
near the interface. According to the analysis given in Chapter 5, the choice between these
scenarios is determined by the value of a parameter
√
Df
PI c PDMS
c
Q =
Dm
PDMS c 0 mK
defined in Eqn. 5.6, in which D PI
f
D PDMS
m
(6.4)
is the diffusivity of free copolymer in the PI drop,
is the diffusivity of a micelle in the PDMS matrix, c PDMS
c
145
is the critical micelle

10 3 0 0.1 0.2 0.3 0.4 0.5 0.6
10 2
t (hr)
10 1
10 0
γ (mN/m)
Figure 6.18: Time t to reach a center of spinning drop tensiometer drop by molecular
diffusion as a function of interfacial tension γ.
concentration in PDMS, and c o m is the initial concentration of copolymers in micelles within
PDMS, given by c o m = co −c PDMS
c , where c o is the total initial concentration of copolymers.
This dimensionless number is the ratio of the flux of copolymer that leaks into the PI drop,
calculated by assuming that the concentration of free copolymer in PDMS is held at the
cmc, to the maximum flux of copolymer that can be provided by copolymer diffusion,
which is the flux of micelles to an absorbing boundary. If Q < 1, micellar diffusion alone
is sufficient to supply the required flux into the drop, and micelles exist throughout the
PDMS matrix. If Q > 1, the concentration of free copolymer at the interface must be
depressed below the cmc to match the flux into the drop, creating a micelle-free region
near the interface at early times.
The partition coefficient K ≡ φ A c /φB c
for the equilibrium distribution of dissolved
copolymer between A and B homopolymer phases can be derived from the Flory-Huggins
146

theory, which yields
ln K ≃ χN c (f A − f B ) + N C
( 1
N A
− 1
N B
)
. (6.5)
The partition coefficient is shown in 6.19 for a series of systems, similar to those studied
here, in which χN C f A = 12 and the symmetric homopolymers have the length N A = N B =
Nf A equal to that of the core block. The partition function is mainly an exponential
function of incompatibility difference between two blocks so that K becomes huge for
asymmetric copolymer.
The ratio c int
f /co of interfacial concentration to bulk concentration shown in Fig. 6.19
is calculated using Eqn. 5.2, and is that which would be obtained in the absence of
micelles. This was calculated using an estimated ratio of copolymer tracer diffusivities
Df
PDMS /Df
PI
= 200. For f 0.55, this ratio very closely approaches the unity because
the partition coefficient K becomes so large, indicating that the copolymer is nearly insoluble
in the PI drop. For f = 0.5, where K = 1, the predicted ratio c int
f /co ≃ 0.93 remains
high because the diffusivity is much higher in the matrix than in the drop. This plot is
directly relevant, however, only at bulk concentrations below the cmc.
The diffusion coefficient of a micelle has been estimated from the Stokes-Einstein equation
D m = kT/6πη I oR, using estimated micelle hydrodynamic radius R from SCFT calculations.
Results for the radii of swollen micelles in two-phase ternary systems and of
unswollen micelles in binary mixtures are shown on the left axis in Fig. 6.20. The estimates
are taken from the series of systems shown in Fig. 4.6, for systems in which
f B χN = 12 with the homopolymers of equal length N A = N B = fN C , which is similar
to the PI/PDMS/IDMS1 system. The predicted radius of a swollen micelle diverges
as f A approaches 1/2, causing the predicted micelle diffusivity to vanish. The assumption
that the copolymer forms spherical micelles is, however, dubious in this limit, where the
equilibrium structure appears to be a bicontinuous microemulsion phase.
147

1
10 12
0.98
10 10
10 8
/ c o
c f
I,int
0.96
10 6
K
0.94
10 4
0.92
10 2
0.9
0.5 0.55 0.6 0.65 0.7 0.75 100
f A
Figure 6.19: The time-independent interfacial concentration and the partition coefficient as
a function of the block ratio f A in the block copolymer. The effect of the micelle diffusion
on c int
f
is not considered.
1000
900
unswollen micelle
swollen micelle
10 1
10 0
R (Å)
800
700
600
500
400
300
10 −1
10 −2
10 −3
10 −4
10 −5
Q
200
10 −6
100
0.5 0.55 0.6 0.65 0.7 0.75 10−7
f A
Figure 6.20: The radii R of unswollen/swollen micelles and the dimensionless ratio Q with
respect to f A . R is taken from the micelle simulation with f B χN c = 12 in Fig. 4.6.
148

Our results for Q as a function of f A are shown in Fig. 6.20, for systems with χNf A = 12
and N A = N B = f A N C . Results for swollen and unswollen micelles were obtained using
micelle diffusivities calculated based on the corresponding micelle radii shown in the same
figure. For unswollen micelles, Q ≪ 1 for all values of f A . This is in part a reflection of the
fact that diffusivity D PDMS
m
of unswollen micelles in PDMS is actually somewhat higher
than the free copolymer tracer diffusivity D PI
f
in PI. For swollen micelles, the diverging
micelle radius causes the micelle diffusivity to approach zero, and thus causes Q to diverge
as f A → 1/2. Despite this divergence, however, we find Q > 1 only for a very narrow
range 0.5 < f A < 0.501, and Q < 1 everywhere else. For f A ≃ 0.6, where we observe an
apparent discontinuity in γ as a function of f A , K ∼ 10 3 , indicating that the copolymer
has become essentially insoluble in PI, and Q < 10 −3 as a result. For f A ≃ 0.5, where this
model yields Q < 1 over a very narrow range of values, it appears that the hydrodynamic
equilibrium diameter in the spinning drop tensiometer becomes small enough that there
actually should be time for copolymer to diffuse to the center of the drop within a few hours.
In this case, however, the whole analysis become dubious, because we do not know what
kind of aggregate the copolymer forms within the PDMS phase near the interface. The
above analysis suggests that all measured interfacial tensions in this experiment represent
the equilibrium values.
6.4.3 Experimental Strategy
From the above results we can propose the general experimental strategy to reach, or at
least mimic, a global equilibrium during experiments.
First, block copolymer should be always added to the matrix for two reasons: to keep
the initial concentration of block copolymer constant due to the much larger volume of
the matrix as addressed by Hu et al. [10] and to reduce the diffusional distance of block
copolymer than the opposite case. As described in Experiment section, we premixed IDMS
149

lock copolymer with the PDMS matrix.
Second, block copolymer should favor the initial phase (i.e., matrix) where micelles
preferentially form. We could simply satisfy this condition either by using block copolymer
with the longer PDMS block or by varying the molecular weight of the PI homopolymer.
Third, it is recommended that both homopolymer phases, especially the matrix, have
low viscosity in order to allow the fast diffusion of block copolymer, and block copolymer
can be added as much as possible so as to reduce the value of Q via c c /c o m.
Fourth, in order to systematically control the spontaneous curvature of a block copolymer
monolayer, the length of the core (PI) block can be fixed to keep the cmc constant
while sequentially increasing the corona (PDMS) block.
6.5 Discussion and Conclusion
We have used a spinning drop tensiometer to measure interfacial tensions of PI/PDMS
interfaces with adsorbed IDMS diblock copolymers. We have characterized three different
types of systems: one with copolymers of varying compositions and nearly symmetric
homopolymers, and two others with PI homopolymers of varying molecular weight and
a nearly symmetric copolymer. For all of the systems studied, the IDMS copolymer is
expected to be more soluble in the PDMS matrix, and to thus form micelles in PDMS at a
lower chemical potential of copolymer than that required to form micelles in the PI drop.
The copolymer was always added to the PDMS matrix.
In these systems, the tracer diffusivity of the copolymer is estimated to be of 200 times
higher in the PDMS matrix than in the PI drop.
Measured interfacial tensions at copolymer concentration above the apparent cmc have
been compared to SCFT predictions that assume swollen spherical micelles in the PDMS
matrix coexist with an excess PI phase. These predictions agreed extremely well with
our measurements for more asymmetric copolymers, with 0.65 < f A < 0.73. In a system
150

with a symmetric copolymer, with f A ≃ 0.5, we observe an ultra-low interfacial tension,
γ ≃ 10 −3 mN/m. This behavior is consistent with theoretical predictions of vanishing
or very low interfacial tensions for symmetric systems, and with many observations of
ultra-low tensions in balanced small molecule systems. However, it had not previously
been directly observed in a polymeric system. For slightly asymmetric copolymers, with
0.5 < f A < 0.6, the observed dependence of γ on increasing f A is much less than predicted
by the SCFT described above. Notably, the measured interfacial tension seems to change
discontinuously with f A at a value near f A = 0.6, where measured values change by roughly
an order of magnitude. Within the narrow range of 0.6 < f A < 0.65, we were unable to
obtain a reliable value of equilibrium interfacial tension, because the drop diameter was
found slowly decrease over very long times, in a manner that depends upon copolymer
concentration.
Experiments in which we varied the molecular weight of PI homopolymer in mixtures
containing a symmetric copolymer yielded very different results for two symmetric copolymers
with significantly different total molecular weights. In experiments that used the
lower molecular weight copolymer (IDMS1) a very low tension was observed at the balance
point and a dependence of γ on PI molecular weight that agreed well with the SCFT
predictions.
Experiments with a higher molecular weight copolymer (IDMS13) showed a dramatically
slower relaxation of the drop size than observed with IDMS1, making it impossible
for us to obtain reliable measurements of equilibrium interfacial tension. It is not obvious
at present why the relaxation of drop size is so much slower for the larger copolymer. The
difference seems to be too large to be accounted for by a difference in a tracer diffusivity
alone.
To determine structure and phase behavior of symmetric mixtures, we also examined
two sets of bulk mixtures with equal volumes of PI and PDMS homopolymers mixed with
151

a symmetric copolymer, over a wide range of copolymer concentrations. The results of
SAXS experiments and optical observations both suggest the existence of a bicontinuous
microemulsion that (for the mixtures with IDMS1) coexists with phases nearly pure PI
and/or PDMS homopolymer at copolymer concentration up to about 30 % copolymer.
6.5.1 Equilibrium vs. Nonequilibrium
In this subsection we discuss the possible equilibrium and nonequilibrium factors that
might cause the seemingly discontinuous change of γ in the spinning drop tensiometer.
First, we consider possible explanations based on equilibrium phase behavior. As previously
mentioned, controlling the block ratio of copolymer is analogous to varying temperature
in the nonionic surfactant system. Thus if we gradually decrease the block ratio
from 0.73 to 0.5 at a copolymer concentration slightly greater than the cmc, it corresponds
to increasing temperature from T < T l where T l is a lower critical end temperature. At
a certain point corresponding to T = T l , we expect the three phase body to continuously
open as the spontaneous curvature of copolymer becomes sufficiently small. In fact, the
opening of a third phase does not cause discontinuous change in interfacial tension at T l
or T u in nonionic surfactant systems. There are several reasons for this. The chemical
potentials of all three components in a three phase body vary continuously with T or f A in
this experiment, even through a critical end point. The interfacial tension of an oil-water
interface with adsorbed surfactant thus also changes continuously. Only a discontinuous
change in the structure of the macroscopic interface could cause a discontinuous change in
γ. This could, in principle, be caused by the appearance of a third phase that wets the
interface. However, it has been observed that this does not occur in mixtures containing
strong nonionic surfactants, and the middle phase does not wet oil-water interface. In
addition, the behavior near a critical end point T = T l is continuous because it is a critical
point, at which the compositions and (presumably) the structures of the water-rich phase
152

and the surfactant-rich phase must merge causing γ ow and γ o,µE to merge and γ w,µE to
vanish. We conclude, on the basis of this comparison to the behavior observed in systems
of small molecule surfactants, that it is unlike that the seemingly discontinuous change in
γ observed in this experiment can be explained by equilibrium thermodynamics.
We now consider what non-equilibrium processes might be relevant to the observed
phenomena. The analysis given in Chapter 5 was an attempt to account for the effects of
limitations on the rate of copolymer transport between phases on the measured interfacial
tension. That analysis ignored any limitations on the rate at which micelles can swell by
absorbing homopolymer as they approach the interface, on transfer of copolymer between
micelles and the surrounding solution, and on the rate at which entire micelles can be
destroyed at or near the interface. Swelling of the micelles, which must be unswollen far
from the interface, is necessary to maintain the equilibrium structure predicted in our SCFT
simulations. Transfer of copolymer between micelles and the solution and/or destruction of
entire micelles is necessary to maintain local equilibrium between the interface and nearby
micelles, i.e., to maintain equal chemical in the interface equal to that in nearby micelles.
Destruction of micelles is unavoidable if there is a non-negligible flux of copolymer into the
second phase.
If micelles had insufficient time to swell as they approached the interface, the chemical
potential of copolymers within the micelles would be higher than that expected in a swollen
micelle. If the chemical potential in the interface nonetheless stayed equal to that in
the resulting unswollen micelles, this would result in a higher interfacial concentration of
adsorbed copolymer, and thus a lower interfacial tension than obtained in equilibrium. This
is essentially the scenario described by Leibler, who assumed equilibrium between micelles
and interface, but did not allow for swelling of the micelles, and found a vanishing interfacial
tension for f A < 0.69 as a result. Our experimental results resemble Leibler’s predictions,
insofar as we obtain an interfacial tension much lower than predicted for swollen micelles
153

for nearly symmetric copolymers. It thus seems possible that the interfacial tension is
reduced by a limitation on the rate at which the micelles can swell.
154

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157

Chapter 7
Micellization and Interfacial
Tension in Viscous Ternary
PS/PB/P(S-b-B) Blends
Abstract
We have measured the interfacial tension (γ) between polystyrene (PS) and 1,2 polybutadiene
(PB) in the presence of poly(styrene-b-butadiene) (SB) block copolymer concentration using
a spinning drop tensiometer (SDT). The critical micelle concentration (CMC) of a
binary system of PS and SB copolymer was independently determined from small angle
X-ray scattering (SAXS) and transmission electron microscopy (TEM). The value of γ
obtained at high copolymer concentrations agreed very well with the equilibrium value
predicted by Self Consistent Field Theory (SCFT). The measured tension continues to fall
to this predicted value and becomes saturated until the copolymer concentration reaches
a value of 6 %, substantially above the measured CMC, < 0.5%. We account for this
discrepancy by considering the kinetics of copolymer diffusion to the interface, and how it
is affected by the bulk concentration of copolymer.
158

7.1 Introduction
This chapter reports the results of experiments on systems containing polystyrene (PS)
and polybutadiene (PB) homopolymers mixed with poly(styrene-b-butadiene) (SB) block
copolymers. Most of the experiments reported here were carried out before the experiments
on the system of PI/PDMS/IDMS in the previous chapter. The choice of a polymer pair
including PS, which is a highly viscous polymer with a high glass transition temperature,
was motivated by a desire to use transmission electron microscopy (TEM) to visualize
micelles at room temperature. For this system we thus used both TEM and small angle
X-ray scattering (SAXS) to independently determine the critical micelle concentration in
binary mixtures of PS and SB copolymers, in addition to measurements of interfacial
tension in ternary samples.
The kinetics of copolymer diffusion in our spinning drop tensiometer are, however,
vastly different in this system than in the PI-PDMS system. In spinning drop tensiometer
experiments, a drop of PB (the less dense material) is surrounded by a matrix of PS. The
diffusivity of copolymer in the PS matrix is several orders of magnitude slower than that
in the PB drop, and also much lower than the diffusivity in either PI or PDMS. The slow
diffusion of copolymer in the PS matrix introduces transport limitations in this system
that complicate the behavior observed in tensiometer experiments, as discussed in more
detail below.
7.2 Experiment
7.2.1 Materials
We have studied several mixtures of polystyrene(PS), 1,2 polybutadiene (PB) and poly(styreneb-
1,2 butadiene) (SB) block copolymer. Characteristics of these materials are listed in Table
7.1 and Table 7.2.
159

Table 7.1: Material Characteristics of Homopolymers
sample code M n (g/mol) a M w /M n N b %1,2 η o (Pa·s) c γo d (PS4K) γo d (PS12K)
PS4K 3900 1.08 48 60
PS12K 12700 1.05 159 3500
PB4K 4000 1.09 57 94 0.1 0.82 1.5
PB5K 5300 1.13 76 94 0.2 0.95 1.67
PB13K 13200 1.06 188 93 1 1.23 1.79
PB26K 26000 1.05 374 95 10 1.35 2.01
a The number average molecular weights determined by GPC and converted using
polystyrene standards. b Based on a reference volume v = 81.5 cm 3 /mol at 140 ◦ C .
[2] c Zero shear viscosity measured at 140 ◦ C with ARES d Interfacial tension measured
with PS4K and PS12K, respectively.
Table 7.2: Material Characteristics of Block Copolymers
sample code M PS (g/mol) a M PB (g/mol) a M w /M n fPS b %1,2
SB1 c 9900 9700 1.08 0.47 88
SB2 15000 8700 1.03 0.59 84
SB3 22000 12600 1.05 0.62 95
SB4 16300 9200 1.04 0.61 5
a The number average molecular weights of PI blocks determined by GPC and converted
using polystyrene standards. b The volume fraction of PS in SB determined by NMR. c
Purchased from Polymer Source.
160

Polystyrene (PS) homopolymer was anionically synthesized in the experiment. Styrene
(Aldrich) monomers were stirred with calcium hydride (CaH 2 ) for a day and purified
with dibuthylmagnesium under vacuum for 4 hours at 298K. Polymerization was initiated
with sec-butyl lithium and carried out in 10 vol % cyclohexane at 40 ◦ C for 4 hours.
After terminated with degassed methanol, the polymer solution was washed with deionized
water ten times followed by filtering it with 0.2-µm filter disk and then quenched in a 3:1
vol mixture of methanol and 2-propanol. The precipitated product was purged under
nitrogen gas for a day and further dried in a vacuum oven for a month. Polymerization
of polybutadiene (PB) homopolymer was similar to the above procedure except for the
following details. Butadiene monomers were successively stirred with dibuthylmagnesium
and n-butyl lithium at 0 ◦ C for four hours each. A two-fold molar excess of purified DIPIP
(1,2-dipiperidinoethane) was initially added to cyclohexane with sec-butyl lithium in order
to promote 1,2-addition. Finally, an antioxidant Irganox was added to PB homopolymer
at a concentration of 0.1 wt %.
Poly(styrene-b-butadiene) (SB) block copolymer was synthesized by sequentially adding
styrene then butadiene monomer. Before adding butadiene monomer an aliquot of PS
block was cannulated out and terminated in methanol to determine the molecular weight.
A small amount of low molecular weight PS homopolymer was observed in the gel permeation
chromatography (GPC) curve in Fig. 7.1. This was removed by dissolving the final
block copolymer in acetone, which is a good solvent for PS but a poor solvent for SB, at 5
wt % solution for 3 hours three times. Afterwards the phase separated SB was quenched
in methanol and dried in a vacuum oven.
Experiments have been carried out on several different types of samples. TEM experiments
have been used to determine the critical micelle concentration (CMC) in binary
mixtures of PS4K and all of the copolymers. SAXS experiments have been used to determine
the CMC in a mixture of PS4K and SB1. The CMC in such a binary mixtures
161

0.3
0.25
0.2
before
1st
2nd
3rd
I (a.u.)
0.15
0.1
0.05
0
19 19.5 20 20.5 21 21.5 22
V e (ml)
Figure 7.1: GPC curves of poly(styrene-b-butadiene) block copolymer before and after
fractionation
is expected to be somewhat above the equilibrium CMC in the PS-rich phase of corresponding
equilibrated two-phase ternary mixture (the situation relevant to an interfacial
tension measurement), in which the freedom to emulsify PB within the core of the micelle
can slightly reduce the free energy of formation of a micelle. Measurements of bare homopolymer
interfacial tension have been carried out between PS4K and PS12K and all of
the PB homopolymers. The measurements of interfacial tension in the presence of block
copolymer were all conducted on samples containing PS4K, PB13K, and SB1. In all but
one measurement, the copolymer was initially mixed into the PS matrix. Disordered emulsions
of PS4K and PB13K compatibilized by SB1 (i.e., blends, in the usual sense) have
also been prepared by mechanical mixing and studied by transmission election microscopy.
Though copolymer SB1 is nearly symmetric (f PS = 0.47), it is expected to be somewhat
more soluble in PS4K than in PB13K, and to preferentially forms micelles in PS4K, as
a result of the lower molecular weight of the PS homopolymer ”solvent”. Flory-Huggins
theory with χ = 0.08 predicts a partition coefficient K = 10 for this molecule between
162

PS4K and PB13K. The other copolymers (which have been less heavily studied) have a
stronger tendency to segregate into PS.
Experimental results for both critical micelle concentrations and interfacial tensions
have been compared with SCFT predictions. Appendix A reviews literature values for the
Flory-Huggins interaction parameter χ(T) for PS-PB mixtures, which are needed as an
input to SCFT. Appendix B reviews the way to estimate the diffusion coefficient of block
copolymer in the homopolymer matrix using the friction factors of the pure homopolymers.
7.2.2 Blending
The PS matrix used in the measurement of interfacial tension was prepared by dissolving
the PS homopolymer and block copolymer in 20 wt % benzene and by subsequently evaporating
the solvent with the freeze-drying method. The homogeneous polymer solution was
rapidly quenched in liquid nitrogen and vacuum dried for three days. The resulting porous
powder of the binary PS/SB mixture was vacuum annealed at 140 ◦ C for a hour before the
spinning drop experiments so as to remove most air bubbles trapped in the freeze-dried
sample and form micelles in the PS matrix.
Disordered emulsions of PS/PB/SB were obtained by mixing an equal volume of the
pure PB homopolymer with the a PS copolymer mixture taken from the matrix of completed
spinning drop tensiometer samples after completion of the interfacial tension measurement.
Mechanical mixing was conducted in a 13 mm diameter, cup-and-rotor miniature
mixer (Mini Max CS-183MMX, Custom Scientific Instruments, Inc.). The samples were
blended at a speed of 320 rpm with three steel balls for 20 minutes at 140 ◦ C under a
nitrogen gas blanket and rapidly quenched in liquid nitrogen to reserve the morphology.
7.2.3 Preparation of SDT Samples
A cylindrical glass tube was vertically installed in a preheated oven shown in Fig. 7.2 after
its one end is sealed with a sliding plug. The plug was equipped with a set of the calibration
163

posts from 0.5 mm to 2.0 mm. Temperature was set to 160 - 180 ◦ C , which is high enough
for the PS matrix to easily flow on the glass wall under small pressure difference. Once a
matrix chunk was loaded, vacuum was pulled through the glass tube for an hour until all
air bubbles were completely removed. This procedure was repeated several times until the
tube was filled with the required volume of PS matrix.
For each sample, a 1 ml gas-tight syringe (Hamilton) containing the PB homopolymer
was prepared separately. The syringe and the glass tube in the heating oven were carefully
aligned using the translational stage shown in Fig. 7.3. After the syringe was plunged into
the middle of the matrix, it was important to wait for a while until the PB drop grew
to a proper size at the tip of a needle due to the volume expansion instead of pushing a
syringe piston. When the needle was withdrawn, a trace of the PB homopolymer always
remained in the matrix and eventually broke into tiny droplets due to Rayleigh instability.
Thus there were always several small drops coexisting with a main drop of interest during
experiments. Finally, a conjugate sliding plug that has a threaded hole at a center was
inserted into the glass tube until the first drop of the molten matrix was squeezed through
the hole, and then sealed with a metal gasket and a bolt.
7.2.4 Interfacial Tension Measurement
The instrumental description and the detailed operating procedure of a SDT are summarized
in Chapter 6. All interfacial tension measurements were carried out at 140 C. Samples
were loaded inside a heating chamber preheated to 140 ± 1 ◦ C and covered with a glass
enclosure to minimize heat loss by convection of air around the rotating tube. Temperature
was measured right next to the rotating tube using a wire-type thermometer.
When the drop diameter d becomes time independent at a constant angular velocity
ω, the interfacial tension γ is calculated from the operating equation[1]
γ = 1
32 (ρ PS − ρ PB )ω 2 d 3 . (7.1)
164

Figure 7.2: The sample preparation for a spinning drop tensiometer in a the heating oven.
The heating tape is removed for clear demonstration.
Figure 7.3: The experimental setup to inject a PB drop into a PS matrix. The region near
a needle of a syringe is blown up at the right corner.
165

where are ρ PS = 0.983 g/cm 3 and ρ PB = 0.860 g/cm 3 [2] are the densities of PS and PB,
respectively. In each experiment on a system containing block copolymer, the apparatus
was allowed to spin at constant angular velocity for 24 hours.
A small amount of the PS matrix was taken with the PB drop after every experiment
for GPC analysis. The cross-linking and thermal degradation of the homopolymers and
block copolymer were negligible.
7.2.5 Transmission Electron Microscopy (TEM)
Samples were microtomed (Ultracut Microtome, Reichert) to thin slices with a thickness of
50 nm using a diamond knife either at room temperature for the binary mixtures, or at -150
◦ C for the ternary mixtures. The resulting sections collected on a 400 mesh copper grid
were stained by vapor of osmium tetraoxide (OsO 4 ) aqueous solution for 30 minutes. The
sample morphologies were observed using JEOL 1210 transmission electron microscope
(TEM) at an accelerating voltage of 120 kV. The PB domains are preferentially stained,
and so PB appears dark in the TEM images.
7.2.6 Small Angle X-ray Scattering (SAXS)
The SAXS measurement were performed for the binary blends of PS/SB in a 6m small
angle beam line at the University of Minnesota. Samples were placed in a chamber under
a helium atmosphere at 140 ◦ C . The integration of isotropic two dimensional scattering
data in an azimuthal direction produced the 1D plots of the scattering intensity I vs. the
magnitude of the scattering vector q = 4π/λ sin(θ/2) where θ and λ are the scattering
angle and the wavelength (1.54 Å), respectively.
7.2.7 Dynamic Mechanical Spectroscopy
The zero shear viscosity η o of each homopolymers was measured through dynamic mechanical
spectrometry with a TA instruments ARES strain-controlled rheometer using 25 mm
166

parallel plates. The PS powder was molded into 1 mm thick disks using a hot press while
liquid PB was directly used. Samples were protected under nitrogen gas.
7.3 Results
7.3.1 CMC in PS4K/SB Binary Mixtures
The CMC of block copolymer in the PS4K matrix was determined using the TEM and
SAXS analysis. To verify the micelle morphology had reached thermal equilibrium, we
prepared binary blend samples by four different methods. First, binary mixtures of PS4K
and SB were flame-sealed in ampules under high vacuum and annealed in an oil bath at
140 ◦ C for a day and quenched in liquid nitrogen or ice water. Second, samples placed
on an aluminum foil in a vacuum oven were annealed for a day and directly quenched in
liquid nitrogen in order to increase the cooling rate of the bulk sample by removing a glass
wall of the ampule. Most samples were prepared in this way. Third, we annealed samples
inside the ARES chamber under nitrogen atmosphere for an hour and quenched it in liquid
nitrogen so as to simulate the actual experimental condition in a sense that the spinning
drop experiments were carried out at atmospheric pressure. Fourth, samples dissolved in
5 wt% toluene solution were cast into a 1 mm thick film on a glass plate, left in the hood
for a week, and vacuum dried for three weeks until there was no change in weight. These
samples were then annealed for 2 days.
Fig. 7.4 shows a series of TEM images of micelles in PS4K/SB1 mixture with increasing
concentrations of block copolymer. By counting the number of micelles ρ m per unit area,
we estimated the volume fraction of micelles φ m as
φ m = 4πR3 ρ m
3h
1
f PB
(7.2)
where a radius of the micelle core R was approximated to be 20 nm and a thickness of the
sample h to be 50 nm + 2R.
167

(a)
(b)
(c)
(d)
168

(e)
(f)
(g)
(h)
Figure 7.4: TEM images of the binary blends of PS4K/SB1 at (a) 0.125 wt% (b) 0.25 wt%
(c) 0.5 wt% (d) 1 wt% (e) 2 wt% (f) 3 wt% (g) 4 wt% (h) 6 wt%. The scale bars are 50
nm.
169

8
7
6
5
φ m
4
3
1
2
1
0
0 1 2 3 4 5 6 7 8
φ c (%)
Figure 7.5: The critical micelle concentration in the binary system of PS4K/SB1 determined
by TEM.
From the plot of φ m vs. φ c in Fig. 7.5, the CMC (φ cmc
c ) was determined to be approximately
< 0.5 wt % by extrapolating φ m to zero. The CMC we obtained for PS4K/SB1 is
much smaller than that reported by Kinning et al. [3] for a very similar system of PS 3.9K
and SB 12K-10K, in which the PB block is predominantly 1,4 addition, for which they
reported the CMC of 2.7 ± 0.2 wt % at 115 ◦ C . Since χ is inversely related to temperature,
we initially expected to get the significantly higher CMC at 140 ◦ C than reported
by these authors. Since the CMC is scaled as e −χf PBN C
, and thus very sensitive to small
changes in χ, we considered the possibility that the difference could be due differences in
the fractions of 1,2 and 1,4 isomers between our copolymer SB1 (which is predominantly
1,2) and the copolymer used by Kinning et al., which contained predominantly 1,4 PB. To
explore this possibility, we also measured the CMC by TEM for mixtures of PS4K with
block copolymer SB4, which contains a predominantly 1,4 PB block, as well as with the
other two copolymers SB2 and SB3 described in Table 7.2.
170

(a)
(b)
(c)
(d)
Figure 7.6: TEM images of the binary blends of PS4K/SB2 at (a) 0.125 wt % (b) 0.25 wt
% (c) 0.5 wt % (d) 1 wt %. The scale bars are 100 nm.
171

(a)
(b)
Figure 7.7: TEM images of micelles in the PS4K matrix (a) 0.5 wt % SB3 and (b) 1 wt %
SB4. The scale bars are 50 nm and 100 nm, respectively.
Nevertheless, as shown in Fig. 7.6 and Fig. 7.7, we found no substantial differences in the
number of micelles that still exist at concentrations less than 0.5 wt % regardless of 1,2
isomers in PB or the length of the PS block. Though it would be difficult for us to rule out
the CMC far below 0.5 wt %, our results do seem to rule out the CMC as high as that found
by Kinning et al. The CMC in the PS4/SB1 mixture has been independently determined
by SAXS. The one dimensional normalized intensity I(q,φ) for various concentrations φ
of SB1 in the PS4K matrix is shown in Fig. 7.8. The difference between I(q,φ) and the
intensity I(q,0) of of the pure PS homopolymer was integrated to calculate the integrated
excess scattering intensity [4]
∫
Q(φ) = 4π
dq q 2 [I(q,φ) − I(q,0%)] (7.3)
If this excess scattering intensity is proportional to the concentration of micelles, the CMC
can be taken to be the concentration at which Q(φ) is linearly extrapolated to zero, as
shown in Fig. 7.9. The CMC obtained by this method was obviously less than 0.5 wt %,
172

log I (a.u.)
1
0.8
0.6
0.4
0 %
0.125 %
0.25 %
0.5 %
1.0 %
2.0 %
3.0 %
4.0 %
6.0 %
0.2
0
0 0.01 0.02 0.03 0.04 0.05
q (Å −1 )
Figure 7.8: The normalized SAXS intensity in the binary system of PS4K/SB1.
180
160
140
120
10 -3 Q (nm -3 )
100
80
60
40
20
0
-20
0 1 2 3 4 5 6 7 8 9
φ c (%)
Figure 7.9: The critical micelle concentration in the binary system of PS4K/SB1 determined
by SAXS.
173

ut could not be precisely determined due to inaccuracy of the baseline correction for small
concentrations of SB1.
From the above results with TEM and SAXS, the highest χ = 0.080 in Fig. A.1 in
Appendix A, which corresponds to the value for 93 % of 1,2 PB, was chosen at 140 ◦ C in
this chapter for the SCFT calculation. It gives φ cmc
c
7.3.2 Bare Interfacial Tension
∼ 0.5% in the binary system.
We have measured bare interfacial tension γ o in the binary homopolymer system of PS/PB
with various molecular weights listed in Table 7.1 in order to test the accuracy of SCFT
predictions for this quantity. With increasing molecular weights of the PB homopolymers,
denoted M PB , γ o approaches the respective plateau value in Fig. 7.10 and Fig. 7.11, in
which the PS matrix was fixed to PS4K and PS12K. Although the dependence of interfacial
tension on χ, which is scaled as ∼ √ χ, is much less than the exponential dependence
of the CMC, SCFT still produces a rather wide range of prediction for γ to cover the
whole experimental results. The upper and lower bounds are obtained with χ = 0.080
and χ = 0.054 that correspond to the value for the highest (93 %) and lowest (7 %) 1,2-
fraction in the PB homopolymer. Both figures commonly show slower increase of measured
interfacial tension relative to the theoretical prediction. Note that the sign of the error in
the theoretical prediction of γ o is opposite to that found in the PI/PDMS system.
7.3.3 Interfacial Tension Reduction
Fig. 7.12 shows the interfacial tension reduction between PS4K and PB13K at 140 ◦ C with
increasing volume fraction of SB1. In all of the measurements but one, the copolymer was
initially mixed in the PS phase at the concentration show. The data point shown by a filled
triangle represents an experiment in which the copolymer was initially mixed with the PB
drop. The numerical values of interfacial tension in this plot are also given in Table 7.3.
The dotted and solid lines are SCFT predictions for two slightly different values of
174

3
2.5
2
γ (mN/m)
1.5
1
0.5
0
0 5 10 15 20 25 30
M PB (kg/mol)
Figure 7.10: Bare interfacial tension between PS4K and PB as a function of the molecular
weights of PB. The experimental results are bound by the upper and lower limits in the
SCFT prediction. The dotted line is given with χ = 0.054 and the solid line is with
χ = 0.080, respectively.
3
2.5
2
γ (mN/m)
1.5
1
0.5
0
0 5 10 15 20 25 30
M PB (kg/mol)
Figure 7.11: Bare interfacial tension between PS12K and PB.
175

χ = 0.08 and χ = 0.064. In these calculations, the CMC was calculated for an equilibrated
two phase ternary system, in which the micelle core swells with PB homopolymer. The
CMC predicted with χ = 0.064 is 4.6 % for this ternary system, and 5.9 % for the binary
system of PS4/SB1 whereas the CMC with χ = 0.08 is 0.83 % for the ternary system
and 1.0 % for the binary system. Note that the predicted CMC is very sensitive to small
changes in χ, but the predicted value of γ above the CMC is nearly insensitive to changes
in our estimate of χ, as noted in Chapter 4.
The interfacial tension measured in experiments in which copolymer is premixed with
PS4K initially decreases with increasing copolymer concentration, and appears to level
off above roughly 6% copolymers. The value of γ obtained above this concentration is
very close to that predicted by SCFT, for either choice of χ. The concentration above
which the tension saturates is, however, far above the CMC of 0.5 % or less that was
obtained for a binary PS4K/SB1 system by both TEM and SAXS. The CMC in a ternary
two-phase system is always somewhat lower than that of the corresponding binary system,
because the freedom of a micelle to swell by absorbing PB homopolymer generally lowers
the free energy of formation of a micelle. The predicted difference between CMC in binary
and ternary systems thus increases the discrepancy between TEM observations and these
interfacial tension measurements. We believe that the discrepancy is the result of transport
limitations, as we discuss below.
We conducted one interfacial tension measurement on a sample in which 4 wt % SB1
block copolymer was added to the PB drop, rather than the PS matrix. The result is
indicated as a solid triangle in Fig. 7.12. In this case we found an interfacial tension was
very similar to the plateau value (0.13 mN/m) found in the other experiments at somewhat
higher concentrations of copolymer, and very similar to the SCFT predictions for the true
equilibrium. In this experiment, we also noted that the drop radius reached its final value
much more rapidly than in experiments in which copolymer was added to the matrix. It
176

2
1.8
γ (mN/m)
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
χ=0.064
0.080
SB1/PS
SB1/PB
0
0 1 2 3 4 5 6 7 8 9
φ c (%)
Figure 7.12: The interfacial tension reduction of PS4K and PB13K vs. the concentration
of SB1 that is premixed with PS4K (circle) or PB13K (triangle). The solid line is the
SCFT prediction with χ = 0.064 and the dashed line is with χ = 0.080.
177

1.4
d (mm)
1.2
1
0.8
0.6
1 %
2 %
4 %
6 %
8 %
4%(PB)
0.4
0.2
0
0 200 400 600 800 1000 1200
t (min)
Figure 7.13: The change in a diameter of the PB drop as a function of time (min) in
spin-up experiments. The data are plotted on a linear scale.
d (mm)
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
1 %
2 %
4 %
6 %
8 %
4%(PB)
0
10 0 10 1 10 2 10 3
t (min)
Figure 7.14: The change in a diameter of the PB drop as a function of time (min) in
spin-up experiments. The data are plotted on a logarithmic scale.
178

Table 7.3: The Reduction of Interfacial Tension
γ b (mN/m)
0 1.23
0.125 1.03 1.05
0.25 0.935
0.5 0.84 0.93
1 0.58 0.60 0.72
2 0.35 0.45
4 0.21 0.23 0.12 c
6 0.13
8 0.14
a wt % of SB1 b interfacial tension of PS4K/PB13K c SB1 was premixed with the PB13K.
φ c
a
is easily noticed in Fig. 7.13 on a linear scale and in Fig. 7.14 on a logarithmic scale.
7.3.4 Blend Morphology
Fig. 7.15 shows TEM images of the morphology of ternary blends of PS4K/PB13K/SB1
with 50:50 vol mixture of the homopolymers at various concentration of block copolymer
premixed into the PS.
However, the average drop size dramatically decreases as φ c increases up to 6 wt %.
With 1 - 2 wt % of SB1, the PB domain size is actually out of the scope of TEM. These
samples with low concentrations of SB1 also were sticky, suggesting macroscopic phase
separation, while those containing more block copolymer were not. We also examined the
morphology of a blend made from PB homopolymer containing 4 wt % copolymer mixed
with pure PS4K. The morphology for this sample is very similar to that found for 1 - 2 wt
% copolymer in PS, with large droplets, despite the low interfacial tension found for this
sample in the spinning drop tensiometer.
7.4 Discussion
In this chapter we have observed that interfacial tension shown in Fig. 7.12 becomes saturated
only above an apparent CMC of roughly 6 % copolymer that is significantly higher
than the true CMC of under 0.5 % obtained by the TEM and SAXS. However, the sat-
179

(a)
(b)
(c)
(d)
Figure 7.15: TEM images of the ternary blends of PS4K/PB13K/SB1 at various concentrations
of SB1. (a) 1 wt% (b) 2 wt% (c) 4 wt% (d) 6 wt%. The scale bar corresponds to
400 nm in (a) and (b), and 200 nm in (c) and (d), respectively.
180

urated value of interfacial tension was in a good agreement with SCFT predictions. In
this section, we consider whether the unexpectedly small reduction in γ at low copolymer
concentrations could be a result of transport limitations. Specifically, we consider whether
the concentration of copolymer at the interface might be suppressed to a value below the
equilibrium value for total copolymer concentrations above the true CMC, and remain suppressed
throughout the time scale of the experiment due to a slow transport of copolymer
from the matrix to the PB drop
Our discussion is based on the analysis presented in Chapter 5. There, we considered
the diffusion of copolymer from a micellar phase (phase I, the PS matrix) into a phase of
initially pure homopolymer (phase II, the PB drop). We found that the time required for
the concentration of free molecules at the interface to first reach the CMC c I c in phase I
depends upon whether there exists an exclusion zone with no micelles near the interface in
phase I during the early stages of the transport process. When no exclusion zone exists,
the concentration of free molecules is expected to reach the CMC quite quickly, in a time
comparable to that required to establish a nearly saturated monolayer. When an exclusion
zone is present, however, the concentration of free molecules will reach the CMC only
when the drop is filled with a homogeneous concentration c I c /K, where K is a partition
coefficient of copolymer between a phase I and II.
In order to provide a plausible explanation for our observation, that is, interfacial tension
remains above the saturation value over a range of copolymer concentrations beyond
c I c , it would be necessary to verify both conditions that there could exist an exclusion zone
under the conditions of interest and that there might be insufficient time to fill the drop to
its final concentration within the 24 hour experiments. In addition, a complete explanation
would also need to show that either condition might not be satisfied at higher copolymer
concentrations, where we seem to observe the equilibrium interfacial tension. In addition,
we provide a brief analysis on kinetics in the case that block copolymer was initially added
181

to the PB drop which thermodynamically resist to micellization in equilibrium.
7.4.1 Kinetics Below the CMC
Below the CMC, the diffusion of block copolymer is well defined by the standard free
diffusion in the semi-infinite composite media.[5] Given the diffusion coefficients of block
copolymer SB1 estimated in Appendix B, which are D I f ≃ 1 × 10−11 cm 2 /s in the PS
matrix and D II
f
≃ 3 × 10 −8 cm 2 /s in the PB drop, and the partition coefficient K =
c I f /cII f
≈ 13 from the Flory-Huggins theory (Eqn. 6.5), the time-independent interfacial
concentration c I f,int of block copolymer relative to the bulk concentration co is uniquely
fixed to c I f,int /co ≃ 0.19 from Eqn. 5.2. It implies that the actual reduction of interfacial
tension obtained during steady state will be about five times smaller than the reduction in
equilibrium, which is reasonably consistent with the result at c o = 0.125 wt % in Fig. 7.12.
7.4.2 Kinetics Above the CMC
Next we consider the micellar system above the CMC and determine whether an exclusion
zone might exist in the PS matrix before a homogeneous solution of copolymer is created
in the drop. According to the analysis in Chapter 5, an exclusion zone should form when
the dimensionless parameter Q defined in Eqn. 5.7 is greater than 1. To calculate Q, we
may estimate the diffusivity of a micelle in PS using the Stokes-Einstein equation D I m =
kT/(6πη o R m ),[6] where η o is the measured viscosity of PS 4K and R m is the hydrodynamic
radius of a micelle. R m was obtained as a function of f A in Fig. 6.20, but the balance
point was shifted to f bal
A
= 0.4 assuming K = 1 at the balance point. As a result, we used
R m = 380 Å, which is somewhat swollen compared with an unswollen micelle (R m = 170
Å).
Fig. 7.16 shows our estimate of Q as a function of block ratio f A of block copolymer
with the fixed homopolymer lengths and the concentration of micelles c o m/c c = 5. For the
present system marked with an arrow, Q is slightly greater than the unity, regardless of
182

10 4 0.4 0.45 0.5 0.55 0.6 0.65
10 2
unswollen micelle
swollen micelle
10 0
Q
10 −2
10 −4
f A
Figure 7.16: The variation of Q as a function of the block ratio f A . The molecular weights of
the homopolymers are fixed and c o m/c c = 5. f A of SB1 is indicated as an arrow. essentially
identical to the CMC.
10 2 0 5 10 15 20 25 30
10 1
Q
10 0
10 −1
c m /c c
Ι
Figure 7.17: The dependence of Q on the micelle concentration.
183

swelling of a micelle core, so that the exclusion zone would open in the PS matrix due to an
insufficient micelle flux. Also, we plot Q as a function of with varying initial concentration
of copolymer c I c/c o m in Fig. 7.17 for the current PS/PB/SB1 system with K = 13. The
resulting estimate for Q passes through unity at c I c /co m = 12, which corresponds to 6.5 %
assuming c c ≈ 0.5%. In light of uncertainties in the determination of the diffusivity D II
f
of copolymer in the PB drop and of the CMC c I c in PB, the exact value obtained for the
critical concentration at which Q = 1 should not be taken overly seriously. The results do
indicate, however, that it is plausible that an exclusion zone may exist at concentrations
for which we observe an interfacial tension greater than the equilibrium value, and may
close at slightly higher concentrations.
We next consider the time to fill the drop. In this system, it is important to distinguish
between the time required for diffusion to establish a uniform concentration of copolymer
throughout the drop, which we will call τ d , and the time required for that concentration to
rise to its equilibrium level c I c/K, which we will call τ eq . Because diffusivities of copolymer
are much lower in the PS matrix than in the PB drop, we find that the dominant resistance
to copolymer transport arises from the matrix, and that as a result, τ d ≪ τ eq . We estimate
τ d < d 2 /kD II
f
≈ 1 hour with a drop diameter d = 0.5 mm and k = 5.8. The drop is thus
expected to contain an essentially homogeneous concentration of copolymer by the end of
a 24 hour experiment.
To roughly calculate the time τ eq required to ”fill” the drop when an exclusion zone
is present (Q > 1), we have used the early-time solution of the transport problem given
in section 5.2.2 of Chapter 5 to calculate the total amount N(t) of copolymer transported
into a cylindrical drop of a specified diameter d per unit length as a function of time. Then
we calculated the time at which the total amount of material transferred corresponds to an
average concentration per length 4N(t)/πd 2 equal to the equilibrium concentration c I c /K.
To describe situations in which Q is very close to 1, for which we expect the exclusion zone
184

to remain narrow, we may use the limit discussed in subsection 5.2.2. This approximation
yields a time
τ eq = π 64
d 2
D I m
( c
I
c
c o m K ) 2
. (7.4)
For d = 0.5 mm, K = 13, and Dm I = 1×10−12 cm 2 /s, an estimated time τ eq spans from ∼ 2
hour for c I c /co m = 1/10 up to 8.4 days for cI c /co m = 1. About cI c /co m = 3, which corresponds
to 2 % as a total, gives the estimate of a day. If we consider Eqn. 7.4 provides the lower
limit of the time to fill the drop due to the fact that the flux into the drop should decrease as
the concentration increases, we can conjecture the drop will be saturated at slightly higher
concentration. It thus appears to be plausible that there exists a range of concentrations
slightly above the CMC in which there exists an exclusion zone and in which there was
insufficient time to fill the drop over the course of the experiment. Because Eqn. 7.4 for
τ eq decreases as 1/(c o m )2 for a drop of fixed diameter, it also appears to be possible that
the time to fill the drop could become accessible at higher concentrations studied in our
experiments. These estimates thus leave it unclear whether the convergence of γ to its
equilibrium value above a copolymer concentration of roughly 6 % is due to the closing of
an exclusion zone as shown in Fig. 7.16 under conditions in which τ eq lies beyond the time
scale of the experiment, or a reduction in τ eq under conditions for which an exclusion zone
is still present. Either hypothesis is, however, sufficient to explain the observation.
Finally, we consider the experiment in which we mixed 4 % copolymer into the PB
drop. As shown in Fig. 7.12, we found interfacial tension equal to the saturation value
obtained at concentrations higher than 6 % when copolymer was added to the PS matrix.
This is a case in which we have added copolymers to the wrong phase, since the solubility
is higher and the critical chemical potential for micellization is lower in the PS matrix than
in the PB drop. By redefining phase I to be the PB drop and phase II the PS matrix,
the analysis of this situation given in section 5.3 indicates that there must be an exclusion
185

zone in the PB drop near the interface due to the constraint of c I,int
f
≤ c II
c K < cI c , but
that micelles may form at the interface in the PS matrix. If micelles form in PS near the
interface during the early stages of transport, interfacial concentration in the PS matrix
is pinned to the value of c II
c , so that interfacial tension is expected to rapidly drop to the
equilibrium value. The criteria for the formation of micelles in PS near the interface are
given by Eqn. 5.21
2 Df
II
π Df
I
c II
c
c o m K + 1 <
cI c
Kc II c
(7.5)
for S ≪ 1 and by Eqn. 5.22
√
D II
f
D I m
c I c
c o mK <
cI c
c II c K
(7.6)
for S ≫ 1. If either condition is satisfied, then micelles will form near the interface because
total flux should be dominated by the one which gives the larger flux. Although the CMC
c I c
in the PB drop is undetermined in this experiment, the first condition can be safely
fulfilled due to very small diffusivity ratio of D II
f /DI f = 1/3000 unless the CMC cI c
in the
PB drop is too close an interfacial concentration Kc II
c
defined in the true equilibrium.
Note that (2/π)(c II
c /c o mK) ∼ 1. Also, one can easily show the second condition is always
satisfied from the fact that Df II/DI
m < 1 and cII c /co m < 1. This result implies that micelles
in the PB phase supply enough copolymer to produce inverted micelles in PS near the
interface thus achieving an equilibrium interfacial concentration of copolymer. This is
consistent with our observation of an equilibrium interfacial tension.
7.5 Conclusion
In this chapter the interfacial tension in a PS/PB/SB system was measured with a SDT,
and the CMC was independently determined by TEM and SAXS. The value obtained for
interfacial tension at high copolymer concentrations agreed well with SCFT predictions,
but a copolymer concentration well above the observed CMC was required to drive the
186

interfacial tension down to this value. An analysis of the kinetics of copolymer transport
suggests that this is because, at intermediate copolymer concentrations, the concentration
at the interface is suppressed below its value at the CMC in the PS matrix during the
diffusion of copolymer from the matrix into the drop.
187

Bibliography
[1] Vonnegut, B. Rev. Sci. Instrum. 1942, 13, 6
[2] Fetters, L. J.; Lohse, D. J.; Richter, D.; Witten, T. A.; Zerkel, A. Macromolecules
1994, 27, 4639
[3] Kinning, D. J.; Thomas, E. L.; Fetters, L. J. J. Chem. Phys. 1989, 90, 5806
[4] Rigby, D.; Roe, R.-J. Macromolecules 1984, 17, 1778
[5] Crank, J. The Mathematics of Diffusion (Oxford:Clarendon Press) 1975
[6] Einstein, A. Ann. Phys. 1906, 19, 289
188

Chapter 8
Summary and Outlook
Throughout this thesis, we have investigated the role of AB block copolymer as an efficient
surfactant in the immiscible A/B homopolymer blends using both theoretical and
experimental tools. We have analyzed the competition between interfacial adsorption and
micellization of copolymer using SCFT and the possible effects of limited kinetics on the
thermodynamic prediction. In this chapter we summarize our critical results in the thesis,
and propose future research directions.
8.1 Summary
Self consistent field theory (SCFT) has been formulated in Chapter 2 for multi-component
polymer systems in a grand canonical ensemble and numerically implemented with a finite
difference method on a one dimensional adaptive grid. The chemical potentials and
macroscopic pressure in homogeneous mixtures are derived in the limit of Flory-Huggins
theory so as to obtain an expression for interfacial tension.
In Chapter 3, the elastic properties of a saturated AB block copolymer monolayer
separating two coexisting A- and B- rich homopolymer phases in A/B/AB ternary systems
have been characterized using SCFT calculation combined with the Canham-Helfrich
theory to calculate the bending energy of weakly curved surfactant membranes. A curved
interface was stabilized in a natural way by adjusting the pressure difference between
189

coexisting phases in mechanical equilibrium. For block copolymers equal or larger than
homopolymers, all elastic constants extrapolated to zero with decreasing χN at the Lifshitz
point at which no distinct boundary can exist between coexisting phases. For several
sets of asymmetric systems in the length of the homopolymers and/or in the statistical
segment lengths, we could determine the balance points by adjusting the volume fraction
f A of the A block within block copolymer. At the balance point, the copolymer monolayer
has the vanishing spontaneous curvature. As the copolymer brush becomes drier,
the effect of such asymmetry in systems on the balance point decreases. In the limit of a
completely dry brush, block copolymer would be always symmetric at the balance point.
The analytical expression for the Lifshitz points (Eqn. 3.13) predicted the balance points
in the asymmetric systems reasonably well.
In Chapter 4, the thermodynamic competition between interfacial adsorption and micellization
of block copolymer has been examined under the condition in which swollen
spherical micelles exist in one homopolymer phase. Unlike an unswollen micelle in a A/AB
binary system, a micelle coexisting with the excess B phase would swell until the spontaneous
curvature is obtained in a micelle. It leads to the slight but nontrivial reduction
in the excess free energy of the micelle formation, which causes the interfacial saturation
to be always preempted by micellization except at a balance point. As a result, interfacial
tension saturated at the cmc remains at a non-negative value. The thermodynamic
behavior of the systems containing a nearly balanced copolymer monolayer can also be
explained within the context of the Helfrich theory. Combining with the simplified cmc
condition that ignores the translational entropy of a micelle, the Helfrich theory predicted
interfacial tension would quadratically vary with the spontaneous curvature of a copolymer
monolayer and vanish at a balance point. This prediction could be independently verified
with the micelle simulation for various symmetric and asymmetric systems.
Transport of block copolymer from the initial phase I to the other phase II has been
190

analyzed in Chapter 5 depending on the existence of micelles and the choice of the initial
phase. Below the CMC, a copolymer concentration at the interface is independent of time
and explicitly determined by the ratio of the free chain diffusivity in phase I and II, and
the partition coefficient K. Above the CMC, micelle diffusion becomes very important
to increase the interfacial concentration up to the CMC. When block copolymer is added
to the wrong phase, the interfacial concentration can be fixed to the value in the true
equilibrium and micelles form in both phases as long as either condition of Eqn. 5.21 or
Eqn. 5.22 is fulfilled.
In Chapter 6, equilibrium interfacial tension between polydimethylsiloxane (PDMS)
and 1,4 polyisoprene (PI) has been measured using the spinning drop tensiometer (SDT) by
initially adding poly(isoprene-b-dimethylsiloxane) (IDMS) block copolymer to the PDMS
matrix. The spontaneous curvature of the block copolymer monolayer was controlled by
varying the PDMS block or molecular weight of the PI homopolymer. Near the balance
point, symmetric block copolymers were able to decrease interfacial tension by three orders
of magnitude to 10 −3 mN/m. The predicted quadratic dependence of interfacial tension
on the block ratio of the PDMS block f PDMS agreed very well with the experimental
results for f PDMS > 0.65. However, for 0.5 < f PDMS < 0.6, interfacial tension was
much lower than the theoretical prediction with a discontinuous change within a narrow
range of 0.6 < f PDMS < 0.65. Using small angle X-ray scattering (SAXS) for the ternary
blends of PDMS/PI/IDMS along the isopleth, we observed the formation of a bicontinuous
microemulsion phase upon achieving ultra-low interfacial tension.
Interfacial tension in the system of 1,2 polybutadiene (PB) and polystyrene (PS) premixed
with poly(styrene-b-butadiene) (SB) block copolymer has also been measured using
the SDT in Chapter 7. The cmc in this system could be independently identified using
SAXS and TEM. While the PI/PDMS/IDMS system is ideal to access equilibrium interfacial
tension for the reasons discussed in Chapter 5 and 6, slow diffusion of copolymer in
191

the highly viscous PS matrix greatly affects our measurements in this system. As a result,
we have observed that interfacial tension becomes saturated to the equilibrium value only
above an apparent CMC that is significantly higher than the true CMC.
8.2 Future Directions
In Chapter 6, the ratio of the length of PI and PDMS homopolymer to that of IDMS
block copolymer maintained consistently less than the unity following several experimental
systems in the literature on a polymeric bicontinuous microemulsion phase. Since the
formation of a bicontinuous microemulsion phase has been observed upon achieving ultralow
interfacial tension in this thesis, the remaining question is whether interfacial tension
can still be ultra-low even in the condition in which the length of homopolymer is longer
than that of block copolymer. We have not found any reference to clearly answer this
question. However, we speculate that it might not occur because the unbinding transition
in the binary A/AB system is unstable if the ratio of the length of homopolymer to
that of block copolymer is greater than 1.1,[1] which guarantees the appearance of the
collapsed lamellar phase in the ternary system. In addition, it is not obvious why a uniform
bicontinuous microemulsion phase appears at much higher copolymer concentration (φ c =
0.3) compared with φ c ∼ 0.1 in the literature. It is probably either due to the difference in
the molecular weight ratio of components, or the difference in χN c of copolymer. Therefore,
it will be of particular interest to measure interfacial tension in the PI/PDMS/IDMS
system by increasing molecular weight of the PDMS homopolymer. It only requires the
minimal syntheses of the PDMS homopolymers since several PI homopolymers were already
prepared in Table 6.1.
Next it is worth further testing the analysis of the effect of copolymer transport on
the interfacial concentration, given in Chapter 5. For systems with low viscosity such
as PI/PDMS/IDMS, a spinning drop experiment is relevant to quantify the interfacial
192

concentration by measuring interfacial tension. For example, by synthesizing a few more
symmetric IDMS copolymers with total molecular weights between 10 kg/mol (IDMS3)
and 20 kg/mol (IDMS13), we can find an optimum molecular weight that can reach an equilibrium
during the experiment because both block copolymer gave significantly different
results in interfacial tension measurements. Also, we suggest that adding block copolymer
to the wrong phase, in which micellization is unfavorable, beyond the cmc would be a better
choice to measure equilibrium interfacial tension. A preliminary experiment has been
reported in Chapter 7, in which SB1 copolymer was added to the less viscous PB drop
rather than the PS matrix. If we apply the same idea to the system of PI/PDMS/IDMS by
adding copolymer to the more viscous PI drop, we would be unable to get ultra-low interfacial
tension because copolymers should be swept to the PDMS matrix. On the other hand,
a thin film experiments could be ideal for highly viscous systems such as PS/PDMS/PSb-PDMS.
Compared with a spinning drop experiment, it has an advantage to carry out
multiple parallel experiments with several thin film specimens. By solvent casting polymer
solutions on silicon wafers using a spin coater, we can construct a bilayer of PS and PDMS
where block copolymer is initially added to either homopolymer phase with concentrations
below and above the CMC. The thickness of each layer can have a range of 100 - 500 nm
depending on the diffusion coefficient of copolymer in the corresponding layer. Forward
recoil spectrometry (FRES) and transmission electron microscopy (TEM) would be good
experimental techniques to identify the interfacial concentration of copolymer and the population
of micelles along the direction perpendicular an interface. This type of experiment
has been performed by Schulze.[3]
Testing the various situations of copolymer transport using a well defined geometry
either in the SDT or in thin film experiments will be greatly helpful to understand nonequilibrium
behaviors of block copolymer in the ternary polymeric emulsions. Furthermore,
the results might be applied to the reactive blends because block copolymer created by the
193

interfacial reaction will follow the kinetic pattern of premade copolymer.[2, 3]
194

Bibliography
[1] Janert, P. K.; Schick, M. Macromolecules 1997, 30, 3916
[2] Jones, T. D. Ph.D Thesis, University of Minnesota, 2000
[3] Schulze, J. S. Ph.D Thesis, University of Minnesota, 2001
195

Appendix A
Flory-Huggins Interaction
Parameter between PS and PB
The Flory-Huggins interaction parameter χ directly affects the thermodynamic properties
of the polymer blends such as interfacial tension and the CMC, which are scaled as γ ∼ √ χ
and φ cmc
c
∼ e −f BχN c
, respectively. Here we review seven different methods to measure χ in
the literature and present all χ equations we have collected for the system of polystyrene
(PS) and polybutadiene (PB) which is used in Chapter 7.
There have been two main approaches to experimentally determine χ: One is to use the
binary homopolymer blends and the other is to use block copolymer melts (or solutions).
The former fits the cloud points of the binary homopolymer blends using the molar Flory-
Huggins free energy of mixing per unit volume (method #1), [1, 2, 3]
∆G M
vkT = [
φA
V A
ln φ A + φ B
V B
ln φ B
]
+ α φ A φ B (A.1)
where ∆G M is a molar free energy of mixing per unit volume, a reference volume v is
usually taken as v = (v A v B ) 1 2 giving α = χ/v the units of cm 3 /mol, and φ i and V i are the
overall volume fraction and the molar volume of a molecule of type i, respectively. The
other method to use the binary homopolymer blends is to fit the inverse structure factor
S −1 (q = 0)(∝ I −1 (q = 0)) at q = 0 obtained from small angle neutron scattering (SANS)
196

using the random phase approximation (RPA) by deGennes [5] (method #2)[11] given as
S −1 (0) = N −1
A φ−1 A + N −1
B φ−1 B − 2χ
(A.2)
where N A and N B denote the length of the A and B homopolymers. If thermal composition
fluctuation is taken into account above the critical temperature, the susceptibility S(q = 0)
is fitted using a crossover function introduced by Belyakov and Kieselev (method #3).[4, 5]
On the other hand, the latter can be categorized into two types: The first type fits
a plot of I(q ∗ ) at q = q ∗ of disordered block copolymer vs. 1/T[7, 8, 11] (or the I(q)
profile in the entire range of q at each temperature)[9] using the RPA of block copolymer
given by Leibler (method #4)[12] or using the theory with fluctuation correction given
by Fredrickson-Helfand [10] near the order-disorder transition temperature T ODT (method
#5). [5, 13, 11] The second type simply relates T ODT to a critical point χN = 10.495 in
the mean field (method #6) or χN = 10.495 + 41.0 ¯N −1/3 with thermal fluctuation where
¯N = Na 6 v −2 (method #7) for a few symmetric block copolymers with different lengths of
N.[13, 11, 14]
Usually χ is assumed to have a functional form of A/T + B. In the case of the homopolymer
blends, it often includes additional compositional dependence of φ A /T,[2, 3]
which normally exerts a minor contribution to χ. Among the above methods, we believe
that χ extracted from the binary homopolymer blends gives the most fundamental
measurement of the monomer-monomer interaction as discussed by Maurer et al. [11] The
equations of χ (or α) for PS and PB reported in the literatures are summarized in Table A.1
with the value at 140 ◦ C based on v = 81.5 cm 3 /mol. Fig. A.1 compares the temperature
dependence of each χ equation for various fractions of 1,2 isomers in PB. Although χ looks
like increasing with the more 1,2 addition in PB, χ is also substantially affected by the
types of the measurement methods.
Interestingly, given the temperature range, all χs from the Flory-Huggins equation
197

Table A.1: Summary of the Interaction Parameters Between PS and PB
Ref System Interaction Parameter χ h (140 ◦ C ) % 1,2
[1] PS/PB α = −0.0009 + 0.750/T 0.075 20 - 25
[2] PS/PB α = −0.001308 + 0.8756/T − 0.02516φ A /T b 0.064 - 0.080 6, 34
[3] PS/PB α = −0.001436 + 0.9401/T − 0.0329φ A /T c 0.054 - 0.080 7 - 93
[4] PS/PB α = −0.0017 + 0.99/T d 0.038 - 0.064 7, 54, 91
[5] SB α = 0.001 + 0.68/T 0.053 7
[7] SB χ = −0.027 + 28/T e 0.041 e n/a
[8] SB χ = −0.021 + 25/T f 0.042 95
[9] SBS/DOP a χ = 0.00659 + 13.6/T g 0.041 7.4, 9.8
a P(S-b-B-b-S) (SBS) triblock copolymer in dioctyl phthalate (DOP) solvent. b For PS
(M w = 2400 g/mol) and 6% 1,2-PB (M w = 2660 g/mol) among 5 blends. c For PS (M w
= 1500 g/mol) and 43% 1,2-PB (M w = 3960 g/mol) among 16 blends. d For PS (M w =
2030 g/mol) and 54% 1,2-PB (M w = 1870 g/mol) among 3 blends. e v is not available. f
Based on v = 77.7 cm 3 /mol. g Based on v = 79.0 cm 3 /mol. h Based on v = 81.5 cm 3 /mol
and φ A =0.5.
(method #1) [1, 2, 3] consistently have the larger slopes A and magnitudes than those from
the RPA of block copolymers (method #4).[7, 8, 9] Frielinghaus et al. pointed out that the
enthalpic contribution of the homopolymer blend is larger than that of block copolymer
in most systems except the case of polyethylene(PE)/poly(ethylenepropylene)(PEP). [6]
Also, Fig. A.1 shows that the thermal fluctuation correction [4, 5] gives χ values that
deviate considerably from the others in both the binary homopolymer blends and block
copolymer, i.e., χ (#1) > χ (#3) and χ (#5) > χ (#4) . It is consistent with Maurer et al. ’s
observation,[11] that is, the Fredrickson-Helfand theory of block copolymer (#5 and #7)
produced χ with much higher slopes relative to the mean field theory. Moreover, they
observed that only χ determined from the measurement of T ODT in the mean field (#6)
was quantitatively close that of the homopolymer blend, but any other theories with the
disordered block copolymer (#4 and #5) caused significant errors in both coefficients in
χ, which may explain why all χ’s of SB block copolymer are much lower than those of the
PS/PB blends. In Chapter 7, we used the χ from ref [3] for comparison.
198

0.18
0.16
0.14
0.12
0.1
ref [2] 6%
ref [3] 93%
84%
61%
43%
7%
ref [4] 7%
ref [5] 7%
ref [7] n/a
ref [8] 95%
ref [9] 7.4%
blend [2],[3]
χ
0.08
0.06
0.04
} flucutuation [4],[5]
} copolymer [7],[8],[9]
0.02
0
2.4 2.5 2.6 2.7 2.8
1000/T (K -1 )
Figure A.1: The Flory-Hugggins interaction parameter χ between PS and PB with various
% 1,2 of PB in the literature. All values are based on a fixed v = 81.5 cm 3 /mol.
199

Bibliography
[1] Rounds, N. A. Ph.D. Thesis, University of Akron, 1971
[2] Roe, R.-J.; Zin, W.-C. Macromolecules 1980, 13, 1221
[3] Han, C. D.; Chun, S. B.; Hahn, S. F.; Harper, S. Q.; Savickas, P. J.; Meunier, D.
M.; Li, L.; Yalcin, T. Macromolecules 1998, 31, 394
[4] Frielinghaus, H.; Schwahn, D.; Willner, L. Macromolecules 2001, 34, 1751
[5] Frielinghaus, H.; Abbas, B.; Schwahn, D.; Willner, L. Europhys. Lett. 1998, 44, 606
[6] Frielinghaus, H.; Mortensen, K.; Almdal, K. Macromol. Symp. 2000, 149, 63
[7] Hewel, M.; Ruland, W. Makromol. Chem. Macromol. Symp. 1986, 4, 197
[8] Owens, J. N.; Gancarz, I. S.; Koberstein, J. T.; Russell, T. P. Macromolecules 1989,
22, 3380
[9] Sakurai, S.; Mori, K.; Okawara, A.; Kimishima, K.; Hashimoto, T. Macromolecules
1992, 25, 2679
[10] Fredrickson, G. H.; Helfand, E. J. Chem. Phys. 1987, 87, 697
[11] Maurer, W. W.; Bates, F. S.; Lodge, T. P.; Alamdal, K.; Mortensen, K.; Fredrickson,
G. H. J. Chem. Phys. 1998, 108, 2989
[12] Leibler, L. Macromolecules 1980, 13, 1602
200

[13] Bates, F. S.; Rosedale, J. H.; Fredrickson, G. H. J. Chem. Phys. 1990, 92, 6255;
Rosedale, J. H.; Bates, F. S.; Almdal, K.; Mortensen, K.; Wignall, G. D. Macromolecules
1995, 28, 1429
[14] Almdal, K.; Hillmyer, M. A.; Bates, F. S. Macromolecules 2002, 35, 7685
201

Appendix B
Estimate of Diffusion Coefficient of
P(S-b-B) Block Copolymer in the
Homopolymer Phase.
In Appendix B, we provide a simple way to estimate the diffusion coefficient of AB block
copolymer in the corresponding homopolymer matrix using the zero shear viscosity of
the pure A and B homopolymers for the system of PS/PB/SB in Chapter 7. The same
argument was applied to other system PI/PDMS/IDMS in Chapter 6.
We first estimate the diffusion coefficient D SB of block copolymer using homopolymer
data. According to recent studies on composition and temperature dependence of the
monomeric friction factor ζ(φ,T) in immiscible polymer pairs such as PS/PI (polyisoprene)
[1, 2, 3] and PS/PMMA (poly(methyl methacrylate)),[4] segmental dynamics are governed
by two independent mechanisms: the inter- and intramolecular interactions. The former
is dominant in the PS/PI system and the latter in the PS/PMMA system depending on
the bulkiness of the monomer segments, that is, how much the backbone rearrangements
of the chain are hindered by the surrounding medium. It was inferred from the fact
that the effective friction factor ζ eff of disordered PS-PI (multi)block copolymer with the
composition φ exhibits a similar temperature dependence and collapses into a single master
curve relative to the glass transition temperature (T −T g ) whereas it does not happen in the
202

PS/PMMA system due to the different temperature dependence of monomer segments.[3, 4]
Since 1,2 PB has a relatively small side group and a similar chemical structure to PI, it
would be reasonable to anticipate that segmental dynamics in the PS/PB system is also
dominated by the intermolecular interaction. With this assumption, the friction factor
ζ i (φ,T) of any molecule of type i in the common effective matrix becomes all identical,
i.e.,
ζ SB (φ,T) ≈ ζ PS (φ,T) ≈ ζ PB (φ,T)
(B.1)
while the friction factor ζi o (T) of the pure homopolymer differs by a few orders of magnitude.
For comparison ζ PS (φ,T) ≪ ζ PMMA (φ,T) in the PS/PMMA system.[4] Therefore,
block copolymer diffusing in the pure homopolymer matrix with φ = 0 or 1 behaves
as if it were imaginary random copolymer of total length N with the friction factor
ζ SB . Furthermore, applying the Rouse mixing rule for the effective friction factor,
ζ eff = f A ζ A (φ,T)+f B ζ B (φ,T) within the mean field approximation,[5, 4] gives ζ SB ≈ ζ eff
using Eqn. B.1.
The Arrhenius mixing rule is often adopted to predict ζ eff (φ,T) from ζi o (T) of each
component,
log ζ eff (φ,T) = φ A log ζ o A (T) + φ B log ζ o B (T).
(B.2)
In fact, Eqn. B.2 usually overestimates ζ eff (φ,T) of the PS/PI system because overall
segmental dynamics is more facilitated by the effect of PI on plasticizing PS than that of
PS on retarding the motion of PI.[2, 3, 4] Nonetheless, it should be at least accurate for
the pure homopolymer matrix, that is, ζ eff = ζi o. The zero-shear viscosity η o in Table 7.1
is directly related to ζi o(T). Since M PS4K < M e,PS (= 13000 g/mol),[6] ζ and D of the
PS4K matrix is given by
η =
ρN av
36m o
Nb 2 ζ
203
D = kT
Nζ
(B.3)

in the Rouse regime, but since M PB13K ≫ M e,1,4−PB (= 1800 g/mol), those of PB13K are
by
η =
5ρN avm o
48Me
2 N 3 b 2 ζ
D = kT 4N e
Nζ 15N
(B.4)
in the reptation regime where N av is Avogadro’s number, m o and b are the molar mass and
the statistical segment length of a monomer segment.[7] As a result, we obtain ζPS o (140◦ C) ≃
2 × 10 −5 dyn-s/cm and ζPB o (140◦ C) ≃ 3 × 10 −10 dyn-s/cm. The corresponding diffusion
coefficient of block copolymer with N = 260 is D SB ≃ 1 × 10 −11 cm 2 /s in the PS matrix
and D SB ≃ 3 ×10 −8 cm 2 /s in the PB drop, respectively. These results are directly used in
Chapter 7. Therefore, the characteristic time τ for block copolymer to diffuse by a distance
of a typical drop radius x = d/2 ≃ 0.25 mm is τ ≃ x 2 /2D PB ≃ 10 4 sec in PB13K and
τ ≃ 10 7 sec in PS4K suggesting that it takes a couple of hours to reach the center of the
PB drop whereas it takes ∼ 10 2 days in the PS matrix.
204

Bibliography
[1] Hamersky, M. W.; Tirrell, M.; Lodge, T. P. J. Polym. Sci.:Part B 1996, 34, 2899
[2] Milhaupt, J. M.; Chapman, B. R.; Lodge, T. P.; Smith, S. D. J. Polym. Sci.:Part
B 1998, 36, 3079
[3] Chapman, B. R.; Hamersky, M. W.; Milhaupt, J. M.; Kostelecky, C. Lodge, T. P.;
von Meerwall. E. D. Smith, S. D. Macromolecules 1998, 31, 4562
[4] Milhaupt, J. M.; Lodge, T. P.; Smith, S. D.; Hamersky, M. W. Macromolecules
2001, 34, 5561
[5] Green, P. F. Macromolecules 1995, 28, 2155
[6] Fetters, L. J.; Lohse, D. J.; Richter, D.; Witten, T. A.; Zerkel, A. Macromolecules
1994, 27, 4639 ‘
[7] Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics (Oxford:Clarendon Press)
1986
205

Appendix C
The Effect of the Uniaxial
Extensional Flow on Interfacial
Tension
Introduction
The interfacial phenomena has been rather widely studied in simple shear flow. Given
homopolymer blends with droplet/matrix morphology, interfacial tension can be obtained
by fitting the elastic modulus of the blend using the model described by Palierne.[1, 2, 3]
Also, interfacial slip, which is one of the interesting phenomena in the shear flow, often
causes anomalously low shear viscosity in the polymer blends.[4] Such an interfacial slip
can be prevented by adding a small amount of block copolymer.[5] Recently, Moldenaers
showed that interfacial tension responds linearly to the shear strain,[2] that is,
γ = γ eq + β ′ ǫ
(C.1)
where γ eq is an isotropic interfacial tension, β ′ is a complex interfacial dilation modulus,
and ǫ is a shear strain.
On the other hand, there are fewer studies measuring interfacial tension in an extensional
flow. Gramespacher and Meissner developed a method to obtain interfacial tension
using the elastic recovery of the blends after extension.[6] Using multilayer samples Levitt
et al. found that total extensional force is always greater than the bulk average of each
206

component due to the existence of interfacial tension and increases with the number of
layers[7] or the interfacial reaction.[7, 8]
In addition, it would be of great interest to examine the effect of block copolymer on
the interfacial property. It is widely known that block copolymer can effectively prevent
interfacial slip in a shear flow.[5] However, the effect of block copolymer on the interfacial
property of polymer blends has not been studied in an extensional flow in which no relative
motion between adjacent phases exists.
In this Appendix, an extensional rheometer (RME) is used to measure interfacial tension
between polystyrene (PS) and i polypropylene (PP) with multilayer samples, which
consist of 32 PS and PP alternating layers. For some multilayer samples, poly(styrene-bethyl
ethylene) symmetric diblock copolymer (SEE) was initially added to the PS layers.
For the rectangular samples used during extension, the macroscopic dimensions will follow
L(t) = L o exp(˙ǫt)
W(t) = W o exp(− ˙ǫ 2 t)
H(t) = H o exp(− ˙ǫ 2 t)
(C.2)
(C.3)
(C.4)
in which L, W, and H are the length, width, and height of the sample, respectively.
Therefore, it is much simpler to analyze the dimensional change for each phase compared
with the blends.
C.1 Experiment
C.1.1 Materials
Molecular characteristics of all materials used in this paper are described in detail in
ref [5]. For homopolymers, polystyrene (Styron 666D, Dow Chemical), denoted PS, and
linear i polypropylene (Exceed 4062, ExxonMobil), denoted PP, have M w = 200 kg/mol
with M w /M n = 2.0 and M w = 140 kg/mol with M w /M n = 2.4, respectively. Both
207

homopolymers were supplied as pellets. Poly(styrene-b-ethyl ethylene) symmetric block
copolymer, denoted SEE, as anionically synthesized in our lab by Jones [12], has M w = 200
kg/mol with M w /M n = 1.05. Block copolymer was initially melt blended with polystyrene
at a concentration of 2.0 wt % using a twin-screw extruder and then pelletized. For the
multilayer samples, PP was coextruded either with the pure PS or with a mixture of PS
and SEE, denoted PS(SEE), into the stacks of 32 layers using a multilayer coextruder
through layer multiplication dies by Zhao.[5, 11]
C.1.2 Sample Preparation
The pellets were directly molded into 60 × 1 × 8 mm rectangles at 200 ◦ C in a hot press
after being vacuum dried overnight. Sometimes final samples had the nonuniform cross
section, which were usually thinner by ∼ 0.1 mm in the middle part, due to the different
thermal expansion coefficient between the polymers and the stainless steel mold during a
water-cooling process. Thus the average values were taken from at least three points in
both width and height.
In case of the multilayer samples, it was crucial to completely remove the residual stress
that resulted from a sudden quench by a double chill roller at the last stage of the multilayer
preparation.[5] Otherwise, the multilayers immediately shrank into irregular shapes upon
melting. For this purpose, the multilayer samples were annealed at 185 ◦ C for 15 minutes
under slight pressure between two aluminum plates separated with shims a bit thicker than
the initial sample heights and then slowly cooled in air. Also, they were trimmed using
a hot wire cutter to prevent a significant delamination. All irregular molten edges were
carefully trimmed with a razor blade in order to get rid of any potential extra stress from
the irregular surfaces.
208

C.1.3 Extensional Rheology
The extensional rheology of the materials were measured with the Rheometric Scientific
(TA instruments) RME which is an extensional rheometer based on Meissner’s original
design.[9] Samples are suspended using heated nitrogen (N 2 ) gas and elongated at a constant
strain rate, ˙ǫ, up to maximum Hencky strain, ǫ = ˙ǫt = 7, using counter rotating belts.
To ensure the reproducibility, all measurements were repeated at least five times unless
otherwise stated. The images of all elongated samples were recorded with VCR to determine
the true strain rate from a plot of log(W(t)/W o ) vs. t where the slope corresponds
to −˙ǫ/2, and where W(t) and W o are the sample width averaged at three different places
of the sample at time t and t = 0, respectively.[13] The force transducer was regularly
calibrated with known weights and its damping system was always cooled with circulating
water during the experiments. Measured total force F(t) was corrected following guidelines
given in ref [10] . Since the maximum force did not exceed 5 cN up to ˙ǫ= 0.1 s −1
for any sample, in most experiments we employed only two lower belts with double side
tapes to minimize the slip between the belt and polymer. The results with two belts were
consistent with those obtained using all four belts.
C.1.4 Data analysis
The extensional viscosity is directly calculated by measuring total force F(t) and by assuming
the affine deformation of samples,
η ǫ (t) = F(t)
A o ˙ǫ exp(˙ǫt) .
(C.5)
The initial cross sectional area A o at temperature T should be corrected based on the
information at T = 298K as
A o = A 298K
o
∑
i
209
( ) ρ
298K 2/3
φ i (C.6)
iρ i

where φ i and ρ i are the volume fraction and the density of the homopolymer of type i. φ i
is determined from the average layer thickness measured using an optical microscopy.[5]
Interfacial contribution ∆F(t) to total force is obtained by
∆F(t) = F(t) − ∑ i
φ i F i (t)
(C.7)
where F i (t) is the extensional force of the homopolymer of type i.[7] Since each sample
has a slightly different dimension, instead it is more convenient to use the stress, denoted
σ(t) = F(t)/(WH). Defining the interfacial stress ∆σ(t) by
∆σ(t) = σ(t) − ∑ i
φ i σ i (t) ,
(C.8)
interfacial tension γ(t) can be determined using the force balance as
(N − 1)γ(t)W(t) = ∆F(t) = ∆σ(t)W(t)H(t) (C.9)
, which leads to
where H T o = H298K o
∆F(t)
γ(t) =
(N − 1)W(t) = HT o ∆σ(t)
(N − 1)
∑ ( )
i φ ρ 298K 1/3.
i
i ρi
exp(− ˙ǫ 2 t)
(C.10)
C.1.5 Shear Rheology
The transient shear viscosity η + (t) of the homopolymers was measured by start-up experiment
using a strain-controlled TA instruments ARES rheometer with 25 mm parallel
plates of 1 mm gap.
C.2 Results and Discussion
In this section, we measure the extensional viscosity, or equivalently the extensional stress
σ(t) = η ǫ (t) ˙ǫ, of the homopolymer and the multilayer so as to extract interfacial tension
using Eqn. C.10.
210

10 6 T = 170 o C
180 o C
185 o C
10 -1 10 0 10 1 10 2 10 3
10 5
η ε (Pa.s)
10 4
10 3
Time (sec)
Figure C.1: The extensional viscosity of the PS homopolymer at different temperatures of
170, 180, 185 ◦ C . The extensional rate is 0.01 s −1 . Curves are averages of five different
runs.
10 6 T = 170 o C
180 o C
185 o C
10 -1 10 0 10 1 10 2 10 3
10 5
η ε (Pa.s)
10 4
10 3
Time (sec)
Figure C.2: The extensional viscosity of the PP homopolymer at 170, 180, and 185 ◦ C .
The extensional rate is 0.01 s −1 .
211

C.2.1 Extensional Viscosity of the pure Homopolymers
Here we present the bulk extensional viscosities of the components in the multilayers at the
various experimental conditions. Fig. C.1 and Fig. C.2 show the extensional viscosity of the
PS and PP homopolymer, respectively, with ˙ǫ = 0.01 s −1 above melting temperature of PP,
which is 165 - 170 ◦ C .[16] Although the viscosity of PS is greatly affected by temperature,
that of PP is nearly temperature independent, because the test temperatures of 170 to 185
◦ C are much higher than the glass transition temperature of PP, -17 ◦ C ,[16] than that of
PS, 100 ◦ C . Hereafter, all material properties are given at T = 180 ◦ C so as to remove
the effect of the crystallinity in PP.
Fig. C.3 and Fig. C.4 show that there is little extensional thickening for the PS and
PP homopolymers at three extensional rates ˙ǫ= 0.01, 0.05, and 0.1 s −1 at 180 ◦ C . The
extensional viscosity agrees with three times the low shear rate transient viscosity 3η + (t)
(solid line). The deviations of η ǫ from 3η + (t) for t ≤ 1 s in both figures due to initial sag
of the samples.[9, 7, 13]
η ǫ of the binary PS(SEE) mixture is given in Fig. C.3. Interestingly, it is observed that
the absolute magnitude of η ǫ is reduced to about half of that of the pure PS homopolymer
with ˙ǫ= 0.01 s −1 and then increases with increasing strain rate. The values at t = 10 s are
summarized in Table C.1 for comparison. The effect of block copolymer on the viscosity
has not been thoroughly investigated in the literature. Decrease in complex shear viscosity
of the homopolymer in the presence of block copolymer has also been found by Jones[12],
and Yurekli and Krishnamoorti.[17] Such reduction in shear viscosity is often found in
Table C.1: Extensional Viscosity at t = 10 s.
˙ǫ(s −1 ) η ǫ (PS) (Pa·s) η ǫ (PS(SEE)) (Pa·s)
0.01 86000 51000
0.05 74000 56000
0.1 75000 63000
212

10 6 10 -1 10 0 10 1 10 2 10 3
10 5
•
PS ε = 0.01
0.05
0.1
PS(PEE) 0.01
0.05
0.1
3η +
η ε (Pa.s)
10 4
10 3
t (sec)
Figure C.3: The extensional viscosity η ǫ of the pure PS homopolymer and the 2 wt %
binary mixtures of PS(SEE) at extensional rates ˙ǫ of 0.01, 0.05, and 0.1 s −1 at 180 ◦ C .
Note that η ǫ of PS(SEE) is greatly reduced from that of PS due to SEE block copolymer.
The extensional viscosity of PS is consistent with 3η + (t), which is given by a solid line. η ǫ
of PS(SEE) is a weak function of ˙ǫ approaching that of the pure PS.
10 6 •
ε = 0.01
0.05
0.1
3η +
10 5
10 -1 10 0 10 1 10 2 10 3
η ε (Pa.s)
10 4
10 3
t (sec)
Figure C.4: The extensional viscosity of the PP homopolymer at extensional rates of 0.01,
0.05, and 0.1 s −1 . Experimental temperature is 180 ◦ C . The solid line represents 3η + (t)
of PP in the LVE regime.
213

1
0.9
W / W o
0.8
0.7
PS/PP
0.6
PS/PP/P(S-EE)
0.5
0.4
10 0 10 1 10 2 10 3
t (sec)
Figure C.5: Shrinkage in width of multilayer samples during annealing in the RME on a
semi-logarithmic time scale.
hyperbranched polymer blends.[19]
C.2.2 Shrinkage of Multilayers
After the residual stress is removed, as mentioned in section C.1.2, there is another subtle
problem to be considered for the multilayer samples. Homopolymer samples retain their
initial configuration in the melt state for a relatively long time due to their high viscosity
and the surface tension force. However, the multilayer samples continuously shrink in
width and increase in height, because interfacial tension between the many layers tries
to minimize interfacial area within multilayers until the inward interfacial tension force is
balanced by the outward pressure force caused by surface tension (Young-Laplace equation)
on the edge of each layer.
The multilayers were annealed inside a RME chamber before extension for about 10
minutes. Fig. C.5 shows the shrinkage in the width of the multilayer samples with or
without block copolymer as a function of time on a semi-logarithmic plot. Although the
214

multilayers with block copolymer shrink slightly faster, there is no substantial difference
between the curves. The true strain rate was obtained by subtracting the contribution of
the shrinkage from the apparent change in width in the image analysis. Interfacial tension
is calculated by replacing the height H o in Eqn C.10 by
H ′ o(t)| t=to =
H o
(W(t)/W o ) t=to
(C.11)
assuming the length of the multilayer is invariant.
C.2.3 Extensional Viscosity of Multilayers
The extensional viscosity of the PS/PP multilayers are given in Fig. C.6 for ˙ǫ = 0.01, 0.05,
0.1, 0.5, and 1 s −1 . Error bars are not indicated for the data with the last two strain
rates because the measurements were carried out only once. Up to ˙ǫ = 0.1 s −1 , there
is no significant extensional thickening behavior. Fig. C.7 plots all extensional viscosities
measured at ˙ǫ= 0.01 s −1 , including the data of the PS(SEE)/PP multilayers. One can
easily notice that the viscosity of the PS homopolymer is almost indistinguishable from
those of the PS/PP and PS(SEE)/PP multilayer even though the multilayers consist of 50
vol % PP homopolymer, which has much lower viscosity. It unambiguously demonstrates
the nontrivial effect of interfacial tension on total extensional force.
C.2.4 Interfacial Tension
Interfacial tension γ(t, ˙ǫ) under the external flow can be obtained using Eqn. C.10. Fig.
C.8 shows the variation of interfacial tension of PS/PP with ˙ǫ= 0.01, 0.05, and 0.1 s −1
as a function of time. In this figure, one can observe a similar pattern for each interfacial
tension curve. That is, it increases from the equal starting value, reaches a maximum,
and quickly drops to zero. It is consistent with the previous observations. [7, 3]. The
initial plateau, indicated as a solid line, is in good agreement with equilibrium interfacial
tension γ eq = 6.48 mN/m, which was measured using a pendent drop method.[15] We do
215

10 6 10 -1 10 0 10 1 10 2 10 3
10 5
•
ε = 0.01
0.05
0.1
0.5
1
η ε (Pa.s)
10 4
10 3
Time (sec)
Figure C.6: The extensional viscosity of the PS/PP 32-multilayer at extensional rates of
0.01, 0.05, and 0.1 s −1 . Experimental temperature is 180 ◦ C .
10 6 10 -1 10 0 10 1 10 2 10 3
10 5
PS
PP
PS(SEE)
PS/PP
PS(SEE)/PP
3η + (PS)
3η + (PP)
η ε (Pa.s)
10 4
10 3
Time (sec)
Figure C.7: Summary of the extensional viscosities of all samples used in this paper at
˙ǫ = 0.01s −1 . Block copolymer reduces the extensional viscosity by a factor of two. Three
times zero shear viscosities in the LVE region are given for the validity of the measurements.
216

not fully understand at present why γ has a maximum at nearly equal time. Vinckier et al.
argued that the layer structures might be destroyed at high strain.[3] However, Eqn. C.10
suggests that it may also be possible for the decrease in interfacial tension to occur when the
interfacial stress ∆σ(t) reaches a plateau when the system reaches steady state. Moreover,
Fig. C.9 shows that the maximum of interfacial tension γ max has a linear dependence on
˙ǫ implying that the interface can be regarded as a two dimensional Bingham plastic in an
extensional flow[18] as given by
γ(t, ˙ǫ) = γ eq + η ex (t)˙ǫ
(C.12)
where an interfacial viscosity η ex (t) is assumed to be only a function of time. This relationship
between γ and ˙ǫ is consistent with the previous study.[18] Laun and Münstedt
reported that interfacial stress between linear low density polyethylene (LDPE) and the
surrounding silicon oil increased linearly with the extensional rate ˙ǫ. Eqn. C.12 also indicates
that equilibrium interfacial tension can be obtained only when ˙ǫ → 0. Fig. C.10
plots η ex (t) = (γ − γ eq )/˙ǫ vs. time. It is not obvious whether or not η ex (t) is independent
of ˙ǫ with the current experimental results. Finally, Fig. C.11 gives the interfacial tension
in the presence of SEE block copolymer. Surprisingly, the maximum interfacial tension is
larger, not smaller, than that obtained without block copolymer. These results are plotted
together in Fig. C.9 showing that interfacial contribution actually increase due to the
presence of block copolymer in the extensional flow.
217

120
•
ε = 0.01
0.05
0.1
100
80
γ (mN/m)
60
40
20
γ eq
0
10 -1 10 0 10 1 10 2 10 3
t (sec)
Figure C.8: Interfacial tension extracted from the multilayer samples as a function of time
with ˙ǫ = 0.01, 0.05, and 0.1 s −1 . The solid line corresponds to equilibrium interfacial
tension γ eq measured with a pendent drop method.[15]
10 3 PS/PP
PS(SEE)/PP
10 -4 10 -3 10 -2 10 -1 10 0
10 2
γ max (mN/m)
10 1
10 0
•
ε
Figure C.9: Maximum interfacial tension vs. the extensional rate. The linear dependence
of γ on ˙ǫ can be easily observed.
218

1200
η ex = ( γ − γ eq
) / • ε (mN.s / m)
1000
800
600
400
200
•
ε = 0.01
0.05
0.1
0
10 -1 10 0 10 1 10 2 10 3
t (sec)
Figure C.10: Interfacial viscosity η ex = (γ − γ eq )/˙ǫ as a function of time for ˙ǫ = 0.01, 0.05,
and 0.1 s −1 . At t → 0, γ → γ eq .
120
100
•
ε = 0.01
0.05
80
γ (mN/m)
60
40
20
0
10 -1 10 0 10 1 10 2 10 3
t (sec)
Figure C.11: Interfacial tension in the presence of block copolymer. It is interesting that
block copolymer increases interfacial tension. Only one experiment was conducted with
˙ǫ = 0.05s −1 .
219

Bibliography
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[2] Van Hemelrijck, E.; Van Puyvelde, P.; Velankar, S.; Macosko, C. W.; Moldenaers, P.
J. Rheol. 2004, 48, 145
[3] Vinckier, I.; Schweizer, T.; Laun, H. M. J. Polym. Sci.:Part B. 2002, 40, 679
[4] Utracki, L. A. J. Rheol. 1991, 35, 1615
[5] Zhao, R.; Macosko, C. W. J. Rheol. 2002, 46, 145
[6] Gramespacher, H.; Meissner, J. J. Rheol. 1997, 41, 27
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