self-assembly, elasticity and orientational order …...self-assembly, elasticity and orientational...
TRANSCRIPT
SELF-ASSEMBLY, ELASTICITY
AND ORIENTATIONAL ORDER IN SOFT MATTER
A dissertation submitted toKent State University in partial
fulfillment of the requirements for thedegree of Doctor of Philosophy
by
Jun Geng
May 2012
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ProQuest LLC.789 East Eisenhower Parkway
P.O. Box 1346Ann Arbor, MI 48106 - 1346
UMI 3510764
Copyright 2012 by ProQuest LLC.
UMI Number: 3510764
Dissertation written by
Jun Geng
B.S., University of Science and Technology Beijing, 2004
M.S., University of Science and Technology Beijing, 2007
Ph.D., Kent State University, 2012
Approved by
, Chair, Doctoral Dissertation CommitteeDr. Jonathan V. Selinger
, Members, Doctoral Dissertation CommitteeDr. Robin L.B. Selinger
,Dr. Antal Jakli
,Dr. Elizabeth Mann
,Dr. Grant McGimpsey
,
Accepted by
,Director, Chem. Phys. Interdiciplinary Prog.Dr. Liang-Chy Chien
, Dean, College of Arts and SciencesDr. Timothy Moerland
ii
TABLE OF CONTENTS
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Orientational Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Lipid Membranes . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3 Topological Defects . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1 Order Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.2 Mean Field and Landau Theory . . . . . . . . . . . . . . . . . 14
1.4 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4.1 Material Frame vs. Laboratory Frame . . . . . . . . . . . . . 22
1.4.2 Important Tensors . . . . . . . . . . . . . . . . . . . . . . . . 23
iii
1.4.3 Thin Film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.4.4 Target Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.5 Computer Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.5.1 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . 31
1.5.2 Molecular Dynamic Simulation . . . . . . . . . . . . . . . . . 32
1.5.3 Coarse-Grained model . . . . . . . . . . . . . . . . . . . . . . 32
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2 Theory and Simulation of Two-Dimensional Nematic and Tetratic
Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3 Mean-field results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4 Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 Deformation of An Asymmetric Film . . . . . . . . . . . . . . . . . . 58
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.2.1 Brief review of non-Euclidean theory . . . . . . . . . . . . . . 62
3.2.2 Asymmetric film . . . . . . . . . . . . . . . . . . . . . . . . . 64
iv
3.3 Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4 Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4 Coarse-Grained Modeling of Deformaable Nematic Shell . . . . . . 77
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 Two vectors Coarse-Grained Model . . . . . . . . . . . . . . . . . . . 79
4.3 Simulation and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 83
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5 Morphology Transition in Lipid Vesicles: Role of In-Plane Order
and Topological Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2 New Results from the Experiments . . . . . . . . . . . . . . . . . . . 94
5.3 Hypothesize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.4 Coarse-Grained Simulation . . . . . . . . . . . . . . . . . . . . . . . . 100
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6 Works On Other Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
v
6.1 Simulation of Generalized n-atic Order . . . . . . . . . . . . . . . . . 108
6.2 Bilayer model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.2.1 Bilayer Potential . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.2.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 112
6.3 Simulation of Stretching a Two Dimensional Nematic Elastomer . . . 115
6.3.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . 115
6.3.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
A Equation of Motion of Particles with Two Vectors . . . . . . . . . . 122
vi
LIST OF FIGURES
1.1 Possible path during the melting of a perfect crystal . . . . . . . . . . 4
1.2 A cartoon of lipid bilayers . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Defects in xy model (n=1) . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Defects in nematic liquid crystal (n=2) . . . . . . . . . . . . . . . . . 10
1.5 Defects in high fold symmetry of n-atic order. θ0 = 0. . . . . . . . . . 12
1.6 A k = +1 radius defect can make the elastomer buckle out . . . . . . 13
1.7 Second order phase transition. The plot is made for Eq. 1.8 with a = 2,
α4 = 4 and Tc = 5. T = Tc (the red / circle line) is the temperature
at which the second order phase transition occurs. . . . . . . . . . . . 18
1.8 First order phase transition. The plot is made for Eq. 1.12 with a = 1,
α3 = −0.2, α4 = 0.3 and T ∗ = 5. T = T ∗∗ = 5.075 (the cyan /
diamond line) is the limit of super heating. T = Tc = 5.067 (the green
/ square line) is the critical temperature at which first order phase
transition occurs. T = T ∗ = 5.000 (the yellow / star line) is the limit
of super cooling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.9 Order parameter φ as a function of temperature for (a) the second
order and (b) first order phase transition. The parameters we used to
calculate this plot are the same as the ones in Fig. 1.7 and Fig. 1.8 . 21
1.10 Material frame and laboratory frame: before and after deformation . 22
vii
1.11 The deformation of a thin film. A similar figure can be found in [22]. 27
1.12 Deformation of a two dimensional spring lattice. A stress-free reference
state exists. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.13 Deformation of a two dimensional spring lattice. A stress-free reference
state doesn’t exist. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1 Schematic illustration of an interacting particle system in the tetratic
phase. The shape of the particles indicates that the rotational symme-
try of the interaction is broken down from four-fold to two-fold. . . . 38
2.2 Numerical mean-field calculation of the order parameters C2 and C4 as
functions of γ (inverse temperature), for several values of κ (two-fold
distortion in the interaction): (a) κ = 0.4. (b) κ = 0.75. (c) κ = 1.5.
(d) κ = 2.25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3 Phase diagram of the model in terms of γ (inverse temperature) and κ
(two-fold distortion in the interaction). The grey solid lines represent
second-order transitions, and the dark solid lines are first-order tran-
sitions. The dashed lines indicate the extrapolated second-order tran-
sitions, which give the cooling limits of the metastable phases. Point
B (0.79,2) is the triple point, and A (0.61,2.2) and D (2,1) are the two
tricritical points. Point C (1,2) is the intersection of the extrapolated
second-order transitions. . . . . . . . . . . . . . . . . . . . . . . . . . 45
viii
2.4 A snapshot of the spins on a triangular lattice in the tetratic phase. The
shape of the rectangles is just a schematic illustration of the symmetry
of their interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.5 Simulation results for the order parameters C2 and C4 as functions of
γ (inverse temperature), for several values of κ (two-fold distortion in
the interaction): (a) κ = 0.3. (b) κ = 0.5. (c) κ = 1. (d) κ = 3. . . . . 49
2.6 The degree of coexistence line β when κ = 0.3. The red line is fitted
from simulation data with β = 0.4912. . . . . . . . . . . . . . . . . . 52
2.7 Simulation results for the phase diagram in terms of γ (inverse temper-
ature) and κ (two-fold distortion in the interaction). The triple point
is at approximately γ = 3.2 and κ = 0.60. . . . . . . . . . . . . . . . . 53
3.1 Schematic illustration of the deformation of a thin shell due to swelling:
(a) before swelling, (b) after swelling without any deformation, and
(c) deformed shell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2 Illustration of the asymmetric swelling experiment. . . . . . . . . . . 65
3.3 (Color online) Illustration of the curved film shapes considered in Sec-
tions 3.3 and 3.4. (a) Partial cylinder. (b) Partial sphere. . . . . . . . 69
4.1 Schematic illustration of our two-vector model for interacting coarse-
grained particles. Each particle has a vector n, which aligns along the
local membrane normal, and a vector c, which has nematic alignment
within the local tangent plane. . . . . . . . . . . . . . . . . . . . . . . 80
ix
4.2 (Color online) Enlarged front and top views of the vesicle for η = 0.3,
with cylinders showing the local nematic orientation c in relation to
the overall shape. In these images the vesicle is opaque, so that only
defects on the near side are seen. . . . . . . . . . . . . . . . . . . . . 86
5.1 Fluorescence microscopy of DPPC vesicles labeled with 2 mol% NBD-
PE. (a) Vesicle above Tm in the Lα phase. (b,d,e) Vesicles cooled below
Tm into the Lβ′ phase. (c,f)Confocal images showing slices through a
crumpled vesicle. (g,h,i) Confocal images of vesicles in the Lβ′ phase in
a fluorescent dextran solution. Some vesicles remain intact and appear
black (g,h), whereas others show leakage and appear red (h,i). Note
that the vesicle in (i) has a clear break in the membrane, as indicated
by the arrow. (Hirst, unpublished) . . . . . . . . . . . . . . . . . . . 95
x
5.2 Polarized fluorescence microscopy images of a single vesicle labeled
with Laurdan in the gel phase. The vesicles are immobilized by partial
fusion onto a mica surface as shown in the confocal image (a) and
diagram (b). Images (c-d) and (e-h) show two different vesicles with the
focal plane slightly above the mica surface. The vesicles are illuminated
by different angles of linearly polarized light (angle indicated in left
corner). The arrows indicate regions of interest where clear tilt defects
can be observed by rotating the polarizer. In (e-h) we also observe a
variation of intensity inside the vesicle, showing variation of molecular
tilt direction for the flattened portion of the vesicle fused on the mica
surface. The focal plane in (c-d) is too far from the surface to observe
this effect. (Hirst, unpublished) . . . . . . . . . . . . . . . . . . . . . 98
5.3 Schematic illustration of our two-vector model for interacting coarse-
grained particles. Each particle has a vector n, which aligns along the
local membrane normal, and a vectorc, which represents the long-range
tilt order within the local tangent plane. . . . . . . . . . . . . . . . . 100
5.4 Coarse-grained simulation of a lipid vesicle. Top-left: High-temperature
Lα phase. Bottom-left and right: Low-temperature Lβ′ phase. Arrows
represent the tilt direction c, black dots represent defects in the tilt
direction, and colors represent distance from the center of mass of the
vesicle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
xi
6.1 Schematic plot of the coarse-grained bilayer model for lipid membrane 111
6.2 Simulations show two single layer with correct direction attract each
other and self-diffusion of the liquid membrane. Particles are marked
with red and green colors to track their diffusion. (a) and (b) are
the initial configurations from different view. (c) and (d) show the
fluctuating liquid bilayer membrane after about 200000 steps. . . . . 113
6.3 Spontaneously formation of vesicles and lamellar phase. The simula-
tion box of all figures are the same (25x25x25). In (a) and (b), particles
number is 10937. In (c) and (d), we have 36191 particles. The position
and orientation are all random at the beginning of the simulation ((a)
and (c)). After about half a million steps, vesicles (b) and lamellar
phase (d) are spontaneously formed. . . . . . . . . . . . . . . . . . . . 114
6.4 A snapshot of the simulated nematic phase on triangular lattice . . . 115
6.5 Strain-stress curve. Strain is calculated as ∆h/h, h is the original
height of the sample, ∆h is the change of the height. . . . . . . . . . 118
6.6 Stretching of the lattice under crossed polarizer. The a,b,c,. . . ,i corre-
sponds to the same symbol in 6.5. The crossed polarizers are in x and
y’s directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.7 A snapshot of the grain boundaries during stretching . . . . . . . . . 120
xii
LIST OF TABLES
4-1 (Color online) Shapes and defect configurations of a vesicle for several
values of the coupling η. The color indicates the distance from the
center of mass of the vesicle, and the black dots indicate the defect lo-
cations. In these images the vesicle is semi-transparent, so that defects
on both near and far sides are seen. . . . . . . . . . . . . . . . . . . . 84
5-1 Shape and defect configuration for simulated vesicle with low trans-
lational viscosity and high rotational viscosity. From first row to the
third row are back, front and right views of the vesicles, respectively.
The color images (left column) represent distance from the center of
mass of the vesicle, and the gray scale images (other columns) represent
the tilt direction, showing the optical intensity that would be observed
with polarized fluorescence microscopy. This vesicle has five +1 defects
and three -1 defects. Note the similarity with the experimental images
of Fig. 5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
xiii
5-2 Morphology and defect configuration for simulated vesicle with high
translational viscosity and low rotational viscosity. From first row to
the third row are back, front and right views of the vesicles, respec-
tively. The color images (left column) represent distance from the cen-
ter of mass of the vesicle, and the gray scale images (other columns)
represent the tilt direction, showing the optical intensity that would be
observed with polarized fluorescence microscopy. This vesicle has only
two defects of charge +1, which is the minimum required by topology.
Note that it is much smoother than the simulated vesicle of Table 5-1. 104
6-1 n-atic order (n=3,4,5,6) on the surface of vesicles. We plot multiple
vectors for one particle according to their n-fold symmetry to facilitate
counting the strength of the defects. The black dots are the particles
with high energy, thus can be used to identify the location of the de-
fects. The near side defects are darker than the far side ones. And the
color of the particles illustrate the distance from the mass center. . . 109
xiv
DEDICATION
To my wife Quan
To my parents
xv
ACKNOWLEDGMENTS
I’m sincerely and heartily grateful to my co-advisor, Dr. Jonathan Selinger, for
recruiting me and for the guidance he showed me throughout my research and the
dissertation writing. Four years ago, I went to his office with my CV to recommend
myself as a member of his theoretical research group. At that time, all I had was
the curiosity to theoretical physics and I had no idea that I could go this far. I
really appreciate his confidence and encouragement to a student without any theory
background. I enjoy every meeting with him and learn a lot from his crystal clear
lectures. Also, I thank him for his support to my decision of my career.
I’m truly indebted and thankful to my co-advisor, Dr. Robin Selinger, for directing
my research and this dissertation. I knew nothing about computer simulation before
I took her computation class. I have learnt so much from her that the simulation
becomes a very important part of my dissertation. Without her, this dissertation is
impossible. Also, I enjoy very much all the meetings with her and I benefit a lot from
her insight of physics. Further more, I’d like to thank her for all the helps and advises
to my personal life.
Also, I’d like to give my truly grateful to Dr. Peter Palffy-Muhoray for all the
knowledge I learn from him in his class and during my research rotations in his group;
To Dr. Antal Jakli, Dr. Philip Bos, Dr. Liang-Chy Chien, Dr. Satyendra Kumar,
xvi
Dr. Qi-huo Wei, Dr. Oleg Lavrentovich, Dr. Deng-Ke Yang, Dr. Jake Fontana and
Dr. David Allender for their excellent lectures.
Finally, I’d like to thank all the group members in the theory group, especially,
Dr. Fangfu Ye, Dr. Mbanga, Dr. Dhakal, Dr. Lopatina and Vianney Gimenez-Pinto
for all their help, discussion and cooperation on my research. And I would also like
to thank Nicholas Diorio and Dr. Taushanoff for their kindly help to international
students.
xvii
CHAPTER 1
Introduction
1.1 General Considerations
Within statistical mechanics, one of the most important issue is the breaking
of symmetry. A certain order, either orientational or translational, or both can be
spontaneously selected by breaking corresponding symmetry in a certain phase. For
example, the transition from an isotropic phase to a liquid crystalline phase breaks
the rotation symmetry of the isotropic state, as shown in Fig. 1.1. Compared to the
isotropic liquid, a liquid crystal is a broken-symmetry phase and has higher order.
One thing we want to understand is what kind of order gets selected? Or equiva-
lently, what kind of symmetry is broken? A phase diagram answers these questions by
charting the phase boundaries between different thermodynamically distinct phases.
One way to draw phase diagrams is to measure the phase transitions by experiments
or computer simulations. Another way is using statistical mechanics to calculate it
by minimizing the free energy.
Also we want to understand what happens if the phase orders in different ways
and different places. In other words,we’re interested in the situation where the con-
tinuous symmetry is broken, thus the elastic energy appears [1]. If order gets stuck
in some places, may make a defect. Even if it doesn’t make a defect, it may still cost
1
2
energy because of the spatially non-uniformity. For example, the density gradient of
a polymer film will make a flat polymer bend in order to the minimize the elastic
energy.
The research in this dissertation explores several specific aspects of these general
issues.
We use both analytical calculation and computer simulation to study a phase
transition between phases with different types of orientational order: nematic and
tetratic, in a uniform two-dimensional geometry. We consider interactions with two-
fold rectangular symmetry, and show that they can produce phases with isotropic,
tetratic (four-fold), or nematic (two-fold) symmetry.
We then investigate the formation of non-uniform curved structures. For instance,
we study the asymmetric swelling of a polymer film with build in non-uniformity.
The asymmetric swelling make one side of the polymer to expand and the other
side to compress. And in this example, there is not a stress-free state. In other
words, the film can not achieve the geometry it wants everywhere. But we can still
calculate optimized configuration with the help of a ”target metric” [2]. We show
that, depending on size, the film can bend into a partial sphere or a partial cylinder.
We also study a more complex interaction among topological defects, in-plane
orientational order and curved geometry. In some situation, the existence of defects
in the in-plane order may be forced by the topology of the surface. For example,
with in-plane orientational order, a zero genus closed surface must have net charge
3
of positive two [3]. The defects can be attracted by highly curved surface to reduce
the generalized Frank energy. Of course, it’s a two way interaction: a defect may also
cause the surface to buckle [4]. For these more complex problems, we use computer
simulations as well as analytic theory. We developed a model for simulating self-
assembling membranes with in-plane orientational order. And we use this technique to
simulate a membrane with tilt order and in-plane nematic order. In the simulation, we
can investigate how those coarse-grained particles self assemble into different shapes
and how the complicated two way interactions among shapes, internal orientational
order and defects changes the morphology of the vesicles.
An outline of this dissertation is as follows. In the rest of this chapter, we provide
an introduction to the related topics and methods of theoretical research in statistical
mechanics of soft materials. We then present several chapters describing the specific
projects of my dissertation research, which have been submitted or published as sci-
entific articles. In Chapter 2, we calculate two dimensional tetratic-nematic phase
transition and compare the result with Monte Carlo simulation; In Chapter 3, we
calculate the deformation of a asymmetric-swollen thin polymer film; In Chapter 4,
we develop a coarse-grained model to study the shape of vesicles with tangent-plane
nematic order; And in Chapter 5, we modified our coarse-grained model to study the
role of topological defects in the formation of complex morphologies during the transi-
tion from a in-plane isotropic phase to in-plane xy ordered phase; Finally, in Chapter
4
Crystal
Plastic Crystal
Liquid Crystal
Isotropic Liquid
Figure 1.1: Possible path during the melting of a perfect crystal
6 we present some other projects that I have begun but have not yet published: a gen-
eralized n-atic in-plane order model for a liquid membrane, a coarse-grained model
that can form a real bilayer lipid membrane and a Monte Carlo simulation of the
stripe formation of a two dimensional nematic liquid crystal elastomer.
1.2 Orientational Order
In this section, we’d like to briefly review the some interesting phase with orien-
tational order in soft matter and the topological defects they can form.
1.2.1 Liquid Crystals
When a crystal with both orientational and translational order melts into an
isotropic liquid and loses all of its order, there are three possible paths of the melting
5
process as shown in Fig. 1.1. The crystal can lose both type of order at the same time
and become an isotropic liquid. On the other hand, between a crystal and isotropic
liquid, there are two possible intermediate phases: plastic crystal and liquid crystal
phase. The former one keeps its translational order and possesses some orientational
degree of freedom, which can be almost free rotation or some jump diffusion between
a finite number of possible orientations [5]. The later one possesses the translational
degree of freedom, which can be without any restriction or restricted in certain di-
mension or range, and keeps the orientational order. Thus, it’s possible to observe
these two intermediate phases during melting of a crystal.
The first observation of the liquid crystal phase is due to Reinitzer in 1888 [6].
Then, several thousands organic materials are found to form liquid crystal phase.
Different phases are also classified, such as nematic, cholesteric, smectic and blue
phase. An essential requirement for mesomorphism to occur is that the molecule must
be highly geometrically anisotropic in shape, like a rod or a disc [7]. Onsager was
the first one to make this intuition into quantitative statistical theory by introducing
the concept of excluded volume [8]. Then, Maier and Saupe developed a molecular
field theory by noticing the analogue between nematic and ferromagnet [9–11]. de
Gennes also contributed to many aspects of liquid crystal physics [12] and won the
1991 Nobel Price in Physics for his work in liquid crystals and polymers.
Besides scientific interest, a big reason that liquid crystal attracts so much at-
tention is because its application in display and other fields. Due to highly tunable
6
properties, for instance, the optical anisotropy, liquid crystal appears in many display,
sensor and other devices. Now, more than one hundred years after liquid crystal had
been found in the heating stage of the laboratory, the LCD becomes a multi-billion
industry. Without liquid crystal, many state of the art devices, such as iPad, iPhone,
will never be possible.
1.2.2 Lipid Membranes
Among all the other liquid crystal phases, smectics are the intermediate phases
that gain their translational degree of freedom in two dimensions and maintain the
order in the third dimension. In other words, at the layer’s normal direction, it’s a
crystal but at the layer’s tangent plane, it’s a fluid. Thus, smectic phase has higher
order, less symmetry than the nematic phase. Many important structure in lyotropic
and biologic system can be modeled as single or multiple layers of smectic phase. For
example, in a lyotropic system, the hydrophilic head-group of amphiphilic molecules
want to contact water while the hydrophobic tail hates water. As a result, they can
self assemble to form bilayer membranes, and with different concentration they form
all kinds of lyotropic liquid crystalline phases including vesicles, lamellar phase.
One typical bilayer membrane is shown in Fig. 1.2. As shown in the Fig. 1.2,
most lipid molecules have two hydrophobic tails (yellow parts) and a hydrophilic
head (green part). Usualy, there are about 20 carbon atoms in one tail, the thickness
of the bilayer is about 5 nm and the area of the head group is about 0.6 nm2 [13].
Thus, one can estimate that the length / width ratio of the molecule is about 5, which
7
Figure 1.2: A cartoon of lipid bilayers
is within the range of the radios that can form liquid crystal phase. Helfrich realized
that one can treat lipid molecules as the director of a nematic liquid crystal and derive
the elastic energy density of the membranes [14]. It turns out to be proportional to the
mean curvature square and proportional to the Gaussian curvature of the membrane.
Many vesicle shapes, including the circular, biconcave, discoidal red blood cell can
be explained by this and related theory. While it’s true that the real biological
membranes have peripheral and transmembrane proteins, filaments of cytoskeleton
etc., and are far more complex than this ideal model, this theory still captured the
most important insight into membrane properties.
8
1.2.3 Topological Defects
Topological defects can appear in systems with broken continuous symmetry. In
crystals, they break translational symmetry and are called dislocations ; in liquid
crystals, they break rotational symmetry and are called disclinations. We’re mainly
interested in defects in orientational order system. Their energy is composed with two
parts: the core energy and the far field elastic energy. The strength of the defects can
be determined by measuring the vector field’s rotation by following a circuit enclosing
its core. A two dimensional disclination can be characterised by the relation between
the vectors’ rotation angle θ and their positional angle φ:
θ(r, φ) = kφ+ θ0. (1.1)
k is called winding number or the strength of the defects and θ0 is a constant and
called the phase of the defects. The gradient ∇θ = k/r, and the elastic energy is
∝ (∇θ)2). This means the elastic energy is proportional to k2 and will decade with
1/r2.
Fig.1.3 shows some possible defects of the xy model with one-fold rotational sym-
metry. The red circuit is to facilitate finding the defect cores and counting the rota-
tions of the vectors. For xy model, if φ rotates by 2π on a circuit enclosing a k = 1
defect, θ will also rotates by 2π.
If we consider the two-fold rotation symmetry, we could plot the vector field of
the nematic defects as shown in Fig. 1.4. The double-heads arrows represent the two-
fold rotation symmetry and because of this symmetry, k = ±1/2 defects can exist in
9
(a) k = 1, θ0 = 0 (b) k = 1, θ0 = π/2
(c) k = −1, θ0 = 0 (d) k = −1, θ0 = π/2
Figure 1.3: Defects in xy model (n=1)
10
(a) k = 1/2, θ0 = 0 (b) k = 1/2, θ0 = π/2
(c) k = −1/2, θ0 = 0 (d) k = −1/2, θ0 = π/2
Figure 1.4: Defects in nematic liquid crystal (n=2)
11
nematic. They’re the most common defects in the nematic liquid crystal because of
they cost the lowest energy.
We can generalize the one and two-fold rotation symmetry to n-fold, which is
called n-atic orders. We plot the most probable defects for n = 3, 4, 5, 6 in Fig. 1.5.
Again the n-atic symmetry is represented by the multiple arrows at a single site.
If we rotate one of the particles that has n-fold rotational symmetry by 2π/n, the
interaction energy between this particle and its neighbors won’t change. The lowest
defect strength of the n-atic order is 1/n. For example, if we follow the red circuit of
Fig. 1.5 (a) counterclockwise for a 2π increment of φ, we will find that the θ angle of
the vectors rotates by 2π/3.
Defects cost energy. Thus, a defect in a subspace can escape to higher dimension
or change the geometry of the subspace they live in to reduce the elastic energy. For
instance, in Chapter 5, we show that k = 1 defects of the xy model can buckle out
and make a bump on the surface of vesicles, also shown in Fig. 1.6. In the liquid
crystal elastomer, the energy of stretching of the film can be dominate and the one
can estimate the shapes by minimizing the bending energy without stretching [15].
Although preferred by entropy, defects are still hard to trap and study. Fortunately
certain topologies like a sphere can trap a fixed net charge of the defects on its surface
due to the Gauss-Bonnet theorem. The interplay of the tangential plane orientational
order, defects and elasticity are very fascinating [16]. We will mainly use computer
simulations to study them. Some results are presented in Chapter 4 and 5.
12
(a) n = 3, k = 1/3 (b) n = 4, k = 1/4
(c) n = 5, k = 1/5 (d) n = 6, k = 1/6
Figure 1.5: Defects in high fold symmetry of n-atic order. θ0 = 0.
13
Figure 1.6: A k = +1 radius defect can make the elastomer buckle out
1.3 Phase Transition
In this chapter, we’d like to discuss how to describe orientational order and the
theory to study the phase transitions.
1.3.1 Order Parameters
As we mentioned previously, we’re very interested in phase transitions in soft
condensed matter. When lowering the temperature, a disordered phase becomes an
ordered phase. Because the disordered phase has higher symmetry than the ordered
phase, we also call this symmetry breaking. To quantitatively study the phase transi-
tion, we need to describe how much different is the ordered phase from the disordered
one. The quantity we use is the order parameter. It can be a scalar (the density
in a liquid-gas phase transition), a vector (the magnetization in the ferromagnetic-
paramagnetic phase transition), or a tensor with different ranks. For example, the
order parameter of nematic-isotropic phase transition is a 2nd rank tensor in three
dimensional space:
Qij =3
2
(
〈vivj〉 −1
3δij
)
. (1.2)
14
i, j = 1, 2, 3, v is a unit vector which points to the same direction as the individual
molecules (v here is not the director, and 〈v〉 = 0). Qij are actually the components
of the real tensor Q = Qijgigj. gi are the contravariant base vectors of the coordinate
lines. But traditionally, in most physics literature they’re just called the tensor. And
the angle brackets denote the ensemble average. Qij is a traceless (because v is a unit
vector), symmetric tensor that has 5 degrees of freedom. In the isotropic state, the
orientation distribution function of the molecules is a constant (= 1/(4π)), so that
one can calculate that 〈vivj〉isotropic = 13δij . This means Q = 0 in the isotropic phase.
The unit eigenvector (n) of the biggest eigenvalue (S) of this tensor is the nematic
director. S can be considered as a scalar order parameter of nematic. To make S = 1
in a perfect nematic phase, we need to add a coefficient 3/2 to Eq. 1.2. Thus another
way to write this tensor is:
Qij =3
2S
(
ninj −1
3δij
)
. (1.3)
We’ll give an example of using a 4th rank tensor in Chapter 2, and other examples
of higher rank tensor order parameters for n-fold symmetry can be found in this
reference [17].
1.3.2 Mean Field and Landau Theory
To quantify the ordering process using order parameters, we can write down the
Hamiltonian H and calculate the partition function. For instance, for the canonical
15
ensemble, the partition function in d dimension space is [1]:
ZN (T ) = Tr e−βH =1
N !
∏
α
∫
ddpαddxα
hdNe−βH . (1.4)
β = 1/(kBT ). The Helmholtz free energy is:
F (T ) = −kBT lnZN (T ) . (1.5)
For a given temperature, we could in principle minimize the free energy of Eq. 1.5 over
all the possible configurations in phase space and determine the phase by calculating
the order parameter. In practice, there are too many degree of freedoms and it’s
too complicated to proceed. Here is where the mean field theory come to the rescue.
Mean field theory treats the order parameter as a spatial constant (thus the ensem-
ble average of all possible local states in phase space is the same) and significantly
simplifies the mathematics. It’s a very useful description if the spatial fluctuations
are not important. It makes qualitatively correct predictions in low dimension (one,
two ,three) and quantitatively correct predictions in higher dimensions. Mean field
theory is so mathematically simple that it’s almost invariable the first approach taken
to predict phase diagrams and properties of new systems [1].
There are many different formulations for different system. I’d like to introduce a
very powerful and simple mean field theory: Landau theory to demonstrate it [18,19].
Landau theory is based on the power expansion of the free energy in terms of order
parameter. It assumes that around the phase transition (first or second order) the
order parameter is small so that one only needs to keep the lowest orders permitted
16
by the symmetry. Landau theory’s free energy is entirely determined by the same
symmetry as the interaction Hamiltonian. (For phase transition without symmetry
breaking, for instance, the liquid-gas first order transition, constructing a Landau
theory is also possible.) In many situations, it can extract important results from
very simple algebraic equations. Let’s start with assuming the exact free energy in
Eq. 1.5 can be approximate by a integral over local free energy density and local order
parameter (φ (x)) gradient:
F (T ) =
∫
ddxf(T, φ (x)) +
∫
ddx1
2c [∇φ (x)]2 (1.6)
c is a phenomenological coefficient. Applying mean filed theory’s assumption that
order parameter φ = φ (x) is a spatial constant (order parameter is the field). This
eliminates the second term in Eq. 1.6 and make F (T ) = V f(T, φ). Usually volume
V is constant or infinite, so we need to only consider the free energy density f .
Expanding the free energy density in Eq. 1.6 into power series of order parameter φ:
f(T, φ) =1
2α2(T )φ
2 + α3(T )φ3 + α4(T )φ
4. (1.7)
The constant drift term of zeroth order in φ won’t change the phase transition and is
neglected. There is no first order term of φ because we usually define that φ (x) = 0
for the high temperature disordered phase without any conjugate field.
A second order phase transition can occur if the cubic term in Eq. 1.7 is not
allowed by symmetry (e.g., the order parameter is a vector) and there is no external
field. For example, the Ising model for ferromagnet-paramagnet transition and the
17
normal to superfluid transition in liquid He4. For the sake of simplicity we only
consider α2 = a(T −Tc), a > 0 a function of T and α4 is a constant. Eq. 1.7 becomes:
f(T, φ) =1
2a(T − Tc)φ2 + α4φ
4. (1.8)
Tc is a constant temperature. To find the local minimal of f , we can calculate the
first and second order derivatives of f respect to φ for a given T :
∂f
∂φ= a(T − Tc)φ+ 4α4φ
3, (1.9)
and
∂2f
∂φ2= a(T − Tc) + 12α4φ
2. (1.10)
Then it’s easy to see from solving ∂f∂φ
= 0 with T > Tc, there is only one solution
φ = 0 and it’s a global minimum (∂2f
∂φ2 |φ=0 = a(T − Tc) > 0). This means at high
temperature(T > Tc), it’s a disordered phase, as shown by the blue / triangle and
green / square lines of the Fig. 1.7. For T = Tc, we got two minimum and one
maximum at the same point φ = 0 as shown by the red / circle line in Fig. 1.7. Tc is a
critical temperature. When T < Tc, there are two minimum separating symmetrically
from φ = 0. This can represent two opposite directions of the bulk magnetization of
Ising model. We can calculate the order parameters from ∂f∂φ
= 0 for T < Tc:
φ = ±1
2
[
a
α4
(Tc − T )
]1
2
(1.11)
The exponent 1/2 in Eq. 1.11 is an critical exponent called degree of the coexistence
curve (β). Mean field calculation value β = 0.5 is slightly bigger than its experiment
value β ≈ 0.3− 0.4 [19].
18
Figure 1.7: Second order phase transition. The plot is made for Eq. 1.8 with a = 2,α4 = 4 and Tc = 5. T = Tc (the red / circle line) is the temperature at which thesecond order phase transition occurs.
19
If the cubic term in Eq. 1.7 is permitted by the symmetry (e.g., the order parameter
is a scalar or second rank tensor like nematic order parameter tensor Qij which can
contract into scalars), a first order phase transition can occur. Again, we only assume
α2 is a function of temperature and Eq. 1.7 becomes:
f(T, φ) =1
2a(T − T ∗)φ2 + α3φ
3 + α4φ4. (1.12)
For a given temperature, we can calculate first and second order derivatives to deter-
mine the minimum:
∂f
∂φ= a(T − T ∗)φ+ 3α3φ
2 + 4α4φ3, (1.13)
and
∂2f
∂φ2= a(T − T ∗) + 6α3φ+ 12α4φ
2. (1.14)
For any given T , all the extremes can be found by solving ∂f∂φ
= 0, and determining
if it’s a maximum or minimum by calculate ∂2f∂φ2 at that location. At very high tem-
perature (T > T ∗∗ = T ∗ +9α2
3
16aα4), there exists only one minimum at φ = 0 as shown
by the blue / triangle curve in Fig. 1.8. The phase is in a disordered state. As the
temperature passing T ∗∗ from high temperature, there will be two minimums: φ = 0
and φ = φo = (√
9α23 − 16aα4(T − T ∗) − 3α3)/(8α4). T = T ∗∗ is the temperature
at which the local minimum φo disappears as shown by the cyan / diamond curve
of Fig. 1.8. If we’re heating an ordered phase to this temperature, the metastable
state at φo can not exist any more no matter how fast you heat because any fluc-
tuation will immediately lead the system to the disordered state φ = 0 (because
20
Figure 1.8: First order phase transition. The plot is made for Eq. 1.12 with a = 1,α3 = −0.2, α4 = 0.3 and T ∗ = 5. T = T ∗∗ = 5.075 (the cyan / diamond line) isthe limit of super heating. T = Tc = 5.067 (the green / square line) is the criticaltemperature at which first order phase transition occurs. T = T ∗ = 5.000 (the yellow/ star line) is the limit of super cooling.
21
(a) (b)
Figure 1.9: Order parameter φ as a function of temperature for (a) the second orderand (b) first order phase transition. The parameters we used to calculate this plotare the same as the ones in Fig. 1.7 and Fig. 1.8
there is no energy barrier anymore). Thus, T ∗∗ is called the limit of super heating.
At T = Tc = T ∗ +α2
3
2aα4
, the ordered phase φo has the same energy as the disorder
phase. Thus, a first order phase transition will occur if the temperature is lower
T < Tc. For equilibrium phase transition, the order parameter will jump from φ = 0
to φ = φc = − α3
2α4, thus it’s a discontinuous phase transition. As shown in Fig. 1.9,
the order parameter of the first order transition is not continuous at Tc, while the
second order transition is a continuous transition. And the disordered phase at φ = 0
will become a metastable phase at lower temperatures than Tc, untill the temperature
decrease to T = T ∗, when the local minimum at φ = 0 will disappear. Below T ∗ any
fluctuation will immediately take the system to the ordered state φo. Thus, T∗ is the
limit of super cooling as shown by the yellow / star curve of Fig. 1.8.
22
Figure 1.10: Material frame and laboratory frame: before and after deformation
We’d like also point out that although Landau theory is a phenomenological the-
ory, we can actually calculate those coefficients (α2, α3, α4, Tc, T∗,. . . ) if we make
proper mean field assumptions and expand our free energy from Eq. 1.5. We can
use this technique to calculate phase transition from molecular potentials as demon-
strated by Maier and Saupe [9–11] and also in our calculation of tetratic-nematic
phase transition in Chapter 2.
1.4 Elasticity
Elasticity will arise if the continuous symmetry is broken.
1.4.1 Material Frame vs. Laboratory Frame
In order to calculate the deformation of a continuum medium, we need to set
up proper frame of coordinates. There are two fundamental ways to describe the
movement and deformation of the continuum points. One is the so called laboratory
23
frame or Eulerian frame. We can imagine the continuum body is made of mass points.
Three of them are shown in Fig. 1.10. O, A, B and O, A, B denote the same points
after and before deformation. Also, the other symbols with a bar are the quantities
for the same points after deformation. The coordinates xi (i = 1, 2, 3) are fixed in the
space (relative to the ”laboratory”) and won’t move with the deformed object. Thus,
the positional vector R will change to R after deformation for the same point O on
the object (symbols with bold font are vectors). On the other hand, we could set
another kind of coordinates which will actually move and deform with the material.
We call this kind of coordinate frame the material frame or Lagrange frame. In
Fig. 1.10, coordinates ϑi (i = 1, 2, 3) are such coordinates. In this kind of frame, the
coordinates are effectively the points’ names. For example, after deformation, point
O moves to O and change its position. But the coordinates (0,0,0) of O in material
frame don’t change because the coordinates itself move with the material. Also, the
arc length OA and OB changed, but the coordinates of A and B didn’t because the
coordination lines stretched the same way as the material.
1.4.2 Important Tensors
As shown in Fig. 1.10, gi (i = 1, 2, 3, the Latin indices will denote number from
1 to 3 and Greek indices will be in the range from 1 to 2 without any further notice)
are the local covariant basis vectors and are defined as:
gi =∂R
∂ϑi. (1.15)
24
The coordinates ϑi are curvilinear coordinates, which means their bases are not nec-
essarily parallel to each other. One can also define the contravariant bases by taking
derivative of R respect to the covariant coordinates. These basis vectors define some
vector fields in the space. And they have the following properties:
gi · gj = δji = δij. (1.16)
δji is the Kronecker δ and its value is 1 if i = j;0 if i 6= j. If a index appears twice
in a single term, Einstein summation convention is assumed. Thus any vector can
be written as a linear combination of the basis vectors: P = P igi = Pigi. A vector
pointing to its neighbor is
dR =∂R
∂ϑidϑi = gidϑ
i. (1.17)
The distance square to its neighbor is
ds2 = dR · dR = gi · gjdϑidϑj = gijdϑ
idϑj . (1.18)
gij = gi · gj (1.19)
gij are the covariant components of the metric tensor. It measures how far away a
point is from its neighbors. Thus, this tensor can be used as a transformation from
the material frame quantity (for example, the coordinates in material frame) to the
laboratory frame. If we define g = det(gij), one can verify that in three dimension
space, the triple product g1 · (g2 × g3) =√g, and that
√g is the volume of the
parallelepiped defined by the three bases. One can calculate the volume of the three
25
dimensional body by a integral over this volume element with respect to the material
coordinates:
V =
∫ √gdϑ1dϑ2dϑ3. (1.20)
Also, the metric tensor can be used to measure the deformation. We can define the
same quantity ds2 for the same point after deformation and measure the distance
(square) change between neighbors:
(ds)2 − (ds)2 = (gij − gij) dϑidϑj = 2εijdϑ
idϑj . (1.21)
We have defined
εij =1
2(gij − gij), (1.22)
where εij are the covariant components of tensor ε = εijgigj, which is called Green-
Lagrangian strain tensor. In term of displacement u = R−R in Fig. 1.10,
(ds)2 − (ds)2 =(dR+ du)2 − (dR)2
=2dR · du+ du · du
=2gidϑi · ∂u
∂ϑjdϑj +
∂u
∂ϑidϑi · ∂u
∂ϑjdϑj
=2gidϑi · gkuk;jdϑ
j + gkuk;idϑi · glu
l;jdϑ
j
=(
ui;j + uj;i + uk;iuk;j
)
dϑidϑj (1.23)
ui;j = ∂ui
∂ϑj− ukΓ
kij and ui
;j = ∂ui
∂ϑj+ ukΓ
ijk are the covariant derivatives. Γi
jk are
the Christoffel symbols of the second kind. If the coordinates lines are straight lines
(basis vectors are not functions of coordinates), all Christoffel symbols are zeros and
26
covariant derivatives become just partial derivatives, so that we’ll have
uij =1
2
(
∂ui
∂ϑj+
∂uj
∂ϑi+
∂uk
∂ϑi
∂uk
∂ϑj
)
. (1.24)
If ϑi are orthogonal coordinates with unit bases (Cartesian coordinates), we’ll have
uk = uk which becomes the same strain tensor defined in [20]. We’d like to point
out that although ε = uijxixj, xi = xi are unit bases of Cartesian coordinates, the
components uij are not necessary the same as εij. Also, if the strain is small, and the
high order term can be neglected, Eq. 1.24 becomes the infinitesimal strain:
uij =1
2
(
∂ui
∂ϑj+
∂uj
∂ϑi
)
. (1.25)
If the elastic body is isotropic, which means the elastic constant is the same every-
where, we can write down the density of deformation energy as an analogy to Hook’s
law:
f3 =1
2Aijklεijεkl, (1.26)
where
Aijkl = λgijgkl + µ(gikgjl + gilgjk), (1.27)
are the contravariant components of the three-dimensional elasticity tensor in curvi-
linear coordinates [21].
1.4.3 Thin Film
Let’s consider a deformation of a thin film, which is a three dimensional body
with the size of one dimension significantly smaller than the other two dimensions.
27
Figure 1.11: The deformation of a thin film. A similar figure can be found in [22].
Fig. 1.11 shows the deformation of a thin film from its original state P to a deformed
state P. Again, symbols with a bar describe a quantity of the deformed state. M
and M are the midplane of the thin film. R and m are vectors pointing to the three
dimension body and the two dimension midplane, respectively. u is the displacement
vector. gi are the basis vectors of the curvilinear coordinates ϑi in the material
frame. n is the unite normal vector of the midplane and aα are the basis vectors of
the midplane. xi are the Cartesian coordinates in the laboratory frame. A metric
tensor in two dimension can be defined as:
aαβ = aα · aβ , aαβ = aα · aβ (1.28)
which are the coefficients of the first fundamental form of the surface. Similar to
Eq. 1.20, the area of the surface can be calculated from the area element√a (a =
28
det(aαβ)). If we want to know how the surface is curved, we can define:
bαβ = n · ∂2m
∂ϑα∂ϑβ= n · ∂aα
∂ϑβ= − ∂n
∂ϑβ· aα. (1.29)
bαβ are the coefficients of the second fundamental form of the midplane. The eigen
value of b βα are the two principle curvatures. The mean curvature of the surface is:
H =1
2b αα , (1.30)
while the Gaussian curvature is
G = det(
b βα
)
. (1.31)
To calculate the deformation of a thin film, we usually want to integrate the energy
density over the thickness. By doing this, a three dimension minimization problem
becomes a two dimension problem embedded in three dimensional space. Proper
assumption should be made to achieve this [23]. We will show our research on this
topic in Chapter 3.
1.4.4 Target Metric
In all previous calculations, we always assume our reference / initial state is a
stress-free state. For example, for a two dimensional spring lattice in Fig. 1.12, if
we replace all the horizontal springs with those that has much longer natural length
(a), the lattice will elongate in horizontal direction (b). Because (b) is a stress-free
state, we can actually use (b) as a reference to define the strain tensor in Eq. 1.24
and calculate the equilibrium state (b).
29
(a) (b)
Figure 1.12: Deformation of a two dimensional spring lattice. A stress-free referencestate exists.
(a) (b)
Figure 1.13: Deformation of a two dimensional spring lattice. A stress-free referencestate doesn’t exist.
30
However, this is not always the case. In many situations, like growing tissues
and leaves, the elastic bodies exhibit very complex configurations even in the absence
of external forces [24]. They don’t have a stress-free configuration. For example,
in Fig. 1.13, one spring of an initially stress-free two dimensional spring lattice is
replaced by another spring with much long natural length (a). This spring system
doesn’t have a stress-free state in two dimension (b). Efrati and others proposed
an alternative definition of strain tensor [25–27]. In the absence of the stress-free
configuration, instead of using initial or final state as a reference state, we can use an
imaginary target configuration, where at any point of the elastic body the stress is
eliminated. And the metric at the target configuration is called target metric. They
call such system ”non-Euclidean” because their internal geometry is not immersible
in the Euclidean space (Fig. 1.13). Then by replacing the reference metric with the
target metric, the strain tensor is defined the same way as in previous definition in
Eq. 1.24. After the new definition of strain, all the other calculations will be the
same. In many cases like Fig. 1.13, the target metric tensor is predefined and this
approach is easier. We will show a calculation of an asymmetric film in Chapter 3 by
using this concept.
1.5 Computer Simulation
With the rapid growth of computation power, computer simulation plays a more
and more important role in all the disciplines that involve mathematical modeling.
Especially in physics, computer simulations provide insight, guidance and deliver
31
results to physicists from Manhattan Project to our daily weather predictions.
1.5.1 Monte Carlo Simulation
The Monte Carlo method is an amazing simulation technique that use pseudo
random numbers to compute complex problem where a deterministic algorithm is
not feasible. The name implies the random nature inherited from the Monte Carlo
Casino in Monaco. It’s widely used in physical science, engineering, games, finance
and business. Comparing to the simple ”what if” method, the Monte Carlo method
made much more efficient by not only drawing random number but actually sampling
the probability distribution. For example, to simulate a system in the canonical
ensemble, we could use the famous Metropolis algorithm [28] to generate a random
Monte Carlo move, which can be a displacement, a rotation et al.. Then the energy
difference ∆E = Enew − Eold between before and after this move is calculated. The
acceptance of the this move depends on ∆E: if ∆E < 0, this move is accepted; if
∆E > 0, the probability of the acceptance of this move is exp(−∆E/T ) (Boltzmann
constant kB is absorbed in T ). Metropolis algorithm’s probability of acceptance of a
random move can be summarize as:
Paccept = min(
1, e−∆E/T)
. (1.32)
In this way, Monte Carlo simulation can correctly replicate the distribution of the
states. Thus, any thermal average of a quantity can be correctly estimated. We use
Monte Carlo simulation to investigate the phase diagram in Chapter 2.
32
1.5.2 Molecular Dynamic Simulation
Molecular dynamic simulation is used to calculate the trajectories of molecules
/ particles by numerically solving the equations of motion for the interacting sys-
tem [29]. A normal equilibrium molecular dynamics simulation corresponds to the
microcanonical ensemble of statistical mechanics, but in many cases, we’re interested
in the properties of a system with fixed temperature. In this situation, we can use
so called Langevin thermostat to add stochastic quantities to reflect fluctuation and
replicate the canonical ensemble (Brownian dynamics) [30]. Comparing to the Monte
Carlo method, molecular dynamic simulation not only can be used to study equilib-
rium system, but also can be used to study non-equilibrium problems and dynamic
processes. In Chapter 4 and 5, we use molecular dynamic simulations to study the
liquid membrane systems with an inexplicit solvent.
1.5.3 Coarse-Grained model
It’s impossible, in terms of space and time scales, to simulate all the details of
atoms and bonds for condensed matter which usually involve more than 1026 degrees
of freedom. Thus, depending on the problem of study, all kinds of ”pseudo atoms”
have been made to represent the interaction between different groups of atoms. This
kind of model is usually called a coarse-grained model. The key of coarse-graining
is to keep the features that are important to the problem but neglect all the other
secondary details. In Chapter 4 and 5, we developed a coarse-grained model to
simulate vesicles with in-plane order. We’re interested in the coupling of in-plane
33
order and the membrane elasticity, so we neglect all the molecular details and even
treat it as a single layer. We keep all the symmetry of the interaction which we believe
is crucial to our study and find quite satisfying results.
BIBLIOGRAPHY
[1] P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics(Cambridge University Press, Cambridge, 1995)
[2] Y. Klein, E. Efrati, and E. Sharon, Science 315, 1116 (Feb. 2007), ISSN 1095-9203
[3] J. Park, T. Lubensky, and F. MacKintosh, EPL (Europhysics Letters) 20, 279(Oct. 1992), ISSN 0295-5075
[4] J. R. Frank and M. Kardar, Phys. Rev. E 77, 041705 (Apr 2008)
[5] J. C. W. Folmer, R. L. Withers, T. R. Welberry, and J. D. Martin, Phys. Rev.B 77, 144205 (Apr 2008)
[6] F. Reinitzer, Monatshefte fr Chemie / Chemical Monthly 9, 421 (1888), ISSN0026-9247, 10.1007/BF01516710
[7] S. Chandrasekhar, Liquid Crystals (Cambridge University Press, 1993) ISBN052142741X
[8] L. Onsager, New York Academy Sciences Annals 51, 627 (May 1949)
[9] W. Maier and A. Saupe, Z. Naturforsch. A 13, 564 (1958)
[10] W. Maier and A. Saupe, Z. Naturforsch. A 14, 882 (1959)
[11] W. Maier and A. Saupe, Z. Naturforsch. A 15, 287 (1960)
[12] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (InternationalSeries of Monographs on Physics), 2nd ed. (Oxford University Press, USA, 1995)ISBN 0198517858
[13] C. Tanford, The hydrophobic effect: formation of micelles and biological mem-branes, Wiley Interscience publication (Wiley, 1973) ISBN 9780471844600
[14] W. Helfrich, Z. Naturforsch 28, 693 (1973)
[15] C. D. Modes, K. Bhattacharya, and M. Warner, Proceedings of the Royal SocietyA: Mathematical, Physical and Engineering Science 467, 1121 (2011)
[16] T. C. Lubensky and J. Prost, Journal de Physique II 2, 371 (Mar. 1992), ISSN1155-4312
34
35
[17] X. Zheng and P. Palffy-Muhoray, electronic-Liquid Crystal Communica-tions(2007)
[18] L. Landau and E. Lifshitz, Statistical Physics, Course of Theoretical Physics(Elsevier Science, 1980) ISBN 9780750633727
[19] L. Reichl, A modern course in statistical physics, A Wiley Interscience publica-tion (Wiley, 1998) ISBN 9780471595205
[20] L. Landau and E. Lifshitz, Theory of elasticity, Theoretical Physics(Butterworth-Heinemann, 1986) ISBN 9780750626330
[21] P. G. Ciarlet, J. Elasticity 78-79, 1 (Dec. 2005), ISSN 0374-3535
[22] H. Stumpf and J. Makowski, Acta Mechanica 65, 153 (1987), ISSN 0001-5970,10.1007/BF01176879
[23] A. E. H. Love, Philos. Trans. R. Soc. London, Ser. A 179, 491 (1888)
[24] E. Sharon, M. Marder, and H. Swinney, American Scientist 92, 254 (May 2004)
[25] E. Efrati, Y. Klein, H. Aharoni, and E. Sharon, Physica D 235, 29 (Nov. 2007),ISSN 01672789
[26] E. Efrati, E. Sharon, and R. Kupferman, J. Mech. Phys. Solids 57, 762 (Apr.2009), ISSN 00225096
[27] E. Efrati, E. Sharon, and R. Kupferman, Phys. Rev. E 80, 016602 (Jul. 2009),ISSN 1539-3755
[28] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller,The Journal of Chemical Physics 21, 1087 (1953)
[29] D. Rapaport, The art of molecular dynamics simulation (Cambridge UniversityPress, 2004) ISBN 9780521825689
[30] D. L. Ermak and H. Buckholz, Journal of Computational Physics 35, 169 (1980),ISSN 0021-9991
CHAPTER 2
Theory and Simulation of Two-Dimensional Nematic and Tetratic Phases
Recent experiments and simulations have shown that two-dimensional systems can
form tetratic phases with four-fold rotational symmetry, even if they are composed
of particles with only two-fold symmetry. To understand this effect, we propose
a model for the statistical mechanics of particles with almost four-fold symmetry,
which is weakly broken down to two-fold. We introduce a coefficient κ to characterize
the symmetry breaking, and find that the tetratic phase can still exist even up to a
substantial value of κ. Through a Landau expansion of the free energy, we calculate
the mean-field phase diagram, which is similar to the result of a previous hard-particle
excluded-volume model. To verify our mean-field calculation, we develop a Monte
Carlo simulation of spins on a triangular lattice. The results of the simulation agree
very well with the Landau theory.
2.1 Introduction
In statistical mechanics, one key issue is how the microscopic symmetry of particle
shapes and interactions is related to the macroscopic symmetry of the phases. This
issue is especially important for liquid-crystal science, where researchers control the
36
37
orientational order of phases by synthesizing molecules with rod-like, disk-like, bent-
core, or other shapes. In many cases, the low-temperature phase has the same sym-
metry as the particles of which it is composed, while the high-temperature phase has
a higher symmetry. For example, in two dimensions (2D), particles with a rectangular
or rod-like shape, which has two-fold rotational symmetry, form a low-temperature
nematic phase, which also has two-fold symmetry. Likewise, if the particles are per-
fect squares, which have four-fold rotational symmetry, they can form a four-fold
symmetric tetratic phase.
An interesting question is what happens if the symmetry of the particles is slightly
broken. Will the symmetry of the phase also be broken, or can the particles still form
a higher-symmetry phase? For example, we can consider particles with approximate
four-fold rotational symmetry that is slightly broken down to two-fold, as in Fig. 2.1.
Can these particles still form a tetratic phase, or will they only form a less symmetric
nematic phase?
Recently, several experimental and theoretical studies have addressed this prob-
lem. Narayan et al. [1] performed experiments on a vibrated-rod monolayer, and
found that two-fold symmetric rods can form a four-fold symmetric tetratic phase
over some range of packing fraction and aspect ratio. Zhao et al. [2] studied ex-
perimentally the phase behavior of colloidal rectangles, and found what they called
an almost tetratic phase. Donev et al. [3] simulated the phase behavior of a hard-
rectangle system with an aspect ratio of 2, and showed they form a tetratic phase.
38
θ1
θ2
Figure 2.1: Schematic illustration of an interacting particle system in the tetraticphase. The shape of the particles indicates that the rotational symmetry of theinteraction is broken down from four-fold to two-fold.
Another simulation by Triplett et al. [4] showed similar results. In further theoretical
work, Martınez-Raton et al. [5,6] developed a density-functional theory to study the
effect of particle geometry on phase transitions. They found a range of the phase
diagram in which the tetratic phase can exist, as long as the shape is close enough
to four-fold symmetric. In all of these studies, the particles interact through hard,
Onsager-like [7], excluded-volume interactions.
The purpose of the current study is to investigate whether the same phase behavior
occurs for particles with longer-range, soft interactions. We consider a general four-
fold symmetric interaction, which is slightly broken down to two-fold symmetry. We
first calculate the phase diagram using a Maier-Saupe-like mean-field theory [8–10].
39
To verify the theory, we then perform Monte Carlo simulations for the same interac-
tion.
This work leads to two main results. First, the tetratic phase still exists up to a
surprisingly high value of the microscopic symmetry breaking (as characterized by the
interaction parameter κ, which is defined below). Second, the phase diagram is quite
similar to that found by Martınez-Raton et al. for particles with excluded-volume
repulsion. This similarity indicates that the phase behavior is generic for particles
with almost-four-fold symmetry, independent of the specific interparticle interaction.
The plan of this chapter is as follows. In Section 2.2, we present our model
and calculate the mean-field free energy. We then examine the phase behavior and
calculate the phase diagram in Section 2.3. In Section 2.4 we describe the Monte Carlo
simulation methods and results. Finally, in Section 2.5 we discuss and summarize the
conclusions of this study.
As an aside, we should mention one point of terminology. The tetratic phase has
occasionally been called a “biaxial” phase, by analogy with 3D biaxial nematic liquid
crystals [5]. However, this analogy is somewhat misleading. In 3D liquid crystals, the
word “biaxial” refers to a phase with orientational order in the long molecular axis and
in the transverse axes, i.e. a phase with lower symmetry than a conventional uniaxial
nematic. By contrast, the tetratic phase has higher symmetry than a conventional
nematic, four-fold rather than two-fold. For that reason, we will not use the term
“biaxial” in this work.
40
2.2 Model
Maier-Saupe theory is a widely used form of mean-field theory, which describes the
isotropic-nematic transition in 3D liquid crystals. In this section, we extend Maier-
Saupe theory to describe 2D liquid crystals with almost-four-fold symmetry, as shown
in Fig. 2.1. For this purpose, we use the modified Maier-Saupe interaction
U12 (r12, θ12) = −U0(r12) [κ cos (2θ12) + cos (4θ12)] , (2.1)
where θ12 = θ1 − θ2 is the relative orientation angle between particles 1 and 2,
and r12 is the distance between these particles. In this interaction, the dominant
orientation-dependent term is cos (4θ12), which has perfect four-fold symmetry. The
term cos (2θ12) represents a correction to the interaction, which has only two-fold
symmetry. If the coefficient κ is small, then the symmetry is slightly broken from
four-fold down to two-fold. (By contrast, if κ is large, then the interaction clearly has
two-fold symmetry and the four-fold term is unimportant, as in classic Maier-Saupe
theory.) The overall coefficient U0(r12) is an arbitrary distance-dependent term.
In mean-field theory, we average the interaction energy to obtain an effective
single-particle potential due to all the other particles,
Ueff (θ1) =
∫
d2r12dθ2ρ (θ2)U12 (r12, θ12) . (2.2)
Here, ρ (θ2) is the orientational distribution function, which is normalized as
ρ0 =
∫ π
0
dθρ (θ) , (2.3)
41
where ρ0 is the number density of particles. To calculate Ueff, we set the x-axis
along an ordered direction (the director in nematic case, or one of the two orthogonal
ordered directions in the tetratic case). In that case, the averages of sin(2θ) and
sin(4θ) vanish by symmetry, and hence Eq. (2.2) becomes
Ueff (θ) = −Uρ0 [κC2 cos (2θ) + C4 cos (4θ)] , (2.4)
where U is the integral over the position-dependent part of the potential, and
C2 = 〈cos (2θ)〉 , (2.5a)
C4 = 〈cos (4θ)〉 . (2.5b)
The resulting orientational distribution function is
ρ (θ) =ρ0 exp[γ(κC2 cos(2θ) + C4 cos(4θ))]
∫ π
0dθ exp[γ(κC2 cos(2θ) + C4 cos(4θ))]
, (2.6)
where we have defined the dimensionless ratio γ = ρ0U/(kBT ).
Note that C2 can be regarded as a nematic order parameter, and C4 as a tetratic
order parameter. In the isotropic phase, the system has C2 = C4 = 0. By comparison,
in the tetratic phase, the system has C2 = 0 but C4 6= 0. In the nematic phase, with
the most order, the system has C2 6= 0 and C4 6= 0.
To determine which of these phases is most stable, we must calculate the free
energy F = 〈U〉 + kBT 〈log ρ〉 as a function of the order parameters C2 and C4. The
average interaction energy per particle is
〈U〉 = −1
2Uρ0
(
κC22 + C2
4
)
. (2.7)
42
The entropic part of the free energy per particle is
kBT 〈log ρ〉 = Uρ0(
κC22 + C2
4
)
(2.8)
−kBT log
(
1
π
∫ π
0
dθ exp[γ(κC2 cos 2θ + C4 cos 4θ)]
)
;
here we have have subtracted off the constant entropy of the isotropic phase. We
combine these terms and normalize by kBT to obtain the dimensionless free energy
F
kBT=
1
2γ(
κC22 + C2
4
)
(2.9)
− log
(
1
π
∫ π
0
dθ exp[γ(κC2 cos 2θ + C4 cos 4θ)]
)
.
Minimizing this free energy with respect to C2 and C4 gives the equations
C2 =1
ρ0
∫ π
0
dθ cos(2θ)ρ (θ) , (2.10a)
C4 =1
ρ0
∫ π
0
dθ cos(4θ)ρ (θ) , (2.10b)
which are exactly consistent with Eqs. (2.5).
2.3 Mean-field results
The model is now completely defined by two dimensionless parameters: γ =
ρ0U/(kBT ) is the ratio of interaction energy to temperature, and κ represents the
breaking of four-fold symmetry in the interparticle interaction. We would like to
determine the phase diagram in terms of these two parameters. As a first step, we
minimize the free energy of Eq. (2.9) numerically using Mathematica. We then do
analytic calculations to obtain exact values for second-order transitions and special
points in the phase diagram.
43
(a)1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
γ
C2,C
4
C2
C4
(b)1.98 2 2.02 2.04 2.06 2.08 2.1
0
0.1
0.2
0.3
0.4
0.5
γ
C2,C
4
C2
C4
(c)1.25 1.3 1.35 1.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
γ
C2,C
4
C2
C4
(d)0.88 0.89 0.9 0.91 0.92 0.93 0.94
0
0.1
0.2
0.3
0.4
0.5
0.6
γ
C2,C
4
C2
C4
Figure 2.2: Numerical mean-field calculation of the order parameters C2 and C4 asfunctions of γ (inverse temperature), for several values of κ (two-fold distortion inthe interaction): (a) κ = 0.4. (b) κ = 0.75. (c) κ = 1.5. (d) κ = 2.25.
44
Figure 2.2 shows the numerical mean-field results for the order parameters C2 and
C4 as functions of γ, for several values of κ. These plots represent experiments in
which the temperature is varied, for particles with a fixed interaction. When the four-
fold symmetry is only slightly broken by the small value κ = 0.4, there are two second-
order transitions, first from the high-temperature isotropic phase to the intermediate
tetratic phase, and then from the tetratic phase to the low-temperature nematic phase.
For a larger value κ = 0.75, the isotropic-tetratic transition is still second-order, but
now the tetratic-nematic transition is first-order, with a discontinuous change in C2.
For κ = 1.5, the two transitions merge into a single first-order transition directly
from isotropic to nematic, with discontinuities in both C2 and C4, and the tetratic
phase does not occur. Finally, for the largest value κ = 2.25, the isotropic-nematic
transition becomes second-order; this behavior corresponds to the prediction of 2D
Maier-Saupe theory with a simple cos 2θ12 interaction.
The numerical mean-field results are summarized in the phase diagram of Fig. 2.3.
The system has an isotropic phase at low γ (high temperature) and a nematic phase
at high γ (low temperature). It also has an intermediate tetratic phase, as long as
the symmetry-breaking κ is sufficiently small. The temperature range of the tetratic
phase is very large for small κ, then it decreases as κ increases, and finally vanishes at
the triple point B. In this mean-field approximation, the isotropic-tetratic transition is
always second-order and independent of κ. The tetratic-nematic transition is second-
order for small κ, then becomes first-order at the tricritical point A. The direct
45
0.2 1 50
0.5
1
1.5
2
2.5
3
3.5
4
4.5
A
D
CB
Isotropic
Tetratic Nematic
κ
γ
Figure 2.3: Phase diagram of the model in terms of γ (inverse temperature) and κ(two-fold distortion in the interaction). The grey solid lines represent second-ordertransitions, and the dark solid lines are first-order transitions. The dashed linesindicate the extrapolated second-order transitions, which give the cooling limits ofthe metastable phases. Point B (0.79,2) is the triple point, and A (0.61,2.2) and D(2,1) are the two tricritical points. Point C (1,2) is the intersection of the extrapolatedsecond-order transitions.
46
isotropic-nematic transition is second-order for large κ, then becomes first-order at
the tricritical point D. Point C is the intersection of the extrapolated second-order
transitions, and represents the limit of metastability of the tetratic phase.
To calculate second-order transitions and special points in the phase diagram, we
minimize the free energy of Eq. (2.9) analytically. For this calculation, we expand
the free energy as a power series in the order parameters C2 and C4, which gives
F
kBT=
γκ(2− γκ)
8C2
2 +γ(2− γ)
4C2
4 −κ2γ3
8C2
2C4
+κ4γ4
64C4
2 +γ4
64C4
4 + . . . . (2.11)
Note that this expression is exactly what would be expected in a Landau expansion
based on symmetry; it is always an even function in C2, but it is an even function of
C4 only when C2 = 0.
To find the isotropic-tetratic transition, we set C2 = 0 in the expansion, because
this order parameter vanishes in both of those phases. The second-order isotropic-
tetratic transition then occurs when the coefficient of C24 passes through 0. Hence,
the transition is at
γ = 2, (2.12)
independent of κ.
For the second-order isotropic-nematic transition, we see that the isotropic phase
becomes unstable when ∂2F/∂C22 = ∂2F/∂C2
4 = ∂2F/∂C2∂C4 = 0, all evaluated
at C2 = C4 = 0. These equations have two solutions, one of which corresponds to
the isotropic-tetratic transition found above. The other solution, representing the
47
Figure 2.4: A snapshot of the spins on a triangular lattice in the tetratic phase.The shape of the rectangles is just a schematic illustration of the symmetry of theirinteraction.
isotropic-nematic transition, is
γ =2
κ. (2.13)
On the nematic side of this transition, we find C4 = κ2γ2C22/[4(2− γ)]; i.e. the order
parameters C2 and C4 increase with different critical exponents. We substitute that
relation into the expansion (2.11) to obtain an effective free energy in terms of C2
alone,
Feff
kBT=
γκ(2− γκ)
8C2
2 +γ4κ4(1− γ)
32(2− γ)C4
2 + . . . . (2.14)
The tricritical point D occurs when the coefficients of both C22 and C4
2 vanish in this
expansion, which is at γ = 1 and κ = 2.
For the second-order tetratic-nematic transition, we cannot use the expansion of
48
Eq. (2.11) because C4 is not necessarily small; instead we return to the free energy of
Eq. (2.9). Anywhere in the tetratic phase we must have ∂F/∂C4 = 0, which implies
C4 =I1(C4γ)
I0(C4γ), (2.15)
where I0 and I1 are modified Bessel functions of the first kind. At the tetratic-nematic
transition, we also have ∂2F/∂C22 = 0, evaluated at C2 = 0, which implies
2− γκ =γκI1(C4γ)
I0(C4γ). (2.16)
These two equations implicitly determine the second-order tetratic-nematic transition
line shown in Fig. 2.3. To find the tricritical point A, we expand the free energy in
powers of C2, for C4 satisfying Eq. (2.15), and we require that the coefficients of
C22 and C4
2 both vanish. As a result, the tricritical point A occurs at γ = 2.2496,
κ = 0.6116 and C4 = 0.4535.
The first-order transition lines in the phase diagram cannot be calculated analyti-
cally; instead they are determined by numerical minimization of the free energy. The
triple point B occurs where the first-order transition lines intersect the second-order
isotropic-tetratic transition of Eq. (2.12). This point is found numerically at γ = 2
and κ = 0.79.
Point C is the intersection of the extrapolated second-order transitions of Eqs. (2.12)
and (2.13), which occurs at γ = 2 and κ = 1. It represents the highest value of the
symmetry breaking κ where the tetratic phase can even be metastable, beyond the
the triple point B where it ceases to be a stable phase.
49
(a)4 5 6 7 8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
γs
C2,C
4
C2
C4
(b)3 3.2 3.4 3.6 3.8 4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
γs
C2,C
4
C2
C4
(c)2.2 2.4 2.6 2.8 3 3.2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
γs
C2,C
4
C2
C4
(d)0.9 1 1.1 1.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
γs
C2,C
4
C2
C4
Figure 2.5: Simulation results for the order parameters C2 and C4 as functions of γ(inverse temperature), for several values of κ (two-fold distortion in the interaction):(a) κ = 0.3. (b) κ = 0.5. (c) κ = 1. (d) κ = 3.
50
2.4 Monte Carlo simulations
So far, the calculations presented in this chapter have all used mean-field theory.
Of course, mean-field theory is an approximation, which tends to exaggerate the
tendency toward ordered phases. In order to assess the validity of mean-field theory,
we perform Monte Carlo simulations for a lattice model of the same system. In this
lattice model, we use the Hamiltonian
H = −J∑
〈i,j〉
(κ cos[2(θi − θj)] + cos[4(θi − θj)]) , (2.17)
summed over nearest-neighbor sites i and j on a 2D triangular lattice, as shown in
Fig. 2.4. This lattice Hamiltonian corresponds to the model presented in the previous
sections if we take the parameter γ = 6J/(kBT ), because each lattice site interacts
with six nearest neighbors.
We simulate this model on a lattice of size 100 × 100 with periodic boundary
conditions, using the standard Metropolis algorithm [11]. On each lattice site, the
spin is described by an orientation angle θ. In each trial Monte Carlo step, a spin
is chosen randomly, its orientation is changed slightly, and the resulting change in
the energy ∆E is calculated. If energy decreases, the change is definitely accepted.
If not, the change is accepted with a probability of exp [−∆E/(kBT )]. Usually, for
a constant temperature, each Monte Carlo cycle of the simulation consists of 10000
trial steps, and 50000 cycles are used for each temperature. However, near phase
transitions, especially near first-order transitions, additional Monte Carlo cycles are
used to eliminate metastable states. The phase diagram is calculated by cooling the
51
system from high temperature with decreasing the temperature in steps of 0.01, or
steps of 0.005 near phase transitions. During the last half of the simulation cycles,
the order parameters are calculated and time-averaged.
To calculate the nematic order parameter C2, we use the 2D nematic order tensor
Qαβ = 2(
〈nαnβ〉 − 〈nαnβ〉iso)
, (2.18)
averaged over all lattice sites. Here, n = (cos θ, sin θ) is the unit vector representing
each spin, and 〈nαnβ〉iso = 12δαβ is the average in the isotropic phase. The positive
eigenvalue of this tensor is C2.
For the tetratic order parameters C4, we use the generalized tensor method of
Zheng and Palffy-Muhoray [12]. We consider the fourth-order tetratic order tensor
Tαβγδ = 4(
〈nαnβnγnδ〉 − 〈nαnβnγnδ〉iso)
, (2.19)
averaged over all lattice sites. Here, we are subtracting off the isotropic average
〈nαnβnγnδ〉iso = 18(δαβδγδ + δαγδβδ + δαδδβγ). To calculate the eigenvalues, we unfold
this fourth-order tensor into a second-order tensor (4×4 matrix), which we diagonalize
using standard methods. The four eigenvalues are 0, −C4,12
(
C4 − (16C22 + C2
4)1/2
)
,
and 12
(
C4 + (16C22 + C2
4)1/2
)
. (In the tetratic phase, with C2 = 0, they reduce to 0,
−C4, 0 and +C4.) Thus, using the previously calculated value of C2, we can extract
C4.
Figure 2.5 shows the simulation results for the order parameters C2 and C4 as
functions of γ, for several values of κ. These results are quite simular to the numerical
52
Figure 2.6: The degree of coexistence line β when κ = 0.3. The red line is fitted fromsimulation data with β = 0.4912.
mean-field results of Fig. 2.2, although the quantitative values of γ, κ, and the order
parameters are somewhat different.
For a small symmetry breaking κ = 0.3, there are two second-order phase tran-
sitions. The order parameter C4 increases continuously at the high-temperature
isotropic-tetratic transition, and C2 increases continuously at the lower-temperature
tetratic-nematic transition. The increase in C2 can be fit to the expression C2 ∝
(γ − γc)β with β ≈ 0.49 as shown in Fig. 2.6; this is consistent with the prediction
β = 1/2 from mean-field theory for the two dimensional Ising model. For a slightly
53
0.2 1 50
0.5
1
1.5
2
2.5
3
3.5
4
4.5
κ
γs
Isotropic
Tetratic
Nematic
Figure 2.7: Simulation results for the phase diagram in terms of γ (inverse temper-ature) and κ (two-fold distortion in the interaction). The triple point is at approxi-mately γ = 3.2 and κ = 0.60.
larger value of κ = 0.5, the tetratic-nematic transition becomes first-order, with an
apparently discontinuous increase in C2 (within the precision of the simulation). For
κ = 1, the intermediate tetratic phase disappears, and there is just a single first-order
isotropic-nematic transition, with apparently discontinuous increases in both C2 and
C4. Finally, for the largest value κ = 3, the isotropic-nematic transition becomes
second-order, with continuous increases in both C2 and C4.
The simulation results are summarized in the phase diagram of Fig. 2.7. This
phase diagram shows a high-temperature isotropic phase, an intermediate tetratic
phase, and a low-temperature nematic phase. The temperature range of the tetratic
phase is very large when the symmetry breaking κ is small, then it decreases as κ
54
increases, and eventually vanishes at the triple point, which is approximately given
by γ = 3.2 and κ = 0.60. Compared with the mean-field phase diagram of Fig. 2.3,
the simulation results show the transitions at lower temperature (higher γ) than in
mean-field theory. This difference is reasonable because mean-field theory always
exaggerates the tendency toward ordered phases.
2.5 Conclusions
In this chapter, we propose a model for the statistical mechanics of particles with
almost-four-fold symmetry. In contrast with earlier work on particles with a hard-core
excluded-volume interaction, we consider particles with a soft interaction, analogous
to Maier-Saupe theory of nematic liquid crystals. We investigate this model through
two complementary techniques, mean-field calculations and Monte Carlo simulations.
Both of these techniques predict a phase diagram with a low-temperature nematic
phase, an intermediate tetratic phase, and a high-temperature isotropic phase. They
make consistent predictions for the order of the transitions and the temperature
dependence of the order parameters, although they do not agree in all quantitative
details.
The main result of this study is that the tetratic phase can exist up to a sur-
prisingly high value of the symmetry breaking κ in the microscopic interaction. We
find the maximum κ = 0.79 in mean-field theory, or 0.60 in Monte Carlo simulations.
Even taking the lower Monte Carlo value, this implies that the interaction in the
parallel direction (1 + κ) can be about four times larger than the interaction in the
55
perpendicular direction (1−κ). Hence, the tetratic phase can form even for particles
with quite a substantial two-fold component in the interaction energy, i.e. for fairly
rod-like particles.
It is interesting to note that our phase diagram is quite similar to the phase
diagram found by density-functional theory for hard rectangles; see Fig. 3 of Ref. [5].
In that theory, the phase diagram shows isotropic, tetratic, and nematic phases, and
the tetratic phase can exist for rectangles with aspect ratio of up to 2.21:1. That
theory shows the same arrangement of the phases, and even the same first- and
second-order phase transitions, with tricritical points on the isotropic-nematic and
tetratic-nematic transition lines. This phase diagram seems to be a generic feature
of particles with four-fold symmetry broken down to two-fold. Thus, we can expect
to see tetratic phases in 2D experiments and simulations, even if the particles are
moderately extended.
As a final point, we note that the nematic and tetratic phases have the symmetry
of the 2D XY model. Beyond mean-field theory, these phases should have only quasi-
long-range order, and the isotropic-nematic and isotropic-tetratic transitions should
be defect-mediated Kosterlitz-Thouless transitions [13], while the nematic-tetratic
transition should be similar to an Ising transition. These deviations from mean-field
theory are not visible in our simulations, and may be difficult to detect in experiments.
Recently, a closely related theory was developed by Radzihovsky et al. [14,15] in the
completely different context of superfluidity of degenerate bosonic atomic gases. That
56
theory exhibits phases corresponding to the isotropic, tetratic, and nematic phases
studied here, with analogous phase transitions described by the same Landau theory.
We would like to thank R. L. B. Selinger and F. Ye for many helpful discussions.
This work was supported by the National Science Foundation through Grant DMR-
0605889.
BIBLIOGRAPHY
[1] V. Narayan, N. Menon, and S. Ramaswamy, J. Stat. Mech., 01005(2006)
[2] K. Zhao, C. Harrison, D. Huse, W. B. Russel, and P. M. Chaikin, Phys. Rev. E76, 040401 (2007)
[3] A. Donev, J. Burton, F. H. Stillinger, and S. Torquato, Phys. Rev. B 73, 054109(2006)
[4] D. A. Triplett and K. A. Fichthorn, Phys. Rev. E 77, 011707 (2008)
[5] Y. Martınez-Raton, E. Velasco, and L. Mederos, J. Chem. Phys. 122, 064903(2005)
[6] Y. Martınez-Raton and E. Velasco, Phys. Rev. E 79, 011711 (2009)
[7] L. Onsager, New York Academy Sciences Annals 51, 627 (May 1949)
[8] W. Maier and A. Saupe, Z. Naturforsch. A 13, 564 (1958)
[9] W. Maier and A. Saupe, Z. Naturforsch. A 14, 882 (1959)
[10] W. Maier and A. Saupe, Z. Naturforsch. A 15, 287 (1960)
[11] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller,The Journal of Chemical Physics 21, 1087 (1953)
[12] X. Zheng and P. Palffy-Muhoray, electronic-Liquid Crystal Communica-tions(2007)
[13] J. M. Kosterlitz and D. J. Thouless, Journal of Physics C: Solid State Physics6, 1181 (1973)
[14] L. Radzihovsky, J. Park, and P. B. Weichman, Phys. Rev. Lett. 92, 160402 (Apr2004)
[15] L. Radzihovsky, P. B. Weichman, and J. I. Park, Annals of Physics 323, 2376(2008)
57
CHAPTER 3
Deformation of An Asymmetric Film
Experiments have investigated shape changes of polymer films induced by asym-
metric swelling by a chemical vapor. Inspired by recent work on the shaping of elastic
sheets by non-Euclidean metrics [Y. Klein, E. Efrati, and E. Sharon, Science 315,
1116 (2007)], we represent the effect of chemical vapors by a change in the target
metric tensor. In this problem, unlike that earlier work, the target metric is asym-
metric between the two sides of the film. Changing this metric induces a curvature
of the film, which may be curvature into a partial cylinder or a partial sphere. We
calculate the elastic energy for each of these shapes, and show that the sphere is fa-
vored for films smaller than a critical size, which depends on the film thickness, while
the cylinder is favored for larger films.
3.1 Introduction
Thin films are three dimensional (3D) objects with one dimension much smaller
than the other two. Such films are not always flat; they can easily form buckled or
wrinkled 3D shapes. Over several years, there has been extensive theoretical and
experimental research to explore the mechanisms for shape selection. This research
is important to understand the formation of biological structures, such as leaves and
flowers, in which thin films assume well-defined shapes with biological functions. It is
58
59
also important for the design of new synthetic materials, which should spontaneously
form desired morphologies [1–4].
Sharon and collaborators have recently developed an important theoretical ap-
proach for addressing shape selection in thin elastic sheets [5–10]. In this approach,
a film is characterized by a “target metric tensor,” which describes the ideal spacings
between points in the film that minimize the local energy. Depending on the mathe-
matical properties of this tensor, there may or may not be any global shape of the film
embedded in 3D Euclidean space that achieves the ideal spacings everywhere. If this
state is not achievable, the film is geometrically frustrated. Its lowest-energy state will
then have residual local stresses and strains, and will generally be curved in a complex
way [11,12]. Sharon et al. have demonstrated this approach experimentally by using
thin films of gels, which can be expanded locally by adding a nonuniform concentra-
tion of a dopant. The gels then relax to the 3D shape that has been programmed by
the dopant concentration profile, in agreement with the geometric calculations.
One limitation of Sharon’s theory is that it assumes the films are uniform across
their thickness. This limitation is significant, because many types of films have some
variation in their elastic properties across their thickness. Indeed, this type of varia-
tion might provide an additional way to design films to form desired structures. The
purpose of this study is to generalize the theory to describe asymmetric films, with
arbitrary variation across the thickness. Through this generalization, we show that
the asymmetry leads to new types of terms in the elastic free energy of the film, and
60
we investigate how these terms change the curvature.
As a specific example to motivate this study, we consider the asymmetric swelling
of a thin film by absorption of a gas or liquid. A schematic view of this problem is
shown in Fig. 3.1. Here, a cross section through the film is illustrated by mass points
connected by springs, with bold lines indicating that a spring is compressed relative
to its stress-free length. Before swelling, in Fig. 3.1(a), all the springs are at the
stress-free length, and the film is flat. After swelling, the intrinsic stress-free lengths
of the springs vary gradually along the thickness direction, as shown in Fig. 3.1(b).
Hence, the film will deform to minimize the energy, as shown in Fig. 3.1(c). Because
the film is 3D, the actual geometry is more complex than the cross section shown in
the figure. It is not obvious whether the film should deform into a partial cylinder
(with mean curvature but no Gaussian curvature) or a partial sphere (with both mean
and Gaussian curvature). In either case, some of the springs will be unable to achieve
their intrinsic stress-free length. Hence, this is a simple example of a geometrically
frustrated structure.
To address this problem, we extend the approach of Sharon et al. to consider a
target metric that depends on position across the thickness of the film. We calculate
the energy of the deformed film in cylindrical and spherical geometries, and find there
is a critical lateral size, which depends on the film thickness and the gradient of the
intrinsic metric. If the lateral size of the film is smaller than this critical size, a partial
sphere is preferred; otherwise, a partial cylinder has lower energy.
61
(a)
(b)
(c)
Figure 3.1: Schematic illustration of the deformation of a thin shell due to swelling:(a) before swelling, (b) after swelling without any deformation, and (c) deformedshell.
62
This chapter is organized as follows. In Sec. 3.2, we briefly review the theory of
non-Euclidean plates, and use it to calculate the energy in the general case where the
target metric is proportional to the distance from the midplane. We then use this
general formula to calculate the energy for a cylinder in Sec. 3.3 and for a sphere in
Sec. 3.4. Finally, in Sec. 3.5, we discuss the results of this study and compare with
previous work on related elastic problems.
3.2 Theory
In the first part of this section, we briefly review the non-Euclidean theory devel-
oped by Sharon et al. for the special case in which the target metric is uniform across
the thickness of the film [7,8]. We then introduce our calculation for the asymmetric
film with a target metric tensor that depends on the distance from the midplane of
the film.
3.2.1 Brief review of non-Euclidean theory
Following Sharon et al., we define the elastic free energy for any configuration
of a material through the following construction. First, we construct a coordinate
system xi, for i = 1, 2, 3, in the material frame. The current 3D position of a point
(x1, x2, x3) is then given by R(x1, x2, x3). As a result, the current spacing between
two nearby material points (x1, x2, x3) and (x1 + dx1, x2 + dx2, x3 + dx3) is given
by (ds)2 = gijdxidxj (using the Einstein summation convention, where Latin indices
range from 1 to 3, and Greek indices range from 1 to 2). Here, the metric tensor is
63
gij = ∂xiR ·∂xjR = ∂iR ·∂jR. By comparison, if we were to cut out a very small piece
of material around the point (x1, x2, x3), and let this small piece relax to its lowest-
energy state, then the spacing between those points would be (ds)2 = gijdxidxj, where
gij is the target metric tensor that characterizes the intrinsic, ideal spacing between
the points. The Green-Lagrangian strain tensor is then defined as the difference
between the current metric and the target metric
εij =1
2(gij − gij) . (3.1)
If the material is isotropic, the most general expression for the 3D elastic energy
density can be written as
f3 =1
2Aijklεijεkl, (3.2)
where
Aijkl = λgijgkl + µ(gikgjl + gilgjk), (3.3)
are the contravariant components of the three-dimensional elasticity tensor in curvi-
linear coordinates [13]. If the target metric gij is independent of the coordinate x3
across the thickness of the cell, then the 3D elastic energy density can be integrated
across thickness to construct the 2D elastic energy density of the plate,
f2 =
∫
f3dx3. (3.4)
This 2D elastic energy density includes a stretching energy (proportional to thickness
h) and a bending energy (proportional to h3). The total elastic energy then becomes
F =
∫
f2√
|g|dx1dx2 =
∫
f3√
|g|dx1dx2dx3, (3.5)
64
(a) Flat film (b) Partial cylinder (c) Partial sphere
Figure 3.2: Illustration of the asymmetric swelling experiment.
integrated over the whole body.
Note that the target metric tensor gij represents an ideal local configuration of
the plate with zero stress and zero elastic energy density. There might or might not
be any global configuration in 3D Euclidean space that has this target metric tensor
everywhere. If such a configuration exists, then it is a stress-free reference configu-
ration, which can be used to formulate Truesdell’s hyper-elasticity principle [14]. On
the other hand, if such a configuration does not exist, then there is no such stress-free
reference configuration. It is still possible to minimize the elastic energy to find the
equilibrium state of the plate, but this equilibrium state must be a frustrated state
with residual local stress. In this case, the object can be called a non-Euclidean plane,
whose midplane has no immersion with zero stretching in 3D Euclidean space.
3.2.2 Asymmetric film
Fig. 3.2 shows our imaginary experiments for asymmetric swelling. Before swelling,
the polymer film is flat as shown in Fig. 3.2(a). Then, if we have some gas vapor
65
passing through the film from bottom to the top. Because of the asymmetric swelling,
the bottom of the film wants to expand more than the top part. As a result, the film
will deform into either a partial cylinder (Fig. 3.2(b)) or a partial sphere (Fig. 3.2(c))
according to different conditions.
In our problem of asymmetric swelling of a thin film, as shown in Fig. 3.1, the
target metric must depend on position across the thickness of the film. If the gas vapor
induces expansion of the polymer film, the expansions at the top and the bottom of
the film are different due to the different concentrations of the gas. Assuming the gas
concentration varies linearly across the thickness, the target tangent vectors can be
written as
gi = gi
(
p+ qx3)
, (3.6)
and hence the target metric is
gij = gi · gj = gij
(
p2 + 2pqx3 + q2(
x3)2)
. (3.7)
In our notation, all symbols with rings represent quantities before swelling ; in par-
ticular, gi are the tangent vectors and gij is the metric before swelling. Likewise, all
symbols with bars represent target quantities after swelling. The coefficients p and
q define how the gas concentration is distributed; p represents a uniform expansion
or compression, and q represents a linear gradient in the expansion factor across the
thickness of the film.
From Eq. (3.1), the strain tensor is defined as the difference between the actual
metric gij and the target metric gij after swelling. To find the actual metric, we must
66
consider a specific configuration of the plate. Following the second Kirchhoff-Love
assumption, we assume that points located on any normal to the midplane in the
initial state remain on that normal in the deformed state, but the distance to the
midplane may change. Thus, a point (x1, x2, x3 = 0) on the midplane before swelling
becomes R(x1, x2, x3 = 0) = Rmid(x1, x2) after swelling, and a point (x1, x2, x3) off
the midplane becomes
R(x1, x2, x3) = Rmid(x1, x2) + ξ(x3)N(x1, x2). (3.8)
Here, N is the unit normal vector to the midplane, and ξ(x3) is the new distance to
the midplane along the normal. Because of swelling, ξ(x3) is no longer equal to x3.
To lowest order for a thin film, ξ(x3) can be written as power series
ξ(x3) = mx3 +1
2m′
(
x3)2
. (3.9)
If we assume that the local volume everywhere remains constant during deformation,
equal to the swelled local volume of the target metric, then the coefficients in this
expansion are constrained to be m = p and m′ = 3q.
Using this configuration of the plate, we can calculate the metric tensor gij as a
power series in x3. From Eqs. (3.8) and (3.9), the first 2 × 2 components of gij can
be written as
gαβ = aαβ − 2mbαβx3 −m′bαβ
(
x3)2
, (3.10)
where aαβ = ∂αR · ∂βR is the first fundamental form of the midplane (i. e. the 2D
metric tensor), and bαβ = N · ∂α∂βR is the second fundamental form of the midplane
67
(i. e. the 2D curvature tensor). The first 2 × 2 compenents of the strain tensor are
then εαβ = 12(gαβ − gαβ).
We can now calculate the elastic energy density of the plate. Based on the first
Kirchhoff-Love assumption that the stress is in the local midplane [7, 15], we can
express the full 3D elastic energy density of Eq. (3.2) in terms of the first 2 × 2
compenents of the strain tensor as
f3 =1
2Aαβγδεαβεγδ, (3.11)
where
Aαβγδ = 2µ
(
λ
λ+ 2µgαβ gγδ + gαγ gβδ
)
. (3.12)
If we assume that the local volume everywhere remains constant during deformation,
then gijεij = 0, and hence the elastic modulus λ → ∞. As a result, the elasticity
tensor becomes
Aαβγδ = 2µ(
gαβgγδ + gαγ gβδ)
. (3.13)
Note that the contravariant components of the target metric tensor in this expression
are the inverse of the covariant components of Eq. (3.7). Hence, the elasticity tensor
depends on x3 as
Aαβγδ =1
p4Aαβγδ
(
1− 4q
px3 +
10q2
p2(
x3)2
)
. (3.14)
Thus, the 2D elastic energy density can be calculated by integrating the 3D elastic
energy density over the film thickness w, and neglecting terms that has higher order
68
than q2, to obtain
f2 =
∫ w/2
−w/2
f3dx3
=1
p4Aαβγδ
(
w
2ε2Dαβε
2Dγδ +
p2w3
24bαβbγδ
+qw3
24ε2Dαβbγδ +
qw3
12aαβbγδ
+q2w3
p2ε2Dαβaγδ +
q2w3
3p2aαβaγδ
)
. (3.15)
To interpret this 2D elastic energy density f2, note that it depends on three tensors
characterizing the local geometry of the midplane: the 2D strain tensor
ε2Dαβ =1
2(aαβ − aαβ) , (3.16)
which gives the difference between the actual metric and the target metric on the
midplane, as well as the 2D metric tensor aαβ and the 2D curvature tensor bαβ. If
there is no swelling, so that p = 1 and q = 0, then f2 becomes the deformation energy
of a symmetric film. In that case, the first and second terms in Eq. (3.15) are the
stretching and bending terms, and the other terms vanish. However, if the film is
swollen asymmetrically, with q 6= 0, then the last four terms provide new couplings
that are permitted by the asymmetry. Two of these terms are odd in the curvature
tensor bαβ , so they favor spontaneous curvature of the film, which is then coupled to
the in-plane strain ε2Dαβ . The last two terms might seem to be higher order in q than
the others, but we will show later that the equilibrium curvature bαβ is proportional
to q, and hence the last five terms are all of order q2.
69
Figure 3.3: (Color online) Illustration of the curved film shapes considered in Sec-tions 3.3 and 3.4. (a) Partial cylinder. (b) Partial sphere.
For further insight into the 2D elastic energy of Eq. (3.15), in the following two
sections we use it to calculate the energy of an asymmetrically swollen film curving
into a partial cylinder or a partial sphere, as shown in Fig. 3.3. In each case, we
calculate the optimal curvature and determine how it depends on the asymmetry
parameter q. We then compare the energies of these two shapes to see which is
favored, as a function of the film parameters.
3.3 Cylinder
We first use Eq. (3.15) to calculate the energy when the film deforms into a
partial cylinder, as shown in Fig. 3.3(a). For this calculation, we use an orthogonal
curvilinear coordinate system (ξ1, ξ2) on the midplane in the material frame, where
ξ1 is the direction around the circumference of the cylinder and ξ2 is the direction
along the axis. The 3D position of any point on the midplane can be written as
Rmid = rc cos
(
C1ξ1rc
)
x+ C2ξ2y + rc sin
(
C1ξ1rc
)
z, (3.17)
70
where rc is the cylinder radius, C1 and C2 are parameters that measure how much
the coordinates elongate or shrink, and x, y, and z are unit vectors in Cartesian
coordinates for the lab frame. From Eq. (3.7), the target metric of the midplane is
aαβ =
p2 0
0 p2
. (3.18)
From Eq. 3.17, the actual metric and curvature tensors (first and second fundamental
forms) of the partial cylinder are given by
aαβ =
C21 0
0 C22
(3.19)
and
bαβ =
−C21/rc 0
0 0
. (3.20)
The elasticity tensor Aαβγδ can be calculated by noticing that the target metric tensor
before swelling is just the identity matrix, independent of position. Hence, the non-
zero terms in the elasticity tensor are A1111 = A2222 = 4µ and A1122 = A1212 =
A2121 = A2211 = 2µ.
We now insert the target metric, actual metric, and curvature tensors into the 2D
elastic energy of Eq. (3.15), and minimize over the parameters C1, C2, and rc. To
lowest order in w, we find that the stretching factors are C1 = C2 = p, the cylinder
radius is
rc =2p2
3q, (3.21)
71
and the corresponding 2D elastic energy density at that minimum is
f2 =29q2
8p2w3µ. (3.22)
These results imply that the curvature tensor bαβ is proportional to q, and that
the strain tensor ε2Dαβ = 0; i. e. there is no strain on the midplane. Note, however,
that there is still nonzero shear strain off the midplane because the metric tensor off
midplane shows anisotropic swelling while the target metric favors isotropic swelling;
that is the reason why the elastic energy is non-zero. In the limit of a symmetric film
with q → 0, then rc → ∞, the film remains flat, and there is only uniform swelling
without energy cost.
Equation (3.22) shows that the elastic energy density of the partial cylinder is
uniform in 2D, independent of position on the midplane, and hence the total elastic
energy is just proportional to the film area. In particular, if the initial shape of
the film is a disk with radius rmax in the material frame, hence radius prmax in the
midplane after swelling, then the total elastic energy is
Fc =29
8πq2r2maxw
3µ. (3.23)
3.4 Sphere
Let us now consider a circular thin film deforming into a partial of a sphere, as
shown in Fig. 3.3(b). For this problem, it is convenient to use polar coordinates
(r, φ) on the midplane in the material frame, where r is the radial displacement from
the center of the circular film before swelling, and φ is the azimuthal angle, which
72
is assumed not to change during swelling and deformation. The 3D position of any
point on the midplane can then be written as
Rmid = rs sin θ(r) cosφx+ rs sin θ(r) sinφy + rs cos θ(r)z, (3.24)
where rs is the radius of the sphere and θ(r) is a monotonically increasing function
of r, to be determined, which describes how the material stretches or shrinks in the
radial direction. (In particular, θ(r) gives the angular position on the partial sphere
up from the (−z)-axis corresponding to the radial position r in the material frame.)
In this coordinate system, the target metric of the middle plane is
aαβ =
p2 0
0 p2r2
. (3.25)
Note that this target metric is equivalent to Eq. (3.18), but in a different coordinate
system. From Eq. (3.24), the actual metric and curvature tensors (first and second
fundamental forms) are given by
aαβ =
r2sθ′(r)2 0
0 r2s sin2 θ(r)
, (3.26)
and
bαβ =
−rsθ′(r)2 0
0 −rs sin2 θ(r)
. (3.27)
In this coordinate system, the non-zero components of the elasticity tensor Aαβγδ are
A1111 = 4µ, A2222 = 4µ/r4, and A1122 = A1212 = A2121 = A2211 = 2µ/r2.
73
The 2D elastic energy of Eq. (3.15) is now a functional of the stretching function
θ(r) and the sphere radius rs. To minimize this energy over θ(r), we expand θ(r) as
a power series in r and minimize over the series coefficients. The leading terms are
then
θ(r) = π − (p− p3w2
6r2s+
3pqw2
8rs)r
rs. (3.28)
To minimize the energy over rs, we must consider two regimes in terms of q, w,
and rmax, the radius of the film in the material frame. In the first regime, where
qr2max/w ≪ 1, the optimum sphere radius is rs = p2/q, and the total elastic energy is
Fs =7
2πq2r2maxw
3µ. (3.29)
By comparison, in the second regime where qr2max/w ≫ 1, the sphere radius is rs =
(prmax)4/3(26qw2)−1/3, and the total elastic energy is
Fs = 4πq2r2maxw3µ. (3.30)
3.5 Discussion
Our calculations in the preceding sections lead to specific conclusions about the
cylindrical and spherical shapes, as well as more general insight into elastic theory
for asymmetric films.
For the specific problem of cylindrical and spherical shapes, we can see that
Eq. (3.23) for the elastic energy of a partial cylinder is between the two regimes
of Eqs. (3.29) and (3.30) for a partial sphere. This result implies that the spherical
deformation is favored for disks of small radius rmax < rcritical, while the cylindrical
74
deformation is favored for disks of large rmax > rcritical. Calculating the actual value
of rcritical requires higher-order terms than we have presented here, and the numerical
result is rcritical = 1.6(wp/q)1/2. To understand this result, note that the partial sphere
has an isotropic deformation in the midplane, which is consistent with the target met-
ric, while the partial cylinder breaks is anisotropic in the midplane and disagrees with
the target, which costs extra energy. For that reason, the partial sphere is favored for
small rmax. By contrast, for large rmax the partial sphere must have extra stretching
in the midplane, and hence it becomes disfavored with respect to the partial cylinder.
More generally, we have shown that the theoretical approach of Sharon and col-
laborators can be applied to asymmetric films. Through this approach, we transform
the 3D elastic energy into the effective 2D elastic energy of Eq. (3.15). This effective
2D elastic energy shows the standard stretching and bending energies, which have
been studied extensively for symmetric films, as well as new terms arising from the
asymmetry. These new terms include a spontaneous curvature term, which is linear
in the curvature tensor and hence favors curvature of the asymmetric film, as well as a
coupling between spontaneous curvature and in-plane strain. These new terms should
provide new opportunities to design synthetic materials that will spontaneously form
desired shapes for technological applications.
This work was supported by the National Science Foundation through Grants
DMR-0605889 and 1106014.
BIBLIOGRAPHY
[1] M. Finot and S. Suresh, J. Mech. Phys. Solids 44, 683 (May 1996), ISSN 00225096
[2] L. B. Freund, J. Mech. Phys. Solids 44, 723 (May 1996), ISSN 00225096
[3] L. B. Freund, J. Mech. Phys. Solids 48, 1159 (Jun. 2000), ISSN 00225096
[4] K. N. Long, M. L. Dunn, T. F. Scott, and H. J. Qi, Int. J. Struct. Changes Solids2, 41 (2010)
[5] Y. Klein, E. Efrati, and E. Sharon, Science 315, 1116 (Feb. 2007), ISSN 1095-9203
[6] E. Efrati, Y. Klein, H. Aharoni, and E. Sharon, Physica D 235, 29 (Nov. 2007),ISSN 01672789
[7] E. Efrati, E. Sharon, and R. Kupferman, J. Mech. Phys. Solids 57, 762 (Apr.2009), ISSN 00225096
[8] E. Efrati, E. Sharon, and R. Kupferman, Phys. Rev. E 80, 016602 (Jul. 2009),ISSN 1539-3755
[9] E. Sharon and E. Efrati, Soft Matter 6, 5693 (Oct. 2010), ISSN 1744-683X
[10] R. D. Kamien, Science 315, 1083 (Feb. 2007), ISSN 1095-9203
[11] A. Hoger, Arch. Rational Mech. Anal. 88, 271 (1985), ISSN 0003-9527
[12] M. Ben Amar and A. Goriely, J. Mech. Phys. Solids 53, 2284 (Oct. 2005), ISSN00225096
[13] P. G. Ciarlet, J. Elasticity 78-79, 1 (Dec. 2005), ISSN 0374-3535
[14] C. Truesdell, The Mechanical Foundations of Elasticity and Fluid Dynamics(Gordon and Breach, 1966) ISBN 0677008201
[15] A. E. H. Love, Philos. Trans. R. Soc. London, Ser. A 179, 491 (1888)
75
CHAPTER 4
Coarse-Grained Modeling of Deformaable Nematic Shell
We develop a coarse-grained particle-based model to simulate membranes with
nematic liquid-crystal order. The coarse-grained particles self-assemble to form vesi-
cles, which have orientational order in the local tangent plane at low temperature.
As the strength of coupling between the nematic director and the vesicle curvature
increases, the vesicles show a morphology transition from spherical to prolate and
finally to a tube. We also observe the shape and defect arrangement around the tips
of the prolate vesicle.
4.1 Introduction
Over the past twenty-five years, a major theme of research in condensed-matter
physics has been the complex interaction of geometry with orientational order and
topological defects. Both theoretical and experimental studies have investigated order
and defects on surfaces of fixed shape, such as colloidal particles or droplets [1–7].
These studies have shown, for example, that a nematic phase on a spherical surface
will form four defects of topological charge +1/2 each, and these defects may be
exploited to develop colloidal particles that will self-assemble into tetrahedral lattices
for photonic applications [8]. Inspired by this potential application, many authors
studied how to control the arrangement of the four half-charged defects in a nematic
76
77
phase. Many effects have been considered, including elastic anisotropy [3], external
field [4], and curvature of the colloidal particles [6].
Further research has investigated orientational order and defects on deformable
vesicles, which serve as simple analogues for biological membranes [9–12]. These
studies show that defects in the orientational order will deform fluid vesicles into non-
spherical shapes. For example, some vesicles have tilt order, which can be modeled
as XY order in the local tangent plane; these vesicles will exhibit two defects of
topological charge +1 each, and can deform into prolate or oblate shapes [11]. Other
vesicles composed of T-shaped lipids or surfactants with rod-like heads may have
nematic order in the local tangent plane [13]. Theoretical studies have predicted that
the four half-charged defects will induce these vesicles to deform into tetrahedra [9].
So far, theoretical studies of orientational order in deformable vesicles have consid-
ered systems that are idealized in several ways: at zero temperature, with no elastic
anisotropy, with only certain couplings between orientational order and curvature,
and with shapes that are slight perturbations on perfect spheres. At this point, the
key question is how the predictions are modified for more complex systems. Com-
puter simulation provides a useful approach to this issue. For example, simulations
can investigate problems where the geometry is not a perfect sphere but rather a more
complex disordered shape, with bumps of positive and negative Gaussian curvature.
One simulation method uses a triangulated-surface model with tangent-plane orien-
tational order; this method has indeed shown complicated tube and inward tubulate
78
shapes [12]. However, a disadvantage of the triangulated-surface model is that the
connectivity of the surface is fixed, unlike experimental systems in which molecules
can detach from and rejoin the vesicles.
4.2 Two vectors Coarse-Grained Model
In this chapter, we develop an alternative method to simulate orientational order
in deformable nematic vesicles, using a coarse-grained particle-based model. This
method allows the particles to self-assemble into a membrane with orientational or-
der in the local tangent plane. The membrane spontaneously selects its own shape,
which may be flat, spherical, or more complex. The model is simple enough to al-
low simulation of large space and time scales. Furthermore, the interaction of the
coarse-grained particles can be correlated with molecular features. Using this model,
we calculate the arrangement of topological defects and the shape of the vesicle as a
function of the interaction parameters. In particular, we find a morphology transition
from spherical to prolate and finally to a tube as the coupling between nematic order
and curvature increases.
To develop an appropriate simulation approach, we are inspired by the coarse-
grained membrane simulations without tangent-plane order by Ju Li and collabora-
tors [14–17]; see also Ref. [18]. In their approach, a membrane is represented by a
single layer of interacting point particles, each of which carries a polar vector degree of
freedom n representing the preferred layer normal direction. The interaction potential
79
Figure 4.1: Schematic illustration of our two-vector model for interacting coarse-grained particles. Each particle has a vector n, which aligns along the local membranenormal, and a vector c, which has nematic alignment within the local tangent plane.
favors association of particles with n vectors lying parallel and side-by-side. In simula-
tions, the particles self-assemble into pancake-shaped single-layer aggregates showing
liquid-like self-diffusion and membrane elasticity. When another term is added to the
potential favoring a slight splay between the n vectors of neighboring particles, the
particles spontaneously coalesce into hollow spherical shells. In this approach, each
particles represents not a single molecule but a large patch of membrane containing
many molecules, and the surrounding solvent is implicit.
To simulate a membrane with tangent-plane order, we define a coarse-grained
point particle with two vector degrees of freedom, as shown in Fig. 4.1. The vector
n again defines the layer normal direction, and a new vector c represents the local
nematic director orientation in the membrane’s tangent plane. Both n and c are unit
vectors and are always perpendicular to each other. The particles interact with each
80
other via an anisotropic Lennard-Jones-type pairwise potential with a cutoff:
uij(ni, nj, ci, cj,xij) = (4.1)
uR(xij) + [1 + α[a(ni, nj , ci, cj, xij)− 1]]uA(xij).
In this expression, the repulsive and attractive parts of the potential are given by
uR =
ǫ(
Rcut − rRcut − rmin
)8
xij < Rcut
0 xij > Rcut
(4.2)
uA =
−2ǫ(
Rcut − rRcut − rmin
)4
xij < Rcut
0 xij > Rcut
(4.3)
Note that the repulsive and attractive terms have exponents 8 and 4, respectively,
in contrast with the exponents 12 and 6 for the Lennard-Jones potential. These
reduced exponents soften the potential and enhance the fluidity of the membrane.
The coefficient α controls the strength of the anisotropic orientational interactions,
which are defined by the function:
a(ni, nj, ci, cj, xij) = 1− Tn − Tc. (4.4)
Tn term contains only the interaction that make the particles self assmble into a lipid
membrane (flat or spherical):
Tn =[
1− (ni · nj)2 − β
]2+[
(ni · xij)2 − γ
]2+[
(nj · xij)2 − γ
]2. (4.5)
In this function, β = sin2(θ0) and γ = sin2(θ0/2), where θ0 represents the preferred
81
angle between the n vectors of two neighboring particles. Tc term contains the inter-
action that superposes the tangential order on the surface of the membrane:
Tc = 2η2[
1− (ci · cj)2]
. (4.6)
η is a constant and characterise the strength of the in-plane order.
Except for the a(ni, nj , ci, cj, xij) term, the interaction potential of Eq. 4.1–4.3
are exactly the same as the potential considered in Ref. [14]. We modified the terms
of n to make the membrane more flexiable. This potential is minimized when the
anisotropic function a(ni, nj, ci, cj, xij) is maximized. If θ0 = 0, the lowest-energy
state occurs when the n vectors of neighboring particles are parallel to each other
and perpendicular to the interparticle separation vector xij . Hence, at low temper-
ature, the particles self-assemble into flat membranes with the n vectors along the
membrane normal. By comparison, if θ0 6= 0, then the n prefer an angle θ0 be-
tween neighboring particles; i.e. they favor a splay. For that reason, the particles
self-assemble into vesicles with some spontaneous curvature, which gives an intrinsic
radius of the vesicles. These results have been reported in Ref. [14] and confirmed in
our simulations.
The new aspect of our potential is the η2 term, which couples the c vectors of
neighboring particles. Because of this term, the lowest-energy state must have either
parallel or antiparallel alignment of the c vectors. Hence, this term favors nematic
liquid-crystal order within the membrane. As the coefficient η increases relative to
temperature, the strength of this nematic order must increase. Note that the c
82
vectors are constrained to be perpendicular to the n vectors, which align along the
local membrane normal. For that reason, the nematic order is only defined within
the local tangent plane to the membrane. Thus, this model provides an opportunity
to simulate nematic order within self-assembled curved membranes. By varying the
parameter η at fixed temperature, we can vary the strength of the nematic order,
and hence the Frank elastic constants of the liquid crystal and the strength of the
coupling between curvature and orientational order.
4.3 Simulation and Discussion
An equation of motion can be derived from the inter-particle potential 4.1, see from
Appendix A. We perform a series of simulations with about 10,000 particles over a
range of η from 0.2 to 0.5, at the same temperature kBT ≈ 0.22ε. Initially, the coarse-
grained particles are placed on a sphere using the random sequential method [19]. The
temperature is then increased and maintained by a Langevin thermostat. Snapshots
of the front, side, and top views of the simulation results are shown in Table 4-1.
The black dots on the vesicles indicate the locations of the four positive half-charged
defects in the tangent plane (except for the largest η = 0.5), and the color indicates
the distance from the center of mass of the vesicle. On the surface of the vesicles,
the c vectors are shown as cylinders in the local tangent plane. The n vectors are
not shown because they are always along the normal direction. The coarse-grained
particles are semi-transparent so that the defects are visible on the near and far sides
of the vesicles; defects on the near side are slightly darker than those on the far side.
83
η Front Side Top
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Table 4-1: (Color online) Shapes and defect configurations of a vesicle for severalvalues of the coupling η. The color indicates the distance from the center of massof the vesicle, and the black dots indicate the defect locations. In these images thevesicle is semi-transparent, so that defects on both near and far sides are seen.
84
From the snapshots in Table 4-1, we can see that changing η changes the nematic
director field and defect configuration as well as the shape of the vesicles. For the
smallest η = 0.2, the four defects align themselves approximately on a great circle.
This great circle is consistent with the continuum prediction of Ref. [3] for nematic
vesicles with a strong elastic anisotropy, so it may indicate that this potential gives
a substantial difference between the effective Frank constants. As η increases to
0.25, the defects move to form approximately a regular tetrahedron. For these small
values of η, the vesicle shape is fairly close to spherical, indicating that the coupling
between director distortions and curvature is not yet enough to substantially distort
the sphere. As η increases further to 0.30, the defects shift further, with two defects
moving to each end of the vesicle. At the same time, the vesicle elongates substantially
between these two poles, forming a prolate shape. For η ≥ 0.35, the nematic order
becomes stronger, so that the defect cores cost more energy. We then observe holes
at the centers of the defects, and the vesicle coexists with a particle gas. The defect
configuration also changes from elongated tetrahedron (η =0.3–0.4) to rectangular
(η = 0.45). Finally, at the largest value η = 0.5, each pair of defects fuses to form a
large pore at each end of the vesicle, thus eliminating the defects and transforming
the vesicle into a tube. The middle of the tube is still swollen, not a perfect cylinder,
presumably because of the spontaneous curvature θ0.
To show the relationship between the local nematic orientational c and the overall
shape more clearly, Fig. 4.2 presents enlarged front and top views of the vesicle
85
(a) Front
(b) Top
Figure 4.2: (Color online) Enlarged front and top views of the vesicle for η = 0.3,with cylinders showing the local nematic orientation c in relation to the overall shape.In these images the vesicle is opaque, so that only defects on the near side are seen.
86
with η = 0.3. In these views the coarse-grained particles are opaque, so that only
two defects at a time (on the near side) are visible, and the defects can easily be
identified as topological charge 1/2. From the front view, we see that the local
nematic orientation c is aligned along the long axis of the prolate vesicle. This
alignment is reasonable, because it allows the c vectors of neighboring particles to
be almost parallel in three dimensions (3D), thus minimizing the interaction energy.
The transverse alignment would have a substantially higher energy for the interaction
of c vectors in 3D. From the top view, we see that the local nematic orientation
c is aligned perpendicular to the separation between the two half-charged defects.
Hence, the director distortion between the defects is almost entirely bend rather than
splay. Furthermore, the top view of the vesicle is not circular but extended along the
average nematic director, perpendicular to the separation between the defects. Thus,
the overall vesicle has a biaxial, potato-like shape.
It is remarkable that the half-charged defects are arranged in pairs, with a pair at
each end of the vesicle, in spite of the usual repulsion between defects. We speculate
that this arrangement occurs because the defects are attracted to the regions of high
positive Gaussian curvature at each end of the vesicle. This attraction to the curved
regions competes with the repulsion between defects to favor an optimum separation
between the defects, which depends on the coupling parameter η. A similar pairing of
defects has been seen in analytic calculations by Kralj [6] for the optimum positions
of nematic defects on colloidal particles with a fixed ellipsoidal shape. Here we see
87
that the pairing occurs even when the shape is not fixed but is free to deform.
Our simulations can be compared with the predictions of Park et al. [9], who
performed analytic calculations of the shapes of deformable membranes with nematic
and general n-atic order (note that n = 2 for a nematic phase). Their theory predicts
that a nematic vesicle should deform into a shape with the symmetry of a regular
tetrahedron, with a half-charged defect at each vertex of the tetrahedron. By con-
trast, our simulations never show the tetrahedral deformation, but only the extended
biaxial, potato-like shape. We speculate that this discrepancy occurs because their
theory is idealized in two ways. First, their free-energy functional considers only the
intrinsic coupling of director variations with curvature via through a covariant deriva-
tive; it explicitly neglects other couplings allowed by symmetry [20]. The interaction
potential in our simulation includes an extrinsic coupling of the nematic director to
the 3D curvature direction. Our earlier simulations showed that the extrinsic coupling
greatly changes the director field on surfaces with fixed curvature [21]; here we see
that it also affects the shapes of deformable membranes. This effect may explain why
the vesicle becomes extended along its long axis and why its ends become extended
in a biaxial way; both of these distortions reduce the interaction energy of c vectors
in 3D. Second, their free-energy functional makes the approximation of a single Frank
elastic constant, while our interparticle potential presumably gives different effective
Frank constants. Shin et al. [3] showed that the anisotropy of Frank constants changes
the arrangement of defects; here it also affects the membrane shape.
88
4.4 Conclusion
In conclusion, we have developed a coarse-grained particle-based model for sim-
ulating self-assembled membranes with orientational order, and we have used it to
study vesicles in the nematic liquid-crystal phase. The simulation results show sur-
prisingly complex vesicle shapes and defect configurations, which arise from features
in the interparticle potential. Thus, the simulation method enables us to explore the
range of phenomena that can occur in soft materials where geometry interacts with
orientational order and topological defects.
We thank A. Travesset for helpful discussions. This work was supported by NSF
Grants DMR-0605889 and 1106014.
BIBLIOGRAPHY
[1] V. Vitelli and D. Nelson, Physical Review E 74, 1 (Aug. 2006), ISSN 1539-3755
[2] a. Fernandez-Nieves, V. Vitelli, a. Utada, D. Link, M. Marquez, D. Nelson, andD. Weitz, Physical Review Letters 99, 1 (Oct. 2007), ISSN 0031-9007
[3] H. Shin, M. J. Bowick, and X. Xing, Phys. Rev. Lett. 101, 037802 (Jul 2008)
[4] G. Skacej and C. Zannoni, Physical Review Letters 100, 1 (May 2008), ISSN0031-9007
[5] M. A. Bates, Soft Matter 4, 2059 (2008), ISSN 1744-683X
[6] S. Kralj, R. Rosso, and E. G. Virga, Soft Matter 7, 670 (2011)
[7] T. Lopez-Leon, V. Koning, K. B. S. Devaiah, V. Vitelli, and A. Fernandez-Nieves,Nature Physics 7, 391 (Feb. 2011), ISSN 1745-2473
[8] D. R. Nelson, Nano Letters 2, 1125 (Oct. 2002), ISSN 1530-6984
[9] J. Park, T. Lubensky, and F. MacKintosh, EPL (Europhysics Letters) 20, 279(Oct. 1992), ISSN 0295-5075
[10] T. C. Lubensky and J. Prost, Journal de Physique II 2, 371 (Mar. 1992), ISSN1155-4312
[11] H. Jiang, G. Huber, R. Pelcovits, and T. Powers, Physical Review E 76, 1 (Sep.2007), ISSN 1539-3755
[12] N. Ramakrishnan and P. B. Sunil Kumar, Physical Review E 81, 1 (Apr. 2010),ISSN 1539-3755
[13] R. Oda, I. Huc, M. Schmutz, S. J. Candau, and F. C. MacKintosh, Nature 399,566 (Jun. 1999), ISSN 0028-0836
[14] G. Lykotrafitis, S. Zhang, S. Suresh, and J. Li, unpublished, 1(2008),http://www.engr.uconn.edu/~gelyko/articles/Lykotrafitis_mem.pdf
[15] P. Liu, J. Li, and Y.-W. Zhang, Applied Physics Letters 95, 143104 (2009), ISSN00036951
[16] C. Zheng, P. Liu, J. Li, and Y.-W. Zhang, Langmuir : the ACS journal of surfacesand colloids 26, 12659 (Aug. 2010), ISSN 1520-5827
89
90
[17] H. Yuan, C. Huang, J. Li, G. Lykotrafitis, and S. Zhang, Physical Review E 82,1 (Jul. 2010), ISSN 1539-3755
[18] P. Ballone and M. Del Popolo, Physical Review E 73 (Mar. 2006), ISSN 1539-3755
[19] B. Widom, The Journal of Chemical Physics 44, 3888 (1966), ISSN 00219606
[20] L. Peliti and J. Prost, Journal de Physique 50, 1557 (1989), ISSN 0302-0738
[21] R. L. B. Selinger, A. Konya, A. Travesset, and J. V. Selinger, The Journal ofPhysical Chemistry B, 111004122647005(Oct. 2011), ISSN 1520-6106
CHAPTER 5
Morphology Transition in Lipid Vesicles: Role of In-Plane Order and
Topological Defects
All membranes have a geometric connection between internal two-dimensional
(2D) order and defects and three-dimensional (3D) shape. To explore this connection
in lipid membranes, we perform coarse-grained simulations on vesicles with internal
xy order and compare the results with experiments of 1,2-dipalmitoyl-sn-glycero-
3-phosphocholine (DPPC), cooling from the Lα untilted liquid-crystalline phase to
the Lβ′ tilted gel phase. At this transition, the vesicles change shape dramatically
from smooth spheres to a disordered crumpled structure, which can be attributed to
topological defects in the direction of molecular tilt. Simulations show the same vesicle
shapes and defect structures, and demonstrate that the final state is determined by
a kinetic competition between curvature changes and defect pair annihilation.
The experiments part of this research is conducted by our cooperator: Professor
Linda S. Hirst and other researchers in her group at University of California, Merced.
5.1 Introduction
In all biological and synthetic membranes, there is a fundamental geometric con-
nection between 2D order and defects within the membrane and the 3D shape of
the membrane. This connection is easily seen in the classic topological problem of
91
92
combing the fur on a dog: it is impossible to comb the fur uniformly without leaving
defects in the combing direction, with a total topological charge of 2. In recent years,
theoretical and experimental physics research has applied this concept to more general
types of order on curved membranes [1,2], including nematic and hexatic liquid crys-
tals [3,4], liquid-crystalline elastomers [5,6], colloidal crystals [7], and superfluids [8].
This research has demonstrated that 3D membrane curvature affects the direction of
2D order, leading to the formation of topological defects. Conversely, 2D order and
defects can modify the 3D shape of the membrane. This connection is important for
materials science, because it offers new opportunities to design materials for directed
self-assembly into desired structures [9]. It is also important for biophysics, because
it explains principles that may influence the shapes of biological membranes.
In this chapter, we present coordinated experimental (by Hirst’s group) and com-
putational studies of the connection between 2D order and 3D shape in lipid vesicles.
On the experimental side, the investigation on giant unilamellar vesicles (GUVs), i.e.
spherical single-bilayer shells, of the lipid DPPC shows that membranes formed from
this lipid undergo a phase transition on cooling from an untilted liquid-crystalline
phase (Lα) to a tilted gel phase (Lβ′). Surprisingly, the vesicles evolve from a smooth
bilayer shell to a highly crumpled surface, producing morphologies far more complex
than expected. Polarized fluorescence microscopy shows that there is a spatial varia-
tion in tilt orientation in the crumpled state, suggesting the presence of extra defect
pairs in the bilayer. To explore the physical mechanisms giving rise to this complex
93
pattern formation, we carry out coarse-grained simulations of a lipid vesicle undergo-
ing a transition from an untilted to a tilted phase. The simulations show that, at the
transition into the gel phase, extra ±1 defect pairs form in addition to the two +1 de-
fects required by topology. If extra defects do not pair-annihilate rapidly enough, the
membrane deforms locally around each defect, producing a crumpled shape. These
deformations stabilize the extra defects and trap the vesicle in a metastable disor-
dered state. Our results may be relevant to a variety of pattern-formation processes
in orientationally ordered materials.
5.2 New Results from the Experiments
To better understand our simulation and compare it to the experiments, we’ll
briefly describe the experiments that are conducted by Hirst’s group.
The GUVs are prepared from DPPC above the melting temperature Tm using an
electro-formation method. Vesicles generated by this method vary in size with an
average radius of approximately 15 µm. They then carry out fluorescence microscopy
on a microscope equipped with polarizing filters, as well as laser scanning confocal
microscopy.
Fig. 5.1 shows examples of the crumpled vesicle shapes generated in the gel phase
of DPPC. These shapes appear somewhat similar to shapes that have been seen in
other experiments on lipid vesicles, but their physical origin is different:
94
Figure 5.1: Fluorescence microscopy of DPPC vesicles labeled with 2 mol% NBD-PE. (a) Vesicle above Tm in the Lα phase. (b,d,e) Vesicles cooled below Tm into theLβ′ phase. (c,f)Confocal images showing slices through a crumpled vesicle. (g,h,i)Confocal images of vesicles in the Lβ′ phase in a fluorescent dextran solution. Somevesicles remain intact and appear black (g,h), whereas others show leakage and appearred (h,i). Note that the vesicle in (i) has a clear break in the membrane, as indicatedby the arrow. (Hirst, unpublished)
95
(a) A wrinkling transition has been reported in polymerized and partially polymer-
ized vesicles, analogous to a glass transition into a quenched state [10, 11]. How-
ever, this phenomenon is different from our experiment because our vesicles are
not polymerized.
(b) Highly scalloped surface topography has been seen recently in vesicles formed
from quaternary lipid mixtures [12]. This complex topography probably derives
from internal membrane phase separation into phases with different intrinsic cur-
vature. However, our work focuses on a much simpler case: a single lipid vesicle
passing through the transition from the Lα to the Lβ′ phase.
(c) Other results have demonstrated small faceted vesicles when vitrified from the
gel phase for cryo TEM [13]. However, these results are observed only in very
small vesicles, 50nm in size, which are much smaller than the GUVs investigated
here.
(d) A recent paper has reported dramatic shape changes in vesicles under high ionic
conditions, resulting from extensive pore formation in the membrane at the gel
phase transition [14]. To see if our vesicles remain intact after crumpling, we
perform a dye leakage assay. Although some vesicles apparently break and allow
dye into their interior (Fig. 5.1 h,i), we observe many examples where the vesicle
remains intact (Fig. 5.1 g,h), indicating that the crumpled surface maintains
a continuous bilayer barrier to the exterior solution. Hence, pore formation is
96
not necessary for the crumpling behavior observed here. Incidentally, the most
crumpled vesicles are typically no longer intact (e.g. Fig. 5.1 i). This leakage is
usually due to a single hole in the vesicle instead of widespread pore formation, as
the bilayer remains smooth in appearance and a gap is usually apparent (Fig. 5.1
i, arrow).
One of the most revealing experiments on this system is to directly image tilt orien-
tation in a crumpled vesicle in the gel phase. Recently Bernchou et al. showed that
polarized fluorescence microscopy with the probe Laurdan can be used to visualize
tilt orientation around a single defect for a lipid gel phase bilayer domain absorbed
onto a flat mica substrate [15]. Molecules tilted in a direction parallel to the polar-
izer direction give a strong fluorescence signal (light state) compared to molecules
perpendicular to the polarizer direction (dark state). As the polarizer is rotated,
molecules with different tilt orientations become aligned with that direction, and
their fluorescence intensity increases. Hence, this technique provides a direct method
for visualizing lipid tilt orientation.
Because the goal is to correlate molecular tilt with membrane curvature, they
must look at Laurdan emission in crumpled membranes away from a substrate. In
the experiment, the challenge is to immobilize vesicles while several images of the
same area are captured at different polarization angles. For that reason, we construct
an observational system in which crumpled vesicles are partially fused onto a mica
surface and the microscope focuses on a plane slightly above the substrate, as shown
97
Figure 5.2: Polarized fluorescence microscopy images of a single vesicle labeled withLaurdan in the gel phase. The vesicles are immobilized by partial fusion onto a micasurface as shown in the confocal image (a) and diagram (b). Images (c-d) and (e-h)show two different vesicles with the focal plane slightly above the mica surface. Thevesicles are illuminated by different angles of linearly polarized light (angle indicatedin left corner). The arrows indicate regions of interest where clear tilt defects can beobserved by rotating the polarizer. In (e-h) we also observe a variation of intensityinside the vesicle, showing variation of molecular tilt direction for the flattened portionof the vesicle fused on the mica surface. The focal plane in (c-d) is too far from thesurface to observe this effect. (Hirst, unpublished)
98
in the confocal image (Fig. 5.2 a) and diagram (Fig. 5.2 b). Examples of two different
vesicles imaged in this way with different polarizer orientations are shown in Fig. 5.2
c-h. These images show clearly that the fluorescence intensity around the edge of the
vesicle varies spatially. Hence, the lipid tilt orientation varies as a function of position
around the vesicle, along with the curvature.
The reason why the images in Fig. 5.2 are less convoluted than those in Fig. 5.1 is
just a matter of selection. Because the polarized imaging system has a greater depth of
field than confocal imaging, it typically gives more visually confusing images. Hence,
they select vesicles with less convoluted contours for the polarized experiment; these
give the clearest images of the vesicle walls.
5.3 Hypothesize
We hypothesize that the crumpled morphologies we observe in DPPC vesicles
are generated by topological defects on the vesicle surface. These defects arise as
vortices in the membrane tilt at the gel phase transition. A vesicle formed from a
lipid with zero tilt in the gel phase should therefore not produce the kind of dramatic
crumpling seen for DPPC. To demonstrate this hypothesis, we compare their results
for DPPC vesicles with vesicles prepared from the lipid sphingomyelin. Sphingomyelin
has been previously demonstrated to exhibit an untilted gel phase [16] below the
liquid-crystalline phase, so it provides a good comparison to DPPC. Observations of
sphingomyelin vesicles confirm that highly crumpled vesicles do not form; the vesicles
remain relatively smooth. Some examples are observed of slightly faceted vesicles, as
99
Figure 5.3: Schematic illustration of our two-vector model for interacting coarse-grained particles. Each particle has a vector n, which aligns along the local membranenormal, and a vectorc, which represents the long-range tilt order within the localtangent plane.
would be expected for a transition to the more rigid gel phase, but no highly crumpled
vesicles can be seen.
5.4 Coarse-Grained Simulation
To model shape evolution of a micron-scale lipid vesicle, coarse-grained or contin-
uum approaches are needed as the number of molecules and associated time scales
greatly exceed the capability of molecular-scale simulation. We essentially need to
superimpose a tilt director field, analogous to that of a smectic-C liquid crystal, onto
a fluid membrane. To achieve this goal, we generalize an earlier coarse-grained model
introduced by Li and coworkers [17–19] to model untilted lipid membranes. In this
model, the membrane is represented by a single layer of interacting point particles.
Each coarse-grained particle corresponds not to a single molecule but to a larger patch
of membrane of order 20 nm2, and the surrounding solvent is implicit. As introduced
in Chapter 4, to describe tilted lipid membranes, we let each particle carry two vector
degrees of freedom: a vector n representing the outward layer normal direction, and a
100
vector c representing the local tilt direction, projected in the plane of the membrane
(Fig. 5.3). We add terms to the interaction potential favoring parallel alignment of
neighboring c vectors, leading to a phase with long-range order of the tilt direction:
Tc = η2 (1− ci · cj) . (5.1)
Other terms of the potential are the same as Eq. 4.1–4.5 and the equation of motion
can be derived from the potential (Appendix A). While this model is too coarse-
grained to capture details of lipid structural changes at the molecular scale during
a phase transition, it is an ideal tool to study the geometric interaction between tilt
order and membrane curvature.
We perform coarse-grained molecular dynamics simulation with Langevin ther-
mostat applied on both translational and rotational degrees of freedom. We impose
the periodic boundary conditions on all three directions of the simulation box. The
system contains 114891 coarse-grained particles and each of them carries 6 degrees
of freedom. The numerical time integration of the equations of motion (Eq. A.3,
Eq. A.4 and Eq. A.5) are performed by using a Adams-Moulton third order method,
which use the same information as the popular Beeman algorithm but even more accu-
rate [20]. To match the time steps for translational and rotational degrees of freedom,
the moment of inertia of both n and c vectors are chosen to be In = Ic = md2. In
the simulation, we have α = 3.1, η = 0.25 and θ0 = 0.015. We get the initial
spherical vesicle by using a random sequential method and then maintain in a high
temperature(kBT/ε ≈ 0.35) for a long time.
101
Figure 5.4: Coarse-grained simulation of a lipid vesicle. Top-left: High-temperatureLα phase. Bottom-left and right: Low-temperature Lβ′ phase. Arrows represent thetilt direction c, black dots represent defects in the tilt direction, and colors representdistance from the center of mass of the vesicle.
102
We begin with a spherical vesicle, with each particle having n pointing radially and
c in a random direction. We then quench the vesicle from the high-temperature phase
without tilt order (Fig. 5.4 top-left) to the low-temperature phase (kBT/ǫ ≈ 0.2) with
tilt order (Fig. 5.4 bottom-left). In the low-temperature phase, the vesicle has multiple
defects in the tilt direction, as shown by the black dots indicating regions of high local
energy. The enlarged view (Fig. 5.4 right) shows that these defects are topological
charges of ±1. Although topology only requires a total topological charge of +2,
the quenched state has many excess ±1 defects. Such disordered textures with many
excess defects are observed in the analogous smectic-A to smectic-C transition in free-
standing liquid-crystal films [21]. At the same time, the vesicle shape becomes more
irregular, with bumps protruding inward and outward. In most cases, the defects
and the bumps occur at the same positions. As in the experiment, this morphology
transition is induced by a change in temperature, rather than by phase separation or
dehydration.
The final morphology depends on the relative viscosities for translational and ro-
tational degrees of freedom. If translational viscosity is low and rotational viscosity
is high, the simulation gives the disordered structure shown in Table 5-1. This result
can be understood because the membrane shape relaxes rapidly in response to the dis-
ordered configuration of defects, before defect pair annihilation can occur. Hence, the
membrane shape stabilizes the quenched-in defects, and leaves a disordered structure
that resembles the experiments.
103
0◦ 30◦ 60◦ 90◦
Table 5-1: Shape and defect configuration for simulated vesicle with low translationalviscosity and high rotational viscosity. From first row to the third row are back, frontand right views of the vesicles, respectively. The color images (left column) representdistance from the center of mass of the vesicle, and the gray scale images (othercolumns) represent the tilt direction, showing the optical intensity that would beobserved with polarized fluorescence microscopy. This vesicle has five +1 defects andthree -1 defects. Note the similarity with the experimental images of Fig. 5.2.
104
0◦ 30◦ 60◦ 90◦
Table 5-2: Morphology and defect configuration for simulated vesicle with high trans-lational viscosity and low rotational viscosity. From first row to the third row areback, front and right views of the vesicles, respectively. The color images (left col-umn) represent distance from the center of mass of the vesicle, and the gray scaleimages (other columns) represent the tilt direction, showing the optical intensity thatwould be observed with polarized fluorescence microscopy. This vesicle has only twodefects of charge +1, which is the minimum required by topology. Note that it ismuch smoother than the simulated vesicle of Table 5-1.
105
By contrast, if the translational viscosity is high and rotational viscosity is low,
the simulation gives the much smoother structure shown in Table 5-2. This result can
be understood because defect pair annihilation occurs quickly, while the membrane
remains approximately spherical. Hence, only two defects of charge +1 remain, and
the vesicle deforms slightly in response to those defects. We speculate that this
mechanism, in which topological defects are trapped in deeply metastable states by
elastic distortions, may occur in many coarsening processes in orientationally ordered
materials.
5.5 Conclusion
In conclusion, we have shown that DPPC vesicles become crumpled at the transi-
tion to the Lβ′ tilted gel phase because of a coupling between membrane curvature and
topological defects in the tilt direction. Coarse-grained simulations of fluid vesicles
with tilt order show the same effect, and demonstrate that the kinetic competition
between curvature changes and defect pair annihilation can determine the final struc-
ture. These results show the importance of 2D order for 3D shape, with potential
applications in soft materials and biological membranes.
BIBLIOGRAPHY
[1] M. J. Bowick and A. Travesset, Physics Reports 344, 255 (Apr. 2001), ISSN03701573
[2] D. R. Nelson, Defects and Geometry in Condensed Matter Physics (CambridgeUniversity Press, 2002) ISBN 0521004004
[3] D. R. Nelson and L. Peliti, Journal de Physique 48, 1085 (1987), ISSN 0302-0738
[4] a. Fernandez-Nieves, V. Vitelli, a. Utada, D. Link, M. Marquez, D. Nelson, andD. Weitz, Physical Review Letters 99, 1 (Oct. 2007), ISSN 0031-9007
[5] C. D. Modes, K. Bhattacharya, and M. Warner, Proceedings of the Royal SocietyA: Mathematical, Physical and Engineering Science 467, 1121 (2011)
[6] C. D. Modes and M. Warner, Physical Review E 84, 021711 (Aug. 2011), ISSN1539-3755
[7] A. R. Bausch, M. J. Bowick, A. Cacciuto, A. D. Dinsmore, M. F. Hsu, D. R.Nelson, M. G. Nikolaides, A. Travesset, and D. A. Weitz, Science 299, 1716(Mar. 2003), ISSN 1095-9203
[8] A. M. Turner, V. Vitelli, and D. R. Nelson, Reviews of Modern Physics 82, 1301(Apr. 2010), ISSN 0034-6861
[9] D. R. Nelson, Nano Letters 2, 1125 (Oct. 2002), ISSN 1530-6984
[10] M. Mutz, D. Bensimon, and M. J. Brienne, Physical Review Letters 67, 923(Aug. 1991), ISSN 0031-9007
[11] S. Chaieb, V. K. Natrajan, and A. A. El-rahman, Physical Review Letters 96,078101 (Feb. 2006), ISSN 0031-9007
[12] T. M. Konyakhina, S. L. Goh, J. Amazon, F. A. Heberle, J. Wu, and G. W.Feigenson, Biophysical Journal 101, L8 (Jul. 2011), ISSN 1542-0086
[13] L. Hammarstroem, I. Velikian, G. Karlsson, and K. Edwards, Langmuir 11, 408(Feb. 1995), ISSN 0743-7463
[14] K. A. Riske, L. Q. Amaral, and M. T. Lamy, Langmuir 25, 10083 (Sep. 2009),ISSN 0743-7463
106
107
[15] U. Bernchou, J. Brewer, H. S. Midtiby, J. H. Ipsen, L. A. Bagatolli, and A. C.Simonsen, Journal of the American Chemical Society 131, 14130 (Oct. 2009),ISSN 1520-5126
[16] P. R. Maulik and G. G. Shipley, Biophysical Journal 70, 2256 (May 1996), ISSN0006-3495
[17] P. Liu, J. Li, and Y.-W. Zhang, Applied Physics Letters 95, 143104 (2009), ISSN00036951
[18] C. Zheng, P. Liu, J. Li, and Y.-W. Zhang, Langmuir : the ACS journal of surfacesand colloids 26, 12659 (Aug. 2010), ISSN 1520-5827
[19] H. Yuan, C. Huang, J. Li, G. Lykotrafitis, and S. Zhang, Physical Review E 82,1 (Jul. 2010), ISSN 1539-3755
[20] L. Shampine and M. Gordon, Computer solution of ordinary differential equa-tions: the initial value problem (W. H. Freeman, 1975) ISBN 9780716704614
[21] C. Zhu, C. Muzny, A. Tewary, D. Link, A. Fritz, D. Coleman, J. Maclennan,and N. Clark, APS March Meeting Abstracts, K1215(Mar. 2007)
CHAPTER 6
Works On Other Topics
6.1 Simulation of Generalized n-atic Order
In Chapter 4 and 5, we developed models to study tangential n-atic orders with
n = 1 (xy order) and n = 2 (nematic order). The n-fold rotational symmetry is given
by different Tc terms shown as in Eq. 5.1 and 4.6. By simply changing this term, we
develop new models of n-atic order with higher rotation symmetry (n = 3, 4, 5, 6).
An generalized Tc term can be written as:
Tc = η2 {1− cos [n arccos (ci · cj)]} , (6.1)
where η is a constant and n here is the number of n-fold symmetry. We perform the
molecular dynamics simulation to simulate n-atic order on a vesicle. The results are
summarised in Table 6-1:
We exam in each case the spatial distribution of defects and resulting deformation
of the vesicle. We find that an initially spherical vesicle (genus zero) with n-atic order
has a ground state with 2n vortices of strength 1/n (see also 1.5), as expected, but
the observed equilibrium shapes are sometimes quite different from those predicted
theoretically [1]. As shown in Chapter 5, for the n = 1 case, we find that the
vesicle may become trapped in a disordered, long-lived metastable state with extra
108
109
n Disclination Vesicle
3
4
5
6
Table 6-1: n-atic order (n=3,4,5,6) on the surface of vesicles. We plot multiple vectorsfor one particle according to their n-fold symmetry to facilitate counting the strengthof the defects. The black dots are the particles with high energy, thus can be used toidentify the location of the defects. The near side defects are darker than the far sideones. And the color of the particles illustrate the distance from the mass center.
110
plus or minus defects whose pair-annihilation is inhibited by local changes in mem-
brane curvature, and thus may never reach its predicted ground state. We expect to
see interesting interplay between defects and vesicle’s geometry when we change the
interaction strength.
6.2 Bilayer model
Real biology membranes are bilayers. Also, in some dynamic process, like fusion,
inner and outer leaflets behave in a different way. Furthermore, the composition of
the two leaflets may be different. If we want to study these kind of problems, we need
a coarse-grained bilayer model.
6.2.1 Bilayer Potential
We develop a single particle, pair wise interaction model that can spontaneously
form bilayer vesicles without any build-in curvature and bilayer lamellar phase with
different concentrations. As shown in Fig. 6.1, each particle carries also a vector
degrees of freedom (n). The inter-particle potential is defined as:
V =N∑
i=1, j>i
uij (ni,nj ,xij) (6.2)
uij (ni,nj ,xij) = uR (xij) + (1 + α (a (ni,nj , xij)− 1)) uA (xij) (6.3)
uR =
ǫ(
Rcut − rRcut − rmin
)8
xij < Rcut
0 xij > Rcut
(6.4)
111
Figure 6.1: Schematic plot of the coarse-grained bilayer model for lipid membrane
uA =
−2ǫ(
Rcut − rRcut − rmin
)4
xij < Rcut
0 xij > Rcut
(6.5)
And rmin = 21
6d, Rcut = 2.55d, d and ǫ are the units of length and energy. α is an
amplification factor and usually chose to be 3.1. Eq. 6.2 to 6.5 are the same as in
reference [2]. This part of formulae ensure the energy well is wide enough to form a
liquid membrane. We modify the orientational part of the interaction:
a (ni,nj, xij) = 1− (pq)2 , (6.6)
p = 1− ni · nj + (ni · xij) (nj · xij) , (6.7)
112
q = 2− ni · xij + nj · xij . (6.8)
The potential energy is minimized if function a is maximized. The biggest number
of a could be 1, which corresponds to p = 0 OR q = 0. As shown in Fig. 6.1, if
two particles are the neighbors in the same leaflet and their n vectors are parallel,
p will be zero; If they are anti-parallel and in the opposite leaflets, q term will be
zero. Thus, there two possible stable states for the coarse-grained particles. Because
the system always want to minimize their surface energies, they will prefer forming
bilayers.
6.2.2 Simulation Results
We perform molecular dynamic simulations with Langevin thermostate [3] at T ≈
0.22ǫ. To demonstrate that this model indeed forms bilayer liquid membranes, we
mark the particles with different color as shown in Fig. 6.2. After a long enough
steps of integration, we found that the two single layers attract each other and form
a bilayer. The initial red-green pattern becomes completely random proves that the
molecules are diffusing within each leaflet and thus form a liquid phase.
Another interesting phenomenon of the lipid molecules are that they can form
a wide range of phases with different concentration in solvent. In this model, the
solvent is inexplicit. However, in our preliminary result, we found that this model
is capable of forming vesicles and lamellar phase with different density as shown in
Fig. 6.3. The interesting part is that there is not a pre-defined radius in the model
113
(a) (b)
(c) (d)
Figure 6.2: Simulations show two single layer with correct direction attract each otherand self-diffusion of the liquid membrane. Particles are marked with red and greencolors to track their diffusion. (a) and (b) are the initial configurations from differentview. (c) and (d) show the fluctuating liquid bilayer membrane after about 200000steps.
114
(a) (b)
(c) (d)
Figure 6.3: Spontaneously formation of vesicles and lamellar phase. The simulationbox of all figures are the same (25x25x25). In (a) and (b), particles number is 10937.In (c) and (d), we have 36191 particles. The position and orientation are all randomat the beginning of the simulation ((a) and (c)). After about half a million steps,vesicles (b) and lamellar phase (d) are spontaneously formed.
115
14 15 16 17 18 19
46
47
48
49
50
51
Figure 6.4: A snapshot of the simulated nematic phase on triangular lattice
when forming vesicles. We speculate that this is a result of minimizing the boundary
of the bilayer and asymmetry of the two leaflets (outer one has more particles than
the inner one).
6.3 Simulation of Stretching a Two Dimensional Nematic Elastomer
This part of work is cooperated with Dr. Fangfu Ye.
6.3.1 Model Description
The nematic elastomers on hexagonal lattices is introduced as a microscopic model
in Ye’s Ph.D dissertation [4]. As shown in Fig. 6.4, in this model, central force springs
(blue lines) connecting neighbor sites form the bonds of an hexagonal lattice. Each
of bonds is attached by a molecule (red lines) and the interactions between these
116
molecules prefer nematic order. There are three part of the Hamiltonian: the elastic
part is
Hel =1
2Kb
∑
b
(Rb − a)2 , (6.9)
where a is the natural bond length, Rb = |Rb| is the distance between two sites and
Kb is the spring constant. Similar to Eq. 2.18, a local order parameter tensor field
can be defined for the molecule as:
Qijb =
√2
(
vibvjb −
1
2δij
)
, (6.10)
where vb = cos θx + sin θy is the unit vector of molecule’s direction. And the inter-
action between nearest neighbor molecules are:
HQ = −1
2J∑
b,b′
γbb′Qijb Q
ijb′ , (6.11)
where γbb′ = 1 if the bonds that the molecules attached to b and b′ are nearest
neighbors and zero otherwise. Again, the Einstein summation convention is applied
if an index appears in a term exactly twice. Finally, we have the coupling term:
Hc = −V∑
b
RibQ
ijb Rbj. (6.12)
This term of energy will favor alignment of the nematic director along the bonds.
And th total Hamiltonian of the system is:
H = Hel + HQ + Hc. (6.13)
They develop this model to study how the nematic phase develops and how the hexag-
onal anisotropy of the lattice affects the elastic and nematic properties of the ordered
117
phase. They show that with large coupling coefficient V and high temperature, there
could exist a hexagonal phase where the molecules point to three directions about
the lattice.
However, we’re mainly interested the instabilities of the low temperature nematic
phase when stretched in the perpendicular directions of the director. Inspired by
Mbanga and other people’s work on elastic instabilities in nematic elastomers [5–8],
we perform this Monte Carlo simulation to study the stripe formation of this model.
6.3.2 Simulation
In the simulation, the Hamiltonian is the same as described previously. The
parameters are: Kb = 100, J = 3, V = 5 and kBT = 0.2. We use a 100x50 triangular
lattice with top and bottom boundary fixed, which mimic the fact when stitching a
film we always hold the boundaries. The temperature is low enough that the nematic
mesogens are always in nematic state during the simulation. The director of the
nematic phase is in the horizontal direction and the lattice is free to relax at the
beginning. As a result, the lattice is stretched a little bit at the beginning of the
simulation as shown in Fig. 6.4.
Another way to visualize the nematic phase is to put it under two crossed polarizer.
We calculate sin 2θ (θ is angle between a nematic mesogen and the crossed polarizers)
to get the light intensity images.
We perform the stretching by hold top and bottom boundaries fixed. After each
step of the stretching, we wait until the energy fluctuation is smaller than certain
118
a
b
c
d
e
f
g
h
i
j
Figure 6.5: Strain-stress curve. Strain is calculated as ∆h/h, h is the original heightof the sample, ∆h is the change of the height.
threshold. Then we perform next stretching. Fig. 6.5 shows the engineering strain-
stress curve of the stretching sample. We still see a lot of fluctuations due to the finite
temperature and sample size although the result is averaged over many snapshots of
the simulation. From a to c (not including c), we have linear response of the lattice
and the strain-stress curve is a straight line. At this stage, we only have a single
domain of the nematic phase. Because the director is set initially in x direction, a, b
are mostly dark under crossed polarizer as shown in Fig. 6.6.
A very interesting phenomenon happened at c, where a small domain of nematic
phase with different orientation starts to grow at the right-bottom corner as shown in
Fig. 6.6 c. Because of newly formed nematic domains have more preferred orientation,
119
Figure 6.6: Stretching of the lattice under crossed polarizer. The a,b,c,. . . ,i corre-sponds to the same symbol in 6.5. The crossed polarizers are in x and y’s directions
the stress suddenly dropped as shown in Fig. 6.5 d. We now in such a regime (d
to f) where the strain can grow without any stress because of the nematic mesogens
reorientate themselves to form new domains and prevent any stress building up. Also,
we note that you need multiple domains to balance the stress and strain. As a result,
we can see the domain boundaries (dark lines) between different domains. Fig. 6.7 is
snapshot of three domains. We can see how the grain boundaries adapt themselves
to fit into different orientations on both sides.
After all the bright domains touch each other and the fixed top / bottom bound-
aries (f), they are stuck in this state. The stress begin to build up again. At g,
the stress is high enough that new grain boundaries can be created and release some
stress (from g to i, the stress almost not changed). Finally, we reach the final state
and there are three stripes left (i, j).
120
10 15 20 25
10
15
20
25
Figure 6.7: A snapshot of the grain boundaries during stretching
BIBLIOGRAPHY
[1] J. Park, T. Lubensky, and F. MacKintosh, EPL (Europhysics Letters) 20, 279(Oct. 1992), ISSN 0295-5075
[2] G. Lykotrafitis, S. Zhang, S. Suresh, and J. Li, unpublished, 1(2008),http://www.engr.uconn.edu/~gelyko/articles/Lykotrafitis_mem.pdf
[3] D. L. Ermak and H. Buckholz, Journal of Computational Physics 35, 169 (1980),ISSN 0021-9991
[4] F. Ye, Elasticity and pattern formation of nematic elastomers (University of Penn-sylvania, 2007)
[5] I. Kundler and H. Finkelmann, Macromolecular Rapid Communications 16, 679(1995), ISSN 1521-3927
[6] S. Conti, A. DeSimone, and G. Dolzmann, Phys. Rev. E 66, 061710 (Dec 2002)
[7] N. Uchida, Phys. Rev. E 60, R13 (Jul 1999)
[8] B. L. Mbanga, F. Ye, J. V. Selinger, and R. L. B. Selinger, Phys. Rev. E 82,051701 (Nov 2010)
121
APPENDIX A
Equation of Motion of Particles with Two Vectors
In this appendix, we’d like to derive a general formular of equations of motion we
used in the simulations of previous chapters. Specifically, in Chapter 4 and 5, we use
a pair-wise potential of the following form:
V =N∑
i=1, j>i
uij (ni,nj, ci, cj,xij) , (A.1)
where n and c are two perpendicular unit vectors associated with a coarse-grained
particle. The equation of motion can be derived from Euler-Lagrangian’s equation
with the constraints that |n| = 1, |c| = 1 and n · c = 0. The Lagrangian for a specific
particle is:
L =1
2mr2 +
1
2Inω
2n +
1
2Icω
2c − V
=1
2mr2 +
1
2Inn
2 +1
2Icc
2 − V. (A.2)
V is summing over all the interacting neighbors. The Lagrangian equation of motion
can be written as:
d
dt∇rL −∇rL = 0 (A.3)
d
dt∇nL −∇nL = λ1∇n (n · n− 1) + λ2∇n (n · c) (A.4)
122
123
d
dt∇cL −∇cL = λ3∇c (c · n− 1) + λ2∇c (n · c) (A.5)
Where in Cartesian coordinates, ∇ab :=∂b∂ax
x + ∂b∂ay
y + ∂b∂az
z. a is a vector and b
is a scaler. The Lagrangian multipliers can be solved by using the constraints.
λ1 =1
2(∇nV · n− Inn · n) (A.6)
λ3 =1
2(∇cV · c− Icc · c) (A.7)
λ2 =1
In + Ic(In∇cV · n+ Ic∇nV · c− 2InIcn · c) (A.8)
Substitute λ1,λ2 and λ3 to Eq.A.4 and Eq. A.5, n and c can be solved as:
n =1
In(2λ1n+ λ2c−∇nV ) (A.9)
c =1
Ic(2λ3c+ λ2n−∇cV ) (A.10)
Also from Eq. A.3, we know as always:
r = − 1
m∇rV (A.11)
Thus, by knowing the second time derivatives we can perform the time integration
and calculate their trajactaries.