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

All rights reserved

INFORMATION TO ALL USERSThe quality of this reproduction is dependent on the quality of the copy submitted.

In the unlikely event that the author did not send a complete manuscriptand there are missing pages, these will be noted. Also, if material had to be removed,

a note will indicate the deletion.

All rights reserved. This edition of the work is protected againstunauthorized copying under Title 17, United States Code.

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.