Energy Minimization in Nematic Liquid Crystal Systems Driven

Home , Energy minimization

ENERGY MINIMIZATION IN NEMATIC LIQUID CRYSTAL SYSTEMS DRIVEN

BY GEOMETRIC CONFINEMENT AND TEMPERATURE GRADIENTS WITH

APPLICATIONS IN COLLOIDAL SYSTEMS

A dissertation submitted

to Kent State University in partial

fulfillment of the requirements for the

degree of Doctor of Philosophy

Jakub Kolacz

December 2015

Except for previously published materials

Dissertation written by

Jakub Kolacz

B.S., DePaul University, USA 2010

Ph.D., Kent State University, USA 2015

Approved by

Dr. Qi-Huo Wei , Chair, Doctoral Dissertation Committee

Dr. Antal I. Jákli , Members, Doctoral Dissertation Committee

Dr. Robin L. Selinger ,

Dr. Elizabeth Mann ,

Dr. Mietek Jaroniec ,

Accepted by

Dr. Hiroshi Yokoyama , Chair, Chemical Physics Interdisciplinary Program

Dr. James L. Blank , Dean, College of Arts and Sciences

TABLE OF CONTENTS

LIST OF FIGURES ...... VIII

LIST OF TABLES ...... XX

ACKNOWLEDGEMENTS ...... XXI

CHAPTER 1 INTRODUCTION ...... 1

1.1 Liquid Crystals in Confined Geometry ...... 1

1.2 Transport Phenomena in Colloidal LC Systems ...... 3

1.3 Active Colloidal Systems ...... 5

1.4 Motivations, Applications and Brief Summary of the Dissertation ...... 6

CHAPTER 2 TOPOLOGY IN NLCS ...... 8

2.1 From Topology to NLCs ...... 8

2.1.1 Defects ...... 9

2.1.2 Homotopy Groups in Sn ...... 12

2.1.3 NLCs: Homotopy in RPn ...... 14

2.1.4 Schlieren Textures ...... 18

2.1.5 Elastic Free Energy ...... 19

2.2 Simulation Methods ...... 21

2.3 NLCs under External Geometric Confinement ...... 27

2.3.1 Pair Annihilation in RPm ...... 29

2.3.2 Spherical Caps on Planar and Homeotropic Surfaces ...... 31

2.3.3 Spherical Caps on 2D Defects ...... 33

iii

2.4 Summary ...... 36

CHAPTER 3 NLC ON CHEMICAL PATTERNS ...... 37

3.1 NLC Drops on Surfaces ...... 37

3.1.1 Contact Angle ...... 37

3.1.2 Fluid Droplets on Chemically Patterned Surfaces ...... 39

3.1.3 Surface Energy of LCs ...... 42

3.1.4 Surface Alignment of NLCs ...... 43

3.2 Materials, Methods and Characterization ...... 44

3.2.1 Materials ...... 44

3.2.2 Contact Angle Measurement ...... 45

3.2.3 Pre-Tilt Angle Measurement ...... 47

3.2.4 Monolayer Self-Assembly...... 50

3.2.5 Cell Fabrication ...... 51

3.3 Characterization of 5CB on SAMs ...... 53

3.3.1 SAM Morphology ...... 53

3.3.2 Contact Angle Results ...... 53

3.3.3 Pre-Tilt Angle Results ...... 55

3.4 Chemical Patterning ...... 56

3.5 Droplet Self-Organization ...... 58

3.5.1 Spincoating ...... 59

3.5.2 Cell Breaking...... 61

3.5.3 Solvent Evaporation ...... 63

3.5.4 Dragged Drop ...... 65

3.5.5 Droplets on Patterned Domains...... 68

3.6 Internal Director Structures ...... 70

3.6.1 Defect Position and Annihilation in Ellipsoidal Caps ...... 72

3.6.2 Spherical Caps of Chiral LCs ...... 74

3.7 Simulations of Internal Director Structure ...... 75

3.7.1 Effect of Contact Angle and System Size ...... 75

3.7.2 NLCs on Circular Domains ...... 76

3.7.3 SAM Surfaces ...... 78

3.8 Summary ...... 81

CHAPTER 4 THERMOPHORESIS IN NEMATIC LIQUID CRYSTALS ...... 84

4.1 Thermophoresis: Background ...... 84

4.1.1 Thermophoresis in Fluids ...... 87

4.1.2 Temperature-Dependent Thermophoresis ...... 89

4.1.3 Applications ...... 91

4.2 Colloids in Nematic Liquid Crystals: Background ...... 94

4.2.1 Colloidal Liquid Crystals ...... 95

4.2.2 Observations of Motion by Elastic Forces ...... 101

4.3 Motivations ...... 103

4.4 Methods ...... 103

4.4.1 Cell Fabrication ...... 103

4.4.2 Gradient Heat Stage ...... 104

4.4.3 Particle Tracking ...... 107

4.5 Control Experiments: Negative Thermophoresis in NLCs ...... 108

4.6 Thermophoretic Motion of Colloids in NLCs ...... 110

4.6.1 Positive and Negative Thermophoresis in 5CB ...... 113

4.6.2 Homeotropically Anchored Colloids in 5CB ...... 115

4.6.3 Theory ...... 119

4.6.4 Additional Observations ...... 122

4.6.5 Elastophoresis vs Traditional Thermophoresis ...... 127

4.7 Summary ...... 128

CHAPTER 5 SELECTIVE POLYMERIZATION OF LC COLLOIDS ...... 130

5.1 Introduction ...... 130

5.1.1 Self-Folding Structures ...... 130

5.1.2 LC Polymers ...... 131

5.1.3 Motivations...... 132

5.2 Methods ...... 134

5.2.1 Materials ...... 134

5.2.2 Cell Filling for LCP Colloids ...... 135

5.2.3 Photomask Patterning ...... 136

5.2.4 Projection Optical Lithography ...... 137

5.3 Results ...... 138

5.3.1 V-Particle Polymerization ...... 138

5.3.2 Discussion ...... 139

5.4 Conclusion ...... 140

CHAPTER 6 CONCLUSION ...... 141

APPENDIX A DERIVATION OF THE YOUNG RELATION ...... 144

APPENDIX B DERIVATION OF Q TENSOR FREE ENERGY FOR

SIMULATIONS ...... 147

REFERENCES ...... 149

vii

LIST OF FIGURES

Figure 2.1 Mapping of two arbitrary loops in real space to their degeneracy space where

blue arrows designate points that are found in the degeneracy space S1 while the red

point signifies a defect point that is not in the degeneracy space...... 11

Figure 2.2 (a) The Ising model where boundary conditions necessitate a domain wall.

There is no way to generate a continuous loop that would contain the discontinuity.

However, confined domains as in (b) can be smoothly transitioned out (c)...... 13

Figure 2.3(a,e) Order parameter space RP1 in the first homotopy group with the first two

winding numbers in each direction shown in red. Forward and backward winding

are distinct as shown in the director representation of a (b) +1/2 defect, a (c) -1/2

defect, a (f) +1 defect and a (g) -1 defect. Schlieren textures of (d) half integer and

(h) integer 1 defects...... 16

Figure 2.4 Order parameter space RP2 in the (i,j) first homotopy group and (k) the first

winding number in the second homotopy group...... 17

Figure 2.5 (a) The degeneracy between a 2D +1/2 and -1/2 defect in 3D degeneracy

space (top) and real space (bottom). (b) A transient defect line from a simulation of a

droplet of LC that transitions from a +1/2 to a -1/2 defect on the surface. (c) A 2D

schematic of the pair annihilation with (d) corresponding 3D defect structure...... 30

Figure 2.6 (a) Radial configuration of a LC sphere with homeotropic anchoring. On

planar surfaces (b,c), the topological constraints force a boojum on the surface with

viii

curvature of the elastic field lines dependent on contact angle. Homeotropic surfaces

(d-g) can have an unstable bulk defect of charge +1 (d,f) that are homotopically

equivalent to a boojum ring located at the three-phase line. Simulation results

showing crossed polarizer images of relaxed spherical caps on a (h) planar

degenerate surface, (i) planar aligned surface and (j) homeotropic surface. The

corresponding director orientations and defects are shown in (k), (l) and (m),

respectively, where the non-blue color signifies areas of high energy cost...... 32

Figure 2.7 Simulation results showing crossed polarizer images of relaxed spherical caps

on a (a) defect spiral with S=1 and ψ=0, (b) defect with S=1/2 and ψ=0 and (c)

defect with S=-1 and ψ=0. The corresponding surfaces, first layer director

orientations and defects are shown for the (d) S=1/2 and (e) S=-1 cases...... 35

Figure 3.1 A droplet sitting on a hydrophilic domain 휸 in a hydrophobic background 휹

as the volume is increased. (a) At small volumes, the droplet takes on the contact

angle characteristic of the underlying domain. (b) When the droplet reaches a

volume that is sufficient to cover the entire domain, the contact line is pinned at the

domain edge until it matches the contact angle on the background, where (c) it again

begins to increase its contacting surface area...... 40

Figure 3.2 (a) A static contact angle drop. Dynamic contact angles were determined by

(b,c) moving the substrate with the stationary syringe tip still in contact, (d) pulling

the syringe tip away to get a receding angle and (e) increasing the fluid of the drop

to get an advancing angle...... 45

Figure 3.3 Contact angle measurements were done by (a) selecting the droplet of interest,

(b) setting a surface baseline and (c) appropriately tresholding the image. (d) After

processing, Matlab determines the radius of curvature and height of the droplet for

accurate determination of the contact angle...... 46

Figure 3.4 (a) Crystal Rotation setup with (b) relevant angles and axes defined. (c)

Transmission vs angle for various pretilts with the center of symmetry labeled and

(d) a transformation figure from pretilt angle to the center of symmetry, where the

dotted line represents a 1:1 transformation...... 48

Figure 3.5 (a) Curve showing the immediate contact angle dependence on immersion

time. (b) Changing contact angle in time for a sample immersed in 1mM HDT for

480s...... 51

Figure 3.6 (a) Reflection spectrum of a tungsten light source from a 25µm glass cell. (b)

Frequency of the light extrema plotted consecutive order with a linear fit...... 52

Figure 3.7 The molecular structure of (a) 5CB, (b) HDT, (c) ODT and (d) MDA where

white, grey, blue, red and gold represent H, C, N, O and Au. (e) 5CB drop on HDT

immediately after being placed on the surface and (f) one week later. (g) 5CB drop

on MDA. (h) Average contact angle values over time for 6 droplets each of HDT,

ODT and a 1:1 mixture of ODT:MDA...... 54

Figure 3.8 POM images of 5CB in a cell made of (a) two MDA surfaces, (b) two MDA

surfaces that had been rubbed antiparallel, (c) two HDT surfaces and (d) two HDT

surfaces that had been rubbed antiparallel. (e) The crystal rotation results (red) and

a best fit for the center of symmetry (blue) for the rubbed MDA surfaces...... 56

Figure 3.9 (a) Micro-Contact Printing procedure: a 1mM solution of ink in ethanol is

coated on a PDMS stamp and allowed to evaporate, leaving behind a thin layer of

ink, then the inked stamp is brought in contact with a gold surface, where the

mercapton molecules form thiol-metal bonds with the gold resulting in a monolayer.

The gold surface is finally placed into a 1mM solution of background, where the

molecules self-assemble on the unoccupied areas. (b) Representative SEM image of

the elastomeric stamp...... 58

Figure 3.10 (a) POM image of a micropatterned substrate with liquid crystal spincoated

on the surface at 3000rpm. (b) and (c) show high-magnification images where it

becomes visible that the liquid crystal is forming a film on the surface...... 60

Figure 3.11 Representative area of (a) positive-patterned droplets where it is visible that

there are three droplets with large contact angles in an array of thin droplets with a

very small contact angle and (b) anti-droplets. (c,d) When cycling the nematic

isotropic phase, a number of the very thin drops collapse down into a crescent that is

confined at the hydrophobic-hydrophilic interface. The broken symmetry appears to

be an effect of the direction of heat flow, which is from the right. (e,f) The

spontaneous collapse of the droplets under the constraint of constant volume is

shown schematically...... 62

Figure 3.12 Representative micropatterned areas after solvent evaporation using

5CB:IPA mixtures of (a) 1:1000 and (c) 1:100 by volume evaporated at room

temperature. (c) Close up of a 1:500 mixture results with (d) corresponding

horizontal imaging. (e,f) Time dependent flattening shown in two images: (e) upon

initial formation and (f) 5 days later. (g-i) A schematic of droplet flattening for our

system under constant volume...... 64

Figure 3.13 The dragged-drop method has three stages: (a) The needle is placed on the

surface and a controlled amount of LC is pushed onto the surface; (b) A moving

surface causes the drop to cover a patterned domain where it gets pinned at the

interface; (c) The movement distorts the drop until it pinches off a droplet. (d) A

schematic of the nucleated droplets as the needle moves away from the plane of the

page showing that the droplets formed on the periphery of the dragged drop tend to

have a smaller contact angle. (e) An optical microscope image of the process. (f) A

SEM image of LC droplets on a chemically patterned surface. The dark areas on the

surface correspond to the MDA ink and the light areas are HDT...... 67

Figure 3.14 (a) An SEM image of droplets on patterned domains. The darker areas on the

pattern correspond to the oxygen in MDA [74]. The blue arrow points to a droplet

that doesn’t completely cover the domain while the red arrow shows a droplet that is

larger than the patterned domain. (b) A POM image shows this effect on the optical

intensity of the droplets. Schematics of a droplet are shown when it is (c) completely

xii

within the domain, (d) completely outside the domain and (e) pinned at the interface.

...... 69

Figure 3.15 (a) A representative area of LC droplets. Each droplet found fell in one of the

following categories: (b) A single defect, (c) a defect with a characteristic closed

loop or (d) three defects...... 71

Figure 3.16 (a) Representative areas of LCs self-organized on µcp ellipses. (b) Close up

of ellipses with 3 and 5 point defects. (c-f) Time lapse images of ellipses where

positive and negative charge colloids move together and annihilate (marked in red).

...... 73

Figure 3.17 (a) Chiral Nematic LC self-localized on 50µm µcp domains with high

magnification (b) POM and (c) unpolarized optical microscope images. While the

large droplets had complex defect knots, smaller drops (d) showed a simpler pattern.

When we cooled the droplet in (c) from the (e) isotropic to the chiral nematic phase,

it underwent (f-g) a slow wrapping of the director field followed by a very abrupt

change into (i) a highly knotted structure...... 74

Figure 3.18 (a) Image taken from experimental results where a single domain is inferred

from 2D analysis of the POM texture. (b) A surface that is generated from analysis

of (a). (c) Final structure of a NLC drop on (b) and (d) a corresponding image taken

from experimental results. (e) An intermediate structure on the same surface with (f)

corresponding droplets taken from experimental results...... 77

xiii

Figure 3.19 (a) Schematic of our model of monolayer adsorption, where an incoming

molecule has a small probability of sticking to an empty site with no neighbors and a

random orientation or a strong probability of sticking to an empty site with a

neighbor and taking on its azimuthal orientation. Simulated surface using sticking

parameter (a) p=0.01, (b) p=0.001, (c) p=0.0001 and (d) p=0.00001. The color of

each region corresponds to the x component of the orientation as designated in the

legend...... 79

Figure 3.20 (a) Surface generated with a sticking parameter p=10-4. (b) The disclination

lines (in red) get pinned at the surface at domain boundaries that satisfy +1/2 defect

charges and (c) the resulting POM texture. (d) Surface generated with a sticking

parameter p=10-5, which shows a generally monodomain surface. (e) The

disclination line does not get pinned at the surface due to a lack of domain

boundaries and instead stays in the bulk of the NLC droplet. (f) The resulting POM

texture, which compares with the experimental texture from literature for NLC

droplets on monodomain SAMs...... 80

Figure 4.1 (a) Thermophoresis in the high Knudsen number limit, where motion is

induced by anisotropically distributed elastic collisions. The red arrows show

molecules with a higher temperature, while the blue arrows are the cooler and more

slowly moving molecules. (b) The mobility of charge carriers in semiconductors also

has inverse temperature dependence. In the case of thermocouples or thermoelectric

xiv

generators, N-type and P-type semiconductors are put together so that they generate

a net current in the presence of a temperature gradient...... 86

Figure 4.2 Thermal Field-Flow Fractionation (ThFFF) is the method in which a

temperature gradient across a flowing liquid in a channel separates molecules by

their size. On the left side particles are dispersed in a medium at a constant

temperature. When a temperature gradient is applied and perpendicular parabolic

flow is induced, like particles will settle into bands depending on their diffusion 푫

and thermodiffusion 푫푻 coefficients and travel at different velocities through the

channel...... 91

Figure 4.3 MicroScale Thermophoresis: (a) In the initial state, particles are evenly

dispersed in the medium. (b) When a laser is used to locally heat the medium, the

particles move away from the heat source proportional to their thermodiffusion

coefficient 푫푻 until (c) they create a void region in the medium. (d) When the laser

is turned off, the particles return into the void region depending on their diffusion

coefficient 푫...... 93

Figure 4.4 Schematic drawings, optical microscope and POM images of homeotropically

anchored colloids in configurations that result in (a,b) dipolar configuration with a

single hedgehog defect and (c) a quadrupolar configuration with a Saturn ring in the

surrounding director field. The optical and polarized optical microscopic images are

taken from the samples used in this work...... 97

Figure 4.5 (a) The heat stage design with color representing local heating and cooling.

(b) An image of the final stage an the optical setup. Thermal images of the stage with

sample thicknesses (c) 25µm, (d) 50µm and (e) 75µm, where the bottom images are

close-up images of the sample surface. (f) The gradient taken from the IR readings

compared to a linear fit between thermocouple readings. (g) The calculated

temperature gradient from IR image at each point in (f)...... 106

Figure 4.6 (a) A cross-sectional schematic of the glass cell filled with colloidal solutions

in the gradient heat stage. The +y axis is defined as pointing toward the cold side of

the cell. Sample trajectories of uncoated 5μm Si colloids moving through (b) a

solution of 85% glycol at 26ºC, (c) the 26ºC region and (d) 30ºC region of a 5CB

cell with the same gradient...... 109

Figure 4.7 (a) Motion of colloids in NLC is driven by thermal expansion of the LC

molecules at the warm side of the cell (thermocouples T2 and T3) and contraction on

the cold side (thermocouples T0 and T1) as well as thermophoretic forces. (b) When

a temperature gradient is induced, motion is dominated by three forces. (c) In the

diffusion regime, the temperature effects are negligible and Brownian motion

dominates. In the thermal expansion regime, the motion is driven by the expansion

and contraction of LC molecules, resulting in motion along +y. Thermophoresis is

the dominant force only when a steady temperature gradient is achieved, resulting in

motion along -y...... 112

xvi

Figure 4.8 (a,b) The displacement of the colloid and the thermocouple readings over time,

which are used to determine the average temperature at the point of measurement.

from positive to negative thermophoresis around 25°C for a gradient of 2.7°C/mm.

...... 114

Figure 4.9 (a) The velocity of various DMOAP-coated silica colloids in a variety of

prepared cells plotted against their respective local gradients and temperatures. The

color of the points represents the local gradient where red points correspond to a

gradient of 3°C/mm and blue points correspond to a gradient of 2°C/mm. There is a

visible dependence on both the gradient and the local temperature. (b) The

thermodiffusion coefficient calculated from the same points, differentiated by high

and low temperature gradients...... 116

Figure 4.10 (a) Sample trajectory of a single DMOAP-coated Si colloid with an

accompanying hyperbolic hedgehog defect in a temperature gradient. (b) By fitting

the velocity at each measured point, we could see a negative temperature

dependence, which can be used to calculate (c) the thermodiffusion coefficient 푫푻.

From the trajectory, we also calculate (d) the mean-squared displacement of the

particle, which is (e) linearized by the velocity to give a value for (f) the local

diffusion coefficient 푫. The uncertainty in velocity, 푫푻 and 푫 is smaller than the size

of the points, as can be inferred from the sample trajectory in (a)...... 118

xvii

Figure 4.11 (a) Temperature dependence of the Order Parameter for 5CB. (b) Drift

velocity of colloids calculated using both the elastophoretic contribution and the

thermophoretic contribution, as modeled by Brenner...... 120

Figure 4.12 (a) The velocity change over temperature of a DMOAP-coated PS colloid

moving through a temperature gradient where blue represents the raw data and red

is a fit based on Equation 4.30. (b) The Soret coefficient calculated for each

temperature...... 122

Figure 4.13 (a) Crossed-Polarizer image of four colloids in the same proximity. The

colloids are characterized by their defect structure: Dipole-Up (DU) has a

hyperbolic hedgehog defect along the +y axis; Dipole-Down (DD) has a hyperbolic

hedgehog defect along the –y axis; the Quadrupolar structures (Q1 and Q2) have

saturn ring defects. (b) The displacement over time when a temperature gradient is

applied. The initial motion along +y and -x is due to thermal expansion effects. The

steady state velocity along –y and +x is due to thermophoresis. In the

thermophoretic regime, the velocity of the colloids (c) along the temperature

gradient and (d) perpendicular to the gradient suggests a dependence on the defect

structure...... 124

Figure 4.14 (a) An example of the geometry of a colloid moving off-axis to the director.

(b) Example trajectories of a Saturn ring-accompanied (left) and hyperbolic

hedgehog-accompanied (right) colloids as they move up the temperature gradient

into the nematic-isotropic interface...... 126

xviii

Figure 5.1 (a) Crossed polarizer images of liquid crystal being aligned by a patterned

photoalignment layer with the ability to (b) generate periodic arrays of defect

structures (unpublished results by Yubing Guo). (c) Arrays of boomerang colloid

particles fabricated using SU8 before they are lifted from the surface and put into

solution [154]...... 133

Figure 5.2 Molecular structure of (a) RM257 and (b) 2-azo where white, grey, red and

blue are H, C, O and N, respectively. (c-f) POM images of RM257 mixed with

photoinitiator on the hot stage and at room temperature (c,d) before polymerization

and (e,f) after polymerization...... 135

Figure 5.3 (a) Custom projection photolithography system in three modes: (b) using a

mirror to pass green light through the photomask onto the CCD camera for focusing

focusing and (d) the final configuration where green light is projected through the

photomask onto the sample for photopolymerization. We show an example of the (e)

mask and (f) cell before exposure...... 137

Figure 5.4 Optical microscope and POM images of V-particles generated using a 50x-5x

setup, imaged through a toluene drop on the surface of the sample...... 139

xix

LIST OF TABLES

Table 3.1 Contact Angles ...... 55

Table 4.1 Properties of Colloidal Microspheres ...... 108

Table 4.2 Colloid Velocities ...... 110

Table 4.3 Comparisons between Elastophoresis and Traditional Thermophoresis ...... 128

ACKNOWLEDGEMENTS

I would like to thank all the people that helped me survive both physically and psychologically over the past few years. First and foremost is my advisor Dr. Qi-Huo

Wei. To occasional frustration, he gave me the freedom to pursue my constantly changing research interest while providing thoughtful insight from his vast background. I would like to thank him for his patience in dealing with “a butterfly that tries to sniff every flower” in research, while applying just the right amount of pressure to keep me relatively on track.

Also, thank you to the current and past members of Dr. Qi-Huo, Dr. Antal Jakli

Dr. Robin Selinger and Dr. Quan Li’s labs: Andrew Konya, Sajedeh Afghah, Yubing

Guo, Dr. Ayan Chakrabarty, Miao Jiang, Dr. Lt. Cdr. Nick Diorio, Muhammad Salili, Dr.

John Harden, Dr. Yannian Li, Dr. Chenming Xue among countless others that helped me with everything from theory to chemical synthesis to experimental design to coding (and would occasionally let me borrow equipment, programs and chemicals).

I would also like to acknowledge Prof. Bingqian Xu and Mengmeng Zhang for their work on AFM measurements of self-assembled monolayers.

I would like to thank Dr. Lt. Cdr. Nick Diorio, Anshul Sharma, Dr. Antal Jakli,

Roberta Rarumy Almeida, as well as all of our accomplices for ensuring intellectually stimulating conversation during our daily lunches.

xxi

I would like to thank my friends at the LCI, most of whom have already left for bigger and better things: Dr. Son Nguyen for countless hours playing ping-pong and discussing the differences between theorists and experimentalists; Andrew Konya for sharing an office that became home to countless ideas for research projects, companies and general intrigue into the human condition; my first-year dinner group of Shuang

Zhou, Lei Zhao and Dr. Son Nguyen for teaching me how to cook for myself; Dr. Lt.

Cdr. Nick Diorio and Anshul Sharma for being Dr. Lt. Cdr. Nick Dioro and Anshul

Sharma, respectively; Dr. Matt Worden for being Canadian; and my entire incoming class for making a supportive and stimulating environment. There are too many good memories to put on paper, but thank you to everyone who made my last few years at the

LCI an unforgettable experience.

I would also like to acknowledge the faculty and staff at the LCI: the superb professor I have had over the years that opened my mind to the field; the administrative staff for magically handling any situation I threw their way (special thanks to Lynn Fagan and Maryann Kopcak); the technical staff for being ever present and helpful.

Finally I would like to thank my parents for never ceasing to ask me when I’d be finished with my dissertation. I could never have gotten this far in life without your constant love and support. And I would literally not be here without you.

Jakub Kolacz

December 2015, Kent, Ohio

xxii

CHAPTER 1

Introduction

Liquid crystals, LCs, are characterized by long-range orientational order in a fluid medium. In a very simplistic way, liquid crystals are fluid molecules whose structure affects local interactions and resultant bulk properties. Such materials are of great intrigue for mathematicians, physicists, engineers, chemists and biologists alike.

Bulk effects of LC orientation and their susceptibility to electric and magnetic fields have been extensively studied and have led to one of the most pivotal developments in modern technology: the Liquid Crystal Display, or LCD [1]. As the market for LCD technology saturates, a push for new science and innovations in LCs is driven by new applications. Fundamental research into surface effects and geometric confinement are leading the way for a new generation of LC devices such as LC sensors

[2] and PDLC (Polymer-Dispersed Liquid Crystal) smart windows [3]. While controlling the behavior of LCs through energy minimization in the presence of an electric field has been the traditional method of utilizing LC devices, recent research is exploring the way that energy is minimized due to changes in surface conditions [4]. This research direction is intimately related to topology and surface chemistry.

1.1 Liquid Crystals in Confined Geometry

Geometric confinement exerts a strong influence on the phase behaviors and microstructures of LCs as the anchoring of molecular orientation by surfaces may induce

frustration and lead to formation of various topological defects [5]. LCs in confined geometries are intriguing systems of great fundamental interest. Nontrivial parameter fields and topological defects are responsible for phenomena ranging from the number of pentagons one can pattern on a sphere to the buckling of a viral capsid into a faceted structure [6]. A large number of studies are devoted to understanding how surface alignment and curvature impact defect location, and consequently how defects and internal director structures can impact soft shapes. This is useful in LC polymer research, where director orientation strongly influences directional contraction and expansion as materials transition from isotropic to nematic states and can lead to folding [7].

LCs in confined geometries are also important to a variety of current technologies. For example, in polymer dispersed liquid crystals (PDLC), micron-sized

LC droplets can be controlled to either scatter or transmit light by varying the molecular orientations with an external electric field [8], a switching mechanism which has been utilized in smart windows [3]. LCs also provide a new platform for chemical and biological sensing [2]. Due to the long range nature of orientational order, the absorption of a small amount of molecules on LC surfaces can change the surface anchoring conditions of the LC molecules and lead to changes in optical properties at the macro- scale.

The sensitivity of liquid crystal sensors is tunable by either controlling material properties or geometric confinement that can influence the ratio of exposed surface area to volume effects of the LC. Microfluidic devices that utilize LCs are becoming popular

because they can easily be incorporated into sensor networks or lab-on-a-chip systems, which are highly desirable for environmental science, forensics, biological warfare detection and point-of-care diagnostics. Such devices need high sensitivity of target molecules, fast response to change, portable size and must be economical and environmentally friendly.

1.2 Transport Phenomena in Colloidal LC Systems

Transport phenomena have long been of interest in biological systems [9]. The movement and absorption of small molecules result in the fundamental mechanisms of life. LC research is intimately related to biological systems due to the liquid crystalline nature of many biomolecules [10]. For example, cell membranes are a specific type of liquid crystal known as lyotropic lamellar lipid bilayers. Understanding the method by which molecules can penetrate these membranes is instrumental in understanding disease and engineering medicine that can appropriately target cells. LC interactions within biological systems are also ubiquitous, as LCs are commonly used as food additives, dyes, detergents, etc.

When particles suspended in isotropic fluids are subject to a temperature gradient, directional motion of the particles can be induced, a phenomenon known as thermophoresis. Thermophoresis in gases is widely accepted as a thermodynamic process based on anisotropic momentum transfer and has been utilized in commercial precipitators [11], which filter fine particles from gas streams. Thermophoretic analogs are also present in some solid systems. The induction of an electric current due to a 3

temperature gradient through a solid is known as the Seebeck Effect and is commonly used to measure temperature in the form of a thermocouple [12] or for energy harvesting in thermoelectric generators [13].

Fluid systems are notoriously difficult to characterize because of the interplay of thermodynamics, hydrodynamics and electrodynamics [14]. Certain systems have been extensively classified and utilized to provide a new avenue for sorting biomolecules and particles by techniques such as microscale thermophoresis (MST) [15] and thermal field flow fractionation (ThFFF) [16].

One medium that remains relatively unexplored under the guise of thermophoresis is anisotropic liquids such as nematic liquid crystals. Since the constituent molecules of nematic LCs possess long range orientational order, the introduction of colloidal particles results in director field distortion and formation of topological charges due to the anchoring conditions set by the colloidal surfaces [17]. The elastic distortion of the director field leads to a myriad of interesting phenomena such as long-range attraction of colloids to interfaces [18], non-linear Stokes drag [19], nonlinear electrophoresis [20, 21] and defect-assisted self-assembly of complex structures [22].

The observation of gradient elastic fields on particles and biological cells shows promise in the areas of cell sorting. Since biological cells communicate and probe their environment through the release of small molecules, there is a great potential in observing the effect of these small molecules on the local elastic environment and vice versa.

1.3 Active Colloidal Systems

Liquid Crystal Polymers (LCPs) have become a very active field in recent years because of their intrinsic anisotropic properties [7]. When anisotropic LC properties are coupled with the solid elastic properties of polymer systems, new behaviors result. For example, when a nematic LCP transitions to its isotropic state, it will contract along the nematic director and expand in the perpendicular plane. By properly tuning the geometry of LC order, such devices make very effective actuators, able to bend or vibrate upon stimulation by heat or light [23-25]. The ability of certain geometries to fold has garnered attention as a method of creating self-folding origami [26].

Self-folding colloids have been a hot topic in metamaterials for almost two decades [26, 27]. Several studies have utilized the ability to use 2D patterning to generate self-folding 3D shapes for objects such as 3D split-ring resonators or antennas for single molecule detection. Recent research has looked at applying active colloids as targeted drug delivery systems, where the adsorption of target molecules can force a colloid to open, releasing its contents into the body. A similar method has been employed for generating self-folding miniaturized surgical tools [28]. Furthermore, the study of synthetically-designed active colloidal systems gives insight into the dynamics of viral capsid formation [29] and the physical nature of capsid interaction with biological systems.

There are many possible applications of active colloidal systems. As our capabilities to fabricate smart active structures increases in ease and decreases in size, the

only limit is our imagination (or Heisenburg’s uncertainty principle). By decreasing the size of active colloids, we can mass fabricate and study bulk motion of these soft machines. For example, a ‘smooth’ colloid that can flow through a porous medium can, by external stimuli, be changed into a ‘sticky’ colloid that will jam. This self-clotting behavior can be useful for wastewater treatment, chemical purification or irrigation systems.

1.4 Motivations, Applications and Brief Summary of the Dissertation

The second chapter of this work will give a brief introduction into the mathematical constructs that can help us understand liquid crystal systems. In NLC’s, this takes the form of homotopy groups that describe the induced orientation of the molecules and defect formation given surface-induced confinement. We will then develop some simulation methods to model NLC systems and use them to probe more complex conditions.

The third chapter will look at the characteristics of droplets confined on surfaces.

We will explore the confinement of NLCs in spherical caps with various surface boundaries. Examining energy minimization in LCs on chemically patterned surfaces allows us to develop a novel method of generating LC droplets confined to a surface for a parallelizable method of printing droplet arrays. We will study the surface tension and anchoring effects that must be taken into consideration when generating such arrays, both of which strongly affect the reproducibility of the optical textures in individual droplets.

The fourth chapter will cover thermophoresis, or energy minimization in systems where colloids are immersed in NLC and a temperature gradient is applied. Although the presence of a temperature gradient offers many phenomena that are dominant in isotropic systems, we will distinguish a thermophoretic force that is specific to anisotropic fluids.

We term this force Elastophoresis, and acknowledge that it is not limited to systems characterized by a temperature gradient. These liquid crystal systems provide a strong improvement on isotropic thermophoretic methods of sorting particles, allowing for much larger thermophoretic mobility and a suppression of Brownian motion. The study of the elastophoretic force can also lead to a new novel method of sorting cells by ‘leakiness’ of a cell membrane or presence of signal molecules, which can alter the local order parameter.

The fifth chapter will explore the ability to generate colloids with nematic order.

We will demonstrate the utility of a custom setup, which we use to selectively polymerize

NLC colloids on the micron scale. NLC polymers are geometrically confined by their director orientation, but can physically distort through the energy minimization process.

This ability makes them ideal candidates for soft transformers and shows potential to open an entire field in the study of bulk properties of active colloids.

CHAPTER 2

Topology in NLCs

In this chapter, I intend to lay the foundation for how we look at nematic liquid crystal (NLC) systems and garner an understanding of why defects arise in NLC systems and the rules governing defect formation and stability. We will first examine how broken symmetries in NLC molecules lead to topological defects and the energy considerations behind defect stability. We will then develop some simulation methods to help us model

NLCs and employ them to shed some light on effects of confining geometries and interplay of boundary conditions in NLC droplets.

2.1 From Topology to NLCs

It is possible to represent each point in real space quantitatively. For example, in standard digital grayscale images, each pixel gets a value between 0 and 255. From this, we can map the each point in real space into a degenerate space of discrete values from 0 to 255. We use a similar method for observing the world in color around us, only each point has three associated wavelengths based on photoreceptive cone cells in the retina of the eye: L (~560nm), M (~530nm) and S (~420nm). The human brain interprets gradual changes of these values in space as solid objects, while discontinuities identify the boundaries of the objects.

Another low-level function of the human brain is to detect orientation in space.

Simple cells in the primary visual cortex detect broken order or anisotropy of objects. In a

2D projection of the visual space we can characterize each line using an angle φ, which allows us to similarly map each line in real space to a hemi-circlular degenerate space. As we will see in this chapter, the hemi-circular degenerate space is classified as 푅푃2 and is used to topologically describe NLCs.

2.1.1 Defects

The symmetry of a medium defines the given degeneracy space. For example, isotropic objects such as spheres (or objects that can be approximated as spheres) all have a degeneracy space containing a single point. As we break the symmetry of a sphere by elongating it and creating an ellipse, it can be mapped by its angle of rotation to a hemi- spherical degeneracy space, topological labeled 푅푃2. If we further break the for-aft symmetry of the ellipse, the angle of rotation can be mapped onto a spherical degeneracy space, 푆2.

Let us consider a system with broken symmetry, which maps onto a degeneracy space. If all the points in real space can be mapped to the same point in the degeneracy space, the medium is said to be homogeneous. For inhomogeneous mapping, the medium can either contain a singularity, which is a topologically stable defect that is not defined in the degeneracy space, or a topological soliton. We will focus here on topological defects in 3D media confined to 0D point, 1D line and 2D wall defects.

Homotopy, as can be inferred from the name, describes equivalence between topological spaces that can be achieved through continuous deformation. We can examine the topological stability of defects using homotopy groups. The notation is as follows:

Π ( ) i 푅 = ℤ = [… , −2, −1,0,1,2, … ] 2.1

푖 where Π푖 is the mapping of an i-dimensional sphere 푆 in real space onto the degeneracy space 푅. An important constraint is that the i-dimensional sphere can only contain points from the degeneracy space. In this example, we have assumed a case where the homotopy group of a degeneracy space is the set of all integers ℤ. Each element of the homotopy group is a class of equivalent topological states. For example, the element 1 would contain all possible configurations in real space that contain a charge 1 topological defect as defined by the boundaries of the i-dimensional sphere and can be continuously transitioned from one configuration to another.

1 Figure 2.1 shows an example of Π1(푆 ), or 1D loops drawn in a 2D space with an

푆2 degeneracy space. In real space, the purple curve is drawn so that it does not contain

1 any defects, and thus is the case of Π1(푆 ) = 0. The curve can be mapped into the degeneracy space as shown on the right. From here, we can also see that the mapped curve can be continuously shifted to collapse down into a point.

1 The topologically distinct state of Π1(푆 ) = 1 is shown by the orange curve.

Here, the red dot corresponds to a point in space that cannot be mapped into the degeneracy space. Furthermore, due to our constraints, we cannot pass the loop through

the defect. Consequently, any loop drawn in real space around the defect will map in the same way onto the degeneracy space, as is shown on the right of the figure.

Figure 2.1 Mapping of two arbitrary loops in real space to their degeneracy space where blue arrows designate points that are found in the degeneracy space S1 while the red point signifies a defect point that is not in the degeneracy space.

The elements introduced in Equation 2.1 are designated the wrapping number. As we can see from Figure 2.1, this wrapping corresponds to the number of times the angle completes a full rotation in the degeneracy space. This is termed the charge of the defect.

The homotopy group also relates the dimensionality of the defect, 푑, to the dimensionality of the medium, 푚, by

푖 = 푚 − 푑 − 1 2.2

1 From this we can see that the dimensionality of Π1(푆 ) is 푑 = 2 − 1 − 1 = 0, or a 0D point defect.

2.1.2 Homotopy Groups in Sn n-vector Model

In its most simplistic form, we can imagine the role of homotopy groups in the n- vector model. The n-vector model is a lattice of classical spins that interact using the

Hamiltonian

퐻 = −퐽 ∑ 푠푖 ∙ 푠푗 2.3 <푖,푗>

n in the name n-vector model designates the dimensionality. The model in n=0 describes a self avoiding walk. n=1 is the Ising model. This begins our topological analysis, since the Ising model is a system of spins on a lattice that can either point up or down. Consequently, we can say that the spin is confined to a 0-sphere S0, which is a pair

0 of points. A property of this system is that Πn(S ) = 0 for any arbitrary dimensional mapping n, but this is deceiving. What this means is that there is no way to generate a loop that would contain a defect without crossing a domain wall. This is schematically shown in Figure 2.2(a). However, the failure of homotopy to describe the system is due to the limitation described in Equation 2.2: 2 − 1 − 1 = 0; in order to observe a line defect in a 2D medium, we would need to generate a 0-dimensional loop around it. If we generate a point defect, it can be transitioned out of the system, as is shown in Figure

2.2(b,c).

This is also the case with a 3D medium of spins, where points and lines are not stable, but walls cannot be described by homotopy due to Equation 2.2: 3 − 2 − 1 = 0.

Figure 2.2 (a) The Ising model where boundary conditions necessitate a domain wall.

There is no way to generate a continuous loop that would contain the discontinuity.

However, confined domains as in (b) can be smoothly transitioned out (c).

n=2 is the case of the XY model, where the spins of each point on a 2D lattice are confined to a 1-sphere S1, or a circle. This vector space can now contain deformations that cannot be transitioned out of the system and was shown schematically in Figure 2.1.

The charge of a topological defect, S, is determined by

휙(푟) 푆 = ∮ 푑푟 2.4 2휋 where ϕ is the sum of rotation angles of the lattice vectors after the completion of a full

1 loop around an arbitrary area in the medium. We can write Π1(푆 ) = ℤ, where the wrapping number Z is the total topological charge. An important property of the set of all real integers Z is that it is additive. For example, two loops around separate +1 defects can be combined by drawing a loop around both defects, wherein the wrapping number would be 2, or have a +2 charge. A +1 and -1 defect in a region are topologically equivalent to no defects in that region. Thus the homotopy group is expressly dictated by the boundary conditions of the medium.

The case of n=3 is known as the Classical Heisenberg Model and is commonly used to describe ferromagnetism, where the spin in 3D space is confined to a 2-sphere S2.

The two relevant properties are

2 Π1(푆 ) = 0 2.5 2 Π2(푆 ) = 푍 A consequence of the former is that defect lines cannot exist in a vector field because any 2D defect in a 3D medium can be smoothed out through the 3rd dimension.

On a spherical parameter space, a closed loop on the surface can be shrunk down into a point without leaving the surface. The latter definition means that, just as in the 2D case, distortions can exist with an integer-value wrapping number and are similarly additive.

2.1.3 NLCs: Homotopy in RPn

LCs are a unique state of matter that exhibits broken symmetry. This property stems from molecular interactions that depend on the structure of constituent molecules and can, in general, be very complicated to describe. In this work we will focus on nematic LCs (NLC), because they can be easily described by homotopy groups [30]. In order to apply topological analysis to NLCs, we must first understand how a collection of physical molecules can be represented as a field.

NLC systems are characterized by the fact that rotations about the molecular long axis are very fast, while rotations perpendicular to this axis are suppressed by neighboring molecules. As a result, NLCs can be described by three degrees of freedom: two that give an average orientation 푛̂ (which, in the smallest number of variables can be

described by θ and φ) and one for the order parameter S, which describes the extent to which molecules are aligned. The director can correspond to either the time averaged orientation of a single molecule or a molecule number averaged orientation at a given time while the order parameter is a quantization of the alignment along the director defined by the average of the second Legendre polynomial

1 푆 = 〈푃 (cos 휃)〉 = 〈 (3 cos2 휃 − 1)〉 2.6 2 2 when only one Euler angle is required.

The director 푛̂ is further constrained by the degeneracy 푛̂=-푛̂, giving it D∞h symmetry and constraining the order parameter space to the real projective spaces RPm.

The properties of these homotopy groups are [31]

0 푖 = 0 2.7 푚 ℤ 푖 = 1, 푚 = 1 Πi(푅푃 ) = { ℤ2 푖 = 1, 푚 > 1 푚 Πi(푆 ) 푖 > 1, 푚 > 0 where ℤ designates integers and is the wrapping number of the system and ℤ2 is a cyclic

m group modulo 2, or in our case the set [0,1] that will be explained later. Πi(S ) is the homotopy group of spheres, with relevant properties shown in Equation 2.5.

2D nematics belong to the group RP1 and can be understood as a director field that is confined to a semi-circle, since rotations by π return the original orientation. In this case, a more rigorous definition can exploit the fact that the entire order parameter hemi- circle can be covered by half of a 1-sphere, so

1 1 1 Π (푅푃1) = ℤ = [… , −1, − , 0, , 1, … ] 2.8 1 2 2 2 where the non-zero elements are shown schematically in the degeneracy space in Figure

2.3(a,e) with the convention that a positive charge wraps in the same direction in real space as in degeneracy space. The dotted line represents the degeneracy between the points found at each end of the space.

Figure 2.3(a,e) Order parameter space RP1 in the first homotopy group with the first two winding numbers in each direction shown in red. Forward and backward winding are distinct as shown in the director representation of a (b) +1/2 defect, a (c) -1/2 defect, a

(f) +1 defect and a (g) -1 defect. Schlieren textures of (d) half integer and (h) integer 1 defects.

Double-wrapping the space to create an integer defect is topologically equivalent

2 to the case of Π2(푆 ) = ℤ as we can see from the similarity between the real space +1 defect in Figure 2.3(f) and the real space orange curve in Figure 2.1.

In 3D, the group is RP2. This means that the order parameter space is defined by a hemisphere where a rotation by π in any direction returns the original orientation [32].

The first homotopy group is a mapping of a 2D ribbon onto the order parameter space and corresponds to the case of line defects. If one attempts to draw a 1-sphere projection starting and ending at the same point confined in one hemisphere as shown in Figure

2.4(b), it must be homotopic to the case of a point on the sphere and can be lifted from the parameter space. This means that the defect can be smoothly deformed out of the system. This is equivalent to our XY model.

Figure 2.4 Order parameter space RP2 in the (i,j) first homotopy group and (k) the first winding number in the second homotopy group.

However, in the real projective plane it is possible to draw a line connecting two antipodal points as shown in Figure 2.4(a), which is a half-integer disclination. Due to the symmetry of the system, this is not homotopic to previous case and cannot be smoothly deformed out of the system. There are some very specific properties that arise from this geometry. For example, we can smoothly transition what we consider to be a 2D +1/2

defect into a -1/2 defect through a twist deformation. In mathematics this forms a finite cyclic group ℤ2, or the set [0,1] as shown in Figure 2.4 where 0 is shown in (b) and 1 is shown in (a). Since the set is additive modulo 2, we get the properties 0 + 0 = 0,

1 + 1 = 0 and 1 + 0 = 0 + 1 = 1. The sum of any two half-integer disclinations will annihilate: since we can smoothly deform a positive half-integer into a negative half- integer, the sum of two of these will generate a closed loop at the equator, which can be smoothly transitioned out of the system.

The second homotopy group of RP2 projects a 2-sphere onto our parameter space, like in Figure 2.4(c). This mapping allows the generation of 3D integer-charge defects in space but the analysis is more complex than the first homotopy group. In the second homotopy group, there is isomorphism to the case of equal-length vectors in space as dictated by Equation 2.10. For order parameters confined to S2 space, any 2D loop drawn on the surface can be collapsed down into a point and pulled from the surface, or escape through the 3rd dimension. However, in the case of a sphere wrapping the vector space, the wrapping number is any integer value. We can define positive integer defects as hedgehogs and negative integer defects as hyperbolic hedgehogs. Half-integer point defects cannot exist.

2.1.4 Schlieren Textures

The imaging of liquid crystal defects between crossed-polarizers is a physical manifestation of the Pontryagin-Thom construction [33]. Schlieren textures for defects

with charge ½ and 1 are shown in Figure 2.3(d,h). Between crossed-polarizers there is degeneracy between the optical activity of molecules rotated by π/2. As a result, the number of dark brushes b can be related to the wrapping number or charge of a defect S by S = b/4.

2.1.5 Elastic Free Energy

Another consequence of the broken symmetry in LCs is a resistance to deformation. One way to describe this resistance is by approximating distortions using elastic free energy. The elastic energy per unit volume is

1 푈 = 퐶 푢 푢 2.9 2 푖푗푘푙 푖푗 푘푙 where uij is the strain tensor, which depends on the deformation and Cijkl is the elastic tensor, which depends on the crystal structure. In the case of liquid crystals, it was reasoned by Frank that the only non-zero constants in the elastic tensor of a liquid crystal system would be the terms corresponding to a splay of the director K11, the twist of the director K22, the bend of the director K33 and a surface saddle-splay distortion of the director K24 [34]. With the corresponding geometries worked out, the Frank elastic free energy for a nematic liquid crystal is given as

1 1 1 2.10 푓 = 퐾 (∇ ∙ 푛⃑ )2 + 퐾 (푛⃑ ∙ ∇ × 푛⃑ + 푞 )2 + 퐾 (푛⃑ × ∇ × 푛⃑ )2 2 11 2 22 0 2 33

1 + 퐾 ∇ ∙ ((푛̂ ∙ ∇)푛̂ − 푛̂(∇ ∙ 푛̂)) 2 24

A common simplification of this free energy neglects the surface term and makes the approximation K11=K22=K33=K. This single K approximation gives

1 푓 = 퐾[(∇ ∙ 푛̂)2 + (∇ × 푛̂)2] 2.11 2 When the bulk energies are in conflict with the enclosing surface, it is also important to look at the elastic free energy cost of the surface, which can be determined using the Rapini-Papoular [35]

1 푓 = − ∬ 푊 (푛̂ ∙ 휑̂)2 푑퐴 2.12 푠 2 where A is the area of the surface, W is the anchoring energy and φ is the orientation angle of the liquid crystal on the surface. The anchoring strength can be considered strong or weak. In the strong anchoring approximation, we assume that the first layer of liquid crystal molecules is aligned exactly with the surface. This will, in general, depend on the size of the system. The condition of weak anchoring is more complex with the first layer of liquid crystal molecules taking on non-trivial orientation.

Aside from orientational effects, surface interactions can also change the spatial distribution of liquid crystals. If the intramolecular bonds are sufficiently strong and the surface alignment is homeotropic or tilted, they can induce smectic ordering, where the liquid crystal molecules align in layers at the surface [36]. These effects may also induce nematic ordering above the nematic-isotropic transition temperature in an effect known as surface-induced order [37-39]. Surface-induced order exists in LC systems where the isotropic phase wets the surface [36].

Under external fields, we can also add additional terms that contribute to energy costs in the system. For example, an applied electric field will lower the energy cost of molecules with positive dielectric anisotropy if they are oriented along the field.

Molecules with negative dielectric anisotropy will align perpendicular to the field. The same energetic considerations can be made for molecules with positive and negative magnetic susceptibilities in the presence of a magnetic field [40].

Coming back to our arguments about topological equivalence, we can now assign energy costs to homotopic director configurations. Since homotopic equivalence does not denote energetic equivalence, we can change configuration within homotopic groups by changing the energy cost of director distortions. We will return to this concept once we introduce LCs in confined geometries.

2.2 Simulation Methods

A simple method of computationally modeling liquid crystal defect textures is using the director formulation. Each point in a simulation is characterized by a vector

푛⃑ = (푛푥, 푛푦, 푛푧) 2.13 where ni is the component of n in the i direction.

Director Dynamics

We initially collaborated with Andrew Konya and Robin Selinger to create a 3D model of LCs in arbitrary geometries using director dynamics.

The liquid crystal medium interacts through deterministic director dynamics with a single K approximation. The directors are placed on a cubic lattice. The torque between two directors is calculated by

휕퐹 휏 푎 = −푛̂푎 × = ∑ (푛̂푎 ∙ 푛̂푏)(푛̂푎 × 푛̂푏) 2.14 휕푛̂푎 푏 and the director is updated in time by

푎 푎 푎 푎 푛̂푡+∆푡 = 푛̂푡 + 휉(휏 푡 × 푛⃑ 푡 )∆푡 2.15

Q-Tensor Formulation

Since the director really points in both the +푛⃑ and −푛⃑ directions, it can more accurately be described by the Q-tensor, defined computationally as

1 푄 = 푛 푛 − 훿 2.16 푖푗 푖 푗 3 푖푗 where δij is Kronecker’s Delta.

Expansion to multiple K’s requires use of a Q-Tensor Formulation. The total free energy in this case is derived in APPENDIX B [41]:

1 1 푓 = (−퐾 + 3퐾 + 퐾 )퐺 + (퐾 − 퐾 )퐺 − 푞 퐾 퐺 + 2.17 6 11 22 33 1 2 11 22 2 0 22 4 1 (퐾 − 퐾 )퐺 4 33 11 6 where

퐺1 = 푄푖푗,푘푄푖푗,푘 2.18

퐺1 = 푄푖푗,푘푄푖푗,푘 2.19

퐺4 = 휀푖푗푘푄푖푙푄푗푙,푘 2.20

퐺6 = 푄푖푗푄푘푙,푖푄푘푙,푗 2.21

Over-Relaxation

There are several deterministic methods of computationally modeling nematic liquid crystals using the Q-tensor. We collaborated with Seyedeh Afghah and Robin

Selinger to explore the over-relaxation method [42], where the director is updated in time using over-relaxation where

1 1 ∆푛휏+1 = 훼(∆푥)2 [− (−퐾 + 3퐾 + 퐾 )퐻휏 − (퐾 − 퐾 )퐻휏 + 2.22 푖 12 11 22 33 1푖 2 11 22 2푖 1 퐾 푞 퐻휏 − (−퐾 + 퐾 )퐻휏 ] 22 0 4푖 4 11 33 6푖 and

훿퐺푗 2.23 퐻푗푖 = 훿푛푖

Monte-Carlo Metropolis Algorithm

We also developed a simulation to look at the more stochastic processes using the

Monte Carlo Metropolis Algorithm.

We first individually generate a random change to each point in our geometry.

The total elastic free energy is calculated before and after the hypothetical change,

allowing us to accept or reject the change with a certain probability using the Metropolis algorithm [43]

푝 = 푒−(퐸푛푒푤−퐸표푙푑)⁄푘퐵푇 2.24

Calculating the derivative at a point by only using the six neighbors generates a computational problem where two independent lattices can exist in the system. As a result, we have to implement a random selection of forward and backward derivatives.

Boundary Conditions for Monte Carlo

By using the Monte-Carlo method, we introduce a few technicalities to the system. Due to the nature of discretization in finite difference methods, it is necessary to use two points to determine a derivative [41]. Since the energy at each point is determined by calculating the derivative of each neighbor, which requires a second neighbor in the calculation, a single monolayer confining surface does not suffice. To resolve this issue, we coded a two-layer confining surface. Because of this, the energy calculated at a point on the surface will have an additional contribution stemming from the second buffer layer, but this does not impact the energy minimization since it enters the energy calculation as a constant.

Simulations Used in this Work

After some initial studies with the Over-Relaxation method, we determined that it did not show significant qualitative differences in liquid crystal textures. Due to its much

lower computational requirements, the simulations in this dissertation were all generated using the Monte Carlo method described in the previous two sections.

Visualization of Director Fields

The visualization of the nematic director was achieved by generating visualization

VTK files [44]. We exported the position, director and energy at each voxel. Using the program Paraview [45], we could import the visualization files and look at a 3D reconstruction of our system. Paraview allows us to (1) look at the bottom-most layer of the liquid crystal to see how the 2D system at the interface behaves; (2) threshold the director visualization by local energy density. By setting a threshold for the minimum energy, we can easily observe 3D disclinations.

Jones Matrix Method for Optical Transmission

In order to compare our simulation results to experimental observations, we needed to convert the 3D director structures into 2D simulated polarized optical microscopy (POM) images. This was achieved by using the Jones Matrix method [46] for calculating transmitted light intensity from a linearly polarized source and initially coded by Andrew Konya.

Our incoming electric field is described by the vector

1 0 퐸̅ = ( ) 2.25 푖푛 0 1 The Jones matrix for our linear polarizer is defined as

1 푃⃑ = ( ) 2.26 0 The angles θ and φ are defined in each voxel along the z axis using the nematic director N

푁 휃 = tan−1 푦 2.27 푁푥 2 2 √푁푥+푁푦 휑 = tan−1 푁푧

Using these angles, we can calculate the effective refractive index along z

2 2 2 2 −1/2 푛푒푓푓 = 푛푒푛표(푛푒 cos 휑 + 푛표 sin 휑) 2.28 and Δn=neff-no.

For convenience we can define the absolute phase change

2휋∙Δ푛∙푑 휓 = 2.29 휆 휆 To get the Jones matrix

cos2 휃 + 푒−푖휓휆 sin2 휃 (1 − 푒−푖휓휆) sin 휃 cos 휃 2.30 퐽휆̅ = ( ) (1 − 푒−푖휓휆) sin 휃 cos 휃 sin2 휃 + 푒−푖휓휆 cos2 휃

The electric field of light after passing through the system is thus given as

퐸휆푥 2.31 퐸⃑ 휆 = ( ) = 퐸̅푖푛(∏푧 퐽휆̅ )푃⃑ 퐸휆푦

Where the intensity is

⃑ ∗ ⃑ 2.32 퐼휆 = 퐸휆 ∙ 퐸휆 We can define specific wavelengths to get the light transmission in an RGB color map by defining λR=650nm, λG=510nm and λB=475nm. 26

2.3 NLCs under External Geometric Confinement

As mentioned earlier, the topology of the director field is determined by boundary conditions on the system. The total topological charge of a surface or volume is determined by the Euler characteristic χ. In 2D, this characteristic can be determined by polygonizing a surface and using the relation

휒 = 푉 − 퐸 + 퐹 2.33 where V is the number of vertices, E is the number of edges and F is the number of faces.

Since convex polyhedra all have 휒 = 2, the same must be true of a sphere, which is the limit as the number of polyhedra goes to infinity. Some other topologically relevant shapes are the 2D disk (휒 = 1), the real projective plane RP2 (휒 = 1), the torus (휒 = 0) and the double torus (휒 = −2). Tori are important because they are a manifestation of holes in a normally convex shape, or handlebodies, where each ‘hole’ has a topological cost of 휒 = −2.

A consequence of the Poincare-Hopf Theorem is that the total topological charge of a surface must equal the sum of topological charges, S [6]

∑푖 푆푖 = 휒 2.34 Since a hollow sphere has a topological charge of +2, tangentially aligned NLC on the surface can have a +2 defect, two +1 defects, four +1/2 defects, or, in general, any combination of defects allowed by topology with a net charge of +2. All of these cases are homotopically equivalent and the configuration is strongly dependent on internally- and externally-induced free energy costs. Free energy cost scales with the square of the

charge [47]. As a result a +2 defect is difficult to maintain at the surface of a drop and will generally break into smaller charges. However placing two +1 defects at the poles is commonly observed, as is four +1/2 defects. The exact energy cost of the defects reflects both the structure and the ratio of elastic constants [6]. For example, a +1 defect has only splay distortion, while a +1/2 defect has both splay and bend.

Bulk NLC in a spherical drop generates interplay between the total topological charge within the volume of the drop and the charge on the very surface. In the simplest case, if the sphere is planar aligned, the topology is limited by the previously explained considerations. The defects are trapped at the surface as boojum defects and the configuration does not necessitate any volume-confined defects inside the sphere.

If we force a drop to be homeotropically aligned there are no forced defects on the surface because the LC is circularly symmetric in the plane of the surface. The total charge drawn inside this confining geometry must be a topological hedgehog, but the actual structure of the defect can be more complex. In the case of homeotropically aligned spheres, the two possible methods are the generation of a +1 hedgehog defect at the center or the generation of a +1/2 defect ring inside the drop. The physical difference between the two is the energy cost of the splay and bend deformation in the NLC, as the radial configuration only has a splay deformation while the axial configuration has both splay and bend.

The transition from volume centered to surface confined defects can be observed by slowly transitioning the surface orientation from hometropic to planar. The defects can

also be strongly influenced by external electromagnetic fields or convective flow at the surface, which can have dynamic interactions with defect charges and positions.

2.3.1 Pair Annihilation in RPm

We first examined strongly anchored NLC droplets on flat surfaces using both mathematical properties and simulation results. These systems are a fascinating due to interplay between 2D constraints and 3D constraints. By randomly orienting the surface, we can confine the first layer of NLCs in the plane of the surface. The only topological constraint comes from the homeotropic anchoring on the curved surface. As a result, the topology of the 2D system is independent of the 2D plane constraints and strictly controlled by higher dimensionality.

If we randomly initialize the directors in the NLC drop and study the surface, we can instantly see the formation of a number of charge S=+1/2 defects, n+, and a number of S=-1/2 defects, n-, on the surface. The Euler characteristic of a disk is χ=1, so we are constrained by

2.35 ∑ 푆푖 = 휒 = 1 푖 meaning that n+=n-+2. Over time, these defects pair-annihilate as is shown in Figure

2.5(c) until two +1/2 defects remain. Due to the topology of the system, these defects actually experience an attractive force that brings them together into a +1 boojum defect on the surface.

Figure 2.5 (a) The degeneracy between a 2D +1/2 and -1/2 defect in 3D degeneracy space (top) and real space (bottom). (b) A transient defect line from a simulation of a droplet of LC that transitions from a +1/2 to a -1/2 defect on the surface. (c) A 2D schematic of the pair annihilation with (d) corresponding 3D defect structure.

If we look at the 3D defect structure, it becomes apparent that from the moment of formation, defect pairs are linked through disclination lines like the one shown in Figure

2.5(b,d). All but one of the disclination lines connect oppositely charged half-integer

defects on the surface. The hybrid disclination lines are a transient feature but they are a fascinating example of what topology can really tell us about a system. As described before, positive and negative half-integer defects can be continuously transformed into one another through a twist deformation, as is shown in Figure 2.5(a). Furthermore, a half-integer point defect cannot exist. Thus we have generated half-integer defects in our

2D nematic that must somehow have connected disclination structures in 3D. However, there is no topological difference between disclinations that connect similar defects from opposite defects. The question of why the system prefers connecting defects of opposite charge cannot be fully understood by energy considerations. Even with a large bend energy cost, the system initially moves through a half-integer-defect-rich stage.

A possible explanation is that fewer points are required to define a half-integer defect (3) than a full-integer defect (4), so in very localized areas, the half-integers will always form first.

2.3.2 Spherical Caps on Planar and Homeotropic Surfaces

Since most NLCs exhibit homeotropic anchoring at an interface with air, the resulting director orientation for a spherical droplet in air is radial, shown in Figure

2.6(a). If we now take a strongly anchored plane and use it to cut the droplet, it is possible to probe the effect of a surface on a drop with varying contact angle.

In the case of planar alignment, the defects are surface-confined, which is independent of contact angle as is shown Figure 2.6(b,c). The position of the defect becomes dependent on the orientation of the surface. For example, with planar degenerate 31

Figure 2.6 (a) Radial configuration of a LC sphere with homeotropic anchoring. On planar surfaces (b,c), the topological constraints force a boojum on the surface with curvature of the elastic field lines dependent on contact angle. Homeotropic surfaces (d- g) can have an unstable bulk defect of charge +1 (d,f) that are homotopically equivalent to a boojum ring located at the three-phase line. Simulation results showing crossed polarizer images of relaxed spherical caps on a (h) planar degenerate surface, (i) planar aligned surface and (j) homeotropic surface. The corresponding director orientations and defects are shown in (k), (l) and (m), respectively, where the non-blue color signifies areas of high energy cost. 32

alignment there is no broken symmetry of the system so the defect ends up at the center.

The case of homogeneous in-plane alignment breaks this symmetry and generates a half-integer defect line perpendicular to the alignment direction in the bulk that terminates on the surface [48] as shown in Figure 2.6(l).

The case of homeotropic anchoring turns out to be more complicated. Figure

2.6(d,e,f,g,m) show a homeotropic surface and a peculiar degeneracy where an apparent net zero charge and a +1 charge are equivalent states. This is because the defect can either be volume-confined or create a boojum ring along the three-phase line [49]. The defect is forced into the three-phase line because that is the highest energy region in the system and in general depends on the surface energy involved, which is manifested as the contact angle.

Tilted alignment can also be modeled at the surface. In the case of tilted degenerate anchoring, the energy minimization is identical to the planar degenerate case until it reaches a threshold that transitions it to a homeotropic texture. The tilted aligned case has a continuous transition

2.3.3 Spherical Caps on 2D Defects

We also simulated liquid crystal drops on defect-patterned surfaces. Defects and spiraling defects are defined on the surface by

푥 − 푥 휋 휑(푥, 푦) = −푆 tan−1 0 + (휓 + 2) 2.36 푦 − 푦0 4

푥 − 푥 √(푥 − 푥 )2 + (푦 − 푦 )2 휋 2.37 휑(푥, 푦) = −푆 tan−1 0 + 0 0 (휓 + 2) 푦 − 푦0 푅 4 , respectively. Here, the in-plane angle φ is dictated by its spatial position as well as an offset ψ. In the case of the spiral, the offset changes as a function of distance from the center of the defect.

Defects with a patterned +1 charge (Figure 2.7(a)) are trivial, as the surface and bulk constraints both necessitated a +1 boojum defect at the center of the cap. The more interesting circumstances occur when the patterned defect does not match the total topological charge of the drop. Since the center of the patterned circle has the greatest influence over local director patterns, with the energy frustration between the top and bottom surface increasing radially, we notice the first layer of the liquid crystal always conforms to the surface at the center. To complete the defect topology enforced by the other enclosing surface, other defects are generated far from the center and the systems often showed stable 3D disclination lines.

Defect patterning gives us radial control over the 3D defect structures in the NLC spherical caps. For example, if we pattern a -1 defect at the center (Figure 2.7(c,e)), we can predict that two twist ½ defects will be generated in the medium with the negative end terminating in the center and the positive end terminating on the periphery. Also, we can predict that a +1/2 defect line would have to be anchored on the periphery and stretch into the center on the drop, due to the radially decaying surface effects in the medium.

This was observed computationally in Figure 2.7(e).

Figure 2.7 Simulation results showing crossed polarizer images of relaxed spherical caps on a (a) defect spiral with S=1 and ψ=0, (b) defect with S=1/2 and ψ=0 and (c) defect with S=-1 and ψ=0. The corresponding surfaces, first layer director orientations and defects are shown for the (d) S=1/2 and (e) S=-1 cases.

2.4 Summary

We have looked at some of the basic properties of nematic liquid crystal systems that stem from their specific symmetry. When a spherical cap of NLC is confined on its curved surface by homeotropic alignment, the resulting structure always exhibits a total topological charge of +1. As a result, any forced charge on the flat surface must be remedied in the bulk. The exact topology of the system is dictated by energy considerations, where we saw that splay, twist and bend distortions in the medium can have different energy costs. These cases were studied using monte-carlo simulations.

CHAPTER 3

NLC on Chemical Patterns

In this chapter we will probe the ability to generate droplet arrays in a parallelizable way. Due to the large surface area of liquid crystal droplets and their localization on surfaces, these systems make for ideal chemical sensors. We will also explore the morphology and defect textures that arise experimentally as a byproduct of our localization process. The defect textures are constrained by the considerations laid out in the previous chapter.

3.1 NLC Drops on Surfaces

3.1.1 Contact Angle

In our analysis of droplets on surfaces, we will consider the macroscopic case, as we did not probe the molecular level interactions. In general, contact angle is a local phenomenon at the three-phase line.

When a liquid phase 훽 comes in contact with a surface 푠 in a medium 훼, the equilibrium state of the system is determined from the free energies. The first measure of the resulting statics is the spreading coefficient 푆, or the difference between the work of cohesion and work of adhesion defined as

푎푑ℎ 푐표ℎ 푆 = 푊푠훽 − 푊훽훽 = Σ푠 − (Σ푠훽 + Σ훽) 3.1 where Σ is the surface tension for the appropriate interactions [50].

In the case where 푆 is positive, the surface is completely wet by the fluid because the attractive force between the fluid and surface is larger than that of the fluid to itself.

In the case where 푆 is negative, the fluid forms a drop on the surface whose static profile is governed by the Young-Laplace equation:

∆푝 = 2Σ훼훽퐻 3.2 where 훥푝 is the change in pressure between the inside and outside of the drop and 퐻 is the mean curvature [51].

In certain systems when fluid drops are in the micrometer range, both gravitational effects on the fluid and line tension are negligible and the profile can be approximated as a spherical cap. One common way to measure the interaction energies between a surface and a fluid is to measure the Young contact angle, 휃푌. The contact angle is related to the individual surface tensions by

Σ훽훼 cos 휃푌 = Σ푠훼 − Σ푠훽 3.3 where 훴푖푗 is the interfacial tension between phase 푖 and phase 푗, respectively. This relation is derived from free energy considerations in APPENDIX A.

Measuring a static contact angle can be a good approximation of the surface effects, but the above treatment of determining the Young contact angle is only accurate for ideal surfaces. In reality, the contact angle with a surface is dependent on whether the interface had been advancing or receding over the surface. This contact angle hysteresis exists due to contamination, surface roughness or inhomogeneity and alteration of the surface by the fluid. 38

Drops on rough surfaces are described in Wenzel’s model [52], where the apparent contact angle 휃∗ is dependent on roughness ratio 푟

cos 휃∗ = 푟 cos 휃 3.4 where, in general, the roughness may vary with orientation.

The Cassie equation describes surfaces that are chemically heterogeneous and for

푛 chemical species each covering an area fraction of 푥푛 takes the form

∗ cos 휃 = ∑푛 푥푛 cos 휃푛 3.5 where 휃푛 is the contact angle measured for a homogeneous chemistry.

Good and Neumann showed that, on a chemically heterogeneous surface with a radius of roughness much smaller than 1µm, the measured advancing contact angle corresponds to the Young contact angle for regions of low energy cost of the surface and the measured receding contact angle corresponds to the Young contact angle for regions of high energy cost. In the literature, the advancing contact angle is often given as an approximation of the Young contact angle on a surface.

3.1.2 Fluid Droplets on Chemically Patterned Surfaces

On a chemically patterned surface, the free energy of a drop varies as a function of position. For an understanding of the phenomenology, we will consider the situation as analyzed by Lenz and Lipowsky [53].

Figure 3.1 A droplet sitting on a hydrophilic domain 휸 in a hydrophobic background 휹 as the volume is increased. (a) At small volumes, the droplet takes on the contact angle characteristic of the underlying domain. (b) When the droplet reaches a volume that is sufficient to cover the entire domain, the contact line is pinned at the domain edge until it matches the contact angle on the background, where (c) it again begins to increase its contacting surface area.

In the simplest condition, we can consider smooth hydrophilic domains 훾 in a smooth hydrophobic background 훿. The morphology of a drop on such a surface is determined by minimizing the total interfacial free energy

퐹 = Σ훼훽퐴훼훽 + Σ훼훾퐴훼훾+Σ훼훿퐴훼훿 + Σ훽훾퐴훽훾 + Σ훽훿퐴훽훿 3.6 where 퐴푖푗 is the surface area between phase 푖 and phase 푗, respectively.

These two surface regions are characterized by distinct Young contact angles

Σ − Σ 훼훾 훽훾 3.7 cos 휃훾 = Σ훼훽

Σ − Σ 훼훿 훽훿 3.8 cos 휃훿 = Σ훼훽

where, 0 < 휃훾 < 휃훿 < 휋.

If we consider that we have a sharp boundary between the two domains, the contact angle on the domain boundary is constrained by 휃훾 < 휃 < 휃훿. In the case of a circular domain of diameter 푎훾, the volume of a drop can be calculated as the volume of a spherical cap

휋푅3 푉(푅, 휃) = (1 − cos 휃)2(2 + cos 휃) 3.9 3 where 푅 is the radius of curvature. For simplicity, it is convenient to define the minimum and maximum volume of a spherical cap confined by the domain boundary:

푎 훾 3.10 푉훾 = 푉 ( , 휃훾) 2 sin 휃훾

푎훾 푉훿 = 푉 ( , 휃훿) 3.11 2 sin 휃훿

The surface areas of a spherical cap are defined as

2 퐴훼훽 = 2휋푅 (1 − cos 휃) 3.12

2 2 퐴훽푠 = 휋푅 sin 휃 3.13

When 푉 < 푉훾, (Figure 3.1(a)) the contact angle is 휃훾 and surface area scales with

푅. When 푉훾 < 푉 < 푉훿 (Figure 3.1(b)), 퐴훽푠 remains constant and 휃훾 < 휃 < 휃훿. Finally in the regime where 푉훿 < 푉 (Figure 3.1(c)), the contact angle is 휃훿 and surface area scales with 푅.

When we consider increasing the total fluid volume in a lattice of 푁 circular domains, the contact angle of all the drops will increase together in equilibrium state A 41

until they reach a critical contact angle 휃∗, wherein the lowest free energy can be reached by having a single drop take on a large contact angle while the other drops have a small contact angle. This state labeled 퐵 is a degenerate state as the large drop can sit on any of the lattice positions, and there can be any number of combinations of large and small drops each constrained by 휃 < 휃훿 as volume increases.

3.1.3 Surface Energy of LCs

In the case of liquid crystals, the free energy can be define as

퐹 = Σ훼훽(휑, 푊푎)퐴훼훽 + (Σ훽푠(휑, 푊푎) − Σ훼푠)퐴훽푠 3.14 where the surface tension of the liquid crystal on a surface is dependent on the pretilt angle 휑 and the anchoring energy 푊푎. This means that the Young relation can be written as

Σ훼푠 − Σ훽푠(휑, 푊푎) cos 휃 = 3.15 Σ훼훽(휑, 푊푎)

A result of surface tension anisotropy is that ampiphilic molecules can wet both hydrophobic and hydrophilic surfaces, with the tilt angle dictated by the lowest energy orientation. In general, both the surface and the fluid can change their orientation to minimize this energy but we will assume the surface is rigid enough that it does not change due to liquid crystal interaction. This is a reasonable assumption when dealing with solid surfaces due to the much lower molecular mobility.

3.1.4 Surface Alignment of NLCs

LC alignment can be achieved using either chemical or physical methods. The physical method creates parallel nanogrooves on the surface. Depending on the length scale, there are two relevant effects that are responsible for LC alignment in nanogrooves.

At the molecular scale, excluded volume interactions would treat the nanogroove peaks as constituent molecules, forcing the alignment through packing considerations. At larger scales we can look at the converse alignment, which would result in large bending energy associated with director undulations over the surface. Thus by aligning parallel to the grooves, the total elastic energy is minimized.

Chemical methods are commonly used to align LCs. This is the case of photosensitive alignment layers, where polarized UV light exposure induces photopolymerization along a certain direction.

The effect of chemical alignment on the NLC molecules is dependent on their structure. A common structure for NLC molecules is a cyanobiphenyl structure, which has been broadly utilized in the LCD industry. We limited our work to the most common cyanobiphenyl: 4-cyano-4’-pentylbiphenyl, known as 5CB.

LCs can also be aligned by molecular monolayers. Systems with alkane-based surfactants show that 5CB molecules can actually insert themselves between the hydrocarbon chains. Hydrocarbon chains can also be affixed to a surface using Self-

Assembly (SA) on gold. The structure of underlying gold influences the structure of Self-

Assembled Monolayers (SAMs) on the surface. Angled deposition of underlying gold can

produce homogenous tilting of long length n-alkanethiols (14+) after SA and result in planar alignment of 5CB molecules along the direction of deposition [48]. It is possible that due to the long length and natural tilt during formation, these systems do not allow molecules to insert into the monolayer, but rather align them along the surface.

Conversely, short-chain n-alkanethiols do not tilt significantly after SA and give homeotropic alignment due to insertion of 5CB molecules between the hydrocarbon chains. Mixing monolayer ratios of short and long chains has been shown useful in inducing specific pretilt angles.

Water also has a unique interaction with 5CB molecules. Although not miscible, liquid crystal molecules macroscopically wet a water surface and induce a strong planar degenerate alignment. This behavior has been observed to be ubiquitous for carboxylic acid-terminated monolayers [54].

3.2 Materials, Methods and Characterization

3.2.1 Materials

Nematic properties were all measured with 5CB, which was purchased from

HCCH. Hexanethiol, octanethiol, decanethiol, dodecanethiol, hexadecanethiol (HDT), octadecanethiol (ODT) and mercaptododecanoic acid (MDA) were all purchased from

Sigma-Aldrich. For our chiral nematic material, we used a ZLI-1132 host liquid crystal with a pitch of 1µm at room temperature provided by Dr. Quan Li.

3.2.2 Contact Angle Measurement

Contact angle measurements were conducted using a home-made horizontal optical setup. We placed a micropositioning stage between a Thorlabs CMOS camera with appropriate lenses and a light source. For static contact angle measurements, the drops were placed onto the surface using a syringe with a flat tip needle that was brought in contact with the surface. We then used a screw to push a very small amount of liquid through the syringe tip. The contact angle that is observable as the droplet slowly expands is the advancing contact angle. The tip is then pulled away and a drop remains on the surface, generating a spherical cap constrained by the static contact angle. Figure

3.2(a) shows an example of the results.

Figure 3.2 (a) A static contact angle drop. Dynamic contact angles were determined by

(b,c) moving the substrate with the stationary syringe tip still in contact, (d) pulling the syringe tip away to get a receding angle and (e) increasing the fluid of the drop to get an advancing angle.

Keeping the syringe in contact with the drop and slowly moving the stage allows for the measurement of advancing and receding contact angles, as is shown in Figure

3.2(b,c). We can also pull the syringe from the surface slowly to look at the receding contact angle like in Figure 3.2(d) or feed more fluid through the syringe to determine the advancing contact angle like in Figure 3.2(e). Angles were measured when the interface first begins to move. The values were measured using ImageJ.

Figure 3.3 Contact angle measurements were done by (a) selecting the droplet of interest, (b) setting a surface baseline and (c) appropriately tresholding the image. (d)

After processing, Matlab determines the radius of curvature and height of the droplet for accurate determination of the contact angle.

To determine the static contact angles, the drop images were input into a Matlab program for profile analysis. The program has three user inputs for generalized use. The first allows a user to select a drop for measurements using a rectangular box ROI, which is useful when multiple drops can be seen in one image like in Figure 3.3(a). The second 46

input sets the base line of the drop (Figure 3.3(b)). Since gold is optically reflective, it is important to separate the drop from its reflection on the surface. The third input sets a threshold for the image (Figure 3.3(c)). This converts the drop into a binary image. The binary image is then inverted and filled. The final pre-processing step uses an in-built function to find the perimeter of the binary drop.

With the coordinates of the perimeter of the drop, the program approaches from the user-defined base line and finds the contour. Since the morphology of a drop with a negligible effect of gravity or line tension can be approximated as a spherical cap, the program fits the profile to a circle (Figure 3.3(d)) in order to find the radius of curvature

푅 and uses the difference between the baseline and the top of the droplet to get the height

ℎ. Finally, the 푅 and ℎ are used to determine the contact angle 휃 by

ℎ 휃 = 2 tan−1 ( ) 3.16 √ℎ(2푅−ℎ)

3.2.3 Pre-Tilt Angle Measurement

The pre-tilt angle of liquid crystal molecules on the self-assembled monolayers were measured by using the Crystal Rotation Method [55]. A cell is placed between two polarizers and a 532nm laser is shown through the setup onto a Thorlabs photodetector as is shown in Figure 3.4(a). The cell is then rotated about an axis that is perpendicular to both the incident light and the rubbing direction in increments of 5o and the readings are recorded. To get a normalization value, the polarizers were aligned parallel so that the maximum transmission was measured. An empty cell was positioned between the 47

polarizers and rotated from -45o to 45o. These measurements were saved as peak waveform for normalization. The same procedure was then followed with the polarizers crossed, as determined by minimum transmission. The cell was finally filled and allowed to sit for about an hour to reduce any slow dynamic processes that resulted from capillary filling. The measurements were done using crossed polarizers at a 45o angle from the rubbing direction. Finally, the cell was tested using a polarized optical microscope to determine if the alignment was monodomain. If the LC was not monodomain, the results were discarded, as polydomain alignment would cancel out the pretilt angle that we were attempting to measure.

Figure 3.4 (a) Crystal Rotation setup with (b) relevant angles and axes defined. (c)

Transmission vs angle for various pretilts with the center of symmetry labeled and (d) a transformation figure from pretilt angle to the center of symmetry, where the dotted line represents a 1:1 transformation.

The total transmission intensity using the crystal rotation method at a rotation angle

휑 is shown in Figure 3.4(b) and mathematically described as

1 1 푇(휑) = sin2 ( 훿(휑)) 3.17 2 2 where 훿 is the retardation of light passing through the cell, given as

2휋 훿(휑) = 푑 ∙ 푓(훼, 휑) 3.18 휆 with 휆 is the wavelength of light, 푑 is the cell thickness and 푓(훼, 휑) being a function of the pretilt angle 훼 and the rotation angle:

1⁄ 1 1 푎2푏2 2 3.19 푓(훼, 휑) = (푎2 − 푏2) sin 훼 cos 훼 sin 휑 + (1 − sin2 휑) 푐2 푐 푐2

1 1 − (1 − 푏2 sin2 휑) ⁄2 푏

with the constants 푎, 푏 and 푐 dependent on the extraordinary 푛푒 and ordinary 푛표 refractive indexes of the liquid crystal

1 1 푎 = , 푏 = , 푐2 = 푎2 cos2 훼 + 푏2 sin2 훼 3.20 푛푒 푛표 as derived by Born and Wolf [56].

To graphically observe the effect of different tilt angles on the crystal rotation results, we input the values of 푛푒 and 푛표 for 5CB into Equation 3.17. For small pretilt

angles, the results are plotted in Figure 3.4(c). In this range, it is possible to create a linear plot relating the center of symmetry and the pretilt angle. This conversion is plotted in Figure 3.4(d), where the dotted line represents a 1:1 mapping of the angles.

3.2.4 Monolayer Self-Assembly

We perform chemical modification of substrates using self-assembly of thiol- terminated molecules. A glass slide was placed in an e-beam evaporator where it was first coated with a 5nm Cr adhesion layers and then 15nm of Au. A very thin layer of gold is used for optical transparency. The surface is placed in an RIE etch machine and Argon plasma cleaned for 30 seconds.

The substrate is then placed in 2mL of a 1mM ethanolic solution of thiol- terminated molecules, where the molecules form thiol-metal bonds with the Au surface.

The process of monolayer formation is a highly complicated process that depends on variable such as molecule chain length and concentration, solvent, temperature and immersion time and has been summarized in a variety of review articles [57-59].

We did some preliminary studies to determine the optimal immersion time for the various monolayers. The contact angle of 5CB on HDT showed some interesting behavior. We first did a calibration curve for smaller immersion times, shown in Figure

3.5(a). The unique feature of the contact angle of 5CB on the surface is that it exhibits temporal decay. For an immersion time of 1 hour, it took about a week for the contact angle to stabilize (Figure 3.7(h)). For the case of much shorter immersion times, the

relaxation time was on the order of hours, as we can see in the case of a 480s immersion in Figure 3.5(b).

Figure 3.5 (a) Curve showing the immediate contact angle dependence on immersion time. (b) Changing contact angle in time for a sample immersed in 1mM HDT for 480s.

3.2.5 Cell Fabrication

Self-assembled monolayers can be used for LC alignment, a method first demonstrated by Abbott et al using angled deposition of Au. We attempt to align these monolayers by rubbing the surface with felt 10 times.

To confine LCs in a thin cell, we use a common fabrication procedure. First, we mix 25um fiber spacers with Norland UV-curable glue. The glue is then applied to parallel sides of one piece of glass. The second piece of glass is brought in contact with the glue with the rubbing direction anti-parallel and then exposed under a UV lamp for 15 minutes.

The cell thicknesses are then measured using an OceanOptics spectrometer

(Figure 3.6(a)), where consecutive peaks in the wavelength spectrum, λ1 and λ2 can give the thickness of the cell by

1 휆 휆 푑 = 1 2 3.21 2 휆2 − 휆1

Figure 3.6 (a) Reflection spectrum of a tungsten light source from a 25µm glass cell. (b)

Frequency of the light extrema plotted consecutive order with a linear fit.

To get a more accurate measurement, we measure the frequency of 10 consecutive extrema and plot them like in Figure 3.6(b). The linear fit of the extrema is used to determine the cell thickness

1 푑 = − 3.22 4 ∙ 푠푙표푝푒

3.3 Characterization of 5CB on SAMs

3.3.1 SAM Morphology

We attempted to use AFM to measure the morphology of the SAM layers.

However, the results were inconclusive, showing noise on the surface that was much larger than the feature size we were examining. Rather than pursuing this line of inquiry, we instead examined the influence of the SAMs on liquid crystals.

3.3.2 Contact Angle Results

We observed the contact angle of 5CB (Figure 3.7(a)) on a hydrophilic SAM:

MDA (Figure 3.7(d)). The MDA surface was strongly wet by 5CB with a contact angle less than 7° (Figure 3.7(g)). This was consistent with carboxylic acid-terminated surfaces in literature [54]. The contact angle of water on MDA was 35°.

We also tested the contact angle of 5CB on a few even-numbered n-alkanethiol molecules pulled from literature. Hexanethiol, octanethiol, decanethiol and dodecanethiol were all strongly wet by 5CB. HDT (Figure 3.7(b)) and ODT (Figure 3.7(c)) had the largest contact angles of the materials we measured, though we did not try longer chain

SAMs. The contact angle of water on HDT and ODT was larger than 100°.

5CB on HDT displayed some unexpected behavior, shown by the red points in

Figure 3.7(h). The initial contact angle was 60 degrees (Figure 3.7(e)) but decreased significantly with time. After about a week, the contact angle stabilized around 35 degrees (Figure 3.7(f)). The contact angle as a function of time for six droplets is plotted

in Figure 3.7(h). The volume of the drop did not noticeably influence the results. ODT did not show this behavior on the time scale we observed, as is shown in blue. The mixture of 1:1 ODT:MDA in green showed a slight decrease in the contact angle in the first three days that stabilized mid-way between the stable contact angle for HDT and

ODT.

Figure 3.7 The molecular structure of (a) 5CB, (b) HDT, (c) ODT and (d) MDA where white, grey, blue, red and gold represent H, C, N, O and Au. (e) 5CB drop on HDT immediately after being placed on the surface and (f) one week later. (g) 5CB drop on

MDA. (h) Average contact angle values over time for 6 droplets each of HDT, ODT and a 1:1 mixture of ODT:MDA.

Table 3.1 shows some relevant contact angles for both water and 5CB immediately after placement.

Table 3.1 Contact Angles

PDMS Glass Gold MDA HDT ODT 1:1

ODT:MDA

DI H20 - 50.1º 61.1º - - 101.8º 93.2º

5CB 49.9º 13.1º 7.1º 17.7º 60.6º 60.1º 49.9º

3.3.3 Pre-Tilt Angle Results

We first fabricated 25µm thick cells from two freshly self-assembled MDA surfaces. The cell was filled with 5CB and observed under the polarized optical microscope (POM), as shown in Figure 3.8(a). From the POM texture, we could observe that the NLC was aligned in domains, which were elongated along the fill direction.

A second cell was fabricated by rubbing the MDA surfaces with felt 10 times before the cell assembly process. The POM image in Figure 3.8(b) shows a constant light intensity through the cell, suggesting that it was monodomain and well aligned. In the case of a monodomain cell, we are able to use the crystal rotation method to determine the center of symmetry 휓푐, which is plotted in Figure 3.8(e). 휓푐 was converted into the pretilt angle induced by the surface using Figure 3.4(d): 1.5°. Planar alignment on this surface is consistent with what is observed on carboxylic acid groups and at the 5CB- water interface [54].

We then created a cell using surfaces with HDT SAMs. Again these showed only a slight bias toward the fill direction (Figure 3.8(c)). We then rubbed the HDT surfaces 55

with felt 10 times and filled the cell. We observed no difference between the rubbed and unrubbed cells, with both appearing highly polydomain (Figure 3.8(d)).

Figure 3.8 POM images of 5CB in a cell made of (a) two MDA surfaces, (b) two MDA surfaces that had been rubbed antiparallel, (c) two HDT surfaces and (d) two HDT surfaces that had been rubbed antiparallel. (e) The crystal rotation results (red) and a best fit for the center of symmetry (blue) for the rubbed MDA surfaces.

There are two important consequences of the tilt angle measurements. The first is that we demonstrated the ability to rub the MDA monolayers to induce uniform planar alignment. The second is that both unrubbed inks showed a highly polydomain structure, meaning that the surface induced alignment must also be polydomain.

3.4 Chemical Patterning

There are three commonly documented methods of chemically patterning surfaces that have been used to self-localize fluids: vapor deposition through a shadow mask,

photolithography of photo-sensitive surfaces and micro-contact printing (μ-CP) [60]. We will focus on the latter method, using an elastomeric stamp to print thiol-terminated molecules on a metal surface.

μ-CP is widely used as a convenient method to obtain patterned self-assembled monolayers. To fabricate master molds, we spin-coat negative resist SU8 on Si wafers and pattern them with contact photolithography. The patterned SU8 molds are then covered with a 10:1 mixture of PDMS precursors and crosslinkers, and then de-gassed and cured in an oven at 65C for 12 hours. The PDMS is then carefully pealed from the

SU8 mold. A representative scanning electron microscopic picture of the fabricated

PDMS mold is shown in Figure 3.9(b).

The process of μ-CP is shown in Figure 3.9(a). The patterned area on the PDMS mold is covered with 20uL of 1mM ethanolic solution of a thiol-terminated molecular ink. After 30 seconds, the rest of the solution is blown off with a stream of air. The stamp is then brought in contact with an Au-coated glass slide for 30 seconds. The ink molecules absorb onto the Au surface via thiol-metal bonds, forming a monolayer. The glass slide is rinsed thoroughly with ethanol and then submerged in a 1mM ethanol solution of thiol-terminated background molecules for one hour.

Stamp time is an important tunable parameter due to the adsorption of SAMs through the gas phase. By changing the contact time of the stamp between 30s and 60s, we saw up to a 17% increase in the size of self-localized droplets compared with the stamp features. NLCs on samples with stamp times less than 30s did not self-organize

into predictable shapes. In general, SAM formation is strongly influenced by diffusion properties of constituent molecules [61].

Figure 3.9 (a) Micro-Contact Printing procedure: a 1mM solution of ink in ethanol is coated on a PDMS stamp and allowed to evaporate, leaving behind a thin layer of ink, then the inked stamp is brought in contact with a gold surface, where the mercapton molecules form thiol-metal bonds with the gold resulting in a monolayer. The gold surface is finally placed into a 1mM solution of background, where the molecules self- assemble on the unoccupied areas. (b) Representative SEM image of the elastomeric stamp.

3.5 Droplet Self-Organization

A variety of methods have been documented for localizing fluid on chemically patterned surfaces, as it proves to be a useful method of generating microlens arrays [62-

64] and waveguides [65]. These methods can generally be broken down into additive or subtractive processes. 58

Additive methods involve some method of generating fluid droplets on the surface. For example, by changing the environmental temperature, pressure and composition, it is possible to force preferential condensation on chemically favorable regions of a sample [66]. One can also immerse the patterned surface into a fluid bulk and lift it out perpendicular to the interface [67]. For more control over surface tension or for devices that are used in fluids, this can be altered to push the sample through a thin film into the target medium [62, 68].

Subtractive methods require the fluid to already be on the surface, and by adjusting the local surface tension they cause breakup of the film. By placing certain fluids on the interface between the patterned surface and a fluid solution in the right quantity, it is possible for a fluid film to spontaneously break up [69]. For devices in a gaseous medium, a common practice is to coat the surface in a thin film and use either a squeegee [70] or absorbent cloth [71] to pull off excess liquid and force the spontaneous droplet formation.

In our system, we found these methods to be either too complicated to be feasible or not reliably reproducible. Therefore we tried several new methods of self-localizing

NLC on chemically patterned surfaces, with fundamentally different physical systems resulting from the dynamics of droplet formation.

3.5.1 Spincoating

We first fabricated a μcp surface with a MDA pattern on a HDT background. We placed a small quantity of 5CB on the surface and used a spincoater to spin the sample 59

using speeds of 1500rpm to 4000rpm. The sample was immediately heated to reduce surface tension. We also mixed the 5CB in small concentrations with isopropyl alcohol

(IPA) before spinning to reduce the total amount of 5CB on the surface.

Figure 3.10 (a) POM image of a micropatterned substrate with liquid crystal spincoated on the surface at 3000rpm. (b) and (c) show high-magnification images where it becomes visible that the liquid crystal is forming a film on the surface.

A representative POM image of the results is shown in Figure 3.10(a). Here the light regions correspond to areas where the alignment is planar on the surface and the dark areas correspond to more homeotropically aligned NLC. By doing some deeper image processing, we can begin to see the defect lines in the HDT background as in

Figure 3.10(b,c).

Unfortunately, we were not able to break the surface tension of the liquid crystal to generate individual droplets in any of our trials.

3.5.2 Cell Breaking

Cells were prepared by connecting one µcp MDA on HDT surface and one glass surface with parallel strips of two-sided tape. We capillary-filled 5CB into the cell and cracked the cell open. The resulting structures fell into two categories: the positive- patterned droplets (Figure 3.11(a)) and the inverted-pattern anti-droplets (Figure 3.11(b)).

We did not delve into the crossover of droplets to anti-droplets in this work.

The positive droplets obeyed the rules outlined in Section 3.1.2, where there were a small number of droplets in the array with the maximum contact angle and the rest of the droplets were relatively flat. Furthermore, we observed that upon cycling the temperature through the nematic isotropic transition by increasing the microscope illumination, we could cause the flat droplets to suddenly collapse into a crescent shape that was pinned at the interface between the hydrophobic and hydrophilic domains, which we can see in the sequential images in Figure 3.11(c,d).

This phenomenon can be understood by considering that, due to the dynamics of the cell-breaking, the contact angle is likely the receding contact angle of 5CB on MDA that is pinned between the domains (Figure 3.11(c,e)). The receding contact angle signifies that the drop is highly strained and unstable, with too small of a volume to comfortably hold its shape. The surface tension finally breaks when a significant temperature gradient is applied across the domain during heating and cooling of the 61

sample causing the droplet, which is still pinned on most of the domain boundary, to buckle into a crescent shape (Figure 3.11(d,f)). This is supported by the fact that all of the

Figure 3.11 Representative area of (a) positive-patterned droplets where it is visible that there are three droplets with large contact angles in an array of thin droplets with a very small contact angle and (b) anti-droplets. (c,d) When cycling the nematic isotropic phase, a number of the very thin drops collapse down into a crescent that is confined at the hydrophobic-hydrophilic interface. The broken symmetry appears to be an effect of the direction of heat flow, which is from the right. (e,f) The spontaneous collapse of the droplets under the constraint of constant volume is shown schematically.

crescents buckle in the same direction and that the heating occurs from the right in Figure

3.11(c,d).

3.5.3 Solvent Evaporation

First, 5CB and IPA were mixed in various ratios and placed onto HDT on MDA

µcp surfaces. As the IPA evaporated, 5CB droplets were left behind on the HDT patterns, representatively shown in Figure 3.12(a-d). Since this is a higher global energy state than if the 5CB had settled on the MDA areas, our solvent evaporation technique must have provided a more complex energy minimization process.

We hypothesize that the first layer of solution separates on the surface, with IPA molecules preferentially sitting on the MDA surface with their alcohol groups pointing into the surface and the 5CB molecules forced to sit on the HDT patterns. The rest of the dynamics becomes dictated by surface tension, generating spherical caps of 5CB on the

HDT patterns as the IPA evaporates. However, more work must be done before we begin to understand this process.

Due to the time dependent contact angle of 5CB on HDT, we also observed the unusual occurrence of droplets initially confined to the pattern spreading out into the background material. The initial contact angle was close to that on the pure HDT surface

(Figure 3.12(g)). Since the contact angle on HDT changes with time, the droplet would attempt to flatten on the surface due to constant volume constraints (Figure 3.12(h)).

When the contacting surface area exceeds the patterned domain, the droplet will very

suddenly spread across the MDA surface to take on the new contact angle (Figure

3.12(i)). We did not observe this effect with ODT ink.

Figure 3.12 Representative micropatterned areas after solvent evaporation using

5CB:IPA mixtures of (a) 1:1000 and (c) 1:100 by volume evaporated at room temperature. (c) Close up of a 1:500 mixture results with (d) corresponding horizontal imaging. (e,f) Time dependent flattening shown in two images: (e) upon initial formation and (f) 5 days later. (g-i) A schematic of droplet flattening for our system under constant volume.

In general, this method showed promise but required precise control of evaporation rate and a careful tuning of the 5CB:IPA ratio for each desired droplet volume based on contact angle, pattern size and coverage.

3.5.4 Dragged Drop

In order to more reliably self-localize NLCs on chemically patterned surface, we took inspiration from water drops rolling down a window. Drops moving at a certain velocity on a non-ideal surface have an asymmetry between their advancing and receding contact angle [72]. This asymmetry results in an increased stress on the drop that eventually forms a rivulet on its receding side and can eventually nucleate droplets in its wake.

A similar method, termed discontinuous dewetting, has also been used to fill well arrays on a substrate using a fluid-coated glass probe [73]. As the probe moves along the surface, it fills subsequent wells due to surface topology, with the fluid hinging on the edges of the walls.

The formation of rivulets on the receding edge of moving droplets is dictated by the capillary number

휂푈 퐶푎 = 3.23 훾 where 푈 is the velocity of the drop, 휂 is the viscosity of the liquid and 훾 is the surface tension of the air-fluid interface. The exact threshold value for droplet distortion varies in literature.

A chemically patterned surface assists in the rivulet formation, causing the receding contact angle to be smaller on the hydrophilic areas. The forced rivulets create instability in the fluid, pinching off droplets as the bulk moves over a surface.

When a fluid drop is dragged across a chemically patterned surface of hydrophilic domains 훾 on a hydrophobic background 훿, the dynamics of the tail end occurs in three stages. First the three-phase line of the drop takes on the characteristic receding contact angle of the 훿-surface (Figure 3.13(a)). Then as the drop is dragged onto the 훾 domain, the tail end gets pinned at the interface until it is stretched to the receding contact angle of the domain (Figure 3.13(b)). If the 훾 receding contact angle is not reached before the drop has moved sufficiently far from the domain, the drop is elongated until it pinches off and a droplet is left on the patterned region (Figure 3.13(c)). An optical microscope image of this process can be seen in Figure 3.13(e).

We also observed the high and low velocity limits of this process, concluding that at sufficiently low speed ~1μm/s, droplets do not pinch off on the surface. The velocity of the moving drop is proportional to the volume of the pinched off droplets. We also noticed that toward the periphery of the dragged drop, the nucleated droplets tended to have a smaller contact angle, as is shown in Figure 3.13(d).

This method not only gives reproducible results, but also allows us to dictate where the droplets self-localize. By adjusting the size of the dragged drop, we can control the diameter within which smaller droplets nucleate. Smaller drops also nucleate droplets with a more consistent contact angle across the pattern. The method is additionally

Figure 3.13 The dragged-drop method has three stages: (a) The needle is placed on the surface and a controlled amount of LC is pushed onto the surface; (b) A moving surface causes the drop to cover a patterned domain where it gets pinned at the interface; (c) The movement distorts the drop until it pinches off a droplet. (d) A schematic of the nucleated droplets as the needle moves away from the plane of the page showing that the droplets formed on the periphery of the dragged drop tend to have a smaller contact angle. (e) An optical microscope image of the process. (f) A SEM image of LC droplets on a chemically patterned surface. The dark areas on the surface correspond to the MDA ink and the light areas are HDT.

flexible because it allows different materials to be concurrently deposited in parallel lines and can be easily incorporated into roll-to-roll processing.

3.5.5 Droplets on Patterned Domains

On cubic arrays of patterned 훾 domains we are able to observe the difference between droplets that are completely contained within the domain and those that are outside of it, as described in Section 3.1.2. The optical transmission of the droplets is related to the contact angle at small thicknesses.

Figure 3.14(a) shows an SEM image that gives us information about the system.

First, we can see that the MDA domains appear darker due to the presence of oxygen

[74]. We can also see from the blue arrow that some droplets that form do not completely cover the domain, and thus take a much smaller contact angle than those that are designated by the red arrow, which are pinned to the domain boundary. We observed the same behavior with POM (Figure 3.14(b)), where the smaller droplets had a very small transmission due to their small height with comparison to larger drops, which take on a contact angle closer to that on the background. The schematic from earlier is reproduced in Figure 3.14(c-d) to illustrate the contact angle in the three relevant regimes: (c) shows a droplet on the ink, (d) shows a droplet on the background and (e) is the case of a droplet pinned at the boundary.

As is shown in Figure 3.14(b), we could achieve regular droplet arrays using the dragged drop method. The ability to make droplets with a constant contact angle is

limited by the size of the dragged drop, since droplets forming at the periphery had a smaller contact angle than those that formed toward the center.

Figure 3.14 (a) An SEM image of droplets on patterned domains. The darker areas on the pattern correspond to the oxygen in MDA [74]. The blue arrow points to a droplet that doesn’t completely cover the domain while the red arrow shows a droplet that is larger than the patterned domain. (b) A POM image shows this effect on the optical intensity of the droplets. Schematics of a droplet are shown when it is (c) completely within the domain, (d) completely outside the domain and (e) pinned at the interface.

3.6 Internal Director Structures

Droplets have to fulfill certain boundary conditions: first, there is homeotropic anchoring at the top curved surface. This means that a radial configuration is maintained far from the center of the droplet. The second is the underlying surface.

Commonly in μcp surfaces, SAMs tend to form as domains. These domains appear to play a significant role in the defect structures that we observe. The individual droplets we observed fall into one of three categories. The first and most common is a propeller structure, which designates a single +1 defect, as shown in Figure 3.15(b). This configuration is isomorphic to a single hedgehog defect at the center of a free-floating liquid sphere. The offset of the defect from the center can be explained by two separate methods. First, surface pinning of the defect on SAM domains and varies from region to region. The second way of understanding this offset is to consider another type of topology: the case of three separate defects (Figure 3.15(d)). Topological analysis suggests that there are two +1 defects on either side of one -1 defect to give a total topological charge of +1.

Second, by looking at the droplets with 3 defects and considering that there are two +1 defects on either side of a -1 defect, it’s possible to get two defects to annihilate near the center of the droplet, leaving the last defect on the periphery.

The second type of POM droplet texture has a propeller accompanied by a characteristic closed loop that has no net topological charge (Figure 3.15(c)). Quasi-2D analysis suggests that if a certain SAM domain forced a specific orientation, then the

adherence to the outside boundary conditions and an already-pinned +1 defect would twist the director field to fulfill all the constraints. The third type of droplet had three separate defects (Figure 3.15(b)).

Figure 3.15 (a) A representative area of LC droplets. Each droplet found fell in one of the following categories: (b) A single defect, (c) a defect with a characteristic closed loop or (d) three defects.

Due to energy considerations, we can determine that the volume effects, scaling with 푟3, will decrease with respect to surface effects, which scale with 푟2. This means that as the droplet size decreases, the defect will be more predictable from the surface.

Since the air interface has a larger contribution than the SAM surface, we understandably observed that droplets smaller than 20μm showed only a single defect in the center. We

also observed that droplets smaller than 5μm showed no optical activity under crossed polarizers.

The more complex defects were most common in 30μm diameter droplets: out of

318 droplets, 6 (2%) had spirals and 23 (7%) had three defects while only 14 (4%) had a single defect located in the center. In 20μm droplets: out of 212, 1 (0.5%) had a spiral and

5 (2%) had three defects while 50 (24%) had a single defect in the center.

As the diameter of the droplets increases, we can imagine that more defects can appear, with the limiting case of large diameter being a film of NLC.

3.6.1 Defect Position and Annihilation in Ellipsoidal Caps

We used the dragged drop method to self-localize 5CB on ellipsoidal chemical patterns. A larger area POM image is shown in Figure 3.16(a). (b) shows ellipsoids with polarizers at 45º to illustrate the shape of the droplets.

In elongated shapes such as ellipsoids, integer-charge defects align along the long-axis. The number of defects is always governed by the topology of the system, necessitating that n+-n-=1. We observed that the more elongated the ellipsoid, the higher the probability of larger number of defects. The dynamics of droplet formation show that defect lines are being pulled from the drop during the dragged-drop process. The confining geometry also creates a larger effective friction for motion of the defects. In an ellipsoid, the frictional force is based on the cross-sectional area perpendicular to the long axis. This means that in an elongated shape, more defects can be trapped without relieving any of the frictional force that is present in spherical caps. 72

Figure 3.16 (a) Representative areas of LCs self-organized on µcp ellipses. (b) Close up of ellipses with 3 and 5 point defects. (c-f) Time lapse images of ellipses where positive and negative charge colloids move together and annihilate (marked in red).

In large-aspect ratio ellipsoids we did not observe any defect annihilation.

However, in thicker shapes such as those shown in Figure 3.16(c-f) we observed defect annihilation at temperatures near the nematic-isotropic transition (~33°C) over the course of 10 hours.

3.6.2 Spherical Caps of Chiral LCs

We used the dragged drop method to localize the Chiral NLC with a room temperature pitch of 1μm on an MDA on HDT pattern. The resulting structures on droplets larger than 10μm diameter were highly knotted, as we can see in Figure 3.17(a).

Images taken with and without polarizers are shown in Figure 3.17(b,c), respectively.

Figure 3.17 (a) Chiral Nematic LC self-localized on 50µm µcp domains with high magnification (b) POM and (c) unpolarized optical microscope images. While the large droplets had complex defect knots, smaller drops (d) showed a simpler pattern. When we cooled the droplet in (c) from the (e) isotropic to the chiral nematic phase, it underwent

(f-g) a slow wrapping of the director field followed by a very abrupt change into (i) a highly knotted structure.

In the droplets smaller than 10μm shown in Figure 3.17(d), we noticed a distorted cross, suggesting that the pitch of the liquid crystal was much larger than the droplet.

The Chiral NLC provided by Dr. Quan Li has negative temperature dependence for the helical twisting power. While heating the sample caused the director lines to unwind based on the initial knotted configuration, cooling the sample always followed the same initial progression, shown in Figure 3.17(e-i). The droplets first formed a spiral that increased in diameter until they reached a certain radius (Figure 3.17(e-h)), where they suddenly wrinkled into a highly distorted structure (Figure 3.17(i)).

3.7 Simulations of Internal Director Structure

We used our Q-Tensor Monte Carlo code from the previous chapter to study the effect of underlying orientational structures on the final liquid crystal texture.

3.7.1 Effect of Contact Angle and System Size

Contact angle played a very significant role on the resulting defect patterns. At large contact angles, the spherical cap surface had a much greater influence and forced a more radial structure. At very small contact angles, the patterned surface played a significant role and was capable generating textures that were extremely frustrated and exotic.

System size was also an important consideration. Larger systems allowed for greater resolution and smaller distortion energy costs in the simulation. This was an effect

we observed in our experimental results, where droplets below a specific size showed less variability in their final texture.

3.7.2 NLCs on Circular Domains

Since our experimental results showed a previously unpublished case of a characteristic closed loop, we attempted to understand how such a case could arise. We reverse-engineered the surface from the POM texture in Figure 3.18(a) by assuming quasi-2D confinement. We first attempted to create a single oriented domain in the background of a planar degenerate surface as in Figure 3.18(b). The results of the simulation, as well as an experimental image for comparison are shown in Figure

3.18(c,d), respectively.

We observed that the orientation of the domain would pin a +1 defect on a position around the domain-background interface depending on the angle and tilt. By changing certain parameters such as the temperature or the tilt of the domain, we were also able to observe a +1/2 defect loop forming in the droplet and straddling the domain, which is shown in Figure 3.18(e). A comparison to our experimental results is shown in

Figure 3.18(f).

Being able to pin the defect, we increased the number of domains. Since having two domains on the surface cannot topologically change the number of defects, we were able to observe some of the desired effects, allowing us to begin to speculate how the

SAMs were influencing the liquid crystal texture.

Figure 3.18 (a) Image taken from experimental results where a single domain is inferred from 2D analysis of the POM texture. (b) A surface that is generated from analysis of (a).

(c) Final structure of a NLC drop on (b) and (d) a corresponding image taken from experimental results. (e) An intermediate structure on the same surface with (f) corresponding droplets taken from experimental results.

The important parameters are the size of the defect with respect to the size of the drop and the distance between domains. When the domains are sufficiently close, the effects cancel and a single defect results.

3.7.3 SAM Surfaces

Modeling SAM Adsorption

There are a number of adsorption models for the formation of SAMs [75, 76]. We took advantage of the sticking coefficient introduced in the Kisliuk model [77] to create

SAM textures that exhibit domain growth.

Monte Carlo SAM Surfaces

We modeled Self-assembled monolayers using a sticking parameter. In our model, there is a higher probability of a molecule sticking to the surface if it is adjacent to a molecule that has already adsorbed on the surface.

Our simulation first places a vector at a random empty site on our surface. If the selected point has no occupied neighbors, it sticks with a probability p and takes on a certain orientation. The orientation is determined by setting a fixed angle θ from the surface, and allowing a random orientation of φ. If the selected point has one neighbor, it occupies the site with p=1 and takes on the orientation of the neighboring point. For sites with multiple neighbors, the orientation is chosen by an equal probability from the adjacent occupied sites. The process is schematically shown in Figure 3.19(a). The number of successfully placed points is recorded as n. To fill the entire surface, we constrain n to be the number of points on our surface, but in general we can reduce this value to generate empty sites with a null vector orientation.

The sticking parameter had a strong influence on the domain size on our surface, which we shown for values of 0.01, 0.001, 0.0001 and 0.00001 in Figure 3.16(b-e). We can compare our domains to good qualitative agreement with SEM images of SAM surfaces found in literature [59, 78].

Figure 3.19 (a) Schematic of our model of monolayer adsorption, where an incoming molecule has a small probability of sticking to an empty site with no neighbors and a random orientation or a strong probability of sticking to an empty site with a neighbor and taking on its azimuthal orientation. Simulated surface using sticking parameter (a) p=0.01, (b) p=0.001, (c) p=0.0001 and (d) p=0.00001. The color of each region corresponds to the x component of the orientation as designated in the legend.

NLCs on SAMs

Using the Ksiliuk-inspired model to create a surface, we could observe what happens when the domains are randomly shaped and oriented.

Figure 3.20 (a) Surface generated with a sticking parameter p=10-4. (b) The disclination lines (in red) get pinned at the surface at domain boundaries that satisfy +1/2 defect charges and (c) the resulting POM texture. (d) Surface generated with a sticking parameter p=10-5, which shows a generally monodomain surface. (e) The disclination line does not get pinned at the surface due to a lack of domain boundaries and instead stays in the bulk of the NLC droplet. (f) The resulting POM texture, which compares with the experimental texture from literature for NLC droplets on monodomain SAMs.

The limiting cases for this work were the case where the p=1, which is a pretilt degenerate surface (shown in Figure 2.7) and the case where p<<1, which yields a pretilt aligned surface (shown in Figure 4.4). In the case where our sticking parameter is large, the result is that a single +1 defect forms at the center. In the case where the sticking parameter is small, the final orientation has a +1/2 disclination loop pinned at opposite ends of the surface running perpendicular to the alignment direction.

Relaxing the simulation with a domain size at least an order of magnitude larger than the individual director size forced local LC domains that pinned defects at the surface. We consistently observed surface pinning of half-integer defects in systems that with many domain boundaries, which was consistent with our experimental observations shown in Figure 3.18(f).

These surfaces showed the culmination of the other two cases we had observed.

There was a variable formation of large-scale domains and defects that result from domain boundaries.

3.8 Summary

We first explored the contact angle and surface pretilt on various self-assembled monolayers. We showed that it was possible to generate droplets with predefined contact angles using mixed monolayers and noted a temporal change in the contact angle with certain long-chain alkanethiols.

By chemical patterning micron-sized hydrophobic and hydrophilic domains, we were able to self-localize droplet arrays of NLC on surfaces. Droplet morphology was not 81

noticeably influenced by the internal director field above a diameter of 3μm, and instead a product of the surface energies of the domains and the dynamics of droplet self- organization. In exploring methods of self-organizing fluids on surfaces, we found it possible to create highly stressed droplets that would collapse in the presence of surface tension anisotropy caused by a temperature gradient. We were also able to use the complex molecular interactions in solvent evaporation to localize droplets on high energy domains. Finally, we developed a reproducible method of generating droplet arrays on chemical patterns that can be easily incorporated into roll-to-roll processing.

We then looked at the topology and elastic energy effects on the NLC director inside the droplets. Since the droplets were confined by a +1 total topological charge, we experimentally explored the director structure to find several homotopic solutions, generating droplets with a single +1 defect, a single defect with a topologically trivial loop and multiple defects that satisfy the relation 푛+1 − 푛−1 = 1. We were also able to predictably generate defects along a single axis by self-organizing the NLC on ellipsoidal patterns. This technique also made it possible to look at the effective frictional force between defects due to elongated geometry and surface conditions, though we did not pursue this line of study.

We finally developed simulations that allowed us to more closely probe the topology of our system. The simulations were verified using experimental and simulation textures from literature, such as the case of surfaces with planar degenerate, planar aligned and homeotropic anchoring. Our simulation results were also consistent with our

own experimental observations. We saw that the textural diversity was related inversely to the contact angle and proportionally to the size.

By reverse-engineering surfaces based on our experiments, we concluded that the existence of aligning domains can generate the textures we observed. We theorize that the

POM textures we observed are based on the SAM micro-structures, which are in turn guided by adsorption dynamics. We used our simulations to look at NLCs on our model

SAM surfaces, which reproduced some of the textures we observed. Since the process is random, we can only statistically reproduce the possible orientation. We could, however, use the droplet texture to infer the underlying micro-structures of the SAM.

CHAPTER 4

Thermophoresis in Nematic Liquid Crystals

Thermophoresis is the general phenomenon that describes the deterministic motion of particles in a medium due to an induced temperature gradient, which is caused by temperature-dependent properties of said medium. In this chapter, we will explore how the temperature dependence of the elastic free energy cost can produce a very large negative thermophoretic force on colloidal particles in a liquid crystal medium.

4.1 Thermophoresis: Background

When particles suspended in a medium are subjected to a temperature gradient, directional motion of the particles can be induced due to temperature-dependent properties of the system. In the literature this effect has been termed thermodiffusion, thermomigration and the Soret effect, but we will use the common terminology of thermophoresis. Thermophoresis has been observed in many systems [79, 80] since J.

Tyndall first documented a dust-free region around a hot body in 1870 [81]. Positive thermophoresis occurs when particles move from a warm side of a medium to the cold side. Until recently, negative thermophoresis, or thermophilic behavior, was rare to find in the literature [79, 80].

In gases, the thermophoresis occurs as result of anisotropic momentum transfer.

This motion can be described by the Knudsen number

푙 퐾푛 = 4.1 푎 which is the ratio between the mean free path 푙 in the gas and the particle size 푎. The free-molecule regime, identified when 퐾푛 → ∞, is the case when gas molecules transfer their momentum when they reflect off the surface of the particle [80]. Figure 4.1(a) shows higher temperature gas molecules in red striking the top surface of a large colloidal particle. Since the molecules at the bottom have a much smaller velocity, they do not impart enough force to balance the position, and there is a resulting net thermophoretic force, 퐹푡푝. This regime is relevant to understanding aerosols and thermophoresis in these systems is utilized commercially to filter fine particles from heated flowing gas by cooling the surrounding pipe [11]. The case where 퐾푛 → 0 is known as the near- continuum regime, which is treated theoretically using continuum fluid mechanics. In this regime, the medium is so dense that understanding the motion requires solving the

Navier-Stokes equation and the behavior is governed by the ratio of the thermal conductivities of the particle and the medium coupled with the appropriate boundary conditions [80, 82].

Certain solids exhibit phenomena similar to thermophoretic motion. The Seebeck

Effect occurs in many semiconductors when a temperature gradient induces a difference in the mobility of either electrons or electron holes in n- or p-doped semiconductors, respectively. The flow generates a local electromotive field

퐸푒푚푓 = −푆퐵∇푇 4.2

where SB is the Seebeck Coefficient [83].

Figure 4.1 (a) Thermophoresis in the high Knudsen number limit, where motion is induced by anisotropically distributed elastic collisions. The red arrows show molecules with a higher temperature, while the blue arrows are the cooler and more slowly moving molecules. (b) The mobility of charge carriers in semiconductors also has inverse temperature dependence. In the case of thermocouples or thermoelectric generators, N- type and P-type semiconductors are put together so that they generate a net current in the presence of a temperature gradient.

The Seebeck Effect has been utilized for both sensing and energy generation.

When a semiconducting wire experiences a difference in temperature between its two ends, it accumulates charge on the colder side. By combining two oppositely doped semiconductors, the cool side will accumulate both the electrons from the n-doped material and holes from the p-doped material, shown in Figure 4.1(b). The accumulation in charge causes the N-type material to repel electrons in conductive material, which are

drawn to the cold interface of the P-type material. This results in a current that is dependent on the material properties and the temperature gradient [83]. The current can either be measured to determine the temperature at one end (with the other end held at a fixed temperature, or cold-compensated) [12] or used to power an electric circuit [13].

4.1.1 Thermophoresis in Fluids

Thermophoresis in fluids is more complicated than in gases or solids, taking into account multiple temperature-dependent properties that are very system-specific.

Anisotropic momentum transfer and electrodynamic effects are overpowered by gradients in mass density and solvation entropy. Phenomenologically, fluids are also more complex due to certain systems that show negative thermophoresis, which has not been experimentally verified in gases [80].

The traditional way of quantifying thermophoretic phenomena is to compare the deterministic motion induced by the gradient to the Brownian motion of the motile particles. The dynamic viscosity, 휂, will counter any force acting on the particles in a medium, 퐹푡표푡, resulting in a steady-state terminal velocity, 푣, described by the Stokes’ drag relation

퐹 푣 = 푡표푡 4.3 6휋휂푅푝 where 푅푝 is the radius of the particle.

The deterministic motion is quantified by assuming a linear dependence of the velocity of the particle on the temperature gradient, ∇푇,

푣 = −퐷푇∇푇 4.4

where the scaling factor 퐷푇 is known as the thermodiffusion coefficient [79]. This value can be compared to the magnitude of the diffusion coefficient, 퐷, for systems with low

Reynolds number:

푘 푇 퐷 = 퐵 4.5 6휋휂푅푝 where 푘퐵 is the Boltzmann constant and 푇 is the local temperature. The ratio of these two values is known as the Soret coefficient

퐷 푆 ≡ 푇 4.6 푇 퐷 Thermophoretic systems with a dominant mass density gradient can be theoretically analyzed using the Navier-Stokes equation [82]. In this case, the thermodiffusion coefficient takes the form

휅푙훽 4.7 퐷푡 = 휅푝 휌푐푝(1+ ) 2휅푙 where 휅푙and 휅푝are the thermal conductivities of the particle and liquid, respectively; 훽 is the thermal expansion coefficient of the liquid, 휌 is the density of the liquid and 푐푝 is the heat capacity of the particle. This model has been verified using laser-heated systems of

1.6µm silica microspheres in propylene glycol for gradients of 0-13,000K/m [84]. The effective force acting on the spheres can be determined from the thermodiffusion coefficient by

Δ푇 퐹 = −6휋휂(푇)푅 퐷 4.8 푡푝 푝 푡 Δ푧 where 푅푝 is the radius of the particle and 휂(푇) is the temperature-dependent viscosity.

Systems dominated by gradients in solvation entropy are generally aqueous solutions where the thermodiffusion coefficient depends on both the temperature- sensitive entropy of water hydration and the entropy of ionic shielding

2 퐷퐴 훽휎푒푓푓 4.9 퐷푇 = (−푠ℎ푦푑 + 휆퐷퐻) 푘퐵푇 4휀휀0푇 where 퐴 is the particle surface, 푠ℎ푦푑 is the hydration entropy of water, 휎푒푓푓 is the effective surface charge density and 휆퐷퐻 is the Debye length [85]. In this case electrophoresis can also be induced by a difference in the Soret coefficients for ions in solution, resulting in an electric field across the medium [86].

4.1.2 Temperature-Dependent Thermophoresis

In aqueous salt solutions, nonlinear dependence of the velocity of polystyrene microspheres on the temperature gradient was reported for larger particles and smaller salt concentrations. These conditions resulted in a decrease in the thermodiffusion coefficient as the gradient increased [87].

While positive thermophoresis is present in all documented fluid systems, comparatively small negative thermophoresis can be present in certain temperature ranges [88-91]. In some systems, the Soret coefficient crosses zero and becomes negative below a defined temperature. In aqueous systems, this usually occurs around 4ºC, where the temperature dependence of water density abruptly changes sign [79]. The point of this 89

low-temperature sign inversion becomes more system specific when dealing with DNA

[85], proteins [92, 93] and surfactants [94], where solvation entropy primarily drives motion [79]. A unique crossover from positive to negative thermophoresis upon increasing temperature in Poly(N-isopropylacrylamide) has also been documented [90].

In these cases, the Soret coefficient is modeled using an empirical fitting function

∗ ∞ 푇 −푇 4.10 푆푇(푇) = 푆푇 [1 − 푒푥푝 ( )] 푇0

∞ ∗ where 푆푇 is the limit of the Soret coefficient at high temperatures, 푇 is the temperature at which systems go from positive to negative thermophoresis, and 푇0 is a scaling factor that depends on the strength of temperature dependence. This dependence is in good agreement with biomolecular systems [86, 94, 95].

In lysozyme solutions, the thermodiffusion coefficient scales linearly with temperature [93]. These systems show temperature dependence in their mass diffusion coefficient, 퐷, that stems from the temperature-dependent viscosity. The hydrodynamic radius, given by

푘 푇 푅 = 퐵 4.11 푎푝푝 6휋휂퐷

remains constant.

For systems that change their direction of motion, the temperature dependence of the thermodiffusion coefficient is

∗ 퐷푇(푇) = 퐴(푇 − 푇 ) 4.12 where 퐴 is a system-dependent constant [94].

4.1.3 Applications

Thermal Field-Flow Fractionation (ThFFF)

Early work on thermophoresis in fluids was driven by the development of thermal field-flow fractionation (ThFFF) [16, 96-100]. Field-Flow Fractionation (FFF) methods allow for the measurement of particle mass, size, density, charge, diffusivity and absorbed layer thickness. ThFFF proves both a fruitful way of separating synthetic polymers from organic solvents and a method of determining the thermodiffusion coefficient for particles ranging from molecules to colloids [99].

Figure 4.2 Thermal Field-Flow Fractionation (ThFFF) is the method in which a temperature gradient across a flowing liquid in a channel separates molecules by their size. On the left side particles are dispersed in a medium at a constant temperature.

When a temperature gradient is applied and perpendicular parabolic flow is induced, like particles will settle into bands depending on their diffusion 푫 and thermodiffusion

푫푻 coefficients and travel at different velocities through the channel.

FFF measurements are characterized by the retention ratio 푅, which is the ratio of the molecules in the medium to the average velocity of the medium. Under the assumption that fluid velocity, 푣, in a channel of height ℎ has a parabolic profile

푦 푦 2 4.13 푣(푦) = 6〈푣〉 [ − ( ) ] ℎ ℎ where 푦 is the displacement from the wall and 〈푣〉 is the average velocity of the fluid.

When in equilibrium, the concentration profile, 푐(푦), is given by

푦 푐(푦) = 푐 푒푥푝 [− ] 4.14 0 휆ℎ where 휆 is the retention parameter [100].

As a result, molecules travel at a velocity dictated by the retention parameter, which is dependent on the inverse Soret coefficient and inverse temperature difference

Δ푇 between the hot wall 푇ℎ and cold wall 푇푐 in ThFFF [16]

−1 퐷 4.15 휆 = [푆푇ΔT] = 퐷푇(푇ℎ−푇푐)

There is an additional velocity distortion factor 휈 that comes from the temperature dependence of the fluid viscosity, giving a retention value of [97, 98]

푉푧표푛푒 〈푐(푦)푣(푦)〉 1 4.16 푅 = = ≈ 6휆 [휈 + (1 − 6휆휈) [coth − 2휆]] 〈푣〉 〈푐(푦)〉〈푣(푦)〉 2휆

The time-dependent mass density can be measured using spectrometric detection to give peaks for each velocity band of particles [98].

Commercial FFF systems are available from Postnova [101].

MicroScale Thermophoresis (MST)

Rather than using a constant temperature gradient, MicroScale Thermophoresis

[15] uses laser heating to generate powerful localized temperature gradients. The local gradient causes particles to move away from the heat source depending on their thermodiffusion coefficient 퐷푇. When the light source is turned off, the particles return depending on their diffusion coefficient 퐷. By looking at the spectrometric intensity, it is possible to look at both the fall time of the optical system, which occurs when all the particles move out of the way of the laser and the rise time when the laser is turned off, giving a measurement of the Soret coefficient.

Figure 4.3 MicroScale Thermophoresis: (a) In the initial state, particles are evenly dispersed in the medium. (b) When a laser is used to locally heat the medium, the particles move away from the heat source proportional to their thermodiffusion coefficient 푫푻 until (c) they create a void region in the medium. (d) When the laser is turned off, the particles return into the void region depending on their diffusion coefficient 푫.

Because the Soret coefficient is dependent on solvation entropy, it is possible to utilize MicroScale Thermophoresis to detect biomolecular interactions. By monitoring the motion of molecules in these systems, it is possible to detect changes in hydration shell, charge or size in the range of small-molecule binding to multi-protein complex interactions [102]. Commercial systems are available from NanoTemper Technologies.

4.2 Colloids in Nematic Liquid Crystals: Background

Liquid crystal media, by their definitions, are fluids that have broken symmetry due to molecular alignment. This symmetry breaking leads to anisotropic bulk properties such as birefringence wherein the electron mobility within aligned molecules shows a strong preference along a specific direction, which is utilized in Liquid Crystal Displays.

One of the consequences of long-range molecular orientation is anisotropy of the viscosity coefficients. In nematic media, resistance to motion becomes dependent on the orientation of rod-shaped molecules with relation to flow [103]. In general, less work is done to slide molecules along their long axis than to push them along the short axis.

As a result of long-range molecular ordering, other properties also show strong directional dependence. Numerous works have classified the anisotropic characteristics of

LC viscosity [104], diffusion [105], thermal conductivity [106], and thermal expansion coefficient [107].

The ordering of liquid crystal molecules also contributes to long-range interactions in the medium mediated by an elastic energy. Objects within this topological

landscape are strongly influenced by forces that are intrinsically different than their counterparts in isotropic media.

4.2.1 Colloidal Liquid Crystals

When a particle is immersed in liquid crystal, the extent to which it distorts the local director field is dependent on the Rapini-Papoular potential, derived from Equation

2.12 as

1 푓 = 푊 sin2(휃 − 휃 ) 4.17 푎푛푐ℎ 2 푎 0 where θ0 is the angle between the easy axis of the liquid crystal and the surface normal and Wa is the polar anchoring coefficient. The de Gennes-Kleman length is defined as the ratio of bulk to surface forces:

퐾 4.18 휆푑퐺퐾 = 푊푎

In many liquid crystal systems, such as nCB on glass, DMOAP or Polyimide, the de Gennes-Kleman length is on the order of microns [108]. This definition is useful in determining to what extent the LC director field is influenced by the presence of an immersed particle. If the size of the object a<<λdGK, there is no disruption of the director field. With sufficiently small particles, such as tracer particles, the diffusion is consistent with self-diffusion of the LC molecules [109]. In the case where a~λdGK, there is weak anchoring of the liquid crystal. Strong anchoring can be assumed when a>>λdGK and is the case where the first layer of liquid crystal takes on the exact configuration of the contacting surface.

The topological arguments we made in Section 2.1 become more nuanced with inclusions in a liquid crystal medium. By our reasoning, we concluded that defects will add and subtract, leading to a final state that is a single defect with the total topological charge of the system. Colloidal inclusions are internally confining surfaces that can be moved by the medium. Thus the colloids, depending on their geometry, can produce further defects in the LC. Furthermore, the colloids can pack together in certain ways or repel to minimize their energy cost in the medium.

Colloids in NLC

There are two common cases for colloids due to their experimental prevalence and theoretical simplicity: homeotropic and planar alignment, where the anchoring direction and the surface normal are parallel and perpendicular, respectively. These two distinct orientations have a fundamental difference in how they interact with a nematic medium.

Planar aligned colloids force the formation of surface-confined point vortices, or the now ubiquitous boojum defects [110]. By generating these surface confined defects, the colloids have no net charge as observed by the medium. They do, however, distort the nematic director proportional to their size.

Homeotropic colloids generate volume-confined defects. In a uniformly aligned volume there is zero total charge, meaning that a +1 topological defect necessitates an additional -1 charge in the bulk medium. In terms of the electrostatic analog, this results in either a dipolar or quadrupolar structure. The dipolar structure is the case where the 96

accompanying defect is a hyperbolic hedgehog. The vector nature of the dipolar structure means there is a spontaneous symmetry breaking in the liquid crystal and the accompanying defect must choose a side along the system’s director alignment direction, which can be seen in Figure 4.4 (a,b).

Another possible configuration for homeotropic colloids is to generate a quarupolar structure, which involves the generation of a +1/2 semi-integer loop commonly called a Saturn ring configuration. This is the case shown in Figure 4.4 (c).

Figure 4.4 Schematic drawings, optical microscope and POM images of homeotropically anchored colloids in configurations that result in (a,b) dipolar configuration with a single hedgehog defect and (c) a quadrupolar configuration with a Saturn ring in the surrounding director field. The optical and polarized optical microscopic images are taken from the samples used in this work.

The difference between these two configurations can be seen as an energy difference. With strong anchoring on micron-sized colloids, a dipolar structure is almost always observed. For example, thick coatings of dimethyloctadecyl[3-

(trimethoxysilyl)propyl] ammonium chloride (DMOAP), which induces strong homeotropic anchoring, result in the formation of dipolar structures. By reducing the concentration of DMOAP, the percentage of the colloids with quadrupolar structures increases. In many strongly anchored systems, the hyperbolic hedgehog is the most common defect formation. However, there are situations, such as when the saddle-splay constant K24 is large, when the Saturn ring is stable even with strong anchoring [111].

When anisotropic colloids are immersed in liquid crystal, they will orient due to the director field. This self-alignment occurs with systems of rod-shaped colloids, where both dipolar and quadrupolar configurations occur [112, 113].

For colloids in motion, it is useful to define the Ericksen number of the system, which is a ratio of the viscous and elastic forces

휂푣푎 퐸푟 = 4.19 퐾 where η is the viscosity of the fluid, v is the velocity of the colloid, a is the length scale and K is the elastic constant of the liquid crystal. In systems with low Ericksen number the flow does not strongly affect the director field.

Brownian Motion

The director configuration around a colloid becomes instrumental in the phenomenology of its motion. Brownian motion, which is a metric for interactions between an object and its surrounding medium in the absence of external forces, is fundamentally different in the presence of a LC medium. The first major difference is the anisotropy of the diffusion coefficient. Due to the anisotropic viscosity [114], homeotropic quadrupolar [115] and planar colloids [116] will show preferential diffusion along the nematic director. Colloids in NLC also exhibit anomalous diffusion [117]. In the dipolar configuration, the field’s memory effects result in colloidal superdiffusion toward the accompanying hedgehog defect. For colloids with planar alignment, Brownian kicks are opposed by an axisymmetric restoring force, which results in subdiffusion.

The diffusion phenomena can be explained by anisotropic viscous drag [118], which opposes motion and is inherently anisotropic due to its basis on the fluid viscosity

[103]. In response to an external stimulus, there is generally a resistance tensor around the colloid. Quadrupolar structures have two viscous drag components, along the director and perpendicular to it. Dipolar configurations are more complex and can generally be described by three drag coefficients.

Sedimentation

Another area where resistance tensors play a pivotal role is in the sedimentation of colloids off-axis. Sedimentation of colloids is unique for two reasons: (1) unlike phoretic phenomena, sedimentation is a force applied on the bulk of a system and not the 99

surface [119]; (2) gravitation does not influence the orientation of the NLC molecules. As a result, the liquid crystal director can be oriented as an angle from the force acting on the colloids. When gravitational force acts at some angle θ to the nematic director, the trajectory of the colloid will travel at an angle φ, where 0<φ<θ [120].

Electrokinetics

Electrokinetic phenomena in LC media are also unique. Colloids experience nonlinear electrophoresis in an alternating electric field [20]. As a result, the net motion of dipolar colloids is directional in an alternating field, toward the accompanying hyperbolic hedgehog defect [21]. The significance of this is that the broken for-aft symmetry of the colloid in the liquid crystal leads to a different velocity depending on whether the field at the colloid is pointing toward or away from the accompanying defect.

Aggregation

In the case of homeotropically anchored colloid pairs, colloids with strong anchoring generate hyperbolic hedgehog defects along the nematic director. These defects eventually mediate the colloids to form chains parallel to the director. Pair separation is linearly dependent on the elastic anisotropy and the inverse of the anchoring strength [121]. There is also a strong dependence on cell thickness and temperature, where, with large enough cells, the force scales to a universal curve [122]. The effective

K constant in this case is consistent with the splay constant K11 [123]. In the case of particles of different sizes, the attractive force is dependent on the arrangement of

100

accompanying hyperbolic hedgehogs, with the greater force resulting when the defect between the particles is joined to the larger particle [124]. In accordance with the electrostatic approximation, colloids in nematic media with hyperbolic hedgehog defects interact by parallel dipole alignment attraction or anti-parallel dipole alignment repulsion.

When one colloid shows a Saturn ring configuration, the forces crosses over from attractive to repulsive as the interparticle distance increases [125].

Forced collisions between colloids can cause a distinct kind of aggregation that does not appear in the static case. By observing collisions in a strong alternating field, it can be observed that when colloids collide with their defects head-to-head, they weave together transverse to the alignment direction [126].

Planar aligned colloids generate a pair of boojum defects on either pole parallel to the director. This leads to interesting aggregation behavior, wherein the colloids form chains of colloids at an angle of 30° to the director [127, 128]. The behavior is mediated by the director field, which attracts when the angle between the director and the shortest particle distance is below 70° and repels between 75-90° [129].

All of these elastically induced intercolloidal phenomena can be explained by the theory of elastic interactions [130], with the more general case of arbitrary colloidal particles also present [131].

4.2.2 Observations of Motion by Elastic Forces

The motion of colloids due to elastic field gradients in liquid crystal systems has been demonstrated in a few select cases in literature. 101

An order parameter gradient can be induced optically in LC systems with azobenzene molecules. When the azobenzene molecules are illuminated by UV light, they change their configuration from trans to cis, breaking the local order of cyanobiphenyl liquid crystal molecules. An incident UV illumination gradient can therefore induce motion as the liquid crystal molecules attempt to minimize their free energy in the system. This system has been implemented as a molecular manipulator to vary the spatial variation of polymer concentration [132].

Two methods have been developed thus far to study thermally induced motion:

(1) abrupt temperature gradients result in anisotropic thermal expansion has been used to drive colloidal motion. The molecules in the warm area of the cell expand while the molecules in the cool area contract, resulting in a flow in the medium and a director reorientation effect that drives motion toward the colder region of the cell [133]; (2) laser heating can induce a large local temperature gradient. Local heating generates an isotropic region in the liquid crystal, attracting colloids in the vicinity to move into the nematic-isotropic interface [18, 134]. For colloids caught in the nematic-isotropic interface, the motion is 2D in the plane of the interface and the drag force is very small due to director perturbations near the bulk [135] and can be dragged with a moving interface if they are sufficiently large [136].

In systems at constant temperature and order parameter, we also see elastically induced motion. Not only are colloids attracted to each other, as stated before, but they are also attracted to defects and disclinations [137]. By properly tuning surface

102

boundaries, stable disclination lines can form and trap colloids in a single track. Colloids are also generally repelled from walls by elastic repulsion [138]. Due to the higher elastic distortion generated by larger particles, levitation increases with size.

Motion along an elastic gradient is not limited to liquid crystal systems. For example, patterned motion of droplets has been achieved by tuning surface rigidity [139].

Live cells are also prone to migration using durotaxis due to structural properties of the extracellular matrix. Most cells migrate to higher rigidity regions [140].

4.3 Motivations

Due to the broken symmetry of liquid crystalline media, we theorized that the energy considerations would be much larger and fundamentally different from those that dominate thermophoresis in isotropic fluids. We therefore intend to combine liquid crystal knowledge under the guise of thermophoresis to aid in the improvement of applications and development of new technology.

4.4 Methods

4.4.1 Cell Fabrication

We first induce strong alignment on colloids to maximize their energy effects in

NLC media. 5um polystyrene colloids are immersed in a dilute solution of octadecyldimethyl(3-trimethoxysilylpropyl) ammonium chloride, or DMOAP. The

DMOAP preferentially coated the colloids. To remove the excess DMOAP, the solution is centrifuged and decanted several times, leaving only a mixture of DMOAP-coated

103

colloids in water. Once the water has evaporated, we place the colloid powder into a vial of 5CB and, using an ultrasonic cleaner, evenly disperse the colloids in the LC.

To provide strong planar alignment of LCs in our samples, we use the cleanroom standard PI2555. First, a dilute mixture of PI2555 is applied onto a clean glass surface and spincoated at 3000rpm. The solvent is removed by placing the cell onto a 90oC hot plate, and then subsequently baking at 250oC for 1 hour. The surface is then rubbed 10 times with the appropriate felt block, as designated by LCI cleanroom procedure [141].

We made thin cells using 25µm fiber spacers, as documented in Section 3.2.5. In the case of homeotropically treated colloids, we can dictate the position of the accompanying hyperbolic hedgehog by having both pieces of glass rubbed in the same direction and filling along the rubbing direction. The cells are capillary filled with the

5CB-colloid solution by placing a small amount of the fluid on the opening of the sample.

4.4.2 Gradient Heat Stage

A custom gradient heat stage was designed and built to measure thermophoretic forces. Two sets of Peltier elements were placed above and below a sample slot. By inverting polarities and connecting the elements on each side in series, we were able to generate a uniform temperature on each side of the stage. The temperatures of the elements were monitored using thermocouples placed in thermally conductive slabs between the Peltier elements and the sample surface. The labeled thermocouples 푇0, 푇1,

푇2 and 푇3 correspond to the temperature readings on the cold bottom, cold top, warm bottom and warm top, respectively. 104

The two sets of Peltier elements were then connected in parallel to a variable current power source. As a result, applying a single current through the stage gave a set temperature gradient that was dependent on room temperature. A schematic of the heat flow for the designed stage is shown in Figure 4.5(a). The final design was constructed from copper, which has a very high thermal conductivity and would thus result in a higher efficiency of heat flow. An image of the heat stage within a home-built optical setup is shown in Figure 4.5(b).

In order to understand the temperature distribution of a cell placed inside the heat stage, we used an IR camera to image the top glass surface. Since glass is opaque in the

IR spectrum and has a well-documented emissivity coefficient close to 1, we were able to measure a reliable temperature of the surface. Since the fluid is sandwiched between two glass surfaces, it can be safely assumed that after some time, the gradient of the glass on the surface should equal that of the fluid in the cell. To test the effect of cell thickness, we looked at the surface temperature profile of three cells with varying thickness: 25µm,

50µm and 75µm, which are shown in Figure 4.5(c), (d) and (e), respectively. From these thermal images we could observe two things. The first is that the gradient was affected by the placement of the thermoelectric elements. This can be seen in the heat bump in Figure

4.5(c), (d) and (e). However, far from the temperature reservoirs, the profile was linear and can be plotted against an assumed linear gradient between the two temperature reservoirs, as is shown in Figure 4.5(f).

105

Figure 4.5 (a) The heat stage design with color representing local heating and cooling.

(b) An image of the final stage an the optical setup. Thermal images of the stage with sample thicknesses (c) 25µm, (d) 50µm and (e) 75µm, where the bottom images are close-up images of the sample surface. (f) The gradient taken from the IR readings compared to a linear fit between thermocouple readings. (g) The calculated temperature gradient from IR image at each point in (f).

Using the temperature at each point in the cell, we could also determine the gradient at each point, shown in Figure 4.5(g). The salient feature in the gradient is that it is much larger on the cool end of the cell and much smaller on the warmer end.

106

We used Figure 4.5(f,g) to convert from the position in the cell to the effective temperature and gradient by scaling the graphs to the size and temperature reservoirs in each individual setup.

4.4.3 Particle Tracking

Due to the relatively slow motion of the colloids and the high memory demand for the resulting large data sets, the particle tracking was done it two separate ways. The first was the generation of videos that were analyzed using the open-source software Tracker

4.87. To maintain high visual resolution, the shutter was set to take an image every few seconds or minutes, depending on the system. While this gave important information about defect location and long-term velocity, it did not provide insight into the Brownian motion of the colloids.

For examining Brownian motion, higher temporal resolution was required.

Particle tracking was done using LabVIEW. We generated a program to take a thresholded binary image of a sphere and, using its circumference, find the center. By determining the position over time, we can determine the mean squared displacement to calculate velocity and diffusion

2 2 2 〈(푥 − 푥0) 〉 = 2퐷푡 + 푣 푡 4.20 in real time and write them to a .txt file for further analysis.

107

4.5 Control Experiments: Negative Thermophoresis in NLCs

We performed preliminary testing to see if elastic effects would play an apparent role in thermophoretic forces in anisotropic media. The temperature gradients used were between 0.002-0.003°C/μm for a total temperature change of approximately 0.01°C across the colloid. A cross-sectional schematic is shown in Figure 4.6(a) with the y-axis appropriately labeled as pointing to the cold side.

The liquid crystal used in our studies was the well-documented 4-cyano-4’- pentylbiphenyl, or 5CB. We first tested colloid motion in an 85% glycol solution, which was viscosity-matched to 5CB. We observed that both silica (Si) and polystyrene (PS) colloids moved against the temperature gradient, consistent with results found in literature. The properties of the Si and PS colloids shown in Table 4.1 were taken from data sheets or literature, where cited. A sample trajectory of a 5μm colloid in the 85% glycol solution is shown in Figure 4.6(b).

Table 4.1 Properties of Colloidal Microspheres

Si PS

Density (g/cm3) 2.0 1.1

Thermal Conductivity (W/mK) 1.4[142] 0.033

Specific Heat (kJ/(kg K)) 0.6[143] 1.3

We also observed that PS colloids, which were lighter than their Si counterparts, had a much smaller effect for the same thermal gradient, as can be seen in Table 4.2.

108

We then tested the colloids in the LC 5CB. Initially we tested PS colloids in the warmer region of the cell, where we expected the elastic gradient would be the highest.

The trajectory of the colloids was along the gradient, showing negative thermophoresis.

Furthermore, we calculated the velocity as being five times greater than what we measured in the isotropic medium for the same temperature gradient.

Figure 4.6 (a) A cross-sectional schematic of the glass cell filled with colloidal solutions in the gradient heat stage. The +y axis is defined as pointing toward the cold side of the cell. Sample trajectories of uncoated 5μm Si colloids moving through (b) a solution of

85% glycol at 26ºC, (c) the 26ºC region and (d) 30ºC region of a 5CB cell with the same gradient.

In the case of Si colloids, we observed that they traveled slower in a similar warm region of the cell. As we would expect from a greater positive thermophoretic contribution, the magnitude of the velocity was smaller for Si colloids.

Finally, we looked at the difference between colloids moving in the warmer section of the LC and those in the colder areas. Due to the nonlinear relationship between 109

the elastic constant and the temperature, we expected to see a much larger effect in the warmer areas of the cell. We observed that the magnitude of the velocity was over three times greater in the warmer area of the cell than in the cold region as can be seen in the red and blue trajectories in Figure 4.6(c,d).

A quantitative summary of the colloidal velocities is shown in Table 4.2.

Table 4.2 Colloid Velocities

Silica Polystyrene

85% Glycol, 푻~ퟐퟔ℃ 5.8μm/hr 2.5μm/hr

5CB, 푻~ퟐퟔ℃ -4.1μm/hr -

5CB, 푻~ퟑퟎ℃ -13.7μm/hr -16.4μm/hr

From these initial studies, we concluded that in NLCs the dominant thermophoretic force was fundamentally different from isotropic systems. There was an obvious change in the sign of the thermodiffusion coefficient and very strong temperature dependence.

4.6 Thermophoretic Motion of Colloids in NLCs

We observed the dynamics of 5µm Si colloids in the presence of a temperature gradient in the low Ericksen number regime. We first began tracking the colloids and applied a temperature gradient. The displacement results for over 4000s are shown alongside the temperature readings from each of the four corners of the stage in Figure

4.7(b). The first effect we see is a fast motion against the temperature gradient that results from thermal expansivity of the medium and scales with time-derivative of the 110

temperature [133]. It shows an effect in both the direction against the gradient and in the perpendicular direction moving the colloid lateral to the center of the cell. We then observed a slower motion along the gradient, which is a quasi-equilibrium effect due to a stable temperature gradient.

The far-left panel of Figure 4.7(c) shows the initially diffusion of the colloids before a temperature gradient it applied. The equilibrating temperature is a result of cycling the gradient on and off. The diffusive regime of homeotropically anchored colloids has been extensively studied in literature [114, 115, 120] and we did not analyze it here.

The central panel of Figure 4.7(c) shows the regime where thermal expansion of the medium drives motion and has also been shown in literature, although not in our particular geometry. As we apply a temperature gradient, the medium on the warmer side of our cell expands. Contraction at the other end of the cell drives fluid flow due to the incompressibility of liquid crystal in a closed volume. As a result, we see a very rapid motion along the temperature gradient, as is documented in literature [133].

The final designation in the colloid trajectory is a slow motion against the temperature gradient driven by thermophoretic forces in the far-right panel of Figure

4.7(c). In general, thermophoretic forces can be broken down, with significant contribution from osmotic flow and thermal effects [144]. Due to the nature of the motion, it appears that elastic energy plays the dominant role, a contribution we term elastophoresis.

111

Figure 4.7 (a) Motion of colloids in NLC is driven by thermal expansion of the LC molecules at the warm side of the cell (thermocouples T2 and T3) and contraction on the cold side (thermocouples T0 and T1) as well as thermophoretic forces. (b) When a temperature gradient is induced, motion is dominated by three forces. (c) In the diffusion regime, the temperature effects are negligible and Brownian motion dominates. In the thermal expansion regime, the motion is driven by the expansion and contraction of LC

112

molecules, resulting in motion along +y. Thermophoresis is the dominant force only when a steady temperature gradient is achieved, resulting in motion along -y.

Figure 4.7(a) schematically shows that the thermal expansion of the medium on the warm side of the fluid drive motion toward the colder region while the free energy cost moves the colloid toward the warmer areas. When a gradient is first applied,

푑푇 푑푇 퐹 ( | ) ≫ 퐹 ( | ) resulting in motion of the colloid toward the colder region. 푑푡 푦′ 푑푦 푦′

푑푇 However, the motion reverses as | → 0. In between the two regions, there is a 푑푡 푦′ coexistence regime that we did not study in this work.

4.6.1 Positive and Negative Thermophoresis in 5CB

From the previous experiments, we concluded that Si had the largest effect stemming from non-elastic forces. Since elastic energy cost scales inversely with the size of the colloid while the motion due to traditional thermophoretic forces doesn’t scale with size for moderate temperature gradients [145], we determined that a crossover from positive to negative thermophoresis might be observed by using smaller Si colloid.

To study this effect, we observed a sample cell filled with a solution of uncoated

1.6μm Silica colloids in 5CB and the rubbing direction along the temperature gradient at several positions along the gradient. From the position, we used the curves from Figure

4.5(f,g) to determine the temperature for each data set.

113

Figure 4.8(a) shows an example of the observed trajectory of the colloid taken at

25.9ºC. At small velocities the colloids exhibited behavior that seems to suggest a stick- slip motion. By taking a linear fit of the data, we determined the average velocity with corresponding error, as shown in Figure 4.8(c). From the average position of the colloid we determined the position in the cell and converted it into a temperature using the transformation from Figure 4.5(f).

Figure 4.8 (a,b) The displacement of the colloid and the thermocouple readings over time, which are used to determine the average temperature at the point of measurement.

(c) The velocity of colloids in different areas of the cell, which shows a crossover from positive to negative thermophoresis around 25°C for a gradient of 2.7°C/mm.

114

For the same gradient at different temperatures, we were able to observe a crossover between negative and positive thermophoresis in 5CB around 25°C for a gradient of 0.0027°C/μm.

4.6.2 Homeotropically Anchored Colloids in 5CB

In order to get a better approximation of the energy cost of the colloids, we used

DMOAP coated colloids, which exhibit strong homeotropic anchoring. Our initial study randomly sampled moving colloids in multiple cells with applied temperature gradients between 0.002-0.003°C/μm. Each colloid was followed over 70-90μm. The results are shown in Figure 4.9.

Figure 4.9(a) shows a 3D plot of the data, where there appears to be a general trend that the velocity increases with both temperature and temperature gradient consistent with our theory. The gradient dependence in this range is much more pronounced. From the velocity of the colloids, we could determine the thermodiffusion coefficient using Equation 4.4. Due to a significant amount of noise in the data stemming from system-specific factors, it became more informative to threshold the data into a high gradient and a low gradient regime, shown in Figure 4.9(b). In this figure, the effect of temperature gradient on the thermodiffusion coefficient is evident.

To eliminate any effect from system inhomogeneity such as variable coating, accompanying director field topology or varying ion concentration in the cell we tracked a single colloid through the entire temperature gradient. Furthermore, we looked at the

115

mean displacement and mean squared displacement of the motion to look at the relation between the thermodiffusion coefficient and the Brownian diffusion coefficient.

Figure 4.9 (a) The velocity of various DMOAP-coated silica colloids in a variety of prepared cells plotted against their respective local gradients and temperatures. The color of the points represents the local gradient where red points correspond to a gradient of 3°C/mm and blue points correspond to a gradient of 2°C/mm. There is a visible dependence on both the gradient and the local temperature. (b) The thermodiffusion coefficient calculated from the same points, differentiated by high and low temperature gradients.

Figure 4.10(a) shows an example trajectory of the colloid, where the black lines indicate a linear fit to determine the mean velocity. A linear fit of the trajectory gives a direct measure of the local velocity in each frame (Figure 4.10b). Using Equation 4.4 we determine the thermodiffusion coefficient at a given point in the sample, defined from the

116

average position of the colloid in each measurement regime. From the average position, we calculate the local temperature from Figure 4.5(f). The results are plotted in Figure

4.10(c). From the trajectory, we also determine the mean-squared displacement, or MSD

(Figure 4.10d). To get an accurate measure of the diffusion coefficient, which is dependent on the linear term of the MSD, we first linearized by the velocity and then linearly fit the data using Equation 4.20 (Figure 4.10e). The results for the diffusion coefficient at a given temperature are shown in Figure 4.10(f). At longer times, the calculated diffusion values are consistent with literature, showing a larger diffusion coefficient along the director than perpendicular to it. The short time deviation is likely an effect of system limitations such as vibration of the stage, which was clearly visible at high magnification. The system was not optimized to observe such minor steps and we did not delve deeply into this regime.

The blue points correspond to the constants along the nematic director and gradient. The thermodiffusion coefficient increases nonlinearly in amplitude as the temperature increases, consistent with our theoretical predictions. The Brownian diffusion coefficient also increases with temperature, which is consistent with results found in literature.

The red points correspond to motion perpendicular to the director and the gradient. As mentioned before, there is a non-zero mean displacement of the colloids.

The amplitude appears to be related to both the x and y position but the exact relation was

117

Figure 4.10 (a) Sample trajectory of a single DMOAP-coated Si colloid with an accompanying hyperbolic hedgehog defect in a temperature gradient. (b) By fitting the velocity at each measured point, we could see a negative temperature dependence, which can be used to calculate (c) the thermodiffusion coefficient 푫푻. From the trajectory, we also calculate (d) the mean-squared displacement of the particle, which is (e) linearized by the velocity to give a value for (f) the local diffusion coefficient 푫. The uncertainty in velocity, 푫푻 and 푫 is smaller than the size of the points, as can be inferred from the sample trajectory in (a).

118

not explored in these studies. The diffusion coefficient perpendicular to the director was also observed to be consistent with the literature.

From the thermodiffusion coefficient, we determined that the colloids are experiencing an effective net acceleration from the temperature gradient. This is unusual as in most cases of colloids in solution there is a viscous drag force that opposes mechanical and thermophoretic forces to give a constant velocity. The mass diffusivity also showed strong temperature dependence.

The error on each point that stemmed from mean displacement and MSD fitting was much smaller than the size of the point. The noise present in the data may be caused by inhomogeneity in the medium due to temperature or mass flux.

4.6.3 Theory

The interaction between a colloid and a nematic liquid crystal medium can be approximated in the weak and strong anchoring regimes. The weak anchoring approximation is valid for colloidal liquid crystal systems in the regimes where

−4 2 푊 < 10 푒푟푔푠⁄푐푚 or 푅푝 > 1푚푚. For strong anchoring the limit is 푊 >

−2 2 10 푒푟푔푠⁄푐푚 or 푅푝 < 10휇푚. In these limits, the free energy cost of the colloid in the fluid can be pulled from literature [146]

푓푠 = 13퐾푅푝 4.21

푊2 푓 = 0.2 푅 3 4.22 푤 퐾 푝

119

where 퐾is the frank elastic constant in the single elastic constant approximation, 푊 is the anchoring energy and 푅푝 is the radius of the particle.

2 2 From the Landau expansion, we can approximate 퐾 ≈ 퐾0푆 and 푊 ⁄퐾 ≈ 훾푆 where 푆 is the order parameter [146], which shows temperature dependence

푇 훼 푆 = (1 − ) 4.23 푇∗

From literature, we input the experimentally fit quantities α=0.18, T*=35.3ºC for

5CB [147] and plot the order parameter in Figure 4.11(a).

Figure 4.11 (a) Temperature dependence of the Order Parameter for 5CB. (b) Drift velocity of colloids calculated using both the elastophoretic contribution and the thermophoretic contribution, as modeled by Brenner.

Within a cell, a temperature gradient across the 푧-axis can be approximated as linear between two temperature baths 푇1 and 푇2

120

Δ푇 푇 −푇 = 2 1 4.24 Δ푧 퐿

Δ푇 푇(푧) = 푧 + 푇 4.25 Δ푧 1 which can be combined with the free energy expression to yield

푇(푧) 2훼 4.26 푓 = 13퐾 푅 (1 − ) 푒푝,푠 0 푝 푇∗

푇(푧) 훼 푓 = 0.2훾푅 3 (1 − ) 4.27 푒푝,푤 푝 푇∗ The total force on the colloid in the presence of a temperature gradient is

푑 퐹 = 퐹 + 푓 4.28 푡표푡 푡푝 푑푧 푒푝 By making the assumption that the anchoring on our colloids is strong and that other thermophoretic forces scale with Brenner’s model, we are able to derive an expression for the total thermophoretic velocity of colloids in an LC medium

2훼−1 퐹푡표푡 푑푇 퐷 푇 4.29 푣푑 = = [퐹푡푝 + 26푅푝퐾0 (1 + ∗) ] 6휋휂(푇)푅푝 푑푧 푘퐵푇 푇 where the viscosity 휂 is also a function of temperature

A 3D plot of the drift velocity as a function of temperature and temperature gradient is shown in Figure 4.11(b).

From our theoretical model, we are able to fit our velocity data to Equation 4.29

푇 2훼−1 푎+푏(1− ) 4.30 푇∗ 푣 = ∗ 휂0−휂 푇

121

where a and b are fitting parameters. The result of the fit is shown in red in Figure

4.12(a).

Figure 4.12 (a) The velocity change over temperature of a DMOAP-coated PS colloid moving through a temperature gradient where blue represents the raw data and red is a fit based on Equation 4.30. (b) The Soret coefficient calculated for each temperature.

In the literature, thermophoretic data is often presented using the Soret coefficient

[79], which is defined in Equation 4.6 as the ratio of the thermodiffusion coefficient to the Brownian diffusion coefficient. Our calculated Soret coefficients for each individual point are shown in Figure 4.12(b).

4.6.4 Additional Observations

Effects of Defect Topology

The previous experiments had been conducted without considering the effect of defect topology on the system. We therefore observed instances where, in the same

122

frame, we could observe multiple colloids. An example of such an instance can be shown by the first frame of a video in Figure 4.13(a), where the applied gradient is along –y.

Figure 4.13(b) shows the colloid displacement over a span of 1306.9 seconds normalized to the original position. From these trajectories, we can parse out the x and y components and look at the long term motion. Figure 4.13(c) shows the motion along the gradient of the three different configurations of colloids after the temperature gradient stabilized. Both quadrupolar structures showed almost identical trajectories, which allowed us to contrast the two dipolar structures.

In the dipolar case, the position of the defect proved very important. When the accompanying hyperbolic hedgehog was on the warmer side of the colloid, the long-term motion was significantly slower that the quadrupolar case. Contrasting with this, when the defect was on the colder side the motion was significantly faster than the quadrupolar standard.

The motion perpendicular to the gradient is shown in Figure 4.13(d). In the case where a hyperbolic hedgehog was the accompanying defect, we noticed that the trajectory was more closely aligned along the director. In the case where we observed accompanying Saturn rings, there was significant lateral motion that could be the result of small inhomogeneities in the temperature profile perpendicular to the temperature gradient. This is an effect of the anisotropic diffusion coefficient, which, for Saturn ring configurations is perpendicular to the director field and results in a smaller resistance to lateral forces.

123

Figure 4.13 (a) Crossed-Polarizer image of four colloids in the same proximity. The colloids are characterized by their defect structure: Dipole-Up (DU) has a hyperbolic hedgehog defect along the +y axis; Dipole-Down (DD) has a hyperbolic hedgehog defect along the –y axis; the Quadrupolar structures (Q1 and Q2) have saturn ring defects. (b)

The displacement over time when a temperature gradient is applied. The initial motion along +y and -x is due to thermal expansion effects. The steady state velocity along –y and +x is due to thermophoresis. In the thermophoretic regime, the velocity of the colloids (c) along the temperature gradient and (d) perpendicular to the gradient suggests a dependence on the defect structure. 124

When both kinds of colloidal systems were in the same frame, we measured a reduction in the velocity if the hyperbolic hedgehog was along the motion of 3% for rapid thermal expansion and 11% reduction for thermophoresis. The difference between these two dipolar configurations can be understood in terms of osmotic flow. In the presence of a temperature gradient, it is common for osmotic flow to drive motion [85]. If we consider that flow is moving toward the warmer region of the cell, they would inherently push the colloids with a greater magnitude if the dipole is pointing along the gradient as opposed to against the gradient, which is what is observed in our measurements.

Colloids Near NI Boundary

We did some preliminary observations of the colloids as they moved into the nematic-isotropic boundary. Previous studies had concluded that defects and colloids will preferentially move into this interface to minimize their energy and work has been done to describe the topological mechanics of this movement [18, 135, 136]. We observed that as a colloid with an accompanying hedgehog moves into the boundary as a sufficiently slow velocity, it first decelerates and changes its defect structure to a Saturn ring. For faster moving colloids, this effect was not observed. The defect conformation change can be explained in two ways. The first is the mechanics of pushing a colloid into a homeotropic boundary.

The second reason is that the bend constant K33 decreases with temperature. As a result, the energy cost of a Saturn ring decreases. Just before entering the interface, the colloid slows down, reducing the influence of velocity on the defect structure. These two 125

effects together contribute to a defect conformation change just before a colloid enters the nematic-isotropic boundary.

Director Off-Gradient

We then oriented the cell so that the rubbing director was off-axis with the temperature gradient at an angle θn=35°, as is schematically shown in Figure 4.14(a).

Figure 4.14 (a) An example of the geometry of a colloid moving off-axis to the director.

(b) Example trajectories of a Saturn ring-accompanied (left) and hyperbolic hedgehog- accompanied (right) colloids as they move up the temperature gradient into the nematic- isotropic interface.

In the cold area of the cell, colloids moved almost parallel to the gradient. As they entered warmer regions of the cell, above 30°C, there was a noticeable motion away from

126

the rubbing direction. As the temperature increased, the angle between the director and the trajectory of the colloid approached 90°. This motion is represented in Figure 4.14(b).

While the trajectory of colloids with hedgehog defects and those with Saturn rings were similar in cooler areas of the cell, as they approached the nematic-isotropic boundary, the trajectories diverged, with the Saturn ring colloids making a 90° turn and moving toward the interface along the director. The difference between the two trajectories can in some part be explained by taking into account the difference in the anisotropic diffusion coefficient between the two defect configurations.

In general, the phenomena observed off-gradient were dependent on the location of the colloids in the cell and there is ongoing analysis into the driving mechanisms.

More work is required to isolate and understand this behavior.

4.6.5 Elastophoresis vs Traditional Thermophoresis

The utility of thermophoresis is based on maximizing the Soret coefficient, as both Thermal Field-Flow Fractionation (ThFFF) and MicroScale Thermophoresis (MST) require a difference between the diffusion and thermodiffusion coefficients. By increasing the Soret coefficient, the measurements done using these methods become more sensitive. Due to the comparatively large Soret coefficient we observed in our system, there is a large potential for using NLCs in ThFFF and MST. NLC systems suppress stochastic diffusion of particles without sacrificing the deterministic motion caused by the temperature gradient. Order of magnitude comparisons are shown in Table

4.3. 127

Table 4.3 Comparisons between Elastophoresis and Traditional Thermophoresis

Elastophoresis Thermophoresis

Diffusion Coefficient ~10-3µm2/s ~10-1µm2/s

Thermodiffusion Coefficient ~-10µm2/sK ~10µm2/sK

Soret Coefficient ~-104K-1 ~102K-1

We also note a possibility of sorting molecules by surface anchoring properties.

The utility of type of sorting is fundamentally different from other phoretic effects, as it specifically probes the interaction of the surface with ordered materials. It may be possible to develope Elastic Field-Flow Fractionation techniques based on elastophoretic phenomena.

4.7 Summary

We explored the thermophoretic forces that act on colloids in NLCs. After observing negative thermophoresis, we concluded that the elastic contribution to the thermophoretic force is dominant in our system. We first observed that colloids move in opposite directions when a gradient is applied in an NLC and a viscosity-matched isotropic fluid. We then did a computational feasibility study and developed a theoretical model that suggested a scaling of the velocity with temperature.

We then tuned our system to observe a crossover from positive to negative thermophoresis. The crossover strongly suggests that the elastic driving force, which increases in the warmer areas, was driving the negative thermophoretic contribution. We

128

then monitored a single colloid for the length of the system and were able to confirm our theoretical model.

We also explored the effect of different defect structures on the motion of the colloids. There was a significant difference in velocity between dipolar colloids that pointed along and against the motion, with faster motion observed when the defect was on the colder region around the colloid. There was also a significant difference between the trajectories of dipolar and quadrupolar colloids. While dipolar colloids moved directly along the temperature gradient, quadrupolar colloids had a tendency to move with a significant lateral component.

Finally, we did some preliminary observations of the colloidal motion when the temperature gradient is off-axis with the director field. This motion is difficult to characterize as it appears to have a more complex dependence on the magnitude of the temperature gradient, the angle between the gradient and the director, the local temperature and the defect configuration around the colloid. Consequently, we were able to observe both a trajectory between the gradient and the director as well as perpendicular motion and crossovers between the two.

129

CHAPTER 5

Selective Polymerization of LC Colloids

5.1 Introduction

Tailoring colloidal interactions is an emerging topic of study that includes the generation of colloids with exotic physical and chemical structures, such as Janus particles, patchy particles and lock-key particles [148-150]. More recent works have suggested that controlled aggregation and self-assembly can also be achieved by placing colloids in liquid crystal. Colloid orientation and motion can be directed because, due to director disruption in a lower-symmetry medium, the system will attempt to minimize its free energy. Most of these studies have been conducted using the specialized particles mentioned earlier, which allow for chemical patterning, and thus different molecular orientation. We demonstrate here the microfabrication of liquid crystal colloids that have their own tailored 2D shapes and director profiles.

5.1.1 Self-Folding Structures

Specific design of quasi-2D structures is being extensively studied as a method of creating transformers. A number of methods have been demonstrated that utilize

130

directional swelling, such as confined gels where there is interplay between active and inactive structures, or faces and hinges.

Faces and hinges have been generated in several ways using lithographic methods. Self-folding cube structures can be patterned in 2D using a distinction between the materials that make up panels and hinges. By properly tailoring the hinges, they can be melted in a reflow process, at which point surface tension pulls the panels together, forming a nm-scale 3D structure [151]. Sub-millimeter origami shapes have also been generated using anisotropic swelling induced by photolithographically patterned polymer trilayers [152].

5.1.2 LC Polymers

In recent years there has been significant interest in the theoretical and computational community on self-folding structures made of Liquid Crystal Polymers

(LCP). LCP networks are composed of molecules that have long-range liquid crystalline order but are connected by flexible chains. These networks can be formed with the LC molecules embedded in either the main chain of the polymer or attached as a side chain

LCP [153].

When the system goes from nematic to isotropic, there is contraction of the polymer along the nematic director and expansion in the perpendicular direction. There are two common methods to induce this transition. Adding molecules that change their conformation, such as azobenzenes, can induce a physical disruption of the liquid crystalline order. Under regular conditions, the azo molecules are in a rod-like trans state 131

that mixes well with the nematic order. When the molecules absorb UV light, they transition into a cis configuration, breaking the long-range LC order. Exposing the system to a high intensity visible light source can return to the original trans configuration.

The second method of inducing a phase transition is through local heating of the polymer. Since the temperatures required are usually very high in liquid crystal polymers, it is common practice to dope the polymer with dye molecules that absorb very strongly at a certain wavelength. Upon sufficient exposure, the dye molecules locally heat the liquid crystal molecules and induce a phase transition [25].

5.1.3 Motivations

Recent advances in our lab have allowed for micro-patterning of liquid crystal director fields in 2D using photoalignment. The precise patterning of defects and defect arrays, as was demonstrated by Yubing Guo, is shown in Figure 5.1(a,b). We wanted to combine this with the ability to create photopatterned colloids such as those generated by

Ayan Chakrabarty [154] in Figure 5.1(c). By controlling both the shape of the photopatterned area and the director structure, we were looking to generate bulk colloids that can change shape in response to external stimuli. This method would also allow us to make a variation on the art of micro-origami that is based on folding and cutting: kirigami.

132

Colloidal Transformers

The benefit of high-density colloids in solution is that the system can retain properties of the colloids while still allowing for dynamic processes that result from fluid flow. By inducing a shape transition in flowing colloids using stimuli such as temperature or light it is possible to create active systems that can change their behavior. For example, we can design colloids that break their symmetry under external stimuli. This could result in a jamming transition when colloids passing through a porous medium suddenly become ‘sticky’ by changing from a rod shape to a V-shape.

Figure 5.1 (a) Crossed polarizer images of liquid crystal being aligned by a patterned photoalignment layer with the ability to (b) generate periodic arrays of defect structures

(unpublished results by Yubing Guo). (c) Arrays of boomerang colloid particles fabricated using SU8 before they are lifted from the surface and put into solution [154].

Kirigami

Origami is the art turning 2D surfaces into 3D shapes using only the folding of paper. Elastomer origami is a rapidly developing field in computational and experimental 133

liquid crystal science [155]. The art of kirigami adds another tool for transforming 2D surfaces into 3D shapes: cutting. By cutting and folding, there is a large myriad of 3D shapes that can be achieved. A benefit of kirigami is the ability to generate a much larger surface area 퐴 for biosensor applications or for promoting chemical reactions, which isn’t limited by 퐴 = 2(푙푤 + 푙ℎ + 푤ℎ), where 푙, ℎ and 푤 are the length, height and width, respectively. Furthermore, it becomes possible to generate quasi-lattice structures that can exhibit more exotic behaviors such as a negative Poisson’s ratio. Such methods have already been applied to graphene systems [156] to generate stretchable transistors and remote acutators [157].

5.2 Methods

5.2.1 Materials

We used RM257 (Figure 5.2(a)) as our liquid crystal monomers. When RM257 is polymerized, it forms a main-chain LCP. We also collaborated with Yannian Li to synthesize the azo-molecule 4,4’-bis[6-(acryloyloxy)hexyloxy]azobenzene (2-azo, shown in Figure 5.2(b)), which could be activated using UV light. We used two photoinitiators.

For the pure RM257, we used the UV activated Irgacure 651. For the case where Brilliant

Yellow was used for surface alignment and when 2-azo was used, we used green-light activated Irgacure 784.

134

5.2.2 Cell Filling for LCP Colloids

In our process, we took inspiration from the literature [25]. For UV polymerization, RM257 is mixed with 3wt% of the photoinitiator Irgacure 651 in a test tube at room temperature. The test tube is heated to 130ºC and stirred vigorously in order to evenly distribute the photoinitiator. The solution is scooped out using a lab spatula and placed onto the edge of the prepared 25μm cells, which are sitting on a heat stage set to

100ºC. The monomer solution fills the cell by capillary action. If the solution is cooled without further processing, the monomers recrystallize. This process is reversible without

Figure 5.2 Molecular structure of (a) RM257 and (b) 2-azo where white, grey, red and blue are H, C, O and N, respectively. (c-f) POM images of RM257 mixed with photoinitiator on the hot stage and at room temperature (c,d) before polymerization and

(e,f) after polymerization.

135

a noticeable change in final material properties.

The cell is then exposed to a 366nm light source, where the solution polymerizes.

If the cell is now cooled to room temperature, it retains its optical alignment, as we can see in Figure 5.2. These films allowed us to test solubility of the polymer for development. We observed that RM257 monomers are soluble in toluene while the polymerized RM257 was insoluble.

For visible light polymerization, preparation was done in a darkroom. RM257 was mixed with 1wt% of the photoinitiator Irgacure 784 at room temperature, then heated to

130ºC and stirred vigorously. After capillary filling thin aligned cells as described above, we exposed the cell to the unfiltered microscope light source. The average power at the center of the halogen light source was measured to be 0.8W/m, with a minimum exposure time of 8min.

5.2.3 Photomask Patterning

A photomask was purchased from Photosciences. We first coated a glass pane with an optically reflective layer of nickel and cut it into 1”x1” squares. We then spincoated the positive photoresist S1818 on the squares and exposed them using a UV photolithography system (Karl Suss MJB4), creating a shadow mask for our nickel etchant. The coated glass was then placed in a nickel etchant for 5 minutes under mild agitation. After checking the samples in order to determine that the pattern had been fully formed, we immersed the samples in a KOH stripper to remove the remaining photoresist. The final mask can be seen in Figure 5.3(e). 136

5.2.4 Projection Optical Lithography

Projection lithography was done using a home-made system, shown schematically in Figure 5.3(a). The added flexibility of the system allowed for the incorporation of a heat stage, which was necessary because the melting temperature for RM257 is well above room temperature. The camera in the center allowed for both the alignment of the

Figure 5.3 (a) Custom projection photolithography system in three modes: (b) using a mirror to pass green light through the photomask onto the CCD camera for focusing (c) using a mirror to pass red light to image the sample during cell filling and focusing and

(d) the final configuration where green light is projected through the photomask onto the sample for photopolymerization. We show an example of the (e) mask and (f) cell before exposure.

137

photomask and imaging of the sample.

Figure 5.3(b-d) shows the three operational modes of the system. The first, (b), shows green light being shown through the photomask onto the CCD camera. This is in order to focus and align the mask, which is shown in (e). (c) is the second mode, which uses a red LED to focus and align the sample, shown in (f). The final mode, shown in (d), is the actual photopolymerization step, where the mirror is removed from the mirror cube and the light from the mask is projected onto the sample.

5.3 Results

5.3.1 V-Particle Polymerization

Initial observations suggested that toluene was actively lifting off the polymer particles from the surface. Therefore, to test the development of the samples, drops of toluene were placed on the surface, which allowed us to image the development of the particles. The left side of Figure 5.4 shows optical microscopy images of V-colloids that were polymerized inside an aligned cell.

By imaging the particles through crossed polarizers, as we show on the right in

Figure 5.4, we can verify that the liquid crystal molecules are aligned along the rubbing direction as was preset by the glass cells before filling. The more brightly colored particles are those where the director is at a 45º angle from both the analyzer and polarizer.

138

Figure 5.4 Optical microscope and POM images of V-particles generated using a 50x-5x setup, imaged through a toluene drop on the surface of the sample.

5.3.2 Discussion

There are many variables to consider when developing a photoexposure system.

The important variables that we tuned are temperature, which determines the diffusion coefficient for the reaction, light intensity, which determines the rate of photoinitiator molecules activation, and photoinitiator concentration, which determines the distance between photopolymerization initiation sites.

We also observed the effect of oxygen on the system. Under certain conditions, such as filling cells of 8µm thickness near the crystallization temperature, the effective viscosity of the monomers is very large. This leads to viscous fingering and small inhomogeneities in the surface cause air bubbles to get trapped in the cell. When we photoexposed these areas, we could see that the polymerization did not occur near air bubbles. 139

5.4 Conclusion

We demonstrated the feasibility of generating NLC colloids with a controlled director field using a custom-built projection lithography system. We were able to generate V-shaped particles whose director pattern was dictated by the local cell alignment at the time of polymerization. We hope to combine this capability with the ability to create defect arrays on alignment surfaces in order to develop new materials such as responsive colloidal fluids or micro-kirigami structures.

140

CHAPTER 6

Conclusion

Ordered media give rise to astounding new phenomena. In order to harness the true potential of such systems, we must first understand how they conform to in different environments, be they geometric confinement or colloidal inclusions. By studying the energy considerations of LC systems, we can see how they behave as a result of interfacial orientationation, confining geometry, chemical patterning and elastic energy gradients.

We first looked at the topology of NLCs as it is influenced by interplay between

2D and 3D confinement and how energy considerations influence the resulting morphology. In the case of a homeotropically aligned convex surface, a spherical cap of bulk NLC will attempt to form a +1 defect at the center of the drop. However, due to 2D confinement on the flat surface, one of several situations can be encountered. While planar degenerate alignment leads to a single boojum defect confined to the surface, broken symmetry of the surface leads to defect lines that enter the bulk of the LC.

141

Homeotropic anchoring pushes the topological defect to the three-phase line, where it becomes a boojum ring.

With an understanding of the mathematical rules governing liquid crystal droplets, we observed the behavior of NLC drops on SAMs. 5CB showed wetting behavior on both aliphatic chains and carboxylic acid groups. On the former, we noticed a peculiar time dependence of the contact angle on HDT which wasn’t present on other aliphatic chains. We also demonstrated that carboxylic acid groups can be aligned by rubbing the surface.

Using μcp with the aformentioned SAMs, we developed a method of self- localizing arrays of LC spherical caps on surfaces through energy minimization. We also looked at the dynamic processes of fluid self-localization to generate metastable states, such as highly stressed droplets and droplets on high-energy surfaces. We then developed a parallelizable method for localizing droplets on chemical patterns. Due to system- specific inhomogeneities on the surface, the observed optical structures were probed using simulations.

We then explored thermophoretic forces that act on colloids in liquid crystals.

Thermophoresis in NLCs differed significantly from isotropic systems and appeared to be dominated by a elastic forces. These forces drive colloids from cold areas to hot areas, which is opposite to most isotropic systems. We were, however, able to also observe a crossover from negative to positive thermophoresis when the local temperature decreased, which suggests that there are regimes where the elastophoretic contribution is

142

negligible compared to traditional thermophoretic forces. To understand the elastophoretic contribution, we developed a theory based on the minimization of the elastic free energy of the system and fit it to the high-temperature regime of our system

(near the nematic-isotropic transition temperature). By comparing our results with isotropic systems that are utilized in Thermal Field-Flow Fractionation and MicroScale

Thermophoresis, we concluded that liquid crystal systems can offer significant advantages because of the large Soret coefficient.

Finally we developed a method of generating aligned liquid crystal colloids with the prospect of bulk systems of switchable colloids. The final work was a step toward the application of energy minimization in director structure and defects in confined NLCs, colloidal motion in NLC media and the switchable nature of LCP colloids. I hope this work provides insight and guidelines for the development of increasingly complex systems such as those composed of active colloids.

143

APPENDIX A

Derivation of the Young Relation

The free energy cost of inserting a fluid droplet between a surface and a medium as shown in (a) above is

퐹 = Σ훼훽퐴훼훽 + Σ훽푠퐴훽푠 + Σ훼푠퐴훼푠 A1 where we can make the simplification that 퐴훼푠 = 퐶 − 퐴훽푠 where C is the total surface area in the absence of the drop. We can thus define the variance of the free energy

훿퐹 = Σ훼훽훿퐴훼훽 + (Σ훽푠 − Σ훼푠)훿퐴훽푠 A2

It can be shown that, in the presence of a high-energy surface and a low energy medium, the smallest surface energy would be achieved when the drop is a spherical cap.

This is not the case when a drop sits at the interface between two fluids, or when a drop on a surface is exposed to the effects of gravity or line tension. Using the geometry of a

144

spherical cap shown in (b), we can determine that the relevant surface areas of the two distinct interfaces are

2 퐴훼훽 = 2휋푅 (1 − cos 휃) A3

2 2 퐴훽푠 = 휋푅 (1 − cos 휃) A4 where the variance is defined by

2 훿퐴훼훽 = 4휋푅(1 − cos 휃)훿푅 − 2휋푅 훿(cos 휃) A5

2 2 훿퐴훽푠 = 2휋푅(1 − cos 휃)훿푅 − 2휋푅 cos 휃 훿(cos 휃) A6

Before we proceed, we must first impose the constant volume constraint

휋푅3 푉 = (2 + cos 휃)(1 − cos 휃)2 A7 3

휋푅3 훿푉 = 휋푅2(2 + cos 휃)(1 − cos 휃)2훿푅 + (3 cos2 휃 − 3)훿(cos 휃) = 0 A8 3 which yields

(2+cos 휃)(1−cos 휃) 훿푅 훿(cos 휃) = A9 (1+cos 휃) 푅

We can then go back to find the variance of the surface area:

1−cos 휃 훿퐴 = 2휋푅훿푅 cos 휃 A10 훼훽 1+cos 휃

훿퐴 = 2휋푅훿푅 1−cos 휃 A11 훽푠 1+cos 휃

145

When we put these expressions back into the variance of the free energy and set it equal to zero, we get the expected result

0 = Σ훼훽 cos 휃 + (Σ훽푠 − Σ훼푠) A12 which is the Young relation

Σ −Σ cos 휃 = 훼푠 훽푠 A13 Σ훼훽

146

APPENDIX B

Derivation of Q Tensor Free Energy for Simulations

The Frank Free Energy of a nematic liquid crystal is

1 1 1 푓 = 퐾 (∇ ∙ 푛⃑ )2 + 퐾 (푛⃑ ∙ ∇ × 푛⃑ + 푞 )2 + 퐾 (푛⃑ × ∇ × 푛⃑ )2 B1 2 11 2 22 0 2 33 We first want to convert the free energy terms into Einstein notation for simple numeric calculation

∇ ∙ 푛⃑ = 푛푖,푖 B2

∇ × 푛⃑ = 휀푖푗푘푛푘,푗푥̂푖 B3

2 (∇ ∙ 푛⃑ ) = 푛푖,푖푛푗,푗 B4

2 (∇ × 푛⃑ ) = 푛푖,푗푛푖,푗 − 푛푖,푗푛푗,푖 B5

푛⃑ ∙ ∇ × 푛⃑ = 휀푖푗푘푛푖푛푘,푗 B6

푛⃑ × ∇ × 푛⃑ = −푛푖푛푗,푖푥̂푗 B7

2 (푛⃑ × ∇ × 푛⃑ ) = 푛푖푛푗푛푘,푖푛푘,푗 B8

Using the relation

(퐴 × 퐵)2 = 퐵2 − (퐴 ∙ 퐵)2 B9

Where we determine the following from definitions

(퐴 × 퐵)2 = |퐴 × 퐵|2 = |퐴|2|퐵|2 sin2 휃 B10

퐴∙퐵 cos 휃 = B11 |퐴||퐵|

147

(퐴∙퐵)2 sin2 휃 = 1 − cos2 휃 = 1 − B12 (퐴)2(퐵)2

And thus

(퐴∙퐵)2 (퐴 × 퐵)2 = (퐴)2(퐵)2 (1 − ) = (퐴)2(퐵)2 − (퐴 ∙ 퐵)2 B13 (퐴)2(퐵)2

Where, if we set 퐴 = 푛⃑ and 퐵 = ∇ × 푛⃑ and realize that (푛⃑ )2 = 1:

2 2 2 (푛⃑ ∙ ∇ × 푛⃑ ) = (∇ × 푛⃑ ) − (푛⃑ × ∇ × 푛⃑ ) = 푛푖,푗푛푖,푗 − 푛푖,푗푛푗,푖 − 푛푖푛푗푛푘,푖푛푘,푗 B14

This gives the Free Energy in terms of the new notation

1 1 1 푓 = 퐾 푛 푛 + 퐾 (푛 푛 − 푛 푛 ) + (퐾 − 퐾 )푛 푛 푛 푛 + B15 2 11 푖,푖 푗,푗 2 22 푖,푗 푖,푗 푖,푗 푗,푖 2 33 22 푖 푗 푘,푖 푘,푗 푞0퐾22휀푖푗푘푛푖푛푘,푗

The Q-Tensor can be derived from the director n by

1 푄 = 푛 푛 − 훿 B16 푖푗 푖 푗 3 푖푗 For index simplicity, we can derive

푄푖푗,푘푄푖푗,푘 = 2푛푖,푘푛푖,푘 = 퐺1 B17

푄푖푗,푗푄푖푘,푘 = 푛푗,푗푛푘,푘 + 푛푗푛푘푛푖,푗푛푖,푘 = 퐺2 B18

휀푖푗푘푄푖푙푄푗푙,푘 = 휀푖푗푘푛푖푛푗,푘 = 퐺4 B19

1 푄 푄 푄 = 푛 푛 푛 푛 − 푄 푄 = 퐺 B20 푖푗 푘푙,푖 푘푙,푗 푖 푗 푙,푖 푙,푗 3 푖푗,푘 푖푗,푘 6 And redefine the free energy in terms of the Q-tensor

1 1 푓 = (−퐾 + 3퐾 + 퐾 )퐺 + (퐾 − 퐾 )퐺 − 푞 퐾 퐺 + B21 6 11 22 33 1 2 11 22 2 0 22 4 1 (퐾 − 퐾 )퐺 4 33 11 6

148

REFERENCES

1. Kim, K.-H. and J.-K. Song, Technical evolution of liquid crystal displays. Npg Asia

Materials, 2009. 1(1): p. 29-36.

2. Woltman, S.J., G.D. Jay, and G.P. Crawford, Liquid-crystal materials find a new order in

biomedical applications. Nature Materials, 2007. 6(12): p. 929-938.

3. Baetens, R., B.P. Jelle, and A. Gustavsen, Properties, requirements and possibilities of

smart windows for dynamic daylight and solar energy control in buildings: A state-of-

the-art review. Solar Energy Materials and Solar Cells, 2010. 94(2): p. 87-105.

4. Xue, C.-Y., S.A. Khan, and K.-L. Yang, Exploring Optical Properties of Liquid Crystals

for Developing Label-Free and High-Throughput Microfluidic Immunoassays. Advanced

Materials, 2009. 21(2): p. 198-+.

5. Kleman, M., Defects in Liquid Crystals. Reports on Progress in Physics, 1989. 52(5): p.

555-654.

6. Lopez-Leon, T. and A. Fernandez-Nieves, Drops and shells of liquid crystal. Colloid and

Polymer Science, 2011. 289(4): p. 345-359.

7. Warner, M. and E.M. Terentjev, Liquid Crystal Elastomers. 2006, Oxford: Clarendon

Press.

8. Coates, D., Polymer-dispersed liquid crystals. Journal of Materials Chemistry, 1995.

5(12): p. 2063-2072.

149

9. Truskey, G.A., F. Yuan, and D.F. Katz, Transport Phenomena in Biological Systems. 2

ed. 2009: Prentice Hall.

10. Pabst, G., et al., Liposomes, Lipid Bilayers and Model Membranes: From Basic Research

to Application. 2014: Taylor & Francis Group.

11. Gonzalez, D., et al., A new thermophoretic precipitator for collection of nanometer-sized

aerosol particles. Aerosol Science and Technology, 2005. 39(11): p. 1064-1071.

12. Kasap, S. Thermoelectric Effects in Metals: Thermocouples. Web-Materials, 2001.

13. Vining, C.B., Condensed-matter physics - Thermopower to the people. Nature, 2003.

423(6938): p. 391-392.

14. Parola, A. and R. Piazza, Particle thermophoresis in liquids. European Physical Journal

E, 2004. 15(3): p. 255-263.

15. Jerabek-Willemsen, M., et al., MicroScale Thermophoresis: Interaction analysis and

beyond. Journal of Molecular Structure, 2014. 1077: p. 101-113.

16. Schimpf, M.E., K. Caldwell, and C.J. Giddings, Field-Flow Fractionation Handbook.

2000: John Wiley & Sons, Inc.

17. Stark, H., Physics of colloidal dispersions in nematic liquid crystals. Physics Reports-

Review Section of Physics Letters, 2001. 351(6): p. 387-474.

18. Gharbi, M.A., et al., Behavior of colloidal particles at a nematic liquid crystal interface.

Soft Matter, 2011. 7(4): p. 1467-1471.

19. Stark, H. and D. Ventzki, Non-linear Stokes drag of spherical particles in a nematic

solvent. Europhysics Letters, 2002. 57(1): p. 60-66.

150

20. Ryzhkova, A.V., F.V. Podgornov, and W. Haase, Nonlinear electrophoretic motion of

dielectric microparticles in nematic liquid crystals. Applied Physics Letters, 2010.

96(15): p. 151901.

21. Lavrentovich, O.D., I. Lazo, and O.P. Pishnyak, Nonlinear electrophoresis of dielectric

and metal spheres in a nematic liquid crystal. Nature, 2010. 467(7318): p. 947-950.

22. Musevic, I., et al., Two-dimensional nematic colloidal crystals self-assembled by

topological defects. Science, 2006. 313(5789): p. 954-958.

23. Kobatake, S., et al., Rapid and reversible shape changes of molecular crystals on

photoirradiation. Nature, 2007. 446(7137): p. 778-781.

24. Ohm, C., M. Brehmer, and R. Zentel, Liquid Crystalline Elastomers as Actuators and

Sensors. Advanced Materials, 2010. 22(31): p. 3366-3387.

25. de Haan, L.T., et al., Engineering of Complex Order and the Macroscopic Deformation

of Liquid Crystal Polymer Networks. Angewandte Chemie-International Edition, 2012.

51(50): p. 12469-12472.

26. Ionov, L., Soft microorigami: self-folding polymer films. Soft Matter, 2011. 7(15): p.

6786-6791.

27. Leong, T.G., A.M. Zarafshar, and D.H. Gracias, Three-Dimensional Fabrication at Small

Size Scales. Small, 2010. 6(7): p. 792-806.

28. Randall, C.L., E. Gultepe, and D.H. Gracias, Self-folding devices and materials for

biomedical applications. Trends in Biotechnology, 2012. 30(3): p. 138-146.

29. Cheng, S., Delineating Structural Characteristics of Viral Capsid Proteins Critical for

their Functional Assembly, in Bioinformatics. 2014, University of Michigan.

151

30. Bredon, G.E., Topology and Geometry. Vol. 139. 1993: Springer Science & Business

Media.

31. Hatcher, A., Algebraic Topology. 2002, Cambridge: Cambridge University Press.

32. Sethna, J., Statistical mechanics: entropy, order parameters, and complexity. Vol. 14.

2006: Oxford University Press.

33. Chen, B.G.-g., et al., Generating the Hopf Fibration Experimentally in Nematic Liquid

Crystals. Physical Review Letters, 2013. 110(23).

34. Frank, F.C., I. Liquid crystals. On the theory of liquid crystals. Discussions of the

Faraday Society, 1958. 25: p. 19-28.

35. Rapini, A. and M. Papoular, Distorsion d'une lamelle nématique sous champ magnétique

conditions d'ancrage aux parois. Le Journal de Physique Colloques, 1969. 30: p. C4-54.

36. Jerome, B., Surface effects and anchoring in liquid crystals. Reports on Progress in

Physics, 1991. 54(3): p. 391-451.

37. Kocevar, K. and I. Musevic, Surface-induced nematic and smectic order at a liquid-

crystal-silanated-glass interface observed by atomic force spectroscopy and Brewster

angle ellipsometry. Physical Review E, 2002. 65(2).

38. Miyano, K., Surface‐induced ordering of a liquid crystal in the isotropic phase. Journal

of Chemical Physics, 1979. 71(10): p. 4108-4111.

39. Sheng, P., Boundary-layer phase transition in nematic liquid crystals. Physical Review

A, 1982. 26(3): p. 1610-1617.

40. Goodby, J., et al., Physical properties of liquid crystals. 1999, Weinheim, Germany:

Wiley-Vch.

152

41. Yang, D.-K. Numerical Simulation and Tensor Representation of Liquid Crystal Director

Configuration. Available from: http://physics101.net/Deng-ke%20--

%20tensor%20representation%20for%20Robin%20class.pdf.

42. Yang, D.-K., K.-U. Jeong, and S.Z.A. Cheng, Structure of liquid crystal droplets with

chiral propeller texture. Journal of Physical Chemistry B, 2008. 112(5): p. 1358-1366.

43. Ruhwandl, R.W. and E.M. Terentjev, Monte Carlo simulation of topological defects in

the nematic liquid crystal matrix around a spherical colloid particle. Physical Review E,

1997. 56(5): p. 5561-5565.

44. Avila, L., et al., The VTK User's Guide (VTK 4.2). 2003, Kitware Inc: USA.

45. ParaView. Available from: http://www.paraview.org/.

46. Jones, R.C., A New Calculus for the Treatment of Optical Systems. Journal of the Optical

Society of America, 1941. 31(7): p. 488-493.

47. Jakli, A. and A. Saupe, One and Two Dimensional Fluids - Physical Properties of

Smectic Lamellar and Columnar Liquid Crystals. 2006: CRC Press.

48. Gupta, V.K. and N.L. Abbott, Using droplets of nematic liquid crystal to probe the

microscopic and mesoscopic structure of organic surfaces. Langmuir, 1999. 15(21): p.

7213-7223.

49. Prakash, G.V., et al., Control of topological defects in microstructured liquid crystal

cells. Optics Express, 2005. 13(6): p. 2201-2209.

50. Harkins, W.D. and A. Feldman, Films. The spreading of liquids and the spreading

coefficient. Journal of the American Chemical Society, 1922. 44(12): p. 2665-2685.

51. Degennes, P.G., Wetting - Statics and Dynamics. Reviews of Modern Physics, 1985.

57(3): p. 827-863.

153

52. Wenzel, R.N., Resistance of solid surfaces to wetting by water. Industrial & Engineering

Chemistry, 1936. 28(8): p. 988-994.

53. Lenz, P. and R. Lipowsky, Morphological transitions of wetting layers on structured

surfaces. Physical Review Letters, 1998. 80(9): p. 1920-1923.

54. Park, M.-K. and C.-H. Jang, Liquid Crystal-based Imaging of Biomolecular Interactions

at Roller Printed Protein Surfaces. Bulletin of the Korean Chemical Society, 2010. 31(5):

p. 1223-1227.

55. Opara, T., J.W. Baran, and J. Zmija, Interferential Method for Determining the

Inclination Angle of Molecules in Plane-Parallel Liquid Crystalline Layers Crystal

Research and Technology, 1988. 23(9): p. 1073-1083.

56. Born, M. and E. Wolf, Principles of Optics. 1989, Oxford: Pergamon Press.

57. Ulman, A., Formation and structure of self-assembled monolayers. Chemical Reviews,

1996. 96(4): p. 1533-1554.

58. Love, J.C., et al., Self-assembled monolayers of thiolates on metals as a form of

nanotechnology. Chemical Reviews, 2005. 105(4): p. 1103-1169.

59. Vericat, C., M.E. Vela, and R.C. Salvarezza, Self-assembled monolayers of alkanethiols

on Au(111): surface structures, defects and dynamics. Physical Chemistry Chemical

Physics, 2005. 7(18): p. 3258-3268.

60. Wilbur, J.L., et al., Microcontact printing of self-assembled monolayers: Applications in

microfabrication. Nanotechnology, 1996. 7(4): p. 452-457.

61. Delamarche, E., et al., Transport mechanisms of alkanethiols during microcontact

printing on gold. Journal of Physical Chemistry B, 1998. 102(18): p. 3324-3334.

154

62. Biebuyck, H.A. and G.M. Whitesides, Self-organization of organic liquids on patterned

self-assembled monolayers of alkanethiolates on gold. Langmuir, 1994. 10(8): p. 2790-

2793.

63. Hartmann, D.M., O. Kibar, and S.C. Esener, Characterization of a polymer microlens

fabricated by use of the hydrophobic effect. Optics Letters, 2000. 25(13): p. 975-977.

64. Wu, M.H. and G.M. Whitesides, Fabrication of two-dimensional arrays of microlenses

and their applications in photolithography. Journal of Micromechanics and

Microengineering, 2002. 12(6): p. 747-758.

65. Kim, E., et al., Fabrication of arrays of channel waveguides by self-assembly using

patterned organic monolayers as templates. Advanced Materials, 1996. 8(2): p. 139-&.

66. Kumar, A. and G.M. Whitesides, Patterned condensation figures as optical diffraction

gratings. Science, 1994. 263(5143): p. 60-62.

67. Chen, W.-C., et al., Fabrication of inkjet-printed SU-8 photoresist microlenses using

hydrophilic confinement. Journal of Micromechanics and Microengineering, 2013. 23(6):

p. 065008.

68. Gorman, C.B., H.A. Biebuyck, and G.M. Whitesides, Use of a Patterned Self-Assembled

Monolayer To Control the Formation of a Liquid Resist Pattern on a Gold Surface.

Chemistry of Materials, 1995. 7(2): p. 252-254.

69. Kim, E. and G.M. Whitesides, Use of minimal free-energy and self-assembly to form

shapes. Chemistry of Materials, 1995. 7(6): p. 1257-1264.

70. Magerl, A. and W.A. Goedel, Porous Polymer Membranes via Selectively Wetted

Surfaces. Langmuir, 2012. 28(13): p. 5622-5632.

155

71. Miller, M.S., G.J.E. Davidson, and T.B. Carmichael, Templated Self-Assembly of Glass

Microspheres into Ordered Two-Dimensional Arrays under Dry Conditions. Langmuir,

2010. 26(7): p. 5286-5290.

72. Snoeijer, J.H., et al., Cornered drops and rivulets. Physics of Fluids, 2007. 19(4): p.

042104.

73. Jackman, R.J., et al., Fabricating large arrays of microwells with arbitrary dimensions

and filling them using discontinuous dewetting. Analytical Chemistry, 1998. 70(11): p.

2280-2287.

74. Lopez, G.P., H.A. Biebuyck, and G.M. Whitesides, Scanning Electron Microscopy Can

Form Images of Patterns in Self -Assembled Monolayers. Langmuir, 1993. 9(6): p. 1513-

1516.

75. Schreiber, F., Structure and growth of self-assembling monolayers. Progress in Surface

Science, 2000. 65(5-8): p. 151-256.

76. Henderson, A.P., et al., A Novel Isotherm, Modeling Self-Assembled Monolayer

Adsorption and Structural Changes. Langmuir, 2009. 25(2): p. 931-938.

77. Kisliuk, P., The sticking probabilities of gases chemisorbed on the surfaces of solids.

Journal of Physics and Chemistry of Solids, 1957. 3(1): p. 95-101.

78. Larsen, N.B., et al., Order in microcontact printed self-assembled monolayers. Journal of

the American Chemical Society, 1997. 119(13): p. 3017-3026.

79. Piazza, R., Thermophoresis: moving particles with thermal gradients. Soft Matter, 2008.

4(9): p. 1740-1744.

80. Zheng, F., Thermophoresis of spherical and non-spherical particles: a review of theories

and experiments. Advances in Colloid and Interface Science, 2002. 97(1-3): p. 255-278.

156

81. Tyndall, J., On Haze and Dust. Nature, 1870. 1(13): p. 339-342.

82. Brenner, H., Navier-Stokes revisited. Physica a-Statistical Mechanics and Its

Applications, 2005. 349(1-2): p. 60-132.

83. Vanherwaarden, A.W. and P.M. Sarro, Thermal Sensors Based on the Seebeck Effect.

Sensors and Actuators, 1986. 10(3-4): p. 321-346.

84. Schermer, R.T., et al., Laser-induced thermophoresis of individual particles in a viscous

liquid. Optics Express, 2011. 19(11): p. 10571-10586.

85. Duhr, S. and D. Braun, Why molecules move along a temperature gradient. Proceedings

of the National Academy of Sciences of the United States of America, 2006. 103(52): p.

19678-19682.

86. Reichl, M., et al., Why Charged Molecules Move Across a Temperature Gradient: The

Role of Electric Fields. Physical Review Letters, 2014. 112(19).

87. Duhr, S. and D. Braun, Nonlinear Thermophoresis beyond Local Equilibrium Criterion.

arXiv preprint cond-mat/0609554, 2006.

88. Sugaya, R., B.A. Wolf, and R. Kita, Thermal diffusion of dextran in aqueous solutions in

the absence and the presence of urea. Biomacromolecules, 2006. 7(2): p. 435-440.

89. Putnam, S.A., D.G. Cahill, and G.C.L. Wong, Temperature dependence of

thermodiffusion in aqueous suspensions of charged nanoparticles. Langmuir, 2007.

23(18): p. 9221-9228.

90. Kita, R., G. Kircher, and S. Wiegand, Thermally induced sign change of Soret coefficient

for dilute and semidilute solutions of poly(N-isopropylacrylamide) in ethanol. Journal of

Chemical Physics, 2004. 121(18): p. 9140-9146.

157

91. Zhang, M. and F. Mueller-Plathe, The Soret effect in dilute polymer solutions: Influence

of chain length, chain stiffness, and solvent quality. Journal of Chemical Physics, 2006.

125(12).

92. Iacopini, S. and R. Piazza, Thermophoresis in protein solutions. Europhysics Letters,

2003. 63(2): p. 247-253.

93. Piazza, R., S. Iacopini, and B. Triulzia, Thermophoresis as a probe of particle-solvent

interactions: The case of protein solutions. Physical Chemistry Chemical Physics, 2004.

6(7): p. 1616-1622.

94. Iacopini, S., R. Rusconi, and R. Piazza, The "macromolecular tourist": Universal

temperature dependence of thermal diffusion in aqueous colloidal suspensions. European

Physical Journal E, 2006. 19(1): p. 59-67.

95. Wienken, C.J., et al., Thermophoretic melting curves quantify the conformation and

stability of RNA and DNA. Nucleic Acids Research, 2011. 39(8).

96. Giddings, J.C., Field-Flow Fractionation - Analysis of Macromolecular, Colloidal, and

Particulate Materials. Science, 1993. 260(5113): p. 1456-1465.

97. Belgaied, J.E., M. Hoyos, and M. Martin, Velocity Profiles in Thermal Field-

Flow Fractionation. Journal of Chromatography A, 1994. 678(1): p. 85-96.

98. Messaud, F.A., et al., An overview on field-flow fractionation techniques and their

applications in the separation and characterization of polymers. Progress in Polymer

Science, 2009. 34(4): p. 351-368.

99. Liu, G. and J.C. Giddings, Separation of Particles in Aqueous Suspensions by Thermal

Field-Flow Fractionation - Measurement of Thermal-Diffusion Coefficients.

Chromatographia, 1992. 34(9-10): p. 483-492.

158

100. Myers, M.N., K.D. Caldwell, and C.J. Giddings, A Study of Retention in Thermal Field-

Flow Fractionation. Separation Science, 1974. 9(1): p. 47-70.

101. Postnova. Available from: www.postnova.com.

102. NanoTemper Technologies. Available from: www.nanotemper-technologies.com.

103. Ruhwandl, R.W. and E.M. Terentjev, Friction drag on a particle moving in a nematic

liquid crystal. Physical Review E, 1996. 54(5): p. 5204-5210.

104. Miesowicz, M., The three coefficients of viscosity of anisotropic liquids. Nature, 1946.

158: p. 27-27.

105. Ilg, P., Anisotropic diffusion in nematic liquid crystals and in ferrofluids. Physical

Review E, 2005. 71(5).

106. Mercuri, F., et al., Thermal conductivity and thermal diffusivity anisotropy in a liquid

crystal. Journal of Thermal Analysis and Calorimetry, 1998. 52(3): p. 739-744.

107. Crawford, G.P. and S. Zumer, Liquid Crystals in Complex Geometries. 1996: Taylor &

Francis Ltd.

108. Lavrentovich, O.D., Transport of particles in liquid crystals. Soft Matter, 2014. 10(9): p.

1264-1283.

109. Pumpa, M. and F. Cichos, Slow Single-Molecule Diffusion in Liquid Crystals. Journal of

Physical Chemistry B, 2012. 116(49): p. 14487-14493.

110. Volovik, G.E. and O.D. Lavrentovich, Topological dynamics of defects: boojums in

nematic drops. Zhurnal Eksperimentalnoi I Teoreticheskoi Fiziki, 1983. 85(6): p. 1997-

2010.

111. Stark, H., Director field configurations around a spherical particle in a nematic liquid

crystal. European Physical Journal B, 1999. 10(2): p. 311-321.

159

112. Juhl, A.T., et al., Ordering of glass rods in nematic and cholesteric liquid crystals.

Optical Materials Express, 2011. 1(8): p. 1536-1547.

113. Tkalec, U., M. Skarabot, and I. Musevic, Interactions of micro-rods in a thin layer of a

nematic liquid crystal. Soft Matter, 2008. 4(12): p. 2402-2409.

114. Stark, H. and D. Ventzki, Stokes drag of spherical particles in a nematic environment at

low Ericksen numbers. Physical Review E, 2001. 64(3): p. 031711.

115. Loudet, J.C., P. Hanusse, and P. Poulin, Stokes drag on a sphere in a nematic liquid

crystal. Science, 2004. 306(5701): p. 1525-1525.

116. Mondiot, F., et al., Stokes-Einstein diffusion of colloids in nematics. Physical Review E,

2012. 86(1): p. 010401.

117. Turiv, T., et al., Effect of Collective Molecular Reorientations on Brownian Motion of

Colloids in Nematic Liquid Crystal. Science, 2013. 342(6164): p. 1351-1354.

118. Diogo, A.C., Friction drag on a sphere moving in a nematic liquid crystal. Molecular

Crystals and Liquid Crystals, 1983. 100(1-2): p. 153-165.

119. Anderson, J.L., Transport Mechanisms of Biological Colloids. Annals of the New York

Academy of Sciences, 1986. 469: p. 166-177.

120. Rovner, J.B., D.H. Reich, and R.L. Leheny, Anisotropic Stokes Drag and Dynamic Lift

on Spheres Sedimenting in a Nematic Liquid Crystal. Langmuir, 2013. 29(7): p. 2104-

2107.

121. James, R. and J.-i. Fukuda, Effect of anchoring energy and elastic anisotropy on

spherical inclusions in a nematic liquid crystal. Physical Review E, 2013. 88(1): p.

010501.

160

122. Kondo, N., Y. Iwashita, and Y. Kimura, Dependence of interparticle force on

temperature and cell thickness in nematic colloids. Physical Review E, 2010. 82(2): p.

020701.

123. Kondo, N., Y. Iwashita, and Y. Kimura, Temperature and Confinement Effect on

Interparticle Force in Nematic Colloids. Molecular Crystals and Liquid Crystals, 2011.

545: p. 1339-1346.

124. Kishita, T., et al., Arrangement dependence of interparticle force in nematic colloids.

Physical Review E, 2010. 81(1): p. 010701.

125. Kishita, T., et al., Interparticle force in nematic colloids: Comparison between

experiment and theory. Physical Review E, 2011. 84(2): p. 021704.

126. Pishnyak, O.P., S.V. Shiyanovskii, and O.D. Lavrentovich, Aggregation of colloidal

particles in a non-equilibrium backflow induced by electrically-driven reorientation of

the nematic liquid crystal. Journal of Molecular Liquids, 2011. 164(1-2): p. 132-142.

127. Mondiot, F., et al., Shape-Induced Dispersion of Colloids in Anisotropic Fluids. Physical

Review Letters, 2009. 103(23): p. 238303.

128. Poulin, P., N. Frances, and O. Mondain-Monval, Suspension of spherical particles in

nematic solutions of disks and rods. Physical Review E, 1999. 59(4): p. 4384-4387.

129. Smalyukh, II, et al., Elasticity-mediated self-organization and colloidal interactions of

solid spheres with tangential anchoring in a nematic liquid crystal. Physical Review

Letters, 2005. 95(15): p. 157801.

130. Chernyshuk, S.B. and B.I. Lev, Elastic interaction between colloidal particles in

confined nematic liquid crystals. Physical Review E, 2010. 81(4): p. 041701.

161

131. Tovkach, O.M., S.B. Chernyshuk, and B.I. Lev, Theory of elastic interaction between

arbitrary colloidal particles in confined nematic liquid crystals. Physical Review E,

2012. 86(6): p. 061703.

132. Samitsu, S., Y. Takanishi, and J. Yamamoto, Molecular manipulator driven by spatial

variation of liquid-crystalline order. Nature Materials, 2010. 9(10): p. 816-820.

133. Kim, Y.-K., B. Senyuk, and O.D. Lavrentovich, Molecular reorientation of a nematic

liquid crystal by thermal expansion. Nature Communications, 2012. 3: p. 1133.

134. Criante, L., et al., Electrically tunable optoelastic interaction range of nematic colloids.

Soft Matter, 2013. 9(22): p. 5459-5463.

135. Abras, D., G. Pranami, and N.L. Abbott, The mobilities of micro- and nano-particles at

interfaces of nematic liquid crystals. Soft Matter, 2012. 8(6): p. 2026-2035.

136. West, J.L., et al., Drag on particles in a nematic suspension by a moving nematic-

isotropic interface. Physical Review E, 2002. 66(1): p. 012702.

137. Pires, D., J.-B. Fleury, and Y. Galerne, Colloid particles in the interaction field of a

disclination line in a nematic phase. Physical Review Letters, 2007. 98(24): p. 247801.

138. Pishnyak, O.P., et al., Levitation, lift, and bidirectional motion of colloidal particles in an

electrically driven nematic liquid crystal. Physical Review Letters, 2007. 99(12): p.

127802.

139. Style, R.W., et al., Patterning droplets with durotaxis. Proceedings of the National

Academy of Sciences of the United States of America, 2013. 110(31): p. 12541-12544.

140. Plotnikov, S.V., et al., Force Fluctuations within Focal Adhesions Mediate ECM-Rigidity

Sensing to Guide Directed Cell Migration. Cell, 2012. 151(7): p. 1513-1527.

162

141. Wall, B. and D. Bryant. LCI Prototype Facility Standard Processes. Available from:

http://www.lcinet.kent.edu/organization/facility/prototype/index.php?content=standard_p

rocesses.

142. Wu, C., et al., Effect of nanoparticle clustering on the effective thermal conductivity of

concentrated silica colloids. Physical Review E, 2010. 81(1): p. 011406.

143. Desnoyers, J.E., et al., Heat capacity and volume of colloidal silica in the presence of

electrolytes and cationic surfactants. Journal of Colloid and Interface Science, 1994.

164(2): p. 483-488.

144. Lazo-Martinez, I.E., Liquid Crystal Enabled Electrokinetic Phenomena, in Chemical

Physics. 2014, Kent State University.

145. Braibanti, M., D. Vigolo, and R. Piazza, Does thermophoretic mobility depend on

particle size? Physical Review Letters, 2008. 100(10).

146. Kuksenok, O.V., et al., Director structure around a colloid particle suspended in a

nematic liquid crystal. Physical Review E, 1996. 54(5): p. 5198-5203.

147. Cui, M. and J.R. Kelly, Temperature dependence of visco-elastic properties of 5CB.

Molecular Crystals and Liquid Crystals Science and Technology Section a-Molecular

Crystals and Liquid Crystals, 1999. 331: p. 1909-1917.

148. Merkel, T.J., et al., Scalable, Shape-Specific, Top-Down Fabrication Methods for the

Synthesis of Engineered Colloidal Particles. Langmuir, 2010. 26(16): p. 13086-13096.

149. Walther, A. and A.H.E. Mueller, Janus Particles: Synthesis, Self-Assembly, Physical

Properties, and Applications. Chemical Reviews, 2013. 113(7): p. 5194-5261.

150. Yi, G.-R., D.J. Pine, and S. Sacanna, Recent progress on patchy colloids and their self-

assembly. Journal of Physics-Condensed Matter, 2013. 25(19).

163

151. Cho, J.-H., et al., Nanoscale Origami for 3D Optics. Small, 2011. 7(14): p. 1943-1948.

152. Na, J.-H., et al., Programming Reversibly Self-Folding Origami with Micropatterned

Photo-Crosslinkable Polymer Trilayers. Advanced Materials, 2015. 27(1): p. 79-85.

153. Lin, D. and D.J. Broer, Liquid Crystal Polymer Networks: Preparation, Properties, and

Applications of Films with Patterned Molecular Alignment. Langmuir, 2014. 30(45): p.

13499-13509.

154. Chakrabarty, A., et al., Brownian Motion of Boomerang Colloidal Particles. Physical

Review Letters, 2013. 111(16): p. 160603.

155. Fuchi, K., et al., Topology optimization for the design of folding liquid crystal elastomer

actuators. Soft Matter, 2015. 11(37): p. 7288-7295.

156. Castle, T., et al., Making the Cut: Lattice Kirigami Rules. Physical Review Letters, 2014.

113(24).

157. Blees, M.K., et al., Graphene kirigami. Nature, 2015. 524(7564): p. 204-+.

164