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Physics Graduate Theses & Dissertations Physics

Spring 1-1-2011

Smectic Liquid Crystal Freely Suspended Films:Testing Beds for the Physics in Thin MembranesDuong Hoang NguyenUniversity of Colorado at Boulder, duong.nguyen@colorado.edu

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Recommended CitationNguyen, Duong Hoang, "Smectic Liquid Crystal Freely Suspended Films: Testing Beds for the Physics in Thin Membranes" (2011).Physics Graduate Theses & Dissertations. Paper 41.

Smectic Liquid Crystal Freely Suspended Films:

Testing Beds for the Physics in Thin Membranes

by

Duong Hoang Nguyen

B.S., University of Arizona, 2003

A thesis submitted to the

Faculty of the Graduate School of the

University of Colorado in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

Department of Physics

2011

This thesis entitled:Smectic Liquid Crystal Freely Suspended Films:Testing Beds for the Physics in Thin Membranes

written by Duong Hoang Nguyenhas been approved for the Department of Physics

Dr. Noel Clark

Dr. Daniel Schwartz

Date

The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above

mentioned discipline.

iii

Nguyen, Duong Hoang (Ph.D., Physics)

Smectic Liquid Crystal Freely Suspended Films:

Testing Beds for the Physics in Thin Membranes

Thesis directed by Dr. Noel Clark

This thesis highlights the unique features of smectic liquid crystal freely suspended films to

study surface effects and two-dimensional physics via three experiments: the first confirms the

popular Saffman-Delbruck theory of the drag on a small cylinder in a thin membrane, the second

measures the surface tension of a freely suspended molecular monolayer, and the third is a high-

speed observation of the coalescence of two thin fluid disks. In the first experiment, we measure

the diffusion constant of thicker domains (islands) in the film. They are perfectly circular and

are readily observed under the microscope, making it possible to track the motion of their centers

with high precision (∼ 10 nm). In the second experiment, the imposed symmetry on a smectic

monolayer helps to detect surface induced asymmetry on bilayer and thicker films via the motion

of 1D interfaces between domains of different thicknesses. Lastly, in the third experiment, the high

contrast of the film under reflection microscopy helps to capture the coalescence at high speed

without introducing much noise at low light levels. Moreover, without the obstruction of any

substrate, an optical trap can be effectively used to manipulate islands to force coalescence rather

than having to wait for serendipitous events.

To my parents, my sister, my wife and my little bear

v

Acknowledgements

When I was a child, I sometimes dreamt I was a descendant of the powerful yet playful monkey

king in “Journey to the West”, a fictional adventure story about him being forced to help a monk

overcome many obstacles during the pilgrimage to obtain sacred sutras from the Buddha, yet he

learned responsibilities, gained wisdom, and in the end, achieved perfect enlightenment. Thirteen

years ago, I got my own journey to get this thesis when my parents, being educated overseas and

having high regards for Western knowledge, asked me to go to Rocky Mountain High School in

America half the globe away from Vietnam.

At my farewell party, I was very moved when Aunt Hoa said on behalf of the whole extended

family that everybody is proud of my going to America to study and hopefully bring back the

ultimate prize, a PhD thesis. All I thought was how I would finish this “mission” quickly and

go back to my parents, my sister, my cousins and my girlfriend. Here I am now with my thesis!

There have been many changes over the past 13 years: my sister got married and had children; my

girlfriend has become my wife and we now have a son; my parents retired from government jobs,

founded companies and have been enjoying playing with their grandchildren. However, I know

one thing is constant: their unconditional love and support for me until the end. I am also very

grateful to chu Lang, who has followed and cared for my progress as a son and has even made

preparations for me to go on the road toward a bright and fulfilling scientific career. My mother-

and sister-in-laws have truly become part of my family, taking care of me, my wife, and my son for

the past two years.

There have been moments of difficulty and loneliness during the journey that would have

vi

been much harder to endure had there not been for many people showing me the warmth of family.

In the last two months of my first year in America, my friend Erin’s family took me into their home

after learning I was not treated well in my original host family. At the University of Arizona, bac

Pham was like a father to us, a group of about 10 Vietnamese students, attended to our needs, and

had us come over to have fun at weekends. For the past 8 years in Boulder, Kelly and Leigh have

been inviting my family to numerous events and never hesitate to lend a hand when we need help.

It is always a pleasure to see their family, especially to watch their children Peter and Emy grow

from preteens to become college and high school students with full of character.

My journey would be in vain without the people who make the thesis happen, among whom

my advisor, Noel Clark, has no doubt been the most important person because he took me into his

world-famous liquid crystals lab, introduced me to the thesis projects and the ideas behind them,

make sure the data are understandable, and organize results in a coherent way with respect to

themselves and to existing knowledge. Moreover, being a true educator, he helps students like me

to understand their projects – not just cranking out cryptic data – and knows when to leave room

for us to pursue independent thoughts and figure things out for ourselves. Even more valuable is

the wisdom that he conveys to us in numerous occasions. He teaches us that there is life beyond a

thesis, that a thesis is more or less a personal account for what we have done, while understanding

the science and codifying this understanding into published papers are more important. When

asked about the key to success at the party to celebrate his election to the National Academy of

Sciences, Noel told us that when he was at MIT and Harvard, there were many people who were

smarter than him and were absorbed by some big question, while he just tackled smaller problems

using freely suspended smectic films, a new system at the time, and read everything related to

liquid crystals. So there was a brilliant scientist admitting he was not the smartest and it was

diligence and gradual insights that brought him to the top of the knowledge pyramid.

I cannot carry out my projects without the help of my research team with crucial contributions

from every member and listing them would make it seem like I do nothing: Matt provides fresh and

interesting ideas, as well as critical data evaluations; Joe’s computer programs make data analysis

vii

much less tedious and enable us to tackle larger data sets; Cheol has a knack of easing all the daily

little troubles out either by suggesting improvements to the experimental setup or by finding the

exact needed equipments; Art know exactly what the lab has and he can build whatever the lab

does not have. Undergraduate assistants are also integral to our research team and I am blessed to

have bright and hardworking Markus, Aaron and Tatyana who take up a large part of the actual

data taking.

I would like to thank NSF and NASA for funding my projects, my thesis committee for

making sure my projects are carried out with sufficient quality. I thank Annett, Barbara, Nikki,

Kathryn in the office who make me forget that there is such a thing as paperwork. Going to work

everyday is much more pleasant with the interactions and mutual support from other people in the

lab: Ali, Christine, Dong, Greg, Robert, Shao, Yongqiang, Youngwoo, Yue, and with learnining

from the experiences of the visitors and the ex-members of the lab: Michi, Loren, Chris, A, Chart,

Guanjiu, Giuliano, Ying, Haitao, Dongki, Huiyu, Nadia, Kirsten. Weekends are also more pleasant

going hiking, watching soccer, seeing movies, playing tennis, drinking beers with Jing, Mark, Zach,

Chenhui, Jacquie, Ethan, Rahul, Clayton, Josh and David (when he comes over from Sweden).

I would have missed my country a lot more had it not been for my Vietnamese friends, among

them: bac Lan, anh Khai, anh Nam, anh Hai, chi Hang, anh Ha, chi Nhan, Cuong, Hung, Tran,

chi Huong.

Thank you all for being part of my journey!

Contents

Chapter

1 Introduction 1

1.1 A bit of history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Smectic A liquid crystal films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Experimental Setup 5

2.1 Making and keeping smectic freely suspended films . . . . . . . . . . . . . . . . . . . 5

2.2 Observing and capturing videos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Measuring film thickness using a camera . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Crossover Between 2D and 3D Fluid Dynamics in the Diffusion of Islands in Ultrathin Freely

Suspended Smectic Filmss 12

3.1 Saffman-Delbruck theory of diffusion in thin fluid films and its extensions . . . . . . 12

3.2 Diffusion of islands in 8CB smectic A films . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 Confirmation of SD/HPW theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Surface Tension of a Freely Suspended Molecular Monolayer 24

4.1 Surface tension of thin smectic films . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2 1D interface motion in smectic films . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3 Determining surface tension difference between between monolayer and bilayer domains 29

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

ix

5 Two-dimensional domain coalescence driven by line tension 37

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2 Model for droplet coalescence in two dimensions . . . . . . . . . . . . . . . . . . . . 38

5.3 Island coalescence in thin smectic films . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.4 Bridge expansion in the viscous regime . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Bibliography 48

x

Figures

Figure

1.1 (a) Thicker circular domains (islands) in a freely suspended smectic liquid crystal

film of 8CB. The color of the islands is due to from the interference of the light

reflected from the top and bottom surfaces. (b) Cross section of an island (thicker

domain) in a smectic A freely suspended film. Molecules orient perpendicular to

the film, but move randomly within each layer. An island is bounded by an edge

dislocation loop, which provides a line tension. . . . . . . . . . . . . . . . . . . . . . 3

2.1 (Top) Making smectic freely suspended films by spreading the liquid crystal across

a hole in a glass cover slip. (Bottom) Side view of an island in a smectic freely

suspended film held across a film holder. . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Reflection microscopy for observing smectic freely suspended films. (Image repro-

duced from Apichart Pattanaporkratana’s original) . . . . . . . . . . . . . . . . . . . 8

2.3 (a) A region of interest (ROI) is chosen in a digitized image of a freely suspended

smectic film to calculate the average brightness for each of the red, green and blue

channels. (b) Brightness as a function of exposure time for each color channel.

(c) After calibration using a black glass, the absolute value for reflectivity can be

calculated and plotted along the predicted curves given by Eq. 2.1. (d) A collection

of 29 reflectivity measurements from the green channel shows that they are well

described by the predicted curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

xi

3.1 Island mobility as a function of radius in smectic films. Experimental data for three

different film thicknesses (N = 3, 4 and 5) and a range of island radii follow the

predictions of the SD-HPW-PS theory (solid curve) and illustrate the crossover from

2D to 3D behavior with increasing island radius. Both mobility and radius are

scaled to be dimensionless. The dotted curves show the predictions of 2D and 3D

theory, valid in the limits a lS and a lS respectively. The inset shows the

diffusion coefficients of islands of different radii in an N = 3 film. The error bars

correspond to the standard deviations obtained by analyzing several subsets at each

radius extracted from a given trajectory. . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 A 35 µm-radius island in a freely suspended smectic A film of 8CB. The film and

island are an integer number of layers thick, and are separated by an edge dislocation

loop (•), which provides a line tension and gives the island its circular shape. The

inset shows in cross section an island of radius a in a smectic A film of thickness h.

The rodlike 8CB molecules are arranged in 2D fluid layers and are oriented along

the layer normal. The low climb mobility of the edge dislocation ensures that the

material in the surrounding film diverts around the diffusing island. 8CB has the

phase sequence: Crystal22C←→ Smectic A

33.5C←→ Nematic40.5C←→ Isotropic. . . . . . . . 16

xii

3.3 Island diffusion in a smectic A liquid crystal film. (a) Three-minute trajectory of a

13 µm-radius island diffusing in an N = 4 layer 8CB film, with the inset showing

a ten-second extract. The trajectory shows both Brownian motion and systematic

drift. The location of the center of the island is determined to an accuracy of

±10 nm. The lower plots show the distributions of displacements measured between

two points along this trajectory separated by time intervals of (b) 0.167 s and (c)

1.67 s. As the time interval increases, the distribution shifts away from the origin

and spreads out, with both the distance of the center of the distribution from the

origin and the variance of the distribution increasing linearly with the chosen time

interval. The rate of change of the center’s position gives the drift velocity while

that of the variance gives the diffusion coefficient. . . . . . . . . . . . . . . . . . . . 17

3.4 Probability distributions of step size (a), displacement in x (b) and y (c) in the island

diffusion trajectories in N = 3, 4 and 5 films at 30 fps framerate. Solid lines are 2D

Gaussian distribution fits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.5 Tests of the analysis method using simulated random walk data with a fixed diffusion

coefficient of 0.5 µm2/s, and drift velocities ranging from 0.05 µm/s to 5.0 µm/s.

The values extracted for both the drift velocity () and the diffusion coefficient ()

are in good agreement with the parameters used to generate the test data. The error

bars correspond to the standard deviations obtained from analyzing a set of different

random walk sequences generated using the same parameters. . . . . . . . . . . . . 21

4.1 Dynamics of a monolayer (N = 1) hole (dark region) containing (N = 2) islands A

and B (bright circular regions) in a N = 2 smectic freely suspended film of 8CB. The

islands appear and grow within the shrinking hole (a-d), eventually merge with the

background film (e-g), and the hole continues to shrink and disappear (h-l). During

this process, material is drawn from the meniscus at the edge into the film. . . . . . 27

xiii

4.2 (a) 1D interface dynamics of N = 1 hole (I1O2) and N = 3, 4 islands (I3O2, I4O2)

in N = 2 films. (b) Radii of hole (2) and two islands () versus time determined

from the video recording in Fig. 1. Black solid circles (4) show the data for the

hole radii corrected for the islands’growth (red arrow) and for the islands’merging

with the background film (orange arrow). Inset: EX-900084 molecule (bulk phase

diagram: Crystal71C←→ Smectic C

79C←→ Smectic A136C←→ Isotropic). (c) The growth

speed of N = 2 islands in N = 1 background (I2O1) is half of the shrinking speed of

N = 1 hole in N = 2 background (I1O2). . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 (a) N = 1 hole shrinks because of an elementary conversion from two N = 1

molecules to one N = 2 pair (each molecule in the pair occupies a place in one

layer). The N = 2 region advances two-molecule-wide distance freed up by the two

N = 1 molecules towards the center of the hole after the conversion. One molecule

width is occupied by the converted N = 2 pair while the other is occupied by a

new pair drawn from the meniscus. (b) Island grows also because of an elementary

conversion from two N = 1 molecules to one N = 2 pair. In this case, one empty

slot is occupied by the converted pair, now residing in the N = 2 island, and the

other by the advancing N = 1 region. Thus for the same conversion rate, the island

boundary moves half the distance compared to that of the hole boundary. (c) Two-

state system in which pairs of side-by-side molecules in state 1 overcome the barrier

EB to be on top of one another in state 2 that is ∆E12 lower than state 1. . . . . . 32

xiv

4.4 (a) Velocities ofN = 3, 4, 5and 6 islands inN = 2 films. The blue (green) band marks

the region that is within 2 standard deviations of the mean velocity measurements

of I3O2 (I4O2) islands. (b) Velocities of I1O2 holes (black solid triangles) and the

growth velocities of I2O1 islands (red open rectangles) scaled by 2, which are in

general much faster than I3O2 and I4O2 velocities. Velocities are determined from

linear fits yielding uncertainties smaller than 0.5%, which for I1O2 are less than the

thickness of the bar circled by the dashed line. The magenta line marks the limit of

v12 when the effect from δP (which makes the velocity more positive) is small. . . . 34

5.1 (a) Hopper’s exact solution for the coalescence of two infinite cylinders with inviscid

exterior fluid [25]. The area bounded by curves is constant and equal to π. The

curves correspond to (from inner to outer along the Y = 0 line)ν = 0.9 (t ≈ 0.0295),

ν = 0.7 (t ≈ 0.147), ν = 0.4 (t ≈ 0.551). (b) Definitions of R, rm, and Rc . . . . . . . 39

5.2 (a) A typical coalescence event (starting at t = 0) of thick pancake-like islands

(bright circular domains) in a thin smectic A liquid crystal film background (dark

region). The film is four layers thick (N = 4, h = 12.8 µm), while both islands have

N = 27. The radius of the top island is 54.9 µm and that of the bottom one is

59.5 µm. Before coalescence, islands are circular due to the line tension of the island

boundaries resulting from the edge dislocation energy. It is this line tension that

drives the dynamics of the coalescence. (b) Image sequence showing the expansion

of the bridge connecting the two coalescing islands. The scale bar is 25 µm in length. 41

xv

5.3 Intensity profiles along the line that goes through the initial contact point of the

islands and is perpendicular to the symmetry axis going through their centers (bot-

tom inset). t = 0 is defined as the time right before a peak shows up on top of

the reference profiles, shown in the left inset as the yellow curves below the that of

t = 0.05 ms (green) and t = 0.29 ms (red). Before the peak in the intensity profile

saturates, the bridge width is measured as the full width at half maximum of the

peak, whereas after it saturates, the width is simply that at the threshold value. . . 43

5.4 Bridge expansion during coalescence. The main figure shows the whole temporal

evolution including the shape relaxation towards the final circular domain, while the

inset focuses on the early dynamics. Without adjustable parameters, the predicted

evolution from Hopper’s model (blue curve) disagrees with the data. The orange

curve shows the fit if the characteristic time tau is used as an adjustable parameter.

In the inset, the cyan curve is the exponential relaxation fit to the data in the later

time, while the red line is the linear fit to the data points after the peak in the

intensity profile has saturated but before the relaxation regime. . . . . . . . . . . . . 45

5.5 Bridge expansion during coalescence events with thickness difference ∆N . The black

line has the slope of 1 in the log-log plots, which shows that the bridge expansion

curves at early time are linear. At later times, the bridge widths relax exponentially

to the final values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Chapter 1

Introduction

1.1 A bit of history

Liquid crystals represent a unique field that transcends many possible gaps in science cre-

ated by differences in time, geography, divisions among subfields, dichotomy between theory and

experiment, or between basic and applied sciences. In 1888, when Friedrich Reinitzer, an Austrian

botanist, sent the first liquid crystal ever examined to Otto Lehmann, a German physicist, a tradi-

tion was started and kept until nowadays in which physicists, chemists, biologists, mathematicians,

theorists, experimentalists, scientists and engineers from different countries continually interact and

collaborate to understand “Nature’s delicate phase of matter” [11].

Interest in liquid crystals has shown remarkable reach and longevity. We are now living in a

time when exciting, large, high-definition liquid crystal displays are a norm, which might obscure

the fact that the field has been around for more than 120 years, longer than most of the current

scientific subfields. The first 30 years saw extensive research of the nature of liquid crystals, an

unfortunate nomenclature implying an intermediary between crystalline solid and the amorphous

liquid [7]. It was Georges Friedel who settled the account and established that liquid crystals are

really distinct states of matter [41].

After a period of quiescent post-war years, in 1958, Wilhelm Maier and Alfred Saupe pre-

sented the first microscopic theory of liquid crystal phases [34]. In 1968, the first liquid crystal

display was demonstrated by RCA [22]. In 1974, the Nobel laureate Pierre-Gilles de Gennes pub-

lished the book “The Physics of Liquid Crystals” that rekindled interest in the field. The following

2

period has seen the discovery of many new liquid crystal phases with interesting properties stem-

ming from polarity, clinicity, chirality and their interplay [37, 28, 26]. Liquid crystals are now part

of the fast growing field of soft condensed matter that is characterized by complexity and flexibility

[14]. One interesting theme in the field has been using soft matter systems as a tunable systems

to study the physics of other systems. Colloids, for example, have been a fruitful model for atomic

and ionic systems [20, 31]. Smectic liquid crystal films are another example of such a model since

they offer clean systems to study membrane physics [61].

1.2 Smectic A liquid crystal films

The layering tendency of smectics make it easy to form single-component homogeneous,

ultrathin freely suspended films that are quantized in thickness (consisting of an integer number of

smectic layers N , which can be selected), and are stable for many hours (or days if not disturbed)

due to its low vapor pressure [69]. Films as thin as N = 2 (≈ 6 nm) are readily made with an area

as large as 10 cm2, making them the thinnest stable substrate-free fluid system for probing surface

effects or two-dimensional hydrodynamics. Smectic A films of rodlike molecules are particularly

useful for our experiments since they have the average molecular orientation perpendicular to the

films, and are liquidlike within each layer. As the result, when observed from above, they are

essentially two-dimensional liquid. Fig. 1.1 demonstrates the thickness quantization with domains

of uniform colors (therefore uniform film thickness) and sharp boundaries (therefor sharp jumps in

film thickness). Although only islands (thicker domains) are shown in Fig. 1.1, it is also possible

to have holes (thinner domains) in freely suspended smectic films.

3

(a)

(b)

Figure 1.1: (a) Thicker circular domains (islands) in a freely suspended smectic liquid crystal film of8CB. The color of the islands is due to from the interference of the light reflected from the top andbottom surfaces. (b) Cross section of an island (thicker domain) in a smectic A freely suspendedfilm. Molecules orient perpendicular to the film, but move randomly within each layer. An islandis bounded by an edge dislocation loop, which provides a line tension.

4

De Gennes said that he loved liquid crystals because they are both ”beautiful and mysterious”

[13]. Indeed, there are many beautiful examples of the textures of liquid crystals already, many of

them are collected by Dierking [15], but we have one more example from a smectic liquid crystal

freely suspended film shown in Fig. 1.1(a), which the Russian painter Wassily Kandinsky would

appreciate its resemblance to his 1926 painting ”Several Circles” [30]. The circles in the film are

thicker domains called islands with the cross section shown in Fig. 1.1(b). These islands are made

of the same material as the film, and once made, have constant thickness. They are separated

from the background film by an edge dislocation loop with a line tension that is responsible for the

island’s circular shape.

Another noticeable feature of the islands is that they come with different colors and brightness

under reflection microscopy. This is due to the dependence of the reflectivity of a thin dielectric

slab on both its thickness and the light wavelength [4], and this dependence will be utilized in our

experiments to measure the thickness of the film and the island.

Chapter 2

Experimental Setup

2.1 Making and keeping smectic freely suspended films

Smectic freely suspended films are made by using the edge of a glass cover slip to spread the

liquid crystal across a circular hole in a film holder made from another cover glass (Fig. 2.1). Only

a small amount of material is needed to make films, so we usually wet the tip of a spatula with

the material, touch the film holder with it, and that is enough to make many (a few tens) films.

Islands are then easily created by blowing obliquely either on the thicker parts of the film, but it

should be done very soon after making films because the meniscus draws in the adjacent thicker

layers very quickly. Using a straw offers more control over the location and direction to direct the

flow. With this technique, sometimes islands can be created out of the meniscus or out of the film

as it relaxes from being bowed. In the latter case, islands are quite monodispersed.

In all of our experiments, we use static film holders made from micro cover glass with typical

size of 18 x 18 mm and thickness of 0.15± 0.03 mm. The process of making film holders start with

gluing 20− 25 pieces of cover glass together to make a stack. We usually place a piece of glass on

top a hot plate heated to about 70C, and put the cover glasses on top of one another with a small

piece of melted wax in between. We then use a copper tube of desired size as a drill bit to make

a hole through the stack, and a mixture of water and 5 µm powder from Buehler as an abrasive

lubricant. Care must be taken while drilling though, since the stack of cover glass can crack easily.

One must push the copper tube onto the stack lightly and slowly without hearing screeching sound.

Going as slow as 30 minutes on a 20-piece stack yields about 50% of useful holders (with one or

6

Spreader

Film holderSmectic film

Island

Figure 2.1: (Top) Making smectic freely suspended films by spreading the liquid crystal across ahole in a glass cover slip. (Bottom) Side view of an island in a smectic freely suspended film heldacross a film holder.

7

two short hairline cracks, no chips) and about 10% of perfect ones. After drilling, the stack can be

disassembled by immersing it in toluene. Individual holders should then be left in toluene overnight

in the fume hood to completely dissolve the wax.

The film holder is then glued onto a rectangular base with a hole that is big enough to avoid

scattering the illumination beam when the film holder is put under the microscope. This base is

held onto a two-axis translation stage featuring two 10 µm Starrett fine adjustment screws. The film

holder is put in a sealed chamber and its temperature can be controlled by Instec STC200 to within

0.1C. Normally, the chamber is mounted on the microscope stage of the Olympus BX51. However,

when the chamber’s orientation needs to be controlled, it is mounted on the Nikon Optiphot-Pol

with adjustment screws for leveling, and the whole microscope is in turn mounted on a rotation

stage so that the chamber can be rotate around a horizontal axis.

2.2 Observing and capturing videos

Smectic freely suspended films are observed under reflection microscopy in which reflected

light from the films goes back through the same objective that acts as a condenser lens to bring

the illumination light onto the films (Fig. 2.2). With this scheme, the brightness goes as N2 for

N ≤ 15 without a background level (films that are infinitely thin appear completely black), thus

offering good contrast for identifying films or islands of different thicknesses.

Besides an observation port for the eyepiece, both the BX51 and the OptiPhot-Pol micro-

scopes have a separate port with a C-mount on which we can mount either Watec’s WAT221S color

CCD camera or VisionResearch’s Phantom v12.1 monochrome CMOS-based high-speed camera

(the exception being the Phantom camera on the Optiphot-Pol due to the heavy weight of the

Phantom). The Phantom v12.1 camera has the maximum resolution of 1280x800 with the speed

of 6,242 frames per second (fps) at full frame and 1,000,000 fps at 128x8. It can be connected to

a computer via the Ethernet and a companion software can be run to transfer the data from the

camera’s 8GB memory to the computer. We can then either save the movie in VisionResearch’s cine

format that preserves all the settings (e.g. capturing speed, exposure time, resolution) or convert

8

lamp

camera

microscope

film

beamsplitter

Figure 2.2: Reflection microscopy for observing smectic freely suspended films. (Image reproducedfrom Apichart Pattanaporkratana’s original)

9

it into other movie formats such as AVI, MOV, or to a series of images in BMP, TIFF, JPG, PNG,

etc. We usually save the interesting movie clips in cine format for archiving purpose, and extract

them into 16-bit TIFF images for analysis. Note that the camera has 12-bit intensity resolution,

so 16-bit image format would preserve all the intensity information with the four least significant

bits set to zero.

The WAT221S color CCD has a few automatic features that need to be turned off such as

auto-balance and gamma adjustment. It also features a dial to adjust the exposure time manually

which is useful for our film thickness measurement setup. We can get the video signal from the

camera either via the BNC or the S-Video output and route the signal to the Panasonic AG-DV1000

MiniDV tape recorder. At this point, we can either record the video to a MiniDV tape or connect

the tape recorder to a computer with a FireWire cable and use VLC media player to view the live

picture via DirectShow and/or record it onto a hard drive. The former is preferred when we need

to archive the data, while the latter is used to get temporary video such as for measuring film

thickness.

2.3 Measuring film thickness using a camera

Rosenblatt and Amer devised a method to measure the thickness of a freely suspended film

from its reflectivity [55], which has been utilized extensively in our lab [8, 45]. The gist of this

method is to align and focus a laser beam onto the film such that after going back through the

condenser lens, the reflected beam is parallel to but slightly off from the incident beam so that it

can be directed with a thin mirror and focused onto a photodetector. The signal from the detector

is proportional to the film’s reflectivity, which is calculated to be:

R =2r2(1− cos 2β)

1− 2r2 cos 2β + r4(2.1)

where r = 1−n1+n , n is the average refractive index of the film, β is the retardance β = 2πnNd/λ, N

is the number of layers of the film and d is the layer thickness.

10

In our experiments, we developed a different way to measure the film reflectivity straight

from the camera’s signal. This offers three advantages over using a laser. First, it is more ver-

satile since we can use almost any camera with manual setting capability on any microscope. In

the experiments that need the rotating stage, setting up a laser reflectivity would be practically

cumbersome. Second, the measurement is more robust against the film’s orientation. With the

laser reflectivity, the base for the film holder should be firmly held on the translation stage since a

small change in orientation can offset the reflected beam enough that the photodetector would read

a different signal. The third advantage is a corollary of the second one: calibration with the cam-

era is more straightforward and less frequent. The typical calibration process for laser reflectivity

involves reading the photodetector signal for many different films (≈ 20) to have a large enough

sample to guess the film thickness for each film such that their reflectivity readings can be put on

a parabolic curve. This involves some arbitrariness of assigning the film thickness, and most of the

time, it needs one of the data point to be from a two-layer film for the assignment to be reliable.

This shortcoming is due to the ineffectiveness of using the black glass as a reference, again because

it is hard to make sure the black glass orients the same way as the film. Using the camera does not

have this difficulty, yielding an absolute value for the film reflectivity.

The procedure starts with using ImageJ to measure the brightness in a specified region of

interest for a series of images taken with different exposure times shown in Fig. 2.3(a). Fig. 2.3(b)

shows the resulting plot of brightness against the exposure time, the slope of which is a measure

of the film reflectivity, and after calibrating against the measurement with a black glass, which is

assumed to have the reflectivity of 0.04, we get the absolute value of the film’s reflectivity. Fig.

2.3(c) shows the reflectivity measurements for three colors plotted at the layer numbers guessed

from the reflectivity for green, and yet the measured reflectivity values for the other two colors still

lie close to the corresponding predicted curves. Thanks to the testing and calibrating efforts from

Markus Atkinson, we now know this procedure is as reliable as the laser reflectivity confirmed by

the fact that the measured reflectivity values of thin films follow the approximating parabolic curve

shown in Fig. 2.3(d).

11

Re

fle

ctivity

0

0.02

0.04

0.06

0.08

0.1

Number of layers0 5 10 15

Collection of measured reflectivities

Predicted reflectivity for green

Inte

nsity (

a.u

.)

0

20

40

60

80

100

Exposure time (s)0 0.005 0.01 0.015

Red channel, k = 3440

Green channel, k = 4960

Blue channel, k = 5380

Re

fle

ctivity

0

0.005

0.01

0.015

0.02

0.025

Number of layers0 1 2 3 4 5 6

Red

Green

Blue

(a) (b)

(c) (d)

ROI

Figure 2.3: (a) A region of interest (ROI) is chosen in a digitized image of a freely suspendedsmectic film to calculate the average brightness for each of the red, green and blue channels. (b)Brightness as a function of exposure time for each color channel. (c) After calibration using a blackglass, the absolute value for reflectivity can be calculated and plotted along the predicted curvesgiven by Eq. 2.1. (d) A collection of 29 reflectivity measurements from the green channel showsthat they are well described by the predicted curve.

Chapter 3

Crossover Between 2D and 3D Fluid Dynamics in the Diffusion of Islands in

Ultrathin Freely Suspended Smectic Filmss

The Stokes paradox, that moving a disc at finite velocity through an infinite two dimensional

(2D) viscous fluid requires no force, leads, via the Einstein relation, to an infinite diffusion coefficient

D for the disc. Saffman and Delbruck proposed that if the 2D fluid is a thin film immersed in a 3D

viscous medium, then the film should behave as if it were of finite lateral size, and D ∼ − ln(aη′),

where a is the inclusion radius and η′ is the viscosity of the 3D medium. By studying the Brownian

motion of islands in freely suspended smectic films a few molecular layers thick, we verify this

dependence using no free parameters, and confirm the subsequent prediction by Hughes, Pailthorpe

and White of a crossover to 3D Stokes-like behavior when the diffusing island is sufficiently large.

3.1 Saffman-Delbruck theory of diffusion in thin fluid films and its extensions

In the early 1970’s, Saffman and Delbruck (SD) considered the problem of calculating the

diffusivity of a protein included in a thin, bilayer lipid membrane, and realized that both the

viscosity of the membrane η and that of the surrounding fluid η′ affect the dynamics of the protein

[58]. In what became a classic paper, Saffman presented a full fluid mechanical description of the

translational motion of an inclusion in a membrane surrounded by another viscous fluid, using no-

slip boundary conditions [57]. Saffman pointed out that the viscosity of the surrounding fluid η′,

however small compared to the viscosity of the membrane η, provides a significant contribution to

the momentum dissipation. The characteristic Saffman length lS = ηh/2η′ is found by balancing the

13

drag from the membrane with that of the surrounding fluid, and is proportional to the membrane’s

thickness h. The Saffman length represents the distance beyond which fluid flow in the membrane

caused by the motion of the inclusion, which normally shows a long-ranged logarithmic decay, may

be ignored [63]. In this picture, an inclusion of radius a may be thought of as carrying with it a

part of the membrane extending a distance lS around it, increasing the effective drag and leading,

in the case a lS , to a finite mobility

µ =1

4πηh

[ln

(2lSa

)− γ

], (3.1)

where γ is the Euler constant.

Hughes, Pailthorpe, and White (HPW) extended Saffman’s model to determine analytically

the mobility of an inclusion of arbitrary radius in a 2D fluid [27]. They predicted that for a lS ,

the mobility would exhibit 3D-like behavior, varying as µ ∼ 1/a but with a prefactor different from

the bulk theory [21]. Petrov and Schwille (PS) derived a simple but accurate approximation to the

rather complicated HPW mobility expression [49]:

µ =1

4πηh

ln(2ε )− γ + 4επ −

ε2

2 ln(2ε )

1− ε3

π ln(2ε ) +c1εb1

1+c2εb2

, (3.2)

where ε = a/lS is the reduced radius and c1 = 0.73761, b1 = 2.74819, c2 = 0.52119, b2 = 0.61465

are constants. The reduced mobility m = µ(4πηh) derived from Eq. 3.2 is shown as the solid curve

in Fig. 3.1, predicting asymptotic 2D behavior m ∼ ln(2/ε) for ε 1, and 3D behavior m ∼ 1/ε

for ε 1.

The SD-HPW-PS picture has been widely used to interpret experimental measurements of

the 2D hydrodynamics of inclusions in fluid membranes [9, 52], and to extract membrane viscosity

[10] or inclusion size [24] from diffusion data. However, in some cases the measurements in the

SD regime give a much stronger dependence of µ(a) on a than predicted, leading to doubts about

the validity of the theory in describing complex membrane systems [18]. Here we exploit the

unique character of fluid, freely suspended films of smectic liquid crystal to extend parameter-free

14

red

uce

d m

ob

ility

, m

=μ

(4π

ηh

)

0.1

1

reduced island radius, ϵ=a/lS0.01 0.1 1 10

N=3N=4N=5theory

D (μ

m2/s

)

0

0.5

1

1.5

2

a (μm)1 10 100

2D

3D

ln(2/ϵ)-γ

π/(2ϵ )

Figure 3.1: Island mobility as a function of radius in smectic films. Experimental data for threedifferent film thicknesses (N = 3, 4 and 5) and a range of island radii follow the predictions ofthe SD-HPW-PS theory (solid curve) and illustrate the crossover from 2D to 3D behavior withincreasing island radius. Both mobility and radius are scaled to be dimensionless. The dottedcurves show the predictions of 2D and 3D theory, valid in the limits a lS and a lS respectively.The inset shows the diffusion coefficients of islands of different radii in an N = 3 film. The errorbars correspond to the standard deviations obtained by analyzing several subsets at each radiusextracted from a given trajectory.

15

quantitative testing of the SD-HPW-PS model into the SD regime.

3.2 Diffusion of islands in 8CB smectic A films

Fluid smectics are particularly well suited for measuring diffusion in 2D because they form

homogeneous, ultrathin freely suspended films that are quantized in thickness (consisting of an

integer number of smectic layers, which can be selected), and are stable for many hours (or days

if not disturbed) [69]. Films as thin as two layers (h = 6.3 nm) can be created with an area

as large as 10 cm2, making them the thinnest stable condensed matter system for probing 2D

hydrodynamics. Smectic A films have positional ordering in the direction perpendicular to the

film, but are liquidlike within each layer, making them ideal for studying 2D physics in general

[69, 38] and 2D hydrodynamics in particular [19].

In our experiments, we observe the diffusion of mobile inclusions called islands, thicker,

pancake-like domains with more smectic layers than the surrounding film, sketched in Fig. 3.2.

These disk-shaped regions are made of the same material as the film, and can be manipulated

using optical tweezers [44, 43]. Moreover, the island radius a can be varied over a wide range

(4 . a . 100 µm) that brackets the typical Saffman lengths in this system (for example, lS = 8.4 µm

in a two-layer film), enabling the study of the crossover region between 2D and 3D behavior.

Islands, once made, have constant thickness and, in 8CB films, a radius that decreases only very

slowly over time, enabling the measurement of translational diffusion coefficients using conventional

video microscopy. The mobility is derived from the diffusion coefficient using the Einstein relation

µ = D/kBT .

While the mobilities of inclusions in thin membranes have been studied experimentally before,

the crossover between 2D and 3D behavior when the size of the inclusion is varied has not previously

been explored. Prasad and Weeks recently reported a crossover of a different nature in soap films,

demonstrating that a particle embedded within the film shows a transition from 2D to bulk 3D

behavior when the film thickness is made much greater than the particle diameter [52].

The liquid crystal used in our experiment is 8CB (4′-n-octyl-4-cyanobiphenyl, Sigma-Aldrich),

16

N = 12

N = 4

Figure 3.2: A 35 µm-radius island in a freely suspended smectic A film of 8CB. The film and islandare an integer number of layers thick, and are separated by an edge dislocation loop (•), whichprovides a line tension and gives the island its circular shape. The inset shows in cross section anisland of radius a in a smectic A film of thickness h. The rodlike 8CB molecules are arranged in 2Dfluid layers and are oriented along the layer normal. The low climb mobility of the edge dislocationensures that the material in the surrounding film diverts around the diffusing island. 8CB has the

phase sequence: Crystal22C←→ Smectic A

33.5C←→ Nematic40.5C←→ Isotropic.

17

y (

μm

)

40

60

80

100

120

140

x (μm)60 80 100 120 140 160 180

(a)

t = 0

t = 3 min

Figure 3.3: Island diffusion in a smectic A liquid crystal film. (a) Three-minute trajectory of a13 µm-radius island diffusing in an N = 4 layer 8CB film, with the inset showing a ten-secondextract. The trajectory shows both Brownian motion and systematic drift. The location of thecenter of the island is determined to an accuracy of ±10 nm. The lower plots show the distributionsof displacements measured between two points along this trajectory separated by time intervals of(b) 0.167 s and (c) 1.67 s. As the time interval increases, the distribution shifts away from theorigin and spreads out, with both the distance of the center of the distribution from the originand the variance of the distribution increasing linearly with the chosen time interval. The rate ofchange of the center’s position gives the drift velocity while that of the variance gives the diffusioncoefficient.

18

which has an in-plane viscosity of η = 0.052 Pa · s [60] at temperature T = 22, and a layer thickness

d = 3.17 nm [12]. The air is assumed to have a viscosity η′ = 1.827 · 10−5 Pa · s [54]. Films are

spread at room temperature across a 1 cm-diameter circular hole in a glass cover slip. Islands are

then created by blowing obliquely either on the thicker parts of the film or on the meniscus along

the edge of the film holder. This process yields islands of 1 µm − 1 mm radius, but we typically

select those in the range 4 − 100 µm for ease of observation. We can consistently make films of

N = 2 to eight smectic layers, with Saffman lengths ranging from 8.4− 33 µm. The reflectivity of

thin smectic films (N . 15) is quadratic in the thickness, giving excellent contrast between islands

and the background film as shown in Fig. 3.2, and providing a convenient way of determining the

number of layers, a measurement typically made using a laser [55, 42]. In the present experiments,

we used the intensity of the green channel of the calibrated video camera signal to measure the

reflectivity.

The films are enclosed in a sealed chamber in order to minimize disturbances from the sur-

rounding air, and are leveled on a goniometer to within 0.5°. This allows us to record the motion

of a selected, isolated island with a video camera for up to ten minutes (acquiring ∼ 18, 000 frames

at 30 frames per second) without its drifting out of the field of view. Because of the lower pressure

in the film meniscus, the selected island gradually shrinks over the course of the experiment. This

enables us to determine the diffusion coefficient for islands of different radii by analyzing sections

of longer video clips in which the radius typically decreases only by a few percent.

The optical contrast between an island and the surrounding background film makes it easy

to identify and track the islands by thresholding the video frames and then using image analysis to

determine the position of the center of the island and its size. A typical trajectory of the center of

an island is shown in Fig. 3.3a. Comparison with video recordings of 10–40 µm-diameter stationary,

circular disks etched on a glass slide using electron beam lithography suggests that the tracking

method is accurate to within 10 nm, far better than the nominal resolution of the imaging system,

which is about 250 nm.

To measure the diffusion coefficient of an island directly from the trajectory of its center, we

19

need to distinguish systematic drift from Brownian motion. Since islands generally have very small

diffusion coefficients [on the order of 0.5 µm2/s, see Fig. 3.1], even drifts as small as 0.1 µm/s may

add a significant [O(60%)] systematic error to a typical 1000-frame trajectory if they are neglected.

Larger islands are especially susceptible to this kind of artifact since their random diffusion is slower

and they drift faster if the film is not properly leveled.

Systematic drift may be analytically separated from diffusion as follows. Let us consider all

displacements between pairs of points on the island trajectory that are separated by a prescribed

time interval. If we shift those displacements so that they have a common origin, they form a

distribution in which the mean displacement corresponds to the net drift in this time interval,

while the variance gives the diffusion [Fig. 3.3b,c]. Assuming that the drift velocity is constant

over the entire trajectory, we can extract the drift velocity and the diffusion coefficient respectively

from the rates of change of the mean and variance with time interval. Fig. 3.4 confirms that the

step distribution is indeed Gaussian, and thus it is valid to take the variance about the mean as

the measure of diffusion.

We verified the accuracy of this drift extraction method using simulated data obtained by

generating random walk trajectories comprising diffusion steps with random length and direction,

themselves obtained from random walk trajectories with a prescribed number of steps of fixed size.

Constant drifts are then superimposed onto the diffusive motion. Both the diffusion coefficients

and the drift velocities are extracted correctly for over more than two decades of drift velocity, as

shown in Fig. 3.5.

Smectic A islands are fluid and are therefore susceptible, in principle, to shape fluctuations.

Since the island position is obtained by computing its centroid, deviations from perfect circularity

of the boundary could be misinterpreted as a change of position. The roughness of the perimeter of

an island of radius a = 10 µm and line tension λ = 80 pN due to thermal fluctuations is estimated

to be on the order of δ '√

(kBT · a)/(πλ) ≈ 10 nm, a variation which is too small to observe in

our system.

20

pro

ba

bili

ty d

en

sity (μ

m-1)

-0.5

0

0.5

1

1.5

2

displacement (μm)0 0.2 0.4 0.6 0.8

N=3

N=4

N=5

pro

ba

bili

ty d

en

sity (μ

m-1)

-1

0

1

2

3

4

5

displacement (μm)-1 -0.5 0 0.5 1

N=3

N=4

N=5

pro

ba

bili

ty d

en

sity (μ

m-1)

-1

0

1

2

3

4

5

displacement (μm)-1 -0.5 0 0.5 1

N=3

N=4

N=5

(a)

(b)

(c)

Figure 3.4: Probability distributions of step size (a), displacement in x (b) and y (c) in the islanddiffusion trajectories in N = 3, 4 and 5 films at 30 fps framerate. Solid lines are 2D Gaussiandistribution fits.

21

ca

lcu

late

d d

rift v

elo

city (

μm

/s)

0.001

0.01

0.1

1

10

ca

lcu

late

d d

iffu

sio

n c

oe

ffic

ien

t (

μm

2/s

)

1

actual drift velocity ( μm/s)0.1 1 10

vcalculated = vactual

D = 0.5 μm2

Figure 3.5: Tests of the analysis method using simulated random walk data with a fixed diffusioncoefficient of 0.5 µm2/s, and drift velocities ranging from 0.05 µm/s to 5.0 µm/s. The valuesextracted for both the drift velocity () and the diffusion coefficient () are in good agreementwith the parameters used to generate the test data. The error bars correspond to the standarddeviations obtained from analyzing a set of different random walk sequences generated using thesame parameters.

22

3.3 Confirmation of SD/HPW theory

We now compare our experimental results with the predictions of the SD-HPW-PS theory.

The mobilities of 7–20 layer-thick islands on 8CB films of three different thicknesses (N = 3, 4,

and 5) are shown in Fig. 3.1. The scaled experimental measurements, plotted directly with no free

parameters, are in very good agreement with theory, collapsing nicely onto the theoretical curve

given by Eq. (3.2). In performing this comparison, we make three key assumptions. First, since

permeative flow in smectics away from the nematic–smectic A phase transition is very slow [50], as

evidenced in this case by the very slow change in island radius versus time, we can, for the purposes

of this measurement, take the excess island layers as having a fixed number of molecules. If the

flow of material through the boundary of the island is negligible on the experimental timescale, the

island may be viewed as a thicker, isolated region bounded by an edge dislocation loop. Second,

since the islands used here are substantially thicker than the surrounding film, and consequently

much more viscous, we neglect fluid flow within the islands, assuming that the islands behave as

solid disks under no-slip boundary conditions. Third, we assume that the mass density and the

viscosity of the liquid crystal material in a thin film are the same as in the bulk.

In an independent measurement of island drift induced by gravity in slightly tilted films,

we observed a terminal drift velocity proportional to the component of the island’s weight along

the film plane and with a magnitude corresponding to a film viscosity of η ≈ 0.044 Pa · s, which

is within 15% of values reported in the literature. We looked for any dependence of the island

mobility on island thickness by measuring the diffusion coefficients of 8- and 17-layer islands of

similar size on a three-layer 8CB film but found no significant difference in their (scaled) mobility.

3.4 Conclusion

In summary, the diffusion of circular islands in ultrathin, freely suspended smectic A liquid

crystal films is in excellent agreement with the SD-HPW-PS theory of Brownian motion of inclusions

in thin 2D membranes immersed in a 3D fluid. The freely suspended film system allows the

23

measurement of diffusion under systematic variation of both the radius of the inclusions and the

thickness of the membrane. The experimental parameters are such that we are able to measure

precisely the mobility of islands with radii both smaller and larger than the Saffman length. The

results confirm that this indeed marks a crossover between 2D and 3D hydrodynamic behavior.

In future experiments, we plan to test the SD-HPW-PS theory over a wider range of reduced

radius ε = a/lS , especially in the small ε regime, which can be achieved either with thicker films

(large lS) or with smaller a. Since freely suspended smectic films have proven to be an excellent

system for investigating 2D diffusion in fluids, we should also be able to use them to explore such

phenomena as rotational diffusion and the hydrodynamic interactions of multiple inclusions.

This work is supported by NASA Grant No. NAG-NNX07AE48G, NSF MRSEC DMR

0820579, and NSF DMR 0606528. We would like to thank Howard Stone for helpful comments

and suggestions, and Youngwoo Yi for fabricating the patterned substrates used for evaluating the

island tracking algorithm.

Chapter 4

Surface Tension of a Freely Suspended Molecular Monolayer

Measurement of the surface tension of the thinnest substrate-free liquid film, a fluid molecular

monolayer freely suspended in air, is reported. A differential technique enables direct comparison

of the surface tension of a single fluid smectic liquid crystal layer in a symmetric environment with

vapor on both sides, to that of bilayer and thicker films, having surfaces with vapor on one side

and liquid on the other. Differences from the bulk phase surface tension γ are small, ∼ 10−3γ,

showing that the end-for-end polar ordering of even strongly asymmetric molecules at the surface

is weak.

4.1 Surface tension of thin smectic films

Fluid smectic liquid crystals (LCs) are three dimensional (3D) phases of rod shaped molecules

organized into periodic stackings of molecular layers, where each layer is a 2D liquid. At the air / LC

interface the smectic layering exhibits a strong tendency for the layers to be parallel to the interface,

and thus for the molecules at the interface to be in the same smectic layer. This organization enables

the formation and stabilization of freely suspended smectic films consisting of an integral number,

N , of smectic layers, where, for some materials N can be as small as N = 1, that is, smectic

films of single liquid layers can be made. Such substrate-free ultrathin fluid molecular films are of

fundamental interest in soft matter nanoscience, serving as fluid systems of reduced dimensionality

for the study of fluctuation, symmetry, and interface effects [69, 39, 47]. Here we show that single

and few layer smectic films can be used to measure the surface tension of a substrate-free fluid

25

molecular monolayer.

In smectic layers the molecular long axes are narrowly distributed in orientation about their

mean orientation, given by the director, n, which can be along the layer normal (smectic A), or

tilted at an equilibrium angle (smectic C). Thus, if the molecular rods are polar, that is their

opposite ends are chemically dissimilar, then the interface of the bulk phase is a structure in which

the principal choice for each molecule is which end to direct out toward the air and which in toward

the liquid crystal. The surface layer then becomes a realization of a 2D Ising model, with each

molecule having a binary variable σ = ±1, giving its end-to-end state and a field applied to reflect

the asymmetry of the environment. By contrast, in the freely suspended monolayer, having air on

both sides makes end-for-end flips energetically equivalent and the system is a zero field 2D Ising

system, with the in-plane interactions determining the states and size of the domains of common

σ [3].

In the context of smectic LCs, a particularly interesting end-for-end chemical dissimilarity is

that of -CHn, -CFm, having a hydroalkyl tail on one end of a rod-shaped molecule and a fluoroalkyl

on the other, since these tails exhibit large surface tension differences in self assembled monolayers:

γ ≈ 22 erg/cm2 for hydroalkyl surfaces, while γ ≈ 6 erg/cm2 for fluoroalkyl monolayers [70]. On

this basis one might expect the surface layers of a material with -CHn, -CFm asymmetry to be

observably polar with the orientation having -CFn at the surface preferred because of its lower

surface energy. This kind of surface polarity should in turn lead to odd-even effects in the surface

tension, as the N = 1 film is by symmetry nonpolar, and the N = 3, 5, 7... layer number films are

forced to have an nonpolar layer at their center.

Mach et al. have probed hydro-fluoro substitution effects in smectic layering by directly

measuring the surface tension of freely suspended smectic A films [32, 33], studying a variety

of molecules of different F-H tail composition for films of two or more layers in thickness, in

the range 100 ≥ N ≥ 2. Single layer films were not stable enough to be studied, as they are

susceptible to rupture due to thermally generated holes [55]. It was found that increasing the

presence of fluoroalkyl groups in the tails decreased the surface tension, from γ ≈ 11 erg/cm2

26

for largely -CFm to γ ≈ 23 erg/cm2 for largely -CHn. Mach et al. also measured γ(N) for a

variety of materials, finding no systematic variation of γ(N) with N to within their 1% uncertainty.

Of particular relevance here is that they studied γ(N) of the -CHn, -CFm asymmetric material

H(10)F(5)MOPP for N ≥ 2. Their data show that for N = 2, 3, 4, 5 films, where any odd-even

effect would be expected to be the largest, the amplitude r of the odd-even variation, measured as

r = [γ(2) + γ(4) − γ(3) − γ(5)]/[γ(2) + γ(4) + γ(3) + γ(5)] is r ≈ (3 ± 10) · 10−4, i.e. too small

to be observed. This motivates making a comparative measurement of γ(1) and γ(2) in materials

with -CHn, -CFm asymmetry, since the largest difference in surface polar ordering can be expected

to be that between N = 1 and N = 2 films.

4.2 1D interface motion in smectic films

We carried out a comparative study of the surface tensions γ(N) for freely suspended films

with N in the range 7 ≥ N ≥ 1 of the -CHn, -CFm asymmetric material 3M-EX-900084 (Fig. 4.2b

inset), by observing the in-plane motion of the 1D interface lines where N changes in the film plane.

Films were stretched across a 3 mm x 1 mm hole and are kept in a hot stage stabilized at a set

temperature within 0.01 K in the smectic C (SmC) phase. They are stable for hours, although

most of the domain dynamics reported here occur within a few seconds to minutes after they are

made. Dynamic behavior is captured by a video camera via a reflected light microscope under

with a 200 W high-pressure mercury lamp as the light source. The optical reflectivity of thin films

(N ≤ 15) is quadratic in the thickness, providing contrast between regions of different N , and a

convenient way of determining the number of layers, a measurement typically made using a laser

[69]. High sensitivity with low dark current imaging of the domains enables the visualization of

one-layer differences in N .

Many hundreds of films were observed and with practice it was possible to draw films with

just a single growing or shrinking island or hole consisting of a 1D interface loop. These events,

about 120 in all, were quantified by measuring the loop radius r(t) vs. time. Fig. 4.2a shows typical

results for r(t) of an I1O2 hole [denoted as IjOk, where j and k give N in the region inside (I) and

27

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

0.5 mm

(j) (k)(l)

island A

island B

0 1 6

8.5 10.73 10.87

11 12 15

18 21.5 t = 24.7 s

Figure 4.1: Dynamics of a monolayer (N = 1) hole (dark region) containing (N = 2) islands A andB (bright circular regions) in a N = 2 smectic freely suspended film of 8CB. The islands appearand grow within the shrinking hole (a-d), eventually merge with the background film (e-g), andthe hole continues to shrink and disappear (h-l). During this process, material is drawn from themeniscus at the edge into the film.

28

|v1

2| (μ

m/s

)

0

50

|v21| (μm/s)0 100

radiu

s (

μm

)

-100

0

100

200

300

400

500

600

700

time (s)0 10 20 30 40

island A

island B|v12| = |v21|/2

|v21|

(b)

(a)

(c)

slope = 1/2

C

O

O CH2C7F15OC6H13O

radiu

s (

μm

)

0

100

200

300

time (s)-20 -10 0

|v24| = 5.4 μm/s

|v23| = 22.4 μm/s

|v21| = 102 μm/s

Figure 4.2: (a) 1D interface dynamics of N = 1 hole (I1O2) and N = 3, 4 islands (I3O2, I4O2)in N = 2 films. (b) Radii of hole (2) and two islands () versus time determined from the videorecording in Fig. 1. Black solid circles (4) show the data for the hole radii corrected for theislands’growth (red arrow) and for the islands’merging with the background film (orange arrow).

Inset: EX-900084 molecule (bulk phase diagram: Crystal71C←→ Smectic C

79C←→ Smectic A136C←→

Isotropic). (c) The growth speed of N = 2 islands in N = 1 background (I2O1) is half of theshrinking speed of N = 1 hole in N = 2 background (I1O2).

29

outside (O) the domain, respectively], and for the radii of I3O2 and I4O2 islands. Fig.4.2a shows

that the loop growth or shrinking velocities are nearly independent of the loop radius, i.e. constant

in time for a given loop, a remarkable common feature over the whole data set. Fig. 4.4 shows

the resulting velocities v12, v32, v42, v52, and v62, measured for I1O2 to I6O2 loops, respectively,

as a function of film temperature T . Here vjk is defined to be positive if the loop motion is in

the direction that moves material from the film and into the meniscus. Thus N = 1 holes and the

N = 3, 4, 5, 6 islands all shrink on an N = 2 film, the shrinking N = 1 hole pulling material onto

the film and the shrinking islands letting material flow off of the film. The I1O2 holes are the only

loops that can pull material onto the film. All other loop motion events remove material to the

meniscus.

Some of the events are more complex and revealing. The sequence of images in Fig. 4.1

shows a typical 1D interface motion scenario following the quick drawing of a film (in ∼ 1 sec).

When first formed, it is N = 2 thick (brighter region) with a N = 1 domain (monolayer hole) in the

middle (dark region). The monolayer hole shrinks, and while it does two N = 2 domains (islands)

appear and grow, eventually merge with the background N = 2 film. The hole then continues to

shrink and disappears. During this process, material is pulled from the meniscus bounding the

freely suspended film, which is opposite of the usually observed tendency of the film to thin [40].

This thinning tendency is due to the negative curvature of interface between the meniscus and the

air, which lowers the pressure within the meniscus and thus draws material from the film.

4.3 Determining surface tension difference between between monolayer and

bilayer domains

The radii of the shrinking monolayer I1O2 hole and of the two growing bilayer (I2O1) islands

are plotted in Fig. 4.2a with the black open triangles showing the hole radius corrected for the

effect of islands’ growth by adding the areas of the islands to that of the hole to account for the

fact that the islands make the hole shrink faster in order to supply material for their growth. When

island A (the bigger island) merges with the background film, the hole decreases stepwise in radius

30

but then continues to shrink with the same velocity, an event that can be corrected for by shifting

the post coalescence data to later time. Again the velocities are nearly constant in time, although

in this case v12 tends to increase when r gets very small.

We analyze these data using the model of Oswald and Pieranski which balances energy change

and dissipation due to interface motion. The IjOk interface velocity for a loop of radius rjk(t), vjk,

is given by:

vjk = ±drjkdt

= mjk [∆P |k − j|d+ (2∆γjk + Ejk/rjk(t))sign(k − j)] (4.1)

defined such that vjk > 0 if the motion returns material to the meniscus, where: (i) the δPd,

∆γ , and E/r terms are effective forces normal to the moving interface, due respectively to the

negative pressure in the meniscus, δP , the difference in surface tension inside and outside the

domain, ∆γjk = γk − γj , and the line tension Ejk of the interface if it is curved [radius r(t)]; (ii)

ηjk is a dissipative coefficient with dimensions of viscosity determined by the dynamics of IjOk

interface motion and by the dissipation for transport of material through the k layer thick film-

meniscus boundary. Observation of a constant loop velocity implies that the crossover radius below

which line tension effects become important, rc = E/|δP (k− j)d+ 2∆γ|, is small compared to the

smallest measurable loop radius. Ignoring line tension the loop velocities are constant and given

by |v| = |δP (k − j)d+ 2∆γ|/η. Thus rc can be written as rc = E/η|v| indicating that line tension

effects should be most evident for slowly moving interfaces.

The solid black curve in Fig. 4.2b is a linear fit to give the monolayer hole (I1O2) shrinking

speed v12, and the solid blue curves are lines of slope v21/2 plotted over the island (I2O1) growth

radii r21(t). Thus, as shown in Fig. 4.3c for events having N = 2 islands on N = 1 holes the island

growth velocity v21 is half that of the hole shrinking velocity v12. This result implies that η12 = 2η21

in Eq. 4.1, since the island and hole interface motion are driven by the same δP and ∆γ. This ratio

can be understood only in the context of considering the molecular scale mechanism of interface

motion. Because a film area of a given N is essentially incompressible the motion of a single IjOk

loop requires a physical process at the IjOk interface whereby in each short time interval some set

31

of molecules is changed from N = j to N = k or vice versa.

Fig. 4.3 depicts local dynamical processes that converts N = 1 to N = 2 to give either

(a) I1O2 interface motion with velocity v12 or (b) I2O1 motion with velocity v21. In each case

the activated event sketched in Fig. 4.3c, converts molecular pairs between metastable N = 1

and N = 2 states, from molecules next to each other to molecules on top of each other, over a

transition state barrier of energy EB. Transitions are driven by the bias ∆E12 = (δPd+2∆γ12)a2,

the energy difference per molecule between the two states across the interface, where a2 is the area

per molecule in a layer: Assuming this two-state system has a trial rate for barrier crossing τt, the

barrier crossing time is τ = τte−EB/kT , the bias ∆E12 induces conversion from the higher to lower

energy state at a rate c = −∆E12/(4kTτ), which is the same for both I1O2 and I2O1. For the I2O1

case Fig. 4.3b shows that the vacancy left by the molecular flipping from N = 1 to N = 2 molecules

is filled by the N = 1 side. Each flip advances the interface by a, giving the I2O1 island growth

velocity v21 = −ca (the minus sign comes from the velocity sign convention). However, for the I1O2

hole shrinking (Fig. 4.3a), the vacancy left by the flipping is filled by the N = 2 side, advancing the

interface by 2a, one from the result of the flip and the other drawn from the N = 2 background,

giving an I1O2 velocity v12 = −2ca. Thus, the ratio v12/v21 = 2 is a geometrical factor that is

required if the interface motion is controlled by the rate of local flip events at the interface. If the

interface motion were instead controlled by dissipation at the meniscus, i.e. limited by the number

of events in which a molecule moves onto the film, then we would have v12 = v21, since, neglecting

curvature, all interface areas would be equivalent. Thus, the “factor of two” indicated that the

interface velocity is dictated by local dynamics, not by the necessity of drawing material from the

meniscus to fill vacancies. Given this, the dissipative coefficient η12 can then be obtained from

the two state model above by equating (∆E12/2kTτ)a = −2ca to v12 = η12(∆E12/a2), yielding

η12 = (2kT/a3)τ . Writing 2kT/a3 as a shear elastic constant G we then get η = Gτ , the standard

Maxwell expression for shear viscosity limiting the shear displacement corresponding the N = 1 to

N = 2 flip sketched in Fig. 4.3, suggesting that η12 should be close to a smectic A shear viscosity.

This can be confirmed in the material 8CB, in which measurements of the ratio η12 = v(R)/[δPd/R],

32

Figure 4.3: (a) N = 1 hole shrinks because of an elementary conversion from two N = 1 moleculesto one N = 2 pair (each molecule in the pair occupies a place in one layer). The N = 2 regionadvances two-molecule-wide distance freed up by the two N = 1 molecules towards the center ofthe hole after the conversion. One molecule width is occupied by the converted N = 2 pair whilethe other is occupied by a new pair drawn from the meniscus. (b) Island grows also because ofan elementary conversion from two N = 1 molecules to one N = 2 pair. In this case, one emptyslot is occupied by the converted pair, now residing in the N = 2 island, and the other by theadvancing N = 1 region. Thus for the same conversion rate, the island boundary moves half thedistance compared to that of the hole boundary. (c) Two-state system in which pairs of side-by-sidemolecules in state 1 overcome the barrier EB to be on top of one another in state 2 that is ∆E12

lower than state 1.

33

in a regime where meniscus dissipation is negligible gives η12 = 0.6 g/cm · s at T = 25C, while the

viscosity for in-plane shear is ηs = 0.5 g/cm · s [51]. Extrapolating the 8CB viscosity to T ∼ 80C

gives an estimate of ηs ≈ 0.1 g/cm · s, nearly that of other SmA’s, such as CBOOA [62] in this T

range, so we will take η12 = 0.1 g/cm · s as an estimate for the I1O2 drag coefficient η12.

We return now to the basic velocity data of Fig. 4.4, which shows that on two layer films,

loops having NI = 1, 3, 4, 5, and 6 all shrink, the NI = 1 (3, 4, 5, 6) cases corresponding to v negative

(positive) since they move material out of (into) the meniscus. Since the sign of vjk depends only on

the sign of the driving energy [δP (|k− j|)d+2∆γsign(k− j)], in which the first (Laplace pressure)

term can only be positive, an immediate conclusion is that for the I1O2 case, ∆γ12 = γ2 − γ1 is

sufficiently negative to cancel the Laplace pressure, make the driving energy negative and enabling

the interface to pull material back onto the film. The velocity of I1O2 is therefore negative. Fig.

4.4 shows that the velocity data for a given T and IjOk have a rather broad spread of values, much

larger that the uncertainty in any given velocity determination from the slope of the rjk(t) data

(Figs. 4.2a, 4.4b inset), and attributable to film-to-film variation in meniscus curvature R and the

resulting variation in δP = 2γ/R. This variation is particularly broad for the I1O2 data where,

because the interface speed v12 is high so that the hole lasts only a few seconds, the films must

be drawn very rapidly. In Fig. 4.4b, since the δP |k − j|d term can only increase vjk, the points

of most negative v for each IjOk are those the least affected by Laplace pressure. For the I1O2

data these consistently fall along a smooth curve (magenta line in Fig. 4.4b) indicating that in this

limit the effects of δP on velocity are weak and that v12 is driven primarily by ∆γ12. Applying Eq.

4.1 we then have v12 = ∆γ12/η12 and, using a typical v12 = −150 µm/s and the estimate above

for η12 = 0.1 g/cm · s, we obtain |∆γ12| = 0.0015 erg/cm2 ∼ 10−4γ, a very small surface tension

difference, in spite of the high velocities of the I1O2 interface. For the IkO2 events the interface

motion is much slower and the films can be drawn more slowly to give larger R, smaller δP , and

smaller variation in δP . If we assume the same mobility for the I3O2 case as for the I1O2, then the

much smaller velocity, some part of which must come from δP , indicates that |∆γ32| is yet smaller

than |∆γ12|. Since the δP |k − j|d term increases with increasing |k − j|, the even smaller vk2’s for

34

v (

μm

/s)

-5

0

5

10

15

20

25

30

35

T (0C)70 80 90

v32

v42

v52

v62

(a)

v (

μm

/s)

-200

-100

0

T (0C)70 80 90

v32

v42

v12

2v21

(b)

Figure 4.4: (a) Velocities of N = 3, 4, 5and 6 islands in N = 2 films. The blue (green) band marksthe region that is within 2 standard deviations of the mean velocity measurements of I3O2 (I4O2)islands. (b) Velocities of I1O2 holes (black solid triangles) and the growth velocities of I2O1 islands(red open rectangles) scaled by 2, which are in general much faster than I3O2 and I4O2 velocities.Velocities are determined from linear fits yielding uncertainties smaller than 0.5%, which for I1O2

are less than the thickness of the bar circled by the dashed line. The magenta line marks the limitof v12 when the effect from δP (which makes the velocity more positive) is small.

35

k = 4, 5, and 6 indicate a substantially growing dissipation for interface motion associated with the

multiple Burger’s vector edge dislocation constituting the larger k interfaces.

Finally, with the help of Fig. 4.3, we discuss the interface structures of the N = 1 and

N = 2 film leading to ∆γ12. As noted in the introduction, a binary variable σ = ±1 can be

associated with each molecule giving its end-to-end orientation (σ = 1 exposes the -CF3 tail end to

the surface and the -CH3 tail end the other layer), with 〈σ〉 = 0 in the N = 1 film and, because of

the polar environment of the surface, 〈σ〉 6= 0 in N > 1 films. We can estimate 〈σ〉 most simply by

considering +1 to -1 single molecule end-for-end flips in the surface layer, estimating the associated

surface energy change per molecule ∆US from the typical van der Waals energy minima employed

in united atom force fields for CF3−CF3 and CH3−CH3 attractions in simulations of fluorocarbon

[59]and hydrocarbon [29]liquids: εCF3 = −58K and εCH3 = −88K per pair. Using the Berthelot

combination rule and assuming, based on the measured ∆γ, that 〈σ〉 ∼ 0 in the adjacent layer, we

obtain ∆US ∼ 43K, leading to an estimate for 〈σ〉 of 〈σUS〉 = ∆US/2kT ∼ 0.06. The corresponding

change in surface tension can be estimated from: (i) the Mach et al. film data in which γ ranges

from γ ∼ 23 erg/cm2 to γ ∼ 13 erg/cm2 as film composition ranges from all hydrocarbon tails to

half hydro- and half fluoro-carbon tails, i.e. dγ/dx ∼ −20 erg/cm2 where x = 〈σ〉 /2; or (ii) from

the decrease of surface tension vs. increasing surface concentration of surface active fluorocarbons

in hydrocarbon fluids, typically dγ/dx ∼ −10 erg/cm2 [35]. The ∆γ corresponding to 〈σUS〉 ∼ 0.06

is then in the range −0.6 > ∆γUS> −1.2 erg/cm2, many orders of magnitude larger than the

observed ∆γ12 = −0.0015 erg/cm2, which corresponds to 〈σ12〉 ∼ 10−4.

It is possible to account for this very small ∆γ12 in several ways. One is taking it to indicate

a very weak response of induced reorientation 〈σ〉, one which would require a strong tendency for

antiparallel ordering of adjacent molecules in the layers, such that only 0.01% of the molecules

are flipped from the antiparallel N = 1 state by the surface effective field. The result would be

the fully interdigitated antiferroelectric ground state sketched in Fig. 4.4. The corresponding

energy cost of flipping a single molecule from an antiparallel ground state, ∆UAF , estimated as

∆UAF ∼ kT ln(−∆γ12/∆γUS) ∼ −6.5kT . This scale is well within the range of change of cohesive

36

energies associated with molecular flips in fluoroalkane-hydroalkanes [56]Furthermore, the fully

interdigitated antiparallel layer structure sketched in Fig. 4.4 has been clearly identified by bulk x-

ray scattering of fluoroalkane-hydroalkane [66]and fluoroalkane-core-hydroalkane [53] smectic liquid

crystals with nearly equal F and H tail lengths, the F-core-H system having a molecular architecture

very similar to that of the molecule studied here. Such antiparallel pairing is also suggested by the

thickness of the surface adsorbed layer of fluoroalkane-hydroalkane solute in hydrocarbon solvent

[35]. Another possibility for the small γ12 is that it is a cancellation wherein, upon going from

N = 1 to N = 2, a decrease in γ due to molecular orientation is opposed by another effect, for

example the expected decrease in the tilt of molecules [23]

4.4 Conclusion

In conclusion, we have prepared freely suspended fluid molecular monolayers of a fluoroalkane-

core-hydroalkane smectic-forming mesogen. Monolayer films are transient, replaced in an few sec-

onds by the bilayer structure via the motion of their mutual interface, the velocity of which can

be used to estimate the surface tension difference between monolayer and bilayer. This difference

is very small, 0.01% of the film surface tension, likely indicating a strong tendency for antiparal-

lel end-for-end ordering of molecules within the layers. This work is supported by NASA Grant

NAG-NNX07AE48G, NSF MRSEC DMR 0820579, and NSF DMR 0606528

Chapter 5

Two-dimensional domain coalescence driven by line tension

The coalescence dynamics of two droplets driven by surface tension has been studied theo-

retically for two-dimensional (2D) systems, and later extended to understand the more complex

three-dimensional (3D) case. Experimentally, when high-speed cameras became available, the 3D

case was the first to be explored due to the lack of a suitable 2D system. This chapter presents

the observations of the coalescence dynamics of disk-shaped domains with 50 µs time resolution

in smectic A freely-suspended liquid crystal films. Right after the coalescence, the system is in

the viscous regime in which the bridge connecting the two coalescing disks expands linearly. The

experimental linear expansion, however, lasts longer than the prediction used in the literature that

includes a logarithmic correction. We show that this logarithmic correction is not needed in our

case, which can be explained by assuming the flow occurs at a length scale equivalent to half of the

bridge gap.

5.1 Introduction

Despite its ubiquitous occurrence in nature, the coalescence of two liquid objects has not

been investigated extensively experimentally due to the lack of suitable image capturing devices.

Although attempts were made to study the falling of drops into liquid surfaces as early as the

late 19th century [64], it was not until the last decade with the advent of sophisticated high-speed

cameras that the field finally took off [36, 16, 1, 2, 68, 65, 5]. It is particularly interesting to

look at the beginning of the coalescence event when the system must be in the viscous regime and

38

crossover to the inviscid one. This crossover can be characterize by setting the Reynolds number:

Re ≡ ρUL/η = 1 where ρ is the mass density, η is the kinetic viscosity, U is the flow velocity and

L is the characteristic length over which the flow occurs. [25, 17, 48, 67, 6].

Hopper pioneered the field by solving analytically the shape dynamics during the surface

tension-driven Stokes flow coalescence of two infinite fluid cylinders with an inviscid exterior liquid

and found that in the viscous regime, the bridge width (the distance between the cusps joining the

two cylinders) expands linearly with a logarithmic correction [25]. Eggers et al. later showed that

Hopper’s 2D result can be extended to 3D systems with the same bridge velocity in the viscous

regime and that it crosses over to the inviscid one in which the bridge expands as the square root of

time [17]. This crossover has been detected recently by measuring the complex impedance between

two liquid hemispheres as they coalesce. The characteristic time is, however, 3 decades longer than

expected and it was explained that previous studies used the incorrect Reynolds number [46].

We present here high-speed observation of the coalescence of flat ultra-thin pancake-like fluid

islands in smectic A freely suspended films. This system has proved to be a useful realization of

2D models to study hydrodynamics since it involves only one component and requires no substrate,

enabling the use of high contrast reflection-mode microscopy.

5.2 Model for droplet coalescence in two dimensions

Hopper has solved for the exact evolution of the boundary of the coalesced cylinder during the

coalescence of two infinite cylinders with inviscid exterior fluid using conformal mapping, a time-

dependent complex function Ω(ζ, t), between the complex ζ plane with |ζ| ≤ 1, and the complex

z-plane describing the two-dimensional cross section of the cylinders [25]. All quantities in the

z-plane are scaled to dimensionless forms: x = x0/R0, y = y0/R0, u = u0η/γ, t = t0/τ , where R0 is

the final radius of the merged cylinder, η is the viscosity of the inner fluid, γ is the surface tension

of the cylinders’ boundaries, and τ = (ηR0)/γ is the characteristic time.

Fig. 5.1(a), which is extracted from Hopper’s paper, shows the progression from the initial

nephroid (sharp cusps) to epitrochoids (rounded cusps). Hopper observed that the shapes of these

39

w=2r2Rc

R

m

(a) (b)

Figure 5.1: (a) Hopper’s exact solution for the coalescence of two infinite cylinders with inviscidexterior fluid [25]. The area bounded by curves is constant and equal to π. The curves correspondto (from inner to outer along the Y = 0 line)ν = 0.9 (t ≈ 0.0295), ν = 0.7 (t ≈ 0.147), ν = 0.4(t ≈ 0.551). (b) Definitions of R, rm, and Rc

40

profiles could be modeled by the mapping ω(ζ) = (ζ − 13ζ

3) with a decreasing |ζ| in the range [0,1]

corresponding to the cusps getting more rounded. However, since this simple function does not

conserve the enclosed area, it is modified as:

z = ω(ζ, t) = B[λ(t)]ω[λ(t)ζ] (5.1)

where B(λ) = λ−1(1 + 13λ

4)−12 , ensuring that the enclosed area is always π.

The shape evolution is now governed by the function λ(t) with λ(0) = 1 referring to the

initial nephroid shape. With changes in variables, ν = λ2 and σ = ζ/|ζ|, the solution is given by

Hopper:

t =1

2π

∫ 1

ν[k(1 + k2)

12K(k)]−1dk (5.2)

z = ω[σ, ν(t)] =1− ν2

(1 + ν2)12

σ

1 + νσ2(5.3)

We can therefore calculate the bridge width from Eq. 5.3:

w = |z(σ = 1, ν)− z(σ = −1, ν)| = 1− ν2

(1 + ν2)12

2

1 + ν(5.4)

which, along with Eq. 5.2, provides the parametric curve (t(ν), w(ν)) which can be used to fit

experimental data. Eggers et al. later extended Hopper’s result to 3D systems beyond Stokes

flow in which the bridge width w ∼ (2γt/πη) ln(τ/t) transitions to w ∼ (2γR0/ρ)1/4√t when the

Reynolds number Re ≡ ργ2t/η3 ∼ 1.

5.3 Island coalescence in thin smectic films

The liquid crystal used in this experiment is 8CB (4′-n-octyl-4-cyanobiphenyl, Sigma-Aldrich),

which has an in-plane viscosity of η = 0.052 Pa · s [60] at temperature T = 22, and a layer thick-

ness d = 3.17 nm [12]. The air is assumed to have a viscosity η′ = 1.827 · 10−5 Pa · s [54]. We make

freely suspended films and generate islands using techniques that have become standard in our

41

t = 0 ms t = 3.6 ms

t = 6.4 ms t = 10.0 ms t = 18.6 ms

t = 0.71 ms 0.00 ms

0.29 ms

0.57 ms

0.86 ms

1.14 ms

1.43 ms

1.71 ms

2.00 ms

2.29 ms

2.57 ms

2.86 ms

3.14 ms

3.43 ms

3.71 ms

4.00 ms

4.29 ms

8.57 ms

12.9 ms

17.1 ms

21.4 ms(a) (b)

R

w

Figure 5.2: (a) A typical coalescence event (starting at t = 0) of thick pancake-like islands (brightcircular domains) in a thin smectic A liquid crystal film background (dark region). The film is fourlayers thick (N = 4, h = 12.8 µm), while both islands have N = 27. The radius of the top islandis 54.9 µm and that of the bottom one is 59.5 µm. Before coalescence, islands are circular due tothe line tension of the island boundaries resulting from the edge dislocation energy. It is this linetension that drives the dynamics of the coalescence. (b) Image sequence showing the expansion ofthe bridge connecting the two coalescing islands. The scale bar is 25 µm in length.

42

lab to get stable films and islands that are suitable for this experiment [44, 43, 69]. We typically

make 3 mm-diameter films of N = 2 to eight smectic layers, and obtain islands with radius in the

range 4 − 100 µm with N ≤ 30. This difference in the thickness of the islands and the film helps

stabilize the islands’ size for up to an hour against the slow draining tendency into the meniscus

(at the edge of the film holder). It also provides good contrast for observing island coalescence (the

contrast ratio between an N = 20 island and an N = 5 film is 10:1) in a range where the thickness

measurements are reliable (for N ≤ 30).

The films are enclosed in a sealed chamber in order to minimize disturbances from the sur-

rounding air, which is not only helpful for coalescence observation, but also for manipulating islands

with the 100 pN strong optical tweezers. This capability is extremely useful since we do not have

to wait for the islands to move closer to each other by random diffusion. When the two islands

do touch, most of the time they do not immediately coalesce unless they are forced together. As

a result, the camera is usually set to a low speed to allow for long video (a couple of seconds) to

be stored in the buffer memory of the camera (Phantom v12.1, VisionResearch). Armed with the

optical tweezers, however, we gain more predictability over the timing of the coalescence events,

thus we can afford to cut down the buffered time to a fraction of a second and set the speed of the

camera as high as 67,000 frames per second (fps) at 512x512 resolution.

A typical coalescence event is shown in Fig. 5.2(a) of two N = 27 islands of roughly the same

size (R1 = 54.9 µm and R2 = 59.5 µm) in a N = 4 film captured at 21,000 fps. Soon after the

islands connect (t = 0), the bridge expands (shown in Fig. 5.2(b) ) as the cusps on either side move

apart. After the cusps disappear (t = 6.4 ms), the merged domain relaxes under the influence of

the line tension of the boundary towards the final circular shape.

During coalescence, the two islands remain along a fixed symmetry axis that goes through

their centers, enabling us to study the dynamics by measuring the intensity profile along the line

that goes through the initial contact point and is perpendicular to the symmetry axis. Fig. 5.3

shows that the intensity profile has a central peak even before coalescence begins, which is an optical

artifact because the pixel at the peak of the scan receives light from more of the surrounding bright

43

Inte

nsity (

a.u

.)

0

5,000

1e+04

1.5e+04

2e+04

2.5e+04

3e+04

3.5e+04

position along the line scan (pixel)0 50 100 150 200 250 300 350

t<=0

0.05 ms

0.29 ms

0.57 ms

0.86 ms

1.14 ms

1.43 ms

1.71 ms

2.00 ms

2.29 ms

2.57 ms

2.86 ms

3.14 ms

3.43 ms

3.71 ms

4.00 ms

4.29 ms

8.57 ms

12.86 ms

17.14 ms

21.43 ms

y = 25,000

threshold

scan

Figure 5.3: Intensity profiles along the line that goes through the initial contact point of the islandsand is perpendicular to the symmetry axis going through their centers (bottom inset). t = 0 isdefined as the time right before a peak shows up on top of the reference profiles, shown in the leftinset as the yellow curves below the that of t = 0.05 ms (green) and t = 0.29 ms (red). Beforethe peak in the intensity profile saturates, the bridge width is measured as the full width at halfmaximum of the peak, whereas after it saturates, the width is simply that at the threshold value.

44

area in the islands compared to pixels that are farther away. We can average the profiles taken from

the images captured before the coalescence and get a baseline to compare the later time profiles to.

The left inset of Fig. 5.3 shows the peaks of the profiles at t = 0.05 ms and t = 0.29 ms (1

and 6 frames after coalescence, respectively) on top of the reference. As the coalescence progresses,

the profile peak gets taller until it reaches a maximum value. After that, these profiles show the

same variation of intensity between that of the inner island and that of the background, allowing

us to choose a threshold value (in this case 18,000) to measure the width of the bridge. However,

thresholding when the peak has not saturated would overestimate the bridge width because of

the high baseline intensity value near the central peak. Therefore early on, we must be careful

measuring the bridge width at half maximum of the peak on top of the baseline (shown by the

arrows in the left inset), while it is more straight forward with the use of thresholding at later time.

5.4 Bridge expansion in the viscous regime

The evolution of the bridge width during the coalescence is shown in Fig. 5.4, with the main

figure showing the complete process while the inset focusing on the early dynamics. Even at the

fastest velocity v = 30 mm/s and largest length scale R0 = 60 µm, the Reynolds number Re =

0.04 1. Therefore, we expect that the coalescence dynamics is governed by Stokes flow in two

dimensions and thus, the Hopper model is an appropriate choice to explain our data. However, the

green curve shows that without adjustable parameters, the characteristic time τ = R0ηs/λ ≈ 1 ms,

where λ is the island boundary’s line tension and ηs is the 2D viscosity, is too short to explain

the early bridge expansion. Allowing τ to be adjustable (orange curve) still fails to capture the

linearity of early time data.

The linear expansion at early time is further confirmed in Fig. 5.5 in which bridge width

expansion curves in log-log plot have slopes of 1 at early time. All the bridge velocities are in

the range v = 24 ± 3 mm/s, independent of the thickness difference between the island and the

background film. This suggests that, similar to coalescence in 3D, the velocity is mainly dictated

by the ratio between λ and ηs, which are both proportional to the number of layer difference ∆N :

45

bri

dg

e w

idth

( μ

m)

0

100

200

time (s)0 0.005 0.01 0.015 0.02

experiment

Hopper model

Hopper model fit

bri

dg

e w

idth

( μ

m)

0

20

40

60

80

100

120

140

time (s)0 0.002 0.004 0.006

y = 2.84·104 x-7.5y = 162-233*exp(t/0.003)

Figure 5.4: Bridge expansion during coalescence. The main figure shows the whole temporalevolution including the shape relaxation towards the final circular domain, while the inset focuseson the early dynamics. Without adjustable parameters, the predicted evolution from Hopper’smodel (blue curve) disagrees with the data. The orange curve shows the fit if the characteristictime tau is used as an adjustable parameter. In the inset, the cyan curve is the exponentialrelaxation fit to the data in the later time, while the red line is the linear fit to the data pointsafter the peak in the intensity profile has saturated but before the relaxation regime.

46

w (

μm

)

10

100

time (s)0.001 0.01

ΔN

7

13

17

17

18

19

20

linear expansion relaxation

y=4*104x

Figure 5.5: Bridge expansion during coalescence events with thickness difference ∆N . The blackline has the slope of 1 in the log-log plots, which shows that the bridge expansion curves at earlytime are linear. At later times, the bridge widths relax exponentially to the final values.

47

ηs = ηd∆N and λ = λ0∆N where the line tension per layer λ0 = 10 pN [71]. As the result, the

ratio v0 = λ/ηs = λ0/ηd = 63 mm/s is independent of ∆N .

Eggers et al.’s extension of Hopper’s model allows for finite viscosity of the outer fluid and

and predicts the bridge velocity to be v = − 1π ln (w/2R0)v0. During the expansion − ln (w/2R0) ≈

1, thus v ≈ 1π (63 mm/s) ≈ 20 mm/s, which is close to the experimental value. However, the

logarithmic correction still changes appreciably during the linear expansion since w is not much

smaller than 2R0.

The long temporal range of the linear expansion regime has recently been observed in the

coalescence of 3D liquid drops in which the crossover time from the viscous to the inviscid regime is

3 decades longer than expected [46]. This could be explained by changing the characteristic length

used in the calculation of the Reynolds number from the bridge radius w/2 to half of the bridge

gap: l = w2/8R, i.e. the flow responsible for moving the cusps is localized within the distance l.

However, this would also modify the expression for the bridge velocity since the the logarithmic

correction resulting from the flow in the region l < r < w/2 goes away, leaving a constant velocity

v = v0/π.

5.5 Conclusion

We have directly observed the coalescence of thin fluid cylinders and found that the linear

bridge expansion in the viscous regime last longer than previously predicted and that there is no

logarithmic correction to the bridge velocity. This can be explained by changing the characteristic

length of flow from the bridge radius to half of the bridge gap. This work is supported by NASA

Grant NAG-NNX07AE48G, NSF MRSEC DMR 0820579, and NSF DMR 0606528

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