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COLLOIDS AT LIQUID-LIQUID

INTERFACES

Diplomarbeit

Michael Nikolaides

Technische Universitt Mnchen

Physik Department

Lehrstuhl fr Biophysik E22

Prof. E. Sackmann

Januar 2001

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1

Contents

CONTENTS............................................................................................................ 1

ACKNOWLEDGEMENTS/DANKE..................................................................... 3

SUMMARY............................................................................................................. 5

INTRODUCTION.................................................................................................... 7

1. COLLOIDS AT INTERFACES ........................................................................ 7

2. SPHERICAL COLLOIDAL CRYSTALS ......................................................... 8

3. ENCAPSULATION OF MATERIALS BY COLLOIDAL SHELLS................. 11

THEORETICAL PART......................................................................................... 13

4. SELF-ASSEMBLY OF COLLOIDS AT INTERFACES................................. 13

4.1. Minimization of the Interfacial Energy............................................................... 13

4.2. Diffusion Limited Self-Assembly.......................................................................... 16

5. INTER-PARTICLE FORCES ........................................................................ 18

5.1. Bulk Forces............................................................................................................. 19 5.1.1. Van der Waals Forces...................................................................................... 195.1.2. Electrostatic Stabilization the DLVO Theory .............................................. 225.1.3. Steric Stabilization........................................................................................... 24

5.2. Inter-Particle Forces on Liquid-Liquid Interfaces............................................. 25 5.2.1. Dipole-Dipole Interaction................................................................................ 255.2.2. Capillary Attraction ......................................................................................... 26

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EXPERIMENTAL PART....................................................................................... 29

6. EXPERIMENTAL METHODS AND SAMPLE PREPARATION.................... 30

6.1. PMMA Colloids and Labeling Process................................................................ 30

6.2. Confocal Microscopy............................................................................................. 31

6.3. Image Processing and Particle Tracking............................................................. 32

7. SELF ASSEMBLY ON THE OIL-WATER INTERFACE............................... 34

7.1. Experimental Setup ............................................................................................... 34

7.2. Attraction to the Interface ....................................................................................35

7.3. Diffusion to the Interface ...................................................................................... 37

7.4. Penetration Height in Each Solvent ..................................................................... 39

8. INTER-PARTICLE FORCES AT LIQUID-LIQUID INTERFACES ................ 41

8.1. Experimental Setup ............................................................................................... 41

8.2. Results..................................................................................................................... 43 8.2.1. Loose Hexagonal Lattice Structure .................................................................448.2.2. Time Evolution of the Loose Lattice............................................................... 468.2.3. Measurements of the Secondary Minimum..................................................... 488.2.4. Effect of Surfactants and Ethanol on the Structure .........................................53

8.3. Discussion ...............................................................................................................54

9. DENSELY PACKED CRYSTAL STRUCTURES ON THE SURFACE ......... 58

9.1. Experimental Setup and Sample Preparation .................................................... 63

9.2. Results..................................................................................................................... 65 9.2.1. Polycrystalline PMMA Structures................................................................... 659.2.2. Carboxyl Latex Single Crystals....................................................................... 68

9.3. Discussion ...............................................................................................................70

10. CONCLUSIONS........................................................................................ 73

11. OUTLOOK TOWARDS POSSIBLE APPLICATIONS............................... 75

12. REFERENCES.......................................................................................... 80

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3

Acknowledgements/Danke

Prof. David Weitz, for giving me the chance to work in his lab. The confidence hehad in my work and the freedom to choose my own research interest were veryimportant factors determining the character of my work. The discussions with himwere an essential contribution in understanding the experimental results andthrough them I always encountered new possibilities to proceed.

Prof. Erich Sackmann, dafr dass er mir die Mglichkeit gegeben hat meineDiplomarbeit in den USA zu schreiben. Trotz der grossen Entfernung von meinemHeimat-Lehrstuhl hat er mir stets sein Vertrauen in mich und in den Erfolg meiner

Arbeit vermitteln knnen.

Andreas, dafr dass er die Brde auf sich genommen hat, mich als Diplomand zubetreuen. Die freundschaftliche Atmosphre und die hervorragende Betreuunghaben ganz wesentlich zum Erfolg dieser Arbeit beigetragen. In Momenten des(Ver-)Zweifelns hat er mir immer wieder einen objektiven Blickwinkel vermittelnknnen. Ohne ihn wre diese Arbeit nicht in dieser Form mglich gewesen.

Toni, for introducing me to the field of colloidal physics, for all the discussions andhelp, theoretically as well as experimentally and especially for the corrections of

my final thesis.

Prof. David Nelson, for introducing me to Thomsons problem, encouraging meand giving me a lot of feedback. He was always ready to help me and answeredall the questions only an experimentalist would ask.

Eric, Megan and Prof. Howard Stone for great scientific discussions and a lot of help.

Ming, for scientific discussions as well as interesting insights into the Americanway of live.

The whole Weitzlab, for the great atmosphere during my year in Harvard.

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Meiner Mutter, dafr dass sie mich immer untersttzt hat und vieles fr ihre Kinder aufgegeben hat. Ohne ihren unermdlichen Einsatz fr unsere Erziehung htte ichdie Mglichkeit, diese Arbeit zu schreiben, sicherlich nie bekommen.

Meinem Vater, meinen beiden Schwestern und meiner ganzen restlichen Familiefr all die Untersttzung und Hilfe ber all die Jahre.

Flo und Olli dafr, dass sie mir gute Freunde whrend des ganzen Studiumswaren und mich nur zwei Wochen hier vom Arbeiten abgehalten haben.

Jean-Michel, for being a good roommate and friend.

Spethi, Gunnar, Bettina und Schneidi fr ihre Besuche hier.

Christiane danke ich fr sehr viel.

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Summary

Colloidal particles confined to the interface between two different phases are of

particular interest as they serve as a model system to answer questions

concerned with fundamental physics, such as crystallization and other phase

transitions. On the other hand for many industrial applications, such as the

production of food, cosmetics or coatings and for biomedical engineering, the

behavior of small particles at interfaces is of great importance. In this thesis three

major topics are addressed, all of them are connected with colloidal particles at

interfaces. The first one is the understanding of the underlying physical processes

involved in the self-assembly of particles at interfaces, as well as the inter-particle

forces between the particles on the interface. In the second part the structure of

two-dimensional crystals in curved geometries is investigated using colloidal

particles at the surface of spherical water droplets as a model crystal. Finally, in

the third part these crystalline, colloidal shells are used as a novel approach to

encapsulate materials, a field with broad applications.

The key to the understanding of the basic physical behavior of particles at

interfaces lies in the interaction potential, which these particles have with one

another. Theoretical models of the interaction potential are well developed for bulk

colloidal suspensions and also for some of the interactions among interfacial

particles. However, there still remains a doubt about the importance of capillary

forces in these systems. In this study a new system has been developed to

provide systematic and quantitative measurements of the interaction potential

between interfacial particles. This system consists of colloidal particles self-

assembled on the surfaces of aqueous droplets in non-polar, organic solvents. As

the colloidal particles used in this work were of the order of one micrometer in

diameter, they were imaged using either confocal or fluorescence microscopy. By

analyzing the Brownian motion of the colloidal particles, distinct properties of the

interaction potential are obtained. The experiments show a deep minimum at

contact and a secondary minimum at separations of several particle diameters,

which is divided from the primary minimum through an energetic barrier of a few

times the thermal energy. The primary minimum is caused by van der Waals

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Summary

6

forces and the secondary minimum is caused by a long-range electrostatic

repulsion together with a long-range attractive force. We present evidence that this

attractive force originates from the interfacial tension between the two phases.

There are different theories for these capillary forces, which are contradictory and

do not agree on whether the interaction will be attractive or repulsive. The results

presented here are the first quantitative measurement of the local form of the inter-

particle potential and they will help to further clarify the situation in the future.

In the second major part of this thesis, the ground state configuration of crystals in

curved geometries, a question originally asked by J.J. Thomson in 1904 is

investigated. The understanding of the interaction potential gained in the first part

is used to build a two-dimensional colloidal crystal on the surfaces of water

emulsion droplets. By adjusting the inter-particle forces, a spherical crystal where

the particles were still diffusing on the interface was built. This makes it possible to

assume that the crystal reaches a thermodynamic ground state after some time.

Computer simulations predict that the symmetry of the crystal, i.e. the defect

structure is dependent on the ratio of the particle separation a and the radius of

the crystal R . For a small ratio of R/a , i.e. a small number of basic units, the

ground state is given by the bucky ball structure, whereas for a ratio above a

certain threshold additional defects will occur. The results from our model system,

the spherical colloidal crystals support the basic features of the numerical

calculations that the geometric disclinations will be screened by additional defects,

arranged in the form of branches sticking out from the center disclination. In

contrast to the assumption in the calculation, there have been three branches

observed instead of five.

In the final chapter of this thesis we present possible applications of this system in

the field of the encapsulation of materials. The crystal order of the colloidal

particles provides a capsule with defined hole size and this makes the construction

of capsules possible that control the diffusion of macromolecules and cells from

the outside to the inside phase and vice versa. First attempts to encapsulate living

cells gave promising results and demonstrated the potential for applications, such

as the encapsulation of food and drugs in combination with a controlled and

delayed release.

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Introduction

1. Colloids at Interfaces

The lateral organization of colloidal particles at interfaces is of interest for a

number of different reasons. On the side of fundamental physics, it sheds light on

the influence of dimensionality on the phase behavior. Pieranski was the first to

demonstrate the formation of two-dimensional crystals from charged colloids at theair-water interface (Pieranski, 1980). More recently other authors used monolayers

of magnetic colloids at the air-water interface to demonstrate a transition from a

liquid to a solid phase (Zahn et al., 1998). Two-dimensional colloidal systems at

the air-water interface have also been used to study two-dimensional melting and

the dynamics of crystal growth (Onoda, 1985). The organization of particles at

interfaces is also significant for a number of industrial applications, such as the

purification of metal from melts, the stabilization of emulsions (Pickeringemulsions) and the stability of foams. There have been many studies of particles

at air-water interfaces in recent years, whereas the important case of colloids at

the interface between two liquids is rather seldom addressed.

In all the examples mentioned above, the understanding and knowledge of the

different contributions to the interaction potential between the particles is critical.

Until 1997 the situation seemed to be understood. The well-known forces between

bulk colloidal particles (Israelachvili, 1991; Russel et al., 1991) are modifiedbecause of the presence of the interface and there will be two completely different

contributions inherent to interfacial particles.

First, the presence of the interface will raise an asymmetric screening cloud

around charged colloids and this leads to a long-range dipole like repulsion (Hurd,

1985), which is responsible for the long range ordering effects observed in many

experiments.

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Introduction

8

Second, the weight of the particles will deform the interface causing capillary

interactions between the floating particles (Chan et al., 1981). These forces will

however be negligible for particles smaller than 5 m in diameter.

Kralchevsky et al. (1992; 2000) were the first to demonstrate another kind of

capillary forces, driven by wetting. These forces are active between particles

confined to a substrate or in a thin liquid film, even down to molecular sized

objects. Like flotation forces they are attractive for two similar particles over a

distance of micrometers. Between particles floating free at a liquid interface,

however, they do not occur.

In 1998 an unexpected, long-range attractive interaction was reported for chargedpolystyrene particles floating at the air-water interface (Ruiz-Garcia et al., 1998),

which is not explainable by either of the known mechanisms of capillary forces.

Recently, other authors suggested a mechanism that is able to explain this

attraction, based on a roughness in the three phase contact line of the particles

with the two solvents (Stamou et al., 2000). However, experimental evidence for

their theory has still to be presented.

All experiments to determine the inter-particle forces are done for colloidalparticles at the air-water interface and are only qualitative. One would expect

similar forces to be involved between particles at liquid-liquid interfaces, but there

is no experimental data available. To our knowledge the data in this thesis

presents the first examinations of this case.

Based on the understanding gained from the experiments, one can proceed and

try to build a spherical crystal out of colloidal particles.

2. Spherical Colloidal Crystals

The question for the minimal coulomb energy configuration of N unit point charges

distributed on the two-dimensional surface of a conducting sphere was originally

asked by J.J. Thomson in 1904 and has since been investigated by many authors

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Introduction

9

(Rubinstein and Nelson, 1983). Originally introduced as a model for the structure

of atoms it is also an interesting optimization problem in computational physics.

Besides that it is similar to problems in other fields such as the arrangements of

atoms in a spherical molecule (Kroto et al., 1985) and the arrangement of protein

subunits in the outer capsid of viruses (Johnson and Chiu, 2000). The shell of

some viruses, which encapsulates the viruses DNA, is built up from proteins,

which are distributed on a sphere. The understanding of the basic physics of this

system might help to better understand the mechanisms involved in the

reproduction of viruses.

The basic geometric situation is clear. The perfect, hexagonal lattice known in flat

geometry is not able to fit a curved topology, a fact already realized by Gauss in

his famous theorem about the topology of a sphere and therefore additional

defects will occur. The easiest curved topology is a sphere and in this case the

easiest crystal is determined by the soccer or bucky ball structure, where all the

electrons (or colloids, proteins or whatever is the basic unit of the structure) are

six-fold coordinated with their neighbors, except for 12 particles with five fold

symmetry (disclinations), which are arranged on the corners of an icosahedron.

This is very good illustrated by the black pentagons on a soccer ball (at least they

were black before the balls became fancy with all kinds of colors (UEFA, 2000)). It

is an interesting and still unresolved question, whether or not the bucky ball is the

thermodynamic ground state configuration for a spherical two-dimensional crystal.

Recent theoretical developments use a continuum elasticity approach to calculate

the ground state of a spherical crystal. In this approach the exact form of the

interaction potential between the basic units of the spherical crystal is not

important for the final symmetry pattern. For example, a change in the potential

from logarithmic to 1/r or 1/r 3 will change the elastic constants used in this

approach, but not the general result of the ground state configuration (Bowick et

al., 2000). Therefore one can use any experimental system, where the particles

are repelling each other to test the numerical predictions.

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Introduction

10

The important parameter for the determination of the ground state of the crystal

structure is the ratio of the separation between the basic units a and the radius of

the spherical crystal R .

Up to a certain ratio of R/a the minimum energy configuration seems to be given

by the bucky or soccer ball structure with its twelve disclinations. If the ratio R/a

increases, simulations show that the situation changes. Direct numerical

calculations for charges (Perez-Garrido et al., 1997) as well as a continuum

elasticity approach (Bowick et al., 2000) predict the occurrence of additional

defects. It is only in the context of this continuum elasticity approach that one can

clearly see that the phenomenon of screening by dislocations is universal and

inevitable for sufficiently large R/a. These additional defects only occur in pairs of

five and seven fold coordinated particles, arranged in lines, representing a

dislocation.

Until today there is only few experimental data available, which could help answer

this almost hundred-year-old question, but colloidal crystals provide a promising

system.

Colloidal crystals can be imaged by means of optical and confocal microscopy andtherefore they are a very popular system to study all different kinds of properties of

crystals. Dushkin et al. studied the dynamics of crystal growth as well as the defect

structure of crystals in three-dimensional colloidal crystals (Dushkin et al., 1993).

In two dimensions, the easiest way to produce a colloidal crystal is the drying of a

colloidal suspension on a flat glass substrate (Denkov et al., 1993). The particles

trapped at the air-water interface will build a hexagonal closed packed crystal, as

soon as they protrude from the thinning liquid film (Pieranski, 1980).

Our goal is to produce a two-dimensional spherical colloidal crystal and image it

with microscopy to directly answer Thomsons problem experimentally. From the

fact that the question is unanswered after hundred years it is almost obvious that it

is not that easy. However in this work real progress toward the assembly of a

spherical colloidal crystal will be presented and there is justified hope to solve this

problem in the near future.

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Introduction

11

3. Encapsulation of Materials by Colloidal Shells

The encapsulation of materials is a huge research area and will have big impact

on industrial and medical applications (Chaikof, 1999). People are interested in

encapsulating all different kinds of materials, ranging from chemicals like

fragrances and flavors in foods to the encapsulation of biomaterials like living cells,

which are very difficult to handle, as they are very sensitive. Applications are as

broad as the variety of encapsulated materials. A controlled release of flavors and

fragrances will use less material or give a stronger taste or odor. Special capsules

might also be used for the targeted delivery of drugs and also for a delayed

release. Encapsulated cells are protected from the immune response, whereas

they can still fulfill specific tasks, for example transplanted pancreas cells could

provide insulin for diabetics.

Most of the approaches so far use either polymers or lipid molecules as

constituents of the capsules (Mhwald, 2000; Caruso et al., 1998). There have

also been experiments to design a capsule by the use of micro mechanical

engineering, which requires complicated and expensive production techniques

(Desai et al., 2000).

In our approach we use colloidal particles as basic unit for our shells and let these

particles self assemble on emulsion droplets. By doing this, we have three major

advantages in comparison to other techniques.

First of all, the self-assembly is very easy, so it is compatible with large-sale

processes.

Secondly, all other methods to make shells require rather harsh chemical

treatments in some step of the encapsulation process, so that there is only a little

chance for biomaterials to survive. In order to encapsulate living cells, the

chemical treatment has to be separated from the encapsulated phase.

Nevertheless the cells need to exchange small molecules with the outside of the

capsule. In a completely closed capsule, the cell will poison itself and die after

some time.

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Introduction

12

The third important point is the efficiency of the encapsulation process. Many

approaches to encapsulate materials first produce the capsules and then fill them.

In the case of polymeric capsules there has been a reversible opening and closing

mechanism introduced, but still, the capsule has to be produced first and then to

be filled. In the case of micromechnical engineering, the capsule is closed after the

filling irreversibly. In general, in the filling stage a lot of material is used in the

solution with the capsules in it, but only a small percentage is encapsulated in the

end.

The fourth advantage of the colloids is that they are available in various sizes and

with many different surface properties, which leads to a tremendous flexibility to

design capsules with engineered properties.

The experiments done during this thesis lead already to the filing of a provisional

patent application at the US patent office (Nikolaides et al., 2000). The foremost

mentioned advantages will certainly proof our technique as a powerful alternative

to the conventional approach.

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Theoretical Part

4. Self-Assembly of Colloids at Interfaces

The self-assembly of colloids at flat interfaces has been studied in literature

(Pieranski, 1980) and the underlying processes are well understood. By bringing

particles from one phase to the interface between two phases, the contributions to

the overall interfacial energy are changed. In some cases, the adsorption of particles on the interface is energetically favorable. The energy balance for the

particles at the interface between two liquid phases will be calculated in section

4.1.

The only mechanism, by which particles adsorb onto the surface of emulsion

drops in the absence of convection, is Brownian motion. At low volume fraction of

colloids in suspension, it will take a long time until a water drop is completely

covered. Under certain simplifications one can calculate the adsorption rate of

particles on a spherical interface from the solution of Ficks equation and thus

obtain an estimate for this time. This will be done in chapter 4.2.

4.1. Minimization of the Interfacial Energy

In this chapter the adsorption of a PMMA colloid at a water-decalin interface is

analyzed in terms of the interfacial energy. Forming an interface between two

different materials will cost an amount of energy proportional to the created area of

the interface. The constant relating the area and the energy is the surface or

interfacial energy between the two materials.

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Self-Assembly of Colloids at Interfaces

14

PMMA/decalin

water/decalinPMMA/water

z

R

water

PMMA

decalin

1- 1

Eo

zozmin = b-a

Edecalin Ewater

Fig. 1: Colloidal PMMA particle at the decalin-

water interface. z denotes the deviation of theheight of penetration from the half and half case.

Fig. 2 Estimate of the surface energy well for a

particle adsorbed on a liquid-liquid interface.

In Fig. 1, a PMMA colloid at the water-decalin interface is shown together with the

three interfacial energies involved. By pure geometrical considerations one can

calculate the overall interfacial energy of a particle at a certain penetration height

(Pieranski, 1980). If z denotes the deviation in the height of penetration from

symmetric immersion and if we define the normalized deviation /o z z R= , then, by

calculating the contributing areas, the three contributions to the overall interfacial

energy are:

( )

( )

( )

2/ /

2/ /

2 2/ /

2 1

2 1

1

PMMA decalin PMMA decalin o

PMMA water PMMA water o

water decalin water decalin o

E R z

E R z

E R z

= +

=

=

Eq. 1

The first line represents the contribution from the interface between the particle

and the decalin, the second line the contribution from the interface between the

particle and the water and the third line is the energy that gets free, because some

of the initial decalin-water interface is destroyed. By introducing the abbreviations

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a and b as //

PMMA decalin

water decalin

a

= and //

PMMA water

water decalin

b

= , the sum of these three contributions

becomes:

( )22/

2 2 2 1o o owater decalin

E E z z a ba b

R = = + + + Eq. 2

As shown in Fig. 2, this energy has a minimum of the value

( )2min 2 2 1 E a b a b= + at min z b a= . Only in the case, where

min1 1 z < < Eq. 3

the particles adsorb on the interface. Particles completely in the oil phase imply

min 1 z , colloids completely in water min 1 z . The energy barrier associated with

the reentry into either solvent is given by

( )( )

min

min

1

1decalin o o

water o o

E E E z

E E E z

= =

= = Eq. 4

for water or decalin respectively. Unfortunately the surface energies associated

with PMMA colloids at a decalin-water interface are not known. To get an idea of

the order of magnitude of these energies, one can take the known values for a

polystyrene sphere at a water-air interface. In this case, the energy barrier has a

height of around 10 6 k B T . This suggests that the particles are really trapped at the

interface and will never leave it again.

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4.2. Diffusion Limited Self-Assembly

In the case that there is no attractive force driving particles toward the surface of awater droplet, one can estimate how long it takes to cover a drop of radius a with

colloids of radius R in the following simple model. Consider the surface of the drop

as a spherical absorber in an infinite medium. The concentration of particles at the

surface of the drop r a= (where r is the radial coordinate measured from the

middle of the droplet) is zero, and the concentration at high distances is C o .

Together with these boundary conditions the concentration is given by the solution

to the well-known Ficks equation in spherical coordinates:

22

10

d dC r

r dr dr =

Eq. 5

The stationary ( / 0C t = ) solution to this equation is:

( ) 1oaC r C r

= Eq. 6

The particle flux toward the surface is given by the product of the gradient of the

concentration and the diffusion constant D :

2( ) ( ) oa

J r D C r DC r

= = Eq. 7

The rate at which particles absorb on the sphere is given by the product of flux at

the surface ( ) J r a= and the surface area of the water drop

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4 o I DaC = Eq. 8

C is related to the volume fraction by /o colloid C V = , and the diffusion coefficient

D is given by Einsteins equation, / 6 B D k T R= . With this one obtains the

constant rate of adsorption:

42 Bk Ta I

R

= Eq. 9

The number of particles at full coverage N p is roughly given by the ratio of the

surface area of the water drop and the cross-sectional area of a colloidal particle22

2

4 2 p

a a N

R R

= = . The time needed to completely cover the drop is then this

number divided by the rate I :

28 p p

B

N aRt

I k T

= =

Eq. 10

A typical experiment in decalin ( )2mPas = , where the radius of the water drop is

a=100 m, the colloids have a radius of 0.7 m and the volume fraction is 0.1vol.

% results in a time of t p= 5.94 10 3s or ca. 7 days.

Although there are some simplifications in this model, like a perfect absorber in an

infinite medium, this calculation still shows that it would take a very long time to

obtain full coverage.

In order to get a full covered water drop on reasonable timescales, one can shake

or vortex the sample and with this increase the hitting rate of particles with the

surface, as described in the experimental section.

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5. Inter-Particle Forces

The behavior of colloidal suspensions is determined by the forces between the

particles (Israelachvili, 1991; Russel et al., 1991). On the one hand, there are

attractive van der Waals forces, which together with the random Brownian motion

will cause particle flocculation. In general there are two different ways to stabilize

the colloidal suspension against flocculation.

First, the attraction can be balanced by using colloids with surface molecules that

dissociate in water. The charged surfaces of these particles together with thecounter ions build a diffuse double layer and the repulsion between the colloids is

then determined by the repulsion between the two double layers. The exact

interaction potential between two surfaces was first calculated by Derjaguin and

Landau in 1941 (Derjaguin and Landau, 1941) and by Verwey and Overbeek in

1948 (Verwey and Overbeek, 1948), hence the theory of the interaction is called

DLVO theory.

A different approach to stabilize the suspension is to graft or adsorb non-chargedpolymers on the surface of the particles. Under the right solvent conditions, these

polymers strongly sterically repel each other, so that the deep minimum of the van

der Waals potential at contact is not accessible. One can think of the polymer

layers sticking out from the surface and being impenetrable. The closest distance

two colloids can approach each other is then determined by the thickness of the

polymer layers on the surfaces. The van der Waals interaction at this distance is

not strong enough to cause significant clustering. However a slight change in thesolvent can induce a significant change in the polymer interactions and cause the

particles to flocculate.

Colloids confined to a two-dimensional liquid interface will exhibit different forces

than colloids in the bulk. Because of the very low solubility of ions in a non-polar

medium, such as decalin, the cloud of counter ions around interfacial colloids will

be asymmetric and the interaction between two colloids will be different than

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Inter-Particle Forces

19

described by the classical DLVO theory. Also, the van der Waals interaction will be

different because of the two different media around the particle.

A distinct interaction, which is not just a variation of the force between bulk

particles but only occurs between interfacial particles are capillary interactions.

They are caused by the deformation of the liquid interface. There are different

theoretical models for capillary forces (Levine and Bowen, 1991; Kralchevsky and

Nagayama, 2000; Stamou et al., 2000) and it is still not clear whether or not there

will be significant capillary interactions between floating particles below 1 m size,

when the deformation of the liquid-liquid interface due to gravity is negligible.

Levine et al. (1991; 1992; 1993) predict a capillary repulsion, whereas Stamou et

al. (2000) predict a capillary attraction between the particles. In contrast to that,

Kralchevsky et al. (2000) predict no capillary interaction between the particles.

Experimentally there are only few qualitative data that hint toward the existence of

capillary interactions in these systems at all. In the experimental chapter evidence

for the existence of an attraction will be presented.

5.1. Bulk Forces

5.1.1. Van der Waals Forces

There are three different contributions to the van der Waals forces which all have a

dependence of 1/r 6

, where r is the separation between two molecules. The first twocontributions only occur in the case of permanent dipoles present in the system.

They originate from the orientation dependence of the interaction energy between

dipoles (the Keesom interaction) and from the induced dipole in a non-polar

molecule by a permanent dipole (the Debye interaction). Their form is given by

(Israelachvili, 1991):

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20

( )

( )

2 21 2

2 6

20

2 6

( )3 4

( ) 4

Keesom

o

Debye

o

u uU r

kT r

uU r r

=

=

Eq. 11

Here r is the distance between the two dipoles, u 1, 2 are permanent dipole

moments of the molecules and is the polarizability of a non-polar molecule.

The third contribution to the overall van der Waals energy is not limited to polar

molecules and therefore occurs in every system, even between two non-polar

molecules. It is for this reason that in literature van der Waals forces are often

considered to only consist of this kind, the so-called dispersion or induced dipole-

induced dipole interaction energy. They are quantum mechanical in origin and the

exact calculation of the interaction energy is rather long. Yet, the physical origin of

the dispersion interaction can be understood with a very simple picture, involving

Bohrs atom model.

The time average of the dipole moment is zero for a non-polar molecule. At any

given point in time, however, there exists a finite dipole moment given by the

actual positions of the electrons around the nucleus. This dipole moment induces

a dipole moment in another nearby molecule and the resulting interaction between

the two dipoles is what raises the attractive force between the two particles. The

result of the exact, quantum mechanical calculation for the interaction between two

molecules across vacuum is (Israelachvili, 1991):

( )

2

2 6

34 4

o Dispersion

o

hU

r

= Eq. 12

Here h is Plancks constant and h is the ionization energy of the molecule.

Summarizing, the van der Waals force between two molecules across vacuum is

the sum of Eq. 11and Eq. 12 and can be written as

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21

6( )C

U rr

= Eq. 13

where C is a constant depending on the different material parameters.

The van der Waals interaction between micron sized colloids in organic solvents

used in this work involves the interaction of many molecules in the particles as well

as in the solvent. An additional complication comes into play at large separations,

when the finite time the polarizing field of the first molecule needs to reach the

second molecule becomes important. This effect is enhanced inside a medium,

where the velocity of light is reduced and these effects are referred to asretardation effects in literature.

Summarizing it is difficult to calculate the dispersion interaction between two

macroscopic bodies in a solvent, but as a first and in many cases sufficient

approximation one assumes pair wise additivity and neglects retardation effects as

well as the effect of the solvent. By integrating the interaction energy 6( ) /U r C r=

for all the molecules in the particle and using the Derjaguin approximation, one

gets the total interaction energy for two spheres in vacuum in terms of the

Hamaker constant 2 2 A C = :

( )1

( )12 2

tot

RU r A

r R=

+ Eq. 14

In contrast to this approach, the Lifshitz theory of van der Waals forces uses a

continuum approach and treats the whole system as composed from two

continuous media, the solvent and the particles, each of them with a polarizability

and a dielectric constant.

The typical range of van der Waals forces is of the order of nanometer separations

between the surfaces of the colloids and in the case of two identical colloidal

particles in a solvent there is a deep minimum in the interaction potential at

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contact. This is the feature of the van der Waals interaction between two spheres,

which is important for the data analysis in this thesis.

For completeness it is necessary to mention that in some cases, when two

different phases interact over a third medium, the van der Waals interaction can

even become repulsive. An example is given by liquid helium interacting with the

metal of a container wall over the gaseous phase (Israelachvili, 1991).

5.1.2. Electrostatic Stabilization the DLVO Theory

With the random thermal motion of the particles, the attraction discussed in the

preceding chapter will cause colloids to flocculate if they are not balanced by

repulsive mechanisms. One possible way to stabilize the particle suspension is to

use particles with surface groups, which dissociate in water. The counter ions in

solution, together with the charged surface constitute an electrostatic double layer.

The surface molecules dissociate in solution despite the fact that this implies the

separation of unlike charges. The reason for this is the higher entropy of this state

and from this point of view, the resulting repulsion between the colloids is entropic.

The repulsion between two surfaces is called double layer repulsion and the range

of the interaction is given by the Debye length -1, given by

12 2 2

i i

i o

e Z

kT

=

Eq. 15

The sum is going over all the different ion species present in the system, i is the

concentration of the i-th ion species infinitely far away from the surface and Z i is

the valency of this ion species. For example in a monovalent solution (such as

NaCl ) at a concentration of 0.1 mMolar , the Debye length is around 1 30 nm = .

This length is easily tunable by the concentration of ions in solution.

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The result of the calculation for the double layer repulsion between two charged

spherical colloids of radius R in solution is given by

2

2

64( ) rel statickTRU r e

= Eq. 16

where is the sum over all the concentrations of ions far away from the surface

and ( )tanh / 4o ze kT = , where o is the surface potential of the particles.

The sum of the van der Waals attraction and the double layer repulsion gives thetotal DLVO interaction potential as illustrated by Fig. 3. When the salt

concentration of the solution is varied, the DLVO interaction potential between the

charged surfaces can be changed significantly. Fig. 3 shows that at too high a salt

concentration, the repulsion mechanism breaks down and the van der Waals

attraction is causing aggregation over all distances. From Fig. 3 it is also evident

that at contact, particles would sit in a deep global, primary minimum. They are

prevented from reaching the primary, global minimum just by the barrier due to thedouble layer repulsion.

increasing salt concentration

r[nm]

V(r)

10

particle surface

particle radius R

primary minimum at contact

secondary minimum

Fig. 3: DLVO interaction potential for different

concentrations of ions in solution. If the secondaryminimum is deep enough, the colloids remainkinetically stable.

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24

5.1.3. Steric Stabilization

The second way to stabilize colloidal suspensions is to add polymers to the

surface of the particles. Theories of the resulting repulsion are very complex and

depend on the coverage of the surface, the quality of the solvent and on the

mechanism how the polymer is bond to the surface.

Analytical solutions are well developed only for the cases of very good and very

poor solvent conditions, but a qualitative description of the stabilization process is

rather simple. Though the potential due to van der Waals forces becomes infinitely

deep as two particles approach each other, the energy of the van der Waals forces

when the surfaces of two particles are separated around 10nm is only of the order

of the thermal energy k B T . By adding the polymers to the surface (assuming the

polymer layers can not remarkable interpenetrate each other) the surfaces are

prevented from approaching each other beyond the thickness of the polymer layer.

At this distance attractive forces are too weak to cause significant flocculation.

Fig. 4 shows the repulsion between the PMMA particles used in this work,

stabilized by a graft copolymer formed of poly-12-stearic acid and methyl

methacrylate (Antl et al., 1986; Schofield, 1993). The steep increase in the

repulsion justifies the treatment of the particles as hard spheres with hard-core

repulsion in many experiments.

W

r[nm]

particle surface

rg + R

polymerelayerthicknessrg

Radius ofthe particleR

Fig. 4: Repulsive potential for the PMMA colloidsused in this work. r g is the thickness of the polymerlayer. The particle is illustrated by the left verticalline, symbolizing a hard core, whereas the repulsionbetween the polymer layers is only near to hardcore.

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5.2. Inter-Particle Forces on Liquid-Liquid Interfaces

5.2.1. Dipole-Dipole Interaction

In Fig. 5 two charged particles at the interface between water and a non-polar

medium are shown schematically.

+++

++ ++ +- --

++---- +

++++ ++ +

- --++

----

r

decalinwater p p

Fig. 5: Two colloidal particles at an interface between water and an organic solvent. The asymmetry of thecounter ionic clouds gives rise to electrical dipole moments and this results in long-range repulsive forcesbetween the particles.

The distribution of charges, resulting from the dissociation of the surface groups is

asymmetric with respect to the interface. Associated with this distribution is adipole vertical to the interface, where one pole is the screening cloud and the other

is the charged particle. Neighboring dipoles repel each other through both, the

water and the oil phase. Pieranski (1980) used the following simple model to

estimate the interaction. The magnitude of a dipole is given by Q 1, where Q =Ze

is the total charge dissociated in the water phase and 1 is the Debye screening

length in water. This yields an interaction potential

2 2

2 3

1( )

4dip dip o

Z eU r

r = Eq. 17

where is an effective dielectric constant at the interface, given by the average of

water ( water =80) and oil ( oil=2), i.e. ( )/ 2 1/ 2 waterwater oil = + . Results from a

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26

more detailed calculation (Hurd, 1985), based on a linearized Poisson-Boltzmann

equation yields the following potential:

2 2 2

2 2 2

2 1 1( )

4 1r

dip dipo

Z eU r e

r r

= + Eq. 18

At short distances r , where the first term in the bracket completely dominates over

the second term this recovers the screened Coulomb potential ( Eq. 16) . However,

for distances greater than the screening length , the first term is negligible small

and one obtains Eq. 17 to a factor of . The resulting interaction is therefore a

long-range repulsion ( 31/ r ) and is responsible for long-range ordering processes

between colloidal particles at interfaces.

5.2.2. Capillary Attraction

Capillary forces between particles at fluid interfaces are important for many

industrial processes as well as for biological systems, such as the interaction

between proteins incorporated into a lipid bilayer. In Fig. 6, the two different

classes of capillary forces are illustrated.

(b) Immersion forces(effect driven by wetting)

(c) Capillary forces negligiblefor R < 5 m

(a) Flotation Forces(effect driven by gravity)

1

2

1 2 RRR

Fig. 6: Capillary forces between interfacial particles. In (a), the situation for floating particles large enoughto deform the interface is illustrated. For particles on a substrate wetting effects become important andcapillary forces are active down to nanometer-sized particles (b). In the case of small (R

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Chan et al. (1981) were the first to present a theory for the interaction caused by

the deformation of the liquid interface by the weight of the particle. These capillary

forces are known as flotation forces. Kralchevsky et al. (2000) developed a theory

for another type of capillary interaction, called immersion forces, which are driven

by the wetting of the particle surface and only occur when the colloids are confined

to a substrate.

From the theoretical treatment of the liquid-liquid interface, one gets an

approximation for the forces between two spheres separated at a center-to-center

distance r at an interface in both cases as:

( )21 2 1 1,21,22 ( ) for r1F Q Q K qr rO q R = + Eq. 19

where the various variables mean: is the liquid-fluid interfacial energy,

1,2 1,2 1,2sinQ r= can be seen as the capillary charge, r 1,2 are the radii of the three

phase contact lines and is the meniscus slope angle. K 1 is the Bessel function of

first kind and q is the capillary length given by 2g

q

= .

Both forces differ from each other through the capillary charge Q 1,2 . When

R 1=R 2 =R and 1k r r q one can derive approximations for both cases as:

( )( )

6

1

21

for flotation forces

for immersion forces

RF K qr

F R K qr

Eq. 20

As shown in Fig. 7 the flotation forces decrease much stronger with the particle

radius R than the immersion forces and in general they are negligible for

5 10 R m < .

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10 8

10 0

10 2

10 4

106

10 -310 -410 -510 -610 -710 -810 -9 R[m]

- W/kT

i m m

e r s i o

n f o

r c e s

f l o t a

t i o n

f o r c

e s

Fig. 7: Comparison between the twotypes of different capillary forces. Theright axis is the energy at contactbetween two particles, where r=2R .Graph after Kralchevsky andNagayama (2000).

In the figure the parameters used for the calculation are: density difference

between the liquids 31 /g cm = , surface tension 40 /mN m = and slope of the

contact line 1,2 60o = .

From this it is evident that neither flotation nor immersion capillary forces act in the

system examined in this work, where small (micrometer diameter) particles float at

liquid-liquid interfaces.

Flotation forces are too small because of the small size of the particles. Immersion

forces require a complete different geometry. The requirement of Youngs

equation to have a fixed three phase contact angle will in our system be fulfilled by

changing the height of penetration in each solvent without deforming the interface.

Consequently, Fig. 6 (c) illustrates the situation in our experiments with no

deformation of the interface and hence no capillary interactions.

Very surprisingly we nevertheless measure an attraction at very long separations

and we will discuss the results of various experiments in the experimental chapter.

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Experimental Part

The results presented in this chapter can be divided in three major categories.

The first one is dedicated to the understanding of the underlying processes of the

self-assembly. It will be shown that no long-range force towards the interface

exists and the physical reason for the particles to go to the interface is given by the

interfacial tension. In the absence of convection only the random thermal motion

brings particles to the interface.

In the second part, the forces between colloidal particles confined to the liquid-

liquid interface are studied. In the course of the experiments the existence of a

secondary minimum at separations of several microns in the interaction was

proven and possible explanations for this will be discussed.

An increase in the coverage density of the colloids can be achieved by an increase

in the rate of particles hitting the surface by simply shaking the samples. This

results in densely packed crystalline structures providing a model system for the

properties of crystals in curved geometries and data obtained will be presented

and discussed in the third major part of the experimental chapter.

At the very beginning, however, some of the used experimental techniques are

presented.

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6. Experimental Methods and Sample Preparation

6.1. PMMA Colloids and Labeling Process

As described in chapter 5 there are two common ways to stabilize colloidal

suspensions. For colloidal particles in organic solvents, the stabilization by double

layer repulsion cannot be used because of the low solubility of charges in oil.

Therefore colloidal particles in organic solvents are stabilized by polymers grafted

onto the particles surface. Polymethylmetachrylate (PMMA) is a polymer

commonly used to make colloids of various sizes. The stabilizing layer consists of

copolymer formed of poly-12-stearic acid and methyl methacrylate and is grafted

onto the surface. As shown in Fig. 4, the repulsive part of the potential is very

steep, so that PMMA particles in organic solvents are a model system for hard

spheres. The PMMA particles used in this work were produced by Andrew

Schofield in the group of Prof. Peter Pusey at the University of Edinburgh in

Scotland. They were originally suspended in dodecane at a volume fraction of

40%. The rhodamine dye was dissolved in a mixture of 25 volume percent acetone

and 75 volume percent cyclohexanone at a concentration of 1mg/ml.

The original PMMA suspension is diluted in dodecane to approximately 15 vol.%

PMMA. To this suspension, 0.47 ml of the labeling mixture is added per milliliter

and the mixture is then vortexed gently. Acetone is a very good solvent for PMMA,

so the particles are swollen by the acetone in the labeling solution and the

rhodamine dye is able to diffuse into the colloids. This process is accelerated by

the vortexing and the entering of the beads into the core of the colloids is indicated

by a color change of the solution from purple to orange.

After the color change the mixture was centrifuged at about 2000 g for ten

minutes. The supernatant solvent was removed and 2 ml of decahydronaphtalene

(decalin) was dribbled in while vortexing. As decalin is a much worse solvent for

the PMMA than, acetone the particles deswell again, keeping the rhodamine dye

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Experimental Methods and Sample Preparation

31

inside. In the following steps the solution was centrifuged and the supernatant

decalin was exchanged with fresh decalin several times, to really get rid of all the

acetone. After five centrifugation steps, the solution was left at rest for three days

to give the remaining acetone time to diffuse out of the particles. After that the

particles were washed again five times with decalin. The success of the labeling

process was directly evident by the fact that the particles were very bright in the

fluorescence microscope.

For the interpretation of the data it is important to mention that very likely some

charges are introduced onto the particles during this labeling process. This

contradicts the common assumption of a hard-sphere repulsion between the

colloids, but after the results in this work did not allow any other interpretation, this

fact was verified by other experiments (Dinsmore and Weeks, 2000).

6.2. Confocal Microscopy

Microscopy is a very common used technique in a lot of scientific applications.With microscopy one gets very accurate two-dimensional information. In order to

get a three-dimensional image however, the depth of focus has to be reduced.

This is realized in the confocal microscope. The basic principle of confocal

microscopy is illustrated in Fig. 8 a (Corle and Gordon, 1996). Only one spot is

illuminated at a time through a pinhole. The detector behind to the pinhole detects

the light reflected from the sample. By scanning the spot over the sample in a

raster pattern, a complete image can be formed. If the sample moves out of focus

as shown in Fig. 8b , the reflected light is defocused at the pinhole and hence does

not pass through it to the detector located on the other side. The result is that the

image of the defocused plane disappears. In our setup we used a fluorescent-

labeled sample and looked at fluorescence instead of reflection.

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32

pinhole

object

focal plane

illuminatingbeam

a) object in focus

pinhole

object

focal plane

illuminatingbeam

b) object out of focus

Fig. 8: Simplified schematic of the confocal scanning microscope showing the sample in the confocal plane(a) and out of focus (b). The reflected light in our setup comes from the fluorescent-labeled sample, whichresults in a much better resolution.

6.3. Image Processing and Particle Tracking

Two different techniques were used to get the data from the microscopic setups.

For all the samples observed in the fluorescence microscope the pictures weregrabbed with a CCD camera and stored on S-VHS videotape (frequency in the

US: 30Hz). Later the interesting parts were digitized using the software program

OpenBox (Schilling, 1999) and further processed in a computer.

The confocal microscope was a Noran OZ laser scanning confocal on a Leica

DMIRB inverted microscope. The used objectives were a Leica 100x oil immersion

objective with numerical aperture of 1.40, a Leica 40x immersion oil objective with

numerical apertures of 1.25. For scans through thick samples a Leica 63x long

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distance air objective with variable numerical aperture was used. The excitation

wavelength of the laser was 488nm and a 590nm long-pass filter was used.

The intensity information was detected by a photo multiplier tube and then directly

saved in digital form on a computer and then further analyzed.

The images consisted from an array of particles, illustrated by bright dots in front

of a darker background and were used to determine the positions of all particles at

all times. If only a few particles were of interest the tracking of these particles was

done with OpenBox. The center of mass of each particle was determined by fitting

a gaussian curve to the intensity matrix from the digitized images. The center of

this curve was taken as the center of the particle. This procedure was repeated for

all particles of interest over a certain time range and with this the time dependence

of all coordinates was known. The further analysis was then done with the data

analysis program IGOR pro.

In order to get the coordinates of many particles at the same time, the program

IDL from Research System was used with routines written by David G. Grier, John

C. Crocker and Eric R. Weeks. First the locally brightest spots were taken. Locally

in this case means that they were the brightest spots in a circular region with givenradius. The center of each particle was then calculated by summing up the

intensity weighed coordinates of all pixels in this circular region and dividing them

by the overall intensity.

( )

( )

iii

center

ii

I rrr

I r=

Eq. 21

This process was repeated for every frame of the videotape and the track of every

particle was obtained. The further analysis was again done in IGOR pro.

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7. Self Assembly on the Oil-Water Interface

7.1. Experimental Setup

As described in chapter 4.1, colloidal particles will stick to the surface of a water

droplet in oil once they touched it, provided the three interfacial energies fulfill Eq.

3. However the question remains what mechanisms brings the particles to the

interface. There are three possible ways one can think of. The first one is an

attractive force over long range, for example electrostatic forces including image

charge forces. The second is given by the thermal random motion of the particles

which will bring them sooner or later to the interface, so they will be trapped, but

as calculated in chapter 4.2, it will take very long for particles to cover the interface

in this case. The third mechanism is given by convection of the suspension.

Convection will be used in later chapters to get very quick dense coverage of the

interface, but in order to sort out whether there is an attraction towards the

interface or not, the occurrence of convection was avoided during the experiments

in this chapter.

In order to investigate the droplet from the beginning the setup in Fig. 9 was used.

Cover slip

microscope slide

glass spacer

deionised water

decalin

to syringe

objective

micropipette

PMMA

Fig. 9: Setup to observe the forces of the particles towards the surface. The micropipette was connectedwith a tube to a syringe, so that a pressure could be applied to create a small water droplet.

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Self Assembly on the Oil-Water Interface

35

For the Leica DMIRB inverted microscope a special holder was designed, so that

a micromanipulator could be mounted to the microscope. The manipulator was

used as a holder for a glass micropipette, which was connected with a tube to a

normal syringe. For the chamber, two cover slips were glued together with 1mm

glass spacer between them. This chamber was left open on one side, so that the

glass micropipette could be inserted into the chamber. The chamber was filled with

the colloidal suspension and the micropipette, which was filled with deionized

water, was inserted and brought into the field of view. After a waiting time to

exclude convective disturbances a small pressure was applied to the syringe, so

that a water droplet was formed in the chamber. The motion of the particles toward

the droplet was then observed and the pictures were analyzed as described inchapter 6.3.

In an alternative setup, the chamber was built from indium tin oxide (ITO) covered

glass and the glass pipette was replaced by a metal needle. This was done to be

able to ground all the components in the system to exclude electrostatic effects. In

the next chapters it will become clear that this is a crucial point, because the

screening length in oil is very long, so that electrostatic charges are far more

important than they are in aqueous solution.

7.2. Attraction to the Interface

The chamber was filled with a colloidal suspension of PMMA particles with 1 m

radius in decalin at a volume fraction of 1% and a droplet of deionized water was

produced with the micropipette as described in chapter 7.1. The resulting motion

of the particles towards the interface is illustrated by the curves in Fig. 10.

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36

80

60

40

20

0

543210 time [s]

distance from the surface [ m]

particle 1

particle 3particle 2

Fig. 10: Tracks of three different particles towards the water droplet. The three shown trajectories were froman ungrounded droplet. If the whole system was grounded, the motion towards the interface disappeared.

The curves show an almost linear behavior, i.e. a constant velocity and one can

explain these tracks as follows.

As a first approximation inertia effects can be neglected. The motion of the

particles is then determined by the equality of the viscous drag force and the outer

force.

( ) 0ext f r F r+ =

Eq. 22

Here f is the fr4iction coefficient and r is the distance from the surface of the water

droplet. A constant velocity .r const = therefore means a constant force over all

distances.

Very near to a big droplet a planar surface is a first approximation, so the almost

constant velocity could be understood by the homogeneous field of a charged

plane.

Before going into too much detail about the interpretation of the acting forces it is

important to mention another point. In a whole series of experiments the behavior

of the same stock solution of colloids was very different from experiment to

experiment. This was in the beginning very surprising and even under the

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assumption of a broad distribution of the size of the charges on the colloids it was

not understandable why in some cases no attraction of any colloids could be

observed.

This contradiction was resolved perfectly when an ITO chamber was built and the

glass pipette was replaced by a metal needle and the whole system was

grounded. When every component of the setup was grounded properly, the

attraction could not be observed in any of the experiments any more.

These experiments show that random static charges in the ungrounded systems

caused the long-range forces. Grounding the system eliminated these long-range

forces.

These results show that the only way particles come to the interface in the

absence of convection is given by their random thermal motion and in the next

chapter experiments to measure the increase of coverage over long times will be

discussed.

7.3. Diffusion to the Interface

The complete coverage of the interface by the thermal motion of the colloids

towards the adsorbing interface takes several days, as estimated in chapter 2. In

this chapter attempts of experiments to measure the increase of the coverage will

be described. To be able to observe the samples over days, one clearly needed to

have a sealed chamber, otherwise the sample just evaporated. The chamber used

was the same as in chapter 7.2 with the difference that the fourth side was alsoclosed. After the injection of a small droplet of water into the suspension of

colloids, the chamber was sealed and the water droplet was observed in the

confocal microscope. A scan of the complete droplet was taken and from this the

number of particles on the surface was extracted using the methods for particle

tracking described in chapter 6.3.

In these experiments gravity cannot be neglected, which was illustrated by the

sedimentation of the particles to the bottom of the chamber. The density of the

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PMMA colloids is 31.44 /PMMA g cm = and the density of decalin is only

30.886 /decalin g cm = . For all the experiments done in pure decalin the particles fell

to the bottom of the chamber before there was a remarkable increase in thecoverage. The way around this is to match the density of the organic solvent with

the density of the PMMA colloids. A commonly used solvent to density match the

suspension is cycloheptylbromide with a density of 31.324 /CHB g cm = . The volume

relation needed to match the densities is therefore / 4.676CHB decV V = .

In the experiments it turned out that in this solvent, however, the relation between

the interfacial energies is no longer in the right range ( Eq. 3) , so that the particles

did no longer adsorb on the surface. All experiments with different solvents to

density match the suspension encountered the same problem, so the only data

available is over relatively short times. In Fig. 11 the increase in coverage for

PMMA colloids with 0.7 m radius in pure decalin is shown. The plateau shows that

no more particles were adsorbed on the interface, as expected, because of the

sedimentation. The slope of the linear increase for small times corresponds very

good to the one calculated by the estimation in chapter 4.2 (Eq. 9) . In the case

illustrated in the graph, this assumption would give a rate of 0.03 particles per

second, which is illustrated by the straight line in the graph.

100

10080604020

# of particles

time [min]

20

Fig. 11: Increase in the number of particles on the interface. The plateau occurs, because all the particles insuspension sediment to the bottom of the chamber after a time of approximately 50 minutes. At the beginninghowever, the rate corresponds very well to the rate expected from the estimation in Eq. 9.

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7.4. Penetration Height in Each Solvent

Once the particles adsorbed onto the interface, the penetration height of theparticles into each solvent can be extracted from the mean square displacement of

the particles on the interface in the following way:

At small times the two-dimensional mean square displacement is given by:

2 4 D x = Eq. 23

Where D is the diffusion coefficient and is given by the relation between the

thermal energy kT and the friction coefficient f as D =kT/f. For a sphere of radius R

in a medium of viscosity , the friction coefficient is given by

6 f R= Eq. 24

In other words, the mean square displacement of the particle is related to the

viscosity of the medium by:

2

6 medium

kT x

R= Eq. 25

For the spheres in our experiments the situation is more complicated. The friction

coefficient for the interfacial particles will definitely depend on the penetration

height in each of the solvents (they both have different viscosities). Given the

theoretical dependence of the friction coefficient on this height of penetration and

the fact that this penetration height is dependent on the interfacial energies, one

could measure the surface energy between the bead and one of the solvents, if

the interfacial energy with the other solvent is known.

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Unfortunately there is no theoretical calculation of f available for our case, but in

Fig. 12 the slope of the msd for short time is shown together with the values as

expected for a sphere completely in water or decalin, respectively.

0.4

0.3

0.2

0.1

0.0

0.70.60.50.40.30.20.1

bead completely in decalin

msd [ m2]

[s]

bead completely in water

Fig. 12: Mean square displacement (msd) for small times. The slope is much higher than in Fig. 16, becausethe observed particles were isolated on the interface. Interaction with other particles could bring down theslope, as discussed later.

The different colors correspond to different particles. As expected, the slope is in

between the two extreme values, indicating that the particle sits on the interface

between the two liquids.

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8. Inter-Particle Forces at Liquid-Liquid Interfaces

Despite the fact that the forces between particles in bulk suspension are well

known and can be influenced in a very controlled way, the forces between colloids

confined to an interface are still not really understood. In this chapter the interface

of a water droplet in oil covered with colloidal particles is observed. From the

behavior of the particles on the interface, information of the inter-particle potential

is extracted. The thermal motion of the particles around their equilibrium position

provides the local form of the potential. The first subchapter will prove theexistence of a primary minimum caused by van der Waals forces as well as a

long-range repulsive force. In the second part the existence of a long-range

attraction will be shown. Based on these results a hypothetical form of the two-

particle potential will be presented.

The insight gained from the experiments in this chapter will be used later to

improve the properties in the case of a densely packed structure.

At the starting point of this chapter however, a short description of the samplepreparation shall be given.

8.1. Experimental Setup

In order to watch the particles self assemble on the interface and to directly

observe cases with dilute coverage of colloids, a special microscope chamber was

designed as shown in Fig. 13. It consists of a Teflon ring and a hydrophobic cover

slip glued to it with vacuum grease.

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Inter-Particle Forces at Liquid-Liquid Interfaces

42

objective teflon ringhydrophobiccoverslip

water

PMMA suspendedin decalin

Fig. 13: Microscope chamber for the observation of the Brownian motion of the colloids directly after theyself-assembled on the surface of the water droplet during the injection.

In this chamber a suspension of 0.1 vol.% of the fluorescent-labeled PMMA beads

in decalin was injected directly on the microscope. After that a water droplet of

0.2 l deionized water out of a Millipore system (resistivity 18.2 M cm) was

injected into this suspension. The chamber was then sealed on top and the

interface between the water drop and the decalin was observed, using

fluorescence microscopy.

In the course of our experiments it turned out that the vacuum grease is slowly

dissolved by the decalin and that for long time studies the presented chamber is

not useful. The design of an alternative chamber is shown in Fig. 14. All parts are

made from hydrophobic glass. Out of a microscope slide thin stripes of glass were

cut and glued to another slide with five-minute epoxy. One side of the chamber

was left open for later filling and the top was covered with a cover slip. The whole

chamber was then sealed with five-minute epoxy. It turned out the epoxy needs at

least 8 hours to dry thoroughly, so that it is not dissolved by the decalin any more.

If this waiting time was not obeyed, after a long time exposure to decalin the

chambers began leaking.

The fourth side of the chamber was left open and as the spacer only had a

thickness of 1mm the capillary force was strong enough to keep the liquid inside

the chamber.

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a)

Cover slip

microscope slideglass spacer

microscope slide

glass spacer

Cover slip

water droplet

b)

Fig. 14: Microscope chamber used for long time studies. The left picture shows the side view, whereas theright picture shows a top view. All the glassware used was hydrophobic.

8.2. Results

By injecting the droplet into the colloidal suspension there were already a number

of particles brought onto the interface. Very surprisingly these particles were

arranged in a hexagonal lattice structure with inter particle separations of up to ten

times the colloids radius and this lattice covered the whole droplet. This structure

collapsed over time into a patch of particles arranged in only one cluster with a gel

like form. Chapters 8.2.1 - 8.2.2 describe this lattice structures and their time

evolution and give an explanation based on van der Waals forces between the

interfacial particles and on electrostatic repulsion. In this interpretation it is

important that a polymer layer, which consists mostly of hydrophobic hydrocarbon

chains and therefore collapses in contact with water, stabilizes the particles. Yet, a

small amount of charges is evidently introduced onto the particles during the

labeling, providing a very weak charge stabilization.

In the case of incomplete coverage the particles were nevertheless arranged in an

ordered structure on the interface and did not diffuse away from each other as

expected for a purely repulsive potential. The harmonic part of the minimum the

outer particles in such a patch explore can be obtained by tracking the thermal

motion of these particles and calculating the probability distribution of the radial

coordinates.

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Chapter 8.2.3 presents measurements of this secondary minimum, indicating a

very long-range attraction. In the discussion part it will be argued, that this

attraction cannot be due to the known kinds of capillary forces described in

chapter 5.2.2. Very recent developments, explaining this attraction will be

discussed in chapter 8.3.

8.2.1. Loose Hexagonal Lattice Structure

Fig. 15 shows part of the top surface of a water drop immediately after the

injection into the microscope chamber. This picture shows the hexagonal structure

covering the interface with an inter-particle separation of around 6 m.

10 m

0.16

0.12

0.08

0.04

0.00

2.52.01.51.00.50.0

msd [m2]

time[s]

Fig. 15: Top surface of the water drop, coveredwith fluorescent PMMA colloids. The inter particlespacing is around 4 times the colloids diameter of 1.5 m.

Fig. 16: Average mean square displacement of theparticles arranged in the hexagonal structure shownin Fig. 15.

The colloids have a diameter of 1.5 m, so the lattice spacing corresponds to aboutfour diameters. The colloids jiggle around their equilibrium positions and from the

mean square displacement information about the local form of the potential can be

achieved. In Fig. 16 the average of the mean square displacements of four

different particles are shown. From the graph it is evident, that the particles move

in a local minimum, which is illustrated by the plateau observed for long times.

From the value of the maximum displacement one can get an estimate of the local

form of the minimum. Approximating the minimum with a harmonic potential of the

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form 2( ) / 2U x k x= and using the equipartition theorem, which gives k B T/2 to every

coordinate, one can calculate k as:

2max

Bk T k x

=

Eq. 26

For the data shown in Fig. 16 this yields a value of 8 2 211.0 10 / 26.6 / Bk J m k T m

= = . This value is in very good agreement with the

value obtained in a later part of this work ( 8.2.3) . In Fig. 17 the same structure is

shown in lower magnification and from this picture the radial distribution function

g(r) was calculated (Fig. 18).

30 m 252015105

1

3

2

g(r)

r [ m]

5.7 m

Fig. 17: Top surface of the water droplet. Theparticles in lattice structure in low magnification.

Fig. 18: Radial distribution function, calculatedfrom the image data.

The radius of the water drop was 300 m. From the radial distribution function one

can see that the preferential distance between the particles is 5.7 m, which does

not necessary mean that there has to be a minimum in the two-particle potential.

This can also occur in the case of a pure repulsive potential between the particles,

because the particles are confined to the interface. In this case however, the inter

particle spacing would depend strongly on the density of colloids on the surface

which was not observed in our experiments. This contradiction can be solved by

the postulation of a long-range attraction, causing a secondary minimum at this

distance. We will show evidence for this fact in chapter 8.2.3.

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8.2.2. Time Evolution of the Loose Lattice

As already mentioned, the regular hexagonal structure described in the precedent

chapter was not stable. Seconds after the injection, first dimers formed and after a

couple of hours all the particles collapsed to a gel like structure. Fig. 19 shows two

different sections of the surface after a time of 5 minutes.

10 m

10 m

Fig. 19: Two examples of the lattice structure after 5 minutes. The big cluster seems to exist already longertime, as the lattice around it already relaxed.

In the left picture, a cluster of two particles builds, leaving a lattice space empty.

The surrounding particle will rearrange a little, but the spacing around the cluster is

bigger than the normal lattice spacing. In the right picture there are seven particles

arranged in a cluster and the surrounding lattice had enough time to rearrange.

The possible explanation that the cluster was already built in the bulk can be

excluded, as in every experiment the suspension was treated in ultrasound for 30

minutes to break up all clusters.

In Fig. 20 the whole droplet can be seen in lower magnification. Again it is evident,that around clusters the density of particles was lowered. Another important

observation is that once two particles touched each other they never fell apart

again. This indicates a very deep minimum in the two-particle potential at contact.

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30 m

Fig. 20: Lower magnification of the water dropletcovered with colloidal particles. Since the injectionof the water a time of 5 minutes has passed and a lotof clusters have already formed on the interface.

The explanation of all the observations made in this and the precedent chapter will

be discussed in chapter 8.3.

In Fig. 21 the final state of the structure is shown in three different magnifications.

All the particles collapsed to a patch of gel like chains. The gel covered only part of

the droplet, because the density of the colloids is higher than in the loose lattice

and the number of particles on the droplet did not increase remarkably during the

experiment as discussed in chapter 4.2. If other solvents were used to density

match the solution, the particles did not self assemble any more, as discussed in

chapter 7.3.

Fig. 21: After several hours all the particles ended up in a gel like patch on the interface. The left two picturesshow a big magnification and one can see the single particles. On the right side it is evident, that the gel onlycovers part of the surface of the water droplet.

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8.2.3. Measurements of the Secondary Minimum

The behavior of the colloids described so far shows two properties of the two-

particle potential. First of all, the existence of very stable clusters shows the

primary minimum at contact between two particles. Secondly, the existence of the

ordered lattice structure shows the existence of a long-range repulsion. As the

particles are confined to the finite interface they would arrange in an ordered

structure, even in the case of a purely repulsive potential, when the density on the

surface is high enough. The inter-particle spacing will then depend strongly on the

density of colloids on the interface. Very surprisingly in the experiments no

evidence for this was found. It appeared that independent of the density the

particles had the same distance, in the case of 0.7 m radius PMMA particles

5.7 m, or they touched each other. If the density of the colloids in suspension was

lowered more and more, the density of colloids on the interface also went down.

But even in the case of very low coverage the colloids still ordered in a loose

lattice structure, only covering part of the droplet.

This is only explainable, if one postulates an additional attraction between the

particles, causing a secondary minimum in the interaction potential. In this chapter

an experimental proof for the existence of such a secondary minimum will be

presented.

In Fig. 22 a case with only seven particles on a water droplet is shown. There were

no other particles near these seven particles on the droplet or in the solution. The

particles kept their stable hexagonal order over more than 5 minutes, after that the

sample was lost because of evaporation. As shown in Fig. 23 the seven particlessat at the bottom of a water droplet of 24 m radius and remained in this hexagonal

configuration and the six outer particles turned approximately 60 degrees around

the inner particle.

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10 m

R=24 m

water decalin

Fig. 22: In the case of very low bulkdensities, also the coverage of the surfacewent down. In this special case, only sevenparticles were arranged in a hexagon.

Fig. 23: The seven particles sat on the bottom of a water droplet of 24 m radius and kept theirstable configuration for longer than 5 minutes.

The tracks of these seven particles are shown in Fig. 24.

Fig. 24: Tracks of the seven particles over a time of 5 minutes. Different colors represent differentparticles. The particles turned around the innerparticle and kept their distance to each other as wellas to the middle particle. The whole system drifted alittle bit, as can be seen by the

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